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extends to an algebra homomorphism of CQ(V)—*A which is uniquely determined and which we shall continue to denote by , g)] 2. We choose k = 2, i.e. 4 algebra of the supergroup itself [8], when gauged, and left- and 4 right -handed generations *'. We thus conincorporating the £^ and Z?(t/i) and their conjugates. struct an SU(7/1) model, whose '•constituent" preon There are still several unresolved issues in the dynamoctet has ics of an internal (quantum) supergauge. It is however e=diag(i,-§;i,i;0,0,0/^), worthwhile to remember the lessons of unitary symmetry (SU(3)), where classification was correct and 73=diag(^-^;0,0;0,0,0/0). useful, even though the dynamics were unclear (and may still be so) — and where there also were problems These preons are heuristic computational aids (like with the statistics (of quarks, before the introduction pre-colour quarks) and cannot be considered as funof colour). In this spint, we have pursued the applicadamental fields (a role played here by quarks and leption of internal supersymmetry as a classification tons). scheme, present dynamical uncertainties notwithstandNote that the quantum numbers in eq. (3) are not ing. the Georgi-Glashow SU(5) assignments, used by Dondi-Jarvis and by Taylor (see bibliography in ref. Our classical treatment [1,4,7] of SU(2/1) yielded [10]), and in which Q, Y G su(5) C su(7). Neither do a Weinberg angle 0W = 30°, X = \g* for the 0 4 couthey coincide with those of the Farhi—Susskind [9] pling, and a mass m^ ~ 250 GeV for the Higgs field. SU(7), where Q, Ye su(7). Our Q, Y involve the U(l) At the time, this was considered an abnormally large in SU(7/1) D SU(7) X U(l). To the extent that the value for m^. More recently, a number of dynamical quantum dynamical role of the supergroup is not unmodels have yielded just such values [9]. These are derstood, the results we now present may be regarded models attempting to explain the emergence of as a specific representation structure in an "orthodox" masses of the order of wiw (~100 GeV), in the connon-simple SU(7) X U(l) GUT. The supergroup would text of unifying theories, where spontaneous breakthen be reduced to the role of an auxiliary mathematidown occurs at 1015 GeV (the "mass-hierarchy" cal construct. problem). All such models involve additional strong interactions at 100-300 GeV. Smce our SU(2/1) We now construct the class I representation [10, presents a simple electroweak unification (i.e. at »zw 13] b=3, with dimension 2 7 = 128. We picky max mass scale) it seems natural that it should realize as a = 2, and the eigenvalues of Y are restricted to the valsupersymmetry the dynamics of ref. [9]. ues [±2, ±4/3, ±1, ±2/3, ±1/3,0], i.e. all values from 2 to - 2 at Ay = 1/3 intervals, but excluding ±5/3. The representation spans the 2 7 state vectors of the 2. SU(7/1) as super GUT. The relevant internal super-GUTis provided [10] by +1 '(x)-<*>(*) = (£'8,+n)<*>. , (C2, d2, e2(00)), c2, d2, e2eR. The manifield 0 obeys a Klein-Gordon-like equation (gabdadh+M2mx)=0. ^ 0 . By doing this we break explicitly at the same time both the dilation and the SL(4,R)/SO(l, 3). We now get and provided A<0 we find a "spontaneous" nontrivial VEV for the tp(x) field, v'9 = (-XI2X^)U2v^ and recover results of the case (i), however, now in terms of only one parameter v0. (hi) Finally, we consider the most elegant possibility, namely to break explicitly only the dilation subgroup of the GL(4, R) group: (f>{x) =vl/l+4(x), vv=
= $. We therefore "drop" them, now dealing solely with the 2-dimensional t, r curvilinear system, with the Schwarzschild radius r, in the coefficients of dt, dr representing the action of the gravitational potentials (or fields), as generated by the static and electrically-neutral spherical mass. Note that all the above would also apply to the (rotating central mass) of the Kerr solution and to the (electrically charged mass) of the Nordstrom solution. We now apply Einstein's Equivalence Principle. It states that locally, it is always possible to gauge away the gravitational potential, i.e. apply an active diffeomorphism which replaces the gravitational potential and the curved manifold by an accelerated flat Minkowski spacetime. This is effectively the content of the Kruskal-Szekeres coordinate transformation (and its inverse), u v A{r) t t
(2.40)
where ( , ) is the scalar product determined by the quadratic form, Q. The nitration by degree on T(V), where F°T(V)
= © T<(V) o
induces a nitration, F"C on the algebra C = CQ(V) so that FqC consists of those expressions which can be written as sums of products involving at most q factors of elements of V. The associated graded algebra is just the exterior algebra AV which is graded commutative. Actually, the "cancellation law" in the multiplication always drops degree by two so that C is a Zs graded (but not graded commutative) algebra where C consists of sums of even products of elements of V and C1 consists of sums of odd products (Atiyah et al., 1964). The elements of C° are filtered by even degrees and the elements of C 1 are filtered by odd degrees in the sense that if two elements x and y lie in Fq(C) and x — y G F"-l(C) (g even) then j - y £ Fq~2(C), and similarly for x and y in Fq(C1) (q odd). Let C be such a filtered graded algebra, so that C is Z2 graded and Z filtered in the above sense, with the associated Z graded algebra graded commutative. We claim that the graded algebra, with a shift of two in gradation degree, inherits the structure of a graded Lie algebra, which we shall call the Poisson algebra associated with C and denote by P(C). More precisely, let us set Pk(C) = Fk+2(C)/Fk(C). If * G Pk(C) and y G Pi(C) we can find x G P* +2 (C) such that x/F*(C) = » a n d y £ P'+ 2 (C) such that y/F'(C) = y, where x is in C° or C1 according as k is even or odd and similarly for y. Since the graded algebra associated with the filtration on C is graded commutative, we know that the expression xy — (— l)*'yx which a priori belongs to Ft+2+,+2(C) actually has a vanishing highest order piece, and hence lies in filtration degree two less, i.e., in F*+ ,+2 (C). We define hl
l
t*,y} = (xy - (-l) yx)/F"+ (C).
Lie algebra OQ(V), the infinitesimal orthogonal transformations relative to the form Q provided that the form Q is nondegenerate, and the bracket of P<> on t h e Pi is the induced action of OQ on A*+2. (More generally, the form Q gives a map of V —> V* and thus from V ® V —* V ® V* which we can consider as Hom(V,V) if V is finite dimensional and contained in Hom(F,K) otherwise. T h e image of P 0 = A2 (10 C V ® V gives a map P ° - > Horn (V,V) and induces an action of Po on the P , which coincides with the bracket.) The subspace P_ 2 is always a central ideal of P(CQ(V)), SO that one can form the quotient algebra. In case dim V = 4, a direct computation shows that f P i , P J = 0, so that one can make a graded Lie algebra out of V © A 2 (^) © A"(V). In general CA<(V),A>(P0] = 0 for i + j > n — 2 if V is an n-dimensional vector space. Indeed, if we choose a basis of V, then any pair of expressions of the form u*, A . . . A vki and D(I A . . . A vit must have a t least two v's in common, and thus their product drops b y a t least two in filtration degree, and the commutator drops by at least four. Thus P_i © P 0 © . . . © Pn-i forms a graded Lie algebra if we quotient out P_ 2 .
K. Tensor products with graded algebras Let A and B be two graded associative algebras. We make their tensor product, A ® B, into a graded associative algebra by setting {A ® B)„ =
ffi
At ® Bi
(2.42)
v+y-n
and (xi ® y,)(xm ® y„) = (-l)imXiXm
® y/yn.
(2.43)
It is easy to check that if, for example, both A and B are graded commutative, then so is A 0 B. For instance, if A = A(V) and B = A(W) then A ® B = A ( F © W). If A = CQl(Vi), the Clifford algebra of F i relative to Qi (which is not graded commutative) and B = CQ2(VZ) then A ® B = C 0l © 0i ,(Ki © V 2 ). (Indeed, the map x © y—> x ® 1 + 1 ® y satisfies (* ® 1 © 1 ® yY = Qi(x) + Qi(y), and hence, by the universal property of the Clifford algebra, gives a map of CQl@Ql(Vi © V%) into A ® B which is easily seen to be an isomorphism.) Let A be a graded commutative algebra and let L be a graded Lie algebra. We make A ® L into a graded Lie algebra by setting (A ® £)„ =
© At ® Lj
(2.44)
i+j-n
(2.41)
Observe that this is independent of the choice of x or y, Indeed, if for example we chose some other x, then x — x has the same parity as x and the filtration degree is two less so that using x or x gives the same answer modulo F* + ! (C). Once we know that the bracket operation on P(C)= ®Pk{C) is well defined, the Jacobi identity is automatic, because it holds for the commutator bracket of three elements x, y, and z when we consider C as a Z2 graded algebra. Also notice that the Poisson bracket is a (graded) derivation of the graded commutative structure of grC. For the case of the Clifford algebra, a direct verification shows that PO[CQ(V)J = A 2 (F) can be identified with the Rev.
583
and [o ® x, b ® y ] = (—l)(deg o)(deg *)„6® [»,„].
(2.45)
A straightforward verification shows that this multiplication is indeed graded anticommutative and that the graded version of Jacobi's identity does indeed hold. As a special case, if L is an old fashioned Lie algebra, we m a y consider it as a graded Lie algebra with L = L0. Then if A is any graded commutative alegbra we can form the graded Lie algebra, A ® L. Thus, for example, A(V) ® L becomes a Z graded Lie algebra for any auxiliary vector space, V, and CQ(V) ® L becomes a Z 2 graded algebra. The technique of multiplying a graded Lie algebra, L, by a graded commutative algebra, A, and, particularly, by the exterior algebra
Mod. Phys., Vol. 4 7 , No. 3, July 1975
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has been extensively employed, first in the implementation of the "supergauge" program in dual models (see references to original papers and recent reviews in our discussion of example II.F), and more recently in supersymmetry (see for example Salam and Strathdee, 1974e). We discuss the latter in some detail in Sec. VI. Notice that the even terms in A ® L which, as usual, form an ordinary Lie algebra, are combinations of products of even terms in A with even terms in L and odd terms in A with odd terms in L. Thus, for instance, if we take A = AV, where V is Minkowski space, Lmm © (A1 (10 ® iodd) 0 (A 2 (F) ® iev. n ) S (A3(V) ® L„dd) ® (A*(V) ® i e v .») is an ordinary Lie algebra. The Lie algebras constructed in this manner in the case of supersymmetry have substantial radicals and possess large dimensionalities as compared with the dimensions of L. If A is a graded commutative associative algebra and B is a graded associative algebra, we can form a graded algebra A ® B. If L denotes the commutator algebra of B, then an immediate verification shows that the commutator algebra of A ® B is A ® L. Suppose that A is a graded commutative algebra which (with a shift of two in the grading) also carries a graded Lie algebra structure so that the Lie bracket acts as a graded derivation of the associative multiplication. Thus we assume that A possesses a graded commutative multiplication and a Lie multiplication such that {.Ak,A{]
(2.46)
and that Ix.yzJ = £x,yjz + (-l)"y£x,zj
(2.47)
For instance, if C is Z 2 graded and Z filtered, as in the case of the Clifford algebra with a graded commutative associated graded algebra A = GrC, then A inherits such a structure as we indicated above. Suppose that B also carries the same structure. Then so does A ® B, where the associative structure is as we defined it above and where [>i ® Si, at ® ft J = (— l) 6 ""([>i,«2] ® 6162 + 0102 ® [ii,& 2 ]).
(2.48)
A straightforward, if rather tedious, computation with the signs shows that Jacobi's identity is satisfied and that the Lie bracket is a derivation of the associative multiplication. If we take A to be a graded commutative algebra and give it a trivial Lie bracket structure (so that all brackets are zero), and take B = L to be a graded Lie algebra and give it a trivial commutative multiplication (so that all associative products are zero), then the above construction reduces to the previous construction of A ® L. Suppose that A = GrC and B — GTD, where C and D are Z2 graded and Z filtered algebras with A and B carrying the inherited structures as indicated. Then C ® D is a Z2 graded algebra and carries the obvious induced filtration whose associated graded algebra is A ® B. A direct verification shows that the induced Lie algebra structure on A ® B is the one described above, i.e., that Gr(4 ® B) ~ Gr.4 ® GrS = C ® D.
As an illustration of this construction, let C be the Clifford algebra of a one-dimensional vector space with nondegenerate quadratic form, so that C is generated by 1 and e with 1 G C and e £ C1 and where e2 = 1. Its associated graded algebra is generated by 1 and e with the associative multiplication 1 • e = e and e2 = 0 and Lie multiplication Q,1J = 0 = L"1,«] and Ce,e] = 2. Let D denote the ring of differential operators on a differentiable manifold, M. We shall think of D as a Z2 graded ring having only even elements and shall filter D by even degrees by letting F2kD consist of all differential operators of degree at most k. Then B = GrD consists of functions on T*M which are polynomials in the cotangent variables. The functions which are homogeneous polynomials in the p's of degree k constitute Bik and the inherited Lie bracket on B is just the Poisson bracket. Then A ® B = Gr(C ® D) is just the Poisson algebra introduced in example G. Indeed, if we let e' — {\/\IT)e and write the even elements in A ® B as 1 ® / and the odd elements as e' ® h —
L2k,
k^
-1
(2.50)
and L _ 1 = L-2+
{«},
(2.51)
where we write an element of Z24+1 as Bz, for z £ Lit. We set LL2A,i2i4-iJ — i_L2k,Lzij CZ L2(k+i)+i-
(2.52)
O,0aT] = 0[z,«>]
(2-53)
with the additional understanding that [£2*,«] = 0 = [u,L2l2
(2.54)
if either k or I = — 1 . (In particular ]~u,if] = 0.)
(2.49)
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186
If k, d ^ — 1 we set L^-2fc+l,i2t4-lJ = 0
(2.55)
Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics and similarly for k = — 1 except that we set C«,te] = z.
(2.56)
This last equation shows that L is generated by its odd terms. Jacobi's identity is automatically verified for any expression involving at least two even terms. An expression involving at least two odd terms vanishes unless it contains exactly one «, and = 0 + [>,z]
585
The spin-conformal algebra W of Wess and Zumino in Sec. I I . H is simple. However, the subalgebra V suggested by Volkov and Akulov (1973) and Salam and Strathdee (1974a) (which is the algebra actually applied in physics to date), Z.o + i i + Li, is not simple. Notice that it is not true that a graded representation of a graded simple algebra is completely reducible. Indeed, we have already seen one example of this phenomenon in Sec. I I . D . Here is another example: for any nondegenerate quadratic form, Q, on an re-dimensional vector space, v, the algebra
= [[«,H,0z] + [>,[«,fe]] and
L = "©
0 = [u,{9w,BzJ] = [_w,Bz~\ — [Ow,z~] = [£u,ew~\,Z~] — [0M>,[tt,0z]]
so that Jacobi's identity holds. This algebra is, of course, very artificial. III. A CRITERION FOR SIMPLICITY We define the notion of subalgebra and ideal of a graded Lie algebra in the usual manner. Thus I is an ideal of L if x G I and y £ L implies [_x,y] G / . We say that / is a graded ideal if the graded components of every element, x, of I belong to I, i.e., if x = @Xj and « £ / then all the Xj G I- We say that L is simple if it contains no nontrivial ideals and if its multiplication is not trivial. We say that it is graded simple if it does not have any graded ideals. We now present a criterion, which appears at first glance to be a bit complicated, but is conveniently verified in practice, which provides a sufficient condition for simplicity. We assert that if L verifies the following conditions then L is simple. (a) Lo contains an element, d, such that \_d,x~\ = kx for any x G -£*, for all k. (b) CL-i.Li] = U. (c) L contains no graded ideals lying entirely in ®~lLj. (d) Lo acts irreducibly on L_i. (e) If k > 0, then [>*,L_i] = 0 implies xk = 0. Notice that condition (a) implies that every ideal must be graded. Indeed, if x = ®Xj lies in I so does [d,x} = ®jxj and so does © j'Xj for any r. For r sufficiently large we can solve for the individual Xj in terms of the j'Xj (s < r), which shows that Xj G I- By condition (d) either 7_i = Z,_i or I-i = 0. If / _ ! = £_! then Lo = lo by (b), but then, by (a), d G / , which implies that Lk = Ik for h ^ 0. Thus L = I. If i_i = 0, then [ 7 0 , i - i ] = 0, which, by (e), implies that I0 = 0. Proceeding inductively, we see that Ik = 0 implies that Ik+\ = 0 and hence that / 3 (B~2Lk, which, by (c), implies that 1=0. This immediately implies, for example, that the symplectic graded algebra, example G of Sec. II, is simple. It does not apply to the Gell-Mann (f,d) algebra which is directly seen to be simple. Also notice that the only role played by i _ i was that it was the first nonvanishing negative component. For instance, if L is a graded Lie algebra with only even components (and thus a classical Lie algebra), we need only replace i _ i by L_ 2 and £ i by i 2 to obtain a useful criterion for simplicity of classical Lie algebras.
Pk[CQ(V)-]/P-2(.CQ{V))
has no graded ideals. Indeed, Pt> is the orthogonal algebra which acts irreducibly on the spaces P , = A i+2 (V). If / were a nontrivial graded ideal then we must have Ij = A ^ 2 ( F ) for some y. Since [_P-i,Pi] r* 0 for any i > 0 we conclude that I0 9^ 0 and hence I0 = P0 and then t h a t Ik = Pk for all k < n — 3. On the other hand, we can construct the algebra L' = L (B{d) where d G Lo' and [_d,x~] = kx for x G Lk- Then L is an ideal in L' and has no complementary ideal, i.e., L' is not completely reducible under L. This same example shows that if we define the analog of the Killing form, f(a,b) = T r ( a d o ) ( a d b) then the analog of Cartan's criterion does not hold. Indeed, it is clear that in any graded algebra we must have f(Li,L}) = 0 for j y± —i. On the other hand, for odd elements we have T r (ad a) (ad b) = 2 T r a d ( [ o , 6 ] ) . For a G P-i and b G Pi the element [o,6] G o(Q) is an infinitesimal orthogonal matrix and hence has zero trace. Thus, although L has no graded ideals, the Killing form / is degenerate. It would be interesting to see which theorems from the classical theory of Lie carry over, and with what modifications, to graded Lie algebras. IV. THE BIRKHOFF-WITT THEOREM FOR GRADED LIE ALGEBRAS In the classical theory of Lie algebras, a central role is played by the universal enveloping algebra. Roughly speaking, if L is the Lie algebra of a Lie group G, then the ring of left invariant differential operators gives the enveloping algebra of L. (We give the more precise, and more algebraic definitions below.) Thus the various Casimir operators lie in the center of the enveloping algebra, and, in general, the universal enveloping algebra is important in representation theory. The first basic structural fact about enveloping algebras is the Birkhoff-Witt theorem. In this section we state and prove the corresponding theorem for graded Lie algebras. Let L be a graded Lie algebra. In what follows, we shall regard L as L0 ffi L\, where Lo is the sum of the even graded pieces and L\ is the sum of the odd graded pieces. This is not necessary, but it saves us some indices. The universal enveloping algebra of L, called UL, is an associative algebra with the following properties: 1. There is a canonical linear map e: L—* UL satisfying
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187
«(*)e(y) - (-l) l '«Cy)c(*) =
e(£x,yj).
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Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics
Proof: (1) The monomials span UL- T O begin with, note that the monomials XK (no ordering restriction on N) span UL, because L generates UL- If i\ > ii, then we can replace /(*)/(y) - (-D"/(>)/(*) = /fl>,y3), «(*«,)«(*,-,) by ± « (*<,)« (*,-,) + t{[xh,XiJ). Similarly, if then there is a unique homomorphism <j>: f £ —» A such that i i £ Xi, then e{x,)t{xi) = ^({[.Xi,xiJ). By repeating these / = <(> o e, or such that operations, we are able to see that any xn can be written as a linear combination of *jif's, where the M's are words. U That proves half the theorem. L
2. Ii f: L—> Ais any linear map from L to an associative algebra, A, satisfying
-
N?
(2) The monomials are linearly independent. We first let V be the free k module with basis {ZM}, where M runs through all words. If i £ / , M = ( M i , . . .*m), and i < i\ (or i = ii and #,-, £ Lo), we shall say i < M and let iM = {i,ii,. ..im).
commutes.
Proposition: The universal enveloping algebra exists and is unique {up to isomorphism). Proof: This follows standard arguments. To construct UL, let TL be the tensor algebra generated by L, and let 3 be the ideal generated by all expressions of the form x ® y - (— l)"y ® x - [x,y~\. Then UL = Tjs. The map t takes x into the tensor x of degree 1. That UL has the right property is easy to check: given / , the universal property of TL says that there is a unique homomorphism \j/: TL —* A which extends / to TL- But \f>(x <8> y — (— l ) " y ® at — Zx,yJ) = f(x)f(y) - {-\)klf{y)J{x) - DeoO = 0, and so ^ takes # to 0. Thus ^ defines 0 on UL- For uniqueness, 1 suppose that VL and e also satisfy the definition. Then property 2 says that we have maps
We shall show that V can be made into an Z-module with X
Scj/Zji/, and hence CM = 0 because the ZM are linearly independent. Everything, therefore, depends on defining the module structure. We need to define XiZu for all i, m. Let a,- = degXi. We assume inductively that if 1{N) < 1{M), then XJZN is defined for all j £ / and XjZu is a ^-linear combination of z,'s with 1{L) < 1{N) + 1, and that if j < i, then XJZM is defined. Then define (where cti = deg xi) XiZM = ziM if i < M; = (—l)" 4 "'*,-^*^) + [*i,*y>iv if M = j'iV with j < i; = i\^,Xi,Xi2zN if M = iN and x< £ L. These expressions on the right are well-defined and continue the inductive hypotheses. Thus XiZM is now always defined.
But then, combining these,
We need to show that XiXjZn— {— i)aia'XjXi^i{= [_XiXj}zffThere are a number of cases. We may always assume i > j (and i > j ii Xj £ Li), by symmetry. We shall not give all the cases here, but enough to make the argument plain. (1) j < N and i > j (or i = j and xt £ Li); then the commutes. By the uniqueness of the extending homoresult comes from the second line of the definition. morphism, ^ o <j> = identity. Similarly,
The Birkhoff-Witt theorem gives more information about the structure of UL .We need a modification of the BirkhoffWitt theorem for ordinary Lie algebras, as given, for example, in Serre (1964). What follows (and most of what preceded) is an adaptation of Serre's treatment.
Zxi,xi]xkZL.
By induction, we know that (*) holds if we permute i, j , and k cyclically. Also, by induction on 1{N), we may assume that XkXlZL = {—l)°"""XlXkZL+
Lxk,Xl]ZL,Vk,l
£ /.
Let {xi} (i £ / ) be a basis of L; for convenience we assume So (*) becomes that / is well-ordered and that the elements of i 0 precede {*\)XiXjXkZL — {—\)ai"'XjXiXkZL= [Lxi,Xj],Xk2zL those of L\. A word of / will be an expression M — (*i, • • • ,im), + (— l)a^ai+a')XkXiXiZL — {—l)'""'+ai''k^n''iXkXjXiZl. with ii < in < • • • < im, and with ij < ik if xis £ L\- (Of course, than xih £ L\, too). We then write i » = (xtj • • • «(*.„). The length of M is 1{M) = m. We know that the versions of (*1) with i, j , h permuted cyclically (*2 and *3, say) are true. To prove (*1), it suffices Theorem. Assume L is free over k. Then the set of monomials to show that some linear combination of (*1), (*2), and (*3) gives an identity. Of course, (*1) must enter nontrivially. XM {where M is a word) is a basis for U{L). Rev. Mod. Phys., Vol. 47, No. 3, July 1975
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587
But (-1)«<«(*1) + (-1)<"'""(*2) + (-1)<"*">• (*3) is of the a Poisson bracket structure for which L ~ Gr 0 t/. I t would form be nice to have an analogous theorem for graded algebras.
0 = Q+ ((-l)""»[[I^*,-l*t]+ (-l)"'"[C»*,^l*y] +
(-l)°<-[£xi,Xk'],Xi])zi.
V. A VERY BRIEF SKETCH OF DEFORMATION THEORY
This last is true, by the Jacobi identity. So the result works for (2).
Here we only outline an illustrative example of the subject in very bare form. We refer the reader to the article of Nijenhuis and Richardson (1966) for a very readable exposi(3) There remain cases where some of i, j , k are equal tion of this example and to their paper (1964) for the presenand the basis element is in Li. For instance, if i = j = k, tation of the theory in its general form. Suppose we start then \jCi,Xj~]xtZL ~ 2xicXkXhZjJ and the result is clear. Thewith a vector space, W, and ask for a description of all possible (classical) Lie algebra structures that W can carry. other cases are similar. This proves the theorem. Thus we are looking for all possible maps /*: W X W —> W The Birkhoff-Witt theorem is usually given in a somewhat which are antisymmetric and which satisfy Jacobi's identity. To say that M is antisymmetric means that different form. We need a number of definitions first. Let XJ\ be the subspace of UL generated by all products of the form e(xi)- • • e(xm), where m < n, let Gr„(f/i)
= Ul/Ur\
and let
Gx(UL) = e Gr«(i/i). In some sense, Gr„(C/i) consists of expressions "purely" of length (»i). Since UL™UL" £ ULm+", we can define (by passage to the quotient) a multiplication on Gr (UL) with Gr m (t/r) CI Grm+„(J7z,). Clearly GT(UL) is generated by the image of e(L) (in UL). In fact, e(L) = UL1. If x or y is in L0, then t(x)t(y) — t(y)t(x) = e([.x,y]) G W , and thus e(x) and t(y) commute modulo Ui}- Thus if we let t~(x) be the image of t(x) in GI(UL),
t~(x)
and
H G Horn (W \W,W)
= Derx (AW).
The Jacobi condition says that C",£iJ = 0 in the F - N algebra of W. Thus the set of all Lie algebra structures consists of the algebraic variety C, of all solutions of t h e system of homogeneous quadratic equations \ji,n2 = 0 in Deri ( A W). Now we are really interested in classifying such structures up to isomorphism. This means the following: L e t A be any nonsingular linear transformation of W. Then A acts on Deri ( A W) by sending ^~—> /tA, where na(x,y) — A/i(A~1x,A~iy). We wish to regard pi and HA as the same. Thus we are interested in the structure of the "orbits" of C under the action of the general linear group, G(V). Deformation theory studies this problem from an infinitesimal point of view: Suppose we are given an analytic curve
e~(y) commute. Similarly, if x and y are in Li, then €~(:r) and t~(y) anticommute.
Mi = M + l
(5-1)
in Deri AW. We wish to find conditions for this curve to lie on C. We also wish to regard as "trivial" a curve of the form
Now let S(Lo) be the symmetric algebra over L, A (Li) the exterior algebra over L\, and T) their tensor product. There (5.2) is a natural map <j> from V to Gr(£7z,). For, according to A»i = PAW, general nonsense, there is a natural map from T(L) to where A (t) is a curve in GL(V) with A (0) = id. Gr(UL) [.T(L) is the tensor algebra over LT\, and the above commutation and anticommutation relations show that it The condition for MI to lie in C is [MI,MI] = 0. Expanding factors through V. Moreover, tj> is onto, since <j>(xi) = t~(xi)in terms of t we get and the elements e~(x>) generate GT(UL). CM„ M I ] = 2 / [ n , v , ] + / 2 ([>i,'PiJ + 2[»,
D
is the trivial one. But because the ordered monomials span U(L), this last statement amounts to saying that the only relation of the form
(5.5)
and, in particular, that \_
£ CMXM — £ CMXM HM)—n i(Jlf)<»
M^(0 = M + C ° » M J H
•
has the left-hand side 0. This is a consequence of the previous theorem.
Thus the first order triviality condition on iit is given as
For ordinary Lie algebras, the filtration on U gives rise to a commutative algebra structure on Gr U and hence also
Thus the possible "first-order" nontrivial deformations correspond to the cohomology space ff'(AW,n,). The second
(Pi — —Dfa.
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(5.6)
588
Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics [/",.?*] = — ig")'P" + ig'^P", [ P ' . P " ] = 0, [/"',X X ] = ig^K' + ig^K", LK*,K'l = 0, [_K»,P>] = -2ig"'D - 2iJ"',
order condition, that fj^i.^ij be a coboundary, is independent of the choice of * i in its cohomology class. If H^AWM
= {0}
(S.7)
then every "first-order" derivation can be extended to a "second-order derivation." Actually one can justify this whole formal theory by use of the implicit function theorem to relate the Hl and H2 to the study of the orbit through p. We refer the reader to Nijenhuis and Richardson (1964, 1966). VI. SUPERSYMMETRY A. The GLA %v and V The first introduction of a GLA as a supersymmetry of space-time (i.e., as a symmetry containing the Poincare group) is due to Volkov and Akulov (1973). They adjoined to the Poincar6 algebra (P a set of four odd generators, behaving like a Majorana spinor. This is isomorphic to the i o © Li ffi Li subalgebra V of the spin-conformal algebra of example 2H which we shall denote by "W. Volkov and Akulov were exploring the hypothesis that the neutrino's masslessness might indicate that it is a Goldstone particle, necessary to the (nonlinear) realization of an exact symmetry of the physical world, a hypothesis suggested earlier by one of the present authors (YN), and which had failed because of the statistics issue (Joseph, 1972; Bella 1973). They therefore had to introduce conserved generators which do not destroy the vacuum, and behave like the physical neutrinos under the Poincare group, i.e., spinors. To preserve Fermi statistics, the new generators were now also required to anticommute. The GLA V thus had fourteen generators (this came out of the requirement of algebraic closure), was not simple, and had to be realized nonlinearly. For ip the neutrino field,
ID,P"2 = iP", [D,K*1= -iK», ID,J»1 = 0,
(6.2)
where J" are the Lorentz group generators, P" the translations, K.* the pure conformal transformations, and D the dilation operator. Note that the indexing for the gradation is provided by doubling the eigenvalue of D, so that the X of Sec. II.H correspond to K", and the W to JP". In the context of adjoining £_i © h\ we are led to add a sixteenth (scalar) operator to the even set. This will be E, corresponding as shown in Sec. II.H to the infinitesimal action of «•'* 1(0 real), acting trivially on the even generators, £E,J»J = IE,P"2 = LE,K"3 = [_E,D2 = 0.
(6.3)
To construct L_i ffi L\ in terms of physical operators, we use the Dirac matrices {T",7"> = H",
° > = (*/2)[7«Tj
(6.4)
and the special matrices 75, /3, and C: 7s = 70717273 = ( l / ^ O e j ^ T ' V W ,
(6.5)
1
7,+ = 07W3" . —7u.~ = C-'y„C, Cyf
(6.6) C+ = C~l,
C~ = — C,
Cys =
= 75C
yfC, (6.7)
We have to distinguish here between *, denoting complex conjugation, and +, which now stands for Hermitian conW> = I + ia{hj>)dj>, (6.1) jugation; ~ denotes transposition. To form the adjoint where a is a constant of dimension (—4), i.e., the fourth spinor we utilize {}, power of a length, and f is a four-spinor (Majorana) anti$ = 4>+$. (6.8) commuting parameter. Considering that the neutrino does We use a Majorana representation, i.e., for p.,v — 1, 2, 3, 0 not seem to fit a Majorana description, the actual physical content would appear rather speculative, but the algebraic we have innovation remains interesting. Apart from the nonlinear y * = _ y < ;
which ensures — 70C = 1; 75 = ip&i. 00
We preserve the energy metric of Sec. II.H, g = 1, gii = g22 = gas = _ ^ a n d g „, = o for p ?± „. F o r the convenience of the reader who is accustomed to the conventions common in the physical literature, we shall rewrite the bracket relations of the observables associated to the spin conformal algebra in terms of a standard basis of Hermitian operators. The physical algebraic system consists in the conformal algebra SU(2,2) for the even gradation [7<",/p»] = +ig»'J>p — jgcp/"-)- ig-pji" — ig<"J»Pt Rev. Mod. Phys., Vol. 47, No. 3, July 1975
190
Note the following reality and symmetry properties: 7o + = 7o J (7o7^) + = 7o7i.; +
(7oo>) + = 7o0>;
(7o750>) = 7o75
Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics C~ = —C;
589
The generators of the subalgebra V are:
(ytC)~ = —ys,C;
(7s7,.C)~ = —TS7MC(6.11) The charge-conjugate spinor is given by
J" e L„,
# - » C f = *', (6.12) P" G U, (6.18) and for our choice of phases >f/c = if/*. We also note that a i.e., the even gradation corresponds to the Poincare algebra. Majorana spinor is one which is equal to its charge conjugate, For massless particles, the helicity X (taken here with the same sign as 712) is the only remaining quantum number in 'I'M = <(IM° and with our phases, ^M = ^/M*, $M = ^~7o (6.13) the little group of the Poincare' group. which reduces the number of its complex components to For p+ = ffl + f r± 0, p~ = f — p3 = 0, p1 = p* = 0 two or real components to four in this choice; we shall on the states, the little group is generated by [ (7 12 , Ja — J*>, transform later to two complex ones, which can then be J'1 - /»')]• identified with the column and row vectors introduced in Sec. II above. From the symmetry properties in Eq. (6.11) In the odd set of (6.16), only the Q* are in the little group. we find, for ^ and x anticommuting Majorana spinors: Using for example the representation (6.10) we find that the only two nonvanishing Qa are Qi and Qt. These are not i>X=Xf; ,bvX = -Xy^; fa„X = -Xo>^; / l 2 (helicity) eigenvectors, and we recombine them into *£y(OVX = iX7,7„^; $y,X = Xyjfr. (6.14) helicity + J and \ operators.
(Qx - *<24)/VZ ^ *, «?, + ie4)v2 s
The odd generators of "W are Q„ and Rp (a,/3 = 1 . . . 4 in Dirac spinor space) and are Majorana spinors, thus involving two complex or four real functions each. The even-odd Lie brackets are the commutators (Wess and Zumino, 1974a; Corwin, Ne'eman, and Sternberg, 1974; Dondi and Sohnius, 1974)
{x,y) = e, for 2P+ = e Zh,x] = x, [A,y] = -y for
-
C".<2 J =
-i(p")*i>Rfi, [JP-.KJ = 0,
=
-WUR,,
Q*->V
and the odd-odd brackets consist of the anticommutators {Q.,Q>) = ~2(yfCUP",
Qi-*-iQ,,
(6.16)
2{y„C)^K-.
Q4-*iQ2.
To conserve parity, we therefore adjoin a 2-space representing states with P~ ^ 0, P+ = P 1 = P 2 = 0. We find,
We can also introduce "adjoint" spinors Q„ and Ra as per Eq. (6.11). This is especially useful in view of further generalizations in which we shall introduce internal degrees of freedom. For su(n), n > 3, the covariant and contravariant representations are not equivalent [ 3 and 3* in su(3), etc.] and this will require distinguishing between Qa and Qa. The bracket relations are {Q«,Rf>} = (75ff„,W" {&.,&> = 2(y,UP", {-»„,#„} = 2(y„).i,K'.
(6.20)
which has (choosing 7j
{Q-,Rf>) = - ( 7 E ^ C ) ^ / " ' - * ( C ) „ „ £ + 2t(7 6 C)„ /) D, (R.^f)
(6.19)
To discuss the representations, we notice that a\ — x + y, oc2 = —i(x — y) define the Clifford algebra d, as can be computed from our defining brackets. I t is a 22 = 4 dimensional vector space with basis at, as, ai<*2, e. Its only irreducible representation is the defining set of 2 X 2 matrices (see for example Boemer, 1963). However Qi and Qt are not parity eigenvectors, as can be seen by using Eq. (6.10) in the parity transformation,
[tf-.e.: = — (757*0 op-fy, lP",R,r\ = - (yty")«iiQp, ZD.QJ = (i/2)Q«, LARJ = - ( * / 2 ) * . , [£,e«] = 3*(7')«*&, [£,*-]
2 / " = h.
This is just the GLA of our Sec. II.A. Its denning 2 X 2 representation acts on a vector space containing one fermion and one boson state (helicities J, 0 or any (n + i ; »/2).
V~,Q.I = -i(f")^2f,
U'Al
y
2i(yl).fD+iiafE, (6.17)
B. Representations of v Working with the conformal group as a symmetry implies massless particles (provided the symmetry is not spontaneously broken by a Goldstone boson). We first construct some physically relevant representations of the subalgebra corresponding to mass zero.
(&+iG0/v2=*', 2J 12 = h'.
(Q, - *e,)/v2 = y',
2P-=e', (6.21)
This time, we pick n = — 1, getting eigenvalues (0,—J) or C—»/2, — (n + l ) / 2 ] for the helicities X. The Fermi states, being helicity eigenstates, have to consist of combinations of the type (^i =F i^/t) of the real components of a Majorana "neutrino," just as we calculated for the Qa in Eq. (6.19). Parity thus consists in complexconjugation, leading to the conjugate space. The bosons thus also can be written as (« ± iv), u a scalar and v a pseudoscalar. We still return to this simplest of all representations when we construct appropriate "superfields," i.e., field representations of supersymmetry. Note however that the findings of Volkov and Soroka (1973) fit within this picture: the massless graviton, with X = ± 2 , gets a companion with X = ± f .
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Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics
Actually, there have been to date very few applications of the full algebra W. Instead, following Salam and Strathdee (1974a), the nonsimple Volkov-Akulov subalgebra V was used in its linear realizations ("supersymmetry"). However, any Lagrangian which is invariant under that algebra, and which is in addition made invariant under the conformal group (by making all masses and all dimensional couplings vanish), will also be invariant under the V? GLA. Taking here the M > 0 case, we use the rest frame to find the "little" GLA. First we note that the Eqs. (6.16) and (6.17) reduce to f{Q.,Qf} = 2dafM (6.22)
n -
The first bracket defines C 4 , the fourth-order Clifford algebra (dimensionality 24 = 16); it is just the algebra of Dirac y" matrices in a Euclidean metric. Its only representation is in four-dimensional matrices (Boerner, 1963). We thus know that all M ^ 0 representations of "O will reduce into four-dimensional subspaces, just as the M = 0 ones worked in doubled two-spaces. To get the "little" GLA in a more familiar form, we diagonalize 70. I n our representation (6.10) 70 = —P2, so that the appropriate unitary operator U will act in p space only, with 1
Ui-p^U- = p,.
(6.23)
This will rearrange our (Majorana spinor) odd operators which has as components Qi=
(Gi+iQ»)/-v2
(6.24)
Qn = (Q, + iQdW
and their Hermitian (here just complex) conjugates Qi , Qn+. Transforming the matrices (—%aif) —> U(—i
[/»,Qi+] = iQu, LJ3\Qul
= -iQi,
C/ 3 1 ,en + ] = -iQi+.
(6.25)
+
The QA-I.II and QA thus form independent 2-spinors. They fulfill {QA,QB+}
= 2BABM,
QA+: Qi+\j,J3,Xp,M), Qi+Qn+\j,J3,Xr,M)
Qii+\i,j*,Xr,M), and \j,j3,Xv,M).
The first two change the spin, j 3 , parity, and statistics of the states according to (6.25) and the eigenvalues of 70. T h e Qi+Qu+ action preserves j 3 and j but inverts the parity. We thus have a 4(2_/'+ 1) dimensional Fock space, with subspaces I i,h, ~xv,M),
I jJn,Xp,M),
I j + i, js, XvVv, M),
I j - i, js, XPvp, M). Notice that fermions and bosons have the same mass. These rest states are then boosted to any p by a Lorentz transformation U(LV). The action of QA and QA+ on the boosted states can be derived from our knowledge of the spinor behavior of the QA and QA+ under Lorentz transformations. An additional result derived by Salam and Strathdee relates to the action of the Qa on two-particle states. Since we have to preserve (6.16),
{QA,QB}
= 0,
{QA+,QB+}
{ Q « a W 2 ) } = 0.
(6.28)
Thus Qa acts as an antiderivation [Eq. (1.4)] (6.29)
We present here an additional diagonalization of the Qa which will prove to be useful in the construction of field representations. It corresponds to diagonalizing —iys £= pitr2 in our representation (6.10)]. The transformed QJ then reduce into two 2-spinors corresponding to (we denote the chiral projection operators by yB and yL) Q* = (1 - t T 6 )/2Q' = yRQ', QL= (1 + = yLQ'.
= 0.
The brackets (6.25), (6.26) together with the angular momentum commutation relations define the little GLA: £ £„ (<* = 4),
pmy,
the cross terms have to vanish,
(6.26)
J'',M
(6.27)
0
where k = 1 is the grading of Q„ £ Endi, and I = 0 or 1 according to whether t>i' £E V is a boson or a fermion.
C7 I 2 ,<3II + ] = Qi+,
C-/31,<2i+] = iQu+,
QA\J,}Z,X„M)=
in analogy to an annihilation operator, we have four possible actions of
^ I »!»!>= |»i'»2)(»i'le«(pi)|»i) + ( - D " I VM'KV*'IQ.(p2) I v,),
V°,Qil = hQi, C^.Qul = -i<2n, [./» G.+] = -hQi+, C7» &i+: = JQn+, Va,Qil = -Qn, Un,Qnl = -Qi, Q11+,
An irreducible representation of the little GLA is thus obtained [Salam and Strathdee (1974b)] operating with the QA and QA+ on the (2j 4- 1) dimensional carrier space of any representation (j,M) of the (Wigner) little group of the Poincare group. Because of (6.26), and taking (j, corresponds here to the "new" J23 direction)
{Q*(1) + Qa<2), Qf,m + <2»(2)> = -2(Y,.C)«0(/> + +
[/",&+] -
Since 70 is diagonal in this representation, the spinors QA and QA+ are parity eigenstates with opposite eigenvalues (note that 17^ will have to be i or —t here).
iyCi/201 (6.30)
The Q' is no longer real, since it is given b y VQ = Q', U+ = U~x. The Majorana condition should then be redefined. We get Q" = C«UQ)+yo)= -cU*Q*
G L-i {d= 2), QA+ G U (d= 2). QA
Rev. Mod. Phys., Vol. 47, No. 3, July 1975
192
= Cya~U*Q* =
-y^U*Q*
Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics and in our choice of c = — 1 , the condition becomes
x's is graded commutative, i.e., %&/ = (~l) 'XjXi. We impose the formal analogs of the group axioms, and obtain what Berezin and Kac call a "Lie group with commuting and anticommuting parameters." If / £ F, then it follows from the right unit axiom that
Q' = UQ = Q'° = -cU*Q* = U*Q*. With 75 real and anti-Hermitian, we thus find ©.*)* = y«>LQ>'* = y»>LQ* = QaL.
(6.31)
The two 2-spinors QaR and Q„L are thus conjugate, and behave like the QA and QA+ of (6.25), except that 7 31 is now the diagonal projection of spin. Thus ^ . G i ' D = -WiL;
LJ*,Q*1 = i d * .
The graded Lie brackets are in general {Q.L,Q*L) = o, {&* = o, {
591
did
(6.32)
and for Af 5*= 0 and rest states
0/(*>y) = f(y) + 2X/(y)xi terms in xt).
+ (higher order (6.34)
The maps / •—> X/' are linear, and can be thought of as the analogs of infinitesimal right translation. If we p u t the obvious gradation on F, then the map / ~—> X/ is a graded derivation of F, and the X' form a graded Lie algebra. In this way one associates a graded Lie algebra with each such "formal Lie group." Conversely, starting with a graded Lie algebra, by use of the analog of the Campbell-Hausdorff formula, Berezin and Kac show how to construct a "formal Lie group in commuting and noncommuting variables" to each graded Lie algebra. The correspondence between the graded Lie algebras and the "formal Lie groups" is functorial in the usual sense.
In applying GLA's as supergauges in the dual models (see references in Sec. I I . F ) , the Berezin-Kac method was used, with Grassmann algebra elements appearing as group parameters. Wess and Zumino (1974a) applied the same technique in order to construct field representations and invariant Lagrangians. Salam and Strathdee (1974a, e) sysFor M ?* 0 and rest, QaR and QaL = (2«fi* c a n t n u s be tematized the approach, which was further developed by treated as annihilation and creation operators in the con- Ferrara, Wess, and Zumino (1974). However, as we shall show, although the method does yield very useful results, struction of representation of states or fields. its foundations are unclear and lack consistency (Ruhl and Yunn, 1974; Ne'eman, 1974). Ruhl and Yunn (1974) and C. Realization on a Grassmann algebra as a Goddard (1974) have recently tried to supply a better set generalized (Berezin-Kac) Lie group; superfields of basic assumptions and have indeed removed some of the Berezin and Kac (1970), following a similar idea of Lazard inconsistencies, except for difficulties with an indefinite (1955) and motivated by problems of second quantization, metric and for the fact that the Minkowski space coordiintroduced the notion of Lie groups with commuting and nates x" are still nilpotent elements obeying the requirement anticommuting parameters. Their idea is the following. Let (a»)" +1 = 0, (with re = 4 in the conventional solution). In G be an (ordinary) Lie group. Then G is a differentiable general, it also seems doubtful whether indeed the genmanifold, and the group multiplication defines a differ- eralized group can be used as a symmetry, except very close entiable map of G X G —> G. The group axioms impose to the identity. We shall now describe the formalism, using some conditions on this map. Let F(G) denote the ring of the notation of the Appendix. smooth functions on G, and F(G X G) the ring of smooth functions on G X G. The multiplication then gives a map We use an iV-dimensional vector space (over the complex from F(G) —* F(G X G), sending the function / of the single field) V ( = A ' F ) , generating a 2N dimensional Grassmann G variable into the function tjtfoi two G variables defined by algebra (*/)(*,y) = / ( * y ) . (6.33) AV = © A'V. while for the M = 0 case, we again see the reduction into two subspaces with P° + P 2 ^ 0, P° - P 2 = P 1 = P 3 = 0 for the first, and the parity-inverted states for the second subspace. The Q\R and QiL are in one subspace, and the QiR and QtL in the other.
The various group axioms, such as associativity, existence of identity, and existence of inverses, can all be formulated in terms of the map <£. Since we can assume that the group coordinates are chosen so that the multiplication is given by analytic functions, we can, without harm, replace the ring of smooth functions by the ring of formal power series, say Fa. Then FQXO can be identified with Fa ® Fa (completed tensor product). This is simply the assertion that any polynomial in two variables can be written- as a sum of products of polynomials in one variable, and hence any formal power series in two variables can be written as a (formal power series) limit of such sums. We can then write all the group axioms as a series of conditions on the map,
The basis vectors of V are vi, t>2, . . ., VN ; since the Grassmann algebra is graded commutative (A2), the elements of V anticommute, Vi A Vj =
( — l ) l l y A Vi.
We shall write this property as {»,,»,} = 0
for any
i, j , »,,, £ V
(6.35)
with multiplication thus being defined by the A operation. We shall also use extensively the elements of A2V = W, resulting from v^j products. In this case, graded commutativity ensures that the elements wa{E.W commute. The Minkowski space coordinates are identified with elements
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Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics
of W, x" £ W = A2V. If we attach a reflection operation R to the Vi G f, »i-» —Vi, the entire AV splits into two parts, rAV = AV<-> + AV<+> |A--FCAV<->
if r is odd,
C AV ( + )
I
if r is even.
"Superfields" are "local" fields, in the variables 8a £ A'V, a:" £ A2V. 8a is a Majorana spinor, 6 = fl« = C7
8a
(6.43)
, + ««, {£«,«<«} = 0
with (a £ V, we have to act with iaQa, where we use t„ rather than ta in order to obtain the necessary tensor construction as in Eq. (6.40). Note that exponentiation by i« follows the Berezin-Kac (1970) method of generating a generalized Lie group. Integration is defined through /<*»<= 0;
(6.36)
fvidvi=
1;
fvtdvj=
{dVi,dVj} = 0.
in the representation (6.10). As to the coordinate, it should be real in any case,
The resulting action is then a commutator bracket, as needed for infinitesimal group action, (6.45)
C««0<»0(»] = n-
(*")*.
Thus V is at least four dimensional. Indeed, a fourdimensional quasi-Minkowski coordinate p t is not a true Minkowski coordinate since (x")N+1 = 0 ] in A2V can be constructed from two 0, 8' £ V, (6.37)
x" = 07*0'
(6.44)
Note that t ^ e(0) and {@,e} = 0, as against Eq. (6.41), require additional dimensions in V.
which amounts to a true reality condition
B*=
Majorana spinors, so that r„0 = iCap. To obtain an infinitesimal translation by a "constant" parameter e„ ^ e„(0)
which, by Eq. (6.9) is Hermitian and real. From Eq. (6.14) we observe that
The action on x" = 8'y'S is thus bound to be i[eaQa,x»]
= ey"8.
(6.46)
Assuming now the existence of a "superfield" 0(*„,0<,), we can use a Taylor series to identify the structure of the infinitesimal operator iaQa, U
6 y 0 ' = — g'y.0
which can be rewritten, using (6.36), as e « " ( 7 V ) . d V = - ».'" ( W ) «*»,>. (6.38) We observe in this expression the (generalized) matrix structure of the A operation between two Majorana-like elements of V. I t is still antisymmetric, because 7 V is symmetric; the antisymmetry is thus derived from (6.35),
- e0 — tffoA) + 0(e>), 30„
where the generalized group element is : U = 1 - ie„Q«
(6.39)
{«»,«/} = 0
This yields the explicit structure
We now turn to the action of the Q„ on these elements. From
&.(AV) = ( iCali 300 - ( *
(6.16)
we know that the doubled action of the Q„ represents a translation in W. We can thus guess that Q« represents such a translation in V, acting in analogy to
(6.47)
. (i«Q«)+ = e.Q„.
and the 7 ° y matrices preserve this feature while taking care of the spinor indices.
{Q.,Qt) = -2(7,C)„„P<-
idx"
[>,,**] =
-
(6.40)
As far as its action on V is concerned, Qa ~ Ta^(d/d8f). Thus Q„ is in V* or in Ai V. Note that for d/d8a, an element r in V*, we are in the larger ffiA, V. Thus d
d
J
dVi
\ dvJ =
0,
[
d
1
— «i [dvi
— Sij.
(6.41)
Qa will thus bracket with 8p as Endi V, {<2«,ft>} = iCat,
(6.42)
where C £ V0 appears as the appropriate metric for
(6.48)
3*„
We now come to one of the difficulties or inconsistencies of this picture. If we regard iQ as a Lie group generator, we get, by Eq. (6.17), C««6«,^Q«] = ««{Q«iGw«s =
P.
AV.
*(T»)«
^iaiy^a^^P".
(6.49)
However, from Eq. (6.14) we know that this vanishes, since ij/y^X = —Xy^. Even if we do not sum over the a and fi indices, we shall at least have vanishing expressions for H = 0, since 702 = 1. This covers in fact the entire little algebra for M ^ 0 [Eq. (6.22)]. We are thus faced with two choices: either the Lie algebra is Abelian, or, as we already noted from Eq. (6.43), e„ and *p have to He in new subspaces of V, which differ from each other and also do not contain the 8a. In these new subspaces, we may be able to ensure nonvanishing of the right-hand side. Indeed, the simplest solution is to add eight dimensions, so as to have different ia and «p' on the right-hand side. The P" are then multiplied by 16 — 4 = 12 new dimensions in A2V.
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Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics Note that all of this is necessary because the superfield 0(zji,0a) is acted upon by a Lie group. However, if we allow for finite transformations, 0a will have "crept" into the new e„, tj subspaces, and our efforts will have been to no avail. Still, we dare not allow Eq. (6.49) to have a vanishing rhs since we would then lose the connection with our starting point, in which Qa acted as the "square-root" of P*. We have by all means to recover Eq. (6.16) or (6.17), even though the information will now be supplied by a commutator. It turns out that we can also add only two dimensions to V, so that N > 6, and disconnect the new dimensions from the spinor indices in e„. Goddard (1974) has constructed this system, using two new dimensions. We denote their basis elements as v5 and i>6. This seems the most economical solution. I t may have been hinted a t by Salam and Strathdee (1973a), but in their solution the number of odd generators would be doubled: (vaQ«) with a = 5, 6. Riihl and Yunn (1974) have pursued this method and come up with 26 generators instead of 14 for V. This results from six for J"", eight for iQ and i'Q, 12 for ee'P„. Goddard has found a way of avoiding the doubling. In the chiral picture (6.30)-(6.32) we see that only cross terms in y" and yL contribute. This is true beyond the rest frame used there. Thus Goddard introduces the matrix g = 1(1 - »7b)»s + 1 ( 1 + hi)ve = yRVs + yLv,
(6.50)
and writes «« = g«etii,
(6.51)
where £0 is a c-number Majorana spinor, i = 1° = C | -
593
The two subspaces of V = Vt + V„ where Ve has ti1_4 as basis, and V, has W6_6, generate subspaces AVe (d = 16) and AV, {d = 11) of AF<+> and AV<~>. Goddard's method utilizes these subspaces for (0„) n and (e„)m, thus allowing only infinitesimal transformations of 0* in Eq. (6.43). T h e Lie group is thus physically applied only very close to t h e identity. We now follow Salam and Strathdee (1974a). Due to t h e anticommuting properties of 0«, any function f{0) must b e a polynomial. Since the monomials 6as8a2• • • 6a„ have to be completely antisymmetric, expanding 4i{x",8a) in powers of da is a finite operation terminating a t « = 4 . The even monomials belong in the A J V + ) , the odd ones in AF» (- >. Altogether,
+ Whf,y,8)Ay(x)
(6.58)
We have altogether (before any subsidiary conditions or equations of motion) eight spinor and eight boson components. Foregoing t h e difficulty about the nilpotence of x", which does not involve (6.58), we find that A (x), F(x), and D(x) axe scalar fields, G(x) is a pseudoscalar, and A„(x) an axial vector field. Besides these Bose fields, we have two (Dirac) spinor fields ^ and x. We can impose a "Hermiticity" condition on the superfield, 0 (*,«)+ = 4>{x,0),
(6.52)
(6.59)
and is real in our (6.10) representation. The y and y are where + implies besides complex conjugation a reversal of Hermitian. Under complex conjugation, 115 and v6 are made the order of anticommuting factors. The Bose fields then make eight real components, and the spinors are Majorana to obey spinors. Had we started with a pseudoscalar
e-« = (g£)+Yo = lw„
L
(6.54)
and iQ = IgQ = i.(^yR + vr/L).&» = 1-5. (6.55) and the Sa fulfill the role of generators of a Lie algebra,
[ 5 « , 5 J = — gay{Qy,Qs}gg! = 2(gy„Cg-)afP"
ty = -tdA + i*F + hysG +
= 2»(T W ,,C)« / U>5»6P*
= 2i(yby,C)afT'-.
(6.56)
dF = §eX - e6V, • SG = — iey^X — ey^, SA, = \iiygy,X — iiygy,
Note that the new (even) Lie generators Sa yield a new set of "translations" 7>. Just as the Sa C LiAV, so is T" in LzAVAV. We can thus "replace" the physical GLA V of Eq. (6.18) by a "generalized Lie algebra" including both 7> and P" for the sake of covariance considerations, £:
{J"',P",Sa,T"):
leiy6y,A',
- \tD, (6.61)
CP»,5J = 0, [T»,SC,'] = 0, [ P " , r " J = 0, L > ' , T X ] = -ig^T" + ig-xZ>. (6.57)
Notice that the numbers of fermion and boson components are always equal, as required by our study of the "little" algebra.
Only Ji" and P" are "physical," in the sense that they do not involve nilpotent elements. We shall return to £ when studying symmetry and unitarity aspects.
In counting components, we did not consider subsidiary conditions, However, A, clearly has to obey a condition reducing it to three components (in the M ^ 0 case). If we
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impose dM„ = 0 we can replace A, in Eq. (6.58) by A, + S„B, B a pseudoscalar field. X„ can also be replaced by X« + fovd"^) „ and D by D — idfd'A. In that case, (6.61) will also have SB = - e y 5 f - *(dxdx)-1erB8x
F = 0, G = 0. Note that the equations of motion for }//, A, and B reduce the (massless) states to one fermion and one boson. The &(x,8) are reducible. One can also work with chiral projections, by imposing conditions
and an additional contribution to the ^ variation,
M l =F iyda&M
W = \tiy-n,b,B. Also, the SX, hA„ SD can now be re-expressed in terms of contributions which involve them only, SA, = \Uyhy,X -
(d>d"/dy,d*)\iiybyJC
SX = iieyar*"(dpA, — d„A„) — %iD.
(6.62)
where, using the representation (6.49) for Qa, which we shall denote as Q(AV), +f£>*757,<2.
(6.63)
This condition cancels three fields, which now make four fermions and four boson components: D = 0,
X = 0,
(6.65)
CSA± = e^t.H, \ WL.R = 7 i . a ( F ± - tB^±)«, U F ± = — «cty
Indeed, the superfield
= 0,.
where Q" stands for the AV representation of Q as in (6.48). These superfields are now irreducible. The scalar (i.e., no spinor or vector index on <j> itself) superfield 4>R is then composed of A-, \f/R, and F— They transform according to
A, = 0.
Such covariant and superinvariant conditions as (6.62) can be constructed from powers of Q(AV), Q(AV) and their chiral projections. Ferrara, Wess, and Zumino (1974), Salam and Strathdee (1974e), O'Raifeartaigh (1974), and Nilsson and Tchrakian (1975) have developed such a calculus. It is based on the application of the 16 elements of the Clifford algebra (6.22) in its AV realization, using Eq. (6.48), thus yielding differential equations. To construct supersymmetric couplings, one utilizes the above method of identifying coefficients of powers of 8. For instance, if
(6.66)
We identify tf>_ = <J>R, + = <j>L, i.e., 4>_ = (<£+)*, though one could also have unconnected projections. Examples of superfields and Lagrangians will appear in our review of the physical examples in which renormalization and other properties were studied. We refer the reader to the above mentioned articles (Ferrara, Wess, Zumino, 1974; Salam and Strathdee, 1974e) for other examples of superfields, both spinorial (<£„, „" etc.) and tensoral {<j>",<£*"' etc.). Furthermore, Capper (1974) has developed Feynman diagrams reproducting the superfield couplings; these are economical when studying the divergences of multiloop diagrams. Considering the physical complications involved in the use of the Grassmann algebra substrate, it may be necessary at some stage to possess a formalism producing the field multiplets directly from the GLA. One can use the (6.32) set, just as the (6.26) subalgebra was used to construct irreducible representations. To construct nonunitary irreducible field multiplets (Salam and Strathdee, 1974e) one applies QR" and QR°* to a "lowest" representation £>(ji,jz) of the proper Lorentz group. Assuming
$3(x,6) = &i(x,8)$2(x,8) we get four submultiplets: two from the action of Qi}, Qi2, Qn (j,0)] and one from their joint action ~ (0.0), plus the original
we can identify A,(x) =
A^Azix),
$3(x) = A $2+
tyiAi,
etc. Note that since the variation of D in (6.61) was only a divergence, the "Dz" component can be used as a Lagrangian density (Wess and Zumino, 1974a). For the case 0i = fa and W^ = 0 one finds, "D," = t"i>->>(-lAd>A + iifX* + i(d„B) 2 +
F2+G2
or £ = i ( V 0 2 + i(d*By + #cty> + 2F> + 1G2 -id„(,Ad„A)
(6.64)
which is indeed an example of a Lagrangian density. The
0y,y,(*)The total dimensionality is thus 4(2^\ -+- l ) ( 2 j 2 + 1). One can also have a supermultiplet with inverted parities by starting with
These representations are however generally reducible. One can extract pieces by contraction with powers of d/dx?, i.e., graded analogs of subsidiary conditions. In constructing irreducible representations, it is important to recall that considering, as in Sec. II.A, the boson and fermion states as forming a two-dimensional graded vector space V; the boson and fermion quantum fields ij>(x) and
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Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics \l/(x) themselves represent Endo V and Endi V operators, respectively. Their GLA brackets with the QL" and QR" are thus fixed by (1.2). Indeed, one may recover the entire (6.61) set, without the c parameters, by bracketing the Q« directly with the fields \fr(x), A(x), etc. Summing up, for Git a GLA generator, CG»,*AA(*)]- - ( - l ) « M . » a . A W > G J .
( 6 . 6 7)
D. Inclusion of internal symmetries Let the indices i, j = 1, . . . , n denote an internal symmetry such as the SU(2) of /-spin, or SU(3). We then have, in addition to Eqs. (6.16) and (6.17), a set (Salam and Strathdee, 1974b) HQ*i,Q<»)
2«„(r,c)^P"
{0«<,G«> = 2 M „ 3 M . Thus, for isospin £SU(2)'] and assuming that the Qai transform as an isospinor (n = 2), we find the symmetry realized over a 16-dimensional carrier space. (The Clifford algebra will have 256 base elements Qai, i{jQai,Qej], etc.. . .) In fact, we can start with any (j,I) multiplet as the lowest state, and construct a representation with 16 (2j-\-1) (21+1) dimensions. The quantum numbers of the states in the case j = 0, 7 = 0 are given by the action of the 2re raising operators only; their graded products form a smaller Clifford algebra C 2n , whose dimensionality is indeed 22"( = 16 for Ispin), which will indeed create the 22" states of the carrier space. This enables us to get their quantum numbers directly; (§,£), A 2 ( | , | ) , A s (l,i), A 4 (i,i). In this case these are just the 16 matrices of the Dirac-Clifford algebra. They reduce to (j,I)p multiplets:
(o,o)+ © (|,|)' e (i,o)- ffi (o,i)- © (*,§)-' © (o,o)+. Going back to the C4» of (6.69) we note that A?Qai will form the Lie algebra 50(8) D 5 0 ( 6 ) ~ 517(4), so that the 16 states can be grouped in 5C/(4) (Wigner) supermultiplets 1©4©6©4*®1. Indeed, we can use a generalization of Eq. (6.32) instead of (6.68): {Qai.Qa) = 0,
a,b=
1, 2
(6.69)
«2.*,PV> = 0
for rest states. Here we have the same number of odd generators in; the results are the same except that A2Q now contains *C<2<«>Qw*J = 5„i 6 ' which is clearly the su(n) algebra, the rest of SO (8) being given by [Q,Q] and [Q*,Q*]. Note that this "little" GLA now has Qai G i - i ; Qai* £ i i ; l.W'GI,. The (6.69) bracket can be generalized for cases where the representation n differs from n*, such as the SU(3) case: {Q.i,Qtf}
constructed the 0(3) case (fitting 6.69) and discussed the totally antisymmetric features of the multiplets, due to the graded commutativity and filtered structure of the Clifford algebra. I t seemed difficult to reconcile with the physical states in the quark model assignments. However, it was soon noted (Wess, 1974) that if one introduces SU(3)coior ®SU(3)oN, the totally antisymmetric representations will indeed contain the observed states whenever the color indices will contract or antisymmetrize to a singlet. Salam and Strathdee (1974b), Dondi and Sohnius (1974), Lopuszanski and Sohnius (1974), and Firth and Jenkins (1974) have further studied the isospin case and written down some of the Casimir operators of that GLA. We shall leave the case of a local gauge symmetry and the problems relating to fermionic charge operators to our discussion of physical applications of supersymmetry.
Restricting the system to rest states, we get a Clifford algebra, C\n, whose dimensionality is 24" and whose matrix representation acts on a 2 2n vector space.
{Q.i,Qu*) = U»*h}M,
595
= 2S**S
VII. APPLICATIONS OF SUPERSYMMETRY
A. General symmetry considerations All supersymmetric models upon "W or its extension by internal degrees of freedom [[as in Eqs. (6.68)-(6.70)] have in common two simplifying features:
L>,e.«D = o H = £
in
The Clifford algebra is now C 8n , d = 2™, acting on a 2 dimensional carrier space. Salam and Strathdee (1974f) have
(7.2)
Conservation is thus guaranteed. In the case of the Ra of (6.15), which do not commute with H, conservation is ensured by d d - [ # " , & , ] = (-757").? — Rt = 0. (7.3) dt dt These examples can be generalized in the following theorem: "A GLA g is conserved if its even subalgebra £ (the Lie algebra) is conserved, and if its odd generators 0 transform irreducibly under £, and contain at least one nonnilpotent generator 0„." Clearly, £Qa,0a2 C £ and does not vanish, so that (d/dt)Qa = 0, leading to (d/dl)Qi = 0 for all i, through the action of £. We now give a preliminary discussion of the role of the Noether theorem (for recent advances see J. Schwinger, 1951; Orzalesi, 1970; Y. Dothan, 1972; J. Rosen, 1974) in the case of a GLA, and in particular for W . From (6.64) as a Lagrangian, and using the variations in (6.61) and the condition (6.62) we find the conserved (spinor-vector) current, /«"(*) = ((7"d*(X(*) -2»(((F(*) +
T i
B(x)y,)y^(x)))a
CWW».
(7.4)
and 6.=
(6.70) s
(7.1)
and
Jd>xj„B(x).
(7.5)
If we use the superfield tj>(8a,x''), we can recover the conservation of Qa and j a " from the solvable generalized Lie
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algebras of Ruhl and Yunn or of Goddard. However, this implies a fictitious nilpotent x* and a Hilbert space over the Grassman elements. The correct answer thus consists in applying the GLA directly. At present, the reinterpretation for GLA of the Noether theorems is in accordance with the following scheme: Symmetry of Action Ct of Lagrangian density £ (up to d"£') of 5-matrix
uau-1 = a, U£U~l = £ + d*£',
USU-1 = 5
I (d/d.)Q = 0,
(3) J" C A;
P" C 2 , 2<«> = 0
[this includes (6.57)]
(4) (p n 2 = 0
• (d/dl)Q = 0
[example: the conformal algebra SM(2,2)].
d,j» = 0
As we can see, the case (6.57) studied by Goddard is in class (3). The O'Raifeartaigh theorem then forbids masssplitting within a multiplet, if at least one state has a discrete m2 eigenvalue for PfPt'\ 1). However, we can deduce the same result directly from Eq. (7.1) for V and any extension by ?, provided Eq. (7.1) holds.
«
Q = f dfx fix) JJ=
expiaQ.
(7.6)
We replace the lhs by a statement at the level of the algebras, [<2,«] - 0, [<2,£] = d„£\ £Qfl = 0} ^
[example: inhomogeneous isl(6,c) with 72 "translations"]
d„f = 0
*~ \Q = S d*-X f{x)
(7.7)
and we now generalize the brackets to include the GLA multiplication. The physical interpretation of this algebraic version is again equivalent to a symmetry: the discrete permutations of field (or superfietd) components produced by the Q charges as generators of the symmetric group in n elements (» = r + 1, 2r + 1, 2r, 2r, 7, for the Lie algebras A„ Br, CT, Dr, G2). This is in analogy to the realization of the discrete symmetry group of parity by the matrices 70 and 1 for spinors. For a GLA, we use the same counting, after first replacing it by the Lie algebra acting on the same bounded homogeneous domain (Sternberg and Wolf, 1975). The inverse Noether theorem yields either a Lie algebra or a GLA, according to whether the conserved currents (or charges) all have integer spin, or contain a subset with halfinteger spin. This results from the same considerations as in the discussions leading to (6.67).
The Coleman-Mandula (1967) theorem has been extended by Haag, Lopuszanski, and Sohnius (1974) to GLA symmetries of the 5 matrix. However, it should be remarked that symmetry breaking according to the Goldstone scheme will tend to violate the requirement of additivity assumed by Haag el al. in their "no-go" theorem. Goddard (1974) succeeds in defining a complex-valued inner product in a quadrupled Hilbert space (one each for "h, i>o, »s A Do, 1), but loses positive-definiteness. In either case, the Coleman-Mandula theorem doesn't apply. B. Improved renormalizability in a Yukawa and <£•> interaction The first example of a supersymmetric interaction was provided by Wess and Zumino (1974b). They added to the free Lagrangian (6.64), •fifree = J (3„/l )2 + i(d>„B)2 + i ^ 3 ^ + 2F 2 + 2G2,
£ „ = ImiFA + GB - | # ) We hope that the methods discussed in Sees. II.J and ILK and the relations between graded and ordinary Lie algebras, as discussed in Sternberg and Wolf (1975) will be used to discuss the Noether theorems from a more geometrical point of view. The GLA V and its extensions (6.68)-(6.70) represent algebras which contain the Poincare algebra 6°, or (P and 5 (the SU(3)ON ® St/(3)Coioi) a s subalgebras. As GLA, they do not come directly under the cases which have been studied and classified by L. O'Raifeartaigh (1965) or under the no-go theorem of S. Coleman and J. Mandula (1967). However, Goddard (1974) has used Eqs. (6.55)-(6.57) to construct the Lie algebra "equivalent" to V, i.e., having the same vector space as carrier space for their representations. According to Levi's theorem, any Lie algebra E can be written uniquely as a semidirect sum (7.8)
E = A + 2, (I)
where A is semisimple, and 2 solvable, i.e., for 2 = 2, 2<»> = [^("-".S*"- 1 )], a commutator bracket, 2<"> = Ofor some n. O'Raifeartaigh then proves that there are four classes of inclusions of
P" = 2 P" C 2, S - P" ^ 0, [2",2 X ] = 0
(7.9)
a mass term (7.10)
and an interaction •£„ = &.F (A2 - B?) + 2GAB - f(A
-
yiB)f].
(7.11)
The terms (7.9)—(7.11) all transform invariantly up to a 4-divergence, under (6.61) as amended through the introduction of the field B [see discussion after (6.61)]. One can also add a term [see &F in (6.66)] £x =
\F.
(7.12)
A and F are scalar fields, B and G are pseudoscalars, and tp is a Majorana spinor. F and G are auxiliary and satisfy the equations of motion, -F = T « ( ^ 2 - B2) + \mA + X/4 -G = igAB + \mB.
(7.13)
Eliminating F and G from the Lagrangian, we find £ = hi^A)2 + Ud„B)2 + £(»8 - m)4> - WU2 + B2) -\gmAiA2 + B2) - ig2iA2 + B2)2 - gfiA - ysB)* - i A Q A + mA + igiA2 - B2)2, (7.14) which represents a nonlinear realization of supersymmetry, corresponding to the elimination of F and G in the linear (6.61), (6.66). We can regroup the part of the "potential"
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which involves the A and B fields only, V = -£(A,B) = WiA + {\/2m)J-\+ lgHA* ~ B2) + \mgA {A' + B>) +i?(A* + B?Y.
597
2
im'B
fZ = 1 - if I r dlk
<
(7.15)
I
—£(A,B) is the "potential" V whose extrema we shall later study in our search for Goldstone-like solutions. Note that the \F term can be eliminated by a shift in A. Salam and Strathdee (1974e) have shown how to derive (7.14) using the superfield calculus. I t results from writing
_
1
V (2rY (k* -f m2)2
1
r-dx
16JT2 J
(7.19)
x'
No diagonal mass is generated for either A or B.
The quadratic divergence of the self-energy cancels out and the remaining logarithmically divergent contribution is proportional to — p*. Similarly, the ip self-energy is pro£ = i ( O 0 8 ( * + * - ) - *Qe(F(*+.) + * ( * . ) ) , (7-16) portional to iy"p,i, and the corrections to the off-diagonal mass terms mFA and mGB cancel. Thus the only mass with 0_ =
6} d* X £ = / dl X iQ{A.V)£ = i(d/d'e)d* X £ +surface term = 0. The relevant, terms in £ are obtained by setting 0 = 0 , yielding (7.14).
K, - gZ*.
(7.21)
No divergent trilinear or quadrilinear interactions are Before we study the effects of renormalization (and dis- generated. Iliopoulos and Zumino (1974) and Tsao (1974) regarding the J2X term at this stage), we already observe have investigated this model in higher orders. For two-loop in (7.14) the expected result of a symmetry: A, B, and <J> diagrams they calculated explicitly the various contributions have related bare masses. The three interactions ($',
Explicit symmetry breaking (in contradistinction to "spontaneous" breaking) is tried by the above authors in the form of a term £ S B = cA
(7.23)
(rather than £\, which was invariant under V). £ S B is not invariant under V, and breaks current conservation, d„j" = <$.
In the one-loop approximation, there is only one renormalization needed, a logarithmically divergent wave function renormalization constant Z, common to A, B, yj/,
(7.24)
However, the entire renormalization program is unaffected, with only finite corrections appearing due to £ S B The masses are now only related by the equation
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mA' + mi
= 2m?
(7.25)
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Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics
derived in the tree approximation. In higher order the equation gets finite corrections.
mixes with the vacuum when transforming under the Qa. Looking at Eq. (6.61) or (6.66) or at a chiral-summed form of the latter,
The £SB term can be eliminated by a simultaneous shift of A and F, A —-> A + a, F —> F + f, with the equations
AA
J 4 / + 2ma + go? = 0 (7.26)
12mf + 2gaf + c = 0
which ensure vanishing of linear terms in A or F. Eliminating / we get a cubic equation for a,
••
Cff>,
AB = -Cytf, A+ = (F+ Gy6) - iui(A + AF = | i C c ^ ,
By,),
(7.32)
AG = hiCyt&l,,
(7.27)
we see that F is the scalar field which is connected to ^ under the transformation
(a3 is the "central" value). Taking in Eq. (7.14) B = fi = 0 and A —> a we have a "potential" — £(a) = V(a)
Note that for didactic reasons we have written (7.32) as A, a discrete transformation (e = 1) involving only the GLA, without going through the Grassmann elements A V. Taking Eq. (7.33) between vacuum states we should get vanishing contributions, except if the vacuum is not superinvariant. In the latter case,
a(2m + ga){(m + ga) + kc) = 0. Taking the limit c —» 0, this has three solutions, ai = 0,
ai = —2m/g,
8„,A* =
a3 = —m/g
V(a) = im?a? + igma? + | g V + ca = W(m+ igay+ ca.
(7.33)
{Q.,M-
(7.28) (Af) = (F). (7.34) Our solutions a, correspond to the stationarity points of V(a). We see that V(a{) = 0, V (a2) = - 2mc/g —> 0, Indeed, Salam and Strathdee (1974c) showed that the as V(a«) = i(w7g 2 ) - c ( m / g ) - ^ | ( m 4 / g 2 ) so that ax and o2 solution (Iliopoulos and Zumino, 1974) of Eq. (7.27) corproduce minima, and a.% is a maximum. This is unstable, responds (for c = 0) to with no possible stabilization through a sign change. From (F) = m>/4g - i \ (7.35) Eq. (7.14) we see that (for c—*0, i.e., vanishing of explicit symmetry breaking) so that the vacuum does break supersymmetry "spontaneously" as in Eq. (7.34), with the massless Majorana \j/ as tmf/ = m 4- , • 0 for o3 Goldstone fermion. For (7.30) they got the equations so that this is a "Goldstone spinor" solution, which is, M(A±i")+ geii>'c"'">(A±*b)(A±>'°) = - ( i V ° > however, unstable. Notice that Eq. (7.25) then requires one of the two bosons to be a tachyon, if the other one is M{F^")+ 2gei'h°>"(A^b)(F^i") = 0. massive. Indeed, from Eq. (7.14) we have to first order in g Diagonalizing the (A '") and choosing an SU(2) symmetric + 3gma — § g2o2; for o3, mj? •• solution (^4+"*) = X-l, the equations reduce to the cubic — » * 2 — igma — i g V ; MB' .2 = for a3, me,2 = — i-m2. \(M+ 2g\)(M + 4gX*) = 0. (7.36) Salam and Strathdee (1974c) have investigated directly the idea of a Goldstone spinor in that same Lagrangian, with similar results.
There are thus three solutions X;, all conserving parity: Xi = 0 ,
X2 = -M/2g,
X3 =
-M/ig.
Xi corresponds to unbroken supersymmetry. For X3 one finds
To include the effects of an internal symmetry, Salam and Strathdee (1974e) have rewritten Eq. (7.16) with cf>+ < ^ ± " > = -(M/4g)a-«, < r V » ) = -(ifcfV8g)S'°, and <£_ =
Summing up the situation with respect to spontaneous supersymmetry breaking, we do not yet have here a stable example where this really occurs ((F) ^ 0). Iliopoulos and and
F± = ( l / v 2 ) ( F ± iG),
(7.31)
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599
(supersymmetry with an Abelian local gauge) that it can be done.
Jmax = 1 multiplet and are both massless. Obviously,- this theory is both superinvariant and electric-gauge invariant.
Ferrara, Iliopoulos, and Zumino (1974) have investigated the Gell-Mann—Low ("renormalization group") eigenvalue equations for the "0 supersymmetric model in (7.14). They find that as implied by Eqs. (7.20) and (7.21), there is no eigenvalue solution other than g = 0, a theory of free fields. Thus the effective coupling increases indefinitely with k2. The result is the same for m —> 0, a °W supersymmetric theory.
The construction of this model involves some complication, with the original Lagrangian appearing as an infinite power series in e and highly nonrenormalizable. However, superinvariance, gauge invariance, and an additional symmetry corresponding to the commutator of Qa with the "electric charge" local gauge generator provide a choice of the latter hybrid gauge such that £, is greatly simplified. In the one-loop approximation, supersymmetry causes various cancellations between divergent contributions and the model is renormalizable. The masses remain equal within the multiplets, and f d'k 1 < W m . = e(l + ie'I), I = / . (7.39)
The improved renormalizability of Lagrangian theories due to supersymmetry has led to the expectation that some otherwise unrenormalizable Lagrangians might become renormalizable when supersymmetry is imposed. There is as yet no example where this has happened. Lang and Wess (1974) and Woo (1974) tried Lagrangians with A*, B*, $A2\p, ipABy$\// V supersymmetric terms. Although there were numerous divergence cancellations, the theory remains unrenormalizable. C. Supersymmetry and Abelian gauges; existence of a Goldstone-Higgs case Wess and Zumino (1974c) have constructed a model theory which appears to involve a minimal set of fields sufficient for the inclusion of an interaction resembling electrodynamics, i.e., a coupling to a conserved charge resulting from an Abelian local gauge. This is the final Lagrangian, after the elimination of several auxiliary fields:
J ( » * k>
In both vacuum polarization and light-by-light scattering there is no need for any special treatment (regularization or other) of the diagrams to ensure gauge invariance of the results. Fayet and Iliopoulos (1974) have added to this Lagrangian a parity-breaking supersymmetric and gauge-invariant term. I t amounts to the appearance of off-diagonal mass terms between A iBj, -Se(AlBi-A1B1).
(7.40)
Diagonalizing, they obtain the fields Ai, -B,: A, = ( 4 , 4 - B 2 )/v2, Br = (Bi - A2)/yJ2,
£ = i E [(<M.) 2 + (d„B
.42 = (A i+
5,)/VZ,
S2 = (B2 - 4 i ) / v 2
(7-41)
with mass terms
- I E
2
2
- | (m2 + £«) (,4V + J,*) - i(m>--
[ V U i + #,- ) + imfi+i]
t=l,2
- IV„.V"
(7.42)
and a quartic self-coupling, -fe 2 C47 - ^2 2 + Bf - BJ)\
+ -XQX
2 - e[F"(^i3>„-^2 -4- B^-Bi
-
+ iX{{Al + 7 B £i)^2 - (A, + e2
T 5 5 2 )^i>]
There are now two cases. Taking £e > 0, we have
i=l ,2
(7.37) Here the charged fields A, B, and \p are given in terms of their real components Hand Ad,rB = Ad„B — ( d ^ B ] {A = ( l / \ 5 ) ( ^ + a , ) ,
B = ( l / v 2 ) ( B i + iB2),
* = (l/VZK^+i^).
(7.43)
We note that the masses of the A, B components are thus no longer equal to the mass of \(/, i.e., supersymmetry appears broken. The role of the Goldstone particle is played by the massless X spinor.
ifiyrfi)
+ ~ \y„V»{ E {A? + BS) + {A^B2 - AzBtf-}. 2
| e ) (I22 + Sj)
(7.38)
case a: case b :
m2 — £e > 0 m1 — £e < 0.
In case a, the origin is an absolute minimum of the A, B fields. Ordinary ("electric") gauge invariance in unbroken, V" is massless and the "electromagnetic" interaction is given by
We observe that beyond the electromagnetic interactions eV'l- E (.A.^.rSi) + i^iy^,2 of the massive charged fields A (0 + ), B(0~), and ^(i), we i=l ,2 have a Yukawa-like coupling of the same strength, of X, ^ , e2 and A or B (as if the electromagnetic field were replaced + - K | i F " [ E {Af + B*)]. (7.44) by the spinor X). The masses of A, B, and
201
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Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics
the direction of Ai and translate that field, Az-*A~i + a, a' = 2(£e - m1)/* > 0 (7.45) which yields modified mass terms (due to the quartic interaction),
D. The Yang-Mills field; fermion number gauges
Salam and Strathdee (1974d) and Ferrara and Zumino (1974) have constructed the supersymmetric version of the Yang-Mills field, i.e., a supersymmetric Lagrangian which is also invariant under the action of a local non-Abelian 2 2 ™ G?i) = 2m'; w (5,) = 2m*; »>(.£,) gauge group (a gauge "of the second kind"). We refer the = 2(|e - »»2) > 0; m*(B2) = 0 ; reader to the original papers and to the review by Salam w>(F„) = { « - » , » > 0. (7.46) and Strathdee (1974f) for the details of the construction. The end result involves n "matter fields" which are realized As for the spinors, the mass term in the Lagrangian by a /max = i supermultiplet each ("scalar" superfields, in the notation of these authors) and one / m a x = 1 superbecomes multiplet (the Yang-Mills superfield), all lying in the ad—\im($i4/i + fa+i) — ieXaiysfc — ifo). (7.47) joint representation of the gauge algebra. In principle, the matter fields are not restricted to the adjoint representation: This will be diagonalized into the new spinor fields: also, they can be massive, whereas the / m a x = 1 superfield 1i = IC(1 + cos^)^i - (1 - cos0)7S^2 - VZ sin/3X], is massless because of gauge invariance. However, there is then no way of introducing a fermion number group. The V2 = J [ ( l - cos/3) 75 ^, + (1 + cosjS)*, + VZ sinffysxj, spinor component of the / „ „ = 1 gauge field is a Majorana f = (v2)-i sinj8(^i+ 7»iW + cos/3X, spinor, which cannot carry a charge since it is its own charge /3 = arctan(ga/»j), conjugate by definition. This can be set right through a mixing with another Majorana spinor, belonging to the and the masses will then be matter fields. For this purpose, the matter fields become m(vi) = «(i)») = 0» 2 +
4,(L—^—~v^,0 a
(.V^
^
x
{V**
j
^
b
a + b + c = 0.
\^[
x c
1 * = — (+v + i+A), VZ
(7-51)
where tv is the Majorana field in the V multiplet, and ipA belongs to a matter multiplet with A and B. The baryonic gauge is given by $ —• e* a <£,
$ * —• e~ *<*<£>*.
The conserved supercurrent is given by j " = Ti{-iV,J[y',y'Jr* -0{A - 755)7"*}-
+
i&A,Bjriy**> (7.52)
All coupling constants in Eq. (7.49) are given by g, the gauge coupling. The theory will be asymptotically free provided the Callan-Symanzik function /3 in the Gell-MannLow eigenvalue problem remains negative (and if the theory remains renormalizable). For the case of n massive matter supermultiplets, and SU{N) symmetry (Ferrara and Zumino, 1974) 0 = - (g716jr2) (3 - n)N.
^? 5
(7.53)
Thus « < 3 preserves asymptotic freedom, and » = 3 yields a finite renormalization constant (/3 = 0).
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Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics Suzuki (1974) has studied the possibility of spontaneous symmetry breakdown and the emergence of masses for the various fields. I t appears that the supersymmetric limit is not realized as a local minimum in every possible direction in the parameter space of independent couplings. Nevertheless, asymptotic freedom will not be ruined by the inclusion of "soft" explicit or spontaneous breaking. "Soft" implies canonical dimensionality less than four. The 0 functions are unaffected, but new and superrenormalizable couplings enter into the renormalization group equation. A search for asymptotic freedom with a massive gauge supermultiplet failed to produce such an example. A model in which the gauge is chiral and thus doubles the gauge supermultiplet provides an alternative way of generating a complex spinor gauge field capable of carrying a baryonic charge (R. Delbourgo, A. Salam, and J. Strathdee, 1974). The theory is symmetric between V — A and V + A. Explicit masses are still forbidden, however, in the matter fields. In Sec. VI [see Eqs. (6.68)-(6.70)] we discussed a "nontrivial" inclusion of internal symmetry (the Yang-Mills and other cases we discussed here being "trivial" in the sense that they do not involve a supersymmetry GLA other than V or W ) . As described in Sec. VI, the main physical result appears to consist of a reproduction of the quark model states when the internal group is taken to be S{7(3) 00 i or X SU(3)ON- The local gauge result in (7.49) might on the other hand be taken to represent an important physical requirement imposed on phenomenological hadron fields in their m —• 0 limit, which would explain saturation at the three-quark level through the requirement that physical states belong to the adjoint representation of the internal algebra. However, this would point to a special role for baryons in 8 as against 10. VIII. RESULTS AND PROSPECTS The actual physical results to date can thus be summed up in the following list: (1) Like every other symmetry, quantum statistics (Bose-Fermi) independence (supersymmetry) implies very strong constraints on couplings and masses. Although all models studied to date are only formal models which do not correspond to reality, perhaps a modification of the method might lead to an explanation of some of the observed regularities in the mass spectrum of the hadrons. These observed regularities, which indicate relations involving small integers between masses of fermions and bosons, have been connected to various aspects of the quark model in a heuristic fashion. Perhaps these relations might emerge from a supergauge symmetry for a more sophisticated Lagrangian. Some better understanding of a relativistic quark model may already be provided by the GLA approach [see Eqs. (6.68)-(6.70)].
(3) We remind the reader that an independent pathway leading to supersymmetry was evolved in the search for a Goldstone role for fermions. Furthermore, other attempts were made to use fermions in spontaneous symmetry breakdown of the supersymmetric models in strong and weak interactions. I t seems that the classical solution for a stable symmetry breakdown cannot be used in the more sophisticated models, but radiative corrections may improve the situation. (4) The straightforward generalization of the Y a n g Mills gauge requires the fermion field to behave like the Yang-Mills vector field under the internal symmetry. If we impose the graded Lie symmetry upon the phenomenological fields, this would explain the appearance of the baryons in an octet, and thus the nonexistence of quarks. This seems very intriguing. On the other hand, one may include the internal degrees of freedom nontrivially in the context of a larger GLA imposed on the fundamental fields. To reproduce the observed multiplets, the internal symmetry has to contain the color variable as well. We thus are led to the most interesting challenge: is there a formalism which will fix uniquely the structure of the fundamental system and its interactions: quarks, leptons, Yang-Mills "color" gauge fields, Higgs-Kibble fields to provide masses, the Weinberg-Salam intermediate bosons, etc. ACKNOWLEDGMENT One of the authors (Y.N.) would like to thank Professor M. Gell-Mann for numerous discussions and collaborative efforts in attempting to understand the origins of supersymmetry and some of its applications. APPENDIX In this appendix we quickly review some of the basic facts concerning Clifford algebras and exterior algebras. Let V be a vector space over a field K. (In the cases of interest to us K will either be the field of real numbers or the field of complex numbers.) We recall that the tensor algebra, T(V), is the graded, associative algebra T(V) = T°(V) + Tl(V) + T*(V) -\
+ V ® V-\
= K + V
,
where T°(F) = K, PiJ) = V, and T*(V) is the space of contravariant tensors of degree k, i.e., Tk(V) is the &-fold tensor product of V with itself. We regard V as the subspace, r ' ( F ) , of T(V). The algebra T(V) has the following "universal" property: let I be any linear map of V into some associative algebra, A with unit. Then there exists a unique homomorphism,
(2) Renormalizability is sometimes improved. In the examples cited the supersymmetry greatly reduces the number of renormalization constants. Previously unrelated types of interactions become connected via a supergauge invariant Lagrangian, which may thus help in unification schemes.
601
N O elements of V lie in I(Q),
let I(Q) be the elements v ® v algebra, T(V)/ will be denoted
and so the
map
V—>T(V)/I(Q) — CQ(V) is injective, and we can regard V as a subspace of CQ(V). If I is any map of V into an associative algebra A with unit satisfying the identity Hv? =
Q(v)i,
then the homomorphism, (p, from T(V) to A must vanish
Rev. M o d . Phys., Vol. 47, No. 3, July 1975
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Corwin, Ne'eman, and Sternberg: Graded Lie algebras in mathematics and physics
on the ideal I(Q), and hence defines a homomorphism, and these generate since we can clearly move from any one which we continue to denote by
REFERENCES The Clifford algebra CQ{V) is generated by 1 and the elements of V. Any graded derivation of CQ(V) must vanish on 1, and is thus determined by its action on elements of V, which can be arbitrary. In particular, any v* in the dual space of V induces a derivation which is determined by sending v to »*(»)• 1. This graded derivation is known as the interior product by v* and is denoted by i„«. Let us denote left multiplication by an element v of V by e„, so that e„w = vw. Then
Aharonov, Y., A. Casher, and L. Susskind, 1971, Phys. Lett. B 35, 512. Atiyah, M., R. Bott, and A. Shapiro, 1964, Topology 3, 3. Bella, G., 1973, Nuovo Cimento A 16, 143. Berezin, F. A., and G. I. Kac, 1970, Mat. Sb. (USSR) 82, 124 (English translation 1970, 11, 311). Boerner, H., 1963, Representations of Groups (North-Holland, Amsterdam), Chap. VIII. Brout, R., 1967, Nuovo Cimento A 47, 932. Cabibbo, N., 1968, in Hadrons and their Interactions, edited by A. Zichichi (Academic, New York). Capper, D. M., 1974, Trieste report, IC/74/66. Chevalley, C , and S. Eilenberg, 1948, Trans. Am. Math. Soc. 63, 85. i,*evw = iv*(vw) = v*(y)w — »(i„«tt>) = v*(v)w — e,i„*w, Coleman, S., and J. Mandula, 1967, Phys. Rev. 159, 1251. Coleman, S., and E. Weinberg, 1973, Phys. Rev. D 7, 1888. so that Corwin, L., Y. Ne'eman, and S. Sternberg, 1974, Tel Aviv Univ. Report TAUP 448-74. iv*e, + e*iv* = v*(v)id. (Al) Delbourgo, R., A. Salam, and J. Strathdee, 1974, Phys. Lett. B 51, 475. This is valid for any Clifford algebra over V, in particular Dolen, R., D. Horn, and C. Schmidt, 1967, Phys. Rev. Lett. 19, 402. Dondi, P. H., and M. Sohnius, 1974, Nucl. Phys. B 81, 317. for the exterior algebra. Dothan, Y., 1972, Nuovo Cimento A 11, 499. Fayet, P., and J. Iliopoulos, 1974, Phys. Lett. B 51, 461. Suppose that V carries a nondegenerate symmetric Ferrara, S., and E. Remiddi, 1974, CERN report TH 1935; (to be published). bilinear form whose associated quadratic form is Q. Then Ferrara, S., J. Wess, and B. Zumino, 1974, Phys. Lett. B 51, 239. we can identify V with V* and write iu for the interior Ferrara, S., and B. Zumino, 1974, Nucl. Phys. B 79, 413. product by an element of V, where iuv = {u,v)\ and ( , ) Firth, R. J., and J. D. Jenkins, 1974, Durham University report, to be is the given scalar product. Applied to the exterior algebra, published in Nucl. Phys. AV, the operator iu is recognized as the annihilation operator Fock, V., 1932, Z. Phys. 75, 622. for fermions, and the operator e» is recognized as the creation Frolicher, A., and Z. Nijenhuis, 1957, Proc. Natl. Acad. Sci. USA 43, 239. operator. In this case the equation (Al) becomes the Gell-Mann, M., and Y. Ne'eman, 1964, The Eightfold Way (Benjamin, familiar anticommutation relation N. Y.). Gell-Mann, M., and Y. Ne'eman, 1974 (unpublished). iuev + e„i„ = (u,v)id (A2) Gerstenhaber, M., 1963, Ann. Math. 78, 267. Gerstenhaber, M., 1964, Ann. Math. 79, 59. for fermions. If we set Gervais, J. L., and B. Sakita, 1971, Nucl. Phys. B 34, 633. Goddard, P., 1974, Berkeley report LBL-3347; to be published in ' ' » = « » + iv and s„ = ev — i„, Nucl. Phys. B. Guillemin, Y., and S. Sternberg, Deformation theory of pseudogroup then structures, Mem. Am. Math. Soc. 64, 1966. Haag, R., J. T. Lopuszanski, and M. Sohnius, 1974, Karlsruhe Unir-ufv + r,ru = 2(u,v), sus» + J„J„ = —2 («,»), versity Preprint. Iliopoulos, J., and B. Zumino, 1974, Nucl. Phys. B 76, 310. and Igi, K., and S. Matsuda, 1967, Phys. Rev. Lett. 18, 625. Jordan, P., and E. Wigner, 1928, Z. Phys. 47, 631. ruS„ + svru = 0. Joseph, A., 1972, Nuovo Cimento A 8, 217. Thus r gives a representation of the algebra CQ(V) on A(V), Kastler, D., 1961, Introduction a electrodynamique quantique (Dunod, Paris). i.e., a homomorphism of CQ(A) into End (A(V)), while s Kodaira, K., L. Nirenberg, and D. C. Spencer, 1962, Ann. Math. 75, gives a representation of the algebra C-Q(V) on A(V). The 536. elements ru and su, as u ranges over V, generate the algebra Kodaira, K., and D. C. Spencer, 1958, Ann. Math. 67, 328-466; 7 End (AT ), since from ru and $„ we can recover e„ and i„, 1959, 70, 145-166; 1960, 71, 43-76; 1961, 74, 52-100. Rev. M o d . Phys., V o l . 47, No. 3, July 1975
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Salam, A., and J. Strathdee, 1974d, Trieste Report IC/74/36, to be published in Phys. Lett. Salam, A., and J. Strathdee, 1974e, Trieste Report IC/74/42 (to be published). Salam, A., and J. Strathdee, 1974f, Trieste Report IC/74/80 (to be published). Schwarz, J. H., 1973, Phys. Rep. 8C, 269. Schwinger, J., 1951, Phys. Rev. 82, 914. Serre, J. P., 1964, Lie Algebras and Lie Groups (Benjamin, New York), Chap. LA3. Souriau J. M., 1970, Structure des Systems Dynamiques (Dunod, Paris). Spencer, D. C , 1962, Ann. Math. 76, 306-445. Sternberg, S., 1964, Lectures on Differential Geometry (Prentice-Hall, Englewood Cliffs). Sternberg, S., and J. A. Wolf, 1975, "Graded Lie Algebras and Bounded Homogeneous Domains," to be published. Suzuki, M., 1974, Berkeley Report LBL 3308, to be published. Tasao, H.-S., 1974, Brandeis University report. Veneziano, G., 1968, Nuovo Cimento A 57, 190. Veneziano, G., 1974, Phys. Rep. 9C, 199. Volkov, D. V., and V. P. Akulov, 1973, Phys. Lett. B 46, 109. Volkov, D. V., and V. A. Soroka, 1973, J E T P Lett. 18, 529 (English translation 18, 312). Wess, J., 1974, Lectures at Karlsruhe Summer School. Wess, J., and B. Zumino, 1974a, Nucl. Phys. B 70, 39. Wess, J., and B. Zumino, 1974b, Phys. Lett. 49B, 52. Wess, J., and B. Zumino, 1974c, CERN report TH 1857; to be published in Nucl. Phys. B. Woo, G., 1974, Cambridge University report DAMTP 74/12. Zumino, B., 1974, in Proceedings of the XVII International Conference on High Energy Physics, London, 1974, to be published. Available as CERN report TH.1901.
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Nuclear Physics B138 (1978) 31-44 © Noith-Holland Publishing Company
AFFINE EXTENSIONS OF SUPERSYMMETRY: THE FINITE CASE Y. NE'EMAN ** Tel-Aviv University, Tel-Aviv, Israel and Center for Particle Theory, Department of Physics, University of Texas, Austin, Texas 78712, USA T.N. SHERRY * Center for Particle Theory, Department of Physics, University of Texas, Austin, Texas 78712, USA Received 31 January 1978
We examine the graded Poincare (GP) Lie algebra of supersymmetry with a view to constructing possible affine extensions of the algebra, i.e. extensions of the GP algebra which contain as a subalgebra the Lie algebra ga(4,r)?). We restrict our attention in this paper to an examination of the finite extensions. We demonstrate explicitly that if we adjoin only a symmetric tensor generator to the GP algebra, then such a generator cannot generate all the deformations, in particular the shear, of the general affine group GA(4, 9?). Similarly, we show that adjoining the supersymmetry generator to ga(4,92) cannot lead to closure of the resulting algebra, even in the trivial case. We further demonstrate that the GLA ga(4/4,92) does not contain the Lie algebra g a ^ , ^ ) represented over the entire superspace upon which ga(4/4,T2) is defined.
1. Introduction Graded Lie algebras [la] ** were introduced in the physics of particles and fields in several contexts. In dual models and strings [2] several infinite-dimensional GLA's were applied as "supergauge" conditions in the construction of the spinning string and were essential in the removal of divergences, etc. In space-time field theory, supersymmetry [3a] *** provided a new generalization of the symmetry concept: * Research supported in part by the United States-Israel Binational Science Foundation. * Reserach supported in part by the VS Energy Research and Development Administration, Grant No. E(4P-1)3992. ** The classification of simple GLA's has been given by KaU [lb]. See also rcf. [lc], *** Wess and Zumino [3b] independently rediscovered the graded Poincare group as a subgroup of their gradcd-conformal SU(2, 2/1). This paper [3b] ushered in the study of supersymmetry. 31
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Y. Ne'eman, T.N. Sherry /Supersymmetry
Bose-Fermi symmetry. The most interesting results have been in the simplification and partial reduction of ultraviolet divergences in various field theories + and in the construction of Goldstone-type models in which supersymmetry is spontaneously broken by a massless spin J~\ fermion *. Higgs-type models in which this fermion is replaced by the acquisition of mass by a / = j field have also been constructed. Supersymmetry has also been applied to gravity, either in92 4 , 4 (V 1 ,6 a ) superspace, first introduced [6] as a representation for supersymmetry, or directly in space-time 9? 4 . Though some insights have been derived in the first approach [7a] ** it is the theory of supergravity [8] which emerged from the second which has supplied the most interesting results. Spectacular progress in the renormalization of gravity [9], positive-definiteness of the Hamiltonian [10], the harmonious cancellation of the difficulties relating to charged/ = y fields [11], etc., represent applications to gravity, aside from the development of supergravity itself and its generalizations [12]. In a recent work, we have suggested a new group theory cum geometry approach to these theories [13]. In discussing gravity as a field theory, one is led to consider GA(4,t72), the general affine group of transformations in a four-dimensional Euclidean (temporarily neglecting the Minkowski structure) space-time. This is the linear subgroup of Einstein's group of general coordinate transformations (GCT) and carries its induced representations [14]. In a future development of the theory, it would seem natural to proceed to a Lie-bracket grading of this group's generator algebra. Moreover, a recent generalization of Einstein's gravity has been suggested [15] in the form of a gauge theory of GA(4,92), the metric-affine theory of gravity. This has been further developed as a theory fitting the hadrons in particular [ 16]. It gives rise to intrinsic currents of spin, dilation and shear (forming together the hypermomentum tensor) which couple to the connection-field (the Yang-Mills field of intrinsic GL(4,92 ), the homogeneous part) in addition to the coupling of the energy-momentum tensor to the (metric) vierbein. To produce intrinsic shear, the hadrons have to belong to the recently constructed [17] band-spinor infinite representations of GL(3,92), embedded in "polyfields" [16] constructed from infinite representations of GL(4,92). As has recently been pointed out [18] such structures produce bona fide bivalued spinorial representations of GL(4,92) and GA(4,92), relevant to gravity theory. Presumably, if we intend to construct the graded spin-extension of GA(4,9?), we should turn to such representations. Embedding the conventional/= A supersymmetry generators of the graded Poincare group in an infinite set with J - f, \ , etc., and looking for the minimal closure with GA(4,92) might yield a four-dimensional analog of the t Wess and Zumino [4a] gave the first direct application to the improvement of renormalizability properties of known theories. For a general review see refs. (l,4b,6). * The first successful "stable" model with spontaneous breakdown was provided by Fayet and lliopoulos [5a]. For more recent comments see ref. [5b]. ** This is a graded-Riemannian geometry. For a superspacc with torsion see ref. [7b].
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V. Ne'eman, T.N. Sherry /Supersymmetry
33
Ramond-Neveu-Schwarz spinning string [2] GLA, perhaps the complete spinningspace-time GLA. However, it seems worthwhile to try first the finite possibilities. The generators of GA(4,92 ), a non-compact Lie group, can be classified in terms of its SO(4) subgroup. We identify this SO(4) group with the Lorentz group, neglecting temporarily the Minkowski structure of space-time. The generators of GA(4,9?) are, in turn: the generators of SO(4), the antisymmetric7M„; the translation generators/^; and a 2nd-rank symmetric tensor TpV representing deformations, either by shear (the traceless J= 2 part) or by dilation (the trace). In this paper we wish to discuss how one should proceed to extend ga(4,92 ), the Lie algebra of GA(4,92), to a finite graded Lie algebra. We shall address this problem from three different angles. To begin with we consider the Z 2 graded spinor extension of the Poincare algebra, GP, i.e., supersymmetry [3]. We adjoin to this algebra a 2nd-rank symmetric tensor generator 7),„ and examine what restrictions the graded Jacobi identity [1] places on its commutation relations with the other elements of the algebra when we require the minimally extended algebra to close on itself. We find that this requires the spin-2 part of TtiV "not to see" the supersymmetry generator, while the spin-0 part behaves as the sumD +E of the scalar generators of the superconformal algebra SU(2, 2/1) [3]. This is consistent with the Cartan theorems [18], stating the inexistence of (finite) spinors in GL(n, 92). Furthermore, the spin-2 part of T^ cannot generate the shear deformations. Approaching the problem from a slightly different point of view, we next examine the Lie algebra ga(4,92), and attempt to extend it by adjoining a Lorentz spinor generator S. Again, postulating closure of the resulting graded Lie algebra, we use the graded Jacobi identities to restrict the commutation and anticommutation properties of S. The result is more startling in this case; the algebra cannot close without the addition of other generators, even in the simplest case when the supersymmetry generators anticommute. We next examine the graded Lie algebra of the group GA(4/4,92). This is the linear subgroup of GCT in the Salam-Strathdee superspace [6,7]. The Bose sector of ga(4/4,92) consists of two ga(4,92) algebras, one generating the affine transformations in x space, and the other generating the affine transformations in 6 space. Our purpose is to find the closure, within ga(4/4, 92), of an algebra containing the generators/, P, and S, and a symmetric tensor T. We identify the smallest closed subalgebra of ga(4/4,92 ) which achieves this result. However, again we find that the symmetric tensor T cannot be identified anymore as a deformation-generating operator. T does generate "orbital" deformations in the space-time sector, but not in the spin sector. Again, this is consistent with the requirement of band-spinors for "intrinsic" deformations. Furthermore, the resulting algebra contains extra generators besides those already considered. We now proceed to exhibit our results explicitly. In sect. 2 we discuss the attempt to extend the GP algebra. In sect. 3 we briefly consider the minimal-spinor-
208
Y. Ne'eman, T.N. Sherry ISupersymmetry
34
extension of ga(4,02), and in sect. 4 we examine the GLA ga(4/4,92 ), and identify the relevant subalgebra. In this paper we use the notation of Corwin, Ne'eman and Sternberg [1].
2. Extension of the GP algebra [3] The generators of the GP algebra are, in turn, the generators of the Poincare Lie algebra, / u „ and P\, and a Lorentz spinor generator Sa. The commutation relations which characterize this algebra are [J»v>J\p] = iVwfv\ [•V' pd
=
-'Wi'
+
inv\Jnp ~ iV^Jvp ~ftvpJtih>
+ iT
li>\pn .
[Pli,Pv]=0,
[P^Sa]=0, {Sa,Sb} = -2(y*QabPK.
(2.1)
This algebra is Z 2 graded,/ u „ a n d / \ belonging to the even gradation and Sa belonging to the odd gradation. The graded Jacobi identity for a GLA is [*. \y.'W = [[*. y\ z] + (-) w l>. [*.']].
(2.2)
where x 6 Lk, y EL{, k = even for even grading = odd for odd grading . We wish to adjoin a symmetric 2nd-rank tensor generator T^v. Thus its commutation relation with ./„„ is given to us as CV» TXp\ ~ -if\n\Tvp + irivXTiip ~ " W 7 ^
+
hvpT^ •
(2.3)
The form of the other commutation relations is open to us, so we write their generic form as
[T^, Sa] =Dfwa
Si, ,
[Ty,v> T\p] = E^v^Pa + ^uvXp /a(3 + G^p01
209
Tap .
(2.4)
Y. Ne 'emtm, T.N. Sherry / Supersymmetry
35
Here we have made use of the remaining property of a graded Lie algebra, namely [Lk, Lt] C Lk+l
;
the commutation relations are consistent with the grading chosen. From their defining commutation relations (2.4), the undetermined coefficients have the following symmetry properties: ,a =-A
A
^a
-a
<*0 - -BnvK
,a^ = C
^=c
C b
b
'- -D
D
Ja
E
nv\pa> ^nu\pa0 and G^^P
are symmetric under (n *-*• v)
and (X *-*• p) and are antisymmetric under (put) *-> (Xp) ,
u
'fivKp
iiv\p
)te that the most general form form D^a
b
D^^
= V^u [a5ab + b(ys)ab] .
(2.6)
Use of the Jacobi identity for [T^, {Sa, Sb}] leads us to the result
<W'
=0
*»»*
= 2a^rhia
'
(2.7) •
(2.8)
All of the Jacobi identities are consistent with the form (2.6) for D^^'. efficient is shown to vanish from the Jacobi identity in the case IV iiv> IVXp> *yil
The F-co-
>
while we also derive the relation G^Xp^VaP = 0 ,
(2.9)
from this same identity. Now, the form of the E- and G-coefficients can be seen from general covariance requirements. Without considering spinor quantities (e.g., 7 matrices), or constant vectors, we cannot construct a quantity such as Ey.va^ which has an odd number of vector indices. On the other hand, the G-coefficient
210
36
Y. Ne'eman, T.N. Sherry / Supersymmetry
could have the form < W " = alV^xpVafl
+ a2rW(r?x
+
lAp")
a3T?a
However, from the symmetry requirements (2.5), we see at once that ax = 0 and a3 ~ -a2 = - c , i.e., G^x,?* = cln^(VxaVP0
+ »?*%") - ThrinfqS
+^ V ) ] .
(2.10)
Furthermore, this form is consistent with the restriction (2.9), and in fact with all the other Jacobi identities. We group together our results as follows: [TpV, P\] = 2ai7MI^Px , [T*v Sa] = TjM„[a5a6 + b(7S)ab] Sb , [T»p. TXp] = 2c[v^T^
- fi^r^,] .
(2.11)
From the second of these equations we see at once that the spin-2 part of T^v, the traceless symmetric tensor 1X2)^ - T^ — \vMi>T does not "see" the supersymmetry generator Sa. On the other hand, the spin-zero part, 71[0)M„ =l4VnvT,where T= rj^Txp, does, as is seen from [7T0V,,, Sa] = ViaSat
+ b(y5)ab) Sb .
From this we see at once that 7X0)M„ is just the sum of the two spin-0 operators D and E of the superconformal algebra SU(2,2/1): nO)„„ = » w ( j « 0 + - j A t f ) ,
(2.12)
where [D, Sa] = %iSa and [E, Sa] = 3/(7 5 ) fl6 S b [1]. Furthermore, from the third of eqs. (2.11) we see that T^ does not generate deformations, even if c were to vanish. The commutation of two deformations should yield the Lorentz generators, and we have already seen that in the third commutation relation of (2.4), the F-coefficients are forced to vanish. 3. Minimal extension of 0 ( 4 , ^ ) The general affine group of transformations is the semi-direct product of the general linear group with the group of translations. Thus, as generators of the general affine group in four dimensions we have the antisymmetric rotation generators J^, the symmetric deformation generators rM„ and the translation generators/^. The Lie algebra of these generators is given by the following commutation relations: Vnv J\p) ~ innpJv\
+
iVi>\Jvp - irlti\JvP - iVvpJfik >
211
Y. Ne'eman, T.N. Sherry [Supersymmetry [•V- P\\
=
37
~iV»\Pi> +iVv\Pfi ,
[•V> TXp) = -huxTpp - in^pTxv + irixvTpp + iVppT^ , IT^, T\p] =iVn\Jvp - hupJ^v +ft*Jup
- hpvJxv ,
[Tuv P\\ = -iVn\Pv - ir)v\Pn , [/>„,/>„] = 0 .
(3.1)
We now consider extending this algebra in a minimal fashion by adding to it a Lorentz spinor generator, which we denote by Sa, though it should not be confused with the generator of sect. 2. Thus, we have \Jpv< Sa\
=
-jiPyvlabSb •
(3-2)
The other commutation relations, using just the closure and the Z 2 grading, take the generic form
[P,i, Sa] = Bpa Sb , [Sa, Sb] = DabaPa + EifJcp
+ Faba0Ta(3 .
(3.3)
However, it is a straightforward exercise to check that these relations are not consistent with the graded Jacobi identities. This follows from the identity P c [Tpu. TKp}} = USC,
] + [TMV, [Sc, T^]]
(3.4)
bi
and from the most general form for Anixva , namely va ApVab
= T?M„[fl5a6 + b(y5)ab]
.
With this form the right-hand side of (3.4) vanishes, whereas the left-hand side gives us j'{Vu\Ovp
- Vnp°\i> + T)X»°np - Vpv°\ji}abSb
>
(3-5)
and this does not vanish.
We are forced to conclude from this result that the assumption of closure of the graded Lie algebra after minimal extension by the addition of a Lorentz spinor generator was false. Already from sect. 2, we can see that this result is possible, for there we had closure but T^ could not generate the deformations. 4. ga(4/4,*X>) and its graded subalgebras In this section we shall be examining the graded Lie algebra of GA(4/4,98 ), and some of its closed subalgebras. GA(4/4,78) can be viewed as the (pseudo) group of
212
38
Y. Ne'eman, T.N. Sherry /Supersymmetry
general affine transformations in a space which has four Bose dimensions and four Fermi dimensions, a space commonly referred to as superspace. This is the linear subgroup of the GCT on superspace. The Bose sector of ga(4/4,92 ) is ga(4,92)i X ga(4,T?) 2 , where GA(4, cR)l is the group of affine transformations on the Bose space, and GA(4,92) 2 is the group of affine transformations on the Fermi space. The S0(4)i subgroup of GA(4,92)i is also defined over the Fermi space; in fact this is the defining feature of the variables of that space. We choose the variables of the Fermi space to transform as spinors under the SO(4)i subgroup. We can realize the generators of the algebra ga(4/4,92) by the differential representations Tuv = /(JCM3„ + xMdv) \ GA(4, f K) 1 acting on the Bose space , ^ = -i9 M s
Lab "
"'"•ST.
Jv a =
««3» GA(4,92 ) 2 acting on the Fermi space
9 -«« d
Q^ =
= *M9<>
odd generators, causing transitions from Bose to Fermi dimensions and vice versa. (4.1)
Rap = ^ 9 M
Here 6a is an anticommuting Majorana spinor, which is thus real by our conventions. We use right-differentiation, so that (4-2)
9 a (Mc) = Bb*ca - M s » . where
From this realization of the generators we find the communication relations of the graded Lie algebra to be: (i) the commutation relations (3.1) for J, T and P; (ii) for the remaining generators [Lai,, Led] - i&adLcb - &cbLad . {Lab. Na] = ftacHb > [J»v Qte] ~ irlv\Qiia ~ iHiikQva »
213
Y. Ne'eman, T.N. Sherry /Supersymmetry [J^v [T^v.
R
a\]
=
>Wv\Ran ~
G\fl] = WV-KQW
[Tfu,, Ra\]
= -ir\v\Ray.
f^afc. 2\c]
=
39
iT
ln*-Rav »
+ iVvxQva
.
- 'Vn\Rav
>
>&acQ\b »
[Z,a6, /? c ^] = -iScbRaX ,
{Qua. Rbv) = 5 ' 5 a 6 ( V + T^) - \ir\nvLba .
(4-3)
and all others vanish. There are in all 72 generators. For our own purposes it will be more convenient to work with a slightly different set of generators. Rather than the 16 generators Lab used above, we use the equivalent set of Lorentz-covariant generators ' uv ~ (S-Onv*~)abLiab >
^n
=
(Cy^ysOabLab.
^s
=
ifysC)abLab
,
F=(C\QabLab.
(4.4)
(Here Cis the charge-conjugation matrix.) These are tensor, vector, axial-vector, pseudo-scalar and scalar generators, respectively, under the action of Lorentz transformations. The commutation relations among this set of generators are [*V> F *p] = 2 toMP Vv\ + Vnx V^p - r)^ Vvp - vvp V^] , I'H'
' v\
~ ~ *•')!»
lVt]LV,VK\ =
>
2(r,MlVll~Vh)1Vv),
214
40
Y. Ne'eman, T.N. Sherry I Supersymmetry
\VIJL,F5\=-2iAll, [All,Fs]=+2iVtl,
(4.5)
and all others vanish. The commutation relations with the other generators of the algebra can be easily evaluated from eqs. (4.3). As can be seen from the 1st relation of (4.5), \iVyj, furnishes a representation of the SO(4)i generators/M„, over the 6 part of superspace. Thus we have 2 representations of 2 SO(4) groups over 6 space:
so(4),... i/*v = -4(c
i6J™bb.
It is straightforward to see that, since det lm" = 0, while det jtu> ¥= 0, these are not equivalent representations of an SO(4) group. We can now identify the GP subalgebra of global supersymmetry, which acts on both sections of the superspace. We denote the generators by script letters, and they are apv = Jp.v + \i\'pv >
?M = Pll,
-CabNt
= -CabNb
- S^W^M
-fr-Va-
(4.6)
These generators satisfy the GP commutation relations of eqs. (2.1). We notice that (7- R)a is one of the spin-j projections oiRail. For the present this is the only projection ofRati that we have need of. We can further identify a 2nd-rank symmetric tensor generator 7^ with respect to 9JUD- I n this there are two arbitrary parameters .4 and B: J HV ~ 1 pV
The commutation relations with the generators of (4.6) are
[ %v,
c5a] = -irjpV [-2A5ab + 2B(y5)ab]6b
[ 7^,
9X] =
-JTJ M 3>„
- iVvX9„ ,
215
+ iVpV(A + fay • R)a , (4.8)
Y. Ne'eman, T.N. Sherry /Supersymme try
+ iVw[9np ~ 3 ' ^ p ] - iVPA9\p ~ J'v^]
41
•
(4-9)
From (4.9) we see that 7MV does not generate the deformations, as on taking the commutation of two 7 transformations we are not left with a pure £ transformation. We further notice that 9 ^ , 5>M, 6a and 7^v do not close to form a subalgebra of ga(4/4,92). In the case A = —\, we need to consider also the generator V^, while in the general case (A ¥= —\) we should also include (y • R)a. The commutation relations of F ^ with 9 , 5", cS and 7 are as follows: \9pv, V\p\ =i[VppVvk +Vu\Vpp -Vp\Vvp WpV,
-VvPVp\]
.
VXp]=0,
[da. V\p] = ~i(.Oxp)ab^b + «y\)abRbp
- fop W ? w O •
(4-10)
Clearly then we also need to include the generator RbP, even in the case A = —\. Its commutation relations are [ 9pv, Ra\]
~ irlv\Rap
I %v. Ra\\
= -hvxRaix
{6a,Rc\}
~ irlp\Rav - WpkKv
- 2(
- ^TsW^cA.) ,
= -Cac^x ,
[Ran, V\p) = -K°*p)acRcp •
(411)
Grouping together the sets of commutation relations (2.1), (4.8), (4.9), (4.10), (4.11) and the first of eqs. (4.5), we see that the42 generators 9 pv> 7^, 9>fl, KM„ and Rafl close to form a graded subalgebra of GA(4/4,92 ). What we have constructed is the smallest graded subalgebra of GA(4/4,92 ) which contains both the GP algebra of simple supersymme try and a second-rank symmetric tensor generator (under Lorentz transformations). However, we have seen explicitly that this symmetric tensor generator does not generate deformations, and so we cannot consider the algebra we have found to be the supersymmetric extension of ga(4,9?).
5. Discussion In this paper we have re-examined the algebra of supersymme try with a view to extending it in the direction of the general affine transformations. The extension of the GP algebra to the graded conformal algebra SU(2, 2/JV) has been known since shortly after supersymmetry itself was introduced [3]. Our attempts in this
216
Y. Ne'eman, T.N. Sherry / Supersymmetry
42
direction have not led to the more interesting physical solutions containing complete deformations. On the other hand, the cases we have examined might well be useful for less ambitious purposes. The general affine groups of transformations are characterized by their 2nd-rank symmetric tensor generators which on commutation yield the generators of their special orthogonal subgroup. Thus, as a first attempt, in sect. 2, we adjoined to the GP algebra a 2nd-rank symmetric tensor. Requiring that the resulting operators close on commutation, and using the graded Jacobi identities, we found that the adjoined tensor generator does not generate deformations. In fact, the analysis showed that the traceless (shear) or spin-2, part commutes with the supersymmetry generators, while the trace, or spin-0, part behaves like a linear combination of the scalar and pseudo-scalar generators of SU(2, 2/1) (i.e., the dilations and chiral transformations). On commutation amongst themselves, the symmetric tensor generators can at most reproduce themselves, or vanish. We examined essentially the same problem, from a different point of view, in section 3. There, we began with the general affine group GA(4,92), and considered adjoining to its Lie algebra a generator transforming as a spinor operator under the SO(4) subgroup of GA(4,92). We demonstrated explicitly that the resulting algebra could not close: not even trivially with the supersymmetry generators anticommuting. We are thus led inexorably to consider larger graded Lie algebras. The example we exafnined in sect. 4 is realized most simply on superspace. It is the GLA of the linear subgroup of the general coordinate transformations on superspace, ga(4/4,92 ). The Bose sector of this GLA consists of ga(4,92 ) ^ X ga(4,92 ) 2 where ga(4,92 ) i acts on the x space and ga(4,92) 2 acts on the 0 space. However, the SO(4)i subgroup of GA(4,92)j also acts on 8 space, defining the 0's as spinor coordinates. Making use of this defining feature of the superspace we identified the GP subalgebra of ga(4/4, 92). This algebra was extended to include the most general 2nd-rank symmetric tensor-generator. For closure, however, other generators of ga(4/4,92) were required. From the resulting commutation relations it is apparent that the symmetric tensor generators do not generate complete deformations. This analysis is telling us that we cannot represent the T^„ generators of GA(4,92) linearly on the 8 space. In actual fact, there is no non-linear differential representation either. This follows from the result that the GCT group on 8 space is generated by a finite-dimensional GLA. There are 64 generators in all, represented as follows with the multiplicities and gradings shown: generators
multiplicity
grading
217
Y. Ne'eman, T.N. Sherry I Supersymmetry
generators
multiplicity
grading
eceb±
24
+i
WcOb~
16
+2
4
+3
WA—
43
the grading operator being 6a d/90fl. The only possibility to represent Tw with the correct covariant structure would utilize 6a6c8b d/b8a, but these generators actually commute. This finding may well have intrinsic importance. On superspace, as it is conventionally defined, we cannot represent the group GA(4,92). Thus, the full potential of gravity cannot be realized on such spaces. As we have seen there are defined on such spaces algebras more general than the GP algebra. It remains to be seen whether or not the algebra we have exhibited possesses double-valued representations of the band-spinor type. In the meantime we conjecture that the algebra does not possess such representations. Thus, we are led to the result that embedding space-time in a larger space in which one specific realization of spin is selected, automatically eliminates the possibility of holonomic spinors, intrinsic shear, etc., and thus restricts gravity in at least its spin degrees of freedom. At any rate, it seems apparent that if we restrict our attention to spin-j-generators alone, we cannot extend the GP algebra in the affine direction. Our next avenue of investigation is to examine GLA's with higher spin supersymmetry generators. In the light of the above discussion we do not restrict our attention to a finite spin. We adjoin to ga(4,9?) an operator which transforms as a GA(4,92 ) spinor. Such spinors have recently been shown to exist [18], and they are infinite-dimensional representations of the group. However, as their structure is not fully known as yet, we shall have to content ourselves with an analysis of GA(3,9?), for which the multiplicityfree spinor representations have been classified [17]. It is an intriguing possibility that the realization of such a group of transformations as the symmetry group of a "superspace" would require a superspace with a countably infinite number of spinor sectors. Such a program of investigation offers more interesting possibilities, and will be reported elsewhere. One of the authors (Y.N.) would like to thank Dr. L. Smalley for raising the issue discussed in this paper. References [1J (a) L. Corwin, Y. Ne'eman and S. Sternberg, Rev. Mod. Phys. 47 (1975) 573. (b) V.G. Katz, Functional analysis and applications (USSR) 9 (1975) 91.
218
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Y. Ne'eman, T.N. Sherry / Supersymmetry
(c) P.G.O. Freund and I. Kaplansky, J. Math. Phys. 17 (1973) 228; M. Scheunert, W. Nahm and V. Rittenberg, J. Math. Phys. 17 (1976) 1626. [2] P. Ramond, Phys. Rev. D3 (1971) 2415; A. Neveu and J. Schwarz, Nucl. Phys. B31 (1971) 86; Y. Aharonov, A. Cahser and L. Susskind, Phys. Lett 35B (1971) 512; J.L. Gervais and B. Sakita, Nucl. Phys. B34 (1971) 633; P. Ramond and J.H. Schwarz, Phys. Lett. 64B (1976) 75. [3] (a) Yu.A. Golfand and E.P. Likhtman, JETP Lett. 13 (1971) 452. (b) J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. [4] (a) J. Wess and B. Zumino, Phys. Lett. 49B (1974) 52. (b) B. Zumino, Proc. 17th Int. Conf. on high-energy physics, London 1974, ed. J.R. Smith (Rutherford Lab., Chilton, Didcot, UK, 1974) p. 1-254; S. Ferrara, Rivista Nuovo Cim. 6 (1976) 105; P. Fayet and S. Ferrara, Phys. Reports 32 (1977) 249. [5] (a) P. Fayet and J. Iliopoulos, Phys. Lett. 51B (1974) 461; B. deWitt and D.Z. Freedman, Phys. Rev. Lett. 35 (1975) 827. (b) B. Zumino, Proc. 1977 Coral Gables Conf., CERN report TH 2293. [6] A. Salam and J. Strathdee, Nucl. Phys. B76 (1974) 477;Phys. Rev. D l l (1975) 1521. [7] (a) R. Arnowitt and P. Nath, Phys. Lett. 56B (1975) 117; R. Arnowitt, P. Nath and B. Zumino, Phys. Lett. 56B (1971) 81. (b) V.P. Akulov, D.V. Volkov and V.A. Soroka, JETP Lett. 22 (1975) 396; B. Zumino, Proc. Conf. on gauge theories and modern field theory, ed. R. Arnowitt and P. Nath, (MIT Press, Cambridge, Mass., 1975); J. Wess and B. Zumino, Phys. Lett. 66B (1977) 361. [8] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214; S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335; S.W. MacDoweU and F. Mansouri, Phys. Rev. Lett. 38 (1977) 739. [9] M.T. Grisaru, P. van Nieuwenhuizen and J.A.M. Vermaseren, Phys. Rev. Lett. 37 (1976) 1662; M.T. Grisaru, Phys. Lett. 66B (1977) 75; S. Deser, J.H. Kay and K.S. Stelle, Phys. Rev. Lett. 38 (1977) 527. [10] S. Deser and C. Teitelboim, Phys. Rev. Lett. 39 (1977) 249. [11] A. Das and D.Z. Freedman, Nucl. Phys. B114 (1976) 271; S. Deser and B. Zumino, Phys. Rev. Lett. 38 (1977) 1433. [12] S. Ferrara, J. Scherk and B. Zumino, Phys. Lett. 66B (1977) 35. [13] Y. Ne'eman and T. Regge, U. of Texas preprint CPT 326, Nuovo Cim., to be published. [14] Y. Ne'eman, Proc. 1977 Bonn Symp. on applications of differential geometry to physics, ed. K. Bleuler and A. Reetz, Lecture Notes in Maths. Series (Springer) to be published. [15] F.W. Hehl, G.D. Kerlick and P. von d. Heyde, Phys. Lett. 63B (1976) 446. [16] F.W. Hehl, E.A. Lord and Y. Ne'eman, Phys. Lett. 71B (1977) 432; Phys. Rev. D17 (1978) 428 ]17] Y. Dothan, M. GeU-Mann and Y. Ne'eman, Phys. Lett. 17 (1965) 148; Y. Dothan and Y. Ne'eman, in Symmetry groups in nuclear and particle physics, ed. F. Dyson, (Benjamin, NY, 1965) p. 287; D.W. Joseph, Un. of Nebraska preprint, 1970, unpublished; L.C. Biedenharm, R.Y. Cusson, M.Y. Han and D.L. Weaver, Phys. Lett. 42B (1972) 257; V.I. Ogievetsky and E. Sokachev, Theor. Math. Phys. (USSR) 23 (1975) 214; Dj. Sijacki, J. Math. Phys. 16 (1975) 298. [18] Y. Ne'eman, Nat. Acad. Sci. USA 74 (1977) 4157; Ann. Inst. Henri Poincare, to be published.
219
Volume 76B, number 4
PHYSICS LETTERS
19 June 1978
GRADED SPIN-EXTENSION OF THE ALGEBRA OF VOLUME-PRESERVING DEFORMATIONS Y. NE'EMAN l Tel-Aviv University, Tel-Aviv, Israel and Center for Particle Theory, Department of Physics, University of Texas, Austin, Texas 78712, USA and T.N. SHERRY 2 Center for Particle Theory, Department of Physics, University of Texas, Austin, Texas 78712, USA Received 4 April 1978
We present an infinite GLA analogous to the Spinning String (vibrating-rotating membranes?) based on SL(3, R) plus two infinite sets of operators S'm (spinors, / = 1/2, 5/2, 9/2...) and E'm (tensors, / = 1,3,5,...), with brackets {S, S} C E, [£•,£"1 = 0, [S,E] = 0. The algebra gsl(3, R). We have constructed a minimal Graded Lie Algebra [1] gsl (3.R)* 1 describing the spin-excitations of sl(3,R), the algebra of SL(3,R), i.e., the covering of the group of transformations preserving a 3-dimensional (determinant) volume-element [2]. This is an infinite-dimensional algebra with generators Jk (k = 1,0, —1) for angular momentum, Th (ft = 2, 1,0, - 1 , - 2 ) for Shear, S'm {/ = (4n + l)/2, m=j, ; - l , . . . - / ; n = 0 , 1 , 2,...} and Em {/ = 2w + l; m = / , / - l , . . . - ; ' ; n = 0,1,2,...}.The spins are | / ( / ) = 1, \J(T)\ = 2, \J(S0\ = / , \J(Ei)\=j, with magnetic numbers k, h, and m assigned to the lower indices. {Jk, Tjj} form sl(3,R) itself, while Sm and Em behave as components of the SL(3,R) bandorsQ)(l/2,0) and 1){\,c) respectively. These are unitary infinite-dimensional multiplicity-free representations [3,4] of SL(3,R), - o o < c <«, c eR. Q(I/2,0) is the only double-valued multiplicity-free bandor [5] in SL(3,R) and it becomes single valued in SL(3,R). In addition to the commutation relations which are thereby defined, closure is achieved with the graded Lie brackets: Research supported in part by the United States-Israel Binational Science Foundation. 2 Research supported in part by the U.S. Department of Energy, Grant No. E(40-l) 3992. *' In our notation g stands for graded, so gsl(3, R) is the graded extension of sl(3,R), and similarly for GSL(3,R) and SL(3,R).
+ (
a) (2) (3)
The structure constants A'Jm, are given by recurrnrti
^
1/?
1/2
rence relations from a single parameter A \A 1/2 which can be absorbed into the definition of S^. The first relation, 1/2 A mm'W + / ' - m ~ ™'X/ + / ' + m+m'+l)]
= t0'-m')(;' + W ' + l ) ] 1 ^mm ; . .+1 (4) m+lm' > is obtained from the (graded) Jacobi identity [1] for [J+l,{Sm,Sm,}] and given/,/' it can be recurrently solved for all A'„\m> in terms otAll.:
i-v 1/+/7 A/r 1 Ai /; =7llL
f2AfK
etc..
The graded Jacobi identity for [T+2,{Sm,$&•}] provides further relations from which we can extract the recurrence relation for the All., 413
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{ah}= {1,2,6,2,1}
for h = {2,1,0,-1,-2}
(12) L4(2/+l)(2/+3)(2/--"" and t(J,c) is given by eq. (6). The commutation relations of gsl(3,R) are fixed by the graded Jacobi identities, by the requirement that it include aside from sl(3,R)a / = 1/2 generator S 1 ' 2 withspinor behavior under the {Jh} so (3,R) and by minimal closure. Proofs will be published elsewhere. A look at equations (1)—(3) and (7)—(11) shows that magnetic numbers are preserved additively. We thus have, aside from the trivial Z2 grading, a Z gradingfl]
*]
(5) This enables us to derive as a first step, 1/2 tV.c) . 1/2 1/2 ASI2 1/2 = r_6]_l ^5/2 1/2 [_2!5!J 2r(l/2,0) / 4 l/2l/2-
and so forth, remembering the symmetry of/l^ m . (eq. (1)). The coefficients t(j, c) have been computed in the construction of the multiplicity-free bandors of SL(3, R),
r
<»*#•*
f,
gsl(3,R)= © £,-,
Jo>To>EoeLo T±2'Et2
E'me.L±2m
[T±2,T;1]=?2y/2Jtl,
h>0,
(9)
[r0,r±1] = + 2 ^ 7 ± 1 ,
and the commutators of sl(3,R) with either of the two operator-bandors S^ or E'm, denoted by B^m, based upon Joseph's formulae [3] for SL(3,R) bandors, [JQ,Bim]=mBlm,
(10)
V±»Bln]**[k(j*m)(J±m+\)]Wlilm±l, [Th,Bim] ~ (- ] > ah [U_m_2~h)\
U+m-2+h)\}
\(j+m+h+2)\ (j-m-h+2)]}1!2
±4 ei
v
E'±leL±2
/
V;\
> ±2a+l
Wj>m ,
a = 0,1, m>2.
Affine supergravity. It seems to us that there are three avenues in which gsl(3,R) might be useful. The first relates to Gravity and Supergravity. Gravitational theories involve GL(4,R). In Einstein's theory, GL(4,R) appears holonomically, as the linear subgroup of the General Covariance Group [6] and determines the covariant derivative (in analogy to the role of isospin in nonlinear chiral symmetry). We have recently shown that this role can be extended to spinors [5], provided they belong to bandors. Several gauge theories of gravities have been based on a GL(4,R) gauge [7]. We have shown [8] that in any of these theories, spinor matter would appear in anholonomic bandors. Thus, to the extent that Einstein's theory is extended to a GL(4,R) gauge, or that we introduce holonomic bandors in the original theory, Supergravity would have to be extended from its Graded-Poincare supersymmetry vacuum limit to a Graded-Linear or Graded-Affine foundation. Our gsl(3,R) is the 3-dimensional model or subalgebra of the sought after gsl(4,R) or ggl(4,R), whose construction requires (as yet unpublished) information on
[rft>r_ft] = -(-i)"2w 0 , for
f
(8)
[JiVTh]=+[3-\h(h±l)]V2Thtl,
[T±2,Tih]=0
eL
Si( 2 a + l ) / 2
(7)
[J0,Th]=hTk,
I€i
(13)
(6)
[J + 1 ,/_i] = -J0 ,
for
where
L4(2/+5)(2/+l)(2/+3)2j For the sake of completeness, we display the formulae for the commutators of sl(3,R) itself, [JQ,Jtl] = ±J±l ,
[2J0,X]=iX
l(j c B
> > ™+h i+2
where 414
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SL(4,R)bandors. To understand the connection with Supersymmetry, we have to identify the operators representing (3dimensional) translations. Trying to adjoin a set Ph (h= 1,0,-1) to the right-hand side of eq. (1), we find it is forbidden by the [J,{S, S}] graded Jacobi identity. However, extending SL(3,R) to GL(3,R) by adjoining the dilation generator D, we find
Note that finite dimensional SL(4,R) extensions of Supergravity can be ruled out, since there is no finitedimensional spinor representation of SL(4R), and it would be impossible to assign a behavior under the shear generators T^" for the Sa. We have investigated elsewhere possible "compromise" solutions [10], but they do not preserve the physical and algebraic properties of shear.
[D,S}=0,
Other applications. SL(3R) has been applied to deformed nuclei [11], and it would seem interesting to check on the predictions from representations of gsl(3,R). The type of excitations arising from a spinextension can be guessed at from the similar extension in Dual Models [12] of the original Veneziano string. Note that this is anyhow a third avenue worth exploring: the Nambu action for the string preserved a twodimensional surface element (the area spanned in space-time by the evolution of the string) and was SL(2,R) invariant. The infinite sets of generators [1] Lm and Gn in the NSR model [13] correspond respectively to Bargmann [14] C° and C\<2 representations. That our gsl(3, R) and a hypothetical gsl(4,R) might respectively provide the basis for similar treatments of an evolving membrane and of an evolving lump appears plausible. One should then check the conditions which have been found to be useful in the case of the string, for example one should determine whether or not the GLAs are canonically representable [12].
lD,S>m]=GJ.s'm,
[A71=0, \D,Ein\=HJ.Ein.
Applying (JDS), (TDS) and (DSE) Jacobi identities we find the only nontrivial result GJ. = G6f ,
Hi = 2GSJ- , V/ .
The constant G can be absorbed in the definition of G. We thus discover a grading for ggl(3R) by the dimension, as in Supersymmetry and the Superconformal GLAs [1 ]. It is a Z-grading, with L( trivial for |/'| > 2 and Jk,Th,DeL0,
S^GLi
V/,m,
4 , € L 2 V/,m. E}m thus has the dimension of a translation, and S^ that of a supersymmetry generator Sa. We should thus connect EJ„ with 3-dimensional translations, with V = {Jk, S^( , E^,} a 3-dimensional analog of supersymmetry. To achieve exact identification, we perform a two-stage group-contraction. The necessary and sufficient condition for such a contraction to a subalgebra V is that V be closed under commutation [9]. We contract first with respect to K1, then with respect to V2 = {Jk, S'm, E'm}. As a result of the second contraction, the Th generators totally decouple from V1, which can then be completely identified with 3-dimensional supersymmetry (or the gradedorthonormal algebra giso(3,R)). At will, we can keep the scaled S3m, £"£, with /' > 2 and enlarge the contracted algebra. In fact, the uncontracted E'm, S'm are the (unitary) translations and Supersymmetry generators of the infinite (homogeneous) superspace GSL(3,R)/SL(3,R), where the graded group manifold stands for a Berezin-Kats formal group. The E'm translations behave unitarily under SL(3,R) as against the conventional nonunitary behaviour of translations in finite spaces.
One of us (TNS) wishes to thank E.C.G. Sudarshan for a number of very helpful discussions. References [1] L. Corwin, Y. Ne'eman and S. Sternberg, Rev. Mod. Phys. 47 (1975) 573. There is no available classification of infinite dimensional GLAs. [2) Y. Dothan, M. Gell-Mann and Y. Ne'eman, Phys. Lett. 17 (1965) 148. (3] D.W. Joseph, Representations of the Algebra of SL(3,R) with A/'= 2, Un. of Nebraska preprint (1970) unpublished, referred to in ref. [11]. [4] Dj. Sijacki, J.M.P. 16 (1975) 298 has completed the survey of the representations of SL(3,R). V.I. Ogievetsky and E. Sakachev, Teor. Mat. Fiz. 23 (1975) 214 for a recent survey of the multiplicity-free set. [5] Y. Ne'eman, Proc. Natl. Acad. Sci. USA 74 (1977) 211.
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[6] B.S. DeWitt, in: Relativity, groups and topology (Les Houches 1963 Seminar) eds. C. and B. DeWitt (Gordon and Breach, New York, 1964). [7] C.N. Yang, Phys. Rev. Lett. 33 (1974) 445; G. Stephanson, Nuovo Cimento 9 (1958) 263; F. Mansouii and L.N. Chang, Phys. Rev. D13 (1976) 3192; F.W. Hehl, G.D. Kerlick and P. Von der Heyde, Phys. Lett. 63B (1976) 446; F.W. Hehl, E.A. Lord and Y. Ne'eman, Phys. Lett. 71B (1977)432. [8] Y. Ne'eman, IHES preprint, to be published in Ann. Inst. H. Poincare; F.W. Hehl, E.A. Lord and Y. Ne'eman, Phys. Rev. D16 (1977).
19 June 1978
[9] R. Gilmore, Lie Groups, Lie Algebras and some of their applications (Wiley, 1974). [10] Y. Ne'eman and T. Sherry, Affine extensions of supersymmetry: the finite case, CPT preprint ORO 329 (Jan. 1978). [11] L.C. Biedenharm, R.Y. Cusson, M.Y. Han and D.L. Weaver, Phys. Lett. 42B (1972) 257. [12] See for example J. Scherk, Rev. Mod. Phys. 47 (1975) 123; P. Ramond and J.H. Schwarz. Phys. Lett. 64B (1976) 75. [13] A. Neveu and J.H. Schwarz, Nucl. Phys. B31 (1971) 86; P. Ramond, Phys. Rev. D3 (1971) 2415. [14] V. Bargmann, Ann. Math. 48 (1947) 568.
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Ann. Phys. (Leipzig) 8 (1999) 1, 3 - 1 7
Quantizing gravity and space time: Where do we stand? Yuval Ne'eman Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel Aviv, Israel 69978, and Center for Particle Physics, University of Texas, Austin, Texas 78712, USA Received 13 July 1998, accepted 23 July 1998 by F. W. Hehl Abstract. We review five possible solutions to the riddle posed by Quantum Gravity: (1) Gravity should stay as a classical theory (L. Rosenfeld); (2) Quantum Gravity requires a formalism which will take the human mind (or the intelligent observer) into account, resolving at the same time the riddle of the collapse of the wave function/state vector in Quantum Mechanics in general (Penrose); (3) Perturbative Quantization; (4) Hamiltonian Quantization (Dirac, Ashtekar); (5) String Theory. We also discuss the quantization of spacetime. Keywords: Theory of gravity; Quantum gravity; String theory
1 Why hasn't it been done yet? General Relativity (GR), Einstein's theory of gravity, was conceived in 1915 and Quantum Mechanics (QM) in 1925, and yet, at the end of this Century we still do not have an established theoretical framework unifying these two theories. Ideally, we would like such a "higher" theory to reduce to nonrelativistic QM for a spacetime with vanishing curvature and infinite velocity of light - and to Einstein's GR when Planck's constant h —> 0. Alternatively, as QM and Special Relativity (SR) have already been consolidated in 1946-1971 within Relativistic Quantum Field Theory (RQFT), the new paradigm should either itself be a RQFT, or reduce to a RQFT for a limiting value of some new parameter. The RQFT program was launched in 1932, but in the beginning the method was plagued by "divergences" - infinities appearing everywhere. These infinities derived from the point-size of the electron or of the interaction region. Even classically, a zero-size electron tends to blow up, since its two halves are at zero distance from each other and their mutual Coulomb repulsion is therefore infinite. This difficulty was resolved in 1946—48 by the method of renormalization, namely an a priori subtraction of these infinite effects. Subtracting two or three such spurious effects was shown to provide an exact theory, namely Quantum Electrodynamics (QED), the most precise theory in physics. An extension of the same method was successful both for Quantum Chromodynamics (QCD) and for the Unified Electroweak theory in 1960—73. In Quantum Gravity, this hasn't worked to date: the number of necessary subtractions would itself be infinite!
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And yet, the present outlook is not as bleak as it was around 1984, when two Berkeley physicists, M. Goroff and A. Sagnotti, having for the first time access to a Crane supercomputer for RQFT calculations, evaluated the vacuum (i.e. just gravitons interacting with gravitons, with no matter present) "two-loop" diagram. RQFT is treated "perturbatively" - i.e. the quantum amplitude for a physical process undergoes a Taylor expansion in some small parameter (here Newton's constant), the assumption being that evaluating just one or two terms will provide a sufficiently precise answer. For the vacuum diagram each term in the expansion adds a closed loop (a particle pair which is virtually created by a graviton and then reannihilates into one; this involves two gravitational couplings, i.e. a factor (GW)2)- F° r that two-loop calculation, however, the computer's answer was "oo"! More precisely, we should say that the computer would never have stopped, since it could not finish the calculation. I stress this aspect because, as you will see, it is relevant to our story and we shall soon be discussing such nonstopping "universal Turing machines" (UTM) (all our computers are UTM). Returning to Quantum Gravity, the infinite answer should have been expected, from dimensional arguments (see below). However, after their 1971 success in quantizing the Yang-Mills local gauge interaction — now the key mechanism behind the Standard Model (QCD -I- the Electroweak theory), the 1975 grand synthesis of particle interactions (excluding gravity) - M. Veltman and G. 't Hooft had tried in 1974 to apply similar methods to gravity and had calculated the "one-loop" vacuum contribution. Due to a fortuitous topological effect, the calculation had shown that amplitude to be finite, and this had created hopes for further "miracles" - which, finally, did not occur. The blame for the nonrenormalizability of the gravitational interaction goes to the dimensionality of Newton's constant GN, which is that of an area — as against the electric charge e, or more precisely the famous a = -^c = 1/137 (and the analogous coupling in Yang-Mills theories) which are dimensionless. This is due to a key difference in the action I = J dAxC\ the Maxwell or Yang-Mills Lagrangian densities CYM — FA*F are quadratic in the field-strengths (or curvatures) F = dA in QED or F = dA - A A A in the Yang-Mills case (A is the potential or connection) whereas the Einstein-Hilbert Lagrangian is linear in the curvature R = dr - r A r, (r is the Christoffel connection): CEH = (8nGN)~x R. Taking the dimension (denoted as []) of lengths [JC] — 1, and since the action is counted in natural quanta and is thus dimensionless [I] — 0, we should have for its 4-density [£] = - 4 . All connections have the dimensions of a derivative, from their role in covariant derivatives D — d-A or D = d-T, i.e. [A] = [r] = - 1 , or [F] = [Rl= - 2 . Thus [£YM] = [FA*F] = -4. On the other hand, as the scalar curvature R in CEH is a contraction of R, we have [R] = —2, so that to achieve [CEH] = - 4 we have to assign [GN] — 2. As a result, every order of Perturbation Theory (PT) will give rise in the renormalization process to a new subtraction counter-term, with a new dimensionality. Thus, aside from the expected conceptual complications, specific to gravity, arising from the identification of the metric of spacetime, in itself, with the gravitational field (whereas in all other interactions spacetime is just the "arena" where the game is played), the nonvanishing dimensionality of GN is therefore "the root of all evil". Note that the Weak Interaction, in its 1957-71 "effective" Cw = GFpjw (GF is
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5
the Fermi constant) "current x current" formulation, was also nonrenormalizable, for a similar reason, though in the opposite direction, \j-Jjw] = - 6 , i.e. [Gp] = 2, until it was replaced by the Weinberg-Salam theory, with a Yang-Mills Lagrangian with spontaneous symmetry breaking. We do know a few things about the Quantum Gravity ("Planck") regime. We may ask - at which energies will gravity become a fully quantum interaction, or at which lengths do we expect spacetime itself to be quantized? A particle behaves least classically, the closer we get to its Compton wavelength IQ = £-c- A star's gravitational field, on the other hand, is strongest when the star collapses to its Schwarzschild radius rs — 2Qf± (about 2 km for the Sun), becoming a Black Hole. Should a particle have IQ — rs, it would be both a black hole and behave quantumwise. This would also be the limit of localizability, since every time we would try to pin down the particle's position, the quantum energy fluctuations would be of the order of that mc2 and a black hole would form. Putting in the numbers, we find lP = 10"32 cm, tP = 10"43 sec, mP = 10~5 g, EP = 1019 GeV. Note that this also implies that the regime of Quantum Gravity (QG) is that of a strong coupling: remember that we often stress the fact that the gravitational attraction between two protons (i.e. two masses with mc2 — 1 GeV each) is weaker by a factor 10~38 than their electromagnetic repulsion. But between two Planck-mass particles, i.e. masses of 1019 GeV each, it will be precisely 1038 times stronger, i.e. the same as the electromagnetic! This strength of the gravitational coupling, however, is a "technical" difficulty (making it hard to apply perturbation methods even if we resolve the renormalization issue) but we are learning to cope with this kind of problem in the case of QCD, with lattice methods, for instance. Of course, the ideal solution would consist in having an appropriate exact (nonperturbative) solution of the QG equations. Such solutions have been constructed in recent years for gravity in two dimensions, a "theoretical laboratory", using String Theory. This is then the problem. For the sake of completeness, I shall sketch eight different approaches which have been tried as the answer. Note that most of the positive advances have occured in the last two methods, i.e. Canonical Quantization "a la Ashtekar" and "String Theory", now renamed "M theory". 2 Pleading "there is no case" We first review three approaches which follow the legal practice of pleading "there is no case", for fundamental reasons. Position I: "Gravity stays classical as a "frame" with the measurement apparatus". The late Leon Rosenfeld, a collaborator of Niels Bohr, suggested at the time that gravity should not be quantized, as it should represent the classical background frame, together with the measurement apparatus, which has to be classical in the Copenhagen interpretation of QM. Rosenfeld's aim was thus to use gravity in order to explain the existence of the classical apparatus in a quantum world. However, the presently favored approach to this issue (Zeh, Zureck and others [1]) is to assume that the world is in a quantum state, but that macroscopic objects decohere as a result of multiple interactions with the environment. We have shown the implications for
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the perception of space time [2]. Moreover, Rosenfeld's thesis was shown to contradict the quantum Uncertainty Relations: it would be possible to use gravity to measure location (e.g. as Leverrier located Neptune from the perturbations in the orbit of Uranus) and velocity simultaneously. Position II: "Gravity cannot be quantized as long as the human mind is not included in the quantum description". In two recent highly readable books, Roger Penrose [3] claims that the route to Quantum Gravity is blocked because something essential is missing in QM, relating to the human mind. When found, it will also resolve the main conceptual difficulty in the interpretation of QM, namely the so-called collapse of the state vector, whenever a measurement is performed. Though this approach is very much R. Penrose's idiosyncratic conception, its sources can be traced to the views of E. Wigner [4] and J.A. Wheeler [5] on QM. I recall debating such views with E. Wigner in Trieste in 1968 and assisting J. A. Wheeler in an exposition of his participatory approach regarding the role of the (human?) observer in quantum reality by providing him with references to Jewish exegetic texts with a similar trend — though without my own adherance to those views. Penrose has no real arguments or definite proposals for the description of Quantum Gravity, but he has gone to great lengths in an exposition of his views as relating to human thinking. Rejecting extra-scientific approaches, he claims to have provided a mathematical proof that human thinking is nonalgorithmic, i.e. that it cannot be reproduced by a computer. Penrose assumes that some physical processes are of a nonalgorithmic nature and could therefore exist in the human brain. These processes would also play a key role in quantum measurements (thus related to a human observer) and in Quantum Gravity, where they would enter through the Quantum aspect and as a background to measurement. Penrose is thus in conflict with the views of the Artificial Intelligence community, since they assume an affinity between computers and the human brain. His first book in this area is dedicated to this debate with the AI community, his second to the mathematical proof that human thinking is nonalgorithmic. The proof is an adaptation of Godel's famous one, inspired by the problematics of the Russel-Whitehead paradox. Since any computer can be reduced to a Universal Turing Machine, let us list all such machines by ordering them (index q). Some Turing machines never stop - e.g. if told to construct an example of a geographical map, which would require n = 5 different colors, in order never to have two different contiguous countries (with a common border of nonzero length) colored with one and the same color. Such a machine is denoted T(q, 5). Ever since the proof of the topological four-colors theorem (a proof which was indeed achieved with the help of a computer) we know that no n = 5 case will be found, i.e. T(q, 5) never stops. The computer which proved that four-color theorem (i.e. "it stopped" for any n > 4), thus proved that T(q, n > 4) never stops; we would denote this computer by A(l, n > 4), since it was the first (q' — 1) to do it and it indeed proved that any T(q, n > 4) never stops. Thus, if A(q', n > 4) stops, T(q, n > 4) does not. Very generally, we thus designate the machines A(q', n) and T(q, n), so that when A(q', n) stops, T(q, n) doesn't.
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Assume our minds work algorithmically, then A(q', n) can describe a human proof of the same, i.e. that T(q, n) never stops. Penrose indeed defines A(q', n) as the set of all such human proofs. Now if the human mind works algorithmically, it emulates a computer and can be identified with some Turing machine A(q', n) = T(q", n). Now take a case q' = q", we get A(q', n) = T(q', n), which contains a contradiction, since for T(q', n) never to stop, A(q', n) should stop, so that it cannot equal T(q', n), which does not stop. Penrose then claims that he has just proven that at least one type of human mind processes cannot be identified with one of the T(q, ri), i.e. it is nonalgorithmic. The status of this proof is not clear. Some logicians have found that it is not rigorous and does not stand up to a careful analysis. Personally, I too do not accept its conclusion, but I am no expert. I assume it will take some time before the Mathematical Logics community reaches a concensus on this matter. Note, however, that there is no indication that all this should be more relevant to Gravity than to any other interaction. Position III: "Quantum Gravity exists only at the 'effective' level in QFT". Here the claim is that Quantum Gravity exists, but not as a fundamental interaction; only Classical Gravity should thus have this status. QG represents an effective theory, induced by the collective quantum fluctuations of the "matter" fields, in the classically curved background. This is a more sophisticated version of position I, in which one assumes that Quantum Field Theory and its radiative corrections generate an "effective" QG, thereby at the same time doing away with the paradoxes of route I. The approach has been suggested and developed by S. Adler, A. Zee, V. Brindejonc and G. Cohen-Tannoudji, Novozhilov and others [6]. The "effective" theory (
(1)
so that the functional variation by dg^ produces a semi-quantized Einstein equation I T g - Ginduced(^v - (1/2) Rgftv + /linducedguv) = ( 1 / 2 ) (0, g\ T^ |0, g) .
(2)
The expectation is that GjndUCed is of the order of G#. However, as stressed by Brindejonc and Cohen-Tannoudji, with the matter Lagrangian containing e.g. QCD, it would appear more plausible to expect at low energy a "Strong Gravity"like force, of the type which might explain Color Confinement [7], rather than anything related to G^.
3 Patching up the (perturbative) quantum field theory treatment The next three routes persist in the construction of a perturbative QFT of gravity, while attempting to cure some key sick feature, in the hope that this will suffice. The proposed cures are applied at various levels: (a) modifying the treatment of spacetime itself; (b) replacing GN or LEH by dimensionally correct expressions; (c) imposing the powerful algebraic constraints of supersymmetry.
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Position IV: "Quantum Spacetime involves Noncommutative Geometry". The idea is to adapt the treatment of spacetime to its discontinuous nature at the Planck length level. One hopes that this will also clean up the infinities, once there are no more "true points". In particular, A. Connes' new mathematical theory of Noncommutative Geometry [8] might be a fitting tool. It is a new extension of the concepts of differential geometry, providing for the treatment of discrete (or quantized) spaces as if they were manifolds, including the construction of the relevant algebra of forms, and a generalization of the de Rham cohomology. I shall return to this last version in my treatment of the next "position". In Connes' approach, a space is defined by an algebra of functions on it A, a Dirac (or translation) operator and a Hilbert space on which it acts. Let us take the simplest version relevant to our subject. A sphere S2 in 3-space R3 is given by the (commutative) algebra of complex functions C over it; this algebra is generated by the coordinates xl. With the Euclidean metric g,y = (5,y we define the sphere S2 by gijx'x' = r2. The algebra C(S2) of complex-valued functions f(xl) can be described through its polynomial expansion f(x') =/o +fxl + (1/2) fyx'x' + ... with the fijk..n carrying symmetrized indices, since the original [x',x'} — 0. If we put all coefficients to zero except for /o, we reduce S2 to a point. If we keep in addition the term linear in x', we get a 4-dimensional vector space. As an algebra, i.e. a system having the operations of addition and multiplication, we arrange things so as to identify it with M2(C), the algebra of 2 x 2 complex matrices, for example. We postulate that the three xl be replaced by x' which are no more commutative; instead, these obey the commutation relations of the generator Lie algebra of SU(2), i.e. they can be written, in terms of the 2 x 2 Pauli a matrices, x' = KO', with K normalized accordingly by r2 — 3K2 and thus [xl, x'} = 2iKs''kxk. The sphere has now become fuzzy- Only the North and South poles can be distinguished (the two states obtained upon diagonalization of x3 = KO3). The /o,/i multiply a basis in M.2{C). If we now preserve in addition a nonzero fy too, we obtain a 9-dimensional vector space (the equation of the sphere removes one of the quadratic terms). We preserve the identification of the x' with the Lie generators of 5(7(2) but we replace the Pauli matrices by the 3-dimensional representation of SU(2), i.e. 3 x 3 matrices. We thus obtain the 9-dimensional complex algebra Mj(C). More generally, for a truncation after fjk... with n — 1 indices, we work with the n x n matrices of the ;' = (n - l)/2 representation of SU(2) and our algebra of functions is the n2-dimensional complex M„(C). It is important here to distinguish between (a) the Lie algebra of SU(2) whose symmetric representations (Young tableaux with all squares along one row only) provide us, for any truncation of the Taylor expansion of the f(x), with the set of appropriate mappings of the xl onto matrices - and (b) the Mn(C) "algebra of functions" resulting from ordinary matrix products of the x'-x'.-.x"'1 or lower. Such products (times their coefficients fij..n-x) do not reproduce themselves in any other representations of the Lie algebra beyond the selected ; ' = ( « - l)/2. This example is just an appetizer, but it shows how one may replace spacetime by fuzzy spaces and avoid localization up to a point. This treatment can at least achieve in spatial evaluations what a cut-off would do in energy or momentum, in a more elegant fashion. Whether or not this might help resolving the difficulties of
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Quantum Gravity is not clear. Very recently, it has appeared that noncommutative geometry might assist in formulating solutions involving other treatments — those we present in our next sections, gauge field theory, canonical quantization or some version of string theory ("M-theory"). Position V: "Quantum Gravity involves a larger and spontaneously broken symmetry of the frames, reducing to the Lorentz group". This method consists in modifying the gravitational Lagrangian so as to get out of the problematics arising from the bad dimensionality of G^. The program was launched by F. Englert and coll. [9]. The idea is to replace Newton's constant by a "soft" expression, namely the square of the vacuum-expectation value (VEV) of a scalar ("dilaton") field, (G^)- 1 = «0| 0 |0» 2 .
(3)
Such a scalar field appears in some variations of Einstein's theory — Weyl's, the Jordan-Dirac-Brans-Dicke theory, etc. It breaks scale invariance and its nonvanishing VEV can be produced through a spontaneous (therefore "softer") breakdown. For instance, one may assume a "Higgs potential" V = -(ji)2 (j>2 + X(t>4 - with a minimum at (0|
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Position VI: "Apply Supersymmetry as an algebraic constraint improving renormalizability". This algebraic constraint has been shown to work miracles in some cases, either reducing the number of "infinite constants" which have to be subtracted (e.g. just one, instead of two, in the Wess-Zumino model [16]) or even making the theory finite with no subtraction at all (supersymmetrized N — 4 Yang-Mills model [17]). Algebraically, these examples belong to a class of theories with maximal helicity ymax _ ^yy^ yy being the number of supersymmetry "flavors". There is thus a possibility that the same "miracle" might occur for N = 8 Supergravity [18]. This was the path we proposed, Gell-Mann and I, in a programmatic address at Aspen in 1976, pointing to N = 8 Supergravity as a holy grail. The theory was indeed constructed in 1978 (it was a true tour de force) by E. Cremmer, B. Julia and the late Joel Scherk [18]. They showed that it is equivalent to N = 1 supergravity in 11-dimensions, up to a Kaluza-Klein dimensional reduction — i.e. the compactification of seven dimensions. That such a compactification can indeed occur spontaneously was shown by P. G. O. Freund and M. A. Rubin [19]. As to renormalizability, what has been proven to-date is only that the amplitudes up to eight loops are finite [20], simply extending the Veltman - ' t Hooft one loop result via N — 8 Supersymmetry. However, the same mechanism, when acting beyond eight loops, might well just yield a similar extension of the two-loop Goroff-Sagnotti infinite result - if the N — 4]™* algebraic constraint should fail. With the 1984 renewed interest in Strings, the N — 8 SUSY program was abandoned, in a pessimistic rejection. The first counter-indication came in 1995, straight from Superstring and Supermembrane [21] theories. Casher [22] has shown that (1) assuming (see our last section) the D = 10 Quantum Superstring is indeed the "Theory of Everything, (TOE) and (2) assuming, as is generally done, that its truncation underneath Planck energy (the supposedly massless sector) is a Relativistic Quantum Field Theory (RQFT), one proves that this truncated RQFT is finite. This RQFT is a version of D — 10 supergravity. However, it has recently been shown that the D = 10 Quantum Superstring is itself the reduction of a D — 11 "M-theory" [23], the theory of a Quantum Supermembrane, which, under truncation, yields our D —11 Supergravity [24]. Applying Casher's theorem to M-theory thus appears to imply the finiteness of the Cremmer-Julia-Scherk model, perhaps justifying our 1976 optimist Aspen program. Or is this starry-eyed optimism?
4 The quantum superstring or supermembrane We now arrive at the two most successful programs, one perturbative, the Quantum Superstring, and one nonperturbative, namely Dirac's Canonical (Hamiltonian) Quantization program, as realized in Ashtekar's version. Position VII: "Replace QFT (i.e. point particles) by strings, membranes or higher dimensional extended structures ("extendons" - or in the present lingua franca "p-branes)". The identification of the Quantum Superstring with a theory of Gravity, is due to Yoneya and to Schwarz and Scherk [25]. Until then, it had been a candidate
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theory of the Strong Interactions and of the hadrons. The Dolan-Horn-Schmid 1966 Finite Energy Sum Rules (FESR) had embodied the idea of a hadron bootstrap in an equation, for which G. Veneziano then discovered a solution, in terms of Gauss' Beta function. It appeared to reproduce the pattern of hadron excitations (for any one flavor) and was shown by L. Susskind and by Y. Nambu to represent the excitations of an elementary string. The model was, however, saddled with a / = 2 massless state, which Yoneya - and Schwarz and Scherk - later identified with the graviton. This evoked little interest until 1984, when it was observed that one of the infinities in the quantization program, the so-called chiral anomaly cancels — provided the internal symmetry group gauged by the vector-potentials of the theory happen to be either E(8) x £(8) or Spin(32). The superstring was then suggested as a "Theory of Everything" (TOE) [26], including gravity - here embedded in supergravity - and all other interactions. What impressed the Particle Physics community most was the uniqueness of the prediction with respect to the internal symmetries - their Lie algebras have to be contained in either one of the algebras of the two above-mentioned groups, both of rank r = 16. However, the theory still lives in a D = 9 + 1 embedding manifold, as imposed by two conditions: the suppression of the dilational anomaly, another renormalization infinity, and that of a (ghost) tachyon state, achieved by supersymmetry. Both relate to properties of manifolds with 8n (n an integer) transverse dimensions. It was, however, soon realized that in the compactification, from 9 + 1 down to 3 + 1, there are six more dimensions which end up just generating internal symmetries, and that this may be done in myriads of different ways, each with a different reduction. Instead of that uniqueness, one now has too much freedom. Moreover, at the level of the original formulation of the theory, one still has a choice between five basic models, yet another source of arbitrariness. Throughout the 1985-1995 decade, the string community's efforts were mainly aimed in three directions: (1) construction of a Superstring Field Theory (SFT), (2) study of 2-dimensional Conformal Quantum Field Theory (2-CQFT) and (3) matrix models of Quantum Gravity in 2 dimensions. The hope was that SFT would be a metatheory, whose vacuum solutions would provide stability requirements which would remove the arbitrariness and select the correct physical vacuum and its TOE. The SFT were constructed, but did not fulfill expectations. The study of 2-CQFT was very instructive. Denoting by £a,oc = 1,2 the worldsheet parameters, by .r"(£a) the embedding manifold coordinates (they become fields over the world-sheet parameters), gap the world sheet metric, G^v the embedding metric, the 1979 Polyakov action [27] is given as / = Jd 2 £(det | - g\)1/2 g^ dax^)
6jjx*(£) G,v(x).
(4)
This action is diffeomorphism-invariant both in the string's world-sheet £a and in the embedding xP. Classically, it is also scale-invariant in the £, but quantum-wise there is a dilational anomaly: the metric gap can always (in 2 dimensions) be diagonalized and written as g — 0(£) diag(l, -1), i.e. a Liouville field
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The Quantum Superstring (QSS) is thus nothing but the 2-CQFT of the string world sheet, while the embedding manifold appears as "internal" symmetry indices - except that one allows that symmetry's space to be curved. The perturbative expansion is over all possible surfaces (for that world sheet). Another useful lesson has been the construction of exact solutions to Quantum Gravity - in 2 dimensions. The embedding manifold is allowed to collapse (by constricting the range of the above "internal" ft, v indices, to e.g. zero or one single value). The 2-dimensional solutions are derived using the method of matrix models, first developed in solving the (number of QCD "colours") N —> oo limit of SU(N) Yang-Mills theory, an important element in the study of color confinement in QCD (G. 't Hooft showed in 1974 that — with a certain approximation — the YM theory becomes a String Theory). Here the main ansatz is the equivalence between a world-sheet (or a lattice approximating to it) and an infinite matrix, used here in the inverse direction from the QCD case. The solutions are interesting - the only quantum gravitational exact solutions known - but only didactically, of course. Note that in the spring of 1998, Maldacena (at Harvard) has just produced very interesting results re QCD and Color Confinement, using the same relationship, and managing to relate QCD quantities - to supergravity in a dual picture, corresponding to a quantum membrane. We shall return to this picture in what follows. The "new wave" in QSS theory started around 1993 and has accelerated since 1995. This is the discovery of duality transformations relating all five basic QSS models, in a chain-wise fashion. Moreover, in one case ("M theory"), the QSS models appear to represent reductions of a D = 11 supermembrane theory, whose truncation is D — 11 Supergravity [18], [23], [24]. All QSS theories thus appear to represent phases of a single metatheory, "M-theory" The term duality carries several different meanings — but all seem to apply here! First, there is Maxwell magnetic-electric duality, which is geometrically identical with the Hodge dual, transforming n-forms into (D — n)-forms. Secondly, as in Dirac's monopole condition, the product of the couplings in the two realizations has to be an integer e/x — N (we absorb the factors JI etc. in the couplings' definition). This then also guarantees that when "e" is strong, "/i" is weak, and vice versa. Dirac-type duality thus provides us with a way of applying PT in strong coupling situations: apply the Dirac duality transformation, then use PT in the physically equivalent but weakly coupled dual realization. Note that this Dirac duality, when applied to the string, transforms r <-» 1/r, with the smallest possible distance ^min — («')1/7- This, therefore, induces a quantization of spacetime in which one can write Ax as a sum of two contributions, A*>(A*)Hei,+-^—
(5)
the first resulting from conventional quantum uncertainty considerations, the second from the string's Dirac duality. There is a third meaning to duality, and it also applies: in the bootstrap S-matrix Dual Models, the forces due to particles exchanged in the "t"-channel, create the particles in the physical "s"-channel - and vice versa. Coleman showed a similar
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example of bootstrap duality in 2-dimensional RQFT, relating the sine-Gordon theory and a Thirring model [28]. The first example of bootstrap duality in QSS theory was the construction of a soliton-like p = 5-dimensional extendon ("p-brane") with its 6-dimensional world-hypervolume, as a solution of the string equations; then, under a Maxwell-Hodge duality transformation, the 5-extendon becomes the basic element of the fundamental theory and it is the QSS (p = 1) which reappears - this time as a soliton-like solution! [21]. Thus, we are faced with a duality combining together all three features, the Maxwell-Hodge formal and geometric, the Dirac strong-weak couplings and the Bootstrap field-soliton descriptions. These developments have been an encouragement for the QSS community - a renewed promise of uniqueness, possibly, plus the possibility that perturbative methods - the QSS main available methodology - might indeed apply everywhere. As a candidate theory of Quantum Gravity, the string has just scored an important success. It was shown [29] that counting the states (energy levels) of the QSS gravitational field for a black hole, reproduces the 1973 Bekenstein formula for the entropy [30]!, derived heuristically at the time, from the intuitive identification of the (ever increasing, classically) area of a black hole with entropy. String theory has thus finally provided a statistical-thermodynamical derivation. Personally, I think the QSS (or its M-theory parent, etc) is a step in the right direction, in the sense that it has a RQFT limit (the string constant - or Regge slope - a' —> 0), a requirement from a more general theory. Also, the Einstein equations do apply, as a condition guaranteeing the preservation of the cancellation of the dilational anomaly [31]. The graviton is accompanied by additional fields, a dilaton and an axial-vector; but these fields may be short-ranged and their action might well be restricted to the quantum (Planck) region. The equations also involve higher-order terms in the curvature, but this is natural for the renormalization process, i.e. in the high-energy short-range region. Note that the QSS is first and foremost a quantum theory, with no natural transition to a classical equivalent. What I, in fact, miss, is an elegant transition to classical General Relativity, rather than to its linearized version. Instead, the graviton state in the theory's Hilbert space continues to exist, even when the embedding manifold is flat, which somewhat lacks elegance — although we know that under certain restrictive conditions, one may recover General Relativity - or at least its linearized version from the interactions of a massless J — 2 field in a flat Minkowski background. Also, I miss a generating fundamental principle, defining the theory. And yet, with so much progress, the answers might well be there, just behind the gate to the new Century. 5 Ashtekar's canonical quantization and the loop representation Position VIII: "Apply (nonperturbative) Canonical Quantization". The method of "Canonical" or Hamiltonian quantization was launched and developed by Dirac in the fifties [32]. Classically, Lagrangian mechanics deal with coordinates x and velocities x (in QFT, fields
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Dirac pointed out that working with the latter method has the advantage of easing the quantization procedure, since all that has to be done is to replace p, q by Hermitian operators and apply Heisenberg's commutation relations. The difficulty resides in ensuring Lorentz invariance, since the Hamiltonian breaks it; this is managed by a system of algebraic constraints which reproduce the original invariances. The difficulty resides in selecting the canonical variables so as to get solvable constraint equations. The method is nonperturbative and thus appealing to the GR (classical) community, used as it is to nonperturbative solutions of Einstein's equation. The open question was which physical variables to select. The dynamical content was explored, still in the fifties, in a famous paper by R. Arnowitt, S. Deser and C. Misner but little progress had been achieved, almost thirty years after Dirac's further launching of the program, specifically for Gravity, when Abhay Ashtekar appears to have found the means of realizing it [33], Ashtekar's apparent success is due to his exploitation and adaptation to (Einsteinian) gravity of two features characterizing Yang-Mills gauge QFT. Feynman's 1959-62 pioneering study of the quantization of the Yang-Mills model was indeed originally initiated as a "pilot project, for Quantum Gravity - borrowing from YM, when working on QG, is thus historically fully justified. Ashtekar thus needed to select variables which would make GR resemble YM theory as much as possible — for instance, using the connection rather than the metric. The second borrowable feature in YM theory relates to topological solutions and complexification. Euclidean YM theory indeed has topological instanton solutions, labeled by an (integer) winding number N: B = (647T2)-1 e"vea J" d 4 x P^r^
= N,
NGZ.
(6)
Topologically they represent divergences of the Chern-Simons 3-forms [34] B — AC. The instanton term can be added to the minimal YM Lagrangian T^ J7^, with an imaginary coefficient deriving from the Euclidean/Minkowski transition - and thus with a consequent complexification of the canonical momenta. Minimalizing the action then implies selecting either self-dual or anti self-dual complex configurations of the field-strengths (corresponding to the two spin-like chiral components of the Lorentz group's massless representations). The same method (and subsequent complexification) is applied in Ashtekar's representation of the gravitational connection and curvature. It starts with the (here Riemannian) connection *Abafi dx^ = 4Ab and the tetrad e° dxM = e°. The connection is complexified and made self-dual AAab = (i/2) e"j4Acd, with a Palatini-like complexified Lagrangian with self-dual curvature, S(eM)
= 5£abcdeaAebA*Fcd.
(7)
Here, as in YM, the complexification corresponds to the addition of B = dC topological terms to the Einstein Lagrangian, with a relative phase ensuring P and CP conservation [35]). This should be followed, after the quantization, by a procedure which would erase any unphysical components originally generated by the complexification. The connection and tetrad are then decomposed as 4 —> 3 + 1 , Hamiltonianwise, working on 2, a 3-dimensional (space-like) manifold. The canonical variables
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are thus (M,V = u,v holonomic, and c,a,b = i,j,k, anholonomic "internal", all spatial directions) the 3-connection A'u = ^Aku (x) playing the role of a potential variable (self-dual), and the densitized triad (canonical momentum like) E]{x), with Poisson brackets {Eui{x),A{{x')} = -id%d{x-xl)
(8)
A'u has 9 components, which should be reduced to 2. This is performed by 7 constraint equations
g,:=Dug; = 0, Vu:= 1^,^=0,
(9)
iik
S:=e FuE'<E] = 0, respectively, 3 Gauss-law-like SO(3) gauge constraints, 3 "vector" constraints corresponding to DiffR3 and one scalar (time-translation) constraint. The Hamiltonian is accordingly constructed from the anholonomic constraints and corresponding surface terms, H(A,E) = i J"z d3x(NuVu - (i/2) NS) 2
u
(10) k
- J ^ d x„{2iN EfA\, + N£v AvkE?E]} The theory has thus been made to look like an SU(2) (i,j,k internal indices) Yang-Mills theory. The next stage consists in finding solutions to the constraints, for variables that obey the quantized commutation relations. The most promising solution appears to have been given by Rovelli and Smolin [36], namely the loop representation, inspired by the Wilson loop [37] formalism, popular with workers in Lattice Gauge Theory methods in (YM) QCD. A loop in 2 is a continuous map y from the unit interval into Z, y : [0,1] —> S,s —> "/"(s), y(0) = y(l). The loop space is the set of all such maps QZ. Given the space of connections A in Z, A e A, one defines a complex function over A x
(11)
This is invariant under the SO(3) gauge. The loop is thus just the integration contour and the Wilson integral (first introduced by Hermann Weyl, but applied by Wilson as the fundamental tool of lattice gauge theory, the loops being laid out as rectangles on the lattice) becomes a way of embedding the YM connection in spacetime. In gravity, another gauge group is the group of local diffeomorphisms, the Einstein covariance group. Contours can be deformed by the spatial diffeomorphisms and are therefore invariant up to knots; more precisely [38], generalized knot classes, which thus characterize the new quantum states. Note that this quantization of the gravitational field, being nonperturbative, does not involve Fock space and thus does not identify "gravitons". One is faced with a problem of interpretation, relating this picture with the more familiar one, based on elementary excitations. The equivalence between the two pictures and
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their mathematical relationship have been investigated [39]. Not surprisingly, in view of the self-duality features of the Ashtekar variables, the reality conditions are realized by a chiral asymmetric treatment, distinguishing between left- and right-chiral gravitons. Other interesting features are the quantization of area and volume. The relevant measurement operators have been constructed, using the loop representation [40], and they indeed display discrete spectra of eigenvalues. It would be interesting to sharpen the distinctions one obtains in this nonperturbative description with the discreteness of spatial features in String Theory. At first sight, it would appear that the two approaches are incompatible: Ashtekar's program applies pure Einstein gravity, without involving anything else — whereas the Superstring is a "Theory of Everything" and because of Unitarity, would involve all interactions, in its perturbative expansion. These differences, however, will appear at Planck energies only, since the truncated superstring is a Supergravity theory, containing Einstein's theory. Moreover, Ashtekar's approach has been extended to Supergravity. Thus, it is possible that only around Planck energies will one be able to decide between String Theory and Canonical Gravity. In any case, we have arrived to the happy state of affairs where rather than having no quantum theory of gravity, we may have one too many. I would like to thank the German Physical Society for the invitation to present this review at its 1998 Regensburg meeting.
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Lett. B 233 (1989) 286 and Phys. Lett. B 242 (1990) 59. C. Y. Lee, Class. Quantum Grav. 9 (1992) 2001 F. W Hehl, J. D. McCrea, E. Mielke and Y. Ne'eman, Phys. Rep. 258 (1995) 1-171 Y. Ne'eman, Phys. Lett. B 427 (1998) 19-25 J. Wess and B. Zumino, Nucl. Phys. B 70 (1974) 39 S. Mandelstam, Proc. Int. Conf. on High Energy Physics, Paris 1982, P. Petiau and J. Porneuf, eds. p. C3-331 E. Cremmer and B. Julia, Nucl. Phys. B 159 (1979) 141. E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B 76 (1978) 409. Y. Ne'eman and M. Gell-Mann, Aspen Institute lecture, June 1976, unpublished P. G. O. Freund and M. A. Rubin, Phys. Lett. B 97 (1980) 233 R. B. Kallosh, P. N. Lebedev Institute, Physics Report 152 (1980) See, for example, Y Ne'eman and E. Eizenberg, Membranes and Other Extendons ("p-branes"), World Scientific Pub., Singapore (1995), Section 5.3 A. Casher, Phys. Lett. B 195 (1987) 50 P. K. Townsend, Phys. Lett. B 350 (1995) 184; E. Witten, Nucl. Phys. B 443 (1995) 85 E. Bergshoeff, E. Sezgin and P. K. Townsend, Phys. Lett. B 189 (1987) 75 T. Yoneya, Prog. Theoret. Phys. 51 (1974) 1907; J. Scherk and J. H. Schwarz, Nucl. Phys. B 81 (1974) 118 M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Vol. I, II, Cambridge University Press, Cambridge, UK (1987) A. M. Polyakov, Phys. Lett. B 82 (1979) 247 S. Coleman, Phys. Rev. D 11 (1975) 2088 A. Strominger and C. Vafa, e-print hep-th 9601029 (1996) J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333 E. S. Fradkin and A. A. Tseytlin, Phys. Lett. B 158 (1985) 316 P. A. M. Dirac, Can. J. Math 2 (1950) 129; Proc. Roy. Soc. A 246 (1958) 326, 333; Phys. Rev. 114 (1959) 924 A. Ashtekar, Phys. Rev. Lett. 57 (1986) 2244; Phys. Rev. D 36 (1987) 1587 See, for example, S. Weinberg, The Quantum Theory of Fields II, Cambridge University Press (1996), sections 23.5-23.6 E. W. Mielke, Phys. Lett. A149 (1990) 345 and A151 (1990) 567; also F. W. Hehl, W. Kopczyriski, J. D. McCrea and E. W. Mielke, J. Math. Phys. 32 (1991) 2169 C. Rovelli and L. Smolin, Nucl. Phys. B 331 (1990) 80 K. Wilson, Phys. Rev. D 10 (1974) 247 J. Baez, ed., Knots and Quantum Gravity, Oxford Science Publications, Clarendon Press, Oxford (1994) A. Ashtekar, C. Rovelli and L. Smolin, Phys. Rev. D 44 (1991) 1740 C. Rovelli and L. Smolin, e-print gr-qc 9411005 (1996)
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C H A P T E R 4: GEOMETRIZATION OF P H Y S I C S TULLIO REGGE Department of Physics, Politechnico di Torino Torino, Italy I forgot when I met Yuval for the first time but certainly the crucial year for our collaboration and friendship was 1978 when we wrote a joint paper {4.3} while at the Institute for Advanced Study in Princeton. Yuval had the starting idea and convinced me that it was a very interesting one, from that day we worked like madmen till the paper was completed. I left the IAS the following year and went back to Italy but our interaction did not come to end. Yuval invited me to Tel Aviv and I visited Israel for the first time. All this happened a quarter of a century ago and I cannot remember all the details but Israel had a very deep and everlasting impact on me. I came back a few years later, this time to Jerusalem, when Yuval was a member of the Knesset. I even had lunch with him in the cafeteria of the Knesset, a rather cramped and informal place. While I was seated I had to be careful not to elbow in the ribs when I turned around the fellow sitting behind me. Only when I left I realized that that fellow was Itzhak Rabin. Yuval got me interested in unified theories, I did not work and even less left my imprint on any such a theories but it still find them fascinating and intriguing. Einstein's dream was the final unified theory and physicists have gone a long way ahead in realizing his dream. We have now a standard model of elementary particles which seems to condense in a compact and unified formalism all we know today about elementary particles. Is it really the final model? In no way I want to sound critical about this splendid achievement but I do not believe that there is a final all encompassing theory of everything. During my last years at the IAS my office was located near the cafeteria of the Institute; as I looked out of the window at noon time I saw Kurt Goedel, a living legend of all times, walking toward his lunch, his timing always perfect. Years before when I still was a young student in Turin one of my professors pulled out of the shelves and showed me with awe the three heavy tomes of the "Principles" of Russell and Whitehead. Years later I was disapponted but also intrigued when I learned that Goedel's theorem had destroyed overnight the hope of condensing all math into four tomes, mathematics has no end and is an evergrowing logical tree sprouting forever marvellous buds. The incompleteness of math is a blessing and not a curse. If Russel and Whitehead had fulfilled their dream their achievement would have been in fact the very boring end of a glorious discipline. For the same reasons I hope that there is no final standard model of elementary particles, I would consider such an achievement a curse and the shameful end of physics. Newton's theory was and it is still rightly considered as one of the greatest achievement of all times. It provided a unified yet very simple theoretical frame for the motion of all celestial bodies. Its greatest triumph was the spectacular discovery of Neptune by Adams and Leverrier based on some irregularities in the motion of Uranus. Years later the same
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Leverrier staged a repeat based on the anomalous precession of the perihelion of Mercury which he attributed to Vulcan, a planet whose orbit was supposed to be very close to the Sun. To his disappointment no such a planet Vulcan was ever seen and we know now that the anomaly is a subtle effect of general relativity. Einstein himself did not believe that general relativity was the end of physics and tried repeatedly to go beyond it. The saga of the standard model of elementary particles began with Gay Lussac law and Avogadro principle, these achievements provided in fact an unambiguous and consistent definition of atomic weights and were the starting point for the work of Mendeleyev. The periodic system was in fact the first standard model, was a spectacular achievement but was also in many ways incomplete and laden with strange irregularities. But just these flaws paved the way for future improvements and for the work of Niels Bohr. In the periodic system atoms played the role of elementary particles, after the work of Bohr and the discovery of nuclear structure physicists were convinced that you could number elementary particles on finger tips, an illusion quickly destroyed by the development of accelerators. The standard model explains today phenomena on an energy range which is billions of times higher than the few eV of Mendeleyev's time but it is by no means simpler than the periodic system. I hope that someday someone will discover flaws in today's splendid model, open a window on underlying arcane structures and be the starting point for a new scientific revolution. I remember vividly since my early high school days a catholic priest who repeated the dictum "errare humanum est" a but added wisely the lesser known end "perseverare diabolicum" b . Science learns form errors and I want the show to go on.
"meaning: to err is human 6 ...to persist in the error is diabolical [Seneca]
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REPRINTED PAPERS OF CHAPTER 4: GEOMETRIZATION OF PHYSICS
4.1
4.2
4.3
4.4
4.5
Y. Ne'eman, N. Rosen and J. Rosen, "On the Origin of Symmetries", in Symmetry Principles at High Energy, Proc. Coral Gables 1964 Conf., B. Kursunoglu et al., eds. (W. H. Freeman Co., San Francisco, 1964), pp. 93-103.
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Y. Ne'eman, "Embedded Space-Time and Particle Symmetries", Rev. Mod. Phys. 3 7 (1965) pp. 227-230.
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Y. Ne'eman and T. Regge, "Gauge Theory of Gravity and Supergravity on a Group Manifold", Rivista Del Nuovo Cimento 1 # 5 (Series 3) (1978) pp. 1-43.
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Y. Ne'eman, "Ghost Fields, BRS and Extended Supergravity as Applications of Gauge Geometry", Proc. XIX Int. Conf. on High Energy Physics (Tokyo 1978), S. Homma, M. Kawaguchi and H. Miyazawa, eds. (Phys. Soc. Japan, Tokyo, 1979), pp. 552-554.
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Y. Ne'eman, "Higher Algebraic Geometrization Emerging from Noncommutativity", in Fluctuating Paths and Fields (Festschrift dedicated to Hagen Kleinert), W. Janke, A. Pelster, H.-J. Schmidt and M. Bachmann, eds. (World Scientific, Singapore, 2001), pp. 173-184.
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93 ON THE ORIGIN OF SYMMETRIES N. Rosen, Israel Institute of Technology J. Rosen, Tel Aviv University Y. Ne'eman, Tel Aviv University and Calif. Inst, of Techn. I. Introduction Since our host has explained that he is so proud of having two Turks taking this kind of approach to the problem of symmetry, one can probably trace this talk to the fact that Israel was under Turkish domination for about 400 years. So maybe the traces of the Ottoman Empire are still very well set. Kursunoglu - I can see them in your face. From Floor - You were getting the credit not the blame. I think this is well understood you are not responsible for what happened. The idea I shall describe is rather less elaborate than some of the talks we have heard on this topic and is made up entirely of handwaving arguments, which make it more difficult to check. The actual origin of symmetries belongs to that theological question which was mentioned yesterday - namely, God's aesthetics. Our goal is to discover what they mean. The internal symmetries of the strong interactions do not have the same universal validity as electric charge (Q). SUo/Z, symmetry1 (or eight spin F) in a certain"polarized" form does characterize the hadrons; some of its components play a role outside the strong interactions: Q in electromagnetism, V1" and W~ (a combination of |AS[=0 and |AS|» 1 operators^) in the weak interactions. Nevertheless, the moment we switch on a non-strong effect, we witness a gradual breakdown of the F-spin components' conservation. This fact, coupled with the emergence of a hopefully parameter-free dynamical theory, have led to an optimistic search for a dynamical origin to these symmetries - in contradistinction to the unavailability of such a derivation for electric charge. Bootstraps may yet yield a successful derivation3. On the other hand, there is a possibility that these symmetries emerge from space-time itself.* Strong interactions *a suggestion that strangeness Is the result of parastatistics has apparently been' Invalidated (S. Llchtman et al, Bui. APS 2.22)
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do carry a specific characterization with respect to spacetime - their short range. We have therefore tried to see whether this fact could be used to generate internal symmetries - and whether these could have anything to do with the physical ones. II. The Space We observe that fully Riemannian space-time C w ^ ?\ is embeddable in a flat 10-dimensional space E ^ Q . J. Rosen has checked^ the need for these 6-additional dimensions and their nature in the various relatlvistic solutions local and cosmological. It has long been known that the Schwarzschild external solution requires only 6 flat dimensions altogether5; the space is then Eg(2 4) with two time-like dimensions out of the total six.'Rosen has embedded several symmetric solutions in the 5 dimensions of an Ec(]_,4)« Less symmetric models of the universe require a higher dimensionality: Godel's universe^ even seems to require all ten, the flat space being then an £20(5 5 ) . We shall assume that this is typical of reality, i.e. space-time is so irregular that we require the full E 10(5,5)' w l t h t h e m e t r i c (_1> _ 1 , -1, -1' -1' 1,1,1,1,1). At'every point of space-time, we may define a 4- dimensional tangent/flat 24(1,3), w i t n t h e Lorentz metric of special relativity. The £-^0(5,5) then contains a space Eg/4 2) orthogonal to the tangent space. We note that infinitesimal generators of rotations (or orthogonal transformations) inside Ew-^ o ) , i.e. the proper Lorentz group generators, and in fact'the whole group, commute with similar generators in E g ^ ^ ) - W e shall make use of this fact to define the symmetries in the latter. III. The Symmetry Working with the Mandelstam representation for the S-matrix, our invariant Lagrangian is replaced by an elementary diagram with two incoming and two outgoing lines. Although these represent states at + infinity, we know that for the strong interactions this corresponds to an extremely small 4-volume in which the interaction occurs. This requires an approperiate modification in the formalism. All hadrons are massive; they can all appear as intermediate states in the various channels. Strong interactions are now believed to occur through forces corresponding to particle exchanges in the crossed channels"; the masses of these particles define a finite, small 4-volume of some (1 Fermi) in which this occurs.
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This is not true of electromagnetic and some weak gravitational interactions, where massless particles neutrinos, photons and gravitons - may be exchanged. Another distinction - in a non field theoretic S-matrix approach - is that all the non-strong Interactions seem to have S-channel forces or arbitrary couplings appearing in processes like pair creation or annihilation^. They are thus not defined through any such range considerations. This may perhaps be related to the smaller amount of symmetry displayed by the hadronic weak interactions. The energies contained inside our small 4-volume are of the order of 10 _2 3 of a gram, and have an extremely weak coupling to gravitation, as is any "intrinsic" curvature within such limits. We are therefore free to believe, to zero order, in Lorentz-invariance of the strong Interactions and work in the tangent E^/^ 3)- T n e S-matrix is then also invariant to orthogonal transformations in the E
6(4,2)' Any quantum number, like Q and B (baryon charge) which is conserved by the long-range forces too, must correspond to an orthogonal (or phase) transformation in a plane orthogonal to the whole C^/^ o\, whatever its curvature. We shall not touch B, a quantity that seems to have some direct relationship to Lorentz space itself, exhibited by the correlation between baryon number and spin character, leading to definite statistical behaviour. As to Q, it could emerge from a plane contained in the E 1Q(5 5) Provi(ie
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shall return to this point and explain why this is a plausible assumption. We thus have two symmetry spaces: an Eg and an E 2 , orthogonal to each other. The E 2 should yield one phase transformation; the Eg defines an algebra Do, isomorphic to A,, the SVty generators' algebra of the Dirac matrices. It has three linearly independent nilpotent operators, i.e. quantum numbers. This is too large a symmetry, with two unobserved quantum numbers, counting the E 2 rotation. On the other hand, we know that there is a certain amount of curvature even within our small strong interaction volume. To first order, it is situated in an external Schwarzschild solution due to the sun; there is also a certain curvature deriving from the cosmology etc. That this first order curvature is strong enough to perturb the orthogonality between the Eg and E2 and local space-time seems somewhat surprising; nevertheless, this might be the mechanism that reduces the 4 quantum numbers of U^ and SU^ to two. If we make this assumption, the effect will be to cancel any symmetry that mixes the one or two extra coordinates required by the Schwarzschild or cosmological curvatures. This touches one space coordinate in both cases and also one time dimension in the Schwarzschild case; it destroys the conservation of the rotation group in Ep (or the V 1 ) and also apparently cancels the validity of one of the three quantum numbers in the Eg six-rotation group. This last point is not clear, as it looks as if it should have reduced R(6) to R(5)j in fact it achieves its effect in the isomorphic SU^, cancelling one of its quantum numbers. We choose our coordinate system in the Eg so that the bad quantum number is H 3 - (6)"* (2Sl6 + 2 2 5 + Z 3 4 ) where the matrices 2 ^ (a,b) through, z
k!
are defined by their elements
i (6j6l-6j*J).
The rotation algebra Do (defining the group R(6)) is thus no more a symmetry of the system; but it does contain an 8-parameter subalgebra commuting with H3 and therefore defining a good symmetry. This is SUo, whose generators in this case correspond to a 6-dimensional reducible representation. To visualize it, take the direct sum of an SU(3) triplet meson (three complex fields like the D + , D°, S° of the eightfold way presentation1, or like K+, K°
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and a complex isoscalar a) and its antiparticles* set. This six-dimensional representation can be transformed by a unitary transformation into a real Euclidean 6-space, with the particles appearing as eigenstates of a diagonalized charge conjugation or R-operator. In this real representation, the SU, matrices are imaginary, hermitian and antisymmetric, i.e. they form a subalgebra of R(6); they all commute with Ho. Up to this point, we have been working in a metric space. Pais 1 0 has shown that to get a meaningful set of dynamical observables, the symmetry's space should be made up of bounded variables, angles, where one can therefore have a finite volume; ordinary energy, momentum etc...are then defined as integrals over both Lorentz and the symmetry space. Dividing by the finite volume of the symmetry space, we get an "average" over it, which is what we measure. This is also why we did not deal with the "mixed" planes. For our SU^ operators, one way to do it is to work in the space spanned by the 8 parameter-angles, all of them real Euler-llke quantities in the Eg (it is possible that differential operators in terms of polar coordinates will give similar results). We thus have to define an isomorphism between the operators in the 6-space and a similar set in this 8-space of angles. This is the physical symmetry ; it generates a representation of SU3/Z0, i.e. the adjoint group, corresponding to the octet model. IV. Conclusion We have followed a general chain of arguments leading to F-spin conservation in strong interactions. In such a derivation, it is the weakness of the gravitational coupling that generates this symmetry in short range phenomena. In fact, we got a larger symmetry than the observed one, and contracted it by assuming that first order gravitational curvature should not be left out and conjecturing that it should have just that observed effect. This is somewhat arbitrary and would imply that strong interactions in interstellar or intergalactic space display more symmetry - and that stellar cores display less of it. This kind of idea could be challenged physically. There may be other ways of testing whether one can really get the restating symmetry from the weakness of gravitation. It would then appear that the strongest interaction is influenced by the weakness of the weakest one. In such a model, it is just because the gravitation is so weak that we can work just with the Lorentz metric and disregard the other dimensions. It is perhaps almost describable
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by the biblical line "out of the mouths of babes comes wisdom".
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References 1. M. Gell-Mann, California Inst, of Technology report CTSL20, (1961), unpub. Y. Ne'eman, Nuclear Physics, 26, 222 (l96l). 2. N. Cabibbo, Phys. Rev. Letters 12, 62 (1964). 3. E. Abere, P. Zachariasen and C. Zemach, Phys. Rev. 132, 1831 (1963). 4. J. Rosen, to be published. 5. E. Kasner, Amer. Journal of Math., 43, 130 (1921). 6. C. Pronsdal, Nuovo Cim., 1^, 988 (l959)-see note (6) p. 990 and references to former attempts. 7. See Witten, Gravitation, (1963) P- 440. 8. G.F. Chew, "Nuclear Democracy", lecture at A.P.S. Pasadena meeting (1963)9. H. Boerner, "Representations of Groups", North Holland pub. (1963), p. 302. 10. A. Pals, Physics 19, 869 (1953).
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Guth - Could you tell how you would get these formulas? Ne'eman - I would think of this idea of short range forces, etc. You have to define what is the thing that is going to be invariant, and I would prefer an S-matrix to a Lagrangian in that case, because it has this easier way of thinking of nonlocality of a certain size. This is one thing that has to be done. The other thing that has to be done is to check what kind of transformations can exist, and you have to know if the rest of physics has already determined exactly what kind of dimensions come out of the fact that space is curved. This is why the younger Rosen was really looking at all the present models to see what kind of dimensions they require. It is not obvious, e.g., if you do not check it, that you would get a time dimension out of it. The simplest one, the Schwarzschlld metric, immediately gives you one time and one space dimension, and there are even funny things happening with that. There is this business (I think Wheeler has investigated it) where you go to the Schwarzschild probe and something happens to you. There is an inversion of the time, if you are in one of these things that passes. If you followed Fred Hoyle's talk at the New York meeting last week, he was discussing the point of view of the observer sitting on one of these imploding masses. Then this observer, at a certain point, really starts to feel that other time. We do not feel it because we cannot go out of a certain universe. But this observer really has a transformation which mixes these other dimensions. So you really have to see what other dimensions there are from dynamical gravitation and cosmology. Then you try to work out the group and see whether it can be brought to one- to- one correspondence with what we have found from phenomology symmetry, and there you check whether you can get anything more than just these effects I was mentioning, like the difference of strong interactions between two extremes of density. Marshak - You talk about the strong interactions having a short range, but also the weak interactions have short range. You have a curious situation where the gravitational Interaction has a long range connected with a (weak) short range and an electro-magnetic which has long range connected with a strong short range, and I wondered to what extent these things would fit in. Ne'eman - I think that it does not because when you are thinking of the Lagrangian way you see the range by the
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vector meson, which goes between, and in that case you are certainly right. But if you are thinking of the Mandelstain graph then a neutrino can go in a diagram or a coupling pair of neutrinos, and neutrinos do not have mass. You can have a second order weak interaction where you have an interference of intermediate states and neutrinos. Marshak - What if you were to mediate the weak interaction just through the vector boson? Ne'eman - In the nonleptonic decay you would have the weak interaction working like electromagnetism and things happening not because of forces in the t-channel. A graph of pair creation in electromagnetism which is also something that is true for weak interaction (and the vector boson is weak) is something happening in the energy channel. Marshak - You made a point about configuration space. There is clearly a big difference between the photon mediating the electromagnetism interaction and the vector boson mediating the weak interaction. Ne'eman - But you have a coupling with a sub-strength and this has a meaning. Marshak - This is just my point. Either you want to make a point about the range without talking about the coupling constant, or vice versa, - in any case, both the weak and the strong. Ne'eman - Yes, this is why I say that I prefer to work in the S-matrix formalism because there this business of the range has a meaning only if the forces correspond to these exchange forces, and this is why you do not treat electromagnetism also as an equal member of a Mandelstam triangle. Kursunoglu - I remember the rumor about Wigner saying something like: if it is good, it is amusing; if not, it is interesting. Let us say your theory is amusing. I know from experience (also from attendance at meetings in general relativity) that anytime you propose an idea by counting dimensions, it is almost a foregone conclusion that the result is wrong Ne'eman - Maybe I agree Kursunoglu - This is of course an exceedingly general remark. Now a specific question is, why did you take short range? Of course the strong interaction is left out in producing your arguments, that the weakness of gravitation does effect the strong interaction so we neglect it and we have the Lorentz group. Why did you take that one? What was the basic motivation in not taking the electromagnetic interaction? Ne'eman - Because of the fact that the electric charge
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is conserved throughout, whereas the thing that is really characteristic of the strong symmetries is that they break down, and isospin is good only insofar as you stay within strong interactions; the moment you go out, isospin breaks down. Electric charges are always good, so it is certainly orthogonal to anything we can think of here. The question is, how can this mechanism come about that has a symmetry and still lets it go? Maybe it is a symmetry which really corresponds to the strong and the weakI do not know. Maybe that is the reason why Cabibbo's angle is what it is. Perhaps it is a certain way to plug yourself into the currents. The obvious thing is that the weak Interaction is not invariant; it breaks down, so something must be happening. In the strong interaction you have this thing of invariant symmetry which disappears the moment you go out of it. Telegdi - You have a nucleon, say, with a pi meson coming out. How is it already not under the weight of a very intense gravitational field? Ne' eman - The answer to this is that apparently there are such things. After all, what do we know about overlapping? All that we ever do is that generally we take 2-dimensional graphs; we think of two nucleons and exchange between them, i.e. we have clouds. But we do not know really what happens when you have a many-body problem in which the particles are apparently under a terrific pressure. Then you have all the compton wavelengths overlapping each other. In that paper of Fred Hoyle, you have the density of 1C-3. You can very roughly see how it happens - you think of a gravitational potential, and ratio between gravitation and strong interactions. Berman - I was wondering why you don't want also to include the weak interactions. Cabibbo has shown that by making a small rotation you can think about the weak interaction as being a Al = 1,0 and with no AS because of the rotation. So then a small perturbation allows us to include both the strong forces and weak forces. Ne'eman - This most recent idea of Cabibbo's is certainly extremely appealing in its simplicity. The whole question then is really how to break symmetry. This model has too much symmetry. There is enough here to have a larger group, certainly even global symmetry could have existed with the number of dimensions we have here. So we have to try to kill some of the symmetry. You will have to kill it slightly more in order to conserve only that part which appeared in Cabibbo's scheme, i.e., where the octet appears with S' or Y 1 being conserved in the weak interaction. That should be the part to remain and other
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operators would have to break down. I want to be very careful and not to get into anything involving thinking of the shape of these particles or the way they rotate or things like that. There has been a long series of attempts in that direction by de Broglie, Bohm, Vlgier, etc. I do not understand how it comes out there, because when you stop to think of these shapes, I think that you again come across these problems of things that do not commute with the Lorentz ordinary transformations or their rotations. One really needs an elegant way where matter densities or something else which will leave the structure as it is but will do the symmetry breaking. Marshak - You have the electromagnetic interaction having this long range. And yet at the same time the electromagnetic current is brought in and seems to have quite a different character from the weak and the strong. Now there is another very interesting point that I think one has very much to understand, and that is that we only deal with charged curents. Now some years ago Okubo and I fooled around with a model where we tried to understand the leptons and the baryons from a unified Heisenberg viewpoint; and in that model, with the electromagnetic field producing a mass difference between the charged and neutral particles, one could automatically create the weak interaction. In a sense, the electromagnetic field may be really different in character from the strong and the weak and may have something to do with the space in just the way that the gravitation does. The long-range electromagnetic Interaction may be performing a very similar role as the long-range gravitation. Generally, in models like this, you only bring in the gravitational, and not the electromagnetic. Then I would be perturbed by the fact that somehow we bring the electromagnetic into the octet, but we do not bring in the gravitational.
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D. W. JOSEPH Generalized Covariance
REVIEWS O F M O D E R N
PHYSICS
VOLUME 37,
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19 6 5
Embedded Space-Time and Particle Symmetries* YUVAL NE'EMANf California Institute of Technology, Pasadena, California The bootstrap yields the strong-interactions symmetry provided the hadron currents participating in nonstrong interactions are first fed into it; the alternative course, in which the bootstrap is required to generate the entire symmetry uniquely, with no a priori information from the nonstrong interactions, seems to lie beyond the present techniques and would also leave unexplained the subsequent emergence of the weaker couplings. We suggest that "internal" symmetries may have a geometrical origin, corresponding to transformations in the global embedding space of the four-dimensional physical Riemann universe. These would be the unitary or orthogonal transformations of the normal subspace, since the latter do transform a small region of curved space-time into itself. This picture fits in with the short ranges of the strong interactions. Cosmological and astrophysical implications are noted. SO-CALLED "INTERNAL" SYMMETRIES T h e conservation laws and t h e dynamics of particle physics have found their most useful formulation in terms of apparently nonkinematical, i.e., " i n t e r n a l " symmetries. These are generalizations of the concept of " c h a r g e , " used in electricity for the last 150 years; they appear as a set of charge currents satisfying equations of continuity under certain limiting conditions. I n practice, the equations really have " s i n k " terms due to some symmetry-breaking interactions. We have used the word symmetries since we m a y relate each conserved neo-charge with the generator of some Lie group, the l a t t e r then representing a s y m m e t r y of the system. This connection, sometimes known as Noether's theorem, has lately been reformulated appropriately, since what we really want is an inverse theorem leading from the observation of a conserved or quasiconserved q u a n t i t y to a symmetry group. Okubo 1 h a d earlier shown in detail why an unrenormalized coupling implies the exist* Work supported in part by the U.S. Atomic Energy Commission. t On leave of absence from Tel Aviv University, Tel Aviv, Israel, and the Israel Atomic Energy Commission. 1 S. Okubo, Nuovo Cimento 13, 292 (1959).
ence of a conserved current; Horn 2 h a s now exhibited the emergence of t h e algebra from t h e current. Another starting point for the d e t e c t i o n of a conserved current has been shown b y Ogievetski a n d Polubarinov* to derive from the o b s e r v a t i o n of particles with unit spin a n d negative parity, t h i s time in a more exact inverted Yang-Mills technique. 4 Cutkosky 6 has obtained a similar result—i.e., t h e g e n e r a t i o n of a symm e t r y from vector-meson couplings—in terms of an approximate a n d idealized bootstrap ( a t h e o r y with no " e l e m e n t a r y " strongly interacting particles, in which all hadrons appear as self-bound s t a t e s of one and t h e same hadron m a t t e r ) . The s y m m e t r y itself shows u p in m a n y different ways—mainly a multiplet structure accounting for t h e various sets of particle states a n d relative coupling strengths or amplitudes for strong r e a c t i o n s (this is in fact the "law of force"—like the c 2 i n t h e Coulomb 2
D. Horn (to be published). * V. I. Ogievetski and I. V. Polubarinov, Zh. Eksperim. i Teor. Fiz. 45, 966 (1963) [English transl.: Soviet Phys—JETP 18, 668 (1964)]. 4 C. N. Yang and R. Mills, Phys. Rev. 96, 191 (1954); also R. Shaw, thesis (unpublished). «R. Cutkosky, Phys. Rev. 131, 1888 (1963).
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REVIEWS OF MODERN PHYSICS • JANUARY
1965
force). The picture is of a large general symmetry respected by the strongest interactions; all other interactions break it by picking out some preferred directions in the algebra space. This enables one to extend the use of the symmetry to all these weaker interactions, taking into consideration the tensor properties of their symmetry-breaking Lagrangians. As things now stand, one feels safe in identifying the group SU(3)/Z(3), represented by the zero-triality representations of SU(3) (the three-dimensional unitary unimodular group) 9-9 and a baryon gauge group U(l) as the dynamical group of the strong interactions. It is broken by: (a) A medium-strong (ten times weaker) interaction behaving like an eighth component (1= F = 0 ) of an SU(3) octet.10 (b) The electromagnetic interaction, behaving like another octet component. (c) The hadrons' leptonic interactions, corresponding to a particular pair of octet components.11 (d) The interhadron weak interactions, whose tensor properties seem to correspond to yet another octet component.12 This established scheme may prove to be only the main part of a larger picture. There are two main motivations for an extension: (a) Trying to account for the appearance of octets; some schemes would get baryon octets from products of baryon and meson triplets. This introduces13'14 either SU(4) orSU(3) X SU(3). (b) Studying the weak interactions. The axial vector currents display a surprisingly small amount of renormalization and thus seem quasiconserved. This involves a larger symmetry now including an octet of parity-changing operators—the minimal16 being SU(3) X SU(3). There have also been some vague indications of the possible existence of AQ/AS=— 1 currents; the parity conserving symmetry would then be16 an i?(8) (eight-dimensional rotations), and the chiral group (i.e., the one containing both parity conserving and parity changing operators) could be either R(9) or SU(8) or R(8)XR(8), etc. The main strong interaction—responsible for the baryon masses since
the chiral symmetry is only conserved in the limit of vanishing baryon masses—breaks the chiral symmetry in a well-defined way; the entire weak Lagrangian would then appear as yet another tensor component under the chiral group. THE ORIGIN PROBLEM
We are thus faced with the following question: Is the strong-interaction symmetry the result of some kinematical relationships of these interactions? This is just what is implied in the bootstrapist's program of deriving the symmetry from such concepts as analyticity and unitarity. Abers, Zachariasen, and Zemach have shown how this should be done in principle17; by taking a simplified situation, Cutkosky6 has indeed demonstrated how a Lie group emerges from the bootstrap assumption. However, the implication goes only halfway through the program, since he has had to assume electric charge conservation as an a priori piece of data. To ensure that the Lie algebra's rank should be r>2 one would have to add another a priori conserved additive quantity, like hypercharge. In fact, in a further treatment aiming at a study of the symmetry breaking, this assumption is essential.18 Moreover, one does not see how a particular group may be picked out; assigning a certain multiplicity to one type of particle can do it, but this is again an a priori insertion. Gell-Mann has demonstrated19 how a symmetry of the strong interactions could emerge from the equal time commutation relations of the weak and electromagnetic charges (or currents' fourth components). This was done just to explain the connection, but could this also be the true sequence? In the above Cutosky-type treatments, we would get the right symmetry by introducing the weak currents as an a priori fact, in addition to the electric and hyper-charges. In Gell-Mann's symmetry generation by recurring commutation we do not get the SU(3) algebra if we start with the weak and electromagnetic currents only; yet it can be done if we further add the hypercharge current. Since this may correspond to an independent "fifth" interaction,20 its inclusion may be justified. The net result of such an approach is that the boot• M. Gell-Mann, California Institute of Technology Synchro- strap—i.e., the kinematics of strongly interacting mattron Laboratory Report CTSL-20, 1961 (unpublished). ter—does indeed generate the hadron symmetry, but 7 Y. Ne'eman, Nucl. Phys. 26, 222 (1961). 8 already there through the See the Appendix in Y. Dothan, Nuovo Cimento 30, 399 only if all the ingredients are weaker—nonbootstrapping21—interactions. The boot(1963). * G. E. Baird and L. C. Biedenharn in Proceedings of Coral strap releases these charges from the scale relationships Gables 1964 Conference on Symmetry Principles at High Energy implied by their various coupling strengths, regener(W. H. Freeman and Company, San Francisco, 1964), p. 58. 10 S. Okubo, Prog. Theoret. Phys. (Kyoto) 27, 949 (1962). 11 N. Cabibbo, Phys. Rev. Letters 10, 531 (1963). " E . Abers, F. Zachariasen, and C. Zemach, Phys. Rev. 132, 18 N. Cabibbo, Phys. Rev. Letters 12, 62 (1964). 1831 (1963). 13 18 See, for example, P. Tarjanne and V. L. Teplitz, Phys. Rev. R. E. Cutkosky and P. Tarjanne, Phys. Rev. 132, 1354 (1963). Letters 11, 447 (1963). 14 ] See, for example, T. Schwinger, Phys. Rev. Letters 12, 237 »M. Gell-Mann, Phys. Rev. 12S, 1067 (1962). M (1964). Y. Ne'eman, Phys. Rev. 134, B1355 (1964). 16 21 M. Gell-Mann and Y. Ne'eman, Ann. Phys. (N.Y.) (to be The bootstrap idea does not cover the particles that do not have strong interactions. The interactions they do possess are published) and papers cited in this work. incapable of producing them in this sense. »Y. Ne'eman, Phys. Letters 4, 81 (1962); S, 312 (1963).
255
YUVAL NE'EMAN Embedded Space-Time and Particle Symmetries
ating them all (and adding a few others) on one and the same scale. From the electric current of the nucleon eN(T3+i)y„N and the so much weaker beta-decay vector currents
we extract e and G* and generate the isospin-hypercharge algebra and its strong currents, Nvy„N,
Ny„N.
An optimistic bootstrap theorist would have more ambituous designs; yet even if his hopes were to be fulfilled, he would still be faced with the problem of explaining the re-emergence of just some of the strong currents, multiplied by a new and smaller coupling, in the weak and electromagnetic interactions. I t seems that the opposite sequence has a better chance of success; we do know how to generate the strong-interaction symmetry provided the weaker currents are already there. We are thus faced with the problem of generating the symmetry relationship of the various currents—i.e., the algebra—from some other source. Is there an alternative to a kinematical generation? We would like to present here the case for a geometrical derivation of the "internal" symmetry. GETTING THE SYMMETRY FROM THE EMBEDDING SPACE Fronsdal22 has treated the problem of a quantized field theory when gravitation is added to the picture. Since the entire field formalism is constructed in a flat space—time, the easiest way to introduce the curvature—i.e., unquantized gravitation—is by invoking the flat embedding space. Joseph 2 ' studied the covariant description of a particle in a curved space-time and concluded that working in the local embedding space provides the only simple answer to his problem. Both noted that the local embedding could also generate additional symmetries of a local nature; those corresponding to rotations in the normal flat space would commute with the ones in the flat tangent space, i.e., with the Lorentz group. We now take up this local symmetry concept, assuming that in dealing with short-range interactions we can regard such a symmetry as a fixed gauge transformation; the action integrals, for instance, can be taken over a limited quasiflat region, at the borders of which all wavefunctions disappear. Unitary or orthogonal transformations in the normal space are operations transforming a small portion of the curved Riemann space—time into itself. Note that we are careful to assume isotropy and homogeneity of the embedding 22 2!
C. Fronsdal, Nuovo Cimento 13, 988 (19S9). D. W. Joseph, Phys. Rev. 126, 319 (1962).
229
space, which would lead to unphysical transformations. We start out directly from a symmetry compatible with the existence of a preferred hyperplane, the physical space time to which all wavefunctions are confined. Moreover, we have not assumed a Yang-Mills type 4 of locally dependent transformations, capable of yielding an over-all invariance of the action with no space time restrictions; this would be inextricably intertwined with gravitation as a field, since the gravitational field fulfills just this function for the first four coordinates.24 Considering that the strong interactions couple massive particles only, we can identify their symmetries with the above invariance over small regions of space-time. Note that this may already be less than the fullest normal-space symmetry at a point. We can therefore try to identify the largest local symmetry derived from the normal-space transformations with the chiral symmetry, broken by the strong interaction producing the baryon masses. One thus starts out immediately with a strong interaction Hamiltonian with particular transformation properties in terms of the thereby a priori broken chiral symmetry. Having refrained from using a locally dependent and "geographically" unrestricted gauge, we have to identify the internal symmetry substrate (giving rise to the chiral group) with that part of the embedding space which is orthogonal to the tangent space throughout our small neighborhood. This implies considering a global embedding space whose abstracted existence at every point yields our symmetry mechanism. From the mathematical papers in this series, we see that the local embedding can always be done in ten dimensions, with an arbitrary metric for the six new dimensions as proved by Friedman.26 The situation is unclear with respect to the global embedding; Penrose26 has shown that it can be extremely restrictive and may fix the signature of the extra dimensions. For a compact spacetime, the global embedding has to have 46 dimensions27 or less; for a noncompact one, the present ceiling is even higher. DIMENSIONS AND THE COSMOLOGY We can now check what the required dimensionality in the embedding space should be for the various candidate chiral symmetries. We note that in any case, the local ten-dimensional embedding would allow the emergence of SU(3) since it is a subgroup of orthogonal transformations in the six-dimensional normal space, provided the latter's metric28 is (H—|—|—|—|—(-) or ( ). An SU(3) deriving from an R (8) «2S R. Utiyama, Phys. Rev. 101, 1S97 (19S6). A. Friedman, J. Math. Mech. 10, 625 (1961). » R. Penrose, Rev. Mod. Phys. 37, 215 (1964), this issue. "28 J. Nash, Ann. Math. 63, 20 (1956). Y. Ne'eman and J. Rosen, Ann. Phys. (N. Y.) (to be published) .
256
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REVIEWS OP MODERN PHYSICS • JANUARY 1965
would require two additional dimensions. As to the chiral symmetries—SU(3) XSU(3) and R(9) both require a 16-dimensional global embedding (we have included here transformations for baryon and lepton number too), whereas SU(8) or i?(8)X-R(8) would imply that it has to be done in 24 dimensions.28 If we allow an analytical continuation, i.e., unitary transformations in the normal space, the dimensionalities drop to 10 for SU(3)XSU(3), 14 for R(9), 13 for SU(8), and20fori?(8)XJ2(8). This then is already one physical implication of such a mechanism; it can be true only if the cosmological picture does correspond to such a global embedding. Unfortunately, the present state of our knowledge of the cosmology does not allow us to check this result. From Rosen's29 actual embeddings, it seems that a minimum of ten dimensions will certainly be required, since even simplified local gravitational solutions require 6-8 or more, and the real world is much less symmetrical than that. SYMMETRY BREAKING We have seen that we start out with a broken chiral symmetry, the unbroken part corresponding to the parity conserving symmetry, i.e., the strong-interaction invariance "internal" group. Further breakdown of this symmetry yields the "fifth"20 and electromagnetic interactions; presumably, this is an effect connected with their space-time spread. The infinite range of electromagnetism, for instance, must badly break the symmetry, which is what we do observe. The weak interaction represents a much more involved situation. Its currents (now including the axial vector part) from another subgroup of the chiral symmetry, containing a piece of the strong interaction subs'J. Rosen, Rev. Mod. Phys. 37, 204 (1964), this issue.
group and a part which was not included in the latter. This complicated picture may be connected with the existence of massless particles participating in this interaction, and with the fact that some transformations involve a space inversion. We shall not study this problem here. If the symmetry breakdown is a result of the spacetime structure (or the range), similar effects should occur within the strong interactions themselves— whenever the curvature is such as to be felt even within a range of 10~~18 cm. Such a situation for a static gravitational field occurs only in extremely dense states, i.e., beyond nuclear density. Situations of that type could result from the last stages of gravitational collapse.80 Again, this is a definite physical implication—though apparently just as difficult to check as our conclusions with respect to the cosmology. CONCLUSIONS It seems to us that this picture thus offers a program of unification; the various interactions all occur as a result of the properties of space time. (Alternative connections of space-time and the "internal" symmetries have been suggested lately31-33; these schemes, however, yield an "internal" symmetry that does not commute with angular momentum.) This is a new type of "geometrodynamics," this time involving all interactions. Whether it is worth pursuing depends upon the chances of discovering further physical implications that would provide an experimental check of such a geometrical picture. 80 See, for example, J. A. Wheeler in Gravitation and Relativity, edited by H. Y. Chiu and W. F. Hoffman (W. A. Benjamin, Inc., New York, 1964), p. 195. » A. Barut, op. cit. in Ref. 9, p. 81. " B. Kursunoglu, op. cit. in Ref. 9, p. 20. "A. Komar, Phys. Rev. Letters 13, 220 (1964).
257
RIY.3TA DEI. NC0.V0 CTMENTO
VOL. 1, N. 5
1978
Gauge Theory of Gravity and Supergravity on a Group Manifold Y. NE'EMAN Department of Physics and Astronomy, Tel Aviv University (*) - Tel Aviv, Israel Center for Particle Theory, University of Texas (*") - Austin, Tex. 78712, USA T. KEGGE School of Natural Sciences, The Institute for Advanced Study (***) Princeton, N.J. 08540, USA Istituto di Fisica delVUniversita - 10125 Torino, Italia (ricevuto il 13 Marzo 1978)
1 2 1 11 20 2_9 36 3_9 40
1. Introduction. 2. Differential geometry and Lie groups. 3. Examples of Lie groups of physical interest. 4. Curvature and covariant derivatives. 5. Lagrangians and field equations. 6 . Supersymmetric theories and the factorization of graded subgroups. 7. Supergravity. Z__Z_ZZ_____Z__ 8 . Conclusions. ~ APPENDIX. - The (Euclidean) Dirac algebra.
1. - Introduction
In recent years, the intriguing properties of graded Lie algebras (G_A) have generated considerable interest [1-4]. In particular, it is widely felt that these structures may provide us with a new principle, unifying gravity with some of the fundamental interactions of particle theory. Such theories (and any gauge theory involving the action of a gauge group with a nontrivial action on space-time) are most naturally formulated by using the concept of (*) Research supported in part by the U.S.-Israel Binational Science Foundation. {**) Eesearch supported in part by the U.S. Energy Research and Administration Grant E(40-l)3992. ("*) Research supported in part by NSF Grant 40768X.
258
2
T . NE'EMAN and
T. REGGB
(graded or ungraded) Grassmann algebras of forms in the context of the theory of Lie groups. Much of the ground work for this kind of theory has been anticipated by OARTAX and is commonplace in the current mathematical literature. However, it is not as broadly known as it deserves among physicists. Part of our discussion is accordingly concerned with the problem of casting unified theories in the language of forms. In order to make the paper as self-sufficient as possible, we rederive known material of the theory of Lie groups, thus also displaying the consistency features of the method. Finally, existing theories are analysed and discussed in the light of the geometrical formalism.
2. — Differential geometry and Lie groups This is a vast subject of which we can only reproduce a few key definitions and results [5]. We also do not attempt to generalize systematically the theory to GLAs [6-7], but shall provide a few examples of interest. Let M be a differentiable manifold with covering Ua and let xa be co-ordinates on Ua- A p-form on J7« is an expression of the kind (2.1)
r]a=JdxtAdxi'A...A&xi'ETlU(xa),
{A} = {A,,...,
A„} ,
w where (a, b represent the gradings of A, B when the variables are elements of a Grassmann algebra) dxiA&xt = — (—1)"* &x*A&xi and the E\*\ are differentiable functions of the xa. the forms rja and r\^ provided that a^-Si
(2.2)
Va =
PIT3--
On Uar\Ug we identify
£>rBv
l I « L t f S Adai-g-. A - . . A d ^ g ; ^ U ( ^ « ) ) •
In this case, the set of all forms r\a defines a unique form r\ on M. In t h e sequel, unless specified, we drop the a-dependence. The A product of differentials can be extended by linearity in an obvious way to generic forms and we have (2.3)
rj^AS"" = (— 1 )*«+'»'f«»/\ T?"1 ,
where TJ"", |<" are p, q forms, respectively, and a, 6 are their GLA gradings. The differential d?f of a p-form is a (p -j- l)-form defined recursively by
259
GAUGE THEOJtT OF GRAVITT AND SUPERGRAVITY O S A GROUP MANIFOLD
3
the properties d(d^, = 0 ,
AQ(x) =
2 d ^ ^ ,
(2.4)
d ^ A f " ) = dTj^Af*" + (— l)'7?'" , Adf'». For any form, we always have d(d?;) = 0. We have adopted here ((2.2), (2.4)) the convention [7] of putting the differential (or the contravariant index) to the left of the derivative (or the covariant index), when summing over « dummy » indices. Note that the operator d behaves as a f = 1, a = 0 element in this product Grassmann manifold of differential forms. Given manifolds M and N, a map •/, M - ^ N, and a form v> on N, the « pull back » k*v* is a p-form on M, defined by the following procedure. If v* is given by v = Jd^A-Adt/-4'NA,..Ar(y),
(2.5)
yeN,
U)
and the map X is realized by y = X{x), then we have (2.6)
X*v - ^ c b * | g A - . . A d a * | g tfU}(A(s)) «}
I t can be checked that dA* v
= A* dv,
(2.7)A*(vAc) =
k*v/\X*o.
The d, A operations are thus co-ordinate independent and are preserved through maps. This property makes them ideal in treating covariant theories. Now we apply the algorithms of forms to Lie groups. (We refer the reader to ref. [2, 6, 7] for the effects of grading as in (2.4).) Let G be a Lie group (or a formal group in the Berezin-Kats sense [8]) parametrized by x1...xc. We use in the sequel elements x,y,s...eG whose l x parameters are x ...x% y ...if, etc. The element 3^ = 0 denoted by e is the identity in G. The product on G is a map G-G^+G and we write z = xy
instead of
« = A(x,
y),
satisfying the associativity condition t(xy) = (tx)y
260
Vt,x,y.
i
T. NE'EMAN and T. REGGE
In co-ordinates we have, for the product element z*, zA=AA(xl...
(2.8)
xc, y1.,.
yc),
yielding the differential forms :2.9)
dzA = d x " V(x, y)A3, + dy v W(x, y)$ .
In the sequel we use the c x 1 row matrices dz = {dz1... &zc} ,
dx = {da;1 ... dxc} etc.
and consider Vj,, W;J as c x e matrices. Briefly, then dz = 6.x V + &y W . Associativity implies t(xy) = {tx)y. This equation can be differentiated in all sets of variables and the results compared. One gets
(2.10)
a)
V(t,x)V(lx,y)
= V(t, xy) ,
b)
W(t, x) V(tx, y) = V(x, y) W(t, xy),
c)
W(tx,y)
=W(x,y)W(t,xy).
As a consequence of (2.10) xve have V{x, e) = W(e, x) = 1. The product on G gives two kinds of natural maps on G, the left and right translations G -^-> G ,
where l(a)(x) = ax ,
G -^-> G ,
where r(a){x) = a» .
Consider now the ( c x l matrix) 1-forms
I
a> = dx W{x~l, x), 7i = dx V(x, x-1) .
\'
I*(a)CJ = w ,
r*(a)7i = n •
"j and 7i are, therefore, called Cartan's left and right invariant forms, respectively, or more briefly the LI and RI forms. By the properties of d and A. any form obtained from W(TI) through these operations is also LI (RI). There-
261
GAUGE THEORY OF GRAVITY AND STJPERGRAVITY ON A GROUP MANIFOLD
5
fore, if we expand (2.13)
dw J = K ^ w * A w r ,
the Ciai= — CiEB must be LI o-forms (that is functions) and are, therefore, constants, the structure constants of the Lie algebra of G. Similarly one has (2.14)
inA = - * C ? „ J I * A » * •
Equations (2.13), (2.14) are referred to as the Cartan-Maurer equations. Under right translations we havp (2.15)
r*(a)w.-= dx V(x, a) Wia^x'1,
xa) =
= dx W(x~\ x) W(a~l, e) V(a-\ a) = tuad(a-*), where (2-16)
ad(a-!) = W{a~\ e) V(cr\ a) = W(a~\ e) V~\e, a^1}..
(In deriving (2.15) and (2.16), use has been made of (2,10).) Similarly, (2.17)
l*(a)n =nV{e, a"1) W(a, a'1) =
nud(a).
By using (2.10) one further finds that (2.18)
ad(o) ad(s) = ad(ax)
so that ad(a) is a cxc representation of G, the adjoint representation. matrix ad(a) is tied to the structure constants by useful identities. Consider the map GxG^-> G. From the previous comments it is that A*(o, A*7i also satisfy the Cartan-Maurer equations. If we use respectively, as co-ordinates on the first (second) copy of (?, we can xy = A(x, y) and (2.19)
A*a> = ai, +
The clear x, y, write
wxzid(y-1)-
In (2.19), wy, wz are the Cartan LI forms written in the variables x, y, respectively. Moreover, x translates y on the left and does not appear in eu„; y translates x on the right and appears in ad(j/ 1 ) - according to (2.15). By inserting (2.19) in (2.13), we find an identity relating 2-forms. Separate identification of the coefficients of da; A da;, da;Ady, dy/\dy produces C'.n ad(2/"1)^ = Ci„ ad(jf *);* *&&-%' , (2.20)
d ad(i/- 1 )^ = Cii,M< a d t y - 1 ) / .
262
6
T . XE'EMAN and
T. REGGE
These identities are very useful in establishing the gauge covariance of field equations. The map a-»• ad^- 1 ) 1 , = cd(a) is also a representation, the «coadjoint» representation of G, which is equivalent to the adjoint in a number of interesting cases ( r stands for the transposed matrix). Tangent vectors transform according to the coadjoint representation. 1-forms are essentially synonymous to covariant vector fields used by physicists. Contravariant fields appear in the mathematical literature as «tangent vectors » and are written in the form (2.21)
*=«•&•
Given a 1-form Q = &xi1qM, we can define the scalar product (V, Q) = ui'qi, as the «values Q(U) of Q on U. Clearly, (c/c.r*) &x" = d". Given two tangent vectors T, U we have a natural bracket operation
[
r TA'
3 Z7"
dx»'
defining a Lie-algebra structure on tangent vectors. If we write coA = oo'J Ax" and introduce Q?A through (2.23)
co>AAl = dl,
or
A?atj.
=
6i,
we can then define tangent vectors DA through (2.24)
^
=
^Sir-
Clearly,
[DA, DB] = CfASDE ,
which shows that the DA form a finite Lie algebra, i.e. by definition the Lie algebra rS of G. In this role, the C*AB reappear in a form more familiar to the physicist. However, (2.25) is fully equivalent in content to the Cartan-Maurer equations. From (2.23), (2.24) it is also clear that the DA are invariant under left translations.
263
GAUGE THEORY OF GRAVITY AND RUPERGRAVITY ON A GROUP MANIFOLD
"
One can similarly introduce a right-invariant vector field SB such that «*(£«) = *J.B, and (2126)
[SA1SB}=-C?ASSE.
The geometrical interpretation of DA is clear when considering the identity Wix-1, x)W(x. e) = W(e, x) = 1. From (2.11) and the second equation in (2.23) we have (2-27)
DA = W'/(x, e) A
.
But now. if yi(x) is a generic function on G, the expression (2.2S)
&y* DAxp(x) = dyA W'/(x: e) ^
v(x)
= J
cx" o cyA Sx"
, , r
, ,, cw cx-!
represents the infinitesimal change By) of ip under an infinitesimal translation to the right in the argument of y>. In this sense, the DA generate right translations. Similarly, the SA generate left translations. Quite naturally then both the LI of DA and the R I of SA can be expressed as (2-29)
[D<,S,] = 0 ,
3. - Examples of Lie groups of physical interest The (Euclidean) Poincare group P is defined as the set of pairs x = {E, x), where EeSOt, a e R 4 . The product rule is, by setting t = (0, t), (3.1)
(Z, z) = ( £ 0 , £ « + * ) = (S, x)(0,
t).
Thus the indices A, B etc. of sect. 2 run in the range 1 , . . . , 1"0. Because of the peculiar structure of P , we use a = 1, ..., 4 and t h e pairs (a, b) with a> 6, instead of A. In this way, the differential form of (3.1), corresponding to (2.9). reads
f dZ = d='€>-f-£d0, ( ds = d 5 t + d* + £ d t . The forms to are obtained as in (2.11) by setting d£", &x — 0 in (3.2) and and then carrying out the replacements ( 0 , t) -»• (2, x) and (if, *) -> {2, *) - 1 -
264
8
T. SE'EMAN and T. REGGE
The result is Q = 5~ l d =•, (3.3)
o = E-1 dx .
Q is thus a skew-symmetric matrix with elements a) 4 ' = — a>ba. Componentwise we have
i
o)"6 = Eca dEcb, oa
= Eca
dx°.
The Cartan-Maurer equations (2.13) read
I
(3.5)
dco06 + ft)acAwc4 = 0 , do" +w"Ao'
=0.
Similarly, the E I forms are (by putting d<9, dt — 0 and replacing (<9, t) by n == d S JET-1,
(3.6)
p = d* — JIX ,
Tangent vectors appear as operators, linear in 3/3£'a'' and djdxa and tangent to the defining variety of SOt with the normalization Tab ~cb —
Xac
A convenient set of L I vectors is -
3
"
has*
T) . = Eca ——
_
~*
3
(3.71 ^°
Clearly, Q{D.)
= w(Da6) = 0 ,
COc'(Z>o») = ^"^«—d c *^%
(3.8)
«u'(DJ
= 3'..
For the E I vector fields we find
(3.9)
*•» Sa
a*» " a * - * " dx"
265
A=V dE"
a.c-./'
GAUGE TUEOR.T OF GRAVITY AND .SUPERGRAVITT OS A GROUP MASIFOLD
9
A general field y(x) can be considered as an element of the Poincare group with its ^"-dependence mentioned explicitly and its dependence on E implicit and by fixing its intrinsic» spin (except that we have to replace the Poincare group P by its covering group). I t is the usual convention (3.1) in defining the Poincare group product that picks left translations (i.e. SA) for co-ordinates and fields: (A, a)(E, x) = (AE, Ax -j- a), which, e.g., for E = 0, x = x" and an infinitesimal P-trans formation A = b1" -f eI'"], a = a? yields (1, a " ) - > ( l , orf'-\- ell"']x''-\- a1*). Had we used right translation, we would have found (1, x1') - > ( 1 , x1'). The entire picture is thus E I and it is natural that the Hilbert-space considerations should lead t o ' t h e {Sab,Sa) set of ('3.9) with «total» angular momentum Sab including both that linear action on field components (« spin») represented by the E'^c/cE™) part, and a complementary «orbital» piece af"{ojcx''1). On the other hand, the procedure for «gauging» a symmetry implies the use of (co-ordinate invariant) forms, which thus have to be the LI forms o)A (and their orthonormal LI algebra DA). The L I set {Dab, D„} is naturally factorized (up to a Lorentz transformation on the Da) and will thus appear in the infinitesimal treatment of gauging [9,10] with only D& giving rise to a gauge invariance of the Lagrangian. The action of Da merges with general co-ordinate transformations on the a:-6pace and a modified Da can indeed be identified with the latter [10]. The formalism can be extended to supergroups. A most interesting example is provided by the graded Poincare group (GP) [11-13]. Consider first a Grassmann algebra A = A+©A~, where .47, A" are, respectively, the even and odd terms in A. We can then form a set GP of triples x=.(E, x, f) with xa e A/, J* e A~, E e SOt. We consider * as an 80t vector, | as a Majorana spinor. We define a product
(3.10)
= {Z, z, f) = art = (EG, Et + x +
fry)0,
U(E)6
+ *)
with t = {0, t, 0). This product is associative provided that (3.11)
v,-1(E)yau(E)
= £" a 6 / or, in brief, u~1{E)yu(E)
=.Ey.
Upon differentiation of (3-10) we get d£ = d£6>-j-£'d0, (3.12)
ds = d £ t + &x -f i df yu{E)d + l£y&u(E)8 + Edt + $iyu{E)dd . dC = du(E)6 + df -f- u{E) &d .
266
,
10
T. JJE'EMAN and
T. REGGE
Bepeating the procedure leading to w we obtain Q = ="-» d £ , (3.13)
o = £->(d*-ifyd£), to = u ' 1 ^ ) d | .
The corresponding B I forms are .'/7' = d £ ="-», (3.14)
p = dx — d £ £ _ I x — i d | y£ — J lyyi""^ 6 f , a = d | —^""cr 0 "! .
In deriving these results we used the identities (see the appendix) $-c<">o°i> =
dulZju-^Z),
(3.15)
Supersymmetric theories are frequently constructed [13,14-16] in «superspace » RlJi. Elements of R4'4 form equivalence classes of GP under right multiplication by 804 elements. Therefore, (~, x, £) ~ ( 5 0 , *, S) and R4'4 is conveniently parametrized by (x, | ) only. GP multiplication on the left-hand side by {0, t, 6), therefore, acts naturally on R4'4 and we have
(3.16)
r = «(©)i+e.
GP thus becomes a generalization of the Euclidean group P on R4. Here t h e metric (3.17)
ds 2 = dxa(x) &xd = oa@o"
is obviously left invariant on P and, therefore, P invariant on R4. Similarly, we have on R4'4, using (3.13), (3.18)
ds 2 = o" X o* + kit X to ,
which is also GP invariant. Notice, however, that on odd superforms the product [7] A (®) is commutative (anticommutative) and on even superforms
267
GAUGE THEORY OF GRAVITY AND SUPERORAVITY ON A GROUP MANIFOLD
11
A(®) is anticommutative (commutative) according to the usual convention (2.3). In this way we retrieve the well-known metric [17,18] of the superspace «vacuur 1 ». Also of great interest are the tangent vectors. The L I tangent vectors are given by
(3.19)
Uah
~ as"
^°
-
ss"'
a*<'
The E I tangent vectors are „3
ex" (3.20)
, -,
3^ ' ~
9
dEh'
-
a ^ + f^
8|'
« • - » •
>-h+\
a*™y * •
By neglecting the a/9 " t e r m s in S^ (spin), we obtain the infinitesimal generators of motions on R4'4. The D-components are related to the « covariant derivatives » of Salam and Strathdee [2]. The Maurer-Cartan equations appear as do>ab + o>acAcoc" = 0 ,
(3.21)
do" +
In deriving (3.19) to (3.21), care must be taken to keep a consistent ordering of Fermi variables in order to avoid confusing results.
4. - Curvature and covariant derivatives Let G be endowed with a set QA of 1-forms (a « e-bein » or «fundamental forms ») generalizing a>A, but not subjected to the Maurer-Cartan equation. We define the curvature RA as (4,1)
B* =
ae'-iCi„e>Ae?
268
12
T. XE'EMAN and T. REGGE
Hence coA has vanishing curvature. p-forms rf is given by (*)
The covariant
derivative of a set of
(Drj)A = &VA—ClaegBAyB •
(4.2a)
I t is convenient to introduce a similar definition of a covariant derivative of a « contra variant» set r)A: (D»?)^ = d7 ? 4 +Cf^o s A7 ? B .
(4.26)
If we vary gA in (4.1) by the infinitesimal 8V, we find BRA
(4.3)
= (msy.
We also have (4.4)
(Dhj)A = (T>(Dr))y = — C'^R"^*
(4.5)
(DR)A
= 0
(4.6)
&WU
= (DiyJ'Ak + ( - l ) V A ( D f h .
,
(Bianchi identities) ,
We next censider two copies of G, {GI; Gv) with co-ordinates (%; y), respectively, and a map A: Gx-^- Gv. Let in general
r(l)Qi = Ftoi + ef a d ( A - » ) ^ .
(Eecall that the dual map X* is denned in (2.6) and ad(a) in (2.16).) I t is often convenient to rewrite (4.7) for infinitesimal gauge transformations. I n this case, we suppose that X{x) is given by kM{%), where X" are infinitesimals. Using (2.11) and (2.20), we then find that the infinitesimal change is given by (4.8)
§gA = D ^ ,
where the r.h.s. is defined by (4.2a). Using (4.1) and (2.20) one verifies that the curvature r(?.)RA calculated from r(k)gA is given by (4.9)
r(X)Ri = R"z ad(A- J (x))^ .
(*) We write D(0) for 6 only when the nature of the group has to be explicitly stated.
269
OAUGE THEORY OF GRAVITY AND SUPERGRAVIT Y ON A GROUP MANIFOLD
13
Similarly, by taking
(4.10a)
riM* = f*cd(rl(x))i =«a(A(*)):'f,,
then (4.106)
{ 1 (Dr(i)C)i = (DC)8cd(A-1(^))f, = ad(*(aO)2(Df), •
Equations (4.8), (4.10) display the covariant nature of D and curvature. Let H eG be a subgroup of G of dimension c'. Let M = GjE be the corresponding homogeneous manifold. In the sequel, we often refer to the case G = P, H = S04, M = R4, or also G = GP, E = SO,, M = R4'4. (? can be considered as a principal bundle with fiber H, base space If. In the above-quoted case, this bundle is trivial, but already for G = SOa, E = S02, M = S2 one finds a nontrivial Hopf fibering, used in dealing with Dirac monopoles. Similar situations will arise in gravity and supergravity, and it is thus better to use from the start the general formal machinery developed for nontrivial bundles. In the following, we treat the conventional case of Bose-type manifolds and refer the reader to ref. [6-7] for t h e (Fermi) graded case, although we shall later develop in detail GP with a graded manifold. Let Uo. be a covering of M, p the projection G •£*• M. p-^Uo) is parametrized by elements (y, g>, a), where yeM, g a eE. We give transition functions
{y,qe,P),
where qa — cpaeqe and yeUa(~\Ug. I n this way, all sets p~x{Ua) are « glued » together into G. On Utt we have a map £«:
y«{y,q*,u)
= q».
Given the L I form coJ on G, %a defines a form ^*w J on p-^U*).
For each a
we suppose given the form
where r*a is a 1-form on M. If the set q*a has t o define a unique form QA on G independent of a, we must have the matching conditions (^.12)
T"* -
270
14
T. NE'EMAN and T. BE&GE
n (4.12) holds, oJ is called a factorized c-bein on the pair (G,H). Let 'S, 2? be the Lie algebras of 0, H, respectively. We take a vector subspace 3? of rS such that # = •#"-)- 3r, 5 0 / = 0. Each element aeS induces a coadjoint transformation on 2? which leaves srif invai'iant:
cd
(a)jfcje.
If. however, .jF is such that c&(a).9rc ^ as well, we say that -M — GjH i s a weakly reductive homogeneous manifold. This happens in particular if H is compact, semi-simple and connected, or even discrete. If weak reducibility holds. cd(«) and ad(a) each reduce completely into two representations of E for a s H. Correspondingly, we split the set of forms coA into sets wB, a>F such that G/(JF) =
0,
O/(3V)
=
0.
These sets are acted upon separately by the two representations in which ;id(a) splits. Clearly we have also [Jf?, &] c 3?.
If, furthermore, [J*, J5"] e J f , M is a symmetric, manifold. If weak reducibility holds, the set of forms oH with eqs. (4.11), (4.12) satisfies the conditions for a Cartan connection on 67 as a principal bundle. In this case the fl'-components of RA represent proper curvature and the P-components are known as the torsion. If [ 5 , 5"] = 0, then the curvature of oB as a Cartan connection coincides with the t?-curvature (4.1). All these conditions are met when G = P , H = S04, 31 = R4, which can be identified as a structure with ordinary Eiemannian geometry. I n this case then (4.11) reduces to f o"> = (5~l dE)ai + (4.13)
r^S^E'",
I Q* = £"" Tc .
Clearly, then
I
where
I
B'" = B*
5dbZca3!cd,
=±EC°-@C,
@'*= dT^-j-T^AT 0 ", 3t' = dTc + T" A T" .
As we saw in (4.9),
271
GAUGE THEORT OF GRAVITY AND SUPERGRAVITT ON A fiROUP MANIFOLD
15
In a Riemannian manifold, the condition 8%a = 0 is imposed and the metric tensor is given by (4.16)
J,T':®T<:,
gltv&y»®&y = c
TC is then identified with the conventional vierbein, and S&c = 0 allows one to calculate r'J from i tt and dxa through a procedure which is essentially the known formula for the Christoffel symbols in terms of the metric tensor. Once T°* is known, the curvature 0tab calculated from (4.15) yields the conventional Eiemann tensor. Some generalized forms of conventional relativity have been proposed, where torsion does not necessarily vanish, according to the original ideas—long forgotten—of Cartan [9,19-22]. These proposals, although far from any physical application [22], are nevertheless intringuing in that they remove in part the disparity of treatment between «internal» and « external» components of the o J , MA multiplets and deal effectively with a Poincare curvature rather than with a Lorentz curvature only, as used by EINSTELX. TO the extent that we might be tempted to give a more fundamental meaning to the group (? itself rather than to a factor GjH, we should certainly move in that direction. However, present theories with torsion do seem to have some drawback in this respect, since torsion does not propagate. While ordinary curvature does not vanish outside of matter, torsion does [22]. This disparity probably has its roots in the factorization hypothesis, which breaks from the very beginning the symmetry of the Poincare group, since it factorizes out in a trivial way the Lorentz-group dependence. (We have used S04 in our calculation for simplicity, but all results obviously carry for S013 with minor modifications.) If we do not assume factorization, the dependence of the c-bein on the Lorentz variables must be dictated by the field equations and boundary conditions. Section 5 contains a discussion on how to a t t e m p t this task and on the difficulties encountered. Now we review the Poincare gauge transformations, before and after factorization. Equation (4.7) is our fundamental starting point, since it realizes a transformation in which- the «y» copies of the group (the transformation parameters) are functions y — X{x) of the «x» copy variables (now used as the background manifold, an extension of space-time). For the P-gauge functions U: U'»(E, x), W(E,
x),
we find the transformations
r (Q')<* = mu)o)<"> = (U-i)acdUcb + (4.17)
272
{U-i)aeeciu,">,
16
T . KE'EMAN and T. REGGE
For infinitesimal transformations, zab = Uab— dab, £a=u",
I
SQ<"> = de" 6 + Qa'Ecb—^'o'"
we get
= DE"" ,
Jfote that D is here the Poincare ccvariant derivative and includes the eac action for o°. Holonomically, it entails derivatives and summations over all ten variables and differentials. However, we show in (5.12) that the full Poincare gauge is too strong as a symmetry of the Einstein Lagrangian. Instead, factorization of the E"* variables according to (4.11), (4.12) reduces the gauge group to 5 = SOt: f {x')ab = (TJ-1)"dU*
(4.19)
+ (Z7- 1 )«T r f Z7*,
[ ( T ')i = ( c r - 1 ) " ^ .
Notice the similarity with (4.13), the factorization equation. The action of an infinitesimal ecb is the same as in (4.18),
I
ST" 6 = de"" + Taa£db — £"rcb = ST"
Dmeab,
= — e"rc.
We thus do not appeal to « gauge the translations», i.e. the ua and £° gauge functions are not used. A gauge theory expressed in terms of forms is automatically covariant, ohat is invariant under the group of general co-ordinate transformations (GCTG). An interesting further adaptation was developed originally by H E H L et al. [22] and by VON DEE H E T D K [10], in the pursuance of the program begun by UTITAMA, SCIAIIA, KIBBLF, [9] and aiming at a derivation of general relativity from gaugelike principles. The new equations are useful in connecting the GCTG action to that of t h e gauge group (E c G in our above treatment). An infinitesimal element of GCTG is given by a tangent vector eM on G. We then have (4.21)
SxM = EM
and
B
»d*»[|d£
where n, m are the gradings of the indices N, M for the case of a graded mani
273
GAUGE THEORT OF GRAVITY AND SUPERGRAVITT ON A GROUP MANIFOLD
17
fold and -where we have defined an anholonomic co-ordinate variation (4.2")
e»=*e»Q''.
The last line can be rewritten as V = deB— 2(e, &oB) ,
(4.24) since
(4.25)
dS' = -l (d^Ada") ( f g - ( - 1 ) - H )
and the scalar product (e, do2) is a 1-form obtained by contracting e" with the second factor das" in doB. Using the
(4.26)
So8 = De8—2(e, £*) = DeB— QEeFR'£F£B'f
4
We refer to (4.2G) as the anholonomized general co-ordinate transformation (AGCT). Whenever R" = 0, the AGCT coincides with an ordinary gauge transformation (4.8). Yet a third way of introducing vector fields on G consists in using the natural extension DB of the set DB given in (2.27). We define the D„ as in (2.23), (2.24): (4.27)
e*(I)B)=d:s.
The commutators of the SB no longer close to form the Lie algebra ^ of G. In order to identify the necessary modifications, we start from the definition of curvature (4.1) in terms of the anholonomic components of fiJ (4-28)
RAz=ieBAQER.B./}
thus yielding (4.29)
< V - i o * A g £ ( C ^ + By/) = 0 .
This entails the (graded Lie bracket) commutation relations ( 4 - 3 °)
0B, DB\ = (CiMt + RV/)DA .
In case of factorization under a subgroup E c G, some important simplifications occur. In particular, fiA= DA for Ae{J^}, where {} denotes the range.
274
18
T. NE'EMAN and
Indeed, for
T. REGGE
Fe{^} Q'(DA)
= to"(DA) - 0 ,
because r* in (4.11) contains only «horizontal» (i.e. 31 = G'H) differentials, orthogonal to DA. Similarly, for 2 ? e p f } o'(DA) = (o'(DA) = ft'.A . Therefore, quite trivially, the commutation relations of the DA, Ae{jSf), are the same as those of DA. In addition, the commutator [D B , DE] for Be{Jf} is equal to the «fiat» case for all E, since R'//= 0 whenever Be{Jf}, V.E. Indeed, according to (4.9a), BA has no vertical components since 3%A is constructed on M = GfH only. The factorized c-bein for the pair [G, B) can be written (4.11) as QA = col + T'-acKr 1 V •
Ae{je),
(4-31) Ae{.¥), where, for simplicity, we neglect the index a and we write (of, for X*G/. AS stated, LA~DA for Ae{St?}. Now we compute explicitly Dr for F e{3F}. We use co-ordinates n? on M = G/E, this being the ease if
J-c
T
p
—
v
ci
(4.32)
so that o*{Tc)=dic,
(4.33)
.4, C e { , F } .
This suggests the form
f DC = ad(9);*Ty(i--i 2 *•/£,) = i W , Ce{Jf},
5 C = Dc ,
where S^ is the E I vector field on E. The quantity in the brackets corresponds to the conventional covariant derivative D™ on M. Indeed, a short computation will show that (4.34) satisfies (4.27): Q'0B)
= 6.1B,
275
VJ.,5.
GAUGE THEORY OF GRAVITY AND SUPERGRAVITY ON A GROUP MANIFOLD
19
Now we apply the vector fields A to a;*, as a specific realization of (4.21). This corresponds to
(by using (4.23)). Applying these fields to (4.26), we obtain (4.35)
AAQ' = -O»{(ZIU
+
R1;*),
which generalizes the global group transformation and exhibits the role of the A in generating when «gauged» ordinary gauge transformations (4.8) for A e{se} and AGCT for A e{3?} as in (4.26). Note, however, that the A do not close to form a graded Lie algebra in a strict sense, unless 1 E { / } . Specifying t o P, we have the AGCT f So06 = De06 — 2{e, R°») , (4.36)
I So" =D£»-2(e ) 5") , which should be compared with (4.18). However, as explained above, (4.2a) ensures that in the factorized case as given b y (4.13) we have (4-3?)
iC,(.V sM =
2CK;„0
= -R(c««(ob) = Rum." = 0 ,
so that (D is still D m ) (4.38a)
So"" = De0"— oe£"2C6
(4.38ft)
8e° = D e " - 2 ( £ , R°),
,
so that e"" of SOi produces an ordinary gauge, whereas t h e translations e" are indeed replaced by an anholonomized co-ordinate transformation gauge. The corresponding Lie-algebra generators A& and vector fields A in the bundle of tangent frames are (4.39)
(4.40)
A , = Dab,
A = ~b°Tr(d,-
S
£«J=- ,
\r"(x)Sif)
,
TVr? = dt
with the commutators (4.41a)
[D», Dcd] = dotDbc + SbcDad—
(4.41&)
[ A M A ] = dbcDa—dacDb,
(4.41c)
[A, A J
276
dacDbd—5bdDac,
20
T. KE'EMAN and T. REGGE
For GP we have for (4.26) before factorization (4.42a)
So0" = de"" + o-'Afi'6 — g^Ac"' — 2(e, R">) = De"" — 2(e, R">),
(4.426)
$o° = d £ " + o"'Ae'— O ' A E " + o y e — 2( £ , 7?") = De" — 2(e,JR*) ,
(4.42c)
So = d e + i(oa*or<"')A£ — . ^ " e A c ' " — 2(e, iJ) = De — 2(e, i?)
and, after factorization of S0t
D{ap]),
(with D =
(4.43a)
So 06 = Be""— gc ed R'c'd"» — Q'e*Rex"'
(4.436)
V
(4.43c)
§
,
= r>e" — 2(e, R°),
Aside from (4.41a), (4.416), we have for the most general factorized case the commutators (4.44a)
[D«,D]
=(aab)D,
(4.446)
[2>„ Dt]
= Ri;cdDei +
(4.44c)
CaS[Da, DB] = R"*I)«
R;;*DC c
+ Kl Sc
+ Rabr>, +
R\7By
(note that, though /? in Dp is an «upper-spinor» index as in | 3 and has to 11 lowered by Cap, we write it as a subindex like a or (ab)) (4.44d)
{D*, Ds) = (y')atD. + R-JaDcd + E ^ £ e + S;'/2>„ .
One last remark about factorization: had we chosen H = G, then factorization means trivially oA = coA, that is empty space and a global symmetry only. If instead H = e, factorization imposes no conditions on gA. All ^-commutators generate curvature terms, which is another way of visualizing the difficulties of providing for an invariant action, as discussed in sect. 5.
5. — Lagrangians and field equations I t is often argued in much of the recent work on unified theories that t h e final goal should be a purely « geometrical» theory. The definition of « geometrical » is at the moment not clearly stated and correspondingly there is room for disagreement as to whether many of the proposed theories really fit the criteria for geometry. In an attempt to narrow the discussion, we
277
GAUGE THEORY OF GRAVITY AND SUPERGRAVITY ON A GROUP MANIFOLD
21
introduce in the sequel a number of interesting geometric criteria: I) The « objects » or fields in the theory are c-beins qA on a convenient Lie group (or supergroup). Note that this generally implies a self-sourced theory with no explicit « matter* fields; alternatively, the geometric criteria characterize only the «generalized-vacuum » limit of the theory. Ila) A Lagrangian density is formed from the c-beins by using the operations D and A only, plus contraction over indices. II&) In addition to Ila), we allow the use of the dual (*) of a form (enlarged definition). Ilia) We consider a preferred subgroup E c G and construct a theory which is gauge invariant under E, but not under the full gauged G. Ulb) In addition to Ilia), we request factorization of the c-beins (reducing the «objects* to fields on the GjE variables) and vanishing of torsion in the G/H manifold. Alternatively, factorization may appear as the result of Ilia). All theories of current interest meet I). Classical general relativity satisfies I), Ila), III6) and does not involve lib). Prom the results of sect. 4, conditions I), III6) are obvious. Ila) follows from the (Trautman) action [21] (5.1)
8A = j firtAe"Ae*e.»«i = /^ai,ATcA*"£«»<* , R'-
fl«
which coincides with Einstein's Lagrangian. (Again, we denote anholonomic indices by Latin lower-case indices, using lower-case Greek indices for the space variables.) Variation of (5.1) proceeds as follows: "we vary first T"*, thus obtaining (5.2)
84 = J " « « * » A T « A T « W =JD(_8r)°>ATeabci ,
where D(ST)°* can be interpreted either as a O-covariant {i.e. Poincare P-covariant) or as an E- ( = Euclidean «Lorentz» S0t) covariant differentiation (the latter is the «anholonomic » conventional one in general relativity) [9, 22]. In the last event, integration by parts and the antisymmetry in (a «-> b) yields the equation (5-3)
J8r'"'Al>TcAr'eatei = 0.
Fow, by definition, the jff-covariant derivative DT C is just the torsion &'. It follows (5-4)
fST'",A^cAT,J£oil.li = 0 .
278
T . NE'EMAN- and
T. KEGGE
Since Sr°* is arbitrary, we get f # « A *'«.»«* = 0 ,
(5.5)
1 ^7\Td
3Tow (5.5) implies @c=0.
Va, 6,
—Si*, 'AT*.
Indeed, if we set
& = uiab r°A TS
with «: r t = - •ulia
and define «"» = £*""'<„, (5.5) implies ucab
—
uacb
^
fc^
a]g0
utab
_
u&a
#
A few manipulations show that u""> = 0 = uc,ab and, therefore, 3$° = JRC=0. The variation in T" then yields (5.6)
(# rt AT')e*/. = 0 ,
Vc.
If we set da"A da;" A da^ = s"*^ di>a, //, r, X, a = 1,...., 4, ?ab
_n
_6
,3Ma
(5.6) can be written as
by using
where A —Vg, and, if we delete the factors &vx, r'f, we get (5-7)
^v£/Jeo;i£'n*o =
0
or (5.8)
SKf — \adif = Ot
which are Einstein's equations for empty space. This method is clearly equivalent to Palatini's in that the vanishing of the torsion follows from the variational principle (and of course from factorization). Tentatively, it would appear possible to assume (5.1) with nonfactorized c-beins (and with Ra» instead of S$ab). One is then faced with two options, neither of which appears entirely satisfactory.
279
GAUGE THEOKT OF GRAVITY AND SUPERGRAVrTT OK A GROtTP MANIFOLD
23
Since there is no factorization, there is no priviledged role of SO, (i.e. H) in P , and it would appear desirable to extend t h e integration to the entire group space (G) rather than to GjH only. The action then takes the form 8A=JR«>/\ec/\e''EllbcaAv,
(5-9)
F
where v is a 6-form including the &E differentials (at least in the factorizable case). None of the obvious forms tried so far for v seems to reproduce Einstein's equations. A second option is to consider (5-10)
SA =
\R^/\Q'/\Q'E^,
where J{i is a submanifold of G or rather a section over G, when this is taken as a bundle. This seems to be unnatural, for Jt* was in the original intention the base space GjH. Furthermore, the very presence of Jl* in the variational principle makes it a dynamical variable, thus also subject to variation. This implies the added complication that the equations describing the embedding of Jlk in G should include arbitrary functions which must be considered as fields, in which case we would have strayed out of our conditions I) for a «geometric » theory. If, however, we decide to overlook these difficulties and try (5.10), we find the field equations
I
R'Ao"—RbAQa
= 0,
R^AQ'e^
=0,
resembling the factorized expressions (5.5), (5.6). Here, though lack of factorization inhibits direct deduction, heuristic arguments do yield R" = 0. Interestingly enough, (5.11) which follows from t h e variation of the fields a"", Q" also suffices to guarantee that the variation in Jt* vanishes trivially. In fact, the differential of the Lagrangian density 2£"»A#iAe*e«»«i
vanishes as a consequence qf (5.11). This fact is not so strange. Any variation of Ji* can be compensated by a change of co-ordinates in G; the embedding functions are thus unchanged, whereas ga, g"" do change. This of course is taken care of by the field equations (5.11). The option (5J.0) thus appears to contain an automatic answer to the threat of «degeometrization » to which we alluded. Returning to the further interpretation of t h e eqs. (5.11) of the unfactorized set, we remind ourselves that t h e factorized forms T " , T° constructed
280
24
T . NE'EMAN and
T. REGGE
through the gauges (-4.11) satisfied R" = 0 and eqs. (5.5). They thus also satisfy (5.11), since oab, o", Ra, R"" can be reconstructed for the factorized case through (4.13), (4.14). Factorization thus supplies a solution for the unfactorized field equations (5.11). Xow we can make the converse conjecture, namely that eqs. (5.11) uniquely yield the solution gA-+rA via (4.11), apart from a generic co-ordinate transformation on G. If this were true, factorization would be a result of (5.11). This would be consistent with our previous deduction from heuristic arguments that (5.11) implies R" = 0, just as can be directly deduced from factorization. The same would now be true of (5.S), i.e. the Einstein equation. The theory, as extended here to forms on all of G, but with the action integral restricted to an arbitrary 4-dimensional submanifold, would still have the same physical content as Einstein's theory. Actually, since the choice V/4 has turned out to be irrelevant, we can choose Jff* = Rl, i.e. identify the submanifold with «space-time ». This then sets our Lagrangian (5.10) to be equal to the factorized (5.1), as a boundary condition o'(ft4) = -zA with o'((?) derivable from o'(R4) through a transformation which is just the inverse of our factorization gauge (4.11). Conceivably, the unfactorized (5.10) may lead to new solutions (with torsion) in the presence of matter fields. However, in the present case, extending the forms to the entire P-manifold could be regarded just as a way of writing down simultaneously all gaugerelated solutions (with appropriate co-ordinate transformations automatically completing the procedure). The form (5.1) of the action brings us to another argument. Clearly, (5.1) is invariant under SO, (i.e. H) gauge transformations, but not under proper local Poincare gauge transformations (that is including translations). I t is instructive to see how this happens. By specializing to a translation-gauge transformation as appears in (4.18), we find (5.12)
[ T{1) Q°" ~ a*", [ T{1)Q"
= du" -f Qa + gal"u? = e" + Du",
where w is the local translation and D is the SO, covariant derivative. Therefore, R"" remains unchanged, but o" does get modified. The change in Ra can be computed directly: (5.13)
T(l)Ra = DT(/) Q" = Do a + D 2 ^ = R" +
R'bu".
Thus a torsionless c-bein acquires torsion under the action of a translation gauge. To no one's surprise, both the Lagrangian and the field equation break
281
GAUGE THEORY OF GRAVITY AND SUPERGRAVTTY ON" A GROUP MANIFOLD
25
the translational gauge symmetry. We have seen in (4.24) how the invariance under general co-ordinate transformations (GCTG) in GjH can be represented as an AGCT translation gauge. Such a gauge corresponds to a translation generator obeying a new algebra with curvatures as structure functions, instead of structure constants. The Lagrangian obeys that AGCT gauge invariance, instead of (5.12). However, the curvatures themselves in (4.24)-(4.30) are determined by the Lagrangian through the equations of motion. What is then the criterion by which we select a subgroup E (in this case SOt), whose symmetry is not broken, out of the whole group G% Clearly, some part of the symmetry must be broken. A Lagjangian built out of curvatures only and invariant under G is by definition a characteristic form and yields only topological information, but no field equation. MACDOWELL and MAKSOURI have recently [23] proposed a curious alternative to the customary Einstein Lagrangian. Their basic approach is to consider a G == 80s, S = SOt theory (actually S03i and S031, respectively, in the physical case) and a factor space Sl = GjE. One, therefore, has curvatures (5.14)
R<* = do* + Q" Ago = — R""
with a, 6 = 1,..., 5. In order to compare this formalism with the previous one, we split the range by calling gai = eg", where e is a constant. We then have
1
i?°» = do"" + Qacf\oc,>— e-QaAeb > a,b=l,...,i,
R° = do" + e a c Ae c ,
In the £->0 limit, (5.15) reduces to (4.15). This limit corresponds to the Inonii-Wigner contraction of SOs into P . I n ref. [23], however, the action integral is chosen just to be (5.16)
8 ^ = =± JR°bAR<dea>cas = = ^ [&>AR«e»«
,
a,.b = 1, ..., 4 ,
Applying (5.15), we get (5.17)
SA = ~
JE^AR'"ecbci
+ f.S«»Ae e Ae'e.» e «— £2 I Q"AQ,'AQ':AQdEatcd = At + A2 + A3 ,.
where
R*» = dea» + g^Ae'"
282
26
Y. NE'EMAK and
T. KEGGE
is the «Poincare» curvature. Now Ax is a topological invariant and yields no contribution to the field equation. A^ is the classical Einstein Lagrangian, A3 is a cosmologies! term. In the limit £^-0, one retrieves the conventional theory and e - 1 is essentially the radius of the de Sitter space, i.e. the vacuum solution when cosmological terms are present. In a way, the MacDowell-Mansouri formulation is more aesthetic in t h a t only curvature components are used and the search for a Lagrangian is narrowed down to the choice of a suitable »S'05-breaking term. A different way of looking at the issue is the following. Can we anticipate the form of a Poincare-breaking Lagrangian that can be derived by contraction from an ansatz involving only curvature terms? In order to answer this question, it pays off to look at the procedure of contraction of a group in terms of the Cartan-Maurer equations. Contraction implies that one can choose the structure constants CA.B£(s) as functions of a parameter £ and that the resulting groups Ge are all isomorphic except at e = 0, where the contracted group arises. We, therefore, have the Jacobi identities (5.18)
C-iBG(s) GiDE(e) + ClBD(e) CB.Ee(e) + ClBE(e) C*aD(e) = 0 .
Differentiating, we find (o.l9)
—^
C"DB(E)
-f CBG(e) — s
0£
h cycl = 0 .
C£
ZSTow the 2-foTm ju.A = —^—I
coBA«
has vanishing covariant derivative at e = 0 (with gA = coJ) as a consequence of (5.19). We have indeed (5.20)
CiBeu>*,\nG = i ^ 2 ? CfDEcoDAo>BAcoGC-lBGu>Bh ^ ? ^ A w
1 oC.BG Copi;0)BAa)DAa)E_
D
=
GiDE + CiB0 i ^ ^ A ^ A c o 0 = 0 .
= ^ Of
C£
We call any form rf with vanishing covariant derivative a « pseudocurvature» form. Clearly (5-21)
CA = C*BDcoBAa>D
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GAUGE THEORY OF GRAVITY AND SUPERGRAVITY ON A GROUP MANIFOLD
27
is a pseudocurvature corresponding to a trivial contraction of 6, where CABD{e) = = f{e)GABD and /(0) = 1- Therefore, rf is really an equivalence class modulo £•* (see the comment at the end of this section). These classes already appear in the Chevalley cohomology on Lie algebras [24]. The existence of a nontrivial class is related to a nontrivial contraction procedure. Indeed for P we have the nontrivial class /x"" = WA &A /u"=0, which corresponds to the contraction of S0t into P. On the contrary, SOb itself has no such class. Thus, on S0h we have the MacDowell-'Mansouri procedure but no pseudocurvature, while on P we have a pseudocurvature but the MacDowell-Mansouri procedure fails: E""Rcd£atcd is not only Lorentz invariant, but Poincare (gauge) invariant as well, and transforms away into a surface integral. The presence of pseudocurvature is. therefore, a warning that it is possible to construct a Lagrangian density directly with /uA without « decontracting» first. Alternatively, nA is the infinitesimal difference between contracted and decontracted definitions of curvature. We should also note that we are interested more generally in all forms /j.A or vA possessing vanishing covariant derivatives. On P we then have {D/u)ab = d//06 + (oatAMtb— tt>'»Ai"0', (5.22a) (Dp)"
=&jj.a-\-MatA/j.i—cDiAluat
and (5.22b)
I
(Dv)o!) = dvoi + a>"Ava(—oiaiAvbt -\-(i>"Ava — w°An, (Dr)„ = dv„ + (oatAvt •
If we find either /uA or vA as a polynomial in <»A, we can replace taA by QA in this polynomial. In this case, it is no longer true that (Dfi)A = 0, (Dv)A= 0, but rather those derivatives become linear expressions in the curvature RA. This provides an interesting argument when building Lagrangian densities. If we have (as in relativity) an action of the kind (5.23a)
A=(BAVA,
then variation in QA gives a contribution of the kind (symbolically)
U =J* {(DS e )^ + R* Sg* | p j = 0 . Integrating the first term by parts we obtain
284
28
T. NE'EMAN and
T. REGGE
or (5.236)
- (Dv)x + R'Jl
= 0.
Equation (5.23) is identically satisfied for o"1 = oA, since RB — 0 and (T>v)A is linear in i? 1 . Therefore, o'1 = w^ can be assumed as «empty space ». The above discussion would lead us to the conclusion that, if G is semisimple, then the MacDowell-Mansouri procedure seems to be ideally suited for the construction of a Lagrangmn. If. on the other hand, G is the result of a contraction, it has in a natural way a pseudocurvature form rA (we can take v„ll= £„i„dQc/\Qd- v„= 0) which can be used in forming a Lagrangian. The C-symmetry in this Lagrangian is broken because pseudocurvature « remembers » which subgroup E in G is not affected by contraction (e.g., SO,, in going from SOs to P). The way in which this breaking occurs is dictated by the structure of the group G itself, rather than by some ad hoc choice of E. It should be emphasized that only the 2-form /uA, with {Dfi)A = 0. can be related directly to the contraction procedure. A generic ps form (ps means pseudocurvature) with (DC)'4 or (Dv) A =0 is still useful in building Lagrangians, but its geometrical interpretation may be less straightforward, involving the use of various group invariants besides the group metric. Finally, we remark that, if fxA, CA, vA, etc., are pseudocurvatures, then
(5.24) h =
CBADiiDt\vB
are also pseudocurvatures. This allows the construction of an algebra of pseudocurvatures of interest in trying to define a suitable Lagrangian formalism. One final comment with respect to the possible use of the trivial-contraction t,A ps curvature (5.21). To apply it to an action such as (5.23a), we first have to lower the index: XA = 9AB^B
,
where we have used the group metric (in a semi-simple group, its inverse g^ also exists). If now we try the action
we discover that this is again a topological invariant and yields no equation of motion, for either semi-simple or contracted G. This is because CA carries no information about a specific E c G, no symmetry breaking.
285
GAUGE THEORY OF GRAVITY AND StXPERGRAVITT ON A GROUP
MAXIF0LD
29
For our further discussion of supergravity, we include here the GP invariant derivatives of multiplets /iA and £A;
(5.25)
(Dfi)' = d/i* + ea'Aju'— Q ,
(Dtu = d c (5.26)
Q*AU
+ e"AC
+ e\\t;a -e°Atb,
(Df). =d£ 0 -e'V\t 4 , DC
=dC + erc 0 -ie o 6 -V
i-a&
6. - Supersymmetric theories and the factorization of graded subgroups A consistent theory of supergravity has been formulated [25,26] in a secondquantized formalism, by finding a set of local graded infinitesimal transformations on a set of fields including gravitation and leaving a Lagrangian invariant (after imposing a constraint equivalent to the vanishing of a generalized torsion). Various attempts have been made to find a geometrical formulation: a presentation of the theory in the context of a Palatini treatment of Einstein's relativity [26, 27], an attempt to derive it as a contraction limit in superspace [28], a partial description using a superspace with constant torsion in its physical vacuum [16] (see also ref. [7,15]) and a generalization of the MacDoweUMansouri treatment of gravity. Other works [32] have pursued the program for a generalized supersymmetry group (including internal degrees of freedom) and for the superconformal group. In this article, we are interested in the supergroup GP and associated subgroups, i.e. the problem of picking H c G. A) The first candidate is E = 80t. I n this case, GP/S0 4 = R4/4, where R is the set of pairs (*, | ) in a Grassmann algebra A, where x° e A+, (* £ A~ (see (-3.16)), i.e. xa is a commuting (I* an anticommuting) variable. Notice, however, that, strictly speaking, each xa contains more information than a real number. A itself is a real space of very high dimensionality, allowed by higher than R4, and containing all nilpotent elements-of the form A~® A~ etc. We refer to this space as «superspace ». Theories using this form of H should have complete local Lorentz gauge invariance and complete covariance under general co-ordinate transformations x' = f(x, | ) , f' = g(x, £). I n this class of theories we have the super-Eiemannian (Vtt in a graded extension of the Schouten notation [22]) Arnowitt-Nath approach [14] and the alternative torsion (C M ) superspace of Wess-Zumino [16]. Theories based on this framework have nontrivial computational tasks to overcome and for this reason iU
286
30
T . NE'EMAX and
T. REGGE
still have some loose ends in their final shape. In particular, the way in which ordinary general relativity is or should be contained in these theories has not been elucidated. In the Vt, version [16] the language of forms is used, b u t only field equations and no Lagrangian are given. In the Vtt theory [14], the standard Einstein tensor calculus is used in the graded version [29], up to the actual writing of a scalar curvature R. The resulting field equations have not yet been investigated in detail (even the listing of all fields has not been completed). A major stumbling block remains the need to use the Berezin integration formalism over the odd variables, a procedure which selects the term of the highest degree in those variables, with aresulting increase in the volume of computations. B) A second choice is H = S0t® A- = GAR. group whose elements are the pairs
(6.1)
This is a particular sub
i(i +
I t is, therefore, a semi-direct product. A mirror image GAL is obtained by considering (6-2)
[Z,(L),
i (1-ty,)^ = 0
instead of (6.1). We have GAxr\GAl = SOt. This choice for H seems to have been overlooked. We remark, however, that GP is a contraction of OSph„ which contains OSp12C as a subsupergroup. The contraction of OSplttC leads to GAR. Those groups appear in the MacDowell-Mansouri version of supergravity [23]. but it seems that their Lagrangian is not invariant under local GAS gauge transformations. Moreover, their forms are written on R* (or possibly our larger sfi since the same formalism would apply), and not on GP/GP^ = J?**. The elements of Jlf are pairs (6.3)
(*,lx)»i(l-*y.)f£ = 0,
*£•<,
with an «analytic » 2-component £t Majorana spinor projection. C) We might also select the sub(graded)group H = GPi, defined as the set of triples * = (S, x, | ) , where now f e A~, but xeA~x A~, that is xe A+ but is a nilpotent element. Obviously this is still a supergroup and also GPj c GP, but GP! is a proper sub(super)group of GP. By obvious considerations we have GP/G?! = R4,
287
GAUGE THEORY OF GRAVrTT i V D SUPERGRAVITT ON' A GROUP MANIFOLD
31
where this is now a purely ordinary manifold on the reals. This is thus the factorization appropriate for a (Euclidean) space-time theory of supergravity. We shall detail it at this stage as a heuristic model. The GP! factorization has no counterpart in ordinary Lie groups since both GP, and GP have apparently the same Lie algebra. This seems a very interesting possibility for we would write forms on ordinary space, but the theory would still have local supersymmetric GP] invariance. The difference between GP, and GP lies only in the choice of the coefficient ring of the element of the Lie algebra. The generic element (6.4)
aVSii+cL'Si
+ dLS
will have the coefficients of the spinors S a, a e AT , those of the Lorentz generators Stj
a'^A*, but the translations S, have nilpotent coefficients in GP 1? (6.5a)
a'eA-
a'S«eGPl,
as against either nilpotent or real ones in GP, (6.5b)
a*eA*,
a'S,eGP.
Similarly, the space-time vector forms o take A+(A~ ® A~) values on the Lie algebra D^D,.^). Q is always J. + -valued and u> has ^"-values on the D of either group. I t is quite conceivable to write an J. + -valued form o as a sum o = oT + o1, where o r is the real-valued and ox is the GP! nilpotent-valued part. We can thus also rewrite for GP the Cartan-Maurer equations (3.21) in the form dco06 -f- (0atAa>a = 0 , (6.6)
do" +aj"tAotr = 0 , dco +} (a>a<>cr°*)a> = 0 , dof + (oat/\o\
+ \ayya<x> = 0 .
288
32
T . NE'EMAN and
x. REGGE
The GP! equations are obtained by putting (6.7)
o, = 0 ,
o = olt
oeGP,,
Thus (3.21) can be used for both GP and GPi provided we restrict the values of o suitably to nilpotent elements for GP,. In a physical situation with curvature, these forms are replaced by a set QA with t h e same scheme of values. We shall develop the GPi choice in more detail. I t is particularly interesting if one looks for pseudocurvature forms. We have no proof that the choice in the following is unique, but it does lead to a new «geometric » derivation of previously known results. By means of (3.21) it can be checked that the multiplet £4 given by
(6.8)
C. = o ,
has vanishing covariant derivative. We propose, therefore, the Lagrangian density (6.9)
%Se = RaiAU
+ B'ACa = R-Q'e'e*,.
+ UMy.ey-Q
= B'"'Q'e'eabf, + 2iRyiQ"y°Q—2i(iyiQ°yaR Factorization implies that Q-» = AlaruA"> + A" &A">, (6.10)
Q* = Ata{'Dal + T< + fy
+ Boc),
where D denotes the GR derivative,
(6.11)
I
Da = da -{• \x°i<j
(T'*, T', T) are forms on !%* and We set here
A"dAa,
m" (6.12)
o
(A,a,a)eGPi.
=
A-\da—\£y&<x),
co = « ( J . _ 1 ) d a ,
289
= + surface terms .
GAUGE THEORY OF GRAVITY AND SUPERGRAVITY ON A GROUP MANIFOLD
33
which satisfies (3.21), so that (6.10) indeed realizes (4.11). The corresponding factorization of the curvatures is given by A'a0l"A"',
R«* =
R" =Ata{Sithab (6.13)
R
+ &' + @y'x — £ay<#"
l
= u{A~ ){@ -j- l@*boaboc) ,
M =(^ —
\&Ma"aabMA).
The factorized 3?-4 are constructed from (T' S , T*, T) by using the same formulae yielding R* from (o's, g', g):
(6.14)
@° = d T a - | - T°'AT' + $ f y ° T , ^
=dT-fi(T' l i 'CT'' ! ')T = D T .
We can now insert (6.13), (6.10) in (6.9). We denote by JS"e the Lagrangian density constructed as in (6.9) and by £?r the same with gA-^rd. There is no A'a, &A"> dependence in £P^ at all. I t is, therefore, possible to choose Ata— dta, d.Ala=0. The final result appears to be rather complicated: (6.15)
— 8 ^ e = — &SCT + 20t«>>®>a.<Ealt>n — 9t°*®'la-'a.%£*n
+
+ 2t[(J?y5y'a) +• ( a y 5 y ' ^ ) ] 3 ? ' — 4 i ^ y 5 y ' ^ a ' + ^ ( a y ' P a ) ( f y « a ) £ a W e — — i^°>(ay'Da) (ay< Da) £a6/e — i i y s ( a y ' D a ) y ' T + ify 5 (ay'Da)y'^ + + 2t^y 5 (fy'a)y'I)a — 2 i D a y s ( f y ' a ) y ' ^ T i ^ f y s y ' D p O a y s y ' a e , * , . . The values of the action have to be real, i.e. in A+. (6.16)
>eA~,
We constrain
0t* e (J.") 2 ,
i.e. no real contribution to the factorized generalized torsion St*, This is a generalization of the
result of GE. Indeed, (6.1-6) implies that the only real allowed contribution to the GE torsion be given by — iry"r, as can be seen in (6.14). This can be considered as a boundary condition taken from the Kibble-Sciama Ut version [22] of GE which fixes the contribution to (real) torsion so that it be proportional to the spin of the spinor fields present (here the T„ field with J = 3). Since the original theory is postulated in GP ; we have also allowed
290
34
T. NE'EMAN and T. EEGGE
residual (nilpotent) contributions in addition in (6.16). Both conditions in (6.16) are satisfied by (6.13) provided a e i " , aeA'xA" as required by G P t . Since the action is given by a d'z integration, we have to define such "-n integral for functions with (nilpotent) values in A~(x)A~. These are now products of either the a.(x), a(x) gauge functions or of (A~ X 4 - )-valued forms: It/=j
(6.17)
.
We choose to define (6.18)
ly = 0
forJT>2.
This choice ensures in (6.15) that (6.19)
jd*x(J?e-
3>T) = 0
and ifr as given by (6.15) with g -* % is GPj gauge supersymmetric in a natural way. This end product which is achieved without further assumptions by our choice of £, shows a formal similarity with the supergravity theory of Freedman, van Nieuwenhuizen and Perrara [23, 25, 26]. The relation to conventional supergravity is by no means trivial. I t is of course obvious that all forms can be taken to have values in [A~]n = reals, A~, [A~]- ( = nilpotent even variables) only, because higher-order terms do not contribute to the action (an [JT] 3 -term in i f must be paired to at least another A "-term since if is even, and this vanishes by (6.17)). Moreover, variations must be carried over consistently with (6.16). This we do by adding a term (6.20)
\A'®°-,
A*<=A-®A-,
to if, where A" is a Lagrange multiplier. Upon variation of T, T°, T 0 ' we find (6.21a)
yafSl
(modulo [A-J3) ,
(6.216)
@"tT>£at,n — 2i!%y>y'T + 2 DA' = 0
(6.21c)
@'fE*„, = — A"r" + AbT*
= 0
(modulo [A-]*) , (modulo [A~]*) .
Equations (6.21) are GVi invariant in this (6.16) « weak» sense. Within this definition we use (6.10) and (6.13) as the gauge transformations, viewing Q — TX(A, alt «)T, R = IX(A, a. a)M. Moreover, since (6.21) is obviously #Z7» local gauge invariant, we may set Aat= <5°\ The remaining gauge functions '<"(.?), a(x) wifh xeR* are of order [A~]* and A~, respectively. Within this choice, the only change in (6.21) which is not obviously of order [A~]3 is
291
GAUGE THEORY OF GRAVITY AND SUPERGKAVITY ON A GROUP MANIFOLD
35
in (6.21'a)
but yaa" — \{yaaci •\-aciya) + i(y°o'f — a"ya) = \{iyie"'iyi
+ d'ay> — 6"v«) ,
yielding A — \iyiytEa"t^"xaa.
+ $&<"ylx
The first term is of order [A~]3 because of (G.21b) and the second is of order [A~]3 because of the Bianchi identity (6.22)
(D^) t = 5?'°T<' + ^ y ' T = 0 .
Thus (6.21a) is weakly GPi gauge invariant. The gauge covariance of (6.21&), (6.21c) requires a change in Aa. If, according to (6.13) and (6.10), &>« = &« + &*'-al + @y"a — \5.ya@Hasix, (6.23)
2
'
x' = x + Da , T*' =x»+ Da6 + x-fa. — \a.y* Da , then (6.24)
A" = A< + - Sl-fia'cM—ixysySZ + - (xaa"ysyeoc)SS^
is a compensating variation for Sta-. That this is indeed so can be checked through a somewhat laborious procedure in which the identities and equations (6*25)
@abx,> = @'»xteal)te = 0
modulo [-A-]*
(but not modulo [A']*) are repeatedly used. Clearly, A' in (6.21) can be deduced from Si* through some linear duality operation, which shows that the relation between these two sets of forms is involutory. If A" — 0, then ^ " = 0 and (6,21) reduces to conventional supergravity. However, this identification is formal, since the latter theory [23, 25, 26] does not restrict the (odd-grading) spinor gauge functions by conditions such as (6.17).
292
T . NE'EMAN and
36
T. REGGE
7. — Supergravity Conventional supergravity rests instead on an action with a Lar*angian density defined as a 4-form, integrated on J?*, a submanifold of GP, in analogy to unfactorized general relativity (5.10):
(7-1)
A = J22'A£. =tj(R*AU J"
+ B*A&.) •
M'
On a generic Mk, the variational equations are [25-27] (7.2) (7-3) (7.4)
R°=0, R<*Q'£abu-2iRYiYeo
= 0,
y°g°R=0.
The equations of motion impose severe restrictions on the components R'B'/. Before we derive these results, it is quite interesting to analyse the relationship between a generic solution to eqs. (7.2)-(7.4) and its restriction to a particular Jf*eQV. I n analogy to the case of ^ 4 c P , we claim that the entire physics is already contained in any single Jt* and that the conventional supersymmetry transformations [25] of supergravity relate the fields on J{k to the fields on any o*ther subvariety Jt*1. Therefore, solving the equations of motion (7.2)-(7.4) on GP is somewhat tantamount to an exponentiation of all ordinary and supersymmetric transformations, just as solving the unfactorized Einstein equations (5.11) amounts to carrying out all possible SOt gauge transformations. Here, however, our Lagrangian has no supersymmetric gauge transformation in the standard sense defined by (4.7), and there is no simplifying factorization of the odd Grassmann variables. The integration of supersymmetries is, therefore, just as difficult as integrating the equations of motion. In this sense, the f" should be regarded as fermionic «times ». This conclusion is strengthened by a close analysis of the commutation relations of DA in GP, in view of equations (7.2)-(7.4) which imply the vanishing of all except a handful of the anholonomic components R'B'^. First, (7.2) asserts that (7.5a)
IffsO,
VB,E.
We next observe that (7.4) implies the vanishing of the 3-form yg'R. Contracting (7.4) with the tangent vectors 0 r , Da, 30) and using (4.27) o-(2),) = o,
293
GATJGE THEORY OF GRAVITY AND SUPEHGKAVITY ON A GROUP MANIFOLD
37
we obtain (with ye acting on the unwritten spinor index of R) y'Rati = 0 .
It follows that (7.56)
Ran = 0 .
We may contract (7.4) next with 0€, jD,, Z)0) and obtain y
= 0.
Taking c^ f and multiplying by y'y' yields y'R» + y'Rc* = 0
and, for / ^ « ^ s , —yR„ = y'Rt, = -yeBc* = y'B.. = o , so that (7.5c)
R„ = 0 .
This proof holds if we replace the spinor index a in Sa with a «vertical » index A = ; (ab). Thus ('•5a)
R(ab)c — R
RiabKcd)
=t=
0 .
We note that only Rcd survives. Turning to (7.3), contracting with 0 t , Da, Dp) and using (7.5c) yields Rap" Sabce =
0 ,
so that (7.5e)
R-j* = 0.
Similarly, (7-5/)
R{Unli?b — RiU'a'" = 0
and contraction of (7.3) with (Bm,Bv,D,t) (7-5(7)
leads to
i? ( ; ( h r = o .
We thus find that all curvature components involving an SOA index (cd) vanish, as would be expected from an S04-factorized theory. The £* variables appear factorized in the expressions for the curvature, at least. To that
294
38
r. KE'EMAK and T. REGGE
extent, we may regard this fact as consistent with the hypothesis of factorization of S0t appearing as a result of the equations of motion. Indeed, we surmise as for general relativity that this does occur here too. Curvature is in any case restricted to R4'4 = GP/JT, i.e. to superspace. The commutation relations of the I)A are thus given by (4.41a)-(4.416) and (4.44a)-(4.44dl). Contraction of (7.3) with (&£-,. Bp, B.») leads to (7.6)
R'a*"e«„-Ki'*e»m.-4iIiJ(ytyt)i
=0
(the dot on the first $ in the last term denotes a « lower spinor»), which leads to a nontrivial relation between the spinor curvature Rmv and the spinor components of the ordinary curvature R"K Eelation (7.6) is the same as eq. (lie) of ref. [30], in which R„, is denoted Hmp and R'^"" is essentially Zmab, the somewhat unwieldy expression given for the variation of the relativity connection field, in the original papers on supergravity [25-27]. That this is indeed R\'Jfb can be seen from (4.43a). Finally, we need to compute (7.7)
E ; r = 2J y #*£;;•».
A possibile source of confusion arises in trying to calculate anholonomic components by using the inverses of the conventional vierbeins instead of the inverse 14 x 14 matrix A'/1 (though the summation over M, JV is now reduced to vector-vector and vector-spinor MN sets only). This explains the occurrence of Q„b,d in ref. [30] as distinct from Rabcd. I t is the Qabci of ref. [30] which coincides with the present Rce°b, while Rabrd of ref. [30] is an auxiliary quantity of no physical and geometrical content. In matching these components with those of ref. [30], care should be taken in setting up a correspondence between our geometrical operators DA (and, in general, any vector field on GP) and the set of second-quantized operators y{x), J?(x), S~ab(x) of ref. [30]. The latter can be written as a GP multiplet ^A(X) with the commutation relations (7.8)
[J*B(x), seE(x')] = S(x, x'){CA.„ + R-M'/)jeA{x)
in analogy to our (4.41a), (4.41b), (4.44a)-(4.44d) set. Note that with t h e vanishing of components in (7.5), (4.44i)-(4.44d) now become (7-9«)
[A,A]
= BZFDu + R^D ,
(7-9b)
Cae[Da,D8]=RZ°aI>c*,
<7-9c)
0*, Be) =iy')**D*,
295
GAUGE THEORY OF GRAVITY AND SUPERGRAVITY ON A GROUP MANIFOLD
39
and (4.43) consist in (the D are GP-covariant derivatives) So04 = De'* — QC£*R'CT—
e'FBca",
(7.10) So = D £ — QcE"jRcd, which are the conventional transformations of supergravity (see (4.42a)-(4.42c) for the variations contained in De" and Be). Supergravity, however, has meaning on a 4-dimensional manifold Mi only which we may take to be |* = 0. (4.43) or (7.10) are then relating the restriction of the forms QA on J(* ( f = 0) before AGCT to the corresponding r e s t r i c t i o n on J?*' (|'° = 0) after AGCT; clearly any covariant theory admits formulae (4.43), but so far only in supergravity the E's'/ components are all functionals of the restriction qA\jf< and (7.10) is a true transformation. I n this sense supergravity is the only known supersymmetric theory with structural group GP.
8. — Conclusions The following facts have emerged from the foregoing discussion. 1) I t is possible and indeed convenient to develop the formalism of gravitational theories as gauge theories on a group manifold. The groups are P or GP for gravity and supergravity, but further enlargements sliould follow the same method. When this is done, the connection forms and the vierbeins appear as a single c-bein on G (c being the dimension of G), which also play the role of connections on G in computing covariant derivatives and curvatures. I n particular, ordinary curvatures and torsions are naturally unified as expected in Einstein-Cartan theories. 2) An action is formed under the assumption that it reduces to the Einstein action in the appropriate cases and be compatible with the vanishing of curvature (« flat » empty space). The notion of pseudocurvature is introduced and related to Chevalley cohomology and to the Segal-Wigner-Inonii theory of group contraction. Pseudocurvatures provide an interesting way of achieving the same ends as the MacDowell-Mansouri [23] procedure, but with methods which are purely intrinsic to G. The method reproduces conventional supergravity [25-26].
296
40
T . NE'EMAX aad
T. REGGE
3) In all these theories, the action is strictly gauge invariant under SOt gauges only. Moreover, the Lagrangian densitv is always a 4-form and is integrated over an arbitrary Jt* variety in G. However, physics is seen to be completely determined by what happens on a single Jt*. i'he transfer of information from any J(* to any other Jf*'cG implies either ordinary SOtgauged symmetry or supersymmetry as an AGCT gauge. In a way, restricting a GP S04-factorized theory to R4/1 includes as partial Rl slices all possible supersymmetry-related conventional supergravity theories. Rl ->- R*1* « determinism » has been realized elsewhere [31]. 4) Factorization is seen to be a key simplifying feature in these theories. We conjecture that, if a Lagrangian is gauge invariant under any E c G, then it is H-factorizable as a consequence of the equations of motion. In this sense, factorization should not be an independent postulate. I t is possible to give a heuristic proof of the factorization hypothesis for solutions infinitesimally close to a factorized one. All these solutions can be reduced to factorized ones by an infinitesimal co-ordinate transformation on G. In the large, however, it may be that there are discrete families of factorized solutions with the same boundary conditions and still topologically distinct. This interesting possibility remains an open question. 5) The procedure can be in principle extended to other groups. I t would also be desirable to extend the theory so as to include «matter » distributions whose energy-momentum, supersymmetry and spin density tensors are nontrivial functions on G. *** The authors would like to thank Prof. J. P R E ^ T K I and the Theory Division of CEEN for their hospitality during part of this work. They would also like to thank Prof. J . A. W H E E L E E and the Center for Theoretical Physics at the University of Texas, Austin, for the hospitality to one of us (TE) during part of the continuation of this work.
APPENDIX
The (Euclidean) Dirac algebra Metric ( 1 , 1 , 1 , 1 ) : (A.i)
yay" + y*y*
(A.2)
a""
=2dttb, = i[yay — y"y"),
297
GAUGE THEORY OF GRAVITY AND STJPERGRAYITY ON A GROUP MANIFOLD
(A.3)
Y°
= -iy*
=
(A.4)
Yt
— y°y1y2y3 = — (
(A.5)
Cy°
=
(A.6)
Ay"
=
-(?)-
-y°TC,
~y^A
(fi = 0 , 1 , 2 , 3 ) .
With the choice A=
(A.7)
— C = y*
we have (A.8)
(v)*
= (yT1 = v i
(A.9)
CT
= — G,
(A.10)
(cyr
(A.11)
(Co°>>)T = Co'",
(A.12)
{Cyt)T
(A.13)
(Cy.yr^-Cy.y,
(A.14)
C2
=cy°, =-Cy,,
=1 ,
and the adjoint ( = charge conjugate) Majorana spinor is defined as if = {Cy)T .
(A.15) Commutators (A.16)
[aab, crcd] = daiabc — d^o"" + d>":aai —
(A.17)
[ya, aci] = 8acyd—badyc
d''iaac,
.
The generators (A.18)
^ ( * )
(A.20)
IS^x),
=
^ - * > A ,
5 rt (f)] = 0
obey the same algebra as (A.16) (A.21)
isu,
Scd] = d.tSt.S.tSu
+ du8^-8uSM
Fierz transformations utilized in sect. 6 (A.22)
41
(&y.x)eab" = iix^YiY'0*)
298
—
iiax^YsY'x)
.
42
T . NE'EMAN and T. REGGE
for t h e case of {jf,«} = 0, a n d (A.23)
(ojy.w) £<">" = —
2i{u1aa',yiy,o})
for [
REFERENCES [1]
See, for example, B. ZUMINO: Proceedings of the XVII International inference on High-Energy Physics, edited by J. R. SMITH (Chilton* Didcot, 1974), p. 254.
[2] [3] [4]
A. SALAM and J. STRATHDEE: Phys. Rev. D, 11, 521 (1975). L. COBWIX, Y. NE'EMAN and S. STERNBERG: Rev. Mod. Phys., 47, 573 (1975). S. FEERARA: Rio. Nuovo Cimento, 6, 105 (1976); P. FATET and S. FERRARA-
[5]
Phy*. Rep., 32 C, 249 (1977). For a previous exposition of the theory for physicists, see B. S. D E W I T T : Relativity groups and topology, in Proceedings of the Les Eouckes 1963 Seminar, edited by C. D E W I T T and B. D E W I T T (New York, N. Y., 1964).
[6] [7]
[8] [9] [10] [11] [12] [13] [14]
F. A. BEKEZIN and D. A. LEITES: DOU. Akad. Nauk 8SSR, 224. (1975); (English translation: Sov. Math. DoU., 16, 1218 (1975)). B. ZcxiN'O: Proceedings of the Conference on Gauge Theories and Modern Field Theory, Northeastern University, Boston, 1975, edited by R. ARNOWITT and P. NATH (Cambridge, -Mass., 1976), p. 255. F . A- BEREZIN and G. I. KATZ: Mat. Sb. (SSSR), 82, 343 (1970) (English translation: 11, 311 (1970)). T. W . B . KIBBLE: Journ. Math. Phys., 2, 212 (1961); D. W. SCIASIA: in Recent Developments in General Relativity (Oxford, 1962), p . 415. P. Vox DER H E T D E : Phys. Lett., 58 A, H I (1976). Yu. A. GOLFAND and E. P. LIKHTMAM: JETP Lett., 13, 452 (1971). J. WESS and B. ZUMINO: Nucl. Phys., 70 B, 39 (1974). A. SALAIT and J. STRATHDEE: Nucl. Phys., 76 B, 477 (1974). P . N'ATH and R. ARN'OWITT: Phys. Lett., 56 B, 177 (1975).
[15] V. P. AKULOV, D. V. VOLKOV aad V. A. SOROKA: JETP
[16] [17] [13] [19] [20] [21] [22]
Lett., 22, 396 (1975)
(English translation, p. 187). J . WESS and B. ZrariNO: Phys. Lett., 66 B, 361 (1977). G. Woo: J*tt. Nuovo Cimento, 13, 546 (1975). P. P. SRIVASTAVA: Lett. Nuovo Cimento, 13, 161, 657 (1975). E. CARTA.V: Compi. Send., 174, 593 (1922); Ann. Ecole Norm. Sup., 40, 325 (1923); 41, 1 (1924); 42, 17 (1925). F. W. HEHL: Thesis, Technical University Clausthol (1970); Phys. Lett., 36 A, 225 (1971). A. TRAUTMAN: Bull. Acad. Pol. Set., Ser. Sci. Math. Astron. Phys., 20, 185, 503, 895 (1972); 21, 345 (1973); Symp. Math., 12 (Bologna, 1973), p . 30. F.
\V. HEHL, P. VON DER H E T D E , G. D. KERLICK and J . M. NESTER.- Rev.
Mod.
Phys., 48, 393 (1976). [23] S. W.MACDOWELL and F. MAXSOURI: Phys. Rev. Lett., 38, 739 (1977). [24] C. CHEVAIXET and S. EILENBERG: Trans. Amer. Math. Soc, 6 3 , 85 (1948). [25]
D. Z. FREEDMAN, P. vox
NIEUWE.VHUIZEN and
3214 (1976).
299
3. FERRARA: Phys.
Rev. D,
13,
43
T. NE'EMAM and T. REOOE [26] [27] [28] [29]
S. D. P. R.
DESER and B. ZUMINO: Phys. Lett., 62 B, 335 (1976). Z. FREEDMA-V and P . VAN NIECTWENHUIZEN: Phys. Bev. D, 14, 912 (1976). NATH and R. ARNOWITT: Phys. Lett., 65 B, 73 (1976). ARNOWITT, P . NATH and B. ZCJJIINO: Phys. Lett., 56 B, 81 (1975).
[30] C. TEITELBOIM: Phys. Rev. Lett, 38, 1106 (1977). [31] L. BRINK, M. G-ELL-MANN, Y. NE'EMAN, P. RAHOND and J . SCIIWARZ: Superspace
Lecture Notes, Aspen Center for Physios, 1977 Snpergravity Seminar (unpublished). [32] A. H. CHAMSEDDINE and P . C. W E S T : Nucl. Phys., 129 B, 39 (1977).
© by Society Italiana di Fisica Proprieta letteraria riservata Diretxore rosponaabile: CARLO CASTAGNOLI Stampato in Bologna dalla Tipografia Compositor! coi tipi della Tipografla Monograi Questo tascicoio 6 stato licenziato dai torchi il 24-VII-1978
300
PROC. 19th INT. CONF. HIGH ENERGY PHYSICS TOKYO, 1978
C6
Ghost-Fields, BRS and Extended Supergravity as Applications of Gauge Geometry Y. NE'EMAN
Tel Aviv University, Tel Aviv We review the methods of Gauge Geometry, including the sequence Lie group/Fiber Bundle/Gauge Manifold. We derive the Ghost Fields and BRS transformations geometrically (and classically), showing that the Ghost fields are nothing but the vertical components of the gauge-potential one-forms. We state the theorem of Spontaneous Fibration of a Weakly-Reducible and Symmetric Group Manifold and apply the method to the derivation of Extended Supergavity (Chiral or Parity-conserving) for any N.
§1.
[D„ -D,']=A,.,.](1-3) In the Principal Bundle P(P, B, re, G, •), with a horizontal base space B in addition to G, and a projection it onto the base space,
Geometrical Introduction
The geometrical treatment of gauge theories' has provided numerous insights and precise results at both the classical and quantum levels. In this note, we present two new applications, one for any Internal Gauge theory, and the other in the realm of the Noninternal ones. In the first, we derive the existence of the Ghost-fields, needed for Unitarity and Renormalization,' as a classical geometrical result on a Principal Bundle. The BRS transformations3 come cut naturally, and one may hope that this approach will produce further insights in the study of the renormalizatjon of Gravity and other theories. Our second example consists in a direct derivation of all allowed "geometrical" theories of Extended Supergravity, using a theorem of Spontaneous Fibration,' in the method of gauging on a Group Manifold which we recently introduced.6" It is instructive to compare a Lie group G, a Principal Bundle P with fiber F=G, and the Group Manifold G. A Lie group G is a space in which finite motions are specified, up to global topological considerations, by its Lie algebra A. The connection forms over G are the Cartan Left-invariant forms w. They provide a rigid triangulation and a vanishing curvature de>-£|>. M]=0.
(1.1)
The Left-invariant algebra Dy is an Orthonormal basis to the w, io'iD'^o)
VpeP,geG:
it(p-g)=it(g)
(1.4)
the transformations of G are vertical by (4). The connection forms
(1.5)
VyeA,
(1.6)
D,\ 0 = 0 .
P is specified by the knowledge of the fields over a section 2" The Group Manifold G is punctually identical to G. However, the connection oneforms p are locally independent.5 In G any direction is a gauge direction (as in G=FaP), but all directions are also curved (as in BczP). The Killing Lie algebra is not isomorphic to A,
[/>f.A']=0t».»-J+(AJ A ' - W i
0-7)
//(Z>j)=(5J.
(1.8)
The curvature R=dp-k[p,p]
(1.9)
is unrestricted. However, the Jacobi identity does give rise to Bianchi identities DR=0.
(1.10)
Gauging corresponds to a local map in the group's right action, with
(|.2)
301
xy=z, <\n=Dl
x—*y, x, y, zeG, oR=[l
R]=D(5p).
(1.11) (1.12)
Supersymmetry (Including Stipergravily)
553
IK.lirZ.~0 (3.6) where the latter condition guarantees that the "flat" Lie group C will satisfy the equations of motion, or in the language of General Relativity, that the vacuum will coincide with a>=<j>+X, D,j ^ = 0 , d,J Z = 0 (2.1)the "tangent manifold." The equations of motion will be 0,, is thus the (horizontal) Yang-Mills potential. E.:=IKm=0 Ek:=R'(5C.ldpk)=0 (3.7) X is the Feynman-De Witt-Faddeev-Popov ghost. It anticommutes just because it is a Eq. (3.4) plays the role of BRS in establishing one-form. A differential <^can be expanded the Quantum version. as 3.2. The spontaneousfibrationtheorem df=S„fdx"+difdyi (The proof is given in ref. 4) =bf+sf (2.2) Given (1) G a semi-simple Group or Supergroup b:=dX"dl„s:=dy'di. (2) G is Weakly reducible and Symmetric" Equations (5)-(6) can be rewritten as (WRS), i.e., there exists a decomposition of *X-1/2[X,X]=0 (2.3) its Lie algebra s$-Bx=0 (2.4) A=F®H, [F, F]cF, £=ty-l/2[<4,0 (2.5) [F, H]CZH, [H, H]czF (3.8) with (3) The complete F is generated by [H, H], 2JZ=Z>Z+[<4, Z] (2.6) and the dimension of F d(F)>(d(H)-1)/3, d(H) the dimension of H. These are the BRS equations. The application to path integrals and renormalization (4) The Lagrangian is F-gauge invariant will be described elsewhere.7 but not /4-gauge invariant, Then, G undergoes Spontaneous Fibration with respect to F, §3. Gauging a Non-Internal Group G i.e., up to global topological considerations, 3.1. Quadratic (simple G) and linear (G con- (classically) tracted) Lagrangians (1) the F variables factorize, in the sense of For a simple G, use the quadratic Lagrangian ref. 6. introduced in ref. 8), (2) all torsions (curvatures in the H direch tion) vanish, L=R'Aui,R , a, be F, Fa A (3.1) R'">=0 (3.9) Let G be Weakly-reducible and symmetric (3) all curvatures become horizontal, namedecomposition of its algebra A=F@H\ [F, F] c f . The /"-metric cah is the most general ly in the decomposition over a cobasis tensor satisfying, [F, H]
§2. Ghosts and BRS Take first an internal gauge, described by P. We may decompose a> with respect to a section I into the sum,
4.1. OSp(N/4) Take C=OSp (A^/4), the Supergroup10 prewhere i implies application of the equations serving a metric n0©(ffi®i7o). where »0 and a„ of motion ("pseudo-invariance9"). stand for the identity in N and 2 dimensions Under contraction over F, H->/.H, Ca is a respectively. 11 6 "pseudo-curvature ' " : We use a, b, • • • for space-time directions; a, /3, • • • and d, /§,••• for spinors, i,j, • • • for C. = U3/,\/3i)|1..(/B»Ap'). C*=0 (3.5) O(N) A'-vectors. DL^D'f'L=0
(3.4)
302
554
Y. NE'EMAN
4.2. Extended left-Chiral supergravity Take Fx: (Jw, Itj, Sai), HASat, Ta). Using (3.2) we find,4
first Right-Chiral Extended Supergravity by taking a decomposition F2: (7 [rt ], Iih Sai), HJiS^t, 7".). Construct Ls and contract //,-+ fiH„ then contract again with fy1-*Xfi[l and find LK (W). Add to (4.16) and find, L^LL+iLR=s„hcHP'Ap"R1"'1 Hil2)
L,=zaiedR w A * W 1 + i(Rlail A /?[..] + # « A f y + * W 1 A *[,,,) (4.1) Contract /Y,-^//, LL(Xi)=sa>elp'Ap''ARUdv +ipcApiAR'w+iaa^p'Ap^AR^i' +iR^Ap1Apai (4.2) where the dash stands for a restriction of (1.9) to F. Both theories violate parity even in the gravitational sector. LL vanishes for the flat C = G and LL(X') obeys (3.6.). Spontaneous fibration guarantees that RUdy=RUiV dy
= R«i>=o
References
(4.3)
ti
R\'. = R\ . i' = Rf!'=Q To recover supersymmetry we still have to contract inside F. We contract over F}, with F,- ( W Uh S«,), ft,(S«i, T„). Taking # , //#, we find (fc denotes a restriction of (1.9)
toF,n£) LL^f*')=eaitdp'ApiA^cil +ip*ApiAR'[a
1. W. Drechsler and M. E. Mayer: Fiber Bundle Techniques in Gauge Theories, Lecture Notes in Physics 67, Springer-Verlag, Berlin-HeidelbergNew York (1977); Y. Ne'eman: Symmetries, Gauges el Verietes de Croupe, Presses de l'Universit6 de Montreal (1978). 2. R. P. Feynman: Acta Phys. Polon. 26 (1963) 697; B. S. de Witt, Dynamical Theory of Groups and Fields, Gordon and Breach, Pub., N.Y. London, Paris (1965); L. D. Faddeev and V. N . Popov: Phys. Letters 25B (1967) 29. 3. C. Becchi, A. Rouet and R. Stora: Ann. of Phys. (N.Y.) 98 (1976) 287. 4. Y. Ne'eman and J. Thierry-Mieg: to be publ. in Ann. of Physics (N.Y.); Avaiable at Tel-Aviv University report TAUP 698-78. 5. Y. Ne'eman and T. Regge: Phys. Letters 74B (1978) 54. 6. Y. Ne'eman and T. Regge: to be published in La Rivista del Nuovo Cimento; Available as University of Texas report CPT ORO 3992 328. 7. J. Thierry-Mieg: "Geometrical Reinterpretation of the Faddeev-Popov Ghosts and BRS Transformations," to be published. Available as TelAviv University report. 8. S. W. MacDowell and F. Mansouri: Phys. Rev. Letters 38 (1977) 739. 9. J. Thierry-Mieg: to be published in Letters Nuovo Cimento. 10. P. G. O. Freund and [. Kaplansky: J. Math. Phys. 17 (1976) 228; V. G. Katz: Func. Analysis and its App. (USSR) 9 (1975) 91.
303
HIGHER ALGEBRAIC GEOMETRIZATION EMERGING FROM N O N C O M M U T A T I V I T Y
Y. NE'EMAN School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel E-mail: [email protected] We review the gradual geometrization which has occurred in fundamental physics from the discovery of special relativity in 1905 to the standard model in 1975. After discussing symmetry and ordinary supersymmetry, we introduce internal supersymmetry. Here the even and odd generators correspond to the form-calculus of a gauge theory with spontaneous symmetry breakdown, with the gaugefieldoneforms occupying the even submatrices and the Higgsfieldszero-forms occupying the off-diagonal submatrices. The Grassmann superalgebra is not (super)-abelian and closes on some semi-simple subalgebra. We study two examples: the electroweak St/(2/l), predicting the mass of the Higgs particle around 130 ± 10 GeV, and P(4R) for Riemannian gravity. Internal supersymmetry does not operate on the physical Hilbert space and as a result of non-commutative geometry, the matter fields in its fibres relate to Z{2) gradings other than that of quantum statistics (chirality in our examples).
1 Introduction - and t h e Physics of T i m e Hagen Kleinert is sixty according to classical clocks, but this is clearly a misinterpretation of the data. Observations show that Hagen and Annemarie have not aged at all. Had they both been born with twins, he with a twin brother, and she with a twin sister, we might have been able to explain our paradox as a complicated extension of the twin paradox, conjecturing that we are now facing the travelling twins, who have just returned and replaced t h e sedentary couple. However, as the Kleinerts never had a twin couple, we must be facing some as yet uncharted and unidentified relativistic effect, perhaps related to some unknown aspect of quantum gravity - coupled a la Penrose 173
304
174
Y. Ne'eman
with the collapse of the state-vector and thus to the Everett-Wheeler ManyWorlds interpretation . . . . With the effect still shrouded in mystery, I have had to go along and behave as if I believed that Hagen is indeed getting older (wiser he certainly is) and I am happy to dedicate this study to his (classical) sixtieth birthday and wish Annemarie and him many happy returns. I shall do my best to be around when Hagen's classical age is 75, and partake in the next Festschrift, but, alas, I cannot make a firm commitment on this matter. 2 S t e p s in t h e G e o m e t r i z a t i o n P r o c e s s Hagen's contributions to Physics in the last decade have mostly been of a geometrical nature, involving Riemannian manifolds with torsion - whether in 3 dimensions and the physics of materials (transitions between phases, e.g. melting [l]) or in 4 dimensions and issues relating to general relativity [2]. We physicists of the late XXth (and hopefully of the early XXIst) century have enjoyed the aesthetics and symmetries of the geometrical representation. Gradually, between 1905 and 1975, it has become the unique language of physics at the fundamental level [3]. I remind the reader that this takeover occurred in the following stages: (a) Minkowski's 1907-08 geometrical reinterpretation of Einstein's (1905) special theory of relativity, namely of the symmetries of Maxwell's electromagnetism, as identified by Einstein, (b) The Einstein-Grossmann application and extension of that model (1911) in a program aiming at reconciling Newtonian mechanics with the above symmetries of electromagnetism, leading to Einstein's construction of the general theory of relativity (GR, 1915) as the new (and fully geometrical) theory of gravity, (c) The construction of a gauge theory, started in 1918 with H. Weyl's (failed) first attempt at a theory of electromagnetism (based on the assumption of local scale-invariance) unifiable with Einstein's gravity (i.e. geometrical). It was followed by his 1928 successful version, in which the geometry is that of a fibre bundle, with fibre group £/(l) realizing local invariance under transformations of the complex phase angle (introduced by quantum mechanics). This was then generalized (1953) to non-abelian groups by C.N. Yang and R.L. Mills and applied (1975) to the S[U(2) <8> [/(3)] fibre group of the standard model. The latter emerged, on the one hand, as a result of our (1961) SJ7(3)flav0r classification of the hadrons and the subsequent (1962-64) discovery of the structural mechanism to which it is due, namely the quark
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Higher Algebraic Geometrization Emerging from Noncommutativity
175
model, followed first (1964-1972) by the introduction of (global) SU(3)coior for the sake of preservation of Fermi-type quantum statistics and then by quantum chromodynamics (QCD), namely local SU(3)co\or, after the discovery (1973) of asymptotic freedom. The geometrical nature of local gauge theories was emphasized (1974) by C.N. Yang and T.T. Wu. (d) Two other developments (also launched in the early twenties) were suggested in the context of further unification: I. Adding dimensions - the program suggested by T. Kaluza (1921) and by O. Klein (1924). II. Following Einstein, adding an antisymmetric piece to the metric or connection. Developments in the seventies in the unification program, namely in NB > 1 supergravity and in superstring (or "M") theory have converged on a fusion of both these features. After the establishment of relativistic quantum field theory by its success (1948) in quantum electrodynamics (QED), the gravitational field of GR, representing the "fabric" of space-time, should by itself be treated as a quantum field of Bose-type. This thereby does not allow a role for an antisymmetric metric by the spin-statistics theorem. The necessary conditions, however, are induced through (1971-73) supersymmetry, which adds fermionic degrees of freedom to any boson. Gravity thereby becomes embedded in supergravity, with the antisymmetric characteristics of (II) represented by the presence of torsion. Finiteness considerations, namely the cancellation of chiral and dilational anomalies, then impose specific higher dimensionalities as in (I). The two programs - torsion and Kaluza-Klein dimensionalities - are thus presently actively pursued in the context of the 11-dimensional Ns — 1 supergravity constructed by E. Cremmer and B. Julia (1978), reducible in 4-dimensions to the Na — 8 maximal or saturated supergravity, a version of supergravity which has been shown to represent the low-energy quantum field theory limit of "M-theory", the state of the art theory of post-Planck level and quantized gravity. [Ns is the "number of supersymmetries", i.e. the dimensionality of the internal degree of freedom, if any, carried by the Lorentz spinor multiplets of supersymmetry generators]. (e) An independent additional geometrical entry is due to Jean ThierryMieg. His 1979 thesis and related articles [4,5] identify the ghost fields of a Yang-Mills gauge theory - as conceived by R.P. Feynman (1962) in order to guarantee off mass shell unitarity and further developed by B.S. De Witt, L.D. Faddeev, and V.N. Popov - with odd elements of the form calculus, the
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Grassmann "supercommutative" superalgebra of the gauge theory. Moreover, the "BRST" constraining superalgebra, linking together physical and ghost fields is seen to coincide with the structural equations of the fibre bundle. (f) The superalgebraic system I describe in the rest of this article derives from the latter. It consists in a nonsupercommutative extended superalgebra of forms, defined over a fibre-bundle whose base-space is split by a Z(2) grading but which does not necessarily coincide with the Z(2) of quantum statistics (in my examples, it will be the Z(2) of chirality). The first model of this type was discovered (1979) by the present author [6] and independently by David Fairlie [7] and involved the simple Lie supergroup 517(2/1) as an "internal supersymmetry", an irreducible algebraic extension of electroweak unification's (spontaneously-broken) local - SU(2) x C/(l) symmetry. The theory has been applied [8] to predict the mass of the Higgs meson, yielding m(H) = 2m{W) in the exact (and unrenormalized) limit, while the inclusion of renormalization effects, as observed in couplings [9], yields as final result m(H) = 130 ± 15 GeV. Note that 9 "events" have been observed at CERN in the fall of 2000 with a Higgs meson mass around 115 GeV. 3 Superalgebras, Supermatrices, and Z(2)-Gradings I first remind the reader of the main definitions and results relating to Lie [10] and to Grassmann (super) algebras [ll]. The first involve the application of a Z(2)-grading on the basis of the Lie superalgebra as a linear vector space; the g{x) eigenvalue also determines the nature of the Z(2)-graded superLie bracket [x, y} and of the relevant super-Jacobi identity. Let the variable E = \ / I represent the two elements of the finite group Z(2). The superalgebra splits into two subspaces, labelled by that grading, E = Vl, g = ]og_1(E G Z(2)g), L =
LQ
+ L\,
g(x G L0) = 0, g{y e I i ) = 1, #.!/})=j(«)®S(y)S2, [x,y} = -(-iyW-°M[y,x}, [x, [y, z} = \x, y}, z) + (_1)«<*>»<») [y, [x, z}} .
(1)
In some cases, there also exists a Z-grading z{La) G Z, where z is a "quantum number" which is additively preserved by the super-Lie-bracket, though the
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nature of that bracket itself is still determined by the Z(2) 2 -binary grading within Z,ZZ> Z{2)2, L = Y,Lii
a n d for a n
y
x
G L°> y e L 6 , [x,y) C La+b .
(2)
i
Lie superalgebras can always be (and generally are) represented in matrix form, organized in quarters Qg according to
I M I A, |
| v« |
|£i|flo|,
It; 1 |,
W
the 5 = 0 and 5 = 1 generators thus spanning the squares along or off the main diagonal, respectively. With this supermatrix acting on a column-vector V split in two by some Z(2)v, the 5 = 0 quarters are the endomorphisms of V, namely AQ = End (V°), £?o = End (V1), whereas the 5 = 1 quarters represent the homomorphisms between the two sectors in V, namely A\ = Horn (Vi, Vb), B\ = Horn (Vo, V\). In supersymmetry, the Z(2)v is again the quantum statistics characteristic and correlates with the statistics of the Hilbert space particles in a supersymmetry study. In the case of a Grassmann (super-commutative) algebra of differential forms, the Z grading and its odd-even partitioning Z(2)j C Z, respectively, represent the total count in the applications of the exterior derivative d (or the number of differentials involved as factors), and the odd/even partitioning to which this degree belongs. If applied to differential forms arising in an anholonomic basis or in a supergroup manifold, the Z(2) grading fixes the exterior (wedge) product according to the rules, dxa A dxb = -(-l)9(«)»Wdx* A dxa, Ff = E a
dx<11
A dxa
*
A
• • • dxafFai.a2...af
F ? AF£ = (-l)Ui-h+9(a)g(b))pb
A
(x) ,
(4)
pa
A third category of Z{2) gradings describes the intrinsic Poincare or Lorentz group Z(2) s -grading of the variables of space-time and its double-covering (spin) and the corresponding exterior-derivative operator, as in the case of the Salam-Strathdee "superspace" of supersymmetry. There would then be a need to characterize algebraic structures by Z(2)3, i.e. yet another Z(2), whose eigenvalues we denote by s(x). As we do not deal here with "classical" [Golfand-Likhtman/Wess-Zumino] supersymmetry, we shall not use the s(x). We now discuss a coupling between superalgebras, in particular the case in which the Lie superalgebra matrices are valued over Grassmann superalgebras
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of differential forms. In these "directly coupled" superalgebras (we use the direct product symbol), the multiplication is fixed by the definition of the Z(2) grading as the base (—1) logarithm of the elements of Z(2), namely the square-roots of the identity, so that for A=A0 + A1, F = F0 + Fi, (A ® F ) 0 = A0®F0®A1®F1, (A ® F)i = Ax
(5)
and the direct product for matrix-elements in two matrices (a®p){a'®p')
= (-l)f{p^a,){aa'®pp'),
(6)
with the sign fixed by the Z{2) eigenvalues of p and a', once a' has to move through p to get to a. In any case, the overall grading h is given by h(a®p)=g(a)f(p)
+ fn(y£rn)^2.
(7)
The simple and semi-simple Lie superalgebras have been classified by V. Kac [12]. 4 The Quillen Super connect ion After I had conceived SU(2/1), Jean Thierry-Mieg and I investigated the possibility that the system of forms (the Grassmann superalgebra) in a YangMills theory, when extended by Higgs fields (i. e. in cases of spontaneous breaking of local symmetry) might generate a nonsupercommutative (or "nonsuperabelian") superalgebra. Such an extended Grassmann superalgebra might sometimes happen to coincide with a simple Lie superalgebra [SU(2/\) in the electroweak case, etc.]. The idea was partly triggered by the composition of the Lagrangian in such models, with a term in the Higgs potential quartic in the Higgs field: such a term could be reproduced by a Lagrangian quadratic in the curvatures, provided these curvatures be taken for a supergroup, in which the even directions g — 0 in the superalgebra's Z(2)g grading are spanned by the original gauge group, while the Higgs fields span the g-odd directions. At that stage, there was no such rederivation for the remaining part of the Higgs potential, namely the spontaneous symmetry-breakdown triggering term, quadratic in the Higgs fields and similar to a mass term - but with the inverted sign. In identifying the Higgs fields themselves with even elements in the Grassmann algebra's form-calculus Z(2)f, we were limited at this stage [13], as we had not dared go beyond Thierry-Mieg's original
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identification of the ghosts as vertical components of the gauge fields, packed into contracted one-forms (in the fibre's direction) and the view in which t h e Higgs fields are ghosts of ghosts, i.e. two-forms, twice vertical. For the groupelements to be fully bosonic and Lorentz-invariant, the parameters would coincide for the even subgroup with its ordinary scalar parameters, while t h e odd part would have the Lorentz-scalar anticommuting ghosts (one-forms in our geometric interpretation of ghosts and BRST). More about our SU{2/\) example. The group is homomorphic with Osp(2/2), whose fundamental representation is 4-dimensional and fits the quarks [14]. Moreover, one is allowed to add one constant real number t o the diagonal quantum numbers; but if by adding one gets integer values for Yw and for the electric charges - the matrix reduces to a 3-dimensional one. The group thus "knows" that quarks have fractional charges while leptons carry integer ones. Note also that since /„, = su(2) and IWIJ>R = 0, we have for the supertrace sTr(I^) = 0; also, as the electrically-charged leptons or quarks are all massive and thus appear both on the right-chiral and left-chiral eigenstates, we also have str(Q) = 0 . Some time later and with a more daring mathematical motivation, D. Quillen [15] postulated his theory of the superconnection, in which the matrix-elements in the odd (g{a) = 1) and even (g(a) = 0) submatrices of a superalgebra are valued over the Grassmann supercommutative zero-forms (/ = 0) and one-forms ( / = 1), respectively, the intertwined coupling thereby ensuring that the total grading be odd everywhere, t = g + f = 1. It was shown that the 1979 electroweak SU(2/1) could naturally be recast in this mold [16]. 5 Noncommutative Geometry: braic Geometrization
The Electro-Weak Higher A l g e -
The third and last step has consisted in reproducing the entire Yang-Mills Lagrangian with spontaneous symmetry breakdown directly from one single invariant; in other words, developing a further generalization which has allowed doing it by squaring one single "curvature", the corresponding generalized two-form. It was provided by a variant of A. Connes' Noncommutative Geometry [17]. These further developments have drawn on a generalization of the concept of parallel transport, as realized by the application of a (covariant) derivative, namely a derivative plus a connection. At the same time, it also resolved a seemingly paradoxical feature of the
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original "internal supersymmetry" interpretation, namely under the action of the g-odd generators of the superalgebra, the absence in the particle Hilbert space of boson to fermion transitions and vice versa. Instead, Hilbert space has carried some other Z(2) grading, unrelated to quantum statistics - chirality in the electroweak case, - so that the endomorphisms induced by the odd generators produce a change of chirality, while the even endomorphisms preserve it. R. Coquereaux and F. Scheck [18,19] were the first to show that this interesting result - namely the interrelationship between physically different Z(2) groups - one in the vector space upon which the transformations are enacted and one in the superalgebra - could be treated as a development of noncommutative geometry (NCG). It was shown that the 1979 electroweak SU(2/1) could naturally be recast in this mold [18-20]. The new arena is a fibre bundle with a non-simply connected base space, namely a direct product of a two-point space Z(2); the points (1, —1) in this realization of Z(2) are labelled LhR B = Z{2)
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Higher Algebraic Geometrization Emerging from Noncommutativity
181
is a discrete move and it will be achieved by a finite matrix, in 5(7(2/1) by /LLQ (same as A6 in SU(3)), which relates e^ —• e'^. It should be anti-hermitian, so that we define the matrix-derivative as T :— ifig- At the same time, however, we need to perform a discrete change in the fibre itself, i.e. transform (/„ = 1/2, /« = -l/2,Yw = - 1 ) - (/„ = /* = 0,YW = - 2 ) , a task for which an appropriate connection is required. It has to resemble A as to its Lorentz properties - i.e. it is a scalar. We also note its quantum numbers in 517(2/1): Iw = 1/2, yi„ = - 1 . This is the Higgs field <J>(x)! Altogether, we shall have yet another new piece in the covariant derivative. In the SU(2/l) internal supersymmetry we have a fibre-bundle with structure group SU(2/1) over a split basis (ML © MR) and get the expression for the overall curvature Tl = dw + w2 = dA + A2 + $ 2 + d$ + A$ + T$ = RYM
+ D$
1 2
+ "V / ",
(8)
where we regroup the terms in their traditional setup, RYM
= dA + A2,
V = [($) 2 ] 2 + [ r $ ] 2 .
D$ = d + A$,
(9)
Squaring that total curvature with its Clebsch-Gordan coefficients and applying T2 = — 1 yields the conventional Weinberg-Salam Hamiltonian HYM
= *RYM
A RYM
,
#(Akinetic = D$2 ,
(10) 2
#(^potential = V* = ~ / * * + W * •
6 Higher Algebraic Geometrization and Riemannian G e o m e t r y One of the macroscopic features of this Universe is its obeying the Riemannian constraint, namely, DpgliU = QPllv=0.
(11)
Following Smolin [22], we have conjectured that this describes the state of affairs at low-energy, arising through the degradation of the basic (highenergy) microscopic state, which is then unconstrained and endowed with more symmetry. Assuming the original and quantum-era Universe to have been affine [23-25] we may be able to throw some light on the symmetrybreaking mechanism. We have conjectured [26] that this symmetry breakdown occurred through a mechanism of the same type studied in this article.
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I have found that the Higher Algebraic Geometrization is provided here by the simple superalgebrap(4,R), a "hyper-exceptional" in Kac's list. The algebra of the homogeneous symmetry group SL(4, R) on the tetrad frames will sit in the even quarters, i.e. AQ, BO in Eq. (3), of the 8 x 8 matrices of the defining representation of P(4, R), along the diagonal. SL(4, R) will be in its covariant representation in AQ, in the contravariant in So. Ai will contain the 10 symmetric matrices (out of 16) in GL(4,R) and B\ will contain the 6 antisymmetric ones. The matrix-derivative will be given by a unit matrix in A\ (or by a Minkowski metric, depending on the issue) and break SL(4, R) down to 5 0 ( 4 ) or 5 0 ( 3 , 1 ) , i.e. to Riemannian geometry. To justify the introduction of the matrix-derivative we have to start with a chirality-split base space - but this is precisely what we have when we take a Dirac spinor (1/2, 0) ® (0, 1/2) or a world spinor [27-29] with this lowest state. We may now write the full "extended curvature" of P(4, R) - including the matrix-derivative piece. It includes the "SKY" [30-32] quadratic SL(4,R) Lagrangian, the kinetic and gauge terms D$+ and J D $ ~ , respectively, for the two Higgs holonomic scalars (one a symmetric tensor in the frame indices, one an antisymmetric), a matrix-derivative generated T $ ~ which will trigger the spontaneous symmetry breakdown, and a term quadratic in the Higgs fields { $ + $ - } . I have described the physical effects in detail in Ref. [26], with results fitting the observed low-energy Riemannian system. References [1] H. Kleinert, Gauge Fields in Condensed Matter, Vol. I: Superflow and Vortex Lines and Vol. II: Stresses and Defects (World Scientific, Singapore, 1989). [2] H. Kleinert, Gen. Rel. Grav. 32, 769 (2000). [3] Y. Ne'eman, Func. Diff. Eqs. 5, 19 (1998); talk delivered at the Intern. Conf. on Diff. Eqs., Ariel, Israel, 1998. [4] J. Thierry-Mieg, J. Math. Phys. 2 1 , 2834 (1980). [5] J. Thierry-Mieg, II Nuovo Cim. A 56, 396 (1980). [6] Y. Ne'eman, Phys. Lett. B 8 1 , 190 (1979). [7] D.B. Fairlie, Phys. Lett. B 82, 97 (1979). [8] Y. Ne'eman, Phys. Lett. B 181, 308 (1986). [9] D.S. Hwang, C.-Y. Lee, and Y. Ne'eman, Int. J. Mod. Phys. A 1 1 , 3509 (1996).
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[10] L. Corwin, Y. Ne'eman, and S. Sternberg, Rev. Mod. Phys. 47, 573 (1975). [11] Y. Ne'eman and T. Regge, Rivista Del Nuovo Cim. 1, 1 (1978); first issued as IAS Princeton and U. Texas ORO 3992 328 preprints. [12] V.G. Kac, Func. Analysis and Appl. 9, 91 (1975); also Coram. Math. Phys. 53, 31 (1977); see also V. Rittenberg, in Group Theoretical Methods in Physics (Proc. Tubingen, Germany, 1977), Eds. P. Kramer and A. Rieckers, Lecture Notes in Physics 79 (Springer, Berlin, 1977), p. 3. [13] J. Thierry-Mieg and Y. Ne'eman, Proc. Nat. Acad. Sci. USA 79, 7068 (1982). [14] J. Thierry-Mieg and Y. Ne'eman, Methods in Mathematical Physics, (Proc. Aix en Provence and Salamanca, 1979), Eds. P.L. Garcia, A. Perez-Rendon, and J.M. Souriau, Springer Lecture Notes in Mathematics 836 (Springer, Berlin, 1980), p. 318. [15] D. Quillen, Topology 24, 89 (1985). [16] A. Connes, in The Interface of Mathematics and Particle Physics, Eds. D. Quillen, G. Segal, and S. Tsou (Oxford University Press, Oxford, 1990). [17] Y. Ne'eman and S. Sternberg, Proc. Nat. Acad. Sci. USA 87, 7875 (1990). [18] R. Coquereaux, R. Haussling, N.A. Papadopoulos, and F. Scheck, Int. J. Mod. Phys. A 7, 2809 (1992). [19] R. Coquereaux, G. Esposito-Farese, and F. Scheck, Int. J. Mod. Phys. A 7, 6555 (1992). [20] Y. Ne'eman, D.S. Hwang, and C.-Y. Lee, in Group 21, Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras 2 (Proc. XXI Inter. Coll. on Group Theoretical Methods in Physics Group 21), Eds. H.D. Doebner, W. Scherer, and C. Schulte (World Scientific, Singapore, 1997), p. 553. [21] A. Connes and J. Lott, Nucl. Phys. B (Proc. Suppl.) 18, 29 (1990). [22] L. Smolin, Nucl. Phys. B. 247, 511 (1984). [23] Y. Ne'eman and D. Sijacki, Phys. Lett. B 200, 489 (1988). [24] C.-Y. Lee and Y. Ne'eman, Phys. Lett. B 242, 59 (1990). [25] C.-Y. Lee, Class. Quantum Grav. 9, 2001 (1992). [26] Y. Ne'eman, Phys. Lett. B 427, 19 (1998). [27] Y. Ne'eman, Proc. Nat. Acad. Sci. USA 74, 4157 (1977). [28] Y. Ne'eman, Ann. Inst. H. Poincare A 28, 369 (1978).
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[29] F.W. Hehl, J.D. McCrea, E.W. Mielke, and Y. Ne'eman, Phys. Rep. 258, l (1995). [30] G. Stephenson, Nuovo Cim. 9, 263 (1958). [31] C.W. Kilmister and D.J. Newman, Proc. Cam. Phil. Soc. 57, 851 (1961). [32] C.N. Yang, Phys. Rev. Lett. 33, 445 (1974).
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C H A P T E R 5: S U ( 2 / 1 ) S U P E R - U N I F I C A T I O N OF T H E S T A N D A R D MODEL A N D N O N C O M M U T A T I V E G E O M E T R Y JEAN THIERRY-MIEG National Center for Biotechnology Information, National Institutes of Health Bethesda MD, USA The derivation of the quantum numbers of the elementary particles from group theory is the guiding principle of Yuval Ne'emans's contribution to theoretical physics. Simply stated, the question would be: is there an equivalent to the Mendeleyev periodic table of the atomic elements at higher energy? Amazingly, Ne'eman stroke twice. In 1961 [l], in parallel with Gell-Mann, he proposed the flavor SU(3) classification of the strongly interacting particles: the hadrons. In 1979 [2], in parallel with Fairlie [3], he discovered the SU(2/1) superunification of the weakly interacting particles: the leptons and the quarks. The comparison with Mendeleyev is quite accurate. The periodic table describes the chemical properties of about one hundred atoms and predicts the missing ones. Its elucidation, the recursive filling of the electron orbitals, came much later. In the same way, the dozens of flavor SU(3) hadrons were only later understood as bound states of quarks held together by color SU(3). Finally, the SU(2/1) super-unification accurately classifies the electro-weak charges of the W and Z vectors, the Higgs bosons, the leptons and the quarks, around 50 states altogether. The difference is that, in the last case, we do not yet have a complete underlying dynamical theory. The problem remains open. Why S U ( 2 / 1 ) The experimental data show that: 1) The weak interactions break parity. They act differently on the left and right chiral states, which must be counted independently. 2) All electrically charged particles are massive. Hence the left and right states electric charges exactly compensate each other. 3) Electro-weak particles occur as multiplets of dimension 3 and 4: the lepton triplet {yL,&i\zR) and the quark quadruplet (uji\uL,dz,\dji) with electric charges (0, —1| — 1) and ( 2 / 3 | 2 / 3 , - l / 3 | - 1/3); U(l) charges (—1, —1| - 2 ) and ( 4 / 3 | l / 3 , l / 3 | - 2 / 3 ) ; and third component of weak SU(2) isospin (1/2, —1/2|0) and (Ojl/2, - 1 / 2 | 0 ) . The fundamental observation of Ne'eman was that if we subtract the charges of the right states from the charges of the left states, that is if we switch sign at each occurrence of the | - symbol, the numbers in each of the six parenthesis just listed add up to zero. In other words, in each multiplet, the super-trace of each charge vanishes [2; 4; 5]. This property is characteristic of a superalgebra: in a Lie algebra, it is the trace, and not the super trace,
317
that vanishes. The second crucial fact is that, whereas no finite dimensional simple Lie algebra of rank 2 or more admits both an irreducible representation of dimensions 3, to fit the leptons, and one of dimension 4, to fit the quarks, SU(2/1) does and in addition with exactly the correct charges [4; 5; 6; 7]. The relevance of the super algebra SU(2/1) in the classification of the electro-weak interactions is therefore ineluctable. Note that in SU(2/1), the leptons {vL,eL\eR) are assigned to the fundamental representation and graded by chirality. To mix left and right states is very unconventional, but on the other hand all the members of a single multiplet are experimentally connected by weak or electromagnetic transition. This is very different from the SU(5) grand unification proposal, which only considers left handed Fermions, but breaks the observed lepton and baryon number conservation by mixing the left electron, the left positron, the left quarks and the left anti-quarks in the same representations. In SU(5), the proton is therefore predicted to be unstable. Despite huge efforts in the 80's, this was not verified experimentally. On the other hand, in SU(2/1), we face a challenge. The transitions between the left and the right states break the usual Yang-Mills framework, which is the only known way to construct a quantum field theory. Therefore, we need a generalization. As happened in the past, some will say that this cannot be achieved. But remember that even the Yang Mills fields were reputed non consistent from 1954 to 1972, when t'Hooft and Veltman solved the problem [8]. And because of this belief, the seminal paper of Weinberg of 1967 on the standard model [9] was not accepted by the community or even quoted until 1972. S U ( 2 / 1 ) : the classical theory. Let us consider the SU(2/1) classical Lagrangian as described in [2; 3; 10]. Note by the way that Ne'eman has currently a physics report on this subject [ll]. If we leave out the supersymmetries, SU(2/1) can be viewed as an embedding of SU(2)xU(l) in U(3), privileging the lepton triplet representation as fundamental, and hence fixing the Weinberg angle to sin2 0 = 1/4. The W and Z bosons are identified as usual as the Yang-Mills fields associated to the Lie subalgebra SU(2)xU(l). The scalar Higgs fields are associated to the odd generators and interpreted as part of the super connection described below. The quartic Higgs potential term appears automatically in the square of the corresponding curvature. Since this term is known, a relation is found between the mass of the W and the mass of the Higgs, which is predicted to be 130 ± 6 GeV [12; 13; 14], just in the current experimental range [15]. An important feature, underlined in [2], is that the direction of the photon appears as the intrinsic square of one of the odd generators, solving the problem noted by Louis Michel: in the standard approach, the direction of the symmetry breaking is unstable. It is fun to realize that in the original papers [3; 2], the quarks were respectively left out and listed as a counter argument: it seemed at the time that SU(2/1) could not accommodate their fractional charges. However, in the following weeks, a marvelous cherry bloomed on the cake. Ne'eman and I [5; 16; 12], and independently Dondi and Jarvis [4] realized that SU(2/1) admits a representation of dimension 4 exactly fitting the quarks. This fact was
318
known in mathematics [6; 7] as a direct consequence of the isomorphism between SU(2/1) and the simple Lie superalgebra OSp(2/2). But this was a great surprise in physics, because previous attempts at embedding SU(2)xU(l) in a larger algebra could not accommodate the quarks without predicting a plethora of new unobserved particles. The existence in SU(2/1) of the quark representation can be seen as the first valid prediction of the theory. Years later, Coquereaux, who was with me at Harvard in 1980-82, and his co-workers [17] rediscovered the SU(2/1) assignments "independently", that is without quoting any of Ne'eman's earlier work. Dwelling on the Connes-Lott non-Abelian differential geometry [18], they added to the connection a constant matrix, representing the discrete transition between the left and right chiral space-time sheets. They showed that this matrix correctly induces the negative quadratic Higgs potential that drives the standard symmetry breakdown. Towards a quantum theory From the classical point of view, the good news is that the SU(2/1) super-unification directly leads to the standard model, with many constraints on the parameters, all corresponding well to the experimental values. But from the quantum field theory point of view, the bad news is that we have exactly recovered the standard model! Since there is no new term in the Lagrangian, it seems that the constraints, in particular the prediction of the Weinberg angle, are not preserved by renormalization. What we would need is a fully quantized gauge field theory incorporating the SU(2/1) symmetry. Indeed, the study by Ne'eman [2] does open an avenue towards quantization. Consider, in addition to the classical Yang-Mills vector fields, the SU(2)xU(l) Faddeev-Popov ghost fields that are needed in the quantum field theory. The ghosts are Fermi scalars. Together with the Higgs Bose scalars, they exactly fill the adjoint representation of SU(2/1), including the grading by statistics. This indicates that SU(2/1) may be represented linearly in the quantized theory, even if it is not a local symmetry of the classical Lagrangian. This observation led directly to the concept of the Ne'eman-Quillen superconnection, introduced by Ne'eman and me in physics [19; 20; ll] and later by Quillen in mathematics [21]. In this approach, the Lie generators are represented by exterior forms of arbitrary odd degree, and the odd generators by forms of even degree. But if we remember that in 4 dimensional space-time, massless vectors carry 2 polarization states, 2-forms are equivalent to pseudo scalars, and 3 and 4 forms have no degrees of freedom, the only new field in the superconnection is a skew symmetric 2-form. The hope is that in some way, this field could 'gauge' the internal supersymmetry and transform it into a dynamic quantum symmetry, without invoking the ultra high energies considered in the SU(5) grand unification or in the models involving gravitation. This problem remains open. Finally, the possible embeddings of SU(2/1) into larger SU(ra/l) super algebras, allowing for several generations (electron, muon...) is discussed in [22; 12; 23; 24]. Then, the superalgebra formalism is applied to strong interactions in [16], and to gravitation in [25]. When a renormalizable SU(2/1) Quantum Astheno Dynamics (QAD) theory is achieved, these other applications will automatically follow.
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I would like to conclude this introduction with a more personal touch. Yuval Ne'eman is a wonderful person to work with. He is very enthusiastic. His knowledge of physics and history seems without bounds. He has a wonderful ability to listen to others and to adapt his explanations exactly to the level of his audience. Hence, it is always possible to understand his ideas and to learn something from talking to him. He can switch subject instantly, and always conveys the impression that he has unlimited time to dedicate to the current topic. In addition, his French, learnt at the Lycee Francais de Port Said, is perfect. According to Yuval, it is better to complete several projects at 90% than a single one at the unreachable 100% efficiency, and this is probably how he manages his multi-universe parallel careers as a soldier, an administrator, a politician, and last but not least one of the most creative physicists of our time. I wish him a happy birthday.
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[1] Y. Ne'eman, paper {1.2} in this collection. [2] Y. Ne'eman, paper {5.1}. [3] D.B. Fairlie, "Higgs fields and the determination of the Weinberg angle", Phys. Lett. B82 (1979) 97. [4] P. Dondi and P.D. Jarvis, Phys. Lett. B84 Erratum B87 (1979) 403. [5] Y. Ne'eman and J. Thierry-Mieg, "Gauge Asthenodynamics (SU(2/1)) (classical discussion)" in Differential Geometrical Methods in Mathematical Physics, (Proc. Conf. Aix en Provence and Salamanca 1979), P. L. Garcia, A. Perez-Rendon and J. M. Souriau, eds. Lecture Notes in Mathematics 836, (Springer Verlag, 1980), pp. 318-348. V.G. Kac, Funct.Anal. 9 (1975) 263, Comm. Math. Phys. 53 (1977) 31. M. Scheunert, W. Nahm and V. Rittenberg, J. Math. Phys. 18 (1977) 155. G. 't Hooft, M. Veltman, Nud. Phys. B 4 4 (1972) 189. S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264. Y. Ne'eman, paper {5.11}. Y. Ne'eman, S. Sternberg and D. Fairlie, Phys. Rept. 406 (2005) 303. S. Sternberg and Y. Ne'eman, paper {5.4}. Y. Ne'eman, paper {5.8}. S. Hwang, C-Y. Lee and Y. Ne'eman, paper {5.10}. DO collaboration, Nature 429 (2004) 638 Y. Ne'eman and J. Thierry-Mieg, paper {5.2}. R. Coquereaux, G. Esposito-Farese and G. Vaillant, Nud. Phys. B353 (1991) 689. A. Connes and J. Lott, Nud. Phys. B18 (1990) 29. Y. Ne'eman and J. Thierry-Mieg, paper {5.7}. Y. Ne'eman and S. Sternberg, paper {5.9}. D. Quillen, Topology 24 (1985) 89-95. Y. Ne'eman, and S. Sternberg, paper {5.3}. Y. Ne'eman, S. Sternberg and J. Thierry-Mieg, paper {5.5}. Y. Ne'eman and J. Thierry-Mieg, paper {5.6}. Y. Ne'eman, paper {5.12}.
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REPRINTED PAPERS OF CHAPTER 5: SU(2/l) SUPER-UNIFICATION OF THE STANDARD MODEL AND NON COMMUTATIVE GEOMETRY
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
Y. Ne'eman, "Irreducible Gauge Theory of a Consolidated Salam-Weinberg Model", Phys. Lett. B 8 1 (1979) pp. 190-194.
325
Y. Ne'eman and J. Thierry-Mieg, "Geometrical Gauge Theory of Ghost and Goldstone Fields and of Ghost Symmetries", Proc. Nat. Acad. Sci. USA 77 (1980) pp. 720-723.
330
Y. Ne'eman and S. Sternberg, "Internal Supersymmetry and Unification", Proc. Nat. Acad. Sci. USA 77 (1980) pp. 3127-3131.
334
Y. Ne'eman and S. Sternberg, "Sequential Internal Supersymmetry", in High Energy Physics 1980, XX Int. Conf., Madison, Wisconsin, L. Durand and L. G. Pondrom, eds. (Am. Inst. Phys., Conf. Proc. 68), Particle and Field Subseries, New York 22 (1981) pp. 460-462.
339
Y. Ne'eman, S. Sternberg and J. Thierry-Mieg, "SU(7/1) Internal Superunification: A Renormalizable SU(7) x U(l) with Factorizable Pomeron", in Physics and Astrophysics with a Multikiloton Modular Underground Track Detector, Proc. "GUD" 1981 Rome Workshop, G. Ciapetti, F. Massa and S. Stipcich, eds. (Serv. Doc. Lab. Naz. Frascati, Rome, 1982), pp. 89-96.
342
Y. Ne'eman and J. Thierry-Mieg, "Anomaly-Free Sequential Superunification", Phys. Lett. B108 (1982) pp. 399-402.
350
J. Thierry-Mieg and Y. Ne'eman, "Exterior Gauging of an Internal Supersymmetry and SU(2/1) Quantum Asthenodynamics", in Proc. Nat. Acad. Sci. USA 79 (1982) pp. 7068-7072.
354
Y. Ne'eman, "Internal Supergroup Prediction for the Goldstone-Higgs Particle Mass", Phys. Lett. B181 (1986) pp. 308-310.
359
Y. Ne'eman and S. Sternberg, "Superconnections and Internal Supersymmetry Dynamics", Proc. Nat. Acad. Sci. USA 87 (1990) pp. 7875-7877.
362
S. Hwang, C.-Y. Lee and Y. Ne'eman, "BRST Quantization of SU(2/1) Electroweak Theory in the Superconnection Approach, and the Higgs Meson Mass", Int. J. Mod. Phys. A l l (1996) pp. 3509-3522.
365
Y. Ne'eman, "Internal Supersymmetry, Superconnections, and Non-Commutative Geometry", in Group Theory and Its Applications, O. Castanos, R. Lopez-Peha, J. G. Hirsch and K. B. Wolf, eds., 5 lectures at XXX ELAF (Latin American School of Physics, Mexico 1995), AIP Proc. 365 (AIP Press, Woodbury, New York, 1996), pp. 311-334.
379
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5.12
Y. Ne'eman, "A Superconnection for Riemannian Gravity as Spontaneously Broken SL(4, R) Gauge Theory", Phys. Lett. B427 (1998) pp. 19-25.
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PHYSICS LETTERS
12 February 1979
IRREDUCIBLE GAUGE THEORY OF A CONSOLIDATED SALAM-WEINBERG MODEL YuvalNE'EMAN1-2 Tel Aviv University, Tel-Aviv, Israel, and Center for Particle Theory, Department of Physics, University of Texas, Austin, TX78712, USA Received 30 October 1978 Revised manuscript received 14 November 1978
We derive the Salam-Weinberg model by gauging an internal simple supergroup SU(2/1). The theory uniquely assigns the correct SU(2)L ® U(l) eigenvalues for all leptons, fixes 6 w = 30°, generates the W±, Z° and ACT together with the Higgs-Goldstone /j_ = 1/2 scalar multiplets as gauge fields, and imposes the standard spontaneous breakdown of S U ( 2 ) L ® U(l). The masses of intermediate bosons and fermions are directly generated by SU(2/1) universality, which also fixes the Higgs field coupling.
1. Introduction. We assume that the SalamWeinberg "standard model" represents a correct description of a Unified Weak and Electromagnetic Interaction, in view of the results of the SLAC [1] e—d polarization experiment [2]. There is however one aspect which appears to mar the picture. This is the lack of aesthetic cohesion and simplicity: the non-simple gauge group (including the uncorrelated quantum numbers of left- and right chiral fermions, the free 0W angle etc.); the need for spontaneous symmetry breakdown, and thus the necessity of an ad-hoc adjunction of a scalar multiplet of Goldstone and Higgs fields; the particular choice of a weak-isospin 7L = 1/2 multiplet for these fields, which does not fit the "extremum" requirement of general symmetry breakdown theory; lack of information with respect to Higgs couplings. Past great syntheses have provided some of the most aesthetic elements in Physics. In this note, we present an attempt to provide such a formulation. Postulating in the lepton sector a simple (super) group [3] gauge, we obtain a unique invariant
derivation of the standard model with sin20w = 1/4, and the precise SU(2)L ® U 0 (l) required eigenvalues (e.g. U0(e^) = - 1 , t/ 0 (e^) = - 2 etc.); a direct gaugederivation of the Goldstone—Higgs multiplet with its particular isospinor assignment, its couplings, and a non-zero vacuum expectation value for the appropriate component. We have adopted the view, emphasized by Wu and Yang [4] and vindicated in the evocation of the Aharonov—Bohm effect and in the calculation of exact solutions (monopoles, instantons, etc.), according to which gauge theories should be treated geometrically, as Principal Bundles PG (P, M, G) of a structure group G (the gauge group) over a base space M (Minkowski space or its subspaces), and Associated Vector BundlesEG (PG,M,R(G), G). The recent identification [5] of the Feynman-DeWitt-FadeevPopov ghosts with semi-vertical components of the connection 1-forms in PG and the accompanying geometrical derivation of the Becchi-Rouet-Stora transformations provides additional vindication of this approach. 2. The Graded Lie Algebra and group manifold. Let G = SU(2/1), the supergroup generated by a Graded Lie Algebra (GLA) isomorphic to the set of GradedTraceless [3] 3 X 3 matricesn A ,A = \ 8
Research supported in part by the United States-Israel Binational Science Foundation. Research supported in part by the U.S. Department of Energy, Grant EY-76-S-05-3992.
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TrG^=0
(2.8) T^o» un 'H^. where x is the chirality. Note that the algebraically im-
(2.1)
Ca,C8DL°.
:
a = 1,2,3
HfDL , [Ha,lib]
1 = 4,5,6,7 =2ifabc(ic (2.3)
l^fl.Mg] = 0
+ " F E R M I ) = mF
(2.4)
{ni,Hf} =
2dijaiia-s/3&ijiis,
(2.5)
CO>«
= 28ab,
Tr(ju8)2 = 2
(2.6)
The diagonal matrices therefore define the Cartan subalgebra basis, -1
1
V3 M3
n
+W
B-Here
in
addition e°(R) = L.
= co«dx"+co fl ,
co8 = co8dx ( '+co 8 .
(3.1)
However, for the Xx- generators, the co'CT would have been anticommuting spin-one mesons, forbidden by the spin-statistics correlation. This is why "internal" GLA cannot generate S-matrix symmetries [ 9 ] , though they can provide useful "supergauges" as in our theory. Spin-statistics do allow however a realization of the u' themselves: the ghosts d" self-anticommute as one-forms but SU(2/1) ensures that the cb' are spinless bosons * ' . Gauging SU(2/1) thus provides a gauge-field multiplet whose composition is exactly that which is necessary for the standard model, including a Higgs-Goldstone multiplet behaving as an IL = 1/2 (eq. (2.4)). We can also calculate 6W, the SU(2) L X U(l) mixing angle. Using the standard notation for the Interaction Lagrangjan, e.g. for the leptons \j/ (or our a of ref. [7])
= 5*). For the L° subalgebra
it is thus as in SU(3), Ti(na,»b)
X = V?,
3. The Gauge bundle and GoMstone-Higgs fields. We gauge G = SU(2/1), working in PG(P, M, n, G, •) where n is the projection and the dot denotes the rightaction of G on P [4,8]. The Lie algebra-valued connection 1-form co = w 4 XA defines the gauge fields. On a section, we also have co' 4 , the semivertical components defining the ghosts [5]. For L°, we have U(2) gauge spin-one fields cog, co8, (a is the space-time index, in fact on a section)
where fAij and dijA refer to SU(3) coefficients [6]. G is thus an internal symmetry based upon a GLA, and the odd (L 1 ) generators X-x (i = 4 7) anticommute even though they are Lorentz-scalars, thus violating the spin-statistics correlation (like the ghosts in renormalization). The normalization of a GLA allows at most two scales (for L°, L 1 ) defined by ordinary Tr (also yielding g A B and TTGQIAHB)
N/3M8»
posed assignments (2.7) thus reproduce the quantum numbers of all lepton multiplets (*»L> e £ , e ^ ; " L ' ^ L ' MR' "L' T L ' T i P ' Actually, t n e third states are eOe^ etc. where eO is an SU(2/1) scalar Fermi conjugation, and the conjugate state is e ^ ( e ° ) _ 1 . Han— Nambu triplets belong to 3 + 1 , 3 + 1, 3 +1 of SU(2/I). Alphons [7] are in 3 + 1 . For SU(m/«), e ° ( m B 0 S E
(2.2) 1
12 February 1979
-1
(2.7)
1 -1
MO (ii 0 is extraneous to SU(2/1) but fixed by orthogonality in U(2/l)). The Group Manifold G is obtained [3] by exponentiation of L° and by multiplication of L 1 by Grassmann parameters. The latter action thus yields bosonic variations which, however, would not be produced by SU(3), since they are obtained by djjA coefficients. We identify the following quantum numbers:
(3.2)
* Alternatively, for non-geometrists, cjjj = e^uf, Cr the Higgs-Goldstone fields. The BRS equation Swj}=D,.u>A yields the representations e|J : 5~1Z)Mand< 191
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Volume 8IB, number 2
PHYSICS LETTERS
to be compared with a
?g{h »31'Wl
RA
+ l>y<'viW*},
(3.3)
we find =
12 February 1979
= i ^ / c b e O A dxT = dw 4 -\cABEuB
and covariant exterior derivatives A B A pE Dp/1 = dpA _ c BEto
2
t g 0 w s 7 * = W 3 , 0W=3O°, sin 0 = l/4, (3.4) which seems close to the present experimental value [1]. The Lagrangian for the gauge multiplet can be written as (imposing symmetry under SU(2/1) X C )
R'=Dco' = Du'
£0 = (FaTaFa°* + F0T&F&aT + FjFia\
R" =
with £(}, e°) a discrete group, and a euclidean metric. The first term contains X04.
/? 8 =i? 8 +4\/3w'A w',
6
(4.3)
become, from (2.3)-(2.5)
4. Spontaneous breakdown of the gauge symmetry. Observing the lepton multiplets, we realize that the masses behave as pig or u 7 under (2.4), violating U(2)w (pia, pig). This implies that the vacuum behaves partly like a pig (C-conserving) component, and thus, for the spinlessu6, X = <0|w 6 |0>*0
A LOE (4.2)
(4.1)
6
and w = w + XWe now turn to the realization of SU(2/1) X C . The SU(2/1) subgroup can be represented conventionally and linearly, and we have noted that (2.8) reproduces the standard-model assumed assignments perfectly. The odd part of the group also acts linearly, with Lorentz-scalar Grassmann elements as parameters (like "ghost" fields). However, in the gauge-fields lagrangian (3.4) we impose a larger and even group, in which the odd matrices pi;- -»• pi/ X e° = juj, where pij- carry even Grassmann elements and represent finite variations. Thus in F>°T <Ja -> e0o(J, etc., and e^e0" = 5£. The actual symmetry group is thus an extension of SU(3), acting non-linearly, with SU(2/1) as a linear subgroup. To break SU(3) we add to (3.4) - / J ¥ I ? W ' W ' which preserves U(2)w but triggers its spontaneous breakdown. Returning now to the 7L = 1 /2 representation, one choice of an SU(2/1) and thusSU(2)L gauge reproduces the "U formalism", reparametrizing the (co4, co5, to7) Goldstonefieldsand absorbing them into the gauge [10], leaving only to6 + x- From this point on, use of eq. (3.4) yields the usual derivation [10] of mass terms for the W* = (coj + ico£)/\/2, since the SU(2/1) covariant derivative on any ci' component happens to coincide with the U(2)w derivative. Indeed, the "curvatures" (CABE are the structure constants)
(4.4)
fia-daifioiAuf (4.5)
where the caret sign denotes restriction to U(2) w . To derive an FljalAF>ivA density, Hodge adjunction (the dual) is used, RA A *RA • This is a nongeometric operation in the Exterior Calculus, and we might extend its definition to the unfactorizable to'. Normally, (lOA
A W B ) * ( W C A 0>D) = CO/1ALO»<:LOvBGJ>'D
d4X.
This implies that in u ^ ^ w ^ ' w ^ u w e a ^ s dx^djc*3 X dxTdx8 the fictitious pair (w^'oo*"') becomes (CJ'O)'), where <1>' is thefieldin this representation. We discuss such duals elsewhere. Note that the dafand d^jj terms in (4.5) reproduce the 0 4 potential when inserted in (3.4). A mixed <2-/term vanishes by BRS. The electromagneticfieldA a remains massless, since R"* contains ( / ^ j W 4 A GO6) which picks out the orthogonal Z°, using (2.4). The Higgs field mass is predicted to be -400 GeV (and X = g2). 5. Fermion masses. It is especially interesting to analyze the appearance of lepton basic masses * 2 , as a direct result of the SU(2/1) universal coupling. We work in the Associated Vector Bundle of the Lorentzspinor i//, forming a 3 representation of SU(2/1). The covariant derivative will couple the wj conventionally, as in (3.3), but will in addition have the co6 term, coupling £ to ^g" by what amounts to an example of the non-linear action of the odd part of the algebra. As to the nature of the interaction, the exterior calculus writes the relevant part of the action A w as / *jA \ coA, with
We assume the existence [7] of a "flavor" component removing the e - M - ? — ... degeneracy, in addition to this pure weak-electromagnetic contribution.
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*U ^hovpofo^AWax"
PHYSICS LETTERS
A ixP A Ax")
(5.1)
Inserting (3.1), we have a4-form, A
w =\feavpo4i7ao>afJia
A (dxv A dxP A dx") (5.2)
which becomes A
hfeavpa^y^a0lxAyi/
v/ =
(5.3) a
X (dx
A
dx" A dx").
In eq. (5.2), (yauaiia) is the full connection oneform on the Lorentz-spinor bundle. In going from (5.2) to (5.3) there is an effective flip of indices, yauApdxP-+Tiy!iuAfidxa.
(5.4)
In the case of the uncontractable w' scalar fields, this can only take place if we have for the connection on this spinorial bundle, W , = $€%<&*
(5.5)
yielding, < 6 ) = k f eavpJ(cb6n6)Htea
A dx" A dx<> A dx") (5.6)
i.e. the appropriate mass term for co^ = w*>' + £_ However, eq. (5.6) also fixes the "Higgs field" coupling, which is thus predicted to be of universal strength ^g. Our introduction of spontaneous symmetry breakdown (4.1) was based on the fact that masses break chirality conservation, that electrically charged leptons are massive and that neutrinos are massless (i.e. a carryover of the 2-component neutrino and of V—A theory). For (u, d') quarks the direction is still the same, assuming a vanishing mass for u and a massive d' (which indeed contains a contribution from s, a heavier quark). We adopt the view that our SU(2/1) and its breakdown, act on quark triplets (u L , d' L , d' R ) etc. in the same way as they act on leptons, through the displacement AU0 = —4/3. However, the $ B which is assumed to carry this AU0in our primitive model [7] would itself couple to co6 since [M B , 0 B ] = —4/3 \ / 3 0 B . It may therefore be impossible with fractional charges to treat a "simple" gauge for weak-electromagnetic unification separately, when dealing with the hadrons, (i.e., without involving color and/or flavor).
12 February 1979
6. Comments. We have not yet studied the possible applications of the SU(2/1) gauge on quantum aspects, and renormalization procedures * 3 . However, from the orbits of SU(2/1) on cJ we know the stability group U(l). Since 7r 2 (SU(2/l))/(U(l)) = JTI(U(1)), we have monopole solutions characterized by integers k€.Z. Also, anomalies may now be cancelled by graded tracelessness. On a previous occasion, a similar consolidation of SU(2)j and U(1) Y in the (simple) SU(3) has led to fruitful applications and unexpected insights. To expect similar developments from SU(2/1) W seems perhaps too much to hope for. However, it appears worthwhile to make this try at the most economic embedding of SU(2) L ® U(l) in a parameter-free "aesthetic" and geometric theory. Failure to do so in the recent past was due to hopes for an early "grand unification" with the Strong Interactions, in a Supreme gauge group. The sequential nature of new quark and leptons seems to preclude such a unification at this stage of our knowledge [7], and we feel justified in attempting to present an "irreducible" weak-electromagnetic unification as a more modest target. Finally, it would be interesting to check the possibility that other such spontaneously broken gauges might exist, perhaps even in strong (flavor) dynamics, with the Higgs mesons corresponding to odd-generator gauge fields. A detailed exposition of the present theory will be published elsewhere. We would like to thank Professors H. Harari, E.C.G. Sudarshan and J.A. Wheeler for their comments, and J. Thierry-Mieg for useful suggestions. References [1] Reported in review by H. Fritzsch, Tokyo (1978) XIX International Conference on High Energy Physics, to be published. [2] For a somewhat less recent review of the experimental situation see H. Harari, Phys. Reports 42C (1978) 2 3 5 309. [3] F.A. Berezin and G.C. Katz, Mat. Sb. (USSR) 82 (1970) 343; * The renormahzability of the theory is manifest when the factorized fictitious CJ£ are used. In the physical theory, CJ<J and i i ' are fields, whereas ui' and u>a are ghosts. 193
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V. Rittenberg, in: Group theoretical methods in physics (Proceedings, Tubingen, 1977), eds. P. Kramer and A. Rieckers (Springer-Verlag, 1978) pp. 3 - 2 1 , is a recent review. [4] T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 3845. [5] J. Thierry-Mieg, to be published in Nucl. Phys. B.; Y. Ne'eman, J. Thierry-Mieg and T. Regge, to be published in: Proc. of the XIX Internat. Conf. on High Energy Physics, Tokyo, 1978; J. Thierry-Mieg and Y. Ne'eman, to be published. [6] M. Gell-Mann and Y. Ne'eman, The eightfold way (W.A. Benjamin, 1964), p. 50.
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[7] Y. Ne'eman, Primitive particle model, to be published. [8] See e.g., S. Kobayashiand K. Nomizu, Foundations of differential geometry, vol. 1 (Interscience, 1963); S. Sternberg, Lectures on differential geometry (Prentice-Hall, 1964). [9] R. Haag, J.T. Lopuszanski and M. Sohnius, Nucl. Phys. B88 (1975) 257. [10] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Proc. of the 8th Nobel Symp., ed. N. Svartholm ed. (Almquist and Wiksells, 1968).
Proc. Natl. Acad. Sci. USA Vol. 77, No. 2, pp. 720-723, February 1980 Physics
Geometrical gauge theory of ghost and Goldstone fields and of ghost symmetries (principal bundle/renormalization/supergroup/spontaneous symmetry breakdown/unified electro-weak interactions) YUVAL NE'EMAN* AND JEAN THIERHY-MlEGt 'Department of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel; 'Center for Particle Theory, The University of Texas, Austin, Texas; 'California Institute of Technology, Pasadena, Cafifomia; and 'Croupe d'Astrophysique Relativiste, Observatoire de Meudon, 92190, -France Contributed by Yuval Ne'eman, November 1,1979 ABSTRACT We provide a geometrical identification of the ghost fields, essential to the renormalization procedure in the non-Abelian (Yang-Mills) case. These are some of the local components of a connection on a principal bundle. They multiply the differentials of coordinates spanning directions orthogonal to those of a given section, whereas the Yang-Mills potential multiplies the coordinates in the section itself. In the case of a supergroup, the ghosts become commutative for the odd directions, and represent Nambu-Goldstone fields. We apply the results to chiral "flavor" SU(3)i X Sipfo and to 5t/(2/l). The latter reproduces a highly constrained Weinberg-Salam model. It has been known since 1963 (1) that a principal fiber bundle provides a precise geometrical representation of Yang-Mills gauge theories. After 1975 (2), this correspondence has been extensively applied to the study of self-dual solutions of the Yang-Mills equation (monopoles, instantons) and of global properties of the bundle, etc. We present here an entirely different domain of applications. First, we reproduce the recently suggested (3-5) identification of the Feynman-DeWitt-Faddeev-Popov ghost fields (6) essential to the renormalization procedure in the non-Abelian case, with local geometrical objects in die principal bundle. This will directly yield the Becchi-Rouet-Stora (BRS) equations (7) guaranteeing unitarity and Slavnov-Taylor invariance (8, 9) of the quantum effective Lagrangian. Except for the "antighost" variation, this quantum-motivated symmetry thus corresponds to "classical" (geometrical) notions, with its dependence on the gauge-fixing procedure (which determines the quantized Lagrangian) limited to section dependence, a mere choice of gauge. We then consider the case of a supergroup (10) as an internal symmetry gauge, generalizing the recently suggested (11) role of Sl/(2/l). We show how the ghosts geometrically associated to odd generators (12) may be identified with the GoldstoneNambu (13) scalar fields of conventional models with spontaneous symmetry breakdown. As an example, we realize the chiral Sl/(3)L X Sl/(3)B "flavor" symmetry (14-16) by gauging the supergroup Q(3) (see refs. 10 and 12). Lastly, we recall some of the more relevant results concerning asthenodynamics (weak electromagnetic unification) as given (11) by the ghost-gauge Srj(2/1) supergroup. Connections on a principal bundle: Gauge (potentials) and ghost fields
restricted to the base manifold M of dimension m=4,so writing (-"(YM) = u\dx»
(a = 1 . . . n, n = 0 , 1 , . . . 3)
[1]
the OJ 1 were identified with the Yang-Mills potentials. In our treatment, the connection o> has m + n dimensions holonomically, ojj (fi = n, i; ft = 0 , . . . 3; i = 1 , . . . n), quite aside from the n components described by the a index and contracted with the abstract Lie algebra matrices \,. We denote the (vertical) projection by TT-.P —- M, the structure group by G, and right-multiplication on P by the dot (-):P X G — P, so that [2) \(p-g)-g = p-(gg)) and for Vx, a neighborhood of x £ M, we get "local triviality" (a direct product) in P. The dot (•) induces a map t horn the Lie algebra A of G into P„, the tangent manifold to P. Thus, for y \ „ Aj,, A,, £ A (a,b,e = 1 . . . n) with
[KM = C'ttK
(3]
we have t:A
,
W
A-XGP,.
One proves that t is a homomorphism of A, with the Lie bracket (LB) operation realized on P , as a Poisson bracket (PB) [£A']LB = [ U ' I P B .
[5]
However, this map ( has no inverse because the image of A (of dimension n) does not span Pt, of dimension (n + m). A linear mapping from P„ to A, the connection o>, is now chosen to provide the missing inverse u-.P,—A,
YXGA,
OJ(X) = X.
[6]
a> is Lie-algebra valued, and belongs to the cotangent manifold •P. It is thus a one-form. If z" are local coordinates over P, one may explicitly write vcGP,,
v = v*(z) — {R,S = 1, 2,
oi = ai's(z)
We start by reintroducing (17,18) the concept of a connection on a principal fiber bundle (P,M,irC,-). Previous authors used definitions in which the connection (a one-form a)"(YM)) was
that
.n + m)
[7]
s
dz \,
oj(o) = o Jco = oiflu"^ = a)a(u)X,,
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact. 720
330
(J denotes a contraction, d/dzR J dzs W(\b) = 5%.
5j). For Xb as v, we have
Abbreviations: LB, Lie bracket; PB, Poisson bracket; BRS, BecchiRouet-Stora.
Proc. Natl. Acad. Sci. USA 77 (1980)
Physics: Ne'eman and Thierry-Mieg Because P , is larger than A, there is a nontrivial kernel H of it}. In other words, to each point p G P . u associates a subspace Hp c P tp. This is known as the "horizontal" tangent vector space at p, and defines an exact splitting of Pt: hEHp~ P*P = VP + "p.
phic to the base manifold M. We fit the z" coordinates to 2 by lifting local i* coordinates from the base M, and a' (group parameters) coordinates from C, by using the maps (TT -1 , r - 1 ) with T a projection onto the fiber to get the equation for 2 : 2:a'(*) = 0, i = 1 , . . . n.
wp(h) = 0 "P
Vp = Im,(A),
= Ker (up)
[8a]
M dx" da' o) = X,da' + *„di«.
One also assumes an equivariance condition [9a)
We now introduce the Lie derivative V„ or convective derivative along a vector field o in Pt [7J. Its action on functions, vector fields, and one-forms reads: ?«/(*) = ° B ^ j j /
[10aJ
V„ v' = [O,O']PB
[10b]
Vo0) = d(v J u) + o J da).
[10c]
[9b]
[14]
We identify the ghost (6) fields C* as (£ is a constant length) ec° = Xfda'
[15]
while (fi'it is the Yang—Mills potential. According to [7], had we taken a topologically trivial P and a global flat section, C"(o) would have coincided explicitly with the Cartan left-invariant one-forms of the rigid group. It would then carry no x" dependence and would not be a true field. However, under a gauge transformation,
The equivariance condition [9a] can be written infinitesimally as Vxn J w = 0.
[13]
We now express the vertical connection form w in this basis
(\)PGVP.
Hp.g = H p .g.
721
&w°(x,a) = De°(x,a) l
so that C°(o) = l/£(a~ butions,
[16]
da)" receives independent contri-
Taking the Lie derivative of [8a], we have VJ(/I
iC = i da' U j C(x,a)) - ± CWt<(*,«).
J oi) = Vj/i J a + h J Vj,w = 0,
We now rewrite fl of [11] in component form. Defining "partial" exterior derivatives
yielding by [9b] h J Vxa) - 0 (i.e., V\o> is vertical). It can thus be rewritten with a linear representation and a factorized z dependence, Vxa>=/(z)[A,a>]LB.
df = sf+d~f; sf^da'j^f;
Vxa)(V) = [X,X']PB J u + \' J V*o> = 0, which vanishes because to(A') = X', a constant. Replacing the last term by [8b] and using [5], we find that f(z) = - 1 . The equivariance condition can thus be stated as
32 = sd + 3 s = s 2 = 0.
[9c]
[11]
and contract it with a vertical vector field A X j n = XJda) + i [ X J w,u] - - [oj.X J a>]. The first term is given by [10cJ, the last two are given by [6] = Vx« + i[X,a>)-i[a;,M, and, when [9c] replaces the first term, the expression vanishes. The curvature two-form is thus purely horizontal, X Jfi= 0.
[12]
This equation is the Cartan-Maurer structural equation of a principal fiber bundle. Up to this point, we have just used textbook geometry. We can now identify the ghost fields. Because we are in P v a gauge choice corresponds locally to defining a section—i.e., a surface 2 in P—locally diffeomor-
331
[18]
[19]
3 is our "ordinary" horizontal d which depends on the section 2 and s is the exterior differential normal to the section. Q can be broken into three pieces—i.e., terms in da' A da>, in da' A dx" and in dx" A dx". Applying [12] implies the vanishing of the first two pieces. By [14,15,18] we have
We now define the curvature two-form (17,18) fi = du + i [OJ.OJ]
lf = dx^J.
Cohomology implies
[8b]
To fix f(z) we take the Lie derivative of [6]:
Vxw = -[X,a)]LB.
[17]
sC = - 1 [C.Cf
[20]
«*" = iD^C*.
[21]
These are the BRS equations (7)for * J a n d C . £~l s is thus the BRS operator. One of us (J.T.-M.) has shown (14) how the covariant quantization path integral, used in summing over all configurations of the potential satisfying BRS, can be given a geometrical form. In this representation, Feynman diagrams involve nonintegrated exterior forms (the ghosts) together with anticommuting Lagrange multipliers (the antighosts). One can then check that the minus sign required by ghost loops, which led to the assignment of Fermi statistics to spin-zero fields C"(x), is indeed just the sign due to self anticommutation of one-forms. Nambu-Coldstone fields When the Lie group C is replaced by a Lie supergroup (10) and the Lie algebra A by a graded Lie algebra (12), some connection one-forms commute instead of anticommuting. For an internal graded Lie algebra, the one-forms obey a Z(2) X Z(2) gradation (18,19) npc
A (<>b = (-l)pq+ABgqb
A vpa_
[33]
722
Physics: Ne'eman and Thierry-Mieg
Proc. Natl. Acad. Sci. USA 77 (1980)
where n?" and t-ib are, respectively, a p-form and a fl-form, the indices a and b represent a basis of a graded Lie algebra, and A and B are their respective gradings. The connections ai< = c;
[23]
thus commute when i represents an odd-grading in the graded Lie algebra (A = 1). 0* is thus a Lorentz-scalar physical Bose field. We have recently conjectured (11) that these fields be identified with Nambu-Goldstone (Higgs-Kibble) fields when the Weinberg-Salam (20, 21) model's Sl/(2) X U(l) gauge group is embedded in the supergroup S(7(2/1). The internal supergroup represents a ghost symmetry (i.e., a symmetry between physical and ghost fields) because it changes the statistics without changing the spins. The Goldstone-Nambu (or, after further spontaneous breakdown, the Higgs fields) thus become in this approach the appropriate gauge fields for the odd part of the ghost symmetry. In the study of Goldstone-type realizations of global symmetries, the Goldstone field corresponded to that part of the invariance group that was not a symmetry of the vacuum and could thus not be realized linearly on single-particle-state multiplets. It is indeed instructive (15,16) to choose as an example the one case of that type we understood between 1960 and 1967: the pion's (and 0" octet) role as the zero-mass Goldstone particle in chiral W(3)ch - Sl/(3) L ® SU(3)B. In the nonlinear picture (22, 23), the vacuum is invariant under the positive parity St/(3) C W(3) charges X + . The remaining 8 of generators [under that St/(3)] corresponding to the axial-vector charges X - is realized nonlinearly. The 8 of 0~ mesons v acts as realizer, expHij.X-)(0,iM = (iMM.
[24]
The 1) axe in fact parameters of the axial generators. We denote the more common parameter of the (linear) vector subgroup by a. For a generic element g of W(3) we get g- 1 expH»; • X") = expH»?' • X") e x p H a • X + ), [25] where TJ —* 7j' is caused by the positive parity part of g - 1 , whereas a is produced by the negative parity element acting on 7), which is itself such an element. The resulting group action is given by g-'M)
= lv',D(exp-ia • X+M
[26]
This action clearly exhibits a Z(2) grading provided by parity. Can we represent it linearly by a supergroup? In ref. 12 we had indeed constructed the relevant/, d coefficient superalgebra explicidy. It now appears as Q(3) in ref. 10. For Q(3)&, take a set of sixteen (6 X 6) matrices, X+ : H" I
X-:|. Am|
M
|«n
[27] |T5
l^m, Ki are SU(3) matrices (14)], and define the brackets,
{x+,x;] = ifmntx-t IXm-Xnl = ifmnt X(
[X~,X~] D : = X^X~ + X*X~ - % (Tr X~X„)/ . = 2d m „,X + ,
[28]
where the dmne are SC7(3) totally symmetric Clebsch-Gordan coefficients for 8 X 8 — %m (14). The symmetric bracket between two odd elements thus differs from an anticommutator (in this defining representation) by a trace. In the adjoint representation, it will again be an anticommutator.
332
We now take this C = Q(S) in P and study the connections. Under the X + generated SU(3) subgroup, we have two octets, a" = £C(x) + dx"
[29]
In the even subgroup, l(x) is a / = 1 octet and C"(x) is the corresponding ghost 8. In the odd piece, the &>' also form an § under X + , but they are commutative, as we have seen in [22] and [23]. Thus, ^'(x) is a 0~ octet of bosons—i.e., physical fields! In fact, we can identify these "exorcized ghosts" as the Goldstone-Higgs multiplet of the theoryl They are accompanied, howevever, by a new type of ghost, the / = 1 Fermi statistics G),(x). The role of the latter is perhaps not entirely understood at this point, but one can already see them in action in one-loop renonnalization group equations: they provide a relatively heavier weighted contribution of ghost type [e.g., in conserving 6W in S 1/(2/1)]. Notice that the entire counting system for such internal supergauges has to be reordered, because the Higgs fields i]' will be coupled universally, thus providing new diagrams of order g3, etc... . We may solve [21] and write G[ = -£s~x D„7)': = e„n*
DUC°: = t„C:
[30]
For a matter field ^, an Sl/(3) triplet in representation [27], the BRS equation is s^» = ^[C^-f: = r",
[31]
where we have defined an effective ghost field r". If ^" is a Lorentz-spinor fermion (the /ff-positive components in fact), r" will be a Lorentz-spinor boson—i.e., a ghost. However, because C is a supergroup, the 4>" fill up only half of a representation like [27]. The other half consists of a triplet I" of Lorentz-spinor opposite parity bosons (i.e., ghosts). Thus the BRS equation becomes sf = £[C,t)u: = \pu,
[32] u
thus relating them to Lorentz-spinor fermions \f/ that complement the r" in making a six-dimensional (and Dirac p* diagonalized) representation of W(3). To have all new ghosts (rB, t") appear as composite and the ip" appear as additional (inverse parity) matter fields, we may write r" = s\[>n, t" - S"1 ^".
[33]
Summing up, we have seen that gauging a supergroup C produces as gauge fields both die vector mesons 0° coupled to the even subgroup G + and a Goldstone-Higgs multiplet n' behaving as A(G~) under A(G + ) itself. At the same time, the theory contains the renonnalization ghosts C and a new set of vector ghosts C'r. Matter fields \j/n and ifr" are split between two analogous representations of G, even though when taken together they fit exactly the quantum numbers of one such representation. In their split assignment, they are accompanied by composite ghost fields that complete the two representations. All of this will be true of SU(2/1) as well. Notice that the resulting gauge Lagrangian in the case of £)(3) (in its physical part) is exactly that of the "flavor" SU(S) of the sixties with phenomenological constituent quark fields and with the I - mesons p, K*, °/UJ° as gauge fields, plus a universally coupled 0~ meson multiplet v, K, ij. This is just the Lagrangian postulated by Gursey and Radicati, which gave rise to SU(6) as its static symmetry (24). There is no U(l) problem!
Physics: Ne'eman and Thierry-Mieg
Proc. Natl. Acad. Set. USA 77 (1980)
Table 1. Kinematics of S[/(2/l) Representation 8, J = 1 S',J = 0 i,J = %
3',./-% 4, J = 'k l
i',J = h
Particles
Ghosts
* 4i . *J,5 «2.9
CJ, Gs„ Gl, Gl C',C2, C3, C" rZ (composite) t\, CR (composite) rlm, r\!3 (composite) ' 8 * . ' R " " (composite)
I , I .T , rp "l.el CR
u¥>,dZ»3 d-B."°,uW
51/(2/1) as t h e ghost theory of asthenodynamics (the w e a k electromagnetic interactions) The idea of a supergroup as an interna] gauge group involving the ghosts of renormalization was first suggested (11) in the context of a basic theory of the unified weak electromagnetic interaction. It reproduces the Weinberg-Salam model (20, 21) in an extremely constrained form, imposed by SU(2/l) D Sl/(2) L X 17(1). The kinematics of SU(2/1) are astonishingly precise in fitting just the observed particle representations of SU(2) L X V(l)u (Table 1). A "family" is thus (3 + 3') + 3 X (4 + 4'). Note that S17(2/1) predicts that the IL = V2 multiplet is in a 4 representation when the charges are fractional and in a 3 when they take on integer values! Similarly, it predicts that the Higgs-Goldstone multiplet is an isodoublet / L = ]k, U = ± 1 . Also, 0\v = 30°, and m , = 245 GeV. The X<£4 self-coupling of the Higgs-Goldstone multiplet is X « 4/3 g2. We refer the reader to the original article (11) and to a recent discussion at the classical level (25).
13. 14. 15. 16. 17. 18. 19. 20. 21.
This work was supported in part by the U.S.-Israel Binational Science Foundation and by the U.S. Department of Energy Grant EY-76-S05-3992. We would also like to acknowledge the support of the Wolfson Chair Extraordinary at Tel-Aviv University. 1. 2. 3. 4.
22. 23. 24. 25.
Lubkin, E. (1963) Ann. Phys. N.Y. 23,233-283. Wu, T. T. & Yang, C. N. (1975) Phys. Rev. D 12,3845-3857. Thierry-Mieg, J. (1978) These de Doctorat d'Etat (Universite de Paris-Sud, Orsay, France). Thierry-Mieg, J., J. Math. Phys., in press.
333
723
Ne'eman, Y. (1979) in Proceedings of the 19th International Conference on High Energy Physics, Tokyo, 1978, eds. Homma, S., Kawaguchi, M. 4 Miyazawa, H. (Phys. Soc. of Japan, Tokyo), pp. 552-554. Faddeev, L. D. 4 Popov, V. N. (1967) Phys. Lett. B 25, 2932. Becchi, C , Rouet, A. 4 Stora, R. (1975) Commun. Math. Phys. 42,127-162. Slavnov, A. A. (1972) Teor. Mat. Fiz. 10,153-161. Taylor, J. C. (1971) Nucl. Phys. B 33, 436-444. Rittenberg, V. (1978) in Group Theoretical Methods in Physics, Proceedings of the 6th International Conference Tubingen, 1977, eds. Kramer, P. 4 Rieckers, A., Lecture Notes in Physics 1979 (Springer, Berlin), pp. 3-21. Ne'eman, Y. (1979) Phys. Lett. B 81,190-194. Corwin, L., Ne'eman, Y. 4 Sternberg, S. (1975) Rev. Mod. Phys. 47,573-604. Nambu, Y. 4 Jona-Lasinio, F. (1961) Phys. Rev. 122, 345358. Gell-Mann, M. 4 Ne'eman, Y. (1967) The Eightfold Way (Benjamin, New York). Weinberg, S. (1968) Phys. Rev. 166,1568-1577. Gell-Mann, M., Oakes, R. J. 4 Renner, B. (1968) Phys. Rev. 175, 2195-2199. Kobayashi, S. 4 Nomizu, K. (1963) Foundations of Differential Geometry (Interscience, New York), Vol. 1, pp. 63-66. Ne'eman, Y. (1979) Symetries, Jauges et Varietes de Croupe (Presses de l'Universite de Montreal, Montreal, Canada). Ne'eman, Y. 4 Regge, T. (1978) Riv. Nuovo Cimenta Ser. HI, 5,1-43. Weinberg, S. (1967) Phys. Rev. Lett. 19,1264-1266. Salam, A. (1968) in Elementary Particle Theory, Proceedings of the Eighth Nobel Symposium, ed. Svartholm, N. (Almquist 4 Wiksell, Stockholm), pp. 367-377. Coleman, S., Weiss, J. 4 Zumino, B. (1969) Phys. Rev. 177, 2239-2247. Joseph, A. 4 Solomon, A. (1970) J. Math. Phys. 11, 748-761. Dyson, F. J. (1965) Symmetry Groups (Benjamin, New York). Ne'eman, Y. & Thierry-Mieg, J. (1980) in Proceedings of the Salamanca (1979) International Conference on Differential Geometry Methods in Physics, ed. Perez-Rendon, A. (Lecture Notes in Mathematics, Springer, New York), in press.
Proc. Natl. Acad. Sci. USA Vol. 77, No. 6, pp. 3127-3131, June 1980 Physics
Internal supersymmetry and unification (Lie superalgebras/fundamental representations/weak-electromagnetic charges/color/quark and lepton assignments) YUVAL NE'EMAN** AND SHLOMO STERNBERG^ Tel Aviv University, Tel Aviv, Israeli (University of Texas, Austin. Texas 78712; and 'Harvard University, Cambridge, Massachusetts 02138 Contributed by Yuval Ne'eman, March 25,1980 ABSTRACT We construct a family of finite-dimensional representations of the superalgebra s\n/aa) that depend on an integer parameter for m > 1 and on a complex parameter, b, for m = 1. We describe some models of elementary particles for s/(2/l), sl(3/l), and sJ(5/l). This involves the choice of the parameter b and the choice of the operators l3(tbe third component of the weak left-handed isospin) and If (the weak hypercharge). These must commute, and are related to the electric charge by the usual formula Q = I3 + % U. In particular, taking I3 to be in its standard form in su(2) c sKJS) c s!(5/l) »>d requiring that U commute with color «u(3) c sl(5) c s^5/l) leaves three nee parameters, two for the choice of V and one for the choice of o. We show that there are just two possible choices of these parameters yielding exactly all 32 quark and lepton charges: the Georgi-Glashow V e su(5), corresponding to U(l,-%) and arbitrary b and f/(0,%) 4 su(5), with b = 2. We provide a general construction of representations of sJ(n/l) consisting exactly of sequences of generations of quarks and leptons. 1. The finite-dimensional irreducible representations of the superalgebra s/(2/l) have been classified by Scheunert el al. (1). Among these representations there is a fundamental family of four-dimensional representations depending on a complex parameter, b. We shall construct a corresponding family of representations of s/(n/l), also depending on a complex parameter, on a space of dimension 2". Our construction will also yield a representation of sl(n/tn) for general values of m (on a space whose dimension is somewhat more difficult to describe), provided that b is a nonnegative integer. In terms of the general description of the irreducible representations of superalgebras given by Kac (2), our representations are "atypical" in the sense that their dimension is smaller than the dimension of a "typical" representation. We begin by recalling the definition of the superalgebras sl(n/m). (We refer the reader to Corwin et al. (3) for the basic general facts about superalgebras and their representations.) Let V and X be complex vector spaces with dim V = n and dim X = m. Let W be the super (or graded) vector space W - V + X, in which V and X are given opposite parity. The superalgebra sl(n/m), or sl(V/X), is the algebra of all endomorphisms of W of supertrace zero. A typical such endomorphism can be written as a matrix of maps
&
trA = tr£>,
11.11
in which A e Hom(V, V), B e Hom(X,V), C e Hom(VJC), and D 6 Hom(X.X). Those endomorphisms with B - C = 0 are even, and those with A = D = 0 are odd. In what follows, it will be convenient to use the identification of Hom(X,V) with
V®X*. In particular, j>®{ for v e V and f e X ' corresponds to the rank-one linear transformation given by
( v ® | ) w = <£,<«>> v.
0 v9£\ /0 .0 0 )' \x8u*
0' 0), /<*.«>»*••
I
0
3127
334
0
\
J
Let S(X) denote the ring of polynomial functions on X*, so that
s(X) = 5 s*(x), 0
in which S*(X) consists of homogeneous polynomials of degree It. Each 1 e X defines a multiplication operator mz on S(X), in which
(mJHv) = (n,x>Hv),
Vvex*.
[1.4]
Also, each { e X* defines a derivation D( (differentiation in the direction £) determined by [1.5a] De1 = 0,
[1.5b]
D(X =
[1.5c]
and
The standard commutation relations Dimx - mxD( = id
[1.6]
hold. Equally well, we could let Fb denote the space of smooth functions defined on some cone in X and homogeneous of degree b. Then D(:Fb —• F* - 1 and mxFh —• Fb+1 and the above commutation relations hold. In particular, if dim X = 1, we can let Fh consist of all multiples of (the formal symbol) xb and define mzxh = xh+1 and Oft* = bxb~l. Again the commutation relations 1.6 hold. dim V
Let A(V) = 9 A*(V) be the exterior algebra of V. Each v e V defines an operation of exterior multiplication, e(v): A*(V) —A* + 1 (V)by e(v)w = vh.u>.
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.
[1.2]
these span Hom(X,V). Similarly, we identify Hom( V,X) with X® V*, etc. When applied to rank-one elements, we obtain the commutator
[1.7]
Also, each u* e V* defines a (super) derivation of A(V), which we denote by <(u*). Thus i(u*): A*(V) - • A* _1 (V) and is determined by i(u*)(o>i ACD2) = i(u*)o>iAo)2 + (-1)** "'
3128
Physics: Ne'eman and Sternberg i(u*)v =
Proc. Natl. Acad. Set. USA 77 (1980)
v e V = A»(V)
[1.8b]
i(u*)c = 0, ce C = A°(V).
The remaining bracket relations are straightforward to verify. In particular, for each integer it, the space
[1.8c]
It is easy to check that e("i)e(v2) + e(v2)e(vi) = 0,
A«(V)®St(X) +
i(uMui) + l(uM"i) = 0,
[1.11]
gives a finite dimensional irreducible representation of sl(V/X) where n = dim V. Here S'(X) is taken to be |0) if j < 0. [Thus, for k = 1 we obtain the basic defining representation of sl(V/X).] In the case dim X = 1, we can, for each be C, consider the representation on the space
and e(v)i(u*) + i(u*)e(v) =
•I 3 -
AKV)9Sk-HX) + . . . + A"(V)®S*-"(X)
A°(V)®Ffc + . . . + A n (V)®F fc -»,
[1.12]
which has the same dimension 2" as A(V). For the case dim V = 2, these are the representations of Scheunert et al. (1) mentioned above. (For dim X > 1 these spaces will be infinite dimensional.) 2. In refs. 4 and 5, the superalgebra s/(2/l) was used as an internal supersymmetry of the standard-model unified weakelectromagnetic gauge. That is, dim V = 2, and dim X = 1. The operators U and I3 were chosen as
[1.9a]
A®Ix + Iv®D,
in which A denotes the induced derivation on A(V) and lx is the identity map on S(X), with similar notations for D and Iv We set, following 1.3,
l
k
0
0 0 0
-V*
0 0
0
[1.9b]
Id- a-***
and the electric charge operator was taken to be I3 + lkU. For reasons of space, we shall write these matrices and similar ones in what follows as
[1.9c]
Because e(i
•16.1-1(2 :i + P and similarly
V = diag(l,l,|2) and I 3 = diag(V2,-V2|0) The value b = % then corresponded to the (Cabibbo-rotated) nonstrange quarks with eigenvalues and particle assignments (up to statistics, see Discussion)
c "Me m - •••'«
U
h
>§, a-'t a
• t a-'fe. a
+p Also,
A? %
0 uT
A1 %
_A*_ -%
%
v2 -v 2 "I 73 <*Z1/3
0 df'3
The value b = 1 corresponds to the anti-lepton assignments 0
A1
Af
[1.10b]
17
h
(eWsDt) • (((n'lSm.) = (i'®u*)®((|Jx>/x + *®S)
2
0 ML
11
Af 1
1
0
V22 --lVk 2 0 V (et) R (•*,)« Mfc
[2.4]
The choices b = 1/3 and b = 0 correspond respectively to the charge-conjugate multiplets (antiquaries and leptons). The representations b = 1,0 are reducible, with A 0 + A 1 and A 0 respectively an invariant subspace. We shall discuss this point later. The same sort of representations b = %, 1, V3,0 will describe any other "generation" of quarks and leptons. 3. We now point out that we can combine the quarks and leptons into a single, eight-dimensional representation of s/(3/l). Indeed, choose
and (i(u*)9mx) • (e(v)
-I, :UL n
335
U = diag(V3>%,%|%)
[3.1]
/ 3 = diag(1/2,-I/2,0|0)
[3.2]
Physics: Ne'eman and Sternberg
Proc. Natl. Acad. Set. USA 77 (1980)
For b = %we get the eigenvalues U
A^ 2
i3
o (•a)i.
A1 1
0
A2 V3
Va
y2 _- v 2 o _ o
y2
-y2
o
uff3
dl"3
d5"3 [3.3]
1
%
(eE)« ("°L)R
H V ("°«k
A3 -%
and the representation restricts under s/(2/l) as indicated. For b = lk we get the conjugate eigenvalues U /3
Af %
o (<*« I/3 )L
Aj -%
-%
Vz (<*Z1/3)R
0
-%
A2 -1
-1
A| -2
- ' A o _ o _ y2 - y 2 o («i / a ) B
A
(UT)L
«°L
el
el
[3.4] We observe that, because b is not an integer, these representations are irreducible. Turning now to a unification of asthenodynamic (weakelectromagnetic) charges with color SV(S)} we choose in s/(5/l) the generators t7 = diag(0,0,141/3,1/3|l)
[3.5]
Is = diag^-yjACOlO)
[3.6]
and and the representation b = 2. This is a 32-dimensional (reducible—see Dicussion) representation with the assignments given in 3.7 in the Appendix (the lower index on the left of a quark denotes color) with the additional quantum number N corresponding to the generator N = diag(l,I,-%,-%,-%|0).
[3.8]
To deal with more than one generation we turn to sl(6/l), b = 5/2. This representation is irreducible because b is not an integer. We choose the generators U = diag(-% -ys.%,%5,%5,%51%) h = diag(1/2,-%,0,0,0,0|0) tf-diag(l,U,-l,-l,-l|0) Y = diag(l,l,-2,0,0,0|0). The associated eigenvalues are given in 3.10 in die Appendix (the lower index 3 denotes color multiplicity), In si(7/l) we would choose 1
U = diag(- A,-y3,%,%,0,0,0|%), b = 3.
N. Note that in their su(5) unification, Georgi and Clashow (6) use N for their weak hypercharge. This provides for an alternative identification of die quarks and leptons in 3.7. These two are the only possible choices of U (up to sign) that give the observed spectrum. Notice that our sl(5/l) representation 3.7 with b = 2 is not irreducible. The subspace A = A 0 + A1 + A 2 is invariant but has no invariant complement. The same is true (e.g., 2.4) as we saw of the subspace A 0 + A1 in b = 1 and of A 0 in b = 0 of su(2/l): Ne'eman's leptons (P°, el, el) were assigned to such a three-dimensional invariant subspace. Indeed, the fact that the fourth state (fg here) drops out for integer-charge multiplets (while fractional-charge ones require all four states) is in itself a remarkable "prediction" of su(2/l). These peculiar reducibility phenomena are an illustration of the fact that representations of simple superalgebras need not be completely reducible. From this point of view, the particles in A3, A4, and A 5 of 3.7 seem "more fundamental," in the sense that starting from the lowest V state el, for example, we can reach all other states, but not when starting with the highest U, i.e., from (eiOz.. The above irreducible subspace A is 16-dimensipnal and if one adopts the Georgi-Glashow choice of 17 - * N, A yields the representation used by Taylor (7). As in *u(2/l), the even-odd gradings J: in A* are correlated with the chiralities throughout our s/(3/l) and sf (5/1). This is nontrivial, because there is no preassigned correlation between quarks and leptons. The correlation subsists in 3.7 with either choice of U. The alternating statistics corresponding to even-odd kinAk imply "ghost" status for one-half of all states in these representations. We shall not dwell on this issue here, because our results are independent of the interpretation of these ghost states—whether as Faddeev-Popov ghosts (4, 8) or some other mechanism (5). It should be noted, however, that the doubling of matter fields in 3.7 and 3.10 provides for a natural realization of Ne'eman's odd (discrete) morphism f used in refs. 4 and 8 to double the dimensionalities of su(2/l) representations. These representations with sequential quark-lepton structure exist for any s/(n/l), with dimensionality 2". The number of generations is 2\ A = n — 5, and the quantum numbers are given by
r^diagff^L.^-.f-U
1
An earlier attempt to unify su(2/l) with color by D. Fairlie, Y. Ne'eman, and Dj. Sijacki in May 1979 failed to provide a general proof of the existence of such representations and their construction. However, these authors proved for *u(5/l) and other candidates that no representations exist with only leptons and quarks (and no antiparticles). The 16-dimensiona] A representation with 0 -» N of 3,8 was noted and abandoned when it proved impossible to retain the unrenormalized value ainVia = V< of «i(2/l), in the as-yet-unproved conjecture of a conserved angle.
336
,
*U4 + A 4 + .d [4 +4.}* time
[3.11]
The same eigenvalues as for si(6/l) occur, but with double the multiplicity. Thus each 2 occurs four times (once in A0, twice in A1, and once in A2), etc., corresponding to four generations of quarks and leptons. 4. Discussions. The maximal even Lie subalgebra of s/(5/I) is sl(5) X C or so(5) X u(l) when reducing to the unitary subalgebra. Aside from U, 13, and the two quantum numbers of the Cartan subalgebra of su (3), we dispose of a fifth generator
3129
[3.12]
\3(4 + 4.))3 times| 4 + A)) 4 + A 2,-y2,(0)(,+3) am., | 0)) hb == diag((y For s/(8/l), A. = 3 and the 8 generations are related by a (seriality) su(3X, subalgebra, st(S/l) D su(8) X «(1), su(8) D SU(2)L X "1(3)* X su(3)„,ior.
[3.13]
Finally, we might speculate that V © X constitutes the primitive field (9). For s/(7/l) this would consist of an isodoublet (ai /3 , alw), two seriality isosinglets a)l% (a = 1, 2) and three color isosinglets G#I( (< = 1,2,3), all fermions, and a boson (ghost) /3|/3. In s//.8/l) the charges involve multiples of e/21: a¥.
«Z 5/7 .
o#;
mdffiT.
3130
Physics: Ne'eman and Sternberg
Proc. Natl. Acad. Set. USA
77(1980)
Appendix A1
A°
u h N
2
1
o
y2
0
1
h N
1
:
R)R
0 0 2
V3
ML
l«P
%
% 0
-v2
(elh
I*R)L U
1
0
0
-%
-%
2UT
3U 2 /
v3
V3
v2
y3 -y2
y2 y3
V3
2«t
A2
/3
v3
V3
-v2
%
-y2
0
0
0
V3
3«?. / 3
i<*Z 1/3
2 <*Z
1/3
3 <*Z
(i<*ii1/3k
1/3
(3
W3)L
A3
0
0
0
N 1 3
idj '
2<*R
WZ ) R
3d; '
A 0 (I«P)L
U h N Y
A° 2 0 0 0
(3
) ,R
<
0
%
%
-1 V2
-y2
-1
-1
A -2 0 0
»l
el
«R
fauilf*)!.
-1
1 -y2 1 1
2 0 1 2
-y2
(2«£/3)R
(3«! /3 )R
0 0 - 2 *°R
[3.7
A2
A1 1 l k 1 1
-y2 -y3 5
-7s 0
<*V)L
v2
1/3
4
U
h N
(**Z
-y3 -y2 -y3
-V3
-V3
1/3
1 3
1/3
-v3 v2
-y3 y2 -y3
-%
U
h
%
%
0 -1 0
0 -1 0
IT
0
0 -1 0
0 2 2
1
1
y2
-y2
2 -1
2 -1
(y 3 ) 3 (y 2 ) 3 o3
o3
13
i3
(D 3
(%)3 O3 -23 O3
(%)3
(-y 2 )3 03 -2s
A3 U h N Y
0 0 3 0
(~%)3 O3 13 23
(%)3 (V3)3 (V2)3 (-V 2 )a; i3 i3 -is -13
(-V 3 )3 (V2)3 -13 I3
(-y 2 ) 3 -is 13
(%)3 0 O3 0 -Is -3 -2a 0 A5 2 "
A4 U
h N Y
("%)3 03 23 03
("%)3 0s O3 23
("V3)3 (V2)3 O3 -13
(-%)3 <-y 2 )3 03 -i3
-1
% -2 1
-1 -y 2 -2 1
0 0 -2 -2
( - V3)3 03 is 03
0 1 2
-1
A6 -2
y2
-y 2
0
-1 -1
-1 -1
0 0
-1
[3.10
5. (0) J. Thierry-Mieg (personal communication) has informed us that a construction similar to our 1.12 has been suggested by P. H. Dondi and P. D. Jarvis. Apparently f or n = 5 they take b = 4 with the N £ su(5) as weak hypercharge as in ref. 7. We have not seen their article, however. Constructions similar to 1.12 were used in "Extended Supergravity" by M. Gell-Mann and Y. Ne'eman (1976, unpublished) as quoted by Freeman (10).
Note Added in Proof. (<) For the cases of reducible representations—i.e, where b = (n — l)/2 is integral, it might be desirable to replace the reducible representation by the direct sum of the irreducible subrepresentation and the corresponding quotient representation. This would have the effect of restoring the symmetry between particles and antiparticles. (ii) The choice b = (n — l)/2 can be independently justified by the requirement that there exist a nondegenerate pairing between A 1 and \"~k. The eigenvalues of diag(l 1/n) would have to be opposite on these two spaces, and this easily implies that b = (n — l)/2. This fact was pointed out to us by O. Gabber. In fact, because A " ® F _ 1 has a natural trivialization under the even part of s/(n/l) we see that the natural multiplication of A.k&Fh~' X \n-kQpb-fa-k) m t o ^ n g p 2 i - n g j v e s a bjUnear pairing when b = (n — l)/2. One can check that this pairing is superinvariant under all of s/(n/l). (iii) If we assume a U of the form given above, and the observed charges, one can deduce that either there is no color symmetry or that the color group is u(3). Details will be presented elsewhere, (to) The Georgi-Glashow choice of U can also be made for arbitrary n ^
This work was supported by the Wolfson Chair Extraordinary in Theoretical Physics at Tel Aviv University, by the United States-Israel Binational Science Foundation, by the United States Department of Energy (contracts DE/AS0278ER04742 and EY-76-S-05-3992), and by the Israel National Academy of Sciences and Humanities. Scheunert, M., Nahm, W. & Bittenberg, V. (1977) J. Math. Phys. 18,155-162.
337
Physics: Ne'eman and Sternberg
Proc. Natl. Acad. Sci. USA 77 (1980) 6.
3131
Georgi, H. & Glashow, S. L. (1974) Phys. Rev. Lett. 32, 438440. 7. Taylor, J. G. (1979) Phys. Rev. Lett. 43,824-826. 8. Ne'eman, Y. A Thieny-Mieg, J. (1980) Proc. Natl. Acad. Sci. USA TJ, 720-723. 9. Ne'eman, Y. (1979) Phys. Lett. B 82,69-70. 10. Freeman, D. Z. (1977) Phys. Rev. Lett. 38,105-108.
2.
Kac, V. G. (1978) in Differential Geometry Methods in Mathematical Physics II, 1977, Lecture Notes in Mathematics, No. 676 (Springer, Berlin), pp. 597-626. 3. Corwin, L., Ne'eman, Y. A Sternberg, S. (1975) Rev. Mod. Phys. 47,573-604. 4. Ne'eman, Y. (1979) Phys. Lett. B 81,190-194. 5. Fairlie, D. B. (1979) Phys. Lett. B 82,97-100.
338
460 SEQUENTIAL INTERNAL SUPERSYMMETRY Y. Ne eman and S. Sternberg Tel Aviv University, Tel Aviv, Israel ABSTRACT The supergroups SU(2/1), SU(5/1) and SU(5=k/l) provide fitting classifications, unifying weak-electromagnetic, color, and sequential flavor respectively. SU(2/1): ELECTROWEAK DYNAMICS Experience with old-fashioned SUC3) has shown that a good classification group should be studied, even if neither its dynamics nor the quantum-statistics of its representations are understood at the time. It has recently been suggestedl>2 that the simple supergroup SU(2/1)=>SU(2)xU(l), applied as an internal supergauge, provide a highly-restricted Salam-Weinberg model. We have proved elsewhere-* that the standard-model matrices, when applied to leptons and quarks sequentially, obey a condition of supertracelessness when left and right chiral fields are assigned to different gradings. This is because SU(2) is traceless, and acts only within the left-chiral sector; and because electric charges Q have to match in both sectors. The U(l) is then a supertraceless matrix U, since it is a linear combination, Q = I3+ JjU. The defining representation is (v£»eL/fe^lj^) with the quantum numbers (-*£,%) in the SNR notation^ (JjUj,^,i3max) characterizing the eigenvalues of the highest I3 state. To commute with the Lorentz group, this is a left-chiral spin-half multiplet. The boson [egl^ has the internal quantum numbers of eg. Several suggestions have been made with respect to the "wrong" statistics and chirality „ , a) adding dimensions to space-time , preferably chiral-fermionic b) treating the entire system as a classification with no statistics implications. For example, a study of anomaly restriction? relating constituents and their composites^ has shown that the SU(n/m) groups fulfill these conditions c) we have shown? that the renormalization-ghosts include an effective field, with"wrong" statistics, for every matter field in the theory. Our [ ] would be the parity-transforms of these ghosts. One of us (Y.N.) may have apreference for this interpretation, but the results of this article do not depend on that choice, except where stated. Note that one might be tempted to forego the complications of a supergroup, and replace it by SU(3) with (v£,e£, (eg) J) and |AJl|=2 generators and currents, and the same 9 ^ 3 0 ° . However, SU(2/1) scores highly in providing us with the 4-dimensional fundamental representation,8 ( u 2£3,d-l{3/[d-lp] L , [ U 2 R 3] L ) , (1/6,1/2) in SNR notation. Indeed, the 4^ representation is characterized^ by + also Center for Particle Theory, U. of Texas, Austin * Wolfson Chair Extraordinary of Th. Physics + supported in part by U.S.-Israel Binational Science Foundation " " " " D.O.E.contract EY-76-S-05-3992 **also Dept. of Math., Harvard University 0094-243X/81/680460-03$1.50 Copyright 1981 American Institute of Physics
339
461 b= H um±A + hi b= 2/3,1/3 respectively for quarks and antiquarks, and b= 0,1 respectively for leptons and antileptons. In the system of representations of SU(n/m) that we use^, integer values for b imply reducibility. SU(2/1) indeed predicts the decoupling of v2 due to the integer charges of the leptons! SU(2/1) also predicts the 1= %, U= +1 assignments of the Goldstone-Higgs multiplet1>2. In the ghost interpretation these scalar fields make a perfect 8_, together with the Faddeev-Popov ghosts of SU(2) x U(l). Indeed, this restriction could be taken as a definition of the theory. In that picture, the (J)^ coupling is determined and the mass of the Higgs field ^ 250 GeV. Treating the full SU(2/1) as a local gauge makes the basic set of leptons or quarks have masses of the order of the W mass. This looked surprising at the time but it is precisely the picture of "technicolor" or "hypercolor" etc.1*). Local SU(2/1) has one main difficulty, relating to the Killing metric g^g = str (MAMg) which is non-positive-definite for the 8th direction g„„ = str (M,,)2 <0. This precludes the straightforward construction of a second-quantized theory. Either we break the symmetry (by using the metric of SU(3), with a certain justification in the ghost picture^) or we treat SU(2/1)/SU(2) x U(l) globally only (this also cancels the heavy fermion masses). SU(5/1): COLOR ADDED In SU(5/1), taking for I,= diag ((^,-^,0,0,0/0)) there are two choices of U. Previous authors11»12 adopted UcSU(5)csu(5/l) i.e. the Georgi-Glashow choice J^N = diag h ( U , 1, -2/3, -2/3, -2/3 / 0)) (1) but this clashes with SU(2/l)cSU(5/l) . We have selected^ hU = diag h ((0, 0, 1/3, 1/3, 1/3 / 1)) (2) which have SU(5/l)^SU(3)color x SU(2/1). The b=2 representation is 32 dimensional and contains precisely two generations (or a generation and its ghosts). It is reducible, and one may select 1 1 > 1 2 \b_ or 31_ (dropping the V R ) . SU(5+k/l): SERIALITY ADDED The representations of SU(n/m) have recently been described ' . They follow the same Clifford-algebra pattern I " 0 " >ftU\ "0" > ft B I " 0 " > » | I " 0 " > « • • . which we used in exploring extended supergravity and discovering the N=8 limit. Here too, we have a restriction on the range of U going from +2 to -2, with +4/3, +1, ±2/3, +1/3 as the only allowed intermediate values. We have proved1^ that these values restrict the color group uniquely to SU(3), or no color at all as an alternative. Our representations b= 2l—, with Umax = 2, for
W • dug H ((&, ^ , 0 ? t t a e s . {££>, i 3 •
tlM ./^»
<» w
are 2->"Hc dimensional. They contain 2 k + 1 generations, or 2 k generations plus their ghosts, depending upon the interpretation (see discussion of e in ref°). Thus k=2 (or k=3) predicts 8 generations. We have provided the exact construction elsewhere 9,3# Half of the generations have inverted physical chiralities.
340
462 Note that for k>l, we now have gmj X ) , so that a quantized local supergauge theory becomes possible. The defining representation describes the primitive constitutents of the theory. For k=2 they have charges, Q = diag ((1/3,-2/3,1/3,1/3,0,0,0/ 1/3)) (5) REFERENCES 1. Y. Ne'eman, Phys. Lett. 81B, 190 (1979). 2. D. B. Fairlie, Phys. Lett. 82B, 97 (1979). 3. Y. Ne'eman, Proc. of Europhysics Conference on Grand Unification Theories and Supergravity (Erice 1980), to be pub. A. M. Scheunert, W. Nahm and V. Rittenberg, J. Math. Phys. ]£, 155 (1977) . 5. E. J. Squires, Phys. Lett. 82B, 395 (1979). J. G. Taylor, Phys. Lett. 83B, 331 (1979). P. H. Dondi and P. D. Jarvis, Phys. Lett. 84B, 75 (1979). 6. T. Banks, S. Yankielowlcz and A. Schwimmer, report WIS-80/19/5-Ph (to be pub.). 7. Y. Ne'eman and J. Thierry-Mieg, PNAS (USA) 77_, 720 (1980). 8. Y. Ne'eman and J. Thierry-Mieg, Proc. (Salamanca, 1979) Int.Conf. on Diff. Geom. Methods in Phys., to be pub. by Springer Verlag (Math. Series) (also rep. TAUP 727-79). 9. Y. Ne'eman and S. Sternberg, PNAS (USA) T]_, 3127 (1980). 10. See papers by M.A.Beg, H. R. Pagels etc. in the Dynamical Symmetry Breaking session of these Proceedings. 11. P. H. Dondi and P. D. Jarvis, Z, Physik C, Particles and Fields 4_, 201 (1980). 12. J. G. Taylor, Phys. Rev. Lett. 43_, 826 (1979). 13. I. Bars and Balantekin, Proc. 9th Int. Conf. on Applications of Group Theory to Physics (Cocoyoc, Mexico 1980), to be pub. 14. Y. Ne'eman and S. Sternberg, ibid, (also report TAUP 133-80).
341
SU(7/1) Internal Superuniflcation: A Renormalizable SU(7)xU(l) with Factorizable Pomeron
Yuval Ne'eman + Tel Aviv University, Tel Aviv, Israel* and University of Texas, Austin, Texas, USA
Shlomo Sternberg Harvard University Cambridge, Mass, USA and Jean Th1erry-M1eg 6.A.R. Observatoire de Meudon, France
Abstract
We present a model based on the supergroup SU(7/1).
I t predicts
8 generations of quarks and leptons, one half of which should have Inverted chlrallties.
I t is equivalent to a reducible anomaly-free representation
1n an SU(7) x U(l) gauge unification theory (GUT).
*
Supported in part by the US-Israel B1nat1onal Science Foundation
+ Supported 1n part by the US DOE, Grant DE-AS05-76ER03992 # Wolfson Chair Extraordinary of Theoretical Physics
342
1.
An Internal Supergauge. The Welnberg-Salam Sl)(2)jX U(l) v model can be embedded1' ' 1n the simple
supergroup
SU(2/1), a highly constraining algebraic structure ' .
Quarks f i t ' '
the fundamental representation £, containing precisely an Isodoublet and two 1sos1nglets with fractional charges fixed by a parameter y
max
(s 4/
3
f o r qtu
R / 3 ' " P • d [ 1 / 3 ' ^K^'
Cablbbo-rotated 2— quark. y
For integer y
»o yields the uncoupled lepton singlet
y
max *
_1 t r i
Plet
*-tvi
» e [ I eR^*
where d
L/R ^ P " 1 " * * 8
the £
the
decomposes Into 2+1., and
vS and the
We f o l l o w t n e
ansatz of ref ' and r e f 6 * ,
1n Interpreting the ghost-statlsties carried by one half the states of an Irreducible representation.
This amounts to adjoining the
e° statistics-
conjugate irrep ' ' , i.e. we use two representations [n_ * n'] , n.' • e°[nj with identical quantum numbers but opposite statistics (and thus different Class ' ) .
After assigning all physical states 1n n_ + n.'
to leptons or quarks,
the wrong-statistics residual states are assigned to a system of generalized Fadde'ev-Popov ghosts (C ( x ) ) . transformation with parameter
These are fields generated by applying a gauge o • AC (x) , A a constant antlcommuting
element, then factorizing out the where
A : s* a (x) = [C(x) , • ( x ) ] a = $ a (x)
* a (x)
are the physical and $ a (x) the ghost states 1n n. + n/ . 8 91 We have extended the superuniflcation to include colour * ' . In
particular, the supergroup SU(7/1) offers a very attractive model which we recently proved
' to be quantum anomaly-free for the locally gauged even
subgroup SU(7) x U ( l ) . Our solution to the statistics Issue thus assigns the internal supersymmetry to the physics of the quantum Lagrangian.
We have since been
able to display the Extended BRS (Curc1-Ferrar1) algebra of the supergroup Itself
' when gauged, and incorporating the new ghost fields.
There are
s t i l l several unresolved Issues 1n the dynamics of an internal (quantum) CO
(0
343
supergauge.
I t I s , however, worthwhile to remember the lessons of Unitary
Symmetry (SU(3)), where classification was correct and useful, even thc-c.the dynamics were unclear (and may s t i l l be so)
-
and where there also
were problems with the statistics (of quarks, before the introduction of colour).
In this spirit, we have pursued the application of Internal
supersymmetry as a classification scheme, present dynamical uncertainties notwithstanding.
2.
SU(2/1) Predictions SU(2/1) decreases the number of arbitrary choices 1n SU(2)j x U ( l ) v
from 10
[3 quark multiplets, 2 lepton multiplets excluding v£,
and Y
for the Goldstone-Higgs $ multiplet,
ey and the couplings
-u 2 * 2 ] to
and the selection of SU(2/1) i t s e l f .
4[y n a x (q). ymxU)
, -u 2 ]
\y
I
and
Treated classically 1 , 4 , 6 *, SU(2/1) yields a Weinberg angle ew « 30° , x • *j g the Higgs Field. for
m..
values
for the
f4
coupling, and a mass m ^ 250 GeV
for
At the time, this was considered an abnormally large value
More recently, a number of dynamical models have yielded just such '.
These are models attempting to explain the emergence of masses
of the order of in. (<\-100 GeV), in the context of unifying theories, where spontaneous breakdown occurs at
10
GeV (the "mass-hierarchy" problem).
All such models involve additional strong Interactions at Since
100-300 GeV.
SU(2/1) presents a simple electroweak unification ( i . e . at m. mass
scale) 1t seems natural that i t should realize as a supersymmetry the dynamics of ref
'.
I t Is worth noting that although SU(3) supposedly have replaced SU(2/1)
with
3_
with 2(vJ,e[ / e°(e^)}, only
could
SU(2/1)
has an Irreducible 4_. Chlral quarks thus definitely select the supergroup.
344
3.
SU(7/1) as Super GUT The relevant internal super-GUT is provided
' by
SU(7/1)» S-J(2/l) E 1 e c t r o w e a k x SU(3) Co1flur x U(l) x U ( 2 ) n a y o u r The e°-paired matter multlplet 1s zj + £ 7 . I t contains 8 sequential generations of where
rP y s
3 x 4?h*s 9 1 x 3? hjS a 1 x l. p h y s ,
represents the physical states of an e°-paired SU(2/1)
representation
n_ 9 n.' .
One half of these g\
generations (the heavier,
presumably) have inverted chiralities ' . As a simple subgroup, SU(2/1)c. SU(7/1) should preserve some form of Appelquist-Carazonne decoupling
, even after being embedded in the
super-GUT. The "constituent" preon octet has quantum numbers
Q » diag. (\, - § ; J . \ ; 0 , 0 . 0/|) 1 I 3 - d i a g . ^ .?
; 0
, o ; o , 0 . 0/0)
J
These preons are heuristic computational aids, (like pre-colour quarks) and cannot be considered as fundamental fields (a role played here by quarks and leptons). Note that the quantum numbers in eq. (1) are not the Georgi-Glashow SU(5) assignments, used by Dondi-Jarvis and by Taylor (see bibliography in ref^), and in which Q, Y e su(5) C su(7). l2)
those of the Farh1-Sussk1nd
involve the U(l) in SU(7/1) ^
Neither do they coincide with
SU(7), where Q, Y e s u ( 7 ) . SU(7) x U(l).
Our Q, Y
To the extent that the quantum
dynamical role of the supergroup is not understood, the model can at the very least be regarded as a specific representation structure in an "Orthodox" non-simple
SU(7) x U(l) GUT.
We use the class I representation * ' 2 = 128_. We pick
y ^ « 2,
b • 3, with dimension
and the eigenvalues of
345
Y are restricted
to the values
[±2,±4/3,±l,±2/3,±l/3,0],
i . e . all values from 2 tc -2
Ay * 1/3 Intervals, but excluding ±5/3.
The representation spans the 2'
state vectors of otha* Grassmann manifold
,
by the set of (£) antisymmetrized products
_ ? .k. Each £* is soanneci k=0 ! A ( r i , , i 2 , . . . i . ]) > of k
"preon" basis vectors from amongst the 1 • 1...7
of !, \ , > e A I.
eigenvalues w of the generators of the Cartan subalgebra of linear quantum numbers -
i1.i2,...ik
The
su(7)
-
the
are given by 7 ^
max
1
u
m
k 1j
u
(2)
i
where <^ are the preon level eigenvalues as 1n (1), W • diag. Uij.b^f'Un/
J
w_ )
J3)
There is a correlation between chirality in SU(2/1) or SU(5+k/l), k=0, but i t 1s lost for M O .
To preserve our c°-pa1r1ng resolution of the
lorentz-invarlance and statistics Issues, we preserve * ' the chiral nature Physical fields 1n A+ « T
of the grading in SU(7/1). and those 1n A" = LA 2 r + 1
4.
are left-chlral
(r » 0 , 1 , . . . 3 ) .
Renormallzability and a Factorlzable Pomeron We have proved 1n ref 10 ' that 1n this representation there 1s a
highly non-trivial cancellation of the SU(7) x U(l) anomalies.
Indeed, they have to vanish, in order not to spoil the
renormallzability of the gauge theory 14 " 16 '. of
Adler Bell-Jackiw
The respective gauge-fields
U(l) and SU(7) are denoted B and V. The triangular diagrams are
(BBB), (BBV), (BVV) and (VVV), but the first two have no anomalies. B gauges the U(l) generated by
S « - 1/6 diag. (1,1;1,1;1,1,1/7).
346
Its
quantum numbers grade the We observe that
A
in j 2 £ or
128',
with
S(Ak) - S(Ak_1) = 1 . A0 is an
Y = -4/7 S + (operator in su(7)), and since
SU(7) scalar, we have S eigenvalues
s = k - 7/2,
going from - 7/2 to 7/2
in unit steps. The value of the (BVV) anomaly is obtained by summing the product of the B charges
sk = - s 7 _ k by the summed squares
C2(su(7))
charges of the A , taken as Irreducible representations of
of the V
SU(7)
with
the appropriate chfral-graded signatures,
(4)
< W - jm=on s2mC2<*2m>'ABVV- m=o I W l The value o f the
c
3(
2^ ^
\J
(VVV) anomaly i s s i m i l a r l y obtained by summing t h e Ak,
— o r d e r Casimlr i n v a r i a n t s f o r the
Ajw ' I
C
A2m
).
A
C 3 (A 2m+1 )
vw" I
m=o
w i t h c h l r a l graded s i g n a t u r e s ,
(5)
m=o
Calculation shows that Agyy * - Agyy - 0
,
AyVV • -Ayyy ' 0
16)
These cancellations are due to a generalization of the cancellation between 5*
and 10 in
SU(5) c S0(10).
I t will be true 17 * for all odd n>5
in
SU(n)o S0(2n). The SU(7) (128 + 128'}
h
gauge 1s asymptotically free in the presence of .
Using the
equation 18 *, we have
8(9) "
l
(c|dj -
g3
approximation to the renormallzation group
2n),
l i C2(J - 1) - J j
i
C2(Ak.J - H)
16ir'
n3 = g •
1 3
*•
n3 g
24/ (7)
(0
347
It Is Interesting that SU(7/1)
picks the maximal numbers of flavours
(16 - 2 x 8) that does not destroy asymptotic freedom for colour SU(3).
This
H
1s also the unique value ("critical QCD ) reproducing the observed factorlzable Pomeranchuk Regge trajectory 1n the analytical S-matr1x
5.
'.
The Fermi on Spectrum We denote the 4 conventional generations of quarks and leptons by
u*, d*. v*, e* (t » 1...4).
To these we have to add 4 predicted chlral-
Inverted sets - with the same Internal quantum numbers. these by 1*. tf. J*. F*.
If we denote
ED has the Internal quantum numbers of
( e p £ , the left-handed positron; E^ * ( e p ^, (E^r/^ ej|, (E^)^ ~ e[
etc.
We are now working on the pattern of symmetry breaking.
References 1.
Y. Ne'e nan, Phys. Lett. 81B., 190 (1979).
2.
D. B. Fa1rl1e, Phys. Lett. 82B_, 97 (1979).
3.
V. G. Kac, Func. Analysis and Its Appls. 9_, 91 (1971); Comm. Math. Phys. 53_, 31 (1977). V. Rlttenberg, in Group Theoretical Methods 1n Physics (Tubingen 1977). P. Kramers and A. Rieckers eds., Lect. Notes in Phys. 79_, pp. 3-21, Springer Verlag, Berlin-Heidelberg (1978). L. Corwln, Y. Ne'eman and S. Sternberg, Rev. Mod. Phys. 47, 573 (1975). Y. Ne'eman, in Cosmology and Gravitation (Erice 1979). P. G. Bergmann and V. de Sabbata eds., Plenum Press, N.Y.-London, pp. 177-226 (1980).
4.
Y. Ne'eman and J. Th1erry-Mieg, 1n Differential Geometrical Methods in Mathematical Physics (Proc. Aix-Salamanca 19/9). P. L. Garcia et al, eds., Springer Verlag, Lect. Notes on Maths. 836, Berlin-Ke1delberg-N.Y., pp. 318-348 (1980).
348
5.
M. Scheunert, W. Nahm and V. Rlttenberg, J. Math. Phys. 18, 155 (1977).
6.
Y. Ne'eman and J. Th1erry-M1eg, Proc. Nat. Acad. Sc1. (USA), 77.. 720
7.
A. B. Balantekln and I. Bars, J. Math. Phys. 22_ 1810 (1981), and Yale preprint YTP 80-06. Class I Irreps have a bosonlc state with highest weight y m a x , Class II a ferm1on1c.
8.
Y. Ne'eman and S. Sternberg, Proc. Nat. Acad. Sc1. 77., 3127 (1980).
9.
Y. Ne'eman and S. Sternberg, 1n High Energy Physics 1980, XXth Intern. Conf. (Madison), L. Durand and L. G. Pomdrom e d s . , Amer. Inst, of Phys. Conf. Proc. 68, (P. and F. Subs, 22), pp. 460-462, NY (1981).
10.
Y. Ne'eman and J. Thierry-M1eg, Phys. Letter (1981).
11.
J. Th1erry-M1eg and Y. Ne'eman, Tel Aviv University report TAUP 140-81.
12.
A. Carter and H. Pagels, Phys. Rev. Letters 43, 1845 (1979). E. Farhi and L. Susskind, Phys. Rev. D20, 3404 (1979). C. Wetterich, Phys. Lett. 104B. 269 (1981).
13.
T. Appelqulst and J. Carrazone, Phys. Rev. Oil., 2856 (1975).
14.
C. Bouchlat, J. Illopoulos and Ph. Meyer, Phys. Lett. 38B, 519 (1972).
15.
D. J. Gross and R. Jacklw, Phys. Rev. 06, 477 (1972).
16.
H. Georgi and S. L. Glashow, Phys. Rev. D6, 429 (1972).
17.
S. Okubo, J. Math. Phys. 18, 2382 (1977).
18.
T. P. Cheng, E. Elchten and L. F. L1, Phys. Rev. D9, 2259 (1974).
19.
A. R. White, CERN report Th. 3058 (XVI Rencontre de Moriond).
349
Volume 108B, number 6
PHYSICS LETTERS
4 February 1982
ANOMALY-FREE SEQUENTIAL SUPERUNIFICATION YuvalNE'EMAN 1 ' 2 Tel Aviv University, Tel-Aviv, Israel3 and University of Texas, Austin, Texas, USA and Jean TfflERRY-MIEG G.A.R., Observatoire de Meudon, France Received 5 November 1981
The supeigroup SU(7/1) defines a model with 8 generations of quarks and leptons. One half of these are chiral-inverted. The model represents an anomaly-free SU(7) X U(l) gauge unification theory (GUT).
1. SU(2/1). It has been suggested [1,2] that the simple supergroup [3] SU(2/1) be applied to constrain the non-simple SU(2)j X U(l)y "standard model" of electroweak interactions. The fundamental irreducible representation [4,5] of SU(2/1) is 4, containing oneI = \, SU(2)j isodoublet, and two7 = 0 isosinglets. The electric charges Q are fractional and depend on one free parameter >>max, the representation's highest eigenvalue for Y, the weak hypercharge, in Q = 7 3 + \ Y. F o r ^ m a x = 4/3 the toternal quantum numbers of the 4 fit q(u R / 3 /u£ / 3 , d £ 1 / 3 / d R 1 / 3 ) , where the oblique bars separate Hilbert subspaces with alternating gradings, and d j / R stands for the Cabibborotated 2nd quark. For integer charges, the 4 automatically reduces, decomposing into 1 + 3 . More specifically, 4 0 W = 0) -> \(y = 0) ®3(v m a x = - 1 ) , fitting precisely an uncoupled p R and£(!>"» e fj/ e R)' In applying a supergroup as an internal supersymmetry, Lorentz invariance and the spin-statistics correlation require the adoption of some appropriate ansatz: clearly, any representation of SU(2/1) should 1 2 3
Supported in part by the U.S.-Israel Binational Science Foundation. Supported in part by the U.S. DOE, Giant DE-AS0576ER03992. Wolfson Chair Extraordinary of Theoretical Physics.
0 031-9163/82/0000-0000/$ 02.75 © 1982 North-Holland
350
be assigned one single spin (here presumably J=\) and chirality; moreover, the even and odd gradings within the representation's Hilbert space correspond to Bose and Fermi statistics respectively. It has been suggested that chiral leptons and quarks involve the pairing of e°-conjugate representations [1,4,6], where Ne'eman's e° (renamed "tilda" in ref. [6]) is a Fermitype (discrete) morphism. e° commutes with the lie supergroup and inverts the gradings and statistics throughout the representation's Hilbert space. To represent quarks, we thus use 4 ©4', where 4 L is assigned [6] to Class I, and 4'R is its e° conjugate in Class II: 4(T8 L ( u R ) / u L , dL/
(1)
©4'(uR/cBR(uL), T3R(dL)/dR) . Similarly, we use 3 e 3' for the leptons. ThecT3 0/0 are / = 3 bosons (i.e., "ghosts", in the usual nomenclature) with the internal quantum numbers of ^ (and inverted chiralities). We have suggested [1] that e° be realized here by the SU(2) X U(l) BRS transformation s (coupled with parity ir},16 (^) = [x, ^].xfl(Jf)(a=l-3,8)isthe/=l,0, y = 0,/=0, Feynman/DeWitt/Faddeev—Popov ghost field, required by unitarity in the quantized Yang—Mills theory. This identification appears particularly fitting, since the SU(2) x U(l) fermionic ghosts x a should be assigned 399
Volume 108B, number 6
PHYSICS
iTTERS
4 February 1982
to a class I, / = 0 octet 8(
400
351
This is also the largest number of flavours (16 = 2 X 8) that does not destroy asymptotic freedom for colour.
Volume 108B, number 6
PHYSICS LETTERS
Grassmann manifold A = 2 j - = 0 A*. Each Ak is spanned by the set of (/,) antisymmetrized products \X.ilti2, ... ik])) of k " p r e o n " basis vectors from amongst the i = 1... 7 of | \ >e A Q . The eigenvalues w of the generators of the Cartan subalgebra of su(7) — t h e linear quantum numbers — are given by
r
t\,H,
-kTj m=l
'k + Yj CO; co*
4 February 1982
Table 1
A0: E R
_
A2- "k> ("f *Dk2> «k2> ER> DR. (°L. U L )R E l N2L, (NR)L'2, D*, U2L> (DR)2L, (UR)2L
(4)
(EL, N L ) R , U R , D R , (DL> U L ) R AS: V]f, .»*,
ff^tl
_
(5R)?.-4, ( E R ^ , a*, U*.
(DR)«
where to,- are the preon level eigenvalues as in (3), A7= (E R £ rV = diag(co 1 ,o; 2 ,...w 7 /Zy com I .
(5)
We have proved [13] that a chiral Z(2)-gradmg of the Hilbert spaces of quarks and leptons will yield supertraceless operators for SU(5/1) D SU(2)t X U(1) Y X SU(3) colour X U C l ^ . This is due to the traceless SU(2)j acting non-trivially on all left-chiral fields and only on these, to the same electrically charged fields appearing with both chiralities, and to all coloured spinor fields being electrically charged. As to N € SU(5), its eigenvalues are the same as those of Y, except for interchanging p R **• e R and u R •** d R . To the extent that i>R is not omitted, it will thus also be supertraceless. The same argument has since been made [14] to explain the inappearance of anomalies in plain GUTs, except that traces replace supertraces, and the nght-chiral particles are replaced by their left-chiral antiparticles. It is worth noting, however, that although SU(3) with 3{»'L, e^, (ejOi.} c o u l d supposedly replace SU(2/1) with 3{f-f, e-/
etc. In table 1 we give the content of Ap h y s = A L ( 1 2 8 ) ffi A R (128'). All quarks appear in 3 colours. 3. Anomalies. We turn to the SU(7) X U(l) Adler Bell—Jackiw anomalies. They have to vanish, in order not to spoil the renormalizability of the gauge theory [15—17]. The respective gauge-fields of U(l) and SU(7) are denoted B and V. The triangular diagrams are (BBB), (BBV), ( B W ) and ( V W ) , but the first two have no anomalies. B gauges the U(l) generated by S = - 1 / 6 diag(l, 1; 1, 1; 1, 1, 1/7). Its quantum numbers grade the A* in 128 or 128', with S(A*) - 5(A*- X ) = 1. We observe that Y = - 4 / 7 S + (operator in su(7)), and since A 0 is an SU(7) scalar, we have S eigenvalues s = k — 7/2, going from - 7 / 2 to 7/2 in unit steps. The value of the ( B W ) anomaly is obtained by summing the product of the B charges sk = — s 7 _ A by the summed squares C 2 (su(7)) of the V charges of the Ak, taken as irreducible representations of SU(7) (the values of C2 and C 3 are given in table 2 with the appropriate crural-graded signatures, ^BW
=
^ B W ( A R ) - -^BWC^L)
3 ^ W = £
S2m C2(A2m)
,
(6)
m=*0 3
'BW
S *2m + l C 2 ( A 2 - + 1 ) m=0
We find BW
X 5+5X
10+1 X
1 =0.
(7)
401
352
Volume 108B, number 6
PHYSICS LETTERS
4 February 1982
Table 2 k 0
1
2
3
s
D
A£
71 -
e
d
1
c2 c3
0
i
n-2
l(n -- 2 ) ( n - 3 )
0
i
n-A
(«-
- G)
G)
3(« - 2 ) ( n - -3) - ( n - 3)(n - 6 ) 2
3 ) ( n - 6) 2
The value of the (VW) anomaly is similarly obtained by summing 3rd order Casimir invariants for the Ak, with chiral graded signatures, A^y
= E
[4]
3
C 3 (A 2m ),
£C3(A2'"+1)>
A^yy=
m<=0
m=0
(8)
(5]
(9)
[6]
yielding
At, VW
'VW
n - 1
-1-2 + 3=0.
The above cancellations of ^ B W anc^ ^ v w a r e ^ u e to a generalization of the cancellation between 5* and 10 in SU(5) C SO(10). It will be true [19] for all odd n > 5 in SU(«) C SO(2«). The SU(7) gauge is asymptotically free in the presence of {128 + 128'} h y s . Using the g* approximation to the renormalization group equation, we have (Cf> = 2«),
[7] [8] [91
[10] [11] 247T2
(10) [12] [13]
References [1] Y. Ne'eman, Phys. Lett. 81B (1979) 190. [2] D.B. Fairlie, Phys. Lett. 82B (1979) 97. [3] V.G. Kac, Func. Analysis and its Appls. 9 (1971) 91; Comm. Math. Phys. 53 (1977) 31; V. Rittenberg, in: Group theoretical methods in physics (Tubingen, 1977), eds. P. Kramers and A. Rieckers, Lecture notes in physics, Vol. 79 (Springer Verlag, Berlin-Heidelberg, 1978), pp. 3-21; L. Corwin, Y. Ne'eman and S. Sternberg, Rev. Mod. Phys. 47 (1975) 573;
[14] [15] [16] [17] [18] [19]
402
353
0
(?)
n -2
-(« -
n
(•)
- a-
(•)
3
n -2
B*
AR
0
3
4)
1
1
0
-1
0
Y. Ne'eman, in. Cosmology and gravitation (Ence, 1979), eds. P.G. Bergmann and V. de Sabbata (Plenum Press, N.Y.-London, 1980), pp. 177-266. Y. Ne'eman and J. Thierry-Mieg, in: Differential geometrical methods in mathematical physics (Proc. AixSalamanca, 1979), eds. P.L. Garcia et al., Lecture notes on Mathematics, Vol. 836 (Springer Verlag, BerlinHeidelberg-N.Y. 1980) pp. 318-348. M. Scheunert, W. Nahm and V. Rittenberg, J. Math. Phys. 18 (1977) 155. A.B. Balantekin and I. Bars, J. Math. Phys. 22 (1981) 1810; Yale preprint YTP 80-06. Class I irreps have a bosomc state with highest weight ymax, Class II a fermionic. Y. Ne'eman and J. Thierry-Mieg, Proc. Nat. Acad. Sci. (USA) 77 (1980) 720. Y. Ne'eman and J. Thieny-Mieg, Phys. Rev. D, to be published. A. Carter and H. Pagels, Phys. Rev. Letters 43 (1979) 1845; E. Farhi and L. Susskind, Phys. Rev. D20 (1979) 3404; C. Wetterich, Phys. Lett. 104B (1981) 269. Y. Ne'eman and S. Sternberg, Proc. Nat. Acad. Sci. (USA) 77 (1980) 3127. Y. Ne'eman and S. Sternberg, in: High energy physics 1980, XXth Intern. Conf. (Madison), eds. L. Durand and L.G. Pomdrom (Amer. Inst, of Phys. Conf. Proc. Vol. 68, N.Y. (1981) pp. 460-462. T. Appelquist and J. Canazone, Phys. Rev. D l l (1975) 2856. Y. Ne'eman, in. Unification of the fundamental interactions, eds. S. Ferrara et al., (Plenum Press, N.Y.London (1980), pp. 89-100. R.N. Calm, Phys. Lett. 104B (1981) 282. C. Bouchiat, J. Djopoulos and Ph. Meyer, Phys. Lett. 38B (1972) 519. D.J. Gross and R. Jackiw, Phys. Rev. D6 (1972) 477. H. Georgi and S.L. Glashow, Phys. Rev. D6 (1972) 429. T.P. Cheng, E. Eichten and L.F. Li, Phys. Rev. D9 (1974) 2259. S. Okubo, J. Math. Phys. 18 (1977) 2382.
Proa Natl Acad. Sci, USA Vol. 79, pp. 7068-7072, November 1982 Physics
Exterior gauging of an internal supersymmetry and SU(2/1) quantum asthenodynamics (Lie supergroups/Crassmann algebra/Becchi-Rouet-Stora algebra/weak electromagnetic unification/Kalb-Ramond field) JEAN THIERRY-MIEC* AND YUVAL NE'EMAN** •Croupe d'Astrophysique Relativists, Centre National de la Recherche Scientifique, Observatoire de Meudon, 92190 France; and tlnstitut des Hautes Etudes Scientifiques, Bures-sur-Yvette, 91440 France Contributed by Yuval Ne'eman, July 27,1982 ABSTRACT A formally unitary Lagrangian model gauging an internal supersymmetry is proposed. The even subalgebra is gauged as a Yang-Mills theory, while the odd generators are gauged—according to Freedman's method—by skew tensor fields, equivalent dynamically to scalar Higgs fields. Chiral fermions are incorporated by following Townsend's construction and form irreducible supermultiplets graded by their helicity. The application to quantum asthenodynamics is discussed.
stone-Higgs ht fields, we find that in SU(2/1), these hf fit in the adjoint representation, together with the Faddeev-Popov ghosts a". At the classification level, thus, S l / ( 2 / l ) is rather promising; moreover, more recently it has been possible (7) to extend the method to further unification with the strong interactions' conjectured color SU(3). The specific selection of S l / ( 7 / 1) as the overall simple unifying group (8) produces an anomalyfree SU(T) renormalizable gauge theory with a prediction of eight generations ( = 16 flavors), half of them chiral-inverted, thus imposing uniquely the "critical" quantum chromodynamics (i.e., with a factorable Pomeranchuk trajectory). Elsewhere, the pion model also can be superunified (9) by using the superalgebra Q(3). The main difficulty feeing internal supersymmetry is to provide a correct interpretation of the odd generators of the superalgebra. Any irreducible representation is made up of Bose and Fermi particles. Thus, if the supergroup is assumed to commute with the Lorentz group, part of those particles violate the spin statistics theorem. We have assumed (1,3) that those ghost states should be interpreted as generalized Faddeev-Popov unitarity ghost states, generalizing to every multiplet the pattern of the scalar Higgs field plus Faddeev-Popov ghost sector. In particular, the Yang-Mills SU(2) x C/(l) vector bosons (W* ,Z°,A° m ) should be completed by a (pair of) isodoublet vector ghost states /3i, (and j3|J. In 1981, we constructed a closed irreducible extended Becchi-Rouet-Stora (BRS) algebra (10) for just those fields, however without being able to explain the role of the /SJ.OSjJ in unitarity. In the new framework presented in this paper, the BRS algebra will be modified, and the ^ ( / S j j will appear naturally as the vector ghosts of a Bose skew tensor field B|,„ gauging g_. In a way, we shall explicitly construct a realization of the operator e^ conjectured by us earlier (3). The price paid is that our new Bose algebra g is reducible with SU(2) x (7(1) as its maximal simple Lie subalgebra. However, two other difficulties of the model are resolved at the same time. The algebra is a Lie algebra and, therefore, should be normalized by traces rather than by supertraces, thus yielding a positive definite metric in the SU{2) x 1/(1) sector and confirming a Weinberg angle of sin 2 6 = Vi- The reducibility of the boson vector algebra is necessary because it implies the vanishing of the Killing metric in the Higgs sector, therefore allowing the use of the SC7(2) X U(l) positive norm in the Higgs and matter sectors. The second difficulty resolved by the present approach is the detailed understanding of the role of the matter ghost fields and of the grading of the physical spinor fields by their helicity. These results are explained in section 6. As yet we have no complete understanding of the symmetry
1. This paper presents a model-theory for the gauging of an internal simple supergroup t S. Its simple generator superalgebra g is exponentiated with parameters supplied by Grassmann's original exterior algebra ft of forms over space-time. The corresponding reducible Lie algebra g is given by the even part of the direct product g ® ft—i.e.,
g = (g+®n+)©(g_®n_).
[i.i]
(The + / - indices denote even/odd gradings in both superalgebras.) We have three aims in presenting these results: (i) to present a novel way in which a gauge supergroup may mix internal symmetry with external action over space-time, a point of physical interest; (ii) to display the resulting system of generalized connections and curvatures, a mathematical result that might have some applications in differential geometry, and to provide a nontrivial example of a Cartan integrable system; and (tit) to suggest an outline for a dynamical realization of a tentative S U (2/ 1) gauge symmetry describing a constrained asthenodynamic (weak electromagnetic unified) interaction (1-3) and to explain some of the more puzzling features of that model: the group metric, grading by chiralities, and identification of states with ghost statistics in the multiplets. 2. It was recently noted (1-3) that the number of independent arbitrary assumptions required by the algebraically nonsimple SU(2) x 1/(1) gauge theory (4, 5) of "unified" weak electromagnetic interactions is greatly reduced by the application of the simple supergroup SU(2/1) D SU(2) x 17(1) as a higher constraining internal symmetry. The five independent multiplets 2(VL. eZ) and l(ej;) selected for the leptons and 2(u^, d{;'"), 1(«B 3 ), l(d*'") for the quarks (in any one generation) in the Weinberg-Salam group assignments are replaced (up to statistics) by the two fundamental irreducible representations (3, 6)ofSC7(2/l): 3(vl,eZ/en) and the fractionally charged 4(ujSVu£\ dC'a/d^'"). Moreover, for integer electric charges, the SU(2/1) irrep 4 -» 3 ffi 1, so that the 3 structure for leptons and the decoupling of »»R are predicted by SV(2/1). In addition, as against the arbitrary selection of an SU(2) doublet for the spontaneous SU(2) x C7(l) symmetry breakdown in GoldThe publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.
Abbreviation: BRS, Becchi-Rouet-Stora. 5 Permanent address: Tel-Aviv University, Tel Aviv, Israel. 7068
354
Physics: Thierry-Mieg and Ne'eman
Proc. Natl Acad. Set. USA 79 (1982)
breaking and cannot confirm our 250-GeV conjecture (3) for the mass of the physical Higgs field. 3. In order to exponentiate a superalgebra g = g + + g", it is always necessary to introduce an exterior algebra of anticommuting parameters. In this way one constructs a Lie group <S (g,ft) = exp (fl+ ® g* + ft" ® g~). This group always admits an underlying Lie algebra g, generally reducible, whose dimensionality depends on fi, i = ft+ ® g+ © a.- ® %dim(g) = ',$dim(g)-dim(n).
=: De<°> =': De(1>
SC = -De
(2)
m
SE = -De
(1
(0)
i2)
(1)
(2)
- [B,e ] - [C,£ ] - [£,e ] = : De(3),
[3.6]
SS = [E,0), a ] + {em,.//} + [e®, 0],
[3.2]
=: De(2)
- {B,e >} - [C,e ]
DG + [B,F] = 0.
4. Along the same lines, an irreducible representation R = R+ + R~ of the superalgebra g will give rise to a representation R of g. Denote by <j> a system of 0,1,2... forms taking their values alternatively in fl+ and R~, 4>(
S=U(0),iM + {e(1,,<«
The Jacobi identity is automatically satisfied. Throughout this paper we use square brackets to denote SU{2) x [7(1) commutation relationships and curly brackets to denote the relationships specific to the superalgebra SU(2/1). They really denote commutators and anticommutators of number matrices, once all exterior products have been evaluated. Such a generalized gauge theory involves a generalized system of connections, skew-symmetric contravariant, and Bose tensor gauge fields AJ, BJ,,, CJ«p. and E\vfa of alternating supergroup gradings, saturating the dimensionality of space-time forms. Under an infinitesimal transformation with parameter i(e", £»,£%,ein,, e^), the gauge fields vary according to
SB = -DE(1» - [B,£<0)]
Dt = 0: DF = 0
S4>=[e(0),<M
However, some real forms of simple supergroups do not admit an underlying superalgebra (11). In the following, we shall work out a model where ft is taken to be the exterior algebra of forms over space-time itself. Therefore, the corresponding gauge theory is highly soldered to the base space. Nevertheless, most of the attractive properties of differential geometry are maintained, and our construction provides a nontrivial example of a Cartan integrable system (12). The generators of g will be denoted as A„ = /i„, Af = m dx", A J" = na dx" A dx", ... (/!„ eg+>M.eg-). The Lie bracket is defined as the exterior product for the forms times the Lie superbracket [fiM,/%}. For instance
SA = -Dem : - -dem -[A,£<°>]
and satisfy the Bianchi identity
The representation is defined by the transformation rules,
[3.1]
[AC,A/} = dx"Adx"{iMi,^} = f^kf.
7069
[3.3]
where D denotes the A. covariant differential with gauge field A£, d is the external differential, A = AJ M« dx", B = lA B'^ ^dx"Adx",C = %CliUI,dx"Adx"Adx",E = \^E' tt dx" A dx" A dx' A dx", and exterior products are implied! These equations define the action of the generalized covariant derivative D. The generalized curvature F is similarly defined,
[4.1]
with the Jacobi identity automatically satisfied. In the SU(2/ 1) system, the connections (Eqs. 3.3) are the generalized gauge fields and the curvatures (Eqs. 3.4) are their field strengths. For the leptons (and quarks) we have a doublet left-spinor (d>)L together with a singlet (two singlets) left-vector-spinor (ijijL, etc., all with Fermi statistics. 5. We now show a free-field Lagrangian such that it has precisely the same physical degrees of freedom as the Weinberg-Salam model. Given a skew p-tensor (j)^, and its exterior derivative (or generalized curl), this is
For a scalar <j>, this is the Klein-Gordon equation; for <£„, this is Maxwell's Lagrangian; for <^„ this is Kalb-Ramond field (13, 14). In N dimensions (N-l space-type dimensions and one time dimension) and for the massless Lagrangian (Eq. 5.1), there is a duality equivalence (15) between p and N-p-2 forms (this is Hodge duality in the transverse dimensions). It can be shown that the number of physical degrees of freedom n for the gauge fields (Eqs. 3.3) is precisely given by adding up the number (^ ) of components of an antisymmetric fc-indices tensor in N dimensions, together with the number of dycontracted components of its complexified (10, 16) geometrical vertical complements [in the direction of the fiber y", in the bundle manifold (9, 17)], ghosts counting negatively. For the forms in Eqs. 3.3, with the fiber-complexified forms denoted by a caret, A" = A% dx" + A"M dyM + A'„ dy* = A° dx" + <x° + 5° B' = 'A B'^ dx" A dx" + B'^ dx" A dyM + BU dx" A dy* + '/* B'n, dy" A < V + B U < V A dy" +
FHF'.CJ/'')
1
ABiit»dyi,Adyfl
^AB^ayAaV+B^d**
F° = dA + lA[A,A}° + Wlldx" + W + ti + V
0' = (DB)' H" = (DCf + lMB,B}a.
[3.4]
These curvatures transform covariantly, SF=[e,f):SF
+ !i C%udx" A dx" AdyM
+ lAC'IL*,dx"Adx"Adyf<
= -[F,e (0) ]
8G = -[G,em]-[F,em] SH = -[H,e(0>] - {G,ew} - [F,e®],
C° = 1AC'luvdx*Adx"AtW
+ [3.5]
355
l
AClMsdx"AdyMAdyN
+ C°lMj,dx"A dyM A dy*+ ^ C%^ dx"Ady"A
df
7070
Physics: Thierry-Mieg and Ne'eman ^C'MNFdyMAd^Ad/
+ +
= (/£ <£c* + i L + + xLL+
AC%NfdyM/\dyNAdyf l
AC^pdyMAdy'iAdf
l
AC%„rdy'ilAdy»Adyf
= 1A c;„„ dx" A d%- A dx> + V4 (r;, + r y
[5.2] etc. Latin letters denote Bose fields; Greek, Fermi ghost fields. We have (per internal index) for N = 4,
"(A) = \JJ ~ (2 X 1) = 2
Stt--lMCf,upf.
- (2 X 6) + (3 x 4) - 4 = 0
n(£) = I 11 - (2 x 4) + (3 x 6) - (4 x 4) + 5 = 0.
[5.3]
Actually, the (j/,j/) could be replaced by anticommuting 8,8 parameters (18) without altering in any way our counting procedure. We note that 6' contains a scalar real h' multiplet, required in the SU(2/1) irreps B'(cf,h') or %'(aa,h.i), in the Curci-Ferrari type of symmetric-complexified algebra of ghosts (16). The higher forms C and E do not contribute to the physical spectrum, nor would the total contribution of the system of nonvanishing ghosts of a higher tensor. 6. Taking the lepton triplet of SU (2/1) as an example, we use the Weyl action for the massless left isodoublet d> (vj,, e Z )• On the other hand, we use the Townsend (19, 20) action for tjrj;, an isosinglet left vector-spinor. We denote by YM>, an auxiliary Dirac spinor two-form and by 0 R an auxiliary right spinor oneform (we use 2-spinor notation):
The natural invariance of the Lagrangian under covariant BRS variations SB^,, = D ^ fl„j is also lost if the constraint is not satisfied: [G,
[6.2]
which have the solution (20) I/
2'B = -Vu ( e ^ ( D ^ + D„KP)))2. [6.3]
= '^a''dILVR.
[7.5]
DJPJP** + D[^P])) = 0
[6.4]
tFM».D[MBJ=°.
The left vector-spinor I/JJ; in Eq. 6.1 thus is seen to represent physically a right spinor vR, thus fitting the right isosinglet fermions such as e^. On the other hand, the formal left vectorspinor i^ij; is a one-form whose vertical complement is given by i,l'=^db»+t\,dyM
[7.4]
The equations of motion are
and the Lagrangian is equivalent up to the equations of motions to a Weyl Lagrangian, in terms of the right spinor vR = E,IL^Aa,AaILA
= (DH (j5, + Jy0XDM(A, + DyK)) +
In the present framework, the constraints manifest themselves already at the classical level. The following method, however, permits a direct transition from ita to 2! ^ considered as classical Lagrangians. Inspired by Dirac s work, we simply introduce a Lagrange multiplier KM, whose equation of the motion enforces the differential constraints and consider the Lagrangian
The YX/1 equations of motion enforce the constraints
[7.3]
where /3„ is the vector ghost in Eqs. 5.2 and S = s + [a, ] of ref. 21. Exactly the same defects plague the vector-ghost sector of the once gauge-fixed Freedman-Townsend Lagrangian SEt = D^ j5„) DM f}v in the covariant quantization formalism (refs. 21 and 22). In the latter case, we have recently proposed, in collaboration with Laurent Baulieu, a solution based on the existence of a secondary gauge invariance of the classical Lagrangian under a transformation with scalar ghost parameter K, SB = [F,K] = D D K . The usual BRS quantization procedure (23, 24) then leads naturally to a modified constraint-free and gauge-invariant Lagrangian for the /3' field: <% = S3 (Bj
^
[7.1]
using the SU(2) X 17(1) symmetric 8y metric, the B J,„ equation of the motion DtfGMW = 0 enforces the constraint
n(B)=(|)-(2x4) + 3 = l n(C) = I :
[6.5] L
The bosonic vector-spinor ghosts i (and x ~) can now be identified with the ghost state with i//R internal isoscalar quantum numbers (1, 3, 9) appearing together with the d>L doublets in 3 or 4 of SU(2/l) [and its symmetric Curci-Ferrari extension forxL-(10)]. It is indeed remarkable that the Townsend Lagrangian thus should explain both the ghost statistics and the chiral inversion in the SI/(2/1) matter multiplets, explaining another puzzling feature brought out by the classifying supergroup. 7. The Interacting Lagrangian. For several reasons, there is no trivial generalization of the Abelian Lagrangian to the nonAbelian case. On one hand, the Lie algebra is reducible, and its Killing metric is nonzero only in the <4° sector, so only the Yang-Mills vector Lagrangian comes out as a natural invariant. On the other hand, if we consider as a Lagrangian for the B^„ the term
l
+ +
Proc. Natl Acad. Sci. USA 79 (1982)
[7.6]
and the system is now closed. At the same time, SE'p is invariant under the nilpotent BRS algebra (refs. 17 and 22) involving fields and ghosts from Eqs. 5.2, and with K as the ghost of KM, Sot = Vi [a,a]
+ 4,)ldy*
SA„ = a „ a
356
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Proc. Natl Acad. Sci. USA 79 (1982)
Sb = 0 SK= -b
We can now construct the covariant differential
Sft = DJ> SKM = - ( f t + ft«) := - f t SB11, = D| ( 1 ^ 1 + [F^,K] = D 1(l ft ] .
K
Townsend's Lagrangian now generaUzes into:
21/2 = e A ^ ( Y ^ (/?.,< + a ,
[7.8]
S7=lA{b,b},Sl=-y-\*{K,b}=-7' ScM = ft y' + {jS'.fe}, SAM = -c M - DJ - {KM,fc} := - c ;
+ MeiV.sy + tf;,,,!,}
SL^ = - f t „ - D[(l Avl - {ft„,K} - % {fl[M,K„]} := - T^„ tfUft]}.
[7.9]
At the linearized level, we recover the Weinberg-Salam spectrum. Note that the closure of the BRS algebra is equivalent, through complexification of the y and projection in the dy, dlj sectors, to the closure of the extended Curci-Ferrari algebra. No additional calculations are required. 8. Let us now construct a BRS algebra (17,22) for chiral spinor fields. We start from a left chiral spin 1/2 R+ multiplet tj>L and write: S^L = 0.
[8.1]
We now introduce a left chiral vector spinor i^J;, its left chiral Bose spin 1/2 ghost i L + , and an auxiliary spin 1/2 fermion TJL, all valued in R~. The nilpotent BRS algebra is uniquely defined as Sx+ = -{b,
+ n^s,*,) [8.51
At the linearized level, we recover Townsend's Lagrangian and propagate a left doublet («/°, eZ) and a right singlet (e^). The dynamical part of the Lagrangian is the generalized Rarita-Schwinger term containing a coupling i/rj- a^ {B v , <j>}, which replaces the usual mass term of the Weinberg-Salam model. 9. The pattern of spontaneous symmetry breakdown is crucial for the application of the model to experiment. Obviously, the tensor Bj„, may not take a vacuum expectation value without breaking at the same time the Lorentz group. However, its central scalar Bose ghost h! = B'Ma dyM A dyN has the quantum numbers of the ordinary Higgs. Furthermore the term (DC)1 contains (B^,,)4, which generalizes in the ghost expansion to (h1)4. Therefore, we conjecture that, in the fully quantized theory, (h% ¥= 0. We remark that a term S3(i//-o-n) = iji-a {h,
The squares are computed using the 50j„ 8,j metric, implying sin20„ = 0.25. This Lagrangian is invariant under the nilpotent BRS algebra given above augmented by the C„ w sector:
SC^D^r^ +
[8.4]
S*„ = 0.
•'A^C^+MB^Byf -VNPI-
[8.3]
and the invariant spinor
2=-I/4(F„(A))2-1/12(D|MB;1)2
B'^ — B„„ + Dl(iK„], C ^
^1 = Dl>. *•] + { ^ . * } + {K|M.D,]*} + [F^.T?] SDiji = 0,
[7.7]
We have explicitly recovered our solution of Townsend's problem as a subcase. Incidentally, the possibility of extending the BRS algebra of Townsend's a model by the inclusion of K^ indicates that Eq. 7.5 might be an admissible counterterm in that theory. The reciprocal is not true. Here, all the fundamental fields—i.e., those which appear in BfB^.ft.fc)—have canonical dimension one. The K„ and its pair of ghosts have dimension zero. In a proper gauge they are expected to decouple, therefore ensuring (i) the renormalizability of the theory by power counting and (ii) its formal unitarity. Our method admits a direct generalization to the C° „p field, whereas the E'^^ has no curl in four dimensions. The complete classical gauge Lagrangian can be written involving the auxiliary two-form £(L^,AM,()
sr^D^
7071
[8.2J
357
+ V3A 8 ). 10. In conclusion, using the dx" generators of the original Grassmann exterior algebra over space-time, we have associated to any simple superalgebra a reducible Lie algebra whose connection defines a set of skew tensor fields. The Bianchi identity is maintained and defines a Cartan integrable system. The corresponding gauge theory is BRS invariant, seems formally unitary and renormalizable provided a set of dimension 0 auxiliary fields, which play the role of Dirac multipliers of the differential constraints, can be consistently eliminated. Applied to Sl/(2/l) quantum asthenodynamics, the model successfully restricts the arbitrariness of the Weinberg-Salam model. It yields a positive definite physical subspace and a consistent interpretation of all ghost states and explains the grading of the quark and lepton multiplets by their chirality. 1. Ne'eman, Y. (1979) Phys. Lett. B 81,190-194. 2. Fairlie, D. B. (1979) Phys. Lett. B 82, 97-100. 3. Ne'eman, Y. & Thierry-Mieg, J. (1980) Differential Ceometrical Methods in Mathematical Physics, Lecture Notes in Mathematics, No. 836 (Springer, Berlin), pp. 318-348. 4. Weinberg, S. (1967) Phys. Rev. Lett. 19,1264-1266. 5. Salam, A. (1968) in Elementary Particle Theory, Proceedings 8th Nobel Symposium, ed. Svartholm, N. (Almqvist & Wiksell, Stockholm, Sweden), pp. 367-377. 6. Scheunert, M., Nahm, W. & Rittenberg, V. (1977) /. Math. Phys. 18,155-162.
7072 7. 8. 9. 10. 11. 12. 13. 14. 15.
Physics: Thierry-Mieg and Ne'eman
Proc NatL Acad. Sci. USA 79 (1982)
Ne'eman, Y. & Sternberg, S. (1980) Proc NatL Acad. Sci. USA 77, 3127—3131 Ne'eman, Y. & Thierry-Mieg, J. (1982) Phys. Lett. B 108, 399-402. Ne'eman, Y. & Thierry-Mieg, J. (1980) Proc. NatL Acad. Sci. USA 77, 720-723. Thierry-Mieg, J. & Ne'eman, Y. (1982) Nuovo Cimento A, in press. Berezin, F. A. & Tolstoy, V. N. (1981) Commun. Math. Phys. 78, 409-428. Cartan, E. (1926) C. B. Acad. Sd. Paris 182,956-958. Kalb, M. & Ramond, P. (1974) Phys. Rev. D 9, 2273-2285. Cremmer, E. & Scherk, J. (1974) NucL Phys. B 72,117-124. Cremmer, E. & Julia, B. (1979) NucL Phys. B 159,141-213.
358
16. 17. 18. 19. 20. 21. 22. 23. 24.
Curci, G. k Ferrari, R. (1975) Nuow Cimento A 30,155-168. Thierry-Mieg, J. (1980) / . Math. Phys. 21, 2834-2838. Bonora, L. & Tonin, M. (1981) Phys. Lett B 98, 48-50. Townsend, P. K. (1980) Phys. Lett. B 90, 275-276. Deser, S., Townsend, P. K. & Siegel, W. (1981) NucL Phys. B 184, 333-350. Thierry-Mieg, J. (1979) Preprint Harvard Univ. Mathematics and Theoretical Physics B86. Freedman, D. Z. & Townsend, P. K. (1981) NucL Phys. B 177, 282—296 Becchi, C., Rouet, A. & Stora, R. (1976) Ann. Phys. N.Y. 98,287321 Baulieu, L. & Thierry-Mieg, J. (1982) NucL Phys. B197,477-508.
Volume 181, number 3,4
PHYSICS LETTERS B
4 December 1986
INTERNAL SUPERGROUP PREDICTION FOR THE GOLDSTONE-HIGGS PARTICLE MASS * !
Yuval NE'EMAN
Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv. Israel and Center for Particle Theory, University of Texas, Austin, TX7S712, USA Received 19 September 1986 Using a seemingly ad hoc but phenomenplogically fitting global internal supersymmetry SU(2/1) D SU(2) X U ( l ) constraining the weak-electromagnetic otherwise arbitrary parameters, we predict for the Goldstone-Higgs particle a mass mH = 2m\v, or 160-170 GeV.
tan 0W =g'/g = 1/VJ, or sin20w = 0.25 .
Weak—electromagnetic unification [1] is based on the non-simple gauge group SU(2) X U(l), with two independent couplings £ and g, uncorrected weakhypercharge quantum numbers for the left- and rightchiral fermions,
W
= U^D
= -1 • ^w(da) = -2 , /3
^ w (UL )=^w(di: /3
^w(4 )=-f
1/3 :
)
Note that the experimental value has come closer to this value recently [4]. The quarks in (2) all fit precisely into one four-component irreducible multiplet of SU(2/1) with eigenvalues
(1)
(tfw,/w,4):
1
(u°,0,0;u0-l,ll,u0-l,i-\;u0-2,0,0).
3 >
(2)
Q = -2M3 - 5\/2tf8 •
(3)
[H6,x} = 0,
removes most of the arbitrariness. The couplings have to obey
xesu(2/l)->x
=Q
(6)
(where [,} is the Lie superbracket) as against su(3), where Xg itself would also have been a solution. We do not claim to have a good understanding of the gauging of a supergroup [5], though much progress has been achieved [6,7] in this direction. In such
* Supported in part by the US DOE Grant DE-FG0585ER4O2O0 and by the US-Israel Binational Science Foundation. 1 Wolfson Chair Extraordinary in Theoretical Physics.
308
(5)
Thus UQ = | for the quarks. Moreover, for integer UQ this four-multiplet becomes reducible and 4 -* 1 + 3, the upper state disconnects and we get triplets fitting the leptons (1) precisely («o = 0)! The supergroup grading follows the chirality assignments. The quantum numbers of
The weak-isospin multiplet selection / w = \ for ("L ^L), (ujj , d£ ), i.e., the left-chiral states, and / w = 0 for e£, ug 3 , d^ 1 ' 3 , i.e., the right-chiral states, appears ad hoc, and so is the adjunction of a Lorentzscalar Goldstone—Higgs multiplet #H(*) w i t h Av = 5". C/w = ±1. The additional couplings X^jj and — CT^H are not constrained by the gauge theory. Some time ago, we pointed out [2,3] that the inclusion of the gauge group in a simple supergroup constraining the low-energy parameters, SU(2)XU(1)CSU(2/1),
(4)
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
359
Volume 181, number 3,4
PHYSICS LETTERS B
a (high-energy limit) "fundamental" interpretation [5—7], the conventional implementation of a supergroup action as relating fermions to bosons (and vice versa) results in a doubling of the multiplets, with half the states representing ghost fields appearing in that formalism [7]. For example, the four (virtual)
4 December 1986
Note that SU(2/1) in either interpretation does not preclude [8] an additional high-energy embedding of SU(2) X U(l) in a GUT such as SU(5) or E(6). In the present ad-hoc low-energy approach, the analogy with SU(6) can be further extended: while the low-energy static SU(6) referred to post-renormalisation "constituent" quarks with masses of some 300 MeV, there was another SU(6) describing almost massless (highenergy) "current" quarks. The two systems were related by the Melosh transformation [12]. A similar though probably more complicated mechanism would relate the low-energy SU(2/I) to the high-energy GUT. We would now like to derive the SU(2/1) prediction for my{, t n e mass of the residual Goldstone—Higgs particle. We have treated this issue previously in a "hybrid" approach [2,3,5], namely using the conventional field theory mechanism with - a 2 ^ + X ^ , feeding into it the SU(2/1) parameters. Since we do not know if the supergroup gauge theory is renormalizable, this appears problematic. However, in the ad-hoc low-energy global symmetry approach, mu is directly related to the masses of the weak interactions intermediate bosons. The superalgebra bracket relations are [*i/<,MB}=2C4££.M£',
strMj4=0,
(7)
where str stands for the supertrace, and HA (A = 1 ... 8) are a basis of 3 X 3 matrices identical to the SU(3) X matrices except for J*8 = 5 ( * 8 - 2 \ / 2 X o ) -
(8)
As a result, we have C
abc=lfabc
(.a>b,C= 1,2,3) ,
C
a8c = VaSc = ° >
C
ai/ = ifai/<
C
8if
C
=
C
iaf = ifiaj
3'/8iy '
iia = dija,
C
i8/
=
(U = 4, 5,6, 7) ,
WiBj >
Cv8 = - i V 3 5 v .
(9)
where/and d refer to the usual SU(3) coefficients. The particle masses are generated by spontaneous symmetry breakdown in the (1$ direction, in the same way as "flavour" SU(3) is broken in the Xs direction. However, ft^ is not a diagonal operator, and has no matrix-elements between a meson and itself (in fer309
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PHYSICS LETTERS B
[2] Y. Ne'eman, Phys. Lett. B 81 (1979) 190; D. Fairlie, Phys. Lett. B 82 (1979) 97. [3] Y. Ne'eman and J. Thierry-Mieg, in: Differential geometrical methods in mathematical physics, Proc. Conf. Aix-en-Provence and Salamanca (1979), Lecture Notes in Mathematics, Vol. 836, eds. P.L. Garcia et al. (Springer, Berlin, 1980) pp. 318-348. [4] P. Reutens et aL, in: The Santa Fe meeting, eds. T. Goldman and M.M. Nieto (World Scientific, Singapore, 1984) p. 270. [5] Y. Ne'eman and J. Thieny-Mieg, Proc. Nat Acad. Sci. USA 77 (1980) 720. [6] J. Thierry-Mieg and Y. Ne'eman, Nuovo Cimento A 71 (1982) 104. [7] Y. Ne'eman and J. Thierry-Mieg, Proc. Nat. Acad. Sci USA 79 (1982) 7068. [8] Y. Ne'eman and S. Sternberg, in: Proc. XXth Intern. Conf. on Hjgh energy physics (Madison, 1980), AIP Conf. Proc. Vol. 68, Particles and fields subser. 22, eds. L. Durand and L.G. Pondrom (American Institute of Physics, New York, 1981) pp. 460-462. [9] F. Gursey and L.A. Radicati, Phys. Rev. Lett. 13 (1964) 173. [10] S. Coleman, Phys. Rev. B 138 (1965) 1262. [11] D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343; H.D. Polilzer, Phys. Rev. Lett. 30 (1973) 1346. [12] H.J. Melosh, Phys. Rev. D 9 (1974) 1095.
mion multiplets it relates the two chiral components of one particle). For the mesons we are thus forced to take the squared % transitions. We shall therefore read offM 2 matrix elements [with i>=
^cA6BcB6A.
(io)
As a result, we find uniquely (H is the residual 0 ^ particle):
mQN±):m(Z°):m(H)=\:2ly/3:2,
(11)
thus predicting m ( H ) ~ 160-170 GeV.
4 December 1986
(12)
References [1] S. Weinberg, Phys. Rev. Lett 19 (1967) 1264; A. Salam, in: Elementary particle theory, Proc. VIII Nobel Symp., ed. N. Svaitholm (Almquist and Wiksell, Stockholm, 1968) pp. 367-377.
310
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Proc. Natl. Acad. Sci. USA Vol. 87, pp. 7875-7877, October 1990 Physics
Superconnections and internal supersymmetry dynamics (Higgs mechanism/electroweak theory/generational unification) YUVAL NE'EMAN AND SHLOMO STERNBERG Ministry of Science, Government of Israel, Jerusalem, Israel; and Department of Mathematics. Harvard University, 1 Oxford Street, Cambridge, MA 02138 Contributed by Shlomo Sternberg, July 23, 1990
ABSTRACT In previous papers we proposed a theory of internal supersymmetry using the superalgebra su(n/l) to give rise to a unified structure that included quarks and leptons in 2" _s generations. In the present paper we suggest that the notion of superconnections as introduced by Quillen provides a natural setting for the dynamics of an internally super-symmetric theory with the Higgs field occurring as the "zero-th order part" of the superconnection. The Higgs mechanism enters quadratkally into the curvature of the superconnection and hence quartically into the Lagrangian. The supercovarlant derivative gives a coupling of the Higgs field to the matter field similar to that put in "by hand" in the Lagrangian of the Weinberg-Salam theory.
where \a\ denotes the degree of a. For example, suppose that A is the (supercommutative) superalgebra of all differential forms on a manifold M and that B = End E, where E = E0 © £i is a supervector space. Thus B0 consists of all "matrices" of the form
In ref. 1 it was noted that the number of independent assumptions required by the Weinberg-Salam SU(2) x U(l) theory can be reduced by assuming that the structure (gauge) group SU(2) x U(l) is associated to the even part of the superalgebra su(2/l) that acts internally on the matter fields (cf. also ref. 2). In ref. 3 it was shown how the basic representations of sl(2/l) occurring in ref. 1 could be extended to sl(n/l) and so give rise to a unified structure that included quarks and leptons in 2"~s generations. These representations have been discussed in terms of Howe pairs and dimensional reduction in ref. 4. It was proposed in ref. 5 that for a theory of internal supersymmetry, the natural "Grassmann variables" to tensor with the internal superalgebra are the differential forms on the base manifold, and an attempt was made to construct a dynamics using a connection associated to this structure. In the present paper we suggest that the notion of superconnections as introduced by Quillen (6) provides a natural setting for the dynamics of an internally supersymmetric theory with the Higgs field occurring as the "zero-th order part" of the superconnection. The Higgs enters quadratically into the curvature of the superconnection and hence quartically into the Lagrangian. The supercovariant derivative gives a coupling of the Higgs field to the matter field similar to that put in "by hand" in the Lagrangian of the Weinberg-Salam theory. A good reference for the material on superconnections and equivariant superconnections has been given by Berlin et al. (7).
If we choose bases of Eo and Ej then we can think of R, S, K, and L as actual matrices. We can then think of elements of A ® B as matrices whose entries are differential-forms forms, but we must remember rules 1.1 and 1.2. For example, if too and
/ 0 K\ I I ,
E0),
L € Hom(E0, E{).
/ 0 EoA V-io O) is an odd element of A ® B. Then rule 1.2 says that /too
0\/
0
A > i \ /
0
AJOAEOA
while / 0 toA/ao 0 \ \L10 0 ^ \ 0 on)
/ 0 -ioiAwA V-EioAftt, ° /
[1.3b]
The minus sign in Eq. 1.3b arises from passing the odd elements of B through the differential forms of odd exterior degree as prescribed by rule 1.2. In this example we can consider the supervector space A®E with grading as in rule 1.1. Then A ® £ is a (left) module for A ® B where we apply the sign rule analogous to rule 1.2. Thus
A0<8B0®A1®Bl,
A®BCEnd(A®£). [11] We can think of A as embedded in A ® B as A ® / and this makes A ® E into an A module where the action is the obvious one. Since A ® £ is a supervector space, End(A ® £) is a superalgebra. It is easy to see that an element of End(A ® £) belongs to A ® B if and only if it supercommutes with all elements of A. In other words,
with multiplication (on homogeneous elements) given by (a ®fr)(a'® b') = (-\)\b\\a'\aa' ® bb\
KeHom(Eu
is an odd element of A ® B. Similarly, if Lm and Lw are matrices of forms of even exterior degree, then
Recall (8) that if A = A0 © Ax and B = B0 © fl, are superalgebras, then the superalgebra A ® B is defined by
(.A®B)i = A1®B0®A0®B1,
SEEnd(£i),
while Bi consists of all "matrices" of the form
Section 1. Generalities
(A<8>B)0 =
RGEnd(Eo),
(::)•
[1.2]
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
A ® B is the supercentralizer of A inside End(A ® £). [1.4] 7875
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Physics: Ne'eman and Sternberg
Proc. Natl. Acad. Sci. USA 87 (1990)
We can define the (odd) operator d E End(A ® £) by
ential forms on M and A(Af, £ ) , the space of smooth £-valued forms. Then A(M, E) is a module for A(M) as before. We can consider End(A(M, £)) and A(M, End(E)) so that
d(a ® e) = da ® e, where da is the usual exterior derivative of the differential form a. We can write this definition symbolically as defining
MM, End(£)) C End(A(M, £)). The analogue of statement 1.4 is MM, End(£)) is the centralizer of MM) in End(A(M, £)).
[2.1]
If, by abuse of notation, we let a denote multiplication by a as an element of EndW ® E) then the supercommutator ofd with a is given by [d, a] = da, [1.5]
In fact, MM, End(£)) is the centralizer of the ring of functions, A°(M), in End(A(M, £)). An odd element, D e End(A(M, £))i, is called a superconnection if
where theright-handside denotes multiplication by da. More generally, for any a<8beA®Bwe have
In other words,
[d, a®b} = da®b.
[1.6]
[D, <*] = <& for alia 6A(M).
D: MM, E)o-> MM, Eh,
D(a A a) = da A a+ ( - l ) W a A I V
d(a®ft) = da®6,
for all a E A(Af) and a E MM, £).
then we can write Eq. 1.6 as [1.7]
The curvature F = F(D) of the superconnection is defined as F = D2.
In particular, if o> is an odd element of A ® B then (d + w)2 = d2 + [d, to] + a,2 = dto +
[1.8]
Note that the right-hand side of Eq. 1.8 is an element of A® B. If we write u> out as a "matrix" / wo L 0 A
[2.3]
Note that for any function/we have [D2, / ] = [D, /]D + D[D, / ] = (df)D + D(df) = [D, df] = ddf= 0. Thus
\Lio " V
F E MM, End(£))0.
then we can write Eq. 1.8 as (d + w)2 =
(
D: MM, £)i-> A(M, £) 0
and
So if, for a E A ® B, we define da by
[d, a] = da for a EA<8B.
[2.2]
[2.4]
The difference Di - D2 between two superconnections is an element of A(M, End(E)) by Eq. 2.3. Hence, in terms of a local trivialization of £, the most-general superconnection can be written locally as D = d + at,
me MM, End(£)h.
[2.5]
We should remember that in Eq. 1.10 the <•> and L terms are matrices of even and odd forms, respectively, but not necessarily homogeneous with respect to exterior degree. Thus, if the base space is four-dimensional then
This means that the local expression for the curvature is given by Eq. 1.8 or, in "matrix" language, by Eq. 1.10. K> is any polynomial (or entire function) of one variable and Str denotes the supertrace, then (cf. ref. 6 or ref. 7) Str(p(F)) is a closed form; i.e.,
to0 = A<j + C 0 ,
Str(p(F)) E A(M) andrfStr(p(F))= 0.
where A0 is a matrix of one forms and C0 is a matrix of three forms and similarly for cut. Also
Furthermore, up to an exact form, Str(p(F)) is independent of the choice of the superconnection; i.e.,
A>1 = ''oi + *oi + Ati>
Str(p(F(D!))) - Str(p(F(D2))) =
where hm is a matrix of functions, Bol is a matrix of two forms, and D0i is a matrix of four forms and similarly for Lw. The A occurring in Eq. 1.10 denotes matrix multiplication where the matrix entries are multiplied via exterior multiplication.
where a(D,, D2) is a differential form that has a simple expression in terms of Dj and D2. Thus, for example, the Chern character corresponds to p(z) = e~z (for all this, see refs. 6 and 7). Now Striab) is antisymmetric in a and b if a and b are both odd elements of End(£). Hence if a and B are odd forms, the expression Str(a CS> a)(*j8 ® b) is antisymmetric as a function of a ® a and B ® b. Furthermore, on the even terms, an expression such as Str(F*F) in the Lagrangian will lead to negative kinetic energy terms for the dynamics (unless £0 or Ei is trivial). Hence we proceed as follows: choose an invariant bilinear form b on the Lie algebra End(£)o- Here
Section 2. Superconnections Now let E —> M be a supervector bundle over an ordinary manifold M. So E = £o © £i, where E0 and E\ are ordinary vector bundles. Let A(M) denote the ring of smooth differ-
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Proc. Natl. Acad. Sci. USA 87 (1990)
invariant means invariant under the "even group" Aut(£0) x Aut(£i). As the adjoint representation of this group is not irreducible, there will be some choices here, beyond overall scale. In the case of eventual interest to us, this amounts to the choice of Weinberg angle. We will see how this choice is made in our theory. Then the Lagrange density for the purely Yang-Mills part of the theory is Y-M(D) = 6(F, *F)
[2.6]
as usual. If we identify End(£h with the Higgs sector, then the Loi entering into "matrix" 1.9 contains, as components of exterior degree zero, sections of End(£)i, that is to say Higgs fields. From "matrix" 1.10 we see that the Higgs field enters quadratically into the curvature, and hence Eq. 2.6 is a polynomial of degree four in the Higgs. For the case of su(n/l) as internal superalgebra, a natural choice of b is as follows: For su(n/l), we have g0 = su(n) © R = su(/i) and gi = C". As a vector space, and also as far as the action of g0 on gi is concerned, we have g0 © gi -» su(n + 1). The difference lies in the bracket of gi x g1 -» g0, one bracket being symmetric and giving a Lie superalgebra and the other being antisymmetric and giving a Lie algebra. Indeed these two structures are related to one another via the notion of a Hermitian Lie algebra (see the first few pages of ref. 9 and cf. also ref. 10). So a natural choice would be to take b to be the Killing form of su(/i + 1), and this was the choice made for the case n = 2 in refs. 1 and 2 for determination of the Weinberg angle. The theory of superconnections can, of course, also be formulated in terms of principal and associated bundles (cf. ref. 7): If g = g0 © gt and G is a Lie group whose Lie algebra is g0. then a superconnection will be a g-valued form on PG of total odd degree (subject to conditions generalizing the
7877
standard ones for connections), where Pc, is a principal bundle with structure group G. If F is a supervector bundle associated to a representation of (G, g) on a supervector space V, then the superconnection form on Pa induces a superconnection D on F. If S is the spin bundle, then we can use D to modify the Dirac operator and so obtain the operator y(D): F ® S -* F ® S. A superinvariant bilinear form on F then gives the matter field contribution to the Lagrangian as (•, -y(D)') on F ® S. Notice that this involves a cubic term that is quadratic in the matter field and of first order in the Higgs field, as in the Weinberg-Salam Lagrangian. This research has been supported in part by National Science Foundation Award DMS-8907995 and by U.S.-Israel Binationa! Fund Contract 87-00009/1. 1. Ne'eman, Y. (1979) Phys. Leu. B 81, 190-194. 2. Fairlie, D. B. (1979) Phys. Lett. B 82, 97-100. 3. Ne'eman, Y. & Sternberg, S. (1980) Proc. Natl. Acad. Sci. USA 77, 3127-3131. 4. Ne'eman, Y. & Sternberg, S. (1982) in Gauge Theories: Fundamental Interactions and Rigorous Results, eds. Dita, P., Georgescu, V. & Purice, R. (Birkhaeuser, Boston), pp. 103142. 5. Ne'eman, Y. & Thierry-Mieg, J. (1982) Proc. Natl. Acad. Sci. USA 79, 7068-7072. 6. Quillen, D. (1985) Topology 24, 89-95. 7. Berlin, N., Getzler, E. & Vergne, M. (1990) Heat Kernels and Dirac Operators (Springer, Berlin). 8. Ne'eman, Y. & Sternberg, S. (1975) Rev. Mod. Phys. 47, 573-603. 9. Sternberg, S. & Wolf, J. (1978) Trans. Am. Math. Soc. 238, 1-43. 10. Sanchez-Valenzuela, O. & Sternberg, S. (1985) Lecture Notes in Mathematics (Springer, Berlin), Vol. 1251, pp. 1-48.
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International Journal of Modern Physics A, Vol. 11, No. 19 (1996) 3509-3522 © World Scientific Publishing Company
BRST QUANTIZATION OF SU(2/1) ELECTROWEAK THEORY IN THE SUPERCONNECTION APPROACH, AND THE HIGGS MESON MASS
DAE SUNG HWANG and CHANG-YEONG LEE Department
of Physics, Sejong University,
Seoul 133-747,
Korea
YUVAL NE'EMAN Wolfson Distinguished Chair of Theoretical Physics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Center for Particle Physics, University of Texas, Austin, Texas, 78712, USA
Received 6 November 1995 A superconnection, in which a scalar field enters as a zero-form in the odd part of the superalgebra, is used in the BRST quantization of the SU(2/1) "internally superunified" electroweak theory. A quantum action is obtained, by applying symmetric BRST/antiBRST invariance. Evaluating the mass of the Higgs field, we exhibit the consistency between two approaches: (a) applying the supergroup's (gauge) value for X, the coupling of the scalar field's quartic potential, to the conventional (spontaneous symmetry breakdown) evaluation; (b) dealing with the superconnection components as a supermultiplet of an (global) internal supersymmetry. This result thus provides a general foundation for the use of "internal" supergauges. With SU(2/1) broken by the negative squared mass term for the Higgs field and with the matter supermultiplets involving added "effective" ghost states, there is no reason to expect the symmetry's couplings not to be renormalized. This explains the small difference between predicted and measured values for s i n 2 ^ , namely the other coupling fixed by SU(2/1) beyond the Standard Model's SU(2) x U ( l ) , and where the experimental results are very precise. Using the renormalization group equations and those experimental data, we thus evaluate the energy E, at which the SU(2/1) predicted value of 0.25 is expected to correspond to the experimental values. With SU(2/1) precise at that energy E, = 5 TeV, we then apply the renormalization group equations again, this time to evaluate the corrections to the above A, the quartic coupling of the scalar fields; as a result we obtain corrections to the prediction for the Higgs meson's mass. Our result predicts the Higgs' mass [170 GeV, according to unrenormalized SU(2/1)] to be as low as 130 i: 6 GeV, using for the top quark mass the recently measured value of 174 GeV.
1. Introduction Some 15 years have passed since the first suggestions of SU(2/1) electroweak theory.1'2 This is an internal supersymmetry or supergauge theory, with the gauged supergroup Q — SU(2/1) containing both the Weinberg-Salam SU(2) x U(l) gauge 3509
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D. S. Hwang, C.-Y. Lee & Y. Ne'eman
group (as the even Lie subgroup G+ C G; we shall use the term "p-even") and its spontaneous symmetry breakdown (Goldstone-Higgs) fields
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BRST Quantization of SU(S/1) Electroweak Theory in the ...
3511
conventional ghosts appearing in the BRST transformations of the W±, Z°, A0 and behave as fermions, the "ghosts" for the g-odd part are bosons and fit the role and quantum numbers of the Higgs field components precisely. As a matter of fact, in this first version the theory can be defined as that theory in which the conventional ghosts and the Higgs fields together make a scalar field octet multiplet. Following the geometrical interpretation4'6 of the ghosts as the "vertical" components of the connection in a Yang-Mills gauge theory's principal bundle, the scalar Higgs field, entering the theory as a "formal" ghost, is identified with a one-form in dy, with y the supergroup manifold parameter — whether it is bosonic or fermionic is unimportant at this stage, since it is contracted within the one-form, e.g.
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3512 D. S. Hwang, C.-Y. Lee & Y. Ne'eman
(a3) Yet a third interpretation for the role of the scalar field was proposed by Ne'eman and Sternberg,13 using Quillen's generalization ansatz, beyond the original constructs of Ref. 7. The scalar field is assigned to the same "w-even, godd" part of the SU(2/1) superconnection, but this time as a zero-form. A similar suggestion was made by Coquereaux et al.,u who based their work on an application of Connes' noncommutative geometry.15'16 The latter technique has the additional advantage of adding the notion of a matrix derivative, for the discrete factor space of the geometry, capable of supplying the negative-squared-mass "trigger" for the symmetry's spontaneous breakdown mechanism. In the present work, we reformulate the BRST quantization of the SU(2/1) theory, interpreting it according to (a3), namely the Ne'eman-Sternberg generalized superconnection approach. The version motivated by noncommutative geometry, including the matrix derivative, will be treated in a sequel to this article.17 In what follows, symmetric BRST/anti-BRST transformation rules are worked out 5 ' 10 by applying the horizontality condition, deriving from Thierry-Mieg's original geometrical interpretation of the BRST constraints, as the Cartan-Maurer structural equations for a principal fiber bundle.4,6,18 This interpretation does not involve new fields or ghosts. A quantum Lagrangian is constructed, through that symmetric BRST/antiBRST algebra.19 Using renormalization group (RG) techniques, we now deal with two features due to SU(2/1), which are still allowed arbitrary values in SU(2) x U(l): the U(l) coupling gi [given the SU(2) coupling <&] and the quartic coupling in the Higgs potential A; alternatively, we may deal with two related quantities, tan^iv = gi/g? and the mass M(H) of the Higgs meson. In both cases we first rederive the SU(2/1) prerenormalization predictions and discuss them in the light of the (a3) superconnection approach: for tw0\y this involves a transmutation between gradings and the related algebraic transition within the Sternberg-Wolf algebra. In the case of the Higgs mass, we learn more about the consistency of the superconnection approach with a kinematical (global) approach to the internal supersymmetry. Having completed this "theoretical" discussion, we move on to the physical realization. The RG techniques are first used to evaluate the variation of the "running" gauge coupling parameter for the SU(2) x U(l) even subgroup. We use the result to fix the value of q2 = Et at which SU(2/1) becomes precise. We can then use the same approach in the opposite direction for the "running" quartic coupling A(
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BRST Quantization of SU(Z/1) Electroweak Theory in the ...
3513
If J is a (2+1) x (2+1) supermatrix, then A, B are 2 x 2 and l x l SU(2/1) g-even submatrices, valued over the one-form gauge fields A = A^dx*, B = Bvdxv; <j>, 4> are respectively 2x1 and 1x2 SU(2/1) g-odd submatrices, valued over the zero-form scalar fields , 4>; in the interest of simplicity in the physical treatment, we use the same symbol to denote both the forms — also multiplets of the Q+ = SU(2) x U(l) even subgroup — and their realizations as SU(2/1) submatrices, at the risk of some confusion. J is thus itself an odd element of U, a product super manifold: J e U~ CU = V(M) ®g. That is to say, it is an odd element within the direct product of V(M), the Grassmann algebra of differential forms over a manifold M, by the abstract superalgebra g = su(2/l) of Q. Returning to V(M) — depending on the context, M might be either just space-time i?4, or some larger manifold, such as the principal fibre bundle P(R,Q), with structure group Q = SU(2/1) (such extended M manifolds will be denoted by a tilde). J is "w-odd" because its A and B elements are to-odd but p-even, while its <j>, $ elements are w-even but p-odd. We now write down any two elements u, u' £ U, with u = (W ® h) and u' = {W'®h'), where W, W are differential forms of Grassmannian Z% uw gradings" |W|, |W'|, and h, h' are SU(2/1) supermatrices of fixed Lie supergroup Z^ "g grading" \h\, \h'\. For the U multiplication, we adopt the canonical convention20 (W®h)- (W ® h') = (-1)MW'\ W AW'®(h-
h!).
(2)
With the above convention and assuming that A, B, C, D £ V(M), we obtain the Ne'eman-Sternberg rule for a product of supermatrices13
(A \D
C\ (A' B) ' \D'
C'\ B'j
A A A' + (-1)I°'IC AD'
A A C + (—1)IJ3'IC A B'
(-l)l^'l£)
(-iyc'lD AC + BAB'
A
A' + B A D'
(3)
We now define the supercurvature as F = dJ + J-J,
(4)
where d = f Q A and d = dxM-^ <8> 1 [1 is the identity for an su(2/l) submatrix]. Here one can see that d acts as a derivation with w-odd, w-odd. For instance, if d acts on a g-odd supermatrix V~ = ( ) multiplied by an arbitrary supermatrix Q,then d(V~ • Q) = (dV~) • Q + (-l)H p ~Hp- • d Q , (5) where \\V~\\ is the total u grading of V~, i.e. the sum (mod 2) of the SU(2/1) "g degree," g = 1 in this case, and the form "w degree" of the entries of V~, namely of C and D in Eq. (3). Thus | | P _ | | = 1 if C, D are valued over w-even forms, and \\V~W = 0 if C, D are valued over w-odd forms (this formulation embodies the fact
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D. S. Hwang, C.-Y. Lee & Y. Ne'eman
that the overall u grading of d is 1 + 0 = 1). From (1) and (4) with the help of the rule (3), we write the supercurvature T in its component form: (dA + AAA + 4>4> d4> + A
J = J + C + C.
(8)
C and C are obtained from J+ by replacing dx1* with dyN, and with dy~M, respectively, i.e. the "vertical'1 components of the J+ (g-even) gauge connection, contracted over the vertical differentials: (ANdyN
L
-\
c- =
0
0
\
=
(c
BNdyNj -U
(AMdyM \ 0
0\
tj'
0 BMdyM
c and t are thus the g-even w-odd one-form anticommuting scalar ghost fields for SU(2) and for U(l) respectively. Note that the M-vertical supermatrices of (9) have no 9-odd submatrices because the corresponding ones in J of (1) are zero-forms
•-(J!)where s = dyN -fa ® 1, S = dyM ^
® 1.
370
=(»°)'
(,0)
BRST
Quantization
of SU(2/1)
Electroweak
Theory in the ...
3515
From the horizontality condition ? = T,
(11)
we obtain the BRST/anti-BRST transformation rules (the differentials on the left count the orders of the vertical forms, with "negative" values for y, i.e. ghost/ antighost charges): (dy)1: 1
(dy)' : (dy)2: (dy)'2: (dydyf:
sJ + dC + C-J
+ J-C
= 0,
(12)
aJ + dC + C-J
+ J-C
= 0,
(13)
aC + CC = 0,
(14)
aC + CC = 0,
(15)
sC + sC + C C + C C = 0.
(16)
We now proceed to decompose the vertical parts of the two-form (7) according to the SU(2/1) g degree, i.e. J into J+ + J~, where j+ = (*
°B\
J- = Q
J),
also noting that d, s and s are g-evea matrices (the identity), whose entries are one-form differential operators, i.e. with odd w grading. We list the even and odd parts separately: 0-even part: sj+
+ dC + C • J+ + J+ • C = 0 ,
&J++dC
+ CJ++J+-C
= 0,
(17) (18)
sC + C • C = 0 ,
(19)
sC + CC = 0,
(20)
sC + sC + C-C + C-C = 0;
(21)
g-odd part: sj~
+C • J~ + J~ -C = 0,
SJ-+C-J-
+ J-C
= 0.
(22) (23)
By introducing an auxiliary 5-even field supermatrix 5, such that
*•'•
«*(oi)-(i:).
<-»
we can fix the remaining BRST/anti-BRST transformation rules, sC = -e-CC-C-C,
(25)
s£ = 0,
(26)
§£ = -£•£
+ £•£.
371
(27)
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D. S. Hwang, C.-Y. Lee & Y. Ne'eman
We note that (17) and (18) are the usual BRST/anti-BRST transformation rules for the one-form gauge field,19 and (22) and (23) are those of the Higgs scalar; in the latter case, the geometrical construction reproduces the conventional transformation rules for matter fields. One can easily check that the nilpotency of BRST/antiBRST transformations is satisfied, i.e. s2 = s2 = 0 and si + ss = 0. 4. Quantum Action and the sin30yv = 0.25 Prediction We follow Baulieu and Thierry-Mieg19 in the construction of a BRST/anti-BRST invariant quantum Lagrangian, adapting the method to the superconnection version (a3), i.e. we add a BRST/anti-BRST closed form to the Yang-Mills classical Lagrangian: CQ = i Tr[:F - r* - ss{J • J*) + as(C • £*)}, (28) where a is a parameter, and * denotes taking duals with respect to the base manifold (the x* variable) for each of the differential form entries W, W in the supermatries (2) and taking Hermitian conjugates for the supergroup supermatrices h, h' themselves. Note the use of the trace (Tr) rather than the supertrace (STr), in contracting the "internal" (g type) indices. The "transmutation" we discussed in (a2) and (a3) of Sec. 1 has replaced the Q algebra's original supermatrices by ordinary numerical U(3) matrices, a transition within the Sternberg-Wolf Hermitian algebra,12 with ICilling metric now given by traces — replacing the supertraces of the SU(2/1) Killing metric (which has no predictive value, since a supergroup then admits "up" and "down" metrics). After a somewhat tedious, but straightforward calculation, one can check that Tr[ss(J--17-*)]=0,
(29)
Tr[ss( J+ • J+*)] = 2 T r ^ • (d£)* + AC • (VC)*],
(30)
where VC = dC + J+ • C + C • J+, and Tr[as(C£*)] = Tr(a£-£*).
(31)
Thus CQ becomes CQ =
^TT[T
• T* - 2 J+ • (d£)* - 2dC • (VC)* + a£ • £*].
(32)
This quantum Lagrangian is fully BRST/anti-BRST-invariant. We now apply this result to the SU(2/1) supergauge. The superconnection J is Lie-superalgebra-valued, J = iftjj'1, I = 1,2,...,8, where / J / ' S 1 are the same as SU(3) A matrices, except for
^ \ 0
0-2/
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BRST Quantization
of SV(Z/1)
Electroweak Theory in the ...
3517
Our above discussion relating to the result that traces, rather than supertraces, supply the invariants, is also relevant here and explains the original (otherwise hard to justify) normalization of the fij according to tr(^ J ) 2 = 2, which yields the prediction sin2 6W = 0.25 (33) as a result of the ratio of coefficients in pz and fj.%.1 The Weinberg angle is given by tanflw = — = — = - ? , 32 9 V3 where gi and #2 are gauge coupling parameters of the U(l) and SU(2) groups, respectively. Thus 0w = 7r/6. As noted in our discussion of the trace, one-form (gauge) fields are J1 = A1, J = 1,2,3, and Js = B, and zero-form fields are JJ = <j>J, J = 4,5,6,7, where V2>± = J*^iJ5, v/2>° = J6 - iJ7, y/2$> = J 6 + ij7. We thus write the superconnection in the form J = i\
^
•
(34)
Here, r 0 = fia, a = 1,2,3, are the Pauli matrices, 1 is a 2 x 2 unit matrix, and Prom (1), (6) and (34), the supercurvature T is given by FA - 75 F B + 2i$$t
y/2(d$ + iA$ +
i^B$) (35)
\/2{d& - i&A - » ^ B # t )
_^F
B
+ 2i$t$
where FA = [d^a + i(A A A)a]ra ,
with
A = AaTa ,
a = 1,2,3,
and FB = dB-1. Introduction of coupling parameters goes as usual. We assign a dimensionless parameter g for the superconnection (i.e. J —» jJ"). We then rescale the squared curvatures term in the Lagrangian by \ : ^ T r ?-r=
|(JFU). A (F A ): + \FB AF£ - (!>*)* A (£>$)* 4- A(# f $) A (***)* ,
where * denotes taking a dual of a differential form, D$ = d$ + ig.A$ + i—=gB$ . v3
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D. S. Hwang, C.-Y. Lee & Y. Ne'eman
Applying the value fixed by the supergroup's anticommutators, we have A s 2y2 .
(36)
If we write C = itijCj,
j = 1,2,3,8,
C =
ifijCj,
£ =
ifijbj,
then the remaining terms in (32) become ~\j+
• (df )* - \dC • (DC)* + ±a£ • £* = -Aj A (dbj)* - do, A [DCJY + ^abj A b) ,
where DCJ = do, + ig[A, c]j,
with A = Ajt,,
j = 1,2,3,8.
The quantum action SQ = / M< £Q is thus given by SQ = y „»)t(Z>M*) - A($t$)2j ,
(37)
where jf=l, 2,3,8 and (i, v are Lorentz indices with metric 9M„=diag(l, - 1 , - 1 , —1). 5. Kinematical and Dynamical Evaluations of the Higgs Mass We note that the SU(2/1) curvature contains, in its main diagonal, both the SU(2) x U(l) field strength and a quadratic term in $, whose square (in the Lagrangian) will reproduce the quartic term of the Higgsfieldpotential £($), with the SU(2/1) fixing the value of the coupling A relative to the SU(2) gauge coupling g2 = g:b \ = 2g2,
L($) = - K 2 * t $ + A ( $ t $ ) 2 .
(38)
The mass of the Higgs field, as related to that of the W bosons gauging SU(2), is given by21 (M(*)) 2 = %(Mw?
= 4(MW?,
Jf (*) = 1MW,
(39)
b The quadratic term in * in the Lagrangian can be generated either by hand or through the Coleman-Weinberg mechanism. Note that the mass squared ratio of Mw and M* does not depend on *c2.
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BRST Quantization of SU(2/1) Electroweak Theory in the ...
3519
where we have applied the value of A as given by (36). The derivation of this result applies the conventional spontaneous symmetry breakdown dynamical calculation, as given for instance in Ref. 21, with the symmetry only providing in addition the value of the quartic coupling, relative to the SU(2) gauge coupling. On the other hand, the above result coincides with that obtained by one of us, 22 in a kinematical calculation, which assumed SU(2/1) to act as a global, low energy symmetry. In that derivation, the mass operator is taken — by the Wigner-Eckhardt theorem — to be proportional in its matrix elements to those of the generator UQ, since we have for the vacuum expectation value v = (0\
= ^2(XKMXNHXNMXK)
= n2J2 |c*w|2 •
N
(40)
N
For the gauge and Higgs boson masses, the K, N indices relate to the adjoint representation of the superalgebra; so does the index "6" of the operator. For the W, Z and photon masses, K = 1,2,3,8 and N = 4,5,6,7; this result then coincides with that of the conventional calculation, in which the squared masses are generated by squaring the isospin hypercharge covariant derivative of
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D. S. Hwang, C.-Y. Lee & Y. Ne'eman
roughly. We may therefore ask: Is there an energy level g2 = E„ at which the symmetry becomes precise? The answer is in the affirmative, with E„ ~ 5 TeV. At this stage in the development of internal supergroup gauging (including the approach originating in noncommutative geometry), we can only speculate about the significance of Et. Presumably, this is the level at which a larger symmetry structure breaks down, yielding our simple unified SU(2/1). Such a structure could perhaps be (1) a larger supergroup as in Ref. 13, including a mechanism reproducing the generation structure, breaking down (with yet another Higgs set?) to yield SU(2/1) (the latter then breaking down at 100 GeV). In the conventional systematics of GUT, we "jump" from 1016 GeV straight down to 100 GeV, but here we have at least one [SU(2/1)] and probably several intermediate algebraic structures; (2) yet another possibility might involve a hypothetical larger algebraic structure unifying our "internal supersymmetry" with the more conventional Poincare supersymmetry, the latter being expected to apply roughly at TeV energies, should it indeed be responsible for the preservation of hierarchy, namely canceling the radiative correction mechanism which would have the 100 GeV spontaneous symmetry breakdown climb up to JSGUT-
Having evaluated E„ we can now invert the procedure, to estimate the renormalization effects for A. We thus assume the supergroup value to hold at the energy E, — 5 TeV and evaluate the correction for A at E ~ 100 GeV. This corrected value can then be used to re-evaluate the predicted Higgs mass, i.e. obtain the value of that mass after the inclusion of renormalization effects. This is then our strategy in what follows. The coefficients of the renormalization group equation depend only on the field contents of the theory, which is the same as in SU(2) x U(l). c We can therefore use the results from the Standard Model. For the gauge couplings, RGE's are given by23
IOT = raF+26iIn^' *' = 1'2,3'
(41)
with
62 =
i2^(-^4 + y)-
c
We have not discussed matter fields in our expression for the action, but they can be introduced, including Yukawa couplings, in the usual manner through the covariant derivative with superconnection for (fermionic) matter fields. See, however, our discussion in Sec. 1, for problems of interpretation at the supermultiplet level.
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where Ng is t h e n u m b e r of generations, a n d
W-TS?B[U-{%*+\*+1*)*]-
<42>
where gt, A denote the top quark Yukawa and Higgs couplings, respectively. We used MATHEMATICA to solve these equations numerically. First, the renormalization of Ow We first obtained the unbroken scale E, = 5 TeV d of SU(2/1) using (41), by finding the scale where g-z = V$9i- Then we solved Eqs. (42) and (43), setting the boundary conditions so that A = 2p| at Es = 5 TeV and Mt = 174 GeV in the low energy range (E ~ 100 to 200 GeV). We assumed three generations (Ng = 3), and used sin2 6W{MW) = 0.229 ± 0.005,23 with a^ 1 = 128.80 ± 0.05, a«l = 29.5 ± 0.6, a r 1 = 8.332, where a" 1 = *f and 4- = \ + -V In solving the equations, we used the relation gt{M) = ^Mt = 2«-Mti where v is the vacuum expectation value of $. We then obtained a Higgs meson mass of 130 ± 6 GeV.e Acknowledgments C.-Y. L. would like to thank G. Kim for helpful discussions on RG calculation. This work was supported in part by the KOSEF through the SRC program of SNU-CTP, and in part by the Basic Science Research Institute Program, Ministry of Education, BSRI-94-2442. References 1. Y. Ne'eman, Phys. Lett. B 8 1 , 190 (1979). 2. D. B. Fairlie, Phys. Lett. B 8 2 , 97 (1979). 3. Y. Ne'eman and J. Thierry-Mieg, in Differential Geometrical Methods in Mathematical Physics (proc. Aix-en-Provence and Salamanca 1979 international conferences), eds. P. L. Garcia, A. Perez-Rendon and J. M. Souriau, Lecture Notes in Mathematics, No. 836 (Springer-Verlag, Berlin/Heidelberg/New York, 1980), pp. 318-348. See also: M. Scheunert, W. Nahm and V. Rittenberg, J. Math. Phys. 18, 146, 155 (1977); M. Marcu, J. Math. Phys. 2 1 , 1277, 1284 (1980). 4. Y. Ne'eman and J. Tbierry-Mieg, Proc. Natl. Acad. Sci. U.S.A. 77, 720 (1980). 5. J. Thierry-Mieg and Y. Ne'eman, Nuovo Cimento A 7 1 , 104 (1982). 6. J. Thierry-Mieg, J. Math. Phys. 2 1 , 2834 (1980); ibid. Nuovo Cimento A 5 6 , 396 (1980). 7. J. Thierry-Mieg and Y. Ne'eman, Proc. Natl. Acad. Sci. U.S.A. 79, 7068 (1982). d
We obtained this value for sin2 $w — 0.229 at the Mw scale. Here the error bounds correspond to the experimental error bounds of sin2 8yy; sin2 0w(Mw) 0.229 ±0.005. e
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3522 D. S. Hviang, C.-Y. Lee & Y. Ne'eman 8. 9. 10. 11. 12. 13.
14. 15. 16. 17. 18. 19.
20. 21. 22. 23. 24.
Y. Ne'eman and J. Thierry-Mieg, Phys. Lett. B 1 0 8 , 399 (1982). C. Y. Lee and Y. Ne'eman, Phys. Lett. B 2 6 4 , 389 (1991); ibid. B 2 6 9 , 477 (1991). G. Curci and R. Ferrari, Nuovo Cimento A 3 0 , 155 (1975). D. Quillen, Topology 24, 89 (1985). S. Sternberg and J. Wolf, Thins. Am. Math. Soc. 238, 1 (1978). Y. Ne'eman and S. Sternberg, Proc. Natl. Acad. Sci. U.S.A. 8 7 , 7875 (1990); and in Symplectic Geometry and Mathematical Physics — Proc. Int. Conf. Diff. Geom. Meth. in Phys. (Marseilles, 1990), eds. P. Donato et al. (Birkhauser, Boston, 1991), pp. 326-354. R. Coquereaux, G. Esposito-Farese and G. Vaillant, Nucl. Phys. B 3 5 3 , 689 (1991); R. Coquereaux, Phys. Lett. B 2 6 1 , 449 (1991). A. Connes, in The Interface of Mathematics and Particle Physics, eds. D . Quillen, G. Segal and S. Tsou (Oxford University Press, Oxford, 1990). A. Connes and J. Lott, Nucl. Phys. (Proc. Suppl.) B 1 8 , 29 (1990). C. Y. Lee, D. S. Hwang and Y. Ne'eman, t o appear in J. Math Phys. M. Quiros, F. J. De Urries, J. Hoyos, M. L. Mazou and E. Rodriguez, J. Math. Phys. 22, 1767 (1981); L. Bonora and M. Tonin, Phys. Lett. B 9 8 , 48 (1981). L. Baulieu and J. Thierry-Mieg, Nucl. Phys. B197,477 (1982); ibid. B 2 2 8 , 259 (1983); see also: R. Ore and P. van Nieuwenhuizen, Nucl. Phys. B204, 317 (1982) for a more geometric version. L. Corwin, Y. Ne'eman and S. Sternberg, Rev. Mod. Phys. 47, 573 (1975). E. S. Abers and B. W. Lee, Phys. Rep. C 9 , 1 (1973). Y. Ne'eman, Phys. Lett. B181, 308 (1986). L. Maiani, in Proc. NATO Advanced Study Institute on Z° Physics (Cargese, 1990), eds. M. Levy et al. (Plenum, New York, 1991). A. Sirlin and R. Zucchini, JVucI. Phys. B 2 6 6 , 389 (1986); H. E. Haber, lectures given at TASI-90 (1990).
378
Internal Super symmetry, Superconnections, and Non-Commutative Geometry Yuval Ne'eman** Sackler Faculty of Exact Sciences, Tel-Aviv 69978 Tel-Aviv, Israel
University
The gauge "internal" symmetries" which span the Standard Model, in the Physics of Particles and Fields, partly involve a mechanism of Spontaneous Symmetry Breakdown (SSB), whose present formulation (by P. Higgs, T. Kibble, P. Anderson and others) was inspired by phase transition mechanisms in the Physics of Condensed Matter. T h e role of the order parameter, breaking the internal symmetry, is assumed to be filled by Lorentz-scalar fields. However, whereas gauge bosons in ordinary gauge theories correspond to connections in Fibre Bundle geometries and thus obey severe algebraic and geometrical constraints, the SSB scalar fields seem arbitrarily "put in by hand". Several suggestions have been made since the Seventies, all aiming at an appropriately constraining geometrization of the scalar fields. In these lectures, I review three such related approaches: (1) internal supersymmetry, (2) the superconnection, (3) non-commutative geometry.
T H E A I M : G E O M E T R I Z A T I O N OF S P O N T A N E O U S S Y M M E T R Y B R E A K D O W N IN G A U G E THEORIES T h e S t a n d a r d Model ( S M ) — t h e 1974 ' g r a n d synthesis' of t h e Physics of Particles a n d Fields—is a Relativistic Q u a n t u m Gauge Field T h e o r y ( R Q G F T ) , w i t h t h e reducible group SU(3)color
® [SU(2)LWI
x
U(1)WY]
as a local gauge g r o u p (we denote by ' L W F t h e Left-chiral W e a k Isospin a n d by ' W Y ' t h e W e a k Hypercharge). T h e r e is, however, a n i m p o r t a n t difference between t h e various c o m p o n e n t s of the gauge g r o u p . T h e 'color' SU(3) gauge group of Q u a n t u m C h r o m o d y n a m i c s ( " Q C D " ) is formally a n u n b r o k e n Wolfson Chair Extraordinary in Theoretical Physics ^ Also on leave from Centre for Particle Physics, University of Texas, Austin, Texas 78712, USA * Member of El Colegio Nacional In contradistinction to General Relativity, gauging spacetime symmetries. © 1996 American Institute of Physics
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symmetry 6 , whereas the Electro-Weak SU(2) x U(l) undergoes a symmetry breakdown, involving an explicit 'spontaneous' mechanism of the Higgs-Kibble type with an a priori presence in the Lagrangian. This particular mechanism was inspired by the Landau theory of phase transitions, in which the free energy is given by an expansion in terms of, e.g., the square of the magnetization vector \M\2 (in the example of the paramagnetic phase transition, which occurs when one crosses the Curie temperature T c ), F = A\ M | 2 + B\ M | 4 + . . . , withal = a(T-Tc),
B>0.
(1)
Above Tc, A is positive and the minimum free energy is attained at | M | = 0 , where F = 0. Below T c , however, with A < 0, the minimum (dF/d\ M | 2 = 0) occurs at \ M \2 = \A \/2B, F - -A2/4B, i.e., at both values M ± ( | A 1/2B)1'2. Replace the (spatial) magnetization vector by an internal symmetry representation vector $ (a complex two-component field in SU(2)xU(l)) and you get the Higgs mechanism. In Relativistic Quantum Gauge Field Theory (RQGFT), all gauge bosons (in an unbroken gauge symmetry G) are massless, to start with. The Higgs mechanism now provides those gauge bosons coupled to the (broken symmetry) quotient G/H, H the surviving subgroup, with masses. Here ('EM' stands for electromagnetism) the gauge fields coupled to (SU(2)LWI X U(1)WY)/U(\)EM, acquire masses, leaving only the photon massless (3). The predictions of the Electroweak model have been beautifully confirmed, except that the theory says nothing with respect to two parameters: the Weinberg angle 6\y relating the ("universal") gauge couplings, "g" for SU(2) and "g' " for U(l), with tan6w = g'/g (experimentally, at 100 GeV, sin26w = .23), and the A coupling in the quartic term of the Higgs (scalar field) potential, in the Hamiltonian density H = | D $ | 2 -{nf
| $ | 2 +A | $ | 4 .
(2)
This is the same as 1), with M *-*• $ , — | A \ •-> —fi2,B *-* A. Thus the minimal values for the SSB case Mmin i-* (0| | $ ° | |0); however, it is the (real) $ J component in $ ° = ( l / \ / 5 ) ( $ 5 + i$°) which has a nonvanishing vacuum expectation value, namely (0|$i|0) = : v = (/i 2 /2A) 1 / 2 and we get for the relevant vacuum expectation value v/\/2. The A coupling is thereby indirectly related to the value of the mass of the surviving degree of freedom of the Lorentz-scalar Higgs field, the quartic providing a contribution A| $ | v2. The phenomenology relates v to the Fermi constant, through GFJ\/2 — g2/8Mw2 = l/(2v2), with the W± mass Mw = gv/2, b
I use the term 'formally' because in the search for a mechanism for color confinement, various constructions, resembling Higgs', have been proposed, involving either the breaking of SU(3), or of its Z(3) discrete subgroup, at times working with an 'effective' Higgs scalar, produced as a condensate, whether of quark-antiquark pairs or of pairs made up of some non-perturbative solutions, such as magnetic monopoles (2,9).
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(M*)
= —~r- •
(3)
Geometry first entered Physics in 1908 (4,5), when Minkowski explained—in a lecture to the joint meeting of the German Natural Scientists' and Physicians' Associations in Cologne—that what his former student Albert Einstein had discovered in 1905, in his momumental paper on the Special Theory of Relativity (SR), simply amounted to the fact that spacetime constitutes a flat pseudo-Euclidean 4-manifold. This also opened the way for Einstein (helped by Marcel Grossmann) to produce in 1915 a theory of Gravity (the "General Theory of Relativity"—GR) from the extension of this geometry to curved Riemannian manifolds. Note that the titles, as selected by Einstein for both his theories, reflect his emphasis on the geometrical gauge invariance principles at their respective foundations, with all the rest—our coordinate frames in space and time—being "relative", i.e., corresponding to some arbitrary gauge choice.0 The relevant groups are the global Poincare group for SR and a superposition of the group of local diffeomorphisms (as a passive symmetry) and the local Loreniz group (as an active symmetry) for GR—to which we would now have added the system of local Lie Derivatives as the active complement {shift and lapse) functions, to local diffeomorphisms, i.e., local translations, not a Lie algebra. Gravity (at the classical level) thus appeared to have opened the door to a geometrical realization of physics. Weyl and Eddington immediately tried to incorporate electromagnetism, with Einstein himself dedicating his last twenty years to this effort. However, between 1915 and 1975—for sixty years— the Physics of Particles and Fields appeared to be a realization of Quantum Mechanics which did not involve geometrical concepts. In the Sixties, Group Theory entered and played an important role, but still, just as a global (i.e., constant parameters) symmetry. Hermann Weyl had tried in 1919—unsuccessfully at first—to extend the geometrical approach to electromagnetism (5). Ten years later, after London had identified the role of the complex phase in the Quantum Mechanical treatment of electric charge, Weyl found local U(l) as the gauge invariance group of QED (7). This was also the first sample Fibre Bundle (FB) geometry to be found to play a key role in Physics, with its second-quantized version (QED) reaching completion between 1946-48. Yet it took till 1953 before C.N. Yang and R.L. Mills (8) extended the theoretical construction to NonAbelian local gauge groups, then till 1971 before G. 't Hooft completed the The titles angered the Marxist philosophers in the USSR, almost to the point of excommunication; Fock had to stress, in the introduction to his book on GR, that this was a correct theory of Gravity, even if its title represented a bourgeois misnomer and should simply be disregarded (6). Note that he also 'protected' Quantum Mechanics, by blaming Bohr's 'bourgeois ideas' for the apparent clash with dialectical materialism in the Copenhagen interpretation.
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quantization process, proving the theory's renormalizability (9) (in both an unbroken Yang-Mills and the SSB-broken mode of the Electroweak)—and till 1973 to discover the theory's asymptotic freedom and thus its applicability t o the Strong Interactions, hence QCD. The study of non-perturbative solutions of these theories' equations (e.g., instantons), brought out the importance of the geometrical approach, as in GR (10). Spontaneous Symmetry Breakdown was suggested in 1967-68 by S. Weinberg and by A. Salam, for the quasi-unification of the Weak and Electromagnetic Interactions. It scored highly when the W and Z mesons were discovered at the predicted masses in 1982-83, after having already passed a first test with the discovery of the electrically-neutral Weak currents in 1973. Scalar (Higgs) fields find their place in a Fibre Bundle geometrical description of the gauge theory, provided certain algebraic conditions hold, between the specific gauge group which is to be spontaneously broken and the wanted residual unbroken gauge symmetry subgroup ("the reduction theorem") (11). Under these conditions, one gets very intriguing non-perturbative solutions, namely magnetic monopoles (12). Specifically, however, the case of the Electroweak SU(2)® U(l) with the specific Higgs field leading to the masses and other parameters of the SM (a spinor under the Weak Isospin SU(2)) does not fulfill the necessary conditions. Thus, it does not fit into the above mentioned nontrivial incorporation into the Fibre Bundle geometry, which therefore cannot be taken as a criterion for a geometrical constraint on the scalar field. Several ideas have been put forward—since the emergence of the SM—all aiming at the obtention of a geometrical origin for the Higgs fields, in local gauge theories with SSB. One of the earliest interests in supersymmetry, for instance, and later in supergravity, derived from a search for an algebraic or a geometrical constraining of the SSB Higgs fields. In practice, the hierarchy paradox, i.e., the instability of SSB when occuring twice at different energies (as required in GUT), appears to involve supersymmetry anyhow, as a dynamically stabilizing device. In (Poincare) supersymmetry, however, the algebraic constraints on the Higgs fields only consist in linking them to some (as yet unseen) fermion fields ("Higgsinos"), which is not what we are after. In Supergravity, scalar and pseudoscalar fields do appear in the extended versions, with N > 4, and in particular for the N = 8 case (with 70 scalar and pseudoscalar fields) (13). Work on the latter, first built in eleven dimensions, with an assumption of spontaneous compactification (14), led to the renewed interest (in the early Eighties) in Kaluza-Klein geometries (15). One basic ansatz for such higher dimensionality embeddings was "Dimensional Reduction", as formulated by Chapline and Manton (16). Here, the Lorentz-scalar fields originate as components of gauge vector mesons corresponding to dimensions which become compactified in the reduction process. In Kaluza-Klein approaches, such dimensions generate the internal degrees of freedom, but in the Chapline-Manton method, the gauge group is part of the input, as a Structure Group of a Fibre Bundle. It is this bundle's base manifold which is assumed to be of a dimensionality higher than 3 + 1 in the pre-reduction
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stage. In these lectures, I shall review a related sequence of such models, involving three different mathematical formulations. (a) Internal Supersymmetry (17,18). (b) Superconnections (19-21). (c) Non-Commutative Geometry (22). Internal Supersymmetry was conceived in two unrelated independent approaches, one (17) stemming from advances in the geometrical interpretation (23) of the ghost fields (24) and the BRST algebra they span (25) (guarantying gauge theory's unitarity); the second approach (18) emerged as a further development of Dimensional Reduction (16). The method of the Superconnection evolved in Mathematics (19) and in Physics (20,21), as a generalization of the Fibre Bundle description of gauge theories, in which the structure group is a supergroup and is spanned by a Grassmann algebra. Non-commutative geometry is an extensive development in mathematics; for our purposes, the innovation is in the construction of gauge theories in which the base manifold involves discrete elements (22).
INTERNAL SUPERSYMMETRY: (1) ALGEBRAICS, (2) MATTER FIELD ASSIGNMENTS Supergroups, superalgebras and supermanifolds entered mathematics and mathematical physics in the Sixties and Seventies. I refer the reader to any of several comprehensive texts (26). In Ref. (27), I have given—without proofs— the key definitions and theorems, including the classification of the SemiSimple Lie Superalgebras, as completed by V. Kac (28) and instructions for their construction. Supergroups are matrix groups, whose elements are valued over a Grassmann algebra A, i.e., the elements are Z(n) (and Z(2)) graded. We denote the Z(2) split as being into w-even and to-odd subalgebras, A = A + © A - . The elements of a Grassmann algebra obey x£ A ' , y G Aj,xye Ki+}; xy= (-lyiyx; dxEAi+1, d2 = Q; d(xy) = (dx)y + (-iy+1x(dy);
(4)
where we have also defined Cartan's exterior derivative (or differential) d. The emergence of commutativity or anticommutativity (when both i, j are odd), as a result of the grading, brings in a correlation with quantum statistics, BoseEinstein (even) or Fermi-Dirac (odd). The supermatrix Q is Z(2)-subdivided into g-even (A, B) and g-odd (H, $ ) parts,
315
383
A,BeA+;
H,$EA",
A+DA"', A " DA*, mGZ+)
Vi = 2 m , Vi = 2m + 1,
Z+=(0,1,2,...).
(6)
Let Abe px p and B be q x q. We define a supertrace and str(Q) = (-l)v[TrA sdet(Q) =
-
superdeterminant:
TrB],
(detA).(detB(Q-1),
srfef(Q 1 Q 2 ) = sdei{Q1).sdet(Q2)
.
(7)
The supermatrix Q acts on a Z(2)-graded "carrier space", in which the first p rows have grading v and the next q rows have (v + l ) m o d 2 grading. We can now describe a superalgebra L. It is a graded linear vector space (dimension n), with a super-Lie bracket preserving the gradings,
i = £fL,-, [ U ; } = -(-l) ij '[/,-,/.-}, [U,-}cii+i,
ft, ft,/*}} = [P.-,/,•},4} + (-l)ijlh. ft, /*}} .
(8)
where we have also introduced an appropriate graded Jacobi identity. To go from the superalgebra to the supergroup, use a Taylor series for exponentiation, with the parameters valued over a Grassmann algebra, the uneven for the <7-even submatrices and the u;-odd for the jr-odd. Thus, Q
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384
Aside from the classical, the list includes (c,d) two infinite sequences of classical hyperexceptional superalgebras P(n), Q{n), n > 3, remotely related to the above. There are also the classical exceptional F(A), G(3) and an infinite family osp(4/2;a). The non-classical "Cartan-type" span the diffeomorphisms of A. Denoting the Grassmann "generators" (the elements of A 1 ) as 6a, 6a , . . . , the algebraic generators are given by "supervector fields" 0a6a • • • [ 9 ^ ] . There are four infinite sequences ti>(n), s(n), s(n), h(n). We shall now turn to the main example (17,18), namely the superalgebra sw(2/l). This is the super-unimodular (sdetQ — 1) case of s/(2/l), which is also homomorphic to osp(2/2) and to w(2). There are 8 parameters and a resemblance to su(3), except for strQ = 0 replacing TrQ = 0 in the algebra. Note that because of the nature of the Killing metric, there is a certain freedom in normalizing the algebraic generators—since it will always be compensated in the contravariant representation. We choose, nevertheless, to normalize by Tr (I%)2 = 2 as is customary for the Pauli matrices. Indeed, the superalgebra's basis LA splits in two: the fli-even matrices of the Lie subalgebra L+ (e.g., sl(m) ® sl(n)
0 0 I , Cs =
(9) The matrices have been normalized by the square of their traces; we shall return to this point in section Superconnections and Higgs Fields, where we d
I used the letter HA in Ref. (17) and further work; however, this tends to generate some confusion with the negative mass-squared parameter in the Higgs potential.
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discuss the algebraic constraints imposed by SU(2/1) on those parameters which are free under SU(2)xU(l). Denoting the g-even directions 1,2,3,8 by a,b,..., and the g-odd 4 , . . . , 7 directions by i, j , . . . , we have for su(2/l) the supercommutators [ia,h] = c*bic
=
2irabic,
[laJi] =
Ciilj=2ifiilj,
{li,li)
= 2diila,
= Ctila
(10)
where the flb,f3ai are SU(3) (totally antisymmetric) structure constants, and dfj are the familiar SU(3) anticommutation coefficients (symmetric in all three indices) in the defining (triplet) representation (29). The difference between the eighth matrix in, SU(3) [diagonal [l/-\/^](l, 1, —2)] and [diagonal [ l / \ / 3 ] ( - l , - 1 , - 2 ) ] in SU(2/1), restricts the symmetry of the C£ to the first two indices.Thus, writing the superalgebra's structure constants as (ABC) for C%B, and marking total antisymmetry by (•• •)" and symmetry in the first two indices by (• • •)+, they are (123)- = 2 i , ( 1 2 8 ) - = 0, (461)" = (571)~ = (452)" = (562)- = (463)" = (563)" = (473)- = (573)- = (468)- = (478)- = (568)" = (578)- = 0 . (471)" = - ( 5 6 1 ) - = (462)- = (572)" = (453)" = = -(673)- =
i/2.
(458)" = (678)" =
^ .
(461)+ = (571)+ = -(472)+ = (562)+ = (443)+ = (553)+ = -(663)+ = -(773)+ = 1/2. (448)+ = (558)+ = (668)+ = (778)+ = - ^ .
(11)
all other (ija)+ = 0. The selection of SU(2/1) as a constraining superstructure for the Electroweak [SU(2) x U(l)] C SU(2/1) is suggested at two different levels, that of the matter fields (leptons and quarks), with chirality providing the Z(2) (/-grading—and that of the Yang-Mills gauge vector and Higgs scalar fields, with a iw-grading, defined by the forms, i.e., by r + , r ~ . We start with the matter fields. Let me remind you of the quantum nunbers of the matter fields. In each generation we have fifteen (or sixteen) fields. Ncoi is the dimensionality of the SU(Z)coior representation and r,y,b are the three color states "red", "yellow" and "blue". The basic correlation here is the Weak Interactions analog of the GellMann-Nishijima rule, Qem
=
lw + Yw/2. 318
386
(12)
Field leptons
"l e-L
en quarks U
R
U
R
uR
"I < <
dl
4 4 d R
dR dR
QBM
1
ILW
1 I3w
YW -1 -1 -2
0 -1 -1
1/2 1/2 0
1/2 -1/2 0
2/3 2/3 2/3 2/3 2/3 2/3 -1/3 -1/3 -1/3 -1/3 -1/3 -1/3
0 0 0 1/2 1/2 1/2 1/2 1/2 1/2 0 0 0
0 4/3 0 4/3 0 4/3 1/2 1/3 1/2 1/3 1/2 1/3 - 1 / 2 1/3 - 1 / 2 1/3 - 1 / 2 1/3 0 -2/3 0 -2/3 0 -2/3
Ncol color 0 0 0
0 0 0
3 3 3 3 3 3 3 3 3 3 3 3
r y b r y b r y b r y b
TABLE 1.
Note that as SU(2)LW acts only on the left-chiral fields, the fact that therefore Tr(A) = 0 in the upper left submatrix of Eq. 5) also implies (since the other diagonal submatrix B — 0, as there is no action on the right-chiral part of the matter fields' carrier space) , we have in fact str(Ii,w) — 0, including of course str(I^r) = 0. The second important point is that whenever the electric charge does not vanish, it has to appear equally in the two chirally-graded sectors, i.e., if there is a uyL, there also has to be a uyR. As a result, str(Qem) = TrA(Qem)-TrB(Qem) = 0. Applying now Eq. 11) we have also str(Yw) = 0, i.e., the sum of the U(l)w eigenvalues of the weak hypercharges vanishes. Thus, the generator algebra of the electroweak SU(2)x U(l), when embedded in the minimal superalgebra, produces supertraceless supermatrices—provided the g-grading is given by chirality. The supergroup SU(2/1) is of rank = 2 and we have just seen that our two diagonal operators are supertraceless and may thus be used as a basis for the Cartan subalgebra of SU(2/1). Summarizing, we may state that the matter fields, leptons and quarks, as fixed by the phenomenology (namely the SU(2)xU(l) assignments), separately and naturally span representations of SU(2/1). Indeed,
Ilf = \C3,
Yw = V3Cs •
319
387
(13)
As a matter of fact, the leptons span the defining three-dimensional representation of SU(2/1), while the quarks span (30) the other fundamental representation, the one which defines the isomorphic OSp(2/2). The representations have been studied and classified (31) by Scheunert, Nahm and Rittenberg (SNR) and are given by two quantum numbers—the SNR choice consisting in the highest Jjy eigenvalue and the Yw/2 of that state (though written in inverse order). Thus the leptons are in (—1/2,1/2), the antileptons in (1/2,1/2) (the Yiy/2 and 1% of el"), the quarks in ( 1 / 6 , 1 / 2 ) , and the antiquarks in ( - 1 / 6 , 1 / 2 ) . The matrices of the 4-dimensional representation are
6=
/0 0 0 \0
0 0 0\ 0 10 10 0 0 0 0/
6=
/0 0 0 0\ 0 0 -i 0 0 i 0 0
,6=
\0 0 0 0/
/ 4/3 0 0 0 1/3 0 & = - /V3 5 0 0 1/3 \ 0 0 0 1_
/
fe =
^
0 -y/2 0 0
/0 0 0 0 0 1 0 0 0 0 - 1 0 \0 0 0 0
V2~ 0 0\ 0 0 0 , & = 0 0 1 0 10/
/
V3 \
0 \ 0 0 2/3/ 0 0 -zV2 0
0 -iy/2 0 \ 0 0 - 1 0 0 0 i 0 0 /
0 iy/2 0 0 \ iy/2 0 0 0
0 0
0 0
0 —i i 0 )
(14)
The (1/6,1/2) representation is not Hermitian, but one can define starHermiticity (31). First, we apply contravariation, i.e., raising the algebraic indices with the Killing metric. For SU(2/1) this metric has the coefficients (A,B) = gAB, (A,B) (8, 8) = - 1 / 3 ,
= 6£,
A,B<= [1,2,3],
(4, 5) = - ( 5 , 4 ) = (6,7) = - ( 7 , 6 ) = i.
(15)
All other coefficients vanish. Another operation we have to define is supertransposition. Denoting ordinary matrix transposition by ~ we have for the supermatrix 5) the supertransposition A E tf B
A~ - $ ' E~ B~
320
388
(16)
and Star-Hermitean conjugation, (QA)* — {QA)T• This defines starHermiticity, i.e., (QA)* - QAThe four-dimensional (1/6,1/2) quark representation has a free parameter, namely a constant which can be added to the main diagonal. However, should this addition of a constant result in integer values for Qem, o n e row and one column will vanish and the four-dimensional representation reduces to the three-dimensional one. Indeed, substracting 4 / 3 from £s reduces it to £s—though sitting in a 4 x 4 matrix, i.e., a picture in which the first row and column describe the non-interacting VR. The supergroup thus predicts that there should be four states when the charges are fractional (quarks, after factoring out SU(3)Coior), whereas the integer charge states (leptonsj should span a three-dimensional representation. Clearly, the SU(2)xU(l) quantum numbers of the matter fields fit beautifully with the SU(2/1) fundamental assignments. And yet, there is a conceptual difficulty: the odd generators should connect bosons to fermions and vice versa, but the chiral Z(2) grading does not reflect a difference in statistics! I had originally suggested a hypothetical answers for this problem (30) but I harbour doubts in this case. The solution (30) would double the representations. In the one where the left-chiral fermion states i>L are physical, the right-chiral states are compound ghost bosons, namely the right-chiral fermions' BRST transforms sipL = [2, TPR] — XR, where H is the FeynmanDeWitt-Faddeev-Popov ghost (24), a Lorentz-scalar fermion. The x a r e bosonic Lorentz spinors, because they result from the product of a spinor physical fermion by a scalar ghost fermion. The SU(2/1) representation thus preserves Lorentz invariance. In the statistics-conjugate representation, the rpn are physical, whereas the %I?L are replaced by the analogous bosonic XLThis doubling solution is thus conceptually perfect, but there are difficulties in including the new bosonic ghost fields XL,XR m the field theory dynamics. Ghost fields were originally introduced to preserve unitarity, by cancelling the unphysical components of a massless Yang-Mills field. Here, there are no unphysical components to cancel, and the new XL,XR might cancel physical fields. This is at the moment a weak point, though it might be possible to resolve it. The other suggestion (32) exploits the theory's chiral nature. The key argument consists in noting that all matter fields listed by the table XII are Weyl spinors, i.e., sitting in either (1/2,0) (left-chiral) or (0,1/2) (rightchiral). The notation we use is the conventional one (JL,JR) for the Lorentz group, in which (J is the angular momentum, K the Lorentz boost generators), JL = —i(J + K),JR = —(J — K). Thus, to the extent that we deal with left-chiral operators, they are only "sensitized" to the first eigenvalue, jx, — \I1 for VL,eL,uL,di and JL — 0 for VR,eR,v.R,dR. In this context, they indeed connect fermions to "bosons", when going from a left-spinor to a rightspinor. Similarly, an operator which is right-chiral would, for instance, turn j — R = 0 into JR = 1/2, i.e., connect a "boson" (the left-spinor) to a fermion (the right-spinor). The odd generators are coupled to /? = j 4 in the Dirac
321
389
Field VR VL
e-L eR UR UL
dL dR
JL
YL
h
0 1/2 1/2 0 0 1/2 1/2 0
1 0 0 -1
0 1/2 1/2 0 0 1/2 1/2 0
1
0 0 -1
n0
JR
1/2 0 -1/2 0 0 1/2 0 1/2 1/2 0 -1/2 0 0 1/2
1/2
YR
-1 -1 -1 -1 1/3 1/3 1/3 1/3
J
Yw
1/2 0 1/2 -1 1/2 -1 1/2 -2 1/2 4/3 1/2 1/3 1/2 1/3 1/2 -2/3
TABLE 2.
algebra (this is not the Dirac bilinear, in which it becomes a scalar), and thus belong to (1/2,1/2) and act precisely in this manner, i.e., connect "bosons" to "fermions" in each column of the Lorentz bracket. As this interpretation implies, the even part of SU(2/1) is left-chiral, behaving as (1,0) and preserving chiralities—while the odd part behaves as (1/2,1/2) and thus modifies the states' chiralities—thus also turning leftfermions into left-bosons and vice versa, and right-fermions into right-bosons and vice versa. For a smooth solution, we then have to separate the Yw eigenvalues into two contributions: the algebraically significant one YLW, carried only by left-chiral particles (as in ILW), and a constant "displacement" characterizing the representation and coinciding with the freedom of displacement for £g as mentioned above. In the above considerations we have applied the superunification machinery to SU(2/1) D SU(2)LWI x U(1)WY, leaving out S£/(3)eo/or- With Shlomo Sternberg (33), we found that S£/(5 + k/l) (k a non-negative integer) reproduces 2* generations of the 1 5 + 1 matter fields of the Standard Model. In particular, SU(7/1) (i.e., four generations) displays renormalization advantages, being anomaly-free (34). The physical implication is thus that there is a fourth generation at higher energies—which is still allowed experimentally, provided the fourth neutrino is very massive (its mass should be larger than 45 GeV). Notice that the representations of the SU(n/m) supergroups have also been applied to nuclei (35) in a supersymmetric extension of the Arima-Iachello 'IBM' symmetry. Now to the gauge supermultiplet. To understand how SU(2/1) emerged from this end, we need to review first the geometrical interpretation of the ghost-fields and of the unitarity-preserving Becchi-Rouet-Stora-Tyutin superalgebra, discovered by Jean Thierry-Mieg in his thesis (20,23,36). He showed that the ghost-fields (24) correspond to the vertical part of the connection in a Principal Fibre Bundle and that the BRST equations (25) are simply the
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Cartan-Maurer structure equations of the bundle, just stating that the curvature two-form is entirely horizontal. However, since this aspect thus relates entirely to the connection, we shall discuss it in the next section. DIGRESSION: THE GEOMETRY OF FIBRE BUNDLES, GHOSTS A N D BRST We use the language of forms on a Principal Fibre Bundle geometry V = {M,G,ir,x), M the m-dimensional base manifold (generally spacetime, or a submanifold of spacetime), Q the (gauge) structure group, ir the vertical projection and x a multiplication on the right by elements of Q, i.e., a map x : V x Q -> V .
(17)
The projection is equivariant under this group action, -^M,
TT-.V ,
Vp€V,Vg,g eG, (pxg)xg'=px
r(p x g) = TT(P) , (gg').
(18)
The multiplication in 19) is in fact a map r, from the abstract generating Lie algebra of G into V*, the tangent manifold to V• Denoting elements of this (n-dimensional) Lie algebra L by A, i.e., [Aa,Aj] = Ccah\c, we have r : L — V, , A — A € V. , [A a ,A 6 ] L B = |Aa,A&
^ ,
(19)
where the "LB" bracket is the abstract Lie commutator, while the "VFB" bracket is the vector-fields bracket in which the differential operators differentiate each other's coefficient functions. The map r has no inverse, since it maps a manifold whose dimensionality is dim(L)=n, into a manifold whose dimensionality is dim('P) = dim(L) -I- d i m ( X ) = n + m. This is where the connection u comes in: it maps back u : -P, — L ,
VA G L ,
w(A) = A ,
u
€ *V .
(20)
w is Lie algebra valued and belongs to the cotangent manifold, i.e., it is a one-form over *V, u> = u^Xadz11, where zR is a coordinate (patch) on V, i.e., (m-t-n)-dimensional. The valuing over the vector-field A is realized through the inner product (or contraction) dzR\ds = SR, where ds differentiates by zs. The connection is thus a map, present everywhere over the bundle and enabling us to recover the (vertical) group generators. Applying the ( m + n ) dimensional exterior differentiation d, we construct the curvature, a two-form
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£l=du
+ -[u,w].
(21)
The Cartan-Maurer structure equations for the bundle simply tell us that Q is pure-horizontal (see a proof in Ref. (36)), AJfi = 0 , i.e., if we draw a section (locally), the two-form will only involve over the base manifold dx11 A dx", fi, v = 1 , . . . , m. Denoting the by ya, the algebraic vector fields (locally) only involve f(x,y)-£z, why the J contraction with the curvature vanishes. This is the gauge invariance of the Maxwell (or Yang-Mills) field strengths. (locally) over a section, we can expand in x and y,
(22) differentials vertical zM which is well-known Projecting
d = d + s, d2 = 0 ~* d2 = 0 , s 2 = 0 , ds + sd = 0 . w = A + E, A = Aydx" , E = Eadya , Q = dA + sA + dZ + sE + \{[A, A] + [A, E] + [E, A] + [E, E]) =
dA+\[A,A]=:F.
sA = DE, 5S = - | [ S , S ] .
(23)
The last two equations, respectively imposing the vanishing of the dx*1 A dya and of the dya AdyP components, are the Cartan Maurer equations—but they are also identical with the BRST equations, once we identify the vertical oneform E with the ghost field (24), though these unitarity equations were derived from physical considerations. The "wrong" spin-statistics of the ghost fields were imposed in field theory in order to get a minus sign for closed loops—no incoming or outgoing ghosts—and thereby to cancel the contribution of the unphysical components of the Yang-Mills field. Here, they are realized through the ghosts' construction as one-forms, contracted with the dy differential, since we exist in x £ M. and not in y £ Q—whereas we do factorize the Ay, field, out of A — A/tdx11, having the Fourier transforms of Ay. acting on Hilbert space to create photons or Yang-Mills mesons. Note that the ghost fields Ea (we display the "coefficient function" of A a ) carry the same representation of the gauge group Q as do the Yang-Mills fields J4°, but they have one spin quantum less (scalars versus vector-mesons) and opposite statistics. Let me now remind you of the way in which the BRST constraints were originally discovered (25). The Yang-Mills gauge invariant Lagrangian is geometrical, though it does involve the metric over M. because of the Hodge dual, LYM = \F*J- ='• \J~2However, as the YM Lagrangian is not positive-definite, the Fourier transform of its inverse, which in RQGFT should have produced the propagator for the Yang-Mills field, fails to do so. This imposes the requirement
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of "gauge fixing", i.e., adding a term CGF = \C2, where C is an operator involving A and its derivatives. To this was added the ghost Lagrangian Cgh — —HMS. Taking as gauge parameter € = (HT), the BRST variation 6B is the gauge variation divided by the (anticommuting) constant T . The gauge causes 6BA — DE , 6BE = — (1/2)[E,H]. The entire quantum Lagrangian Cqu — C-YM + £GF + £gh becomes invariant, provided we impose SgS — C and select ME — 6BC, plus impose 6B2 = 0 when applied to either A,E. To this, we of course add the matter Lagrangian. This is the vector bilinear in the spinor field ipj^ii, basically a 3-form *j, with the dxpdp of the free Lagrangian replaced by D, the covariant differential, plus the (scalar) mass term, when there is one. Under BRST, the ip transforms as under the gauge group, with the ghost in the parameter, which is what we referred to in the last paragraphs of the previous section. Note that for chiral fields (as happens in the electroweak case) the D one-form acts separately on each chirality, whereas the scalar 0-form in the mass term connects the two chiralities. This will be the key to the understanding of the use of non-commutative geometry, which we discuss in the last section. Returning now to the entire system, the BRST equations replace gauge invariance, as they represent the symmetry of the complete quantum Lagrangian. In the geometrical interpretation, the BRST operator 6B now becomes s, the vertical piece of the exterior differentiation. The original BRST, as we showed, had in addition a variation for the antighost, clearly non-geometrical. The system was improved within R Q G F T by Curci and Ferrari (37) who developed an elegant "BRST-antiBRST symmetric" formalism. We showed (38) that this could be geometrized by doubling the structure group (or using the—mutually commuting—left-algebra and right-algebra of group generators, i.e., working with the double bundle V = (M,Q®Q, ir, x ) . Beaulieu and Thierry-Mieg then showed how to construct the quantum Lagrangian from this geometrical foundation (39). After this long digression into the geometry of gauge theories, we now return to the supergroups. S U P E R C O N N E C T I O N S AND T H E H I G G S F I E L D S A. The One-Form Solution. In Ref. (17), the gauging of SU(2/1) follows the usual convention. It is thus performed by an octet of vector-mesons, the four of SU(2)xU(l), namely W ± , Z ° , J 4 ° and four 5-odd K*-\ike fermionic new 1~ ghosts. As an octet of one-forms, notice that the overall grading (g x w) is odd for the g-even part and even for the <7-odd. In addition, this "YM" multiplet is accompanied as usual by a similar scalar formal ghost octet. However, whereas the ghosts for the g-even part are indeed the conventional ghosts appearing in the BRST transformations of the W±,Z°, A0 and behave as fermions, the "ghosts" for the g-odd part are bosons and fit the role and
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quantum numbers of the Higgs field components precisely. As a matter of fact, in this first version the theory can be defined as that theory in which the conventional ghosts and the Higgs fields together make a scalar field octet supermultiplet. The overall grading is the same as for the gauge multiplet, considering that fields (as one-forms) and ghosts are just different components of the same one-forms. Following the geometrical interpretation of the ghosts as the 'vertical' components of the connection, in a Yang-Mills gauge theory's Principal Bundle, the scalar Higgs field, entering the theory as a 'formal' ghost, is thus identified with a one-form in dy, with y here the supergroup manifold parameter—whether bosonic or fermionic is unimportant at this stage, since it is contracted within the one-form, e.g., $ ' = $*NdyN. The 'horizontal' components (in dx) of this 5-odd part Q~ of SU(2/1), however, appeared in the beginning to constitute an embarrassment for this first interpretation, since such fermionic vector-mesons are themselves ghost fields—which are not present in the conventional formalism. Their role as ghosts in a new formulation was, however, subsequently clarified in Ref. (38). Indeed, the presence of new ghost fields makes it possible, in this version of the theory, for the renormalization group equations to implement additional conservation laws, beyond those of Q+. B. The Two-Form Solution. In Ref. (20), and more successfully in Ref. (40), the scalar field is a 'ghost-for-ghost', in the 5-odd part Q~ of the supergroup. In this version, the BRST equations are replaced by the ghost-antighost symmetrized system, geometrically realizing the Curci-Ferrari construction (37), i.e., doubling the structure group manifold (Q
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grading, the remaining numerical matrix belongs to U(3). Sternberg and Wolf have studied algebras under both commutators and anticommutators (41) and this transition operates within such a system. This is relevant to the evaluation of 6w in SU(2/1), as we shall further see. C. The Zero-Form Solution and SU(2/1) Constrained Couplings. Yet a third interpretation for the role of the scalar field was proposed by Ne'eman and Sternberg (21), using Quillen's generalization ansatz, beyond the original constructs of Ref. (20). The scalar field is assigned to the same w-e\en part of the SU(2/1) superconnection, but this time as a zero-form. We note that the negative mass-squared in the SSB mechanism is the only parameter in the Lagrangian, out of three which are left free by S U ( 2 ) x U ( l ) , which is not fixed by SU(2/1) by itself. The other two are the coupling A of the quartic potential ( $ $ ) 2 , determined by the SU(2/1) gauge universality and tan#w = g'/9, which appears likewise determined, though its value depends on the normalization of the algebra's defining 3 x 3 representation, a fact to which we return in what follows. Anyhow, it should be noted that both A and tan $w are only given up to possible quantum radiative corrections. Since we still put in "by hand" the negative-squared-mass term of the Higgs Lagrangian, as a "trigger" for the spontaneous symmetry breakdown, the "spontaneity" is limited to the physical model, while the actual algebraic realization is not spontaneous. The internal "supergauge" is thus broken explicitly, a fact which leads us to expect its couplings to be susceptible to renormalization through radiative corrections. Note that the multiplet of Higgs fields, appearing in the g-odd submatrices of the superconnection, has an additional advantage in this zero-form solution: it has no vertical components, i.e., it does not add new ghosts; the scalars are rather more like matter fields. The SU(2/1) curvature contains, in its main diagonal, both the SU(2)x U(l) field-strength and a quadratic term in $ (resulting from the C," symmetric coefficients), whose square (in the Lagrangian) reproduces that quartic Higgs potential. This fixes the value of the A coupling relative to the SU(2) gauge coupling g, i.e., A = 2g2. The mass of the Higgs field, as related to that of the W bosons is thus
(M*? = f(Mw? = i{Mw)\ M$ = 2MW ,
(24)
where we have applied the value of A as given above. The derivation of this result applies the conventional spontaneous symmetry breakdown dynamical calculation, as given for instance in Ref. (3), with the symmetry only providing in addition the valve of the quartic coupling, relative to the SU(2) gauge coupling. On the other hand, the above result coincides with the one obtained in a kinematical calculation (42), which assumes SU(2/1) to act as a global, low-energy symmetry. In that derivation, the mass operator is taken— by the Wigner-Eckart theorem—to be proportional in its matrix elements to
327
395
those of the generator ^6, since we have for the vacuum-expectation value v = (0|$ 6 |0). For Dirac fermions, this operator connects the two chiral states, (V'LlCelV'fi) 7^ 0, provided both exist. For bosons, as it is not diagonal, we have to evaluate M 2 , {VK\M2\T)K)
=^2(r]K\C6\VN)(VN\<:6\m)
= n2^2\cK6N\2
N
.
(25)
N
For the gauge and Higgs boson masses, the K, N indices relate to the adjoint representation of the superalgebra; so does the index "6" of the operator. For the W, Z and photon masses, K = 1,2,3,8 and N = 4,5,6,7.This result coincides with that of the previous (conventional SSB) calculation, in which the squared masses are generated by squaring the isospin-hypercharge covariant derivative of
Bw = TT/6 .
(26)
With the symmetry thus explicitly broken (plus the difficulties with the matter field ghosts) there is room for radiative corrections and subsequent renormalization of A and 9 (and as a result also of M $ ) . In a recent calculation (43), renormalization group (RG) techniques were used, first to evaluate the variation of the "running" gauge coupling parameter for the SU(2)xU(l) even subgroup. We used this result to fix the value of q2 = Es at which tan 9\y can be expected to reach the SU(2/1) value of .25. Assuming that this represents an energy level at which SU(2/1) is precise, the same approach was
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used in the opposite direction for the "running" quartic coupling X(q2) to evaluate the correction to the Higgs mass. With the "top" quark mass at 170 GeV, the theoretical SU(2/1) value of M$ = 2MW = 170 GeV is lowered to 130 GeV. The same publication also contains the appropriate symmetric BRST-antiBRST equations and the complete quantum Lagrangian. D. The Superconnection Solution. Note that to the extent that we do not require a justification of the sin 2 9w = 1/4 value, we might take a different view of the zero-form solution and apply a formalism in which the entire superconnection has to be totally odd; the g-even subalgebra is indeed w-odd as a one-form, and the g-odd SSB piece is w-even, in order to have an odd total grading. This is in fact the approach of Ref. (21). This is also the most fitting approach for the applications of non-commutative geometry, and was used by Coquereaux et al. (44,45), while extracting the essentials in the application by Connes and Lott (22) of Connes' (46) and Quillen's non-commutative geometry (19). We note that the latter technique has the advantage of adding the notion of a matrix-derivative, for the discrete factor-space of the geometry, capable of supplying the negative squared-mass ^trigger" for the symmetry's spontaneous breakdown mechanism.
NONCOMMUTATIVE GEOMETRY As physicists, we have used matrices whose elements were themselves operators from the very first steps of R Q F T and especially since the successes of RQGFT. Mathematically, some of this was delicate going; it has only been since the advent of Quantum Groups that the full care and power of mathematics have been brought to bear on such calculations. Quantum Groups extend algebra by valuing the matrices over further algebras. Non-commutative geometry (NCG) is a closely related field, in which the notion of spaces and manifolds are extended to include discrete structures, matrix spaces. Basically, one is led to develop non-commutative versions of the calculus of forms (for non-commutative differential geometry), etc. I refer the reader to the mathematics literature (19,46). In this review, I shall only deal with the application of NCG to the SM (22,47), where it reproduces (44), the results which we have derived from SU(2/1). Coquereaux, Esposito-Farese and Scheck have analyzed the relationship and sketched it all out in Ref. (45). Note that the last five years have witnessed a flood of studies applying NCG to different issues, such as Quantum Gravity, non-commutative lattices, etc. Basically, it is the grading of the carrier space of the matter fields which has been reinterpreted in the NCG approach. The ^L and I^R subspaces of the carrier spaces in the representations (—1/2,1/2) (leptons, three-dimensional) and (1/6,1/2) (quarks, four-dimensional) we described in Sec. Internal Supersymmetry: ..., are generated "naturally" by replacing spacetime M by two copies of it X = ML U MR; in other words, spacetime M is multiplied by a
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set of two points. This is not very different from the usual way in which we visualize locally internal degrees of freedom, by direct multiplication by M, except that here the "internal" space is just two points. Our supermatrices thus describe the action in either copy (in the diagonal submatrices) and the distance between the two copies (in the off-diagonal submatrices). Nothing is said about a superalgebraic structure, creating the impression that we do not have to worry about the statistics, which might represent an important conceptual advantage, if mathematically consistent. Perhaps what is taken for granted is the chiral interpretation (32). As a matter of fact, as summarized by Cammarata and Coquereaux in the Bariloche lectures (48), it seems that the structure is so loose as to just reproduce the SM, with no added algebraic constraints, but with a conceptual elegance which is missing in the usual derivation. Other authors (49) assume a much stronger constraining—though also up to renormalization effects (and without worrying about the statistics of the matter fields). I am somewhat worried, indeed, about disregarding the statistics question, since the matrices of Ref. (48) do contain 5-odd elements (generalized forms over a discrete system) realizing the flf-odd matrices of the algebra—which is thus our SU(2/1). The mathematical construction which we discussed in section Internal Supersymmetry: ..., with respect to the statistics issue is in fact an intricate entanglement of SU(2/1) with the Lorentz group—just the issue which in the non-commutative geometry approach is being tackled by the above X base space, doubling spacetime. The two solutions appear either identical or complementary. Hence, we have adopted an alternative stance and have studied the SU(2/1) constraints—as further strengthened by the matrix derivative term from NCG, now providing a geometrical interpretation of the — /i2$<$ SSB trigger in the Higgs potential (50). To discuss the "two-copies version" of spacetime, we note that the set (two points, multiplying spacetime proper) is given as {L, R}, say ML = M ® L, MR = M ® R. The NCG is de facto here a nonlocal differential geometry, since L and R cannot be connected by infinitesimal transformations. Basically, we are working in an Associated Vector Bundle, V = : [ML © M H , S U ( 2 ) X U(l),(1/2,-1)© (0,-2)], where we have left out the projection and right-multiplication. Remembering that the matter Lagrangian involves the Covariant differentiation D (as discussed in section Digression: ...) acting on the spinor fields ipL, ipR within the current three-form *j (to make a fourform), i.e., ensuring parallel transport, we now face in addition the problem of parallel-transporting between the two copies of spacetime. This is done by a matrix ^6 which connects e^ to en, i.e., the equivalent of differentiation in the continuous part is done by this matrix-derivative. To have a multiple of the identity when squared, so as to supply the (—/i2) coefficient of $ 2 in the Higgs potential, we need to assume a VR and a 4 x 4 matrix "T", with yet another C6-hke contribution relating VR to VL (and thus, massive neutrinos). The minus sign arises through the need for an i for the (discrete) form calculus to be faithful, as we shall soon see. Incorporating it in T, we have T 2 = — y?
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and d is replaced by d + T. Moreover, whereas D = d+W or D = d+Z in the y-even parts (i.e., within ML or MR themselves), with W, Z as one-forms, here we add to the connection a scalar field 4>, capable of relating the SU(2)xU(l) representations (1/2, —1) in L to (0, —2) of R. The matrix-derivative T being a spacetime scalar, there is also no (j. index and no differential contracting with the $ , which is thus a plusform. The complete covariant derivative is thus D = d + T + A j - e v e n + ^ j - o d d , with overall grading everywhere odd, as prescribed in the superconnection solution. The generalized curvature T of 23) is now Jr = dA + d$ + T$ + A2 + A$ + $2 = fA
+ DA$ + T
(27)
The action will therefore consist of F2 = TA2 + (DA$)2-tx2$2
+ \$4.
(28)
We now move on to the actual calculus, following Ref. (48). The idea is to reproduce the ^-grading of the superalgebra matrices in the previous sections, as a "natural" result of a discrete form calculus. Let x be the coordinate function which 'detects' L and y the function which 'feels' R: x{L) = l,x(R)
= 0;
y(L) = 0,y(i2) = l . xy = yx = 0, x2 = x, y2 = y ; l(I) = l,l(i2)=l)a: + y = l .
(29)
and an arbitrary element of the commutative and associative algebra generated by x and y can be written as Ax -f fiy, X,fi complex numbers. It is represented by the matrix [diagonal(A,/z)]. This algebra is thus equivalent to C @C. Introducing a differential 8, 62 = 0, 61 = 0, 6(xy) = (6x)y + x6y. From x + y = 1 we deduce 6(x + y) = 0,Sx = —6y; from x2 = x, we get (8x)x •+ x8x — 8x, (6x)x — (1 — x)6x, (6y)y = (1 — y)8y. Thus, we have a Grassmannian, in which Q° = Cx © Cy, fl1 is spanned by x6x, y6y, etc. The even Cl2p are spanned by [diagonal(Aa;(5a;) 2p ,//y(^y) 2p )]. The odd Q2p+1 are spanned by elements ax(8x)2p+1 + 0y{6y)2p+x and occupy the off-diagonal supermatrix elements
( 0 ia \
\i0 o ; • In addition, we get the relations x(8x)2px
=
x(6x)2py 2p
l
x{8x) + x x(8x)
2p+1
y
x(8x)2p, = 0, = 0,
= x(6x)2p+1
331
399
.
(30)
Altogether, we have the supermatrix of SU(2/1). In the C a m m a r a t a Coquereaux approach (48), the statistics issue is not mentioned—simply because the formalism is not carried through the superalgebraic structure and into the Hilbert space of its representations' particle states, staying only at the above geometrical level. Only the SU(2)xU(l) even subgroup is considered as a Hilbert space symmetry and the identification of the diagonal matrices of the SU(2)xU(l) charges does not carry over to the entire SU(2/1). Indeed, as long as the full battery of 27,28) has not been invoked, it is perhaps possible to claim that one only gets an elegant geometrical rederivation of the entire Weinberg-Salam Lagrangian—including the Higgs fields and their SSB •potential. However, the construction and application of 27,28) puts to work— at least—gl(2/l), and it would seem impossible to evade the statistics issue, which may already have been imposed by the local invariance guaranteed by the superconnection in 27). Returning to the Grassmannian ft, the 2x2 matrix representation looses the infinite p-grading, replacing it with the Z(2)upgrading, coinciding here with the (/-grading. The Higgs fields are in ft1 G J2 - • As a matrix, they are written as
*="=(£?)• with a curvature FJJ = 6H+H2, which can be shown to yield Fu = [[d>+
't Hooft, G., Nucl. Phys. B 190, 455 (1981). Seiberg, N., Phys. Rev. D 49, 6857 (1994). Abers, E.S., and Lee, B.W., Phys. Reps. 9C, 1 (1973). Ne'eman, Y., in Mathematical Physics towards the 21st Century, Eds. Sen, R.N., and Gersten, A. (Ben-Gurion University Press, Beersheva, 1994), p. 59. 5. O'Raifeartaigh, L., idem, 74. 6. Fock, V., The Theory of Space, Time and Gravitation (Pergamon Press, MacMillan, New York, 1964). See the Introduction to the 1955 Russian edition, available in this translation.
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7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
27.
28. 29. 30.
31. 32. 33. 34.
35. 36.
Weyl, H., Zeit.f. Phys. 56, 330 (1929). Yang, C.N., and Mills, R.L., Phys. Rev. 95, 631 (1954); 96, 191 (1954). 't Hooft, G., Nucl. Phys. B 33, 173 (1971). Polyakov, A., Phys. Lett. B 59, 82 (1975). See for example P.J. Hilton, An Introduction to Homotopy Theory (Cambridge University Press, 1953), theorem 4.1. 't Hooft, G., Nucl. Phys. £ 7 9 , 276 (1974). Polyakov, A.M., JETP Lett. 20, 194 (1974). Cremmer, E., Julia, B., and Scherk, J., Phys. Lett. BB 76, 409 (1978); Cremmer, E., and Julia, B., Nucl. Phys. 5 1 5 9 , 141 (1979). Freund, P.G.O., and Rubin, M.A., Phys. Lett. B 97, 233 (1980). See for example T. Pirani and S. Weinberg, Physics in Higher Dimensions (World Scientific, Singapore, 1985). Kerner. R., A.I.H. PoincareQ, 143 (1968); Chapline, G.F., and Manton, N.S., Phys. Lett. B 120, 105 (1983). Ne'eman, Y., Phys. Lett. B 8 1 , 190 (1979). Fairlie, D., Phys. Lett. B 82, 97 (1979). Quillen, D., Topology 24, 89 (1985). Thierry-Mieg, J., and Ne'eman, Y., Proc. Nat. Acad. Sci. USA 79, 7068 (1982). Sternberg, S., and Ne'eman, Y., Proc. Nat. Acad. Sci. USA 87, 7875 (1990). Connes, A., and Lott, J., Nucl. Phys. (Proc. Suppl.) B 18, 29 (1990). Thierry-Mieg, J., J. Math. Phys. 2 1 , 2834 (1980); ibid. R Nuo. Cim. A 56, 396 (1980). Feynman, R.P., Acta Phys. Polon. 24, 697 (1963); DeWitt, B.S., Phys. Rev. 162, 1195 (1967); Faddeev, L.D., and Popov, V.N., Phys. Lett. B 25, 29 (1967). Becchi, C , Rouet, A., and Stora, R., Comm. Math. Phys. 42, 127 (1975); same authors, Ann. Phys. (NY) 98, 287 (1976). Corwin, L., Ne'eman, Y., and Sternberg, S., Rev. Mod. Phys. 4 7 , 573 (1975). See also textbooks, such as J. Wess and J. Bagger, Supersymmetry and Supergravity (Princeton University Press, 1983), or West, P., Introduction to Supersymmetry and Supergravity (World Scientific, Singapore, 1986). Ne'eman, Y., in Cosmology and Gravitation: Spin, Torsion Rotation and Supergravity, Eds. Bergmann, P.G., and de Sabbata, V. (Plenum Press, New York and London, 1980), pp. 177-226. Kac, V.G., Fund. Anal. 9, 263 (1975). Gell-Mann, M., Phys. Rev. 125, 1067 (1962), formulae (4.10). Thierry-Mieg, J., and Ne'eman, Y., in Methods in Mathematical Physics, Eds. Garcia, P.L., Perez-Rendon, A., and Souriau, J.M. (Springer Verlag, Lee. No. in Maths. 836, Berlin, 1980), pp. 318-348. Scheunert, M., Nahm, W., and Rittenberg, V., J. Math. Phys. 18, 155 (1977). Ne'eman, Y., "SU(2/1) as a Chiral Symmetry", in Proc. Sudarshan Anniversary Workshop, to be published. Sternberg, S., and Ne'eman, Y., Proc. Nat. Acad. Sci. USA 77, 3127 (1980). Ne'eman, Y., and Thierry-Mieg, J., Phys. Lett. B 108, 399 (1982); Ne'eman, Y., Sternberg, S., and Thierry-Mieg, J., in Physics and Astrophysics with a multikiloton Modular Underground Trace Detector, Eds. Cignetti, G., et al. (Frascati Lab. Pub., 1982), pp. 89-92. Balantekin, A.B., Bars, I., and Iachello, F., Phys. Rev. Lett. 4 7 , 19 (1981). Ne'eman, Y., and Thierry-Mieg, J., Proc. Nat. Acad. Sci. USA 77, 720 (1980).
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37. 38. 39. 40. 41. 42. 43. 44.
45. 46. 47.
48.
49.
50.
Curci, G., and Ferrari, R., R Nuo. Cim. A 30, 155 (1975). Thierry-Mieg, J., and Ne'eman, Y., II Nuo. Cim. 7 1 A , 104 (1982). Beaulieu, L., and Thierry-Mieg, J., Nucl. Phys. B 197, 477 (1982). Lee, C.Y., and Ne'eman, Y., Phys. Lett. B 264, 389 (1991). Sternberg, S., and Wolf, J., Trans. Amer. Math. Soc. 238, 1 (1978). Ne'eman, Y., Phys. Lett. B 181, 308 (1986). Hwang, D.S., Lee, C.Y., and Ne'eman, Y., TAUP report N233 (1995). Coquereaux, R., Esposito-Farese, G., and Vaillant, G., Nucl. Phys. B 3 5 3 , 689 (1991); Coquereaux, R., Haussling, R., Papadopoulos, N.A., and Scheck, F., Int. J. Mod. Phys. A 7, 2809 (1992). Coquereaux, R., Esposito-Farese, G., and Scheck, F., Int. J. Mod. Phys. A 7, 6555 (1992). Connes, A., Publ. Math. IHES62, 257 (1985); ibid., Noncommutative Geometry (Acad. Press, eng. edition, 1994). Connes, A., "Essay on Physics and Noncommutative Geometry, in The Interface of Mathematics and Particle Physics, Eds. Quillen, D., Segal, G., and Tsou, S. (Oxford UP, 1990), p. 10. Cammarata, G., and Coquereaux, R., "Comments about Higgs fields, noncommutative geometry and the Standard Model" lectures at VI Simposio Argentino de Fi'sica Teorica de Particular y Campos, Bariloche (1995). Kastler, D., Rev. Math. Phys. 5, 477 (1993) (parts 1,11); Kastler, D., and Schucker, T., "A Detailed Account of Alain Connes' Version of the Standard Model (parts HI, IV",to be pub. (hep-th/9501077); same authors, Teoret. Mat. Fiz. 92, 522 (1992). Hwang, D.S., Lee, C.Y., and Ne'eman, Y., TAUP report N236 (1995).
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14 May 1998
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PHYSICS LETTERS B
yi
ELSEVIER
Physics Letters B 427 (1998) 19-25
=
=
^
=
=
A superconnection for Riemannian gravity as spontaneously broken SL(4,R) gauge theory Yuval Ne'eman Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, Israel 69978 ' and Center for Particle Physics, University of Texas, Austin, TX 78712, USA Received 29 August 1997; revised 9 February 1998 Editor: M. Dine
Abstract A superconnection is a supermatrix whose even part contains the gauge-potential one-forms of a local gauge group, while the odd parts contain the (0-form) Higgs fields. We demonstrate that the simple supergroup P(4,R) (rank = 3) in Kac' classification (even subgroup SL(4,i?)) provides for the most economical spontaneous breaking of SL(A,R) as gauge group, leaving just local 50(1,3) unbroken. As a result, post-Riemannian SKY gravity is made to yield Einstein's theory as a low-energy (longer range) effective theory. © 1998 Elsevier Science B.V. All rights reserved.
1. Superconnections and the electroweak SU(2 / 1 ) as a model The superconnection was introduced by Quillen [1] in Mathematics. It is a supermatrix, belonging to a given supergroup 5, valued over elements belonging to a Grassmann algebra of forms. The even part of the supermatrix is valued over the gauge-potentials of the even subgroup G c S , (one-forms B^dx* on the base manifold of the bundle, realizing the "gauging" of G). The odd part of the supermatrix, representing the quotient S/G = H c 5, is valued over zero-forms in that Grassmann algebra, physically the Higgs multiplet
Wolfson Distinguished Chair in Theoretical Physics. 0370-2693/98/S19.00 © 1998 Elsevier Science B.V. All rights reserved. PR S0370-2693(98)00326-8
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under the supergroup represents symmetry between bosons andfermions. Here, however, though the superconnection itself does fit the quantum statistics ansatz, this is realized through the order of the forms in the Grassmann algebra, rather than through the quantum statistics of the particle Hilbert spacel (both the W*, Z° A^ on the one hand, and the Higgs field
B D
A C
B' D'
' AAA' CAA'
+ (-l)*BAC'
(-l)"AAB'
+ BAD'"
+ (-l)dDAC
{-\)cCAB'
+
DAiyi
(1)
where n = a,b,c,d are the respective orders of the n-forms A, B, C, D in the Grassmann algebra. The next instalment came from Connes' noncommutatitive geometry (NCG), generalizing to discrete geometries some geometrical concepts (such as distances) till then defined only for continuous spaces. Connes and Lott [11] used the new formalism to reproduce the electroweak theory, providing it with a geometric derivation: the base manifold is Z2<& Mi/X = ML ffi MR, where Z 2 is a discrete space containing just two points L, R representing chiralities and A/ 3 / 1 is Minkowski spacetime 3 . NCG defines a space by the functions and Hilbert space states living on it and the operators acting on that Hilbert space. Here, parallel-transport within ML (or within MR) is performed by D = d + 6(G), with B standing for the relevant gauge potentials and G = SU(2)W X U(l)Y. Moving, however, from a state sitting over a point in ML (say v[(x)) to one sitting over a point in MR (say eR(x)) requires a scalar "connection"
Our 1982 solution nevertheless did include a fitting scalar field, within the extended "ghost'"-system corresponding to the forms being taken over the entire bundle. The authors of Ref. [11] worked on Euclidean M*, instead of M 3 / 1 for technical reasons.
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Y. Ne'eman/Physics Letters B 427 (1998) 19-25
21
apparent difficulty with the non-spin-statistics grading of the matter fields and explains how the grading can be related to chiralities instead. Moreover, the parallel-transport operator is found to require an additional "matrix derivative" SH, relating "twin" states in ML and MR, such as eL and eR, etc. (this is the role of /3 in the Dirac y^ calculus). With this additional term, the curvature-squared Lagrangian S A *R for 5 = SU(2/1) contains the complete Weinberg-Salam Lagrangian. Indeed, R = RG + (1/2)[
2. Riemannian gravity as deriving from a broken SKY affine gravity The interest in deriving Einstein's Riemannian theory through the spontaneous symmetry breakdown of a non-Riemannian theory stems from quantum considerations. First, the quantization of gravity implies spacetime quantization at Planck energies (where the Compton wavelength is also the Schwarzschild radius, (h/2TTmc) = 2Gm/c2). This quantization, in itself, represents a departure from Riemannian geometry. Secondly, the addition in the Lagrangian of terms quadratic in the curvatures renders the theory finite (the new terms dominate at high-energy and are dimensionless in the action); however, it is nonunitary, due to the appearance of p ~ 4 propagators. These are present because of the Riemannian condition Dg^v = 0, relating the connection r to the metric gpbV (the Christoffel formula). Thus r = dg and R = dT+ ( 1 / 2 ) J T = (d)2g + (dg)2 and R2 will involve p 4 terms in momentum space and thus p ~ 4 propagators. These can then be rewritten as differences between two 5-matrix poles - one of which is then a ghost, due to the wrong sign of its residue. We refer the reader to the work of K. Stelle in the seventies [17] and the more recent exact results of Tomboulis [18]. It seems therefore worth trying to reconstruct gravity so that the Riemannian condition will only constrain the low-energy end of the theory, as an effective "weak" result between matrix elements in that regime. The connection and the metric would thus be a priori two entirely independent fields and there would be no p~~4 propagators, thus making it possible for the theory to be unitary. Such a suggestion was made at the time by L. Smolin [19]. We have investigated a model [20-23] in which the high-energy theory, i.e. prior to symmetry breakdown, has as its anholonomic (gauge) group the metalinear SL(4,R) or GL(4,R) 4 . Our model contained the Stephenson-Kilmister-Yang (SKY) Lagrangian [24-26] plus a term linear in the curvature, and proved the Yang-Mills-like renormalizability and BRST invariance of the quantum Lagrangian. Whether the theory is unitary is not known at this stage, because although the classical Lagrangian is free of p4 terms in the kinetic energy, there remained a residual p 4 term in the gauge-fixing term of the quantum Lagrangian [22,23]. This may or may not be harmless, but it might still be possible to replace that dangerous term by an expression which would be guaranteed not to yield poles. In such theories, (a) the G = SL(4,/?)-invariant R(G) A * R(G) SKY Lagrangian has to have its symmetry broken by a Higgs field corresponding to an 5L(4, R) multiplet containing a Lorentz-scalar component, to ensure that F = 50(1,3). In the algebraic structure we use (the superalgebra of S = P(4,R)), this includes a metric-like symmetric tensor (a,b = 0 , . . . 3 are anholonomic indices supporting the local action of S and its subgroups)
The conformal SV(2,2) = 56X4,2) or the super-deSitter OSp(4/l) of McDowell-Mansouri would still be Riemannian.
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Y. Ne'eman/Physics Letters B 427(1998) 19-25
Minkowski-trace). (c) Any remaining components of i>H{x) should acquire masses and exist as free particles. In the P(4,R) model,
3. The simple superalgebra p(4,R) The defining representation of the generating superalgebra of the P(4,R) supergroup is an 8 X 8 matrix, divided into quadrants. / and IV carry the sl(4,R) algebra, with / in the covariant representation XI (a,b = 0,1,2,3 and the tildes indicate tracelessnes trXj;=0) and IV in the contravariant, i.e. /V= — IT (T indicates transposition). In the off-diagonal quadrants, 11=2}," carries the 10 symmetric matrices of gl(4,R) and ///= X\" carries its 6 antisymmetric matrices. There are thus altogether 31 generators, of which 15 Qf are even, representing the action of sl(4,R) and 16 Nb are odd, of which 10 are the symmetric N+ = T, and 6 the antisymmetric N~ = M, exhausting the set of generators of gl(4,R) (we use the notation of Ref. [27], i.e. the T, M are the shears and Lorentz generators, respectively). We shall also have occasion to use the nonsimple completion gP(4,R) in which the SL(4,R) even subgroup is completed to GL(4,R), without any change in the P(4,R) itself. The simple superalgebra is thus given as,
/
-
Si
I
II\
"
"
4
,///
ivj
in
=
si:
= =
>
-
\Q-ab r
"
Mab
(2)
ab
and Qab:={X~"b)^-{Xl)]v,
N:b: = (Xi;)n,
Na-„: = (Si;)in
(3)
To formulate the super-Lie bracket, we choose to replace the two-index (vector) notation by a single (matrix) index, as in SU(2) or SU(3) usage. We select an SU(4) basis (4X4, ' V matrices) in which the i = 1... 8 correspond to setting the SU(3)\i matrices in the upper left-hand corner of the v matrix with that index and define similar matrices for the rest. Since we are dealing with SL(4,R) rather than SU(4), we have to multiply the real matrices by V— 1, thus making these generators noncompact. With O; denoting the Pauli matrices, and [<7i]it2 denoting a cr, matrix placed in the [1,2] rows and columns of the v matrix, we have a basis, "l=!'Al=*'l>i]l,2> "2 =
A
2=[o-2]l,2
"4=»A4
=
"3
«l>l]l,3>
"6 = ! ' A 6 = ' ' t a l k s '
=
«A3 = »'l>3]l,2
"5 =
A
5=[°2]l,3
=
A
7=[°"2]2,3
"7
* g = iA g = ( i / , / 3 ) d i a g ( l , l , - 2 ) "9= ~ v
n
=
~ - =i[o-i]2A,
"13= " V,
='[o-l]l,4.
=»!>i]3,4'
=[arl]l,4
"10= - -
vn= - - = [o- 2 ] 24 "14= ~ -
=
[°"2]3,4
- - = (i/v/6 )diag( 1,1,1,-3)
(4)
406
Y. Ne'eman/Physics Letters B 427 (1998) 19-25
23
Using the definition of the fijk (totally antisymmetric) and dijk (totally symmetric) coefficients of su(3), generalized to su(4) and corrected by the factors V— 1 for the symmetric matrices in the su(4) basis when changing to sl(4,R) as indicated above, we get coefficients fijk and dijk whose symmetry properties are thus reduced to the first two indices only. We can now write the Lie superbrackets as, [Qt,Qj]=2iflJkQk, [Q?,N;] = 2ifijkNk+ , [Q?,N0+\ = 0, [Qf,Nj~] = 0 [Qj,N;] - 2dijkNk+ , [fi?>o + ] = 2 < , [Qj,Nf] = 2idijkNk {N+,N7}=2dijkQ£,
{N0\N-}=2iQ?
(5)
4. The superconnection, supercurvature and the Lagrangian At this stage we set up the relevant superconnection a la Quillen, as an ad hoc algorithm (we shall later discuss the possibility of generating it from the matter fields' fiber bundle, by using a Connes-Lott type of product base space). The superconnection will thus be given as
rffl
#*-fy,
*[*->2J'
-rix-,
(6)
The nonvanishing v.e.v. field <j>(x) =
Ho
M
r)
<"
The resulting (generalized) curvature is then, R = R(G) + +{
(8)
where $>+,
^(^2 + ^4)
—j— = „0
(g) There is no {<J>+,0Q} term, so that the 9 traceless components of 4>+ do not acquire mass as free particles. Instead, they become the longitudinal components of the G/F= SL(4,R)/SO(4)-gauging piece of the connection, which acquires mass under the spontaneous symmetry breakdown. This elegant "metamorphosis" is very similar to the acquisition of longitudinal components by the W and Z mesons in the Electroweak theory.
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Y. Ne 'eman / Physics Utters B 427 (1998) 19-25
5. The matter Lagrangian and Connes-Lott like geometry We now discuss a Connes—Lott like derivation. We stick to the chiral Z2 grading, i.e. to the product space Z2 ®M 3,1 =ML®MR as base space. The matter fields will consist of world spinor manifields [27-31], the (spinorial) infinite-component representations of the double-coverings GL(4,/f) and 3Z(4,/?), which, for several decades, were wrongly assumed not to exist - in the General Relativity literature - even though a well-known algebraic theorem states that the topology of a Lie group is that of its maximal compact subgroup, i.e. 50(4) c SU4,R) and accordingly SU(2) X SU(2) c 51(4,/?). We refer the reader to the relevant literature, e.g. Chapter 4 and Appendix C of Ref. [31]. The appropriate choice is the manifield based on ^({,0) ®<2r(0,j), where 31 denotes the SZ(4,/?) irreducible representation (applying the deunitarizing automorphism [27] J / ) and (|,0), (0,|) denote the lowest representations of the SO(A) subgroup, here a nonunitary representation of SL(2,C\ namely a Dirac spinor. We refer the reader to the literature - see Figs. 3, 4, 5 of Ref. [31] - for a detailed discussion of this field. Obviously, for a massless field, (j,0) and (0,^) respectively form the fibres over ML and MR, with G = SL(4,R) as a common structure group. The odd connection bridging parallel-transport between points on the bundles constructed over ML and MR has to contain (e.g. in a Yukawa-like term) a i\,\) y"-supported scalar 0 J at least. However, considering the structure of the manifield (see Figs. 4, 5 in Ref. [31]), the N+ generators with their (1,1) action and the
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
D. Quillen, Topology 24 (1985) 89. 3. Thierry-Mieg, J. Math. Phys. 21 (1980) 2834. Y. Ne'eman, Phys. Lett. B 81 (1979) 190. D.B. Fairlie, Phys. Lett. B 82 (1979) 97. Y. Ne'eman, J. Thierry-Mieg, Proc. Nat. Acad. Sci. USA 77 (1980) 720. V.G. Kac, Func. Analysis and Appl. 9 (1975) 91; also Comm. Math. Phys. 53 (1977) 31; see also V. Rittenberg, in: P. Kramer, A. Rieckers (Eds), Group Theoretical Methods in Physics (Proc. Tubingen 1977), Springer Verlag Lecture Notes in Physics 79, Berlin, Heidelberg, New York 1977, pp. 3-21. L. Corwin, Y. Ne'eman, S. Sternberg, Rev. Mod. Phys. 47 (1975) 573. J. Thierry-Mieg, Y. Ne'eman, Proc. Nat. Acad. Sci. USA 79 (1982) 7068. Y. Ne'eman, S. Sternberg, Proc. Nat. Acad. Sci. USA 87 (1990) 7875. Y. Ne'eman, S. Sternberg, Proc. Nat. Acad. Sci. USA 77 (1980) 3127. A. Connes, J. Lott, Nucl. Phys. (Proc. Suppl.) 18B (1990) 29. R. Coquereaux, R. Haussling, N.A. Papadopoulos, F. Scheck, Int. J. Mod. Phys. A 7 (1992) 2809. R. Coquereaux, G. Esposito-Farese, F. Scheck, Int. J. Mod. Phys. A 7 (1992) 6555. N.A. Papadopoulos, J. Plass, F. Scheck, Phys. Lett. B 324 (1994) 380. N.A. Papadopoulos, J. Plass, Mainz preprint MZ-TH-95-11. F. Scheck, hep-th/9701073, January 1997. K. Stelle, Phys. Rev. D 16 (1977) 953; Gen. Rel. Grav. 9 (1978) 353. E.T. Tomboulis, Phys. Lett. B 389 (1996) 225. L. Smolin, Nucl. Phys. B 247 (1984) 511. Y. Ne'eman, Dj. Sijat'ki, Phys. Lett. B 200 (1988) 489. C.-Y. Lee, Y. Ne'eman, Phys. Lett. B 233 (1989) 286. C.-Y. Lee, Y. Ne'eman, Phys. Lett. B 242 (1990) 59. C.-Y. Lee, Class. Quantum Grav. 9 (1992) 2001. Stephenson, Nuovo Cimento 9 (1958) 263. C.W. Kilmister, D.J. Newman, Proc. Cam. Phil. Soc. 57 (1961) 851.
408
Y. Ne'eman/Physics Utters B 427 (1998) 19-25 [26] [27] [28] [29] [30] [31]
C.N. Yang, Phys. Rev. Lett. 33 (1974) 445. Dj. Sijacki, Y. Ne'eman, J. Math. Phys. 26 (1985) 2457. Y. Ne'eman, Proc. Nat. Acad. Sci. USA 74 (1977) 4157. Y. Ne'eman, Ann. Inst. H. Poincare A 28 (1978) 369. Y. Ne'eman, Dj. Sijacki, Int. J. Mod. Phys. A 2 (1987) 1655. F.W. Hehl, J.D. McCrea, E.W. Mielke, Y. Ne'eman, Phys. Rep. 258 (1995) nl, 2.
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C H A P T E R 6: S P I N O R R E P R E S E N T A T I O N S OF GL(N,R) CHROMOGRAVITY
AND
INGO KIRSCH Jefferson Laboratory of Physics, Harvard University Cambridge MA, USA A topological description of matter in a generic curved spacetime presupposes the knowledge of spinors (existence, representations on particles and fields etc.) of the group Diff(n,R) of General Coordinate Transformations (GCT). In the conventional non-topological approach (Lorentz) spinors are introduced in an anholonomic way, i.e. via a bundle of cotangent frames which transform under the Lorentz group. In other words, the Diff(n,R) group is enlarged to the Diff(n,M)<8>SO(l,n — i) one. The spinorial fields are holonomic Diff(n,R) scalars, and have non trivial 5 0 ( 1 , n — 1) representation properties only. In contrast, holonomic, i.e. curved-space spinors do not rely on the introduction of anholononic coframes which represent an additional geometrical structure. The properties of such spinors are encompassed by the concept of world spinors.a World spinors are (unitary) half-integer representations of the double-covering group Diff(n,R) of general coordinate transformations. The question of the existence of such representations reduces to that of the existence of a double-covering of its maximal homogeneous linear subgroup GL(n, R). The quotient space Diff(n,R)/GL(n,R) is simplyconnected and as such irrelevant for the covering question. There are three issues {6.6} which have to be clarified a priori when dealing with spinorial representations of GL(n,R). First, the existence of a double-covering of GL(n,R) {6.2} can be shown by applying Iwasawa's theorem: Any semi-simple Lie group can be decomposed into the product of a maximal compact Lie group K, an Abelian group A and a nilpotent group N. The maximal subgroup K of GL(n,R) is the orthogonal group 0{n) which is known to have the covering group 0(n) ~ Pin(n). The existence of the doublecovering GL(n,R) is thus guaranteed by taking also into account the simple connectedness of the groups A and N. Second, despite the possibility of embedding the linear group GL(n, R) into the complex group GL(n,C), the same is not true for its double covering GL(n,R). As shown in {6.6}, there does not exist an embedding GL(n,R) —> GL(n,C). As a consequence, there are no finite spinorial representations of GL(n,R). This does however not exclude the existence of infinite-dimensional spinor representations of GL(n,R), i.e. of an embedding into a group of infinite complex matrices. Manifields are the corresponding infinite-component spinorial and tensorial fields. Moreover, it turns out that the world spinor quantum particles are characterized by unitary infinite-dimensional SL(n — 1,R) representations. a
T h i s does not exclude the possibility of an anholonomic introduction of spinors of the linear group GL(n, R) as in the Metric-Affine Theory of Gravity [l], see below.
411
Third, an extension of particle representations to the corresponding field ones implies, due to their unitaiity, that the boost generators possess a non-vanishing intrinsic part and boosting a "particle" described by a world spinor field would raise its spin endlessly. To avoid such an unphysical behaviour, one applies the so-called "deunitarizing automorphism" {6.6} to the linear algebra gl(n,M), i.e. one identifies the finite unitary representations of the maximal compact subgroup SO(n) with the nonunitary representations of the physical Lorentz group SO(l,n — 1). This ensures a correct particle-field matching: GL(n,R) is represented non-unitarily with respect to the Lorentz group but unitarily when restricted to the (particle states) stability group SL(n — 1,R). When reduced with respect to the Lorentz subgroup, world spinors reduce to an infinite sum of spinorial representations such as e.g. (^©§©|©...). Physically, these representations can be thought of as a sequence of particles with half-integral spin which are connected by the action of the nine shear generators of the linear group. These non-compact generators are themselves related to the time derivative of the energy quadrupole operator. Thinking of a hadron as an extended structure, representations of the linear group resemble a band of rotational excitations of deformed nuclei. Indeed, representations of the special linear group SL(n,R) (n — 3,4) [2],{6.4} have been used in {2.3, 6.5} to classify the hadronic spectrum. The infinite-component fields, "manifields" associated with spinorial representations of the linear group have a broad range of applications in both classical gravity and high energy physics. In gravity, there are two interesting applications of holonomic world spinors: (a) nonlinear world spinors in Einstein gravity which are linear upon restriction to the Lorentz group, (b) world spinors in a pure affine or topological theory of gravity. Another application of manifields opens up in the framework of the Metric-Affine gauge Theory of Gravity (MAG) [l]. A metric-affine spacetime is equipped with a bundle of coframes which transform under the linear group. For the description of matter, it is thus natural to establish anholonomic spinors of GL(n,M) in the tangent bundle. These spinors describe matter fields with hypermomentum which are not only sources for a spin current but also for a shear and dilation current; for details see Ch. 7 of this book and references therein. Ne'eman's interest in space-time properties of matter started back in 1965 when he realized, together with Dothan and Gell-Mann {2.3}, the relevance of the SL(3,M) group for description of hadronic matter excitations as well as its connection to gravity. Moreover, at this time they discovered the decontraction formula (inverse of the Wigner-Inonii contraction) that is an algebraic way of describing interacting and/or non-perturbative regimes. Strong gravity [3; 4] was an intermediate step in an attempt to describe hadronic recurrences by a gravity-like theory, that culminated in "Chromogravity" {6.7}, a QCD based IR approximation in terms of colorless field variables. The QCD bound states of a given flavor are described by Gy.v ~ g^A^A^, G^p ~ da^A^A^A^,, etc. variables (A^ is a gluon field, a = 1, 2 , . . . , 8; gab and dabc are the Cartan metric and the totally symmetric d-coefficients of SU(3), respectively). The QCD variation of these composite objects yields, in the IR ap-
412
proximation, the familiar form of the GCT group, and indeed an action of the infinite GCT algebra is realized in the space of these colorless fields {6.11, 6.12}. The G^v field obeys a riemannian constraint D^G^ = 0, where Da is the Chromogravity covariant derivative, and plays a role of a pseudo-metric, "chromometric". One can construct, along the lines of Einstein's gravity the corresponding "cliromocurvature" tensor RPa^v and an action in terms of R+ R2, that in interaction with matter manifields yields a Regge trajectory type of hadronic spectrum with J ~ m2 {6.9}. There are several approaches to the construction of field equations for manifields. In {6.3} a Lorentz-invariant wave equation for manifields is constructed which is based on a subclass of multiplicity-free SL(4,R) representations. Mickelsson [5] has constructed a truly GL(4,R) invariant Dirac-type wave equation however with a rather unclear physical interpretation. In [6] a Dirac-type infinite-component equation is considered from the point of view of building it up from physically well-defined Lorentz subgroup components. In [7] an SL(3, R) invariant Dirac-type equation is constructed by embedding the SL(3, R) invariant vector operator X^ into SL(4,M). This operator is an extension of Dirac's 7 matrices. Despite these promising attempts, world spinors have not yet found their way into a commonly accepted theory. String/M-theory has not made use of world spinors so far, mainly because a completely satisfying metric independent formulation has not yet been found. Also a possible application for the description of M2-branes in M-theory as envisioned by Ne'eman {8.5} has not come to reality. Nevertheless, world spinors are the relevant objects for a topological formulation of gravity with matter as well as for a metric-affine extension of Einstein gravity and should not be lost out of sight.
413
References [1] [2] [3] [4] [5] [6] [7]
F. W. Hehl, J. D. McCrea, E. W. Mieike and Y. Ne'eman, Phys. Rept. 2 5 8 (1995) 1. Dj. Sijacki, J. Math. Phys. 16 (1975) 298. Y. Ne'eman and Dj. Sijacki, Ann. Phys. (N.Y.) 120 (1979) 292. Y. Ne'eman and Dj. Sijacki, Phys. Rev. D 3 7 (1988) 3267. J. Mickelsson, Comm. Math. Phys. 88 (1983) 551. I. Kirsch and Dj. Sijacki, Class. Quant. Grav. 19 (2002) 3157. Dj. Sijacki, Class. Quant. Grav. 21 (2004) 4575.
414
REPRINTED PAPERS OF CHAPTER 6: SPINOR REPRESENTATIONS OF GL(N, R) AND CHROMOGRAVITY
6.1
6.2
Y. Ne'eman, "Gravitational Interaction of Hadrons: Band-Spinor Representations of GL{n,Rf, Proc. Nat. Acad. Sci. USA 74 (1977) pp. 4157-4159.
417
Y. Ne'eman, "Spinor-Type Fields with Linear, Afhne and General Coordinate Transformations", Annales de I'Institut Henri Poincare, Sect A 28 (1978) pp. 369-378.
420
And in Dynamical Groups and Spectrum Generating Algebras, A. Barut, A. Bohm and Y. Ne'eman eds. (World Scientific, Singapore, 1988), II, pp. 846-855. 6.3
6.4
A. Cant and Y. Ne'eman, "Spinorial Infinite Equations Fitting Metric-Affine Gravity", J. Math. Phys. 26 (1985) pp. 3180-3189.
430
Dj. Sijacki and Y. Ne'eman, "Algebra and Physics of the Unitary Multiplicity-Free Representations of SL(4, R)", J. Math. Phys. 26 (1985) pp. 2457-2464.
440
And in Dynamical Groups and Spectrum Generating Algebras, A. Barut, A. Bohm and Y. Ne'eman eds. (World Scientific, Singapore, 1988), II, pp. 808-815. 6.5
Y. Ne'eman and Dj. Sijacki, "SL(4, R) Classification for Hadrons", Phys. Lett. B 1 5 7 (1985) pp. 267-274.
448
And in Dynamical Groups and Spectrum Generating Algebras, A. Barut, A. Bohm and Y. Ne'eman eds. (World Scientific, Singapore, 1988), II, pp. 816-823. 6.6
6.7
6.8
6.9
6.10
6.11
6.12
Y. Ne'eman and Dj. Sijacki, "GL(4, R) Group-Topology, Covariance and Curved-Space Spinors", Int. J. Mod. Phys. A2 (1987) pp. 1655-1669.
456
Dj. Sijacki and Y. Ne'eman, "QCD as an Effective Strong Gravity", Phys. Lett. B 2 4 7 (1990) pp. 571-575.
470
Dj. Sijacki and Y. Ne'eman, "Derivation of the Interacting Boson Model from Quantum Chromodynamics", Phys. Lett. B250 (1990) pp. 1-5.
475
Y. Ne'eman and Dj. Sijacki, "Proof of Pseudo-Gravity as QCD Approximation for the Hadron IR Region and J ~ M2 Regge Trajectories", Phys. Lett. B276 (1992) pp. 173-178.
480
J. Lemke, Y. Ne'eman and J. Pecina-Cruz, "Wigner Analysis and Casimir Operators of SA(4, R)", J. Math. Phys. 33 (1992) pp. 2656-2659.
486
Y. Ne'eman and Dj. Sijacki, "Chromogravity: QCD-Induced Diffeomorphisms", Int. J. Mod. Phys. A10 (1995) pp. 4399-4412.
490
Y. Ne'eman and Dj. Sijacki, "Inter-Hadron QCD-Induced Diffeomorphisms from a Radial Expansion of the Gauge Field", Mod. Phys. Lett. A l l (1996) pp. 217-225.
504
415
6.13
Y. Ne'eman, "Nuclear Physics Implications of the Spin 2 Multiplet", in Symmetry Principles at High Energy, Proc. Fifth Coral Gables Conf. 1968 (W. A. Benjamin, New York, 1968), pp. 149-151.
416
513
Reprinted
from
PTOC. Natl. Acad. Sci. USA Vol. 74, No. 10, pp. 4157-4159, October 1977 Physics
Gravitational interaction of hadrons: Band-spinor representations of GL(n,R) (gravitation/linear group/double-valued representations/hypermomentum)
YUVAL NE'EMAN Department of Physics and Astronomy, Tel-Aviv University, Israel; and European Organization for Nuclear Research CH 1211 23 Geneva, Switzerland
Contributed by Yuval Ne'eman, July 22,1977 ABSTRACT We demonstrate the existence of double-valued linear (infinite) spinorial representations of the group of general coordinate transformations. We discuss the topology of the group of general coordinate transformations and its subgroups GA(nR), GL(n,R), SL(nr) for n = 2,3,4, and the existence of a double covering. We present the construction of band-spinor representations of GL(n,R) in terms of Harish-Chandra modules. It is suggested that hadrons interact with gravitation as band-spinors of that type. In the metric-affine extension of general relativity, the hadron intrinsic hypermomentum is minimally coupled to the connection, in addition to the coupling of the energy momentum tensor to the vierbeins. The relativistic conservation of intrinsic hypermomentum fits the observed regularities of hadrons: 5(7(6) (~ spin independence), scaling, and complex-/ trajectories. The latter correspond to volumepreserving deformations (confinement?) exciting rotational bands.
SL(3,R) had been suggested (2) as excitations in quasi-orbital angular momentum in the quark model, for a description of the observed "Regge trajectories." Because GL(3,R) is the "little g r o u p " for time-like momenta in the general affine group 7 := GA(4,R), the representations used in ref. 2 can now be reinterpreted as spin-excitations and used for band-tensors. Similar band-tensors had been used for GL (4,R) in ref. 3, in a description of the spinning top. Our new band-spinors will represent nucleons, etc., including their high-spin excitations. The current view of hadron dynamics is based on a quark field with a color-gluon mediated super-strong and confining interaction ("QCD"). Bandors represent an intermediate picture between these "fundamental fields" and a phenomenological rendering. They do include some part of the gluon action, because SL(3,R) is characteristic of excitations induced by volume-preserving stresses—perhaps the confining interaction itself. They are in fact a somewhat less sophisticated "string" or "dual model." In two other articles, we shall present this physical idea in more detail (4). In particular, we shall show that the band-spinor description fits well into a recent generalization of general relativity, the metric-affine theory (5, 6). This can b e regarded as a GA(4,R) gauge, in which the vierbein is coupled minimally to the energy-momentum tensor, and the affine connection is similarly coupled to the intrinsic hypermomentum tensor whose components are the spin, dilation, and shear currents. Note that these three quantities correspond to the observed regularities of the quark model: Sl/(6) (i.e., spin-independence), scaling, and the Regge trajectories. The rest of this note is dedicated to the mathematical issue— i.e., the existence of double-valued representations of S and its subgroups.
1. I n t r o d u c t i o n This note deals with two issues—one mathematical and one other physical. Mathematically, we present a new type of double-valued representation of the group of general coordinate transformations (Einstein's general covariance group) G and of its linear subgroup S : = GL(n,R), the general linear group in n dimensions (n > 2) over which the representations of G are built. Alternatively, our new representations can be regarded as representations of another group S, not included in G, realizing a global or gauged symmetry of matter fields. Our double-valued representations are infinite and of discrete type and reduce to sequences of double-valued representations of the Poincare group P . For time-like momenta, they reproduce rotational excitation bands—e.g., with spin
7-A + M + . . . . J 2 2 2 W e accordingly have termed them "band-spinors," and more generally (for both single- and double-valued cases) "bandors." Note that it had always been assumed in the folklore of general relativity (and often written in texts) that GL{n,R) has no double-valued or spinorial representations; the existence of band-spinors is thus a nontrivial addition. We provide here the existence proof and a general construction, details being treated elsewhere. Note that one source of the prevalent belief that there are no spinors ("world "-spinors) stems from an unwarranted extrapolation of a theorem of E. Cartan (1). As can be seen in the text, Cartan was aware of the restriction of his proof to spinors with a finite number of components. Physically, we suggest that hadrons interact with gravitation as bandors. Single-valued discrete infinite representations of
2. T o p o l o g i c a l c o n s i d e r a t i o n s : T h e covering g r o u p of SL(nB) a n d GL(aB) We are studying the groups,
o ? D n n f l
[2.i]
e 3 3 D ? D O [2.2] in which # is the unimodular linear SL(4R) and & is the special orthogonal SO(4). We do not enter into the further structure induced by the Minkowski metric at this stage. At various stages we shall also deal with the same groups over n = 3 and n = 2; we shall then use the notation S3, ^ 3 , etc . . . . Because our aim is to find unitary representations of G, ? , S, and cf that reduce to double-valued unitary representations of V and 0 , we have a priori two candidate solutions: (i) S D 0, S3 D ©a, ? D ¥
The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.
(«) < ? D ? D S D ? D 0
y D¥ 4157
417
4158
Physics: Ne'eman
Proc. Natl. Acad. Sci. USA 74 (1977)
in which the bars denote double-covering of the relevant groups. In the first case, we would be dealing with single-valued representations of G and its subgroups, and 0 would be contained through its covering 0. In the second case, all groups would display the same bivaluedness as 0, and we would have to go to their respectivejcoverings to find a single-valued representation containing &. Because
has constructed the unitary representations of SL(2R), because this is the double-covering spin(3)(+—) of the 3-Lorentz group (+1, —1, —1) and even though only single-valued representations of SL(2R) are required for this role, he has also constructed (Id) multivalued linear representations of that group. The representations
C},fc = ±,<, = ± + «*
03 = Sl/(2)
are bivalued representations of spin(3)(+—) = SL(2R) as can be derived from Bargmann s formula
it is enough that we show that #3 t> S17(2) to cancel solution («')•
We introduce an Iwasawa decomposition (7) of S. For a noncompact real simple (all invariant subgroups are discrete and in the center) Lie group J5B, it is always possible to find % = Ji-A-M
U(b) = exp(4ilh) U(a)
= M n » = |l|
[2.3]
2
i = - (*idi - x2d2), S 2 = - (xjd2 + x2di), 23 = - ^ ( * i d 2 - x 2 d i )
[2.10]
[2i,Z2] = - i S s , [2 3 ,Si] = i'S2, [S 2 ,2 3 ] = iSi
[2.11]
[2.4] with commutation relations
Applying Eq. 2.3 to £3, 3i is 03. Because this is maximal and unique
with S3 generating the compact subalgebra (eigenvalues m in ref. 9). However, when using the same algebra as the double covering (10) of SO(l,2), the identification in terms of the (completely different) (1,-1,-1) space is given by
^3 2)03
and we are left with solution (ii) only. Applying Eq. 2.3 to S, S = 0AsMs
[2.5] 2i = i(x0di + Xido), S 2 = -i(x 2 d 0 + x0d2), S 3 = <(*1d2 - x2 di)
we also have I = 0AsMs
[2.12]
[2.6 J
Now the groups A and J\[ in an Iwasawa decomposition are simply connected, and_AAf = AN is contractible to a point. Thus, the topology of ^ is that of 0. The same result has been shown to hold (8) for <S when the L4 is Euclidean or spherical and holds under some_weak conditions for any L4. By the same token, § has the topoloy of 0(n,R), the double covering of the full orthogonal (which includes the improper orthogonal matrices, with det = —1). S and S thus have two connected components. _ For n > 3, ^ is thus completely covered by <£, the double covering. However, 0(2) and SL(2R) are infinitely connected. ^2 < fa
[2.9]
for two elements lying over the same element of SL(2R). We take; = 1. Note that in reducing SL(4R) to SL(2R), the generators are represented on the coordinates by
in which 3i is the maximal compact subgroup, A is a maximal Abelian subgroup homeomorphic to that of a vector space, and N is a nilpotent subgroup isomorphic to a group of triangular matrices with the identity in the diagonal and zeros everywhere below it. The decomposition is unique and holds globally •KC\A = AC\M
[2.8]
[2.7]
in which of is the full covering. Topologically, solution (ii) is thus realizable. The singlevalued unitary (and thus infinite-dimensional) irreducible representations of £ correspond to double-valued representations of <£ and reduce to a sum of double-valued representations
of©. This being established, it is interesting to check a second source of confusion at the origin of the statements found in the literature of general relativity and denying the existence of such double-valued representations. This is based upon an error in the statement of a theorem of Cartan (9): "The three linear unimodular groups of transformations over 2 variables \SL(2C), SU(2), SL(2R)] admit no linear many-valued representation." As can be seem from Cartan's proof of this theorem (9), it holds only for SL(2C) and SU(2). Moreover, Bargmann (10)
with the same commutators and the same role for 23. We stress this correspondence because it clarifies some additional aspects connected with arguments (11) against the existence of bivalued representations of £4. 3. The SL(3R) band-spinors: Existence The unitary infinite-dimensional multiplicity-free representations of SL(3R) are characterized by j0 (the lowest;) and c, a real number, ZK$3\)o,c)
[3.1]
the ladder representations (2) corresponding to j 0 = 0 and j 0 = 1; —« < c < <=. We provide here a construction, based upon the "sub-quotient" theorem for Harish-Chandra modules ^12). We return to the Iwasawa decomposition of Eq. 2.6 for ,£3 ^ 3 = ©3^1 N
[3.2]
and taking first S3 define Ma, the centralizer of A in "H—i.e., in 03. This is the set of all u £ f t such that (o ^ M^aao-i = a) [3.3] for any a £ A • The elements of A span a 3-vector space, and M3 thus has to be in the diagonal. Because det(./M3) = 1, the elements of 03 belonging to M3 are the inversions in the 3 planes: (+1,-1,-1), (-1,-1-1,-1), and (—1,-1,-1-1). Together with the identity element, they form a group of order 4, with a multiplication table mi m 2 = m3, m 2 m 3 = mi, m 3 mi = m2, ml = 1. It appears Abelian in this representation. _ Returning now to ^3 and 0a, we look for Ms C 03- The in-
418
Proc. Natl. Acad. Sci. USA 74 (1977)
Physics: Ne'eman
T h e covariant derivative of a band-spinor field SI"* will be given by
versions are given in SU(2) by exp(iir
±1)
[3.4]
D„ *
AM
=
SU(Z)/M3
and
5
+ r „ / ( G \ ) H 18 * ?
[4.1]
1 5 9
2'2'2'
G"T is an infinite dimensional representation of S, and T ^ T is the usual affine connection.
[3.5]
The representations p (jo, c) of Q3 are given by jo for a representation of the Ms group of "plane inversions" in SU(2), and c for the characters of^yt, because M is represented trivially. The representations of $3 will thus be labeled accordingly; from Eej. 3.5 we see that they will be spin-valued representations of ©3. Because AM = AM, univalence is guaranteed. Note that t h e only multiplicity-free bands* are ft(Ss,; 0, c), 2) (^3; 1, c) with —=0 < c < » and D(£3; V2, 0). 4. CA(4R)
= J„ *
J
can now be used to induce the representations of ^ 3 . Note that #s/Q3
3
in which a, fl runs over the sets a, aXp, aXppo, • • — i . e . , spins
T h e subgroup (Qs C <^3 @3 := M$
4159
We would like to thank Professors B. Kostant and L. Michel for their advice. This work was supported in part by the United States-Israel Binational Science Foundation.
GL(4R)
Because the representations of 2 = 0) ® D(jo = 0, Jo = V2) with ( A j « = 1, Aj< > = 1) noncompact action. Each (/('', j ' 2 ' ) level of a band-spinor field satisfies a Bargmann-Wigner equation (16) for ;' = | ; " ' | + | ; ' 2 ' | . * Following our original search (1969-1970) with D. W. Joseph for spinor representations of SL(3,R) (and while we were still unaware of the general-relativity taboo), this result was found by Joseph and proved (13). It has recently been reconfirmed (14) after having been questioned (15).
419
1. Cartan, E. (1938) Leqons sur la Theorie des Spineurs II (Hermann Editeurs, Paris), Article 177, pp. 89-91. 2. Dothan, Y„ Gell-Mann, M. & Ne'eman, Y. (1965) Phys. Lett. 17, 148-151. 3. Dothan, Y. & Ne'eman, Y. (1965) in Symmetry Groups in Nuclear and Particle Physics, ed. Dyson, F. J. (W. A. Benjamin, New York), pp. 287-310. 4. Hehl, F. W., Lord, E. A. & Ne'eman, Y. (1977) Phys. Lett., in press. 5. Hehl, F. W., Kerlick, G. D„ & von der Heyde, P. (1976) Zeit. Naturforsch. Teil A 31,111-114, 524-527, 823-827. 6. Hehl, F. W., Kerlick, G. D. & von der Heyde, P. (1976) Phys. Lett. B 63,446-448. 7. Iwasawa, K. (1949) Ann. Math. 50, 507-558. 8. Stewart, T. E. (1960) Proc. Am. Math. Soc. 11,559-563. 9. Cartan, E. (1938) Lecons sur \a Theorie des Spineurs I (Hermann Editeurs, Paris), Articles 85-86, pp. 87-91. 10. Bargmann, V. (1947) Ann. Math. 48, 568-640. 11. Deser, S. & van Nieuwenhuizen, P. (1974) Phys. Rev. D 10, 411-420, Appendix A.' 12. Harish-Chandra (1954) Trans. Am. Math. Soc. 76,26-65. 13. Joseph, D. W. (1970) "Representations of the Algebra ofSL(3R) with A; = 2" (University of Nebraska preprint, unpublished, referred to in ref. 15, 16p. 14. Ogievetsky, V. I. & Sokachev, E. (1975) Teor. Mat. Fiz 23, 214-220; English translation pp. 462-466. 15. Biedenharn, L. C , Cusson, R. Y., Han, M. Y. & Weaver, O. L. (1972) Phys. Lett. B 42,257-260. 16. Bargmann, V. & Wigner, E. P. (1946) Proc. Natl. Acad. Sci. USA 34,211-223.
Ann. Inst. Henri Poincare,
Section A :
Vol. XXVIII, n° 4, 1978, p. 369-378.
Physique
theorique.
Spinor-type fields with linear, affine and general coordinate transformations by Yuval N E ' E M A N (*) Department of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel and I. H. E. S., 91440 Bures-sur-Yvette, France
ABSTRACT. — We demonstrate the existence of bivalued linear (infinite) spinorial representations of the Group of General Coordinate Transformations. We discuss the topology of the G. G. C. T. and its subgroups GA(nR), GL(n, R), SL(nR) for n = 2, 3, 4, and the existence of a double covering. We demonstrate the construction of the half-integer spin representations in terms of Harish-Chandra modules. We give D. W. Joseph's explicit
matrices for j 0 = -, c = 0 in SL(3R), which will act as little group in 2 GA(4R).
1. INTRODUCTION AND RESULTS Einstein's Principle of General Covariance imposes two constraints on the equations of Physics in the presence of gravitational fields : a) a smooth transition to the equations of Special Relativity; note that we require a formulation of the Equivalence Principle in Field Theory [7]. Operationally, « locally, the properties of « special-relativistic » matter in a non-inertial frame of reference cannot be distinguished from the properties of the same matter in a corresponding gravitational field [2] ». (*) Research supported in part by the United States, Israel Binational Science Foundation. Annates de VInstitut Henri Poincare - Section A-Vol. XXVIII, n° 4-1978.
420
370
Y. NE'EMAN
b) under a general coordinate transformation x1* —> x", the equations are general-covariant, i. e. form preserving. This article relates to (b), i. e. it deals with representations of the Covariance Group < spinors I e. g. - © - © - © . . . I thus somewhat resembling a band of rotational excitations over a half-integer spin deformed nucleus. We shall therefore use the term band-spinor (or « bandor » ) for these infinite spinor representations, so as to distinguish them from (finite) coventional spinors. Historically, spinors were « fitted » into General Relativity [3] [4] after their incorporation into Special Relativity through the Dirac equation. It was noted that they behaved like (holonomic, or « world » ) scalars under (S, their spinorial behavior corresponding only to the action of a physically distinct local Lorentz group =§?E, with ^nJ?E = 0
(1.1)
Both ifE and conventional spinors thus required the introduction of a Bundle of Cotangent (or Tangent) Frames E, i. e. an orthonormal set of 1-forms ( « vierbeins »; a = 0, 1, . . . , 3 the « anholonomic » indices)
e" = ^(x)dxf
]
ea = nabe>> nab = the Minkowski metric J
(1.2)
with the L 4 (general affine) or U 4 (Riemann-Cartan) manifold metric given by &*,<% = g„v
(1.3)
Band-spinors are « world » spinors, and thus do not require E for their definition. Contrary to what is stated in most texts on General Relativity, the introduction of E should indeed not be construed as resulting just from the world-scalar behavior of spinors. E represents a further geometrical construction corresponding to the physical constraints of a local gauge group of the Yang-Mills type, in which the gauged group is the isotropy group of the space-time base manifold. We can thus even introduce band-spinors in the vierbein system [5]: the isotropy group would then Annates de Flnslitut Henri Poincare - Section A
421
SPINOR-TYPE FIELDS WITH COORDINATE TRANSFORMATIONS
371
have to be enlarged from =SfE to ^ E , i. e. the theory would have to realize a global General Affine (#") symmetry as its starting point, & =
X
J,
& ^ 0>
(1.4)
where J represents the translations. The quotient of J^ by J is ^ E (i. e. S acting on the anholonomic indices). Note that one source of the prevalent belief that there are no <€ spinors ( « world » spinors) stems from an unwarranted extrapolation from a theorem of E. Cartan [6]: « It is impossible to introduce spinor fields, the term « spinor » being taken in the classical Riemannian connotations; i. e., given an arbitrary coordinate system x", it is impossible to represent a spinor by any finite number N of components ux, so that these should admit covariant derivatives of the form (a, ft are spinor indices, \x\ are vector indices) C
L\u a = dfa + T£{x)up
(1.5)
with the TMf as specific functions of x ». As can be seen, Cartan was aware of the restriction of his proof to a finite number of spinor components. Our band-spinors *¥" indeed do admit covariant derivatives as in (1.5), in the « world » (holonomic) system,
D^S = dx* + r^Grji*
(i. 6)
where a. runs over the sets a, a x k, a x Xpa, . . . , i. e. spins ;
._ 1 5 9 '~2'2'2'
" "
G" is an infinite dimensional representation of ^, and iy v is the usual affine connection. 2. TOPOLOGICAL CONSIDERATIONS : THE COVERING GROUP OF SL(nR) AND GL(«R) We are studying the groups, <€ => & r> 0 =>
(2.1) (2.2)
where SP is the Unimodular Linear SL(4R) and & is the Special Orthogonal SO(4). We do not enter into the further structure induced by the Minkowski metric at this stage. At various stages we shall also deal with the same groups over n = 3 and n = 2 ; we shall then use the notation %, Sf3, etc. Since our aim is to find unitary representations of <£, #", (S, y which Vol. XXVIII, n° 4 - 1 9 7 8 .
422
372
Y. NE'EMAN
reduce to bivalued unitary representations of SP and &, we have a priori two candidate solutions : (a) (b)
& => G 4
Z>
^ 3
#
=3
#
jjr
03,
# =>^ !D
3
<9
#
where the bars denote double-covering of the relevant groups. In the first case, we would be dealing with single-valued representations of
(2.3)
where C/f is the maximal compact subgroup, si is a maximal Abelian subgroup homeomorphic to that of a vector space, Jf is a nilpotent subgroup isomorphic to a group of triangular matrices with the identity in the diagonal and zeros everywhere below it. The decomposition is unique and holds globally Jfnrf = ^ n /
= # n J f
= {l}.
(2.4)
Applying (2.3) to y 3 , X is &3. Since this is maximal and unique, ^ 3 *
&i
and we are left with solution — b) only. Applying (2.3) to Sf, we also have
Sf = (9sJ^s
(2.5)
9 = ®slsjfs
(2.6)
Now the groups si and Jf in an Iwasawa decomposition are simply connected, and slJf = siJf is contractible to a point. Thus, the topology of £P is that of (9. The same result has been shown to hold [8] for <8 when the L 4 is Euclidean or Spherical and holds under some weak conditions for any L 4 . By the same token, # has the topology of 0(n, R), the double covering of the full Orthogonal (which includes the improper orthogonal matrices, with det = — 1). <S and # thus have two connected components. Annates de I'lnstitut Henri Poincare - Section A
423
SPINOR-TYPE FIELDS W I T H COORDINATE
TRANSFORMATIONS
373
For n ^ 3, y is thus completely covered by y, the double-covering. However, 0(2) and SL(2R) are infinitely connected. ^2<^2
(2-7)
where y2 i s m e m u covering. Topologically, solution (b) is thus realizable. The single-valued unitary (and thus infinite-dimensional) irreducible representations of y correspond to double-valued representations of £f and reduce to a sum of double valued representations of 6. This being established, it is interesting to check a second source of confusion at the origin of the statements found in the literature of General Relativity and denying the existence of such double-valued representations. This is based upon an error in the statement of a theorem of E. Cartan [9]: « The three linear unimodular groups of transformations over 2 variables (SL(2C), SU(2), SL(2R)) admit no linear many-valued representation ». As can be seen from Cartan's proof of this theorem in ref. [9], it holds only for SL(2C) and SU(2). Moreover, Bargmann [70] has constructed the unitary representations of SL(2R), since this is the double covering Spin(3) (+ __) of the 3-Lorentz group (1, — 1, — 1); and even though only single-valued representations of SL(2R) are required for this role, he has also constructed (§ 7 d) multivalued linear representations of that group. The representations C* h = ~, q = \+s2
(2.8)
are bivalued representations of Spin (3) ( + __, = SL(2R) as can be derived from Bargmann's formula V(b) = exp (4ilhn)V(a)
(2.9)
for two elements lying over the same element of SL(2R). We take / = 1. Note that in reducing SL(4R) to SL(2R), the generators are represented on the coordinates (holonomic variables) by, Si = - ( X A - x2<32), S 2 = - ( * i d 2 + x2d1), £ 3 = -l2(^2
-x2dt)
(2.10)
[Z3, Z J = iL2 , [22> Z 3 ] = iZ,
(2.11)
with commutation relations [Ex, Z 2 ] = -
ffi3,
with E 3 generating the compact subalgebra (eigenvalues m in ref. [70]). However, when using the same algebra as the double-covering [10] of SO(l, 2), the identification in terms of the (completely different) (1, — 1 , - 1 ) space is given by, Si = i{x'0d[ + x\d'0),
E 2 = - i{x'2d'0 + x'08'2), Z 3 = i(x[d2 - x'2d\)
Vol. XXVIII, n° 4 - 1 9 7 8 .
424
(2.12)
374
Y. NE'EMAN
with the same commutators and the same role for S 3 . We stress this correspondance because it has led to some additional confusion and arguments [77] against the existence of bivalued representations of £f2, and with it 5 V 3. THE SL(3R) BAND-SPINORS : EXISTENCE The unitary infinite-dimensional representations of SL(3R) were introduced [12] in the context of an algebraic description of hadron rotational excitations ( « Regge trajectories [75] » ). A construction was provided ( « ladder representations) for the multiplicity-free | A; | = 2 bands, where j is the &3 spin. Such representations are characterized by j0 (the lowest j) and c, a real number, ®(f3;Jo,c) (3.1) the ladder representations corresponding to j 0 = 0 and j0 = 1. We shall not dwell here upon the physical context of shear stresses in extended structures, connected with ref. [12], and we refer the reader to the first part of ref. [J], for that purpose. However, it was a result of this physical context that the author noted with D. W. Joseph the possible existence of similar bivalued representations, i. e. band-spinors. Joseph provided [14] a construction for @>( P3 ; - , 0 I and proved that together with the subsets 3){^3; 1, c), 3){^3; 0, c), — oo < c < oo, this formed the entire set of | A j | = 2 multiplicity-free representations. The latter result was recently confirmed by Ogievetsky and Sokachev [15], after having been put in question [76]. We shall provide here a different construction, based upon the « subquotient » theorem for Harish-Chandra modules [77]. We return to the Iwasawa decomposition (2.6) for !?3 $>3 = &3stfjV
(3.2)
and define M3, the Centralizer of s/ in JT, i. e. in <S3. This is the set of all a e (93 such that {aeJi3\aaa~l = a) (3.3) for any aestf. The elements of s4 span a 3-vector space, and Ji3 thus has to be in the diagonal. Since det (Ji3) = 1, the elements of &3 belonging to J43 are the inversions in the 3 planes: ( + 1, — 1, — 1), (— 1, + 1, — 1) and (— 1, — 1, + 1). Together with the identity element they form a group of order 4, with a multiplication table m^m2 = m3, m2m3 = m1, m3mx = m2, ml = 1. It appears Abelian in this representation. Returning now to P3 and (S3, we look for .M3 c Q3. The inversions are given in SU(2) by exp (inojl), which yields the Non-Abelian group ^ 3 : (±iff„, ± 1 ) . Annates de I'lnstitut
425
(3-4) Henri Poincare - Section A
SPINOR-TYPE FIELDS W I T H COORDINATE
375
TRANSFORMATIONS
The subgroup 3.3 <= £f3 SL, =
Ji3s4Jf
can now be used to induce the representations of 5^3. Note that P3/M3 = S U ( 2 ) / ^ 3 .
(3.5)
The representations p(j0, c) of E3 are given by ; 0 for a representation of the Jt3 group of « plane inversions » in SU(2), and X for the characters of s4, since Jf is represented trivially. The representations of £P3 will thus be labelled accordingly; from (3_.5) we see that they will be spinvalued representations of M3. Since s$Jf = stJf, univalence is guaranteed.
4. THE SL(3R)
BAND-SPINORS
CONSTRUCTION
OF S > ( i , 0 Following our original introduction [12] of infinite-dimensional singlevalued representations of SL(3R), we now turn to our algebraic point of view. The five non-compact generators of £f3 are isomorphic to a multiplication of the symmetric X matrices of su(3) by yf — 1, and behave like a j — 2 representation under the compact (93 (the antisymmetric X matrices). They can thus mediate transitions between | Aj | = 2, 1, 0 levels of the compact subalgebra. In the following analysis we shall deal with a highly degenerate subset: the multiplicity-free | Ay | = 2 representations. Although several treatments have appeared since [15] [16] we choose to reproduce the results of D. W. Joseph's unpublished 1970 work [14]. In Joseph's notation, the ^3 generators are chosen to be : b=2
0 0 0 0 —i 0
i 0
£±
5
=
0 f±-=
0 i 0
0 -1
-1 0
0
+i
0
—i 0
±1
+1
0
i
. !~I\ 0 0
0 ±i . 0
f±±'=
0 0 0 2i 0
i 0
0 0
±1 0
±1
0
- i
(4.1)
i (4.1)
and using Capital letters for their Unitary Representations, H+=H,
E : = E_,
F
+
=F,
Ft = - F _ ,
Ft+=F_
(4.2)
the SL(3R) matrices are produced by
a = £A + ^e + + f_e_+£/ + f + / + -K-/- + £+ + / + + + £--/-£* = £, «=£_, £* = £, « = -£_, « + =£__ Vol. XXVIII, n° 4-1978.
426
(4.3)
376
Y. NE'EMAN
The commutation relations are, [h, e±] = ±2e±, [h,f±±]= [e±,f±±]
[e+ ,eJ\ = h
±4/±±, = 0,
[e±,f±]
[e± , /+] = ^ 6 / , [ / + + , / _ _ ] = -2&, [ / ± ± . / ± ] = 0,
[h,f±]=±2f±,
[h,f] = 0
= y/4f±±,
[e±,f]
=
y/6f±
I (4.4)
[e± , h +] = V 4 / * [ / + , / - ] = £,
[f,f±]
= j6e±,
[ / ± ± , / T ] = 2e ± [f,f±±]
= 0.
By imposing the | Aj | = 2 requirement upon the generator matrix elements and making use of (4.4), Joseph found a unique half-integer spins solution: E ± |;, m > = J(j
+ m){j ± m + 1) | j , m ± 1 >
H | j , m > = 2m |;, m > < ; + 2, m + 2 | F + + | ; , m> = x/0" + ™ + 4)(j + m + 3)(j + m + 2)(; + m + 1 ) t (j + 2, m + l | F + | ; , m >
= V0" + ™ + 3Xi + m + 2)0' + w + 1)0' -m + l)t < j + 2, m | F | j , m > = y/fy + m + 2)0' + m + l)(j - m + 2)(; - m + 1) t < j + 2, m - l | F _ | J , m >
(4.5)
= 27(7 + m + 1)(; - m + 3)(; - m + 2)(j - m + l ) t (j + 2,
m-2\F__\j,m) = J{j-m + 4)(; - m + 3)0' - m + 2)(; - m + Tj t 2 (2j + 3) + c2
4(2; + 5X2/ + 1X2/ + 3)2 4n + 1 n = 0, 1, 2. . . ;
: 2 -> 0
All other matrix elements vanish. This describes &[ - , 0 I. The same method showed that the only other representations in that set were the previously derived [12] band-tensors 0(0, c)
mi c)
GO < c < oo .
Annates de I'lnstitut
427
(4.6) Henri Poincare - Section A
SPINOR-TYPE FIELDS W I T H COORDINATE TRANSFORMATIONS
377
This degenerate set of representations corresponds to the SL(3R) case of a recently discovered class of representations of semi-simple Lie groups in connection with the study of the enveloping algebras and the A. Joseph ideal [18] [19].
5. GA(4R) AND GL(4R) Since the representations of <€ are those of # , the physical states can be described by induced representations of # over its stability subgroup and the translations. The stability subgroup is GL(3R), and we can thus use the product of our representations of ^ 3 by the 2-element factor group 0(3)/SO(3), since ^ 3 will have the topology of 0(3). Further complications will arise as a result of the local Minkowskian metric r\ah of (1.2). The representations we developed fit the case of timelike momenta. We shall study the other possibilities in another publication. For the construction of fields, we should use ^ 4 . Our analysis in section 3 can be repeated for this group; the MA will correspond to a product of two sets + (
-, j(,2) = 0 j e ^ U 1 ) = 0,.#>= ^ \ with (A/(1> = 1, A/ 2 ) = 1) non-
compact action. Each (;'(1), / 2 ) ) level of a band spinor^eW satisfies a Bargmann-Wigner equation [20] for j = \j(1)\ + | / 2 ' | . The covariant derivative of a band-spinor field 4** will be given by eq. (1.6). We shall deal with the field formalism in a future publication. ACKNOWLEDGMENTS
We would like to thank Professor B. Kostant for indicating to us the « sub-quotient » theorem and for pointing out the work of A. Joseph. We would also like to thank Professor L. Michel for advice and criticism, and to acknowledge a conversation with H. Bacry. We thank Professor D. W. Joseph for sending us his unpublished (1970) results. REFERENCES [1] P. VON DER HEYDE, Let. al Nuovo Cimento, t. 14, 1975, p. 250. [2] A. EINSTEIN, The Meaning of Relativity, 3rd ed. (Princeton, N. J. 1950). [3] H. WEYL, Zeit. f. Physik, t. 56, 1929, p. 330. [4] V. FOCK, Zeit. f. Physik, t. 75, 1929, p. 261. [5] F. W. HEHL, E. A. LORD and Y. NE'EMAN, Phys. Lett., t. 71 B, 1977, p. 432. See also
Phys. Rev., t. 17 D, 1978, p. 428. Vol. XXVIII, n" 4-1978.
428
378
Y. NE'EMAN
[6] E. CARTAN, Lecons sur la Theorie des Spineurs, Hermann & C Edit., Paris 1938, article 177. [7] K. IWASAWA, Ann. of Math., t. 50, 1949, p. 507.
[8] T. E. STEWART, Proc. Ann. Math. Soc, t. 11, 1960, p. 559. [9] Ref. 6), article 85-86. [10] V. BARGMANN, Ann. of Math., t. 48, 1947, p. 568. [11] S. DESER and P. VAN NIEUWENHUIZEN, Phys. Rev., D 10, 1974, p. 411, Appendix A. [12] T. DOTHAN, M. GELL-MANN and Y. NE'EMAN, Phys. Lett., t. 17, 1965, p. 148.
Y. DOTHAN and Y. NE'EMAN, in Symmetry groups in Nuclear and Particle F. J. Dyson, ed. Benjamin, 1965.
Physics,
[13] G. CHEW and S. FRAUTSCHI, Phys. Rev. Lett., t. 7, 1961, p. 394.
[14] D. W. JOSEPH, Representations of the Algebra of SL(3R) of Nebraska preprint, Feb. 1970, unpublished.
with | A/ | = 2, University
[15] V. I. OGIEVETSKY and E. SOKACHEV, Theor. Mat. Fiz., t. 23, 1975, p. 214, English
translation, p. 462. See also Dj. SIJACKI, J. M. P., t. 16, 1975, p. 298 and Y. GULER, J. M. P., t. 18, 1977,
p. 413. [16] L. C. BIEDENHARN, R. Y. CUSSON, M. Y. HAN and O. L. WEAVER, Phys. Lett., t. 42 B,
1972, p. 257. [17] HARISH-CHANDRA, Trans. Amer. Math. Soc, t. 76, 1954, p. 26. See also J. DIXMIER, Algebres Enveloppantes, Gauthier-Villars pub., Paris, 1974, §9.4-9.7. [18] A. JOSEPH, Ann. Ecole Normale Superieure, t. 9, 1976, p. 1. Also, Comptes Rendus, t. A 284, 1977, p . 425 and several recent Bonn and Orsay preprints by the same author. [19] W. BORHO, Sem. Bourbaki 489, Nov. 1976.
[20] V. BARGMANN and E. P. WIGNER, Proc. of the National Acad, of Sci., t. 34, 5, 1946, p. 211. (Manuscrit recu le 5 septembre 1977)
Annates de VInstitut Henri Poincare - Section A
429
Spinorial infinite equations fitting metric-affine gravity A. Cant3' and Y. Ne'emanb)
Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, IsraeF1 (Received 12 December 1984; accepted for publication 12 June 1985) Two different approaches are used to construct infinite-component spinor equations based on the multiplicity-free irreducible representations of SL(4,R). These "manifleld" equations are SL(2,C) invariant; they exist in special relativity, and can directly be coupled to gravitation in the metricaffine theory, i.e., in Einstein's general relativity with nonpropagating torsion and nonmetricity. In the first approach the maximal compact subgroup SO(4) of SL(4,R) is "physical." A vector operator X * is constructed directly in the infinite-dimensional reducible representation J^dlsc(i,0) _®^dlsc(0,J). In the second approach, SL(2,C) and a vector operator y* are embedded directly in SL(4,R) via the Dirac representation. A manifleld equation is then constructed (in a manner analogous to the Majorana equation) by taking an infinite-dimensional irreducible multiplicityfree representation of SL(4,R), spinorial in y„ in the (ji,j2) reduction over SO(4). Both manifields can fit the observed mass spectrum.
I. INFINITE COMPONENT FIELDS
Relativistic quantumfieldtheory exploits the concept of a local field as the fundamental dynamical object, with the particle aspect emerging as the offspring. The particles span unitary irreducible representations of the Poincare group ISO(3,l) and its double covering ISO(3,l). Fields, on the other hand, transform asfinite—andthus nonunitary—representations: of GL(4,R) when tensorial, or of SL(2,C) for spinor fields. The latter group appears here as the double covering of the Lorentz group, i.e., SL(2,C) = SO(3,l ((quantum probabilities do not involve phases and thus allow the double covering). The nonunitarity of the representations [or non-Hermiticity of the relevant matrices of the Lorentz or GL(4,R) algebras] does not matter physically: the Lagrangian's Hermiticity requires the addition of the complex conjugate expression, and the non-Hermitian parts of the Noether-theorem-generated densities cancel.1 As a result, the special Lorentz transformations in SL(2,C), for instance, have _only orbital components with the pieces S d3x(i/icr0ii/> + H.c.) canceling. Boosting an electron state thus contributes only to the kinetic energy. The same type of cancellation occurs for the (noncompact) deformation generators in GL(4,R). The compact subalgebras of GL(4,R) or SL(2,C) being the only ones to contribute to the physical currents and generator observables, why do we need the full groups altogether? The action and its Lagrange density have to be globally invariant under the (active) Poincare group. When we include gravity we require invariance under the (passive) general covariance group (the diffeomorphisms A with local dependence of the transformations). The latter is realized nonlinearly over the linear subgroup GL(4,R); we thus have to use "world tensors" and the equivalence principle can be fulfilled in the easiest manner by keeping them in special a|
Present address: Department of Theoretical Physics, Research School of Physical Sciences, Australian National University, Canberra ACT 2601, Australia. ""Also on leave from University of Texas, Austin, Texas 78712. c) Wolfson Chair Extraordinary in Theoretical Physics. 3180
J. Math. Phys. 26 (12), December 1985
relativity, too. This involves regarding GL(4,R) as "GL(3,1;R)," i.e., introducing the Minkowski metric linearly and identifying accordingly the SO(3,l) subgroup, with the special Lorentz transformations given by symmetrical matrices that do not belong to the SO(4) maximal compact subgroup of GL(4,R). Alternatively, one may define x4 — ix° (the "Pauli metric") and identify the orthogonal matrices of the compact SO(4) with the physical Lorentz group, as we demonstrate in (5.3). One can then ask, in either case, for global Lorentz invariance and ensure that this be manifest invariance. For spinor fields with a finite number of components, the transition to A does not exist [there is nofinitespinorial representation of SL(4,.R )] and the spinor components are invariant under GL(4,.R ). Global (active) covariance under the double-covered Poincare group is formally ensured by SL(2,C). For all fields SO(3,l) = SL(2,C) C ISO(3,l), the double covering of the Poincare group is the global covariance group in the final result. This is thus the covariance group of special relativity both for particles and forfields.Of course there is the additional advantage of a smooth manifestly invariant classical fields' limit, where the particle aspect does not enter, and neither does unitarity. Infinite-component fields, however, as they correspond to unitary representations of SL(2,C) and to Hermitian infinite matrices of the sl(2,C) algebra will, in contradistinction, yield "internal" contributions to the special Lorentz transformations. In this case, the boosts will excite the spin variable, too, and may thus contribute to the potential energy (i.e., connect to a different mass). Such infinite-component fields werefirstintroduced by Majorana,2 who used the only two irreducible representations of SL(2,C) for which an invariant (linear)first-orderwave equation of the form3 {*"
0022-2488/85/123180-1 OS02.50
430
© 1985 American Institute of Physics
3180
nuclear, or other excitations, following the reintroduction of Majorana's work by Fradkin. 5 However, difficulties arose due to the presence of a continuous set of solutions with spacelike momenta, in addition to the discrete spectrum— which is itself not realistic sine? states of higher spin have a smaller rest mass. Dirac 6 recently rediscovered these equations and further developed the formalism. II. GRAVITY: THE EINSTEIN, EINSTEIN-CARTAN AND METRIC-AFFINE THEORIES (i) Einstein's theory is Riemannian, i.e., it precludes the propagation of either torsion or nonmetricity. Only the metric field gpV{x) propagates. Alternatively, we may use the tetrad fields e° (x), with g^)
= et{x)ei(x)nab,
(2.1)
where t)ab is the Minkowski metric ( + 1, — 1, — 1, — 1). In the above, the Latin indices a,b represent components of the four-vector representation of the anholonomic group. In Einstein's theory with spinor matter fields,7 or in Einstein-Cartan theory, 8 this is SL(2,C) acting on the local frames. The tetrad fields had to be introduced 7 in gravity after the discovery of the electron's spin, in order to cope with half-integer spin fields. In differential geometry they describe a general moving frame, i.e., a set of one-forms d" denned over some region U of space-time: 0° = e%dx».
(2.2)
At each point x = x e U, the 0" serve as local "coordinates," inertial at 8?. From the principle of equivalence, i.e., a smooth transition to special relativity, when the gravitational field is extinguished, we now get a requirement of local SL(2,C) = SO(3,l) invariance of the locally inertial coordinate system at each point: the frame is orthonormal. The spinor field carries a (J,0) e (0,ij representation of this (anholonomic) local Lorentz group, but is invariant under the dhTeomorphisms (general coordinate transformations). Ordinary tensor fields vary under the (passive) action of the (holonomic) diffeomorphism group A and its affine [GA(4,Rj] and linear [GL(4,R)] subgroups but are scalar under the anholonomic Lorentz group. To recapture their variation under the active anholonomic transformations of the local Lorentz group (and thus to satisfy the principle of equivalence) they have to be contracted with the tetrads:
X( e -'£M-4£r.
(2.3)
They would then become world scalars (i.e., invariant under the holonomic A). General relativity with spinors is thus rewritten in a manner which makes A act trivially on all fields. This treatment was presented in most textbooks as if it was required by the (erroneous) assumption (to which we return later) that there can be no world spinors, i.e., that the diffeomorphism group has no double covering A. In any case this is irrelevant for the Dirac field, as there are indeed no finite-dimensional unitary bivalued representations of SL(4,R), GL(4,R), or GA(4,R), or of the diffeomorphism group A. Finite spin fields are thus treated anholonomically 3181
J. Math. Phys., Vol. 26, No. 12, December 1985
only, as objects belonging to the tangent manifold, Minkowskian for a theory obeying the equivalence principle. So the discovery of half-integer spin did not modify Einstein's theory, but it required reexpressing the gravitational field in terms of tetrads rather than the metric, the latter now appearing as a higher construct. (ii) In Einstein-Cartan gravity, as developed by Sciama, Kibble, Trautman, and Hehl, 8 space-time is allowed to carry torsion, as well as curvature. Applying the Poincare group double covering as a local gauge on the anholonomic indices, curvature is seen as the field strength of the SL(2,C) Lorentz connection
= dvv/b - d^b
+ »„<>- - »„%<, (2.4)
V
:
= d^l - a»< + » / « < - «vVv-
Holonomically, torsion introduces an antisymmetric piece in the Einstein connection Tflpv, in addition to the symmetric Christoffel symbol, V v = («~ , ).''S; V = j ( r „ ? - r v £ ) .
(2.6)
Considering gravity heuristically as a gauge theory of the Poincare group, one would thus have expected to deal with two gauge fields (both with spin J = 2), i.e., cof for SL(2,C) and e° for the translations. In the Einstein-Cartan version of gravity, varying the Lagrangian with respect to both yields the two equations R^~\g^Rpp=kE^
k: = &irc-*G
(2.7)
(Einstein's equation) and, with S^: = S^^ + gPf, SVJ S^=kS.ltvp
(2.8)
(Cartan's equation), where £^ v is the energy-momentum density tensor and S^vp the angular-momentum density tensor. Einstein's (and the Einstein-Cartan) Lagrangian for the gravitational field is linear and contains only one derivative (from 2.4). Ths is why (2.8) is just an algebraic equation and only implies a substitution of torsion by spin. The Einstein equation (2.7) contains curvature (2.4) and through it the connection co and through (2.6) the torsion S, as can be seen by writing the holonomic expression for the connection for a four-dimensional Riemannian differential manifold with torsion
r „ / s g"*£; a da gpy - grSsa%), (2.9) K%- = 8°8?Sl + 8°S?Sl -
8ZS?Sl,
with ^Sv,:= -&v„=0.
(2.10)
The first term in T is the Riemannian connection, and the second is the torsion contribution. In fact, Eq. (2.9) results from substituting (2.6) and (2.8) in (2.5) and solving for I \ Substituting spin for torsion in (2.7) simply adds a term quadratic in spin on the right-hand side.8 Here, D^ is the covarA. Cant and Y. Ne'eman
431
(2-5)
3181
iant derivative, with connection T and Q^ is the nonmetricity tensor. Thus, even though we have allowed torsion, it does not propagate. It is confined to the regions where the spin density exists. The effective theory is thus still Einstein's, except for the spin-spin term to be added to E . (iii) The metric-affine theory 9 allows the most general differentiable manifold Lt, with a connection (allowing parallel transfer) and a metric (allowing local measurements of angles and distances). Expression (2.1) does not vanish, and T^f in (2.9) acquires an additional term in the parentheses, 5 Qafiy The local gauge group on the anholonomic indices is GL(4,R), deforming the tetrad frames. We use a gravitational Lagrangian in which the connection is now this complete affine connection, and with a new term added, J?{g,dg,r,dT) 0*0,
= ( - detg)1/2(g^R^%+PQaQ"),
Qa: = \Qa/
(2.11)
(Qa is known as the Weyl vector), we get as a third field equation, Qa=kra,
(2.12)
where Y„ is the scale current, a reducible component of Y aft ,, the hypermomentum tensor density (det - g)1 /2 Y Pp" V =
- - i ^ - X ".p iP, d[dyt/>)
where the X ^ are the matrices of the GL(4,R) algebra. Thus nonmetricity Q does not propagate, and is confined to the regions of nonvanishing deformation-current or scale-current density. The energy momentum tensor density acquires a new term quadratic in the scale current.
III. THE DOUBLE COVERING OF GL(4,R) The anholonomic group acting on the local frames has thus been enlarged to form SL(2,C) = SO(3,l) in Einstein's theory with spinors and in Einstein-Cartan theory, or to GL(4,R) in the metric-affine theory. 9 The (erroneous) universal impression among physicists that GL(4,.R ) possesses no double covering10 seemed to restrict the application of metric-affine gravity to bosonic matter. The existence of a double covering GL(n,R) was realized in physics in 1977.11 This implied the existence of spinortype fields transforming (whether fermonic or bosonic) as "bandor" 1 2 1 3 unitary infinite-dimensional representations of the (meta-) linear, affine, and diffeomorphism groups; under reduction of these covering groups to the covering group of the orthogonal subgroup SO(3) the fields decompose into representations of SU(2) = SO(3). It had been conjectured' 2 that hadrons with their Regge excitation bands could be described by such bandor irreducible unitary representations of GL(3,R) C GL(4,R). It was now proposed13 that such a description should also fit their interaction with gravity. The physical interpretation of the GL(n,.R ) currents was clarified and it was suggested that in metric-affine gravity, spinormatter fields indeed appear as infinite-dimensional unitary representations of the anholonomic GL(4,R) acting on the tetrad indices.14 3182
J. Math. Phys., Vol. 26, No. 12, December 1985
The term polyfield or manifield was suggested. It was also pointed out that since the diffeomorphism group is realized through (nonlinear) group coordinates over the linear GL(«,R) subgroup, manifields could also be considered as providing for world spinors, 11 i.e., holonomic spinors, whether in Einstein or in affine gravity. 15 In this role the representations correspond physically to the double covering of the Greek-indexed coordinate (holonomic) linear group GL(4,R) C A = Dlff(4,R), in contradistinction to the above anholonomic GL(4,R) acting on the tetrad (Latin) indices, in'the metric-affine theory. We thus have three gravitational roles for such manifields: (a) anholonomic spinor matter fields in the metricaffine theory, 14 (b) holonomic world spinors in "classical" Einstein gravity, 11 and (c) holonomic world spinors in affine gravity. 15 Mickelsson 16 has constructed a wave equation fitting case (c). His equation is GL(4,R) invariant; when the gravitational field is extinguished, it preserves global GL(4,R)invariance, i.e., it does not obey the principle of equivalence. On the other hand, it could fit in an affine theory with a basic non-Minkowski microscopic structure of the space-time manifold, perhaps with macroscopic spontaneous breakdown to Minkowski space-time. Such models have only been discussed qualitatively 15 to date. Another (technical) reason why we do not favor a GL(4,R) invariant equation is that—as we shall see—the "bandor" representations do not allow the construction of such an equation. In this article, we propose two distinct ways of meeting case (a). The manifield equations we construct are of the form (1.1). Although they involve unitary representations of GL(4,R), they are only SL(2,C) invariant and thus have a good equivalence-principle limit. They can be used as more infinite-component field equations in special relativity and conventional tetrad gravity, or [role (a)] as spinor matter manifields in metric-affine gravity. Our equations are in close analogy with the Dirac equation, and, as for the Dirac case, the gravitational field enters through the inverse-tetrad fields $X °da^X°D„t
= j,X°(e-')*(
+»„),*,
where Da is the anholonomic covariant derivation and co^ is the connection. In Einstein gravity with Dirac spinor fields we have »„ = « „ " * * ,
(3.2)
with Xbc a finite-dimensional nonunitary matrix representation of the sl(2,C) algebra. In metric-affine gravity Aab is a unitary infinite-dimensional matrix representation of the gl(4,R) algebra. When gravitation is introduced, the Xab take the six SL(2,C) (nonlinear) values for Riemannian spacetime, or the full 16 (matrix) values for metric-affine gravity. In the next section we shall summarize the properties of multiplicity-free representations of SL(4,R); Sees. V and VI discuss the formation of wave equations according to two quite distinct approaches. In each case we propose infinitecomponent fields. These manifields may thus provide the correct mode through which the sequences of hadron excitations interact with gravity. 1314 Both fit role (a) but only the manifield (5.8) may fulfill role (b). A. Cant and Y. Ne'eman
432
(3.1)
3182
Then, in the spherical basis,
IV. SL(4,R) AND ITS REPRESENTATION The unitary irreducible representations of the group SL(3,R) have been constructed and listed. 1718 Those of G = SL(4,B) have been studied 15 - 171920 though a complete description is still lacking. The representations of G, which are multiplicity-free on reduction to the maximal compact subgroup A: = SO(4) = SU(2)xSU(2), have a particularly simple form and were constructed explicitly.15,19 We use the basis (jW\x = 1,2;i = 1,2,3 j for k, with
[J!*JjM]=iS„WF-
Ji"
h m)
[j[2),ZJk]=ielkmZjm, +
8lkelmntf\
We also have the spherical basis, given in terms of the above Cartesian basis by
y'l
k
m.
m
k
(4.10)
m2 [yi(yi + l ) - « . K ± l ) ]
(4.3)
Ik
k
\m\
m'2
l/'i \m.
l
[ Ji K ZaP ] = ccZa„
[ jff>, Zaf)
H
k
J k
1 h\(
V — m[ a
(kk\
k
mj
1k \
\ — m2 P
mj (4.11)
\z\kk)
= _ / ( _ lf'+A[(2j[
The commutation relations (4.2) become
) •
\ml m2 + , ) • are given by
while the noncompact operators Za/3
Zu
+ Z22) + ;(Z21 ± Z 1 2 )].
pi
l)]1
[J2U2 + 1 ) - m2{m2±
Z 0 .±i = +(1/ A /2)(Z 3 1 +/Z 3 2 ), = ± 2[(Zn
1 m
\m,±
\m.
X<J'Ji\ \Z\ \jji). The reduced matrix elements are
±,
J2
1
Ji
L/i
n
-i-lfZ±l0=+{l/j2)(Zl3±iZ23),
h m
m2l
(4.2)
= -i(8jmeiknj^
">1
)
(4-1)
[jll\Zjk]=ieiJmZmk,
Jl
-
k
J6
The remaining (noncompact) matrices in sl(4,R) transform as the irreducible tensor operator Z of type (1,1) under K:
[ZipZkm]
\Ji
+ l)(2y2 + l)(2 7l + 1)
x(2; 2 + i)] 1 / 2 (p 1 + / > 2 - 2 [ v ; ( y ; + i)
\=pZae,
- y'i(y'i + i) +k(k
[/i'.^]=(2-«(a±l)),"ZB±w
;:
= &Za±h0(l-8±hJ,
(4.4)
*(» i X
+ i) -kik
+ i)])
o i)
and clearly they are nonzero only for the four possibilities [J%.Zafi]=Tl2'a.0±i.P{l-8±1J,)
(a,/3 =
0,±l),
with the remaining ones following from the so-called "sl(4,R) condition" [ Z U ) Z _ , _ 1 ] = -(jP+j?\
(4.5)
It is convenient to introduce here too, the basis used by Mickelsson16 for gl(4,R): Lrs = e„ - e„,
A„ = ers + esr,
rj = 1,2,3,4,
(4.6)
where e„ is the 4 x 4 matrix with 1 in the r,s position and all other elements zero. The L„ span k = so(4), and we have [Lrs>L,u] =8stL„
—8„LSU —8SUL„
+SruLs[,
[Lrssl,u ] = < M „ - 8„ASU + SSUA„ - 8„AS„ [Ars,A,u ] = 8„ L„ + 8„LSU + 8SUL„ +
(4.7)
8mLst.
2/yf,
M„ = Ltl -Lki=
{ijk j a cyclic permutation of j 123).
3183
k):
7'i° =/»i + L
7.° = 0,
or
2ijfK (4.8)
To construct multiplicity-free representations we take the subspace VoiL 2(K) with orthonormal basis
r1
k=k±i-
Strictly speaking, from (4.9), the values of jltj2 should only be 0,1,2 But at this stage we can formally continue (4.10) and (4.11) to half-integer values of juj2 as well. The sl(4,R) condition (4.5) must be rechecked. One can proceed to find the complete set of all the unitary irreducible multiplicity-free representations SL(4,R) (see Ref. 15 and 19). We shall only need some of these representations. First, we have that class, belonging to the discrete series, which is spinorial: i.e., double valued for SL(4,R), and quadruple valued for SO(3,3) [note that SL(4,R) = SO(3,3), but single valued for SL(4,R). Their K content has the structure of a triangular lattice {pi,p2 are Casimir invariants): D"^l/,0,
We also put Lk = L,j +Lk,=
y'I=y'i±i.
J = [(2y, + l)(2/2 + l ) ] 1 ' 2 / ) ^ , D&,. (4-9) J. Math. Phys., Vol. 26, No. 12, December 1985
(4.13)
7° = 0, Pi=
— 1- i. i. 5 —-•
j?=p,
+ l.
/>2 = 0,
| y ' i - A l >/>i + iSecond, we want to mention the ladder series12 of tensorial bandors, i.e., single-valued for SL(4,R), with K content as follows19: A. Cant and Y. Ne'eman
433
3183
Id
(0,0;/.2):[(0,0),(1,1),(2,2),...
sl(4,R)->sl(2,C), defining representation—^ ,0) ® (0,J),
(4.14)
(5.3)
adjoint representation—^ 1,0) m (0,1) e 2( \ ,\) e (0,0). P\=
— 1,
p2eR.
The second of these, rather surprisingly, turns out to be relevant for manifield equations (see Sec. VI). This representation was constructed in Ref. 20 in solving the strong-coupling model for the nucleon, for the value p2 = 0. V. SL(2,C)-INVARIANT WAVE EQUATIONS We now turn to a consideration of some wave equations appropriate for the gravitational interactions of hadrons. The general type of equation we have in mind is (in momentum space, with no gravitational field yet present) {X*Pll-K)*iP) = 0, (5.1) where tp takes its values in a Hilbert space V carrying a unitary multiplicity-free representation ir of SL(4,R), and K is an SL(2,C)-invariant operator on V, possibly a function of P1=PI" Pn tthis generality is sometimes needed when we look for realistic mass spectra—see Sec. VI). The X ** {fi = 0,1,2,3) are linear operators on V. We demand only SL(2,C) invariance as discussed in Sec. Ill, so t h e X ** transform as an SL(2,C) vector. Physically, we want an equation which provides a kind of "extended" Dirac field. At this stage we are confronted by various choices, namely, (a) which is the "physical" Lorentz subgroup of SL(4,R) and (b) how is it embedded? These points are by no means trivial, as we shall see. Our embedding of K = SO(4)—as well as SO(3,l)—in SL(4,R) has been the natural one, described by the Lie algebra branching rules sl(4,R)—>-so(4) or sl(2,C): defining representation—>• (i, i), (5.2) adjoint representation—>•( 1,0) © (0,1) e (1,1). Since our representation TT of SL(4,R) is K finite, i.e., on reduction to A'it contains the representation (jltj2)> but a finite number of times, it is most natural 1516 to take the quantum numbers (ji,j2) to refer to thephysical Lorentz group. This means using x4 = ix°. The non-Hermiticity of these "physical" Lorentz generators does not affect the physics, as explained in Sec. I. For this solution the Lorentz boosts will again be purely orbital and contribute to the kinetic energy only. All of this is perfectly respectable, since only finitedimensional representations of K are involved. But if we had taken directly the SL(2,C) subgroup then 16 TT would not contain any finite-dimensional representations of SL(2,C); this case is usually ignored. An important property of embedding (5.2) is that we must look outside sl(4,R) to find the required K vector X **. This is our first approach to wave equations, which is further discussed in this section. We refer to (5.2) as the natural embedding. It is based on an automorphism proved in Ref. 19. However, there is a second approach, suggested to us by the case of the Majorana representations of SL(2,C). There is an embedding of SL(2,C) in SL(4,R) obtained via the Dirac representation: 3184
[We shall show later that SO(4) cannot be so embedded.] Now everything is quite different: we have two linearly independent SL(2,C) vectors y^tf* in SL(4,R). So we can obtain automatically an SL(2,C)-invariant equation suitable for our purpose simply by taking an irreducible representation TT of SL(4,R). It is important to realize that (5.3) does not provide a direct embedding of SO(3,l) in SL(4,R). Instead, SO(3,l) is embedded in SO(3,3) and SL(2,C) in SL(4,R) = SO(3,3). We shall discuss this possibility (the Dirac embedding) in Sec. VI. Let us comeback to embedding (5.2). The condition that X * can be a AT vector is [Lrs,X,]=Ss,Xr-Sr,Xs.
(5.4)
To express (5.4) in a form convenient for applying angular momentum algebra, we define the quantities XAB (A,B = ± i ) b y
Then we see that the XAB transform like the canonical basis for the A"-vector representation [\, J): L Jo > XAB J
=
•" XAB,
[ j 0 , XAB J = B XAB, (5.6)
[j± >XAB\ = XA±lB,
[j± ,XAB J
=XAB±,.
It is well known from the theory of Lorentz-invariant wave equations 21,22 that, in a candidate representation TT of SL(4,R), the matrix elements of XAB are given by 23 h
\J, \m.
mi
= (-DJ X
pi \-m'2
m2
\-m\
i h ) <7'i j'i)
B
m2)
A mj
(5.7)
(Jiji)-
They are nonzero only for the four possibilities j[ =j\ ± \ , ji =h + i- It is immediately clear that, among the representations (4.13), the only possible unitary multiplicity-free spinorial representation TT of SL(4,R) that admits a K vector is the (reducible) combination suggested in Ref. 15. We have that w = ^ d i s c (i,0) ® ^ d i s c (0,i)
(5.8)
(so in each case />, = — \, p2 = 0), with the K content shown in Fig. I. Here the dark (white) circles refer to •0di*:(O,J){.0di!C(LO)), and the only nonzero matrix elements of XAB are between the K representations (j,j + J) and (j + \,j) (j = 0,1,2,...), i.e., across the diagonal. We want to remark here that our multiplicity-free representations do not allow the existence of a SL(4,R) vector. The proof is given in Appendix A. Now the operators Xk in Mickelsson's wave equation 16 do transform as SL(4,R) vecA. Cant and Y. Ne'eman
J. Math. Phys., Vol. 26, No. 12, December 1985
434
-H
possible, as in (6.39) or (6.42). Spins coupled to zero eigenvalues ofX° are excluded,21 as they would have (infinite) unphysical masses. This may imply a need for subsidiary constraints. Since (5.8) belongs to the double-covering SL(4,R), this manifield, though constructed so as to couple anholonomically to gravity, may also have a holonomic version.
>
+,
i,".'/,)
/
x*
* ,
>
VI. A WAVE EQUATION BASED ON THE DIRAC EMBEDDING In this section we want to study the possible Lorentzinvariant wave equations obtained by considering suitable representations IT of SL(4,R), where SL(2,C) is embedded according to (5.3). First of all we shall write down the (Lie algebra) embedding explicitly, directly using results of Ref. 25, where a general study was made of those real Lie algebras containing sl(2,C) and a vector operator. The starting point is embedding of the compact algebras
+
i
'k-
^
f
j
i
i_
.•—'
^
K
su(4) D su(2) e su(2), FIG. 1. Action of X&>) on &'lx({,0) ffi 3>iix(0,\).
provided by the Dirac representation (^,0) © (0,^). We use the fact that, 25,26 ifg0 is a real form of sl(4,C), obtained via the Weyl "unitary trick" from the involutive automorphism s (say) of su(4), then sl(2,C) is a subalgebra of g0 if and only if
tor. However, his representations of SL(4,R) are not multiplicity-free, and the argument of Appendix A is no longer valid. For our equation we write the reduced matrix elements in (5.7) as 24
s(X,Y) = (Y,X),
(6.2) X-^NXN'1
bj = (j+L,j\\X\\j,j+i), 7 = 0,1,2 The a, and b} can be arbitrary complex numbers. Thus, strictly speaking, we have a family of wave equations, each one described by a particular choice of these coupling constants (assumed nonzero). As far as SL(2,C) properties are concerned, each such system is an infinite set of decoupled equations for successively higher half-integral spins. Each constituent (j,j + i) ^± (j + \,j) in general has the 2] + 1 spins: 2j + \, 2j — \,...,\. The gravitational field, in the form of the noncompact shear operators Z a/3 , will couple between these constituents, and also throw up new K representations so that altogether we recover the representation (5.8) of SL(4,R). Although we are not concerned here with the Lie algebraic properties of the vector operator X", we note 21 that the Lie algebra generated by the X * and k will be, for almost all choices of a}, by
N--
T
N~l.
{ °
i02
)
2
(6.3)
K-io o)One can check that s satisfies (6.1). But how do we know that the resulting real form is sl(4,R) [and not su(2,2) or su*(4), for example]? The reason is that 25 U-1NU
= I,
where
" =- ( '
*
u
e t/(4),
(6.4)
and thus s = a '0a, where 6,a ae the automorphisms given by e-.X-^-X, a:X-*U-lXU.
(6.5)
Since s is conjugate to 0, and 0 clearly gives the real form sI(4,R) with the Cartan decomposition
Including the Zae will no doubt generate an infinite-dimensional Lie algebra. The mass spectrum, too, depends on the choice of a,, by Two equations for which the quantities a]t byj = 0,1,2,..., coincide clearly have the same spectrum. The spectrum is given by
sl(4,R)=& ' © / > ' = so(4) © {real symmetric matrices],
(6.6)
we see that we indeed have a reahzation of sl(4,R). Our Cartan decomposition is given by g0 = k ®p,
(5.10)
if K in (5.1) is a constant. More realistic mass spectra appear
= -NX
Note: su(4) consists of skew Hermitian matrices. The matrix JV€ SU(4) may be taken to be
sp(4,C) e sp(24,C) e - e sp(2(2y+ l)(2y + 2),C) ®
3185
(6.1)
s:su(4)—>-su(4),
(5.9)
(A a nonzero eigenvalue of X°),
® su(2).
From Ref. 25 we have the following result: sl(2,C) is embedded in sl(4,R), and 5 is the (outer) automorphism
«, = U y + JII*lly + l../>.
m = K/A.
V(X,Y)esu(2)
(6.7)
where the maximal compact subalgebra is
J. Math. Phys., Vol. 26, No. 12, December 1985
A. Cant and Y. Ne'eman
435
3185
ia B y 0 -/? ;6 0 y a,b € R; fry e C •=a-1k' -7 0 -ib P 0 -y -]3 ia, [isomorphism to so(4)]; while the noncompact generators are (c 8 T 8
—c i/ rj —c —8 \fi - r -S c which means that our realization is P=-
ceR;
T
go=
S,T,6,y s C = a-'p'
=
(6.8)
= Up'U
o^Bo2 Jjij) I A'B G gl,2'C,; Re Tr(^ ' = °} = a~'(^M
[{-
Uk'U'
(6.9)
= ^ [sl(4'R)l ^"
(6.10)
This realization is somewhat strange, but we can go over to the more familiar one by applying the isomorphism a. We shall always do this since we want to compare our embedding with the more familiar case of (5.2). Our embedding proceeds via sp(4,R) as follows. For the compact algebra, su(2) e su(2) C usp(4) = u(4) n sp(4,C),
(6.11)
where sp(4,C)= {X<=sl{4,C)\BXB-l=
-XT\
(6.12)
and
V 0
aj
Then sl(2,C) is embedded in sp(4,R) (see Ref. 25). We take the automorphisms of usp(4) given by s:X-*MXM-\
(6.13)
where
J)eUSp(4) (M2= -I).
M=i(f
(6.14)
The resulting real form g has Cartan decomposition g = k $ p, where
k=i
(6.15)
— ia
P
~P
— ia 0 id
id 0
id 0 ia
-P
'
° \ a,rfeR;/?e C
" -ia)
•
C/fc
(6.16)
(
•
and
P =
b
r
-r
-b -8 ic
— ic
-6
ic 8 -b
:•;
-r
b )
5
1
\ b ,c e R
y,8 e R
Hereg is a realization of sp(4,R) contained in our realization g0 of sl(4,R); the maximal compact subalgebra k is isomorphic to u(2). Clearly s, as given by (6.2), is an extension of s, because if X e usp(4) s(X)= =
-NXTN^' +NBXB~1N-'
= MXM''=~s(X)
(since NB = M).
The isomorphism a given by (6.5) takes g to sp'(4,R)= ( X € s l ( 4 , R ) | 5 ' X 5 ' - 1 =
-XT\,
(6.17)
Qp-
r
whertB' = UTBU, (6.18) the more familiar realization. We notice that so(4) cannot be embedded in sl(4,R) via the Dirac representation, since any two maximal compact subalgebras of sl(4,R) are conjugate under some automorphism, and so(4) is aleady embedded via the natural representation (J ,J). It is interesting to see how this result appears if we ask the general question: which real forms g0 of sl(4,C) contain so(4) embedded via the Dirac representation? We discuss this in Appendix B: it turns out that su(2,2) and su*(4) are the only possibilities.
J. Math. Phys., Vol. 26, No. 12, December 1985
A. Cant and Y. Ne'eman
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Coming back to our embedding sl(2,C) C sp(4,lR) C sl(4,R), we write the sl(2,C) generators of rotations and Lorentz boosts as
where the metric is g"" = d i a g ( - 1 , - 1 , - 1 , 1 , 1 , 1 ) . The sl(4,M) generators are identified as follows: Q >' = iy'y'=
-Hk
(ijk: cyclic permutation of {1 2 3)),
e» = liyy=(//2)r',
Using the Dirac matrices in the form
'-0° .0- r-C- .-"0-
(6.27)
Q'5 = hrY= ~\F'> g" 6 = -iiy",
'-'
Q"s= -yfy°
we introduce another vector operator, given by
(6.28)
= (i/2)y°\
46
Q = -If(6.21)
where
Then, for sp(4,R), we see that £ has been } H k,iy°) and j? has basis [F k',iyk'}. The remaining generators of sl(4,R) are iy0', y5 ek, iyk' &p. Note that sl(4,i?) contains two vector operators as expected from (5.3). But only one of these—in this case /y—belongs to sp(4,B), once the skew-symmetric form B is fixed. Under the isomorphism a:g0—>-sl(4,R) we have (in the notation of Sec. IV) iy°'^Mu
iy°'^M},
y5—-M2,
[
'
'
for the compact generators and F ->Z 12 ,
F —* — Z 32 ,
1
/V —Zl3, if-+-Z3„ 'V"-*Z„, if'-*-Zw
F3—» — Z 2 2 , if-+-Z2i,
(6.23)
if-*-Z21,
for the noncompact ones. Notice that in our setup the physically relevant su(2) subalgebra is that spanned by L: in the approach taken in Refs. 15 and 16 it is that spanned by L + M. Also, the maximal compact subalgebra &ssso(4) has no physical role; though it is still mathematically relevant in the study of the representations of sl(4,R). Again we stress that y*, y^' are not vectors under the so(4) subalgebra, but under the noncompact subalgebra sl(2,C)=sso(3,l). It is also clear that sl(3,R) does not fit into our scheme in such a way that its maximal compact subalgebra so(3) is spanned by L. The Lie algebra sl(4,R) is isomorphic to so(3,3). We can easily write the so(3,3) generators in terms of Dirac matrices as follows. Introducing the notation ym = y\f,f,
- if,y°, - il),
m = 1,2,3,4,5=0,6,
These formulas are analogous to Barut's 27 four-dimensional realization of so(4,2); the only difference is that he takes y4 to b e y 5 a n d g m " = d i a g ( - 1 , - 1 , - 1,-1,1,1). It is interesting to compare our approach with Barut's theory 27 ' 28 of the hadron spectrum using SO(4,2): in both cases sl(2,C) is embedded via the Dirac representation. Barut was led to so(4,2) by the well-known properties of the hydrogen atom, which has a so(4) kinematical symmetry. We have the spectrum-generating algebra sl(4,R) s so(3,3). Kihlberg 29 has, in fact, suggested using so(3,3) for hadrons, with the maximal compact subalgebra so(3) $ so(3) interpreted as the sum of spin and isospin algebras. In our approach, however, using SL(4,R), we have the gauge group of gravity naturally appearing. This is why we can speak of the gravitational interaction of hadrons. Now we can produce Lorentz-invariant wave equations of the form (5.1), suitable for the description of the gravitational interactions of hadrons. One can say that our equations are extensions of Dirac's equation, since we used the Dirac representation of sl(2,C). If wefixthe vector operator to be y1", then the equation is parity invariant. The parity operator P is essentially25 the M of (6.14); it singles out the real form sp(4,R). Parity invariance means that iy° e k. In the same way charge conjugation C is essentially the N of (6.3); it gives the real form sl(4,R) and charge conjugation invariance means that iy° e k. We have enlarged the sp(4,R) algebra—whose ladder representations give the Majorana equations—to all of sl(4,R), by taking the algebra generated by all the products of y matrices (not just the commutators [y", yv], which close on sp(4,i?)). Another way 9 of obtaining sl(4,i?) from Dirac's equation (y"Pli-M)^p) = 0
(6.29) 5
is to let the mass term Mbe proportional to y , and then take commutators of the y* and M. We can now take one of the unitary irreducible representations of SL(4,R) given by (4.13) to obtain the Lorentzinvariant wave equation
(6.24)
W r > , - * ) # ) = o,
(6.30)
we put
where ip takes its values in the Hilbert space V of the repreem" = ir"Y"(6.25) sentation jr. We could take K to be v(M) as the simplest approach, or even a general Lorentz-invariant operator-valued Then we have the commutation relations of so(3,3), function of p2. m pq p mp q mp \Q ",Q \ = g" Q ""* — g Q "* — g" Q + g^'O"', Since the physical spin su(2) subalgebra is that spanned (6.26) by L, the spin content for each representation ir is easily 3187
J. Math. Phys., Vol. 26, No. 12, December 1985
A. Cant and Y. Ne'eman
437
3187
obtained: for each K representation (ji,j2) appearing, we have (2/2 -f 1) copies of the SU(2) representation j Y . Clearly, in the present context of the Dirac embedding (5.3) the appropriate spinorial representations TT are those that contain K representations (jl,j2) with half-integer j v There are thus two candidate multiplicity-free representations (the method can, of course, be extended to non-multiplicity-free representations as well) with lowest spin ^: (i) 3}i,M{\ ,0) with spin content (J) & 3® s 6(§) © log) e - , (ii) ^" add (i , i ; p2),
(6.31)
p2 e R, with spin content
2(i) e 4® e 6 © e ™ .
(6.32)
0
K=/3I
(/?eR),
[},/«)) 9l[Um\
ffi3(§,/<3»)
(6.33)
(in the notation of Ref. 21). We do not know what the labels lw, lm,... are. The first term in (6.33) may be the Majorana representation {i ,0j. We are primarily interested in the mass spectrum of (6.3). Since if belongs to the maximal compact subalgebra k of sl(4,/J), there will be a discrete spectrum of rest masses (i.e., those corresponding to timelike momenta, p2 > 0). It is easy to calculate the mass spectrum in a given case. First, we observe that, since Mi = T-lMxT,
m = P/A
T =-
fl
0 /
0 0
1
0 -1
0 -(
K = (ap2 + I3)l,
a(f) = 2jl
So, choosing/; = (m,0,0,0), we have (ir(r°)m -am2-0)ip(p)
1
= 2Tj, T- .
m[A + (A2-4a/3)1
/2
]/2a,
(6.40)
which gives a better mass formula; in particular if fi = 0 we get m
(6.41)
=A/a
and the mass is linear in A. The observed Regge spectrum m 2 ~ y , with daughter trajectories is obtained by taking K=\a{p2fn
+/?)/.
(6.42)
We observe that of the two "spinorial" equations (6.31) and (6.32), it is the ladder example that has nonsingular ir(y°), symmetric charge-conjugate (or negative-energy) states and can describe [with (6.42)] the physical mass spectrum. Its coupling to gravity is purely anholonomic and does not involve the double covering of SL(4,R) and A. APPENDIX A: LIMITATIONS ON X> AS SL(4,R) FOURVECTOR
(6.35)
\XcD\Zap\XAB}
(6.36).
Thus for a unitary multiplicity-free representation w of SL(4,R) we see from (4.1) that the spectrum of niy0) is given by A = 2m2 = 2j2, 2(j2 - 1),..., - 2j2,
= O,
i.e., the spectrum of rest masses is given by 27
:2/(m
(6.39)
We can calculate the commutators [Zae,XAB\ in the spherical basis most simply by applying the Wigner-Eckart theorem for the tensor operator Z acting by commutation on the vector representation. Then the matrix elements are
we have, from (4.8), (2)
a,0<=R.
0
(6.34) / - 1 0
[A a nonzero eigenvalue of Tr{y°)].
But this decreases as A increases; states of higher m2 and thus higher spins _/', have a smaller mass as in the Majorana equation. It may be more realistic to take instead
In this appendix we shall show that, for the multiplicityfree representations other than (4.14), no SL(4,R) vector X v can be constructed (apart from the trivial case X " = 0). If X v is to be an SL(4,R) vector, then as well as (5.4), we must have [A,S,X,]=SS,X,+S„XS. (Al)
where 1 0 -i 0
(6.38)
then the spectrum of rest masses is given by
rdlsc
Since I'y 6 k, integer values of j 2 as in ^ (i ,0) may involve self-charge-conjugate states for zero eigenvalues A of iy°. ladd The representation ^ ( | ,\ ;p2) on the other hand is symmetric in positive and negative energy states, like Dirac's spinor. The Dirac embedding (5.3) is an embedding SO(3,l) C SO(3,3) or SL(2,C) C SL(4,R). This is why the spinor nature of the equation and particles is not correlated with the single or double valuedness of the SL(4,R) representation. For gravity, the Dirac embedding produces an anholonomic spinor and cannot be utilized for a holonomic ("world") spinor (see our discussion in Sees. I—III). We would also like to know the SL(2,C) and Sp(4,E) reduction, but this is not readily available from our infinitesimal approach. Certainly we have a direct sum of (infinitedimensional) unitary irreducible representations: for example, we conjecture that the SL(2,C) decomposition of 2>iiac(\ ,0) is
(6.37)
where j 2 goes over all the su(2) X su(2) representations (j1,j2) 3188
that occur in ir. Note that for half-integer j 2 the equation is indeed symmetric in positive and negative energy states, like Dirac's equation. For integer j 2 , -niy0) will have one zero eigenvalue for each value of j 2 . The mass spectrum depends on the form of K. If we take
J. Math. Phys., Vol. 26, No. 12, December 1985
" - a : i)(-'D i a «"•
so that (Al) becomes [Zae, XAB ] = 2/( - 1)^ + B {(I - A {A + a)) X(i-B(JB
+ P))}1/2XA
+ a,B +
(A3)
using the 3/ symbols tabulated in Ref. 23. This result can also be obtained directly from (4.3) and (A1) if we use the relation A. Cant and Y. Ne'eman
438
fi,
3188
iZ„ = \ Sit(Akk - Ai4) - (Atj + eijk Aki). (A4) Now if we take the commutator [Z0O,X1/2l/2 = — iXi/21/2 ]. f° r example, we see that the matrix elements f ,
;
' ^ | [ZmJW2W2]
|y +
i
y
'\
(A5)
are zero because Z never couples (jitj2) to itself in the representations (4.11) we have constructed except for (4.14) with P2/O. Thus
IJ V
j + \\x 1/ + 1 J\_Q n' \X,/2'/2\m J""'
(A6)
and since this is true for each direction of coupling in Fig. 1, we see that XAB=0: no SL(4,R) vector exists for our wave equation (5.8). APPENDIX B: DIRAC EMBEDDING OF so(4) C sl(4,C)
Suppose that g0 is a real form of sl(4,C) for which the maximal compact subalgebra k contains so(4) embedded via the Dirac representation. Theg0 arises from some involutive automorphism s of su(4) such that s(X,Y) = [X,Y), V(X,Y) e su(2) e su(2). There are two possibilities. (a) If sJC->MXM~l (inner) then (Bl) gives fal 0 \ a -P1 1 M=
io
(Bl)
Bl)
and so k--
ie si
AJeu(2);
Tv(A + B) = Q
=ssu(2) a su(2) e center of k and the real form is su(2,2). Since M e USp(4) we have in fact an embedding so(4J C sp(2,2) with &=susp(2) e usp(2). (b) If s: X-+NXN ~ \ this gives (ad2- 0 \ = B2 = - 1 ) soAW= - | a | 2 = /. Clearly k = usp(4) and this time the real form is su*(4). So we can embed so(4) in either of these real forms; these possibilities do not concern us here. [Note that sl(4,R) would have to come from JVN = / in (b). This never happens.] ACKNOWLEDGMENTS
We thank Professor S. Sternberg for his advice. One of the authors (A.C.) is grateful to the Wolfson Chair Extraordinary of Theoretical Physics, Tel Aviv University, for its kind hospitality andfinancialsupport, and R.
3189
Kerner of the Departement de Mecanique, Universite de Paris, VI, for helpful discussions. The other one of us (Y.N.) was supported in part by the U.S. Department of Energy Grant No. DE-AS0576ER03992.
J. Math. Phys., Vol. 26, No. 12, December 1985
'See for example, Y. Ne'eman, Algebraic Theory ofParticle Physics (Benjamin, New York, 1964), Chap. VIII. E. Majorana, Nuovo Cimento 9, 335 (1932). 3 D. T. Stoyanov and I. T. Todorov, J. Math. Phys. 9, 2146 (1968). "SeeRef. l.Chap.X. ! D. M. Fradkin, Am. J. Phys. 34, 314 (1966). 6 P. A. M. Dirac, in Invited Papers, Tracts in Mathematics and Natural Sciences, edited by M. Hamermesh (Gordon and Breach, New York, 1971), p. 1. 7 S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972), Chap. XII, Sec. 5, p. 365. 8 For a review, see F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976). "F. W. Hehl, G. D. Kerlick, and P. von der Heyde, Phys. Lett. B 63, 446 (1976); Z. Naturforsch. A31, 111, 524, 823 (1976); E. A. Lord,Phys. Lett. A 65,1 (1978); F. W. Hehl, G. D. Kerlick, E. A. Lord, and L. L. Smalley, Phys. Lett. B 70, 70 (1977); F. W. Hehl and G. D. Kerlick, GRG 9, 691 (1978); F. W. Hehl, E. A. Lord, and L. L. Smalley, GRG 13, 1037 (1981). '"See, for example, Ref. 7, p. 365; also, Ref. 8, p. 401, last paragraph of Sec. 1., etc. "Y. Ne'eman, Ann. Inst. H. Poincare A 28, 369 (1978). 12 H. Dothan, M. Gell-Mann, and Y. Ne'eman, Phys. Lett. 17, 148 (1965). 13 Y. Ne'eman, Proc. Nat. Acad. Sci. USA 74, 4157 (1977). 14 F. W. Hehl, E. A. Lord, and Y. Ne'eman, Phys. Lett. B 71, 432 (1977); Phys. Rev. D 17, 418J1978). I5 Y. Ne'eman and Dj. Sijacki, Ann. Phys. (NY) 120, 292 (1979); Proc. Nat. Acad. Sci. (USA) 76, 561 (1979); 77, 1761 (1980). 16 J. Mickelsson, Comm. Math. Phys. 88, 551 (1983). 17 Y. Ne'eman, Found. Phys. 13,467 (1983). Note that the SL(«,« ) have been studied intensively in mathematics (though their universal coverings have not). See, for example, D. A. Vogan, Jr., Representations ofReal Reductive Lie Groups (Birkhauser, Boston, 1981); B. Speh, Math. Ann. 258, 113 (1981). ,8 Dj. Sijacki, J. Math. Phys. 16, 298 (1975). "Dj. Sijacki and Y. Ne'eman, J. Math. Phys. 26, 2457 (1985). 20 Y. Dothan and Y. Ne'eman, in Resonant Particles, edited by B. A. Munir (Ohio Univ. Athens, Ohio, 1965), p. 17. Reprinted in Symmetry Groups in Nuclear and Particle Physics, edited by F. J. Dyson (Benjamin, New York, 1966), p. 287. '"1. M. Gelfand, R. A. Minlos, and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications (Pergamon, Oxford, 1963). M A. Cant and C. A. Hurst, J. Aust. Math. Soc. Ser. B 20, 446 (1978). 23 A. P. Yutsis, I. B. Levinson, and V. V. Vanegas, Theory of Angular Momentum (Israel Program for Scientific Translations, Jerusalem, 1962). 24 H. J. Bhabha, Rev. Mod. Phys. 17, 200 (1945). 23 A. Cant. J. Math. Phys. 22, 870, 878 (1981). 26 J. F. Cornwell, Rep. Math. Phys. 2, 239 (1971). 27 A. O. Barut, in Springer Tracts in Modern Physics, Ergebnisse der Exakten Naturwiss, Vol. 50 (Springer, Berlin, 1969), pp. 1-28. 28 A. O. Barut and R. Raczka, Theory of Group Representations and Applications (Polish Scientific Publishers, Warsaw, 1977). 2, A. Kihlberg, Ark. Fys. 32, 263 (1966). 2
A. Cant and Y. Ne'eman
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Algebra and physics of the unitary multiplicity-free representations of SL(4,R) Dj. SijaCki Institute of Physics, P.O. Box 57, Belgrade, Yugoslavia
Y. Ne'eman a)b| Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel (Received 13 March 1985; accepted for publication 2 May 1985) The systematics of the multiplicity-free unitary irreducible representations of SL(4,R) are restudied, and an amended list is presented. An automorphism essential to the physical application for particles and fields in Minkowski space is described.
I. PHYSICAL APPLICATIONS Present knowledge about the unitary irreducible representations (unirreps) of the SL(n,R) groups is incomplete. In particular, very little is known about the unirreps of the double-covering groups SL(n,R). The cases n = 2,3,4 are important in physics.' SL(2,R) and its multiple coverings appear in numerous problems such as classical and quantized strings, projective transformations, integration over random surfaces, etc. The case n = 3 has been utilized to classify the excitations of deformed nuclei and of hadronic states lying along Regge trajectories. SL(4,R) plays a role in the strong coupling theory and in various dynamical spectrum generating algebras. A particularly useful application for any n relates to the representations of diffeomorphisms on n-dimensional manifolds.2 These are realized nonlinearly over the linear representations of the SL(«,R) subgroups.3 Diffeomorphisms appear in the theory of gravity, in hydrodynamics and magnetohydrodynamics, etc. In recent years, with the realization of the existence of a double covering of the diffeomorphisms,4 to be represented nonlinearly over SL(4,R) in the case of space-time, a program was launched, aimed at the construction of SL(4,R) or even "world" spinors as spinorial "manifields" and their wave or field equations. These would appear as a superposition of matter fields, representing, for instance, phenomenological hadrons with their system of excitations, in special relativity and in the corresponding transition to three possible versions of general relativity with an active local invariance group of the tetrad frames ("G-structures"): (a) the tetrad formulation of Einstein's theory,5 as in the case of finite spinors ("anholonomic" application of a local Lorentz invariance), (b) the Einstein-Cartan picture,6 in which the spinor fields in addition supply a spin source term to Cartan's (algebraic) torsion equation (nonprogagating local torsion), and (c) the metric affine picture,7 where the anholonomic invariance is further extended to SL(4,R), with the manifields supplying a shear source term8 to the (algebraic) nonmetricity equation (nonpropagating local nonmetricity). Alternatively, dropping the anholonomic treatment alaJ b|
Wolfson Chair Extraordinary in Theoretical Physics. Also at the University of Texas, Austin, Texas 78712.
2457
J. Math. Phys. 26 (10), October 1985
together, the manifield can be used in two approaches as a holonomic world spinor: (d) Einstein's Riemannian general relativity, and (e) a tentative affine theory, 9 with propagating torsion, curvature, and nonmetricity, but where only torsion or curvature is not confined. Cases (a)-(c) have recently been resolved through appropriate wave equations. 10 A tentative gravitational Lagrangian has been proposed 1 [ for case (e) and a holonomic equation has been constructed. 12 Case (d) has also been recently resolved.13 All of these require knowledge of the unitary irreducible representations of SL(4,R), or at least of the multiplicity-free unirreps [in which any representation of the maximal compact SO (4) subgroup will appear at most once in the reduction over that subgroup]. Some such representations (the "ladder" class) were constructed in connection with dynamical groups. 14 The first comprehensive study of the entire system of representations was published by Kihlberg.15 This has been followed by further results due to Sijacki 16 and Borisov.17 The present authors published a supposedly comprehensive catalog of all multiplicity-free unirreps. 9 Unfortunately, some of the representations listed fail the test of fulfilling the algebraic commutation relations, 18 as was pointed out by Friedman and Sorkin,19 who published what was purported to be a corrected list. We find, however, that in as much as Ref. 9 was incorrect through overlisting, Ref. 19 erred through underlisting. Considering the importance of the issue, we have now surveyed these same systematics once again, with more insight and some hindsight. Hopefully, this article will thus supply a "final" catalog of the multiplicity-free unirreps of SL(4,R). As for the rigorous mathematical results, we point out that the unitary duals of GL(3,R) and GL(4,R) have been determined by Speh.20 _ In Sec. II we outline the relations between the SL(4,R), SL(4,R), and SO(3,3) groups and their maximal compact subgroups, we give the relevant commutation relations, and state the procedure for the construction of all unirreps. In Sec. Ill we find the two quotient groups of SO (4) from which all multiplicity-free unirreps are to be obtained, and list the corresponding group generator matrix elements. In Sees. IV and V we discuss the irreducibility and unitarity properties of the SL(4,R) unirreps. In Sec. VI we exhibit a
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© 1985 American Institute of Physics
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list of all SL(4,R) multiplicity-free unirreps, and make a connection and comments to previous work. Finally, in Sec. VII, we present a deunitarizing automorphism and make use of it to lay out a basis for the SL(4,R) field structure.
shear transformations. The SL(4,K) commutation relations are now given by the following relations: [Mab, Mcd]=
- /(i?oc Mbd - rjad Mbc -VbcMad
+VbdMac),
II. SL(4,K) GROUP STRUCTURE AND REPRESENTATIONS
[Mab, Tcd ] = - i(Vac Tbd + Vad Tbc
The SL(4,1R) group is a 15-parameter noncompact Lie group. The space of the group parameters is simply connected. The maximal compact subgroup of this group is SO (4), the double covering group of the SO(4) group. There is a four-element center of SL(4,K), which is isomorphic to Z 2 ® Z 2 . The factor group of SL(4,R) with respect to a twoelement (diagonal) subgroup Z 2 of Z 2 8 Z 2 is isomorphic to SL(4,R), i.e.,
[Tab, Tcd\=
—
SL(4,R)/Z?~SL(4,R), while the factor group of SL(4,R) with respect to the whole center Z 2 ® Z 2 is isomorphic to SO(3,3), i.e., SL(4,R)/[Z 2 ®Z 2 ]~SO(3,3). These relations are summarized by the following diagram of exact sequences: 1 1 I I 1 - » Z f ->• Z2®Z2 -+ Z2 -+ 1 I 1
->• Zd2 ->• SL(4,R) I SO(3,3) i 1
I -+ SL(4,R)
->• 1 •
I SO(3,3) I 1
The maximal compact subgroups of the groups SL(4,R), SL(4,R), and SO(3,3) are the groups SO(4) ~SU(2) 9 SU(2), SO(4)~[SU(2)®SU(2)]/Z2, and SO(3) s SO(3), respectively. The relations between the maximal compact subgroups are given by the above diagram, with each group substituted by its maximal compact subgroup. Uj\,j2 are the Casimir labels of the SU(2)» SU(2) group, then in an arbitrary representation Z 2 ® Z 2 is represented by (1,( — )1Jl} 8 {1,( — )2jl) a n d Z 2 is accordingly represented by [ 1,( - f' = ( - )2J'}.
(2.2)
Vbc Tad — t]bd Tac),
- i(r/ac Mbd + rjad Mbc
+ VbCMad +VbdMac), We will conduct the study of the multiplicity-free unitary irreducible representations of the SL(4,M) group in the basis of its maximal compact subgroup SO (4). In this way one has on one hand an advantage of carrying out a rather straightforward calculation, and on the other hand, of applying immediately the most general mathematical theorems on the completeness of the results, which refer to the case when the unirreps of a noncompact group are analyzed in the basis of its maximal compact subgroup. The SO (4) ~SU(2) ® SU(2) subgroup is generated by J\i} = i eiJk Mjk + i T„,
J? = \ eijk Mjk -
2
T0i,
(2.3)
where i,j,k = 1,2,3. The remaining nine (noncompact) operators transform with respect to SU(2) 8> SU(2) as the components of the (1,1) irreducible tensor operator Z. We will write them in the spherical basis as Za p, a, fi = 0, + 1. The minimal set of the SL(4,R) commutation relations now reads as follows: [JX\J[£\=8pqJ±^\
/>,
[Wa.ll]=pZa,g,
(2.4)
[jm,Za^]=(2-a(a±\))il2Za±uli, [J^,Za^]=(2-P(p±\))lllZa_,±i, [Z+1,
+l
, Z _ 1 , _ , ] = -[J[? + Jg>).
The remaining commutation relations can be obtained by making use of the Jacobi identity. All (multiplicity-free) unirreps of a noncompact group can be constructed explicitly by the following three-step proLet Qab, a,b = 0,1,2,3, be the SL(4,R) generators. The cedure, which is based on the work of Harish-Chandra. 21 SL(4,R) commutation relations read (1) One determines the matrix elements of the group generators in the basis of all homogeneous Hilbert spaces [Q.b.Qc* ] = 'gbcQa* - 'ga<,Qcb, (2-1) over the maximal compact subgroup and its quotient groups. where for the structure constants gab one can take the invar(2) One determines all sublattices of the maximal comiant metric tensors: either 5 o 6 = ( + l, + l, + l,-(-l) with pact subgroup labels that are invariant under the action of respect to the SO (4) subgroup or the noncompact operators. Each invariant lattice deterVob = ( + 1, — 1, — 1, — 1) with respect to the Lorentz sub- mines the basis of the representation invariant Hilbert space. group SO (1,3) of the SL(4,R) group. The metric tensor gab (3) One determines the constraints on the representation is SL(4,R) covariant. The antisymmetric part (when gab labels by imposing a condition of Hermiticity on the genera= r)ab)ofQab,\.t.,Mab = Qlab], form the six Lorentz genertors in each Hilbert space corresponding to the above invarators, while the remaining nine symmetric operators, i.e., iant sublattices, starting with the most general unitary, positive definite Hilbert space scalar product. Tab=Q{ab)> generate the relativistic (four-dimensional) 2458
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Dj. Sijacki and Y. Ne'eman
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III. MATRIX ELEMENTS OF THE SL(4,R) GENERATORS
/I'll-'1
Let the group elements k of the maximal compact subgroup SU(2) g SU(2) be parametrized by two sets of Euler angles, i.e.,
lm,
(<*„&, y,)DZ„Ja2,J32,
y2)},
Ytf»i^(l32,
= (((2/1 + l)(2/ 2 +l))
m2l J
J\ '"•l, +± 1
h \ mJ'
•U2U2+I)
y'i lirii
j \ — m'\{
Jl
lm,
h m2 +
.}•
h \ m2l \ji — m2
j'l
xf
1
h \(
j'l
x
y2)}.
(3.3)
{j\J2\\Z\\jlJ2)
= -i(-f+J\(2j[
+ \W2 + \)
X(2/-1 + lK2/2 + l)) 1/2 X{e1 + ie2-i[j\U;
+1)
-JiUi + 1) +J2W2 + 1) -J2U2 + 1)]) Vo
0
0 / vo
0
0/'
where e = e, + te2, e„e 2 e R is the SL(4,R) representation label. The 3-y symbol
vo 0 oJ' with half-integer entries is to be evaluated by taking the corresponding expression for integer entries and continuing it to the half-integer ones. In the basis (3.2) we find
(02,72)},
/ >I
[SU(2)®SU(2)]/SU(2)
{2U+\)"2Di^(ri,/3,y2)}.
J
\nti
j
m2
I j Ira, m2
i
A-
(3.2)
The SU(2) ® SU(2) group generator matrix elements are well known, while the matrix elements of the noncompact generators were determined in Ref. 9, by making use of the decontraction formula. They agree with the ones obtained in a general analysis of SL(4,R) unirreps.22 In the basis (3.1) we find
J
J
) lm,i mJ j
,2
and
)}
h
= ( -
(3.1)
II J
) = (y,(y, + i) m2l -m.l/n. + l))1
7
1/2
XDJi„ „(0i. YiWim
) , m2l
Jl
m2
Pi
[SU(2)/U(1)] ® [SU(2)/U(1)]
)1
2
h
and
Thus, we recover the multiplicity-free Hilbert subspace of the first possibility. The third possible homogeneous vector space is defined over the quotient group [SU(2) ® SU(2)]/ SU(2), where the elements of the "denominator" SU(2) group are parametrized by the Euler angles (a,/3,y) = {avfSx —f}2,a2). In this case, the Casimir labels./', andy2 are equal mutually, i.e.J =j\ =j2. The Hilbert space basis is given by the Wigner functions, f(2/+ 1)1/2 XDim,mSri,P,Yz)\P =P\ + Pi\- Hereby we have listed all possibilities (with nontrivial maximal compact subgroup labels), and therefore, in order to determine all multiplicityfree unirreps of the SL(4,R) group, one has to look for solutions in just two homogeneous vector spaces given by quotient groups of the maximal compact subgroup SU(2) 8 SU(2), with corresponding basis vectors as follows:
h
m2l
\j' lm,
— m2(m2 + 1)) 1/2
(((2/, + l)(2/2 + l))U2Dl6mt(P1>rimm2(02,r2)}. The second possible homogeneous space is defined over the quotient [SU(2)/U(lj] ® [SU(2)/U(1)]( with the corresponding basis given by the vectors i((2/, + l)(2/2 + \))mD'imt(Plt
)^m m2i
Ji
Jt22
M ,
Ira,
h
J{ll \Ji - lm,
where U\,j2) are the SU(2)» SU(2) Casimir labels. The labels (nltn2) determine an additional multiplicity of the (juj2) values, besides the assumed (2/, + l)(2y2 + 1) multiplicity corresponding to the (mltm2), |m,|
A=mi\j>
m2l
jw\J> Ira,
k (ai, /?„ ri,a,. P2, y2) = kt[au /?,, Y,)k2(a2, P2, Yi)The first possible homogeneous vector spaces are those defined by the whole SU(2) g SU(2) group. The complete set of Wigner's D function provides a basis, i.e., {((2/, + l)(2/2 + \))mDim<
j
)
•
UU+D-m^m^l))'
•
{j(j+\)~m2(m2±l)Y
Ira, + 1 m-
lm,
mn,2 +± 1/
and
J. Math. Phys., Vol. 26, No. 10, October 1985
Dj. Sijacki and Y. Ne'eman
442
2459
\m\
m2 I
a,p
| m,
< y ' i - i 7 2 + i | | 2 | \hh) = - / ( - )*• ~ '( + ie2 +j\ - y 2 - l)(y,(y2 + l))" 2 .
m2l \ —m,
x
a
m,/
( i i i)°" , i z M A
(3 4)
-
_/((27' + l)(27 + l))" 2
{j'\\Z\\j)=
X(e, +
7, (0,0), 7.(0,1) = 7 . (1,0),
fe2-J[/(y'+l)-7U+I)]),
where e = e, + /
We will treat separately the above two cases. In the first case, (3.1) and (3.3), one has a priori a general lattice of all UIJT) Points
\(JUJ2)\JLJ2 = 0,2,1,^,2,...). In this case, owing to the 3-y symbols, the reduced matrix elements of the noncompact operators vanish if y'[ = y , and/or y2 =y 2 . Thus, the most general lattice of the (y, ,y2) points splits into eight sublattices. Since only the four possibilities (y [,j2) = (y, ± 1,72 + 1) occur, the (y1(y2) content of each sublattice is determined by 7, +7' 2 (mod 2), y, —j2 (mod 2), and the "minimal" (y'„y2) value. The eight such lattices we label explicitly as follows (cf. Fig. 1): /_, =7,(0,0), L2 = m,i), L4 = Z , ( y ) = L ( y ) , L7 = L[0$,
It is obvious now that, owing to the (y',)I/2 and/or (y'2)1/2 factors in (4.2), the lattices L (0,0) and L (0,1) are invariant, i.e., the reduced matrix elements (y, — ly 2 — 1| \Z\ \j^j2),
L3=L(0,1)
L5=L(i,0),
with the SL(4,B) commutation relations satisfied for every e„e 2 s R. Whenever e2 ^ 0 these two lattices are irreducible. In the case of the lattices L (j,0) and L (|,0), the reduced matrix elements
= L[1,0),
L 6 = L(0, 2 ),
7, (0,0;y, -y 2 >y, 0 ),
7io = 2,4,6,...,
•^ (j>0;7l ~j2^Jl0l>
7l0
L[l,0;ji-J2>jw),
7'io=1.3,5
=
2i2'2'-"'
•^ (j>0;7i —72>7io)' 7io = 2>i'T>-" • In a common notation these lattices read
(4.1)
L8 = L(l,0).
L (7'o.°;7'i -72>7'o), (4.4)
In order to determine which of these sublattices are invariant under the action of the noncompact generators, we will make use of the following explicit forms of the reduced matrix elements:
7o = 2.1>2> 7i +72=7'o(mod 2).
(4.3)
+ ie2 +y, +y2)(y,y2)1
<J1+1J2-M\Z\\JJ2> = -H- P - V , + ie2 - 7 , +j2 - l)«y, + l)yj 1 / 2 , (4.2)
To satisfy the commutation relations of the SL(4,R) group for these sublattices it is necessary that e, = 1 —y',0>e2 = 0. In the case of the lattices L (0,1) and 7, (0,3), the reduced matrix elements (y, — ly 2 — 1| |Z||y,y 2 > and O'i — O2 + 1| \Z\ \j\J2) vanish for the edge points (0,y2), but the <7, + ly"2 — 1| \Z\ |7,7 2 ) elements do not vanish for the points (7„0). These lattices are not invariant, and the algebra commutation relations are not satisfied. In this case one can again constrain the e„e 2 labels and find invariant sublattices. If we take e, = 1 —j20,j2o = J,l,|,...,e 2 = 0, then (y, + ly 2 — 1| \Z\ \jij2) vanish for all points with y2 —7, =y 20 , and we thus find another set of triangularlike infinite irreducible sublattices 7, (0,0; y2 -y,>y 20 ), y20 = 2,4,6,..., L (0,2;y2 -7'i>72o). J20 = L (0,l;y2 -j^ho),
M&-.
y'20 = 1.3,5,...,
£(0.$72-7'i>7'2o). 720 = 1,1$,.-. • In a common notation these lattices read •MO,7o;72-7'i>7o)> 7'o = i.l»i 7'i +7'2=7o(mod 2). FIG. ! . £ , , / = 1,2
8, sublattices [Eq. (4.1)].
(4.5)
The SL(4,R) commutation relations are satisfied for the la-
J. Math. Phys., Vol. 26, No. 10, October 1985
Dj. Sijatki and Y. Ne'eman
443
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bels corresponding to the sublattice points provided that e, = 1 — j20,e2 = 0. Finally, if we take el = l,e2 = 0, then both <;, - lj2+ 1| \Z\ \jj2) and <;, + 1 y2 — 11 \Z\ \jj2) vanish for all points on the line j , =j2, and (ji- 1J2- 1| \Z\ \J1J2) vanishes for/, =j2 = 0. Therefore, we find an additional invariant sublattice L (0,0;y'i =j2 =j),j = 0,1,2,.... The corresponding SL(4,R) representations are a special case of the representations corresponding to the lattices of (4.7).
(/,*)=
(j\\Z\\j)=
Wj-l))"2{el
+
-i(2j+\)(el+ie2),
J2
j'i
= 0,2,4,...),
Hi)={(j,j)\j
=
(4.7)
V. UNITARITY The next question we want to discuss is that of the unitarity of the multiplicity-free representations of SL(4,M), or in other words, the Hermiticity of the corresponding generators. Since SL(4,R) is a noncompact group, its unitary representations are necessarily infinite-dimensional. Unitarity is a matter which depends on the Hilbert space one is working in, i.e., it depends on the corresponding scalar product. In order to obtain all multiplicity-free unirreps of SL(4,R), we start with the most general scalar product of any two functions/ andg,
Kli
;) "
2461
(5.2)
where K(JJ,J2) are the matrix elements of the kernel. The positive definiteness of the scalar product, i.e., (/,/) > 0 for every /yields K[Ji.J2)>0, (5.3) and the Hermiticity of the scalar product, i.e., (/, g) = ( g , / ) * implies K[Jl.J2)=K*Ul.J2l(5-4) The Hermiticity of the noncompact operators reads in the spherical basis as follows:
(4.6)
i,ti,-}-
L/i \m,
--"UiJi)
Z2e = {-r-0Z-a.-p(5-5) Making use of this condition, and of (3.3) and (5.4), we arrive at
O + l I \Z\ \j)= _/((2/+3)(2/+l)) 1 / 2 (e 1 + / e 2 - . / - l ) .
L(0)={U,])\J
\JI
m2J
'e2+j),
One can see immediately that (j — 11 \Z\ \ j) vanishes when j = 1, for every e„e 2 and that one has an invariant lattice of points L %\,jx =j2 =j), j — j,j,j,... • The same matrix element vanishes for / = 0, provided e1 = e2 = Q. However, owing to the existence of a nontrivial reduced matrix element (y'j \Z\ \j) in this case, one can explicitly verify that the SL(4,R) commutation relations are satisfied for e, = 0 and an arbitrary value of e2. Thus in the second case we find two irreducible invariant lattices for e2 e R,
(5.1)
where/c(£ ',k jisascalarproductkernel,k,k' e SU(2) ® SU(2), and dk is an invariant SU(2)®SU(2) measure. We have shown in Ref. 9 that for the most general multiplicity-free SL(4,R) representation, the noncompact operator matrix elements take on the following form:
It is rather straightforward to check in the case of the 1.(0,0) and £(1,0) lattices that the sublattices of points y'i —A> — 1, — 2, — 3,... or j 2 —ji> — 1, — 2, — 3,... split under the action of the noncompact operators into a lattice of the form (4.3) or (4.4) plus an additional sublattice, and that for the latter the positive definiteness of the corrresponding Hilbert space scalar product [determined by (5.3) and (5.6)] is not satisfied. There are, therefore, no more irreducible invariant sublattices correponding to (3.1) and (3.3). In the second case, (3.2) and (3.4), one a priori has a halfline-like lattice of points {(j\,j2) = [j,j)\j = 0,J,1,|,...J. The explicit form of the reduced matrix elements (3.4) is given by
jdk'dkf*[k')K(k',k)g(k),
(el + ie2-i[j'l(j[
+ l)-y'i(y'i + l)
+J2U2 + 1) -J2U2 + l)]My';.72) = ( - e, + ie2 - i[j[(j[
+ 1) -ji(ji - 1) (5.6) +J2U2 + 1)-J2(J2+ l)WJi,J2)This equation provides us with two cases, i.e., e, = 0, e2 e R and e ^ O , e2 = 0. It is at this stage that we inspect the unitarity of the representations, as well as the positive definiteness of the scalar product, for each of the irreducible sublattices found in the above, (4.3)-(4.5) and (4.7). The SL(4,R) representations corresponding to the invariant lattices L (0,0) and L (1,0) of (4.3), are already unitary with a trivial kernel for the scalar product, K{JUJ'2) = 1 for every jltj2, provided e, = 0, e2 e R. These unirreps of SL(4,R) form the principal series, which we denote by Dp'(0,0;e2) and D p*(\.0;e2). In the case e, ^ 0 , e2 = 0, from (5.3) and (5.6), we find that there is a solution for e, if k , | < l - l/i - A |
and | e 1 | < 2 + y 1 + 7 2 ,
(5.7)
for every (jvj2) point of a given lattice. We find by inspection that 0 < |e,I < 1 for the lattice L (0,0), and that there are no solutions for the lattice L (1,0). These unirreps form the supplementary series, and we denote them by /J supp (0,0;e,). The matrix elements of the kernel are now
F
U' +J2 + e> + 1)H1 - *.)H|./i -j2\ +e, + 2)r{2 - e,) n^ +J2 - e, + i)r(i + e,)r(\j\ -j2\ _ e , + 2]T[2 + e,j
J. Math. Phys., Vol. 26, No. 10, October 1985
(5.8)
' '" Dj. Sijacki and Y. Ne'eman
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For the SL(4,R) representations corresponding to the irreducible lattices of (4.4) and (4.5) to be unitary, a nontrivial kernel is required for the scalar product. The unitarity condition (5.6) can then be satisfied for any e, = 1 — j l 0 , y'io = i.1.1.-.., or e, = 1 -Ao.72o = £.1,1,-, and e2 = 0, and that the positive definiteness condition is also satisfied by the scalar product. The corresponding unirreps form the discrete series of multiplicity-free unirreps of the SL(4,R) group. We denote them by Z>disc(l —e^O), and by Ddisc(0,l - <»,), et = 2 ,0, - 1, - 1,..., and they correspond, respectively, to the irreducible lattices of (4.4) and (4.5). For the discrete series, (5.6) yields K{J
j ) =
"
r(jl +j2 + gi + ' ) ^ ( | y i - k \ + g| + 2) r (y, +j2 -e1+ \)r(\j, -j2\ - e, + 2) XK(rDmU\),min{j2)),
(5.9)
where/r(min(/,),min(/2)) is either/c(l — e,,0)or/r(0,l — e,). The SL(4,R) representations corresponding to the irreducible lattices L (O)andL (1) of (4.7) are, as we have already stated, unitary for e, = 0 and an arbitrary e2 s K. This result follows from an explicit verification of the SL(4,R) commutation relations. These representations form the ladder unirreps, and we denote them by Z>ladd (0,e2) and £>ladd (2;e2). The irreducibility of the Hilbert spaces in which we have defined the multiplicity-free unirreps of SL(4,R) is guaranteed by construction—none of them possesses an invariant subspace under the action of the group generators. The second-order Casimir operator for SL(4,R) is defined by C2 = Qab Q"° = - 4 + *(*, + ie2f. For the principal and the ladder series it is given by C2=-4-}ei,
(5.10) (5.11)
while for the supplementary and the discrete series it takes, respectively, the values C2=-A
+ \e\
(5.12)
and C2= - 4 + i e ? ^ - 4 + i ( 7 0 - l ) 2 ,
(5.13)
wherey^ = ^,1,^,2,... is e i t h e r ^ ory20. VI. SUMMARY OF SL(4,R) MULTIPLICITY-FREE UNIRREPS We have parametrized, in this work, the unitary irreducible representations of the SL(4,R) group in terms of the parameter e = ex + ie2. The representations are defined in Hilbert spaces which are symmetric homogeneous spaces over certain quotient groups K' of the maximal compact subgroup^ = SU(2) ® SU(2), i.e., in the spaces L 2(K ')of squareintegrable functions over K' with respect to the invariant measure over K'. We have considered the most general scalar product (5.1) of the Hilbert space elements with, in general, a nontrivial kernel K. The K' representation eigenvector labels, which define a basis of the SL(4,B) representation Hilbert space, are constrained to belong to certain irreducible lattices L. Therefore, we denote the unirrep Hilbert spaces by H (K ',K,L ). 2462
There are, besides the trivial representation, four series of multiplicity-free unirreps of the SL(4,R) group. Principal series: Dpr (0,0;e2) and D"'(l,0;e2), et=Q, e2 s R. They are denned in the Hilbert spaces H (K \,K,L ), where K[ = [SU(2)/U(1)] ® [SU(2)/U(1)], K(JVJ2) = 1, Vji>J2> a n d the irreducible lattices are, respectively, L (0,0) andL (1,0) [cf. (4.3)]. The generator matrix elements are given by (3.3), and the Casimir invariant is given by (5.11). Supplementary series: Z>supp(0,0;e,), 0 < |e, | < 1, e2 = 0. They are defined in the Hilbert spaces H{K[,K,L ), where K J = [SU(2)/U(1)] ® [SU(2)/U(1)], K[J\J2) is nontrivial and given by (5.8), and the irreducible lattice is L (0,0) [cf. (4.3)]. The generator matrix elements are given by (3.3) and (5.2), and the Casimir invariant is given by (5.12). Discrete series: Z) disc (l - e,,0) and O diso (0,l - e,), et = 1 —j0Jo = i> M>e2 = 0- They are defined in the Hilbert spaces B(K{,K,L), .where K{ = [SU(2)/U(1)]«[SU(2)/ U(l)], K[jltj2) is nontrivial and given by (5.9), and the irreducible lattices are, respectively, L {j0,0;jx —j2>Jo) a n < i L (°./o;y'2 —J\>Jo) [cf- (4.4) and (4.5)]. The generator matrix elements are given by (3.3) and (5.2), and the Casimir invariant is given by (5.13). Ladder series: Z)ladd(0;e2) and Dladd (2;e2), e, = 0, e2 e R. They are defined in the Hilbert spaces H(K2,K,L ), where K2 = [SU(2) ® SU(2)]/SU(2), K(J1,J2)
= K(J,J) = 1, Vy, and
the irreducible lattices are, respectively, L (0) and L (1) [cf. (4.7)]. The generator matrix elements are given by (3.4), and the Casimir invariant is given by (5.11). Let us comment briefly on the previous work on SL(4,K) multiplicity-free unirreps. The ladder series, with e2 = 0, were obtained by Dothan and Ne'eman, 14 and in this work e2 was constrained by the algebraic structure of the physical model they considered. The general ladder series, e2 5^0, were obtained by Mukunda, 23 by means of an analytic continuation in the SU(4) labels, and by Sijacki,16 who solved the commutation relations explicitly. Kihlberg 15 failed to check the SL(4,R) commutation relations in the multiplicity-free case, when continuing the SO (4) labels to the halfinteger values; in addition he did not consider the homogeneous space over the K 2 group and thus obtained only the e2 = 0 ladder unirreps. Friedman and Sorkin 19 realized the importance of checking the commutation relations for the continued values (or equivalently checking whether the relevant sublattices are invariant), and made an attempt to find all multiplicity-free unirreps. However, they did not actually solve their relevant equations (A.6)-(A. 17): instead, they made use of our result 9,18 according to which only (y'i + l y ' 2 + l l |Z|y,y' 2 )and
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The Noether theorem determines the following structure for the total angular momentum:
The SL(4,R) multiplicity-free representation labels used in previous work are given in terms of the labels of this work by fl = e2 (Ref. 23), -q = e2 (Ref. 16), (a„a2) = (e2, - e,) (Ref. 15), (p1,p2) = (-el, - e2) (Refs. 9 and 18), and k= - 4 ( e , +ie 2 ) 2 (Ref. 19).
Mah =
The SL(4,M) generatorsM ab , Tab,a,b = 0,1,2,3 of (2.2) can be rearranged according to the following set: / , = \eijk M]k, N,j= T0i, K, =M 0,, Tv, TM, where i,j = 1,2,3. The SL(4,R) commutation relations now read IJJ]
ea» = e»x •
=ietjkNk,
-ieiJkJk,
JiJjk] =i£ijiT,k
+i€ik,Tj„
T
,pTk,\ = -i(8tkejlm
K„Nj]=
-HTij+8,Tm), -HS.N.+S^Nj),
ff„TJk]=
-i&jK.+S^Kj),
^,.,7-00]=
-2iN„
-2X„
\Em
TIJ,Too]=0. The compact operators are Jt and Nt, while the remaining ones K,, Tu and Tw are noncompact. Note the following subgroups: SO (4)
:
J.Jt,,
SO(l,3)c~SL(2,C)
:
/,,*„
(7.2)
: J„T9SL(3,R) The commutation relations (6.1) are invariant under the automorphism /,-•/„ Ni-+iKi,
Kt-* w „ 1T— -T , ti a
(7.3)
' 00"_ *'-'o0'
As a result of this automorphism, the vector spaces carrying unirreps of SL(4,R) can also realize the action of SL(4,R)^, except that some of the latter group's matrices will not be unitary, having been multiplied by V — 1 in (7.3). This is essential for most physical applications. Indeed, ordinary tensor fields carry finite nonunitary representations of SL(4,R), and Dirac or Bargmann-Wigner spinor fields carry finite nonunitary representations of SL(2,C)C SL(4,R). In both cases, the physical generators of Lorentz transformations—the boosts—are entirely orbital. This can be seen in the following way. 2463
-da
(7.5)
(7.6)
J, =eiik(J(?+ •!?)• (7-7) The So,, on the other hand, represent the noncompact special Lorentz transformation generators Kt. In any finite-dimensional representation of SL(4,R) or SL(2,C), they are given by anti-Hermitian matrices. For example, in the fourdimensional defining representation, they are given by the real symmetric matrices Eol + Ea, where
('•!) +8„eJkm + Sjkeilm + Sj,e,km )Jm
K„TJk] =
Nl,T00]=
(7.4)
and Sab is the matrix representation of Mab on the <j> vector space. These matrices are unitary for the S,-,, which belong to the compact subgroup SO (3); they represent the Jtj in Eq. (2.3),
Ji,TO0]=0,
d:
+ h.c,
-Sab >
Ji,Ki]=ieijkKk, K,,Kj] =
+ yab"}da^
is the canonical energy-momentum tensor, Jf is the Lagran gian density, <j> t n e field, Sab ** is the intrinsic spin tensor density
='e,,kJk,
J„Nj]
0b" - xtea»)
where h.c. denotes the Hermitian conjugate expression, a^ is a spacelike hyperplane,
VII. THE DEUNITARIZING AUTOMORPHISM .:/
J
J>
= S±SB
(A,B are the row and column indices). The first bracket in the expression (7.4) for Mab is the orbital angular momentum, the second is the spin. As a result of the addition of (Sab)+, the M,y indeed contain both orbital and spin angular momentum, but the intrinsic spin piece cancels in M 0i. The physical boost for all known physical fields is entirely orbital and contributes to the kinetic energy only. In unitary representations of SL(4,R), the boost possesses a nonvanishing intrinsic piece and raises the mass or potential energy, connecting the particle to a higher excited state. To avoid this unphysical result, we identify the physical generators instead in SL(4,R)JS,. Here M ^ is given by the finite non-Hermitian matrices of (iTol). It is this deunitarizing automorphism which allowed the authors in Refs. 9 and 12 to claim that the SO (4) compact subgroup matrices can be used for the Lorentz SO(l,3). In Ref. 10, both possibilities were investigated, the unitary SL(4,R) case representing hypothetical particles obeying a Majorana-like equation. Note that the V — 1 in the automorphism can also be absorbed in the space-time manifold, with x4 = ix°. This is the "Pauli metric" in which the metric is indeed Euclidean and thus SO(l,3)->- SO(4). In the general affine9 approach to particle physics and gravity, the fundamental symmetry is that of the GA(4,R) group. In studying the unirreps of this group in the space of quantum states we have found that for hadrons, SL(3,R) (see Refs. 18 and 24) is the relevant "little group," from which
J. Math. Phys., Vol. 26, No. 10, October 1985
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one induces the GA(4,R) unirreps. This stability group has to be represented unitarily, since its states form a basis of the quantum mechanical Hilbert space. In the GA(4,R) representations on fields, the SL(4,K) homogeneous subgroup is actively realized in the space of the field components, and its representations thus define the general affine fields. The two pictures have to merge for the stability subgroup, so that the SL(3,R)C SL(4,R), when represented on fields, has to be unitary. This SL(3,R) is generated by the / , and TtJ, and we observe that these operators are indeed unaffected by the deunitarizing automorphism (7.3). ACKNOWLEDGMENTS One of the authors (Dj.S.) wishes to thank the Wolfson Chair Extraordinary in Theoretical Physics of Tel Aviv University for its hospitality and financial support. The other author (Y.N.) was supported in part by the U.S./Israel Binational Science Foundation.
'Y. Ne'eman, Found. Phys. 13, 467 (1983). Y. Ne'eman, "Representation of the group of difFeomorphisms in the physics of particles, gravity andfluids,"to be published in Ann. NY Acad. Sci. (Proceedings of the 1984 Tel Aviv International Conference on Collective Phenomena). 3 A. B. Borisov and V. I. Ogievetsky, Teor. Mat. Fiz. 21, 329 (1974). 4 Y. Ne'eman, Ann. Inst. H. Poincafe, 28, 369 (1978). 2
2464
5
See, for example, S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972), Chap. XII, Sec. 5. See, for example, F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976). 7 F. W. Hehl, G. D. Kerlick, and P. von der Heyde, Phys. Lett. B 63, 446 (1976). 8 F. W. Hehl, E. A. Lord, and Y. Ne'eman, Phys. Lett. B 71,432 (1977) and Phys. Rev. D 17,418J1978). 5 Y. Ne'eman and Dj. Sijacki, Ann. Phys. (NY) 120, 292 (1979). 10 A. Cant and Y. Ne'eman, "Spinorial infinite equations fitting metric-affine gravity," Tel Aviv University report TAUP N156-84 (to appear in J. Math. Phys.). "Dj. Sijacki, Phys. Lett. B 109, 435 (1982). 12 J. Mickelsson, Commun. Math. Phys. 88, 551 (1983). 13 Y. Ne'eman, "World spinors in Riemannian gravity," Tel Aviv University report TAUP N157-85. 14 Y. Dothan and Y. Ne'eman, in Resonant Particles, edited by B. A. Munir (Ohio U., Athens, Ohio, 1965), p. 17; reprinted in Symmetry Groups in Nuclear and Particle Physics, edited by F. J. Dyson (Benjamin, New York, 1966), p. 287, "A. Kihlberg, Ark. Fys. 32, 241 (1966). 16 Dj. Sijacki, Ph.D thesis, Duke University, 1974; Ann. Israel Phys. Soc. 3, 35 (1980). "A. B. Borisov, Rep. Math. Phys. 13, 141 (1978). ,8 Y. Ne'eman and Dj. Sijacki, Proc.Nat. Acad. Sci. USA 76, 561 (1979), and, more especially, 77, 1761 (1980). "J. L. Friedman and R. D. Sorkin, J. Math. Phys. 21, 1269 (1980). 20 B. Speh, Mat. Ann. 258, 113 (1981). 2, Harish-Chandra, Proc. Nat. Acad. Sci. USA37,170, 362, 366, 691 (1951). 22 Dj. Sijacki, "The continuous unitary irreducible representations of SL(4,R)," Institute of Physics, Belgrade, preprint. 23 N. Mukunda, in Non-Compact Groups in Particles Physics, edited by Y. Chow (Benjamin, New York, 1966). 24 Dj. Sijacki, in Frontiers in Particle Physics '83 (World Scientific, Singapore, 1984). 6
Dj. Sijafiki and Y. Ne'eman
J. Math. Phys., Vol. 26, No. 10, October 1985
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PHYSICS LETTERS
Volume 157B, number 4
18 July 1985
SL(4,R) CLASSIFICATION FOR HADRONS * Y. NE'EMAN
u
Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
and Dj. SIJACKI Institute of Physics, P. O. Box 57, Belgrade, Yugoslavia Received 16 April 1985 It is suggested that the complete spectrum of baryon and meson resonances for each flavour is given by spinor and tensor infinite-component systems based on unitary irreducible representations of SE(4,R). This is a shell-model-like dynamical geometrical symmetry, presumably resulting from the quark structure and QCD, with possible connections to gravity. The fit to observations is excellent.
We submit that the complete spectrum of resonances for each hadron flavour is described by infinite-dimensional linear unitary irreducible representations (unirreps) of the simple metalinear group SL(4, R). This very restrictive classification is presumably a dynamical-geometrical symmetry of the quark model and QCD, in the same general sense that the nuclear shell model is believed to be generated by meson exchanges between nucleons. However, it is also possible that the SL(4, R) scheme might reflect some more fundamental considerations, relating to a deeper geometrical link between gravity and the strong interactions. A discussion of this speculative possibility is included in the context of a separate letter [1] dealing with hadrons in the presence of the gravitational field. According to QCD, the observed spectrum of hadrons represents the set of stable and metastable solutions of the Euler—Lagrange equations for a secondquantized action, constructed from quark and gluon fields. The parallels are with chemistry, where the ele* Supported in part by the US DOE Grant DE-AS0576ER03992 and by RZNS (Belgrade). 1 Wolfson Chair Extraordinary in Theoretical Physics. 2 Also on leave from University of Texas, Austin, TX, USA.
ments and compounds, with their excited states, are known to represent the solutions of Schrodinger's equation, with nuclei, photons and electrons as constituents — or as mentioned above, with the interpretation of nuclear structure in terms of nuclear forces involving mesons and nucleons. In each of these precedents, however, it has not been possible to use the fundamental dynamical model for actual calculation beyond the relevant "hydrogen atom" level, and the scientific disciplines of molecular-, atomic- and nuclear chemistry-spectroscopy have had to pursue their independent courses, the conceptual breakthroughs at their foundation notwithstanding. In hadron physics, the experimental exploration of the hadron spectrum goes on even though theory has moved away to the constituent level, except for the "bag model" approximate calculations. What is needed and offered here is a shell-model type of theory encompassing the entire system of hadron states, with sufficient predictive power and a plausible conceptual connection to the fundamental level. We submit that this scheme indeed fits the observed spectrum remarkably. It also compares very favourably with the known alternatives from the sixties — the "symmetric" quark model [2], the Veneziano representation, and the SO(4,2) scheme [3]. 267
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Volume 157B, number 4
PHYSICS LETTERS
Spatial deformations. The first contribution to the description of the pattern of excitations in angular momentum / was the Chew—Frautschi plot, based on the general dynamics of Regge trajectories and recurrences. Within the on-mass-shell iS-matrix methods of the sixties, motivated by the bootstrap idea, this was followed by an attempt at a direct description of the entire pattern of trajectories, namely the Veneziano model. Though this scheme did reproduce a general (m2 ~ / ) pattern, its theoretical constraints created contradictions with the observed spectrum, in addition to the difficulties relating to the required dimensionality (D = 26). The model was later abandoned and used to construct the off-mass-shell string or superstring "manifield" (an infinite superposition of fields), still a speculative program. Present work on strings emphasizes the colour and flavour aspects and the massless gauge fields, including gravity, rather than the description of the hadron spectrum. In a first attempt to provide for an algebraic classification that might then also point to a specific mechanism for the originating dynamics, it was proposed [4] that the Regge trajectories with \AJ\ = 2 recurrences correspond to the multiplicity-free (in J) "ladder" representations of SL(3, R), the group of linear unimodular transformations of real three-space. This fitted the sequences/=0, 2,4,... and 1,3, 5,..., but it was not known at the time whether or not SL(3, R) had any similar double-valued spinorial representations \, f,... or \, \ In fact even the question of the existence of the double-covering group SL(3, R) had not been investigated. The authors of the SL(3, R) proposal also suggested that the dynamical symmetry might originate in (inertia!) gravitational considerations arising from the temporal behaviour of the extended spatial spread of these hadrons (quadrupoles). It was pointed out that such pulsations are known to cause the spectrum of rotational and vibrational excitations of deformed nuclei, where similar sequences of angular-momentum levels are indeed observed and that the connection between the algebraic scheme and the dynamics could be tested directly, through the saturation of commutators in a current-algebra fashion. This was indeed investigated [5]. The model of ref. [4] was later given a relativistic formulation [6] through an embedding in SL(4, R), the group we propose here. 268
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18 July 1985
Confinement. Strings appear to reproduce some of the features of quark confinement in hadrons. The Nambu string lagrangjan represents the measure for the surface generated by the rotating string; it is thus SL(2, R) invariant. An evolving confined "lump" (the bag) would similarly involve SL(4, R) invariance [7]. Field covariance. It is now generally thought that relativistic quantum field theory (RQFT) should serve as a methodology for both phenomenological and fundamental levels. Indeed, 't Hooft has suggested that what is fundamental in one energy regime would have to be considered as composite in a higher region, with some inter-regional constraints on the corresponding anomalies. This is yet a third strong motivation to work with SL(4, R), since ordinary tensor fields and finite-dimensional vector spaces carry linear non-unitary representations of SL(4, R) (further extended non-linearly to carry representations of the general covariance group A, in the presence of gravity — see ref. [1]). Investigating the availability of spinorial representations of SL(3, R) for the model of ref. [4], we proved [8] the existence of the double-covering or meta-linearjroups SL(3, R) C SL(4, R) = SO(3,3). (Note that SO(3,3) D Z(2) ® Z(2) as center.) Applying the Principle of Covariance "backwards" together with the Principle of Equivalence, we have the fields carrying first the full double-covering A of the covariance group (the diffeomorphisms), then the doublecovering of the affine group ADSA(4,R)= [SL(4,R)XT4] D [SO(l,3)XT 4 ] (1) finally reducing to the local Poincare" group (T4 stands for the abelian translations). Under this reduction, the stability subgroup of SA(4, R) is SL(3,R) ® T'3 and the SL(3, R) results of ref. [4] emerge as a trivializationof thisT3, thus {SL(3, R) X T4} D {SO(3) X T4} = SU(2) X T 4 . The formalism of RQFT, the relativistic extension of the quadrupole pulsations algebra and the evolvingbag SL(4, R) measure characterizing QCD — all three approaches thus lead to SL(4, R) either independently or as different aspects of the same dynamics. Multiplicity-free unirreps. The SL(4, R) group is the double-covering group of the 15-parameter noncompact group SL(4, R) of Minkowski space transformations: x'a = Xabxb, det(Zaj)= 1, a,b = 0,1,2,3. The SL(4, R) generators [6]
(2)
Volume 157B, number 4
18 July 1985
PHYSICS LETTERS
Q\=/daM [x°0^(x) - JSfc/e/Cx)) 5|-
+ intrinsic part]
4
(3)
3
obey the following commutation relations:
2
\Qab> Qcd\ = "7*c Qad - mod Qcb • (4) I where BtfQc) is the local stress energy—momentum 0 tensor of hadronic matter, and nab is the Minkowski 3 4 5 J, metric (+1, —1, —1, —1). The antisymmetric part Afaj = F j& *• {(/1./2)} content of representations flladd(i): crosses; Q\ab] generates the metric-preserving Lorentz sub0°*%, 0): solid dots andD
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Volume 157B, number 4
PHYSICS LETTERS
^-.disc 4R(i.O),Dg5R(0,i) (9) •2e4L3R(2)®2XLS3R(i^2). each SL(3, R) unirrep appears infinitely many times, and
DfuR&Oi):
m = { $ , i (21)2. (!) 2 , ( ¥ ) 3 . -..}• (10)
Physical identification of the Lorentz group. In ref. [1] we present an automorphism sfi. of the si (4, R) algebra which allows us to identify the finite (unitary) representations of the abstract SO(4) compact subgroup (//, Nj) with non-unitary representations of the physical Lorentz group (//, K{), while the infinite (unitary) representations of the abstract SL(2, C) of (//, Ki) now represent (non-unitarily) the compact (/,-, JV,). The SA(4, R) stability subgroup SL(3, R) is unaffected. The unirreps of the abstract SL(4, R) are thus used as non-unitary representations of the physical SL(4, R) and thereby avoid a disease common to infinite-com-
18 July 1985
ponent equations. In these theories the Lorentz boosts possess a unitary intrinsic part [3,12], whose action is to excite a given spin state to other spins and masses, contrary to experience. The non-unitarity of the intrinsic parts here cancels their physical action precisely as in finite tensors or spinors, the boosts thus acting kinetically only. Field equations. For the tensorial (meson) representations, the simplest choices are either Z)ladd(0;e2) or i^ a(Jd (|;e2) with a Klein—Gordon-like (infinite-component) equation for the corresponding manifield * ( x ) (d^d" + m2)^(x)=0.
(11)
We fit all mesons in D***^), reflecting the quark structure. Spinor (baryon) manifields obey a first-order equation [13] with infinite X^ matrices generalizing EHrac's, except for the requirement of anti-commutation [12]. The X„ behave as a Lorentz four-vector (5, ^) and we are forced to use the reducible pair of SL(4, R) unirreps Z? disc (|, 0) © D^sc(0, i ) for the manifield ¥ ( * ) ,
Table 1 Assignment of N and A states aX
/ ^ ( i 0) (3.0)
D^O, \)
r 1+
5 31
r
(f,2)
1+ 3-
s+ 5 7-
I
9+
T
(.1,3)
1+ 2 3-
1
r
N(940)
A(1116)
(0,^)
N(1440) N(1520) N(1680)
A(1600) A(1690) A(1820)
d.i)
N(1710) N(1700) N(2020) N(2190) N(2220)
A(1800)
(2,f)
N(2100) N(2080)
13+ •y
N(1535)
A(1405)
12
N(1650)
A(1670)
i2
N(1675)
A(1830)
12
N(2090) N(1720) N(2200) N(1990) N(2250)
A(1800) A(1890)
r i2 2 2+
A(2110) A(2100) A(2350)
2 92
(3,i)
12
1+ 2
A(2325)
5-
y
z+
2 9+
T
2
a+
2-
11-
1-
2 2.2
u+
N(2600) N(2700)
2
U.2
a
> Undedjned are uncertain, one and two star states of ref. [ 14].
270
451
A(2020)
PHYSICS LETTERS
Volume 157B, number 4
(iA-M3" - K) *(x) = 0 , ( 12 ) where in momentum space K is a function of p2 and quantum numbers of appropriate subgroups. The AfM operators only connect the l/i - /2I = { states across the two unirreps. These are thus the only physical (propagating) states in *(x), all others decouple. For the A(1232) system we use the same pair of intrinsic, unirreps now adjoining an explicit four-vector index as in the Rarita—Schwinger field, (Lr w 9"-K)*p(x) = 0 .
18 July 1985
{0l,/2)}={(iO),(!,l),(|,2),...} ®{(0,i),(l,!),(2,f),...},
(15)
D^iO)®^"^,!),^: {0l,/2)}={(l,2-),(U),(3,f),...} ® {(2. 1). (4.2), (1.3),...}.
(13)
(16)
The SO(4) states (15) and (16) when reorganized with respect to the SL(3, R) subgroup, form an infinite sum of Regge-like A/ = 2 recurrences with the / content
The physical SO(4) multiplets projected out by Lorentz-invariance in eqs. (12), (13) for the fermions and given by our selection for the bosons in (11) are thus
2, 2> 2 '
(17)
ladd belong to ^SL3R(2).
fl^Ciea), *•
The former states while the latter ones are projected out of Z> S L| R (|,a 2 ) by the field
(14)
{(/W2)} = { ( i i ) , (1,1). CM),-}, Table 2 Assignment of A states.
/^(Mv
Jp
a.?)
12 3+ 2
(2, |)
Odisc(0,^V
jP
A
A (1620)
(in
1+ 2 32
A(15S0)
(|,2)
1+ 2 32 5+ 2
A(1910)
A (1232)
1
1-
A (1900)
3+ 2 52
A (1600)
2+
A(1950)
12 3+ 2 52
A(2150)
2
(3,f)
A
(4, |)
2
(f,3)
A (1920) A(1930)
2
£+ 2
A (2420) (|,4)
1+
A(2350) A (2390) A (2400)
2 32 5+ 2 7— 2 9+ 2 112
A (2750) A (2950)
13 + 2 152
T
132 15+ 2
A(2200) A (2300)
2
2
29 2 11+
2
2
2+ 2
1+ i+ 2 2-
1"2
2+
A(1940) A(1905)
1-
2+
2 92" 11 +
A (1700)
271
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Volume 157B, number 4
PHYSICS LETTERS
o o
equations (13). Note that we have thus achieved the goal of a fully algebraic model in terms of the total angular momentum /. We find it necessary to use parity-doubling, the actual spectrum displaying approximate exchange-degeneracy features. The parity of states within an SL(4, R) unirrep is determined by the parity of the lowest-/ state. We thus assign all hadron states of a given flavour to the wave-equation-projected states corresponding to parity-doubled SL(4, R) unirreps, i.e. s.t. their lowest-/ states have opposite parities. Assignments. The experimental data is taken from ref. [14]. We follow the mass ordering. Table 1 covers all known N and A states, while the A states are presented in table 2. Note that the J = \ A-states come from the / = 0 part of the ... (3, -5) ... explicit index in "9u of (13), while the other A-states come from the / = 1 part; the discrepancy in mass diminishes with increasing 0i./2)- To save space, we only present in table 3 the first three S0(4) states for SU(3) flavours. The few remaining levels are as follows:
o o o
noOn r- \t> oo
3
I
.350) [892)
M
18 July 1985
* ^
o o oo rON h-
"7r-system": rolP*
{(!. ?): JP = 2", A(2100); Jp = 3 + , A(2050)} , "K-system":
O O
{(!, | ) : / ' = 2-, K(2250); Jp = 3 + , K(2320)} ,
35
"p-system": {(I, i): Jp = I", p(2150); Jp = 4 + , 5(2040) ;
—£
(§, §): / ' = 5", p(2350); Jp = 6+, 8 (2450)} , "K*-system": fO 00
O O o © O oo
{ ( f , i ) : ^ = 4 + ,r(2060)},
o"\ M
*s
M
"co-system":
a-i
{(a. f) : jP = 2+ , e(2150); Jp = 4 + , h(2030) ;
Is
(I,!): / ' = 4+, e(2300); ft, »*): Jp=6+, r(2510)},
¥< I +
I +
O H N < < I
"0-system":
Io +H I(s +m I+ +w
{(if),^=2 + ,g T (2240)}. The new-flavour particles fit into the SL(4, R) unirreps as well, and their assignment is left to the reader. Note the new grouping in non-Regge SO(4) multiplets (see fig. 2).
£4 272
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Volume 157B, number 4
PHYSICS LETTERS
18 July 1985
the slope of the Regge trajectory for that flavour, and n is defined by (6) (see fig. 2). We illustrate this mass formula for the best known system of N resonances (fig. 3). For the average masses of the (J, f) unirreps of mesons belonging to Dl3AA(\,ei), m2 = m20 + 0/a{)(j-k), (19) while (at least) for the lowest SO(4) states (\, \) of opposite parity we find the following mass formula: m2(0-) + m2(l+) = m2(0+) + m2(l-).
In contradistinction to (12) and (13), eq. (11) does not contain any operator matrices. This may explain a rather small mass splitting of SO (4) meson states, reflected in the mass formula (19). Main predictions. We list here a few firm predictions testing the theory. For the N-system, we predict the existence of a (1, f) state Jp = -§+ at ~1660 MeV; we claim that the (f, 2), Jp = \~ N(l 700) should be at ~1850 MeV; and we predict three new (|, 3) states: f+ at ~2240 MeV, \~ at ~2350 MeV and | + at -2450 MeV. For the A system we predict a (1, | ) , Jp = § + state
(15) according to Fig. 2. Physical Z>disc(§-, 0) or (0, £•) states (: the mass formula (18) with m0 = ( a ' ) - 1 = 1 GeV.
Masses. We find a striking match between the (J , mass) values and the wave-equation-projected SL(4, R) unirrep states. Moreover, a remarkably simple mass formula fits these infinite systems of hadronic states. For N and A (and the higher spin A) resonances we write m2 = m20 + (l/o£)(/i+ f2-\-\n),
(20)
(18)
whereOTQis the mass of the lowest lying state, a'f is
(|,3)
13 2
• theory x experiment
N(2700)?
JJ_
N(2600)
2
(4,2)
_2_ 2
/N12200)
J_
/
/
f N(2I90) '
2
/
(-f-.il
JL
/
2
/ /
^(1680) , ^ ( 2 0 0 0 ) ? ^
_3_ 2
/
("2-.01
_l_ 2
* i /* i N(939) N(1440) i
0
1
-
I
2
/N12080)?
/
I
/
*U N07IO)? i
3
4
5
6
7 m2[GeV2]
Fig. 3. Predicted C ^ ^ - j , 0) states versus experimentally observed N-system (a' = 0.96 GeV — ').
273
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Volume 157B, number 4
PHYSICS LETTERS
at ~ 1 7 5 0 MeV, and a (f, 2), Jp = \~ state at ~ 1 9 5 0 MeV. In the meson case we point out two states belonging to (|, | ) SO (4) multiplets. In the 7?'-system we predict Jf = 1 + , H'(~l 290), and in the T-system Jp = 0", 77b(~9435). We note that a model based on a dynamical SO(3,2) symmetry derived from a first-quantised spatially extended relativistic oscillator has been recently suggested [ 1 5 ] . The main conceptual difference is in our (RQFT) field covariance derivation. References [1] Y. Ne'eman and Dj. Sijaclci, Phys. Lett 157B (1985) 275. [2] S.D. Piotopospescu and N.P. Samios, Ann. Rev. Nucl. Part Sci. 29 (1979) 339. [3] A.O. Barut and A. Bohm, Phys. Rev. 139B (1965) 1107; A.O. Barut, Springer Tracts in Modern Physics, Vol. 50 (Springer, Berlin, 1969) p. 1.
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[4] Y. Dothan, M. Gell-Mann and Y. Ne'eman, Phys. Lett 17 (1965) 148. [5J L. Weaver and L.C. Biedenham; NucL Phys. A185 (1972) 1. [6] Dj. Sijaclci, Ph. D. Thesis, Duke University (1974). [7] Dj. §rja£ki, An. Israel Phys. Soc. 3 (1980) 35. {8] Y. Ne'eman, Ann. Inst Henri Poincare' 28 (1978) 369. [9] Dj. §ija£ki and Y. Ne'eman, Algebra and physics of the unitary multiplicity-free representations of SL(4, Ft), Tel Aviv University report TAUP N158-85, to be published. [ 10] Y. Ne'eman and Dj. §ija2ki, Ann. Phys. (NY) 120 (1979) 292. [11] Dj. SijaSki, J. Math. Phys. 16 (1975) 298. [12] D.Tz. Stoyanov and I.T. Todorov, J. Math. Phys. 90 (1968) 2146. [13] A. Cant and Y. Ne'eman, Spinorial infinite equations fitting metric-affine gravity, Tel Aviv University report TAUP Nl 56-85, J. Math. Phys., to be published. [14] Particle Data Group, Rev. Mod. Phys. 56 (1984) No. 2, Part II. [15] A. Bohm, M. Loewe and P. Magnollay, University of Texas report DOE-ER-03992-576 (1984).
International Journal of Modern Physics A, Vol. 2, No. 5 (1987) 1655-1668 © World Scientific Publishing Company
GL(4, R) GROUP-TOPOLOGY, CO VARIANCE AND CURVED-SPACE SPINORS* VUVAL NE'EMAN** Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel and DJORDJE SlJACKI Institute of Physics, P 0 Box 57, Belgrade, Yugoslavia Received 7 May 1987
The existence of a topological double-covering for the GL(n, R) and diffeomorphism groups is proven. These groups do not have finite-dimensional faithful representations. Common mistakes in the literature are corrected. A theorem concerning an SL(n, R) inner automorphism is proven, allowing for the construction of the corresponding infinite-component spinor fields. The effect of covariance on the Hilbert space is explained.
1. Introduction The existence and structure of spinors in a generic curved space have been the subject of more confusion than most issues in mathematical physics. True, to the algebraic topologist the problem appears to have been answered long ago, with the realization that the topology of a noncompact Lie group follows that of its maximal compact subgroup. This perhaps the reason for the low priority given by mathematicians, in the case of the linear groups, to the study of the representations of their double-covering, for instance.1 As a matter of fact, some highly intriguing issues have recently been noticed.2 With no hint or help from the mathematical literature, and with several deeply engrained old errors in the physics texts, no wonder that so much confusion should still permeate some of the more recent work in this area. The issue is an important one for the physicist, however, and we shall make one more effort to clarify it. The physics literature contains two common errors. For fifty years, it was wrongly believed that the double-covering of GL(n, R), which we shall denote GL(«, R) does not exist. Almost every textbook in general relativity theory, upon reaching the subject of spinors, contains a sentence such as "... there are no representations of GL(4, R), or even 'representations up to a sign', which behave like * Supported in part by the US DOE Grant DE-FG05-85ER40200, by the US-Israel Binational Science Foundation, and by RZNS (Belgrade). * Wolfson Chair Extraordinary in Theoretical Physics. * Also on leave from the University of Texas, Austin, Texas, USA. 1655
456
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Y. Ne 'eman & D. Sijacki
spinors under the Lorentz subgroup" (this excerpt is taken from one of the best known texts, perhaps rightly considered as the most complete and the most "physical" amongst available treatises on gravity). Though the correct answer has been known since 1977, 3 " 6 the same type of statement continues to appear in many new texts, as exemplified by a recent otherwise very comprehensive review of Supersymmetry, Supergravity and Superstrings.7 A second source of confusion is generated by errors in the identification of the embedding map GL(4, R) -* GL(4, C). An example of this kind of error is provided by a recent article in this journal. 8 GL(4, C) does possess finite spinorial representations, and an error in the identification of its physical GL(4, R) subgroup seems to have given the impression that GL(4, R) does have finite spinorial representations. In what follows, we shall prove that the embedding GL(4, R) -> GL(4, C) does not exist; neither can we embed GL(4, R) in a "double-covering of GL(4, C)", since there is no such thing for the latter, a simply-connected group. Indeed, both the above textbook quotation and the review citation 7 would have been correct had the word "representations" been replaced by "finite representations". The forbidden embedding [GL(4, R) -/• GL(4, C)] implies the inexistence of finite spinorial representations for GL(4, R). The third reason for the overall confusion concerns the unitarity of the relevant spinor representations. In dealing with noncompact groups, it is customary to select infinite-dimensional unitary representations, where the particle-states are concerned. For both tensor or spinor fields, however, finite and nonunitary representations are used (of GL(4, R) and SL(2, C) respectively). We shall show that the correct answer for spinoral GL(4, R) consists in using the infinite unitary representations in a physical base in which they become nonunitary. 9 In recent years, the unitary infinite-dimensional representations of the doublecoverings GL(n, R) and SL(«,R) have been classified and constructed for n — 3, 1 0 n = 4; 9,11 the case n = 2 has been known for many years. 12 Field equations have been constructed for such "manifields" within Riemannian gravitational theory and for Einstein-Cartan gravity, 13 including the case of "world spinors", 14 and for affine 15 ' 16 or metric-affine gravity. 17 SL(4,R) manifields have also been used in classifying the hadron spectrum. 18 SL(10, R) manifields have been applied to the embedding of the superstring in a generic curved space. 1 9 - 2 0 In Ref. 15, leptons were assigned to nonlinear realizations of SL(4, R). In what follows, we shall also discuss the way in which the general covariance group (the diffeomorphisms) is realized nonlinearly over the Poincare-covariant Hilbert space, in the presence of a massless spin-2 state. This is a crucial point in understanding the superstring and its spectrum, with its relevance to quantum gravity. 2.
Existence of the Double-Covering GL(«, R)
The basic results can be found in Ref. 21. Theorem 1 Let g0 = k0 + a0 + n0 be an Iwasawa decomposition of a semisimple Lie algebra g0 over R. Let G be any connected Lie group with Lie algebra g0, and le. K, A, N be
457
GL(4, R) Group-Topology, Covariance and Curved-Space Spinors 1657
the analytic subgroups of G with Lie algebras k0, a0 and n0 respectively. The mapping {k,a,n) -* kan {keK,aeA,neN)
(2.1)
is an analytic diffeomorphism of the product manifold K x A x N onto G. The groups A and N are simply connected. Any semisimple Lie group can be decomposed into the product of the maximal compact subgroup K, an Abelian group A and a nilpotent group N. As a result of Theorem 1, only K is not guaranteed to be simply-connected. There exists a universal covering group Ku of K, and thus also a universal covering of G: GuczKuxAxN.
(2.2)
For the complex case we have, Theorem 2 Let g be a semisimple Lie algebra over C, gH the Lie algebra g considered as a Lie algebra over R. Let J be the complex structure on gR which corresponds to multiplication by j on g. Let q be any compact real form of g and let a be any maximal Abelian subalgebra of q. Then the algebra h = a + ia is a Cartan subalgebra of g. Let A be the set of roots of g with respect to h and let A+ be the set of positive roots with respect to some ordering of A. If n+ denotes the space
I 9' considered as a real subspace of gR, the following direct decomposition is valid gr = q + Ja + n+ . Let Gc be any connected Lie group with Lie algebra gR and let Q, A* and N+ denote the analytic subgroups of Gc with Lie algebras q, Ja and n+, respectively. Then the mapping (<j, a, n)-*qan,
qeQ,
aeA*,
neN+
is an analytic diffeomorphism of Q x A* x N+ ontoG c . The groups A* and N+ are simply connected. Thus here again it is the compact subgroup (Q here) whose topology will determine
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the topology of Gc [Gc]„ ^QuxA*xN+.
(2.3)
For the group of diffeomorphisms, let Diff(n, R) be the group of all homeomorphisms / of R" such that / and f'1 are of class C1. In the neighborhood of the identity Vr>t = L e Diff(n, R)| lg(x) -x]<e,
[^(x)
- <5<~] < e, \x\< r
i, k = 1,..., n | (2.4)
Stewart22 proved the decomposition Diff(w, R) = GL(n, R) x H x Rn
(2.5)
where the subgroup H is contractible to a point. As a result, as O(n) is the compact subgroup of GL(w, R), one finds Theorem 3 0(n) is a deformation retract of Diff(n,R). As a result, there exists a universal covering of the Diffeomorphism group D3f(n, R)u ^ GL(n, R ) „ x H x J i , .
(2.6)
Summing up, we note that both SL(n,R) and on the other hand GL(n,R) and Diff(n, R) will all have double coverings, defined by SO(n) and 0(«) respectively, the double-coverings of the SO(n) and O(n) maximal compact subgroups. In the case n = 4, we have the homomorphism between SO(3) x SO(3) and SO(4). Since SU(2) so 3
< >-z(2T
(2J)
where Z(2) is the two-element center {1, — 1}, we have [SU(2)Vx SU(2)] S O ( 4 ) ^ L J V JZ(2)" ^lr
(2-8a)
where Z(2)d is the diagonal discrete group whose representations are given by {1,(-1)2^=(-1)^}
(2.8b)
where j t and j2 are the Casimir labels of the two SU(2) representations. The full Z(2) x Z(2) group given by the representations
459
GL (4, R) Group- Topology, Covariance and Curved-Space Spinors
{1,(-1) 2J '}®{1,(-1) 2 ^}
1659
(2.9)
is the center of S0(4) = SU(2) x SU(2), which is thus the quadruple-covering of SO(3) x SO(3) and a double-covering of SO(4). SO(3) x SO(3), SO(4) and SO(4) = SU(2) x SU(2) are thus the maximal compact subgroups of SO(3,3), SL(4,R) and SL(4, R) respectively. We have the exact sequences
Z{
—» Z.2
1
1
i
i
X
^ 2 ~>
Z2
I A
-1
1
- S T ( 4 , R H • SL(4,R)-» 1
i
I
SO(3,3)
SO(3,3)
1
i
1
1
(2.10)
3. The SL(4, R) - S L ( 4 , C) Embedding In the usual approach to the Poincare symmetry based formulation of the parity invariant spin4 (Dirac) fields one makes use of fields which transform with respect to the Lorentz group SO(l, 3) ^ SL(2, C) according to a direct sum of two irreducible representations, <MiO)©(0,i).
(3.1)
SL(2,C)-SL(4,C)j {(iO)©(Q,±)}-4j
<•'
One may now further embed
assigning \j/ to a 4-dimensional SL(4,C) representation reducing to (3.1) under the SL(2, C) subgroup. Adjoining the translations, we can also embed the Poincare group ISL(2, C) in the general affine transformations group, GA(4, R) = T 4 © GL(4, R) more specifically in the corresponding double covering group GA(4, R), where the SL(2, C) group which acts nontrivially in the space of field components is enlarged to the GL(4, R) group. A natural question to ask8 is whether the simple subgroup SL(4, R) of the GL(4, R) can be embedded into a finite complex matrix group; had that been possible, the spinorial representations of the latter would then provide us with the
460
1660 Y. Ne'eman & D. Sijaiki
appropriate curved-space generalization of the, say, Dirac field. Unfortunately, this is not possible Ref. 8 notwithstanding, as we shall demonstrate. A glance at the classical semisimple Lie group Dynkin diagrams tells us that we need to investigate two possibilities: either one can embed the SL(4, R) algebra at(4, R) in the Lie algebra of the appropriate noncompact version of the orthogonal algebra <30(6) of the Spin(6) group, or in the at (A, C) algebra of the SL(4, C) group. In the first case, the appropriate noncompact group is Spin(3,3) ^ SO(3,3) which is, as we have already seen, isomorphic to the SL(4,R) group itself. As for the second option, we consider first the problem of embedding the algebra <^(4,R)->
(3.3)
referring to the relevant groups later. The maximal compact subalgebra ao(4) of the at(4, R) algebra is embedded into the maximal compact subalgebra <JU(4) of the at (A, C) algebra. at(4,R)-><,t(4,C) u do(4)
u
.
(3.4)
du(4)
There are two principally different ways to carry out the <JO(4) -»^u(4) embedding (see Ref. 13 and Appendix B). Natural (\, \) embedding *o(4)^[>u(4)] orth .
(3.5a)
In this embedding the oo(4) algebra is represented by the genuine 4 x 4 orthogonal matrices of the 4-vector (j, j) representation, i.e. antisymmetric matrices multiplied by the imaginary unit. The SL(4,C) generators split with respect to the naturally embedded oo(4) algebra as follows
si^ee^^e^©^,
(3.5b)
where c and nc denote the compact and noncompact operators respectively. The 6C and 9K parts generate the SL(4, R) subgroup of the SL(4, C) group. The maximal compact subgroup SO(4) of the SL(4, R) group is realized in this embedding through its (single valued) vector representation (i,i). In order to embed SL(4,R) into SL(4, C), one would now have to embed its maximal compact subgroup SO(4) in the maximal compact subgroup of SL(4, C), namely SU(4). However, that is impossible ( i . i ) # {(i,0)©(0,2)} : SO(4) appears here in its defining vector representation, and according to Theorem 2.2 in Ref. 21, all maximal compact subgroups of a
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GL(4, R) Group-Topology, Covariance and Curved-Space Spinors 1661
connected semisimple Lie group are connected and conjugate under an inner automorphism. The alternative could have been to embed SO(4) in a hypothetical double covering of SU(4)—except that SU(4) is simply connected and thus is its own universal covering. We, therefore, conclude that in the natural embedding one can embed SL(4, R) in SL(4, C) but not the SL(4, R) covering group. Dirac {(*,0) © (0,*)} embedding In this case the <JO(4) -> <JU(4) embedding is realized through a direct sum of 2 x 2 complex matrices, i.e.
The SL(4,C) generators now split with respect to the ou(2) © du(2) algebra as follows 3l3lc©lw©lIK©4c©4ce4«©4nce6c©6IK.
(3.6b)
It is obvious from this decomposition that in the ^(4,C) algebra there exist no 9-component noncompact irreducible tensor-operator with respect to the chosen ou(2) © «u(2) ~ do(4) subalgebra, which would, together with the 6C operators, form an at(4, R) algebra. Thus, we conclude that this type of <JO(4) embedding into <JU(4) c <j
= iQbcQad -
iQadQcb -
(4-l)
where for the structure constants gab one may take the relevant invariant metric tensor: either the Euclidean a„» = d i a g ( + l , + l , - H , + l )
(4.2a)
with respect to the SO(«) subgroup, or the Minkowskian iU = diag( + 1 , - 1 , - 1 , - 1 )
(4.2b)
with respect to the Lorentz subgroup SO(l, n — 1) of GL(n, R). The metric tensor gttb is GL(n, R)-covariant.
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1662 Y. Ne 'eman & D. Sijacki
Taking gab = rjah, the antisymmetric generators Jab
=
(4.3a)
Qla.b]
generate the physical Lorentz subgroup in n-dimensional Minkowski spacetime. The traceless symmetric operators (4.3b)
Tab = Q{a,b} - (l/n)iu(T. act as "shears" and deform, though all the while preserving the n-volume. The trace
(4.3c)
is the dilation operator. The gl(n, R) commutators are: Mrti Jed J
=
~ iVlacJbd ~ ^ad^bc ~ ^ be Jad + '/w'oc)
Uab> led] — ~ I Wac^M + ^oi^bc ~r)bc^od'~
VbdJoc)
(4.4) [Tab* Tcdl ~ + iOlacJbd + factor + ^bc^ad + f w ' a c ) LD,Jah-]=0;
[D,ra6]=0
Together with the translations Pa, a = 0, 1, ..., n — 1, these generate the general affine group GA(n,R) and its double-covering GA(n,R), as semi-direct products GA(«,R) = T„©GL(n,R).
(4.5a)
Without the dilations, this reduces to the group SA(n,R)=TB@SL(n,R)
(4.5b)
with the additional commutators [.Qab,PJ=-i9acPb (4.5c)
[Pa.n] = o. In what follows, we concentrate on the algebraic simple part <j/(n, R) and the group SL(n, R). Denoting the space-like co-ordinates by i, j = 1, 2,..., n - 1 we regroup the Jab and Tab in the following subsets:
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GL (4, R) Group-Topology, Covariance and Curved-Space Spinors
Jij
angular momentum
K( = J0i
Lorentz boosts
Ty
shears
1663
(4.6)
TQQ
The relevant subgroups are SO(M — 1) SO(n)
(spatial rotations), generated by the Jtj the maximal compact subgroup, generated by the JSj & Nt
SO(l, n — 1) the Lorentz subgroup, generated by the JtJ & Kt SL(n — 1, R) R+
the "(n — 1) volume"—preserving group, generated by the JtJ & Ttj a one-parameter subgroup generated by T 00 .
The commutation relations for at(n, R) now read: Ui)>Jki~\ = '(<Wji - SuJjk ~ djkJu + djiJik)
lK„Kj-] = -iJtJ Uij, r « ] = W* TM + du Tjk - dJk Tu - Sj, Tik) Ua, T00\ = 0 LT„, r „ ] = -i(6ikJj, +
+ SqToo)
[Kl>7Jk]=-i(^Ar4 + i t t ^ )
464
1664 Y.Ne 'eman & D. Sijacki
lNl,TJIJ=-t(6uKk
+ dlkKj)
[K«,r 0 0 ]=-2iiVi
W,T00l=-2iKi [r ( ,,T o o ]=0.
(4.7)
The compact operators are Ji} and Nh while the remaining ones Kt, Ttj and T00 are noncompact. It is now straightforward to establish the following theorem. Theorem 5 For any SL(n, R), n > 3 group there exists an inner automorphism ("deunitarizing automorphism") generated by stf = exp(5T00), which leaves the R+ (g>SL(n - 1,R) subgroup intact, and which acts on the SL(4, R) generators in the following way stJys/'1
= Ji},
s/Tutf-1 = T„,
sSToost'1 = T00, (4.8)
s/Njs*-1 = iKj,
s/Kjj/-1
= iNj,
stDsf'1 = D.
For theories based on the SA(n, R) space-time symmetry, the stability subgroup (little group) in the case of the (Hilbert space) representations on states is T^-^ © SL(n — 1, R), where SL(n — 1, R) is invariant with respect to the automorphism. The applications to hadron spectroscopy18 are achieved for the SA(4, R) symmetry upon a trivialization of the T3' Abelian part of the corresponding stability group. The unitarity of the SA(n, R) representations requires, when T^t -*• 1, the unitarity of the SL(n — 1,R) representations, and these representations are infinite-component ones. For the SA(n, R) representations onfields,the relevant group acting in the space of field components is SL(», R). The compatibility of the SA(n, R) representations on states and on fields is achieved provided the SL(n, R) representations when restricted to the SL(n — 1, R) subgroup ones are unitary. However, had the whole SL(n, R) been represented unitarily, the Lorentz boost generators would have a Hermitian intrinsic part; as a result, when boosting a particle, one would obtain another particle—contrary to experience. The deuniterizing automorphism s/ allows us to start with the unitary representations of the SL(n, R) group, and upon its application to identify the finite (unitary) representations of the abstract SO(«) compact subgroup (Jy, Nt) with nonunitary representations of the physical Lorentz group (Jy.Kj), while the infinite (unitary) representations of the abstract SO(l,n — 1) group of (J0-,Kf) now represent (nonunitarily) the compact (7y, Nt) generators. The non-Hermiticity of the intrinsic boost operator parts here cancels their physical action precisely as in finite tensors or spinors, the boosts thus acting kinetically only. In this way, we avoid a disease common to infinite-component wave equations and to spectrum generating groups. With ordinary finite tensor fields corresponding to nonunitary representations of GL(4, R), the Lorentz subgroup is indeed realized as finite nonunitary representations,
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1665
isomorphic to the unitary representations of SO(4). By analogy, J. Mickelsson 23 was drawn in one model to identify the physical Lorentz subgroup with the compact SO (4), in using unitary infinite representations of GL(4, R), though stressing the unphysical nature of his identification. With the automorphism si, we get the correct interpretation of the assignment,9 confirming Mickelsson's intuitive choice of the finite SO(4) representations as nonunitary representations of the physical SO(l,3). 5. Poincare and GL(4, R) Representations and Covariance Let us consider first the Poincare group in a 4-dimensional Minkowski space-time, and let us concentrate on the m = 0 representations. In the case of these representations on states, the appropriate little group is E(2) = T2' © SO (2), where the Abelian subgroup Tj is generated by the operators X^ = Kx — J2 and Xt = K2 + J l 5 while the SO(2) subgroup is generated by J3 (note Jt = zijkJjk). In the usual particle physics applications T2' is represented trivially, and the particle states correspond to the 1-dimensional helicity states J3 |A> = A|A>, A = 0, + | , + 1, When parity is conserved, for A # 0 one works in a 2-dimensional space {|A>, | — A>}. Each state |A>, A > 0 can be obtained as a direct product of A — 1, (A — £) states 11> and the lowest state |1>, ( | j » when A is integer (half-integer) respectively; and analogously, when X < 0, from the states | —1>, (|— ^>). An arbitrary element of the Poincare group TA © SO(l, 3), (a, A), aeT^AeSO(l,3) is represented in the Hilbert space of vectors {\p,k}; p1 = 0,p0> 0} as follows r(a,A)\p,X>
= e"«A'>-dW{L-1(Ap)>lL(p)}|Ap,A>,
(5.1)
where A is the SO(l,3)-»SO(l,3) image of A, L(p) is the appropriate boost for p = L(p)p, p = (£,0,0,£), and da) is the corresponding SO(2) representation. Let us turn now to the m = 0 tensorial representations of the Poincare group on fields. In contradistinction to the case of Poincare representations on states, where the representations are induced from those of the little group E(2), the Lorentz subgroup SO(l, 3) is acting nontrivially in the space of field components. Its representations, say j)UiM determine the allowed types of fields. However, as for the physical applications of the Poincare symmetry, it is crucial that the particle and the field descriptions match. Let us start with the field components which correspond to the two (due to parity) helicity states | ± A>. The simplest, but by no means the only way to describe these field components in the momentum space is by making use of the totally symmetric product of the two polarization vectors e£(p) which correspond to the helicity A = + 1 states. The T2' Abelian subgroup of the m = 0 little group E(2), is now (as a part of the Lorentz group) represented nontrivially and for instance, acts on the two basic polarization vectors as follows *?(?)-<£(?)+
rf±(p)p<°>|(;
p(OV = (£,0,0,£),
**eR.
Moreover, the action of an arbitrary Lorentz group element A is given by 24
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(5.2)
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Y. Ne'eman & D. Sijacki
A;ev±(Ap) = exp( + i% ) A)) e ±(p) + ^±(p,A)^;
p2 = 0 , P o > 0 .
(5.3)
For a product of k polarization vectors, one finds a sum of X factors, linear in p^. It is clear now that in the case of tensorial m = 0 Poincare representations on field, the basic objects are the classes of polarizations {e»v...(P) + cfe,l..pM + cie?P...Pv + •"} instead of the polarizations themselves, and that the physics should not depend on the specific selection of a class element. We now invoke the covariance requirement, i.e. we make use of the GL(4, R) linear tensorial representations, which support the nonlinear representations of the general co-ordinate transformation group Diff(4, R) (the diffeomorphisms of R4). On account of the covariance and Hermiticity, we find that (in the configuration space) the basic objects are the potentials M„v„...(x)} = A w ...(x) + dJyp,.Xx) + dj„...{x) + d>„v...(x) + ••• -
(5.4)
which transform inhomogeneously. The nontrivial appearance of the SL(4, R) representations is evident in the case of the helicity 2 potentials fy,v(x) (g„v(x) — ^v + h^(x) H ), transforming with respect to the 10-dimensional GL(4,R) representation CD, which reduces to the two irreducible Lorentz representations (1,1)©(0,0). In order to form scalars out of the potentials, one first constructs the appropriate tensor fields (field strengths), determines their irreducible components and consequently obtains a certain number of invariants. The coupling of the matter current J"vp-- to the potentials has to be independent of the choice of an element from a class, i.e.
{A„,.M + P„K...(P) + PJ„..XP)
+ --}J"V"-(P)
=
A^.XPU^"
•(/>),
and thus Vfi
PflJ"
-ip) = 0,....
(5.5)
Demanding the full covariance, we obtain immediately (off-shell) that the massless fields (particles) are coupled to the conserved currents, i.e. d^w-ix) = 0
(5.6)
The relevant examples are: the helicity ± 1 potential /4„(x) is universally coupled to the (Yang-Mills) current J"(x), the helicity 2 potential /i„v and ^M[vp] are coupled to the energy-momentum 9^ and the spin £"'v''1 currents respectively, while the helicity ± 3 potential AMvp) is coupled to the shear current TMvp\ in a general affine theory of gravity.15
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GL(4,R) Group-Topology, Covariance and Curved-Space Spinors 1667
It is interesting to note that most authors in developing a Poincare field theory, at some point implement GL(4,R) covariance and define the theory over the whole space-time, without being aware of it. So far we have not considered the spinorial massless fields. The fundamental distinction between tensorial and spinorial fields is in the fact that the GL(4, R) (or SL(4, R)) group does not have finite spinorial representations. In other words, the intermediate step of the group chain Diff(4, R) => GL(4, R) => SO(l, 3) is for the finite spinorial representations artificial, and thus finite spinors "live" in the flat space only. It is true that for the supergravity "gravitino" with helicity ± f, one has a potential ^a„(x) with the \i index supporting covariance, and the corresponding conserved current; however, there exist no fully holonomic (world) formulation of the supergravity theory and the gravitino is indeed introduced in the flat tangent space at each point of the R* base space. It has generally been assumed that the mere presence of a massless helicity ± 2 state in the Hilbert space induces the effective couplings of Einstein's theory and should, therefore, realize covariance (Diff(n, R)) over the Hilbert space (nonlinearly). Actually, a closer perusal of Eqs. (5.2)-(5.6) shows that covariance (or at least GL(n, R)) is also part of the input. This should be born in mind when dealing, for instance, with the Scherk-Schwarz re-interpretation of the string as a theory of gravity. 25 As we have stated in the Introduction, there exist infinite-dimensional spinorial representations of the GL(4, R) (or SL(4, R) group). Moreover, these representations provide a Hilbert space in which one can realize the (intrinsic) spinorial representations of the double covering of the group of diffeomorphisms Diff(«, R).11> 1 4 ' 1 6 The simplest SL(4,R) spinorial representations, which support the world spinors, are the multiplicity-free representations.9 These representations, when reduced with respect to the SO(4) maximal compact subgroup of the SL(4,R) group, contain each subgroup representation at most once. The representation labels of the SO (4) ~ SU(2)
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1668 Y. Ne'eman & D. Sijacki representations, we have developed a field-theoretical formalism 14 ' 19 which generalizes appropriately the tetrad formulation of general relativity. The infinite (tetrad-like) frames E^x), the "alephzeroads", carry a GL(4, R) index A and a Diff(4, R) index M, and connect the flat (local) and the world (curved) spinors. References 1. B. Speh, Mat. Ann. 258 (1981) 113. 2. J. Rawnsley and S. Sternberg, Am. J. Math. 104 (1982) 1153. 3. Y. Ne'eman, in GR8 Proceedings of the VIII International Conference on General Relativity and Gravitation, University of Waterloo (Canada), 1977, p. 262. 4. Y. Ne'eman, Proc. Nat. Acad. Sci. USA 74 (1977) 4157. 5. Y. Ne'eman, Ann. Inst. Henri Poincare A28 (1978) 369. 6. Y. Ne'eman, Found. Phys. 13 (1983) 467. 7. For recent examples, see: (a) M. F. Sohnius, Phys. Reports 128 (1985) 39. See p. 192. Reprinted in Supersymmetry and Supergravity, ed. M. Jacob (North Holland, Amsterdam, Oxford, New York, Tokyo)/(World Scientific, Singapore, Philadelphia, Hong Kong) 1986, pp. 1-166; see p. 154. (b) R. M. Wald, General Relativity, (University of Chicago Press, Chicago, London, 1984) p. 359. (c) M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, (Cambridge University Press, Cambridge, London, New York, 1987) p. 225. For examples antecedent to the discovery of GL(n) spinors, see (d) S. Weinberg, Gravitation and Cosmology (John Wiley & Sons Inc., New York, London, 1972) p. 365. (e) F. W. Hehl, G. Kerlick and P. v. d. Heyde, Z Naturforsch. 31a (1976) 823-827; see p. 823. 8. M. W. Kalinowski, Int. J. Mod. Phys. A 1 (1986) 227. 9. Dj. Sijacki and Y. Ne'eman, J. Math. Phys. 26 (1985) 2457. 10. Dj. Sijacki, J. Math. Phys. 16 (1975) 298. 11. Dj. Sijacki, Ann. Israel Phys. Soc. 3 (1980) 35. 12. V. Bargmann, Ann. Math. (Princeton) 48 (1947) 568. 13. A. Cant and Y. Ne'eman, J. Math. Phys. 26 (1985) 3180. 14. Y. Ne'eman and Dj. Sijacki, Phys. Lett. 157B (1985) 275. 15. Y. Ne'eman and Dj. Sijacki, Ann. Phys. (NY) 120 (1979) 292. 16. Dj. Sijacki, Phys. Lett. 109B (1982) 435; Dj. Sijacki, in Frontiers in Particle Physics, eds. Dj. Sijacki et al. (World Scientific, Singapore, 1984), p. 382. 17. F. W. Hehl, G. D. Kerlick and P. von der Heyde, Phys. Lett. 63B (1976) 466; Z. Naturforsch. A31 (1976) 111, 524, 823; E. A. Lord, Phys. Lett. 65A (1978) 1; F. W. Hehl, J. D. McCrea and E. W. Mielke, "Weyl space-times, the dilation current and creation of gravitating mass by symmetry breaking" to be published in the Proceedings of the Hermann Weyl (1986) Commemorative Conference. 18. Y. Ne'eman and Dj. Sijacki, Phys. Lett. 157B (1985) 267. 19. Y. Ne'eman and Dj. Sijacki, Phys. Lett. 174B (1986) 165. 20. Y. Ne'eman and Dj. Sijacki, Phys. Lett. 174B (1986) 171. 21. S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York and London, 1982), Theorems 5.1 (p. 234) and 6.3 (p. 239). 22. T. E. Stewart, Proc. Am. Math. Soc. U (1960) 559. 23. J. Mickelsson, Commun. Math. Phys. 88 (1983) 551. 24. S. Weinberg, Phys. Rev. 134 (1964) B882; Phys. Rev. 138 (1965) B988. 25. J. Scherk and J. H. Schwarz, Nucl. Phys. B81 (1974) 118.
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QCD as an effective strong gravity Dj. Sijacki '-2 Institute of Physics. PO Box 5 7, Belgrade, Yugoslavia
and Y. Ne'eman '-3-4-5 Sackler Faculty ofExact Sciences, Tel-Aviv University, Tel-Aviv, Israel Received 15 March 1990 We approximate QCD in the IR region by the exchange of a dressed two-gluon phcnomenological field OuAx) = B"ldB'lnal,, >k» a color-SU(3) metric. The model (1) produces color confinement, (2) explains the successful features of the hadronic string. (3) predicts the spectrum of baryons and mesons with their Reggc trajectories, (4) justifies the interacting boson model of Arima and Iachello in nuclear physics, (5) "predicts" scaling.
1. Introduction The non-pcrturbative features of QCD have made it difficult, in almost any situation, to apply the theory exactly - except for some vacuum configurations, studied especially to prove confinement [ 1 ], yet without completely conclusive results to date, and for the (perturbative) asymptotic domain. Various approximation schemes have been tried, ranging from the non-relativistic quark model (NRQM) to the bag (BM) and to the Skyrme-Witten models (SWM). Their validity is generally restricted to one or two individual features of the theory, missing all others by a wide range [2]. Before the rise of QCD and throughout the earlier stages in the evolution of the theory, an ad hoc "strong gravity" hypothesis [ 3 ] was tried, in which the f° me1
Supported in part by the USA-Israel BNSF, Contract 8700009/1, and by the FDR-Israel GIF, Contract 1-52.212.7/ 87. 2 Supported in part by Science Foundation (Belgrade). 1 Also on leave from Center for Particle Theory, University of Texas, Austin, TX 78712, USA. * Supported in pan by the USA DOE Grant DE-FG05-85 ER 40200. 5 Wolfson Chair Extraordinary in Theoretical Physics.
son (with/=2 + and a mass of 1270 MeV) was given a central role as the "strong graviton". In the light of the f°'s quark-antiquark structure, its postulated gauge-field nature can at most be regarded as "phcnomenological". Moreover, the results were inconclusive. We mention this "f/g two-graviton model" because it shares superficially with our present proposal a certain kinship with gravity - entirely ad hoc in the "f/g model" but QCD-derivcd in ours. In what follows, we present an approximation which reproduces most features of the strong-coupling region. The model (1) produces color confinement dynamically; (2) explains the successes of the hadronic string, another approximation to QCD; (3) predicts the complete structure of the hadron spectrum for baryons and mesons, including Rcgge trajectories; (4) explains ihc low-energy nuclear physics spectra and "predicts" the highly successful Interacting Boson Model ("IBM") of Arima and Iachello [4]; (5) predicts scaling. At the same time, all results of asymptotic freedom (the UV end) remain unhindered, i.e. the fit to the NRQM stands. Our basic ansatz is weaker than the full QCD "dogma" that hadron observable states are colorSU(3) singlets. We only assume the proven satura-
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B. V. (North-Holland)
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lion properties, i.e. that the color-singlct configurations are the lowest lying ones. 2. Gluonium-induced effective gravity If the hadron lowest ground states are colorless (our assumption) and in the approximation of an external QCD potential (in analogy to the treatment of the hydrogen atom in the Schrodinger equation), the hadron spectrum above these levels will be generated by color-singlet quanta, whether made of dressed twogluon configurations, three-gluons Every possible configuration will appear. No matter what the mechanism responsible for a given flavor state, the next vibrational, rotational or pulsed excitation corresponds to the "addition" of one such collective colorsinglet multigluon quantum superposition. In the fully relativistic QCD theory, these contributions have to come from summations of appropriate Feynman diagrams, in which dressed n-gluon configurations are exchanged. We rearrange the sum by lumping together contributions from n-gluon irreducible parts, n = 2, 3 so and with the same Lorentz quantum numbers. The simplest such system will have the quantum numbers of gluonium, i.e. « = 2. The color singlet external field can thus be constructed from the QCD gluon field as a sum (t\ah is the SU(3) metric, daU is the totally symmetric 8 X 8 X 8 -• 1 coefficient) B"l,Bhl,r}ah + BauB'\B^dah, + ....
(1)
In the above, B"u is the dressed gluon field. It will be useful for the applications to separate the "flat connection" Na„, i.e. the zero-mode of the zero-mode of the field. Writing for the curvature or field strength /• "Vv — §uBa„~duB"n— \f"hcB 'MBL„, we define Ba„ = N%+A«„_ so that F(N) =0,
(2)
i.e. or. in form language, dN=N*N.
version of the (8 + 4)-dimensional inhomogeneous {SU(3)®M}, M is Minkowski space, after spontaneous vibration [6] and contraction of the holonomic SU (3) indices. Eq. (3) is therefore also the BRS equation for the ghosts C, when we replace d-*s and N->C. We can now rewrite the two-gluon ("gluonium" ) configuration as G„v(x)=B\B>\t}ah,
(4)
which looks very much like a spacctime metric. It is an effective spacetimc metric representing some of the geometric features induced in the spacetime basemanifold of the color-SU (3) principal bundle. We assume that (4) is the dominating configuration in the excitation systematica and note that Lorentz invariance forces this metric to obey a ricmannian constraint D„G^„=0, where D„ is the covariant derivative of the effective gravity (the connection will be given by a Christoffel symbol constructed with the metric (4)). The separation of the flat part of Bafl in eq. (2) reproduces the separation of a tetrad ?%(-<) =5"„+f,Ax) m t 0 l n e Hat background piece and the quantum gravitational contribution. As a result, G\,„(.v) itself can be separated similarly. We note two points: (1) Out of the ten components of GM„ in (4), the six that survive after the four riemannian constraints have spin/parity assignments Jp=0+, 2 + . This suggests a relationship with the IBM model systcmatics [4], in which the fundamental excitation was selected with these quantum numbers, to fit the phenomenology. Note that the absence of dipolar excitations is in itself an indication that a gravity-like force is involved [7]. (2) An effective ricmannian metric induces einstcinian dynamics. However, our correspondence is between low-energy (IR) QCD, with its strong coupling, and the high-energy (UV) strong coupling region of our effective gravity (and not with the weak coupling newtonian limit). This includes the curvature-quadratic counterterms generated by the rcnormalization procedure and corresponds to the (effective) invariant action,
A„v = - J d4.r J^G iaR^R-'-P^
(3)
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+ yK^R) .
a
Note that (3) implies that N M is the Cartan left-invariant form of SU(3) in a soft-group-manifold [5]
(5)
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This is the Stelle [ 8 ] action, when used for true gravity. It was shown by Stelle to be renormalizable, a feature befitting our present application, since QCD is renormalizable and any piece of it should preserve the finiteness feature; but (5) is not unitary, which makes it unutilizable for true gravity - but which befits the present application: a "piece" of QCD should not be unitary, QCD being an irreducible theory. Stelle's main result, however, was to show that renormalizability is caused by p~* propagators. But p~* propagators are dynamically equivalent to confinement! [9]. Such propagators were generally introduced in QCD ad hoc. Here, they stem from our basic premise. Note that although we only assumed that the lowest states are color-singlets, the p~ 4 propagators will cause any colored state to be bound and confined; adding a quark or gluon to a color-singlet hadron will polarize the vacuum, creating pairs until the configuration becomes colorneutral. In recent years, quadratic lagrangians like (5), with, in addition, torsion-squared terms, have been investigated classically in the context of the Poincare gauge theory of gravity [10]. The exact solutions display, aside from the newtonian potential M/r, a component behaving ~ r2, dominating the strong-field limit and originating in the curvature-squared terms as in (5). There is one more feature that relates to eq. (5). It has been shown [ I I ] that the string, as a gravitational theory, is equivalent to an action such as (5), i.e. with quadratic countertcrms. From a different viewpoint, the embedding of the string in curved "target" spacetimes has been interpreted [12] as a series of constraints on the manifold's geometry due to the necessity of preserving the cancellation of the conformal anomaly. Such constraints are regarded in string theory as replacing Einstein's equation in fixing the geometry of the target space - and their lowest terms are also those of (5). As a result, our ansatz explains the good fit of string theory in its original hadron version, in reproducing the IR region features: color confinement, string flux-tubes etc. One more comment relates to "f/g gravity" [ 3 ] and the f°. That hypothesis was marred by the need to write a geometric einsteinian equation for a massive graviton, a doubtful procedure. Here, the G^ effective directly QCD-induced metric field is massless because of QCD gauge invariance!
3. Classical and quantum algebraic structure The gluonium external field G"„„(x) transforms under Lorentz transformations as a (reducible) second-rank symmetric tensor field, with abelian components, i.e. [G„„ G/K,]=Q. Algebraically,
where the quadrupolar excitation-rate is given by Tliv = rial^dk~[a'1),+
(k)a\+(k)exp(2ikx)
+ a\(k)ah„(k)e\p(-2ikx)]
,
(6)
for (infinite) gl(3, R) non-compact excitation bands [13], whereas Ull„ =
nuh\&k~[ctaS{k)a»Ak)
+ a%(k)a"„+(k)],
(7)
generates finite u(3) spectral multiplets. We have made use of the canonical transformation [«%(£) + ^jVVxp(iA*) ]-<*"„(*:), [a\ + (k) + \N\exp(-ikx))^a°fl+(k)
(8a) .
(8b)
Using [«%(*), a \ + (A:')]= £ 5 a/ 'rf^(/c-/f), one verifies that the operators T„„ and U„„ together with the operators Sll„=nat,jdk~{aau + (k)ahAk) -a«u{k)a'S(k)],
(9)
close respectively on the gI(4;R) and u(l, 3) algebras. Note that the largest (linearly realized) algebra with generators quadratic in the a„ + , a„ operators is sp( 1, 3; R), where the notation " 1 , 3" implies a definition over Minkowski space.
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elsewhere [ 18 ] the application of our "G„„" effective strong gravity ansatz to nuclei. The u( 1, 3) SGA of (10) is related to both the highly-successful IBM u(6) symmetry and to the sp(3, R) analysis [19] with its (Elliot) su (3) and DGN [ 13 ] si (3, R) subgroups for deformed nuclei [20].
sp(l,3,R) — u( 1,3)— . - - . .-gl(4,R)-»—•-tl0((T)SO(l,3)-
>-so(l,3).
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su(l,3)-> ••• sl(4;R)-.-.• tg(a)so(l,3)(10)
4. Hadrons
The gl(4; R) algebra represents a spectrum generating algebra for the set of hadron states of a given flavor [14,15]. We now return to the expansion in eq. (1). The si (4; R) is generated by gluonium. What about threegluon andrc-nucleonexchanges? The corresponding algebras do not close and generate the full Ogievetsky algebra of the diffeomorphisms in Minkowski space [ 16 ], the four-dimensional analog of the Virasoro algebra (the algebra of diffeomorphisms on the circle). Had we considered the entire (infinite) sequence when writing eq. (1), we would have generated this diff(4; R). The maximal linear subalgcbra of diff(4; R) is gl(4: R). The remaining generators (i.e. diff(4;R)/gl(4,R)) can be explicitly realized in terms of the gl(4, R) generators [17], both for tensors and for spinors. In our case, this would involve functions as matrix elements of the representation of our generators T„, and SMV in eqs. (6) and (8). As in general relativity, the entire "G„„-covariancc" can be realized in terms of the invariant action given in eq. (5). But diff(3. I; R) can also be represented linearly. It will then involve infinite, ever more massive, repetitions of the representations of sl(4;R). In cither way, we find that using si (4; R) takes care of the entire sequence in eq. (1). The inhomogeneous versions of the algebras in eq. (10), i.e. their scmidirect product with the translations t4, are relevant to the Hilbert space spectrum of states (sec section 4). In the case of u( 1, 3) in eq. (10), when selecting a time-like vector (for massive states), the stability subgroup is the compact u(3) with finite representations - as against the noncompactgl(3;R) forsl(4; R). This fits with the situation in nuclei, where symmetries such as the u(6) of the IBM model [4] are physically realized over pairs of "valency" nuclcons outside of closed shells. There is a finite number of such pairs, and the excitations thus have to fit within finite representations. We present
Dynamically, we have discussed in section 2 the role of gluonium excitations in generating the transition to a hadronic excited level, from any given hadron stale. To this we can now add scaling symmetry: the general linear group GL(4, R) decomposes into its unimodular SL(4, R) subgroup, the R + of scale transformations and parity II: GL(4, R) = [ri((T)SL(4,R)](X)R + . This scaling symmetry corresponds to the observations in dcep-inclastic photon-nucleon scattering experiments. Color confinement too manifests itself algebraically at several levels. The si (4; R) subalgcbra preserves the four-dimcnsional measure, a geometric realization of confinement as a dynamical four-volume-preserving rotation-deformation-vibration pulsation mechanism. When dealing with the hadron Hilbert space states, momenta come in and the translations thus have to be adjoined to the algebra. Here we get sa(4; R) = U (ff)sl (4; R). The massive states of the hadron spectrum are then classified according to the stability subalgcbra, here sa(3;R)=t: 1 (a)sl(3;R). The t, quantum numbers are trivialized, as is done with the formal translations t2 of the cuclidcan two-dimensional stability subalgebra, for the massless states in the Poincare group representations. Hadron states are then characterized by the si (3; R) subalgebra, whose infinite representations correspond to Regge trajectories [13]. These preserve the measure in threespace: a Regge trajectory described by such a representation corresponds to a given "bag". Spinors are taken care of by the infinite representations of the double-covering groups S A ( 4 ; R ) = T 4 ( < T ) S L ( 4 ; R ) . We collect all hadronic field configurations obtained by successive application of the quantum gluonium field sl(4: R) dynamical algebra, into an infinitecomponent field (manificld). These manifields are subject to the following constraints: (i) Owing to Da(i,,p(x)=0, the wave-equations have to be
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Lorentz covariant; (ii) The lowest manifield components should fit the basic quark system field configuration in its Lorentz content; (iii) They should transform according to non-unitary representations of the SO(l,3)cSL(4;R) in order to meet the experimental fact that a boosted particle keeps its spin quantum number. These natural requirements of our gluonium picture determine uniquely the selection of manifields and their equations [14,15], while the "non-unitarity" condition on SO(l, 3) representations is achieved by making use of "A-deunitarized" SL(4; R) unitary irreducible representations [21 ]. For mesons, we take a manifield 0, ( • + M2)0=O, transforming according to the ladder representation D( J, j )A. For the three-quark octet configuration wc use the manifield f, (\x"bM—M),P =0 transforming according to the spinorial multiplicity free SL(4;R) representation [D(|,0)( + ) D(0, j) ] A . For the decuplet three-quark configuration wc take a manifield Vp fulfilling (ix"^A/)V„ = 0. It transforms according to the spinorial multiplicity-free representation [D(^,0)( + ) D(0,l)lJ'(X)(j,l)Wc find a good fit with the experimental data. In fig. 1 we present a generic Regge daughter-trajectories structure for baryons and refer to refs. [14,15] for details. This classification was suggested and re-
Fig, l.
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alized phenomenologically [14] and formulated using field theory and QCD [15] as an ad hoc ansatz. This article purports to supply solid dynamical foundations deriving from what appears to be an extremely versatile approximation for QCD.
References [ 1 ] C.G. Callan, R.E. Dashcn and D. Gross, Phys. Rev. D 17 (1978)2717;D 19 (1978) 1826. [2] D.B. Kaplan, Phys. Lett. B 235 (1990) 163. [31 C.J. Isham, A. Salam and J. Strathdec, Phys. Rev. D 8 (1973) 2600; D 9 (1974) 1702; Lett. Nuovo Cimenio 5 (1972)969. [4] A. Arima and F. Iachello, Phys. Rev. Lett. 35 (1975) 1069. [ 5 ] Y. Ne'eman and T. Regge, Riv. Nuovo Cimento Ser. 3, 1 (1978) No. 5. [6] J. Thierry-Mieg and Y. Newman, Ann. Phys. (NY) 123 (1979)247. {7] See e.g. C.W. Misncr, K.S. Thome and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973), §36.1. [8] K.S. Stelle, Phys. Rev. D 16 (1977) 953. [9]Seee.g.J.Kiskis, Phys. Rev. D 11 (1975) 2178; G.B. West, Phys. Lett. B 115 (1982) 468. [ 10 ] P. Bacckler and F. W. Hehl. in; From SU (3) to gravity, eds. E. Gotsman and G. Taubcr (Cambridge U.P., Cambridge, 1985) p. 341. [ 11J S. Deser and A.N. Redlich. Phys. Lett. B 176 (1986) 350. [12] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 158 (1985) 316. [ 13] Y. Dothan, M. Gell-Mann and Y. Ne'eman, Phys. Lett. 17 (1965) 148. [ 14] Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 157 (1985) 267. [ 15 ] Y. Ne'eman and Dj. Sijacki, Phys. Rev. D 37 (1988) 3267. [16] V.I. Ogicvetsky, Leu. Nuovo Cimento 8 (1973) 988. [ 17] Y. Ne'eman and Dj. Sijacki, Ann. Phys. (NY) 120 (1979) 292. (18 ] Y. Ne'eman and Dj. Sijacki, to be published. [ 19] S. Goshen and H.J. Lipkin, Ann. Phys. 6 (1959) 301; D.J. Rowe. The shell model theory of nuclear collective states, in: Dynamical groups and spectrum generating algebras, eds. A. Bohm, Y. Ne'eman and A.O. Barut (World Scientific, Singapore, 1989) p. 287. [20] O.L. Weaver and L.C. Bicdenham, Phys. Lett. B 32 (1970) 326; R.Y. Cusson et al.. Nucl. Phys. A 114 (1968) 289. [21 ] Dj. Sijacki and Y. Ne'eman, J. Math. Phys. 26 (1985) 2457.
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Derivation of the interacting boson model from quantum chromodynamics Dj. Sijacki l - 2 Institute ofPhysics, P.O. Box 57, YU-11 001 Belgrade, Yugoslavia
and Y. Ne'eman 1-3-4-5 Sackler Faculty ofExact Sciences, Tel-Aviv University, 69978 Tel-Aviv, Israel Received 4 July 1990 The strongly-coupled IR region in QCD is approximated by the exchange of a phenomenological field G„,(x) representing a color-neutral pair of dressed gluons. The GK, acts formally as ariemannianmetric, i.e. D„G„,=0. As a result, the surviving quanta are color neutral and have Jr=0+, 2*, with symmetric couplings to nuclear matter - hence the IBM paradigm. The theory fixes the structure of the IBM hamiltonian. It also generates the Sp(3, R) set of symmetries of highly deformed nuclei.
1. Introduction The interacting boson model [ 1 ] has been very successful as a dynamical symmetry "in correlating as well as providing an understanding of a large amount of data which manifest the collective behavior of nuclei" in the words of a recent comprehensive review [ 2 ]. The model's point of departure is the observation that the two lowest levels in the great majority of even-even nuclei are the 0 + and 2 + levels, with relatively close excitation energies, realized by proton or neutron pairs. The model postulates a corresponding phenomenological U(6) symmetry between the six states in (0,2). Iachello has often compared the IBM to the 1961 postulation of SU(3) in 1
2 3
4
5
Supported in part by the USA-Israel BNSF, Contract 8700009/1, and by the FDR-Israel GIF, Contract 1-52.212.7/ 87. Supported in part by Science Foundation (Belgrade). Also on leave from Center for Particle Theory, University of Texas, Austin, TX 78712, USA. Supported in part by the USA DOE Grant DE-FG05-85 ER 40200. Wolfson Chair Extraordinary in Theoretical Physics.
particle physics (a "flavor" symmetry in the present dynamical picture) from purely phenomenological considerations and an identification of the observed patterns. In that example, the true "strong" interaction is now known to be generated by quantum chromodynamics, a force induced by "color" SU(3). It is flavor-invariant, and thus explains the approximate flavor-SU(3) invariance at the historical departure point, while the flavor degrees of freedom themselves stem from other origins. Note that with the (0 + , 2+) excitations in nuclei already appearing in the presence of a single nucleon pair (above a closed shell), there is little justification in relegating these features purely to "collective" mechanisms; rather, it seems one has to look for an explanation in QCD itself.
2. The need for a "strong gravity" The conventional long-range binding mechanism due to QCD, i.e. the exchange of quark-antiquark pairs (mesons with spins 0 and 1) does not generate quadrupole excitations. Skipping the 1~ dipole is generally intimately connected with the (tensor)
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gravitational potential [3]. The 2 + mesons such as the f°(1270) are represented by tensor fields and could thus do it, and an "f/g two-graviton" (or "strong gravity") model was indeed current prior to the discovery of QCD [ 4 ]. The idea was abandoned when it was realized that the spin-2 mesons are also quark-antiquark pairs; moreover, their larger mass would make them very short-ranged and in-effective for the generation of the (0, 2) excitations in the presence of pions, etc. In the present article, we suggest a QCD-derived foundation for the IBM; we also point to a possible related mechanism in nuclei, generating the Sp (3, R) symmetry and its SU(3) and SL(3, R) subgroups.
3. A "strong gravity" from QCD With QCD as the basic strong interaction, it should indeed also be involved in the dynamics and symmetries of nuclei, complexes in which quarks appear already organised - to a large extent - in color-neutral nucleons. In a recent study [5] in particle physics we have identified a mechanism through which the action of QCD in the IR region (at "long range", i.e. ~ 1 fm) includes a component emulating the UV (high energy, short range) region of quantum gravity - i.e. when gravity becomes a strong force. We remind the reader that the present view is that all baryons and mesons represent systems in which the color-carrying quarks and antiquarks are organised in color-neutral "confining" configurations. In our study we have noted that the basic exchanged quantum for color-neutral systems is generated by quanta of the "effective" color-neutral field Gllv{x)=Ba)l{x) Bb„{x) t]ab, the sum of all diagrams representing the exchange of two dressed gluons; a, b=\,..., 8arecolor-SU(3) indices, #%(*) is a color gluon vector potential, the color gauge field; n,v=0, ..., 3 are spacetime indices, r\(j is the SU(3) KillingCartan group metric. No matter what the mechanism responsible for a given hadron state, the next vibrational, rotational or pulsed excitation will correspond to the "addition" of one such collective colorsinglet multigluon quantum superposition. The dominating configuration is the two-gluon system (/„„(*) which formally looks very much like a spacetime metric. It constitutes an effective spacetime metric
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representing some features induced by QCD in the IR region. Note that there exist at present models in which gravity itself is an effective force, induced by some other interaction [6]. Here we adopt this view not for true gravity, but as a description of part of the IR region of QCD. This is an important point, because Bap(x) is not a gauge-invariant expression in QCD - but GM„(x) will induce another "local gauge invariance", that of the diffeomorphisms, the covariance group of this effective strong gravity. This ensures that three-gluon, four-gluon, etc. exchanges will also preserve that invariance. Since, aside from Diff(4, R), Lorentz invariance has to subsist locally (in the tangent), this metric has to obey a riemannian constraint T>aGlll/=Q, where D a is the covariant derivative of the effective gravity, with the connection given by a Christoffel symbol constructed with this effective metric. As a result two features emerge, the first applying to hadron physics [ 5 ], the second to nuclear structure, We first summarize the conclusions regarding hadrons.
4. Hadrons An effective riemannian metric induces the corresponding einsteinian dynamics. However, as it is the strong coupling region of QCD that is represented by this geometry, one has to add the high-energy sector of this effective gravity, i.e. the curvature-quadratic quantum counterterms that will be generated by the renormalization procedure. The result is an invariant action in which the Einstein-like lagrangian R is accompanied by a parametrized combination of the allowed quadratic terms I,m = -^A*x%f^G(aRlirR'",-PR2
+ yK-2R) .
(1) This is the Stelle [7] action, when usedfor true gravity. It was shown by Stelle to be renormalizable, a feature befitting the present application, since QCD is renormalizable and a piece of it should preserve the finiteness feature; but (1) is not unitary, which also befits this application: a "piece" of QCD should not be unitary, considering that QCD is an irreducible
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theory. Stelle's main result, however, was to show that renormalizability is caused by 1 /p4 propagators. But lip4 propagators are dynamically equivalent to confinement! In the applications of QCD to the IR region, such propagators were always introduced as an ad hoc assumption (note that there is as yet no definitive proof of color confinement, except in a lattice approximation of spacetime). Here, we derive them from (1). Moreover, it has been shown [8 ] that the presence of the quadratic terms in (1) induces a potential ~\/r+r2. It is important to remember that 1 /p4 propagators do not represent a Klein-Gordon equation; the direct linkage between the mass of the exchanged meson and the range of the force it propagates, a concept heuristically derived from the Yukawa potential, this linkage is completely lost. At most, a 1 /p4 propagator can be regarded as the difference between two Yukawa forces, the one due to a particle and the other to a ghost. We emphasize this point because the range of our effective strong gravity with its 1 /p4 propagator is not given by the mass of a hypothetical on-massshell "glueball", also made of two gluons mainly. Such a glueball - if it exists - will add a small contribution to the exchange of the/ 0 . In our case we shall have to use other means to estimate the range of the effective strong gravity. Another application of our ansatz to particle physics is an understanding of the successful application of the quantum string to hadron physics in 19671973. We now know the closed string to contain quantum gravity; since hadron physics emulates gravity in a certain sector, no surprise that the string should have provided a good qualitative picture, at least - including flux-tubes as actual strings. The K~2 coupling of the R term in true gravity is Newton's constant. In the present context of an effective analog to gravity, the string tells us that the relevant K~2 is given by the slopes of Regge trajectories in particle physics, about (1 GeV)2 per unit of spin. Such trajectories are also predicted directly by the present ansatz, when the GL(4, R) effective gravity algebra that we discuss in the next paragraphs, in the context of nuclei, is used as a Spectrum Generating Algebra.
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5. Nuclei It is the physics in this (strong coupling) IR region that is also relevant to nuclei: it is the non-perturbative nature of QCD in this region which causes the difficulties in both hadron and nuclear physics. Now note that out of the ten components of (/„„ the six that survive the four riemannian constraints have spin/parity assignments Jp=0+, 2+. It is mainly this feature, suggesting a relationship with the IBM systematics, that we apply in the present work. Algebraically, G^ix) carries the ten-dimensional (non-unitary) irreducible representation of GL(4, R). In true gravity, this is a geometric group, the linear subgroup of the co variance group Diff (4, R). Here it is a dynamical construct, except for the geometric Lorentz subgroup. The non-relativistic subgroup of SL(4, R) (the traceless piece, whose algebra includes shears, aside from Lorentz transformations) is SL( 3, R). Under this group, the 0 + and 2 + states span together one irreducible six-dimensional representation. The couplings to the "effective" gravity are given by the SL(4, R) group; they will thus be SL(3, R) invariant. There is thus full justification, in this picture, for the IBM postulate of a U(6) symmetry between the defining states! Note also that this SL( 3, R), obtained from the basic QCD fields, takes on a geometric interpretation, once we use GMV as a formal metric field. In that picture, SL(3, R) predicts the conservation of three-volume, i.e. incompressibility, for nuclei and hadrons (where it justifies the "bag" model as an approximation of hadron dynamics). When applying "effective gravity" to nuclei, it is natural to assume that closed shells assume the role of "vacua", as rigid structures. "Graviton" excitations should then be searched for in the valence nucleon systematics. In this sector of even-even nuclei the GM, quanta can indeed excite nucleon pairs; the overwhelming preponderance of proton-proton and neutron-neutron over proton-neutron pairs can be fully explained in terms of the Clebsch-Gordan coefficients in the direct channel. Dynamically for one pair, we assume that the pairing force itself is due to the exchange of a "strong graviton" between the two nucleons. The paired system then displays further excited states with the absorption of additional such quanta. The picture now is of an external field supplying these quanta, perhaps like the role of the elec-
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tromagnetic field in the hydrogen atom in the Schrodinger equation treatment. It is thus natural that proton pairs and neutron pairs should have the same energy difference between 0 + and 2 + , since these are due to the same flavor-independent component of QCD - precisely for the same reason that the Eightfold Way (flavor) SU (3) invariance is due to the flavor-independence of QCD. In estimating the amplitude for such pair excitations, we note that the gluonium exchange here is dominated by a pole in the direct channel at k4=4m2, m the nucleon mass. Remembering that "effective strong graviton" excitations are also seen in hadrons, where they cause the A 7= 2 sequences of resonances along Regge trajectories, we get a value for the effective 1 /K2 in eq. (1). It is given by the slope of the Regge trajectories, roughly 1/K 2 ~ 1 (GeV)2. We now return to the hamiltonian corresponding to eq. (1), as translated into our effective gluonium dynamics. The curvature R corresponds in true gravity to terms dP+P-P, the Christoffel formula gives r ~ G - ' 3 G a n d / ? ~ a ( G - ' a G ) + (C-'eG) (G-'6G). Here G stands for G^, b and b + represent the destruction and creation of a six-dimensional "strong graviton" quantum, H=jp$dk{Cdk2/K2)(b 2
2
+ C2{k /K )(b 4
+
4
+
4
+
+Alk (b +A2k (b +A3k (b
+
+
-b){b
+
-b)(b
+
-b)
-b)(b
+
-b)(b
-b)
+
-b)(b + -b)(b
-b)
+
-b)(b
values are roughly in the right ballpark. The values will decrease for larger M. Our (2) is of course equivalent to the IBM hamiltonian with higher order terms.
6. Symmetries of deformed nuclei Now to the system of quadrupole-generated symmetries [9] of strongly deformed nuclei, closing on Sp (3, R). The components of G^„(x) are abelian, i.e. [GM„ Gpa] =0. Algebraically, at the classical level, GMP and the Lorentz group generators form the algebra of Ti0(ff)SO( 1, 3), and inhomogeneous Lorentz group with ten tensor "translations" ((a) stands for a semidirect product). In the quantum case, we can write, GMV= T^„+ UM„ where TM, = n.t j" d £ [ a % + (k) a\+(k)
+
-b)}.
(2)
The coefficients C; and A, respectively contain dynamical information relating to the y and a, fi terms in (1), an approximation for the non-linear effect of the s/—G, the reduced matrix element for the coupling to the nucleon pair, etc. M is a mass parameter that takes care of the dimensionality. We select M to be of the order of the impacted system, i.e. the valence nucleons, M~ 20 GeV. For k2 we use the dispersion relation result mentioned above, i.e. ~4 (GeV)2. Using our string-Regge result (1 /K2 ) ~ 1 (GeV) 2 , and assuming the C coefficients to be of the order of unity, we get for the first terms
478
exp(2ifcc)
+ a%(/c) <**„(&) exp(-2ifcx)]
(3)
and Ultp = r]ab\dk~[ct% + + a%(k)a\
-b)
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+
(k)abAk)
(k)].
(4)
This time we use the creation and annihilation operators a% + , aby of the QCD gluon itself, which can be regarded somewhat like a tetrad field with respect to G„„ as a metric. For this to fit the formalism, we have to separate out the "rigid" piece (analogous to e'f,=d'f, + h '„ in the tetrad case). Here this is the "flat connection" N"^ i.e. the zero-mode of the field. We then use the canonical transformation: [a%(*) + lJV% exp(iAa)]-»o%(fc),
(5a)
[a% + (fc) + iJV%exp(-ifcc)]->aV(fc) • ( 5 b ) Using [a"M(k), a*„ + (k')]=5ab6^8(k-k'), one verifies that the operators T„„ and U„„ together with the operators S„„ = i u j d £ [ a V ( * ) «»,(*) -a\(k)ab„+(k))
(6)
close respectively on the algebras of GL(4, R) and
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U( 1, 3). Note that the largest (linearly realized) algebra with generators quadratic in the aM+, a„ operators is the algebra of Sp(4, R). This algebra contains both previous ones:
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Sp(3, R) generators of Rowe [9], as expressed in terms of shell-model harmonic oscillators. References
Sp(4,R).SO(l,3)
»U(1,3) -SU(1,3) »GL(4,R) -*SL(4,R) >T10(ff)SO(l, 3)-T,((7)SO(l, 3)-»
[ 1 ] A. Arima and F. Iachello, Phys. Rev. Lett. 35 (1975) 1069; Ann. Phys. 99 (1976) 253; 111 (1978) 201; 123 (1979) 468. [2]D.H. Feng and R. Gilmore, in: Dynamical groups and spectrum generating algebras, eds. A. Bohm, Y. Ne'eman and A.O. Barut (World Scientific, Singapore, 1989) p. 209. [3] See e.g. C.W. Misner, K..S. Thome and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973) §36.1. [4] C.J. Isham, A. Salam and J. Strathdee, Phys. Rev. D 8 (1973) 2600; D 9 (1974) 1702; Lett. Nuovo Cimento Ser. 3,1 (1972)969. [ 5 ] Dj. Sijacki and Y. Ne'eman, Phys. Lett. B 247 (1990) 571. [6 ] See e.g. S.L. Adler, Rev. Mod. Phys. 54 (1982) 729. [7] K.S. Stelle, Phys. Rev. D 16 (1977) 953. [ 8 ] P. Baeckler and F.W. Hehl, in: From SU (3) to gravity, eds. E. Gotsman and G. Tauber (Cambridge U.P. Cambridge, 1985) p. 341. [9] S. Goshen and H.J. Lipkin, Ann. Phys. 6 (1959) 301; D.J. Rowe, in: Dynamical groups and spectrum generating algebras, eds. A. Bohm, Y. Ne'eman and A.O. Barut (World Scientific, Singapore, 1989) p. 287; G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 47 (1981) 223; J.P. Draayer and K.J. Weeks, Phys. Rev. Lett. 51 (1983) 1422. [ 10] Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 157 (1985) 267, Phys. Rev. D 367 (1988) 3267.
(7)
The GL(4, R) algebra represents a spectrum-generating algebra for the set of hadron states of a given flavor [10]. In the case of U(l, 3), when selecting a time-like vector (for massive states), the stability subgroup is U(3), a compact group with finite representations - as against the non-compact SL(3, R) for SL(4, R). This fits with a situation in nuclei in which the symmetries are physically realized over pairs of "valency" nucleons outside of closed shells, as in the case of IBM: there is a finite number of such pairs, and the excitations thus fit within finite representations. In nuclei the relevant symmetries in (7) correspond to the nonrelativistic subgroups, i.e. to the Sp(3,R) and the related T5(o-)SO( 3), SL(3,R) and SU(3). Moreover, averaging over the color SU(3) degrees of freedom, which are summed over anyhow, in (4)-(6), and integrating over the d£, we get the
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Physics Letters B 276 (1992) 173-178 North-Holland
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Proof of pseudo-gravity as QCD approximation for the hadron IR region and J~M2 Regge trajectories 1A3
Y. Ne'eman
Stickler Faculty ofExact Sciences, Tel-Aviv University, Tel-Aviv, Israel
and Dj. Srjacki
M
Institute of Physics, P.O. Box 57, Belgrade, Yugoslavia Received I October 1991 We prove the approximation in which the IR region of QCD is dominated by the exchange of a two-gluon effective metric-like field G„„(x) =/?22}£9*£ (i/^a colour-SU(3) metric), "gauging" pseudo-diffeomorphisms. We derive the equations of motion for the effective pseudo-gravity. Aside from yielding p ~" propagators, indicating confinement, we obtain linear J~ M2 Regge trajectories.
We have shown that this corresponds to p~A propagators, a feature considered in field theory as an indication of confinement (see, e.g., ref. [2]). Another result [3] is the emergence of the "IBM" [4] and other quadrupole-generated approximate symmetries in nuclei [5,6] and the SL(4, R) systematics in hadrons [7] - also geometrically related to confinement. In this context, we had originally identified the kinematics [8 ] but had had to assume an ad hoc ansatz for the SL(4, R) invariance of the related QCD contributions; we now have a dynamical justification for that ansatz. Various other gravity-like features, which had been assigned in the sixties [9] to the action of the f°( 1250; Jp=2+) - now considered as a "run-of-the-mill" quark-antiquark bound state - including an ad hoc "strong gravity" postulate [10], now derive naturally from QCD. As a matter of fact, we borrowed the name "strong gravity" from that hypothesis [10] and used it to describe our QCD-derived theory in our original presentation [1]. To avoid confusion, however, we shall hereafter replace it by "pseudo-gravity", to indicate that it is in the nature of a gravity-like component of a very different force.
1. QCD generated (pseudo-)gravity in hadrons The fact that hadrons in their strong interactions exhibit in the IR region features resembling gravity has been known since the sixties. For example, observe the ease with which dual models and the string, originally a theory of the hadrons and of the strong interactions - the first such theory reproducing the phenomenological Chew-Frautschi plot of linear Regge trajectories, abstracted from observations could be reinterpreted as a theory of gravity. We have recently suggested [ 1 ] that this feature is due to a component of QCD in the interaction between (zerocolour) hadrons, namely the exchange of a two-gluon effective gravity-like "pseudo-metric" field (t]ab a colour-SU(3) metric, B%(x) thegluon), GvAx)=B%Bbl,r]ljb.
(1)
Supported in part by the USA-Israel BNSF, Contract 8700009/1, and by the FDR-Israel GIF, Contract 1-52.212.7/ 87. Also on leave from Center for Particle Theory, University of Texas, Austin, TX 78712, USA and supported in part by the USA DOE Grant DE-FG05-85 ER 40200. Wolfson chair Extraordinary in Theoretical Physics. Supported in part by the Science Foundation (Belgrade).
In the present work, we first provide a precise definition of the approximation of QCD in the hadron
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IR region by the action of that pseudo-metric. We then show that the equations of motion for this (effective) pseudo-gravity, aside from yielding p - 4 propagators, as proven in ref. [ 1 ], also generate J~ M2 linear Regge trajectories (see, e.g., ref. [ 11 ]). We know of no other direct derivation of both of these features from an approximation of QCD. Note that the dynamics conjectured in the sixties, namely Yukawa exchanges of pions etc., could not reproduce linear J~MZ trajectories; this later was the main success ushered in by the Veneziano model [12].
2. Local pseudo-diffeomorphisms as the gauge group In ref. [ 1 ] we defined the two-gluon effective gravity-like "potential" of eq. (1), with the SU(3)colour gluon,
B%=N%+Al, Wt-^Nl^fLN^N^.
(2)
Af£ is thus the constant component, yielding a vanishing field strength. GMV acts as a "pseudo-metric" field, {.passively) gauging effective "pseudo-diffeomorphisms", just as is done by the physical Einstein metricfieldfor the "true" diffeomorphisms of the covariance group. This can be seen by evaluating the variation of the pseudo-metric under SU (3 )coi0Ur; the homogeneous SU (3) variation 5t naturally cancels for this SU(3) scalar product, but the gradient term survives
6 February 1992
As a result, Guv(x) transforms as a world tensor, for "artificial" local pseudo-diffeomorphisms £„. It was shown long ago [13] that any spin-2fieldwill behave and couple like the graviton; it will stay massless because of Lorentz invariance, conservation of the energy-momentum tensor and Einstein covariance relating to these pseudo-diffeomorphisms. Thus, the local SU(3) colour gauge variations contain a subsystem ensuring that the G„„ di-gluon indeed acts as a "pseudo-metric" field, precisely emulating gravity. This provides proof and precise limitations for our original conjecture [ 1,3]. Note, however, that in ref. [1 ] we used the term "gluonium" for G„,{x) - an unfortunate choice, since it evokes the idea of a (massive) bound state; various theoretical estimates actually show that such a resonance would have a ~ 1.5 GeV mass. The dynamical role of that resonance represents a Yukawa short-range exchange force. This has nothing to do with the action ofG^ix), an effective potential whose action resembles that of gravity, i.e. of a long range force. We showed that the dynamically dominating terms in the lagrangian are those, quadratic in curvature, generated as counterterms by the renormalization procedure. These terms create the p~* propagators [ 14]. G^x) isa riemannian metric, because it preserves the Lorentz group, so that D„G„„ = 0.
(4)
3. Metric-affine formalism For the di-gluon we thus have &
Wlai,iNl+A'M)(Ni+Ab,)}
= tiab(^eaN»+N''ufcb
+ dlle<'Ai+Aal,d„eb).
The terms involving the constant JVJ, JV J can be rewritten in terms of effective pseudo-diffeomorphisms, defined by ^nabeaNi,
8 e G^ = 9 , 6 + 8 ^ „ .
(3)
As to the terms in /1£, ,4 J, integration by parts yields
But taking Fourier transforms, i.e. the matrix elements for these gluon fluctuations, we find that these terms are precisely those that vanish in the IR region. 174
481
The effective lagrangian for this (IR) region in QCD can be written as L=LMCF, 6V, e, r)+Le(e, de, r, dr). LM is the matter lagrangian and is given in terms of matter manifields ¥ [7,15] and their gradients dY, and of the "pseudo-tetrad" e%(x) and "pseudo-connection" rABft{x) derived from G^(x). L% is the pseudo-gravitational lagrangian. This L is written in a Palatini "first-order" formalism, with formally independent tetrad (or metric) and connection. The manifield V(x) is an infinite-component spinorial field [7,15], i.e. an 'W-deunitarized" [ 16 ] infinite representation of SL(4, R), the double-covering of SL(4, R). One way of "physically" generating such a field is through the Salam-Strathdee [17]
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Volume 276, number 1,2
mechanism. These authors showed how a non-linear realization of a group G over linear representations of its subgroup H<=G could be replaced by a linear representation of G through an iterative application of the realizer-field creation operators. In our case, we may start with y/(x) and
= exp{G"'«[r^:sl(4,R)/so(3, l)]}y>,
(7b) i.e. with both symmetric (AB) and antisymmetric [AB] pairs,
A
8if,a)Y=[-t (x)QA-a B(x)
B
VA ]Y.
rABMeB„-(fi~v),
RAM. = dMeA
A
fi BM, = d,.r
c
A
Br+r BMr c,-<jt~i>).
(7a) (7b)
The Noether currents resulting from this SA(4, R) in variance are the energy-momentum (8a) and hypermomentum (8b), 6A»-T"
1 57_M
e=det(e"%) ,
e be bU
—-
(8a) (8b)
Bn
(5)
where Si is a representation and T^ are the nine traceless shear generators, symmetric in {(i,v). Aside from T / , the 15 si (4, IR) generators K,/ include the Lorentz algebra MJ, antisymmetric in [ji, v]. The second manifield, 0(x)=L(Gflv) fl(x) is a (tensorial) boson manifield whose lowest SO (3, 1) submultipletis (5, | ) . In the absence of gravity (here pseudo-gravity), the matter lagrangian would be LM = 'PiX>'cl.Y+ d"0 6„0, invariant under global SL(4, R), as would be true for any tensor field by construction. In ref. [ 7 ] we have constructed the equations obeyed by Y and 0 (see also ref. [ 18]). The Hilbert spaces of Y and 0 are given by the representations of SA(4, R) [ 8 ]. In our LM we have dropped the boson manifield 0(x) for the sake of simplicity. (Pseudo-) gravity enters through the replacement O^D,,, where the index "A" denotes a local frame: £>^=dM—rAB„VAB, t)A = fSA*rbll with rthe connection and fiA"• e„B=8AB, e the pseudo-gravity tetrad; VAB is the si(4, R) algebraic generator in the tangent frame. We use D for the full covariant derivative with si (4, R) connection, e^^x) and rABll(x) can be taken as gauge fields for§A(4, R): A
6 February 1992
with the symmetric (AB) pairs in (8b) denoting shear currents and the antisymmetric pairs [AB] representing angular momentum. We refer the reader to ref. [19] for a discussion of shear currents; they can be orbital, e.g. time derivatives of gravitational quadrupoles [ 5 ] or intrinsic, when matter is organized in manifields. The shear currents are not conserved in Minkowski space, and Einstein's theory does not preserve a local SL(4, R) acting on the tetrad frames, as in (6). Here, however, we choose to start with a more general approach, provided by metric-affine gravity [19,20]. Variation with respect to eA M and rA B>1 yields the two equations
Y>vnA^-EA"=eeA'', dd„eA
hvnA
67?
(9a) Vft
eTAB9L„ 66„r B ,
nA
6L0
(9b)
AVfi
where the 77are the canonically conjugated momenta and the E represent the gravitational contributions to the momentum and hypermomentum current tensor densities, EA " = eA "Lfi-RBAMB A
A
E B"=e „nB"».
•",-RBCA*nCB '" ,
(10a) (10b)
All expressions can be rewritten holonomically, i.e. with "curved space" indices//, v,... only, i.e. in terms of the fields
(6)
As in gravity, the corresponding field strengths are the torsion (7a) and the (generalized) curvature
•-*\ABe"
(11) 175
482
Volume 276, number 1,2
r\v=aA°e\rA
PHYSICS LETTERS B
Bn •
e= v /-det(<7^„)
(11 cont'd)
For our considerations, it is more convenient to use as independent dynamical variables the equivalent set ( 5 is Cartan's torsion tensor, the antisymmetric part of T; Q is the non-metricity tensor)
o__pa
o
—fS n
instead of the (G^„, f^) set. The linear connection can be expressed in terms of these variables as pa
L
_
IflavAixPy
fjtp — 2 V J
ppv
(13a)
X(9 a G^),— S CtfiGyg+Qafiy) , a
a
1
a
a
A "'piI1,=S l,d' ltd\+d tld^d%-d vS%6\ (13b) and explicitly, ^
fiV — \fi
v \
' 2
W//I*
6 „
j, + O
iip )
(13c) {„"„} is the Christoffel symbol. The 64 components of r are replaced by 24 and 40 in 5 and Q respectively; the new variables are tensor quantities. The most general first-order gauge lagrangian for affine gravity can thus be written as Ls =
La{G,dG)+Ls(S,dS)+LQ(Q,dQ)
(14)
Note that the simplest such lagrangian is the GL(4, R) or SL (4, R) scalar curvature tensor. If differs from Einstein's in that the original curvature 2-form R^A B (prior to contraction) has all 16 (or 15forSL(4, R)) combinations, whereas Einstein's has only the 6 antisymmetric [AB]. Since R = dT+ J [J", r], we see from (13c) that this will contain terms such as { } 2 , S2 and Q2. Variation of the total lagrangian L=La + LS+LQ+LM with respect to G, S and Q yields the equations of motion,
5G„
5L„ 8GU
"+P" ff ),
(15b)
= VCG(Jff'"'-J''(7''-rJ'"'ff),
(15c)
SO*.'
8(2^"
&"" is the symmetrized energy-momentum tensor, 27,/'' and 4 , " " are the spin and shear currents, antisymmetric and symmetric in ^«-> v respectively.
where p
6 February 1992
= J-G9""
(15a)
176
483
4. Equations of motion for the effective (pseudo)gravity At this stage we return to our hadronic theory and apply this formalism to that sector in the IR region of QCD. Here we have only one propagating field, namely G„„. We are thus led to a situation analogous to the Einstein-Cartan theory, in which spinor fields contribute to the torsion currents, but there is no propagating torsion piece of the connection (13c), independent of G^. Torsion, and here non-metricity as well, exist pointwise in the matter distribution, but do not propagate from it through the vacuum as a free wave or via any interaction of non-vanishing range. The pseudo-gravity lagrangian (14) therefore does not contain dS and dQ. Eqs. (15b), (15c) thus become purely algebraic equations, relating torsion to spin and non-metricity to shear with some proportionality factors given by the couplings in Ls and LQ: o
a y
aiy
a
• ya
Q
°~A
" — A " + A"
(16)
To simplify the expressions we replace S and Q by linear combinations S^ (the contortion tensor) and 2Mp"proportional to 1^" and JMP". Thus Ls+LQ
2 = l,s T*-iip '£""
,+
IQ2A» °A«\
(17)
As a matter of fact, selecting for L g the generalized SL (4, R) scalar curvature, and substituting as in (16) generates (17). This then becomes an effective addition to the matter energy momentum tensor - once we segregate the pure riemannian part of the Ricci tensor on the left-hand side of (9a) and thus move
Volume 276, number 1,2
PHYSICS LETTERS B
(17) to the right-hand side. After this rearrangement, we recover eq. (4). The effective action for this IR (zero-colour) hadron sector of QCD, written as a pseudo-gravitational theory, with matter in SL(4, R) manifields, then becomes / = \ d'xJ^G
-\lQ2Apn"A"\)Hlll,=0,
[\a(p2y-\ls2fsM,xM\ -i/eVoV7'Mfl/.»(/>)=0.
(18)
The first three terms constitute the Stelle lagrangian [14] and provide the p~* propagators. The fourth and fifth terms in (18) are spin-spin and shear-shear contact interaction terms. The a, b, c are dimensionless constants; la, ls and lQ have the dimensions of lengths; from our knowledge of the hadrons [ 1 ] we estimate them to be of hadron size ~ 1 GeV. Following Stelle [21 ] we obtain the field equation (; denotes D, i.e. purely with the { }), aR^".„-(a-2b)R.,KV +
{\a-2b){R„-{RGll¥)R>''.n
~2aR" Rm„, + 2bRRtiV
x
- / ( / + 1)-C 2 1 ( 3 , B ) ,
-clo^R^-iRG^) = {0^ .
1) ,
(22a)
j
Tn T-x-*T, 'T'j^M, JM j - CI (3,R,
$(Rliv-LRG^)(aR«iR,x-bR2)
- \Z„XZ"\G^-H„XJ"\GU„
(21)
For pseudo-gravity, we may regard (20) and (21) as the dynamical equations above the theory's "vacuum", as represented by hadron matter itself. The equations represent the excitations produced over that ground state by the pseudo-gravity potential. In these expressions, we have factored out the SL(4, R) group factors (the bilinear forms in the algebra's generators). The factors/^ andfQ represent the residual part of the configuration space integrals. Eq. (20) is like an equation for the //„„(.*) field in an external field of hadronic matter. Previously [8], we showed that the rest frame (stability) "little" group is SL(3, R) <= SL(4, R). Taking a hadron's rest frame (U=l,2,3) Af„ i M" x ^M i J M i J ^J(J+
x
+
(20)
which becomes in momentum space
(-aRtil/R>"' + bR2-cld2R
+ /j%„*r*" f f + /e J zU»d'<''
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(19)
It is the IR region of QCD for which we are using pseudo-gravity. This is the strong coupling limit of QCD, corresponding to low energies and "large" distances. Thus, the corresponding sector in a gravitylike theory has to be its high energy limit; for true gravity, we know that the weak Newtonian coupling of macroscopic physics reaches in the Planck region strong interaction strength. Thus, our correspondence is between the low energy IR sector of QCD and the UV strong coupling sector of pseudo-gravity. In eq. (19), this implies dominance by the R^R"" and R2 terms and we can neglect the R term (riemannian, with only { } as connection). We linearize our theory in terms of Hli„(x) = Gfl„(x) — r\^, where n^ is the Minkowski metric. Taking just the homogeneous part, as required for the evaluation of the propagator, we get for the H^ field the equation of motion
(22b)
where C2 is the si (3, R) quadratic invariant. As a result, we find that in a rest frame, all hadronic states belonging to a single SL(3, R) (unitary) irreducible representation (i.e. one value of Cl\iw) lie on a single trajectory in the Chew-Frautschi plane, i.e. (J+{)2=(a'm2)2 {cf)2=[{2/a)Vs2fs
+ ai, + lQ2fQ)]->,
a
o = T + 7_2 r , i-2 r ^si(3,n) • *
(23a) (23b) (23c)
's Js~r'Q JQ
a' is the (asymptotic) trajectory slope. Reality of a' requires^/a>0 and/g/a>0. For J = 0 eq. (23) implies: (mj=0)2=-(2/a)lQ2fQC2HX„). Thus eq. (23) is meaningful only if C2i(3,R) <0. Indeed, this is the case for the (relevant) unitary irreducible SL(3, R) representations (cf. eqs. (6.3), (6.4) and (8.7) of ref. [22 ]). Neglecting a slight bending at small m2, i.e. the a2, term, we finally obtain the linear Regge trajectory 177
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(24)
6 February 1992
References
5. Conclusions As derived here from QCD-generated pseudogravity, the J~M2 linear Regge trajectories embody the spin-2 nature of the effective di-gluon, dominating the IR sector of hadron interactions. The (M2)2 (like the p - 4 propagators indicating confinement) results from the typical (curvature)2 lagrangian of a gauge theory, when the connection r is replaced by the ~G~'bG affine connection {} of a riemannian spin-2 theory. The J2 results from the direct substitution, in the linear curvature scalar (characterizing spin-2 gauge theories) of non-propagating torsion by spin, through the algebraic Cartan equation of a riemannian theory (higher powers of the curvature will yield corrections proportional to JA, etc.). This then fixes J2~ (M 2 ) 2 . Long ago [5], we noted the existence of a link between Regge trajectories and what we then thought was plain gravity; we can now conclude that these moments of inertia become relevant to strong interactions because QCD emulates gravity in this sector. In nuclei [ 3 ], the missing dipolar excitations (see, e.g., ref. [23]), the 2 + + 0 + ground state of the IBM symmetry [4], the quadrupolar nature of the SL(3,IR)( SU(3) andEucl(3) sequences [6] - all of these features again characterize the action of a gravity-like spin-2 effective gauge field. Overall, the evidence for the existence of such an effective component in QCD seems overwhelming. No surprise, therefore, that the string should fit both true gravity and the hadrons with their strong interactions [12,24].
178
485
[ 1 ] Dj. Sijacki and Y. Ne'eman, Phys. Lett. B 247 (1990) 571. [2] J. Kiskis, Phys. Rev. D 11 (1975) 2178; G.B. West, Phys. Lett. B 115 (1982) 468. [3]Dj. Sijacki and Y. Ne'eman, Phys. Lett.B250 (1990) 1. [4] A. Arima and F. Iachello, Phys. Rev. Lett. 35 (1975) 1069. [5] Y. Dothan, M. Gell-Mann and Y. Ne'eman, Phys. Lett. 17 (1965) 148. [6] L. Weaver and L.C. Biedenharn, Phys. Lett. B 32 (1970) 326; J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128, 562; G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 47 (1981) 223; J.P. Draayer and K.J. Weeks, Phys. Rev. Lett. 51 (1983) 1422. [7] Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 157 (1985) 267. [8] Y. Ne'eman andDj. Sijacki, Phys. Rev. D 37 (1988) 3267. [9] P.G.O. Freund, Phys. Lett. 2 (1962) 136; M. Gell-Mann, Phys. Rev. 125 (1962) 1067, footnote 38. [10] C.J. Isham, A. Salam and J. Strathdee, Phys. Rev. D 8 (1973)2600. [11] S.C. Frautschi, Regge poles and S-matrix theory (Benjamin, New York, 1964). [ 12 ] G. Veneziano, Lett. Nuovo Cimento A 57 (1968) 190. [ 13] S. Weinberg, Phys. Rev. B 138 (1965) 988. [14] K.S. Stelle, Phys. Rev. D 16 (1977) 953. [ 15 ] Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 157 (1985) 275. [ 16] Dj. Sijacki and Y. Ne'eman, J. Math. Phys. 16 (1985) 2457. [17] C.J. Isham, A. Salam and J. Strathdee, Phys. Rev. 184 (1969) 1750. [ 18 ] A. Cant and Y. Ne'eman, J. Math. Phys. 26 (1985) 3180. [19] F.W. Hehl, E.A. Lord and Y. Ne'eman, Phys. Rev. D 17 (1978)428. [20] F.W. Hehl, G.D. Kerlick and P. v.d. Heyde, Phys. Lett. B 63(1976)446. [21 ] K.S. Stelle, Gen. Rel. Grav. 9 (1978) 353. [22] Dj. Sijacki, J. Math. Phys. 16 (1975)298. [23] C.W. Misner, K.S. Thome and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973) sect. 36.1. [24] G. Cohen-Tannoudji and N. Zenine, Z. Phys. C 49 (1991) 159.
Wigner analysis and Casimir operators of SA(4,A?) Jiirgen Lemke,a),b) Yuval Ne'eman, c)d) and Jose Pecina-Cruz Center for Particle Physics, University of Texas, Austin, Texas (Received 25 February 1992; accepted for publication 2 March 1992) In theories involving gravity, including QCD-generated gravitylike effects in hadrons, SA(4,/?) plays a role. Its single Casimir invariant and that of its SA(2,R) and SA(3,.R) subgroups are evaluated. The group orbits are studied and the unitary irreducible representations are classified.
I. INTRODUCTION The affine groups, both the general affine GA(4,R) and its unimodular ("special") subgroup SA(4,iQ, with their double-covering groups GL(4,/J) and SL(4,i?) appear as symmetries of the spectrum of particle states in various gravity-related theories. The following list is not exhaustive. (a) Theories in which space-time is no more Riemannian, above Planck energies.1 In such theories, the primordial local symmetry is either the conformal group or its homothecy subgroup, i.e., the Poincare group combined with dilations, or alternatively, GL(4,/J) (which also includes dilations) or its SL(4,iJ) subgroup (excluding the dilations). Here, we are interested in the latter case. The fields then carry nonunitary representations of 5^(4,7?) and the particle Hilbert space is that of SA(4,/{). Under spontaneous symmetry breakdown, the local gauge group reduces to the Lorentz group and the Hilbert space becomes that of the Poincare group. Similar situations arise in metric-amne theories of gravity.2 (b) Einsteinian gravity, when interacting with hadron matter, in a phenomenological description in which quarks and gluons are replaced by baryons and their excitations. Such a description3 involves manifields, i.e., deunitarized4 infinite-dimensional representations of SL(4,.R), the double covering of the special linear group. The Hilbert space here is then that of SA(4,.R). (c) This formalism can be extended (for any fields) to fit a semiquantized description for particles under the effect of gravity, i.e., particles in a curved space. The Hilbert space group is then defined by the group of diffeomorphisms, induced over SA(4,i?). (d) An approximation to QCD in the (confinement) IR region5 which emulates (Riemannian) gravity, with
applications in particle6 and nuclear7 physics. Here again matter is represented by SL(4,i?) manifields, with states classified by SA(4,«). The field-particle algebraic relationship follows the prescriptions of relativistic quantum field theory, which at the classical level, at least, contains the tools for a smooth transition to general relativity. The principle of covariance, for one, requires the fields to carry the action of the group of diffeomorphisms. This action will generally be represented nonlinearly, over the linear subgroup SL(4,i{) or over its double-covering group SL(4,i?). Therefore, even in special relativity, before the introduction of the gravitational field or of curved space, the fields carry nonunitary representations of SL(4,i?) D SO (1,3) or (for spinors) SO(l,3) = SL(2,C). The Hilbert particle space symmetry, on the other hand, is determined by the principle of equivalence, i.e., it is that of the special theory of relativity, i.e., the Poincare group & =_SL(2,C) X j^? 4 . Similarly, in the affine situation, when SL(4,.R) replaces SL(2,C), we obtain as Hilbert space of particle states that of SA(4,R)=SL(A,R)X^4
The elements of SA(4,iJ) are given by 5 X 5 matrices (the Mdbius representation): A=[
Pp
),
LeSh(4,R),
pctf*.
(2)
In a work treating the invariants of real lowdimensional Lie algebras, Patera et al.% evaluated the Casimir invariant of SA(2,/J) (named AS40 in their list). Defining the elements of the Lie algebra in the 3 x 3 matrix form,
Supported in part by the "Deutsch Akademischer Austauschdienst" and by a graduate scholarship of the Land Nordrhein-Westfalen. Permanent address: Institute for Theoretical Physics, University of Cologne, D-5000 Koln 41, Germany. c, Wolfson Chair Extraordinary in Theoretical Physics, Tel-Aviv University, Israel. •"Supported in part by the USA DOE Grant No. DE-FG05-85 ER 40200. b)
J. Math. Phys. 33 (8), August 1992
IL
II. THE CASIMIR INVARIANT OF SA(W,/7) AND THE GROUP ORBITS
a)
2656
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(3)
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© 1992 American Institute of Physics
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Lemke, Ne'eman, and Pecina-Cruz: Wigner analysis of SA(4,R)SA(4,R)
where, as displayed in the 3 x 3 matrix, les\(2,R) is a traceless 2 x 2 matrix and p is a column vector (the momenta). The Casimir invariant is quadratic in the translations/)' and is altogether of cubic order (we denote the dimensions by numbering them from 0 to TV— 1, in analogy with Minkowski space): C(2) = t(p0)2-rPy-s(p')2.
(4)
The basic advance in the study of the Casimir invariants of the affine and related groups followed the work of Sternberg.9 Rais, !0 Perroud," and Demichev and Nelipa12 showed finally that the SA(N,R) have a single such operator [and the GA(N,R) have none] which, using the Cartan-Weyl basis in the related gl(N,R),
(4)1=81^
From (4) we have C(2)=q(p0)2, the product of the squared "energy" by q, a fake translation momentum of SA(1,/J) and really a component of the SL(2,i?) shear, pointing in the 0 direction. If we now use the coadjoint matrix Ik
0 \ (7b)
(o k~)
to rescale the p° momentum by a factor A, we see that the q will be rescaled by a factor k~2, thus preserving the invariance of the Casimir operator.
(5a)
is given by
III. THE PROJECTIVE REPRESENTATIONS AND COHOMOLOGY
(5b) which is equivalent to the determinant 2
N 1
C{N)=det(p,Ep,(E) p,--,(.E) - p)-
Orb^^—fO}.
The basic construction follows Wigner's 13 classical treatment of the Poincare group's Hilbert space and projective representations [for quantum mechanics (QM)]. Let H be a Hilbert space with scalar product (,) and H the corresponding projective Hilbert space, i.e., //:=[a¥|*e#,ae%?*].
(5c)
This involves powers of E [the basis in the related gl(N,R) algebra] going from 0 to TV— 1. Thus C(TV) is a polynomial of degree TV in the translations p and of degree TV(TV—1)/2 in the s\{N,R) generators; altogether it is thus of degree TV(TV+1 )/2. The expression (5b) or (5c) automatically takes care of the tracelessness of the s\(N,R) generators, i.e., the diagonal generators appear in combinations EJ—E\. The group SA(N,R), acting on the space of momenta, has two orbits: Orb, = {0},
C !)•
(6)
For the null orbit, i.e., when we select states for which all TV components of the momenta vanish, the Casimir invariant vanishes, since it is a homogeneous symmetric polynomial of degree TV in the momenta. In the second orbit [which, incidentally is invariant under the entire GL(N,R)], for small values of the momenta, the invariance of the Casimir operator implies that the eigenvalues of the SL(N,R) homogeneous operators must grow fast. In an example treating SA(2,R) and due to Sternberg, putting the nonvanishing TV-vector p at rest (i.e., />°=£0,/>'=0), the "little group" (the stability subgroup) consists of matrices
J. Math. Phys., Vol. 33,
(8)
Measurable quantities, in QM, are invariant under phase transformations. Physical systems are therefore described by elements *efiT, the "ray" representations. For the same reason, should the group acting on space-time possess a double-covering group, the latter may act as an isomorphism (up to a phase) on the representations [e.g., Spin (TV) instead of just SO (TV), the geometrical orthogonal group]. An SA(4,/?) transformation A on space-time (in the above sense) induces a transformation p(A):H->H. This has to be an element of the set U(H) of unitary operators on H, as long as SA(4,/J) is assumed to be a symmetry of the system. Each homomorphism p from SA(4,i?) to U(H) gives rise to a projective representation ir(p) = p . In the opposite sense, according to Wigner's theorem, each projective representation p of SA(N,R) can be obtained from a representation p of a group G, i.e., we can find a group G and homomorphisms p. and p such that the following diagram is commutative and that both sequences are exact sequences: 1
_
K
1
_
£/(i)
G
i
SA(N,R)
Z
U{H)
Pi
i. 8, August 1992
_
U(H)
-
1
-
1 (9)
IP
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Lemke, Ne'eman, and Pecina-Cruz: Wigner analysis of SA(4,R)SA(4,R)
providing an interesting model for primordial fermion fields (in fact manifields). This picture has been studied in Refs. 3 and 20. It fits all applications mentioned in our introductory comments. Note that C ( 3 ' ) = 0 , and as a result C(4) = 0 as well, since the multiplyer of (p 0 ) 4 is precisely the C(3') Casimir invariant of the stability subgroup defined by p°. Case IIB. The fake momenta p'=/=0. We can select a frame in which only p'° does not vanish, a fake energylike component. C(3') ~ (p° ) 3 = (m')3, m' a mass-like eigenvalue. The new little group is SA(2,i?)". Again, the "translations" are fake momenta p". We can have two cases. Casell B 1. All components of p" = 0 and C(2") = 0 . In that case, we get again both C(3") = 0 and C(4) = 0 . The effective little group is SL{2,R) (i.e., the double-covering, in an infinitely covered group). The unirreps have been classified by Bargmann 15 and are useful in a variety of physical contexts. Case IIB 2. p"=£0, C(2") ~ (p")2= ( m " ) 2 . The little group is SA(1,.R) as in (7a), with one fake momentum p'". Again we have two possibilities. Case IIB 2a. p'"=0, C ( 1 " ' ) = 0 . This is a scalar representation. As a result, C(2") = C(3') = C ( 4 ) = 0 . Case IIB2b. />'"=#), C(l'")=q = m'" [see (7a)]. Note that here C(2") = ( m " ) 2 m " \ C(3') 2 = (m')\m") m'" and C(4) = ( m ) 4 ( m ' ) 3 ( m " ) 2 m " ' . To summarize, we have five classes of representations: I, II A, II B 1, II B 2a, and II B 2b, which are illustrated in the following diagram:
The groups G and K are determined by the second cohomology of the Lie algebra of SA(N,R), i.e., %"2(sa(N,R)).H For 7V= 1, we have <%*2(sn(N,R)) = {0} and G is the covering group of SA(1,.R), which is SA(l,i?) ~(R,+ ) itself. Hence G=SA(l,R) and K=\. For 7V=2, the two-form dtt dt2 is left-invariant and closed, but the one-forms f, dt2 or —t2 dtj are not leftinvariant. Thus dim J^ 2 (sa(2,i?)) = 1 and G is the central extension of the universal covering group of SA(2,R) (an infinite covering15) by 3). Moreover, the lift of a certain projective representation is uniquely determined, since JTl(s&(2,R)) = 0. For # = 3 , 4 we obtain JT 2 (sa ( # , £ ) ) = 0 and G is equal to the universal covering group 16 ' 17 of the group SA(N,R). For N=4 we_have K=-l,l, i.e., G is the double-covering group SA(4,J?). 4 IV. INDUCED REPRESENTATIONS The two orbits of (6) provide for a classification of the unitary irreducible representations (unirrep) of SA(4,/?). We have a hierarchy of stability subgroups over which the unirrep is constructed as an induced representation a la Wigner and Mackey. The four-vector p either vanishes, p=0 (case I) and C(4) = 0 or it doesn't, p=i±0 (case II) and C(4)~(/>°) 4 =m 4 . Case I. Physically, it is useful to think of this case as the very low frequency limit of a massless particle, with its Regge excitations. The little group is SL(4,.R). The unirreps of this group have been classified.4'18 They are rather unphysical in that the Lorentz subgroup will appear in unitary infinite representations, the unirreps of Gelfand and Yaglom.19 These contain all spins, and the action of the Lorentz boost on a state with spiny connects it with they'-t-1 andy — 1 spins. Particles here are thus not characterized by definite spins, as phenomenologically required. These representations are also known as "infinite spin" representations. Still, there are problems in physics in which the SO( 1,3) CSL(4,i?) is not the physical Lorentz group, and these unirreps may then prove useful. Note also that we do not encounter this difficulty with the fields and manifields, since these are constructed with the deunitarizing automorphism jrf.4 In a nonunitary and finite representation, the Lorentz boosts stay antiHermitean and cancel. Case II. The little group is SA(3,/?)'. This affine group consists of the semi-direct product of the spatial SL(3,/?) with a "fake" set of three "translation" momenta p', in fact representing contributions of the spatial shears to the 0 direction. We now have two subcases. Case II A. All three components p' = 0. The effective little group is then SL(3,i?). The unirreps are induced over this subgroup; they can be reduced to infinite discrete sums of spins, fitting the hadron situation and also
J. Math. Phys., Vol.
SA(A,R)
-
SL(4,R):
I
-
SL(3,/J):
HA
«-
SL(2,.R):
IIB\
t SA(3 ( /{)' t SA(2,i?)"
•
(10)
T SA(l,i?)
IIB2a,IIB2b
Moreover, C(4) = 0 ,
for
LII A,II B 1,11 B 2a;
C(4) = (m) 4 (m') 3 (m") 2 m'",
for
II B 2b.
(11)
V. DYNAMICAL CONSIDERATIONS At first sight, the Casimir invariant (5b) appears to constrain the masses and spins in a wrong manner, as in the Majorana19 infinite equation: the higher the spin, the lower the mass; this is the opposite of what we observe in hadron phenomenology and of what is assumed in the Chew-Frautschi plot for a Regge trajectory. However,
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Lemke, Ne'eman, and Pecina-Cruz: Wigner analysis of SA(4,R)SA(4,R)
considering that in the general case, including the most useful case I I A the invariant vanishes [as seen in (11)], the value of (m)4 stays unconstrained in all but case II B 2b. Instead, constraints on the value of the masses may be derived dynamically,5 rather than kinematically as in (11). It is remarkable that an evaluation based on the pseudo-gravity approximation for QCD in the IR region does reproduce the linear correlation between (m)2 and the spin j .
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446 (1976); F. W. Hehl, E. A. Lord, and Y. Ne'eman, ibid. 71, 432 (1977). Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 157, 267, 275 (1985). "Dj. Sijacki and Y. Newman, J. Math. Phys. 26, 2457 (1985). 5 Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 276, 173 (1992). 6 Dj. Sijacki and Y. Ne'eman, Phys. Lett. B 247, 571 (1990). 7 Dj. Sijacki and Y. Ne'eman, Phys. Lett. B 250, 1 (1990). 8 J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, J. Math. Phys. 17, 986 (1976). 9 S. Sternberg, Trans. Am. Math. Soc. 212, 113 (1975). °M. Rais, Ann. Inst. Fourier 28, 207 (1976). 'M. Perroud, J. Math. Phys. 24, 1381 (1983). 2 A. P. Demichev and N. F. Nelipa, Moscow Uni. Phys. Sem. 35, 6 (1980). 3 E. P. Wigner, Ann. Math. 40, 149 (1939). "V. Bargmann, Ann. Math. 59, 1 (1954). 5 V. Bargmann, Ann. Math. 48, 568 (1947). 6 Y. Ne'eman, Ann. Inst. Henri Poincare A 28, 369 (1978). 7 Dj. Sijacki, J. Math. Phys. 16, 298 (1975). 8 Dj. Sijacki, in Frontiers in Particle Physics 83 (World Scientific, Singapore, 1984). 9 E. Majorana, Nuovo Cimento 9, 335 (1932); I. M. Gelfand and A. M. Yaglom, JETP (Russian version) 18, 703, 1096, 1105 (1948). °Y. Ne'eman and Dj. Sijacki, Phys. Rev. D 37, 3267 (1988). 3
ACKNOWLEDGMENTS We are grateful to Prof. S. Sternberg for his advice and comments. We would also like to thank Prof. A. Joseph and J. Patera for their advice, Prof. A. Bohm for his support of JL at the University of Texas at Austin. JL is grateful to Prof. F. W. Hehl for his constant support and advice. 'Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 200, 489 (1988). F. W. Hehl, G. D. Kerlick, and P. von der Heyde, Phys. Lett. B 63,
2
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International Journal of Modern Physics A, Vol. 10, No. 30 (1995) 4399-4412 ©World Scientific Publishing Company
CHROMOGRAVITY: Q C D - I N D U C E D D I F F E O M O R P H I S M S
YUVAL NE'EMAN* Sackler Faculty of Exact Sciences, Tel-Aviv University, DJORDJE SIJACKlt Institute of Physics, PO Box 57, Belgrade,
69978 Tel-Aviv,
Israel
Yugoslavia
Received IS March 1995 We expand the QCD gluon field BJ around a constant "pure gauge" solution N£ in the IR sector, i.e. BJ(x) = N* + -AjKx), and provide a mathematical definition for an "IR limit" in which the frequencies of the fluctuating field A*(x) vanish. We prove that in this limit, SU(3) c o | o r gauge transformations become equivalent to space-time diffeomorphisms. A gravity-like contribution is then shown to emerge from the overall "n-gluon exchange" component in the expansion of the generating functional of QCD Green functions, with the two-gluon term acting like the metric field in gravity. This QCD-induced "chromogravity" provides an effective long range action, i.e. longer-ranged than the contribution of quark-antiquark (meson) exchanges. We conjecture chromogravity to be responsible for many of the features of the hadron spectrum and of color confinement, issues for which there is as yet no proof in QCD (including lattice calculations), beyond general qualitative arguments. The method exhibits a smooth transition to the perturbative and semiperturbative treatment of high energy hadron scattering, including the emergence of the Pomeranchuk trajectory.
1. I n t r o d u c t i o n The adoption of QCD and its incorporation in the Standard Model were the outcome of the success of asymptotic freedom (AF) in fitting the scaling results of the deep inelastic electron-nucleon scattering experiments (uthe quark parton model"). At the same time, there was the fact that color SU(3) could provide an explanation for the (otherwise) "paradoxical" key algebraic features of the successful nonrelativistic quark model (NRQM): "wrong" spin statistics of the baryon ground state [56 in SU(6)flavor> , p i„], plus zero triality of the entire SU(3)flavor (eightfold-way) physical spectrum. AF indeed also provides a successful perturbative treatment for the "ultraviolet" (UV) region, e.g. high energy electroweak hadronic interactions, corresponding to the current quark aspects of NRQM. There is also a fair understanding of hadronic strong interactions in the "hard" and •Wolfson Chair Extraordinary in Theoretical Physics, Tel-Aviv University. Also on leave from Center for Particle Physics, University of Texas, Austin, USA. ^Supported in part by the Science Foundation (Belgrade). Also supported by the Wolfson Chair Extraordinary in Theoretical Physics, Tel-Aviv University.
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"semihard" regimes, using the leading and double log approximations in perturbative QCD and Reggeized diagram techniques. 1 Nothing of the sort, however, has emerged to date in the "infrared" (IR) frequency antipodes. Any theory describing compositeness is expected to provide a "hydrogen atom" foundation, i.e. at least two basic results: (1) The evaluation of the energy (binding or resonating) of the simplest compound state, given the constituents' masses and spins and given the couplings (fixed or running) postulated at these constituents' level. (2) Evaluating the composition and energy levels of higher excitations of this system and of other analogous bound states. In QCD, the huge investment of talent and of man and computer years notwithstanding, nothing of the sort has been reached. With highly sophisticated use of enormous computing power, the best that has been achieved2 has been in the nature of a "correction" to the naive and most simplistic quark model. Given the (UV) masses of the u and d quarks (i.e. about 5 and 9 MeV respectively) and given as, i.e. "running" values of the coupling to gluons, one would have hoped to be able to reproduce the masses of the proton, neutron, pion, rho, plus the gNN* and 9NNP couplings. Instead, mp has had to be added to the input, to provide the scale of hadron masses (or of a constituent quark), the theory being as yet unable to provide this fundamental energy scale. Once this is done, the calculation indeed provides an improvement over simplistic quark counting. Instead of mnlmp = 3/2, one obtains values fitting observations to within a few percent. Alternatively, issues related to the IR region continue to be treated by using a combination of the constituent NRQM 3 and Regge systematics, 4 with the parameters again taken from experiment. 4,5 Both methodologies can be derived from QCD, provided one is situated in a very specific regime: heavy quarks, justifying static potentials for NRQM, and high energy scattering with certain approximations for Reggeism. There is no known way of associating them — in this QCD context — with the "soft" (IR) regime corresponding to constituent (nonheavy) quark physics. Other such approximations consist in more remotely related methods, such as the Skyrme-Witten, Kazakov-Migdal and other models, in which QCD is replaced by a scalar force and/or reduced dimensionalities — or on calculations based on lattice methods. The first can only reproduce general qualitative results, the latter are limited to the lowest energy levels.5 A major effort based on the application of lattice methods to a "light front" frame treatment has recently been launched by Wilson and collaborators, 6 in the hope of supplying an answer to the quest for a QCD derivation of the energy spectrum of hadrons (or of the constituent quark model). As for color confinement, very little has been achieved beyond Wilson's original demonstration in lattice QCD. The conjectured mechanism of "dual superconductivity" in which chromomagnetic monopoles are assumed to form
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Q CD-Induced Diffeomorphisms
4401
condensates and expel the chromoelectric field has still not been proven in QCD, where even monopoles have not been obtained as clean gauge-invariant solutions. True, a gauge in which the non-Abelian parts of SU(3) co i or are made to vanish — so that the gauge group SU(JV) is replaced by U(1)® JV_1 — does display appropriate U(l) monopoles, whose generating singularities must still exist somewhere, even after the gauge is modified when the non-Abelian part of the group is restored 7 (although we do not know whether these singularities preserve their character — as long as the relevant solutions have not been formulated in a gauge-invariant manner). Examples of confining theories with an explicit "dual superconductivity" mechanism (the monopole condensate providing a Higgs mode) have recently 8 been constructed. It is in view of this state of affairs that other algorithms have been suggested for the application of QCD to the IR region and to the hadron spectrum. The authors of this paper have conjectured9 (a) that gluon exchange forces (with the gluons in color-neutral combinations) make up an important component of interhadron interactions in the "softest" region (where they compete with qq meson exchanges) and in the intrahadron mechanism of color confinement; (b) that the physical role of this component is to produce a longer range force, with many of the characteristics of gravity, starting with the basic mathematical foundation, namely invariance under diffeomorphisms; (c) that the geometric features which characterize gravity-like forces in general, in this specific case play a key role in the color confinement mechanism. The basic statement made in Refs. 9 and 10 relates to the simplest n-gluon exchange, namely that of the two-gluon system G,„,(x) = {K)-2gabBl(x)Bbv{x)
(1.1)
[K has the dimensions of mass, fi, u,... are Lorentz four-vector indices, a, b,... are SU(3) adjoint representation (octet) indices, gab is the Cartan metric for the SU(3) octet, and J3£ is a gluon field]. We have suggested that GM„(a;) fulfills the role of an effective (pseudo) metric, with respect to the (pseudo) diffeomorphisms in point (b), in the same manner that the physical metric (through its Christoffel connection) "gauges" the true diffeomorphisms. Some of these assumptions were proven in Ref. 10; one purpose of the present work is to complete these proofs, thus putting our original hypothesis on more solid mathematical foundations; the other aim is an application to the understanding of some features (such as the Pomeranchuk trajectory), which were provided for in (1967-1973) duality and have since been partly traced within the QCD description of elastic high energy scattering. Our new algorithm is seen to go over smoothly into these descriptions when transiting between the relevant regions of QCD. Before starting our calculations, we should also mention that several other groups have independently been searching for a similar algorithm, i.e. the extraction of gravitation-like geometrical components from QCD. Lunev 11 D. Z. Freedman and
492
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collaborators and the Hehl group 12 have used different routes in the search for a QCD foundation generating such a conjectured chromogravity algorithm.11 In Ref. 10, we proved our point (b), namely the conjecture with respect to the emergence of effective chromodiffeomorphisms for which (1.1) is the effective metric. We start here by repeating that argument; we then complete it by checking the infinitesimal algebra's commutators. The next sections deal with the more complex structures deriving from n-gluon exchanges, thus completing the mathematical base for our conjecture (b); the results of the dynamical study in Ref. 10 then underpine conjecture (a) as well. The gluon color SU(3) gauge field transforms under an infinitesimal local SU(3) variation according to 6,B* = 3Me° + B£{At}»ec = d^
+ ifScBbec
(1.2)
(we use the adjoint representation {Xt}t = ifbc)- To deal with the nonperturbative IR region, we expand the gauge field operator around a constant global vacuum solution iV°, d^K
- dvN* = iftJ^K,
B; = NZ + A ; .
(1.3a)
(i.3b)
Such a vacuum solution might be of the instanton type, for instance. Consider, for example, the first nontrivial class, with Pontryagin index n = 0. Expand around this classical configuration, working, as always for instantons, in a Euclidean metric (i.e. a tunneling solution in Minkowski space-time). At large distances the instanton field is required to approach a constant value 9abN^dve" = dv{gabN;tb).
(1.3c)
Note that in the instanton system, such a constant value would arise from a null solution B£ = 0 ("pure gauge") through the application of a local gauge transformation involving a gauge function c"(x) linear in x, yielding a topologically nontrivial object. In what follows, we preserve the definition (1.3c) and the linear gauge; should a gauge transformation adjoin a new rc-dependent variation, we choose to include it in the A^(x) component of (1.3b). Returning to the "curved" pseudometric (1.1), we can now replace K by the "flat" density,
gabBiBj "" ~ [det(ft,tiV;JV»)]i/« '
{1A)
a A. Salam et a/.13 had postulated a "strong gravity" force prior to the emergence of QCD, perhaps related to gravity itself. 14 At a later stage, such a force was assumed to represent part of a "Lorentz extension" SL(6,C) = {[SL(2,C) ® SU(3)]} (double parenthesis: tensor envelope) of SU(3) c o i o r , i.e. of QCD. 1 5
493
Chromogravity:
QCD-Induccd Diffeomorphisms
4403
thus yielding a nonsingulax dimensionless Euclido-Riemannian "metric." Its color SU(3) infinitesimal gauge variation is given by *.GM„ = 6c{gab(NZ + Al){Nbv + Abu)} = 9ab(d„ea Nt + N^dvth + d^Al + i9ab {f^By
+
Bt + fbcdB%BUd) .
A^e") (1.5a)
The last bracket vanishes, since it represents the homogenous SU(3) transformation of the SU(3) scalar expression in (1.1) ihcd{BlBl + BlBbv)ed
(1.5b)
(or, more technically, due to the total antisymmetry of fabc in a compact group). With JV°, Nb, representing constant fields, we rewrite the terms in which they appear as a new infinitesimal variation, ft. = Va>,eaNb .
(1.5c)
We note that the expansion (1.3b)-(1.3c) implies that the N° is by definition that part of B£ which yields (constant) surface terms in an integration, whereas the A^{x) "fluctuation" does not contribute. As a result, we may safely integrate by parts the terms in A", Abv in Eq. (1.5a), thus getting 9ab(eadttAbu + dvAa^b),
(1.5d)
an expression whose Fourier transform vanishes for A; —» 0, i.e. in the infrared sector. We shall return and provide a generalized definition of this "IR limit." Meanwhile, as a result, we can write in this limit, hG»» = 9*t» + &ft. = d^aGav)
+ M^G?.),
(1.6)
where we have changed over to the £" variable of (1.5c), and where we can reidentify 6{ as a variation under a formal diffeomorphism of the R1 manifold. Equation (1.6) simulates the infinitesimal variation of a "world tensor" GM„ under Einstein's covariance group, x" -* x" + £". £" thus has to be defined as a contravariant vector; G^v of (1.1) is invertible, thanks to the constant part N*, in (1.3b), using a Taylor expansion to evaluate the inverse GM"(x). Note that as the n, v indices are "true" Lorentz indices, acted upon by the physical Lorentz group, the manifold has to be Riemannian: only Riemannian manifolds — with or without torsion — have tangents with orthogonal or pseudo-orthogonal symmetry groups. Thus D9G^ = 0,
(1.7)
preserving only the 2 + , 0 + components of G^,. To complete this proof, we now evaluate the commutator of two such variations, [S(1,S(1}G^=S(3GI1V,
494
(1.8a)
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Y. Ne'eman & D. Sxjaiki
and verify that 6 M := (^€iM)€a + (WivKZ
- (^6M)€i " ( ^ ) t f
(l-8b)
indeed closes on the covariance group's commutation relations. In Refs. 9, 10 and 16 we have dwelt upon the dynamical applications of this treatment, especially in reproducing the composition and energy spectrum of hadron states. We have also pointed out features (such as p - 4 propagators) indicating confinement. The results reproduced in particular the spectrum of Regge excitations. 10 Remembering that the enthusiasm for the string as a hadron strong interactions paradigm (1967-1973) had originally been due to its success in reproducing just this feature, we note that our gravity-like component induced by QCD — which we accordingly rename "chromogravity" — also yields these Regge systematics and thereby adds considerable credibility to QCD itself — in this, its least transparent sector. Moreover, we shall show that yet another essential feature of Dual Models, namely the HarariFreund structure of the Pomeranchuk trajectory — a high energy feature which has been rederived in perturbative QCD — can also be understood in terms of the chromogravity component, thus bridging these apparently disconnected methodologies. 2. The C h r o m o g r a v i t y I R Limit In Sec. 3, the definition of our "IR limit," which we based on the vanishing of the four-momenta of the "fluctuating fields" A°, Ahv in (1.5d) — after an integration by parts in which only the constant fields N* contribute to the surface terms — will be extended so as to include similar terms with vanishing momenta in all many-gluon zero color exchanges. This can be taken as an operational definition, sufficient for our general purpose. Recapitulating, we can write a generic IR state, carrying four-momentum k, as follows: oo
\iR,k)= ^2 fm(ki,k2,...
,km)6kM+kl+...+km\hk2
• • • km),
(2.1)
m=l
where \kik?•••km) represents a state of m soft gluons (fc< » 0 , i = 1 , 2 , . . . , m ) . Integrating by parts (with surface terms again appearing only for the constant parts N°), the matrix elements of the terms in A*, Abv become in this IR approximation <#R,
k'\gab(ead^Ai
+C
U ^ ^ I R ,
k),
(2.2a)
an expression that is proportional to the soft one-gluon momentum, and t h a t vanishes for fc —• 0, i.e. in the infrared sector. As a result, when changing over to the fa variable of (1.5c) and reidentifying S( as a variation under a formal iZ4 diffeomorphism, we get (1.6). For the sake of completeness, however, we note that in general one has to consider expressions of the form (0iR)*,|O(AJ,a^)*€G^|0ni,fc> •
495
(2.2b)
Chromogravity:
QCD-Induced Diffeomorphisms
4405
We evaluate such expressions, in this IR approximation, by inserting a complete set of states, and retaining only the soft virtual quanta. To gain some insight, however, into the meaning and justification of this selective summation, we remind the reader that such an IR approximation of QCD can also be thought of as the first step, the "zeroth approximation," in a strong coupling regime — for an expansion in which the "small parameter" represents the number of "hard," or nonsoft, virtual quanta, held in the evaluation of any physical quantity. Fried has indeed shown 17 that by making use of the Fradkin representation 18 for relevant Green's functions, one has a continuous family of "soft," or IR approximations, which maintain gauge invariance. This is then a consistent gauge-invariant (strong coupling) IR approximation, with dressed gluon propagators, which incorporate the iteration of all relevant quark bubbles, each carrying all possible internal, soft gluon lines. The consistency of this IR approximation requires one to consider only those QCD variations that connect IR gluon configurations mutually. Let us consider the expression for the A = B — N variation, which follows from Eq. (1.2), i.e. 6€A% = 3Me° 4- ifabcAbtiec. The left hand side of this expression is a difference between two soft gluons, implying that the IR matrix elements of its partial derivative are soft. Thus, we find the following "IR constraint" on the QCD gauge parameters: <# R , fc'ld,V + ifa6cA* dpec\4>m, k) « 0.
(2.3)
3. n - G l u o n Fields Our treatment is nonperturbative; we use, however, the formal expansion provided by the generating functional of Green's functions, for a classification of the contributions making up the overall nonquark component. The expansion involves all possible color singlet configurations of gluon fields. We rearrange the sum by lumping together contributions from n-gluon irreducible parts, n = 2 , 3 , . . . , oo and with the same Lorentz quantum numbers. Thus, QCD "gluon-made" operators which mutually connect various hadron states are characterized by color singlet quanta. The corresponding color singlet n-gluon field operator has the form • Ba" (31) where
dw
=
ffajaj
=
OOJOJOJ ,
,
"•01(12
d
(3)
"010J03
d(n)
(3.2) — d
1 nblClH
v
...
"oiaa---a„
Q^n-A^n-i *n-4Cn-4
X d c „_ 4 6„_ 3 a„_ : I S i , "- 3 < : "- : , dc n _30 n _ 1 a„ ,
496
" > 3 ,
4406 Y. Ne'eman & D. Sijatki B* is the dressed gluon field, gaia, is the SU(3) Cartan metric, and d a j 0 , 0 3 is the SU(3) totally symmetric 8 x 8 x 8 - * 1 tensor. It was shown by Biedenharn 19 that the set of all di"l a ... a „ tensors, n = 1,2,..., can be used to form, together with the group generators, a basis of all SU(3) invariant operators. In this case all such higher rank operators can be expressed in terms of two invariant operators. In our case, the set of all G)?^...^ operators, n = 1,2,..., form a basis of a vector space of colorless purely gluonic configurations. Moreover, in our case, in contradistinction to the ordinary group theoretical situation, these field operators are also all functionally independent. The QCD variation of the GJ."^-^,, field is given by
+ BZ\dli2ea>.-B;:+---
+
BaM\BZl---dltne«")
d£)ai...an(g^fr,tB'fllB%..-B?n
+
Here again the homogenous terms vanish as before due to the fact that di"aj •••<»„ is totally symmetric. Applying the decomposition (1.3), we can now rewrite the n-gluon configuration transformation as
+ ^ . , . , „ (Ai\d„2t°*N% • • • JVJ : + • • • + A;J jvyjivji • • • » M B * - » ) + (1 ~ i = 2
n)
+ ^ o , . . - , (S^c-M^A-|JV2 • • • JV£ .+ • • • + Al\A-£NH • • • A f t l } 0„ n <°" ) + • • • + x rfSi...... (OMI «ai *l\ • - AZ + Al\ dn
£
° ' ' - ^ + - + *l\ Al\ • • • 0m. «°n ) •
As to the terms in A*1., i = 1,2,..., n, the considerations we mentioned prior to the integration by parts in (1.5d) hold here. Thus, applying integration by parts here too, we get -4!i,..^(a«(^i)€«».--JVJ:+--- + -<£U-.. ( «
a ,
flM-(^l1)w;i---«^)-(!«<
^ . K ^ i W J • •• N% + • ••+a^A»\Ai*)N^
=2
»)
• • • *£:!«•") - • • •
- 4 " U " - - . ( « * , » i . i ( ^ - - - ^ : ) + « B a » M W i - - - ^ : ) + • • • + « " " » » • - M ; I ^ s •••))•
497
Chromogravity:
QCD-Induced Diffeomorphisms
4407
But taking Fourier transforms — i.e. the matrix elements for these gluon fluctuations — we find that these terms are precisely those that vanish in our definition of an IR region, as discussed in the previous section. The terms involving the constant connections JV£j, i = 1,2,...,n, can be rewritten in terms of effective pseudodiffeomorphisms,
where {/Ji/i2 • • • Mn} denotes symmetrization of indices, and An—I) _ j(n) AT<>1 MOj . , . ATOn-j o„ J SMlMJ'"Mn-l — "aioj.--a„-'*^,-i» M j Vn-lC '
/o 4\ \°m*)
generalizing our results as derived for G$ = GM
generalizing the n = 2 case in (1.8), i.e. an infinitesimal nonlinear realization of the Diff(4, R) group in the space of fields {G^,... Mn \n = 2,3,...}. 4. £< m ) Operators Let us consider an oo-dimensional vector space over the field operators {G(")|n = 2,3,...},i.e. V(G«\GV\...)
=
V(G%M,G$(12li3,...).
We can now define an infinite set of field-dependent operators {L^m^\m = 0,1,2,...} as follows: 1/1
T^p
r(m)P vxv%-~vm+\
±J
" " , a a -1 6(ga2bB>) ~ 9a^"> 6{ga2bB)) ' — /f( 3 )
R a i Ra»
_ uj ( m + 2 ) pal Dt, — aia2"-a m + 2-" p , 1 -«-' l /j
R a i R°a
= A
D«m+1 K m + i cf
£
u
Db\ '
0(gam+3b-tf^)
The Lli action on the field operators {G(n*|n = 2,3,...} reads
r (0)^(3) _ fp ^ ( 3 ) , cp ^ ( 3 ) „ ^(3) •^"l '-'1*11*21*3 — °UiUr"lf»2M3 ^ °uj , J /ill'lM3 + V^MlWl t
(4-2) r(o)pMn)
— ftp /rj(")
n
J-A/J r>( )
j .
L>U1 <-r/11^2---/Xn — °/ii<-r"lM2--Alii ~ Va,-rMl''lM3"Mn T
498
4-/?/'
r^
^" t >„< J /*lM2—/*»-ll'l >
4408
Y. Ne'eman & D. Sijaiki
The [$£
action on the field operators {(?("> |n = 2,3,...} reads
TWenW
- xp nW
Uv^v^yjM1MJM3
— 0
r(DPG(n)
_ «
4,/JP nW
+ f;prW
j4,'Ji^«^jtJM» ^ ° / * a " W i " * / ^ "*" 0J*3L*/*iJ«2''i»'a i G
( " + D
. «
(4
G("+l)
-3)
n
44- AP / ^ v + l ) T • • • -r o # J „ ( - T / 1 1 ^ , . . . / i n _ l V 1 ^ , ,
In the general case, L\^^...Vm+l, {G^\n = 2,3,...} reads r(m)p T(m)p
n(2) a(3)
• L 'l'll/j"-l/ m + l ( -'filfl2M3
m = 0,1,2,..., action on the field operators
_ cp M2+m) _ rp
—
cp
n(2+m)
n(3+m)
°/n""l>'a---' / m+l/-'3/*3 i f.p
, f-p
^(3+m)
(4.4) r ("»)/>
/"(»)
n m
_ sip M + )
>Xp v
+ lP-a"-/*n ' T
n m
n( + )
/ia
T U/Jll*Jjtl^j...M„_ii/,i/2--l/>„+i i
Let us now consider the algebraic structure defined by the {Z/ m )|m=0,1,2,...} operators Lie brackets. For the L^ operators themselves we find [ L (o) j L (o) ] c L (o) ) i.e.
In the most general case, for the brackets of L^ and Z/m) we find [£<'>, £ (m) ] C l ( l + m ) ,
(4.5)
and, more specifically, rr(')pi
r(m)p a
-i _
m+1 riv-T-i. £Pi + m ) P a3 ^ r Pl rr ( '{l+m)p
"l+JOt+l—^m + I
«=1 i+1 2_^,VVJ
• L '«'i«^—«'i-i<'io'a—« r m+i»'j+i"-»'i+i •
499
V^-"V
ChromogTavity: QCD-Induced
Diffeomorphisms
4409
We have constructed an c© component vector space, V = V ( G ^ j , G$Mll3,...), over the n gluon field operators, as well as the corresponding algebra of homogenous diffeomorphisms, diffo(4,il) = {Lr™^.. Vm+1\m = 0 , 1 , 2 , . . . }; the vector space V is invariant under the action of the diffo(4,i?) algebra. Let us point out that there exists a subalgebra of the entire algebra when m values are even, i.e. one has the following structure: ^(evenJ^Ceven)"! Q £,(even) )
[ i M ^ W j c i H d ) ,
(4.7)
[z,(° dd ) 1 L(° dd )] c Z,( even ). Moreover, the space V splits up under the action of the subalgebra of the L^ even ) operators into an even n space and an odd-n. space of n gluon field operators, i.e. V(G^,G^,...)=V{G^2\G^,...)®V(G^,G^,...).
(4.8)
Let us define the dilation-like operator (chromodilation) D as a trace of 0)
L<, ', i.e.
D = Lf>p .
(4.9)
This operator commutes with the LL 'P operators, [£,40>']=0,
(4.10)
and belongs to the center of the gl(4, R) chromogravity subalgebra generated by the Lv operators. On account of the chromodilation operator one can make the decomposition (4.11) gl(4,/?)=r®sl(4,i?), where D corresponds to the subalgebra r, while the basis of the sl(4, R) subalgebra is given by T(o)P
= L(o)P
_
}_6PD
The commutation relation of D with a generic diffo(4, R) operator reads [D, 4 r ^ . . . , m + 1 ] = mL\?k..^+l
(4
12)
L)J^...Vm+l (4.13)
and thus the chromodilation operator D provides us with a Z+ grading. This grading justifies and/or explains the m label used for the L)$£...„m+1 operators. The chromodilation operator D counts the number of single gluon fields in a multigluon configuration, as seen from the following commutation relation:
[Aci:L-,.]=4i-N.
500
(4.i4)
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Y. Ne'eman & D. Sijaiki
Note that the essential point we have demonstrated is that the set of colorless many-gluon exchanges includes a gravity-like component. True, we have no way — at this stage — of evaluating the quantitative weight of this contribution from the parameters of QCD. What therefore remains as a working hypothesis is the assignment of gravity-like observational features (including those features which had caused Salam and others 1 3 - 1 8 ' 2 0 to believe in a strong gravity contribution extraneous to QCD) to this chromogravity component's contribution and the derivation of the chromogravity quantitative parameters from this phenomenological approach. 5. Conclusions Clearly, the J = 1 Yang-Mills gauge of QCD includes, in a certain (properly defined) IR limit, a component simulating local diffeomorphisms. This component is gauged as a Riemannian geometry, i.e. d la Einstein, with an "effective" "(chromo)metric." It is important to stress that the GM„(x) of (1.1) is not the digluon "glueball," yet another massive state in the hadron spectrum, which lattice calculations put somewhere between 1.5-2.5 GeV. 21 We are dealing here with a force emulating gravity, a force component whose long range is protected by the pseudodiffeomorphisms gauge, within limits set by its emergence at the colorless and vanishing frequencies end of the spectrum only. In a study of nuclear excitations, we have pointed 22 to the fact that this range, longer than that of the pions or rho mesons of the nuclear binding, might therefore induce the basic J = 2 + , 0 + excitations of the Arima-Iachello interacting boson model. 23 In the analytically continued S matrix, there is thus an effective pole at J = 2, M2 — 0, emerging in that "IR limit" — i.e. an effective contribution simulating such a pole. There is no conflict with the Froissart bound, since we are dealing with a massless gauge field "gauging" diffeomorphisms, i.e. an interaction with a locally conserved current, in which unitarity manages to overcome the problematics of high spin exchange, as in gravity or in supergravity, where the exchanged 7 = 2, J = 3/2 poles are protected by current conservation and the local gauge invariance. As a matter of fact, the emergence of a J = 2 + gauge component within a J = 1~ gauge theory should not come as a surprise. We remind the reader that the truncated massless sector of the open string reduces to a J = 1" Yang-Mills field theory while the same truncation for the closed string reduces to a J = 2 + gravitational field theory. Considering that the closed string is nothing but the contraction of two open strings, the analogy with our findings in the present paper is clear. The field theory corollary has even been used by the Bern-Kosower group with their string-generated method, to evaluate ("true") gravitational amplitudes. 24 This also throws some light on the emergence of the J = 2 + pole in closed strings, within the context of dual models, where such poles partly found their realization (massive, off the intercept) on the Pomeranchuk trajectory. In its Ereund-Harari
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Chromogravity: QCD-Induced Diffeomorphismi
4411
dual structure 25 and with the Harari-Rosner identification of quark lines in the dual bootstrap, 26 this "vacuum" trajectory is generated by the non-quark-matter (i.e. gluonic) component of the interhadronic chromodynamic interaction. More recent studies in the context of QCD have pointed to the two-gluon exchange as the main component of that vacuum trajectory, in its dominance of the high energy scattering region.1 We note that our extension here, of a notion we defined in the IR region, into the domain of high energy scattering, goes over smoothly into representations which were derived directly in that regime, using semiperturbative QCD. If we go back to the pole dominance considerations of the sixties, dispersion relations showed that a J = 2+ pole, by dominating the matrix elements of the hadron energy-momentum tensor (and sitting on the vacuum trajectory), would couple "universally," i.e. like a graviton. 27 This is analogous to the manner in which one sees that the p, LJ J = 1~ poles — dominating the electric current of the hadrons — couple "universally," i.e. like a photon. All of this implies that the identification of a J = 2 + "chromograviton" carries with it a long list of previously noted hadronic gravity-like correlations. However, this is not the only pole in the J = 2+ sector of the analytical 5 matrix, since gluons also make up a J = 2 glueball, as mentioned above (1.5-2.5 GeV); there is thus possible mixing between several gluonic structures. In addition, we have the quark-antiquark states — the f° meson at M = 1.27 GeV (Salam's candidate "strong graviton" at the time) and the f0' at M = 1.525 GeV. We conjecture that all of these mix. Future work should be directed in part to a better understanding of these issues. We mentioned earlier several other approaches leading to some form of "chromogravity." 11,12 The most "mature," by Freedman et a/., uses gauge-invariant variables (the chromoelectric fields Ex). They can thus derive geometric features [GL(3, R) invariance] but naturally cannot get diffeomorphic-like gauge transformations (due to the gauge invariance of the formalism). We should note that this GL(3, R) invariance fits well with the applications of our algorithm in the classification of the hadron spectrum 12 and with the phenomenological identification 28 of Regge trajectories with unitary (infinite component) representations of SL(3, R), as extended to include spinorial ones, after it was shown that these do exist 29 ' 30 and might even describe the phenomenological coupling of hadrons to ordinary gravity (as "world spinors" 31 ). References 1. L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rep. 100, 1-150 (1983). See Chapter 3 in particular. 2. F. Butler, H. Chen, J. Sexton, A. Vaccarino and D. Weingarten, Phys. Rev. Lett. 70, 2849 (1993); D. Weingarten, Nucl. Phys. (Proc. Suppl.) B34, 29 (1994). 3. F. J. Yndurain, in The Theory of Quark and Gluon Interactions (Springer-Verlag, New York, 1983). See for example Sec. 4.10.
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4. S. Godfrey and N. Isgur, Phys. Rev. D 3 2 , 189 (1985); S. Capstick and N. Isgur, Phys. Rev. D 3 4 , 2809 (1986). 5. See examples in N. Isgur, Proc. XXVI Int. Con}, on High Energy Physics, Vol. I, ed. J. H. Sanford, AIP Conf. Proc. 272, 33 (1993). 6. S. J. Brodsky, G. McCartor, H. C. Pauli and S. S. Pinsky, Particle World 3, 109 (1993); K. G. Wilson, T. S. Walhout, A. Harindranath, W.-M. Zhang, R. J. Perry and S. D. Glazek, Phys. Rev. D49, 6720 (1994). 7. G. 't Hooft, Nucl. Phys. B190, 455 (1981). 8. N. Seiberg and E. Witten, Nucl. Phys. B426, 19 (1994). 9. Dj. Sijacki and Y. Ne'eman, Phys. Lett. B247, 571 (1990). 10. Y. Ne'eman and Dj. Sijacki, Phys. Lett. B276, 173 (1992). 11. F. A. Lunev, Phys. Lett. B295, 99 (1992); V. Radovanovic and Dj. Sijacki, Class. Quantum Grav. 12, 1791 (1995). 12. M. Bauer, D. Z. Freedman and P. E. Haagensen, Nucl. Phys. B 4 2 , 147 (1994); E. W. Mielke, Y. N. Obukhov and F. W. Hehl, Phys. Lett. A 1 9 2 , 153 (1994). 13. R. Delbourgo, A. Salam and J. Strathdee, Nuovo Cimento 49, 593 (1967). 14. F. W. Hehl, Y. Ne'eman, J. Nitzsch and P. von der Heyde, Phys. Lett. B 7 8 , 102 (1978). 15. A. Salam and J. Strathdee, Phys. Rev. D8, 4598 (1978); C. Sivaram and K. Sinha, Phys. Rep. 51, 11 (1979). 16. Dj. Sijacki and Y. Ne'eman, Phys. Rev. D47, 4133 (1993). 17. H. M. Fried, Phys. Rev. D27, 2956 (1983). 18. E. S. Fradkin, Nucl. Phys. 76, 588 (1966). 19. L. C. Biedenharn, J. Math. Phys. 4, 436 (1963). 20. P. Caldirola, M. Pavsic and E. Recami, Nuovo Cimento B48, 205 (1978). 21. See for example Y. Liang et aL, Phys. Lett. B307, 375 (1993), and Refs. 4-9 quoted therein. 22. Y. Ne'eman and Dj. Sijacki, Phys. Lett. B250, 1 (1990). 23. A. Arima and F. Iachello, Phys. Rev. Lett. 35, 1069 (1975). 24. Z. Bern, D. C. Dunbar and T. Shimada, Phys. Lett. B312, 277 (1993). 25. P. G. O. Freund, Phys. Rev. Lett. 20, 235 (1968); H. Harari, Phys. Rev. Lett. 20, 1395 (1968). 26. H. Harari, Phys. Rev. Lett. 22, 562 (1969); J. L. Rosner, Phys. Rev. Lett. 22, 689 (1969). 27. P. G. O. Freund, Phys. Lett. 2, 136 (1962). 28. Y. Dothan, M. Gell-Mann and Y. Ne'eman, Phys. Lett. 17, 148 (1965). 29. Y. Ne'eman, Ann. Inst. Henri Poincare 28, 369 (1978). 30. Y. Ne'eman and Dj. Sijacki, Int. J. Mod. Phys. A 2 , 1655 (1987). 31. Y. Ne'eman and Dj. Sijacki, Phys. Lett. B157, 275 (1985).
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Modern Physics Letters A, Vol. 11, No. 3 (1996) 217-225 ©World Scientific Publishing Company
INTER-HADRON QCD-INDUCED DIFFEOMORPHISMS FROM A RADIAL EXPANSION OF THE GAUGE FIELD
YUVAL NE'EMAN Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, Israel and Center for Particle Physics, University of Texas, Austin, TX 78712, USA DJORDJE SIJACKI Institute of Physics, Belgrade, Yugoslavia
Received 31 October 1995 In a previous work,1 we proved that a subset of the QCD local SU(3)coior gauge transformations, involving color-neutral gluon field operator products, reproduces effective local diffeomorphisms. That proof involved integrations by parts, with possible surface terms. We present an alternative proof, which does not involve integrations, based on a radial expansion. It also provided new insight into the structure of the gluon fields.
1. Introduction In a recent publication1 we have provided a mathematical derivation in which we demonstrate the emergence of "effective" diffeomorphisms as a subset of the co/or-SU(3) local gauge transformations in inter-hadron interactions. That proof, however, involves an integration by parts, with the possibility of complications as a result of surface terms. In this letter, we show that the proof can also be derived directly, using an expansion in terms of distances. Replacing the low-frequency limit by the (conjugate) large-distance limit results in avoiding integrations and surface terms. We had originally suggested2 that the spacetime-geometrical nature of QCD confinement — as discussed in ad hoc theories such as the "Bag Model" — might induce Regge systematics based on SL(4, R) or SA(4, R), with relevant stability subgroup SL(3,.R),3 preserving a fixed volume. This would also fit with observations from the phenomenology and with an algebraic description of gravitational-like quadrupolar pulsations,4 even though unrelated to gravity. The approach was further strengthened by field theory covariance considerations, also involving an SL(4,.R), this is related to true (gravity-extendable) covariance. Between the two •Wolfson Distinguished Chair in Theoretical Physics. 217
504
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Y. tfe'eman
& Dj. Sijacki
groups, a transformation is involved which relates to the one between current and constituent quarks.5 Finally, we looked for the originating mechanism within QCD itself6 and provided a first mathematical derivation of "chromo-diffeomorphism",7 later generalized tnRef. 1. Spacetime-geometrical QCD algorithms have recently been explored by several authors.8-15 In these approaches, however, gauge invariance is stressed, so that the pseudo-gravitational entities ("metric", etc.) are constructed out of the fieldstrengths rather than the potentials. As a result, they cannot exhibit diffeomorphisms, though they may yield GL(3, R) or SL(3, R) invariances, presumably related to confinement. Note that many of these results are presently limited to SU(2) as the color gauge group, with proofs exploiting the equality of dimensionalities between this group's adjoint representation and the antisymmetric tensor representation of spatial SL(3, H). 8-11 Recent studies have also identified the boundary conditions (in terms of curvature or torsion) which characterize these pseudo-gravitational Lagrangians14'15 in three and four dimensions. As against these treatments, we have chosen to work with variables displaying gauge variations (the four-potentials or gluons), thus providing for a source of (effective) diffeomorphism. As a result, (a) it is the SL(3,i?) invariance which then becomes implicit, as the stability subgroup of SA(4, R) — to the extent that the effective "chromo-gravity" might take on affine features. Hadron spectroscopy does display a good fit between this dynamical algorithm and phenomenology, with respect to both classification and energy-spacings in Regge sequences.4,5,7 Other physical results — to date — consist in (b) a smooth transition1 to the ("semi-soft") domain of hadron high-energies, in which the same color-neutral gluon combinations have provided a model for diffractive scattering16'17; (c) an understanding of the relationship between the quark current-to-constituent mass-growth and the slope of the Regge sequence, in terms of chromo-graviton self-energies; and (d) a dynamical derivation of the J = 2+, 0 + ground state in even-even nuclei, the algebraic foundation for the successful "IBM" SU(6) spectrum generating group in nuclei.18'19 This appears as a long-range excitation of the chromo-gravitational quanta, somewhat similar to a Van der Waals effect in QED. Note that it is important to distinguish between all of these "inside QCD" approaches and their precursors. Before the advent of QCD, Salam and collaborators had in fact suggested that the strong interactions might altogether be due to a "strong gravity", in which the f0, J = 2 + meson at 1235 MeV played a graviton-like role.20 After the adoption of QCD, he raised the possibility of the existence of such an interaction21'22 in addition to QCD, by gauging SL(6, C) D SL(2, C)Lor. ® SU(3)coi.. The present authors were then involved in an alternative ansatz, for a similar emergence of a strong interaction contribution from geometric considerations, together with ordinary gravity. It was based first on Poincare gauge theory,23 then on affine gravity,24 exploiting the newly established existence of (infinite-component) spinorial representations25"28 of the SL(n,R) C GL(n,i?) C A(n, R) C diff (n,R) without which3 could not have been used. In contradistinction
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Inter-Hadron
QCD-Induced Diffeomorphisms
219
to all of this, present approaches point to spacetime-geometrical effects due to QCD itself. 2. Chromo-Diffeomorphisms — The Original Derivation We had originally conjectured5 that (a) color-neutral gluon exchange forces make up an important component of inter-hadron interactions, or in another nomenclature, of soft QCD interactions; (b) this component produces a longer-range force, with many of the characteristics of gravity, including (and "protected" by) the basic mathematical criterion, namely invariance under diffeomorphisms; (c) the simplest such n-gluon operator product, that of a two-gluon system G„„(s) = {K)~2gabB%{x)Bhv{x),
(1)
where K has the dimensions of mass, /*, i/,... are Lorentz four-vector indices, a, b,... are SU(3) adjoint representation (octet) indices, gab is the Cartan metric for the SU(3) octet, B° is a gluon field which fulfils the role of an effective (pseudo) metric, with respect to these (pseudo) diffeomorphisms, in the same manner that the physical metric (through its Christoffel connection) "gauges" the true diffeomorphisms. In Ref. 1, the proof of this conjecture then follows a three-step argumentation. The first step consists of the definition of an % limit" and the demonstration of the presence of effective ("cftromo") diffeomorphisms, as a class of QCD gauge functions, in the color-gauge transformations of the above GMI/. The gluon color-SU(3) gaugefieldtransforms under an infinitesimal local SU(3) variation according to S.BI = 6>» + gBl | ^ j % « = d„e- + igfabcBb^
(2)
(using the adjoint representation {^}? = — ifbac = ifabc; g being the QCD coupling constant). The gauge field operator is then expanded around a constant vacuum solution (pure gauge) of the instanton type, Bl = Nl + A%, dMN; - dvN; - igfahcNlNl
(3a) - 0.
(3b)
By further defining a constant (flat) Euclidean metric, one can now replace K in (1) by the "flat" density, "" " [det(flBkJV-JV*)]V4 W thus yielding a nonsingular dimensionless Riemannian "metric" for either signature. Its color-SU(3) infinitesimal gauge variation is given by ^ M t + A£d„e») + iggab{fZiB;edBl
506
+ fhedBlBlsd}.
(5)
220
Y. Ne'eman & Dj. Sijacki
The last bracket vanishes, since it represents the homogeneous SU(3) transformation of the SU(3) scalar expression in (1), or, more technically, due to the total antisymmetry of fai,c in a compact group. It is at this stage that our proof in Ref. 1 involves an integration by parts, replacing the terms of type A*dueh by terms — duA*£b, and defining the "ir" region through dvA% = 0. Although it is plausible to assume that the only surface term in integrating B^dvcb would come from iV* rather than from A£, justifying that proof, it is good to know that this argument can be replaced by a more heuristic approach, foregoing the integration altogether. 3. The r-Expansion At this stage, one of the two sets of conditions is assumed: (a) a Euclidean metric, to adapt to quantum solutions (such as instantons), in which case one discusses the limit of large \x\; (b) alternatively, in Minkowski spacetime, one uses a nonrelativistic picture. Working in polar coordinates (r,6,<j>,t), one expands in powers of r, having fixed t = 0, taking the limit of large i— which again is the same as large \x\. Since large r will also imply large |x| for the Euclidean case, one mostly uses that definition for the limit at which the gauge field is expanded, thus covering both applications as large | i | limits. Expanding £ ° in powers of r, one now writes, £ ~ • •• c_2(0,4>)r~2 + c-ir'1 + N + c^r + c2r* .
(6)
One now projects, out of this summation, states (or sometimes values of 0,
(7)
In the Euclidean case, the vacuum solution N^(x) is then indeed of the instanton type, and can be written, for instance, as
N,(X) = iv;(s)^ = _ML ? [r/- 1 (x)a^(x)].
507
(8)
Inter-Hadron
QCD-Induced
Diffeomorphisms
221
Only A^{x) preserves an z-dependence in the above limit and we have seen that it consists in negative powers of x. Thus, •§;&% ~ 0 and as a result, in the Euclidean case — and effectively in Minkowski spacetime too (for spherical symmetric situations) one has the defining constraint, ft-BJsO.
(9)
Fourier-transforming, in Minkowski spacetime, to momentum space, one has a summation / dkvB^ ^ 0, which, when applied to gluon Hilbert space "on mass shell" states, projects out those boundary conditions in which only the kM = 0 states contribute, i.e. the low frequency (ir) gluon regime. We can now go back to Eq. (5). From t h e analysis of the r-expansion constituents, we now conclude that the terms involving N£ in (5) dominate those involving A^ at large distances. This is the basic term where we no longer depend on the integration by parts. With JV", N*, representing — in the r —• oo limit —constant fields, one now rewrites the terms in which they appear as a new infinitesimal variation,
We shall return and complete the definition of this "ir limit". Meanwhile, as a result, one can write in that limit, «eG„„ = d^v + d„^ = d^'Gau) + d^i'G^),
(10b)
where we have changed over to the fff variable and where we can re-identify 6$ as a variation under a formal diffeomorphism of the R* manifold. Equation (10b) simulates the infinitesimal variation of a "world tensor" G,,„ under Einstein's covariance group, x" —<• x" + f . £* thus has t o be defined as a contravariant vector. Note that GM„ in (1) is invertible, thanks to the constant part N£, in the long range limit, and using a Taylor expansion we can evaluate the inverse Gy,v{x). As the H, v indices are "true" Lorentz indices, acted upon by the physical Lorentz group (or by SO(4) in the Euclidean case), the manifold has to be pseudo-Riemannian or Riemannian: only these manifolds — with or without torsion — have tangents with pseudo-orthogonal or orthogonal symmetry. Thus D„Gllv
= Q.
(11)
To complete this proof, one now evaluates the commutator of two such variations, foi'^lG/ji/ = %£>„ , (12a) and verifies that
6* = (^6M)€J + VMS - (a,&M)ff - («M6v)ff •
508
(12b)
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Y. Ne'eman & Dj. Sijaiki
The definition of this "ir limit", which is based on the vanishing of the gluon fourmomenta, is now extended so as to include similar terms with vanishing momenta in all many-gluon zero-color operator products. This can be taken as an operational definition, sufficient for general purposes. To draw an easier physical picture for the following considerations, place them in the context of Minkowski spacetime. One can then write a generic "ir state", carrying four-momentum k, as follows: oo
l&r.fc) = 5 3 / m ( f c l , & 2 , - . - ,fcm)£fc > ki+fc2+"-+fcJ fc lfc2'--fcm>> n»=l
(13)
where \kik2 • • • km) represents a state of m soft gluons (hi « 0, i = 1,2,... ,m). One can now verify the effect of taking the variation (5) (or (10b)) — as a typical result in this formalism — between such states. For the residual component in (5) one writes (&,
fc'|
+ M j e * ) | t f l r , *) •
(14)
Our conditions will indeed make the terms in N dominate over those in A. As a result, we can change to the £" variables of (12b) and re-identify 6^ as a variation under a formal R1 diffeomorphism. To identify the subset of color-SU(3) gauge functions which produce appropriate N£ and fit our ir condition, we apply (9) to the variation in (2). d^cBtf
= a„S„e0 + igjabcdvBlec
+ igfabe{AbM + Nfrd„ec
^d„dvea+igf\cNldvec = D^(d,ea),
(15a)
where one drops the 3„J9£ term because of (9) and the term in A^ since it is diminishing fast, as compared to the term in N*. Thus, the condition amounts to the vanishing of the long-range covariant derivative, i.e.
W O = ~i9fabcK(d^
(15b)
•
To solve these conditions, make use of a similarity transformation that diagonalizes (in an 8 x 8 color-representation space) the fbcNb matrix: fabeN^ -> d ie 1 b U2f beN*U- c = (UfN^U- )^? and e° -» U^e . (15b) is solved by 6V(t/£)° = e-isWfW1)^)*"
,
(15c)
and finally, ' ^ • = -^/^-),
( e )
"""""""'"-'"-
509
1 )
-
<15d)
Inter-Hadron
QCD-lnduced
Diffeomorphisms
223
where the integration constant is fixed by requiring that (Ue)a < 1 (consistency of the calculation). This is indeed the long-distance limiting value for the class of gauge functions yielding "pure gauge" fields of the instanton type, i.e. fields tending to a constant value iV£ at large distance. 4. n - G l u o n O p e r a t o r P r o d u c t s The second step in the proof consists of showing that the infinite algebra of chromodiffeomorphisms is realized within the set of all multi-gluon color-neutral operator products, thus generalizing the two-gluon construction. We have given a detailed proof in Ref. 1, except that it depends on an integration by parts, as in the twogluon case. Here again, our r-expansion can fully replace the integration by parts, making it free from the possible effects of surface terms. The set of all possible color-singlet configurations of gluon fields is rearranged by lumping together contributions from n-gluon, n = 2 , 3 , . . . , oo and with the same Lorentz quantum numbers. The corresponding color-singlet n-gluon field operator has the following form: G & , - * . = d&,-anB%BZ
•••&£ ,
(16a)
where d(3> u
aio2a3
=d "'010203 >
(16b) ti^
—A
t
nblCld
"OiOj-.-On — " a j o j b i »
<. "0.6303
• • nblt-*Cn-*H 5/
xjb"-jC"-3dc„.3a„.Ia.,
•. u
C„_40n-3<»«_J
n>3,
doio2a3 is the SU(3) totally symmetric 8 x 8 x 8 - » l tensor. We showed in Ref. 1 that the set of all Gj»™/*2.../*» operators, n = 1,2,..., forms a basis of a vector space of colorless purely gluonic configurations. These field operators are also all functionally independent. The QCD variation of the G ^ 2 -M» ^ e ^ ^ give11 by ^ G ^ - M .
= d i ? „ 2 . . . a n ( a M l £ a > B % - B £ + B%d„e" + B;\B;I
• • • d^e"")
+ 9a2TUstBl\BU
•••BZ
+ ---
+ d&2...tt„(g^fr,tB^sB%
• • • BH + • • • + f~rfT«B?xB%
...B£ • • • B'^e*.
(17a)
Here again the homogeneous terms vanish as before due to the fact that di?a a -a„ is totally symmetric. Applying the decomposition (3), one can now rewrite the ra-gluon configuration transformation as
510
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Y. Ne'eman & Dj. Sijacki
+ NZN% • • • 0„„£»«) + di%...an(A^d^NS
• • • N£ +
+ A J l J V ^ J V ^ . . . ^ e - ) + ( l « i = 2,...,n)
+ Al\Al\.--d^).
(17b)
As in (10), the terms in JVJJ*, * = 1,2,... , n, dominate over the corresponding terms in A£*, again yielding variations which can be cast in a diffeomorphic mode,
ScG^...^
= ^ ^ - V . . ^ } = hG(:L^
,
(18a)
where {//1M2 • • • /*n} includes symmetrization of indices, and c(n-i)
=//(")
TV"1 AT"2 ••• A T 0 " - 1 * - 0 "
flSM
generalizing the results as derivedforG $ = GMV[det(p06JV*JV*)]x/4. A subsequent application of two SU(3)-induced variations implies 1*.,.*«M* ) «-M.=*«,G1 , J > «"-M..
i-e- [^.^]4"L-Mn=^3 4 " L - ^ (19)
generalizing the n = 2 case in (12), i.e. an infinitesimal nonlinear realization of the Diff(4, R) group in the space of fields {G^M2...Mn \n - 2 , 3 , . . . } . The third and last step in the proof consists of the construction of the operators of the chromo-diffeomorphism algebra, following Ref. 1. We do not reproduce it here, as it does not involve integration by parts and is unmodified by our present use of the r-expansion. The resulting generators are given by j (m)p
__ j(m+2)
Aol po2 . .. 6 o m + i
°
f 201
"" +1 S(gam+2bB^p) acting on the G^ fields of (16a), with the latter making up an oo-dimensional vector space V(GW, (j( 3 ),...). The L^ form a Z+-graded algebra, with the grading given by a dilaton-like operator, counting the number of gluon fields in a multi-gluon configuration. The bracketing of L^ and X/m) preserves that grading [£W|x(m)]CdL('+m)>
(21)
the L\!$£...v„+l, m = 0,1,2,... spanning the diffo(4,i?) algebra of homogeneous diffeomorphisms.
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Inter-Hadron QCD-Induced Diffeomorphisms
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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
14. 15. 16. 17. 18. 19. 20. 21.
22. 23. 24. 25.
26. 27. 28. 29.
Y. Ne'eman and Dj. Sijacki, Int. J. Mod. Phys. A, in press. Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 1 5 7 , 267 (1985). Y. Dothan, M. Gell-Mann and Y. Ne'eman, Phys. Lett. 17, 148 (1965). J. Lemke, Y. Ne'eman and J. Pecina-Crnz, J. Math. Phys. 33, 2656 (1992). Y. Ne'eman and Dj. Sijacki, Phys. Rev. D 3 7 , 3267 (1988); D 4 7 , 4133 (1993). Dj. Sijacki and Y. Ne'eman, Phys. Lett. B 2 4 7 , 571 (1990). Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 2 7 6 , 173 (1992). F. A. Lunev, Phys. Lett. B 2 9 5 , 99 (1992); B 3 1 1 , 273 (1993); B 3 1 4 , 21 (1993). D. Z. Preedman, P. E. Haagensen, K. Johnson and J. I. Latorre, CERN-TH.7010/93, unpublished. P. E. Haagensen and K. Johnson, Nucl. Phys. B439, 597 (1995). M. Bauer, D. Z. Freedman and P. E. Haagensen, Nucl. Phys. B 4 2 , 147 (1994). V. Brinjedonc and G. Cohen-Tannoudji, CEA Saclay report DAPNIA/SPhN 95 17, April 1995. D. Singleton, "Exact Schwarzschild-like solution for Yang-Mills theories", Univ. of Virginia preprint (1995); "Exact Schwarzschild-like solution for SU(7V) gauge theory", preprint hep-th/951097 (1995). F. W. Hehl, E. W. Mielke and Y. N. Obukhov, Phys. Lett. A 1 9 2 , 153 (1994). V. Radovanovic and Dj. Sijacki, Class. Quantum Grav. 12, 1791 (1995). L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rep. 100, 1 (1983). E. M. Levin, in QCD 20 Years Later, eds. P. M. Zerwas and H. A. Kastrup (World Scientific, 1993). A. Arima and F . Iachello, Phys. Rev. Lett. 35, 1069 (1975). Dj. Sijacki and Y. Ne'eman, Phys. Lett. B 2 5 0 , 1 (1990). C. D. Isham, A. Salam and J. Strathdee, Phys. Rep. D8,2600 (1973); D 9 , 1 7 0 2 (1974). A. Salam, in Fundamental Interactions in Physics, eds. B. Kursunoglu et al. (Plenum, 1973), p. 55; also in Five Decades of Weak Interactions, Ann. NY Acad. Sci. 294, 12 (1977). C. Sivaram and K. Sinha, Phys. Rep. 5 1 , 11 (1979); A. Salam and C. Sivaram, Mod. Phys. Lett. A 8 , 321 (1993). F. W. Hehl, Y. Ne'eman, J. Nitsch and P. V. D. Heyde, Phys. Lett. B 7 8 , 102 (1978). Y. Ne'eman and Dj. Sijacki, Ann. Phys. (N. Y.) B120, 292 (1979). Y. Ne'eman, Proc. 8th Int. Conf. on Gen. Rel. Grav., ed. M. A. McKiernan (Univ. of Waterloo, 1977), p. 262; Proc. Nat. Acad. Sci. USA 74, 4157 (1977); Ann. Inst. Henri Poincare A 2 8 , 369 (1978). Y. Ne'eman and Dj. Sijacki, Int. J. Mod. Phys. A 2 , 1655 (1987). Dj. Sijacki and Y. Ne'eman, J. Math. Phys. 26, 2457 (1985). Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 1 5 7 , 275 (1985). Y. Ne'eman, in Spinors in Physics and Geometry, eds. A. Trautman and G. Furlan (World Scientific, 1987), p. 313.
512
SYMMETRY PRINCIPLES AT HIGH ENERGY FIFTH CORAL GABLES CONFERENCE JANUARY 24-26, 1968
Arnold Perlmutter, C. Angas Hurst, and Behram Kursunoglu EDITORS Center for Theoretical Studies University of Miami
1968
W. A. BENJAMIN, INC. New York Amsterdam
513
CHAIRMAN'S REMARKS: NUCLEAR PHYSICS IMPLICATIONS OF THE SPIN 2 MULTIPLET Yuval Ne'eman Tel-Aviv University Tel-Aviv, Israel I would like to follow Professor Teller's example—assume that he set an example for the morning chairraan-and make a comment which I would expect people to kill afterwards. This comment is one that has come to me in trying to answer the usual question about "what is useful in high energy physics?" I was thinking about the following problem: In a very simple-minded and naive picture of nuclear physics, there is the pion attractive force, and then there is the barrier which is called the hard core. One says that the deuteron, for instance, is held together by the pions, but the nucleons are held at some distance from each other by the fact that there is a hard core. In particle physics language, we have translated this by saying that this hard core is the co or other vector mesons. We now know that we have beyond that the spin-2 multipletthe f , for instance, which certainly has the right quantum numbers for nuclei(n-n,p-p) and provides an 149
514
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NE'EMAN
attractive force. So, in a way, we have perhaps here something that looks like a barrier that could be crossed, and afterwards, we would get again a bound state. Obviously, that would mean that you have an isotope of deuterium (via Ap exchange) , which would be at a very different energy. Now this is a very naive picture because we know that at these distances that picture may be very far from the relativistic treatment. But, on the other hand, some qualitative effects which are connected with the existence of this further attractive force should appear somewhere, and they would be the kind of thing that you would have to make by squeezing somehow the nuclei beyond this barrier. Perhaps you could see some effects in scattering or some of the moments might show...I suggest just that, because of the fact that the knowledge of the existence of the spin 2 mesons is a relatively new thing (we have been with them for about a year or two only) that we should send the news down to the low energy physicists' region and have them see what kinds of effects could be expected, from this additional piece of nuclear force. This is just a suggestion for comments in the same mood as Professor Teller's discussion yesterday. I don't know whether anybody has any immediate comments. If there are, Professor Breit suggested that we should not postpone the discussion because afterwards it becomes forgotten. If not, we'll go on to the first speaker.
515
SPIN 2 MULTIPLET
151
Breit - As I was saying just before the meeting, it would seem that whether one will see any special effect like an isotope or not, that at all events a higher spin meson at a short distance might affect the so called tensor force and therefore might effect the mixture of the triplet ~JD1 wave function to the S, wave function and that could perhaps be seen especially in the photo disintegration of the deuteron. In that reaction things do not check as well as they should; there are discrepencies even at quite low energies.
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C H A P T E R 7: M E T R I C - A F F I N E GRAVITY FRIEDRICH W. HEHL Institute for Theoretical Physics, University of Cologne Cologne, Germany Before we characterize shortly the reprinted articles of Yuval Ne'eman and his collaborators on metric-afnne gravity (MAG), we would like to explain some fundamental notions: • Coframe: Suppose that spacetime is a four-dimensional continuum in which we can distinguish one time and three space dimensions. At each point P, we can span the local cotangent space by means of four covectors, the coframe i? a = e^dx1. Here a, (3, • • • = 0,1,2, 3 are frame and i,j,--- = 0,1,2,3 coordinate indices; dx% provides a basis for the coordinate coframe. This specification of spacetime is the bare minimum that one needs for applications to classical physics. • Linear connection: In order to be able to formulate physical laws, we need a tool to express, for instance, that a certain field is constant. If the field is a scalar 0, there is no problem, the gradient di, if equated to zero, will do the job. However, if the field is a vector or, more generally, a world spinor (see Chapter 6, in particular {6.2}, and also [l; 2]) or an arbitrary tensor field ip, we need a law that specifies the parallel transfer of tp from one point P to a neighboring point P'. This law can be implemented by means of a linear connection Ta13 = T^dx1 ("affinity"). The field Tal3(x) with its 64 independent components has to be prescribed before the parallel transport of a world spinor or a tensor field tp can be performed and, associated with it, a covariant derivative be defined (whose vanishing would imply that the field is constant). The linear connection Ta@(x), shortly after the advent of general relativity, was recognized as a fundamental ingredient of spacetime physics (see [3]). The law of parallel transport embodies the inertial properties of matter. If one interprets coframe da and connection YaP(x) as gravitational potentials, then this framework for spacetime developed so far can be reconstructed as gauge theory of the affine group A(4,R) = R4,E>GL(4,R), i.e., of the semidirect product of the translation group R 4 with the linear group GL(4,R). • Metric: Experience tells us that there must be more structure on the spacetime manifold. Locally at least, we are able to measure time and space intervals and angles. A pseudo-Riemannian (or Lorentzian) metric gij = gji is sufficient for accommodating these measurement procedures. If gap denotes the components of the metric with respect to the coframe, we have g^ = e^e^g^. Nowadays there exists a definite hint that the conformally invariant part of the metric, the light cone, is electromagnetic in origin (see [4; 5; 6; 7]), that is, it can be derived from premetric electrodynamics together with a linear constitutive law for the empty spacetime (vacuum). Hence the metric, or at least its conformally invariant part, emerges in an electromagnetic context. Nevertheless, for general relativity and its gauge-theoretical extensions
517
a la MAG, the metric is (provisionally) considered to be a fundamental field and is as such a further gravitational potential. • Metric-afRne gravity (MAG): Accordingly, the coframe i? a , the linear connection Ya^, and the metric ga@ control the geometry of spacetime. The metric determines the distances and angles, the coframe serves as translational gauge potential (see also [8]), whereas the connection provides the guidance field for matter reflecting its inertial properties and it is the GL(4,R) gauge potential. The relations between gap, d01, and Ta^ have to be found out by means of the field equations of gravity. A Lagrangian field theory of gauge type describing matter and gravity within the geometrical framework mentioned, has been called MAG [9]. It is a framework for gravitational gauge theories. General relativity (possibly in its teleparallel version) is the simplest case. The next case is the Einstein-Cartan theory, a viable gravitational theory (see [10]), then the Poincare gauge theory with propagation metric and Lorentz connection follows (see [ll]) and eventually we reach the full metricafhne theory, see the article {7.8} and [12; 13]. Incidentally, simple (N = 1) supergravity is the Einstein-Cartan theory with a massless spin 3/2 Rarita-Schwinger field, the "gravitino", as source (see [14]). • Hypermomentum current: Suppose now that metric, coframe, and linear connection are basic field variables, then matter is embedded in such a metric-affme spacetime (via minimal coupling, e.g.). Typically we have fermionic matter that is described by world spinors. Similar as in general relativity, a hydrodynamic description of matter is possible by means of a hyperfluid [15]. The variational derivatives of the matter Lagrangian with respect to the field variables (the gravitational potentials), SL/d'd0' =: E Q , 6L/dga/3 =: aa/3, and 8L/5TaP =: Aap are the currents that represent the sources of gravity, namely the energymomentum currents (the canonical Noether current S Q and the metric Hilbert current aa/3) and the so-called hypermomentum current Aap (or rather the intrinsic part therefrom). Its trace A := A 7 7 is the dilation (or scale) current, the tracefree current ft.ap := A"^ — \ 5%A carries SL(4, R) charges. The SL(3, R) subset of these charges was mentioned by Yuval Ne'eman in his letter to us (see below) as those that were discussed earlier by Dothan, Gell-Mann, and Ne'eman (paper {2.3}) in the context of a dynamical group that could generate Regge trajectories in hadronic physics. It was Yuval Ne'eman who suggested to relate the hypermomentum current, that is, an object representing a gravitational source current, with measurable quantities in nuclear and particle physics, see his letter below and the chromogravity approach in Chapter 6, particularly {6.7} and {6.11}. We understand the GL(4, R) = R 1
518
article {7.2} basically conveys the message that the coframe, if parallel transfer is taken care of, can be understood as translational potential within MAG. In {7.3} a symmetry breaking mechanism is discussed leading from the affine to the Poincare group, see also [16; 17; 18]. {7.4} and {7.5} address the BRST transformations and the renormalizability of a metric-afrine model with spontaneous symmetry breakdown. In {7.6} it is remarked that the coframe is not degenerate (i.e., it can be inverted) in MAG. {5.12} builds a bridge between MAG and the superconnection discussed in Chapter 5. In {7.7} it is shown that the appropriate test "particle" for MAG is an object with a spin precessing (for measuring torsion) and with possible pulsations, i.e., mass quadrupole excitations (measuring the shearing of the light cone). {7.8} was already advertised as a thorough review on gauge theories of gravity. • Presently MAG is an appropriate and consistent framework for the formulation of gravitational gauge theories. So far it was not yet successful in singling out the one correct classical gauge theory of gravity other than the Einstein-Cartan theory. But we expect that the exact nature of the interrelationships between connection and metric on the one side and between hypermomentum and energy-momentum on the other side will be decisive for the further development of gravitational theory.
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References [1] I. Kirsch and Dj. Sijacki, Class. Quant. Grav. 19 (2002) 3157. [2] Dj. Sijacki, Class. Quantum Grav. 21 (2004) 4575. [3] L. Mangiarotti and G. Sardanashvily, Connections in Classical and Quantum Field Theory, World Sientific, Singapore (1999). [4] F.W. Hehl and Yu.N. Obukhov, Foundations of Classical Electrodynamics — Charge, Flux, and Metric. Birkhauser, Boston (2003). [5] D.H. Delphenich, On the axioms of topological electromagnetism, arXiv.org/hepth/0311256. [6] Y. Itin, Phys. Rev. D70 (2004) 025012. [7] C. Lammerzahl and F.W. Hehl, Phys. Rev. D70 (2004) 105022. [8] Y. Itin, Class. Quant. Grav. 19 (2002) 173. [9] Tensorial (bosonic) sector of MAG: F. W. Hehl, G. D. Kerlick, and P. Von der Heyde, Phys. Lett. B63 (1976) 446. Spinorial (fermionic) sector: Y. Ne'eman, paper {6.2} in this collection. Review: F. Gronwald and F.W. Hehl, On the gauge aspects of gravity, in: International School of Cosmology and Gravitation: 14 th Course: Quantum Gravity, held May 1995 in Erice, Italy. Proceedings. P.G. Bergmann et al. (eds.). World Scientific, Singapore (1996) pp. 148-198. [10] A. Trautman, The Einstein-Cartan theory, in: Encyclopedia of Mathematical Physics, J.-P. Franchise et al. (eds.). Elsevier, Oxford, 13 pages, to be published (2005) [http://www.fuw. edu.pl/~amt/ect.pdf]. [11] M. Blagojevic, Gravitation and Gauge Symmetries, IOP Publishing, Bristol, UK (2002). [12] Yu.N. Obukhov et al., Phys. Rev. D56 (1997) 7769. [13] E.W. Mielke and A.A. Rincon Maggiolo, Gen. Rel. Grav. 35 (2003) 771. [14] T. Ortm, Gravity and Strings, Cambridge Univ. Press, Cambrige, UK (2004). [15] Yu.N. Obukhov and R. Tresguerres, Phys. Lett. A184 (1993) 17. [16] E. A. Lord and P. Goswami, J. Math. Phys. 29 (1988) 258. [17] R. Tresguerres and E.W. Mielke, Phys. Rev. D62 (2000) 044004. [18] R. Tresguerres, Phys. Rev. D66 (2002) 064025.
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Handwritten letter, here typed in latex, uncorrected:
The Institute for Advanced Study Princeton, New Jersey 08540 Y. Ne'eman School of Natural Sciences 4 February 77 Dear Prof. Hehl, I happened to read your Rev. Mod. Phys article on U(4) theory and looked up your (+ colleagues) "hypermomentum" papers in Zeit. fiir Naturf. I noticed your remark about the role of traceless-hypermomentum in Elementary Particle Physics as being still unknown. We have indeed come across such currents. See Dothan, Gell Mann, Ne'eman, Physics Letters 17, 283 (1965) in our discussion of the SL(3,R) subset, which can be constructed in quark field theory from the time-derivatives of the gravitational quadrupoles. The model was tested successfully in some nuclei by Biedenharn & coll. a few years later. It is interesting that the 50(3) subgroup gives an "L" type operator which does not act on spins, presumably because <SX(4, R) doesn't. One way of getting this L for quark fields is described in M. Gell-Mann, Phys. Rev. Lett. 14, 77 (1965). I thought you might be interested. Yours sincerely Yuval Ne'eman P.S.: We realized these currents carry null charges, because of the equs of motion. In an (L,4,g) they could carry a real "charge".
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REPRINTED PAPERS OF CHAPTER 7: METRIC-AFFINE GRAVITY
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
F. W. Hehl, E. A. Lord and Y. Ne'eman, "Hypermomentum in Hadron Dynamics and in Gravitation", Phys. Rev. D17 (1978) pp. 428-433.
525
Y. Ne'eman, "Gravity is the Gauge Theory of the Parallel-Transport Modification of the Poincare Group", in Differential Geometrical Methods in Mathematical Physics II, Bonn 1977, K. Bleuler, H. R. Petry and A. Reetz, eds. Lecture Notes in Mathematics 676, (Springer Verlag, 1978), pp. 189-215.
531
Y. Ne'eman and Dj. Sijacki, "Gravity from Symmetry Breakdown of a Gauge Affine Theory", Phys. Lett. B200 (1988) pp. 489-494.
558
C. Y. Lee and Y. Ne'eman, "BRST Transformations for an Affine Gauge Model of Gravity with Local GL(4, R) Symmetry", Phys. Lett. B233 (1989) pp. 286-290.
564
C.-Y. Lee and Y. Ne'eman, "Renormalization of Gauge-Affine Gravity", Phys. B242 (1990) pp. 59-63.
569
Lett.
E. W. Mielke, J. D. McCrea, Y. Ne'eman and F. W. Hehl, "Avoiding Degenerate Coframes in an Affine Gauge Approach to Quantum Gravity", Phys. Rev. D48 (1993) pp. 673-679.
574
Y. Ne'eman and F. W. Hehl, "Test Matter in a Spacetime with Nonmetricity", Class. Quant. Grav. 14 (1997) pp. A251-A259.
581
Y. Ne'eman, "Gauge Theories of Gravity", Acta Physica Polonica B29 (1998) pp. 827-843. (Issue dedicated to A. Trautman).
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PHYSICAL
REVIEW
D
VOLUME
1 7,
NUMBER
2
15
JANUARY
1978
Hypermomentum in hadron dynamics and in gravitation F. W. Hehl and E. A. Lord Institute for Theoretical Physics, University of Cologne, D-5000 Cologne 41, Federal Republic of Germany
Y. Ne'eman** Tel-Aviv University, Tel-Aviv, Israel and Center for Particle Theory, Department of Physics, University of Texas, Austin, Texas 78712 (Received 31 August 1977) The infinite unitary irreducible spinor representations of the SL(3,K) algebra of hadron excitations are embedded in a global GA(4,R) with intrinsic dilation, shear, and spin pieces in its hypermomentum current (i.e., the affine generalization of angular momentum). When gauged over a space with a local Minkowski metric, GA(4,K) reproduces the metric-affine theory of gravity, in which the intrinsic hypermomentum is coupled to the connection, and the energy-momentum to the tetrad.
the components of N a s ladder operators 1 ) were interpreted a s bands of L excitations superimposed on the total quark spin, thus somewhat resembling the observed structure of the Regge trajectories. Some further physical understanding of the orbital N operators i s provided by their nuclear applications. 8 These involve a computational approximation in which one assumes that the spatial charge distribution i s the same as that of m a s s . The band structure appears to fit observations roughly, but the commutator i s far from saturated by the lower states, a fact which i s possibly due to the approximations. 7 For hadrons 8 the algebra reproduces the Chew-Frautschi plot 2 L=a + fiE2 asymptotically (i.e., for large L), and using the same approximations a s in the nuclear c a s e , yields plausible values for the electric radii.
I. THE SU3Jt) ALGEBRA OF HADRON EXCITATIONS It was suggested by Doth an et al.1 (whom we r e fer to as DGN) that if "• • • long sequences of fairly well-defined levels should emerge from an experimental study of baryons and mesons, one might very well wish to describe them by means of a noncompact algebra," and that the excitations involved might be related to s t r e s s e s causing deformations in an extended structure. The rotational bands in deformed nuclei with A J = 2 were cited as analogous: The appropriate algebra here i s the Lie algebra of SL(3,iJ), generated by the three orbital angular momentum operators and by the five time derivatives of the energy quadrupole operators, which generate shearlike deformations. It was suggested that a AJ = 2 relation for the Regge trajectories 2 could arise from a similar mechanism. In the study of extended structures the notion of infinite trajectories generated by noncompact spectrum-generating algebras (SGA) has since been further exploited in other directions, for example in dual models and strings. 3
The above scheme involves an "orbital" interpretation of the generators of SL(3,fl). The N generate (volume-preserving) shear strains. The operators L and N correspond to orbital hypermomentum charges [see Eq. (3.10) below]. Some of the experimental evidence s e e m s , however, to call for direct J excitations. If one plots the most recent mass-squared values of hadrons 8 against their spins, then in the corresponding Regge trajectories of given parity there s e e m s indeed to be a A J = 2 rule at work. 10 The reason for the orbital interpretation of the SL(3,ii) generators in DGN was chiefly that no half-integer representations of SL(3,.R) were known at that time. It is easy to see that SL(3,fl) has no finitedimensional half-integer representations, since the fundamental triplet representation has / = 1.
The model presented in DGN will be briefly summarized: The generators of SL(3,R) consist of three angular momentum operators T generating the compact SO(3) subgroup, and five noncompact generators N, which transform under SO(3) as a n / = 2 representation. Thus, the N connect different SO(3) representations at A
The unitary infinite-dimensional irreducible representations of the principal s e r i e s for St,(n,R) were described by Gelfand and Graev 11 in a functional form that i s inappropriate in the present context. One of the authors enlisted the help of Joseph who proved 12 that there e x i s t s a 428
17
525
17
HYPERMOMENTUM IN HADRON DYNAMICS AND IN GRAVITATION
with a completely symmetric set of s - 1 tensor indices and a single Dirac index, satisfying certain identities coming from the subsidiary conditions of the field equations. In momentum space, ^i is obtained from the D 1 / 2 representation of the subgroup GL(3,.R) of GA(4,ii). 16 The compact generators/ [M „] (fi, v= 1,2, 3) will be direct sums of spin matrices. The dilation operator / p p will also not connect different spins, while the noncompact generators /(,,„> - j6llvfpr' will connect spin s with spin s± 2. That is, we extend the little group SO(3) of (P, generated by/ [ ( J 1 / ] , by introducing six extra generators. When dealing with lightlike momentum, GL(3,i?) would arise from a similar extension of the null-plane little group E2 of
half-integer representation B 1 / 2 (/= a, I , . . . ) but that there is no multiplicity-free D 3/2 representation. The theory was further developed by BiedenharneiaZ. 8 andbyOgievetskiiandSokachev 13 , who supplied a refined construction of S>1/2. Recently, one of us 14 gave a detailed discussion of the bivalued representations of the group of general coordinate transformations also from a topological point of view. Having thus reviewed the fundamental importance of the SL(3,fl) transformations for hadronic matter, we combine it with scale and Poincare transformations, thereby arriving at the general affine group GA(4,i?). II. THE GENERAL AFFINE GROUP GA(4,«) This group is the semidlrect product of the general linear group GL(4,fl) and the translations t. Its Lie algebra is defined by the commutation relations [/aB,//] = 6aS/rS-6r7a6,
(2.1)
[ / « , / / ] = B«7 r -
III. THE CANONICAL HYPERMOMENTUM CURRENT-INTRINSIC AND ORBITAL
The affine group GA(n,R) can be derived by contraction from the semisimple group GL{n+l,R). The contracted group has (n +1)2 generators, n2+n of them generating GA.(n,R). The r e maining n +1 (which we denote by ea and e), together with the translation generators/„, generate an w-dimensional Heisenberg algebra [fa>eB] = 6a6e, with e commuting with the entire contracted group. 15 This derivation of the affine group indicates that some of the Casimir operators and labeling characteristics of GL(5,ii) would be expected to be preserved in GA(4,fl). We now consider the infinitesimal action 64> = xaaa0+\aB(xBda+faB)4,
429
We now consider a simple special-relatlvlstlc Lagranglan model Involving fields (or polyflelds)
(?r J 38£/a y 0)
(3.1)
and that associated with GL(4,.R) Is the hypermomentum current11 ,la
(2.2)
of the group GA(4,R) on fields <J> in a space-time with a local Minkowski metric. The existence of the metric singles out the Poincare^ subgroup
yi", = Ai", + Alih,
(3.2)
which consists of an orbital piece Ai'^-x'S,*
(3.3)
and an intrinsic piece A, >* = -*»//<#>•
(3.4)
The currents satisfy the conservation law 3,V =0
(3.5)
and the quasiconservatlon law17"19
a>r,"• = - ,
where i/>, is a unitary spin-s representation of the Poincare subgroup. The components of if, can be characterized, for instance, as quantities il>aB...
where au Is a symmetric tensor defined by the response of the Lagrangian density to strain:
526
(3.6)
F. W. H E H L , E. A. L O R D , A N D Y. N E ' E M A N
430 \ilOl'
= 6£,-\il(x>B,+6ll)£.
tum. The trace of the same equation shows that the dilation current (T( =T,,*') is not, in general, conserved. In the domain of asymptotic freedom, we would have an approximate scale invariance (CT,' = 0) which then leads to a conserved dilation current. The divergence of the intrinsic dilation current would then be the trace of the energy-momentum tensor 3tA* =T„*. Intrinsic GL(4,fl) invariance, associated with the conservation of the intrinsic hypermomentum currents, may well be an approximate symmetry of the asymptotic freedom regime in quantum chromodynamics. We know that scaling arises as a logarithmic approximation, and a similar situation may describe spin independence [observed approximate SU(6)] and the A./= ±2 excitation bands. We then have a unified description of these three phenomena17-18; they are manifestations of a single current, the hypermomentum current. This suggests a link with gravitation, since the intrinsic hypermomentum current is coupled to the linear connection of space-time In a very natural generalization of Einstein's theory. In the spirit of current algebra, this determines its matrix elements, just a s the coupling to the metric field determines the matrix elements of the energy-momentum tensor. This generalization is the metric-affine theory of gravitation. 17 " 19 In the following section we show how the metric-affine theory arises as a gauge theory of GA(4,fl).
(3.7)
The charges associated with the currents are the momentum P^fd'xE,0
(3.8)
and the total hypermomentum T i ' = f d\
T,j0
(3.9)
consisting of orbital hypermomentum A.'s j d'x A,-"^- jd3x
x'E?
(3.10)
and intrinsic hypermomentum
A.'^Jrf'x A,'° = - jd3x * 7 . V
17
(3.11)
[For an ordinary Dirac field, the quantities A'""' are time derivatives of the energy quadrupoles. Such an interpretation is no longer possible for polyfields. ] Under the assumption of canonical equal-time commutation relations for 0, the intrinsic hypermomentum generates the intrinsic GL(4,iJ) s , and the three-space components of total hypermomentum and linear momentum generate the subgroup GL(3,R) consisting of dilations, shears, and rotations of the matter fields. In the spirit of current algebra, a reasonable hypothesis is that the hypermomentum and momentum of hadronic matter obey these same commutator algebras, and that the hadronic currents satisfy (3.5) and (3.6). GL(3,fl) commutes with P0, and can therefore be considered as an approximate rest symmetry—we have no trouble with "no go" theorems. 20 Note that it is the existence of the infinite-dimensional spinor representations of GL(3,fl) that enable us to extend the concept of intrinsic spin to intrinsic hypermomentum, for fermionic matter. We now have an alternative interpretation of the Regge trajectories, in which the quark D 1 / 2 is interpreted as the sequence (
IV. THE AFFINE GAUGE THEORY WITH LOCAL MBSKOWSKIAN STRUCTURE The metric-affine gravitational theory is based on a space (L4,g) in which the components of the metric g(j and the connection r , / (not necessarily symmetric) are regarded as 74 independent fields in a variational principle. The gravitational Lagrangian density Is a scalar density V constructed from these components and their derivatives. The derivatives of "matter fields" occurring In the rest of the Lagrangian density are covariant derivatives constructed from the connection r (> *. Thus we have a minimal coupling hypothesis that universally couples the connection to matter. Only gauge fields (electromagnetism, gluons, etc.) are not coupled to the connection. Alternatively, the metric-affine theory can be arrived at by generalizing a global affine group GA(4,fl) to a gauge group, over a metric spacetime with a local Minkowsklan structure. To establish the notation, consider first the usual Yang-Mills theory of an unspecified Lie group G, with generators/, satisfying [fA,fB]= cAB%. Consider the action of an Infinitesimal element n = ixAfA of the gauge group G combined with an
527
17
HYPERMOMENTUM IN HADRON DYNAMICS AND IN GRAVITATION
infinitesimal coordinate transformation x ' - x 1 ' = xx - I 1 . For a set of polyfields or fields
the same way, the affine gauge theory derived here is the metrlc-affine theory. Rewriting (4.7) in terms of the parameters fi„ B , we find just the behavior of the anholonomic components of a field 0, of a tetrad, and of a connection, under coordinate transformations and space-time-dependent linear tetrad deformations. As shown by one of us, 19 the metrlc-affine gravitational theory can be formulated as a theory Invariant under such tetrad deformations. If the tetrad is chosen orthonormal, we find that
(4.D
Introduce the connection one-form for G (r,dx' = Tt*fAdx'), the corresponding covariant derivative operator d{
431
(4.2)
and the gauge fields
F i>s [d () d ( ]=3 ( r,_3 y r i + [r j > r,]. In terms of the parameters X = \i - ^Vit the transformation laws 6
(4.3)
vC£
-g
Let £ be a Lagrangian density, dependent on <j>, r , , and a metric glt (and derivatives of these quantities). Invariance of £ under coordinate transformations and space-time dependent G transformations leads to the identities (4.5)
where 5£
-g
ten
6S
ai- o r ?
6£/or, a
B
are the canonical energy-momentum and the canonical intrinsic hypermomentum current of the field 0, which are now defined dynamically as the currents that couple to the gauge potentials of GA(4,iJ). In a Mlnkowski-space approximation with eta = 6j a , Tin 8 =0, the transformation law of
6r, = -d,\ + t'F„.
2^
(4.8)
we have (4.4)
U"»iil-c«cr),lc,=oI
6£/6e,a,
(4.6)
6V/&eia=-2kS=ixai,
We now simply take G to be the 20-parameter group GA(4,R) whose Lie algebra is defined by (2.1), and identify the translational part of the group with the operation of parallel transport in space-time. This means that the connection of GA(4,ii) becomes a Cartan connection.21 We obtain a tetrad e, a = r j a and an anholonomic linear connection TlaR. Algebraically, the identification of the translations with parallel transport is expressed by | a = - \ a (i.e., ji a = 0). Since linear momentum, unlike angular momentum, has no intrinsic part, we also s e t / o = 0 for the field 0. Then the equations (4.4) become
(4.9)
61)/6r,aB=2fe^A8°".
The holonomic description is obtained by choosing e,a- 6,01 and takingg {] and r , / as the independent variables. Defining torsion and nonmetriclty to be S , / = r [ U ] * and Q ( r t s - ? , # , » , respectively, the connection can be written
r M *={},}-M,/ + *Q,A
(4.10)
M
ijn = -Siik + Sm - s »(y -Quuli • The tensor Mut = -Mfki is the contortion. The spin current and the intrinsic dilation + shear current are coupled to contortion and nonmetri city, respectively, in this formulation:,
64> = (A?/S + ^ V a ) < / , ,
&V/Mkll =-2kS^g£>.Zink
(4.7) B
,
(4.11)
Br I B = « , > - ( - v y x ; + « * F r 4 B » ) , where V a is the covariant derivative operator associated with the homogeneous part of the group GL(4,R). We have precisely an affine generalization of the Poincare gauge theory. 5 ' 22 The Poincarg gauge theory, with a particular choice of the Lagrangian for the gauge fields, is identical with the U4 gravitational theory of Sciama and Kibble.23 In
(See Ref. 17 for details; compare also Ref. 24.) With the gravitational Lagrangian V =f^g(R + PQiQ') (Qi=kQn,k), nonmetricity does not propagate outside matter, so that the comments of Hayashi 25 (reproducing the Einstein-Weyl dialogue) will not apply. It is interesting to note that an affine-metric
528
432
F. W. HEHL, E. A. L O R D , A N D Y. N E ' E M A N
17
theory o f / - g r a v i t y is p o s s i b l e , in which the spinor fields a r e nonlinear r e a l i z a t i o n s of global GA(4,i?) r a t h e r than infinite-dimensional l i n e a r r e p r e s e n t a t i o n s . The Goldstone bosons associated with the spontaneous breakdown of GA(4, fl) s y m metry to Poincare' s y m m e t r y would have spin two and spin z e r o , and give r i s e to a m e t r i c . Howe v e r , such a scheme is not consistent with the p r e s e n t approach: It is an alternative possibility for linking the metric-affine theory with particle physics in which Eq. (3.6) is interpreted as a " p a r t i a l conservation of s h e a r and dilation c u r r e n t s . " There would be a formal r e s e m b l a n c e to the work of Ogievetskii and Borisov, 2 8 except that the affine g r o u p h a s a different interpretation. Since their affine group i s g e n e r a t e d by the linear p a r t of the infinite gauge algebra of the coordinate transformations, it could not have the dynamical role that they assign to it; t h e r e a r e no conserved Noether c u r r e n t s for such t r a n s f o r m a t i o n s . (Note that a s y m m e t r y with Goldstone-type spontaneous breakdown c o r r e s p o n d s to a limit in which c u r r e n t s a r e conserved though the vacuum is not invariant.) In the Sciama-Kibble theory, the Riemannian
s p a c e - t i m e of E i n s t e i n ' s theory is generalized to a U 4 , so a s to i n c o r p o r a t e the s p i n - c u r r e n t dynamically a s a s o u r c e of torsion. T h e r e now a p p e a r to exist s i m i l a r phenomenological a r g u m e n t s for a corresponding t r e a t m e n t of t h e i n t r i n s i c dilation and s h e a r c u r r e n t s that give r i s e to nonm e t r i c i t y . We hope that this note h a s clarified the theoretical and phenomenological consequences of this possibility, and shown how the m e t r i c affine t h e o r y of gravitation w ith its ( i 4 , g) s p a c e time would then provide an a p p r o p r i a t e minimal coupling.
•Research supported in part by the United StatesTsrael Binational Science Foundation. tResearch supported in part by the U. S. Energy Research and Development Administration Grant No. E(4P-1)3992. *Y. Dothan, M. Gell-Mann, and Y. Ne'eman, Phys. Lett. 17_, 148 (1965); Y. Dothan and Y. Ne'eman, in Symmetry Groups in Nuclear and Particle Physics, edited by F. J. Dyson (Benjamin, New York, 1966), p. 287. 2 G. Chew and S. Frautschi, Phys. Rev. Lett. 7, 394 (1961). 3 G. Veneziano, Nuovo Cimento 57A, 190 (1968); see also G. Veneziano, Phys. Rep. 9C_, 199(1974); J. Scherk, Rev. Mod. Phys. 47, 123 (1975). i M. Gell-Mann, Phys. Rev. Lett. 14, 77 (1965). 5 We take the conventions of F . W. Hehl, P. von der Heyde, G. D. Kerlick, and J . M. Nester, Rev. Mod. Phys. 48, 393(1976). The holonomic (coordinate) indices i, j , k,. .. and the anholonomic (tetrad) Indices a, p, y,... run from 0 to 3. Greek indicesix, v,... are spatial indices 1, 2, 3. We use (ij) = \(ij +ji) and [ij] s i('J — ji) for symmetrization or antisymmetrization of indices. s O. L. Weaver and L. C. Biedenharn, Phys. Lett. 32B, 326 (1970); Nucl. Phys. A185, 1 (1972); R. Y. Cusson, ibid. A114, 289 (1968). 7 A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. II, Nuclear Deformations (Benjamin, Reading, 1975), pp. 410-412. 8 L. C. Biedenharn, R. Y. Cusson, M. Y. Han, and O. L. Weaver, Phys. Lett. 42B, 257 (1972) and Errata. For a recent set of SL(3,J?) xSU(6) assignments, see for ex-
ample Dj. Sijacki, ibid. 62B, 323 (1976). 'Particle Data Group, Rev. Mod. Phys. 48, SI (1976). U L excitation would have produced many more recurrences of the same parity at intervals A J = 1. For the nucleon (939; | + ) one does see a state (1810; f + ) in addition to the "traditional" (1688; | + ) , but it could just belong to some other representation. Note that the fit of Regge parametrizatlon in the scattering region is best understood as the continuation of a trajectory, and not of a system of bifurcations resembling a cosmic-ray shower. " I . M. Gelfand and M. I. Graev, Izv. Akad. Nauk SSSR, Ser. Mat. 17, 189 (1953). 12 D. W. Joseph, University of Nebraska report, 1970 (unpublished). 13 V. I. Ogievetskii and E. Sokachev, Teor. Mat. Fiz. 23, 214 (1975) [Theor. Math. Phys. (USSR) 23, 462 (1975)]. See also Dj. Siijacki, J . Math. Phys. 16, 298 (1975); and Y. Guler, ibid. 18, 413 (1977). U Y. Ne'eman, Proc. Natl. Acad. Sci. (USA) (to be published) . 15 Thus the ea behave like abstract "coordinates." 16 Note that the usual comment found in many textbooks and reviews that "the group GL(4,«) [orGA(4,fl)] has no spinor representations," relates to /int'ie-dimensional representations. A corresponding detailed discussion is given in Ref. 14. " F . W. Hehl, G. D. Kerlick, and P. von der Heyde, Z. Naturforsch. 31a, 111 (1976); 31a, 524 (1976); 31a, 823 (1976). 18 F. W. Hehl, G. D. Kerlick, and P. von der Heyde, Phys. Lett. 63B, 446 (1976).
ACKNOWLEDGMENTS This p a p e r was finalized while we enjoyed the hospitality of the Institut des Hautes Etudes Scientifiques in B u r e s - s u r - Y v e t t e , and we all owe p a r t i c u l a r thanks to P r o f e s s o r L . Michel for inviting u s . We would like to acknowledge i n t e r esting c o n v e r s a t i o n s with S. Coleman, T. Regge, and C. N. Yang, and a l s o with P . von d e r Heyde and L . Smalley. E . A. Lord is grateful to the Alexander von Humboldt Foundation for the award of a fellowship.
529
17
H Y P E R M O M E N T U M IN H A D R O N D Y N A M I C S A N D IN G R A V I T A T I O N
"E. A. Lord, Univ. of K&ln report, 1977 (unpublished). 20 S. Coleman and J. Mandula, Phys. Rev. 15J9, 1251 (1967); R. Haag, J. T. Lopuszanskl, and M. Sohnlus, Nucl. Phys. B88, 257 (1975). 21 See, for example, S.Kobayashl, Transformation Groups in Differential Geometry (Springer, Berlin, 1972); also Y. Ne'eman and T. Regge (unpublished). 22 P. von derHeyde, Phys. Lett. 58A, 141 (1976); Z. Naturforsch. 31a, 1725 (1976); see also Lett. Nuovo
433
CImentol4, 250 (1975). D. W. Sclama, in Recent Developments in General Relativity (Pergamon, Oxford, 1962), p. 415; T.W.B. Kibble, J. Math. Phys. 2, 212 (1961). M L. L. Smalley, Phys. Lett. 61A, 436 (1977). 25 K. Hayashl, Phys. Lett. 65B, 437 (1976). M A. B. Borlsov and V. I. Oglevetskli, Teor. Mat. FIz. 21, 329 (1974) [Theor. Math. Phys. (USSR) 21, 1179 (1974)]. 23
530
GRAVITY IS THE GAUGE THEORY OF THE PARALLEL - TRANSPORT MODIFICATION OF THE POINCARE GROUP
Yuval Ne'eman * Tel-Aviv University, Tel-Aviv, Israel
Abstract
We prove that only the Dynamically - Restricted Anholonomized General Coordinate Transformation Group reproduces Einstein's theory of Gravitation directly when gauged. This amounts to a Modified Poincare" group where translations are replaced by Parallel transport.
We also explain the role of GL(4R) and explore the Modified Affine Group.
Using the Ogievetsky theorem, we present several No-Go theorems restricting the joint application of Conformal and Affine Symmetries.
1.
Introduction: Gauge Theories
The first local gauge invariance principle (LGIP, or just "gauge") to be suggested [Weyl, 1919] dealt with dilations, and was introduced as an addition to Einstein's Gravity.
H. Weyl was looking for a geometrical derivation of Electromagrat-
ism, which would thereby also "unify" it with Gravitation.
His first theory invoked
dilation invariance, and failed at the time since macroscopic evidence appeared to be clearly in disagreement with such a postulate.
This particular theory has re-
cently been revived at the quantum level as a gauge invariance with "spontaneous breakdown" [Englert et. al., 1975].
The geometrical derivation itself was revived
after the advent of quantum mechanics as a U(l) gauge [Weyl, 1929] i.e. a locally dependent phase for complex charged matter fields instead of scale invariance. would now render it as a Principal Bundle B
as
The gauge transformations are given by
the set of Bundle automorphisms whose action onW. point x «OT- invariant.
We
is the identity, i.e. leaving a
They thus act only in the fiber above that point, and can
be written as g(x), g c G.
They belong to the "stability group" of translations in
This abstract "internal" gauge invariance was H. Weyl's second definition, and it won wide acceptance.
Three decades later, it served as a model for the (G = SU(2))
local Non-Abelian internal gauge of C.N. Yang and R.L. Mills [1954; see also Shaw 1954].
*
The method was further generalized [Ne'eman 1961, Gell-Mann 1962, Salam and
Partially supported by the U.S. - Israel Binational Science Foundation.
531
190
Ward 1961] to SU(3) and in p r i n c i p l e t o any Semi-simple group [Gell-Mann and Glashow 1961, Ionides 1961]. In recent y e a r s , t h i s SU(3) universal (and therefore gaugelike) coupling which i s indeed observed in the coupling of hadrons t o massive vectorPC mesons (the p , u, <j>, i|), Y, with J = 1 ) has t o be regarded as a pole-dominance approximation for phenomenological vector f i e l d s [Gell-Mann, 1962]. On t h e o t h e r hand, an SU(2)
-
x U(l) LGIP involving a subgroup of t h a t SU(3) but acting on lep-
tons and on SU(3) invariant quarks as well i s favored as a Weak and Electromagnetic Unified Gauge [Weinberg 1967, Salam 1968] (though other groups are s t i l l p o s s i b l e ) , and an SU(3)
.
LGIP i s believed to represent the quark-glueing [Nambu
1965,
Fritzsch and Gell-Mann 1972, Weinberg 1973] (and confining?) p a r t of the Strong Interactions.
Those applications have become serious candidate
dynamical t h e o r i e s
since the achievement of G. ' t Hooft and M. Veltman f t Hooft 1971, ' t Hooft and Veltman 1972, Lee and Zinn-Justin 1972] in completing the renormalization of the Yang-Mills i n t e r a c t i o n [Feynmann 1963; De Witt 1964, 1967; Faddeev and Popov 1967; Fradkin and Tyutin 1970; Veltman
1970], including the case of "spontaneous break-
down" [Higgs 1964a, 1964b, 1966;
Englert and Brout 1964; Guralnik e t al 1964; Kibble
1967] of the local gauge coupled with a Goldstone- Nambu r e a l i z a t i o n of the global symmetry [Goldstone 1961, Nambu and Jona - Lasinio 1961].
For the Strong I n t e r a c t -
ions, renormalization has also led to the discovery of Asymptotic Freedom [ P o l i t z e r 1973, Gross and Wilczek 1973] which seems p a r t i c u l a r l y f i t t i n g for short range quark i n t e r a c t i o n s , and appears to support the SU(3)
1
gauge idea.
We review t h e high-
l i g h t s of a Yang-Mills type gauge. The dynamical variables in a B(#T, G) gauge theory may include matter f i e l d s (quarks) q (x) which are generally represented as sections of a vector bundle E a s sociated to B, E = B xG A(G) where A(G) i s the ( 3 x 3 for quarks) appropriate r e p r e s e n t a t i o n of G on q a : ( M g 0 ( x ) ) , q(x) o g = ( b ( g o ( x ) g ( x ) ) ,
A(g - 1 )q(x))
The covariant derivative in E involves matrix connections ( p o t e n t i a l s ) PCX) = p^
A
(x) XA dx u
(1.1)
where X. i s the Lie-algebra of G in the A(g) r e p r e s e n t a t i o n . t i v e in E i s then (D q ) a = d q a - ( p ) \ q b
The covariant deriva-
(1.2)
532
191
and t h e dynamical theory i s derived by t h e replacement
V - (Vba
<
(1 33
-
known as a "minimal" or "universal" coupling.
SS
Indeed, with a free Lagrangian
= - q a (YM 3 p + ») q a
0
the unwanted c o n t r i b u t i o n due t o 3 — - %
Y
U , -1 . Cg \
(1.4) g f 0
> a b g
q
\
i s cancelled by "I
r +
P
(g P 8
"1 +
g
j
•>
dg)
For an i n f i n i t e s i m a l transformation
(1.5) (A(g)),
a
u
= <$a + ( i c t X . ) D
a
t h e unwanted 3 a A
AD
y
term a r i s e s in 8
, / " i - ^ CXJ. 3 q b = 3 a A A b 3(3 q a ) » V
V
where J
J /
A
is the Noether current, satisfying a covariant conservation law
D JA = 0
(1.6)
The curvature
R
= (d p - p . p) = (dp - y [p, p]) = (RA X A )
(1.7)
similarly satisfies the Bianchi identity
(DR) = 0
(1.8)
The equations of motion are
(D *R )
= *J A
(1.9)
where * stands for the duals
533
192
The
*R A = ^ E R A nv 2 0Tyv ax
(1.10a) V '
*J A = 1 e JA pat 6 vyox v
(1.10b) '
equations of motion can be used to turn (1.6) into a non-G-covariant conservat-
ion law for a new current
d J^ = 0
(1.11)
where J ' w i l l include contributions from the p more problematic in Gravitation.
p o t e n t i a l s themselves.
This w i l l be
Connections, covariant d e r i v a t i v e s and curvatures can also be introduced i n B i t s e l f , where they w i l l r e g u l a t e t h e i r own gauge invariance (no " s o u r c e s " ) . The matrices X, w i l l now belong to the adjoint r e p r e s e n t a t i o n s , CX
A ° = " CABC
The d e f i n i t i o n s are RA = d p A - h PB - PC CBCA = 3spB . p C RBCA
(1.12)
(Dp)A
= dp
(1.13)
(DR)A
= 0
A
-p
B
.p
C
CBCA
(1.14)
using contractions with v e c t o r - f i e l d s D., D
A = AA " \
pA
C1'15)
and with the r e s u l t i n g commutator (from double contraction of ( 1 . 1 2 ) ) , [D
where C
A'
V -
_ = 0 in/ft
( c C AB
+ R
°AlP
D
C
C1.16)
but not in "Superspace" 31
as we s h a l l l a t e r s e e .
Notice t h a t in the adjoint representation, (1.13) can also be w r i t t e n as Dp = dp - [p, p] and i s not equivalent to R. This i s due to the antisymmetry of CBC or (-X B ) C in the (B,C) i n d i c e s , as against ( A g ) ^ in (1.2) for (p B X^)^ q where t h e r e i s no such link between B and b . The antisymmetry implies a f a c t o r 2 in contracting with (% dx y . dxW) as against the curl dp.
534
193
2.
The F i r s t Step.-
Gauging t h e ( i n t r i n s i c ) Lorentz Group
We f i r s t r e t u r n t o Gravity when R. Utiyama [1956] attempts t o derive t h a t t h e o r y from a Gauge P r i n c i p l e .
Since not much was known a t t h e time about t h e renormaliz-
a b i l i t y of Yang-Mills LGIP t h e o r i e s , t h i s was i n t h e main an a e s t h e t i c u r g e .
Utiyama
gauged the (homogeneous) Lorentz group G = SL(2,C) = :L using t h e equivalent of conn e c t i o n one-forms
P
3
= P
1J
dxU
( i , j = 0 , 1 . . . 3 i n a local frame; y = 0 , 1 . . . 3 holonomic
However, t o reproduce E i n s t e i n ' s theory i t appeared t h a t he had t o introduce c u r v i l i n e a r coordinates, and a s e t of 16 "parameters" A.
V
x
( ).
(2.1)
a-priori
These were i n i t i a l l y
t r e a t e d as given functions of x and l a t e r became f i e l d v a r i a b l e s , t o be i d e n t i f i e d with orthonormal vector f i e l d s A, r e c i p r o c a l t o a v i e r b e i n frame, p 1 (Aj^ = 6*
the p
(2.2a)
thus arising as vierbein fields, with (r\. . in the Minkowski metric) n. . P 1 p •" = g
(x)
Still, the relationship of p
(2.2b)
, to the Christoffel connection r
was incomplete,
since the formula he derived was forced by an arbitrary assumption to select only (ji v) symmetric contributions to r
. A s we shall see, this role of the Connection
("Affinity" in holonomic - "world tensor" - language) as a Gauge Potential has since been perfected.
However, it contrasted sharply with the physical intuition of work-
ers in Gravitation [e.g. Thirring 1977] who regard the metric (or vierbein) as the Gravitational Potential, and consider the Connection as the analogue to the Field Strength in Electrodynamics. Sciama [1962] and Kibble [1961] continued Utiyama's project.
Although they were
aiming at a full Poincarfi gauge (G = ISL(2,C):=P), their main achievement consisted in clarifying the Lorentz gauge. They showed that this consisted only in the stability group overJtl
, i.e. the "internal" action of H = SL(2,C) = :L, which we generally
describe as the Spin of the Matter fields (though it does not include contributions to physical spin due to the holonomic - "Greek" - indices of gauge fields, curvatures etc., i.e. in particular, the photon or the Yang-Mills' fields own spins).
This
"Latin" or "anholonomic" spin s. . gives rise to a new interaction term, in which it is minimally coupled to the connection p
^.
535
194
A
S
= h ^ / ' V -l"13**^
(2.3)
where *S-i is a dual three-form *S.. = i
S.. T dxv - dxp - dxa
E
(2.4)
Indeed, t h i s a r i s e s when we perform the replacement « . " 3 H-A. P D 1
u '
1
(H)
y
, »
D
(H)
= 8
y
lj
+hP
y
£..
li
(2.S)
lj
where ^f. . i s a representation of the Lorentz generators, appropriate for a c t i o n on the $ matter f i e l d in<&M0|i, 3 i|i)
3
-__«
£„•-:.,„"
(2.6)
u The factor e = det p
1
a r i s e s i n the replacement d x |—• e d x of the matter a c t i o n However, the v a r i a t i o n of the action by <5 p i i also receives a c o n t r i b u t i o n -3 2 from the Einstein free action (X: = 8irc G, G being Newton's constant; [K] = [L ] in " n a t u r a l " units)
measure.
A= i
F ^ - P ^ ^ ' i J U - i r l ^ ^ i J
C2.7,
so that one has a new equation of motion (besides Einstein's) involving three-forms Rk - pZ e... . = - i\ *s.. ljkS. 2 ij
(2.8)
In these equations, R J and R are the curvature two-forms, with R1* = dx" - dxv R--ij = dp^ r.i
JV
j V „ i
, i
+
p 1 k . p k £ = % P k . P l R^- « i
k
„(L)
i
,k
(2.9) *
n
i <••> m i
R = dx -* dx R- =dp + p , ~p = Dv ^p = %p >. p R, (2.10) In conventional nomenclature, R i s the Riemannian curvature and R the Cartan " t o r s ion". For empty space, (2.8) becomes R = 0 and solving (2.10) for p . (in D p ) then produces the Christoffel symbol formula. However, when (Latin) spinning matter i s present, solving for p , w i l l produce in addition an antisymmetric contribution to the Christoffel connection.
536
195
C o n t r a c t i n g R w i t h two v e c t o r - f i e l d s we f i n d , k
Rr:
U
= A.
= i.^,
V
A.
k
R
= ( A . , A . , Rk)
A . , (dp
= a::
+ p
k
+ ^(p.
(2.11)
- P1))
x
6.1 - p.
k
l
6.
)
Thus, R
ij
k
= a
k + h
ij
(p
i
k
j " pj
k
i:)
C2 12)
-
The doubly contracted exterior derivative ii:: holonomity", [Schouten 1954; Hehl et. al the tangent space) we can lower th the
has been called "the object of An-
1976a],
Using the Minkowski metric (in
index, and remembering that the antisymmetry
of the Lorentz generators imposes
p. . . = I k 3
- p. . . I j k
we can extract p. . ,
(2.13)
,
p. . . =: a.
. , - a. ,
. + a.
. . + R. , . - R. . .
- R. . ,
(2.14)
The last three terms, making up together the "Contortion tensor" K. . . , vanish for k i J K R = 0 and represent the contribution of ("Latin") spinning matter when present. i j k
-
jki
K
i j "
i j k
Inserting the last expression for R
in (2.10) into (2.8), and replacing the
holonomic index in S..
s::
k
= p-
k
s..
V
(2.15)
we g e t t h e e q u a t i o n of m o t i o n ,
R. . i ]
k
- 6k i
R i
. ]
x
- Sk R.
X
= : T. .
1 1 1
i j
k
= k s.
k
. l
(2.16)
j
T:: is sometimes named the "Modified Torsion". We can also contract the upper il k (naming) index of the torsion tensor R in (2.11)
537
196
R
= A. k
y v I f we now i n s e r t R
R pv
- A. K
p p
J
p v
R. . x j
( 2 . 1 2 ) we f i n d
P = .
h
P .
(p
P) = .
p
h
(K
P - K
p
)
(2.17)
Holonomically, torsion thus corresponds to the antisymmetric part of the connection. Note that these are not the indices which are antisymmetric in the anholonomic connection due to the Lorentz gauge.
Returning to the equation of motion we derived,
we note that Eq. (2.16) being algebraic (due to 2.17)
rather than differential (due
to the particular choice of the Einstein Lagrangian which is linear in the canonical momenta), the connection potential p
•* does not propagate.
Instead, like a gauge
connection in a current-field identity [Lee et al 1967] it is replaced by the spin-cur2 rent itself, so that (2.3) becomes a spin-spin term with very weak coupling k , a contact term.
Sciama and Kibble thus rediscovered Cartan's modification [1922-25]
of Einstein's Relativity.
At the same time, this can be regarded as a "first order"
or Palatini [1969] formalism for that theory (independent variations for p It then differs from it by that k
s 1 s. term only [Weyl 1950].
and p
J
).
This theory, further
analyzed by Hehl [1970] and by Trautman [1972] is known as the Einstein - Cartan
-
Sciama - Kibble theory (or U. theory), and is thus indeed derivable in its spin-torsion parts from a Lorentz gauge.
538
197 3.
D i f f i c u l t i e s in Gauging
T r a n s l a t i o n s ; Pseudo-Invariance
The attempt to reproduce Gravity had of course to come t o grips with the main p a r t of the theory - the universal coupling of the Energy-Momentum t e n s o r - c u r r e n t t o the g r a v i t a t i o n a l p o t e n t i a l [ i . e . to the metric g t o the p
or i n a vierbein formalism,
of ( 2 . 2 ) ) . Indeed, varying p l i n (2.7) y i e l d s E i n s t e i n ' s equation for
empty space, R1^ » p k e . . , „ = 0
(3.1)
which becomes, in holonomic language, a f t e r some manipulations, _ Jj R 6U
RV
=o
(3.2)
In the presence of matter we have R l j - p k e . . . „ = k *t„ K
where *t
i]kfc
(3.3)
1
*•
'
i s the energy-momentum current 3-form *t
= 1 e. l
6
for the density t
pJ
-
k p
-
m
t1 ,
P
i]km
(3.4)
*•
^
. .
The Sciama-Kibble approach f e l l short of a t t a i n i n g t h i s goal by a gauge p r i n ciple.
Kibble noted t h a t the Lorentz-gauge invariance having been ensured by the
covariant derivative ( 2 . 5 ) , the remaining unwanted gradient term corresponding to t r a n s l a t i o n s i s a homogeneous term, in c o n t r a d i s t i n c t i o n to the Yang-Mills case, 6(D v
(L)
U
W
= Jse i j f.. lj
(D v
(L)
U
3 5V D ( L ) I(U V
I|0L *-
its removal i s achieved by a multiplicative application d i t i v e construction.
r a t h e r than by the usual ad-
Indeed, taking
k = \ ^ Du(L} •
D
V = " ** k V
+
\ «" \
V
^
yields 6Dk* = h e l j £±. D ^ - e 1
k
D^
Kibble thus a t t r i b u t e d to (the vector field) A.
539
(3.6) v
the r o l e of a t r a n s l a t i o n gauge
198 field, with £ w as the translation parameter.
This fitted an analysis of the action
of the Poincare" group on fields, in which the intrinsic Lorentz action was given anholomic indices, but where all the rest (both orbital angular momentum action and translations) was incorporated in the General Coordinate Transformation and represented holonomically,
^
6* = h eiJ f.. *
- e * z
V
vx
V
• eP
(3.7)
with in addition
S
ili = -lv
3 * +h
E1^
f- • *
(3.8)
The separation (3.7) in which the "orbital" action of E was incorporated in £
appeared as e
and
corresponds indeed to the Fiber Bundle picture, in which the
gauged group is the stability subgroup of P, the Poincare' group.
However, the as-
signment of E,v to coordinate transformations precluded any form of gauging for translations.
A variation 6 iji had been introduced so as to reproduce
V
= *'(x) - Hx) = S* - 5U 3 y * - i|i(x)
(3.9a)
Si|i(x) = if'Cx*)
(3.9b)
i . e . a resetting of the value of the argument to i t s original value x, after the action of a Lorentz transformation, in view of the latter 1 s simultaneous effect on the coordinates (its orbital action). Sometime
using 6 4 in his interpretation, Kibble remarked that one could also
regard (3.5) as involving a translation-gauge field (A, Dv = &vV 8 k k y
+
0VP - Si/) 3 + «£ h P k k ' y k y
i j
v
v
- 6, )
f. IJ
(3.10)
where the second term could correspond to 3 as the 6 tji algebraic generator in (3.8) multiplied by i t s gauge field.
A recent
attempt
to pursue this idea [Cho 1976]
appears to have failed due to the difficulty of expressing e = det p
in that inter-
pretation. A
tetrad field is defined by erecting at every point x a frame of vectors
r/(x) 3r x (x) Pu
v
(X) := (
X
3x y
)
(3.11) x = X
540
199 and its variation under a coordinate transformation
« Pp
i
(xj - - 3 U i" P v
xv -*• x
+ £
is given by
1
(3.12)
which i s indeed the inverse of the £ v a r i a t i o n of A, in ( 3 . 5 ) . However, one can replace the l i n e a r connection p J corresponding to gauging the Lorentz group, by a Cartan connection [Kobayashi 1956, 1972, Trautman 1973], in which the Bundle S t r u c t ure Group G i s the Affine (or Poincare") Group. For the Poincare" group t h i s means having in the Bundle " i n t r i n s i c " t r a n s l a t i o n s and altogether 10 connections. By choosing the origin in that f i b e r , one can make the t r a n s l a t i o n - c o n n e c t i o n coincide with the frame, except that we now have an anholonomic t r a n s l a t i o n gauge, with variation, 6 p X (x) = 3 e 1 + p V H P ]
X
.
£j - e 1 . p j = D ( P ) e J V V
X
(3.13a) v
or for the forms 6 p 1 = de 1
+
pij
-
j E
- p j - e i j = DCP> e 1
(3.13b)
Such a translation-gauge was indeed suggested by Trautman [1973] and by P e t t i [1976]. I t yields the universal coupling of eq. (3.3) through Noether's theorem or the Bianchi i d e n t i t i e s . The d i f f i c u l t y i s t h a t the Einstein Lagrangian i t s e l f i s not Poincarfi-gauge invariant [e.g. Ne'eman and Regge 1978a]. Under the t r a n s l a t i o n gauge (3.13) we find terms in D(P)' e i a r i s i n g from £•. in ( 2 . 7 ) . I n t e g r a t i o n by p a r t s then J
4
makes the action produce a v a r i a t i o n proportional to the t o r s i o n R . (The Action i s of course t r i v i a l l y t r a n s l a t i o n - i n v a r i a n t . ) One way out of t h i s dilemma i s to abandon the concept of invariance for a weaker "pseudo-invariance", holding only a f t e r the application of the equations of motion. This was done somewhat half-heartedly in Supergravity [Freedman e t a l 1976; Deser and Zumimo 1976; Freedman and van Nieuwenhuizen 1976] so emphasized by C. T e i t e l boim [1977], and generalized to gravity by J . Thierry-Mieg [1978]. Indeed, applying (2.8) for empty space ( i . e . R = 0) a f t e r the v a r i a t i o n makes E i n s t e i n ' s free Lagrangian invariant under (3.13). However t h i s i n t e r p r e t a t i o n does not guarantee the poss i b i l i t y of exponentiation to a f i n i t e gauge, i . e . group action. In a d d i t i o n , J p - 0 under the t r a n s l a t i o n gauge, which has to be modified so as to f i t R1 = 0, with no gauge mechanism to provide for the new S p . Moreover, the i n t e r p r e t a t i o n f a i l s when spinning matter i s present.
541
200
4.
The P a r a l l e l Transport Gauge
(AGCT)
The next step in solving the mystery of the t r a n s l a t i o n gauge i s due t o von der Heyde [1976; see also Hehl e t al 1976a]. Returning to Kibble's H concept (eq. 3.9a - 3.9b) he noticed t h a t with space being already "curved" due to the Lorentz gauge, the transport term had to involve p a r a l l e l t r a n s p o r t , i . e . the covariant der i v a t i v e D l Ji|j r a t h e r than 3 i|i. Moreover, to preserve the Poincare' group appartenance of the t r a n s l a t i o n generators, the operator D. of (3.5) should be used, with the anholonomic ("Latin") indices covering P r a t h e r than j u s t L. This s o l u t i o n thus combines the idea of 10 connections, including the vierbeins p , with t h a t of p a r a l lel transport. 7" ,m, .. , , i j ~ ..m n . Y. .m ,n &o i|i (x) = h (e f i ; j ) n * - (e D R ) n $
, . -. (4.1)
We have seen t h a t e1-* = 6 1 S3 e p v , but for e k y
k y e = eM
V
k
, , _, (4.2)
PV
This i s due to the " f l a t n e s s " of the fiber (parameters e1-1 = eWV) as a g a i n s t the cur4 vature induced by the Lorentz gauge inTTZ . Equation (4.1) can be i n t e r p r e t e d as an active Lorentz transformation followed by a passive r e s e t t i n g of the coordinate frame to the o r i g i n a l value of x. We can convince ourselves of the r o l e of - e D. as a translation-gauge by noting t h a t the e n t i r e 6 i|i transformation amounts to a t r i v i a l action on the base space, a conclusion which would s t i l l be t r u e in the p r i n c i p a l bundle when taking the p (x) for i|/(x), except for the gauge term. Thus as the homogeneous p a r t of an i n f i n i t e s i m a l PoincarS transformation in the extended bundle with G = P, i t should be considered as a gauge transformation. The i n t e r a c t i o n Lagrangian i s s t i l l produced by the replacement
\
v
\ I—• Dk
c4-s>
and the Equivalence Principle to maintained, independently of the existence of microscopic t o r s i o n . Indeed, S p e c i a l - r e l a t i v i s t i c matter i n a n o n - i n e r t i a l frame i s always l o c a l l y equivalent to the same matter i n a g r a v i t a t i o n a l f i e l d [von der Heyde 1975], Note also t h a t in t h i s derivation, the appearance of curvature i s n a t u r a l , due to our improved understanding of geometry: Utiyama and Kibble had t o make a jump to c u r v i l i n e a r coordinates, whereas the Fiber Bundle p i c t u r e t e l l s us t h a t curvature i s nothing but the base-space effect of gauging a group i n the Fiber. Indeed, even the electromagnetic U(l) or the modern SU(3) gauges induce curvature in space time (the F u v a ) .
542
201
We have r e c e n t l y g e n e r a l i z e d t h i s approach
[Ne'eman and Regge 1 9 7 8 a , b ]
, showing
t h a t t h e Supersymmetric ( " l o c a l " ) t r a n s f o r m a t i o n s o f S u p e r g r a v i t y c o r r e s p o n d t o a s i m 4 l i a r p a r a l l e l - t r a n s p o r t action in Superspace with a f u r t h e r r e s t r i c t i o n t o W . We now a n a l y z e t h e g e o m e t r i c and a l g e b r a i c s t r u c t u r e o f t h e
parallel-transport
gauges. To u n d e r s t a n d t h e s e gauges and i n d e e d t o a n a l y z e t h e e n t i r e p r o b l e m o f g a u g i n g a " n o n - I n t e r n a l " g r o u p , i . e . a group w i t h some a c t i o n on s p a c e - t i m e , we r e v e r t t o a new m a n i f o l d .
N o t i n g t h a t i n g a u g i n g P , t h e F i b e r was L w i t h 16 d i m e n s i o n s , and t h e
base-spaced!
had 4 d i m e n s i o n s , we o b s e r v e t h a t t h e Bundle d i m e n s i o n a l i t y was 10, t h e
same as t h a t o f t h e P o i n c a r S g r o u p . I n S u p e r g r a v i t y , w i t h a 1 4 - d i m e n s i o n a l g r o u p , 4/4 w o r k e r s i n S u p e r s p a c e 31 , an 8 - d i m e n s i o n a l m a n i f o l d , found t h a t t h e y had t o r e s t r i c t t h e p u r e gauge group t o L = S 0 ( 3 . 1 ) . manifold
Adding, we f i n d a g a i n 8 + 6 = 14, t h e g r o u p
dimensionality.
I n t h e formalism we r e c e n t l y d e v e l o p e d w i t h T. Regge [ 1 9 7 8 a , b ] f o r t h e g a u g i n g o f n o n - i n t e r n a l g r o u p s , we work i n t h e Group M a n i f o l d . (A = i ,
Generalized curvatures R
[ i j ] i n P) a p p e a r as t h e n o n - v a n i s h i n g r i g h t - h a n d s i d e o f t h e C a r t a n - M a u r e r
e q u a t i o n s f o r L e f t i n v a r i a n t forms u , when such forms a r e r e p l a c e d by a " p e r t u r b e d " A s e t p (a t e n - b e i n ) (see ( 1 . 1 2 ) ) , A I B E „ d p - J s p - p C B
A
A 1 B E D A = R = % p - p R g E
dp A - h p B . p E (C B E
™ „ -, (4.4a)
D
E
A
+ RBE *) = 0
(4.4b)
For an o r t h o n o r m a l b a s i s of v e c t o r f i e l d s D„ o r t h o g o n a l t o t h e p , o
p A ( D B ) = 6A
« A CD^' 1 -) = 6g
RA—*
0
(4.5)
p-+(0
[
V V = tCBE A + RBE *) DA
[DB*1-'
D
E'I"]
= C
(4 6)
-
BEA°A
C4.7)
(4.7) i s the L e f t - i n v a r i a n t generator algebra. " s t r u c t u r e f u n c t i o n s " i n s t e a d of
I n ( 4 . 8 ) we have an a l g e b r a w i t h
constants.
We can now a l s o c a l c u l a t e t h e v a r i a t i o n o f D p : (6D) E = (6D )E T ^ =
B [e
DB, DE] = e B (CBE A
+R
BE
A 3 °A=
A
• R^
^ E B
543
A
) DA
(4.8a)
(4
"8b>
202
and s i n c e t h e p r o d u c t p representation p
D_ i s i n v a r i a n t , we can d e r i v e t h e v a r i a t i o n s o f t h e
adjoint
from t h o s e o f t h e c o - a d j o i n t D„ ( t h e d i f f e r e n c e i s i m p o r t a n t when
t h e group i s n o t s e m i - s i m p l e , which i s t h e c a s e f o r P , GP, E x t e n d e d GP, GA(4R) e t c . b u t n o t f o r t h e Conformal S U ( 2 , 2 ) , Graded-Conformal S U ( 2 , 2 / 1 ) o r E x t e n d e d G. Conformbe a l SSU U((22,,22 / N ) . The f a c t o r (-1) t a k e s c a r e o f t h e g r a d i n g i n c a s e o f a Graded ( o r Super) Group. S ( p E D E ) B = (6 p ) B E DE + ( - l ) b e p E (a D ) E B = 0 (6P)BADA=-(-l)bepE a
B
A
p
=-(-l)
b e
p
E
A
(C B E
RBE
+
A
) DA
(CBEA+RBEA)
(4.8c)
D
I f we t r e a t e (Z) as a l o c a l gauge (Z i s 1 0 - d i m e n s i o n a l f o r P) , we h a v e t o a d d t h e n e c e s s a r y g r a d i e n t t e r m . Summing o v e r t h e B i n d e x we g e t , (we l e a v e o u t t h e TGI f o r s i m p l i c i t y , D J i s t h e c o v a r i a n t d e r i v a t i v e d e f i n e d o v e r t h e group G) • C
6p
A
J A
= de
E
- p
B ,_
- e
A
(C EB
or a l s o (see d e f i n i t i o n following
_
A-
_(G)
) = Dv Je
+ Rgg
A
E
- p
B
e
n
R^
A
gradings
,.
„.
(4.9)
(4.17))
Sp A = D C G ) e A - 2 ( e , RA)
(4.10)
( D C G ) n ) A = dn A -
(4.11)
B P
. nE CBE
We have shown t h a t a l l t h i s i s unchanged when a subgroup H ( t h e L o r e n t z group L f o r b o t h P and GP) i s f a c t o r i z e d o u t i n t h e group m a n i f o l d ( s o t h a t we a r e l e f t w i t h ? f t 4/4 f o r P and JR, » i.e. " S u p e r s p a c e " , f o r GP, as b a s e s p a c e s M). T h i s a l s o c o r r e s p o n d s t o H b e i n g gauged, a s i n s e c t i o n 2 . I n t h a t c a s e , d e n o t i n g by E, F t h e i n d i c e s i n A t h e r a n g e o f G/H , and by A, B * H , p c o n t a i n s o n l y dx d i f f e r e n t i a l s (x-«M = G/H) A F and w i t s e l f , p o n l y dx d i f f e r e n t i a l s , DA=DAL-X-
(4.12a)
PA(DBL'1*) = /
(DBL-:-) = 6AB,
pF(DB1"1-) = 0
(4.12b)
which a l s o i m p l i e s Rgj
K
= 0 ;
I,J,K e
G
; A,B
« H
544
; E,F
•« G/H
(4.13)
203 Similarly, for holonomic indices, since the only "perturbed" forms are constructed of M differentials, K
R
= 0
;
Q.R < H
;
V,U < G
;
Y,Z «
G/H
(4.14)
To f u r t h e r o u r u n d e r s t a n d i n g o f t h e s e p a r a l l e l t r a n s p o r t g a u g e s , we a n a l y z e t h e e f f e c t of a general c o o r d i n a t e t r a n s f o r m a t i o n factorization
i n G/H) on o u r one forms
Sx
K, .VJe11 ) = Dx —rr p 3x
where we have d e f i n e d ( s e e
=e
U
after
(4.15)
, V r3eK U 3 K = dx {—w + e — g p v 3x 3x
K
or
p
= e
. K .,, U 6p = 6(dx p
e
( e i t h e r i n the G-manifold,
p
K
, U V 3 U + dx E — p 3x v
U E
3 ~^T
PU
K, i
3x
(4.2))
K
(4.16)
u
Since . O w 3 - d x ) (—g P v 3x K we can r e g r o u p t h e t e r m s i n 6p , A
K
dp
, , , V = - h (dx
K
3 K. — Pu ) v 3x
6p K = de K - 2 ( e , dp K )
(4.17)
where the scalar product parenthesis represents contraction with the second factor in the two-form. Also,' . K
n (G)
<5p = D v
K
I
s + p
= D v "" E +
U - e py
(E,
- 2 dp
J _
K
C:J
J
- 2
(E,
, K, d p ) =
+ p - p C T - •* =
D"-"-1 E " - 2(E, R") = D C G ) eK - p1 e
, .
(4.18)
RJJK
(4.19)
The algebra of parallel transport operators in G is thus in fact an algebra generating "anholonomized" (see (4.16)) General Coordinate Transformations (AGCT) on the G manifold. That gauge invariance is thus guaranteed by the General Covariance
545
204 of the Lagrangian. Indeed, it is this gauge which reproduces the General Covariance Group, rather than GL(4R)-gauging, as commonly believed.
5.
Dynamically Restricted AGCT gauges
We can construct the D for the factorized case. These correspond to translations in the quotient space G/H. From (4.S), (4.12b) p B CDE) = 0 ,
p F (DE) = 6*
(5.1)
we f i n d ( s t i l l u s i n g t h e i n d i c e s a s i n ( 4 . 1 3 ) -
D
D
E=AE
Y
(H)
Y0
YCH)
<5-2>
"h
- H
*
(4.14))
p
^
3x"
Y
A<{H}
A S
(S-3>
A A
*
where D^ ' is the H-covariant derivative, p is the post-factorization form on M itself and S. is the Right-Invariant algebra (of left-translation), which commutes L X A with the D." ' and has structure constants - C n c . For the Poincare group with ab H the S0(3.1) variable to be factorized, ..ij ,_
..
,„-l , , ij
(S J (=,x) = (= 01
dH) J + P
kl, ,
„lj
(x) = J
„ki
s.
(H,x) = E k l p k (x)
rr.
..
(5.4) (5.5)
The pi;l(x) are the connection potentials we introduced in (2.1) and used in Section 2, while we have used 0 one-forms.
J
and 0
in these last equations to denote the pre-factorization
Our previous discussion of the parallel-transport or AGCT gauges holds
for either set. For the parallel-transport modified translation gauge we thus get the variations, . ij n (P) ij 6 p = D e - p
k 1 D ij e R,.
. i k 1 D i n (P) i 6p = Dv e - p e R^
,c ,.. (5.6) ., ,. (5.7)
where , (P) ij J ij ik A Jkj kj „ ik d^J n J = dn J + p " A rT - p"J * n
546
(5.8)
205 n (P)
i
,i
ik ^ k
D"- •" n = dn + P
k ^ ik
n -P
,_ „, .
n
(5.8b)
Compare (5.7) with (3.13b) and with (3.12)! The parallel-transport gauges((5.6)-(5.7)) introduced by Von der Heyde [1976; see also Hehl et al 1976a] (in space-time; here generalized to the Group manifold), are still "semi-trivial", since they only reproduce General Covariance.
Note that
one very important point is guaranteed: we realize that AGCT form a group and can be exponentiated, since they are just a subset of the Group of Diffeomorphisms. Now once a Lagrangian is introduced, it will yield equations of motion.
These
Y
equations will restrict the values of the R_T components in (4.19), (5.6-5.7). For instance, we have seen in (4.13) and (4.14) the results of the Lagrangian being gauge-invariant under
a subgroup H (the Lorentz group in P and GP).
First, the
parallel transport generators in the H direction coincide with the Lie-Algebra L.I. generators, so that H-gauging is "conventional".
Secondly, applying the equations
of motion produces the cancellations (notation as in (4.13))
R
AB J " °
R
IJ E = °
<5-9)
which makes the Dynamically Restricted AGCT gauge for translations coincide for p itself (the vierbein p A ii p , the connection p
J
in Gravity), with an ordinary translation gauge (but not for ).
In Supergravity, where H is also the Lorentz group,
D.R. A.G.C.T. translations thus also look like an ordinary gauge for p not when acting on p
or p , the spinor potential.
itself, but
Supersymmetry D.R. A.G.C.T.
also produce a variation involving R -1 for both vector and spinor variations [Ne'eman and Regge, 1978a,b]: , ij r,(GP) ij c d _ <5pJ=Dv J e J - p e R , cd 6P 1
= D< GP > e 1
6p a
= D(GP)
a e
Jij
- p
c a. e R • co
ii
rr. ... (5.10) ^
(5.11)
- pC sd Rcda
(5.12)
The components R • •* in (5.10) are essential to the "local supersymmetry" transformations of Supergravity [Freedman et al 1976; Deser and Zumino 1976; Freedman and van Nieuwenhuizen 1976] D (GP)
^ij
= D (P)
^ij
(5_13)
_(GP) i _(P) i - i D v ' n = D*" ' n + P Y n r.(GP) a
,a
,
DK ' n = dn + h
, ii
,_ , .. (5.14) ij»
(p J a J )
A
a
, , ij ..a . ij
n - h (a J p)
547
n
,_ . _.
(5.15)
206 The action in supergravity is given by A
=l |m4
(Rij - ^
+
Ra.ra)
(5.i6)
4 with, on a generic fn. , the equations (anholonomic spinor indices are not explicited) Ra = 0
Rlj pk
(5 .17)
e
ijkl"
2 i
p
^ 5 ^
C5
= °
Y1 p 1 R = 0
- 18)
(5.19)
from which one derives R
I J
i s 0
R
am l j
e
VI J
ijk* "
' «
R
R
ag=0;
R
*k i j « i j . l =
R
ia=°'
4i
R
mk
%
[ij]K=0
VK
{ S 20
-- ^
(5
V
-20b)
a Equation (5.20b) and the e
variation in (5.10) are essential to supergravity.
Indeed, the Supersymmetry transformations e of Supergravity, which were derived directly, posed the problem of what we now know is a Dynamically Restricted AGCT, before it had ever been raised in Gravity, although the survival of the e transformation in (5.6) is completely analogous.
In both theories, p
does not propa-
gate and is extracted from R = 0 in terms of the other potentials, which tended to hide the physical importance of either (5.6) or (5.10). We still have to discuss one more aspect of these theories.
Working in the
Group Manifold, how come we only useffl? for the integration in either (2.7) or (5.16)? First, the reduction of the base space to the quotient of G (P or GP) by its subgroup H (L in both cases, although other such subgroups exist for GP):
if 95 ,
S*C and 9~ are the Lie algebras of G, H and G/H, the conditions may be
(c);
(a) weak reducibility: (b) a symmetric manifold:
\&C , T ] C 7 [ "3s , ?" ] c f*C
(c) an ideal:
[ "7 , 7 ] C. 5*
For G = P and H = L, all three hold, but for G = GP and H = L, we have (a) and 2 for H = L 8 A (left or right handed supersymmetry) we have (a) and (b); for
H = GP (supersymmetry with only nilpotent elements in the ring of parameters) we have (a), (b) and (c).
Each case induces a different theory, with ordinary Super-
gravity corresponding to H = 1.
The MacDowell-Mansouri [1977] version of de Sitter
548
207 Gravity follows (a) and (b). 4 4/4 The homogeneous spaces P/L =Jfl , GP/L = JR. correspond to the "factorized" theories.
We conjecture that if a Lagrangian is H gauge-invariant, then it is H-
factorizable as a consequence of the equations of motion.
A heuristic proof of
this hypothesis exists for solutions infinitesimally close to a factorized one. All such solutions can be reduced to factorized ones by an infinitesimal coordinate transformation on G.
However, discrete families of factorized solutions with the
same boundary conditions but topologically distinct may exist in the large. This explains restricting the action integral t o % in Gravity. In Super4/4 gravity, factorization reduces us to tK . However, physics is seen to be comp4 letely determined by what happens on a simple^? . The transfer of information from 4 —4 4/4 4 anyiR. to any otherm. in31 corresponds to our AGCT gauges. Partial Wl slices correspond to all possible supersymmetry-related conventional Supergravity theories.
549
208 6.
GL(4R) and Affine Gauges
In trying to reproduce Gravity as a gauge theory, several authors GL(4R).
43-46") •* gauged
This group seemed to fit that role, judging from the fact that in holon-
omic ("world tensor")
coordinates, the covariant derivative is
D
GV
$
(6.1)
p
is the GL(4R) matrix representation corresponding to the world-tensor field <j> .
Indeed, world tensors are classified by the finite irreducible (non-
unitary) representations of GL(4R). However, as proved by DeWitt [1964a] this has nothing to do with a type gauge.
Yang-Mills
We are dealing with the General Coordinate Transformation Group, and
its structure constants do not correspond to a GL(4R) gauge. shown, they correspond to an AGCT translation gauge.
Indeed, as we have
However, as a group, the
G.C.T.G. is represented over its linear subgroup, which happens to be GL(4R).
This
is true of any such non-linear group, owing to the role played by the Jacobian det471 erminant. We refer the reader to De Witt's text J and to the work of A. Joseph and A. I. Solomon [1970], who, in working out the theory of Global and Infinitesimal Nonlinear Chiral
transformations, explained the construction of representations
and covariant derivatives for such non-linear (and non gauge-factorizable) groups. (In Chiral symmetry, Isospin is the linear subgroup). One more general point about GL(4R).
It had always been assumed in the folk-
lore of general relativity (and often written in texts) that GL(nR) has no doublevalued or spinorial representations.
E. Cartan [1938] is referred to for this
prevalent belief, in two of his theorems.
As can be seen in the text, one theorem
refers explicitly to spinors with a finite number of components. is an overstatement:
valued representations". the covering group
SU(2) is of course compact and simply connected;
SU(2) = SO(3)
of
S0(3), where spinors are bivalued.
has SU(2) as compact subgroup, and thus has the same topology. SO(1.3)
The other theorem
"the three Unimodular Groups in two dimensions have no multiit is SL(2C)
Indeed, SL(2C) =
is the covering group of the Lorentz group, and Lorentz bivalued represent-
ations become single valued here.
Now it is true that SL(2R) = SO(1.2) and the
bivalued representations of SO (1.2) become single-valued in SL(2R), which may explain the error in Cartan's theorem.
However, SL(2R) has like S0(2) an infinite covering,
and we can find in Bergmann's analysis [1947] double-valued representations of SL(2R), which become single valued in SL(2R) = SO(1.3), etc. In a recent study [Ne'eman 1977, 1978], we have proved the existence of doublevalued representations of SL(nR), GL(nR) and the G.C.T.G. in n. These reduce to
550
209 infinite direct sums of S0(n) or 0(n) spinors.
They are single valued in SL(nR),
For such "polyfields", (6.1) can be used, provided the G v
GL(nR) and GCTG(n).
are infinite-dimensional. These band-spinors or bandors are all known for SL(2R) [Bargmann 1947] and SL(3R) [Joseph 1970; Sigacki 1975].
They have now also been listed for SL(4R)
[Sigacki 1978]. Note that SL(4R) = S0(3,3) and some of these representations had been included in a study of SO(3.3) by A. Kihlberg [1966]. Gauging GL(4R) [Yang 1974] prior to the introduction of bandors implied that spinor matter fields would not be minimally coupled therein.
Note that most of
these theories did not really exploit GL(4R) anyhow, and added metric restrictions [Mansouri and Chang 1976] which reduced GL(4R) to SO (1.3) or alternatively reduced GA(4R) - the Affine group in 4 dimensions, i.e. GL(4R) x T. - to Poincare SO(1.3) x T.. However, we shall further discuss one consequence of starting with a larger group which is generally disregarded: the representation structure. We now study the result of a GL(4R) gauge, in the context of a GA(4R) mixedgauge (ordinary for GL(4R), D.R.AGCT for the translations). It is [Hehl et al 1976b, 1977a] in the Metric-Affine theory and in its Spinor version [Hehl et al 1977b, 1978] and gauge [Lord, 1978] that the actual enlargement of the sets of connections, curvatures and currents are used, rather than an immediate restriction to Einstein's theory.
The spinor matter field is now a polyfield, i.e.
an infinite representation of GL(4R), with physical states given by GL(3R) bandors (this is the little group). One such bandor is 25 (^,0) which reduces under the 5 9 13 % 8 j 8 j 8 y- e •••
spin to the sum
The connections now include in addition to those of P, ten p ^ symmetric in (i,j).
The D..., generators in the (flat) group space generate shear (for traceless
D-. ...) and scaling (for the trace).
We thus enlarge the angular momentum current
tensor into the hypennomentum tensor, with shear, scale and spin currents in its intrinsic part:
h
ab U = s ab y
+
* "ah h "
+
^
V
where n , is the Minkowski metric, hv is the scale (or dilation) current and F , v is the shear current. Note that the "orbital" part of hypermomentum can be reduced to the set of time-derivatives of gravitational quadrupole moments [Dothan et al 1965; Hehl et al 1977b]. The Noether currents of the theory are given by . V
*a
"
e
-1
5%
,,
6F"a
...
C6.3)
V
551
210
h
ab
=
-e
<6'4)
T-^b
The field equations are (-C is the gravitational field Lagrangian)
^
= - 2< e t /
a
C6.5)
f^ab = 2 e h ab P
<*•«
Choosing the free action
^
M
J
"?[ab]+8^
^
(6-7)
where r, 1 O % = 4 PVff
C6-8)
we find that equation (6.6) becomes again algebraic in relating connections to J hypermomenta, and the p does not propagate. Note that the holonomic p - .. corresponds to the Non-metricity tensor, P r
i = "D
g
(6-93
which appears in the identity, p
CT
£ gaT Aa0Y
f h 3 g„ - g
A a 3 Y := S a « 6 6 Y + 6 a VtJT
v
p
T
6B
Sy
-
U
T
V
R';E -
&a 6 B T
V
h D g„ )
(6.10)
«Y
(6.11)
y
When no polyfields are present, there is no non-metricity.
In the presence of
intrinsic hypermomentum, non-metricity exists but is confined to the region where that matter exists, without propagating over intermediary regions.
Again, the
linearity of the Einstein Lagrangian in derivatives preserves the Riemannian properties of space-time.
Macroscopically, one can always define a local Minkowski
metric.
552
211 7.
Extending the PoincarS Group: No-go Theorems
There are three main ways in which one has extended the Poincare' group:
- Conformally, into the simple group Con(4R) = SU(2.2) = SO (4.2) - Linearly into the Affine group
GA(4R)
- Spinorially, into GP In fact, the latter extension can also be performed for SU(2.2), extending it thus further into SU(2.2/1) or SU(2.2/N).
We have recently shown [Ne'eman and Sherry,
1978] that GA(nR) can be similarly extended into infinite-dimensional graded Lie Groups
g GL(nR).
Although we have constructed these graded-Affine groups for n =
2,3 only as yet, it appears plausible that
g GL(4R) should also exist.
There is one important point we should note when gauging a group G larger than P.
Although we may afterwards introduce constraints which will reduce the theory
to Einstein's General Relativity, there are still traces of the larger group
G ? P.
For example, the matter fields physical states have to fit in unitary representations of G.
In our case, these would be Polyfields (with either integer or half-integer
spins).
In Conformal Relativity resulting from gauging the Conformal group [Englert
et al 1975; Hamad and Pettitt 1976, 1977; Kaku et al 1977], these would be Mack's [1977] Unitary representations of the Conformal group. Ogievetsky [1973] has proved that in a holonomic representation of Con(4R) U GL(4R) generator algebras, closure occurs only over the entire analytical General Coordinate Transformations Group A.
This is due to the commutators of the Special
Conformal Transformation generators K
and the Shears
S,
.., which keep generating
operators
X
m n r s . l X2 X3 X4 \
with ever-increasing powers (m, n, r, s ) . etsky, 1974; Cho and Freund, 1975] theories.
In more recent work [Borisov and Ogiev-
this theorem has been applied to Gravitational
We would like to note the following theorems that can be drawn from Ogiev-
etsky's:
(1) Assuming a theory to be (globally) invariant under Con(4R):= C and GL(4R):=G reduces it to a trivial S-matrix.
[-£ , C ] = 0
,
[X
Indeed, we find that if the Lagrangian Jo obeys
, G ]= 0
(7.1)
then [ £ , [c,G] ] =
[X
, A ] =
0
553
(7.2)
212 so that we have an infinite number of active-Symmetry Noether theorems. (2)
Gauging both C and G imposes a trivial S-matrix.
This results from
(7.2) because a local gauge includes the case of a constant (global) gauge. These theorems are not modified by spontaneous breakdown via a Goldstone mechanism, since this still yields all global Noether currents. A Higgs-Kibble mechanism breaks the local gauge group but preserves the global conservation laws.
Thus, only a Higgs mechanism breaking the A gauge down to global
(or local) P invariance can release the S-matrix from triviality. It is important to remember that the Ogievetsky algebra is a representation of the Diffeomorphisms, but as such is purely a holonomic construct with no (active) Symmetry connotation.
Symmetries and their local extension as Gauges are entirely
anholonomic.
Acknowledgements We would like to thank Dr. J. Thierry-Mieg for an enlightening discussion.
554
213 References
Bargmann, V. 1947, Ann. Math. 48_, 568-640. Borisov, A. B. and V. I. Ogievetsky, 1974, Teoret. i. Mat. Fiz. 11,
329-342.
Cartan, E. 1922, Com. R. Ac. Sci. (Paris) 174, 593; 1923, 24, 25, Ann. Ec. Nor. Sup. 40, 325; 41, 1; 42, 17; 1938, Lecons sur la Theorie des Spineurs II (Hermann Editeurs, Paris) articles 85-86, pp. 87-91, 177, pp. 89-91. Cho, Y. M. 1976, Phys. Rev. D14, 2521-2525. Cho, Y. M. and P. G. 0. Freund, 1975, Phys. Rev. D12, 1711-1720. Deser, S. and B. Zumino, 1976, Phys. Lett. 62B, 335. De Witt, B. S., 1964a "Dynamical Theory of Groups and Fields" in Relativity, Groups and Topology (Les Houches 1963 Seminar), ed. C. and B. De Witt, Gordon and Breach (N.Y), pp. 587-826.
See in particular article
13_, pp. 688-689. - 1964b, Phys. Rev. Lett. 12, 742. - 1967, Phys. Rev. 162, 1195; 162, 1239. Dothan, Y., M. Gell-Mann and Y. Ne'eman 1965, Phys. Lett. jJ, 148-151. Englert, F. and R. Brout, 1964, Phys. Rev. Lett. 13, 321. Englert, F., E. Gunzig, C. Truffin and P. Wlndey 1975, Phys. Lett. 57B, 73-77. Feymman, R. P., 1963, Acta. Phys. Polon. 24, 697. Faddeev, L. D., and V. N. Popov, 1967, Phys. Lett. 25B, 29. Fradkin, E. S., and I. V. Tyutin, 1970, Phys. Rev. D2_, 2841. Freedman, D. Z., P. van Nieuwenhuizen and S. Ferrara, 1976, Phys. Rev. D13, 3214. Freedman, D. Z. and P. van Nieuwenhuizen, 1976, Phys. Rev. D14, 912-916. Fritzsch, H., and M. Gell-Mann, 1972, in Proc. 16th Intern. Conf. H.E.P., J.D.Jackson and A. Roberts,eds., NAL pub. (Batavia), 2_, pp. 135-165. Gell-Mann, M., 1962, Phys. Rev. 125, 1067-1084. Gell-Mann, M. and S. L. Glashow, 1961, Ann. Phys. 15, 437. Goldstone, J., 1961, Nuovo Cim. 19, 154. Gross, D. and F. Wilczck, 1973, Phys. Rev. Lett. 30, 1343. Guralnik, G. S., C. R. Hagen and T. W. B. Kibble, 1964, Phys. Rev. Lett. 13, 585. Hamad, J. P., and R. B. Pettitt, 1976, J. Math. Phys. 17, 1827-1837; - 1977, in Group Theoretical Methods in Physics (5th Int. Coll.) J. Patera and P. Winternitz, eds., Academic Press (N.Y.), pp. 277301. Hehl, F. W., 1970, "Spin und Torsion in der allgemeinen Relativitatstheorie, oder die Riemann-Cartansche Geometrie der Welt", Clausthal Tech. Univ. Thesis.
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214 References - continued:
Hehl, F. W., P. v. d. Heyde, G. D. Kerlick and J. M. Nester, 1976a, Rev. Mod.Phys. 48_, 393-416. Hehl, F. W., G. D. Kerlick and P. v. d. Heyde, 1976b, Phys. Lett. 63B, 446-448. Hehl, F. W., G. D. Kerlick, E. A. Lord and L. L. Smalley, 1977a, Phys. Lett. 70B, 70-72. Hehl, F. W., E. A. Lord, and Y. Ne'eman, 1977b, Phys. Lett. 71B, 432-434. Hehl, F. W., E. A. Lord and Y. Ne'eman, 1978, Phys. Rev. D17, 428-433. von der Heyde, P., 1975, Lett, al Nuovo Cim. 14, 250-252. von der Heyde, P., 1976, Phys. Lett. 58A, 141-143. Higgs, P. W., 1964a, Phys. Lett. U_, 132; - 1964, Phys. Rev. Lett. 13, 508; - 1966, Phys. Rev. 145, 1156. •t Hooft, G., 1971, Nuel.Phys. B33, 173 and B35, 167. •t Hooft, G., and M. Veltman, 1972, Nucl. Phys. B50, 318. Ionides, P. 1962, London University Thesis. Joseph, A. and A. I. Solomon, 1970, J. Math. Phys., 11, 748-761. Joseph, D. W. 1970, "Representations of the Algebra of SL(3R) with
Aj = 2",
University of Nebraska preprint, unpublished. Kaku, M., P. K. Townsend and P. van Nieuwenhuizen 1977, Phys. Lett. 69B, 304-308. Kibble, T. W. B. 1961, J. Math. Phys. 2, 212-221; 1967, Phys. Rev. 155, 1554. Kihlberg, A. 1966, Arkiv f. Fysik, 32_, 241-261. Kobayashi, S. 1956, Canad. J. of Math. 8, 145-156. Kobayashi, S. 1972, Transformation Groups in Differential Geometry, Springer Pr. (Berlin). Lee, B. W. and J. Zinn-Justin, 1972, Phys. Rev. D5, 3121, 3137 and 3155. Lee, T. D., S. Weinberg and B. Zumino 1967, Phys. Rev. Lett. 18, 1029. Lord, E. A. 1978, Phys. Lett. 65A, 1-4. MacDowell, S. W. and F. Mansouri 1977, Phys. Rev. Lett. 38, 739. Mansouri, F. and L. N. Chang 1976, Phys. Rev. D13, 3192-3200. Mack, G. 1975, DESY 75/50
report.
Nambu, Y. 1965, in Symmetry Principles at High Energy (Coral Gables 1965), B. Kursunoglu, P. Perlmutter and I. Sakmar, eds., Freeman Pub. (San Francisco) pp. 274-283. Nambu, Y. and G. Iona-Lasinio, 1961, Phys. Rev. 122, 345. Ne'eman, Y. 1961, Nucl. Phys. 26_, 222-229. Ne'eman, Y. 1977, Proc. Natl. Acad. Sci. U.S.A. 74_, 4157-4159. Ne'eman, Y. 1978, Ann. Inst. Henri Poincare 28. Ne'eman, Y. and T. Regge 1978a, Rivista del Nuovo Cim.
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215 References - continued:
Ne'eman, Y. and T. Regge 1978b, Phys. Lett. Ne'eman, Y. and T. Sherry 1978 Ogievetsky, V. I. 1973, Lett, al Nuovo Cim. £, 988-990. Palatini, A. 1919, Rend. Circ. Mat. Palermo
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Petti, R. J. 1976, Gen. Rel and Gray. 7_, 869-883. Politzer, H. D. 1973, Phys. Rev. Lett. 50, 1346. Salam, A. 1968, in Proc. Eighth Nobel Symposium, N. Svartholm ed., Almquist and Wiksell pub. (Stockholm) 367-378. Salam, A. and J. C. Ward 1961, Nuovo Cim. 20, 419. Schouten, J. A. 1954,Ricci Calculus, Springer Pub. (Berlin). Sciama, D. W. 1962, in Recent Developments in General Relativity, Pergamon Press (N.Y.-Warsaw) pp. 415-440. Shaw, T. 1954, Thesis. Sijacki, Dj. 1975, J. Math. Phys. j ^ , 298. Sijacki, Dj. 1978, to be pub. Teitelboim, C.
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Thierry-Mieg, J. 1978, to be pub. Thirring, W., 1978, in this volume. Trautman, A. 1972, Bull. Acad. Polon. Sci., Ser. Sci. Mat. Ast. et Phys. ^ 0 , 185 and 583. Trautman, A. 1973, Symposia Mathematica 12, 139-162. Utiyama, R. 1956, Phys. Rev. 101, 1597. Veltman, M. 1970, Nucl. Phys. B21, 288. Weinberg, S. 1967, Phys. Rev. Lett. 1£, 1264. Weinberg, S. 1973, Phys. Rev. Lett. 31, 494. Weyl, H. 1918, Sitzimgsberichte d. Preuss. Akad. d. Wissensch. Weyl, H. 1929, Zeit. f. Phys. 5j6, 330. Weyl, H. 1950, Phys. Rev. 77, 699. Yang, C. N. 1974, Phys. Rev. Lett. 33_, 445. Yang, C. N. and R. L. Mills 1954, Phys. Rev. 96_, 191.
557
Volume 200, number 4
PHYSICS LETTERS B
21 January 1988
GRAVITY FROM SYMMETRY BREAKDOWN OF A GAUGE AFFINE THEORY * YuvalNE'EMAN '-2 Sackler Faculty of Exact Sciences, Tel Aviv University. 699 78 Tel Aviv, Israel
and Djordje SlJACKI Institute of Physics, P.O. Box 57, 11001 Belgrade, Yugoslavia Received 7 September 1987
We construct a gauge field theory based on GA(4, R) space-time symmetry containing spinorial and tensorial (infinite-component) matter manifields. We break the dilation symmetry and thus trigger a spontaneous breaking of the SL(4, R)/SO(l, 3), also generating Newton's constant. The resulting "large-scale" space-time (as compared to Planck length) is of Riemann-Cartan type and in the flat limit we recover special relativity.
1. Introduction. A relativistic quantum field theory (RQFT) of gravity may involve the gauging of GL(4, R) [1-3], to the extent that the answer can and should indeed be given within RQFT [4]. For a Yang-Mills or other Lie-algebra-valued connection A =Ajkj cbc'' (such as the spin-connection) with dimensionless algebraic generators A,, the dimension dim(Aj) = 1 so as to match that of the spatial derivative in the expression for the covariant derivative D^d^—A,,1 A,-. However, for diffeomorphisms, or in gauging translations with the metric g),Ax) or tetrad e°(x) as gauge fields, A, is itself replaced by d„ or D„. Thus the inverse tetrad b/(x) acts as a dimensionless multiplier bf D„=D a . The dimensionless nature of the metric field g^,{x), essential to its kinematical role via the equivalence principle, is at the bottom of most of the difficulties in the quantization procedure. On the one hand, although the Einstein (or Einstein-Cartan) lagrangian eahcdRab A ec A ed involves a four-form, its dimension is 2, forcing the newtonian constant to have dimension 2, causing the need for an infinite number of counter-terms for that lagrangian. Moreover, this scale dependence imposes on the theory a requirement of finiteness, rather than renormalizability, an extremely strong condition. On the other hand, when terms quadratic in the curvature are added, since the connection is not an independent field and has the form dg, we get contributions ~(dg)4, (d2g)2 or (dg)2{d2g). Normally, since renormalizability implies cutoff independence ( = scale invariance) and a dimensionless action, terms of order 4 in the lagrangian dominate the high-energy region. In this work we indeed revert to renormalizability rather than finiteness. We assume that terms of order 4 dominate the high-energy region and provide the key to a renormalizable theory, with the terms of dimension 2 dominating the low-energy and macroscopic regime; this scheme [5] is similar to YM with spontaneous symmetry breakdown, where the dimension-four, F2 and 04 terms dominate at high energy, and the —n2
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
558
489
Volume 200, number 4
PHYSICS LETTERS B
21 January 1988
[p2(p2-m2)]-<=m-2[(p>-m2)-l-{p2)-1], i.e. a wrong sign for one of the poles, a ghost. This may be remedied by adding terms that would make the connection r independent of the metric and produce an analog to YM. A recent calculation has however shown that the most general lagrangian quadratic in Poincare" curvatures and torsions, when linearized and having selected the coefficients of the various Poincare irreducible pieces so as to remove all ghosts, tachyons, frozen states and p* ghosts, ends up containing nothing but the original massless graviton [6]. (Torsion)2 terms are of dimension 2 and no help in this matter. Moreover, as long as the theory stays riemannian, the metricity condition. D^„=0
(1)
makes the spin-connection depend on the metric (and matter fields). The solution could be to gauge the entire GL(4, R) instead of the Lorentz group, for the homogeneous part. The theory is then affine and non-metric at high energy, with eq. (1) emerging as a low-energy result [ 5 ]. Similar answers seem indicated [7] in the hamiltonian method of quantization of gravity and in yet another approach [8]. 2. General affine gauge theory. We gauge [ 1 -3 ] the general affine group GA (4, R) = T4 @ GL( 4, R), a semidirect product of translations and the double-covering GL(4, R) of the general linear group GL(4, R) *', generated by Qab. Here GL(4, R)=R+®SL(4, R) => R + ®SO(l, 3), where R+ is the dilation subgroup. A general affine metric gab transforms as a 10 under global GL(4, R): {gab} = {AacAb\d\,Aj'eGL(4, R)}; the Minkowski metric rj is_defined in a given "flat" GL(4, R) gauge and is a Lorentz-subgroup invariant only. Raising and lowering GL(4, R) indices with gab thus corresponds to the usual result when one goes to the flat gauge. The antisymmetric operators Q[ab] = HQ»b—Qba) generate the Lorentz subgroup SO(l, 3), the symmetric traceless operators (shears) Q{ab) = HQab+Qha) — \gabQc generate the proper 4-volume-preserving deformations while the trace Q=Qaa generates scale-invariance R + . Qlab] and QUb) generate together the SL(4, R) group. The gauge potentials are the tetrads eafl and connections rabfi* The antisymmetric r^b^t traceless symmetric r(ab)u and the trace raat„ parts correspond, respectively, to the local Lorentz, shear and dilation transformations. The corresponding field strengths are the torsion RaM„=d/4ea„+rabltebl,-(p.^v), and generalized "curvature" Rabnp=d,rabi>+rabJ"acv- (p.+->v). Let 0 be a generic matter field with a global-general-affine matter lagrangian i? m (0, 30). The total gauge-invariant lagrangian reads if =if m (0, 30, e, r) + SCg(e, de, r, dr), where r enters £fm through the covariant derivative D,,=d„—\raflc Qcd. Variation with respect to 0 yields the matter-field equations 8Jz?m/80=O. Variation with respect to e",, and rabll implies two (gravity) gauge field equations with the corresponding momentum and hypermomentum (angular momentum and deformation) currents as sources -h£ejhe%=ed"a=h<£Jhe%,
- 6 J S y 8 r % = *r*„=8J2' B1 /8r%,
(2)
a
where e=det(e w ) and (quasi) conservation laws: Dtl(eea")=e6b"Rl'!lf,+er'l>'cR^atl,
T>^eY"/) = +ehlted>'a.
(3)
The two gravity equations can be rewritten in the form D„7t/"-e/=e0/,
(4)
D„7rV"-eV=d"V,
where TtJu'=dS£%lbbveall = 2b£e%ldR''vtl, nab"v=d<ejddvrha„=2d<ejdRbaiul,
(5)
Our conventions are: a, b,... are anholonomic (local) GA(4, R) indices; /i, v, ... are holonomic (co-ordinate) indices; the Minkowski metric is rj^,,—diag ( + 1, —I, —1, —I) fi=c=l.
490
559
Volume 200, number 4
eJ'=ea^t-R"a„vr-R"caXr,
PHYSICS LETTERS B
€V = e W -
21 January 1988
(6)
3. GA(4, RJ or SA(4, RJ matter fields. The SA(4, R) unirreps [9-HJ are induced from the corresponding little group unirreps. The little group turns out to be SA(3, R) ~ =T 3 @SL( 3, R), and thus we have the following possibilities: (i) The whole little group is represented trivially, and we have a scalar state, which corresponds to a scalar field
(7)
For the SL(4, R) spinorial fields we take an infinite-component field if/ which transforms with respect to an sf deunitarized unirrep belonging to the principal series of representations: 491
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(c2,d2,e2;({0))®Dtl4JR)(c2,d2,e2;(0,{)),
c2,d2,e2etR, while (J, 0) and (0, ^) denote parity-conjugated spinorial representations. The manifield y/ satisfies a Diraclike equation (igahXadh-M)V{x)=Q,
(8)
where^ a is an SL(4, R) four-vector acting in the space of our spinorial manifield. We construct Xa in the following way: first we embed SL(4, R) into SL(5, R), and then select a pair of (mutually conjugate) principal series representations which contain in the SL(4, R) reduction our spinorial representations. Let the SL(5, R) generators be Qag, d6=0, 1, 2, 3, 5. We define Xa = Q{sa\> a=0, 1, 2, 3 and thus arrive at the sought-for SL(4, R) four-vector. The transition to the GL(4, R) symmetry is straightforward. Note that the Xa operators constructed in this way yield upon commutation the SL(2, C) generators (generalizing a Dirac y-matrices property). 6. Generalaffine lagrangian. (As basic fields we take e/(x), rb>1a(x). y/(x), 0(x) a scalar field q>(x), as well as GA(4, R)-nonlinearly-realized SL(2, C) Dirac field C(x). The most general first-order gauge field lagrangian takes the form ¥i = £?i(e)f, RIMa,Rhtll,a) = eLi(Rcf, /?&•/) • If it contains at most second derivatives of the basic field variables, it consists of terms quadratic in torsions and curvatures and of the scalar curvature. Owing to the dimensional coupling constants, the scalar curvature and the (torsion)2 terms are not allowed by dilation invariance. We circumvent this fact by making use, besides the well known
b„e°, T V , d „ / y , y/A, dvi//A,^A, d„
e[
=
-(l/4/J 3 )^ [ a ( 6 „^,i? [ a ( * , l r W 1 -(l/4K0/f,. [ f t ,c r f l /? ( a [ * W , -(l/4'c 2 )/? ( a M ^ 1 i? ( ^ W I + wisahXae„ "D„^+ i*-»e/e 6 '(D„0) + (D„0) + b
+
2
+ 0l° yaeb>'A£+W
+
{g^e/erf^d,* 2
0)
(9)
R isjhe general-affine scalar curvature, a,, a2, pu fi2, /?3, K,, K2, fi, X^, X, Xv are dimensionless constants, DM is a GL(4, R) covariant derivative; say for the spinorial manifield y/, it is given by ^WA = (SABdfl-iTh/(Qat')
B A
)¥B
.
where (Q/) / is the appropriate representation of the GL(4, R) generators and .4, B runs over the 0'i. ji) SO(l, 3) components of the appropriate GL(4, R) representation, and A^ is the GL(4, R) nonlinear covariant derivative in the space of SO(l, 3) Dirac field f. 7. Symmetry breaking. We now consider a breaking of the GA(4, R) which preserves the Poincare symmetry. 492
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In the case of a Weinberg-Salam type theory, the Yang-Mills action is scale-invariant, so that one starts with a lagrangian invariant with respect to a direct product of an internal SU(2) L xU(l) group with the dilation group. Introducing for the Higgs-Goldstone field an explicit (mass)2 term which breaks scale-invariance, and using the wrong sign, one triggers a spontaneous breakdown of the internal gauge symmetry. Thus "spontaneous" breakdown of the internal symmetry is really achieved with the help of an explicit breaking of the "external" dilation group. In our case, however, the dilation symmetry is one of the keystones of the GA(4, R) gauge field theory and the symmetry breaking has to be more subtle. We find that the deunitarizing automorphism is of crucial importance for the SL(4, R) symmetry breaking. Had we started with a manifield <j> which transforms with respect to an SL(4, R) unirrep, we would not have a Lorentz-symmetry preserving tadpole for this field since the representations of the Lorentz subgroup in the reduction are infinite-dimensional. However, for our j^-deunitarized SL(4, R) unirreps the physical Lorentz subgroup is represented nonunitarily and our tensorial manifield
GN = Vl67tv2v .
(10) 2 2
2 2
2
A
The tensorial manifield potential reads K(0)=Ay 0 +A 0 (0 ) , where <j> = 1A$i
[^+2A,(S0S0«J
<>A=0,
(11)
and if A<0 we are in a situation of a spontaneous symmetry breakdown of the SL(4, R) group. Our tensorial manifield transforms as D|E(4,IR)(C2, d2, e2; (0, 0)) which does not contain in its decomposition an SL(3, R) scalar. On account of &/, the only subgroup scalar is the Lorentz scalar (0, 0). Applying an appropriate SL(4, R)/SO(l, 3) transformation (within an irreducible space) we have Y.B*B =
(12)
The spinorial manifield mass becomes Af(v) =/">;.
(13)
(0, 0) is the lowest component of 0; At the next level we find three representations: (2, 0), (0, 2) and (1, 1). The first two of these cannot b£ obtained from (0, 0) via a single Lie algebra application, while |(1, 1) > =Qia.b) I (0,0) >, i.e. (1,1) ~SL(4, R)/SO(l, 3). Thus the (1,1) components of <j> are the Nambu-Goldstone fields which become the longitudinal components of the shear connections. By making use of a gauge transformation exp(iQi°h)S{ah){x)), Ziab)~(1, 1) we find 493
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The shear potential mass matrix is given by <<2
JW^M(16+ *>) g _\ I)2 (I _l
l
b)\f,
(14)
where (in the spherical basis) a, b= +1, 0, - 1 . For the dilation potential mass we find M2{r°a)l)v?.
(15)
The vacuum symmetry is SO(l, 3), and we find Af2(/W>=0 •
(16)
For the Higgs fields we get Ml(EA) = -Uv;2 , ^ = 0',,A)^(1, 1), A<0.
(17)
Assuming the dimensionless constants to be of order 1, we find that v'^, is of the order of the Planck mass Afpi, and thus all nonzero-mass modes have masses of order MPI. However, other values of K,, K2 could diminish the mass of the shear connections. At low energies or at distances / » / P i the potentials rlab)/1 and raatl do not contribute and we obtain eq. (1). Moreover, on account of v^0 the global GL(4, R) symmetry too is broken down to Lorentz invariance. Thus when the macroscopic gravitational field is "turned off" we arrive at the Minkowski space-time of special relativity - and not at an affine space-time as in the unbroken SL(4, R) case [16]. The principle of equivalence is thus effectively fulfilled and we recover Einstein's gravity over macroscopic distances. At the same time, the Planck scale is a "soft" feature, and horizons occur macroscopically only. At very short distances, the theory is scaleless and potentially renormalizable. References 11J F.W. Hehl, G.D. Kerlick and P. von der Heyde, Phys. Lett. B 63 (1976) 446. [2] F.W. Hehl, E.A. Lord and Y. Ne'eman, Phys. Lett. B 71 (1977) 432; Phys. Rev. D 17 (1978) 428. [3] Y. Ne'eman and Dj. Sijacki, Ann. Phys. (NY) 120 (1979) 292; Proc. Nat. Acad. Sci. (USA) 76 (1979) 561; Dj. Sijacki, Phys. Lett. B 109 (1982) 435. [4] A. Casher, Phys. Lett. B 195 (1987) 50. [5] L. Smolin, Nucl. Phys. B 247 (1984) 511. [6]R. Kuhfuss and J. Nitsch, Gen. Rel. Grav. 18(1986) 1207. [7] A. Komar, Phys. Rev. D 30 (1984) 305; P.G. Bergmann and A. Komar, J. Math. Phys. 26 (1985) 2030. [8] J.W. Moffat, J. Math. Phys. 25 (1984) 347. [9] Dj. SijaJki in: Frontiers in particle physics '83, eds. Dj. Sijacki et al. (World Scientific, Singapore, 1984) p. 382. [10] Dj. Sijacki and Y. Ne'eman, J. Math. Phys. 26 (1985) 2457. [ 11 ] Dj. Sijacki, SL(n, R) Spinors for particles, gravity and superstrings, Conf. on Spinors in physics and geometry (Trieste, 1986); Ann. Isr. Phys. Sc. 3 (1980) 35. [ 12] C.J. Isham, A. Salam and J. Strathdee, Ann. Phys. (NY) 62 (1971) 98. [ 13] Y. Ne'eman and Dj. Sijacki, Intern, i. Mod. Phys. A 2 (1987) 1655. [14] Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 157 (1985) 267, 275. [ 15] A. Cand and Y. Ne'eman, J. Math. Phys. 26 (1985) 3100. [16] J. Mickelsson, Commun. Mat. Phys. 88 (1983) 551.
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BRST TRANSFORMATIONS FOR AN AFFINE GAUGE MODEL OF GRAVITY WITH LOCAL GL(4, R) SYMMETRY C.Y. LEE ' and Y. NE'EMAN ' • " Center for Particle Theory, Department of Physics, University of Texas, Austin, TX 78712. USA Received 6 October 1989
BRST transformations are constructed for the fields in an affine gauge model of gravity with spontaneously broken local GL(4, R) symmetry, as a step in the quantization procedure. The invariance of the quantum action under these transformations holds under general gauge fixing conditions.
1. Introduction So far in quantum gravity, no consistent model, both unitary and renormalizable, has been found. Recently Ne'eman and Sijacki [ 1 ] proposed a model for quantum gravity that reproduces Einstein's gravity in the "low energy" region (lower than Planck energy), and is possibly unitary and renormalizable. Stelle [2 ] has already shown that a gravity lagrangian involving quadratic curvature terms as well as the Einstein term is renormalizable. The theory was, however, not unitary, mainly because of p4 terms in the graviton propagator, arising from the interdependence between connection and metricfields.Such independence arises in the absence of torsion and non-metricity [3], Other models such as Yang's [4] may be renormalizable, but appear unsuitable for the retention of Einstein gravity as the low energy effective theory. The basic feature of the model in ref. [ 1 ] is that it has an enlarged symmetry as compared with the Einstein, Einstein-Cartan or Poincare gauge theory (PGT). This enlarged symmetry, the group GL(4, R) gauged on the local frames, leaves the connection and the metric as independent fields. The problem of p 4 terms thus disappears. With a qua' Supported in part by DOE Grant DE-FG05-85ER4O2O0. Also, Wolfson Chair Extraordinary in Theoretical Physics, Tel Aviv University, Tel Aviv 69978, Israel. 3 Supported in part by GIF research contract I-S2-212-7/87.
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dratic lagrangian similar in structure and dimension to the Yang-Mills case, there also appears to be a fair possibility of renormalizability. The local GL(4, R) symmetry is of course not what we observe in the macroscopic or low energy limit, and the symmetry has to be broken by a Higgs mechanism [5], reducing to local Lorentz symmetry. In the above, GL(4, R) denotes the double-covering of GL(4, R) and reduces to that group for finite (tensorial) field representations. The spinorial double-covering proper exists only in infinite matrix representations [6] and the corresponding infinite-component fields ("manifields"). At first sight, it would seem that we do not need the enlarged local GL(4, R) symmetry, to make the metric and connection fields independent. In PGT [7] with a local Lorentz SO(l,3) symmetry, the metric and connection fields are indeed already independent if torsion is present. However, the torsion terms are of dimension <sf=2, and do not contribute to the high energy limit, where the (squared) curvature terms with d=4 dominate; the connection is thus again effectively metric-related, with p" terms just where they are damaging. Moreover, the (squared) torsion terms with d—2 dominate in the low energy region, and it is thus in macroscopic gravity that we would have to observe the independent - and both propagating - connection and metric, thus contradicting observations. This effect would subsist even if we were to salvage the high energy limit, multiplying the torsion squared terms by
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triggering a Higgs mechanism breaking the (homothecy) 0 ( 1 , 3) down to SO( 1,3). A recent study [8] indeed appears to indicate that no PGT model can be renormalizable if one imposes unitarity. Replacing SO (1, 3) by GL (4, R) we introduce additional degrees of freedom. Moreover, the riemannian metricity condition does not hold for spacetime in the very high energy region, K,„„ = D , ^ * 0 ,
(1)
thus guaranteeing the independence of metric and connection fields where it counts. In ref. [1], the general affine group GA(4, IR) was gauged and anholonomic indices were used throughout for the curvature and torsion (squared) terms. This formalism is equivalent to the method that we adopt in this paper, namely working with two "parallel" gauge groups: the general coordinate transformation (GCT) Dlff(4, R) and the GL(4, R) transformation of the local frames. We use holonomic (greek) indices for GCT and anholonomic (latin) indices for GL(4,R). The BRST transformation was introduced to insure unitarity in the renormalization of gauge theories [ 9 ]. In this work, we apply the method and construct the BRST transformations for the spontaneously broken affine gauge theory of gravity, as a basis for further investigations of its renormalizability and unitarity properties. In section 2, we introduce the BRST transformations for the various fields of the theory and show their BRST nilpotentcy. In section 3, we discuss the invariance of the quantum action under the BRST transformations for general gauge fixing conditions. In section 4, we conclude with a discussion of results and prospects.
28 December 1989
with all indices contracted. Here, e is the determinant of the tetrad e^, and
(3)
where X is a vector field. We define the tetrad 1-forms 6" dual to the frame Ksuch thatj?"^) =d%. The local frame K transforms under GL(4, R) as follows: V = VA, /leGL(4,R) le.e'a=Y.Aba^-
(4)
Connection and tetrad 1-forms transform under GL(4,R) as follows [10]: (o'=tAdA+A-lwA,
(5)
d'=A-'e.
(6) a
a
Identifying the connection co bll and the tetrad e M as the fields in co% =w"hlt Ax" and d" = eaM dx", the variations of the connection and the tetrad under an infinitesimal GL(4, R) transformation A = 1—A can be written as follows (a comma is a partial derivative and a semi-colon the relevant GL(4, R) covariant derivative). 8gp&>%,=: — A"(,i/t+A2/«u_bll — A.A&;a^ = — Xabll, 8**5.=***$. •
2. BRST transformations
(7) (8)
of torsion squared terms)
The variation under an infinitesimal GCT x' =x—£(x) is given by a Lie derivative L4 [11]. Thus we can write the variations of tetrad and connection fields under both Dlff(4, R) and GL(4, R) as follows:
-I-(a combination
8eatl =
In ref. [ 1 ], the invariant lagrangian was given by ^nv=
of curvature squared terms) + (matter and Goldstone-H iggs terms) ] ,
L(ea„+bIB,eall
=
(9)
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h(aab/i = L4toa6„ +8gpW%
5(v / ^L) = ( ^ N / ^ L ) .
(18)
For the BRST transformations of gauge fields, we know from the Yang-Mills case that we replace the transformation parameters with the corresponding ghosts and a constant anticommuting parameter; in our case <J" with CA and k% with C%A. Here, C and Cb are anticommuting ghost fields and A is a constant anticommuting parameter. For the BRST transformations of ghost fields, we consider first their variations under the gauge groups, applying the above replacements. We then check the nilpotency of these transformations. For the coordinate ghost C, considering as a contravariant vector, we write its variation as follows:
(10)
The torsion and the curvature tensors are defined as follows: (11) (12) where square brackets indicate antisymmetrizations of the indices n and v, with the following variations: 8 ^ % = L^T"^ + 8gP 7*% = £ T^L+Z ,MTax„+£ ,„TaflX+A'jT./lt,,
28 December 1989
(13)
8^3,^1/ = Lf ^!A/JI- + 8gP/?%„
ACsLjC'si'Cj-W.
(19)
For the GL(4, R) ghost C%, as a scalar under GCT (14) and as a group element of GL(4, R), and recalling the transformation property of a GL(4, R) element To deal with covariant and contravariant indices //, underaGL(4, R) transformation A =1—A, namely for both holonomic and anholonomic systems we deH' —A ~ 'HA, we write its variation under the GCT fine, in addition to the tetrad eaM, the local metric and the GL(4, R) as follows: gab=(ea,eb) and the coordinate metric gll„ = (eM,e„). Parentheses indicate inner products, though SCJ^LeCl + SipCJ not preserved in this non-riemannian geometry. The = ZxC%j +AJCi -ldbC°d. (20) relationship between these metrics and the tetrad is given by After replacing all transformation parameters with b a b ghost fields and a constant anticommuting parameSuv = (
5e=eedJ6ea„
=«i*(05u+£V3+ii*J),
hhC=-CCjA, bBC% = {-C^C%Ji-C%C j,)A .
(16)
(24)
It is straightforward to check the following nilpotency properties:
where we introduced a frame basis field (inverse tetrad) ef. The last term always survives and induces an undesirable trace term at the end: 8(eL) = (fVL),,+Ai
(23) d
(25)
(17)
If we use y/g instead of e, we end up with a total derivative:
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8|w%, = 0,
(26)
8BC=0,
(27)
5BC5, = 0 .
(28)
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5 B2 5 Bl F(/I,0)
3. Invariance of the quantum action under the BRST transformations
=5 82
To quantize a gauge theory we have to fix a gauge [5]. Thus, in addition to the "classical" i^nv, the quantum lagrangian contains a gaugefixingterm and a corresponding ghost term. We have to insure the invariance of the quantum action under the BRST transformations. Since i^nv is already invariant under BRST (as a gauge transformation), we only need to check the variations of the gauge fixing term and the ghost term. Because of the Higgs mechanism of spontaneous symmetry breakdown (SSB), we may need a't Hooft type gauge, containing both gauge and matter fields. We thus consider a quantum lagrangian with a general gaugefixingterm, •SSff= 4 v -'2F(A, 0) 2 - CMC,
&F(A,
&BJ0 °B,0T
gT
OBZ&B,?
(32)
Except for the third and the sixth, all other terms vanish because ordinary functional derivatives commute with each other, while the BRST variations anticommute. The proof is thus complete if the variations of both A and 0 themselves are nilpotent under BRST. As we saw in the previous section, A satisfies this condition. For matter fields 0,
(29)
(33)
8|0 m = 5B2 [ C ^ ^ - i C K Q * ) ™ * , ] ^ ,
Here, we picked a component of a GL(4, R) multiplet 0m (countably infinite [6] for matter spinors and for an additional bosonic Goldstone-Higgs field breaking SL(4, R) in ref. [ 1 ]) and Qba is the appropriate representation for the generators. The transformation rule for the matterfieldsis the conventional
(30)
and the variations of antighosts are schematically expressed as follows: bBC=F{A,
SF(A, 0) 5 (SFjA, 0) B,0 , V 5,4 S B l ^ + 80
&F(A, 0) &F(A, 0) 8 A &BlA + 5 82 08 Bl /l 5A&A Bl 50 8/1
where A denotes the tetrad and the connection, 0 denotes matter (including Goldstone-Higgs) fields, C and C represent ghosts and antighosts, respectively. As usual M is given by the variation of the gauge fixing function F(A, 0): SBF(A,
28 December 1989
(31)
50 m =^0„a + ie c m) m ,,0n,
Notice that for each gauge fixing condition we need an antighost; also, we need the same number of independent gauge fixing conditions as the number of generators in the symmetry groups. We will thus have 4 coordinate and 16 GL(4, R) ghosts and 20 corresponding antighosts. The invariance of the quantum action is guaranteed if the gauge fixing function is nilpotent under BRST. We check
(34)
where ta= -k% and &, = Qba. After performing the second BRST transformation, all the terms cancel straightforwardly except for the following two terms: 8l0 m = [iCa
(35)
.
But we can rewrite the last term as follows [denoting the GL(4, R) indices with greek letters a, 0, etc. ]: -C»C'(fi,G*)«/*/=-ICoClQ*,
QeUt,
= -^CaCC^(Qy)ml^.
(36)
Using the structure constants of GL(4, R) [11] explicitly, Va,= C'.-e * i rfc = W , - < i W ,
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one can see that the two terms cancel.
Acknowledgement
4. Discussion
One of the authors (C.Y.L.) would like to thank Professor C. DeWitt and Professor G. Hamrick for helpful discussions.
We have thus shown that the quantum action with a general gauge fixing function is invariant under the BRST transformations up to a total derivative as in the Einstein case [12]. Anticipating the difficulties one can expect in the next stage, we note that if we use a 't Hooft type gauge, we may be able to remove mixing terms between the connection and matter fields, as in the ordinary case of a spontaneously broken Yang-Mills gauge. However, if we strictly adopt the lagrangian given in ref. [ 1 ], we will encounter difficulties due to the Einstein-like curvature scalar term: it will yield a term linear in the connection, after SSB. This might be obviated by the exclusion of the curvature scalar term. The Einstein gravity required in the low energy region, might possibly be induced by the torsion squared terms, in a macroscopic teleparallelism structure. Finally, we comment on the equivalence between the affine gauge formalism in ref. [ 1 ] and our formalism. Since anholonomic indices are used in most of the cases in ref. [ 1 ], one may wonder whether the lagrangian given in ref. [ 1 ] is still invariant under our BRST transformations. The answer is positive due to contractions of all indices. For example, RahcdRdabc is a typical term in the lagrangian of ref. [ 1 ]. It can be written in our notation as Ifh^R^eJebpgAfig1"'. Recalling that all indices in our fields behave as tensor indices, the holonomic under GCT and the anholonomic under the local GL(4, R) (except for the ones belonging to the connection), it is not difficult to see that the above type of terms are indeed invariant under our transformations.
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References [ 1 ] Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 200 (1988) 489; for earlier metric affine models involving some degree of non-metricity, see F.W. Hehl, G.D. Kerlick and P. van der Heyde, Phys. Lett. B 63 (1976) 446; F.W. Hehl, E.A. Lord and Y. Ne'eman, Phys. Lett. B 71 (1977)432. [2]K.S. Stelle, Phys. Rev. D 16(1977) 953. [ 3 ] F.W. Hehl, P. van der Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48 (1976) 393. [4] C.N. Yang, Phys. Rev. Lett. 33 (1974) 445. [5] For a review, see e.g., C. Itzykson and i. Zuber, Quantum field theory (McGraw-Hill, New York, 1980). [6] Y. Ne'eman, Ann. Inst. H. Poincare, 28 (1978) 369; Y. Ne'eman and Dj. Sijacki, J. Math. Phys. 26 (1985) 2547; Intern. J. Mod. Phys. A 2 (1987) 1655; Phys. Lett. B 157 (1985)275. [7] For a review see e.g. F.W. Hehl, in: Cosmology and gravitation, eds. P.G. Bergmann and V. de Sabbata (Plenum, New York, 1980); see also F.W. Hehl, Y. Ne'eman, J. Nitch and P. van der Heyde, Phys. Lett. B 78 (1978) 102. [8] R. Kuhfussand J. Nitsch, Gen. Rel. Grav. 18 (1986) 1207. [9] C. Becchi, A. Rouet and R. Stora, Phys. Lett. B 52 (1974) 344; Ann. Phys. 98 (1976) 287; I.V. Tyutin, unpublished. [ 10] M. Spivak, Differential geometry, Vol. II (Publish or Perish, Berkeley, 1979). [11] B.S. DeWitt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965). [12] For example see G. 't Hooft, in: Trends in elementary particle theory, Lecture Notes in Physics, Vol. 37, eds. H. Rollnik and K. Dietz (Springer, Berlin, 1975); K. Nishijima and M. Okawa, Prog. Theor. Phys. 60 (1978) 272.
Volume 242, number 1
PHYSICS LETTERS B
31 May 1990
RENORMALIZATION OF GAUGE-AFFINE GRAVITY Chang-Yeong LEE ' and Yuval NE'EMAN
1A3
Center for Particle Theory, University of Texas, Austin, TX 78712, USA Received 15 March 1990
We outline a proof ofthe renormalizability of a non-riemannian model, based on gauging GL(4, R), in which Einstein's gravity dominates the low-energy region through a Goldstone-Higgs spontaneous symmetry breakdown mechanism. Whereas in other models of gravity, renormalizability, when proved, is the result of 1/p4 unitarity-violating propagators, in the present case it follows instead from the Yang-Mills-like features of the theory.
1. Introduction Einstein's gravity is perturbatively nonrenormalizable [ 1 ]. A perturbatively renormalizable relativistic quantum field theory (RQFT) of gravity, if it exists, should therefore differ from Einstein's in the quantum region (i.e. Planck energy). It should, however, reduce to Einstein's in the low-energy (long range Ax^lnan<*) region. True, a finite theory of quantum gravity may meanwhile emerge from the string; however, even then, the sub-Planck energy sector should be describable by a renormalized relativistic quantum (point-local, not string) field theory [2]. The main obstacle to the renormalizability of Einstein's theory consists in the dimension d=2 of Newton's constant and thus in the need for new counterterms in each order of the perturbation series. Alternatively, the difficulty can be blamed on a complementary aspect, namely the d=2 of the EinsteinHilbert lagrangian, linear in the curvature. This should be compared with the dimensionless coupling and d=4 curvature-quadratic lagrangian of a YangMills gauge theory, guaranteeing cut-off independ1
2
3
ence of S-matrix amplitudes and restricting the renormalization counter terms to a finite set. In gravity, the addition (to the scalar curvature) of d=4 terms, quadratic in the curvatures, has been shown [3] to lead to a power-counting renormalizable theory. Renormalizability is due to the 1 jp4 graviton propagators, a result of the riemannian constraint D„g/a,=0. This relates the connection r"^ to the metric g^, so that the (37") and (TT) in the curvature J? become ~ (d^g) and (dg)2, making theR 2 type terms yield 1 /p4 propagators. However, quartic propagators can be shown to contain ghosts, and this type of renormalizable theory ends up being nonunitary. Other models, involving, in addition to Einstein's, all possible quadratic Lorentz-curvature and torsion terms, have been analyzed [4-6]. Several combinations were shown to be unitary; however, in every such case, the l/p4 propagators cancel and the resulting power-counting renormalizability is lost.
2. GL{4, R) gauge theory with spontaneous symmetry breakdown In a model that was recently suggested [7], the lagrangian has d=4 and the coupling is dimensionless. The theory is based on gauging the linear group GL( 4, R), i.e. a lagrangian consisting of some linear combination of the irreducible components of the SKY [8] lagrangian, quadratic in the GL(4, R) curvatures,
Supported in part by the USA DOE grant DE-FG0585ER402O0 and by the USA-Israel Binational Science Foundation contract 87-00009/1. Also Wolfson Chair Extraordinary in Theoretical Physics, Sadder Faculty of Exact Sciences, Tel-Aviv University, TelAviv 69978, Israel. Supported in part by GIF research agreement 1-52.212.7/87.
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X W ^ t e , , , SK> emx, e„K, S\),
PHYSICS LETTERS B
(1)
where J ' is a reduction tensor, i.e. a product of metric tensors g m (x) and their inverses gK(x), tetrads emk(x) and frames enK(x) and Kronecker symbols S"^ a, are parameters defining the linear combination; £=det£ w Spontaneous symmetry breakdown (SSB) is then triggered by two Goldstone-Higgs fields: (1) a dilation ff(x), anSL(4, R) scalar; (2) amanifield
31 May 1990
Nevertheless, the non-compact nature of the gauge group could still cause a failure of unitarity. We hope, however, that unitarity can be assured through a proper selection of the GL(4, R)-irreducible pieces of the lagrangian.
3. BRST transformations in the background field formalism An appropriate set of BRST transformations was constructed [16]. For our proof, we adopt the "background field" (BF) formalism [17,18]. The gauge fields ("Y") are split, Y=A + Q, i.e. into a "background" part A and a "quantum" part Q. The BRST variations S B I ^ - D C , 5BC= - [C, C], Cthe ghost field, are then reproduced in two different manners which we denote as types I (5') and II (8"): 8'A = -DC,
8M = 0, 6 " Q = - D C ,
a
8"C=8 B C.
( C V V A + C V "
d
+ C dw
= eV*Agai(x)^0,
8'C=-C/C,
Note that 8'(A + Q) = 8"(A + Q) = 5BY; 8'C=8"C = 5BC; here D=3 + Y, D= 6-M;/stands for the gl(4, R) algebra structure constants. Type I variations leave <£# (the gauge-fixing) and <£&. (the ghost lagrangian) separately invariant, i.e. an invariance preserved by the quantum counter terms. Recapitulating the BRST algebra [16], 5BWV= -
KXll„ = Dxg/lp=Dx{gab(x)-eaM-ebr}
8'Q=-CfQ,
M
- C V
b/l-C
d
„WdM) ,
a
8BC b=( — C CabjL—CajC b) ,
due to the "pre-metric" gab(x) which is an SL(4, R) gauge-dependent symmetric tensor, until it becomes the LE effective metric, as a result of SSB. The components of
570
SB^%=-(CV^+CV^+CV^) ,
8 3 ^ " " = C x G ' i \ - CjG^-
C/}^+
kCx/}ilv,
8BC=-C^C;,.
(2)
8B = 0 for all fields and ghosts. We add to (1) the gauge-fixing and ghost lagrangians, ^ = - ( l / 2 a ) (//""LVA,,)
(H^a)"^)
-(l/2^)/f'"'D / ,D J ,{(D^^)(p j ,/7' K 7 )^ f f }, b
v
(3)
^x=-C aH" ^J>"co
a bv
-//""D^D^CDpS"^}, implying the antighost variations
(4)
31 May 1990
PHYSICS LETTERS B
Volume 242, number 1
BBCba=-(l/a)H"vDMCOb
the quantum fields, ghosts and antighosts Qt = 8 (—i In Z) /SJQJ, etc., in the presence of the sources; Z(At, Jl, Ct, Ct) the argument of the logarithm in (7) is the generating functional; the arrow t implies a "multiplet" (e.g. Ct denotes the set Ca6, C ) ; the •^t, Ct, Ct are source terms for quantum fields, ghosts and antighosts, At, Lt are source terms for 8" Q\ and 8"Ct. The measure A imposes integration over the Qt, Ct, Ct. The quantum action Jd 4 x2?=E in the exponential in (7) is given as
(5)
8 B C=-(l/0)flWD*f a -
In the above, we have redefined the G"" fields so as to incorporate in them the J~g density factor. With the A/Q split this yields (the upper case Greek letters Q, E, H are the A, while the lower case Greek letters co, e, t] are the Q, this r\ should not be confused with the Minkowski metric, sometimes denoted by the same letter)
E(At,Qt,Ct,Ct,Kt,Lt) = | d*x[%nv(co+ W, e+E, ri+H) l/
g V
s (7""=H""+rj»",
+ JS»f( W, H, Deo, D7) + g&(Ct, At, Dd" Qt) + (*w) •,ba^"coabp+ (Ae) *a8"e%+ ( A , ) ^ S ^ " "
G""G„=lH'"H„=3''a.
(6)
-L^'C^-L^"^ .
Eqs. (2) and (5) also define the type II for co, e, r\ and the ghosts, since 5" Q=8B7. Note that the ^(x) had no kinetic term in (1) and only acquires it in S£g. The frame and lower-indices pre-metric are defined by Taylor expansions.
(8)
Another useful expression is the reduced effective action in which we resubtract <£#, r=r-|d4x^f.
(9)
The Slavnov-Taylor identities are expressed as
5
4. Effective action and Slavnov-Taylor identities The background field effective action r is written
^S? + 6 '*-S + 6 ' c '
8T 8T 8(3t 8At
A A E, H; co, e, n, Ct, Ct, At, L\)
' +8'C 5 ' r 8
= 0,
as
6 r
5T sr ^ 8Ct ' SLt "~ '
(10b)
= Jp(fl,£,ff;Jt,Ct,Ct,AT,LT) f/*"D
(10c)
= - i l n j A'co,e,r],Ct, Ct)
7f^D,D 0 D
(lOd)
Xexpf i j d 4 x[i?(fl, E, H; co, e, //; CT, Ct; At, Lt)
Eq. (10a) results from type I invariance, the other three from type II.
-Jt-Qt-ft-Ct-Ct-Cr
+Jw-co+Jc-e+Ji-ri +
a
5. Proof of renormalizability. Removal of divergent contributions
b
C"a-C b+Clt-C'+C aC'b+C-C'
We assume that P has been successfully renormalized up to («— 1) loops, i.e. all divergences have been removed by a redefinition of the fields, parameters a, and BRST transformations. We then proceed to nloop order and prove that the divergent parts in
—fwd)—Te-e—Jg-tj
-CbA\-CvV-ZbaC%-Zv-C'.
(7)
Here Qt, Ct,
61
571
Volume 242, number 1
P=Kj[(Kwyba+Cba-(H'"'Dti)]
r' ( n ) can be similarly removed. r' ( n ) is split first into finite and divergent parts: 1
(»)—' ( « )
+ i
{o)abv+Q"bv)M{i,rl,E,H)
(11)
(n)
+K2(Ke)"aF\(€,tj,E,H)
Eqs. (10) have to be satisfied to n-loop order; the new divergent contributions from w-loops have a pole in e in the dimensional reduction regularization method (working in (4—e) dimensions and should thus obey (10) separately. Werewrite (10) forr' ( d n) :
S'AV ^ h +5'2f S
1
+
+k2LyCv-N,(e,ri,E,H)
(12a)
/sr>wsrfn)
5Gt Am)
VSQTA s*t
which can be removed by redefinitions e%-+c%-K2F'lt{e,E,Ti,H)
(14b)
:
8"£%-8"e%
(12b)
-K2
(12c)
8C»
(13)
^ S - e V ^ ^ I ^ + ^ l ^ ) ^ , (14a)
+ 5CT ASLT/ V s e t A 8LT =o,
.
The divergences in XP can all be removed through field renormalizations or by renormalizing the BRST transformations. For example, the K2 term yields a divergent contribution (we read out Xfrom eq. (12b) andPin(13))inr' ( d n ) of (12b'):
8Ct
sr^ysrn
K3[(Ks)"x+Cv-H»"DA]-T/U,ri,E,H)
+XlL"aC'b-N(9,ri,E,H)
+8'
set
" -n5(^w)^
31 May 1990
PHYSICS LETTERS B
[(S)--+(PM- «*>
For the first term in P we use the redefinitions H"T)X
=o,
(12d)
co-*co-Kl((o+Q)M(e,r],E,H),
where P' coincides with the expression for P in eq. (9). Eq. (12b) can be reexpressed in the form Xrfn) —0, where Xcontains the variations ofP" as coefficients of the variations of rfn). One checks that X2=0. As a result we can write
5"oi^5"co—Ki\ (o)+Q)
)-+(SH
'SM\ 8e
(14e)
+ 8"coM>. For the third term,
r?H)=G(Al,Q\)
v
ni»-,nf-K3T'
(12b')
(€baEba,t^,H^),
8"n"v^?,"t]>i"-K3
d(G)=4, and to satisfy type II invariance G(A\, Qt) = G(Al + Ql). To satisfy type I, it has to coincide with 7inv in (1), up to changes in the a,. d(X)=d(r)-d(Q)-d(K)=d{r)-d(C)
(14d)
(14f)
(f>-(£>4 <•«
The X terms are renormalized through the redefinitions
-d(L),
Cab->C%-XiCabN{t,r,,E,H),
and we see by inspection that d{X) = 1 and ghost number N#, = 1, so that d(P) = 3, N& (P) = -1. From (12c), (12d), Kl and Ct in P are restricted to the combinations (Kwyba+Cba(H'"'DJl) and (Kg)"x+ Cv{Hf"rD>_). This constrains P to the form (Af, N, N' are arbitrary scalar field Junctionals, Fa. vector in both tangent and world indices and T a world tensor) 62
572
a
(15a)
a
&"C b^S"C
b
(15b) ,
C-,C"-X2C 'N'(e,ri,E,H),
(15c)
31 May 1990
PHYSICS LETTERS B
Volume 242, number 1
75=4+ £
-2.2
8
- ^
+ e
. ( ^ )
8
-
e + e
. ( » )
8
dv=Sv+Ny(co)+Ny(Cb)+Nv(a)+Nv(0)-4.
. ,
(19)
(15d)
The theory is renormalizable if all vertices (i>) in the action have nonpositive d^ this ensures that the divergences can be removed by a finite number of counterterms, since higher order diagrams will not generate higher degree divergences. In the case of our action, dv is indeed non-positive. We conclude that our GL(4, R) quadratic lagrangian with SSB is renormalizable; this is due to its similarity to the Weinberg-Salam model, i.e. a Yang-Mills gauge theory with SSB, and is achieved without quartic propagators for the gravitational field of the theory. It remains to show that a unitary theory can be extracted from this model.
The divergences contributed to rfn) by I^V(A\ + 2 t ) of (1) are removed by a redefinition of the a, in that equation. As a result, the entire divergent piece of r' ( „) has been removed, and our proof by induction from (n— 1) to n is complete.
6. Proof of renormalizability. Finite set of counterterms The degree of divergence of an arbitrary Feynman diagram is
D=4A+Y^-injej
(16) References
where A is the number of independent loops, S„ is the number of derivatives acting on internal lines at the vertex V, 77, the number of internal lines in the diagram corresponding to the field j and 8j is the negative power of the momenta in the relevant propagator fory". Topologically, with Kthe number of vertices, we have the identity A=TjITj+l — v. We can thus reexpress D as Z>=4 + £ (4-0,)77,- £ (4-<J„)
(17)
in the BF method, the 1PI diagrams with external "Q" lines and those with internal "A" lines will not contribute to the effective action. Hence, nJ = \'ZjNJ, where N" is the number of internal lines./' leaving the vertex v. As a result, D=4-
£ ( 4 - 4 - * Z (4-6»,)tfj-) .
(18)
Our effective action yields values of By. 2 for coabM and C f e 4 for rfv and C"; ea„ does not propagate, but to make room for torsion-squared terms (though they do not appear essential to the model; note that a condition of zero torsion would be quantum-stable) we can assume a value 4. If we include terms (Da)2, ( D 0 ) 2 and a2R, &R we find values 2 for a(x) and for
[1] M. Goroffand A. Sagnotti, Phys. Lett. B 160 (1985) 81. [2] A. Casher.Phys. Lett. B 195 (1987) 50. [3] K.S. Stelle, Phys. Rev. D 16 (1977) 953; Gen. Rel. Grav. 9 (1978)353. [4] D. Neville, Phys. Rev. D 18 (1978) 3535; D 21 (1980) 867. [5]E. Sezgin and P. van Nieuwenhuizen, Phys. Rev. D 21 (1980) 3269; D 22 (1980) 301. [6]R.KuhfussandJ.Nitsch,Gen. Rel.Grav. 18 (1986) 1207. [7] Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 200 (1988) 489. [8] G. Stephenson, Nuovo Cimento 9 (1958) 263; C.W. Kilmister and D. J. Newman, Proc. Camb. Philos. Soc. 57 (1961)851; C.N. Yang, Phys. Rev. Lett. 33 (1974) 445. [9] Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 157 (1985) 267, 275. [10]J.Mickelsson,Commun.Math.Phys. 88 (1983) 551. {11 ] A. Cant and Y. Ne'eman, J. Math. Phys. 26 (1985) 3100. [12] Dj. Sijacki and Y. Ne'eman, J. Math. Phys. 26 (1985) 2457. [13] Y. Ne'eman, Ann. Inst. H.Poincar6 28 (1978) 369. [14]Y. Ne'eman and Dj. Sijacki, Intern. J. Mod. Phys. A 2 (1987) 1655. [ 15 ] C. Lee, Renormalization of a gravity model with local GL (4, R) symmetry, University of Texas CPT preprint (February 1990). [16] C. Lee and Y. Ne'eman, Phys. Lett. B 233 (1990) 286. [17]B.S.DeWitt,Phys. Rev. 162 (1967) 1195; J. Honerkamp, Nucl. Phys. B 36 (1971) 130; G. 't Hooft, Nucl. Phys. B 62 (1973) 444; B.S. DeWitt, in: Quantum gravity 2, eds. C. Isham et al. (Clarendon, Oxford, 1981). [ 18] L.F. Abbott, Nucl. Phys. B 185 (1981) 189; Acta Phys. Pol. B13 (1982)33; D.G. Boulware, Phys. Rev. D 23 (1983) 389.
63
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PHYSICAL REVIEW D
VOLUME 48, NUMBER 2
15 JULY 1993
Avoiding degenerate coframes in an affine gauge approach t o quantum gravity Eckehard W. Mielke Institute for Theoretical Physics, University of Cologne, D-509SS Koln, Germany J. Dermott McCrea* Department of Mathematical Physics, University College, Dublin 4 and Dublin Institute for Advanced Studies, Dublin 4, Ireland Yuval Ne'emant Raymond and Beverley Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel Friedrich W. Hehl Institute for Theoretical Physics, University of Cologne, D-5092S Koln, Germany (Received 30 December 1992) In quantum models of gravity, it is surmised that configurations with degenerate coframes could occur during topology change of the underlying spacetime structure. However, the coframe is not the true Yang-Mills-type gauge field of the translations, since it lacks the inhomogeneous gradient term under gauge transformations. By explicitly restoring this "hidden" piece within the framework of the affine gauge approach to gravity, one can avoid the metric or coframe degeneracy which would otherwise interfere with the integrations within the path integral. This is an important advantage for quantization. PACS number(s): 04.50.+h, 11.15.-q, 12.25.+e
I. INTRODUCTION It should be recognized by now that "...gravity is that field which corresponds t o a gauge invariance with respect t o displacement transformations," as Feynman [1] put it. On a macroscopic scale, gravity is empirically rather well described by Einstein's general relativity (GR) theory which resides in a curved pseudoRiemannian spacetime. In a first order formalism, one introduces a local frame field (or vielbein) ea = e ' a 6\ , which is expanded via the tetrad coefficients e'a in terms of the coordinate basis di : = d/dx', together with a coframe field or one-form basis -d13 = ej& dxi , which is dual t o the frame ea with respect to the interior product: e^fi® = e * a e j " = 6% . Except for the Introduction and unless explicitly mentioned, our approach is completely metric independent. Quite often, the coframe i? a is advocated as the translational gauge potential, although it does not transform inhomogeneously under local frame rotations, as is characteristic for a connection. The Einstein-Cartan Lagrangian [2] is given by
ated with the Lorentz connection one-form V"13 = r ^ and r)a0^i : = y ^ d e t oap\ tap-i6 is t h e Levi-Civita tensor. Because of the orthonormality chosen, the local metric components read oap : = diag(l, —1, —1, —1). In constructing macroscopic viable gravitational Lagrangians, the fundamental length £ needs to be identified with t h e Planck length ^pianck- Vacuum G R can be consistently recovered by imposing the constraint of vanishing torsion Ta : = jDt?a via the addition of the Lagrange multiplier term fj.a ATa t o (1.1). Then only the BelinfanteRosenfeld symmetrized energy-momentum current occurs as a source of gravity, the contribution of the matter spin being subtracted out. One avenue of quantizing gravity is to consider t h e functional integral fv'dVT
e x p ( i J Vfec),
(1.2)
where a summation is understood over all inequivalent o
a
o
coframes # : = •&" Pa, Lorentz connections T := V*13 Lap , and spacetime topologies, as well. Since in G R t h e o
* Deceased. ^ n leave from the Center for Particle Physics, University of Texas, Austin, Texas 78712.
Lorentz connection Taff is constrained by Ta = 0, t h e Lagrangian in (1.1) will become the Hilbert second-order Lagrangian and an integration over all coframes is sufficient. In any case, this summation will also involve degenerate ( d e t e j ^ = 0) or even vanishing coframes. This instance would induce the breakdown of any length measurement performed by means of the concept of the metric g = oap i? a <S>tfPand would signal t h e possible occurrence of a topology change, as argued in t h e paper of Horowitz [3]. This also gives a flavor of some of the
0556-282 l/93/48(2)/673(7)/$O6.O0
673
a0
VEC = -^R A^A^Va0yS, where Ral3
(1.1)
= R^a^ is t h e curvature two-form associ-
48
574
© 1993 The American Physical Society
MIELKE, McCREA, NE'EMAN, AND HEHL
674
conceptual difficulties [4] not encountered in the quantization of internal Yang-Mills theories on a fixed spacetime background. Degenerate coframes are not only restricted t o the realm of quantum gravity. In Ashtekar's reformulation of canonical GR cf. Ref. [5], the "triad density," i.e., more precisely, the tangential two-form - # a , is in fact allowed to become degenerate as a classical solution of Hamilton's equations. Moreover, in a first-order formulation of topological three-dimensional (3D) gravity [6], we uncovered a "dynamical symmetry" in which coframe and Lorentz (rotational) connection become related [7] to each other
Thus we obtain i
48 = (
I, as is required for the
1
action of the affine group on flat affine space. The Lie algebra a(n, R) consists of the generators Pa, representing local n-dimensional translations, and the a L p, which span the Lie algebra gl(n, R) of n-dimensional linear transformations. Their commutation relations are (2.2) (2.3)
via tiA = cT*A = §i) A B C T B C , where A, B, C = 0,1,2 (or = 1,2,3 for Euclidean signature) and c is a suitable con-
(2.4)
o
stant. For TAB -+ 0, a degenerate coframe (triad) will occur in this model. By regarding the coframe and Lorentz (rotational) connection as part of the Cartan connection
r = #ApA + rBCLBC,
(1.3)
Witten [8] could show that the 3D Hilbert-Einstein Lagrangian can be absorbed in a Chern-Simons term for (1.3), thus facilitating the proof of the finiteness of the corresponding 3D quantum model. Degenerate coframes, however, tend to jeopardize the coupling of gravity to matter fields, as exemplified by Dirac or Rarita-Schwinger fields. The basic reason is t h a t the local frame ea, even if it still exists, is not invertible any more; i.e., the relation ea\<&0 = £% , which is needed in the formulation of the matter Lagrangian, would then be lost. In this paper, we want to resolve this riddle by demonstrating explicitly that # a is only part of the dimensionless translational gauge potential, if we use a Yang-Mills-type gauge approach to the affine group, which includes the Poincare group of elementary particle physics as subgroup. Thereby we also clarify the subtle relationship between infinitesimal translational gauge transformations and four one-parameter subgroups of diffeomorphism of spacetime, both regarded as acting actively.
In the flat n-dimensional affine space R", t h e rigid affine group A(n, R) := it n g:GL(n, R) acts as the semidirect product of the group of n-dimensional translations and n-dimensional general linear transformations. Thus it is a generalization of the Poincare group P := R4 & SO(l,3), with the pseudo-orthogonal group S O ( l , n — 1) being replaced by the general linear group GL(n,.R). In the following, we will work in a Mobius-type representation [9, 10]. T h e A(n, R) is that subgroup of GL(n-t-l, R) which leaves the n-dimensional hyperplane :=
A(n,R)
j x = (*) = i(A
e Rn+1\
(2. 5)
Here Lap := L^a^ are the generators of Lorentz transform a t i o n s , ^ , ^ := £( a ,a) — (1/n) gagL"1^ represent shears, whereas T> := L 7 7 is the generator of dilations. III. THE AFFINE GAUGE A P P R O A C H In a matrix representation analogous to (2.1), we can write the affine gauge group [11] as A(n,R)
= j(A(0z)
r(
x) 1
) | A(x) e
gC(n,R),
T(X) eT(n,R)\.
(3.1)
Following a Yang-Mills-type gauge approach, we intro-
and require t h a t it transforms inhomogeneously under an affine gauge transformation: JI-'W
? = A~l{x)V
A(x) +
A~1{x)dA(x), A(x) e A(n, R). (3.3)
Since we regard it as an active transformation, it is formed with respect to the group element
invariant :
A-i{x)=(A-Hx)
jjsGL(n+l,fi)|
A e G L ( n , . R ) , reR"\.
Lap = ( Lag +/La/3 + — ga0 V \ .
duce the generalized affine connection [12], cf. Ref. [13],
II. T H E R I G I D A F F I N E G R O U P A(n, R)
R"
If there is a spacetime metric g = gap ida
(2.1)
575
-A-H*Mx)y
which is inverse to A(x) € A(n,R). affine curvature is given by
(3>. 4)
The corresponding
48
AVOIDING DEGENERATE COFRAMES IN AN AFFINE GAUGE . . . pa
R:
fa
fts
fts
=dr + TAr L
L
r
r
_ / dr< > + r ( « A r< > dr< > + r w A r< > \ -{ o o ) (3.5) and transforms covariantly under the affine gauge group [14], i.e., R
A
^
R = A'1(x)RA(x).
affine p-form \& = I . I as (3.7)
T
Only by imposing the gauge r ' ' = 0, would one recover the covariant exterior derivative D := d + T^ with respect to the linear connection. If we insert (3.2) and (3.4) into the inhomogeneous transformation law (3.3), it splits into T<-L'>' = A~l(x)r<-L'>A(x)
^
A-1(x)dA(x),
+
(3.8) and r
rffl' = A - ' ( x ) r i " + A - l ( * ) i M * ) .
^ » )
(3.9) The local translations T(X) automatically drop out in (3.8) because of the one-form structure of r ' r ' . Thereby (3.8) aquires the conventional transformation behavior with the exterior derivative d (and not the covariant one) of a Yang-Mills-type connection for QC(n,R). Thus we can identify T ^ ' = V = Ta^ La0 with the linear connection. Because of the covariant exterior derivative term DT(X) := dr(x) + r W r ( x ) in (3.9), the translational part r ' T ) does not transform as a covector, as is required for the coframe t? := t?" Pa, i.e., the one-form with values in the Lie algebra of Rn. However, we may follow Trautman [15] and introduce a vector- •valued zero-form £
£
A
^
= 0, would imply van-
IV. R E D U C T I O N T O A C A R T A N C O N N E C T I O N
=
( n )
=
( i
a
) > which
r\T)a=eia-Di$a,
(4.1)
which, for Di£a — 6f, makes contact with the approach of Hayashi et al. [18]. In a recent paper [19] on t h e Poincare^ gauge approach, the £a are kinematically interpreted as "Poincar6 coordinates"; note t h a t in Eq. (2.14) of that paper, vielbein and translational connection are identified opposite to our notation. Observe also t h a t we do not have to put the "Poincare1 coordinates" £ a to zero in order to obtain the affine gauge transformation law (3.12) of the coframe. The reason is t h a t the local translations are now "hidden" in the invariant transformation behavior of the exterior one-form 1? under (passive) diffeomorphisms. Note also that in our approach, in contrast with that of Sexl and Urbantke [20, p. 381], we do not need t o break the affine gauge group kinematically via DT(X) = 0. An attempt to motivate t h e translational connection (4.1) from the theory of dislocations can be found in Ref. [21], whereas Hennig and Nitsch [22] provide an explanation in terms of jet bundles. Since f = €aPa aquires its values in the "orbit" (coset space) A(n, R)/gl(n, R) =» Rn, it can be regarded as an affine vector field (or "generalized Higgs field" according to Trautman [23]) which "hides" t h e action of the local translational "symmetry" T{n,R). If we required t h e condition [24] D(. = 0 ,
(4.2)
the translational connection T^ would, together with the coframe »9, be soldered to the spacetime manifold cf. [25], and t h e translational part of t h e affine gauge group would be "spontaneously broken" cf. [26]. The stronger constraint of a "zero section" vector field £ = 0 would
sa
PS'
transforms as f
teleparallelism models with Ra ishing torsion [17].
Our key relation (3.11), in components, takes t h e form
5i-(D*;r(T)).
A
If r( T > vanished throughout the manifold, t h e vector field £ would represent a four-dimensional version of Cartan's generalized radius vector [16]. T h e integrability condition is, in this instance, given by t h e vanishing of the translational part of the affine curvature (3.5), i.e., a R(T) ~ DrV) + Rf)a^)Pa = 0, which, for = (T
(3.6)
T h e covariant exterior derivative D := d + V acts on an
VW
675
= J 4 _ 1 ( I ) £,
reduce the generalized affine connection T on the affine bundle A(M) to the Cartan connection [9]
i.e.,
^-A-'CxJK-rCx)]
(3.10)
under an active affine gauge transformation. Then tf := r ( T ) + D£ transforms as a vector-valued A(n,R), as required: A-'(x)
$' =
(3.11) one-form
A-1(x)ti.
under
the
(3.12)
576
-(s»)
(4.3)
on the bundle L{M) of linear frames. Because of Eq. (3.12), this is not anymore a connection in the usual sense. However, thereby we would recover the familiar (metric-) affine geometrical arena [27] with nonmetricity, torsion, and curvature, as is summarized in Table I. In the anti-de Sitter gauge model of gravity of Stelle
676
MIELKE, McCREA, NE'EMAN, AND HEHL
48
TABLE I. Affine geometrical arena with nonmetricity, torsion, and curvature. Potential Metric g""3 Coframe Connection r „ "
tfa Re13
Field strength Q0"3 = Dgal3 T" = £>i?<* = dTa? + r 7 " A I V
and West [28], t h e £" parametrize t h e coset space SO(2,3)/SO(l,3). The coframe •da and t h e Lorentz cono
Bianchi identity DQ"e = 2Ay'" g^i DTa = R-,a A i F £>.&/ = 0
(CE + 6U)T = \D(UJ + e\Vj) + e\Ra0] La0 , (5.6)
o
nection Va/3 = —T^a can then be derived from t h e original SO(2,3) connection via a nonlinear realization of t h a t group involving t h e £ field. Such a Cartan connection arises not only from a reduction of (anti-)de Sitter bundles [29], but also from conformal G structures [30].
and (.Ct+6U)0 = ]pea - ( t V + e j i y * ) ^ + £JT Q ] Pa . (5.7)
V. AFFINE GAUGE TRANSFORMATIONS VS ACTIVE DIFFEOMORPHISMS The affine gauge transformations in (3.3) are finite transformations. If we expand them up t o first order according to A(x) = l+ujLae
+ --- ,
r{x) = 0 + ea Pa + • • • ,
(5.1) (5.2)
we obtain from (3.8) and (3.9), respectively, ^ - . r ^ = ( © < * / ) £",» + • • • ,
(5.3)
<5A-ir
(5.4)
Incidentally, for t h e "product" of Lie generators we use the Lie brackets of Sec. II, since we work in the adjoint representation. I t is gratifying t o note t h a t the leading exterior covariant derivatives reveal, in particular, that the translational connection T^ is really the "compensating" field for infinitesimal local translations s in the Yang-Mills sense. Let us compare this result with the "diffeomorphism" approach, which was orginally developed for the Poincare subgroup of t h e A(n,R): In essence, the translational part of the transformation U = \ + e + u = l + eaPa+wjLc'0
(5.5)
is embedded as an n-parameter subgroup of the infinitedimensional group of active diffeomorphisms of spacetime [31]. In order t o calculate t h e effect on the linear connection and the coframe, one has t o consider the action [32] of the Lie derivative Ce with respect t o the vector field £ together with an infinitesimal frame rotation parametrized by u>. Since Ce = tc := (e\d+de\) holds for geometrical objects which are invariant under changes of the basis, a straightforward calculation yields
577
The "annoying" linear connection term in (5.6) and (5.7) can be dismissed by going over to t h e parallel transport version of Hehl el al. [2] and Ne'eman [33] in which, instead of Pa = —dat t h e covariant derivative components Da := ea\D are adopted as generators of local translations: Then the infinitesimal transformations read ft = 1 - ea Da + u>J La(, = n - e J l V L"0 . (5.8) Since this amounts t o a redefinition u> := ui—e\ Ta® La0 of the parameters of t h e infinitesimal linear transformation, we can simply read off, from (5.6) and (5.7), t h e new results (C; + 6*)T = \DUJ>
+ &\ Ra0] L
a 0
,
(5.9)
and (C€ -r fc)tf = [Dea - w0a^ + ejTa]
Pa .
(5.10)
In this parallel transport version, the leading covariant derivative pieces are t h e same as in t h e affine gauge approach. In particular, the "hidden" translational piece in the affine transformation (3.12) of t h e coframe gets thereby "uncovered" in (5.10). In the end, is it "...somewhat a matter of taste...," as Nester [34] p u t it, whether or not one prefers the parallel transport interpretation of translations over the affine gauge approach? One could argue that the Pauli-type curvature and torsion terms in the infinitesimal transformations (5.9) and (5.10) violate the spirit of t h e principle of minimal coupling, a cornerstone of a conventional Yang-Mills-type gauge approach. These terms also show up in the commutation relation [Da , D0] = -Ta0iDy
+ Ra0-ys IPs
(5.11)
for the operator Da of parallel transport, if applied to a zero-form. Because of the torsion and curvature terms on the right-hand side, a softening [35] of the Lie algebra
48
AVOIDING DEGENERATE COFRAMES IN AN AFFINE GAUGE . . .
structure cannot be avoided in such a diffeomorphismtype approach. Using the covariant derivatives (or Lie derivatives) has t h e advantage of being physically meaningful as a parallel transport, as explained in Ref. [2], once we put up a frame, and, in a corresponding first order approach, these "nonminimal" structures do not touch the explicit form of the Lagrangian. However, they are algebraically less useful because (5.11) is not a Lie algebra any more. Moreover, as we will show below, the affine gauge approach lends itself to an important resolution of degeneracy problems in quantum gravity. VI. AFFINE G A U G E A P P R O A C H TO Q U A N T U M GRAVITY W I T H TOPOLOGY CHANGE Now we may return to the question of the proper meaning and range of validity of the functional integral (1.2) in quantum gravity. T h e lesson learned from our affine gauge approach is that, instead of the coframe, a summation over the true translational connection r ' T ) is more I
vkcc = -•&«"'
A
677
akin to a quantum Yang-Mills-type approach. Moreover, it would cause no problems if t h e functional integration were not only performed over T ^ ' = 0, as in a Yang-Mills theory, b u t over r ( T ) = 0 as well, keeping the coframe t5, by definition, nondegenerate. In effect, we may now consider the functional integral
JvT exp(if
VE C ),
(6.1)
where summation is understood over the generalized affine connection. Because of Eqs. (3.8) and (3.12), t h e Einstein-Cartan Lagrangian is gauge invariant also with respect to the full, but "hidden," affine gauge group .4(4, R). In order to study the possibility of a degenerate or even vanishing translational connection, it is instructive to insert the representation (3.11) of t h e coframe into the EinsteinCartan Lagrangian (1.1) amended by a cosmological term (-hcoa/t2)vIn the gauge T ( T ) —* 0, we find then t h e truncated expression
( r ( 7 > + D O A (r
- ^ f r ( r ( T ) Q + • D « Q ) A ( r ( T ) " + Dt")A ( r ( T ) 7 + °C) A (r (T)5 + DftTh&s
—*
-^R010 A r AC m
V evaw + ^eoe
-&*[(*"
°
+
the Bianchi identity DR & = 0 for the Riemann-Cartan curvature, which is valid in the Riemann-Cartan framework of (6.2) with vanishing nonmetricity. Thus the occurrence of a vanishing translational connection in the functional integral (6.1) is without harm, because of the survival of a quadratic and linear curvature Lagrangian for the linear connection. In the case that A cos = 0, except for the dimensional coupling constant, the truncated expression (6.2) resembles the Stephenson-Kilmister-Yang (SKY) Lagrangian [3638], which is known to be perturbatively renormalizable [39, 40]. Moreover, for the constant vacuum condensate £<> = H ^ 0 and zero otherwise, the Lagrangian (6.2) reduces "spontaneously" to the Euclidean 3D topological gravity model: oAB
[VABC R
(6.2)
in which the dreibein can be identified with part of the o
o a
X r
^a/37*
I f M " ^ A D? A D?)^,] .
In order to separate off the boundary term, we employed
V f e c c o = jp
A DC A V
o
A RoCH2
+%* tiABc V
A
hB *
h°Hi]. (6.3)
578
o
Lorentz connection via dA = TQA and TA = RQA- Other directions of symmetry breaking, such as £<> = H ^ 0 , lead t o Minkowskian models of 3D gravity [41] cf. [42]. For a time-dependent H = H{t), one would obtain, in addition t o the curvature-dependent H2 + H* potential in (6.3), a kinetic term for H and could analyze, following Giddings [43], the instability of t h e £a = 0 solution. Thus, in the gauge TL = 0, we obtain a means to analyze the instability not only for the diffeomorphism invariant solution i/ =. 0, but also for the true translational invariant solution T T = 0. It remains to be seen, if also the signature of the physical spacetime has a dynamical origin in such a framework, as is suggested by Greensite [44], cf. [45]. In quantum gravity, a vanishing translational connection r ( r ' may be accompanied by a topology change of the underlying spacetime manifold. T h e rich spectrum of possible topological structures in quantum "geometrodynamics" has been outlined in Ref. [46]. Our affine gauge approach exactly leads us t o the more detailed mechanism devised by Horowitz [3]. According to (3.11), the vanishing of the translational connection converts the
MIELKE, McCREA, NE'EMAN, AND HEHL
678
coframe components into tfa = D£a. At each of the boundaries 8MA w S3[JS'3 of the topology changing spacetime manifold M* we may also adopt the gauge r( L > = 0 for the linear (or Lorentz) connection. In the vicinity of that boundary, i?Q = d4a
and
g = oa0 d£a ® d^
[1] R.P. Feynman, Lectures on Gravitation, compiled by F.B. Morinigo and W.G. Wagner (California Institute of Technology, Pasadena, California, 1962/63). [2] F.W. Hehl, P. von der Heyde, G.D. Kerlick, and J.M. Nester, Rev. Mod. Phys. 48, 393 (1976); see also F.W. Hehl, in Cosmology and Gravitation: Spin, Torsion, Rotation, and Supergravity, Proceedings of the International School, Erice, Italy, 1979, edited by P.G. Bergmann and V. de Sabbata, NATO Advanced Study Institute Series B: Physics, Vol. 58 (Plenum, New York, 1980), p. 5. [3] G.T. Horowitz, Class. Quantum Grav. 8, 587 (1991). [4] C.J. Ishara, in Recent Aspects of Quantum Fields, Proceedings of the XXXth International Universitatswochen fur Kernphysik, Schladming, Austria, 1991, edited by H. Mitter and H. Gausterer, Lecture Notes in Physics, Vol. 396 (Springer, Berlin, 1991), p. 123. [5] E.W. Mielke, Ann. Phys. (N.Y.) 219, 78 (1992). [6] P. Baekler, E.W. Mielke, and F.W. Hehl, Nuovo Cimento B 107, 91 (1992). [7] This is analogous to the connection representation in which the "triad density" is represented by the functional derivative ^i9_B := S/SAB with respect to the canonically ±
[10] [11]
in Fig. 1 of Fief. [47]. In summary, the advantage of the affine gauge approach is that the coframe, even for vanishing translational connection, remains nondegenerate almost everywhere.
(6.4)
represent the (fiat) Minkowskian spacetime in terms of the four-dimensional Cartesian coordinate system {£"}• Obviously, this solves also the vacuum Einstein-Cartan theory with zero action. For generic £a, the (inverse) tetrad components e^ = 9, (f will be nondegenerate almost everywhere. In order to accomplish a spatial topology change, we may choose £° to be the "height function" of Morse theory. Then, d£ is a timelike covector and we can apply the "trouser world" construction as is described, for example, by Konstantinov and Melnikov
[8] [9]
48
conjugate Ashtekar variable AB. This provides a mapping from the Hamiltonian constraint of gravity with a cosmological term to the Chern-Simons three-form [7] of the Ashtekar-Sen connection, see B. Brugmann, R. Gambini, and J. Pullin, Nucl. Phys. B385, 587 (1992). E. Witten, Nucl. Phys. B311, 46 (1988/89). S. Kobayashi, Transformation Groups in Differential Geometry (Springer, New York, 1972). E.W. Mielke, Geometrodynamics of Gauge Fields — On the Geometry of Yang-Mills and Gravitational Gauge Theories (Akademie-Verlag, Berlin, 1987). In a fiber bundle approach, one introduces first the bundle of affine frames A(M) := P(Mn,A(n,R),ir,S) where TV denotes the projection to the base manifold and S the (left) action of the structure group A(n, R) on the bundle. Active affine gauge transformations are the vertical automorphisms of A(M) . Similarly as the diffeomorphisms of the base manifold M " , they form the infinite-
579
ACKNOWLEDGMENTS We would like to thank Jorg Hennig (Clausthal) and Yuri Obukhov (Cologne/Moscow) for constructive comments on a preliminary version of this paper. E.W.M., F.W.H., and Y.N. were supported by t h e GermanIsraeli Foundation for Scientific Research &c Development (GIF), Jerusalem and Munich, project 1-52-212.7/87. Y.N. was supported in part by DOE Grant No. DEFG05-85-ER40200.
dimensional group A(ji, R) := C°°(A(M) XAd A{n, R)). The group QC{n, R) := C°°(^(Af) x A d GL(n, R)) of linear gauge transformations and the group T(n, R) := C°°(A(M) XAd Rn) °f local translations are subgroups of A(n, R). Taking the cross section in the associated bundle is abbreviated by C°° and Ad denotes the adjoint representation with respect to GL(n, R). Because of its construction, the group of local translations T{n, R) is locally isomorphic to the group of active diffeomorphisms Diff(n, R) of the manifold, cf. V.I. Ogievetsky, Lett. Nuovo Cimento 8, 988 (1973), and S. Sternberg, Ann. Phys. (N.Y.) 162, 85 (1985). The infinite-dimensional group Diff(n, R) contains the (n + ro2)-dimensional group A(n, R)H of holonomic affine transformations as a subgroup, cf. P.G. Bergmann and A.B. Komar, J. Math. Phys. 26, 2030 (1985), which is generated by the fields Pi = —dt and Vj — —x'dj. Note that differentiable coordinate transformations, which leave exterior forms invariant, are regarded as passive diffeomorphisms. [12] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1963), Vol. I. [13] L.K. Norris, R.O. Fulp, and W.R. Davis, Phys. Lett. 79A, 278 (1980); V.N. Ponomariev, A.O. Bravinsky, and Yu.N. Obukhov, Geometrodynamical Methods and Gauge Approach to the Theory of Gravitational Interactions, in Russian (Energoatomisdat, Moscow, 1985). [14] Our matrix formalism, cf. [10] and K.L. Malyshev and A.E. Romanov, "Aspects of T(3) & 50(3)-gauge theory of dislocations and disclinations I and II," in Russian, Leningrad preprints LOMIP-2-90 and P-5-90 (1990), and references therein, is a spacetime generalization of the socalled motor calculus of R. von Mises, Z. Angew. Math. Mech. 4, 155 (1924). [15] A. Trautman, in Differential Geometry, Symposia Matematica Vol. 12 (Academic, London, 1973), p. 139. [16] E. Cartan, On a Manifold with an Affine Connection and the Theory of General Relativity (Bibliopolis, Napoli, 1986). [17] Otherwise there also exists, for example, the nontrivial solution Ta = anaff(p and Ra(l = an""3, with the dimensionful constant o.
48
AVOIDING DEGENERATE COFRAMES IN AN AFFINE GAUGE . . .
[18] K. Hayashi a n d T . Nakano, P r o g . Theor. Phys. 3 8 , 491 (1967); K. Hayashi and T . Shirafuji, ibid. 6 4 , 866 (1980); 80, 711 (1988). [19] G. Grignani and G. Nardelli, P h y s . Rev. D 4 5 , 2719 (1992). [20] R.U. Sexl a n d H.K. Urbantke, Gravitation und Kosmologie, 2nd ed. (Bibliographisches Institut, Mannheim, 1983). [21] G- Sardanashvily and M. Gogbershvily, Mod. Phys. Lett. A 2 , 609 (1987). [22] J. Hennig and J. Nitsch, Gen. Relativ. Gravit. 1 3 , 947 (1981). [23] A. T r a u t m a n , Czech. J. Phys. B 2 9 , 107 (1979). [24] K.A. Pilch, Lett. M a t h . Phys. 4, 49 (1980). [25] P.K. Smrz, J. M a t h . Phys. 2 8 , 2824 (1987). [26] L. O'Raifeartaigh, in Differential Geometry, Group Representations, and Quantization, edited by J . D . Hennig, W . Liicke, and J. Tolar, Lecture Notes in Physics, Vol. 379 (Springer, Berlin, 1991), p . 99. [27] F . W . Hehl, J . D . McCrea, E.W. Mielke, and Y. Ne'eman, Found. P h y s . 1 9 , 1075 (1989); P h y s . R e p . (to be p u b lished); J . D . McCrea, Class. Q u a n t u m Grav. 9, 553 (1992). [28] K.S. Stelle a n d P.C. West, Phys. Rev. D 2 1 , 1466 (1980). [29] S. Gotzes and A.C. Hirshfeld, Ann. Phys. (N.Y.) 2 0 3 , 410 (1990). [30] J . D . Hennig, in Conformal Geometry and Spacetime Gauge Theories, Proceedings of t h e 2nd International Wigner-Conference, Goslar, 1991, edited by H.D. Doebner et al. (World Scientific, Singapore, 1992). [31] T h e plus sign is in accordance with our earlier conventions for active transformations.
679
[32] R . D . Hecht, F . W . Hehl, J . D . M c C r e a , E . W . Mielke, a n d Y. Ne'eman, P h y s . Lett. A 1 7 2 , 13 (1992). (33) Y. Ne'eman, in Differentia! Geometrical Methods in Mathematical Physics, edited by K. Bleuler, H.R. Petry, a n d A. Reetz, Lecture Notes in M a t h e m a t i c s , Vol. 676 (Springer-Verlag, Berlin, 1978), p . 189. [34] J. M. Nester, in An Introduction to Kaluza-Klein Theories, Proceedings of t h e Workshop, C h a l k River, O n t a r i o , 1983, edited by H.C. Lee (World Scientific, Singapore, 1984), p . 83. [35] T . W . B . Kibble a n d K.S. Stelle, in Progress in Quantum Field Theory: Festschrift for Umezawa, edited b y H. Ezawa a n d S. Kamefuchi (Elsevier, N e w York, 1986), p. 57. [36] G. Stephenson, Nuovo Cimento 9, 263 (1958). [37] C.W. Kilmister a n d D.J. N e w m a n , P r o c . C a m b r i d g e P h i los. Soc. 5 7 , 851 (1961). [38] C.N. Yang, P h y s . Rev. Lett. 3 3 , 445 (1974). [39] K.S. Stelle, P h y s . Rev. D 16, 953 (1977). [40] C.-Y. Lee a n d Y. N e ' e m a n , Phys. L e t t . B 2 4 2 , 59 (1990); C.-Y. Lee, Class. Q u a n t u m Grav. 9, 2001 (1992). [41] T . T . Burwick, A.H. Chamseddine, a n d K . A . Meissner, P h y s . Lett. B 2 8 4 , 11 (1992). [42] G. Grignani and G. Nardelli, P h y s . L e t t . B 3 0 0 , 38 (1993). [43] S.B. Giddings, P h y s . Lett. B 2 6 8 , 17 (1991). [44] J. Greensite, Phys. Lett. B 3 0 0 , 34 (1993). [45] A.D. Sakharov, Zh. E k s p . Teor. Fiz. 8 7 , 375 (1984) [Sov. Phys. J E T P 6 0 , 214 (1984)]. [46] E.W. Mielke, Gen. Relativ. Gravit. 8 , 175 (1977). [47] M.Yu. Konstantinov a n d V.N. Melnikov, Class. Q u a n t u m Grav. 3 , 401 (1986).
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Class. Quantum Grav. 14 (1997) A251-A259. Printed in the UK
PII: S0264-9381(97)77948-6
Test matter in a spacetime with nonmetricity Yuval Ne'emanf and Friedrich W Hehlf f Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, Israel 69978 and Center for Particle Physics, University of Texas, Austin, TX 78712, USA $ Institute for Theoretical Physics, University of Cologne, D-50923 KOln, Germany Abstract. Examples in which spacetime might become non-Riemannian appear above Planck energies in string theory or, in the very early Universe, in the inflationary model. The simplest such geometry is metric-affine geometry, in which nonmetricity appears as a field strength, side by side with curvature and torsion. In matter, the shear and dilation currents couple to nonmetricity, and they are its sources. After reviewing the equations of motion and the Noether identities, we study two recent vacuum solutions of the metric-affine gauge theory of gravity. We then use the values of the nonmetricity in these solutions to study the motion of the appropriate test matter. As a Regge-trajectory-like hadronic excitation band, the test matter is endowed with shear degrees of freedom and described by a world spinor. PACS numbers: 0450, 0350K, 0420J
1. The case for metric-affine gravity Even though Einstein's treatment of spacetime as a Riemannian manifold appears to be fully corroborated experimentally, there are several reasons to believe that the validity of such a description is limited to macroscopic structures and to the present cosmological era. Indications [1] from the only available finite perturbative treatment of quantum gravity— namely the theory of the quantum superstring—point to non-Riemannian features on the scale of the Planck length. On the other hand, recent advances in cosmogony, i.e. in the study of the early Universe as represented by the inflationary model, involve, in addition to the metric tensor, at the very least a scalar dilaton [2] induced by a Weyl geometry, i.e. again an essential departure from Riemannian metricity. Allowing minimal departures from Riemannian geometry (i.e. from a V4 manifold) would consist in allowing torsion (i.e. a I/4) and nonmetricity (i.e. an (L4, g)). Andrzej Trautman, to whom this paper is dedicated on the occasion of his 64th birthday, has made important contributions to the study of the first suggestion [3, 4], namely the possibility of a spacetime with torsion Ta ^ 0. In this work, we would like to sketch some of the features relating to the second possibility, namely to the assumption that spacetime is endowed with nonmetricity^ Qap ^ 0. We have recently reviewed [6] the class of gravitational theories with such geometries, the metric-affine gauge theories of gravity (or 'metric-affine gravity' MAG for short). As in any gauge theory, the geometrical fields of gravity are induced by matter currents. In § This is a 'positive' paraphrasing of the more conventional 'negative' assertion, namely that spacetime does not fulfil the Riemannian metricity constraint Qap = —Dgap = 0 [5], a wording influenced by our Einsteinian conditioning. 0264-9381/97/SA0251+09S19.50
© 1997 IOP Publishing Ltd
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Einsteinian gravity, it is the symmetric (Hilbert) energy-momentum current which acts as a source for the metric field and the Riemannian curvature. In MAG, we have, in addition, the spin current and the dilation plus shear currents inducing the torsion and nonmetricity fields, respectively. Both spin and dilation plus shear are components of the hypermomentum current, symmetric for dilation plus shear and antisymmetric for spin. And yet, there is a rather profound difference between these two physical features. Special relativity (SR) is synonymous with Poincar6 invariance, which includes the Noether conservation of angular momentum. It would then be relatively straightforward, at the level of relativistic quantum field theory (RQFT), to constrain the kinematics so as to do away with the orbital part of angular momentum and thus obtain a conserved spin current. However, even this is unnecessary, since through its Pauli-Lubafiski realization, spin itself is related to the density of a Poincare group invariant and is thus an 'absolute' of SR. The conservation of shear, on the other hand, is not a characteristic of SR and would require the homogeneous Lorentz group—or its double-covering group SL(2, C) = 50(3, 1)—to be embedded in the larger SX(4, R). Such a switching from Poincare to affine A(4, R) = R4 2> 5L(4, R) is, however, implied in our having given up the Riemannian metricity condition, since we have thereby also lost the presence of the pseudo-orthogonal group as the local symmetry of the tangent manifold, i.e. the local Lorentz frames and with them the equivalence principle, with the direct transition to SR. Indeed, this is the 'meaning' of our basic non-Riemannian ansatz, namely that we are studying phenomena and situations in which there is no conventional 'flat' SR limit—either in small dimensions, when approaching the Planck length, or in the early Universe, during inflation, within Planck times from the 'seeding' vacuum fluctuation 'event'. Presumably, it is then through a spontaneous breakdown of the local A(4, R) symmetry below Planck energies, down to Poincare invariance, that SR and the Riemannian metricity condition set in (see [7, 8] for such examples). Alternatively, we might be dealing with situations in which the dynamics have led to boundary conditions generating shear currents—quadrupolar pulsations of nuclear or hadron matter in small dimensions [9], for example, or the Obukhov-Tresguerres hyperfluid [10] in macroscopic configurations. 2. World spinors as matter fields The unavailability of local Lorentz frames poses no problem in the context of boson fields. The latter are conventionally represented by tensors, i.e. linear field representations of SL(4, R). These become world tensors in the transition from special to general relativity, i.e. nonlinear realizations of the group of local diffeomorphisms Diff(4, R), carried linearly through the SL(4, R)H holonomic linear subgroup. In RQFT, the fact that tensor fields are built to carry the action of a group larger than that allowed by SR, is taken care of through subsidiary conditions, etc. Thus, the (symmetric) energy-momentum tensor's 10 components a'j, as defined in GR through the action of SL(A, R), e.g. through ffy :=2(-gr1'2SL/Sgij,
(1)
are a good example of a 10-dimensional SL(4, 7?)-irreducible multiplet then reducing under SO(l, 3) (or SL(2, C)) into 9 + 1—the seggregation of the ' 1 ' being assured through the removal of the trace, indeed a Lorentz scalar. In any case, boson fields are naturally constructed so as to be capable of carrying the action of SL(4, R), instead of the Lorentz group, whether in a local frame or holonomically. This is not true of the conventional fermion fields we use to represent matter. These are spin |7| = \ field representations of the double covering of the Lorentz or Poincare groups, i.e. of Spin(l, 3) = SL(2, C) or of R4 2> SL(2, C) and can only carry—at best— nonlinear realizations of A (4, R) H C Diff(4, R). Linear action can nevertheless be realized,
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A253
through the use of infinite-component manifields, linear field representations of the double covering of the linear, affine and diffeomorphism groups [11]. Such fields can be used in a Riemannian context and even in SR, as well as in our present metric-affine geometry. In the former, they are particularly suited for the description of hadrons and nuclei, composite objects displaying excitation bands [9]. These phenomenological features have no other description in the framework of an effective field theory. Fermionic hadrons or nuclei are then assigned to spinor manifields—world spinors in GR—and boson excitation bands to the related boson manifields ('infinitensors'). As to the non-Riemannian scenarios of superPlanckian energies or of the early Universe, spinor manifields enter naturally in the context of quantum superstring theory. As field representations of SL(4, R), world spinors can also be assigned to a local SL(4, R)A anholonomic frame, as well as serving holonomically and carrying the action of Diff(4, R) D SL(4, R)H. The different spin levels in a world spinor are related by the \SJ\=2
(2)
spin-raising and spin-lowering action of the gravitational field. In its absence, i.e. in SR, world spinor manifields reduce to an (infinite) direct sum of Lorentz spinor fields— a reduction similar in principle to what happens to the a'-* tensor in our example above; moreover, the anholonomic spinor manifields can be assigned to the more elegant multiplicity-free representations. World spinor manifields, however, cannot stay in such representations; transvection of an anholonomic spinor manifield into a world spinor, using countable-infinite vielbeins, destroys the multiplicity-free feature (see [12], also chapter 4 and appendices Cl-6 in [6]), as exemplified by the Mickelsson equation [13]. 3. Geometrical fields, currents and equations of motion We denote the frame field by ea = e'a 3,-
(3)
and the coframe field by &p=e/dxj.
(4)
The GL(4, /f)-covariant derivative for a tensor-valued p-form is D = d+r\//o(L°»A,
(5)
where p is the representation of GL(4, R) and Lap are the generators; the connection 1-form is VaP = Tia^ dx'. The nonmetricity is a 1-form Qap := -Dgap,
(6)
the torsion and curvature are 2-forms 7"*:=D# 0 ,
(7)
/?/ := d r / - r\/ A r / .
(8)
The Weyl 1-form Q--=iQyr,
(9)
when subtracted from the nonmetricity, yields the traceless nonmetricity Zap •= Q*P - QgaP-
(10)
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The Bianchi identities are DQaf,=2R(aP), DTa = RYa A &y, DRj = 0.
(11) (12) (13)
As regards the physics, Qap, Ta and Ra& play the role of field strengths. We now turn to the source currents for the above fields. These will depend on the Lagrangian (^ is a matter manifield), itot = LM{g„p, dgafi,iT, dtf". T / , d r / , V, D¥). which can be rewritten in a covariantized form as
(14)
Lm = W f t t f , Co*. * ° . r " , Rj, *, D*). (15) Separating the Lagrangian Ltot = VMAG + L into geometrical VMAG and matter L parts, the matter current 3-forms are then given by the Euler-Lagrange functional derivatives (denoted by S) of the material piece L. We have the canonical energy-momentum current S a := SL/S&" = dL/d&a + D(dL/dTa),
(16)
the hypermomentum current A"/, := 8L/SVJ
= (L%*)
A
(3L/3(D*))
+2gpy(3L/dQar) + i?a A {3L/dTp) + D(BL/8Rap), (17) and also a related 'current', which is a 4-form, the (symmetric) metric energy-momentum, which we used in (1) as an example of a tensor which reduces under SR, namely or"" := 2SL/Sgaf) = 2dL/dgafl + 2D{dL/dQap).
(18)
Then the field equations turn out to be [6] 8L/8V = 0 (matter), DM"^ - mafl = aafl (zeroth), DHa -Ea = T,a (first), a a a DH f, - E fs = A f, (second), where we have used the canonical momenta ('excitations'), Af" := -2dVMAG/dQan, a (3-form) momentum conjugate to the metric field, Ha:=-dVuAG/ZTa,
(19) (20) (21) (22) (23) (24)
a (2-form) momentum conjugate to the coframe field and H"f> := -dVMAG/dRafi, (25) the (2-form) momentum conjugate to the GL(4, 7?)-connection. The currents m°',E a ,E a p are, respectively, components of the metric energymomentum, of the canonical energy-momentum and of the hypermomentum currents, contributed, via VMAG* by the gravitational fields themselves—the so-called vacuum contributions. Diffeomorphisms and GL(4, R) invariance yield two Noether identities [6] which, given in their 'weak' form, i.e. after the application of the matter equation of motion (19), become DS a = {ea\T?) A E , + (e„J V ) DA% + »a A E , - gpyaaY = 0.
A A
"r - i ^ a J G / i y V ,
584
(26) (27)
Test matter in a spacetime with nonmetricity
A255
4. The OVETH spherically symmetric vacuum solution The search for exact solutions to the field equations of MAG is still in its infancy. First, Tresguerres [14] and, subsequently, Tucker and Wang [15] treated simplified situations, in which only gravitational dilation currents represented a departure from Riemannian geometry. Recently, a vacuum solution (i.e. with L = 0) has been found by Obukhov et al [16] ('OVETH'), in which the selection of VMAG> however, is such as to provide for (gravitational) sources of shear, dilation and spin. Most recently, a static vacuum solution with axial symmetry was added to the set [17] ('VTOH'). In terms of irreducible components, in a metric-affine spacetime, the curvature has 11 parts, the torsion 3 and the nonmetricity 4 (see appendix B in [6]). A general quadratic Lagrangian (signature — h + + ) can thus be written as 1 VMAG
2K
-a0Rap
A r]aP - 2kr, + Ta A ( J2
a\n:
(28) Here K := 2n£planck/'(he) is the gravitational and X the cosmological constant, r\ is the volume 4-form, r\ap := *(&a A ftp) and ao-3. ^i-4> c 2 -4, Wi_6, Z1-5 are dimensionless coupling constants. The antisymmetric and symmetric components of the curvature are denoted by Wap := R[ap] and Zap := /?(tt/g), respectively. The OVETH solution belongs to a somewhat simplified Lagrangian with w, = 0 ,
(29)
Zl = Z2 = Zi = Z5 = 0 , w
Y
i.e. preserving in VMAG only one component Zap := Ry gap/A from the symmetric part of the curvature, namely the trace, Weyl's segmental curvature. In addition, the following constants do not occur in the solution and can be put to zero: a\ = a3 = b\ = b2 = c2 = 0.
(30)
This leaves in VMAG terms involving two parts of the nonmetricity, a shear 0)
Qap =
l{^aeli)\A-\gapA),
(31)
with A:=tfa(e'5J£(,/,),
(32)
and the dilation (33) Torsion appearing in VMAG is restricted to its vector part, (2)7.0
=
^ "
(34)
A T <
with (35)
T:=e/,}Tl>.
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Y Ne 'eman and F W Hehl
Taking polar Boyer-Lindquist coordinates (t, r, 9, ) = (6, 1, 2, 3) and a Schwarzschild type (static, spherically symmetric and Minkowski-orthonormal) coframe, with one unknown function f(r), &° = fdt,
d2 = rd6,
!?'=(l//)dr,
i?§ = r sin<9 d 0 ,
(36)
i.e. a metric ds2 = - f2 dt2 + dr2/ f2 + r2 (d92 + sm2 9 d(j>2) = oap&a ® &»,
(37)
where we have also used the local Minkowski metric oap = diag(—1, 1, 1, 1), then the triplet of 1-forms Q, A, T should have the structure Q = u(r)f
dt,
A = v(r)f dt,
T = r(r)f
dt.
(38)
The exact solution is given by the functions / = yj\ - QxM/r) u = koN/fr,
+ (Ar 2 /3«o) + zA [K(k0N)2/2aQr2], v = kiN/fr,
z = k2N/fr,
(39) (40)
with M and N arbitrary integration constants and the couplings ^0.^1.^2 given by combinations of the ao, a2, bi, C3, C4 dimensionless couplings in the Lagrangian. In addition, bs, is constrained by a condition relating it to the five other couplings [16]. The nonmetricity is thus given by Q^ = (l/r)[k0Noa'1
+ piN^e^j
- \oa^)]dt
(41)
and the torsion by Ta = (k2N/3r)
&a A dt.
(42)
The integration constant M is the Schwarzschild mass, k$N a dilation, k\N a (traceless) shear and k2N a spin charge. For N = 0 and a 0 = 1> one recovers the SchwarzschilddeSitter solution in GR. For our purposes, we note that the vacuum solution's nonmetricity, which will couple to a test particle's shear, in this spherically symmetric case, represents a 1/r potential. 5. The VTOH axially symmetric vacuum solution Still using the gravitational Lagrangian (28), with the simplifications (29) and (30), and the same polar coordinates, VTOH [17] posit a Kerr-type solution #6 = &'1 &2 =
(A/B)l/2(dt-josm20d(j)), =(A/By1/2dr, (43)
(B/f)1/2d6,
V1 = ( 5 / / r 1 / 2 sin 9 [-j0 dt + (r2 + j2) d], where A = A(r), B = B(r, 9), f = f{9), and jQ is a constant. The 1-form triplet Q, T, A of section 4 now generalizes to expressions involving three functions u(r,9), v(r,9), r(r,9) appearing in the third and fourth irreducible components of nonmetricity and in vector torsion, QuH = [«(/•. 0)oap + -9v(r, 9) (* ( o « w J - \oaP)} &\ Ta = \r{r,e)&a
A j? 6 ,
(44) (45)
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Test matter in a spacetime with nonmetricity
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with the solutions, u = k0Nr/(AB)1/2,
(46)
1/2
v =kiNr/(AB) ,
(47)
r = k2Nr/(AB)1/2,
(48)
and A = r2 + j 2
2
- 2KMV - (X/3a0)r2 (r2 + j2) + z^(.k0N)2/(2a0),
2
2
(49)
B = r + j cos 0,
(50)
/ = l + (V3a o )7o 2 cos 2 0.
(51)
Here the £o»&i,&2 are functions of the couplings flo. #2, t>3, C3, c 4 (the same ones as in section 4) and, again, the same constraint relates bt, to ko, k\, k2, c 4 . Physically, M and jo represent the Schwarzschild mass and the Kerr angular momentum. For vanishing jo, we recover the OVETH solution of section 4. 6. The test particle in the OVETH and VTOH solutions The Noether identities (26), (27) already provide important information with respect to the behaviour of test matter in MAG. Obukhov [18], generalizing a corresponding result [19, 20] from Riemann-Cartan spacetime, has recast equation (26) in the form (quantities with a tilde denote the Riemannian parts), D [E„ + £j>v {ea\OTpY)\
+ A^ A (
eaOTpy)
= r ^ A (eaJ Ryp),
(52)
where <>Tpy := Tpy — TpY denotes the non-Riemannian part of the connection. The expression on the right-hand side of (52) represents the Mathisson-Papapetrou force density of GR for matter with spin r^r := AiM. For A^Y = 0, the equation of motion becomes DYia = 0, i.e. without dilation, shear and spin 'charges' the particle follows Riemannian geodesies, irrespective of the composition of VMAG- Thus, we have to use as test matter only configurations which carry dilation, shear or spin charges, whether macroscopic or at the quantum particle level. At the latter, the hadron Regge trajectories provide adequate test matter, as world spinors with shear. In the world spinor equation, when written anholonomically, the GL(A, R) Lie-algebravalued connection raP[p(Lap)]fi/M, acting on the component ,i'N(x), parallels the action of the same GL(4, R) generators (symmetric gy(aLyp) for shear and dilations) in the expression for the original Noether current of hypermomentum for the matter Lagrangian, Aaf, = [(Lap)NMVN] A [ 3 L / 3 ( D * ) M ] + • • •, where they enter in writing the variation of the matter field. This is just a reflection of the universality of gauge couplings, in which gauge fields are coupled to conserved currents. An identity (equation (3.10.8) in [6]) expresses the components of the connection 1-form as a linear combination of the components of the Christoffel symbol (of the first kind), the object of anholonomity C := d&a, the torsion T" and the nonmetricity Q"P, Tya/S = ^{ygpa}
+ C{ypa) — 2"(y^a) + Q{ypa)],
(53)
where the {} are Schouten braces [21]. It is through this replacement that we get in the matter equation (19) the action of the nonmetricity field. This can also be rewritten as rap = [V4-terms] + [C/4-terms] + \Qap + (e [a J Qp]YWY.
587
(54)
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Y Ne'eman and F W Hehl
Turning now to the two vacuum solutions with nonmetricity, and leaving out dilations (ko = 0), we find, with G := ^ / V , 0)
Qaf> = 4Gr(AB)^2
( ^ ) J - \0afi)
ff°,
(55)
or
( ( 3 ) e) a ,
Gr
(AB)'/2
3i?°
2d1
2#> 2&l 2# 5
-#6 0 0
\ Particularly simple are the diagonal elements, (3)
2# 2 0 -0° 0
2&3 0 0
(56)
-tf°
e6o/3 = ( 3 ) en = (3) 0is = (3)e33 = -Gr (dt - jo sin2 9 dcj>) / (r2 + j 0 2 cos2 (?),
(57)
which reduce to the spherically symmetric — Gdt/r for y'o = 0. This is then the static potential entering, via (54), the world spinor equation, a 1/r potential with a centrifugal cut-off at small r. Applying this potential to a Dirac or Bargmann-Wigner equation, written for any component in the 'flat' equation (4.5.1) of [6], we get in the spherically symmetric case, a hydrogen-like relativistic spectrum, thereby superimposed on every state in the diagonal, with multiplicities growing with the Bargmann-Wigner spin value. The resulting world spinor is thus much more populated, but not yet in the off-diagonal sectors of figure 3 in [6]. In the axial-symmetric case, jo # 0, and we thus have, in addition, the \&J\ =2 action, reaching into the off-diagonal sectors and partially filling them. The energy spectrum will follow. The generic world spinor corresponds to SA(4, R) representations in class IIA (see appendix C5 of [6]), i.e. with no kinematical constraint on the mass spectrum. Let us take, for instance, a (hadronic) linear M2 = a Jo + b as our free world spinor. We shall now have, in the simplest (lowest state J = \) case, a superposition of the (Dirac-relativistic) hydrogen-like gravitational excitations due to nonmetricity, onto the linear spectrum of the flat limit ('free') manifield. Next, we would have to solve the Bargmann-Wigner equation for / = | in this hydrogen-like potential, etc.
References [1] Fradkin E S and Tseytlin A A 1985 Phys. Lett. 158B 316 Callan C G, Friedan D, Martinec E J and Perry M J 1985 Nucl. Phys. 262 593 Gross D 1988 Phys. Rev. Lett. 60 1229 Gross D and Mende P F 1988 Nucl. Phys. B 303 407 Gross D and Mende P F 1987 Phys. Lett. 197B 129 [2] Guth A 1981 Phys. Rev. D 23 347 Guth A 1993 Proc. Natl Acad. Set, USA 90 4871 Linde A 1982 Phys. Lett. 108B 389 Linde A 1990 Phys. Lett. 249B 18 La D and Steinhardt P J 1989 Phys. Rev. Lett. 62 376 [3] Trautman A 1973 On the structure of the Einstein-Cartan equations Differential Geometry (Symposia Mathematica 12) (London: Academic) p 139 [4] Trautman A 1980 Fiber bundles, gauge fields, and gravitation General Relativity and Gravitation. One Hundred Years after the Birth of Albert Einstein vol 1, ed A Held (New York: Plenum) ch 9, p 287
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Test matter in a spacetime with nonmetricity
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[5] Hehl F W, Kerlick G D and Heyde P v d 1976 Z. Naturf. 31a 111 Hehl F W, Kerlick G D and Heyde P v d 1976 Z Naturf. 31a 524 Hehl F W, Kerlick G D and Heyde P v d 1976 Z. Naturf. 31a 823 Hehl F W, Kerlick G D and Heyde P v d 1976 Phys. Lett. 63B 446 Hehl F W, Kerlick G D, Lord E A and Smalley L L 1977 Phys. Lett. 70B 70 Hehl F W, Lord E A and Ne'eman Y 1977 Phys. Lett. 71B 432 Hehl F W, Lord E A and Ne'eman Y 1978 Phys. Rev. D 17 428 [6] Hehl F W, McCrea J D, Mielke E W and Ne'eman 1995 Phys. Rep. 258 1 [7] Ne'eman Y and Sijacki Dj 1988 Phys. Lett. 200B 489 [8] Tresguerres R Weyl-Cartan model for cosmology before mass generation Proc. Relativity Meeting 1993, Relativity in General (Solas, Asturias) ed J Diaz Alonso and M Lorente Pdramo (Gif-sur-Yvette: Edition Frontier) p 407 [9] Dothan Y, Gell-Mann M and Ne'eman Y 1965 Phys. Lett. 17 148 Dothan Y and Ne'eman Y 1966 Symmetry Groups in Nuclear and Particle Physics ed F J Dyson (New York: Benjamin) p 287 [10] Obukhov Yu N and Tresguerres R 1993 Phys. Lett. 184A 17 Obukhov Yu N 1996 Phys. Lett. 210A 163 [11] Ne'eman Y GR8 ed M A McKiernan University of Waterloo (Canada) p 262 Ne'eman Y 1977 Proc. Natl Acad. Sci., USA 74 4157 Ne'eman Y 1978 Ann. Inst. H. Poincare A 28 369 Ne'eman Y and Sijacki Dj 1987 Int. J. Mod. Phys. A 2 1655 Sijacki Dj and Ne'eman Y 1985 J. Math. Phys. 26 2457 See also Budinich P and Trautman A 1988 The Spinorial Chessboard (Berlin: Springer) [12] Cant A and Ne'eman Y 1985 J. Math. Phys. 26 3180 [13] Mickelsson J 1983 Commun. Math. Phys. 88 551 [14] Tresguerres R 1995 Z. Phys. C 65 347 Tresguerres R 1995 Phys. Lett. 200A 405 [15] Tucker R W and Wang C 1995 Class. Quantum Grav. 12 2587 [16] Obukhov Yu N, Vlachynsky E J, Esser W, Tresguerres R and Hehl F W 1996 Phys. Lett. 220A 1 [17] Vlachinsky E J, Tresguerres R, Obukhov Yu N and Hehl F W 1996 Class. Quantum Grav. 13 3253 [18] Obukhov Yu N 1995 Private communication [19] Meyer H 1982 Gen. Rel. Grav. 14 531 [20] Hehl F W 1985 Found. Phys. 15 451 [21] Schouten J A 1954 Ricci Calculus 2nd edn (Berlin: Springer)
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Vol. 29 (1998)
No 4
ACTA PHYSIC A POLONICA B
GAUGE THEORIES O F GRAVITY* Y. NE'EMAN ** Raymond and Beverly Sadder Faculty of Exact Sciences Tel-Aviv University, Tel-Aviv, Israel 69978 and Center for Particle Physics, University of Texas Austin, Texas 78712, USA
Dedicated
(Received November 7, 1997) to Andrzej Trautman in honour of his 64th
birthday
The relatively simple Fibre-Bundle geometry of a Yang-Mills gauge theory — mainly the clear distinction between base and fibre — made it possible, between 1953 and 1971, to construct a fully quantized version and prove that theory's renormalizability, moreover, nonperturbative (topological) solutions were subsequently found in both the fully symmetric and the spontaneously broken modes (instantons, monopoles). Though originally constructed as a model formalism, it became in 1974 the mathematical mold holding the entire Standard Model (i.e. QCD and the Electroweak theory). On the other hand, between 1974 and 1984, Einstein's theory was shown to be perturbatively nonrenormalizable. Since 1974, the search for Quantum Gravity has therefore provided the main motivation for the construction of Gauge Theories of Gravity. Earlier, however, in 1958-76 several such attempts were initiated, for aesthetic or heuristic reasons, to provide a better understanding of the algebraic structure of GR. A third motivation has come from the interest in Unification, making it necessary to bring GR into a form compatible with an enlargement of the Standard Model. Models can be classified according to the relevant structure group in the fibre. Within the Poincare group, this has been either the R4 translations, or the Lorentz group SL(2, C) — or the entire Poincare SL(2, C) x R4. Enlarging the group has involved the use of the Conformal SU(2, 2), the special Affine SA(4, R) = SL(4, R) x R 4 or Affine A(4, R) groups. Supergroups have included supersymmetry, i.e. the graded-Poincare group (r» = 1. . .8 in its extensions) or the superconformalSU(2, 2/n). These supergravity theories have exploited the lessons of the aesthetic-heuristic models — Einstein-Cartan etc. — and also achieved the Unification target. Although perturbative renormalizability has been achieved in some models, whether they satisfy unitarity is not known. The nonperturbative Ashtekar program has exploited the understanding of instantons and self-dual solutions in QCD, in the complexification and in the selection of new variables. Note that supergravity involves Lie Derivatives as supertranlations, and several models have treated local spacetime translations similarly. The reduction of the larger groups, down to Poincare, has involved spontaneous fibration and spontaneous symmetry breakdown. In this context, noncommutative geometry may allow for further geometrization. PACS numbers: 11.15. - q , 04.50. + h Presented at the Workshop on Gauge Theories of Gravitation, Jadwisin, Poland, September 4-10, 1997. Wolfson Distinguished Chair in Theoretical Physics. (827)
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NE'EMAN
1. Historical b a c k g r o u n d This article constituted the opening "introductory" lecture at a workshop held in Jadwisin in September 1997 and honoring the 64th birthday of our distinguished colleague and friend, Andrzej T r a u t m a n . I noted in my lecture that 64 is 100 in numerical base 8, and t h a t the choice of t h a t base, motivated by Trautman's characterization of The Spinorial Chessboard (a spacetime feature) happened to resonate in my case, even though my eightfold way counted internal degrees of freedom. In some ways this similarity yet difference, this dichotomy, is also reflected in this title and in the subject matter of this meeting — the 'authentic' gauge theories having their gauge group acting on internal degrees of freedom, whereas in the case of gravity, the group acts on spacetime. The first gauge field theory was introduced — in an a t t e m p t to merge Electrodynamics with General Relativity — by Hermann Weyl in 1919 (gauge group R1, for scale [1], a noncompact spacetime feature), then withdrawn and reformulated in 1929 with the compact gauge group U ( l ) acting on t h e complex phase of the electron wave-function [2] (the model for all future internal degrees of freesom), after F. London had identified t h a t feature. It played an important role in the construction of Q E D and particularly (in the form of the derived Ward-Takahashi identities) in the renormalization procedure. Note t h a t Emmy Noether had meanwhile also published her two theorems, establishing the algebraic linkage between the gauge group and the conserved current in the first — and the actual coupling of the gauge potential to this current in the second. In 1953, Yang and Mills [3] generalized the gauge mechanism to SU(2), and thereby to any compact non-Abelian gauge group. In this case — as in many others — physics and geometry developed (independently) along related lines and the physical gauge theory paralleled the emergence of fiber bundles as geometrical constructs, a fact which was only realized in the Sixties — Andrzej being one of the pioneers who m a d e the connection. Weyl was both mathematician and physicist and it is not surprising that his gospel spanned both disciplines: C a r t a n , Chern, Eckmann, Ehresmann, Hirsch, Hopf, Lichnerowicz, Pontrjagin, Steenrod, Whitney, W . T . Wu are some of the names on the mathematical side. Physics has fully repaid t h a t debt, first in 1984 when Sam Donaldson and Michael Friedman used the exact solutions of physical gauge theories to make serious advances in the classification of 4-manifolds, and again in 1994 when N a t h a n Seiberg and Edward Witten's solutions for supersymmetruc gauge theories were applied to the mathematical program.
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Gauge Theories of
Gravity
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It was only in the Sixties t h a t some particle physicists — E. Lubkin, L. Susskind and others became aware of the mathematicians' efforts and results. Yang himself, with T . T . Wu, explored the relationship and wrote a "dictionary": gauge theory = fiber bundle, gauge group = fiber group, field potential = connection, field strength = curvature, etc. [4]. To any student interested in learning about fiber bundles and yet preserve a physicist's view, I particularly recommend Yang's contribution to the 1977 Marshak Festschrift ''Fifty Years of Weak Interactions" [5]. In physics, though the Yang-Mills theory was originally a physical model with no direct application, by 1975 it had become the mold for the entire Standard Model — the gauge theory of SU(3) c o i o r for Quantum Chromodynamics (the "Strong Interactions") and the spontaneously broken gauge theory of SU(2) x U(l) for the Electroweak interactions. This was the consequence of a successful renormalization program, started by Feynman in 1958 [6], continued by B. DeWitt, A.A. Slavnov, J . C . Taylor, L . D " Faddeev and V.N. Popov, B.W. Lee and J. Zinn-Justin and others, completed by "t Hooft in 1971 [7], with added final touches by Becchi, Rouet and Stora ("BRS") [8]. Thierry-Mieg [9] provided in 1979-80 an elegant geometrical interpretation to the unitarity-guaranteeing BRS construction. Note t h a t the reason Feynman started this program was the difficulty he was experiencing in his a t t e m p t to quantize and renormalize General Relativity. Feynman took up the Yang-Mills model as an easier pilot program for gravity... We see t h a t Gauge theories and General Relativity were very close from the start and throughout their evolution. The progression from Feynman to 't Hooft describes the acquisition of a perturbativc solution. This was practically all t h a t had been needed in Q E D . In non-Abelian QCD, however, this turned out to be good for the high energy (UV) sector (due to asymptotic freedom) but useless for the low-energy (IR) region. One answer to this problem was the discovery of exact solutions, both for the fully symmetric case (instantons [10]) and for the broken symmetry case (monopoles [11]). In the first part of this article, I shall review the mathematical and physical characteristics of Gauge Theories in general. I shall then analyze the various motivations for the construction of Gauge Theories of Gravity and the possible algebraic routes, after which I shall discuss results.
2. G e o m e t r i c a l s t r u c t u r e A Principal Fiber Bundle is a manifold V(M,G,7r,»). M is the base manifold (generally flat spacetime), Q is the gauge group, ir the projection 7T : V -¥ M; • is the right-multiplication of V by Q. For p, p' 6 V, g, g' G Q
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Y. NE'EMAN
we have a verticality condition, n(p • g) = n(p) (the group acts only on the fiber, staying above the same point in M, i.e. there is a well-defined vertical direction) and equivariance (p • g) • g' = p » (gg1)., i.e. the group product is faithfully mapped. The latter (right-multiplication) is realized as a map from the abstract infinitesimal Lie algebra A of Q onto the tangent manifold V*. VA £ /I, with [Aa,A(J = ifa bcXc, we have t : A ->• A 6 V*, the abstract Lie bracket becoming mapped into a differentiation bracket [Aa,A(,]/ = Xa(Xbf) - Xb(Xaf). Thus [Aa~Ab] = [Aa, A&] Note that if the dimensionalities are dimM = m, dimQ — k, dirnV = m + k, the map t is from k onto k -f- m. To have an inverse, this map needs a structure which is aware of which is the correctly parallel-transported vertical direction at any point of V. This is the connection u> : V* -> A, VA £ yl,u;(A) = A. The gauge-potential or connection is thus a one-form, acting by contraction, i.e. at a point p, ua(Xb) = u}a\Xb = &b • Defining the field-strength (or curvature two-form) as Q = du> + |[u;,a;], the BRS equations reexpress [12] the Cartan-Maurer structure equations stating the horizontality of the curvature, AJi? = 0.
3. Physical characteristics Given a "free" Quantum Field Theory Lagrangian for a matter field ip, namely, C^{ip,dii), we obtain the full Lagrangian as (£^)d-*-D + CYM{&), with D ~ d+uaXa and CYM = i?A*/2. The Bianchi identity is DQ = 0 and the equation of motion is D*Q = *j. Infinitesimally, the gauge group acts on the fields via 8^ = iaa (Xa)3k^k. Note that the other invariant bilinear in the curvatures, Q A Q is a topological invariant and corresponds (up to a numerical factor) to the exact solutions (instantons) of the symmetric theory [10]. They tu, out to be 4-divergences — here of the axial vector unitary singlet current, i. i important point in the physical hadron theory. One can also thus freely at:--' them (with some factor 0) to the Lagrangian — except that they may viok e P or CP ("the 6 problem" in QCD). This entails special features, all relating to the couplings: (a) The couplings are universal, i.e. they are given (in *j(d —> D)) for the connection wa by the matrix element of the corresponding algebraic generator, in the if)* representation of A, c^; = < -tpi | Aa | ^ >• If f° r a n y symmetry the coupling is a Clebsch-Gordan coefficient, here it is a specific one, the a index specifying the adjoint representation of the algebra. For Abelian Q this is a constant number, representing the large-distance value (Ze for electric charges, whatever the measured system); for non-Abelian Q we have an extra factor, making it into a running coupling g3ai = cj^/(<72) and with f(q2) possibly diverging at some "large" distance (around 1 fm, for
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QCD). In gravity, the coupling is to the matrix-elements of the density of the generator of translations in an Abelian invariant subgroup of the Poincare group, a matter related to the Equivalence Principle. (b) The coupling is also dimensionless, due to the structure of the Lagrangian, quadratic in the curvatures — thus a four-form, whose integration over four-dimensional spacetime yields a dimensionless action, fitting the Quantum Postulate, with no need for the coupling to contribute dimensionalities. This feature is essential for a program of perturbative renormalization — otherwise one would need new counter terms at every order of the perturbative expansion. (c) These couplings are to conserved Noether currents. Renormalization effects are forbidden (for dimensionless couplings) if the relevant interactions obey the same symmetry . This is how the CVC — conserved vector current nature of the Weak Interactions was first identified by Gershtein and Zeldovich, and later by Feynman and Gell-Mann, through the equality of the Weak vector couplings in neutron beta-decay and in muon decay. The Strong Interactions, which should have renormalized the coupling in neutron decay, respect the isospin symmetry generating the Weak current (up to the Cabibbo term). Similar algorithms were found for all Lie-group-generated symmetries of the hadrons (such as the Goldberger-Treiman relation, etc). They were especially important in 1958-1975 (the era when QFT was taboo) because they could be given the form of a dispersion relation [12]. Again, there is a similarity with the Equivalence Principle — in Gravity, the coupling is to a conserved current of the Poincare group — a symmetry respected by all known interactions (and expressing itself in the Eotvos experiments' precision nonrenormalization results). (d) The potential (or connection) can be gauged away by a local active gauge transformation (of the relevant internal degree of freedom). This feature (corresponding to an acceleration replacing a gravitational potential for gravity, i.e. the Equivalence Principle) entails local phase effects in QED or QCD, but becomes nominal for broken symmetries like the electroweak gauge. Using the path-integral formulation for the quantization (v is the group volume, AUJ the measure in the space of connections),
z=(v{g)y1
f[Auyfcd4x
one may define canonical variables
where (r, s, t are space indices, 0 is time) the momentum ££ = -j|4- = c- st (~)0r
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Weyl already introduced what we now know as the "Wilson loop" (a holonomy) Tw := t r P e *'*', a gauge invariant quantity, therefore a possible observable. Anticipating on the next sections, we note that it is also invariant under the diffeomorphisms of the integration loop, which is why it is useful in Ashtekar's canonical treatment of gravity. In QCD, the Wilson loop is the basic tool for computation, in what is now known as lattice gauge theory — the loop being selected around a rectangle. Using two spacelike and two timelike sides and making the timelike ones tend to infinity, we have Wilson's lattice proof of color confinement. Note that in Hamiltonian quantization, the Bianchi identity becomes a constraint.
4. Gravity seen as a twisted and deformed gauge theory The story here sounds like a Freudian Oedipus' (or Electra's) complex . Gravity was the mother of all gauge-like theories, with an interplay between two "gauge groups" (part of the mystery) — the diffeomorphisms Diff(R4) ("the Principle of Covariance") and in addition, the Lorentz group SL(2, C) on the local frames (a factor which became more explicit after Dirac's equation for the electron and its inclusion in GR through local frames in 1928). Also, it is to the local Lorentz group we turn when we want to implement the Equivalence Principle and replace a potential by an acceleration. Remember that it was because of Gravity's GR that the Weyl and the Yang-Mills gauge theories were born and that it was also because of gravity that Feynman launched the quantization program for the YM model; yet when we now go into details, we shall draw a picture showing gravity to be like a caricature of a gauge theory — thus also motivating the search for a different presentation (yet preserving the macroscopic predictions). First — covariance. Is this really a gauge group? For one thing, it does not have an active mode. Example: a change of scale is a diffeomorphism, and GR is indeed passively invariant under such a transformation {i.e. changing the unit from centimeters to inches), but it is not invariant under an active physical invariance, such as a doubling of all distances. The forces would really weaken, whereas in Weyl's scale-invariant 1919 theory (or in Englert's modern version), they would not. One reason is that Newton's constant has dimensions. In Englert's theory [13], there is no such constant, it is replaced by a scalar field (whose vacuun expectation value happens to have that value, but could take any other). Secondly, mathematically, diffeomorphisms appear equivalent to "gauging the translations". Again, although this route has been explored by Cho and others, I do not consider this as a valid mode because the translations d^
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are not covariant and we would not be able to perform active displacements with them. The covariant operators are the covariant derivatives with frame indices (here u becomes r and Q is R) Da = e^(<9M - i"^.). However, the Da, under commutation do not make a Lie algebra. This kind of translation is algebraically known as a Lie derivative. It was discussed in Ref. [14]. We called it an anholonomized general coordinate transformation (AGCT) in Ref. [15]. In supergravity, the spmorial displacements consist in such Lie derivatives, as we showed in that work. As a m a t t e r of fact, gauging the "modified Poincare algebra", with translations replaced by the A G C T would be a conceptually clean answer, but this also means t h a t the group we are "gauging" is not a Lie group with a Lie algebra. Its translations subalgebra has four generators — but structure functions instead of structure constants. As a result, even the variations of the gauge potentials are not the usual 5ua = Dea; instead, one has an additional piece e\Ra. We shall return to this approach in the sequel [16]. The Principle of Covariance is thus not really a physical gauge principle, but it is certainly mathematically useful. Equivalence, on the other hand, has many of the attributes of a gauge theory (e.g. universality, a potential t h a t can be gauged away) but no mathematical derivation. Our third point, indeed, is that the Lorentz subgroup SL(2,C) C Diff(/? 4 ) (the overline denotes the double covering group) is indeed actively implementable. And yet the dynamical theory, as expressed — our points (a, b) — by the Noether content of the coupled conserved current is not that of the Lorentz group. On the contrary, the relevant current is the energy-momentum tensor, i.e. the density of the generators of translations, the quotient of t h e Poincare group by t h a t same Lorentz group! And .yet in the implementation of our point (d), i.e. gauging away the potential, we do have to use the local Lorentz group! The fourth point relates to the dimensionality of Newton's constant, the theory's coupling. Again, with a Lagrangian linear in the curvature, i.e. a two-form, we need to assign to the coupling a dimensionality of the inverse of a squared length. This will impact heavily on the perturbative renormalization program. All of this is due to the Einstein-Hilbert Lagrangian, linear in the curvature. Whereas in the YM field equation D'Q = *j, the Q and j relate to the same algebraic generators, in GR the Q (now R) is the field-strength of rotations and the current j is t h a t of the translations. In the EinsteinC a r t a n version, we have yet another cross-eyed equation relating the torsion — the field strength of translations — to the spin, i.e. the Lorentz group current.
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We may gain some consolation from a 1977 demonstration by MacDowell and Mansouri [17]. They showed that if you undo the Wigner-Inonii contraction, replacing translations by rotations into the fifth dimension (the contraction consisting in having taken the radius to infinity) and write a "somewhat YM-like" 5th component Eabcd$Rab ARcd of a topological RAR, you will have an "aesthetic" understanding of our problem. The curvatures into the 'old' spacetime dimensions now have an extra term Rah' = Rab - ( l / 2 ) T a 5 A r5b. When you now reimplementate the contraction, ra5 —* ea and the additional term is 77^ = eabcd^c A ed. The quadratic Lagrangian yields a quadratic topological (instanton-like) invariant, a cosmological term coming from squaring the nab — and the Einstein Lagrangian ?/„(, A Rab — which has thus acquired a 'respectable' progenitor (gauging the de Sitter group in an almost YM fashion..). The advantage of this presentation is that it can directly be extended to Supergravity — the contraction holding for OSp(l/4) (as SO(3, 2) = Sp(4, R); alternatively, for the other de Sitter group SO(4,1) = Sp(2, 2), we would thus have OSp(l/2,2)). 5. Motivations for a Gauge Theory of Gravity — conceptual simplicity The first motivation to look for a more YM-like alternative was explorative, a search for conceptual simplicity, for a YM-like interpretation of GR. It started almost immediately after the YM paper, mainly in the contributions of Utiyama [18], Kibble [19] and Sciama [20]. One result was the renewed interest in torsion, appearing in all treatments based on the Poincare group, since it is the field-strength of the translations. Here there was an encounter with the veterans of the 1920-1950 Einstein-stirred search for a unification betwen GR and Electromagnetism. Other than Weyl and his gauge approach, there had been Eddington, Cartan, Mme Tonnelat, Stueckelberg, Finkelstein, Rodichev, Ivanenko and collaborators, Pellegrini and Plebanski, etc. Torsion with its antisymmetric indices plays an important role in all this. Before we leave this subject, it is interesting to note that Einstein's quest found its tightest solution in N=2 Supergravity: between the J = 2 graviton and the J = 1 photon, two J = 3/2 gravitinos (i.e. one charged complex field) are what was needed to construct an irreducible theory of GR plus EM [21]. However, when this was found, the stir was minimal. After Einstein's death, there were not many left who would disregard the Strong and the Weak interactions in an effort for unification. This phase ended up in (1) a rebirth of the Einstein-Cartan theory, as a mild gauge-like facelift to GR, described in [15] — and (2) in the Pomcare Gauge Theory, quadratic in the curvatures and torsions [22]. This includes the option of a more drastic piece of surgery, namely teleparallelism [23], ex-
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ploiting mostly the possibility of replacing the Einstein-Hilbert linear term by a quadratic squared-torsion term — the particular irreducible component known as the Weitzenbock invariant. Another conceptually valid version is the use of the "modified" Poincare group — with Lie derivatives (AGCT) replacing the translations — providing an elegant physical interpretation — including an understanding of what in gravity is not simply gauge-like. This approach is particularly appropriate for treatments which include supergravity, since that is the nature of the relevant supersymmetric transformation [24]. Note that in supersymmetry a way was also found to return to an orthodox Lie superalgebra — by adding auxiliary fields. Another interpretative result was the method of gauging on the group manifold. With Regge [15] and Thierry-Mieg [24], we showed that starting from two copies of the group manifold, taken both as base space and as fiber (the copy which becomes the base manifold is allowed to curve, the one in the fiber is rigid — as usual in a fiber bundle), but using a Lagrangian breaking the group symmetry (in the gravity case the group is the Poincare group and the Lagrangian is locally only Lorentz-invariant) there occurs a process of spontaneous compactification and factorization. Thus, on mass shell, the fiber reduces to the Lorentz subgroup and the base manifold reduces to the space of translations, i.e. spacetime. This methodology works also for supergravity. An alternative approach is to have both the Einstein linear term and terms quadratic in the curvatures — with different interpretations, including the possibility of new contact interactions [14] or a contribution to the Strong Interactions with confinement [25]. Two schools, Hehl's group in Cologne and Trautman, Kopczynski, Tafel etc., in Warsaw, have led these movements, with Trautman [26] providing the most thorough analysis of a U(4) geometry as the Riemannian V4 geometry is indeed replaced here by a U(4), i.e. with the inclusion of torsion. The equations of motion now involve asymmetric tensors, closer to the canonical derivations (first Noether theorem) for the relevant currents. G^" is an asymmetric Einstein tensor, T,iVp is the torsion, S^ the canonical energy-momentum current tensor, T^VP the spin angu;ar momentum current tensor. The equations of motion are, Kj
—
t\^-j
598
,
-
.
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with the Noether currents, now as defined by their couplings (2nd Noether theorem), SC °'p
2JHV a»v
= =
v. srr pp
= = = =
-1 i±v
Kp
SK„S ' O*" -VpiT^" SC
-TVP
(2)
V^ + 2 5 ^ ,
(i/2)(r^-r,/), C P 4 . 9A
_ C
P _L C P
_ CP
where we have also defined the symmetric energy current a and the various torsion tensors T and 5 and the contortion K. There are two possible direct routes to extend this formalism (still without invoking supersymmetry), according to whether one embeds the Poincare group in the homothetic [1] and conformal [13] groups SL(2, C) X J?3'1 C SU(2, 2) (the homothetic is the middle step inwhich one only adds the R1 of dilations) or in the affine [30] groups, SL(2,C) X R3'1 C GL{4,R) x R4, with the special affine SL(4, R) x R4 as middle step. Either route has been used and we shall return to affine geometry when we look at perturbative quantization, where it has been applied. The interest in these possibilities simply followed the search for a more general outlook [28]. There appeared to be an inherent difficulty in the affine case, in which the Lorentz SO(1,3) is replaced by the linear SL(4,i?), because it was (wrongly) assumed in the GRG community that, while 5'0(1,3) = SL(2,C)i.e. there is a double covering to the Lorentz group — hence the existence of spinors — there is no double covering group for SL(4,R). I broke this superstition in 1978 [29] and constructed (infinite-component) linear and affine spinors [30] and even world spinors [31] on which the diffeomorphisms are represented nonlinearly over their linear subgroup, just as for tensors. Note that in Metric-Affine Gravity, we have new components to the curvature, deriving from the nonmetricity Dg^v = <3M„ ^ 0. Starting from gauge-like considerations, Yang constructed such a model (coinciding with Stephenson's and Kilmister's, the SKY model [32]), but this model, taken macroscopically, does not approximate to Newtonian gravity. 6. Motivations for a Gauge Theory of Gravity: perturbative quantization When the renormalization program for the Yang-Mills field achieved its goal in 1971, it was natural that Veltman and 't Hooft should turn to gravity [33]. The first answer was a nice surprise — considering the various reasons which predicted failure, mainly the dimensionality of the coupling.
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It turned out that the one-loop vacuum contribution (gravitons interacting with gravitons) is accidentally finite, due to a topological identity. However, adding matter — scalar, spinor or vector — leads to infinities [34]. This line of theoretical experimentation was not pursued — had it, supergravity might have been discovered somewhat earlier, since N = l supergravity can be regarded as gravity plus J = 3/2 "matter". Due to the supersymmetric algebra, as was later proved by Kallosh, anything true of gravitons can be generalized to the entire supersymmetric multiplet. Hope for another set of miracles was finally dashed when Goroff and Sagnotti [35] using a supercomputer, managed to evaluate the two-loop vacuum diagram and found it to be infinite. Meanwhile, Stelle had shown [36] t h a t a curvature-squared term in the Lagrangian would make it finite — its dimension frees it from the need for a dimensional coupling; moreover, it would also ensure t h a t it dominate the linear term in the high energy regime. Such terms would certainly be generated in the renormalization procedure, even if the original Lagrangian were the linear one. However, Stelle also showed the theory to be non-unitary, due to its p~4 propagators. These propagators are created because of the Riemannian nature of the theory, i.e. the dependence of the connection on the metric (resulting from Dg^, = 0). As a result, r ~ dg, R ~ d2g + {dg)2 and £ ~ (d2g)2, (dg)4, d2g{dg)2, all producing p~4 terms in the inverse Fourier transform. Such propagators can be simulated by a difference between two poles p~4 ~ (l/i> 2 ) — [ l / ( p 2 — ™2)L one of which has to be a ghost. Tomboulis [37] has recently provided a proof of the breakdown of unitarity at the nonperturbative level. To cope with this issue, we have assumed [38] t h a t the fundamental (high energy) theory is an affine or metric-affine gauge like model, such as the SKY Lagrangian. An appropriate Higgs field causes the local SL(4, R) symmetry to break spontaneously (presumably at Planck energies), reducing to SL(2,C) and the low-energy theory is then Riemannian (and Einsteinian, for instance if we prepare a
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bosons to fermions and vice versa, whereas in this construction, the matter fields are [VL, e j , e# so t h a t the odd generators only connect different chiralities. However, when taking the boson multiplet and working with forms, the action fits the statistics ansatz, since the W, Z, A vector mesons in the even p a r t make one-forms, while the Higgs in the odd p a r t are zeroforms. With J. Thierry-Mieg and S. Sternberg we worked hard throughout 1980-1990 to overcome the interpretative issue. Meanwhile, however, the mathematician D. Quillen published his Hheory of the superconnection" [42] and we understood t h a t this was what I had constructed, a superconnection. One should count together both parities, t h a t of the Grassmann elements over which the supermatrix is valued, and t h a t of the generating superalgebra. The even parts are the Grassmann-odd Yang-Mills connections and are superalgebraically even, the odd parts are Grassmann-even Higgs fields but they are superalgebraically odd. The overall parity is thus odd, as befits a (super) connection, still a connection. We reformulated the theory, therefore as a Quillen superconnection [43]. Meanwhile, another advance had happened in Mathematics, namely Connes' Noncommutative Geometry [44]. Connes and Lott applied it t o the electroweak theory and reproduced the Weinberg-Salam model geometrically [45]. Soon afterwards, Coquereaux, Scheck and collaborators [46] showed t h a t by modifying some steps in NCG, the same geometric derivation yields our superconnection! Already in 1980, we had shown [47] t h a t such supergroups (and their superconnections) reproduce the Higgs-Kibble model of spontaneous symmetry breakdown in other examples. At the recent Marcel Grossmann VIII in Jerusalem I presented [48] such a superconnection for the model we discussed here, namely a SKY affine Lagrangian whose SL(4, R) is spontaneously broken to SL(2, C), i.e. Einsteinian gravity as the low energy theory of a fundamentally "post-Riemannian" affine high energy theory. The relevant supergroup is the double-covering of the simple (rank 3) P ( 4 , R), whose even subgroup is SL(4,.R) I recommend this extension of gauge theories over noncommutative geometry in any case, for its aesthetic characteristics [49]. 7. N o n p e r t u r b a t i v e (canonical) q u a n t i z a t i o n In Section 3 we noted the existence of an exact (i.e. nonperturbtive) solution to the YM theory, namely the instanton.This is the Chern-Pontrjagin topological invariant, characterizing a bundle manifold, v = g ^ - J i ? A Q. This is a four-divergence u = dG, that of a Chern-Simons three-form G ~ t r { w A i ? - ( 1 / 3 ) W A U A W ) } . With m a t t e r fields, this is the term generating the chiral anomaly. As an invariant and a constant, it can be added to the
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Q C D Lagrangian, with some coefficient 9. It does, however, generate violations of parity P, because it is a pseudoscalar — due to the e^Pa of the four-form in Minkowski spacetime. In the physical t e r m , this is compensated by the second epsilon tensor, t h a t of the Hodge duality. Violating P, it also violates CP (and T). These problems can be resolved by assigning a small value to 0 (experimentally, 9 < 10~ 9 ) or inserting t h e imaginary unit i, but Q C D would appear more natural if one would not have to make arbitrary assignments. Once we have both terms in the Lagrangian, minimizing the action implies putting either one of f2±i*f2 to zero, i.e. selecting self-dual or anti-selfdual fields and connections. The imaginary unit here is related to **/ = —/ and the overall result amounts also to complexification. We now turn to gravity. In striving to achieve a nonperturbative quantization, Ashtekar [50] has emulated the YM model, using self-duality eigenfields as canonical variables, thus also enacting a complexification of the model. The Chern-Simons terms here are [51] Crr = -{PAR+{l.3)rArAr) for Lorentz curvatures and Cn = {l/k2){-daAd-da-daAdbAi^). The"instanton term" here is the divergence dC, to be added to Einstein's Lagrangian. As a result one gets the constraints, the vector Xa = D{Ea\ (i,j = 1..3), corresponding to the Gauss constraint in the YM case, the three X{ = F^El guaranteeing invariance under 3-dimensional diffeomorphisms — and the scalar (time-evolution) Hamiltonian constraint X = eahcFijaElEJc. Ashtekar's program has succeeded in generating q u a n t u m gravitational states, realized through the loop quantization as an observables' representation These are the Wilson loops we discussed in Section 3, as applied by Rovelli [52] and Smolin [53]. The program's difficulties are partly in our inexperience with the interpretation of nonperturbative states, where we cannot count quanta. It is interesting that the success in gravity has already produced new applications in YM theory, in the search for a proof of color confinement in Q C D . Some ten years ago, we conjectured [54] t h a t Q C D itself produces something resembling gravity, in its IR region, and t h a t it is this gravity-like component which generates color confinement, a geometrical feature. We were indeed able to prove t h a t such a component does exist [55,56], using semi-perturbative and algebraic methods. Independently, D.Z. Freedman, K. Johnson and several collaborators have been attacking the same problem nonperturbatively, applying variables inspired by Ashtekar's [57]. Let me mention that Seiberg and Witten have recently supplied a mathematical proof of confinement [58], but this depends crucially on the presence of supersymmetry, beyond SU(3) co ioi-
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8. Unification as a motivation With all other interactions quantized and sitting in a gauge theory mold, it seems obvious that gravity should be reformulated in a similar mold, if we are to end up with a unified theory. The drive for unification goes both ways — physicists working on internal degrees of freedom have always looked for a spacetime origin — with the Kaluza-Klein approach as the simplest unifying mechanism. Right after the advent of (flavor) SU(3) we already tried this route [59]. With SU(6), combining spacetime spin with "internal" unitary spin, the prospects appeared good at first for an algebraic unification. Such hopes were dashed by the various no-go theorems of the Sixties, culminating in the Coleman-Mandula (very negative) formulation. Hope was born again after the discovery of supersymmetry and the options left open by the HaagLopuszariski-Sohnius theorems. The theorems, however, did not restrict gauge groups, and the result was Supergravity. There was the additional hope that the new algebraic constraints would tame at least some of the divergences, as had happened in the Wess-Zumino model. Of the two original formulations of supergravity [60,61], the Freedman et al. dealt with gravity in the classical manner, whereas the Deser-Zumino presentation was influenced by the algebraically more elegant Einstein-Cartan formulation, as discussed by Kibble, for example (including the "first order" approach). Unification is maximal with the N = 8 model, which we were hoping, Gell-Mann and I, would also enjoy improved renormalizability, as indeed later happened with the related N = 4 Yang-Mills theory (which is simply finite, no radiative corrections at all). N = 8 supergravity was constructed in 1978 by Cremmer and Julia [62], as a 4-dimensional reduction of N = 1 in eleven dimensions. This approach calls for a Kaluza-Klein interpretation and there have been interesting leads for spontaneous compactification. Yet the answer has to await the verdict as to the theory's renormalizability — which is still not known. For some years (1984-1995) this model was abandoned, due to pessimistic evaluations of its chances. Supergravity itself (but not the 11-dimensional model) is just the QFT (in 10 dimensions) obtained when truncating Superstring Theory beneath Planck energies — this was the view throughout the above period. Recently, this picture has veered again in the direction of the 11-dimensional model, which was shown to emerge from the truncation of a supermembrane in that dimensionality [63]. In the last two years, interest has grown enormously, with the discovery of dualities which relate all Superstring theories to this "M Theory" [64]. Should this be the answer, we could rest from our search for a gravitational gauge theory . . . My own suggestion is to restrain the excitement at this stage and continue in our quest.
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D.Z. Freedman, P. v. Nieuwenhuizen, S. Ferrara, Phys. Rev. D13, 3214 (1976). S. Deser, B. Zumino, Phys. Lett. B62, 335 (1976). E. Cremmer, B. Julia, J. Sclierk, Phys. Lett. B76, 409 (1978). E. BergshoefF, E. Sezgin, P.K. Townsend, Phys. Lett. B189, 75 (1987). See for example J.H. Schwarz, Phys. Lett. B367, 97 (1996).
606
C H A P T E R 8: STRINGS, B R A N E S A N D O T H E R E X T E N D O N S DJORDJE SIJACKI Institute of Physics Belgrade, Serbia Strings, subsequently membranes and other extended objects, "extendons", i.e., "branes", opened up a new era in the development of particle physics and gravity. Note that, contrary to its appeal, the term "extendon" {8.4} never caught enough attention and the majority of researchers settled rather for "brane" or "p-brane", where "p" refers to the number of spatial dimensions (p = 0 for particle, p = 1 for string, etc.). The whole subject started as a theoretical/mathematical attempt to describe certain phenomenological features of strong interactions, primarily in the sector of hadronic recurrences of a given flavor, and nowadays, after a couple of "revolutions", it represents a very promising path towards the ultimate "theory of everything". It is somewhat paradoxical that, at the primordial stage of string theory, some researchers perceived its emerging as a step to expel the field theory approach from the physics of strong interactions and, eventually, from particle physics altogether. The string/brane field theory came back as a boomerang revealing the profound geometric foundations of the basic interactions governing our world. It offers a possibility to unify gravity, a gauge theory of a spacetime symmetry, with the rest of the basic interactions that are related to internal gauge symmetries. This unification takes place within a rather unique quantum field theory: it is unitary, renormalizable/finite, anomaly free, etc. and based entirely on spatially extended objects and higher dimensional (D = 11) spacetime symmetries. With his refined feeling for algebraic and geometric structures, Yuval Ne'eman was one of the pioneers in conducting research and organizing meetings in various fields (higherdimensional embeddings, gauging and group manifold structure, super symmetry, supergravity, infinite (super)algebras, duality diagrams and category theory etc.) that represent important aspects of the contemporary "brane" physics. His contribution to the field certainly exceeds the results that could be presented by his selected research papers. Quantum gravity, one of the outstanding problems of contemporary physics, is a principal challenge for string/brane theory. Its solution would harmonize quantum theory and general relativity, the two basic pillars of 20th century physics. The string relevance to quantum gravity was initiated by the fact that the Einstein theory of gravity scattering amplitudes are obtained from the "graviton"-scalar amplitudes of the string model in the zero-slope limit. In the conventional lagrangian formulation for superstrings, the curved 2-dimensional (locally reparametrizable) string world sheet R2 is embedded in a flat 10dimensional Minkowski spacetime M 1 ' 9 (Poincare invariance). On the other hand, macroscopic gravity is described classically by Einstein's theory, corresponding to a generic curved Riemannian R4 manifold (general covariance). Thus one is faced with an apparent difference in the manifest symmetries of th^se two theories. This difference poses not only an aca-
607
demic question, but it is crucial for numerous practical questions such as nonperturbative gravitational solutions (Schwarzschild) etc. Yuval Ne'eman was particularly interested in the formulation of generic curved target-spacetimes for theories of supersymmetric extended objects {8.3, 8.4}. The first term in the expression of the super p-brane action reads [1]
where i = 0 , 1 , . . . p labels the coordinates £ l = (r, a, p,...) of the brane world-volume with metric 7^, and 7 = det^ij), the target space is a supermanifold with super-space coordinates Za{?) = {Xm(?), 6 Q ( f ) ) , Uf = diXm - ®TmdiQ, m = 0,1, • • • , D - 1, a = 1, 2, • • • , 21 2 J, and Tm are the corresponding D-dimensional spacetime gamma matrices. Note that ©a transforms with respect to the fundamental spinorial representation of the Spin(l,D — 1) ~ 5 0 ( 1 , D — 1) group. It is straightforward to generalize the p-brane action, in the bosonic case, such as to be invariant w.r.t. the group of general coordinate transformations by replacing the flat space quantities by the corresponding curved space ones (Xm -> X™, rjmn - • gfhn, etc.). In the bosonic p-brane case the SO(l, D — 1) group is replaced by the Diff(D,R) one, while in the super p-brane case, the Spin(l,D — 1) group is to be replaced by the covering group of the general coordinate transformations, D i / / ( D , R ) . There are no finitedimensional representations of the Diff(D,M) group for D > 3, the Diff(D,B.) group is a group of infinite matrices, and thus one can not proceed in a straightforward manner as in the bosonic case. The standard procedure to study spinorial point particle/field theories in a generic curved spacetime RD (coordinates xm) is to describe them w.r.t. the double-covering group Spin(l,D — 1) of a local (flat) cotangent Minkowski spacetime M1,D~l. Such a procedure is meaningless in the super p-brane (p > 1) case. Note that, the expressions appearing in the lagrangian, efl(^1) = diXm = dXm/d^1 are generalized (rectangular) "£>-ads". In analogy to general relativity with tetrads, the p-"volume" plays the role of the curved spacetime (holonomic coordinates) and the embedding Z?-dimensional Lorentz indices fulfill the roles of (anholonomic) spacetime D-ad indices. In the curved spacetime case, one has additional fields e^(Xm) = dXm/dXm ("frames over frames") that cannot be used to perform required fiat-to-curved transition of the T matrices. There are two embeddings for curved target-spacetime superbranes: Restricted curving. In this case the Diff(D,'R) group is restricted to a subgroup that preserves the orbits of Spin(l,D — 1) when acting simultaneously on both even and odd sectors of the superspace coordinates Za = (Xm, ®a). In other words, no linear transformations are allowed other than Spin(l,D — 1) and a restricted set of non-linear ones leading to manifolds carrying the action of Spin(l,D — 1). Here, both spinors and the Dirac 7 matrices are the usual ones of the Spin(l, D — l) group. The first term of the super p-brane
608
action [2] reads
Here, the target space is a supermanifold with super-space coordinates Za = (Xm,@a), where rh = 0 , 1 , . . . , D - 1 and a = 1,2,..., 2 ^ 1 . Furthermore, Ef = (diZ&)E%(Z), where E-- is the supervielbein and a = (m a) is the tangent-space index. In the standard superspace formalism one tends to describe 0 a as a "world" fermionic coordinate, but this time in a very restricted sense only. Generic curving. In this case one makes use of the true, infinite-dimensional, "world spinors" of the Diff(D,R) group. The minimal option is to replace Xm by Xm, as in the bosonic case, and Qa by a world spinor 0 ^ , A = ^ , . . . ,oo. A world superspace option is achieved when infinite representations of Diff(D, M) are used for both bosonic and fermionic sectors, Za —> Z1 = (XM,0A). Moreover, a Dirac-like equation and the corresponding generalizations of the 7 matrices for Diff(D,R) spinors, as constructed recently [3] for D = 3, are required for expressions such as E™ = diXm — iQ (^Thiff))AB^i®' • The generic curving formalism for superbranes paves a way to explore the full complexity of the generic structure of the curved quantum spacetime and to grasp its mysteries still to be discovered.
609
References [1] E. Bergshoeff, E. Sezgin and P.K. Townsend, "Supermembranes and eleven-dimensional supergravity" Phys. Lett. B189 (1987) 75. [2] M.J. Duff, "Supermembranes", Lectures given at the Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 96), Boulder, 1996, hep-th/9611203. [3] Dj. Sijacki, "SX(4,R) embedding for a 3D world spinor equation", Class. Quant. Grav. 21 (2004) 4575.
610
REPRINTED PAPERS OF CHAPTER 8: STRINGS, BRANES AND OTHER EXTENDONS
8.1
8.2
8.3
8.4
8.5
8.6
Y. Ne'eman, "The Two-Dimensional Quantum Conformal Group, Strings and Lattices", in Proc. Int. Symp. on Conformal Groups and Structures, Clausthal (1985), H. D. Doebner and A. 0 . Barut, eds. Lecture Notes in Physics 261, (Springer Verlag, 1986), pp. 311-327.
613
Y. Ne'eman and Dj. Sijacki, "Spinors for Superstrings in a Generic Curved Space", Phys. Lett. B174 (1986) pp. 165-170.
630
Y. Ne'eman and Dj. Sijacki, "Superstrings in a Generic Supersymmetric Curved Space", Phys. Lett. B174 (1986) pp. 171-175.
636
Y. Ne'eman and Dj. Sijacki, "Curved Space-Time and Supersymmetry Treatments for p-Extendons", Phys. Lett. B206 (1988) pp. 458-462.
641
E. Eizenberg and Y. Ne'eman, "Classical Lagrangian and Hamiltonian Formalisms for Elementary Extendons", II Nuovo Cimento 102A (1989) pp. 1183-1197.
646
R. Brustein, Y. Ne'eman and S. Sternberg, "Duality, Crossing and MacLane's Coherence", Israel J. Math. 72 (1990) pp. 19-37.
661
611
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THE TWO-DIMENSIONAL QUANTUM CONFORMAL GROUP, STRINGS AND LATTICES
* Yuval Ne'eman S a c k l e r F a c u l t y o f Exact Sciences Tel A v i v U n i v e r s i t y , Tel A v i v , I s r a e l and University
* + ++
o f Texas, A u s t i n ,
Texas++
Wolfson C h a i r E x t r a o r d i n a r y i n T h e o r e t i c a l Physics Supported i n p a r t by the U . S . - I s r a e l BNSF Supported i n p a r t by t h e U.S. D e p a r t m e n t o f Energy Grant No. DE-FG05-85ER40200 and by the Center f o r T h e o r e t i c a l P h y s i c s , U n i v e r s i t y o f Texas, A u s t i n
613
312 1.
Conformal Groups and t h e V i r a s o r o
Algebra
T h i s conference c e n t e r s on the Conformal Groups. dimensional
pseudo-Euclidean spaces, t h e r e are
pseudo-orthogonal like
and
t
conformal
Lie groups S 0 ( s + l , t + l ) ,
timelike
dimensions.
transformation
sense o f an angle as w e l l equivalent
In 2 - d i m e n s i o n s
as i t s m a g n i t u d e , i t
s
space-
(x,y), defining a
Cauchy-Riemann d i f f e r e n t i a l
the
can be shown t h a t
diffeomorphism w = f ( z ) ,
This i s because the c o n d i t i o n s
it
is
for z = x+iy.
f o r i s o g o n a l i t y t u r n o u t to be j u s t
equations
2-
finite-dimensional
for metrics with
as an i s o g o n a l mapping w h i c h conserves
to an a n a l y t i c a l
The 2 - d i m e n s i o n a l
Over any b u t
the
.
conformal
qroup plays an i m p o r t a n t r o l e i n 2 2) d i m e n s i o n a l quantum f i e l d t h e o r y ' . Because o f the above c o r r e s p o n dence between conformal and d i f f e o m o r p h i s m g r o u p s , one may i n f a c t d e f i n e two a n a l y t i c a l d i f f e o m o r p h i s m groups Ag and A | c o r r e s p o n d i n g t o the v a r i a b l e s z and 7 i n the above n o t a t i o n . These groups can 3) be d e s c r i b e d a l g e b r a i c a l l y , using a method due t o O g i e v e t s k y ' . One expands the i n f i n i t e s i m a l z'
= f(z)
m
= -i
6z(z)
= z + <5z
i n powers o f L u
variation
(1.1)
z , so t h a t d e f i n i n g m+1 z
(me/)
>— az
(1.2)
we have ,m , m )z = i I Cn
(Im
«Z
and the Lm form the c l a s s i c a l (m-n) L
tLm'LmJ
m+n
infinite
Ogievetsky
(1.3)
m ,n e 7
an a l g e b r a p l a y i n g an i m p o r t a n t r o l e i n c l a s s i c a l Analytical
algebra
E i n s t e i n Covariance group and i t s
physics,
In Quantum Mechanics, the commutation r e l a t i o n s deformation
(e.g.
double-covering
through the i n s e r t i o n o f a c e n t r a l
(1.3)
the
').
undergo a
element i n the
algebra.
The g r a d i n g L „ , a scale o p e r a t o r , i s e s s e n t i a l . The r e s u l t i s t h e 5) V i r a s o r o a l g e b r a ' , d e r i v i n g from both the e q u a t i o n s of motion and the boundary c o n d i t i o n s straints
and r e p r e s e n t i n g the a l g e b r a i c
f o r the Veneziano s t r i n g .
t h e vacuum as the h i g h e s t L
| o >= o ,
system o f
The spectrum i s g i v e n by
conputting
weight
n e Z , n > o . (1.4)
o > =
o>
and using the lowering L
to construct the entire set of s t a t e s .
614
The
313 V i r a s o r o a l g e b r a commutation r e l a t i o n s
[Lm»Ln> = <"""> L m + n
+ d{
T7
(n,3
are (d i s a r e a l
number . d > 1)
-m)V-n} (1.5)
[Lm,d] = o and h e r m i t i c i t y L
n" -
L
determines
-n
d-6)
The r e p r e s e n t a t i o n s dual model the s p a c e .
that
are then c h a r a c t e r i z e d by ( d , v ) .
' ' i; was the Regge i n t e r c e p t and For the r e p r e s e n t a t i o n
owski-like metric
d
In the
Veneziano
the d i m e n s i o n a l i t y
t o be u n i t a r y ,
( o r 24 t r a n s v e r s e d i m e n s i o n s ) .
of
d=26
f o r a Mink-
I n the
superstring
of Ramond and Neveu-Schwarz and i n the more r e s t r i c t i v e v e r s i o n o f 8) Green-Schwarz ' , d=10. The correspondence between 2 - d i m e n s i o n a l Conformal
symmetry and the A n a l y t i c a l D i f f e o m o r p h i s m s was e x p l o i t e d 2) by B e l a v i n e t a l ' f o r the study o f 2 - d i m e n s i o n a l systems i n S t a t i s t i c a l Mechanics and has produced a u n i f y i n g a l g e b r a i c t r e a t m e n t f o r a v a r i e t y o f problems i n the Physics o f Condensed M a t t e r . The work was f u r t h e r developed by F r i e d a n e t al ' and Goddard e t a l '. With d i n ( 1 . 5 ) t a k i n g up a s e t o f values w i t h i n a n o t h e r a l l o w e d r a n g e , o < d < l (see e . g . ( 1 . 7 ) ) , one reproduces the c r i t i c a l b e h a v i o u r p a r a meters f o r the f o l l o w i n g p r o b l e m s , d model 1/2
Ising
7/10
tricritical
Ising
4/5
three-state
Potts
6/7
tri-critical
three-state
Potts
In these p r o b l e m s , the two a l g e b r a s d.
The v a l u e s o f v ( z ) v
the f i e l d and v " ' are s i m p l e l i n e a r
= v(z)
- v(z)
gives i t s
combinations of i r
(and a p o s i t i v e - d e f i n i t e (a continuum)
z and 1)
+ v ( 7 ) = tA ' gives
r e l a t e d t o the c o n f o r m a l anomalies d>l
(for
spin.
' and v ' ~ ' . '.
FQS '
Critical
showed t h a t
d = 1 - 6/(m+l)(m+2)
(m > 1 ,
values
d
is
unitarity
requires
either
t h e a b o v e , g i v e n by
m eZ)
v = { [ ( m + 2 ) p - ( m + D q J 2 - l } / 4 ( m + l ) (m+2) ,
like
of
exponents
The v a l u e o f
H i l b e r t space s c a l a r p r o d u c t )
and v >o or d i s c r e t e
(J s p <m
are u s e d , w i t h common
t h e " s c a l i n g " dimension
1
,
(1.7)
p.q e 7)
12) Goddard ' has emphasized the natureof the "group multiplication" for & , due to the group action (1.1) z" 6 z'(z) = z"(z'(z))
(1.8)
615
314 This is thus composition, and m=±l,o is an s u ( l , l )
z' - m^— b*z+a* Similarly, <-h L n ' k n -n n z.
A is non-abelian.
The subalgebra with
corresponding to the projective
, |a|2 - |b|2 = 1
there is an i n f i n i t e sequence of 4 u ( l - l ) L
transformations
„ • S- L n> » o n n
(1.9) algebras
With
- f az"+b a / n b*z n +a*
(1.10)
616
315 2.
Local Currents and A f f i n e Kac-Moody Algebras 13} c o n s i s t e d i n a system o f dynamical
Current a l g e b r a s
variables,
the l o c a l c h a r g e - c u r r e n t d e n s i t i e s
o f t h e SU(3) g e n e r a t o r s , o r o f t h e i r chi r a l 14) SU(3) x SU(3) e x t e n s i o n , c o n s t r a i n e d by an a n g u l a r c o n d i t i o n '
' due t o
Lorentz-invariance
time commutation [j°(x,t)
, j°(x',t)]
= i
f
[j°(x,t)
, j-Vx'.t)]
= i
f
ab
C
«3(x-x')
j°(x,t) ,-i
C 3i{6J(x-x')}
" S c h w i n g e r " term i n ( 2 . 2 )
term i n ( 1 . 5 ) .
currents
S3(l-1)
ab
+ i The
The l o c a l
obeyed e q u a l -
relations, (2.1)
x,t) (2.2) etc.
i s o f the same n a t u r e as t h e
The r e p r e s e n t a t i o n s were c l a s s i f i e d
'.
and C. M. Sommerfield c o n s t r u c t e d a c a n d i d a t e dynamical
H. Sugawara theory
i n c l u d i n g the commutation r e l a t i o n s between the components o f local
', the
c u r r e n t d e n s i t i e s and the components of the 0 WV energy-momentum
tensor. the
central
They c o n s t r u c t e d
the H a m i l t o n i a n d e n s i t y as a b i l i n e a r
in
currents yv
(x) = A { j * ( x )
j*(x)
+ jf(x)
j*(x)} (2.3)
+ B
%
<Jp(*> V * >
iO,
IJ^(R.t), j ^ J T ' . t ) ]
i
}
jjtf+t'.t)
f ab
(2.4)
wi th j°(it,t)
J d3x e l k x
j°(x,t)
and a simple r e p r e s e n t a t i o n
is
(2.5)
given by
j ° ( j r , t ) -* -A a (' t- )-e1' * ( A , an i u ( 3 )
(2.6)
m a t r i x) ,
a
The mathematical s t r u c t u r e of these a l g e b r a s was d e s c r i b e d i n on} Ref.19. C l e a r l y , t h i s was an i n f i n i t e L i e a l g e b r a ' with a close resemblence to f i n i t e L i e a l g e b r a s . The m a t h e m a t i c i a n s V. 6 . Kac and 21) R. Moody ' i n d e p e n d e n t l y took up the s i m p l e s t c l a s s o f i n f i n i t e L i e algebras, that of Affine
infinite-dimensional
g r a d i n g , such as the one p r o v i d e d by " - L " i n o ' mJ
(1.5)
(2.7)
me Z
m
algebras, admitting a 7
i.e. through a scale-operator (from (1.2)) L D = -L_ o == i z
3_ 3Z
(2.8)
617
316 and w i t h "manageable" growth i n the d i m e n s i o n a l i t y
o f t h e graded sub-
spaces dim{Vn]+1}/dim{Vm}
These are t h e A f f i n e very much .
m9"1 ,
g< 2
(2.9)
Kac-Moody a l g e b r a s , and t h e y resemble
from the c i r c l e S' i n t o a s i m p l e f i n i t e
L i e group G:
K : S' -*G This
i.e.
(2.10)
is p r e c i s e l y S'
: z et
z = e1 *
z-
(2.4)
The Kac-Moody group K i s g e n e r a t e d by t a k i n g smooth maps
,
,
g(z)
( 2 . 6 ) , as |z|=
1
(2.11)
0<4><2IT
eG
the m u l t i p l i c a t i o n 9 : 8 92(z)
(2.12) rule
here i s p o i n t w i s e
= gx(z)
.
g2(z)
(2.13)
K i s a l s o known as the " l o o p group" o f G. a
g = expl-i
0 (z)
multiplication,
Given the a l g e b r a o f
G,
(2.14)
T } a
- 1 ^abC Tc
' W
(2.15)
w r i te
g = 1 - i o a ( z ) T. a = 1 - i K ° axn )
"
:
- ^
^l^
Q
]
z
T
-n
(2.16)
n e 7
a.-n (2.17)
= 1 ^c Cn
and f o r h e r m i t i a n T a we f i n d t h e u n i t a r i t y n
condition (2.18)
-n
Q u a n t i z i n g from Poisson to L i e
brackets
[ T j . T j ] = 1(H) f a b c Tc + 0(H 2 ) [T'.TJ]
= i
fabr
T*
+ k m 6m
m n c m+n 181 \?) The Sugawara model ' yields '
6
ab
(2.19)
m,-n
[L m ,T a ]= -n T3. (2.20) m' n M=n 22) Ramond and Schwarz ' had tried to classify all possible dual model
618
317 gauge a l g e b r a s .
T h e i r c l a s s i f i c a t i o n a s s i g n e d the odd ( i . e .
onic)
to n o n - t r i v i a l
generators
only U ( l )
representations
and S U ( 2 ) , but f o r d e f i n i t e l y
o f G. This
unphysical
fermi-
allowed
values o f d .
e v e r , once the odd g e n e r a t o r s were assumed tobehave t r i v i a l l y as was done i n the new s u p e r s t r i n g o f Green and Schwarz, i t p o s s i b l e to take an a r b i t r a r y anomalies i n t e r f e r e s generated)
currents of (2.4)
"currents"
(2.3).
However, i t
group G.
Still,
with renormalizabi1ity
t y p e , and i n t h e energy-momentum
There are c h i r a l , c o n f o r m a l and mixed a n o m a l i e s . for G = S 0 ( 3 2 ) / ? / 2 \ .
t h i s i s a l s o t r u e f o r G = Eg x E g . the " l o w - e n e r g y "
of
(Yang-Mills
was r e c e n t l y n o t i c e d by Green and Schwarz
anomalies cancel
under G,
seemed
the existence
b o t h i n the
How-
' that
J. Thierry-Mieg
The c a n c e l l a t i o n s
f i e l d theory approximation.
all
' showed t h a t also occur
in
see t h a t
the
Cartan s u b a l g e b r a o f t h e above two c a n d i d a t e G make up the o n l y
two
even u n i m o d u l a r l a t t i c e s
i n 16 ( E u c l i d e a n ) —
owskian).
We s h a l l
dimensions —
_
Green and Schwarz f u r t h e r showed
'
t h a t f o r these G, the
p e r t u r b a t i o n expansion o f t h e r e l e v a n t s u p e r s t r i n g finite.
The o r i g i n a l
method o f i n t r o d u c i n g
t h e o r y was based on a t t a c h i n g the i n t e r n a l t r e m i t i e s of "open" s t r i n g s . diagrams26,27'
( o r 18 Mink-
„
("type I")
is
t h e group G i n a s t r i n g quantum numbers t o t h e e x -
Considerations
a t the l e v e l
of
"tree"
c o n s t r a i n G t o the sets o f S O ( n . R ) , USp(2n) and U ( n ) .
The l a t t e r a r e , however, u n s u i t a b l e due t o t h e appearance o f i n loop c a l c u l a t i o n s .
This c l a s s i f i c a t i o n
anomalies
thus does n o t a l l o w
the
e x c e p t i o n a l a l q e b r a s , i n c l u d i n q Eo. 28) I t was o n l y t h r o u g h the development by F r e n k e l and Kac ' o f a c o n s t r u c t i o n f o r the u n i t a r y i n f i n i t e r e p r e s e n t a t i o n s o f t h e K o f ( 2 . " 9 ) , using s t r i n g o p e r a t o r t e c h n i q u e s p o s s i b l e t o use the e v e n - o r t h o g o n a l S 0 ( 2 r , R ) , the t r a c e l e s s A
'
*
',
Lie a l g e b r a s
that i t D
( g e n e r a t i n g the S U ( r + l )
became
(generating etc.)
and the e x -
c e p t i o n a l s o f the E„ f a m i l y ( r = 6 , 7 , 8 ) f o r G. A r a t h e r p r o m i s i n g 30 31) p h y s i c a l model has now been developed * ' , using e i t h e r S 0 ( 3 2 ) / , , 2 ) or E a x E Q .
The l a t t e r
group has g r e a t advantaaes
in i t s
fit
phenomenology and has t h e r e f o r e a t t r a c t e d g r e a t e r a t t e n t i o n Several
t e x t s on Kac-Moody alqebras have appeared i n
y e a r s 33,34) ' ; .
619
with '.
recent
the
318 3.
Integral
Lattices
In the above c o n s t r u c t i o n , t h e t h e o r y o f I n t e g r a l L a t t i c e s an e s s e n t i a l
role.
A lattice space.
plays
is a p e r i o d i c
"Cubic" l a t t i c e s ,
a r r a y spanning the e n t i r e
d-dimensional
i n any number o f dimensions d ,
a b e l i a n L i e groups i n L i e group t h e o r y .
Lattices
and the c u b i c a r r a y s can be s e g r e g a t e d . The s i m p l e s t n o n - t r i v i a l
integral
arelike
can be decomposed
They are denoted 2 . lattice
is
the l a t t i c e
of
o f t h e a l g e b r a A2 ( g e n e r a t i n g S U ( 3 ) , S L ( 3 , R ) , S L ( 3 , C ) , e t c . ) v e n t i o n i s t h a t the l e n g t h o f the r o o t v e c t o r s is normalized to (p1) u =1 ( t h i s
= 2.
(six
in t h i s
We g i v e an enumeration o f r o o t
The conexample)
lengths:
i s the r o o t a t the o r i g i n ; as a l a t t i c e , we count
o r i g i n o n l y once, not as we do i n c o u n t i n g the a l g e b r a i c Uj=o (no o d d - l e n g t h v e c t o r s ) u2=l,
u2=6.
For the A
in
roots
the
roots),
general,
u2=n(n+l). The Dynkin diagram f o r the A
necting n dots.
the d i m e n s i o n a l i t y roots
is a s t r a i g h t
line
Each d o t r e p r e s e n t s one fundamental of the Cartan s u b a l g e b r a .
segment c o n root,
In A - ,
the
spanning
fundamental
are the I - s p i n r a i s i n g and U-spin r a i s i n g o p e r a t o r s , w i t h an
angle of 2^/3 between them. In theDynkin d i a g r a m , t h e n dots o f A 2 are a l l marked "p = 2 " , i . e . they a l l r e p r e s e n t v e c t o r s w i t h equal norms.
The l a t t i c e
has a d e t e r m i n a n t , t h a t o f t h e Cartan m a t r i x
s c a l a r p r o d u c t s between fundamental
roots.
For A2 as
cos 2ir/3 = s i n Tr/ 6 , t h e m a t r i x has " 2 " i n the d i a g o n a l the o f f - d i a g o n a l For the A
element.
Thus t h e d e t e r m i n a n t w i l l
in general, a l l
v e c t o r s are again 2 ^ / 3 . use o f a s i n g l e
of
and " - 1 " f o r
be equal
the angles between c o n s e c u t i v e
to
3.
fundamental
This i s coded i n t o t h e Dynkin diagram by the
l i n e t o connect the dots. In o t h e r a l g e b r a s , w i t h
a n g l e s , one has double l i n e s l i n e f o r 57r/g i n G 2 .
f o r 3-n/q i n the B n , C
Between fundamental
other
and F 4 , or a t r i p l e
root vectors
corresponding
t o unconnected d o t s , t h e angle i s i r / 2 and the m a t r i x gets no c o n tribution . Integral
lattices
angles o f 2TT/ 3 .
A use o n l y " s i n g l e - l a c e d " Dynkin d i a g r a m s ,
Thus o n l y
the A n , D
and E g , Ey, Eg c o n t r i b u t e
r e p r e s e n t the s e t o f "component l a t t i c e s " For the Dn ( g e n e r a t i n g S 0 ( 2 , n ) , e t c . ) Ey,
density of l a t t i c e
we reach 24 d i m e n s i o n s .
det D n =4, f o r a l l
Eg the d e t e r m i n a n t s are r e s p e c t i v e l y
i s a unimodular l a t t i c e .
until 3,2,1.
n.
We thus
The d e t e r m i n a n t , i n c i d e n t a l l y ,
p o i n t s per u n i t volume.
the o n l y component l a t t i c e s
620
For Eg,
l e a r n t h a t Eg gives
the
The A n , Dn and E g , g are
( i n any number o f d i m e n s i o n s )
by v e c t o r s of norm 2.
i.e.
and
generated
319 A lattice's
dual A* i s a n o t h e r l a t t i c e ,
g e n e r a t e d by a l l
whose s c a l a r p r o d u c t s w i t h t h e component l a t t i c e s ' an a b s o l u t e value o f one o r z e r o .
For A 2 , t h i s
vectors
f o r quarks and a n t i q u a r k s .
lattice
of a l l representationsof
entations
l i e on A^.
have
means a d j o i n i n g the
The new l a t t i c e
Ap i s i n f a c t t h e
A2, i . e . a l l states
The same w i l l
vectors
root vectors
in all
repres-
be t r u e o f a l l A , D* and E*,
except t h a t we know from L i e a l g e b r a t h e o r y t h a t a l l
representations
of Ep are generated by t h e a d j o i n t , so t h a t ER i s s e l f - d u a l . One i s 35) interested ' i n u n i m o d u l a r l a t t i c e s , and i n p a r t i c u l a r ^ even ones ( i . e . such t h a t have u 2 r + i = ° » r = o , 1 , . . . ) . Even u n i m o d u l a r l a t t i c e s e x i s t o n l y ( f o r E u c l i d e a n spaces) i n d = 8 k , k = 1 , 2 , . . . For Mink36) owski t y p e spaces ' w i t h m t i m e - l i k e dimensions o u t o f a t o t a l o f d , the s i g n a t u r e unimodular
is
i=
d - 2 n , and has to be a m u l t i p l e
o f 8 t o a l l o w even
lattices,
d = 8k + 2n
(3.1)
I have noted that this is precisely the condition
for the spinors
in that space to allow both Weyl and Majorana conditions result from the dimensionalities
'.
These
of Clifford algebras for the various
metrics. Returning to the even unimodular l a t t i c e s , we observe that
there
is one such lattice in k=l (this is E g ) , two in k = 2 , namely Eg x E g and D l g (generating S 0 ( 3 2 ) / 2 „ ) .
In k=3, there are 24 such
lattices.
Of these, 23 correspond to various semi-simple or other direct product 38) Lie a l g e b r a s , and o n e , the Leech lattice ' A, , the only lattice up to 2 24 dimensions whose shortest vector has norm p = 4 . Note that in 32 o
dimensions there are more than 10 unimodular l a t t i c e s . The number u , of vectors for each norm in the Leech lattice is 39) 2 la ' ( h e r e p = 2v) u2v(AL) = % | ^ ( o u ( v ) - T M ) (3.2)
6
O,,(D) i(v)
i s t h e sum o f t h e e l e v e n t h powers o f t h e d i v i s o r s
i s Ramanujan's u 4 = 196,560
function.
This y i e l d s
; ufi = 16,773,120
; ug = 3 9 8 , 0 3 4 , 0 0 0
Conway 4 0 ' d e f i n e d the group " ^ O " ( o f o r d e r of a l l E u c l i d e a n congruences
o f v, and
(3.3)
8,315,553,613,086,720,000)
( f i x i n g the o r i g i n )
o f A. .
This o r d e r
d e r i v e d as a p r o d u c t o f |t0|
= u 4 x 93,150 x 2 1 0 x | M 2 2 |
where | M 2 2 | i s
(3.4)
t h e o r d e r o f t h a t Mathieu g r o u p , one o f t h e (now known
621
is
320 to be) 26 " s p o r a d i c " s i m p l e f i n i t e g r o u p s . By t a k i n g the q u o t i e n t o f •0 by i t s c e n t e r { 1 , - 1 } ( t h e r e f l e c t i o n - p a r i t y ) , Conway got y e t another simple " s p o r a d i c " g r o u p , "•1"
<3-5)
= "^"/2(Z)
Using "• 1" and the properties of the Leech lattice, Griess and Fischer d i s c o v e r e d 4 1 ' the "Monster" or "Friendly Giant" F,, the last and 55 largest of the "sporadic" simple finite groups,of order ^10 . It contains 20 of the sporadics as quotients of F,, by some subgroups ("the Happy F a m i l y " ) ; 5 are clearly not contained (the "Pariahs") and for one (Jj, of order 175,560, the "wicked dwarf") it is still not known whether or not it is contained in F,. The construction of F, 42) has already been used to suggest a new superstring model in Physics ' following a general "encouragement" in Ref.24. By taking the quotient of the Leech lattice by its double M = A L / 0 , one defines a 24-dimensional vector space over a field of r
C Ai
characteristic groups
4
2,
Fg.
' p of order p
For any prime p , t h e r e are n+
, with p
linear
"extraspecial"
characters
and ( p - 1 )
i r r e d u c i b l e c h a r a c t e r s o f degree p n , one f o r each
faithful
primitive
p
root of u n i t y . Here we use the e x t r a s p e c i a l group Q, o f o r d e r 24+1 12 2 . I t thus has an i r r e d u c i b l e r e p r e s e n t a t i o n VQ o f o r d e r 2 (note t h a t t h i s
i s the d i m e n s i o n a l i t y
in 26-dimensional
Minkowski
o u t o f Q and ( - 1 ) .
space).
o f a Weyl-Majorana A finite
real
group C i s
spinor
constructed
The F r i e n d l y G i a n t F 1 i s d e f i n e d as a group con-
t a i n i n g an i n v o l u t i o n a , w i t h C as c e n t r a l i z e r
(i.e.
the elements
commuting wi th a ) , Fl =
(3.6)
Fj and C have a representation of dimension 196,884, denoted B. module B is endowed with the structure of a commutative nonassociative algebra, b
l
9
B b2 * b3
' ^
b
ie
B
(3,7)
w i t h a symmetric nondegenerate a s s o c i a t i v e b i l i n e a r 9g i s preserved by a . Note the s p l i t
form.
The p r o d u c t
In f a c t F^ can be d e f i n e d as F j = A u t ( B , 0 g ) .
(in
f a c t , B can be c o n s t r u c t e d as the
dim B = 1 9 6 , 8 8 4 = 24 fi 9 8 , 2 8 0 6 9 8 , 2 8 0 9 300 The sum o f the f i r s t
This
two subspaces
i n between q u o t a t i o n marks)
"B^"=
(the conventional V, 9 V2 i s 24 x 2
second and t h i r d dim (V 2 ® V-j) = u . ( A . ) .
622
union)of (3.8)
n o t a t i o n we p u t .
The sum o f
the
They are l i n k e d by a p a r i t y -
321 l i k e morphism exchanging " B j " = Vj
with "B2" = V
The l a s t
subspace
V . , dim V4= 2 4 ( 2 4 + l ) / 2 , corresponds to a symmetric S0(24,R) or t o the number o f t r a n s v e r s e ( i . e . p h y s i c a l ) symmetric it
(massless)
tensor
can i n f a c t be s p l i t
l e a v i n g dim " B * " = 299.
(like
g
tensor,
components o f a
) i n 26 Minkowskian
dimensions;
i n t o 2 9 9 9 1 , by e x t r a c t i o n o f t h e
trace,
Note a l s o t h a t i f w e remove t h a t s i n g l e t
( 3 . 8 ) we f i n d dim B = 196.883 = 47 x 59 x 7 1 , the p r o d u c t o f
from
three
primes. There a r e v a r i o u s ways o f c o n s t r u c t i n g A - , e i t h e r 24-dimensional
l a t t i c e s ( t h e Niemeier l a t t i c e s ) 45) various products ' of Eg. R e t u r n i n g to Kac-Moody a l g e b r a s characterize lattice
(2.19),
the r o l e o f the i n t e g r a l
',
from the o t h e r
o r by
we can now
lattice
generating i t ,
spanned by the r o o t diagram of the g e n e r a t i n g L i e
( 2 . 1 5 ) . F r a n k e l and Kac in d - d i m e n s i o n a l
;
combining
used " v e r t e x o p e r a t o r s "
(pp is
space, z i s a Mandelstam-type i n v a r i a n t
i.e.
the
algebra the momentum energy
variable) V(p.z)
= zP/2 exp(p" I
±«"
n
zn)
zP"
alJ
•
U
(°> x
n>0
,
u r
i
1
x exp(-pM l n>o
„-n\
v z n
U
_ i pv Hq
(3.9)
) e
In the a d a p t a t i o n t o Kac-Moody a l g e b r a s , the p^ are i d e n t i f i e d the l a t t i c e momenta i s
root vectors.
This means t h a t
t h a t o f the l a t t i c e
space!
KH
:
S" -
of G i n t o
K.
the
subalgebra Moreover,
the
of H,
H
(3.10)
a l r e a d y spans the e n t i r e K. embedding space i s or o f the
the d i m e n s i o n a l i t y o f
Only t h e C a r t a n
H(G)c G p l a y s a r o l e i n the a f f i n i z a t i o n r e p r e s e n t a t i o n o f the a f f i n i z a t i o n
with
For s t r i n g s ,
the d i m e n s i o n a l i t y
thus the d i m e n s i o n a l i t y o f the C a r t a n
lattice.
623
of
the
subalgebra,
322 4.
Physical
Applications
The p i c t u r e i n which s p a c e - t i m e i s embedded i n the Cartan s u b a l g e b r a o f a s i m p l e group was a l r e a d y p r e s e n t i n Extended Cremmer J u l i a
and the l a t e Joel Scherk
by w o r k i n g out N=l S u p e r g r a v i t y
'
Supergravity.
c o n s t r u c t e d N=8 S u p e r g r a v i t y
i n D=11 d i m e n s i o n s .
They were
sur-
p r i z e d to d i s c o v e r t h a t t h e t h e o r y possessed a symmetry under a noncompact form o f E , .
When i n the D = l l m a n i f o l d we assume t h a t space-
t i m e i s 3- or 5- d i m e n s i o n a l , the i n t e r n a l correspondingly.
J u l i a and T h i e r r y - M i e g
t r u e f o r any s p a c e - t i m e dimension
symmetry becomes Eg o r
E,
' havechecked t h a t t h i s
is
d
D= 11 S u p e r g r a v i t y has SL(11,R) as c o v a r i a n c e
group, i . e .
a l g e b r a i s A , Q . When we reduce s p a c e - t i m e t o d d i m e n s i o n s , covariance
involves
symmetry w i t h
rank
A. -.erA.Q, l e a v i n g t h e p o s s i b i l i t y
the
Lie
its
o f an i n t e r n a l
10-d+l.
T u r n i n g to s u p e r s t r i n g s , we h a v e r e c e n t l y seen the o f the a n o m a l y - c a n c e l l i n g q u a l i t i e s
o f r16
exploitation
(the S0(32)/j(2)
C a r t a n sub-
a l g e b r a ) and r „ x r Q ( i . e . ED x ED) i n t h e P r i n c e t o n " h e t e r o t i c " 8 30 31) model * '. I t i s c o n s t r u c t e d by w o r k i n g on t h e l i g h t - c o n e , and s e g r e g a t i n g the r i g h t - m o v e r s sets o f f i e l d s dimensional
The r i g h t - m o v e r s
a
S ( t - a ) , with i = l . .
a=1..8.
i s bosonic and 2 6 - d i m e n s i o n a l , w i t h e i g h t
X ^ i + a )1 and s i x t e1e n Y ^
4.,^
-
v
"internal" X^i+a),
J. ^ i
J. ^ x
i
V
= " «m+n,o ^
'
-2in(r+a)
(4.1) J
U .P l-}l«
( t h e f a c t o r •? i s due to the dependence on (x+o) X
, J
only)
i s made to p a r a m e t r i z e a compact s p a c e , a t o r u s .
topological X(o)
configurations
= x + 2 a'p
T
There are
p
+ 2 NRa
are q u a n t i z e d i n R
X1
o f equal r a d i i E x1
+ /FITR
+
..
(4.2)
units.
The t o r u s
T is
i=l
el.
n.
"
The momenta
"maximal", a product
R = ( a ' ) 2 = /%, i d e n t i f y i n g
y
stable
where (R i s a r a d i u s )
winds N times around the m a n i f o l d as a runs f r o m o to T. circles
left-moving
transverse
n
"
I
The
and
I = 1,..,16,
1^1
n^o
^ 4
t r e a t i n g t h e two
make a t e n -
s u p e r s t r i n g , w i t h e i g h t t r a n s v e r s e bosonic X 7 ( x - a )
e i g h t Majorana-weyl sector
from the l e f t - m o v e r s ,
asymmetrically.
points according (4.3)
1
where the e.,- a r e the s i x t e e n fundamental l a t t i c e , w i t h norm
624
r o o t v e c t o r s o f an Eg x Eg
of to
323
(ej)2
= 2
(4.4)
This implies t h a t 16 . . 9 e ij = I i ej
•
det
9
=
l
(4
-5)
(although a more " n a t u r a l " i d e n t i f i c a t i o n would be to regard the e- as i n v e r s e " t e r a d s " ) . The momenta are 16
P1
=I
n- e-
(n. i n t e g e r )
(4.6)
1=1
momenta me span the l a t t i c e v e c t o r s . the mo i ..e. the numbers N 1 . The s t r i n g mass o p e r a t o r i s 1
7
a • M2
They must equal the winding
N + (N-l) + I I ( P 1 ) 2
(4.7)
I N counts the r i g h t movers, N the l e f t movers,
i.e.
N = p aQ , N = p : a Q : The subtracted unit comes from normal ordering considerations and Lorentz invariance. In addition, one has to constrain
N = N - 1 + { { (p 1 ) 2
(4.8)
implying t h a t the u n i t a r y o p e r a t o r s h i f t i n g a to o+A in X , X , X does not a f f e c t p h y s i c a l s t a t e s in the s p e c t r u m . This o p e r a t o r i s exp 2 i AlN-N+1-j- £(p ) ] . The c o n s t r a i n t removes the l e f t - m o v e r tachyon from the physical H i l b e r t s p a c e . The s t a t e s |i or a> R x o ^ | o > L span N=1,D=10 s u p e r g r a v i t y . The s t a t e s | i or a> R x a _ 1 | o> L and | i or a> R x |p >L reproduce the N=l, D=10 super-Yang-Mills m u l t i p l e t , with gauge group E g x Eg ( o r $ 0 ( 3 2 ) ) . As the l a t t i c e has u =16, t h e r e are 16 n e u t r a l v e c t o r mesons ( p l u s t h e i r supersymmetric p a r t n e r s ) , c o r r e s p o n d i n g to a U(l) isometry of the t o r u s . The u2= 480 o t h e r r o o t v e c t o r s complete Efi x E fi . The model i n c l u d i n g i n t e r a c t i n g s t r i n g s , i s Lorentz and E g x E„ i n v a r i a n t . The one-loop diagrams a r e u n i t a r y (due to t h e s e l f - d u a l l a t t i c e ) and f i n i t e . The hexagonal anomalies are c a n c e l l e d . 42) Chapline ' has suggested a model in which the g r a v i t a t i o n a l and Yang-Mills p i e c e s of the s u p e r s t r i n g are c o n s t r a i n e d by the a c t i o n of a f i n i t e group. The s t r i n g i s basedon the Leech l a t t i c e A. . Here t h e mass o p e r a t o r i s (1 = 1. . . 2 4 ) , the t r a n s v e r s e dimensions) M2 = I a 1 J - 1 + i c n>o " n n
l(p1)2 I
(4.9)
625
324
with u 9 as in ( 3 . 2 ) and ( 3 . 3 ) . We see t h a t for M^=o we have t h e 24 s t a t e s g e n e r a t e d by a*-. For M = 1 , we g e t c o n t r i b u t i o n s from c^2 , 24 s t a t e s a
-l
a
_i
•
30
s t a t e s ( o r 299+1)
2
j(p ) , i . e . the 196.560 s t a t e s for u 4 momentum v e c t o r s , a l t o g e t h e r 196.884 s t a t e s , thus spanning the a l g e b r a B of ( 3 . 8 ) . Supersymmetry i s generated by the morphism exchanging. Vp and V 3 . The B 1 s t r u c t u r e f i t s a 26-dimensional g r a v i t i n o . F-. and a of ( 3 . 6 ) c o n s t r a i n the e n t i r e system. One cannot use c o m p a c t i f i c a t i o n to a t o r u s , as t h i s w i l l l e a v e no p o s s i b i l i t y of having two " o r d i n a r y " s p a c e - t i m e curved t r a n s v e r s e dimensions. The d e t a i l s of the model have n o t been worked out to d a t e .
626
325
References 1.
2.
See, f o r example, E. T. Copson, Theory o f F u n c t i o n s o f a Complex Variable,
Oxford Un. P r e s s , pp. 1 8 0 - 2 0 1 , London
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15. R. Dashen and M. G e l l - M a n n , Phys. Rev. L e t t . 16. M. G e l l - M a n n , D. Horn and J . Weyers, P r o c . Intern. p.479
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351-362,
Volume 174, number 2
PHYSICS LETTERS B
3 July 1986
SPINORS FOR SUPERSTRINGS IN A GENERIC CURVED SPACE " Yuval NE'EMAN
12
Stickler Faculty of Exact Sciences, Tel-Avw University, Tel Aviv, Israel
and Djordje SIJACKI Institute of Physics, PO Box 57, 11001 Belgrade, Yugoslavia Received 20 March 1986 The embedding of the superstnng in a generic curved space involves the use of world-spinors behaving according to the (infinite) unitary representations of SL"(10, H), the double-covering of the linear group on 3tw The necessary construction and lagrangian are provided
Introduction Closed, unonented, type I superstnngs [1 ] may provide a finite theory [2] of quantum gravity The hope that this is indeed the quantum version of the gravitational field is based upon the positive fit at the level of on-mass-shell amplitudes [3], but the picture is otherwise still somewhat obscure Indeed, in the conventional lagrangian formulation [1,4,5] for strings or superstnngs, the world-sheetTZ2 (locally-reparametrizable, or "generally-covanant" as a curved two-space with coordinates f **, /z = 0, 1) is embedded in aflat .©-dimensional Minkowski manifold 9/? 1*0-1 On the other hand, macroscopic gravity is described classically by Einstein's theory, corresponding to a curved nemannian9€ 4 manifold True, such an92 4 can always be embedded locally in 9# 1,0-1 fori) > 10 (though this cannot be guaranteed to be feasible globally for D< 23 [6]) but the residual local dimensions should then curl up "a la" Kaluza—Klein, so that for/) =10, for example, 9# I' 9 will be replaced by 92 1 0 3 9 ? 4 However, the attempts to embed the string in a curved manifold have encountered three fundamental difficulties (a) The fermionic 0(f) frame-fields required by supersymmetry and constructed at any pomt f of the world-sheet as spinors in9/f 1>°-1, cannot be embedded (ref [7] notwithstanding) in a curved generic nemannian92-°, since there exist no finite-dimensional spinonal representations of GL(Z>,R) on the one hand, and on the other hand one cannot apply the usual tetrad formulation, as we shall explam The current discussion of the more restncted program of Kaluza—Klein spontaneous compactification of six dimensions in 97? 1>9 hopefully leading to the observed spinor fields m flat physical 97£ 1>3 avoids this issue by concentrating on Ricci-flat vacuum solutions This would be consistent if spinor fields were made to appear only in the solutions, but not in the equations, which have to be general-covanant (b) The "critical" dimensionality [4] of the embedding space D = 26 (for strings) or D = 10 (for superstnngs) is modified [7,8] in the presence of generic curvature, thus destroying the essential conditions for a ghost-free finite theory (c) Without spinors we also lose supersymmetry, essential as a constraint for the removal of a tachyon [4] whose unwanted presence now makes the theory unphysical * Supported in part by the US DOE Grant DE-FG05-85ER40200, by the US-Israel Binational Science Foundation, and by RZNS (Belgrade) 1 Wolfson Chair Extraordinary in Theoretical Physics 2 Also on leave from the University nf Texas, Austin, TX 78712, USA
0370-2693/86/$ 03 50 © Elsevier Science Publishers B V (North-Holland Physics Publishing Division)
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We would like to suggest solutions for all three problems, based upon the (anholonomic) application of the doubly-covered groups of diffeomorphisms and superdiffeomorphisms in the "tangent" at f ** In this letter we use the (infinite) spinonal representations of the double-covering GL(10, R), whose existence was pointed out some years ago [9] and which we have recently constructed [10] for D = 4 and utilized [11] for the construction of world-spinorsinTZ"*, thus answering the quest in (a) while realizing the principle of general covanance In a subsequent letter [12], we resolve problems (b) and (c) by embedding GL( 10, R) in the doubly-covered real-form GQ(10,R), a supergroup generated by the superalgebra q(10) [13] We shall show that under these conditions, the curving of 9ft * >9 -* 9? 1 ° preserves supersymmetry (l e no tachyons) and the resulting critical dimension (I e no ghosts) both on and off mass shell Note that the possible emergence of fmite quantum gravity (and even supergravity) in one sector of a large string-unified theory is especially interesting after the recent demonstration [14] of the existence of residual infinities in Einstein gravity. The latter theory is thus apparently just the low-energy limit of a more elaborate structure, fitting the quantum regime The obvious alternative of quadratic lagrangians provides a renormalizable theory, but it appears to suffer from ghosts It is therefore important to thoroughly explore the possibility of deriving quantum gravity from superstrings Frame-fields in the Green-Schwarz model In the Polyakov [15] formulation of the Green-Schwarz [1,5] quantized superstrmg, every point f of the evolving-string world-sheet (greek indices) carries a global D-dimensional (latin lower case indices) "Pouicare"' supersymmetnc frame (Xm (f), 0 a a (f)), m = 0, 1, , D — 1 representing the components of a bosonic D-vectoi frame and a = 1, , 2^D~2^2 standing for the fermionic components of a real (Majorana) chiral (Weyl) spinor frame Note that the "Majorana plus Weyl" condition for a Minkowski metric exists only for D = 2 mod 8 The index a = 1,2 stands for the (somewhat trivial) spinor index in the sheet dimensionality Note that the expressions appearing in the lagrangian
^ ( f ) = Mr'Wif,,. = aMJP»(0, *«(0 = 3 W V f , = aM0*(r'),
0.2)
are generalized (rectangular) "tetrads" with the latin indices supporting the corresponding action of the tendimensional local Lorentz group SO(l ,9) or its subgroups In the analogy to general relativity with tetrads, the two-sheet plays the role of curved space—time (holonomic, "greek" coordinates) and the embedding ten-dimensional Lorentz boson or fermion indices fulfill the role of the "latin" (anholonomic) tetrad-indices Curvature and spinors In the attempts [7,8] to introduce curvature in the embedding, going from 9ft 1>9 to 92 1 0 , one replaces r)mn -> G^jf(f)> a curved metric Ref [7] provides a full series of additional replacements, necessary for an "effective" field theory action in curved space, and the equations of motion indeed reproduce the Einstein equations in the low-energy limit However, the method fails for the spinors, a result that leads to some confusion in ref [7] Indeed, in C^^(f) the m, n are GL(10, R) indices The frames in eq (1) are necessary for the definition of the action of the Lorentz group on spinors If we replace the flat m, n indices by curved ones, in, n, the spinors now have to carry the action of GL( 10, R), which is impossible for finite spinors Moreover, the corresponding transition for the relevant gamma matrices y -*y cannot be performed by an ordinary tetrad-like matrix, since the two sets of "coordinates" now correspond to fields over fM "e™" = 9X m (f)/dXm (f) Our X m (f) are in the flat tangent at ?*\ and there are no "tangents to the tangent", frames over frames It is possible that a physical description could be found, in which the f of the sheet would be directly (holonomically) embedded in a curved 9 2 1 0 with coordinates xm But finite spinors would again require the definition of a flat tangent X"1^), but no ;T" (?) In our solution, the frame in the tangent comes directly under the linear action of GL(10,R), and of the nonlinear action of the double-covering A of the differmorphisms, thus representing infinite spinors [11] Following the equivalence principle, these reduce in the_Minkowski limit needed for the definition of particle states into a sum of spinonal unitary representations of SO(l,9) or of the relevant stability subgroups of the PoincareISO(l,9), whether SO(9) (massive case) or SO(8) (massless case) 166
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PHYSICS LETTERS B
3 July 1986
GL(10,R) group structure The GL(10,R) group is the double-covering of the non-compact group GL(10,R) of the °tK l>9 space linear transformations The maximal compact subgroup of this group is SO(10) — Spin(lO), the double-covering group of the SO(10)group There is a four-element center of SO(10) which is isomorphic to Z 4 The factor group of GL(10,R) orSO(10) w r t a two-element subgroup Z 2 of Z 4 is isomorphic to GL(10, R) or SO(10) respectively LetHmn,m,n = 0 , 1 , , 9 be the GL( 10, R) generators The GL(10, R) commutation relations read \Hmn>Hkl}
=
^SnkHml ~ Vmflkn
(3)
'
where for the structure constants gmn one can take the invariant metric tensor either hmn = (+1, +1, , +1) w r t the SO(10) subgroup or r>mn =(+1,—1, , — l ) w r t the ten-dimensional Lorentz subgroup SO( 1,9) of the GL(10,R)group The metric tensor is GL(10,R) covanant The antisymmetric operators (when gmn = rimn)Lmn = H[mn] generate the metric-preserving Lorentz subgroup, the traceless symmetric operators Tmn = Hrmn\ — ^ *?,„„//' (ten-shear) generate the (non-trivial) ten-volume-preserving transformations, and the trace D = Hkk generates the dilation subgroup The traceless part H^n ofHmn, i e Lmn and Tmn, generates the SL(10, R) group with the commutation relations (3), where Hmn is substituted for Hmn In terms of L and T, the SL(10,R) commutation relation are [L, L]CL,
[L, T] C T,
[T, T]CL
(4)
ST(10, R) is the double-covering group of the SL(10,R ) group, SL(10, R)/Z 2 - SL(10, R) In the (1 + 9) notation the SL(10, R) generators are the compact/,, (angular momentum), and W, = T0l, and the noncompact K, =L0l (boost), T„ (nine-shear), and TQQ,I,/ = 1,2, , 9 The relevant sl(I0, R) subalgebras are the maximal compact subalgebra so(10) / , ; andN r the Lorentz so(l,9) Jt andK,, and sl(9,R) / ( / and Tt] The commutation relations (3) are invariant under the "deunitanzing automorphism" T T * ' t f - V ^ xr ^OO^T'oo. D-D, Nk-nKk, Kk-nNk ^ (5) This automorphism allows us to identify the finite (unitary) representations of the abstract SO(10) compact subgroup (/„, vV() with non-unitary representations of the physical Lorentz group (J K,), while the infinite (unitary) representations of the abstract Lorentz group of (/„, Kt) now represent (non-umtarily) the compact (./„, TV,) The m i=- 0 stability subgroup non-abehan part SL(9, R) of the inhomogeneous GL(10, R) group GA(10, R) (containing the ten-dimensional Poincare group) is unaffected by sti., and we use its unitary representations to characterize the particle states The unitary irreducible representations of the abstract GL(10, R) are thus used as non-unitary representations of the physical GL(10, R) and thereby avoid a disease common to infinite component equations (for the four-dimensional case cf refs [10,11 J) The SL(10, R) group can be contracted (a Ja Wigner—Inonu) w r t its SO(10) subgroup to yield the semidirect-product group T J 4 @ SO(10) The T 5 4 is an abehan group generated by 54 operators Umn, which form a second rank symmetric operator (••) w r t SO(10) The commutation relations are
[J,J] CJ,
[J, U]CU,
[U, U]=0
(6)
SLflO, HJ unitary representations Owing to the fact that dilations commute with the SL(10,R) subgroup of theGL(10,R) group, the essential part of the GL(10, R) unitary representations is given by the SL(10,R ) unitary (finite-dimensional) spinonal and tensorial representations An efficient way of constructing explicitly the SL(10, R) unitary representations is based on the decontraction formula [16], which is an inverse of the Wigner— Inonu contraction (cf refs 117,18] for SL(n, R),« = 3,4) According to the decontraction formula, the following operators Tmn =pUmn + h(U-U)-l'2[C2(SO(10)),
Umn) ,
(7)
together with/ m M satisfy the SL(10,R) commutation relations The parameter p is an arbitrary real number, and C 2 is the SO(10) second-rank Casimir operator 167
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PHYSICS LETTERS B
For the representation Hilbert space we take the homogeneous space of L 2 functions of the maximal compact subgroup SO(10) parameters The SO(10) unirrep labels are given either by the Dynkin labels (Xj, X2 > > ^ s ) o r by the highest weight vector which we denote by [Af j , Af2, , M 5 ] The SL(10, R) commutation relations (4) are invariant w r t an automorphism defined by s(L) = +L, s(T) = -T, which for real matrices becomes transposition symmetry (cf ref [19] for SU(3)) This enables us to define an "s-panty" to each SO(10) unirrep of an SL(10, R) representation. In terms of Dynkin labels (generalizing the S L ( « , R ) , n = 3 , 4 [20,21] we find j = (_)*l + M + *3+(*5 -*4~e)/2
(8)
;
where e = O(-H) if X5 - X4 is even (odd) The 54 representations of SCT(IO), I e (20000) = aa, has y(20000) = +1
(9)
A basis of an SO(10) unirrep is provided by the Gel'fand—Zetlin pattern characterized by the maximal weight vectors of the group chain SO(10) 3 SO(9) 3 3 SO(2) We write the basic vectors as | ^ | > , where (m) corre-
'(«)'
sponds to SO(9) 3
(*).
3 SO(2) subgroup chain weight vectors When necessary, we generalize the basis to | [Af] Ho (m) accommodate both the group action to the right and left, and work in the homogeneous space over SO(10)® SO(10) The (k) and (m) labels correspond to the two subgroup chams The 54 abehan generators{U mn } = { ^ / D ) D ' } of T 5 4 @ SO(10) commute, mutually, and thus in the most general case we find 4
l/£ ) a J=Sp<'Wn ) ](W), w
,=o
00)
In)
where
A(w)-(S
?({«})
[aa]\
(M) \ (*) are the SO(10)-Wigner functions,£({a}) e SO(10), and (k;) are all "sublabels" for which s-panty is +, as required by (9) For the ten-shear we obtain from (7), (10), and the orthogonality properties of the D-functions
/ (*') (m )
, , I (*0\ (w)
/ MI [an] [M]\/[M'] 00 (m)/\(*')
W]\ (*)/'
(11)
where the reduced matrix elements are
W]\.
(W)
V)
2,<-(M i=l
mmmm)1121
[p(0)+Hc2W))-c^wm ([M(k'j'
[DO] [M)\ (0)
[an] [M]
))
N([M}) is the representation dimension, and ( . ) are the SO(10) " 3 ? " symbols. For the representation parameters we could have a prion p(°), p W , , p( J ) G C Imposing the hermiticity condition (unitary representations) we obtain several series of (infinite) unitary spinorial representations e g principal series when all p® are pure imaginary, and all [M1] and [M] half integer Let us concentrate on the simplest-multiplicity free (each SO(10) irrep appears at most once) spinorial representations [22] These are obtained from (11) by setting (k't) = (&,) = (0), i = 0, , 4 andj>W, ,p< 4 ) = 0, and by an analytical continuation of [M] labels to the half-integer values. By making use of the SO(10) Clebsch—Gordan series and the s-panty we find the following simplest unitary spinorial systems (belonging to discrete series) charac168
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3 July 1986
tenzed by their SO(IO) irrep dimensionality content T) dlsc (16)={16, 144,720,2640, 7920, } ,
={ 5 60, 3696',8800,15120,
},
cZ)disc(672)={672, 1440,11088, } ,
(12)
each of them supplemented by a system of conjugated states, e g q> dlsc (16) = {16, T44,720, 2640, 7920, }
(13)
The "ladder" tensonal unirreps can be most easily obtained as the limiting cases of the totally symmetrized 2nboxes (n -*• °°) products of the SU(10) unirreps applied to the "vacuum" consisting of either singlet or one-box state •,an,DDDD, ,D,OOD,DDDDD, Thus we find q > l a d d ( l ) = { l , 5 4 , 660,4290,19305, } ,
^ ^ ( l O ) ={10, 210', 1782',9438,37180, }
(14)
Field equations The above constructed unitary SL(10,R) representations define the corresponding SL(10,R ) covanant infinite-component fields — "manifields" There are two crucial physical requirements to be fulfilled at this point First the intrinsic Lorentz generators represented on fields must be non-umtarily represented Otherwise, the Lorentz boosts would excite a given state to other spms and masses contrary to experience We resolve this by applying the deunitanzing automorphism &l(5) to the SL(10, R) unitary representations The non-umtanty in the intrinsic boost parts now cancels their physical action precisely as in finite tensors or spinors, the boost thus acting kinetically only Second in the absence of gravity, by the principle of equivalence, only the Lorentz group SO(l, 3) C SL(4, R) survives (or, in fact, the double covering of the Poincare" group when we adjoin the translations as well) We generalize this requirement, namely, that for D = 4 our SL(D, R ) manifields satisfy only SO(l, 3) covanant equations [23], to D = 10 and SO(l, 9) covanance For the tensonal representations the simplest choices are either 1) kdd(l) or
(15)
Spinor manifields obey a first-order equation (cf ref [23] for D = 4) with infinite Tm matrices generalizing Dirac's The T m behave as SO(l, 9) ten-vectors, and we are forced in the simplest case to use the reducible pair of SL(10, R) representationscDdlsc(16) ®
(16)
=
The Fm {(rm)g} operators connect various SO(10) spinonal states of an SL(10, ) representation, eg T ® 16 D 16 © 144, r ® 1 4 4 D l ! e i 4 4 ® 720, T ® 720 D 144 e 720 ® 2640, T® 2640 D 720 © 2640 © 7920, General covanance for manifields Ref [11] provides a detailed discussion of the covanant derivative for manifields Applying it to the superstring we get D^$ = (9^ - Tj"„Wmn})$, where rflmn is the connection and {Hmn} the appropriate (spinonal or tensonal) infinite representations of SL(10, R ) For "spinor manifields, rei [11] also introduces appropriate infinite " t e t r a d s " ^ ^ ( f ) and metric G^gift = E2A(^){S}ABEgB(i),v/heTeB = r° is the infinite generalization of Dirac's -yu in the adjoint i// = ^ t y0 The curved space spinor manifield is then * ^ ( f ) = •*A(X)EAA(X), with EA^ the inverse "tetrad" J n describing the dynamics of the superstring in curved space, the spinor frames da(£) have to be replaced by A ^ iX) Note, however, that when quantizing the lagrangian, using covanant quantization, we have to go first to the equivalence principle limit ^ ( f ) -»• ^ ( j " ) yielding a reducible infinite sum whose first level has the qauntum numbers of 0"(f) provided we use for
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+ g»*f r ( *t ^\E\{Ym
itT^t&D^lPW*
r°)CJE-C(r«(e-l)» fBpvPaDT{E%
x ( ^ t r o ^ c ^ ^ ^ ^ - i ^ ^ - i ^ ^ - i ^ ^ d _p3)£g*0 9i^r?aTA^ + ,
*D)gm
(n)
where 92^jf/- is the9? 10 curavture and p 3 the y5-like matrix over the 72 2 world sheet Since thex m variable does not enter our expressions in the superstring, the "decad" (or ten-bein) e"^ and its inverse (e-1)jjj have to be expressed in terms of the E4 or its inverse We assume that the fundamental information or the curvature of gravitational field is introduced through the infinite frames on the double-covering Thus we write
^=^44-
ie-l£-K$I$KZ,
(18)
where the K"^ and K"^ are the constant transition matrices between the finite- and infinite-dimensional representations of SO(l,9) (where the^4 index denotes a reducible infinite sum) andSL(10, R) (with .4 running over the com ponents of the genume irreducible representation) respectively The g"*n can be denved from the decads in eq (18) Note that the K transition is a homomorphism with the Z 2 kernel of the double-covermg center Note also that the frame Xm transforms under SCT(1, 9) like the first level of{Q)ladd(10)}-^ It is only when we shall impose [12] the farther requirement of curved space supersymmetry that we shall have to replace Xm by the mamfield, in addition to the 6" -»• ^A replacement in (17) References [1J MB Green and J.H Schwarz, Phys Lett B 109 (1982) 448,B 136 (1984) 367 [21 MB Green and J H Schwarz, Phys Lett B 149 (1984) 117, B 151 (1985) 21, J Thierry-Mieg, Phys Lett B 156 (1985) 199 [3] J ScherkandJ.H Schwarz, Nucl Phys B 81 (1974) 118 [4] M.Jacob, ed , Dual theory, Physics Reports reprint book (North-Holland, Amsterdam, 1974) [5) J H Schwarz, ed , Superstrings (World Scientific, Singapore, 1985) [6] A Friedman, Rev Mod Phys 37(1965)201 [7] E S FradkinandAA Tseytlin, Phys Lett B 158(1985)316 [8] C Lovelace, Phys Lett B 135 (1984) 75 [9] Y Ne'eman, Ann Inst Henri Poincare A 28 (1978) 369 [10] Dj SijackiandY Ne'eraan.J Math Phys 26(1985)2457 [11] Y Ne'emanandDj Sgacki.Phys Lett B 157 (1985) 267, 275 [12] Y Ne'emanandDj Syacki.Phys Lett B 174(1986) 171 [13] Y Ne'eman, in Cosmology and gravitation, eds PG Bergmann and V de Sabbata, Nato Advanced Study Inst Ser B 58 (Plenum, New York, 1980) pp 177-226 See in particular p 192 [14] MH GoroffandA Sagnotti,Phys Lett B160(1985)81 [15] AM Polyakov,Phys Lett B 103 (1981) 207, 211 [16] Y DothanandY Ne'eman, in Symmetry groups in nuclear and particle physics, ed F J Dyson (Benjamin, New York, 1966). [17] Y Ne'emanandDj Syacki.J Math Phys 21 (1980) 1312 [18] Y Ne'eman and Dj. iigacki, Ann Phys (NY) 120 (1979) 292 [19] LC v Biedenharn,Phys Lett B 28 (1969) 537 [20] Dj Sijacki, J Math Phys 16 (1975) 289 [21] Dj Suacki, Ann Isr Phys Soc 3 (1980) 35 [22] Dj Sijacki, Lecture Notes in Physics, Vol 201 (Springer, Berlin, 1984) p 88 [23] A Cant and Y Ne'eman, J Math Phys 26(1985)3180
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SUPERSTRINGS IN A GENERIC SUPERSYMMETRIC CURVED SPACE * Yuval NE'EMAN ' 2 Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel Aviv, Israel
and Djordje SIJACKI Institute of Physics, P O Box 57, II 001 Belgrade, Yugoslavia Received 20 March 1986 Supersymmetry (off-shell) is necessary and sufficient for the preservation of the value of the critical dimension and the cancellation of anomalies when the superstrmg is embedded in a generic curved space This is achieved by embedding the doubly-covered general linear (spinonal curved space) group GL(10, R) in GQ(10, R), a non-simple real form derived from the exponentiation of the hyperexceptional superalgebra q(10)
Introduction In the preceding letter [1] we suggested a solution to the problem raised by the spinor frame fields 0a(f), (a = 1, , 32) on the Green—Schwarz superstrmg [2] 92 2 world sheet (coordinate f", ^ = 0,1), when it is embedded in a generic curved space 9? 1 0 (coordinate xm =0, ,9) In the usual technique in general relativity, a spinor would be defined in the local tangent (= flat embedding 9/?1-9, coordinate xm) space at x™, where a local Lorentz group can act on it (this is known as "anholonomic" action) However, m the string formalism, this method is used up prior to curving of the embedding space, smce the original spinor field is indeed already defined as a local (fiber) frame-field on the tangent to the curved string (the base manifold) at the string coordinate f »* Indeed, the coordinates xm (flat) or xm (curved) do not appear in the formalism, having been replaced for the flat case by the action of their isotropy group on that vector frame-field X m (f) as a fiber on the local tangent to the curved string at f M in an associated bundle What is thus required is a further step, the (generic) curving of that tangent (at f **) itself Hence the complication for all but those cases in which the curved manifold carries the double-covering SO(l,9) or at least that of an SO(l, 3) 9 G subgroup Were it not for the spinor, generic curving could have been achieved (as in ref [3]) by replacing Xm (f) by X™Q;), a world-vector (= "holonomic") carrying finite linear representations of GL(10, R) and non-hnear representations of the general covanance group A (the analytical diffeomorphisms) In other words - changing the structure group of the bundle from SO(l,9) to A In our solution, we therefore indeed replaced the action of SO(l, 9) by that of GL(10, R) and A, the double-covering of the general linear and general covanance groups, as the new bundle structure groups For spinors, where the double-covering is required, these can, however, only be infinite-dimensional We thus replaced the 0fl(f) by *' 4 (f) infinite frames transforming accordmg to the unitary infinite dimensional representations of the linear subgroup SL(10, R) As explained in ref [l],in * Supported in part by the US DOE Grant DE-FG05-85ER40200, by the US-Israel Binational Science Foundation, and by RZNS (Belgrade) 1 Wolfson Chair Extraordinary in Theoretical Physics 2 Also on leave from the University of Texas, Austin, TX 78712, USA
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the equivalence principle (flat) limit, necessary for the definition of particle states and for covanant quantization, these frames reduce to infinite reducible non-unitary representations ^ ( f ) of the SO(l,9) subgroup, an infinite direct sum of finite conventional spinor frames, the lowest level of which is identical to the 8A (f) There are, however, two further known difficulties to overcome, for the superstnng to be embeddable in a generic curved space Beyond the fitting in of the spinors, we have to preserve the supersymmetry they support, otherwise we cannot get nd of an unphysical tachyon state [4] Moreover, we also have to preserve the critical role played by the dimension/) = 10 in removing the conformal anomaly and ensuring the segregation of ghosts (in analogy to the role of the BRS equations m gauge theories) In Lovelace's calculation [5] for an S^" 1 compactification with SO(/V) symmetry of the original flat "intemalizable" dimensions for the bosonic string, the sum of the free field conformal anomaly in the residual [D — (N — 1)] flat dimensions, plus that contributed by the string within the compactified subspace (equivalent to an 0(N) sigma model in 9? 2 , or N- 2 effective "flat" dimensions) results in a critical dimensionality [D - (N - 1)] + [N - 2] =D- 1, so that the Liouville scalar field kinetic term has as coefficient y = — [26 — (D - 1)] or (27 — D) The effect is a non-conservation of the Virasoro charges and the ghost state does not decouple In a similar calculation [3] with generic curvature but no spinors the coefficient y = —[26 — (D - | a'9?)], where T2 is the embedding curvature For the Neveu—Schwarz—Ramond model [4], ref f3] got y ~ —[10 — (D - a'TC)] These examples display the importance of preserving the constraints of supersymmetry in the generic curved case Superalgebra choice We seek to find a superalgebra for a superstnng in a generic curved space which would reproduce the Green—Schwarz superstnng particle content in the flat limit, and which would allow us to find an off-shell supersymmetnc formulation Thus, we pose the following requirements on the superalgebra (a) In the even (bosonic) sector it should contain the SL(10, R) algebra of ref [ 1 ], (b) in the m = 0 flat-space limit it should contain the superalgebra of the string stability group, (c) in the m ^ 0 flat-space limit it should contain the superstnng particle states, and (d) it should ensure an off-shell cancellation of the anomalies We require both (b) and (c), since in contradistinction to the superstnng where the spectrum is generated via a successive application of the spectrum-generating algebra operators, we construct the entire spectrum by a single application of the noncompact supergroup unitary representations The requirement (a) is met by the following simple Lie superalgebras [6] classical sl(m/n) and msl(n), classical hyperexceptional p(w) and q(«), and non-classical w(n), s(n) andTfn) A straightforward inspection shows that sl(/w/«), w(n), s(n) andT(n) violate our requirement (d) since they are realized in spaces with an unequal number of bosonic and fernuonic degrees of freedom The superalgebra p(n) does not allow for a non-trivial central extension, which is essential for our applications, and cannot meet requirements (b) and (c) The superalgebra msl(n) has twice as many generators as compared to q(tt) on the one hand, and on the other hand it is not a-pnon clear that if requirements (b) and (c) are satisfied for some msl(n) representation that (d) would be satisfied as well Finally, we are naturally led to select the q(n) superalgebra for the superstnng in curved space The simplest possibility to meet requirement (a) is provided by the superalgebra q(10) &Q(10, R) supergroup The Q(10, R) supergroup is generated by the operators of the simple q(10) superalgebra This supergroup can be enlarged to the non-simple group of general linear supermatncesGQ(10,R) on an 9ft1-9/10 superspace with real cttL 1»9 bosonic subspace The odd (symmetric) brackets for the GQ(10, R) algebra are given (in contradistinction to the Q(10, R) case) by the anticommutators The even (bosonic) operators of the GQ(10,R) algebra generate the GL(10, R) group We take the double covenng GQ(10, R) of the GQ(10, R) supergroup, with the corresponding 5L(10,R) even subgroup [1] The 0(10, R) generators are aa <8 \g, where aa, a = 0, 1 are the relevant Pauh matnces and the \a, a = 1, 2, ,99 are the traceless generators of SL(10, R) For G*Q(10, R) we relax the traceless condition for the even subalgebra, as we shall do here for the odd (ferrruoruc) generators as well for convenience only For the even and odd generators we take respectively H
mn=a09Xmn'
F
mn = °l • Xmn>
0)
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where (Km„)pi/ - igmpg„n, m, n,p, q = 0,1, ,9, and for gmn one can take the invariant metric tensor either 5 m n =(+l^+l, , + l j w r t the SO(10) subgroup or i\mn = (+1,-1, , - l ) w r t the ten-dimensional Lorentz subgroup SO(l, 9) of the GL(10, R) group The GQ(10, R) brackets are U*nm>HkH
=i
KkHml-SmlHkn\
Wmn,Fkl] = i ^ f e F m / - gm,Fkn),
{Fmn,Fk,} = i f r ^ , +^ lfI ^ Jbl ) (2)
_ There are two relevant subgroup chains of the GQ(10, R) even part GL(10, R)3§L(10, R) 3 §5(10) 3 §0(2) ® §0(8), GT(10, R) 3 SL(10, R) 3 §0(10) 3 §0(9)
(3)
The traceless symmetric components of both Hmn(ten-shear) and Fm„(ten-supershear), l e #£„„} and Frmn\, transform w r t SO(10) 3 §0(9) 3 §0(8) and §0(10) 3 §0(2) ® §0(8) respectively as 5 4 3 4 4 © 9 ©1 3 3 5 y ©8y ©1 ©8y ©1 ©1,
54 3 (1, 35y) © (2, 8y) © (2, 1) © (1, 1)
(4)
The antisymmetric components of both //OT„(ten-Lorentz) and Fmn(ten-super Lorentz), l e H,mn-i and F<mn\, transform w r t §5(10) 3 §0(9) 3 §0(8) and §0(10) 3 §0(2) ® §0(8) respectively as 45336®9328®8y®8v©l,
45 3 (1,28)® (2,8y) ® (1,1)
(5)
Graded algebra GQ(10, ft) and the spin-statistics theorem The GQ(10, R) algebra is a graded algebra, l e it is defined by the brackets with appropriate antisymmetry or symmetry properties and by the graded Jacobi identity requirements The graded brackets for GQ(10, R) are given by commutators and anticommutators However, even though the odd generators are a subject of anticommutation relations they do not transform w r t either the even subalgebra GL(10, R) or the maximal even orthogonal subalgebra §0(1, 9) as spinors Indeed, (4) and (5) reveal their tensonal character in the defining (1, 9/10) X (1, 9/10) representation This fact is due to the non-existence of finite SL(10, R) spinors, and the way how §0(10) is embedded in SL(10, R) It is only for the lnfinitedimensional representations of GQ(10, R), defined in the superspace of SL(10, R) infinite tensors and spinors [\], that the odd operators connect bosons (fermions) to fermions (bosons), I e behave as physical spinonal operators, and thus the spin-statistics theorem is recovered Contraction, quotient space and the stability group Let us split the even and odd generators of GQ(10, R) according to 2 + 8 notation, i e w r t the "GQ(2, R)" X CQ(8, R) subgroup Hmn = {H^Ji^,Ha1 ,Hla], Fmn = ^a&>Ft],Fal,Fla},(x,^-0,9 and;,/ =1,2, ,8 If we rescale the generators in the GQ(10, R) algebra, Ha& - ^HaB=Ke,Ho,-*effo1 =H'0t,H,Q+eHl0 =H\^Htl ^H^F^+eF^F^F^^ eFal = Fal, Fla ^ eF,a = Fja, F -*• Fv, and take e -»• 0 (Wigner—Inonu contraction), we obtain the GQ(10, R ) ^ superalgebra It is straightforward to check that this superalgebra contains the flat-space limit stability super subalgebra generated by the even operators H,%] j , H,9,,, H^Q , Hg9, and the odd operators FQ, , F,'0, F9,, F/9 Here the H^ 1 generate the §0(8) subgroup while fr[vj and#[ 9 ,j generate the S0(9) subgroup Let us define s
i = *FL>
s
s al
hFL>
a=0 9
<>
l= 12
'>
>8
( fi )
The anticommutators read
tt«. V
= {5
< » ' V = °> # « • V = 2(°oho
+
"3*9)-/.'
(7>8>
where a0 and a 3 are the Pauli matrices, and where h0 and h9 are the eigenvalues of the operators 5(^00 +#99) and j(// 9 9 - H'QQ) respectively For ajmitary (infinite-dimensional) §E(10, R) representation, hQ and h9 are real The super subalgebras con taming the SO(8) or S0(9) subalgebras correspond to the m = 0 or m ^ 0 supersymmetry stability algebras respectively, and we denote them by ssa(8) and ssa(9) These supersymmetry stabihty algebras generate respectively the stability groups SSG(8) and §SG(9) of the homogeneous superspaces GQ(10, R)/ SSG(8) and GOOO, R)/SSG(9) 173
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The flat-space limit spectrum Owing to the fact that the operators//^ and/fgg commute with all generators of the GQ(10, R)con contracted group, in the GQ(10, R) unitary representation space, we can renormalize the relevant odd operators and haveh 0 ,hg = 0, ±1 in (8) For the choiceh0 = l,fc9 = 0, we Find them =£ 0 supersymmetry Chfford algebra with the brackets given by (7) and by For the choice h0 = 1, h9 = ± 1, we find the m = 0 supersymmetry Chfford algebras with the brackets given by (7) and respectively by or
(W=0>
teh-V=4*v
(11)
The correspondence to the superstnng spectrum of states [4] is now evident Note that the 16-dimensional representation of the SSG(8) supergroup corresponds to the m = 0 superstnng Fock space ground state, while our operators/^r 9,i and S0l yield, in the representation space, results corresponding to the action of the superstnng spectrum-generating operators a£. and$*, respectively Frames, finite and infinite In the treatment of gravity with world-spinors [7], we constructed manifields over xm genenc nemannian CKD space in two steps First, anholonomically (in the tangent frame), the manifields, XM(xm), A(xm), where M and A represent an infinite set of components descnbing the two infinite direct sums of finite non-unitary representations of the SO(l, D - 1) Lorentz group (for M) or its double-covenng (for .4) §0(1, D ~ 1), which we obtam by reducing one infinite irreducible deurutanzed [1] representation of SL(Z>, R) D §0(1, D - 1), tensonal or spinonal respectively These are thus fiber-bundles whose base manifold is 92° and structure group SO(l,£> - H Injhe second stage, we "holonomize" these manifields into world-tensors and world-spinors For a finite m(xm), the anholonomic equivalent is obtained through the action of a "Z)-ad" ("Dbein") etlix*) field The soldering form em(x™) - eZfx^dx"1 thus solders the base manifold to the tangent, and
= E^(x)XM,
XM(x)=E®(x)XM(x),
^A(x)=E^(x)^2{x),
V2{x) = E*(x)
(12)
We use the same E for the countable-infinity "alephzeroad" (or "°°-bem") frames and their inverse, as we shall always wnte the indices For a Riemann—Cartan manifold Z>~£^=0,etc
(13)
The alephzeroad frames contain information about curvature and represent the gravitational field, since we can wnte
4 w = * f * ) * m > % # w = * f c s 4 . «*(*>=***#<*)*«. *c>
<14)
where the K^, Kn are constant transition matnces between the finite- and infinite-dimensional representations of SO(l, D - 1) and SL(Z), R) respectively Indeed, the gravitational field is most fully described by the EA (x), since they relate the double-coverings The relevant transition matrices thus realize a homomorphism, with the Z2 in the center of §S(1, D - 1) and SL(D, R) as kernel, e%(x) = KXE$WK*,
etc
(15)
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For the string and superstring, the base-manifold is the 9? 2 world-sheet, and all above frame-fields and alephzeroads are functions of J*1 The embedding xm variable on92 10 does not appear in the problem, so that the information on curvature will have to be given through the matrix elements of £^(f), and the decads ("ten-beins") e | j (?) W1U De derived from them The lagrangian. Using the Green—Schwarz superstring [2] as embedded in a generic curved 9 ? l u in eq (30) of ref [3] and corrected for the spinors in eq (17) or ref [1 ] , we can now write the supersymmetnc version,
JC-^^AjOT + 2^^OT9M^(*tr0)y4^(r"(e-1)J )f p„p oJ D T (f|* 5 )/:|^ +^MVr(^tr0)/4^(r'"(e-1)^)fz)M(f| **) (^r0)cE§(r"(e-l)*)%pvPaDT(E%*D)gm~ x(*tr°) c ^(r [ H ,,(e- 1 ^( e - 1 )f f ( e -^)g(i-P3)£j^a l ,jr*a T ^44+
,
(16)
where 9?^~g-y is the 9Z10 curvature (derivable from ^2jg^X£- using K% etc ) The pM are 7-matrices on the world sheet 922 and p 3 the 75-hke matrix The above expression for the lagrangian is incomplete in two senses On the one hand we have included only the lowest pieces in terms of the curvature effects, as in ref [3] On the other hand, the full GQ(10, R)-invanant lagrangian requires the knowledge of the unitary infinite-dimensional representations, as yet unknown Presumably, the representations Q)ladd(10) and rDdlscll6) ® ^ ^ ( I B ) of the GL(10, R) even subgroup do not yet span a complete representation space for GQ(10, R) However, their flat-space limit reproduces the superstring spectrum Since there are no representations of GL(10, R) with lower quantum numbers, any additional representations appearing in the reduction of a GQ(10, R) representation will not add to the lowest states of the physical spectrum Further terms should bring in the spinonal components of the curvature, derivable from c^~iSZ:B an<^ c o n " spiring with the tensonal terms in order to provide for supersymmetry and the preservation of the cancellation of anomalies We have to ensure that the solutions of this theory will indeed be nemannian, or of the Riemann—Cartan type with non-propagating torsion (at least m the 92 4 submamfold) This can be done by inserting m the lagrangian terms with multiphers [8] s t the equations of motion provide the necessary constraints on Gjg and G»», equivalent toeq (13) The theory indeed preserves supersymmetry off-mass-shell, due to a special feature of the q(D) algebras Our GQ(10, R) embedding indeed fulfills two conditions on the one hand, it has vanishing supertrace, and on the other hand, the bosonic and fermionic representations of the G"E(10, R) bosonic subgroup are correlated by tr
bose"=tr f e r r m "
[1] [2] [3] [4] [5] [6] [7] [8]
07)
Y Ne'emanandDj Sijacki, Phys Lett B 174 (1986) 165 M B Green and J H Schwarz, Phys Lett B 109 (1982) 448, B 136 (1984) 367 E S Fiadkinand A A Tseytlin, Phys Lett B 158 (1985) 316 M Jacob, ed , Dual theory, Physics Reports reprint book (North-Holland, Amsterdam 1974), J H Schwarz, ed , Superstrmgs (World Scientific, Singapore, 1985) C Lovelace, Phys Lett B 135 (1984) 75 Y Ne'eman.in Cosmology and gravitation, eds , P G Bergmann and V De Sabbata, Nato Advanced Study Inst Ser B58 (Plenum, New York, 1980) pp 177-226 See in particular p 192 Y Ne'emanandDj Sijacki, Phys Lett B 157 (1985) 275 FW HehlandGD Kerhck, Gen Relat Grav 9(1978)691, FW Hehl.EA Lord and L L Smalley, Gen Relat Grav 13(1981)1037
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CURVED SPACE-TIME AND SUPERSYMMETRY TREATMENTS FOR/»-EXTENDONS * Yuval NE'EMAN '-2 Sackler Faculty ofExact Sciences, Tel Aviv University. 69978 Tel Aviv, Israel
and Djordje SlJACKI Institute of Physics, PO Box 57, 11001 Belgrade, Yugoslavia Received 22 December 1987
The possibilities of embedding p-extendons (strings, membranes, etc.) in a generic or constrained curved space-time with supersymmetry are studied for a quantum test object in an external background and for a superextendon field theory. The generic case is detailed, using infinite-component world-spinors.
1. Curved space-time and supersymmetry The structure of curved superspace is still very illdefined. Three difficulties have caused this state of affairs: (A) the need to explore supermanifolds as a topological novelty [ 1 ]; (B) Minkowski space-time supersymmetry's s,vISO(l,£>— 1) non-inclusion [2] as a subgroup in the tangent isotropy-isometry IOSp(l,£>— l/r-N) group of the "natural" superpseudo-euclidean R 1D_l/rA 'superspace of Nath and Arnowitt [3] (r is the spinordimensionality, TVis the "internal" index, iV< 8 for D=4); (C) the fact that curved ("world") spinors are relatively unfamiliar infinite-component objects [4]. Issue (A) appears to have been given satisfactory answers. Issue (B) was resolved by defining the group elements [5] of swISO(l,D— 1) and identifying the quotient superspace M 1 °- 1/ '' A '=s A .ISO(l,Z>-1)/ SO(\,D- 1)xSO(/V) constructed explicitly in refs. [5,6] and implicitly in ref. [7]. With this definition
' Supported in part by the US DOE Grant DE-FG05-85ER40200 and by RZNS Belgrade. Wolfson Chair Extraordinary in Theoretical Physics. Also on leave from the University of Texas, Austin, TX 78712, USA.
458
of the "quasi-flat" superspace - in fact already curved [2] in the riemannian framework [3] - there are three ways to allow further "curving" in the spinorial sector as well as in the vectorial, considering that the covering groups of both the diffeomorphisms and the linear, in the reduction Dlff(D,R) GL(AR) SOU, D— 1), exist only [4] as groups of infinite matrices: (a) Restricted curving, i.e. staying with finite tangent space SO( \,D— 1) spinors, but restricting further curving of M ' ° ~ " ' " t o such as can be described by that "diagonal" subgroup of Diff(M ID - " r N) that preserves the orbits of SO( 1 ,D— 1) when acting simultaneously on both even and odd sectors of superspace. In other words, allow no linear tranformations other than SO( \,D— 1) and adjoin a restricted set of non-linear ones leading to manifolds carrying the action of SO( l,D— 1). This method inherited from supergravity, has been used extensively in the attempts to curve the "target space" in superstrings [ 8,9 ] and in supermembranes [10]. It allows the highly restricted rheonomic curving undergone by superspace in supergravity [11] in which the group parameters are constrained so that the odd coordinates are not gauged over. The supertranslations act anholonomically as Lie derivatives ("anholonomized" general coordinate transformations of ref. [5]), i.e. as part of the curved-space modified structure group [12]
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acting as an effective fibre in the appropriate principle bundle, with the bundle's vertical projection projecting only onto space-time 11(g) <-xeRLD~', rather than onto M lz) -' / '- Jv . (b) Generic curving, i.e. using infinite Diff(D,K). We have initiated this treatment for the superstring [13] and found a possibility [14] of preserving supersymmetry in the curved superspace (we studied in detail the D = 10 case). This is the method we shall use here mostly. (c) Non-linear curving: it might be possible to use finite SO(l,D— 1) spinors and represent the quotient Dlff( AR)/SO( 1,-D- 1) non-linearly over the SO( 1 ,D— 1) subgroup, following the pioneering work of Ogievetski et al. [15]. this method has not yet been investigated for curved superspace.
[18], and for/>=2 see ref. [19]). Owing to the fact that the action (1) is manifestly curved-target-space covariant, the algebra of constraints should coincide with the flat-target-space algebra (when evaluated in the particular gauge g,m = >1mm ^*,^ 2 ...A P + I =0). Indeed, the recent result forp= 1 reproduces the classical Virasoro algebra for generic curving [ 20,21 ], i.e. {L±(a),L±(a')}PB = ±[L±(o)+L±(<j')]dad((T-cT-)
,
(2)
where *-'± \Q\ gmnt A,f,a )
s\(P%± a„X>glV + WA», X(Pn±^X^
)g'""
+ daXUfl,) = 0,
(3)
m
while P'f, =8/8(6TA' ) are the canonical momenta. In the case of (flat) Minkowski space bosonic string light-cone formulation, one expands Xm, Pm, and the Virasoro operators Lk, k= 0, ± 1, ± 2,..., in terms of the SO(D-2) raising (k<0) and lowering (k>0) oscillator operators ask,s=l,2,...,D — 2. One can achieve a manifestly covariant formulation by introducing the ghosts and antighosts and making use of BRST invariance. In this case one has the "special covariantized" oscillators a™ of the SO( 1,-D— 1) Lorentz group. For a bosonic string embedded in a generic curved space we find it more natural to make an expansion in terms of the GL(JD,IR) oscillators A'k",k=0,±l, ±2,..., which satisfy the "general-covariant" commutation relations
2. Super p-extendon generic curved-space mechanics *1 Recently, Bergshoeff et al. [10] have constructed an action for a p-extendon. The bosonic sector of this theory can be generalized for a generic curved target space to read
5= J" d"+'f(i J^y y'^)d.X%X"gmn (X)
+
h
(pliyt''
X 0|,,+ i X
-'"+'dhX'"'d,2X^...
"* '^m,,ii2.../ii / ,+ i (-* ) I •
[A'?,A?]=kdk+,,0g**. ( 1 )
Here, i=0,1, ...,p labels the coordinates £'(T, a,p,...) of the extendon world-volume with metric yu, and 7=det(y,y); m = 0,1, ...,D— 1 labels the target-space coordinates x* with metric g^, and Amirfl2...„-,„+, is a (p+l)-form characterizing a Wess-Zumino-like term. This action generalizes the action proposed by Lovelace [17] and by Fradkin and Tseytlin [8]. There are no a priori constraints on gA/i, and the main difficulties are due to non-linearities (for p— 1 see ref. *' We prefer the term p-extendon for the generalized quantum extended body, to "p-brane" [16] and hope it is not too late for the new term to be adopted.
(4)
When using A'k, instead ofa'k, one finds, starting with the second level, a different organization of states: the trace part corresponding to the symmetric Z)-dimensional Young tableau of SL(Z),IR) is henceforth not extracted, whereas it would not have been included in SO(l,£>—1). The higher the level of SL(D,R), the more differences will come in. For a curved-space formulation of extendon mechanics, one can apply either generic or constrained curving. Generic curving [13,14] is based on the anholonomic application of the Diff(Z),R) or superDiff(D,R) and their infinite-dimensional (world) spinorial representations. A smooth transition to a flat-space (or quasi-flat-superspace) p-ex459
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tendon formulation can be achieved provided the relations between p, D and N are as determined in ref. [16]. The smallest superalgebra which allows a generic curved-space embedding for a p-extendon, through a non-linear realization of the superDiff(Z),R) group, is a contraction of the algebra of the GQ(AR) group [14]. The p-extendon super coordinates are now (Xa,^FA), where the jQ and A indices correspond respectively to infinite-dimensional bosonic (tensorial) andfermionic (spinorial) Diff(Z),R) non-linear representations which are linear with respect to the GL(DJR) subgroup. We note that in the case of infinite-dimensional GQ(D,R) representations the spin-statistics correlation theorem holds. The relevant (oo-dimensional) GQ(D,R) =GL(AR) representation theory is yet to be developed, and we list here some relevant GL(AR) representations with the first few SO(l,D— 1) representations appearing in their reduction. For the p= 1, D= 10 superstring the "vectorial" and the two spinorial chiral representations are [13]
Here, the target space is a supermanifold with superspace coordinates Z £ = (A"*, 9&), where m = 0,1,..., D-\ and a=l,2,...,2 ( °- 1 > / 2 . Furthermore, Ef = {diZL)E'l(Z), where E'l is the supervielbein and L=(m,a) is the tangent-space index, w = 0,l,...,D-l and a=l,2,...,2<°- | ) / 2 . In the standard superspace formalism one tends to describe G& as a "world" fermionic coordinate, but this time in a very restricted sense only. The superspace (p+1 )form B=
(5)
For the p= 2, D= 11 supermembrane, the "vectorial" and spinorial representations are [ 16 ] D(ll) 3 {11,275,2717,16445,72930,...} , (6)
Restricted curving, the alternative approach to the superextendon curved-space mechanics for the p= 1 case, was proposed by Witten [9], and was further developed in the cases ofN=2 supergravity [22] and super-Yang-Mills [23] background fields. This target-space supergravity-like superspace formulation has been generalized to the p=2 supermembrane (as well as to the general p) case [10], and consequently further investigated (see ref. [19] and references therein). The p-superextendon action reads
- i -1 )JZr7+ Jj^yji x£f/ + V^ + ,...L 2i ,J.
£
"' 2 "' +,
<8>
sigma-model 0=0 [24] or GBRST = 0 [25], and the
D( 10) =5(16,144,720,2640,7920,...},
D(32) =, {32,320,1760,7040,22880,...}.
JpTYy.EL'EL2-El"+'B^'-^
is the potential for the (p+2)-form H—dB. These extendons do not influence the appropriate ©-dimensional supergravity background (see refs. [9,22] for p= 1), evolving through an a priori defined "external" supergravity, provided H is a closed form [9-11 ]. In that respect this "curved-space" superextendon formulation is in the same class as the
D(10) =.{10,210,1782,9438,37180,...}, D( 16 ) = {16,144,720,2640,7920,...}.
26 May 1988
E E
^ n(?)
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"gravity from strings" [26] approaches. Calculations for a superspace superextendon are done on the tangent (anholonomic) quasi-flat coordinates and in fact, do not involve curved-space spinors. In supergravity or in the gauge Poincare or superPoincard space-time formulations we have fibre bundles with only curved space-time as base manifold and a flat Lorentz fibre enlarged by anholonomic supertranslations as Lie derivatives, without involving curved spinor coordinates altogether. In the case of curved superspace superextendon mechanics, however, it is the (p+l)-dimensional extendon world-space which plays the role of base space, while the {Xm, Ba) coordinates label "points" of the flat or quasi-flat fibre (locally tangent to the base). There is then no room for additional "tangents-to-the-tangent" in terms of the dynamical variables £'. The ZM= (Xm,Ga) coordinates thus do not parametrize locally a generic [superDiff(Z),R) determined ] curved superspace, a fact which severely restricts various superspace theories [9,10,19,22,23, 27,28 ]. A spinorial interpretation of G& for a generic curved space-time subspace is erroneous in such treatments. The physically interesting allowed cases are as follows: (a) The target space structure group is
Volume 206, number 3
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PHYSICS LETTERS B
s,vISO( 1,D — 1); the test superextendon "lives" in a quasi-flat superspace determined by the supergravity tangent space. (b) The target space structure group is sISO(l, Pi-l)XOSp(P 2 /(22)X...xOSp(P3/63) with superDiff(AR) realized non-linearly over it; the superextendon is embedded in a definite non-trivial supergravity solution and does not distort it by its presence. (c) The target space structure group is superDiff(AR), realized either linearly or non-linearly, over its GQ(D,R)con supersubgroup; the superextendon is embedded in a generic curved superspace and we recover our previous generic formulation. It is interesting to point out that for the total Grassmann manifold making up generic curved superspace R l -°- | / r N we find, as r corresponds to a countable infinity,
formulation [13,14] was emphasised in thefieldtheory context as well [ 32 ]. An interesting possibility is to remove the explicit background dependence by shifting the field and using a purely cubic action [33,34], This allows us to concentrate on the superextendon superfield and to impose superDiff(Dp.) covariance as a requirement. For a bosonic extendon field theory, curved-space covariance creates no difficulties. In the string field theory case, one expands the string field &(X(o)) in terms of the oscillators A'l'. Such an expansion does not seem to be the most suitable for a general curved space-time, i.e. for generic non-perturbative solutions. It is natural to expect that the curved-space extendon constraints satisfy the same algebras as in the flat case [eq. (2) forp= 1 ], and to expand the extendon field in terms of operators
dim(R G r ; l s J=2 N °= C ,
At,
(9)
fc,- = 0, ±1,...,
i.e. it is of a continuum cardinality even if we do not include ordinary R UD~' in the even part.
,
i=l,2,...,p,
(11)
which generalize those from (4), with appropriate changes for the ghost and antighost fields. Thus, the extendon field reads
3. Super/>-extendon generic curved-space field theory
Superextendon field theory is suitable for the study of non-pertubative effects, and hopefully for the incorporation of covariance and supersymmetry. The dynamical equations occur in the target space, though simultaneously constrained by the mechanical evolution represented by the world-hypervolume of the extendon. This overcomes the difficulties of the previous geometrically ill-defined construction of a curved tangent frame for the target manifold. In the following discussion we assume a superextendon field theory action of the form S=^(0*QBRST®+I0*0*0),
K=(k\,k2,...,kp)
=( l
lA*
\*>0/=l
(12)
/
An immediate consequence is that all \D(D+\) components of the "graviton" component field in D dimensions are treated on the same footing. We now turn to the superextendon field theory. In a Green-Schwarz-like construction, superspace is now forced on us by the (XM(£), f^d)) substrate. We list the physically interesting possibilities admitted by the fermionic sector in that superspace. The target superspace superextendon curved variables (XM(£), f,A(0) are constrained by "internal" superextendon mechanics. Furthermore, contrary to the case of the mechanics of a test extendon, in the field theory one can define flat tangent superspace frames, etc., to the target superspace. The generic curved target superspace superextendon field theory is defined by the anholonomic application of superDiff(D,R) infinite-dimensional linear or non-linear representations over the GQ(AR )<:<>„
(10)
whatever the precise definitions of / and * for the open and closed superextendon configurations; for p= 1 there are two formulations [29,30]. The background fields are contained in the BRST operator [31] which enters the quadratic part of the action. Ghosts are inessential for the present discussion. A need for a background-independent superextendon
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subgroup [14]. The tensorial and spinorial lowest component fields are given for p= 1 and p= 2 by (5) and (6), respectively. The basic input for actual computations is in this case the spectrum due to superextendon "internal" mechanics. Another option apparently available would consist in treating the curved target superspace superDiff (D, R) kinematically only, and developing an appropriate group manifold theory [6] with the s^ISOU, D— 1) structure group. However, a preliminary analysis indicates that in this case one is bound to make use of superextendon fields which have explicit external space-time indices, i.e. &A{X({o}), ¥({(T})), 0 [AB\X({a}), P({<x})), for which a correct physical interpretation would be lacking. Finally, there is the possibility of first developing a NSR-like spinning-extendon mechanics, as a foundation for a superextendon field theory. Aside from the difficulties due to the curving of the extendon world-volume for p= 1 this would also yield a restricted curved-superspace formulation, with manifest space-time supersymmetry algebra appearing onshell only.
References [ 1 ] FA. Berezin and D.A. Lcites, Dokl. Akad. Nauk. SSSR 224 (1975)505; B. Kostant, Lecture Notes in Mathematics, Vol. 570 (Springer, Berlin, 1977) p. 177; M. Batchelor, Lecture Notes in Physics, Vol. 94 (Springer, Berlin, 1979) p. 458; A. Rogers, J. Math. Phys. 21 (1980) 1352; 22 (1981) 443, 939; B. DeWitt, Supermanifolds (Cambridge U.P., Cambridge, 1984). [2] Y. Ne'eman, Trans. N.Y. Acad. Sci. 38 (1977) 106, and references therein. [3] P. Nath and R. Arnowitt, Phys. Lett. B 56 (1975) 177; B 65(1976)73. [4] Y. Ne'eman, Proc. Natl. Acad. Sci. USA 74 (1977) 4157; Ann. Inst. H. Poincare A 28 (1978) 369; Y. Ne'eman and Dj. Sijacki, Intern. J. Mod. Phys. A 2 (1987) 1655. [5] Y. Ne'eman and T. Regge, Riv. Nuovo Cimento 1 N5, Ser. 3(1978).
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Y. Ne'eman and T. Regge, Phys. Lett. B 74 (1978) 54. J. Wess and B. Zumino, Phys. Lett. B 74 (1978) 51. E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 158 (1985) 316. E. Witten, Nucl. Phys. B 266 (1986) 245. E. Bergshoeff, E. Sezgin and P.K. Townsend, Phys. Lett. B 189(1987) 75. A. D'Adda, R. D'Auria, P. Frd and T. Regge, Riv. Nuovo Cimento 3 N6, Ser. 3 (1980). FW. Hehl, P. v.d. Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48 (1976) 393. Y. Ne'eman and Dj. Sijaeki, Phys. Lett. B 174 (1986) 165. Y. Ne'eman and Dj. Sijacki, Phys. Lett. B 174 (1986) 171. V.O. Ogievetskii and I.V. Polubarinov, JETP 48 (1965) 1625; C.J. Isham, A. Salam and J. Strathdee, Ann. Phys (NY) 62 (1971)98. A. Achucarro, J.M. Evans, P.K. Townsend and D.L. Wiltshire, Phys. Lett. B 198 (1987) 441. C. Lovelace, Phys. Lett. B 135 (1984) 75. H.J. De Vega and N. Sanchez, Phys. Lett. B 197 (1987) 320. M.J. Duff, Class. Quantum Grav. 5 (1988) 189. J. Maharana and G. Veneziano, Nucl. Phys. B 283 (1987) 126. R. Akhoury and Y. Okada, Phys. Rev. D 35 (1987) 1917. M.T. Grisaru, P. Howe, L. Mezincescu, B.E.W. Nilsson and P.K. Townsend, Phys. Lett. B 162 (1985) 116. J.J. Atick, A. Dhar and B. Ratra, Phys. Lett. B 169 (1986) 54. C.G. Callan, D. Friedan, E.J. Martinec and M.J. Perry, Nucl. Phys. B 262 (1985) 593; C.G. Callan, I.R. Klebanov and M.J. Perry, Nucl. Phys. B 278(1986) 78. T. Banks, D. Nemeschansky and A. Sen, Nucl. Phys. B 277 (1986)67. S. Deser and A.N. Redlich, Phys. Lett. B 176 (1986) 350. B.FW. Nilsson, Nucl. Phys. B 188 (1981) 176. R.E. Kallosh, in: Second Nobel Symp., Phys. Ser. TI5 (1987)118. E. Witten, Nucl. Phys. B 176 (1986) 291. H. Hata, K. Itoh, T. Kugo, H. Kunimoto and K. Ogawa, Phys.Rev.D35 (1987) 1318, 1356. M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. E. Witten, in: Second Nobel Symp., Phys. Ser. Tl 5 (1987) 70. H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B 175 (1986) 138. G. Horowitz, J. Lykken, R. Rohm and A. Strominger, Phys. Rev. Lett. 57(1987)383.
IL NUOVO CIMENTO
VOL. 102 A, N. 5
Novembre 1989
Classical Lagrangian and Hamiltonian Formalisms for Elementary Extendons (*). E . EIZENBERG and Y. N E ' E M A N ( * * ) ( * * * )
Sackler Faculty of Exact Sciences, Tel Aviv University - Tel Aviv, Israel (ricevuto il 30 Marzo 1988)
Summary. — We study the description of d-dimensional extended structures in D-dimensions when no field is introduced other than the extendon X„(0 and its metric gah(£). PACS 12.90 - Miscellaneous theoretical ideas and models.
1. - Introduction. In recent years there has been a rise of interest in theories of extended objects (extendons) (18). The properties of the simplest of them—strings—are well established (9). It is thus natural to generalize the results to higherdimensional extended objects, in particular to membranes. (*) Supported in part by US DOE Grant No. DE-FGO5-85ER40200. (**) Wolfson Chair Extraordinary in Theoretical Physics. (***) Also on leave from Center for Particle Theory, University of Texas, Austin, TX 78712, USA. O P. A. COLLINS and R. W. TUCKER: Nucl. Phys. B, 112, 150 (1975). 0 P. S. HOWE and R. W. TUCKER: / . Phys. A, 10, L 155 (1976). (3) M. J. LOWE and D. J. WALLACE: Phys. Lett. B, 93, 433 (1979); F . DAVID: Phys.
Lett. B, 102, 193 (1980). (4) J. HUGHES, J. Liu and J. POLCHINSKI: Phys. Lett. B, 180, 370 (1985). (5) E. BERGSHOEFF, E. SEZGIN and P. K. TOWNSEND: Phys. Lett. B, 189, 75 (1986). (6) E. BERGSHOEFF, E. SEZGIN and Y. TANII: preprint IC/87/107. O M. J. D U F F , P. S. H O W E , T. INAMI and K. S. S T E L L E : Phys. Lett. B, 191, 70 (1986);
M. J. D U F F : CERN preprint TH4797(86). (8) Y. N E ' E M A N and D J . SIJACKI: Phys. Lett. B, 206, 458 (1988). (9) There are several thorough reviews on the theory of strings, e.g., J. SCHERK: Rev. Mod. Phys. 47, 123 (1975); Dual Theory, edited by M. JACOB (North Holland, Amsterdam, 1974); Superstrings (The First 15 Years of Superstring Theory), 2 vols., J. H. SCHWARTZ (World Scientific, Singapore, 1985). 76 - // Nuovo Cimento A.
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E. EIZENBERG and Y. NE'EMAN
In 1976, Collins and Tucker f) studied the classical and quantum mechanics of free relativistic membranes based on the Nambu Lagrangian. A year later Howe and Tucker (2) constructed a locally supersymmetric and reparametrizationinvariant action for a spinning membrane. In( 3 ) perturbative renormalization properties of membranes were investigated. These models were then used to study a surface which represents the interface between two states. More recently, Hughes, Liu and Polchinski (4) have constructed a Green-Schwarz-type action for a three-extendon propagating in flat six-dimensional space-time. Finally, Bergshoeff, Sezgin and TownsendC) proposed an action for a supermembrane coupled to D = 11 supergravity. The Hamiltonian formulation of the supermembrane was given by Bergshoeff, Sezgin and Tanii( 6 ). Starting from the action given in ref. (5), higher extendons were also constructed. Some types of superstrings were then derived by correlated dimensional reduction of the world volume and the space-time (7). In the theories developed in ref. ("), the existence of a super three-form related to supergravity is assumed. It represents a geometric feature existing in D > 4. A question then arises, whether an «elementary» extendon theory can be constructed, i.e. without adding any external field and using only the extendon field Xffl itself or at most both the field XJ£) and the world-volume metric gab(Q. In pursuing this idea, we start (sect. 2, 3) with the description of extendons using only the extendon field Xffl. This includes an investigation of a Nambu-type action in both Lagrangian and Hamiltonian formalisms. Adding the metric gab(0 we then try, in sect. 4, to generalize the Polyakov formalism to the extendon case.
2. - Nambu-type actions. Lagrangian formalism. Our task is to investigate a (d - l)-dimensional vibrating object (extendon) in .D-dimensional space-time. It is completely described by a c£-dimensional hypersurface X^a), p = 0,..., D - 1 and a = 0,..., d - 1. £° are internal variables of the object: £° = T its proper time, and a set (£\ f2,..., ?d_1) defines a point of the vibrating object. We assume a Minkowski metric r]1" for the /^-dimensional space-time. The natural requirement from the equations of motion of the object is their invariance with respect to the reparametrization of the hypersurface X„(f), i-e. to an arbitrary coordinate transformation on the hypersurface: £"—>f'a. Note that the «classieal» equations of a vibrating object 32X„(f)
*-i 92XU(£)
do not satisfy this condition. Indeed, reparametrization Ea-^%'a
647
transforms
CLASSICAL LAGRANGIAN AND HAMILTONIAN FORMALISMS ETC.
1185
eq. (1) into dEfd?
32?*
—5—5- dfdbx^ +—— dbxm = o,
9?' a 3£ 9£9r° which is not equivalent to (1). To generalize eq. (1) we need, in analogy to general relativity, to replace the derivative da by the co variant derivative Da. With this purpose in mind let us investigate the metric #S of the hypersurface XJ& induced by a metric YJ of a Ddimensional Minkowski space, that leaves the interval invariant: ds2 = n"' dZ,dX v = v^daX^bX^ade
= (v^daX,dbXv)
d£°d?6 = gS, #°
The Christoffel symbol is given by (2)
n % = I(deSfo + dbgfc - dagt) = ^daXtf)
3bdcXX0.
The covariant derivative for the scalar X^O is defined as (3a)
DaXtf) = daXtf),
while the covariant derivative for any vector is given by (36)
DaVb =
daVb-r^Vc,
where rcab = gliri%, and the inverse matrix to the metric gin. Replacing the derivatives da in eq. (1) by the covariant derivatives Da we may represent the equations of motion in the covariant form (4a)
DaDaX,(0 = 0
or
(46)
&aadbxtf) - gtnb dex,(S) = — - — da[Wetgin\dax,(01 = o. Vldet^r)
For the case of a string (d = 2) eqs. (4) follow from either the Nambu or the Polyakov Lagrangians. In passing, we note that only D-d equations among the D equations (4) are independent. Indeed dhX»{Z)DaDaXtf) = diX^gZidadcXM)
- g?nriacdfX,m
=
— y i » U b,ac
648
gbfyin1
h,ae) — « •
1186
E. EIZENBERG a n d Y. NE'EMAN
This is the Weierstrass condition for parametrization invariance of t h e Lagrangian which has eq. (4) as its associated Euler equations. Let us determine the class of Lagrangians that depend on the variables of the hypersurface only (i.e. Xffl and their derivatives) and leading to the equations of motion (4). The action S is the scalar integral
(5)
S= JdrJride B = jGdfl.
Here di2 is the infinitesimal volume
(6)
Q = Vldet (9.^(0S 6 X"(0)| I I d«°, a=0
and G is a scalar function of the field variables XJ£) and their derivatives (remembering that gtj, = daX^) dbX>i(0). To get the equations of motion, which are the system of second-order partial differential equations, we have to choose the integrand G that contains X^Q and their first derivatives only. The only scalar in Minkowski space is the scalar product (*), hence G must be composed of v^W1, where v and w are products of X^) or its derivatives. Since, however, they transform according to the rule
(7)
Xtf) = X^Z')}=XK®',
3X„(0 d?
(XW)\(w \ di'h ) \ dta
it is impossible to combine them into a nontrivial invariant with respect to an arbitrary reparametrization £ =
L=
^{-lf-l\daXJ^dbX<^\,
where a is introduced to settle units. There is one more possibility for constructing L, namely, one can represent it
(*) According to a theorem proved by Weyl (10) every invariant depending on (d + 1) vectors X and daX in a /^-dimensional vector space is expressible in terms of the scalar products daXlldbX",XtllXi' and X^X". (10) H. WEYL: The Classical Groups: Their Invariants and Representations (Princeton University Press, Princeton, N.J., 1946).
649
CLASSICAL LAGRANGIAN AND HAMILTONIAN FORMALISMS ETC.
1187
as the sum of two terms (9)
LY[d^
= CdQ + dbAbYidSa = GdQ, 0=0
where Aa is some function and C is a nonscalar function of 3aZfl(£)c>&.X>(£). The simplest choice for the scalar G is the curvature R:
(10)
R = gt,[d,rlik - dkrla+r\krz - rwU =
= afcz»a^,[(aia;Z")oiafczv) - o^x-oo^x)] 1
V(-l)
d_1
dt tVC-D'-'detflr"3*^(0 dkdkX»(0] -
detfir"
V(-l) d_1 det^'
3, [Vt-DMetflr"9*^(0 S*9 i Z"(0].
The first term on the r.h.s. of (10) includes the second derivatives in a nonlinear form, so we cannot gauge them away. We are thus left with eq. (8) as the only possible Lagrangian. For the relativistic string (d = 2) L becomes the Nambu Lagrangian. An infinitesimal variation of the path traced by the system during its evolution gives the following equations of motion:
subject to the edge conditions (12)
SL
=0
(o # 0),
(since we vary F keeping the initial and final positions of the system fixed: 3Zfl(^° = T0,T1) = 0, %XJ£aM = Q,ic) is arbitrary). Introduce the following quantities: (13)
^ 3 4 1 '
Then the equations of motion become (Ha)
3aP« = 0
650
1188
E. EIZENBERG a n d Y. NE'EMAN
and P%£a = 0,7i;a^0)
(12a)
= 0.
It emerges from definition (13) and eq. (9) that (14)
Pl =
1 SL2 2L 8daX"(0
L S(detgin) 2det
2(detgm)
Ab"(S%dcX^
Sgfc SdaX^)
+ SacdbX,(0) =LcfbdbXM)
=
LdaXtf).
Here Aah = (det<7m)<7fB6 is a cofactor of g™b. Substituting this expression into eq. (11a) yields the equations of motion (4c)
3 o (L5«Z,(a) = 0 .
Classically it is always possible to choose a coordinate system so that at any given point of the world hypersurface, rcab = 0 (locally Lorentz orthonormal frame) and g^ is Minkowskian. In this case eq. (4c) reduces to (1). To find the energy momentum current one performs the variation of S in a different way: the initial, but not the final position of the system is kept fixed, but only actual motions of the system are allowed. We have d-l
(15) SS = 2 j 6=0
r I
SL S3bXSO
d-l
8Z„(Odr =
II d?°
\o=0,a#6
= £/( ff dA[P^SX,(0 b=0J
d-l
(16) 8A'(?)
r /
d-l
\a=0,a*b
\
El n q r a 6=0"' \ 6=0, 6#a
Carrying out the integration on a surface (3 C) which encloses some region (C) on the hypersurface, results in d-l
(i7)
r I
d-l
-. = s f n nPt= [nndaPz '-'Use, \»=».»«
/
(O \ s - °
/
as a corollary of the equations of motion (11a). It follows then from eqs. (16), (17)
651
1189
CLASSICAL LAGRANGIAN AND HAMILTONIAN FORMALISMS ETC.
and (12a) that
if the integration in (16) is performed on any surface between the initial and final positions of the system. We identify, now, ^ with the total momentum of the vibrating object and conclude that this quantity is conserved
(is) a,,*, = J f n n doPi=- s (n n d*n= = 1' f ( R zb) W 1=1
= o) - P%?=*)] = o.
\ 6 = 1, b4=a
n
3. - Nambu-type action. Hamiltonian formalism. In the normal case when eq. (13) can be solved so that the derivatives doX^O may be expressed in terms of P° the Hamiltonian is defined as the Legendre transform of L: (19)
#0CP°) = 3oX«(0PI - (a„X>«)),
where daXJ& = 30ZM(P?). The necessary and sufficient condition for the inversion of eq. (13) is
(20)
det
S2L 8d0X-(Q8d0XXZ)
#0.
Let us show that for the Lagrangian (8) the determinant of the second derivatives of L is equal to zero and therefore the Legendre transform of L does not exist. It follows from (14) that
(2i) — £ — [(- ly-WLP*]=TT^ 6d0X (£)
Kdetr) &xjm =
od0X\i)
det a™
SP°
L
Sd0X"(E)
= (- lY-WPtl* + —
652
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E. EIZENBERG a n d Y. NE'EMAN
Thus
(22)
m^F^f
iK)
L
-detgm
8d0X"®
SdoX*($
L
*L
Sd0X^OSd0XXO
[Aa0 3a Xtf)] - (det g*) 3° X£Q 3° Xtf) =
= VA°° = 2
AM-0bdaX^O3bXXO-(detg^)d0X,(Od0XXO:
0,6=1
a0 oi>
here A ' (23)
is the cofactor of gfh in ( - l)aAa0(a, b ¥= 0). A00'06 #c6 =
Aa0-0b = gabAfM -
g0bAa0.
Finally, we obtain (24)
F „ = fl-00^, - 3aX,(0 a a X v (a) = F „ .
It is easy to see that K e r F ^ O . Indeed (25)
F^XXQ
= gm(3aX,(0 - gab &X0)
= 0.
Therefore (26)
det(F„ v L) =- dct 6
*L = 0. 8d0XWS3oX"(Q
The impossibility of solving eq. (13) for 3°Xll(£) means that the variables P° and XJX) cannot be considered as independent and the motion is restricted to a surface(*) in the phase space P% XJig) of dimension D s =S 2D - dim (kerF) = 2D — d. ^-constraints determining the surface can easily be
(*) In order to prove the last equality it is convenient to choose a particular coordinate system
daX^o = s/fSl~sjta-s°a), daXJ£) is a D-vector whose nonzero elements equal lif,u = a = 0or — 1 if ft = a ^ 0. Then daXffl is a Z>-vector with nonzero element equal 1 for p. = a. The metric of the hypersurface is determined by g1^, and -1 F^=
if ft = v > d - 1,
\ 0
otherwise.
653
CLASSICAL LAGRANGIAN AND HAMILTONIAN FORMALISMS ETC.
1191
found. Indeed P° P°" = L2 3° Xa(0 3° Xm
(27a)
=
(v - I)"" 1 00 % A a?
and (276)
PldaX^)
= L&XJ& daXHQ = Lgobgba = L%.
Therefore (28)
(a#0)a> 0 = P»Pi5 + ^ 1 r - A 0 0 - 0 ,
(The sign ~ means «weakly equal» according to Dirac, that is the equality sign holds on the 2(D - d) hypersurface of the full 2Z)-dimensional phase space.) Equations (28) are independent in the sense that the rank of the matrix (3
(29)
where the c'° are arbitrary functions of P° and da^X^O-
H0~L(d°Xtf)d0X>(O-l)
It follows from (19) that
= 0,
and hence HQ can be expressed as a linear combination of the constraints 4>a. Inserting this result in (29) we obtain H = ca0a.
(29a)
(*) The matrix $4>J8 Tbpl consists of d rows labelled by the index a and Dx d columns labelled by the pair of indices (a, p). For any specific /* the determinant of the d x d minor is 4" = 2(P°")d-2[(P0")2 - i l i L
A 0 0 '*3 1 -Z''(0 3kX"(0]
= 2(P°nd-l(A')"".
Since
(A'rG^(A%=pip^-
( - l)d a*
at least one of zFH=0.
654
( - l ) * - 1 00 A 4=0 ar
1192
E. EIZENBERG a n d Y. NE'EMAN
For any phase-space function / the change in time is described by the equation (30)
30f~{f,H},
where the Poisson bracket is defined as follows: rf-i
(31)
¥ Sh SX"(0 8P°(0
{f,h}
«/ Sh 8P»(Q 8X"(Z)
From this definition there emerges (32a) (326)
{Z"(Q,Pj?(5)}«.=fo = isst-KS - $'), R*>*"«), Px°tf)}^.
= */• 9^-.*"-i(§ - §') •
Forming the Poisson bracket with two ^-functions we obtain (33a)
{*a(0, # 6 «')}^ r » = [s6(0 a^ - *„(£') 3r.] 8d-\Z, - §')
for a, 6 # 0
(336)
{*0({), *„«')}*>=?* = [*„«') 9? - *>o«) 3f] ^ - 1 (« - I')
for a =£ 0
and (33c)
{%(0,
+2
(-1)°
A 00,a 6 ( ? ) [ f p o ( 0 9 f t +
0b{O
gr]
Sd-1^
_£,)
We conclude that i) the constraints (28) are obeyed for all time and there is thus no additional (secondary) constraint, and ii) all constraints are of the first class (the rank of the matrix {<Pa,®b} is zero).
4. - Polyakov formalism. In a more general treatment, Polyakov investigated a Lagrangian which includes two independent fields, X^O and the metric gab (independent of the gTb) on the world-sheet of the string. The Polyakov Lagrangian is (34)
LPol = ±
^-detggabdaXtf)
dbX^),
(o, 6 = 0,1).
The minimun of the action for this Lagrangian is attained for the induced metric: 9ab = p(?, v)g'ai,, where p(r, a) is an arbitrary scalar function.
655
CLASSICAL LAGRANGIAN AND HAMILTONIAN FORMALISMS ETC.
1193
A simple generalization of the Polyakov Lagrangian to vibrating objects of other (d=£2) dimensionalities is impossible. This becomes obvious when considering the scale transformation of the metric (35)
gab->g'ab = *gab,
which yields the transformation of the Lagrangian (36)
L Pol ^L'=L Pol A w - 2)/2 .
This would imply that the metric which minimized the action is degenerate 9ab = 0, in the case d>2. Another way of showing the impossibility of generalizing (34) to d ¥= 2 is the following. Multiplying the equation of motion for gab (37)
daX<£) dhXHg) - ±gabg°fdeX^)
dfX"W = 0
by gab and taking the trace one gets (38)
gabdaXtf)dbX^)(l-fj
= 0.
The only solution of (37) and (38) is (39)
daXJ& dbX^(0 = 0
for d * 2
(in the case of the string, eq. (38) reduces to an identity, thereby allowing a nontrivial metric). Consider two modifications of the Polyakov Lagrangian which allow generalization to d¥=2. Adding to the r.h.s. of (34) an additional term (40)
Lh = Lm + ^ a ( V - detg - V ~ det/i)
is equivalent to solving a variational problem under the constraint on the metric gab: (41)
V-det# = V - d e U ,
656
E. EIZENBERG and Y. NE'EMAN
1194
where h is an arbitrary metric hab{XJ<S)', T, a-), a is a Lagrange multiplier. The minimum of the action is reached on the surfaces (42)
daXlt(OdbX"® = g,ab
lid
det(deXJOdfX^)) deth
and thus (43a)
U = ^ ( - dethf'2-M [- det (S0Z„«) dbX»(0)Y/d, Lh = —
(436)
\/-deth
det (daXtf)
dbX"(0)
lid
deth
For the string (d = 2) eq. (43a) coincides with the Nambu Lagrangian. The choice h = gm in eq. (436) leads to the generalization of the Nambu Lagrangian to the case (*) d =A 2. To get another modification, let us examine the reason for the appearance of the factor AW_2)/2 on the r.h.s. of (36) (and the subsequent impossibility of generalizing the Polyakov Lagrangian). It is caused by the fact that the inverse metric gab enters eq. (34) in the first power while gab is raised to power d/2. This suggests an idea of introducing a Lagrangian in which gad and gab enter with the same power, e.g., (44)
Ld = j ^ -
2
dbX^O)dB
y-detgisfdMS)
(a,b =
0,l,...,d-l)
guaranteeing the invariance of the Lagrangian with respect to the scale transformation. The equation of motion for the metric gab (45)
daX0) dbX"W - ±gabge%X^)
dfXH& = 0.
Providing the minimum of the action leads to a generalization of the Nambu Lagrangian. Therefore, the equation of motion for the world-hypersurface XJig) takes the familiar form (4). It is easy to see that there are no constraints for the Lagrangian Lh, eq. (43). Indeed, S2Lh
= ^V-det<7sr°V*0,
sd0xmsd0xxo N
(*) One could get the same result assuming from the very beginning the Lagrangian L =LPol + ^jL[y/-detg-
^-det(aoX„(a36X"(9)].
657
1195
CLASSICAL LAGRANGIAN AND HAMILTONIAN FORMALISMS ETC.
for any nondegenerate metric g. The momentum of the vibrating object is
(46)
p, = ^ = ^v^^wa,
and the Hamiltonian for the system is Hh = ± V-det
(47)
-^=(V-detj7-V-detfe) =
d3X>(0) --jf-PAX>(Q
+
+ ± V - d e t ^ ( - ^ - A dtX(0 djX"® - ^ (V-detflr- V - d e t f c ) . The time evolution of any function f(P, X) can be evaluated through dof(P,X) = {f,H}. In the case of the Polyakov Lagrangian the momentum is also given by eq. (46). The Polyakov Hamiltonian is found by taking a = 0 in eq. (47). The absence of constraints in the case of the second generalization of the Polyakov Lagrangian is not so evident. The momentum P^ is gT
(48)
P
"
=
i
8d X^O
=
Nd^ V -
detgdVFrig-'g^r-^g^d.X^),
where gin = (3aXM(£) 3,,X'"(£)) is the induced metric introduced in sect. 1. We investigate the degeneration of the matrix F =
^ SdoX^SdoXAZ) in d w 0a 0b -d\~Tg[(Tr(g-'g )y - g g daX^)dbX''(O(d Ndd
+
- 2) + (Tr(g-ig«)r-^g00vJ
Denote Trig-'g™) = a. Then (49)
F,, =~[(d-
2) S % ( £ 3°XX0 + «g°%„].
658
1196
E. EIZENBERG a n d Y. NE'EMAN
Let u'' be a vector in KerFuv
dLa
F„,u" = (d - 2) dQXM) d°Xv(0 W + < W = 0 .
Then
^-^d0xxod°xxOu\
ua
*g
Multiplying the previous equation by 3X"(£) one gets 3 C X"(0 u, = - ^—g^Oc «g"'
daXtf) dcX'(dg°bdbX>(d
uv = d-2 <x#',00
•tfagZsfb(dbX"Wull).
Thus
^ + agf" ^(fVt/
36X"(0M„
= 0.
Using the equation of motion (45) for the metric g we get 3„X^)u^
~00 ~
c
#
or (50)
w uAdax^ + dg ~^s^d bx^)\
=o
Thus any vector in the KerF^ has to be orthogonal to every da.X>(£). Indeed, substituting a¥=0 into (50) we get (51)
M„3^0Z"(O = 0.
For a = 0 we get from eqs. (50) and (51).
uud0X'(d[i
+ ^—^) = o.
And thus uAX<(0
659
= 0.
CLASSICAL LAGRANGIAN AND HAMILTONIAN FORMALISMS ETC.
1197
Hence using (49) w e g e t
a.
a
The L a g r a n g i a n Ld is t h e r e f o r e free of c o n s t r a i n t s . T h e expression for t h e Hamiltonian in this case is
(52)
N
Hd=
Pf-^P&XHg)-^-^.
V-detc/ 00
9
Note added in proof. Although reparametrization invariance ostensibly allows classically equivalent Lagrangians L = \detg\ll2f(gabdaXl,dbXli), our solution is nevertheless unique in its preservation of Weyl scale invariance and of the equations of motion (4a).
•
RIASSUNTO(*)
Si studia la descrizione di strutture estese d-dimensionali in .D-dimensioni quando non viene introdotto nessun campo se non 1'estendone Xffi e la sua metrica gab(0(*) Traduzione
a cura della
KjiaccmecKHe JlarpaiiJKCB npoTHiKeiiHbix Macnm.
Redazione.
H raivnijibTOHOB
{popviajimvibi
JJJIS
3.ieivieHTapiibix
Pe3H)Me (*). — M M HccnenyeM onHcamie d-Mepm>ix npoTaxeimbix CTpyKTyp B DH3MepeHiwx, Korjja He BBOHHTCS HHKaKHx nojieH, KpoMe nojia npoTHxeHHoft l a c r a u H X^) H MeTpHKH gab(0(*)
nepeeedeno peda.Ku,ueu.
660
ISRAEL JOURNAL OF MATHEMATICS, Vol. 72, Nos. 1-2, 1990
DUALITY, CROSSING AND M A C L A N E ' S COHERENCE BY
RAM BRUSTEIN,* YUVAL NE'EMANn AND SHLOMO STERNBERG* Tel Aviv University,** Ramat Aviv, Israel
ABSTRACT
It is shown that MacLane's rectangle, pentagon and hexagon identities in category theory, when applied in particle physics to duality diagrams or to rational conformal field theories in two dimensions, yield the necessary physical algebraic constraints.
§1. Introduction In a recent series of papers [1-5] on rational conformal field theories in two dimensions, certain constraints were introduced on matrices, the knowledge of which determines the theory. These constraints are associated with certain transformations of graphs. An examination of their form shows that they are similar to constraints introduced by MacLane [6] many years ago in category theory. In the present article we explain the nature of these constraints and their relation to MacLane's theory. This will then allow us to apply MacLane's results (in a recently extended form [ 16], [17]) to determine a generating set for these constraints. The content of this paper was obtained in the mathematical physics seminar at Tel Aviv University in June and the early part of July of 1988 and circulated in preprint form at that time. It was presented in a slightly revised form at the conference on Hopf algebras at the University of Beersheba in early January 1989. We begin with a brief introduction. In the diagrammatic description of particle interactions one may regard a diagram such as f
Present address: Theory Group, Department of Physics, University of Texas, Austin, Texas, USA. n Supported in part by DOE grant no. DE-FG05-85ER40200. % Also Department of Mathematics, Harvard University, Cambridge, Massachusetts, USA. t% Supported by US-Israel BSF grant 87-00009/1. Received June ] 8, 1989 and in revised form November 22, 1989
19
661
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R. BRUSTEIN, Y. NE'EMAN AND S. STERNBERG
Isr. J. Math.
(l.D 1
3
as indicating that particles 1 and 2 combine to form an intermediate particle which decays into particles 3 and 4. "Crossing" refers to the assertion that there is a relation between the interaction described by the above diagram and the interactions associated to the diagrams (1.2)
and
(1.3)
In its simplest form the assertion relates the analytic continuation of the amplitudes of these processes when expressed in terms of Lorentz invariant combinations of the four momenta of the interacting particles, the "Mandelstam variables" s, t, and w; cf. [7]. In the 1960's a systematic study of crossing in the context of "dual models" was begun [8]. With the advent of the Veneziano model [9] and its reinterpretation as a quantum string theory, duality was considered as a basic ingredient in the formulation of strong interaction physics [10]. In some of the early papers on dual models [11] one sees the "crossing transformation"
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DUALITY, CROSSING AND MACLANE'S COHERENCE
21
applied to an internal portion of a more complicated graph. Thus, for example, the above transformation, when applied to the graph
a
\
2
b
yields the graph
Recently [1-5], crossing transformations on graphs have reemerged in the context of finding constraints on two-dimensional rational conformal field theories. Roughly speaking, one associates to each crossing transformation a matrix which is, essentially, the matrix of a monodromy operator on a space of meromorphic sections of a line bundle over a Riemann surface whose knowledge determines the theory. Now one may pass from one graph to another by two different sequences of crossing transformations. Each of these routes will correspond to an expression in the basic matrices. Equating the expression corresponding to two different routes then provides a constraint on the basic matrices. It then becomes important to determine a set of generating relations from which all the others follow. In examining the nature of the generating relations one is struck by their similarity to the basic constraints introduced by MacLane in his celebrated paper [6] on tensor products in categories. Recently, MacLane's results have been extended by Joyal and Street to the case of "braided monoidal categories" [17]; see also [16]. The purpose of the present article is to explain the relation of the crossing constraints to MacLane's theory. (We gather from a footnote in [1] that Witten has also observed the similarity to MacLane's theory.) Moore and Seiberg [1] have presented a set of generating relations somewhat different from the ones presented here.
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R. BRUSTEIN, Y. NE'EMAN AND S. STERNBERG
Isr. J. Math.
2. Constructions on graphs In order to avoid confusion arising from differences in terminology between mathematicians and physicists we make the following definitions. A graph consists of two sets, E, its set of edges (or one-dimensional objects) and N its set of nodes (or zero-dimensional objects). Furthermore to each edge, e, is associated a subset {«,, n2) of Ncalled its boundary nodes. We say that nx and «2 are incident to e. (We do not, at this stage, exclude the possibility that nx might equal n2.) In the physics literature, nodes that are incident to more than one edge are called vertices. But in the mathematical terminology the word vertex refers to all nodes. So we will avoid the use of the word vertex altogether, and speak of internal and external nodes. We will consider graphs where every node is incident to either three edges (the internal nodes) or to one edge (the external nodes). In physics language this means that we are considering graphs similar to the Feynman diagrams occurring in a 0 3 theory. External edges (those attached to external nodes) will be called legs. We can consider two types of constructions: Joining: We can pick one leg from each of three graphs and join them. So
\
goes to
iff
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DUALITY, CROSSING AND MACLANE'S COHERENCE
23
(In this figure X, Y and Z stand for various graphs whose internal structure we have not drawn.) Or we can pick two legs from one graph and join it to one from another. So
\
goes to
Or we can join three legs of a single graph. Crossing: If (1.1) occurs as a subdiagram of a graph, then it is replaced by (1.2) or (1.3) (with all the other connections unchanged). An illustration of this operation was given in the preceding section. Suppose that we consider connected graphs with one leg with an exterior node marked. At the risk of mixing metaphors, we will call such a graph a rooted graph, and the marked leg the root. Then joining becomes a binary operation, J, on rooted graphs, by joining the two roots to the trivial graph, consisting of a single edge and two nodes, and then marking the remaining node. Thus
/A/ L> 665
24
R. BRUSTEIN, Y. NE'EMAN AND S. STERNBERG
Isr. J. Math.
goes to
AX, Y)
Notice that the operation J does not introduce any new loops (cycles). In particular, if Xx and X2 are trees (have no cycles), then J(X{, X2) is again a tree. Indeed, let nx and n2 denote the number of nodes in X{ and X2 and ex and e2 the number of edges. Then n = «, + n2 and e = ex + ez + 1 where n and e denote the number of nodes and edges in J(XU X2). So if «, = e{ + 1 and n2 = e2 + 1, then « = e + 1. 3. Cyclically ordered graphs For any finite set, A, let S(A) denote the group of all one-to-one transformations of A onto itself. So if A has n elements, the group S(A) is isomorphic to the symmetric group Sn = S({ 1, 2 , . . . , n }). Suppose that A has « elements. By a cyclic order on A we mean an element, s, of S(A) which has order n. There are thus (« — 1)! different cyclic orders on A. By a cyclically ordered graph we shall mean a graph for which a cyclic order has been chosen at each (internal) node. When we perform the operation J on two cyclically ordered graphs, Zand Y, we must decide which of the two cyclic orders to put on the joining node. If x denotes the root (the marked leg) of Zand y denotes the root of Yandj denotes the new leg, we must choose between the cyclic orders {j, x, y) and (J, y, x) at the new node. We will denote the first choice by J(X, Y) and the second choice by J(Y, X). We have thus proved The join operation defines a nonsymmetric binary operation, J, on cyclically ordered graphs. If X and Y are trees, then so is J(X, Y). PROPOSITION.
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Vol. 72, 1990
DUALITY, CROSSING AND MACLANE'S COHERENCE
25
Let us consider the basic four leg graph
>
<
and let G = G{ >~< ) denote 5(Legs( >—< )), so G is isomorphic to S4 if we label the legs as in (1.1). There are three unordered graphs corresponding to (1.1), (1.2) and (1.3) and the group G acts transitively on this three-element set (the action of S4 on unordered partitions of {1, 2, 3, 4} into two unordered subsets). Each of these unordered graphs gives rise to four cyclically ordered graphs. Thus 1
4
2
4
1
3
(X) (X) (X) 2
3 A
1
3 B
2
4 C
2
3
(X) 1 D
4
are the four cyclically ordered graphs corresponding to (1.1). The group G acts transitively on the twelve-element set of all cyclically ordered graphs. The element (13)(24) fixes the graph A above. (If we were to consider oriented cyclically ordered graphs, where an orientation is chosen on each edge, the (13)(24) would not act trivially and the orbit of G would contain twenty-four distinct possibilities.) If >—< is an internal subgraph of some unordered graph X0, then we have the two graphs Y0 and Z0 associated to X0 by crossing, and G acts transitively on this three-element set as before. Let us choose a cyclic order on X0. We denote the corresponding cyclically ordered graph by X. Conversely, if Xis cyclically ordered, we let X0 denote the corresponding unordered graph. Suppose that X is a cyclically ordered graph, >-< is a four-legged subdiagram of X and 6 E G ( >-^C ). We let bX be the cyclically ordered graph given by cyclically ordering the nodes of bX0 as follows: each node of bX0 other than the two nodes corresponding to the nodes of >-< comes from a unique node of X0. On each such node we put the old cyclic ordering as in X. At the nodes corresponding to those of the crossing subgraph, > ^ , we put the cyclic ordering given by the action of G as described above. We can now describe "moves" on cyclically ordered graphs: Move: For a cyclically ordered graph, X, pick a four-legged subgraph, >—<.
667
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R. BRUSTEIN, Y. NE'EMAN AND S. STERNBERG
Isr. J. Math.
Then pick an element & £ G'( >—<) and obtain the cyclically oriented graph bX by the procedure given above. The mathematical problem that then arises is: describe the relations among the moves. That is, describe those sequences of moves that will lead from X back to the same graph X. We must be precise about what we mean by "the same X": Notice that the various legs of the graph retain their identity at each move. So after a sequence of k moves, we obtain a graph Y together with an identification of the legs of A'with the legs of Y. We say that "Fis the same as X" if there is a one-to-one map (of the edges and modes) of X onto (the edges and nodes of) Y which preserves all incidences and cyclic orders, and which reduces to the given identification on the legs. We can also consider a more restricted class of moves defined as follows: Suppose we consider a four-legged graph >—< . We claim that a cyclic order on the nodes of >—< induces a cyclic order on the legs of >—< :
C^KJ
induces
I ^K
In other words, the cyclic orders (12m) and (ra34) induce the cycle (1234). (Notice that if we choose the nodes in the opposite order we get the same cycle, (3412) = (1234).) Notice that (1234) carries 1
4
2
4
3
3
1
A
2
(1234X4
and the cycle associated to (1234)^4 is again (1234). Notice also that (1234)2 = (13)(24) which carries A into itself, as is to be expected. Restricted moves: Let Xbe any cyclically ordered graph. Each four-legged subgraph, >—<, is cyclically ordered, and hence determines an element, a, of G{ >—<). Apply this preferred element a to Xso as to move to aX. Then choose another four-legged subgraph etc. These are the restricted moves.
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Vol. 72, 1990
DUALITY, CROSSING AND MACLANE'S COHERENCE
27
4. Restricted moves on rooted trees In this section we will show that a cyclically ordered rooted tree is a scheme for forming tensor products. (In more technical language it is a binary word in the sense of [ 12, p. 161 ] but where we don't allow for the empty word.) We shall also see that a restricted move on a rooted tree is just an associativity operator in the sense of MacLane, and hence we can apply Theorem 3.1 of [6] to determine all the relations among restricted moves on cyclically ordered rooted trees. Suppose we are given a rooted tree. We can regard the cyclic order on the nodes as a "sorting procedure" on the remaining legs as follows: Start from the root (the marked leg). At its juncture, mark the first edge encountered in the cyclic order by L and the second edge by R. This is to be interpreted as follows: All legs connected to the edge marked L will be placed to the left of all the legs attached to the edge marked R. If the edge we just marked L is not a leg, continue to its other joining node and repeat the procedure of marking the first edge encountered in the cyclic order by L and the second by R. Do the same for the edge marked R if it is not a leg. Continue. Now consider each interior node as a tensor product and hence each interior edge as a parenthesis system. Thus, for example,
f f is marked as
R
and corresponds to the "tensor producf'expression ((JC®JO®Z)®(W0H).
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R. BRUSTEIN, Y. NE'EMAN AND S. STERNBERG
Isr. J. Math.
Here is another way of saying the same thing (in the language of [ 12, p. 161 ]). Any cyclically ordered rooted tree can be built from smaller trees by the operation J unless it is already the trivial tree consisting of a single edge. So we let this trivial tree be denoted by and write X®Y
instead of J(X, Y).
Thus we have proved: There is a bijective correspondence between cyclically ordered rooted trees and binary words. The correspondence can be established by representing the tree by successive application of the join operation. THEOREM.
In short: every cyclically ordered rooted tree is a binary word in the sense of [12, p. 161] with no e0. Now consider the four-legged graph
2
3
where we have marked the leg 4. It corresponds to (1 ®2)<8>3. The operator a = (1234) applied to this graph gives
1
2
which corresponds to 1 ® (2
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DUALITY, CROSSING AND MACLANE'S COHERENCE
29
(2) A "pentagon identity" which relates the two paths going from 1®(2®(3®4))
to ((Kg) 2)® 3) 0 4
one in two moves and the other in three moves. In our present presentation we would draw MacLane's pentagon identity as
3
See, in relation to the pentagon identity, the papers [1] and [4]. MacLane was interested in the following application: suppose we postulate a tensor product on a category, C. That is a functor from C X C to C denoted by <8>. An associativity isomorphism is then a natural transformation
a=a(A,B,C):A®{B®C)^(A®B)®C such that a(A,B,C) has a two-sided inverse. The question was: what are the conditions on these a's so that there is a unique natural transformation between any two «-fold tensor products? His answer, Theorem 3.1 of [6], is that the pentagon identity (together with the naturality) is necessary and sufficient. We shall describe a different application of MacLane's result in the next section.
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A four cycle and any transposition (two cycle) generate S4. On any rooted tree each four-legged subdiagram, >—<, has a marked edge as we have observed, and hence a unique transposition in G( >—<), the transposition of the two legs not joined to the marked edge. This transposition clearly corresponds to an instance of commutativity transformation in the tensor product interpretation. Indeed the diagram
cy<j corresponds to (1 <S> 2) ® 3, and thus interchanging 1 and 2 is a commutativity operator. Hence we can read from MacLane's paper the hexagon identity relating associativity and commutativity. In terms of our four-legged graphs, MacLane's hexagon identity is
cX) 1
2
3
3
^
2
V^>
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DUALITY, CROSSING AND MACLANE'S COHERENCE
31
Theorem 5.1 of [6] asserts that the hexagon identity together with the pentagon identity and naturality determine all the relations on moves on cyclically ordered rooted trees. In particular the braid relations follow as MacLane proves in [6]. 5. Moves on labelled graphs Let the group G act on the set P. Let F(P, L) denote the set of all functions from P to L (i.e. the set of all "labels" of elements of Pby elements of L). Then G acts on F(P, L) by the rule (bf)(p) = f{b~xp). We need another notion. Let K be a finite set. A vector bundle E — K is a rule which associates a vector space Ek to each kEK. Suppose we are given an action of G on K. Then an action ofGonE (consistent with the given action on K) is a rule which associates to each bEG and to each fE.K a linear transformation b:Ef^Ebf so that we get an action of G on E. Thus the identity of G gives the identity transformation for each island £> — Ebf-* Ecbf equals
Ef^Ecbf.
Suppose that all the Ef have the same dimension and have been identified with the fixed vector space, V. Then b: Ef^-Ebf can be identified with a linear transformation
A(f,b):V^V. The consistency condition that says we have a group action then becomes (5.1)
A(bf,c)A(f,b)=A(f,cb).
If Vhas a preferred basis the A's become matrices and (5.1) becomes a matrix equation. Now consider a cyclically ordered graph JTin which we have labelled all the edges by labels from a finite set, L. Let >—< be a four legged subdiagram of X and bEG( >-^n). Then each edge, e, of the graph Wother than the new central connecting edge corresponds to an edge of X, and hence carries a label. Thus we label all these edges according to our standard rule, (bh)(e) = h(b-le).
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For the central edge we proceed as follows: We consider the vector bundle over the space of labels on the legs of >—< whose fiber is CL. We assume that we are given an action of G on this vector bundle. In other words, to each 6 E (7( >—<) and to each label/on the legs of >—< we are given a matrix A(f,b)qp, p,q^L satisfying (5.1). Let {bX)q denote the graph bX with the label q on the new central edge and with all the other edges labelled as above. Then we move X into ZqA(f,b)qp(bX)q. We have moved a labelled graph X into a linear combination of labelled graphs. Thus our moves are on the space of linear combinations of labelled graphs. Notice that after a finite number of steps the sum will be over various labels, but not over labels of legs. So we once again obtain consistency relations, since we get from one labelled graph to a sum of others with the same external labels by several routes, and the two sums over the internal labels must be equal. This imposes a set of constraints on the matrices A, above and beyond (5.1). Similarly, we may consider the constraints arising from restricted moves. In either event the generating relations were described in the preceding section. 6. Applications to rational conformal field theories Rational conformal field theories are characterized by (i) the central charge c of the left and right Virasoro algebras (in mathematical language, the choice of an element in H2(g) where g is the Lie algebra of Fourier polynomial vector fields on the circle), (ii) the finite set of primary fields and their conformal weights (hi, /z,), and (iii) by the structure constants CIJK of the operator product algebra (O.P.A.) [13]. Factorization of the n -point function and the structure of the conformal families leads to the construction of conformal blocks [13]. (For simplicity we consider minimal theories only. The extension to other cases is not hard. We do not distinguish between fields and their conjugates to avoid complicated notation.) For example, the 4-point function can be written as (4>i(zu Zi)
(6.1) ~Zcijpcpk!l(J. P
\i
k
) (z)X"c.c." UP
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DUALITY, CROSSING AND MACLANE'S COHERENCE
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where by "c.c." we mean the corresponding expression with the bar variables and z is the cross ratio Zx — Z2 Z 4 — 2 2 Z —
.
z, — z 3 z 4 — z 3
The /'s are the holomorphic conformal blocks of the 4-point function. The blocks span a vector space of meromorphic functions. Equation (6.1) is described pictorially by k i
k
We can now make contact with the formalism introduced in the previous sections. We see that we can associate to each holomorphic (antiholomorphic) conformal block a 4-leg labelled graph of the kind discussed previously. Similarly we can associate a labelled graph to the holomorphic blocks of the npoint function. The legs of the graph usually correspond to some primary fields while the internal edges correspond to summation over descendants of a given conformal family. Crossing: The completeness of the s- and /-channel blocks of the 4-point function leads to the relation [14, 1, 5]: (6.2)
cnpcpUl(] \1
]) (Z) = 1A(2
]) cnqcqXj(l
4/p
Otlpq
q
\\
4
)
\l
(l-z)
\)q
which can be described graphically by 2
3
\
4
>-r^-?41
4
/—r-( '2
\ \
The matrices A are therefore the matrix representation of the associativity operator a. (The matrices A are called F i n [1].) The commutativity matrices are obtained by the same reasoning. They are diagonal matrices whose entries depend on the V s .
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For example, the matrix of basic commutativity operator discussed in Section 4, (1 ® 2)® 3 — (2
(6.3a)
(6.3b)
= ^(*'-*'-*^-
\ 3 2 /??
(The matrices C are called SI in [3].) The matrices A and C obey the consistency condition (5.1), in particular they act on the labels of the graphs as elements of S4. A special case is the braiding matrix B\ according to (5.1) it is given by B = CAC. (The matrices B are called R in [3].) However C2 may not be equal to the identity, since it follows from (6.3) that C2 is a scalar matrix consisting of a possibly non-trivial phase factor. (We wish to thank Prof. MacLane for pointing out an error in our original treatment at this point.) Hence the original MacLane coherence theorem involving commutativity does not directly apply. But a more recent generalization of MacLane's theorem does apply. A braided monoidal category, cf. [16] and [17], is a category with a tensor product defined axiomatically together with an associativity, a, and a commutativity, c, but where c1 is not assumed to be the identity. One assumes the pentagon identity and two hexagon identities — the hexagon identity of Section 4 for c and the same identity for c~l. (Of course these reduce to one and the same identity if c2 = 1.) The coherence theorem of Joyal and Street [17] then asserts that a diagram build up from instances of a and c using tensor products and composition commutes if and only if the associated braids are equal. (We wish to thank Prof. MacLane for referring us to the papers [16] and [17] and to the use of the Joyal-Street coherence theorem at this juncture.) We have to justify the fact that the same matrices A and C represent the associativity and commutativity operators inside higher-order graphs. For this we need to know that conformal blocks with descendants as external legs transform with the same A and C matrices. In addition we need to know that we can isolate a 4-leg graph inside a higher order and operate on it with operations defined on conformal blocks of the 4-point function. The latter point can be proved by using the operator product expansion in the n -point function and by summing and desumming over descendants [15]. The first
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point can be easily proved in the case that there is no mixing between the blocks, i.e. no integer differences between any of the /z,'s. This is done by using the linear relation between blocks with external descendants and external primaries [13]. In the case that mixing does occur the blocks do not form an orthogonal basis to begin with [14]. In many cases there exists an extended algebra which allows the separation of the mixed blocks. With this in mind the identification of operations on holomorphic blocks with our general formalism is complete. The demand of coherence is equivalent to the demand that the n -point function be well defined and compatible with the O.P.A. The matrices^ and Ctherefore obey MacLane's pentagon and two hexagon identities, one involving C and the other involving C~l. The equations resulting from the pentagon identity are
r 4/sp \5 4/rq \q 3/pt
\5 sin
\5 4
sq
They can be obtained from the pentagon identity in Section 4 in the following way: The marked leg is labelled 5. The internal edges of the top 5-legged graph are labelled r and 5. Equation (6.4) is obtained by equating the coefficients of the conformal block of the 5-point function with internal indices q and t for each q and t separately. Similarly, the equations resulting from the hexagon for a and c are obtained by equating the coefficients of the block 1 2 3 4. of the 4-point function. The marked leg in the hexagon diagram in Section 4 is labelled 4 and the internal edge of the top 4-legged graph is labelled p. Then the hexagon identity yields
,
\2
1/w
\3
2/«
\1
3/«r
\2
IJPP
\3
\}pr
\1
Alrr
and the corresponding equation with C _ 1 . Due to the coherence theorem, these equations in addition to the defining relations (5.1) are the complete set of equations obeyed by the matrices A and C. We have checked that the Ising model (which is the simplest case with nontrivial y4's) is a solution of the hexagon and pentagon equations.
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7. Concluding remarks (1) We have indicated the applications of the results of Sections 2-5 to conformal field theories in two dimensions. But it seems to us that there might be other areas of applications of the types of problems discussed, in particular to string theories and to statistical mechanics. (2) There are some immediate mathematical problems that we have not dealt with here but hope to study in the future. The most pressing, of course, is to find the structure of the solutions to the matix equations described in Section 5. In addition, one would want to extend the results of Section 4 to the case where loops are allowed, so that MacLane's theorem does not immediately apply. Also, one might want to consider more general types of graphs, not necessarily those with three edges incident to every internal node. (3) It would be of interest to explore the relations of the problems studied here to other areas of mathematics, for example, the relation to knot theory and the "category of tangles"; cf. [18] and the references given there. Also the relation to the work on quantum groups as developed in the Leningrad school. In fact, it was a lecture by Prof. Faddeev at the Landau conference at Tel Aviv which stimulated the interest of one of us (YN) in the current topic. It would also be interesting to try to understand if there is a "continuous analogue" of the constructions in Sections 4 and 5, where "crossing" is replaced by some form of surgery or cobordism. For example, the idea of replacing the deleted edge in the crossing operation by a vector bundle is highly reminiscent of a blowing up operation as it appears in birational equivalence; cf. [19], for example. ACKNOWLEDGEMENTS
We would like to thank J. Bernstein and S. Yankielowicz for helpful discussions. As indicated above, we also wish to thank Prof. MacLane.
REFERENCES 1. G. Moore and N. Seiberg, Phys. Lett. 212B (1988), 451; Comm. Math. Phys. 123 (1989), 177. 2. C. Vafa, Phys. Lett. 199B (1988), 91. 3. K.-H. Rehren and B. Schroer, Nucl. Phys. B312 (1989), 715. 4. E. Verlinde, Nucl. Phys. B 300 (1988), 369. R. Dijkgraaf and E. Verlinde, Nucl. Phys. (Proc. Suppl.) B5 (1988), 485. 5. R. Brustein, S. Yankielowicz and J.-B. Zuber, Nucl. Phys. B313 (1989), 321. See also the survey by Gawedski. Sem. Bourbaki, Nov. 1988, expose 705.
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6. S. MacLane, Rice Univ. Stud. 49(4) (1963), 28. See also G. M.Kelly, J.Alg. 1 (1964) 397, and J. D. Stasheff, Trans. Am. Math. Soc. 108 (1963), 275. 7. S. Mandlestam, Phys. Rev. Lett. 112 (1958), 1344. 8. R. Dolan, D. Horn and C. Schmidt, Phys. Rev. Lett. 19 (1967), 402. 9. G. Veneziano, Nuovo Cimento 57A (1968), 190. 10. Y. Nambu, in Symmetries and Quark Models (Proc. 1969 Wayne State Conf.), R. Chand ed., Gordon and Breach, New York, 1970, p. 269. H. B. Nielsen, XVth Int. Conf. High Energy Phys., Kiev (1970). L. Susskind, Nuovo Cimento 69A (1970), 457. 11. K. Kikkawa, B. Sakita and M. A. Virasoro, Phys. Rev. 184 (1969), 1701. H. Harari, Phys. Rev. Lett. 22 (1969), 562. J. L. Rosner, Phys. Rev. Lett. 22 (1969), 689. 12. S. MacLane, Categories for the Working Mathematician, Graduate Texts in Mathematics, Springer-Verlag, Heidelberg, New York, Berlin, 1971. 13. A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B 241 (1984), 333. 14. V. S. Dotsenko and V. Fateev Nucl. Phys. B 240 (1984), 312. 15. S. Yankielowicz, private communication. R. Brustein, Doctoral Thesis, Tel Aviv University, 1988. 16. P. Freyd and D. Yetter, Advances in Math. 37 (1989), 156; Coherence theorems via knot theory, J. Pure Appl. Algebra, to appear. 17. A. Joyal and R. Street, Braided monoidal categories, Macquarie Mathematics Reports, 1986. 18. V. G. Turaev, Invent. Math. 92 (1988), 527; also LOMI Preprint E-6-88. 19. V. Guillemin and S. Sternberg, Invent. Math. 97 (1989), 485.
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C H A P T E R 9: VARIOUS TOPICS IN A S T R O P H Y S I C S JOHN BAHCALL School of Natural Sciences, Institute for Advanced Study Princeton NJ, USA Very few outstanding scientists have the leadership skills to found an enduring program in their area of specialty. It is even more exceptional when the program maintains a high international standing. Yuval Ne'man founded the modern school of particle and mathematical physics in Israel and the school is nourishing. Moreover, he did the exceptional twice. Yuval also founded the Israeli program in astronomy and astrophysics, a second enduring and flourishing school. I was privileged to observe Yuval in his pioneering work, beginning in the 1960's and continuing to this day, to establish astronomy and astrophysics as a major research activity in the small country of Israel. As the reader can readily imagine, Israel had, and has, many more urgent and more practical demands on resources than establishing a program in astronomy and astrophysics. Yuval's task was made even more difficult because there was practically no tradition for astrophysics being pursued in an active research context in the Israeli universities. In addition, Yuval's own scientific activities were largely focused on particle physics and mathematical physics. Prom the time that Yuval became president of the rejuvenated Tel Aviv University in 1971, he set as a personal goal the establishment of a thriving program in astronomy and astrophysics. Unlike most of his contemporaries among the leading particle physicists, Yuval had a detailed knowledge of the most exciting things that were occurring in modern astrophysical theory and in observational astronomy. In everything he did, Yuval led by example. I remember him taking me on one of his many trips to outpost kibbutzim in the late 1960's, where he delivered an evening popular lecture to the settlers on Big Bang Theory and its relationship to particle physics. This inspiring lecture took place decades before it became fashionable for particle theorists to use the early universe as a laboratory to test their latest ideas. In founding astronomy and astrophysics in Israel, Yuval insisted that the country have both a major observational program and a strong core of young theorists. Starting either theoretical or observational astronomy as a new research discipline in Israel required great imagination, incredible stubbornness in the face of many practical obstacles, and enormous attention to detail. Yuval had imagination, stubbornness, and mastery of detail, all in very generous portions. Among the almost infinite list of things that Yuval was doing while serving as President of Tel Aviv University, he found time to be personally involved with persuading some of the most talented young Israelis studying abroad or working in other fields within the country to convert to astronomy. Founding the Wise Observatory required Yuval to raise what in those days was a large sum of money at the same time that he recruited the scientists to build and commission the telescope. There were many personnel
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and bureaucratic difficulties that had to be overcome. He solved all the problems with his personal charm and scientific leadership. He even served briefly as Director of the Wise Observatory. Only Yuval, with his unique combination of scientific prestige, personal charm, and research insights could have accomplished the task of launching an enduring Israeli program of high international standing in both observational astronomy and theoretical astrophysics. This achievement is made all the more remarkable by the fact that Yuval's mastery of fundamental physics and his remarkable mathematical skills are not well matched to the detailed phenomenological basis of progress in astronomy. He appreciated the details despite the fact that his own talents lay primarily in more abstract work. In the first article reprinted here, 'Expansion as an Energy Source in Quasi-Stellar Radio Sources', 1965 {9.1}, Yuval suggested the possibility that the energy of the newly discovered quasars could be due to expansion from an inhomogeneity left-over from the general expansion of the universe, a suggestion made independently by I. D. Novikov. Yuval's short article has two characteristics that exemplify Yuval's scientific work: 1) He addresses big problems; 2) He exploits his deep knowledge of physical theory. In the present case, Yuval used his understanding of general relativity to propose a solution to the quasar energy problem that did not require, like some of the contemporary proposals, negative energy or negative mass fields that were inconsistent with the theory of relativity. Nature chose to fuel the quasars by a different mechanism than Yuval proposed. In fact, in a follow-up paper in 1967 written together with Gerald Tauber, 'The Lagging Core Model for Quasars' {9.2}, Yuval showed that his proposal was difficult to reconcile with the basic facts of quasar phenomenology. We now know that the mechanism preferred in the real world for producing quasars is accretion onto a massive black hole. In the last decade, the Hubble Space Telescope has revealed the presence of massive black holes in the cores of many nearby galaxies. Nevertheless, the picture of a rapidly expanding relativistic source is in some ways reminiscent of the idea of inflation, which is the modern paradigm for the formation of the Universe. The paper reprinted here on 'unconventional and pathological cosmological models' {9.3} shows the openness with which Yuval approaches science (and indeed all intellectual questions). He explores a wide range of possibilities. One of the most interesting aspects of this paper is how it reveals the enormous transformation in cosmology that has occurred in the three decades since Yuval wrote his review of the subject. Today, the greatest fear of cosmologists is that the 'standard model', which agrees so precisely with available observational data, will turn out to be successful in describing the enormous amount of new data that will be available in the next decade. If this happens, we will make very little progress. Science thrives when a conflict is revealed between observation and theory. At the time of Yuval's review, a number of senior cosmologists raised doubts about standard cosmological ideas and about possible discrepancies with basic ideas of an expanding universe. Interestingly enough, the fundamental puzzles for current cosmological theory (the nature of dark
682
matter and dark energy and the smallness of the cosmological constant) were not important themes at the time of Yuval's review. The last two astronomical papers reprinted here, 'Inflationary Cosmology, Copernican ReleveUing, and Extended Reality' {9.4} and 'Heuristic Methodology for Horizons in GR and Cosmology' {9.5}, demonstrate attention to the current issues in fundamental cosmology and show Yuval's insightful, original approaches to different current questions of interest. Yuval has subsequently noted that the defect in classical theory discussed in the abovementioned paper cancels out with the inclusion of Bekenstein entropy and the ensuing (quantum) Hawking radiation. As the black hole dissipates, its mass decreases, the Schwarzschild radius tends to zero and is never crossed in the local frame just as it isn't in the distant one. The anomaly Yuval discusses may be related to the discussion of a Holographic Universe considered by Susskind, 't Hooft, Hawking, and others. I close with a personal remark. I consider myself lucky and honored to have had Yuval as a mentor and friend and as a source of inspiration, scientific and personal, for more than forty years.
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REPRINTED PAPERS OF CHAPTER 9: VARIOUS TOPICS IN ASTROPHYSICS
9.1
9.2
9.3
9.4
9.5
Y. Ne'eman, "Expansion as an Energy Source in Quasi-Stellar Radio Sources", Astrophys. J. 141 (1965) pp. 1303-1305.
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Y. Ne'eman and G. Tauber, "The Lagging-Core Model for Quasi-Stellar Sources", Astrophys. J. 150 (1967) pp. 755-766.
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Y. Ne'eman, "Unconventional and Pathological World Models", Trans. Int. Astronomical Union, X V I A (1976), Part 3 (D. Reidel, Dordrecht/Boston, 1976), pp. 151-157.
702
Y. Ne'eman, "Inflationary Cosmology, Copernican Relevelling and Extended Reality", in Examining the Big Bang and Diffuse Background Radiations, M. Kafatos and Y. Kondo, eds., Proc. IAU168 Symp. (Kluwer Academic, Dordrecht, Netherlands 1996), pp. 559-562.
709
Y. Ne'eman, "Heuristic Methodology for Horizons in GR and Cosmology", in Gravitation and Cosmology, Supplement to 6 (2000) pp. 30-33. (Special issue dedicated to I. Khalatnikov 80th birthday).
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Reprinted from THE ASTROPHYSICAL JOURNAL Vol. 141, No. 4, May 15, 1965 Copyright 1965 by the University of Chicago Printed in U.S.A.
EXPANSION AS AN ENERGY SOURCE IN QUASI-STELLAR RADIO SOURCES* YTJVAL NE'EMANJ California Institute of Technology, Pasadena, California Received January 11, 1965 ABSTRACT The Greenstein-Schmidt model of some quasi-stellar radio sources distinguishes between low-density filaments where radio and redshifted optical radiation is emitted, and a small dense core responsible for continuum radiation and variability. It is suggested here that the system's energy originates from expansion in the dense core; this expansion is viewed as an inhomogeneity of the general expansion of the Universe in conventional general relativity.
In a recent study Greenstein and Schmidt (1964) analyzed 3C 48 and 3C 273 and reached a plausible model consisting in the main of the following points: (a) The observed redshifted lines are emitted by low-density filaments spread over a region of 1-10 pc; radio emission also originates from an extended region, (b) The optical continuum is emitted from a core of some 109 Mo, with radius < 1 pc. The observed light variations originate in this source. To allow for their period of ~ 1 0 years, the radius may yet have to be much smaller than 1 pc. Hoyle, Fowler, Burbidge, and Burbidge (HFB 2 1964) emphasized the difficulties in accounting for an energy emission of 1060 erg, spread over a period of the order of 106 years as required. Nuclear processes are not adequate to insure hydrostatic equilibrium, rapid collapse setting in and resulting in lifetimes of < 104 years or thereabouts. It has been pointed out, however, that rotation may help in achieving longer nuclear burning (Fowler 1964); if actual calculations do provide for an extended release of nuclear energy covering periods of 106 years, the problem of the quasar's energy may indeed have been solved. The above-mentioned authors (HFB2 1964) have, however, assumed the need for an alternative energy input and suggested that gravitational collapse be the answer. Two difficulties then arise: the same rapid rate of the implosion that invalidated the nuclear energy solution, and an increasing difficulty in the effective release of the large energy gained through increased binding. To cope with these difficulties, Hoyle and Narlikar (1963) have postulated a negative-energy C field, coupled negatively to gravitation; its existence generates a repulsive core, well inside the Schwarzschild radius. In the proper time of the collapsing mass, the Schwarzschild radius is crossed in a very short time, and * Work supported in part by the U.S. Atomic Energy Commission. t On leave of absence from Tel Aviv University, Tel Aviv, Israel, and the Israel Atomic Energy Commission, Tel Aviv, Israel.
1303
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YUVAL NE'EMAN
Vol. 141
the collapse is soon stopped inside and reversed. The superstar oscillates between this internal limit and its outside original size, the amplitude decreasing gradually owing to energy release. This release occurs every time the material happens to lie outside the Schwarzschild radius (HFB 2 1964). Such a theory is in conflict with Einsteinian general relativity, since it assumes we are witnessing oscillations whose period is infinite in our coordinate system. According to any analysis of the nature of the Schwarzschild throat (Fronsdal 1959; Kruskal 1960; Fuller and Wheeler 1962; Bergmann 1964), only "one-way traffic" is allowed: the collapsing mass reaches the Schwarzschild radius at t = «> only. After the rapid catastrophic collapse, it will settle in an infinitely slow final progression to this radius, emitting very redshifted radiation. Such a situation may be in accordance with the hypothesis of "hidden masses" with respect to some of the problems studied in HFB 2 (1964), but it invalidates the idea of collapse as a superrich and long-lived source to explain the quasars. An alternative suggestion of negative mass (Hoffmann 1964) would be difficult to reconcile with quantum theory and particle physics. We would like to suggest a mechanism that would have some of the advantages of the Hoyle-Narlikar theory, without violating our understanding of Schwarzschild singularities while also dispensing with negative energy or negative-mass fields. By going over to the coordinates (Kruskal 1960),
u=
[ ii^ - 1 1 2 exp (ibdcosh GS0 -
' = h^~ lT CXP (l^O Sinh (Tm*)' P = f
) exp f - — ^ ) = a transcendental function of (M2 — »2) ,
the Schwarzschild metric becomes ds* = f(-dv2
+ du2) + rUv?.
Kruskal has shown that the physical universe lies in the v > \ u | quadrant of the uvplane. Causality can never be violated; but the borders of the physical region include both T = — °° and T = + °° situations. As recently emphasized by Bergmann (1964), this coordinate time is proportional to the natural time for an observer who maintains a roughly constant distance from our object. We suggest that the 109 Mo we are discussing are in an expanding state, gradually leaving the neighborhood of the Schwarzschild radius; in a Kruskal diagram this is a rise from the T = — co region. Cosmologically, what is implied is an inhomogeneity in the original expansion of the Universe. Our massive object has been lying near the singularity throughout periods of the order of 1010 years, finally emerging into observability as a result of sufficient expansion. In its own coordinate system it has been expanding very rapidly from a superdense state in the v < — [ u [ quadrant, between the true r = 0 singularity hyperbola and the Schwarzschild radius. This is just what we generally get for the Universe as a whole, in our own coordinates. In a distant object like the quasar the only implication is that extremely large energies must have accumulated from strong interactions in the superdense situation, with no outlet for the decay of the various mesons. Having attained a stage in its slow expansion when densities of its various layers fall beneath a certain limit, it is now freeing these mesons and may be emitting intense cosmic radiation. Mesonic decays provide for a continuous injection of high-energy particles, photons, and neutrinos into the surrounding medium. It should be noted that expansion should
N o . 4, 1965
QUASI-STELLAR RADIO SOURCES
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make the optical continuum effectively less redshifted than the emission from the lowdensity region. This effect should be stronger than the extra redshift expected from gravitation. We are indebted to E. Schucking, A. Schild, and J. Bardeen for valuable criticism. REFERENCES Bergmann, P . G. 1964, Phys. Rev. Letters, 12, 139. Fowler, W. A. 1964, Rev. Mod. Rhys., 36, 545. Fronsdal, C. 1959, Phys. Rev., 116, 778. Fuller, R. W., and Wheeler, J. A. 1962, Phys. Rev., 128, 919. Greenstein, J. L., and Schmidt, M. 1964, Ap. J., 140, 1. Hoffmann, B. 1964, Gravity Research Foundation prize essay. Hoyle, F., Fowler, W. A., Burbidge, G. R., and Burdidge, E. M. 1964, Ap. J., 139, 909. Hoyle, F., and Narlikar, J. V. 1963, Proc. Roy. Soc. {London), A273, 1. Kruskal, M. D. 1960, Phys. Rev., 119, 1743.
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T H E ASTKOPHYSICAL JOURNAL, Vol. ISO, December 1967
THE LAGGING-CORE MODEL FOR QUASI-STELLAR SOURCES YUVAX NE'EMAN Tel-Aviv University, Tel-Aviv, Israel,* and Southwest Center for Advanced Studies, Dallas, Texas AND G E R A L D TAUBERf
Tel-Aviv University, Tel-Aviv, Israel Received March 29, 1966; revised June 13, 1967 ABSTRACT The model in which quasi-stellar sources are assumed to consist of lagging massive cores due to inhomogeneities of the original cosmological expansion is investigated. I t is found that in the approximation of spatial isotropy the resulting expansion is too fast to account for the observed data, except in the case when a cosmological constant is introduced. However, if the expansion is followed by oscillations of the core, it is possible to account for the observed pulsations. For this purpose a phenomenological equation of state is constructed. I t is possible to choose the constants so that these oscillations do not penetrate the Schwarzschild sphere. These solutions can then be joined to a non-isotropic exterior solution imbedded in a Friedmann universe. To obtain complete fitting, solutions with non-isotropic pressure should be used. INTRODUCTION
It has been suggested (Novikov 1964; Ne'eman 1965) that initial inhomogeneities in the cosmological explosion might have resulted in lagging cores; quasi-stellar sources (QSS) could then represent the delayed expansion of these objects. The present study begins with a rough survey of the limitations imposed by the field equations and equations of state on such cores. We find that the usual approximation of a spherically symmetric core with a homogeneous and spatially isotropic interior leads to severe restrictions. Our study concerns itself mainly with time-dependent, locally isotropic models within the general relativistic framework. In this approximation we find the following: a) For p = 0, A = 0, neither for the open nor closed universe can the constants be adjusted to give the right order of magnitude for the radius of the expanding core. b) For an equation of state of the form p/c2 = p but A = 0 it is possible to adjust the constants, but the law of expansion is of the formRc^ct (as is also the case in the pressure-free model), i.e., it cannot describe a "lagging" core. c) The presence of a cosmological constant A—either in (a) or (b)—gives the correct expanding solution of galactic mass, but the value of the cosmological constant required is different from the one corresponding to an expanding universe. This gives rise to discontinuities of both the pressure and A at the boundary. The difficulty for the pressure can be settled by a better approximation where p = p(/,t); however, if A should then remain discontinuous we may be led through the incorporation of the discontinuity into the energy density to some model in which matter is not conserved in these regions. This may correspond to some more sophisticated topology for the universe. Note that whether or not QSS turn out to fit the characteristics of cosmological expanding cores these investigations should throw some light on the possible characteristics of any objects created by such a local lag in the cosmological expansion. * Permanent address. t On leave of absence from Western Reserve University, Cleveland.
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YUVAL NE'EMAN AND GERALD TAUBER
In the second part of this article we study pulsating cores with A = 0 instead of expanding ones. We construct a phenomenological equation of state which will roughly reproduce the observed "pulsations" of the quasi-stellar sources. A mixed polytropic equation insures that we get oscillations which leave the object outside of its Schwarzschild radius throughout. Moreover, by satisfying certain conditions on the constants, we have a nonnegative pressure inside. The pressure at the boundary can be made to vanish at the maximal extent of the oscillation. This enables us to produce in part a smooth matching to an exterior Schwarzschild solution which is imbedded in the Friedmann universe. A study based upon p — p{r,t) and such a mixed polytrope should follow in order to test the model correctly, since it is evident that homogeneity and isotropy defeat the model's purpose We do not deal in this article with the actual processes of energy generation. PHYSICAL REQUIREMENTS
We have to account for the following: (a) an energy generation of about 104G ergs sec -1 for periods longer than T ~ 106 years ( = 3.1013 sec); (b) the existence of irregular fluctuations with a rough period T of several years, though some variations may occur even within months or days, i.e., 106 sec < T < 108 sec. The fluctuations lead to a classification of all models into two types: those in which they are due to a statistical repartition of "flares" over a relatively large region R > 102 pc, and those where they represent irregular pulsations of a dense central core. We assume here with Greenstein and Schmidt (1964) a "pulsating" model; this implies a source whose physical size is smaller than 1 pc, perhaps very much smaller; we assume 10 u cm < R < 1018 cm. To account for a total energy output of 1069 ergs (for an activity of 105 to 106 years) to 1063 ergs (if the activity goes back to "big bang" days), we have to consider concentrations of at least 107 to 1011 Mo, respectively (taking a ratio of 1:100 between radiated energy and original mass). In fact, the energy of the magnetic field itself is of the order of 1062 ergs, which may imply that the larger mass should be considered anyhow (Moffet 1965). EMISSION DURING EXPANSION THROUGH A SCHWARZSCHILD RADIUS
The nature of the Schwarzschild throat has been investigated by various authors (Fronsdal 1959; Kruskal 1960; Fuller and Wheeler 1962; Bergmann 1964). To visualize the possibilities, we use Kruskal coordinates; it is easiest to follow casual sequences by looking at a Kruskal diagram. In Figure 1, we follow the history of a gravitational collapse from the point of view of a distant observer. We note that outgoing light is cut off when the imploding body crosses the r = 2m line, but this happens at t = oo ; actually, we shall never observe this cutoff. We shall only gradually lose touch with the collapsing star which will seem to be settling down in an infinitely slow final progression toward r = 2m, emitting extremely redshifted light. If entire collapse to r = 0 is a matter of a few hours in the system's own coordinates (Hoyle, Fowler, Burbidge, and Burbidge 1964), only a finite number of quanta can be emitted before the r = 2m line is crossed, which implies the actual disappearance of our star. This picture justifies in another context the various conjectures of the above authors with respect to "hidden mass" in galaxies. Figure 2 represents the situation for an expansion from a singularity—which is the picture suggested for quasi-stellar objects in the lagging-core model. We note that our observer has recorded the expansion since its very start, even though the expanding core seems to be emerging from an impossible past as long as it is produced from within the Schwarzschild radius. Actually this implies the existence of a certain amount of matter and a finite number of photons on their way to our observer at the beginning of our time scale, wherever we draw this to.
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FIG. 1.—Gravitational collapse viewed from a distance (r and t belong to observer)
FIG. 2.—Expansion viewed from a distance (r and / belong to observer)
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The core model assumes that the central part of a QSS indeed represents matter in an expanding state, with radial motions corresponding to a progression away from the Schwarzschild throat, rather than into it. There is then no limitation on the allowed densities, and no horizon cutting it off. Following Greenstein and Schmidt, one then assumes that the observed redshifted (and generally non-fluctuating) lines are emitted by low-density filaments spread over a region of 1-10 pc, with a steady radio emission also originating from extended regions. On the other hand, the optical continuum, much radio emission of wavelength shorter than 31 cm (Dent 1955; Moffet 1965) and perhaps some fluctuating lines (Wampler 1965; Burbidge 1965) are emitted from the core itself. The main observed light variations originate in this region. Large concentrations of matter are basically unstable; however, a number of models have been developed in which final collapse is postponed or avoided by rotational motion, subconcentrations, etc. Such models cannot make use of the main characteristic of a large mass—its gravitational potential energy. Indeed, this is generally no loss, as the actual possibility of a direct re-emission of the gravitational energy during and mostly at the end of the collapse, in the form of a rebounce or of high-energy particles (and neutrinos) ejection, is hindered by the increasing closeness to the Schwarzschild radius; gravitation becomes so strong as to stop anything from coming out. It is somewhat frustrating to observe that this same factor intervenes to stop the collapse at densities which do not allow the more direct means of energy generation to take place, such as the production of mesons in strong interactions. The core model represents the only situation where the Schwarzschild radius and strong gravitational fields do not forbid such high densities and energy emission even when the massive object is entirely within its Schwarzschild radius. RELATIVISTIC TREATMENT OF A CORE
The most general line element with spherical symmetry in co-moving coordinates is of the form ds2 = - e V r 2 - e"(dd2 + sin2 0) + e'dl2, (D where X = \(r,f), etc. (r is the radial variable and t denotes the time). The resulting field equations (Landau and Lifshitz 1962) can be readily integrated either if the pressure p = 0, or, at most, is a function of the time, i.e., p = p(t) only. In either case, v is then only a function of the time and we can set ev — 1, which amounts to a recalibration of the time t. We also find that (Tolman 1934a, b) 6
(dn/dr)2
6
if2"
"
'
(2)
where/ = f(r) is as yet an arbitrary function of r. The remaining field equations can be integrated provided we impose an additional relation between the pressure p and density p. We now suppose that this model represents an inhomogeneity within an ordinary homogeneous isotropic expanding universe for which the line element is give by ds2 = c2dt2-R2(
) [dl2 + l2(dd2 + sin2 6)],
^
(3)
where I is the dimensionless radial variable I = r/n and R — R(t). The appropriate field equations, valid for r > TQ, the outer boundary of the inhomogeneity, are then (Tolman 1934a) 1 fdR\2
8TTG
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MODEL FOR QUASI-STELLAR SOURCES
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Across the boundary r = ra we also have to satisfy the junction conditions (Synge 1960) which here reduce to ~, ,
OTTG
,
.
Gi1 = —4— p — A = (continuous)
(6)
and (7)
where la = ra/r0. As has been shown by Bonnor (1956) in the pressure-free case (and is certainly true in the more general situation), the conditions (7) do not determine the functions X and JX uniquely but give us the possibility to satisfy a similar set of boundary conditions at, say, r = n < ra. For the region r < r»we can thus choose a homogeneous expanding model of the form (3) but with different constants which is separated from the exterior uniform expanding universe r > ra by a transition region n, < r < ra for which the more general form (1) holds. In this model the interior region (r < n) contains the expanding core of the QSS, while the transition region, which can be made as small as we please, contains the radiating envelope. (This interpretation allows us not only to change the sign of k between the two expanding homogeneous regions but also to take into account more complicated processes in the transition region. A similar approach has been used by Bonnor [1956] to account for condensations in pressure-free models of the universe.) We shall now consider several specific models and see whether the conditions pertaining to an expanding core of proper radius R in a time of the order 1014 < T < 1017 sec can be met. This is represented by the dashed region in Figure 3. P R E S S U R E - F R E E CASE
In order to avoid any discontinuities of the pressure whatsoever at the boundaries r = n and r = ra, we shall take p = 0 throughout. (Clearly, this is not too satisfactory and we shall consider examples of different equations of state below.) In this case the remaining field equation can be integrated to give 1 r t=
lJ
d c'2
(f-l+2Fe-^+^Ae")^+H{r)'
W
where F and H are additional arbitrary functions of r and where we have introduced the cosmological constant A. The simplest case arises if we set/ = 1 and take A = 0. In fact, this is the well-known solution of Oppenheimer and Snyder (1939) corresponding to k = 0. A short analysis shows that this solution leads to a radius R(t) of the form R(t) = (at +
b)1's,
where the constant a is related to the mass"M..For a mass M — 107 Afo this gives a = 1022 cm3/2/sec and is therefore inconsistent with our requirements. In the general case, for which A = 0, the integral (8) can be written in the form (Bonnor 1956) V V k = l, o2>0, e"/2 = — ( 1 - c o s i / ' ) , t = —A\l>-smf)+H, (9a) a' ca6
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YUVAL N E ' E M A N A N D G E R A L D T A U B E R
a2<0,
k = - \
e"/2 = - ^2 ( c o s h f - D , a
t = — -6 ( s i n h i A - ^ ) + H, ca
(»«>)
where we have set 1 — p = a2. The corresponding solution valid either in the interior (r < n) or exterior (r > ra) region is given by k= l
R = K (I-cos
6),
k = - l ,
R = K (cosh 6-1),
/ = —(0-sin 0)+C,
(10a)
K. * = —(sinh 0 - 0 ) + C .
(iob)
The constant K is related to the mass of the core through K Logl0Rcm
tr-
1
2GM 37TC 2
ior k = 1
present
and
K = —r— 4C2
for k = — 1
conjectured size of universe
10' 102 103 104 105 106 Kf
10s 109 10'° 10" 10'2 10° 10* X)'5 70'" 10'7
Log,0tsec FIG. 3.—Study of lagging cores with p = 0. E: models with k = 1; H: models with k — — 1; m: core mass of 106 MQ; M: core mass of 1011 MQ; M': core mass of 1018 MQ. Shaded area represents QSS requirements.
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MODEL FOR QUASI-STELLAR SOURCES
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(the difference in the two cases arising from the different radial factor in [3]). The boundary conditions (7) can be satisfied provided we take
ab = lb{\ + Wy'\
4'{rb,t)=6(i),
Fh = lb*K ,
H„ = C,
and
H'b = 0.
The solutions given by expressions (9a) and (10a) are oscillating ones with a period T = wK/c and a maximum distance Rm = IK. For masses of the order of 10 6 -10 n Me, the values of K which are obtained are not sufficient to give either the desired radius or time necessary to describe the expanding core (see Fig. 3). Only metagalaxies of 10171020 Mo may have the required size after 1013 sec; they pass through this stage while collapsing after their maximum expansion. Even for k = — 1 (eqs. ([9b], [10b]) a simple calculation shows that the effective R~ ct rate of expansion cannot reproduce the data (see Fig. 3). MAXIMAL-PRESSURE CASE
It is clear that pressure-free disturbances cannot be used for a realistic description of the delayed explosions. In fact, Zel'dovich (1961) has demonstrated that the maximum concentration of matter is achieved by an equation of state of the form P resulting from a vector field. Such an equation of state might be correct at the early stages of the explosions when the density is at least of the order of 1015 gm/cm3. If we also assume that p = p(t) only then the distribution is not only homogeneous but also isotropic; the field equations (4) and (5) pertaining to the inside distribution can be readily integrated. However, as the pressure in the exterior region (r > ra) vanishes (or at most takes those values for a weak gas), it is impossible to match the boundary condition (6), and we arrive at a discontinuity of the pressure. Nevertheless, the model is of some interest, since it predicts solutions which attain a maximum distance before collapsing to the singularity. In the absence of the cosmological constant A we find in analogy to (10) the solutions k = l,
R=Asm6,
/ = ( — J ( 0 - § sin 2 0 ) ,
(12a)
k=—l,
R=Asinh6,
/ = ( — j ( J sinh 0 - 0 ) ,
(12b)
where the constant A is given in the two cases by
--omom *•*-' - -=(^)(f) *"—*•• i?i being an arbitrary distance marking the boundary of the distribution at t = 0. Although it is now possible to fit the required radius R, a simple numerical analysis shows that the law of expansion is of the form
R~ct and thus unable to fit the data.
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YUVAL NE'EMAN AND GERALD TAUBER EFFECT OF COSMOLOGICAL CONSTANT
So far we have neglected the effect of a cosmological constant A which can be introduced into the field equations. For example, in the pressure-free case equation (8) reduces to t=
\/xdx
,
T I , 3 „3 •(x -x -; V (*
r-j—^TT.,
03}
+ V)112
where we have introduced the notation r - x = (Ac2/3)1/2,
e"'*=ax,
a2 = ^ ( l - / 2 ) , A
T= ^-. a*A
The corresponding solution of equations (4) and (5) leads to (14)
- - / - , (y3-y \/ydy + y)112'
where 7?=Rv K
I5y
, 2
r=(Ac2/3)1/2
R2 = —
re
U
(Aec
/6)
v
I/2
=
7
STGPQRQ3 ^
^
.
49
-2
The period r = (Ac /3) leads to a value for A of the order of 10~ cm for r ~ 107 yr. and A ~ 10~65 cm -2 for T ~ 1010 yr. Since we want expanding solutions only, we must choose 7 > -£j. At the boundary r& we have
so that for A ~ lO"60 cm' 2 we have r > •& provided that the mass M > (31/2/81) 1033 gm which is reasonable. (The present value for Acos ~ 10~64.) The analysis for the maximal-pressure case is similar and leads essentially to the same results. It is possible to obtain the required values for the radius and time to describe an expanding core, but the needed value of A (for the interior solution) is much smaller than the currently accepted value of A„ for an expanding universe (Tolman 1934a)— other than A = 0, so that we may have a discontinuity at t = 0. PULSATING CORES
Since the lagging core (except in the presence of a cosmological constant) gives too fast an expansion to fit the experimental data, we shall now assume that the expansion after a time t is followed by a pulsation of the core. It is well known (de Sitter 1931) that in the pressure-free case there exist solutions which vary between zero and a finite maximum radius. This is also true for our maximalpressure case (both in the presence and absence of a cosmological constant). However, the presence of the Schwarzschild singularity forces the solution to expand to its maximum radius Rm and then to collapse back into the singularity R = 0 without being able to oscillate further. Of course, it is not clear whether at these high densities other processes might not set in, which prevent its reaching the Schwarzschild radius (Thorne 1965). This puts a severe limitation on the possibility of constructing a truly oscillating solution and on the accompanying equation of state. On one hand, both limits of the oscillation must be at a finite distance (the lower one larger than the Schwarzschild radius of the density distribution), and on the other hand neither the pressure nor the density must become negative in the region of interest. Tolman (1934a) has shown that for such solutions to exist within the framework of a homogeneous and isotropic cosmology, the pressure must increase during expansion. This can only be possible if the expansion is accompanied by an additional process, say, the accumulation of formation
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radiation, or the existence of a non-isotropic region at the boundary which acts as an elastic buffer. The pressure would then be sufficiently high at maximum to bring about a reversal of the motion. We then arrive at the following phenomenological model of the pulsating core: There exists a homogeneous region i? min < R < i? max in which the solution oscillates (the core proper) which is joined smoothly to a finite region, isotropic but not homogeneous, in pressure and density (the core envelope). For the sake of convenience, we shall also assume that in that enveloping region the pressure falls from its maximum value pmar_ to p = 0 at the outer boundary which can then be joined again smoothly to an exterior Friedmann universe. It is clear from this discussion that the onus of the existence of oscillating cores has been placed on the non-homogeneous region. It is certainly possible, in principle, to find solutions of the time-dependent field equations satisfying given boundary conditions and only mathematical difficulties and the lack of a satisfactory equation of state have so far prevented the exhibition of such solutions. (We follow Bonnor (1956) in the insertion of such a buffer layer.) PHENOMENOLOGICAL TREATMENT OF OSCILLATING SOLUTIONS
In the absence of an "exact" equation of state describing the interaction of the individual particles composing the "fluid," recourse can be made to a phenomenological relation between the pressure and density. Such a relation can take many forms, as long as certain conditions are satisfied. At high densities the most rigid equation of state is of the form
while at low densities a polytropic term could be added to the linear term - ^ = a p + /3p<"+1>/*. Even more generally, a power series like ci might be used. Instead of postulating arbitrarily an equation of state, we shall assume a solution for the density (and pressure) which gives rise to oscillating solutions and then see to what equation of state this leads. A possible form of the density p might be P = po(a0 + a1vp + a2vg + . . . + anv„)--, »5) »" where v — R/Ro is a dimensionless distance; a0,ffli,etc., constants to be determined; and p < q < . . . n integers which we shall fix. The pressure is of a similar form, except that the constants are no longer arbitrary, but must be so chosen as to satisfy equation (5). It follows that the pressure will be of the form 4 = -r~[ ao(n - 3) + ai(n - 3 - p) v" + a2(n - 3 - q) v" + . . .]
(i«)
with the notation described above. As before, the remaining field equation (4) can be reduced to a simple quadrature cJ [D(a0 + alv"+aiv"+. where we have set D = (8irG/3c2)Ro2p0.
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The simplest case (for which n = 2) can be shown to lead to non-acceptable solutions; but n = 4, p = 2, and q = 4 will lead to a simple oscillating solution, provided a2 is taken negative. If, for simplicity, we also let Oo = a\T)\a%\ we find
which will oscillate between the limits a— 1
a\
The density (15) then becomes
p=
S(t + a i x ~ | f l 2 1 * 2 )'
(19)
which will be positive within the region of interest, provided that 4 > a> 1 . On the other hand, the pressure (16) becomes ji = f^(^-a (-j1x — aiX + + 3\a 3 | ai2\x'), | x 2 J,
(20)
which will be positive and is found to increase as x increases. These last two equations, then, not only give the density and pressure as a function of x, but may also be considered as the parametric equation of state of the distribution. It is not difficult to see that elimination of x between equations (19) and (20) leads to 3T =
(2D
where we have defined -f
|a2|,
X = Oij3 = a\a2\,
The negative sign for the square root in equation (21) has to be used in order to agree with the values of the pressure and density at the limits. It can also be seen that for large values of the density (21) reduces to
in agreement with experience. We still require that our oscillating solution be always outside the Schwarzschild. radius characteristic of the distribution. Now, the definition of a Schwarzschild radius is only meaningful if we are dealing with an exterior solution smoothly joined to an interior one. Einstein and Straus (1945) have shown that it is possible to join a static exterior Schwarzschild field (in its conformally Euclidean form) ds*= - f l + | L Y ( a ' t t 2 + a V + < f e 2 ) + S | T W ^ r : | 2 ; cHP \ 2rJ (1 + m/2r)2
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MODEL FOR QUASI-STELLAR SOURCES
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to a pressureless homogeneous isotropic expanding universe of the form (3) provided t h a t
¥^-'-0+&)'(7iT 1
Here, M — (4ir/3)p0rQ is the mass which is a constant for a pressure distribution of m a t ter (Tolman 1943a) and n is the value of r a t the boundary. Inspection of equation (23) shows that the gravitational mass m is just the total mass M in units of length multiplied by (—g)1/2 evaluated at the boundary
= h3(l + l h 2 ) - 3 — Y-
m=— -V-g
< 23a >
Consequently, the Schwarzschild radius rs is given by D = — VbHl+iv^-Wo, 4
(24)
which must be smaller than R.
' ' -
1 V
"
< ^ ) « >
resulting in a condition on a and /3. In applying this approach to our problem, we are assuming that the amount of m a t t e r contained in the outer non-homogeneous region which is joined smoothly to the exterior Schwarzschild metric (22) is small compared with the matter contained in the interior oscillating region, so that the Schwarzschild radius of the combined distribution is still given by equation (24). However, we shall show that it is possible to satisfy an even more stringent condition than the one implied by equation (24). Taking % as the value of v at the maximum extension (18a) % = yw, we have to satisfy the inequality
Neglecting the multiplicative factor on the right-hand side and replacing 2(a — 1) by a, we arrive at the more stringent condition Dy < 2 .
(25a)
Substituting for y — <x//3 and taking a = 4 as the most unfavorable value for the inequality to hold, condition (25a) is certainly fulfilled as long as j8 > 2D
or
| o21 > 2 .
(25b)
Thus, it is possible to have an oscillating solution between finite limits such that the oscillation takes place outside the Schwarzschild radius of the distribution. (In fact, (eq. [25b] is more stringent than necessary and can therefore take account of the fact that the actual Schwarzschild radius is larger than rs given by eq. [24].) There still remains enough freedom in the constants tofitthe observational data. The period T is given by r =
—^<"Scp cD
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(26)
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YUVAL N E ' E M A N AND GERALD T A U B E R
Taking, for example, R0 = 1017 cms and p0 ~ \ X 10~~7 gm/cm 3 , we obtain a total mass M of the order of 10 u Mo, D = ^ 5 and a period MO/ exceeding T
= 5 yr.
(26a)
GENERAL DISTRIBUTION
Taking the blue stellar objects (Sandage 1964) to represent radio-quiet QSS and estimating their number at ~10 4 , we would get a rough figure of ~ 1 0 8 light-years for the average interval between inhomogeneities of the over-all geometry. Such intervals are large enough for the effects of those local inhomogeneities to be extremely weakened and leave the general uniform Friedmann universe as a good approximation, as long as only general features are discussed. It is worth noting that a lagging-core model with T = 1017 sec may have an evolutionary aspect which would explain the somewhat steeper count-luminosity curve observed for the QSS relative to the radio galaxies (Veron 1965). One of us (Y. N.) would like to thank Dr. Novikov for making available to him several manuscripts relating to this problem. He would also like to express his gratitude to the University of Miami and NASA for enabling him to present this paper at the Conference on Observational Aspects of Cosmology. REFERENCES Bergmann, P. G. 1964, Phys. Rev. Letters, 12, 139. Bonnor, W. B. 1956, Zs.f. Ap., 39, 143. Burbidge, E. M. 1965, Miami Conference talk (December, 1965). Bent, W. A. 1965, Abstract presented at the Miami Conference (December, 1965). Einstein, A., and Straus, G. 1945, Rev. Mod. Phys., 17, 120. Field, G. 1964, Ap. J., 140, 1434. Fronsdal, C. 1959, Phys. Rev., 116, 778. Fuller, R. W., and Wheeler, J. A. 1962, Phys. Rev., 128, 919. Greenstein, J. L., and Schmidt, M. 1964, Ap. J., 140, 1. Hovle, F., Fowler, W. A., Burbidge, G. R., and Burbidge, E. M. 1964, Ap. J., 139, 909. Hoyle, F., and Sandage, A. 1956, Pub. A.S.P., 68, 301. Kruskal, M. D. 1960, Phys. Rev., 119, 1743. Landau, L. D., and Lifshitz, E. M. 1962, The Classical Theory of Fields (Cambridge, Mass.: AddisonWesley Publishing Co.). Moffet, A. T. 1965, Abstract presented at the Miami Conference (December, 1965). Ne'eman, Y. 1965, Ap. J., 141, 1303. Novikov, I. D. 1964, Astr. Zh., 41, 1075; transl. in Soviet Astr.—.A.J. 8, 857 (1965). Oppenheimer, J. R., and Snyder, H. 1939, Phys. Rev., 56, 455. Sandage, A. 1961, Ap.J., 133, 355. Sitter, W. de. 1931, B.A.N., 6, 141. Synge, J. L. 1964, Relativity—the General Theory (Amsterdam: North-Holland Publishing Co."). Thorne, K. S. 1965, Science, ISO, 1671. Tolman, R. C. 1934a, Proc. N.A.S., 20, 169. . 1934S, Relativity, Thermodynamics and Cosmology (Oxford: Clarendon Press). Veron, P. 1965, Abstract presented at the Miami Conference (December, 1965). Wampler, E. J. 1965, Abstract presented at the Miami Conference (December, 1965). Zel'dovich, Ya. B. 1961, / . Exper. Theoret. Phys. (U.S.S.R.), 41, 1609; transl. in J.E.T.P., 14, 1143 (1962). Copyright 1967. The University of Chicago. Printed in U.S.A.
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COSMOLOGY
47.
151
COSMOLOGY (COSMOLOGIF.)
PRESIDENT: M. S. Longair VICE-PRESIDENT: 1. D. Novikov. OKGANIZING COMMITTEE. G. do Vautoulcurs, L. Gration, W. II. McCroa. G.C. McViiiic, C. W. Misncr, H. Nariai, Y. Nc'ciiian. D. \V. Sciaina, Ya. I). Zcl'dovicli.
6.
U N C O N V E N T I O N A L A N D P A T H O L O G I C A L WORLD MODELS
(Y. Ne'eman) In this section, we survey recent work in some of the more speculative approaches to Cosmology. Such attempts derive their motivation from the following premises: (1) trying to account for observational data in cases where there appear to arise difficulties in a straightforward utilization of the laws of General Relativity (GR) as applied to an assumed homogeneous and isotropic cosmology (i.e. the familiar Friedmann universe, FU). We list the main difficulties involved (although several of these may just be due to observational error): (a) apparent anomalies in red shift (z)/magnitude and similar counts, especially for quasars. (b) anomalous values of z in apparent quasar-galaxy associations, and within apparent groups of galaxies. (c) the large power output of quasars, in a cosmological interpretation of their z values; also, some radio-interferometry reports of faster-than-light velocities within some multiple systems. (d) the apparent high degree of isotropy of the 2.7 K Background Radiation (BR). (e) theoretical difficulties in accounting for the creation of the galaxies. (f) unexplained numerical coincidences, many of which were pointed out by Dirac and Eddington in the thirties. A new one is the equality (within 1-2 orders of magnitude) between the energy density produced by starlight, and the value for BR. (g) the emergence of an infinite-density singularity in the past, a situation in which we have no understanding of the laws of physics. (2) continuing the exploration of the wealth of cosmological solutions allowed by GR, and attempting at the same time to define the additional topological constaints which have to be imposed so as to exclude unphysical situations. This may in turn generate the need for new types of measurements. (3) in the absence to date of a theory of Quantized Gravitation, evaluating the possible effects of quantization, in an ad-hoc approach.
702
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COMMISSION 4 7
(4) attempting to fit in some of the symmetries (including their 'spontaneous' breakdown) of the Physics of Particles and Fields: T, CP, scale-invariance, the charge-space symmetries of the Strong Interactions, etc. The data mentioned in (1) cannot be considered at present to be in strong contradiction with GR; moreover, there exists, t o date, no alternative theory accounting for all the difficulties listed in (1). It is plausible to assume that future observations, together with further work on (2) (3) and (4) will improve the fit between theory (incorporating GR) and observations. Some of the developments we review here may contribute to this improved application of Covariant Cosmology. However, it seems both useful and healthy to explore in addition other possibilities, and we shall touch upon two non-Einsteinian attempts which present new versions of a Steady State model (thus answering in particular point ( l g ) ) : Segal's stationary universe ('Chronogeometry') and the Hoyle-Narlikar most recent reformulation of the original Steady State model. A. Doing away with our Singular Past by going beyond General
Relativity
Both non-Fisteinian models we mentioned refute the Big Bang ami thus attribute the 2.7 K BR to starlight (the effective Olbers sky). Both replace GR covariance by the assumption of conformal symmetry (i.e. the Poincare' group, scale invariance and 'pure' conformal transformations), with GR becoming a local approximation. In Segal's chronogeometry (Segal, 197 2, 1975), z is not due to a Doppler shift, and there is no expansion of the universe. We are back at Hinstein's original static closed model as far as cosmology is concerned (there is no corresponding dynamical theory appended to date). The universe has a radius ~ HQ' and finite volume, (and thus no Olbers paradox) and the red-shifts are due to an 'aging' of light as first suggested by Shamir and Fox (1967). Aging involves a distinction between two time-axes, T and t, t = tan T. 'Local' time, as used in our special-relativistic physics is r, and coincides with T (global time) for small values. The scale of T is derived from ( H 0 ) " ' . The theory claims to provide a better fit to z-magnitude counts, with a red-shift distance-squared law (the density distribution function is z 1 ' 2 ( 1 + z)~ 2 dz). Some of the difficulties we listed in (1) are alleviated, but not resolved: e.g. a very different z within a 'quintet' of galaxies (such as the W 172 chain) cannot be interpreted as representing very high relative velocities, but it now uniquely implies different distances and an illusory grouping. One important test of this model requires the existence of very old galaxies, since there has been no evolution from a BB. The order of magnitude fit between the values of the age of the Universe as given by nucleosynthesis, stellar evolution and H n is here an approximation due to the tan T distribution. The 2.7 K BR is assumed to represent starlight after very many scatterings and circumnavigations of the Universe. There is no direct explanation for the (perhaps incorrect) order of magnitude insufficiency of starlight density. Since there is to date no dynamical theory, a most important test would be the theoretical one of proving the dynamical stability of this static system. In the laboratory, experiments of the type suggested by Shamir and Fox (1967) should provide clearcut answers, when the necessary precision is reached. In the Hoyle-Narlikar (Hoyle 1974, 1975; Doyle Narlikar 1974) ('conformal gravitation') Steady-State model, masses are just local values of a massless scalar field coupled to massless matter fields, although it is not clear whether or not this should involve observable effects. The possibility of a change in the union's mass from mlt = mr lo m^ ~ 200 me has been suggested (lloyle, 1974). Such changes would imply very different processes in the early universe! On the other hand, the observed expanding universe is assumed to represent just one 'region' where the mass field has positive values. The preceding era had negative values for that field; however, it is assumed that the gravitational interaction was still the same, i.e. the coupling to the mass field also changes sign under time-reversal. We are reminded of a model with a CP inverted universe in the preceding phase, suggested in 1967 by F. Schucking. The usual singularity of the Friedmann model is replaced by the zero-mass interface of the two regions. The difference of a factor 1 0 - 5 0 between the observed density of BR and that which could be due to starlight is explained by the contribution of hydrogen burning in the previous era.
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Additional regions with negative values of the mass-field coexist with our region, in non-communicating space-sectors. One can speculate about the nature of the static (Newtonian) contribution of gravitation between sectors. As to horizons, (he vanishing of masses (and of Schwarzschild radii) ensures that they should not appear in the early stages of our era. This present version of the Steady State model is not yet a complete and consistent theory, and much remains to be done in order to fix on definite exact predictions which could allow a proper Popper-test. There are in addition several theories of gravitation which differ from GR (see the analysis of C. Will, I*>72; Thorne, Lee and Lightman, 1973) but their cosmological implications have not been explored. It is possible that some of these theories might also yield non-singular cosmologies. One such possibility has been pointed out by Trautman (1973) for an Einstein-tartan theory of gravitation. Here the singularity may be averted by all particle spins being correlated in the first I 0 " 1 0 s of the Universe (see also Raychaudhuri, 1975). In the oscillating FU model suggested by Golt (Gott, 1974), we also have a T (and CI') inverted past era (or region). However, because of the Ihcrmodynamical arrow of time, the 'inhabitants' of that era also think of themselves as residing in an expanding universe, rather than a contracting one. If CPT is conserved, this is consistent, provided they redefine matter as antimatter (Ne'eman, 1968, 1970. Aharony and Ne'eman, 1970, 1973). In addition, there are'spacelike' solutions representing a tachyon (Bilaniuk, Deshpande, and Sudarshan, 1962) universe, where one is doomed to experience the 'inexorable How of space' instead of our own feeling of the inexorability of the passage of time (Ne'eman, 1975). This is because our own always-allowed rest-frames (leaving only At ¥=()) are replaced by the availability of time-arrested frames (with only A . v ^ O ) . All of these regions are joined together analytically by getting rid of the singularity through the introduction of an ad hoc negative pressure term in the earliest moments ( I 0 ~ 4 0 s) of our own universe. Such a negative pressure would just stand (Parker and Pulling, 1973; see also Beckenstein, 1975) for as yet little explored effects (or an effective viscosity, as in some of the other models we shall mention), in the same spirit as the suggested emergence of an effective A cosmological constant from quantum effects (Sakharov, 1967). A positive non-zero A acts as a negative energy-density and can also cancel the singularity. Having noted the attempts at cosmological completeness under CPT, we should add that an important insight into a necessary connection with the Physics of Particles and Fields has come from attempts to check the cosmological implications of those models of Particle Quantum Symmetry-Breaking in which the theory is made invariant, whereas it is the vacuum which is assumed to break the symmetry ('spontaneous' symmetry-breaking like in Heisenberg's ferromagnet). In particular, Kobsarev, Grun and Zeldovich (1974) have shown that the isotropy of BR (up to \0~3) makes it rather implausible to believe that the C'P violation (or T violation) discovered in K meson decay could be 'blamed' on the vacuum. If that were true, we would have observed a domain structure in the universe, with 'walls' (i.e. anisotropics). However, this argument is proved as yet only in the simplest models. B. Phenumenological
Rehabilitation
within the Framework of Einsteinian
Law
The task of fitting the observational material (optical, radio, y. X-ray and microwave) to a conventional Friedmann universe (homogeneous and isotropic) is covered by the other reviewers. We shall only mention that although the various items in the data have still not reached the required confidence level, the impression one receives is that the normal range of Robertson Walker metrics will contain a model that will fit the general structure of the post HR universe. As to the earlier phase, this reviewer tends to adopt some of the more recent versions of the approach suggested by Misner (1968) of a homogeneous but anisotropic model later evolving into the present FU. The original model was received with scepticism, since it appeared the 'mixing' might be insufficient or too slow (Doroshkevich et al., 1971). The analysis by Belinskii and Khalatnikov (I 975) of an improved version in which the singularity is cancelled through an assumption of viscosity (effectively, a negative pressure, violating only the stronger of Hawking's criteria) following Matzner and Misner (1972) and Murphy (1973) indicates that such an evolution is highly plausible.
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As a philosophical digression, we note the failure of the 'Copernican Principle' as it was applied to the universe, namely that 'we do not occupy a privileged position in space-time' (Bondi, I960) or that 'we may assume that our view of the universe is not a preferred picture' (Kllis, l l )71). This was in fact the second, and more speculative principle, with the first just implying 'that local physical laws are the same everywhere in the Universe' (Kllis, 1971a). In all recent treatments (including the newest version of the Steady State) except for Segal's, and especially in treatments respecting GR, we are led to at least two different phases in the history of the last 1 0 - 2 0 billion years, and we may be observing in our distant past just that border region between the two eras. Our present conditions are only typical of the second phase; indeed it is only roughly typical of it, since the density of matter has changed tremendously within that phase itself. In the method of physical science, such symmetry assumptions as the second Copernican Principle, make excellent starting points for the unifying synthesis stage in the construction of a theory. They tend to be premature in the exploratory stage, where it seems more appropriate to adopt Hamlet's statement that 'there are more things in Heaven and Earth, Horatio than are dreamt of in your philosophy'. Returning to the pre-BR universe, present studies have exploited the homogeneous (but anisotropic) models of GR, as classified by Bianchi, Schuking and Behr (Bianchi, 1897; Schiiking, 1962; Behr, 1962, 1965a, 1965b; Fstabrook et al., 1968; Jantzen, 1975). Generally speaking, the anisotropy (such as having three different functions of t multiplying d.v2, dv2 and dz 2 in the metric of type I) presumably generates elongated light cones that manage to sweep away the horizons expected in the isotropic case. Treatments introducing macroscopic concepts such as the viscosity parameters still require a check of relativistic microscopic consistency, as a guarantee against non-local (and non-relativistic) effects. We leave the detailed discussion of the Bianchi types and of quantum effects to other reviewers, and treat in the broadest lines recent studies of pathologies. These apply topological and other methods in an attempt at improving our understanding of the structure and role of the primordial singularity, and the implications of time-ordering or of causality. C
Pathologies
The study of topological pathologies appearing in relativistic world model has gone through three main phases to date. In the first phase, investigations dealt with the question of the singularities' inevitability for the physical universe. After some optimisim generated by the work of Lifshit/. and Khalatnikov (1963), the more pessimistic (and fatalistic . . .) theorems of Hawking and Penrose (see Hawking and Kllis, 1973, ch. 8) settled the issue. Singularities are unavoidable, under some relatively mild conditions, e.g. the existence of a non-negative energy density ('the energy condition'), as measured by any observer, the inexistence of closed time-loops allowing a long-lived observer to return to his own past, as in the Gddel (1949) universe (i.e. an expression of causality), universal expansion, and a geometric generic condition. In the second stage, it was noticed that singularities did not always imply an infinite density of matter (Shepley, 1969). Some singularities appeared to represent catastrophic world-line features: rather than an infinite squeeze of the observer, they might just cause him to cease to exist after a given moment. Such vanishing (when a certain parameter intrinsic to the trajectory, the 'generalized affine parameter', would reach some given value) is called 'incompleteness' (or 'confined' incompleteness for that particular type, where the world-line is confined to one part of the model). This 'topological death' of the subject may take the form of an infinite spiral which never crosses the fatal moment [Taub-NUT space: (Taub, 1951; Newman et al., 1963)|. Another characteristic of this particular type of incompleteness reflects itself in non-llausdorff properties of the 'Schmidt-completion' (Schmidt, 1971, 1973) of that World-Model, a particular type of extension of the physical universe in which one has been able lo adjoin to space-time some unphysical 'missing pieces'. 'Hausclorff means that any two distinct points can respectively be embedded in two disjoint open subspaces (this is not true of the fork-point on the two branches of a half-opened zipper, for instance i.e. a zipper is non-llausdorff). For an observer's world-line this 'zipper behaviour' is not a healthy state and
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can he regarded as another type of split-personality (Majicek, 1971). However, there are less inconvenient non-Hausdorff models, and one may end up allowing some such classes in a physical model (Miller and Kruskal, 1973). On some space-time completion, this behaviour can be just a way of identifying 'confined incompleteness' singularities of the embedded physical model. The study of singularities and understanding of the connections between material and world-line types requires the invention of new and ever more powerful topological tools (Schmidt, 1971; Eardley, et al., 1972; Sachs, 1973. Duncan and Shepley, 1974, 1975), and we are still very far from achieving an understanding based on sound deductive methods. Meanwhile, the third and more recent phase has seen several inductive programs. With the availability of a complete classification of all homogeneous but anisotropic world-models (Bianchi-types) and a certain familiarity with these cosmologies in the context of the Misner program of 'mixmaster' isotropization, it became possible to search for characteristic singularities in these models. This was achieved in one way: singularity splits in two (with different evolutions). However, Eardley (1974) has looked at the 'lagging core' model astrophysically, and claims that such white holes can only be very short-lived (it would require M ~ It)' 7/l/(.* !o survive for I 0 7 yr, until the end of the period of recombination). They would attract photons and matter and would become black holes. Since no such black holes are known to lie around, there were probably no large while holes, according to Eardley, and the pre-BR universe cannot have been too inhomogeneous (C'arr, 1975) [quantum tunneling effects will have evaporated away lumps of less than I0 1 5 g only (Hawking, 1974; Page, 1975)1. Note that according to a recent study by Hawking (1975), black and white holes are indistinguishable to an outside observer, due to quantum statistical effects. See also Narlikar- Apparao (1975) for new white hole prediction. This problem is also related to the conjecture that GR forbids the existence of 'naked' singularities, i.e. singularities which arc not hidden by matter. Yodzis ct al. (1973) produce a counter-example, so that the conjecture cannot be true in its crudest form. One general conclusion from the above works is the importance of the equation of state. Actually, the final answer seems to depend only in part on the topological properties, and the physical conditions are an essential input. A further illustration of this state of affairs can be derived from a comparison between I he opposite results found by Collins and Hawking (I 973) on the one hand, and Belinskii and Khalatnikov (1975) on the other hand. Both studies were searching for an answer to the same problem: can the present 1 models in which the matter content of space-time is a perfect fluid, but with the fluid flow vector not normal ('lilted') to the 3-surfaces of homogeneity. The matter may thus move with non-zero expansion, rotation and shear (King and Ellis, 1973; Ellis and King, 1974; Collins, 1974). The Bianchi classification does not segregate world-line singularities from the others, and examples can apparently be found in all models. All non-lilted models (perhaps with the exception of type IX) have a matter singularity. At least some type V models have an 'intermediate' singularity, where the matter density does not diverge (though it may do it elsewhere in the model). There are as yet no clear-cut gains in understanding, as a result of this inductive method, beyond the first example (Shepley, 1969), although much effort has gone into the construction of 'tilted' models. Another inductive program has dealt with velocity-dominated singularities in irrotational dust cosmologies (Eardley et al., 1972). In these El.S models [including the Tolman-Bondi (Bondi, ll>47) spherically symmetric types] an iterative approach yields some information about 3-space at the time of the singularity (including 'dating', i.e. if there is more than one singularity, we learn about I heir relative timing). This represents an advance in the treatment of inhomogeneous models (see also Szekeres, 1975 for a generalization) in which there is more than one BB (Novikov, 1964; Ne'eman, 1965), i.e. a universe with large while-holes. ELS do indeed find one explicit model in which the singularity is 'regional' in the beginning, i.e. most of the universe is Eriedmann-like, while there is 'a lump' in one region for a while. In another example, the original Eriedmann universe has evolved naturally out of a 'chaotic' anisotropic model. The Cambridge team proved that the set of self-isotropization models is of measure zero, i.e. a Eriedmann universe is a most improbable outcome of dynamical evolution, and we
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exist by yet another Darwinian evolutionary chance-mutation, at the cosmological level. The Landau Institute team have just found just the opposite result: all possible anisotropic beginnings lead to the same inevitable Friedmann universe. This contradiction is due to the different physical assumptions: the Landau Institute team introduced viscosity and thereby removed the singularity altogether. This approach seems justifiable for the early universe; however, when the isotropic stage is reached there is probably very little viscosity left. Thus an improved model should lie in between the two calculations, and a more reliable answer should thus await the next iteration. It should also be interesting to check on the actual class of Friedmann models reached (since the Cambridge calculation points out the low probability of a model with escape velocity). In an entirely different line of work, some progress has been made in the study of the intrinsic global topology of space-time in view of explaining its causal structure. Hawking, King and McCarthy (I c>75) have suggested a topology determining the causal, differential and conformal structure of space-time, replacing a suggestion of Zeeman (1964). Zeeman had noted that the group preserving time-ordering along paths of free particles is the Poincare group and scale invariants. The new topology preserves such partial ordering (time-ordering, or along a null line) along arbitrary Feynman paths, i.e. interacting particles. This selects the full conformal group, perhaps at (he price of replacing continuity by countability in space-like directions, a point to be further studied. Yet one more line of topological investigation deals with the global geometry of the universe. For Friedmann world models, k = +1 (closed, spherical) fits only S3 (the 3-dimensional sphere) or 7''1 (real projective 3-space). For k =0(flat)only W3 (real 3-dimensional) is allowed; fork = —1 only H"' (open). For anisotropic models, the possibilities are much more numerous (see review in Criss et al., 1975). It is dangerous to exclude altogether inhomogeneous and anisotropic spaces, because they could look like Friedmann models locally (Fllis, l ( )7l). Our last remark relates to a social context. In recent years we have been exposed to strong criticism from both outside and inside the scientific establishment, for not caring enough about the 'relevance' of our work. Both Astronomy and Relativity Physics are considered as relatively parasitic in that utilitarian approach, and Cosmology is in an even worse position. However, a recent study (Tipler, 1974) discusses the gravitational field of a rapidly rotating infinite cylinder (van Stockum, 1937). This has closed time-like lines, as in the Godel universe (Godel, 1949). It seems possible to preserve these lines even for a very long finite cylinder, and we may have here a formula for a time-machine. This should certainly prove the usefulness of cosmology.
RLI LRF.NCLS Aharony, A. and Ne'eman, Y.: 1970, Lett. Nuoro Omento 4, 862. Aharony, A. and Ne'eman, Y.: 1973, in G. Iverson et al. (eds.), Fundamental Interactions in Physics and Astrophysics, Plenum Press, N.Y.-London, p. 397. Hehr, C. G.: 1962, /.. Astrophys. 54, 268. Bchr, C. G.: 1965a, 7.. Astrophys. 60. 286. Hehr, C. G.: 1965b, Astron. Abhandl. Hamburg Stemwartc 7, 249. Bckenstein, J. I).: 1975, I'hys. Rev. D I I, 2072. Belinskii, H. A. and Khalatnikov, I. M. 1975,./. l.xp. Vieorct. rhys. (V.S.S.R.) 69,401. Bianehi. I..: 1897, Mem. Soc. Ital. I)c. Sci. (Dei XL) (3) II, 267. Bondi, It.: 1947, Monthly Notices Roy. Astron. Soc. 410, 107. Hondi. II.: 1960, Cosmology, Cambridge University Press. Carr, H. J.: 1975, Caltcch OAP-389. Collins, C. H.: 1974, Comm. Math. I'hvs. 39, 131. Collins, C. 11. and Hawking, S. W.: 1973, Astrophys. / 180. 317. Criss, T. H., Malzncr, R. A., Ryan, M. P. and Shepley, L. C: 1975, in G. Shaviv and J. Rosen (eds,). General Relativity and Gravitation, .1. Wiley and Sons, N.Y. Toronto, and Israel Univ. Press, Jerusalem, p. 33. Doroshkevich, A. G.. Zel'dovich, Ya. H., and Novikov, I. IX: 1971,/ Kxp. Vieoret. Phys. (V.S.S.R.) 60, 3. Duncan, D. P. and Shepley, L. C: 1974, Nuovo Omento H, 24, 1 30. Duncan, I). P. and Shepley, L. C: 1975,/ Math. Phys. 16, 485.
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Eardlcy, D. M.: 1974, Pltys. Rev. Letters 33. 442. Eardley, D. M., Liang, E., and Sachs, R.t 1971,/ Math. Phys. 13. 99. Ellis, G. I. R.: 1971a, in R. K. Sachs (ed.), General Relativity and Cosmology, Academic Press, N.Y.t.ondon, p. 104. Ellis, G. I. R.: 197 lb, O H . Rel. Grav. 2, 7. Ellis, G. I . R. and King, A. R.: 1974, Comm. Math. Phys. 38. 119. Estabrook, l\ »., Wahlquist, H. D., and Behr, C. G.: 1968,7. Math. Phys. 9. 497. Gbdel, K.: 1949, Rev. Mod. Phys. 21, 447. Gott 111, J. R.: 1974, Astrophys. J. 187, 1. Hajicek, P.: 1971, Comm. Math. Phys. 21, 75. Hawking, S. W.: 1974, Nature 248, 30. Hawking, S. W.: 1975. Caltech OAP-412. Hawking, S. W. and Ellis, G. F. R.: 1973, Vtc large Scale Structure of Space Time, Cambridge University Press. Hawking, S. W., King, A. R., and McCarthy, P. J.: 1975, Caltech OAP-405. Hoyle, I-.: 1974, in K. Brechcr and G. Setti (cds.). High Energy Astrophysics and its Relation to Elementary ParticlePhvsics, Mil Press, Cambridge and London, p. 297. Movie, I-.: 1975, Astrophys. J. 196, 661. Movie. !•., and Narlikar, J. V.: 1974,/lcri'oH at a Distance in Physic-sand G>stnology, W. II. Freeman and Co., San Francisco. Jant/.en, R.: 1975, Princeton University senior's thesis. King, A. R. and Ellis, G. F. R.: 1973, Comm. Math. Phys. 31, 209. Kohsarov. Grim, and Zcl'dovich, Ya. B.: 1974. I.ilshitz, E. M. and Khalatnikov, 1. M.: [963, Advan. Phys. (Phil. Mag. Sup.) 12, 185. Malzncr. R. A. and Misner, C. W.: 1972, Astrophvs. / 171. 415. Miller, .1. G. and Kruskal, M. D.: 1973,7. Math. Phvs. 14, 484. Misner, C. W.: 1968, Astrophvs. / 151,431. Murphy. G: 1973, Phys. Rev. D 8. 4231. Narlikar, J. V. and Apparao. K. M. V.: \915, Astrophys. Space Sci. 35, 321. Ne'eman, Y.: 1965, Astrophys. J. 141,1303. Ne'eman, Y.: 1969, Proc. Israel Acad. Sci. 13, I. Ne'eman, Y.: 1970, Int. J. Thcor. Phys. 3, 1. Ne'eman, >'.: 1975, m Connaissance Scientifiqtic cl Philosophic. Academic Royale de Belgitiue, p. 193. Newman, E. T., Tamburino, L. and Unti, T. J.: 1963,/ Math. Phvs. 4, 915. Novikov, I. 1).: 1964, Astron. 7.h. 41, 1075. Page, I). N.: 1975, Caltech OAP-419. Parker, L. and 1 idling,S. A.: 1973, Pltys. Rev. U7, 2357. Raycliaudhuri, A. K.: 1975, Pltys. Rev. D 12, 952. Sachs, R. K.: 1973, Comm. Math. Phvs. 33, 215. Sakharov, A. D.: \961.Dokl. Akad. Nauk. 177. 70. Schmidt, B. G.: 1971, Gen. Rel. Grar. I, 269. Schmidt, B. G.: 1973, Comm. Math. Phys. 29. 49. Schi'tcking, E.: 1962, Hamburg Relativity Seminar. Segal. I. E.: 1972, Astron. Astrophvs. 18, 143. Segal. I. E.: 1975, Proc. Nat. Acad. Sci. USA, 72, 2473. Shamir. .1. and Fox (Opher). R.: 1967, Nuovo Cimento 50, 371. Shapley, L. C: 1969, Phvs. Letters 28A, 695. vanStockum. !.: 1937, Proc. Roy. Soc. Edinh. 57, 1351. Szckeres, P.: 1975, Comm. Math. Phys. 41, 55. I home, K. S.. Lee, I). I.., and Lightinan, A. P.: 1973, Phvs. Rev. D 7, 3563. Tipler, L. J.: 1974. Phys. Ret: D 9, 2203. Trautman, A.: 1973, Nature 242, 7. Will. ('.: 1972, Physics Today 25, no. 10, 23. Yod/is, P., Scitert, II. J., and Muller zum llagen, II.: 1973, Comm. Math. Phvs. 34, 135. Zecman, E. C. J.: 1964, / Math. Phvs. 5, 490.
M. S. LONGAIR President of the Commission
708
INFLATIONARY COSMOGONY, COPERNICAN RELEVELLING A N D EXTENDED REALITY
YUVAL Wolfson Raymond Tel-Aviv
NE'EMAN Distinguished Chair of Theoretical Physics and Beverly Sackler Faculty of Exact Sciences University, Tel-Aviv, Israel 69978
and Center for Particle Theory, Physics Department University of Texas, Austin, Texas 78712 ABSTRACT. "Eternal" Inflation has relevelled the creation of universes, making it a "routine" physical occurence. The mechanism of the Big Bang, from the conditions triggering it, to the eventual creation of the entire matter content of the resulting universe, involves no singular physical processes. However, causal horizons, due to General Relativity, separate the newborn universe from the parent universe in which it was seeded as a localized vacuum energy. The new universe's expansion only occurs "after" infinite time, i.e. "never", in the parents frame. This forces a reassessment of "reality". The two universes are connected by the world line of the initial localized vacuum energy, originating in the parent universe. Assuming that the parent universe itself was generated in a similar fashion, etc., an infinite sequence of previous universes is thus connected by one world-line, like a string of beads. 1. Copernican relevellings in Inflationary C o s m o g o n y Inflation [1-3] was conceived as the solution to paradoxes within conventional Friedmann cosmology and in the interface with Standard-Model inspired unified gauge (or string) theories. As spin-off, it yields in addition a mechanism for the Big-Bang, in the form of a "de Sitter model" exponential expansion, locally triggered by a large vacuum-energy in a microscopic region. This is the quantum field theory version of the "cosmological constant", in which the latter represents a quantum vacuum energy density (localized), e.g. a fluctuation of the Higgs field responsible for spontaneous symmetry breakdown of the G U T (at !OieGeV, in the presently experimentally favoured "minimal supersymmetric" version), i.e. of an inflaton field. Another unexpected bonus consists in the energy-conserving features of that same mechanism, in creating the full particle content of the universe (a null total energy throughout the process, with the gravitational binding energy cancelling the mass). Several aspects of Inflation represent a Copernican relevelling. First, the Big-Bangoriginated universe is very much larger (beyond the observational horizon at 1.5 X 10 10 light-years) than the observable "village". Secondly, it is eternal [2]: (a) our Big-Bang was born in an existing universe and (b) flatness ensures a quasi-eternal expansion. Thirdly, the "Big Bangs" are "normal" phenomena and occur stochastically, provided the conditions 559 M. Kafatos and ¥. Kondo (eds.), Examining the Big Bang and Diffuse Background Radiations, 559-562. © 1996IAU. Printed in the Netherlands.
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560 we described (large vacuum energies) happen to materialize. There are thus infinitely many "universes" - using the term to imply something like our observable universe plus its unobservable embedding (resulting from the same Big Bang). Very roughly, this is a return to the Bondi-Gold-Hoyle (1948) Steady-State universe, yet on an infinitely grander scale - and with continuous creation replaced by creation of spacetime and matter in discrete "bursts".
2. Classical General Relativity Horizons Stretch t h e T w i n P a r a d o x In this presentation, however, I shall discuss yet another revolutionary feature, which I recently pointed out [4]. This is a new conceptual watershed in our understanding of time, and even more so of "reality". We shall see that even though the "parent" and "offspring" universes are not disconnected (the offspring being the outcome of a "vacuum fluctuation" in a tiny region of the parent), the newborn will never develop and never 'exist' - within the eternal time frame of the parent! And yet it will exist in its own time frame, an existence with a time-stretch spreading over billions of years - years that will never "come" for the parent's clocks. This leads to surprising metaphysical conclusions about our idea of reality, which has to be replaced by a new "surreality". We develop these points in the next sections. Before this, however, we review in this section the seeds of this conceptual revolution, as they already appear in the simplest problems in classical GR. Horizons, as produced by GR, have been thoroughly studied by Penrose and described in the literature [5,6]. The conceptual issue we discuss here is present classically in the simplest Schwarzschild horizons. After the discovery of the quasars, Hoyle et al.[7] suggested that their energy originate in the gravitational collapse of very massive stars. It was then realized that in the formation of a black hole, the collapsing matter never really reaches its Schwarzschild radius, in the reference frame of a distant outside observer A. For a quasar this is of the order of 10 16 cm, thus yielding a density of 10 -4 <7/cm 3 with very little chance for nuclear reactions to be initiated. This led to the suggestion that quasars are (extremely dense)white holes, rather than such rarefied black holes [8]. Returning to the collapsing star case, its matter accumulates as a shell close to the Schwarzschild radius, gradually becoming infinitely red-shifted, with time-dilation causing it to emit less and less all the time. The whole of A's 'eternity' then corresponds to one hour, in the reference frame of an observer B in the collapsing star, falling into the black hole. This is a common GR extension of the twin paradox of Special Relativity. In B's frame, however, things happen very fast: the Schwarzschild radius is reached and crossed within that hour and after another comparable stretch of B's time, he (or she) disappears in the r = 0 (classical) singularity. A metaphysical problem then arises - namely 'when' does this last half-hour of the collapsing star occur? Clearly, half an hour after the end of (our) time! The issue disappeared, however, when, as a result of the work of J. Bekenstein [9] and of S. Hawking [10], it was realized that quantum black holes, unlike the classical ones, evaporate away through quantum tunneling and through pair creation at the microscopic level. This causes a gradual shrinkage of the Schwarzschild radius and the vanishing of the horizon. Thus, the issue is only marginally present in black hole physics. The constraints fixing the size of this contribution do not allow us to go into the actual formalism and we refer the reader to the publications listed in ref. [4]. We also recommend gaining insights through the study of the Schwarzschild solution in KruskalSzekeres coordinates [11] and through the latter's adaptation to de Sitter geometry by Gibbons and Hawking [12].
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561 3 . N o n - o v e r l a p p i n g T i m e - E x t e n s i o n s in " E t e r n a l " Inflation We now come to the related conceptual revolution with respect to time, in the context of Eternal Inflationary Cosmology [2]. The model assumes that the first stage (lasting some 10~ 3 5 sec) of a Big Bang follows a de Sitter Model (i.e. an exponential expansion), triggered by a large quantum vacuum energy-density A = < 0|V($)|0 >; V is the potential of the inflaton, e.g. the 'upper' Higgs at EGUT = 10 16 GeV. The scale function S(t) is then given by S(t) = exp(Ht), with Hubble constant H — (87rGA/3c 2 ) 1/2 . For this stage to last for a brief instant only and then to transit into the Friedmann model we observe, the vacuum energy 'trigger' has to correspond to a 'false' vacuum (e.g. the symmetric $ = 0,V = 0 solution for the quartic potential of the Higgs field), reached through a supercooling-like unstable procedure and easily replaced (through tunneling) by the true vacuum and Friedmann's slowed expansion. The 'falseness' of that vacuum is a necessary but not a sufficient condition as a 'gracious exit' from the inflationary regime proceeds through the merger of 'bubbles' of 'true' vacuum, forming inside the prevailing 'false' vacuum - a merger which has to overcome the exponential growth of the interbubble intervals. In the latest version [13] this is achieved by assuming Einsteinian gravity to represent the low-energy (long-range) regime of an Affine [14,15] or Conformal quantum gravity. Newton's "constant" is then G = (16x < 0|CT|0 > ) _ 1 / 2 , o{t) the (Brans-Dicke like) dilaton field. In the Planck-energy regime of the de Sitter stage, G isn't yet 'frozen' at this present value; the increasing a and decreasing G then decrease S(t), letting the bubble-merger process catch up. Let us follow the birth of a new universe [16,17]. A vacuum fluctuation occurs (e.g. as the energy concentration in a topological defect, such as a cosmic string). The dimensions of this trigger could be as small as 103 - 109 Planck lengths. At the end of the inflationary stage it will have reached the size of an orange - and 10 10 years later (in its own frame B) it will look like our observable universe. Outside observers A will just note the creation of a tiny black-hole like object, with only the very beginnings of an expansion, lasting in this state "forever", i.e. while t - » o o - very much like the case of the Schwarzschild horizon above. Our entire universe is an A frame and will never see the transformation of that tiny false vacuum region into anything else. However, for an inside frame of reference B, we have the birth of a de Sitter universe, a Big Bang, followed by the exit phase, then evolving into a new Friedmann (flat) universe - and perhaps, some 10 10 years later, astronomers discussing horizons and concepts of reality. Note that classically, the new universe would have involved a singularity (a time-like half-line) due to the Penrose theorem - except that quantum tunneling makes it now possible for that budding 'world' to escape the theorem. In one such solution [16], the new universe starts with a configuration which, classically, would make it recollapse without inflation, a true black hole; would thus have reached its ordained singularity in the future. Instead, however, it quantum-tunnels into an exponentially inflating solution whose classical singularity would have lain in the past, thus avoiding the singularities altogether. It then goes on to make a universe, with the latter carrying no singularity 'blemish' and being in no way different from its parent, "our" present universe. Presumably, this is also how the universe we live in came into being, with an eternal lifetime and with no singularities. We should thus extend the Principle of Covariance to all such universes. They are all eternal except that this is meaningless within our present conceptual framework: the new universe will never exist, in our frame A, in all our time; and yet it is as good as our own universe, will have (in its B frame) galaxies, suns, astronomers and physicists. So where and when does it exist? Note that the time variables in the two frames overlap before the "happy event" which triggers the birth of a universe. They then separate, B going it by itself,
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562 observing A fading away, flashing out its eternity in the infinitely red-shifted environment of the new Big Bang.. 4. Transcendant t i m e and Surreality There is, presumably a countable infinity of such "eternities", branching out from each other, then separating, with the offspring, eternally "incubating" - without ever being born - in the parent universe's reality. [Semantically, real as against abstract implies existence in spacetime as perceived in the user's frame, which justifies our discussion of the effects of the above de Sitter horizons on reality.] And yet, beyond the parent's eternity, there is another full-fledged universe, the offspring, flourishing and "realizing itself. This new picture calls for our conceptual framework to admit "surrealism", i.e. "existence" beyond our subjective space and time [4j- Note that in the direction of the past, there is one world-line tying together all past eternities. (This construction misses 'brother' or 'cousin' universes, selecting only the line of direct descendance). We may use a "transcendant- time variable" r (A refers here to a frame in the n~th universe, distant from the point where the n+l-th universe will be born), r = J2A*,nez(n)" arctan[tanh(tAn)] for a linear sequence. Z(n)" denotes all integer values between 0 and n. This time-resembling variable spans surreality; the genealogy of universes can be represented by (r, J2 tn). References 1. A.H. Guth, Phys. Rev. D 2 3 (1981), 347; A. Linde, Phys. Lett. B108 (1982) 389; A. Albrecht, and P.J. Steinhardt, Phys. Rev. Lett, 48 (1982) 1220. 2. A. Linde, Phys. I e « . B 1 7 5 (1986) 395. 3. Recent reviews of Inflationary Cosmology: A.H. Guth, Proc. Nat. Acad. Sci. USA, 90 (1993) 4871; A. Linde, in Gravitation and Modern Cosmology 1991, A. Zichichi, ed., (New York: Plenum Press); P.J. Steinhardt P.J., Class. Quantum Grav., 10 (1993) S33. 4. Y. Ne'eman, Found, of Phys. Lett., to be pub. ; also "Inflation, the Top and all that" (Erice lectures, May 1994) to be pub. ; "Time after Time in Eternal Inflation" PASCOS 94 contribution, to be pub. 5. R. Penrose, Phys. Rev. Lett. 14 (1965) 57. 6. S.W. Hawking and G.F.R. Ellis, The Large-Scale Structure of Spacetime, Cambridge Un. Press, Cambridge (1973). 7. F. Hoyle, W.A. Fowler, G.R. Burbidge and E.M. Burbidge E.M., Ap. J. 139 (1964) 909. 8. I.D. Novikov I.D. Astr. Zh. 41 (1964) 1075; Y. Ne'eman Ap. J.141 (1965) 1303; Y. Ne'eman and G. Tauber Ap. J. 150 (1967) 755. 9. J.D. Bekenstein, Lett. Nuovo dm. 4 (1972) 737; Phys. Rev. D 7 (1973) 2333. 10. S.W. Hawking, Comm. Math. PhysA (1975) 199; Phys. Rev. D 1 4 (1976) 2460. 11. M.D. Kruskal M.D., Phys. Rev. 119 (1960) 1743; G. Szekeres, Pub. Math. Debrecen 7 (I960) 285. 12. G.W. Gibbons and S.W. Hawking, Phys. fle^D15 (1977) 2738. 13. D. La and P.J. Steinhardt Phys. Rev. Lett. (1989) 62 376. 14. Y. Ne'eman and Dj. Sijacki, Phys. Lett.B200 (1988) 286. 15. F.W. Hehl, J. Dermott McCrea, E. Mielke and Y. Ne'eman, "Metric Affine Gravity", to be pub. in Physics Reports. 16. E. Farhi, A.H. Guth and J. Guven, Nucl. P/n/s.B339 (1990) 417. 17. W. Fischler, D. Morgan and J. Polchinski, Phys. Rev. D 4 2 (1990) 4042.
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Gravitation & Cosmology, Vol. 6 (2000), Supplement, pp. 30-33 © 2000 Russian Gravitational Society
HEURISTIC M E T H O D O L O G Y FOR HORIZONS IN GR AND COSMOLOGY Yuval Ne'eman Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel Aviv, Israel Center for Particle Physics, Physics Department, University of Texas, Austin, Texas, USA
Abstract I use an intuitively helpful methodology based on the approximation of Special Relativity with Newtonian Gravity (SRNG) to improve one'B understanding of dynamical horizons and their algebraic characterization, from black holes to Cosmology. I derive both Friedmann's flat-case cosmology and the Inflationary model from such considerations. I then comment on the present observational results.
1
Horizons and Equivalence Principle (local) considerations
The Schwarzschild metric da2 = c2dt2 - S3(t){[dr2/(l
- kr2)} + r2[d92 + sin26d
(1)
is the solution of Einstein's equation R^ - ( 1 / 2 ) ^ + &*G/c*)\giu,
= (SnG/c*)^
(2)
for an energy-momentum tensor representing a spherical mass (the "Kepler problem"). Note that when still in Cartesian coordinates, x already represents a (physically) curvilinear system, the diffeomorphic active transform x(X) of the flat Euclidean coordinates. X (capital letters will indicate flat coordinates). Writing it, as is usually done (1), in polar coordinates, represents yet another diffeomorphism though passive, or formal, t{x), r{x), 9{x), <j>{x). These are thus curvilinear polar coordinates. Due to the spherical symmetry of the right-hand-side of (2), the 9,
= A(r)cosh(t/2ra), = A(r)sinh(t/2r,), = [(r-r.l/r^xe* = 2r,tanh~1(v/u), \v/u\ < 1; = 2r,tan~1(u/v),\v/u\ >1
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(3)
Heuristic Methodology for Horizons Yuval Ne'eman
31
with the metric ds2 = 4rje- r / r ' {-dv2 -f- du2) + r2{d92 + sinHd*2) (4) The new coordinates u(t, r), v(t, r) thus represent a flat Minkowski 2-dimensional manifold, with the t-dependence of u, v and the r dependence making it an accelerated frame. We apply a further (Fourier) transformation and reidentify the pseudo-plane with Minkowski (one-space and one-time) momentum space. In this flat manifold, the right-hand quadrant u 2 — v2 > 0 holds timelike momenta (i.e. real particles), the upper quadrant u 2 — t>2 < 0 contains spacelike momenta (tachyons) and the two t = ±oo, r = r, asymptotes represent the light-cone. The infinite time-dilation displayed by a collapsing star in its approach to the Schwarzschild radius (as observed from outside and at a certain distance) is thus the Kruskal-Szekeres transform of an attempt to accelerate (in flat Minkowski spacetime) a massive particle towards the speed of light. The three sectors - namely the "outside Schwarzschild solution", the "precise black hole" and the "inside Schwarzschild solution" in the u,v plane are thus the Equivalence Principle flat transforms oft,r and relate the above three Schwarzschild solutions to the three orbits of the u, v pseudoplane Poincare group (massive particles, photon-like massless particles, tachyons). All of these considerations undergo some "radiative corrections" caused by Bekenstein-Hawking radiation. We may now formulate a generalized theorem, as a conjecture, namely that all dynamical horizons in General Relativity ore locally related through the Equivalence Principle to the orbits of the Poincare group in flat Minkowski space.
2
Global considerations
The approximation represented by the combination of Special Relativity with Newtonian gravity, which we shall denote by SRNG has often been fruitfully utilized as an insight-providing methodology (it was also Einstein's 1907-1911 ad hoc physical paradigm). Special Relativity is important here especially through its giving rise to the concept of the invariant rest-mass M and of its energy EQ = Mc2. In his pre-GR evaluation of the deflection of light in a strong gravitational field, this allowed Einstein to assume that Newton's force of gravity couples to energy rather than to mass - and to generalize by coupling it to light via its energy E = hv. The "exercise" yielded a solution amounting to one-half of the subsequent General Relativity prediction, which was observationally validated by Eddington's 1919 Principe Island Solar Eclipse expedition. We apply the approach to the definition of a Black Hole, or more generally, of a GR horizon. This limiting state of collapse should correspond to the absolute value of the (negative) gravitational potential energy becoming so large as to cancel the entire potential energy, namely the invariant rest-mass energy: Mc2 - nGM2/R0 = 0 (5) where n is a numerical factor of the order of unity and R0 is the stellar radius corresponding to this situation and for which we solve, obtaining R0 = nGM/c2. In precise GR, one obtains n = 2. Another such approach, involving somewhat less than the SRNG approximation, has been known ever since Mitchell (1784) and Laplace (1796) followed it, in conceiving the (theoretically possible) existence of Black Holes - without having recourse to GR. In that calculation, one poses the problem of the evaluation of the escape velocity from a massive star and one then asks what should the star's radius be (for a given mass) for the escape velocity to be incrementally larger than the velocity of light, light is thus effectively confined and the star occluded. Equating the centrifugal force to the gravitational attraction, we have in general mv2/R — GMm/R2 and for the case v = c we get for the (future, for Mitchell and Laplace) Schwarzschild radius, R, = GM/c2, i.e. again precisely one half of the GR value.
3
Friedmann Cosmology, the de Sitter model and Inflationary Cosmogony
In Cosmology, S(t) is now the "radius" or "scale function" of the Universe, and the Friedmann (grandson)/Robertson-Walker spherically symmetric metric is written as ds2 = c2dt2 - S2(t){[dr2/(1
- kr2)] + r2\de2 + sin2d
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(6)
32
Early Universe, Quantum Gravity and Cosmology
Defining V = S', A = S", where each "prime" sign stands for one differentiation by the time variable, Priedmann's application of Einstein's equation (1) to cosmology produces (V/S)2 + kJ/S2 2(A/S)+
= (87rG/3c2)[p(t) + A]
{V/S)2 + Jfe^/S2 = {8nG/c2)[-p(r,t)
+ \)
(?)
It is relatively easy to identify the pieces in transiting to our SRNG approximation. We multiply the first (energy) equation by (MS2)/2 and transfer to the left the gravitational potential term from the right-hand side, replacing it on the right with the (fc/2)Afc2. We also replace M = (47r/3)/>53/c2 and again have the interpretation "kinetic energy + (negative) gravitational potential energy (up to a numerical factor of order 1) = — (k/2)Mc?. We can now, e.g., derive the value of the critical energy density and pc by making that same M = 0 (and/or k = 0) assumption. As a matter of fact, for this last purpose, we could even take the energy-equation incrementally, galaxy by galaxy, avoiding the numerical factor. Thus, writing mv2/2 — GMm/S = 0 and defining a "Hubble factor" as H(t) = v(t)/S(t) we may replace t; = HS and get, pc = (3c2/8nG)H2
(8)
Returning to the first Friedmann equation with k = 0, M = 0, we assume that the Universe emerges from a minimal seed (a vacuum fluctuation) and thus has p(r, t = 0) = 0. We get the de Sitter model, i.e. a constant Hubble factor H, H2 = (87TAGV3C2 (9) which implies the differential equation S'(t) = HS(t) and the Inflationary exponential solution S = em
(10)
Since the early 1980-s, there appears to be a concensus in favour of Inflationary Cosmology [1] - and going further, from philosophical considerations, in favour of the Eternal version (due to A. Linde [2]) with a suitable Cosmogony. This is a return to the philosophically satisfactory (but phenomenologically disproved, with the Penzias-Wilson discovery of the 3K background radiation) Steady State Model, suggested in 1948 by F. Hoyle, H. Bondi and T. Gold. It was the first concrete realization of a "steady state" model, to my knowledge preceeded only by Gersonides' - Rabbi Levi ben Gerson, astronomer and astrologer to the Pope in Avignon around 1300, inventor of the sextant ("Jacob's staff) and Popperian demolisher of Ptolemy's epicycles, thus "clearing the decks" for Copernicus. Levi ben Gerson reached the conclusion of an eternal and continuous creation, just from intuitive conceptual arguments; he was also the only astronomer before the 18th century who estimated correctly the distances to the nearest stars. Yet there was a precursor to Eternal Inflation with the de-Sitter type mechanism. The idea that matter enters the Universe at some specific points, rather than 'everywhere', as in the Hoyle et al. theory, this idea was raised qualitatively by Ambartsumian in the fifties. Later, when the first quasars were discovered, Igor Novikov [3] and I [4] independently suggested that these were lagging cores of the cosmological expansion. Working with the late Gerald Tauber [5], we actually developed this idea, using a de Sitter mechanism for each "bang". Our work was not in vain, as it was used and quoted by Harrison, in developing his version of the much-quoted Harrison- Zeldovich density fluctuations. Of course, thinking of the quasars, we had to force all "cores" to expand into one-connected geometry. The present picture repeats the structure but on a scale larger by a factor of a million at least, probably much more - and with each core expanding into its own geometrical arena, with a minimal connectedness to the rest. The above derivations involving a vanishing invariant rest-mass resolve a key issue in Cosmogony: how is matter created without violating energy-conservation. Pair creation had allowed for the creation of electrical and other charges without violations of the corresponding conservation laws, but it forced us to assume that huge amounts of antimatter have been created; the mystery of their abode was then resolved by the Nishimura-Weinberg idea of annihilation into the photons of the background radiation, with a smallish numerical advantage for "matter" (related to the CP violation) generating as a residue the present "matter" universe. The zero-mass cosmology allows for the creation and existence of the "observed" 10100 fermion particles without violating the conservation laws related by Noether's theorem to the Poincare group - namely energy, or linear and angular momenta.
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Heuristic Methodology for Horizons Yuval Ne'eman
4
33
The present picture and problematics
Very recently, measurements based on accurate observation of supernovae explosions [6] have been found to indicate that the rate of the universal expansion is on the increase. This has to involve the cosmological constant, which is unique in its repulsive action, in the gravitational context. Whether the actual mechanism represents some more sophisticated and as yet uncompleted "gracious exit" from inflation to a flat Friedmann model (to be reached in some distant future - or never), or if, alternatively, we are already in the flat Friedmann cosmology but will always continue to be affected by inner repulsive forces - in both cases the accelerated expansion is due to the presence of a sizable (vacuumenergy) cosmological constant component in the energy-density, of the same order, i.e. A = p(x). As against the 1950-1990 observational upper bound of A = O.Olp we now tend to admit A — p. This is just one side of the coin. The other is a profound clash with Effective Relativistic Quantum Field Theory. That theory is our main scaffolding - and although changes are not impossible, it can be assumed that it will remain so. And yet, ERQFT requires the A to take on values larger by a factor 10100! As a matter of fact, this paradox is one of the two incongruities which may usher in further revolutions, just as Lord Kelvin's two dark clouds (the null result of the Michelson-Morley experiment, and the failure of equipartition in the spectrum of black body radiation) ushered in Relativity and Quantum Mechanics. Either A is too large - or it is too small.
5
Dedication
This in absentia contribution is dedicated to Isaak Khalatnikov, colleague and friend, whether in Moscow or in Tel-Aviv. Upon this, his eightieth birthday, I wish him a happy and creative continuation, during the next 40 years he may expect according to Jewish tradition. I would also like to thank Prof. Khlopov for enabling me to present this contribution at the last minute.
References 1 A. H. Guth, Phys. Rev. D23 (1981) 347; A.D. Linde, Phys. Lett. B129 (1983) 177; D. La and P. Steinhardt, Phys. Rev. Lett. 62 (1989) 376. 2 A.D. Linde, Phys. Lett. B175 (1986) 395. 3 I.D. Novikov, Astr. Zh. 41 (1964) 1975. 4 Y. Ne'eman, Ap. J. 141 (1965) 1303. 5 Y. Ne'eman and G. Tauber, Ap. J. 150 (1967) 755. 6 G. Goldhaber, personal communication; see also C.J. Hogan et al., Scientific American, January 1999, 28-42 and in the same issue, M.A. Bucher et al., p. 43-48.
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C H A P T E R 10: F O U N D A T I O N S OF P H Y S I C S ALONSO BOTERO Departamento de Fisica, Universidad de Los Andes Bogota, Colombia As Yuval Ne'eman is fond of recalling, Kelvin's lecture at the Royal Institution in 1900, "Nineteenth-Century Clouds over the Dynamical Theory of Heat and Light", served as an ominous overture to the great physics revolution that was to take place in the forthcoming years. From two innocent "dark clouds", namely, the undetectability of the ether and the unaccountabihty of the black-body spectrum, emerged the two great theories of the twentieth century - Relativity and Quantum Mechanics. With over a century after Lord Kelvin's lecture, both theories have certainly proved their worth at dispersing more than a cloud at the phenomenological level. At the foundational level, however, the skies are not as clear. The challenge that both theories have posed on our classical intuition with respect to notions of reality, causality, locality, and individuality has been so radical, that we are still coming to terms with the ultimate implications of this new world-view. In this chapter, we present a selection of papers by Yuval Ne'eman and collaborators on some of these outstanding foundational issues, classified into three main categories. The Arrow of Time: The problem of the arrow of time (See e.g., [l]), although long-standing in the philosophy of science, becomes especially acute in the light of the new time arrows that have emerged from empirical observations in biology, cosmology, astrophysics, and microphysics. The reconciliation of these arrows, together with the classical thermodynamic arrow, is still a question of considerable debate. The papers in this chapter begin with a general review of the problem of irreversibility, aptly titled "The Arrows of Time" {10.1}. Emphasizing the variety of interpretations that are possible for the direction of time, five distinct "arrows" are here identified: Thermodynamic/Statistical, Cosmological, Evolutionary, Physiological, and Microscopic. The four subsequent papers explore in more detail some of these arrows and their interrelationships, starting with the microscopic time arrow, as first implied by the K° —> 2ir CP-violation [2] experiment (under the assumption of CPT invariance). Consequences of unitarity on the ranges of the CP-violation parameters [3] \rj+-/r]00\ and Ree/|?7_| | are presented in {10.2}. The next two papers, {10.3}, and {10.4}, explore the relationship between the microscopic and cosmological time arrows: under the assumption of CP-violation and CPT- invariance, an oscillating universe admits a time-reflection symmetry about the moment of maximum expansion together with label exchange between matter and antimatter, so that no absolute direction of time can be defined from microscopic physics in this context; on the other hand, full CPT violation in an oscillating universe allows for the unambiguous definition of a microscopic arrow. Finally, {10.5} conjectures a resolution of the paradox posed by
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the coexistence of the Thermodynamic arrow (increasing entropy, i.e, "disorder") and the Evolutionary arrow (increasing order, apparently decreasing entropy) by drawing an analogy with black-hole physics: a net entropy increase under gravitational collapse (increased order in matter) ensues after properly accounting for the entropy increase in the associated gravitational degrees of freedom. Quantum Non-Locality: The quantum-mechanical feature of Entanglement, first identified by Schrodinger [4] and popularized by the celebrated EPR-Bohr exchange [5; 6], has by now been well-established experimentally [7], even under space-like separations in the order of kilometers [8]. At the foundational level, however, entanglement still remains a great mystery. Particularly puzzling is the "peaceful coexistence" [9] between entanglement, non-locality and relativistic causality of information transmission. The papers {10.6}, {10.7} and {10.8} in this chapter explore the connection between entanglement and geometry. Any notion of physical correlation necessarily implies the comparison, at different points in space-time, of physical properties (spin, charge, flavor, color, etc.), transforming under the action of specific gauge groups. The natural geometric setting for non-local comparison is therefore not space-time itself, but rather the principal fiber bundle manifold (FBM) over space-time with the relevant gauge group as the structure group. As shown in {10.6}, the vertical direction in the FBM, devoid of any intrinsic causal structure, suffices to implement correlation and indeterminacy at the classical level. The idea is expanded in {10.7}, where further evidence for the relation between non-locality and geometry is given in terms of other non-local or global aspects (i.e., Aharonov-Bohm effect, topological charges, etc.). Finally, {10.8} emphasizes the essential (albeit at times, implicit) role played by closed space-time loops in the FBM approach and their relation to the operational realization of non-local effects. Collapse and the Classical/Quantum Interface: The Copenhagen interpretation axiomatically grounds quantum mechanics on the "measurement process", through which quantum observables are given unambiguous operational definitions in terms of macroscopic effects [6]. This standpoint relegates two scientifically legitimate questions to a realm lying outside of what can be deduced from the axioms of the theory: a) Wherein lies the division between microscopic and macroscopic? and b) Is there, and if so, what is, the physical process that accompanies the so-called "collapse of the wave function" (i.e., the irreversible, non-unitary, connection between pre- and post- measurement states)? These two questions are explored in the remaining papers of the chapter. The first paper {10.9} shows how, from the point of view of decoherence, the Planck dimensions (mass, length) provide two scales to distinguish the microscopic from the macroscopic with regards to localizability. Regarding the second question above, {10.10} and {10.11}, provide an "Ohmic" solution within quantum mechanics, by producing an explicit mechanism by which a single branch in an Everett-type [10] description can be realized without ap-
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pealing to ancillary collapse processes. The mechanism is a generalized phase-transition scenario, resulting from spontaneous symmetry breaking in the measurement interaction, together with a decoherence condition natural to large-iV systems, i.e., macroscopic objects, providing the required "thermodynamic" limit.
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References [1] The Nature of Time, edited by T. Gold and D. L. Schumacher Cornell University Press, Ithaca (1967). [2] J. H. Christenson, J. W Cronin, V.L. Fitch, and R. Turlay. Phys. Rev. Lett. 13 (1964) 138. [3] T. T. Wu and C. N Yang, Phys. Rev. Lett. 13 (1964) 380. [4] E. Schrodinger, Naturwissenschaften 23 (1935) 807-812 and 823-828. Translated in Quantum Theory and Measurement, J. A. Wheeler and W. H. Zurek (eds.), Princeton University Press, Princeton (1984), pp. 152-167. [5] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47 (1935) 777. [6] N. Bohr, Phys. Rev. 48 (1935) 696. [7] A. Zeilinger, Rev. Mod. Phys. 71 (1999) S288. [8] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 81 (1998) 5039. [9] A. Shimony, International Philosophical Quarterly, 18 (1978) 3. [10] H. Everett III Rev. Mod. Phys, 29 (1957) 454.
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REPRINTED PAPERS OF C H A P T E R 10: FOUNDATIONS OF PHYSICS
10.1
Y. Ne'eman, "The Arrows of Time", in Proc. Israel Academic of Sciences & Humanities, Section of Sciences, No. 13 (Jerusalem, 1969), pp. 1-13.
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M. Gronau and Y. Ne'eman, "Consequences of Unitarity in Some Models of CP Violation", Phys. Rev. D l (1970) pp. 2190-2191.
736
A. Aharony and Y. Ne'eman, "Time-Reversal Symmetry and the Oscillating Universe", Int. J. Theor. Phys. 3 (1970) pp. 437-441.
738
A. Aharony and Y. Ne'eman, "Time-Reversal Violation and the Arrows of Time", Nuovo Cimento Letters 4 (1970) pp. 862-866.
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Y. Ne'eman, "The Sign and Micro-Origin of Complexity as Entropy", Found. Phys. Lett. 16 (2003) pp. 389-394.
748
Y. Ne'eman, "Classical Geometric Resolution of the Einstein-Podolsky-Rosen Paradox", Proc. Nat. Acad. Sci. USA 80 (1983) pp. 7051-7053.
754
Y. Ne'eman, "The Problems in Quantum Foundations in the Light of Gauge Theories", in Honour of the 75th Birthday of John A. Wheeler, Found. Phys. 16 (1986) pp. 361-377.
757
Y. Ne'eman and A. Botero, "Can EPR Non-Locality be Geometrical?", in The Dilemma of Einstein, Podolsky and Rosen — 60 Years Later, Int. Symp. Honoring Nathan Rosen (Haifa, 1995), A. Mann and M. Revzen, eds. (I.P.P. and Israel Phys. Soc. Publ., Jerusalem, 1996), pp. 42-48.
774
10.9
Y. Ne'eman, "Localizability and the Planck Mass", Phys. Lett. A186 (1994) pp. 5-7.
781
10.10
Y. Ne'eman, "Decoherence Plus Spontaneous Symmetry Breakdown Generate the "Ohmic" View of the State-Vector Collapse", in Symposium on Foundations of Modern Physics 1993, Quantum Measurement, Irreversibility and the Physics of Information, P. Busch, P. Lahti and P. Mittelstaedt, eds. (World Scientific, Singapore, 1994), pp. 289-302.
784
Y. Ne'eman, "Classical to Quantum: A Generalized Phase Transition", in Proc. Int. Conf. Microphysical Reality and Quantum Formalism, Urbino, Italy (1985), F. Selleri, ed. (Kluwer Academic, 1988), pp. 145-151.
798
10.2
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Proceedings of the Israel Academy of Sciences and Humanities Section of Sciences, No. 13
THE ARROWS O F T I M E 1 by
YUVAL N E ' E M A N
INTRODUCTION
This lecture deals with a scientific Altneuland. The mystery of time, its nature and irreversible character were among the earliest questions to be raised by thinking man. Old man Chronos or the worm Uruboros represented partial answers in the age of mythology. The problem was then transferred to the philosophers, who tackled it as best they could, that is, by reflecting hard and trying to fit it into whatever scheme of things they favoured. With the engineer's approach provided by Carnot, it became a scientific issue. Indeed, by the end of last century, irreversibility was generally considered as a solved problem, thanks to Carnot's Second Law. In 1905, to the world's—and the philosopher's—profound surprise, time turned out to be less rigid and immutable than had been thought: relativity displayed its plastic nature, and the Twin Paradox still sounds like a fairy tale to most people. The arrow of irreversibility then became an issue again, as a result of the development of cosmology following Einstein's reconstruction of the Theory of Gravitation in 1916. But it was not until the summer of 1964 that a completely new challenge was provided with the discovery of possible irreversibility at the fundamental interactions level. More than ever, it is now clear that irreversibility and the existence of a time-arrow constitute a scientific problem, to be tackled jointly by physical experimentation and mathematical construction, the twin methods of physics. THE
FIVE
ARROWS
It is not unusual to see articles relating to the 'arrow' of time. I have used the plural form—because at the present stage we have in fact 1
Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR grant number EOOAR68-0010, through the European Office of Aerospace Research.
1
723
Yuval Ne 'eman five different arrows. I shall devote the first part of my paper to their nature and possible interrelations, all conjectural and requiring further calculation, observation and experimentation; the rest will deal with the latest problem, that of Microscopic Time Irreversibility as displayed by the Fitch-Cronin effect. The arrows are: A. B. C. D. E.
Thermodynamical (or statistical) Cosmological Evolutionary Physiological Microscopic THERMODYNAMICAL
TIME
The statistical derivation of irreversibility is well known. The most common way of illustrating it is the double-box experiment: we have a box divided by a partition into two halves. The three pictures in Fig. 1 show the molecules of a gas in this box at various stages.
Fig. 1
We are told that all three mirror free motion of the molecules. There is no doubt in our mind that the time-sequence goes from left to right and exhibits the situation after removal of the partition, with the gas originally inserted in the left half alone. The opposite could only be true if some external agent (a piston) was providing energy to thrust the gas back to the left. We would not be able to guess the timesequence if the number of molecules were small:
Fig. 2
In the large-statistics case, the probability that the molecules accumulate in one half of the box as a result of their random motions is practically nil. In the other case, the probability is high enough and there is no way to tell which direction corresponds to the time-sequence. For a given temperature (T) interval Tx to T2, a macroscopic thermodynamic 2
724
The Arrows of Time (idealized) reversible transition is characterized by the entropy change AS
r,
T
(1)
(rov)
(Q is the heat flow into the system) which is the same for all 'paths', namely, sequences leading from Tx to T2. For an irreversible process between the same states, the entropy change [Tl dQ
is now larger than the path integral. For a closed system, there is no dQ and (1) vanishes, and irreversibility along the transition makes AS > 0
(3)
in other words, the entropy of an isolated system always increases. This famous result of the second law of thermodynamics then leads to the expectation that an isolated system evolves until it reaches equilibrium. Classical statistical mechanics show that S is proportional to the logarithm of a state's phase-space volume, that is, it represents the probability of a transition to that state. In an isolated system, random motion brings about the most-probable (or 'most-random') distribution at the end. This is the state of highest entropy. COSMOLOGICAL TIME
At this point we turn to cosmology. In 1929, Hubble established his velocity-distance law, showing that Slipher's observation of red shifts in the light from distant galaxies bespoke an expanding universe. It turned out that an expanding—or a contracting—universe should indeed have been anticipated from the equations of General Relativity. The question arises: could the universe just as well have been in a contraction state? Is there, in fact, a contraction stage, either before the high-density singular state which was in existence some 10 billion years ago, or after the present expansion has reached its maximal extent and reverses itself (in some 30 billion years according to one recent estimate)? Will the astronomers in Anno Mundi 60.109 observe blue shifts? According to one line of thought, 2 they will not. They will think of themselves as being in the year 20.109, will observe red shifts and will have their time-arrow reversed! 2 T. Gold (1962) Recent Developments in General Relativity, New York-Warsaw, p. 225; M. Gell-Mann, Proceedings of the Temple University Symposium on Relativistic Astrophysics and Elementary Particles, Philadelphia, 1967 (in press).
725
Yuval Ne 'eman The reasoning is based upon the consideration that the current expansion will make the state of maximal extent B have a very high entropy.
If the final collapsed state C is similar to the original dense state A in our past, R ' (the 'radius' of the universe)
** max
-
1
1
^.^
B
1
A
1 Fig. 3
\ C
^ f(time)
its entropy Sc is smaller that the largest-extent state. An observer will, therefore, place his thermodynamical arrow from the dense state to the diffuse, and time will run A-+B C-*B There is a local criterion which is supposed to fix this arrow at every point: because of the expansion, the sky is dark and cosmological space cold (3 ° K according to recent data), so that heat flows all the time from any body (say, a star) to outer space. The number of photons emitted rises with time. In a contracting universe, the sky would become lighter (if the universe is infinite, it would be white and the temperature of outer space would reach 5000° K), and any system would have to absorb more photons than it emits. According to T. Gold, this then provides the local and global time-arrow and ensures that only expanding universes can exist. For all we know, we are now perhaps 'really' in a contracting universe and our time-direction is inverted— except that we do not know what 'really' means, since the statistical arrow points as we feel. This approach is purely conjectural, and may prove completely erroneous. Tolman has shown 3 that general relativity provides timesymmetric models for the universe. Oscillating cosmologies have been shown to be thermodynamically reversible; moreover, a sequence of such pulsations does not have to stop out of 'exhaustion' upon reaching 3
R.C. Tolman (1934) Relativity, Thermodynamics and Cosmology, Oxford, p. 426.
4
726
The Arrows of Time a state of maximal entropy. The gravitational energy enables the model to continue for ever, the entropy increasing for ever. In such a picture, we would have Sc > SB, with no time-reversal at C, and cosmology would then have no preferred time-direction. The question has not been studied in detail for models containing both matter and radiation and allowing for their interaction—and with the present data for the cosmological parameters. Gold's view corresponded to one variant of the Steady State Theory, which now seems excluded anyhow. It would be worthwhile making a good physical study of the issue now. EVOLUTIONARY
TIME
It is axiomatic that the evolutionary process—physical and biological— corresponds to an ever-decreasing entropy. This is made possible by a permanent supply of energy—just as we could reverse the sequence in the double-box by pumping the molecules into one of the two compartments. On earth, we mainly benefit from a flow of negative entropy from the sun. The discrepancy which appeared to exist between the age of the universe, or of the earth on the one hand, and, on the other, the time needed for a sequence of random mutations to bring about the development of the highest life-forms now seems to have almost gone. Periods of more intense cosmic radiation may be all that is needed to bridge the gap left. If, however, we continue back into astrophysical evolution and nucleosynthesis, it seems that we require something similar—an external source—to explain the original condensation (that is, 'ordering') processes which conduce to the formation of galaxies and galactic clusters. Calculations 4 have shown that random fluctuations in a homogeneous dust would tend to vanish rather than develop into concentrations. The alternative is to have order in the original boundary conditions; 5 such a cosmological model corresponds to one of the possible explanations 6 of the quasar phenomenon, with inhomogeneity present before the cosmological expansion. The issue is again very far from being clarified, for it appears that such inhomogeneities might even cause topological separation and something like a many-sheeted universe, with the quasars representing matter 'poppingin' from other sheets. I think that we may expect real progress in both the cosmological and the evolutionary aspects to follow the introduction of a much-needed mechanism (or an equation) capable of describing the singular states A and C and the passage from contraction to expansion. 4 W.B. Bonnor (1956) Z. Astrophys., 39:143. 5 R.P. Feynman (1965) The Character of Physical Law, London, p. 108. 6 Y. Ne'eman (1965) Astrophys. J., 141:1303.
5
727
Yuval Ne'eman PHYSIOLOGICAL
TIME
This is a problem which belongs to the realm of cybernetics and information theory. The concept of 'memory' is now in common usage, and it is probable that further developments in the study of perception and the future theory of consciousness will provide, for the first time, a scientific description of physiological time. It has been claimed that a connection with thermodynamics and perhaps cosmology will come about via information theory; the definiteness of the past would then correspond to greater information due to converging world-lines and higher order. As yet, this is mainly qualitative, and one wonders how physiological time will point the same for a man in deep space as for a man on earth, or even for Shadrach, Meshach and Abed-nego in the furnace. MICROSCOPIC
TIME-REVERSAL
The mathematical formalism required for the operation of time-reversal in Quantum Mechanics was furnished by Wigner 7 in 1932, following a theorem by Kramers 8 dealing with the degeneracy in fermion systems. The time-reversal transformation consists in an antilinear operator K which complex-conjugates anything on its right Kxj/(x, t) = ip*(x,-t) Kia^!
+ a2ii2) = a*^* + a*2xj/*2 ^ a ^ ? + a2\j/*2
(4) (5)
and a unitary transformation U U+ U = 1
(6)
T= UK
(7)
< ^ > * = <
(8)
T|^>-*
(9)
so that time-reversal is and
or alternatively
The study of microscopic time-reversal is closely linked to two other operations: space-reflection P and charge-conjugation C. The CPT
7 8
E.P. Wigner (1932) Nachr. Ges. Wiss. Gottingen, 32:35. H.A. Kramers (1930) Verh. K. ned. Akad. Wet., 33:959.
6
728
The Arrows of Time 9
theorem has been proved in field theory and seems to hold for any local Hamiltonian (at all events, no CPT violating theory has yet been constructed from a local Hamiltonian). As we shall now see, since 1964 there has been at least one 1 0 conclusive experimental proof that CP is violated, plus a possible, though still somewhat marginal, second 11 experiment. This amounts to T violation if CPT holds; the latter invariance may, however, still prove to be only an approximate symmetry. The existing uncertainty in CPT may yet mean that CP breaks down, while T invariance holds. 12 Ever since the discovery of parity (P) violation, physicists have been checking and rechecking whether C, CP, T or CPT holds. The structure of the Atonous ( = 'weak') Hamiltonian JK>A, as it crystallized after 1957, required CP invariance, and the 1964 experiment 10 by Fitch and collaborators was intended to check on CP invariance in neutral K decay. Since all other checks had confirmed that (
g^>J^A^>-1^~i=
+J^A
(10)
it was expected that neutral kaons would merely offer one more confirmation in their decay. CP
V I O L A T I O N IN Kl
DECAYS
The neutral kaon system is the most sensitive to any kind of asymmetry involving C. This is because K° is the only metastable particle which could mix with its antiparticle, provided that /-spin and strangeness non-conservation are allowed. The C or CP eigenstates would be superpositions of K° and K°, K, = J^(K°
K2 =
+ CP K°)
(11)
±{K°-CPK°)
where CP | K ° > = J J | J K 0 > , | J / | = 1 and we can choose Y\ = 1. The main decay product of a neutral kaon is a pair of pions (n+n~ 9
G. Liiders (1954) K. danske Vidensk. Selsk. Skr. (Mat. Fys. Medd), 2 8 : 5 ; idem (1957) Ann. Phys., 2 : 1 ; W. Pauli (1955) Niels Bohr and the Development of Physics, New York. 10 J.H. Christenson, J.W. Cronin, V.L. Fitch & R. Turlay (1964) Phys. Rev. Lett., 13:138. 11 M. Schwartz et al. (1967) Phys. Rev. Lett., 19:987; J. Steinberger et al. (1967) Phys. Rev. Lett., 19:993. 12 Y. Achiman (Tel Aviv Univ. preprint).
7
729
Yuval Ne'eman 0
or 7t°7i ). This is even under CP, because of the necessary evenness of the wave function under total inversion (for 2n° it is just P = (— 1)J = 0 , C = +1). Another prominent decay product is a system of three pions (7t+7t-7c° or n°Tt°n°). It is mostly odd under CP, because C = + 1 for 3n° or for 7t+7c~7i:° in the lowest angular momentum combination L = / = 0 (L is the orbital angular momentum of n° relative to the centre of mass system of the 7r+7c~ pair; / is the relative angular momentum within 7 r + 7 i ; " ) ; P = — 1 from the intrinsic parities. States with CP = + 1 exist for 3n in configurations with I = L =1, that is, they are relatively suppressed by higher centrifugal barriers. We thus predict <^
1
|^|2TT>#0
\jfw\3ny~0
(12)
(K2\jfw\2n>=0 (,K2\jfw\3n}^0 which seemed, indeed, to hold. With the difference between two- and three-body phase space, one expects Kt to be short-lived and K2 long-lived. And, in fact, one observes a short-lived K°s and a longlived Kl with the decay rates,
r
< K » all modes = ( U 5 5 ± 0.019) x 1 0 " s e c -
where 'all modes' actually stands for 27t, the only detected mode to date; and F
^
all modes = («>.9 ± 1.0) x 1 0 ' s e c -
which includes T{K°L -> TT* e± ve) = (7.64 + 0.44) x 106 sec" 1 T(K°L -> n* n* vM) = (5.30 + 0.38) x 106 sec" 1 T(K°L -» 3TC°) = (4.60 + 0.50) x 106 sec" 1
r ( K ^ 7 r + 7r-7r°) = (2.34 ± 0.13) x 10 6 sec _ 1 In 1964, the additional mode T{K°L-+n+n-)
= (3.15 ±0.17) x 10 4 sec" 1
was detected, implying a violation of CP.
730
The Arrows of Time MICROSCOPIC
I R R E V E R S I B I L I T Y , A S S U M I N G CPT
INVARIANCE
The CP violating K£ -> In is characterized by <^|7t+7t-> 1+-
=
(13)
\1+-
(Ks\n+n-y
and (KL\n"n°y
i
= P7oo ?
>7oo
(14)
The present experimental values 13 are |f/ + _| =(1.96 + 0.05) x 10" 3 4>+. = 78° + 15° The value of rj00 is still not well established. The first experiments found |foo
-O*10"3
but very recent values are (3.81 + 1.0) x 10~ 3 ; (3.2 + 0.6) x 1 0 " 3 ; + 0.7N and even |2 ' ) x 10 3. This is an important parameter for the identification of the CP-breaking mechanism, since | r]00 | •=/=• \ r\ + _ | involves | /1 = 2 contributions. To describe the two non-orthogonal actual superpositions of K ° and K° corresponding to Kg and Kl, we use a non-unitary transformation, 14 W =
nsl(\
»s Hi ~ £ s) -nll(l-BL)
+£S)
KZ'a+eJ
(15)
on the spinor
-A =
K°
(16)
with n
siL= {2(1 + | £ S / L | 2 V
(17)
The phenomenological Hamiltonian can be written as a sum T + iM, 13 J.W. Cronin (1967) Proc. 1967 Int. Conf. on Particles and Fields, Rochester, p. 3. The most recent data were announced at the Chicago APS meeting, 1968. 14 We follow the review by T.D. Lee & C.S. Wu (1966) A. Rev. Nucl. Sci., 16:511.
731
Yuval Ne'eman where T and M are Hermitian 2 x 2 matrices in that space. M is the mass matrix, Y the reaction (decay) matrix, dip i-f = (M - iTW
(18)
at
After diagonalization, the two eigenvalues of M — iT are i s~^7s\
i ™L--yL
m
(19)
Diagonalization yields equations for e s / i (r y , M y ). If C P T holds £s =
£L
= £
(20)
which results from the equality of the two diagonal matrix elements in the pre-PF-diagonalization stage, when \jj represents the particle-antiparticle pair (16). The parameter e is directly connected with timeirreversibility since it turns out, in first order, to be proportional to the off-diagonal matrix elements connecting K° to K° and vice versa, £=
Imr12 + UmM12 Ay + 2i'Aw
(21)
where A
7 = 7s ~
7L
*
Am = ms-
mL
(22)
Using yL x 1.7 x 10~ 3 y s
>
— 2Am Ay
we find arge«45°or225°
(23)
To observe the irreversibility, we can, for instance, compare the decay of a K ° beam with its somewhat idealized microscopic reconstitution. (We have to invent an experiment equalizing phase space both ways, if this can be done at all.) To first order, the fractional number of K° mesons remaining in the beam is, at time t, R(t) = (£-.Re £)[>-*' + e~ •'•'] + Ree[e-{i(ys+yi-)"iAm)t+ For any reversible process with R(t) = e-y< 10
732
c.c.~] (24)
The Arrows of Time we get for the idealized production (starting at t = — oo and complete at t = 0) the symmetric curve P(t) = eyt
Fig. 4
For the production of K°, we would get from C P T P(t) = (i + Ree) [ e w + e"'] - R e e [ e »-'*">' + c.c.']
(25)
as can be computed from K°(t)>
~-
(1-e)
V^
and (K° \ tf | x> = <x 12? \ K°)
« ° H > *
(1 + e)
V^
| Ks°(t) > + | Kl(t) > }
(26)
where x ~ x, leading us to use {|Ks°(-t)> - | * * ( - ' ) > }
(27)
We see that (24) and (25) contain a reversible (f-even) part (28) corresponding to s -»0 and an irreversible part (t-odd), Rirr(r) = Ree[e- f * <1,s + yI - ) ' iAm), + c.c. - e~v'1- e - ''-']
(29)
Fig. 5
11
733
Yuval Ne'eman With this novel type of irreversibility, we could create a new macroscopic function derived from l=Y.imndm,n
(30)
m, n
where Imn= | < m\jtr-
\n > | 2
(31)
is the rate of T violating transitions, mediated by the 2fC~ T-odd part of the Hamiltonian, and dmn is the appropriate weight in the macroscopic configuration. The difference between R(t) and P( — t) is another microirreversibility function, which can perhaps be used to construct entropylike variables.15 F A I L U R E OF A COSMOLOGICAL I N T E R P R E T A T I O N
When the CP (or T?) violation was first discovered, attempts were made to connect it with cosmological effects. The idea was that a long-range force could be responsible for a mass difference between K ° and K ° (thus violating CPT rather than CP). For instance, it could have a potential energy proportional to hypercharge, and connect K1 with K2 via The potential energy should transform like the 4th component of a 4-vector (for a -f field) and vary linearly with the K2 laboratory energy. For a spin J, A£~
1
J\-v
and
This was checked experimentally; it turns out that the effect displays no energy-dependence. For a non-local long-range J = 0 field, one can show 1 6 (j)+- ->90° +
12
734
The Arrows of Time
i
experiment
«7
-2/
$1 Fig. 6
current or of the 3rd component of /-spin in the universe due to the cosmological expansion — yields results which are inconsistent with the stability of K-mesons as observed, and so on. 17 This takes care of one aspect of the possible connections between the microscopic and cosmological time-arrows. But we still have to check whether the existence of a microscopic irreversibility described by I(t) does not clash with the reversal of the cosmological time-arrow at B in Fig. 3, as suggested by Gold and Gell-Mann (above, n. 2). One attempt, made by Schucking,18 denies that such a reversal ever takes place and achieves cosmological CPT in variance by postulating the existence of an antiuniverse. This should be distinguished from the usual assumption of antimatter galaxies in our own universe. Thanks to the CP violation as observed in Kl -* In or in K°L -* n + + e± + v^ or n+ + /z* + vj,, we may now hope some day to be able to distinguish between galaxies and antigalaxies. Take, e.g., the result of Steinberger et al. (above, n. 11) 8=
r + + r_
= (+2.24 + 0.36) x 10"
where r + is the rate for (e+ n~ve) and T_ for e~ n+ ve. Assuming that we communicate with the physicists in some distant galaxy, we simply have to ask them to make a Kl beam (which will be long-lived anywhere) and observe the asymmetry. By fixing the sign of 5 as positive, we can now indicate to them that the more-numerous leptons are positrons, and so forth. Read in Hebrew 19 March 1968
17 S. Weinberg (1964) Phys. Rev. Lett., 13:495. 18 E. Schucking, Proceedings of the Temple University Symposium on Relativistic Astrophysics and Elementary Particles, Philadelphia, 1967 (in press).
13
735
2190
CHIANG,
PHYSICAL
REVIEW
D
GLEISER,
VOLUME
1,
AND
NUMBER
HUQ
7
1
1 APRIL
1970
Consequences of Unitarity in Some Models of CP Violation* M. GRONATJ
Department of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel AND Y. NE'EMAN
Department of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel and Center for Particle Theory, University of Texas, Austin, Texas 787'1Z\ (Received 11 June 1969; revised manuscript received 10 December 1969) Using the model-dependent assumption of lit saturation of the unitarity sum, we find estimates for | ijoo/i»+-1 and Ree/ ] ij+_ | for a certain class of theories. Two models are tested, and 11700/1;+-1 is found to be different from the originally estimated value, which was based on Ree as input.
T
HE directly measured CP-violation parameters in different results for models where that was done the KL° —» 2TT and KL° —» irlv decay experiments originally. Following the standard phenomenological analysis are ij + _, TJ0O> a n d Ree.1 In most theories of CP violation 4 which have been proposed to explain the KiP-dtc&y of the K°-K° system, let us denote the matrix element experiments, the only quantity rigorously predicted by of the decay of K" into a two-pion standing-wave the theory is the order of magnitude of the symmetry- state with isospin / by |^r|e'*^. The short- and longlived neutral-kaon states are written in terms of the breaking effect. Only the so-called superweaklike eigenstates of hypercharge as 2 models find strict values for |i70o/'7+-|> #+_, <j>oo, and Ree. When dealing with other theories, it thus appears I-K^°> = [ 2 ( 1 + N 2)}~1/2{ (I+60) I #°> as if much freedom is allowed for the actual values of ± ( l - e „ ) | £ ° > } . (1) these parameters, and estimates are sometimes based 4 Unlike Wu and Yang, who choose c/><>=0, we use the on the use of the experimental value of one of them as phase convention in which e is real. This is the phase 0 input (Ree in general). convention in which the CP-violation phases fa are 6 It is the purpose of this note to show that the unitarity measurable quantities. condition8 can be used to provide estimates for | T) 00 /IJ + _ | Using the approximate | A / | =\ rule for Ks°—> 2?r as well as Ree/|i; + _| in the framework of a certain and the smallness of the observed CP-violation effect, class of CP-nonconservation models. This, of course, is one finds in conflict with the use of Ree as input and leads to q + _«e„-W0 o +(M2|/Mo|VZ)
>?oo~ eo+#o - ( M 21V2/1A „ I )4>2ei{, where 5=57r+52—50. ' J. S. Bell and J. Steinberger, in Proceedings of the Oxford International Conference on Elementary Particles, 1965, edited by M. Alston-Garnjost (University of California Press, Berkeley, Calif., 1967). 4 T. T. Wu and C. N. Yang, Phys. Rev. Letters 13, 380 (1964). 6 G. Charpak and M. Gourdin, lectures delivered at the Matscience Institute, Madras, India, 1966 and 1967, p. 54 (unpublished).
736
1
CONSEQUENCES
OF
The unitarity condition3 can be written in the following form: «o(r s +2iAAf)=E (n\T\KsT(n\T\KL°),
(3)
n
where the sum extends over all open channels (including phase space), AM=ML—MS, and T s is the total decay rate of Ka<>. We now assume that the 2w contributions dominate the right-hand side of Eq. (3). This condition is obviously satisfied in any model in which CP is violated only (or predominantly) in the 2ir decay mode of K°. From this model-dependent assumption, one obtains «o«0„+|^2/^o|^2.
(4)
Denoting by r (y) the ratio between the 1=2 and 1=0 C-P-conserving (violating) amplitudes, we obtain from Eqs. (2) and (4)
., + _«<£ 0 (l+ry-f«+2- 1 V"), Vo^toCL+ry+i-l^ye"),
(5)
UNITARITY
IN---
2191
In a mode presented by Delaney and Welling,' CP violation is introduced in the effective three-particle weak interaction of pseudoscalar mesons, namely, only in the 2ir decay mode of K°. In this model, the | A / | = £ and | A/1 > J CP-violating amplitudes are found to be of comparable strength: y -1.2.
(7)
The authors, who seem to overlook the consequences of the unitarity sum rule, use the charge asymmetry parameter e0 as input, 8 and obtain |WT+-I=1-5.
(8)
On the other hand, we obtain from Eqs. (5)-(7) |W?+-!<0.3,
(9)
which seems to be inconsistent with the present experimental information.1 In a model suggested by Yun, 9 CP is explicitly violated only in the two-pion decay mode of the K meson, through C nonconservation in the electromagnetic interaction. The relative strength of the | A / | > J to \AI\ = J K—* 2ir amplitudes is smaller for the CP-violating transitions than for the CP-conserving ones: y
These relations provide estimates of the ratios | »jqo/i7+_ | and e 0 /|r) + _| in terms of a single model-dependent parameter y. For the irir phase shifts and the parameter r, which measures the validity of the | A / | =i rule in Thus, while Yun obtains—again taking e0 as input— Ks° —> 2?r decays, we may use the following values 6 : h„oA+-l=l-23, (11) 8t-80 = -(S0±2O)°, which is substantially distinguishable from the superweak prediction,2 we would expect [from Eqs. (5), (6), Let us now examine two such models of CP violation and (10)] the equality |>jool = \v+-\ to hold up to 6% in this theory. which have been proposed recently. 'R. M. Delaney and D. J. Welling, Phys. Rev. 176, 1841 6 J. Cronin, in Proceedings of the Fourteenth International Confer- (1968). 8 ence on High-Energy Physics, Vienna, 1968, edited by J. Prentki
and J. Steinberger (CERN, Geneva, 1968), p. 281. The quantitative conclusions drawn from Eqs. (S) are insensitive to possible deviations from the above-quoted values.
The three measured values of the charge asymmetry (Ref. 1), although consistent with each other, still provide only a crude estimate of eo. »S. K. Yun, Phys. Rev. 178, 2439 (1969).
737
International Journal of Theoretical Physics, Vol. 3, No. 6 (1970), pp. 437-441
Time-Reversal Symmetry Violation and the Oscillating Universef AMNON AHARONY and YUVAL NE'EMAN Department of'Physics and Astronomy, Tel-Aviv University, Ramat Aviv, Israel Received: 16 April 1970
Abstract The expressions for the fractional number of K°'s and K0,s in a neutral kaon beam are discussed with reference to time-reversal asymmetry. The suggested relation between the sign of Ree (eis the Lee-Wu T-violation parameter) and the cosmological arrow of time if CPT is broken is further clarified.
1. Introduction In a previous article (Ne'eman, 1970), the experimentally established violation of CP symmetry in the decay of the long-lived K meson (Christenson et ah, 1964)—and the further possibility of CPT violation—were studied in the context of time-symmetric oscillating models of the universe. It was shown that the current assumption, according to which the contracting phase of the oscillation is reinterpreted as a time-inverted expansion, cannot be retained at all if CPT is violated; if only CP is violated, the assumption is allowed and involves inverting the definitions of matter and antimatter. To describe the evolution of a K°-K° complex, the Lee-Oehme-Yang formula (Lee et al., 1957) was used. This formula predicts the number of neutral K mesons remaining in the beam at any time t. In the present article we refine the argument and apply it to a different set of formulae which emphasize the observables involved. In Section 2, the formulae for the fractional number of ^"'s and K0,s in a neutral kaon beam are discussed, in an ordinary and in a time-inverted coordinate schemes. In Section 3 these formulae are used for defining a relation between the cosmological arrow of time and the behaviour of the microscopic K°-K° system. t Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under AFOSR grant number EOOAR-68-0010, through the European Office of Aerospace Research. 437
738
438
AMNON AHARONY AND YUVAL NE'EMAN
2. Time Reversal Asymmetry in K°-K° Distinguishing Formulae We first consider at t = 0 a beam of pure K°'s (strangeness = +1). For t > 0, these particles will decay via the weak Hamiltonian eigen-states Ks and KL, thus forming at time t—aside from the decay products—a number of K°'s. Using the parametrization of Lee and Wu (Lee et al., 1967), with 8 the CPT non-invariance parameter (S = 0 if CPT is conserved) and e the 3"non-invariance parameter, a direct calculation gives for the above beam (Aharony, 1970) RK\K°, t) = i((l - 4 Re 8) exp ( - y t t) + (1 + 4 Re 8) exp ( - y s t) + + [(1 + 4ilmS)exp(iAmt)
+ c.c.]exp[-i(yL
+ ys)t]}
(2.1)
K
R °(K°, t) = i ( l - 4 Re e) {exp (-y s 0 + exp (-y^. 0 - 2 cos /lm(x x e x p [ - K n + n)>]}
(2.2)
RK°(K°,t) and RK°(K°,t) are, respectively, the fractions of # ° and of K° particles in the beam at the kaon's proper time t. yL and ys are the inverse lifetimes of KL and of Ks, and Am is their mass difference. For a beam initially made of pure K0,s, Rk\R°,t) and Rs°(K°,t) will be given by the same formulae, except for a change in the signs of e and of 3. (All expressions are to first order in e and S.) We now wish to consider the same beam in a time reversed coordinate system. The expressions for reversed time, —t, are obtained from those for t by applying the time-reversal operation T. As shown in the previous article (Ne'eman, 1970) and by Zweig (1967), the equation of motion
ip = (M-ir)<[,
(2.3)
for the two-dimensional state-vector iji describing the K°-K° complex (M and Tare the 2 x 2 mass and decay matrices) is transformed under Txo i d ijjT* = (M* - /T*) >fsT*
(2.4)
where t' = —t. Therefore, $T*{t') will exhibit a time evolution similar to that of ifi(t), except for the transformation €->~e,
S-+S
(2.5)
Since equation (2.1) describes the fractional number of ^T°'s in a beam beginning at t = 0 with pure K°, we can deduce that the fractional number of K°'s in the time-reversed coordinate system, described by PK°(K°,t'), will have the same time dependence [note that equation (2.1) involves only 8, which does not change under T]: PK\K°,~t)
= RK\K°,t)
739
(2.6)
TIME-REVERSAL SYMMETRY VIOLATION
439
Similarly, we obtain from equations (2.2) and (2.5) PKXKn,-t)
= (l+SRe€)RK°(Ka,t)
(2.7)
3. A Relation Between Microscopic and Cosmological Arrows of Time In a time-symmetric oscillatory model, we would expect every physical situation to repeat itself after a time T, T being the oscillation period of the universe. Thus, the fractional number of K° and of K° particles in the universe at the points A and B of maximum contraction (Fig. 1) should coincide. We can in fact restrict ourselves to a given volume element and discuss a subsector of the universe containing initially (at A) a beam of pure K° particles. At times t> 0, the beam will decay, leading to a decrease in the number of K0,s and an increase in the number of K0,s, described by equations (2.1)-(2.2). These formulae give the time-evolution only for short times,
Bit = r) Figure 1
since they are based on the Wigner-Weisskopf approximation (Lee et ah, 1957). Still, if the beam exists at times larger than T/2, then we must assume that the decay products are contracted to reproduce the initial K° beam, since we demand complete identity of the physical states at A and at B. By the time symmetry of the oscillating model, we may assume that the behaviour of the beam at the time (T — t) approaching the point B is the same as at the time —t, given by eqs. (2.6)-(2.7). The left-hand sides of these equations were derived as the results of a decay process in the timeinverted contracting universe, but they may be reinterpreted as the fractional numbers ofK°'s and K°'s needed at the time (r — t) in order that the beam will end at the time r as pure K°. Equation (2.6) thus represents a complete symmetry of the fractional number of K0,s in the beam with respect to the time T/2. Note that this symmetry is independent of CPT invariance, since it holds for any value of S. There is no such symmetry in equation (2.7); the fractional number of ^ ° ' s in a beam of K0,s will reveal a symmetry with respect to T/2 only if Ree = 0, hence if T is conserved! If T is not conserved for the K°-K° system, as it now seems to be established (Casella, 1968, 1969; Achiman, 1969), we may use (2.7) to define
740
440
AMNON AHARONY AND YUVAL NE'EMAN
the direction of the arrow of time. There are several possible ways to determine Re e experimentally, e.g. by measuring the charge asymmetry in KL leptonic decay (Schwartz etal., 1967) or by other experiments measuring the overlap of the states KL and Ks. In most of these experiments one has expressions with combinations of e and of S, but Ree can be deduced from them. A direct measurement of Re e is presented by equation (2.2) (Aharony, 1970): One has to measure the number of K0,s in a beam initiated by K° and determine the coefficient of equation (2.2). Such experiments have been discussed by Crawford (Crawford, 1965). In a time inverted contracting world this experiment will give a different sign for Re e. Note, that we have yet no theoretical way to relate the sign of Re e with the oscillation phase of the universe (Zweig, 1967). Still, if the definitions of matter and antimatter are agreed, the sign of Ree is fixed by them since it is related to the difference between the fractions of K° and of K° in KL and Ks, and is thus determined by the experiments mentioned above. (These differences involve Re(e + S) and Re(e —8), and thus fix both Ree and ReS.) Because of this ambiguity, it is interesting to consider a beam ending (or starting—in the time-inverted contracting world) at time t = 0 as pure £°. Using the rule following equation (2.2) and equation (2.5) we find Pk\K°,-t)
= RK\K°,t)
(3.1)
One might thus think that there is no distinction between the two oscillation phases if we invert the definitions of matter-antimatter. But this is not so, since in this case we shall have no symmetry between PR°(K°,t') and PK°(K°,t) unless S = 0. Thus, only if CPT is conserved, one regains a complete symmetry if one defines K° as K°, and vice versa. One might try to avoid the difference between the decay formulae in the expanding and in the time-inverted contracting phases, by assuming that in the time-inverted contracting universe one deals with different kinds of basis states |is:0'> and \K0'}, instead of \K0} and \K0}, for which the physical behaviour is similar to equations (2.1)-(2.2). But it is easy to check that no combination of \K°) and \K0} will give the same behaviour unless Ree = 0. 4. Conclusion If both CPT and Tare not conserved, an experiment counting the fractions of K°'s and of K°'s in a K° beam will distinguish between the expanding and contracting phases of the universe oscillation, thus forming a relation between microscopic and cosmological arrows of time. If CPT is conserved, this distinction may be resolved by inverting the definitions of matter-antimatter. Even if CPT is conserved, there remains the question of the behaviour of a beam of AT°'s beginning at t = 0 and remaining—through the maximum
741
TIME-REVERSAL SYMMETRY VIOLATION
441
expansion phase at T/2—until the end of a period, T. Had it been possible to conserve such a beam, including information concerning its behaviour at short times, an asymmetry is due to appear at the time T — t. References Achiman, Y. (1969). Lettere al Nuovo Cimento, 2, 301. Aharony, A. (1970). Lettere al Nuovo Cimento, 3, 791. Casella, R. S. (1968). Physical Review Letters, 21, 1128. Casella, R. S. (1969). Physical Review Letters, 22, 554. Christenson, J. H., Cronin, J. W., Fitch, V. L. and Turlay, R. (1964). Physical Review Letters, 13, 138. Crawford, F. S. (1965). Physical Review Letters, 15, 1045. Lee, T. D., Oehme, R. and Yang, C. N. (1957). Physical Review, 106, 340. Lee, T. D. and Wu, C. S. (1967). Annual Review of Nuclear Science, 17, 513. Ne'eman, Y. (1970). International Journal of Theoretical Physics, Vol. 3, No. 1, p. 1. Schwartz, M. et al. (1967). Physical Review Letters, 19, 987. Zweig, G. (1967). Paper presented at the Conference on Decays of .ST Mesons, PrincetonPennsylvania Accelerator, November 1967, Unpublished.
742
A H A R O N ? , et al. 7 Novembre 1970 Lettere al Nuovo Cimenio Serie I, Vol. 4, pag. 862 866 A.
Time-Reversal Violation and the Arrows of Time (*). A.
Department
of Physics
and
Y.
Tel-Aviv University
AHARONY
Astronomy,
Tel-Aviv
University
- Uamat
Aviv
NE'EMAN
University of Texas
- Ramat - Austin,
(riccvuto il 28 S e t t e m b r e
Aviv Tex. 1970)
1. - Introduction. The problem of the relations between the macroscopic arrows of time has been discussed by many authors. Most of them discuss relations between the thermodynamical arrow of time, defined in the direction of increasing entropy, and the cosmological one, defined in the direction in which the universe expands (x). The discovery of OP-violation in the decay of the neutral kaon (2), with the assumption of OPT-conservation, imply a violation of time-reversal symmetry T in this decay. It has recently been established that, even without the assumption of OPT-in variance, the experimental data indicate T-violation in this decay (3). There thus seems to exist a « microscopic arrow of time », related to the behaviour of the neutral-kaon beam (4), and the question of its relations with the macroscopic arrows of time arises. In this letter we briefly discuss these relations. The details of the argumentations and calculations will be published elsewhere (6-8). (*) Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR grant number EOOAR-68-0010, through the European Office of Aerospace Research. (') T. GOLD: in Recent Developments in General Relativity (New York, Warsaw, 1962), p. 225; M. GELL-MANN: comments in Proceedings of the Temple University Panel on Elementary Particles and Relalivistic Astrophysics (1967). («) J. H. CHRISTENSON, J. W. CRONIN, V. L. FITCH and R. TTTRLAY: Phys. Rev. Lett., 13,138 (1964).
(') Y. ACHTMAN: Lett. Nuovo Oimento, 2, 301 (1969); R. S. CASELLA: Phys. Rev. Lett., 21, 1128 (1968); 22, 554 (1969). (*) G. ZWBIG: paper presented at the Conference on Decays of ~K-Mesons, Princeton-Pennsylvania Accelerator (1967); Y. NE'EMAN: paper presented at the March 1968 Session of the Israel Academy of Sciences. (') A. AHARONT: Lett. Nuovo Oimento, 3, 791 (1970). (') A. AHARONT and Y. NE'EMAN: to be published in Intern. Journ. Theor. Phys. (1970). (') Y. NE'EMAN: Intern. Journ. Theor. Phys., 3, 1 (1970). (B) A. AHARONT: Microscopic irreversibility in the neutraVkaon system and the therm.odynamicaZ arrow of time (TAUPS 148-70, 152-70). 862
743
TIME-REVERSAL VIOLATION AND THE ARROWS OF TIME
863
2. - Microscopic arrow of time. The K°-K° system is usually described by a two-dimensional state vector, whose evolution in time is determined by the Wigner-Weisskopf approximation (9). The decaying eigenvectors of the total Hamiltonian are found to be j \K°sy = \Kly + (e + d)\Kl> , (1)
1 \J%> =
\X%>+{B-d)\E»>,
with \Kl) and |JTj> the GP eigenstates, e the T-violation parameter and 6 the GPTviolation parameter (10). All the expressions for experimental results featuring OPor T-violation for the K0-K° system (e.g. the fractional number of K°'s in an initially pure K° beam (5-6), the charge asymmetry in K° leptonic decay ('), etc. ( 10,11 )) involve e and <5. Prom the definitions of e and S it may be shown (6-7) that under the transformation of time-reversal T -e ,
Therefore, many experiments concerning GP- or T-violation will give different results in a direct and in a time-inverted co-ordinate scheme. For example, the fractional number of K°'s in a beam initiating as pure K° in a time-inverted co-ordinate scheme P^'(K°, — t) will be related to the same quantity measured in a direct scheme R*°(K«, t) by (6) (3)
PK°(K°, — t) = (1 + 8 Re e) i?K°(K°, t) .
Similar results for other experimental quantities may be easily derived. Thus, if T-symmetry is violated, namely if Re e ^ 0, the measured sign of Re e may be used for the definition of the direction of a microscopic arrow of time.
3. - Relation with the cosmological arrow (e-7). The conjecture linking together statistical and cosmological time arrows implies intrepreting the contracting phase is an oscillating cosmological model as a timeinverted expansion (1). Roughly, in a contracting universe entropy would have to increase the universe changing from a disordered spread-out state to a contracted, ordered, state, which is not conceivable. Consider now an oscillating (symmetric) cosmological model. In such a model, if T is the oscillation period, there must exist a complete identity of physical behaviour at time —t and at time (T — t). Therefore, in a contracting phase one should observe (") T. D . L E E , R . O E H M E and C. N. Y A N G : Phys. Rev., 106, 340 (1957). (") T. D . L E E and O. S. W u : Ann. Rev. Nucl. Sci., 17, 513 (1967). (") For a review, see J . STEINBERGER: i n Proceedings of the Topical Conference on Weak (Geneva, 1969), C E R N 69-7.
744
Interactions
864
A. AHAEONY and Y. NE'EMAN
different physical laws than in the expanding phase, due to the asymmetry in the microscopic behaviour discussed above. E.g., the fractional number of K°'s at time t in a beam starting at time 0 as pure K° will be _RE°(i£°, t), whereas in the contracting phase this number will be P*-°(K°, t).
4. - Relation with the thermodynamical arrow (8). We describe the kaon's decay as being due to the interaction between the kaonic system, which has the three basis states |0>, a-JO) and <4|0> (|0> is the vacuum, a% creates K°), with a thermal bath which contains all the possible final decay states, e.g. 2iz, TTIV, etc., in thermal equilibrium. The Hamiltonians of the kaonic system, the bath and the interaction are respectively (4)
Hi — hwxiala,! + aza2) ,
(5)
5? = 2 ^ * ^ * .
(6)
j,=-«n»wi'*+u i rk
(bTlt creates an eigenstate of the strong Hamiltonian in the decay channel r with quantum numbers k). The reduced density matrix of the kaon system is now defined as (7)
oa=TiBS,
where g is the total density matrix, and the trace is taken over the states of the bath. The time evolution of g0 is given by the Wangness-Bloch master equation (12). If we start at t = 0 with 0
0\
(8) "
e(0)
(the first row and column stand for |0>), then a t time t we have
(9)
&,(*)
(••) K. WANQNESS and F. BLOOH: Phys. Rev., 89, 728 (1953).
745
865
TIME-REVERSAL VIOLATION AND THE ARROWS OF TIME
with
i^=AQ
(10)
<-!:
(11)
(12)
fl„
— jiAi+
iQ(Y — T r Q ) ..
dco
r 2 ^r(^)(9rt3r*)c 0 - m j r (exp [ - phw] + 1) , to — (oK + le
= 2n ^ # r («>*) exp [ - /SfttoJ R e
(g*kg£)mK,aic da>
2P
2 J V r M e x p [ - phm] I m
J < (.Wr(eo) is t h e d e n s i t y of s t a t e s in t h e c h a n n e l r, ft = l/JcsT). F o r t e m p e r a t u r e s low c o m p a r e d t o 10 12 degrees, exp [—f}Ra)K] t o u n i t y . I n this case,
Am — - ( / £ — ys) (e -
is small
( ^ ^ )
0 -<°jr
compared
6)
(13) A„ = | Am — ~(YL—
YS)
(e + S) ..
where ys, yL are t h e inverse lifetimes of K^, KL a n d Am their m a s s difference. Therefore An, Azl a n d Q m a y b e r e g a r d e d as small p e r t u r b a t i o n s , a n d eq. (10) s o l v e d . F o r a n y initial condition t h e solution a p p r o a c h e s t h e s a m e equilibrium, 0 (14)
0
exp [— fih
0 exp
[—flftoix]/
W e n o w define t h e inverse e n t r o p y
Sa(*)
(15)
Ka(t)=
Tr[eo(ln
e
„-ln
e a e a )]
I n s e r t i n g t h e solutions of (10) for a p u r e K° a t 1 = 0 w (16)
.
find
Ka(t) = K°a(t) [1 + E e eK\(t) + R e 8K2a(t) + I m 8K3a(t)] .
K°a{t) a n d —Ree-K\[t) are given in F i g . 1 a n d 2 for T = 300 °K. T h u s , m i c r o s c o p i c irreversibility results i n an oscillation in t h e e n t r o p y , s u p e r i m p o s e d on t h e o t h e r w i s e regularly m o n o t o n i c increase. T h e t o t a l e n t r o p y c h a n g e in t h i s d e c a y is huiEjT, unaffected b y t h e T-violation.
746
866
A. AHARONT and Y. NE'EMAN
In a time-inverted co-ordinate scheme one obtains equations similar to (10), except for a change in the sign of e in eq. (13), which causes a change in the sign of the second
Wig. 1 . - I n v e r s e entropy for K°-decay w i t h T a n d C^PZ'symmet^y(J'i/ys=1.602•10-^ Amlys=0A69(.")).
F i g . 2 . - T h e oscillatory p a r t i n t h e e n t r o p y t u n c tion due t o T-violation ( R e e = 1.42 • ! ( > - ' ( " ) ) .
term in (16). Thus, entropy will increase in both time schemes, and will have the same total change, but will show different features in its time dependence. These features are directly related to the sign of Re s.
5. - The CPT-symmetric case. In all expressions we discussed, the change of the initial state from K° to K° results in a change in the signs of both e and 8. Therefore, if OPT is conserved (<5=0) the behaviour of matter in a direct time scheme will be the same as that of antimatter in an inverted scheme. In this case, the definition of the microscopic direction of time is not absolutely defined. But if <5 # 0 it is clear that no such ambiguity exists; the behaviour of a K° decaying in a direct time scheme is clearly different (e.g. in the features of t h e entropy time dependence) from that of a K° decaying in an inverted scheme.
(") A. B A R B A R O - G A I / H E R I ,
S. E .
DBRENZO, L. R. P R I C E , A. RITTENBERG,
N. B A R A S H - S C H M I D T , C. B R I O M A N , M. R O O S , P . S O D I N G a n d C. G. W o r n : Rev. Mod.
747
A.
H.
Phys.,
ROSENFELD, 42, 87 (1970).
Foundations of Physics Letters, Vol. 16, No. 4, August 2003 (©2003)
THE SIGN A N D MICRO-ORIGIN OF COMPLEXITY AS ENTROPY (CONJECTURED RESOLUTION OF A PARADOX)
Yuval Ne'eman School of Physics and Astronomy Raymond and Beverly Sackier Faculty of Exact Sciences Tel Aviv University, Tel Aviv 69978, Israel Received 14 May 2003 Evolution produces ever more ordered matter, while also increasing its complexity all the time. There are various ways of measuring complexity, such as Kolmogorov's algorithmic complexity, drawn from information theory, and identified with entropy, enchancing inrreversibility in harmony with the second law of thermodynamics. On the other hand, however, the creation of order should have reduced entropy; quoting Schroedinger, it represents "negentropy." To resolve this apparent contradiction we first review a similar set up (though with a totally different interaction) occurring in black holes, a model in which the physics are now explicit and fully understood at the quantum level. Key words: black holes, entropy, evolution, complexity, string theory. 1.
REVIEWING BASICS ABOUT THE SECOND LAW A N D THE ARROW OF TIME
That irreversibility and the physical arrow of time are related to statistical/probabilistic considerations was one of the main achievements of XlXth century physics, with the second law of thermodynamics as its abstract formulation. Any textbook will have a P, V diagram of the Carnot cycle and a drawing of a cylinder-piston set up in the cycle's four stages [1], with an assumption of an ideal gas. With the two adiabatic transitions simplifying the issues, and with the indices H, C standing for "hot" and "cold," denoting the two absolute temperatures 389 O894-9875/03/080O-038W0 C 2003 Plenum Publishing Corporation
748
390
Nc'eman
T of the isotherms and the relevant heats (or energies) entering the system (or leaving it, if negative) at these temperatures, it is relatively easy to prove that {QH/TH) — {Qc/Tc)- If the cycle is operated as an engine, QH is positive (heat is supplied and the cylinder's expansion produces work outside), while Qc is negative (the piston pushes the molecules into a corner while heat is withdrawn). The ideal gas assumption is essential in classical thermodynamics. Physics-wise it covers a physical region in which the Standard Model fundamental interactions responsible for the short-ranged (nuclear) forces (strong and weak) have been integrated out, together with the contribution of electromagnetism to atomic and molecular bindings, leaving us with just kinetic energy, plus in some problems, weak chemical potentials. All other parts of the fundamental Hamiltonian are included indirectly through the masses, angular momenta and various structural elements at the molecular level. It is under such an assumption of no-interaction conditions that the textbooks also present elementary illustrations demonstrating the relationship between entropy and the arrow of time, or irreversibility. These are transitions which would be allowed by the first law (conservation of energy) and yet do not occur-whereas they do occur in the inverse direction. The most obvious example is the transition between two arrangements (as to organization and location) of a cloud of ideal gas molecules in a closed room at different instants. Illustrations show in one (marked [D]) a very dense cloud of molecules, all bunched in one small corner of the room-while the other [S] shows them spread out all over the room, i.e., over a much larger volume. [D] is the relatively more ordered set up (also requiring more information for a description), [S] is the less ordered one. Transitions D —• S are commonly seen, caused by the random directions and velocities of the molecules, whereas S —> D is never observed, the chances of a random distribution (in size and direction) of velocities leading the entire cloud into one corner being practically null. Given the two photographs, we would bet that D was the earlier take, S the later one. This assumes that chance be the only intervening factor, naturally as a result of the molecules' random motion.
2. EVOLUTION, COMPLEXITY AND THE APPARENT PARADOX The picture became more opaque when other and apparently disconnected time-arrows were noted on the physical scene: (1) cosmological (linked to the universal expansion [2]), (2) the radiation arrow, (3) Treversal quantum symmetry violation, derived indirectly first, from the discovery of CP violation in K^ —> 2n and recently verified directly, a non-statistical arrow, (4) the evolutionary drive, fully generalized to
749
Complexity as Entropy
391
include everything from cosmogony to epistemology, (5) the cognitive inner human sense of duration, (6) black-hole irreversibility, etc.. In What is Life? [3] E. Schroedinger discussed the evolutionary arrow, in the context of his pioneering approach, which gave birth to molecular biology. He noted that evolution generates order and structure and thus appears to diminish entropy. In other words, evolution generates negative entropy. However, a different approach to the evolutionary time-arrow centers on complexity, as the essence of what is achieved by evolution. Kolmogorov (followed and expanded by Chaitin and Bennet) suggested a quantitative algorithmic description for complexity [4-6], namely the length of the shortest program describing the system completely. Algorithmic complexity has been one choice, but there are improvements, such as the Lempel-Ziv version [7]. In most treatments, however, complexity appears to fit very naturally in the role of a positive contribution to entropy, moreover increasing with time. A paradox has thus emerged, a contradiction between Schroedinger, with our conventional notion of entropy as disorder on the one hand and, on the other hand, the methods developed in information theory for the description of complexity and the second law. 3.
BLACK HOLE ENTROPY AS A PILOT MODEL
To obtain the correct interpretation for the evolutionary arrow we focus first on a pilot model, provided by gravitational irreversibility-as represented by matter falling into black holes. This issue has been probed in depth and is now useful as a model, thanks to it's being fully understood. Although the interaction is totally different, it will be possible to draw relevant conclusions. The black-holes case developed in three stages: (a) The treatment of black holes encountered issues of irreversibility, which as formulated by Christodolou and Ruffini [8], Penrose and Floyd, Hawking, Bekenstein, Carter and Bardeen, appeared to involve a new discipline, with a marked resemblance to thermodynamics. (b) J. Bekenstein [9] postulated that this is so, indeed and wrote his formula SBH = -A/4, linking the entropy S to the area A of the black hole's envelope, based on dimensional and heuristic arguments. (c) S. Hawking verified [10] the identification with entropy and showed that this entails inclusion of quantum effects. Once this mode is opened, a "generalized" thermodynamics is obtained, with the possibility of defining an effective temperature for the black hole, etc. As a result of pair creation, for example, the
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black hole can produce Hawking radiation when the outside temperature is lower than the black hole's effective temperature; (d) Using String theory to describe quantum gravity, A. Strominger and C. Vafa [11] and independently J. Maldacena [12] studied an example with 5 noncompact dimensions and N — 4 supersymmetry (this can be type II String theory on K3 x S1 or also Heterotic String theory on T®); similar results have been obtained for TV = 8 (i.e., maximal) Supergravity. Back to the N = 4 example, such black holes can carry an axion charge Qx, or an electric charge QE, or both. Extremal black holes (BPS-saturated states) vrith either charge missing have degenerate zero-area horizons. Counting the bound states of BPS solitons of the supergravitational field with nonvanishing charges of both types, the logarithm of the degeneracy for large axion charge Qx and minimal electric charge QE yields an entropy given by SBH = 2TT^QX(Q%/2) + 1, which reproduces the result obtained for this case from Bekenstein's formula, i.e., SBH — 27T-y/Qx(QEp/2. This is 'energy microstates' counting, i.e., orthodox statistical arguments. At this point we have to note that the example selected by the string theorists is constrained by supersymmetry, i.e., this is a sourceless situation, with the N-wise extended supergravitational field supplying both the "matter" fermions and the bosonic gravitational radiation. Both categories are involved. The authors have thus provided a deeper perspective on gravitational irreversibility by exposing the reductive micro-origin of its contribution to entropy, i.e., this time including gravitons and their collective instanton states. We note that this result corresponds to the action of gravity in its strongest attractive phase, and thus in a sector which is very far from the ideal gas of thermodynamics. We have brought up the case of black hole entropy in the hope of learning about entropy in evolutionary processes. Indeed, here too, at first sight, it would seem that order is generated, as all increments of mass are converging onto the singularity at the center of the black hole or rather more precisely, accumulating on its surface, as seen in finite time by an outside (distant) observer. Thus, as for [D] in our previous discussion, and as with evolution according to Schroedinger, a black hole also appears to represent the generation of order, i.e., negative entropy. However, as we have in this case the fundamental Strominger-Vafa results, we can check the micro. We find that the mistake in the black-hole case has been in following the role of matter solely-as befits the ideal gas case-whereas here the spinor-matter part of the energy-momentum tensor current density is coupled to the gravitational field in the Hamiltonian. Thus, we should also consider radiation, namely the gravitational field itself, with the tensions it exerts in its strong binding.
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The states counted by Strominger and Vafa are solitons and include topological realizations of the gravitational field in its binding action. Thus, there is indeed a negative contribution to the total entropy coming from the "improvement" in the orderliness of the nuclei and electrons now imprisoned within the black hole, or even better, now stuck on its envelope (in view of the holographic interpretation of the conservation of quantum information [13]). As against this negative increment, there is a (larger) positive contribution originating in the tensions created within the black hole's highly energized gravitational field quanta. That this should require counting at the quantum level is not surprising, as the involvement with the quantum level is already present in the quantum tunneling required by Hawking radiation. Alternatively, we may regard the quanta of the gravitational field as part of the (second-law compensating) environment, or as the moving piston contracting the volume available to the gas, in the simplest example. Bekenstein's component is then the preserver of the second law. Note that in cosmology, we had indeed assumed [14] that the dense state [D] being the most ordered, with the lowest entropy, a contracting universe would produce negative entropy, in violation of the second law. We had then speculated that a collapsing universe might invert its time-arrow and would then become an expanding one [15]. We see now that this picture was incomplete, as it ignored the contribution of the main actor in cosmology, namely the gravitational field itself. 4.
EVOLUTIONARY GENERATED COMPLEXITY AS A N ADDITION TO ENTROPY
Returning now to (generalized) evolution, we are clearly again in non"ideal gas" regions. We can use the insight provided by the black hole case. The Hamiltonian as in the previous case, contributes through binding components, ranging from the role of quantum chromodynamics in nucleosynthesis to the biophysical contributions (mostly electromagnetic) making up nature's own genetic engineering. As with the area of the envelope of a black hole in the Bekenstein formula, we have to identify a "time-arrowed" quantity characterizing the action of the evolutionary drive. Moreover, as against Schroedinger's view in What is Life? [3], according to which evolution represents negative entropy, because it produces order, we note that this negative entropy, however again relates to matter, while the complexity function will represent the positive entropy produced by the tension within the binding fields. This conjecture could be verified in nucleosynthesis, where the physical interactions (e.g., the Bethe cyclic set of reactions making a helium nucleus plus some neutrinos from every 4 hydrogen nuclei) are well understood in the physics of particles and fields, either in terms of
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the Standard Model, or in low-energy pion-nucleon physics. The latter appears more relevant to the Bethe reaction and easier to evaluate, rather than having to account for confinement as an intermediate stage. Working at the hadron level may also be related to the way nature avoids a clash with Zamolodchikov's c-theorem [16]. Once a useful and informative level of complexity has been properly defined, it should thus be appended as additional positive entropy to the second law, in an extension of thermodynamics (or better to a related formulation in Shannon's Information Theory) on top of Schroedinger's order-generated negative entropy and more than cancelling it, a price fixed by the second law. Acknowledgement. I am endebted to Dr. Eshel Ben-Jacob for some material on complexity and to Prof. Eliezer Rabinovici for pointing out the difficulty relating to the c-theorem. REFERENCES 1. See, for example, D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics (Wiley, New York, 1993), pp. 605-628. 2. B. Gal'or, Modern Developments in Thermodynamics (Wiley, New York, 1974). 3. E. Schroedinger, What is Life? (Cambridge University Press, 1967). 4. See, for example, M. Li and P. Vitanyi, Introduction to Kolmogorov Complexity and its Applications (Springer, 2002) (trans, from Chinese). 5. J. Chaitin, J ACM 22 (1975) 329-340. 6. C. Bennett, "Logical depth and physical complexity," in The Universal Turing Machine: a Half-century Survey, R. Herken, ed. (Oxford University Press, 1988), pp. 227-258. 7. A. Lempel and J. Ziv, IEEE Trans. Info. Theory 22 (1976) 75. J. Ziv and A. Lempel, IEEE Trans. Info. Theory 24 (1978) 530. 8. D. Christodolou and R. Ruffini, Phys. Rev. D 4 (1971) 3552. 9. J. D. Bekenstein, Phys. Rev. D 5 (1973) 2333-2346. 10. S. Hawking, Comm. Math. Phys. 43 (1975) 199-220. 11. A. Strominger and C. Vafa, Phys. Lett. B 379 (1996) 99-104. 12. J. Maldacena, "Black Holes in String Theory," Princeton University Ph.D. thesis. 13. L. Susskind, J. Math. Phys. 36 (1995) 6377-6393. 14. Y. Ne'eman, Intern. J. Theoret. Phys. 3 (1970) 1-6. 15. A. Aharony and Y. Ne'eman, Intern. J. Theoret. Phys. 3 (1970) 437- 441. 16. A.B. Zamolodchikov, JETP. Lett. 43 (1986) 730.
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Proc. Natl. Acad. Sci. USA Vol. 80, pp. 7051-7053, November 1983 Physics
Classical geometric resolution of the Einstein-PodolskyRosen paradox (quantum mechanics/nonseparability/gauge theory/fiber bundle) YUVAL NE'EMANt Tel Aviv University, Ramat Aviv, 69978, Israel; and Institute for Advanced Study, Princeton, NJ 08540 Contributed by Yuval Ne'eman, July 7, 1983
ABSTRACT I show that, in the geometry of a fiber bundle describing a gauge theory, curvature and parallel transport ensure and impose nonseparability. The "Einstein-Podolsky-Rosen paradox" is thus resolved "classically." I conjecture that the ostentatiously "implausible" features of the quantum treatment are due to the fact that space-time separability, a basic assumption of single-particle nonrelativistic quantum mechanics, does not fit the bundle geometry of the complete physics.
EPR nonlocal action The typical EPR experiment deals with a composite state (AB ...) at the origin O that then decomposes, the constituents A, B, etc., thus going their separate ways. The composite state carries a certain irreducible representation D^Lp of the group G, appearing in the Clebsch-Gordan decomposition of the product D A x D B X . . . DA ® DB (
Fiber bundle geometry reproduces nonlocal quantum effects semiclassically
i *-'comp-
[l]
A measurement of the observable go, a quantum number of G or of its generator algebra g (for a Lie group) performed on the constituent A constrains through Eq. 1 the values of the same quantum number at B, C, etc. A simple example of an EPR situation is provided by II decay (20). The two y rays should have their spin polarizations adding up to zero (and to negative total intrinsic parity) when observed, whatever the AB distance. For example, once the helicity of A is measured and found to be + 1 , a measurement of B will have to yield - 1 with certainty. In the contending realism view, this would be understood classically as indicating that the physical allocation of helicities had actually preceded both measurements and is part of the underlying reality searched for by Einstein and his collaborators. The probabilistic answer given by quantum mechanics is then supposed to represent an epistemological uncertainty (i.e., lack of knowledge), rather than an intrinsic indeterminacy. However, experimental verification of the existence of a fundamental indeterminacy has been provided, e.g., through the falsification of Bell's inequalities ( 2 1 24), which had indeed assumed the existence of such an underlying reality. We are thus forced to accept nonseparability or apparent action at a distance. When one usually adds here the postulate of special relativity (a somewhat inconsistent procedure in treating nonrelativistic quantum mechanics, to say the least), nonseparability appears to imply in addition acausal behavior, because there is no way for the results of a measurement in A to propagate fast enough so as to affect a (laboratory frame) simultaneous helicity measurement in B.
Those features of quantum mechanics that appear to disagree with a classically conditioned physical intuition are of two types: features relating to the act of measurement (collapse of the statevector) and nonlocal features. The latter include the AharonovBohm (1) effect; monopoles both of the Dirac type (2) in electromagnetism and of the 't Hooft-Polyakov type (3) in YangMills (YM) theories (4, 5); other YM structures such as instantons (6, 7), merons, and such; and Einstein-Podolsky-Rosen (8) (EPR) simulated action at a distance. In recent years, the Aharonov-Bohm (1) effect [first tested experimentally by Chambers (9)] was given a semi-classical geometric realization, as a topological global effect (10). Electromagnetic gauge invariance is realized geometrically on a fiber bundle (11, 12). Such a manifold (13-15) corresponds to a nontrivial juxtaposition of a (vertical) group space (or group representation space), the fiber at each point of the (horizontal) base space (generally space-time). The relevant group constituting the fiber is in this case the phaseinvariance group. (This is in fact the only data with a quantum origin, the rest of the description being entirely classical.) The fiber bundle is a locally trivial juxtaposition, yet allowing for a nonsimply connected geometry to arise globally as in the case of a Mobius strip, where a twist is introduced globally only. In Yang's treatment of the Dirac monopole (16), the global effect is achieved by the removal of a point at the origin of the R 3 base space. It is this nontrivial structure that creates nonshrinkable loops or n-spheres and results in discrete topological effects (winding numbers) reproducing both the Aharonov-Bohm effect itself (12) and monopoles (16), instantons, merons, and such. In recent years, several papers have reported this geometric realization of YM gauges (4, 5). In the present note, I point out that the geometry of fiber bundles (13-15, 17-19) provides in addition a classical geometric realization of EPR nonlocal action or nonseparability of the composite wave function.
A point that does not appear to have been generally discussed in the study of the paradox relates to the local definition of the angular momentum generators themselves (in nonrelativistic quantum mechanics, this is of course, axiomatic) or of the momenta in the original EPR gedanken experiment. How can these algebraic generators be defined in two different places and still be assumed to represent the same entity? How do we know that the two helicities can still be added to zero? Could
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Abbreviations: EPR, Einstein-Podolsky-Rosen; YM, Yang-Mills. t Also Ministry of Science and Development, Jerusalem, Israel; and on leave from the University of Texas, Austin, TX 78712. 7051
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not some difference in normalization, for instance, have crept in along the trajectories of the y quanta, so that +1 at A would not cancel - 1 at B? Are we certain that the z direction at A is indeed parallel to the z at B? Is a spin-scalar at the origin still a scalar at A or B? It is surprising that, with so much attention given to the nonlocal effects, considerations of that type were generally not envisaged. Yet it is precisely to answer these queries—posed in a different context—that YM gauges were invented (4, 5), following Weyl (25). It is thus not surprising that we should now be able to claim that the geometry generated by such gauges (11, 12) does contain a topological mechanism realizing EPR nonseparability. Whereas the AharonovBohm effect, monopoles, instantons, and such result from those discrete nonsimply connected global situations, EPR reflects parallel transport, another fundamental characteristic of fiber bundles (13-15, 17-19).
picture quantum mechanics as implying that "God plays dice"— or better, roulette. Should we then visualize the state vector |A) flipping randomly between the states of DA, the parallelism provided by the connection constrains the states |B), |C), etc., and forces them to flip in unison with |A). In other words, the gauge field a% and its curvature or field strength /£„ ensure that our quantum number go is the same everywhere and that all fibers flip together in the quantum roulette. This is nonseparability, ensured by gauge invariance. (tit) A measurement of |A) is thus a measurement of the entire V. (iv) In a relativistic frame, relativistic quantum field theory being causal, gauge invariance will again ensure nonseparability and this without violating causality. Comments
EPR is classically realized in a fiber bundle Separability is a natural assumption in a picture in which space (or space-time) is uncorrelated with the relevant EPR observable, as it is in nonrelativistic quantum mechanics. However, in a fiber bundle geometry, the manifold is constrained so as to preserve parallelism whatever the magnitude of the base-space interval. Translation from the origin, where the composite state carries the irreducible representation D$mf of G (in most examples—the scalar representation) has to be generated by the covariant derivative 0 ^ = 3 ^ + oi\X.u. Here d^ is the space (or d space-time) , represents the horizontal components of dx1* the Lie algebra valued connection &>, and A represents a basis of g. We work in the product basis of Eq. 1, so that D^ can act both on the composite state and on the constituents. The manner in which the connection co at any point p(x,y) of a principal bundle P(M,G, 77,') maps the tangent manifold P* onto the abstract (or global) Lie algebra g of G has been presented elsewhere (26). x is a coordinate on M, y is a coordinate on the fiber G. In a field theory, g is given by the Noether theorem as a space integral over P divided by the volume of G. The map provided by
[2]
17 is a projection (defining the vertical direction) from P onto M V p G P, V a, a' £ G -> ir(p'a) = ir(p)
[3]
(p'a)* a' = p»(aa').
[4]
Note that t realizes the Lie bracket (LB) operation on P* as a Poisson bracket (PB) V L, V £ g, t([L, L']LB) = [t(L), t(L')W
[5]
Returning to the EPR experiment, we observe that an associated vector bundle V(P,M,G,DA x DB x ..., IT,') realizes the following features naturally: (t) G acts on V equivariantly (26) so that both the composite state in DSLP (0) at the origin x = 0 and the separating constituents DA(x), DB(x'), etc., transform parallelly and together under the action of G. The constraint (Eq. 1) is thus maintained whatever the magnitude of the intervals AB, etc (ii) Antiparaphrasing Einstein, we
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Paradoxes appear when a description is incomplete. The woman in Houdini's trick—reappearing alive and whole after being sawed in two—poses a paradox only as long as we are not informed about that opening underneath her box that enables her to remove herself from the action of the saw. Nonrelativistic singleparticle quantum mechanics is an approximation; it is incomplete both in being nonrelativistic and in separating out a single particle. Einstein considered separability as a plausible approximation of attenuation by distance. However, this can no more be taken at face value when the true description of nature involves manifolds such as fiber bundles, in which space-time is only the base-manifold. Various schools of thought exist with respect to the incompleteness of quantum mechanics. Some claim that what is missing is a representation of the observer's mind processes; others require the adjunction of an infinite replication of alternate worlds and such. What I suggest here is that—to the extent that EPR is concerned—the paradox arises only because of the nature of the single-particle approximation (separability), an idealization that is contradicted by the geometric structure of the (gauge theory) complete physics. This geometric resolution of the EPR paradox fits observables protected by a gauge principle. This is true of momenta and of angular momenta, where gravity plays that role. It is also true of the exactly conserved quantum numbers of color-Sl/(3) in quantum chromodynamics and of the (spontaneously broken) degrees of freedom of quantum asthenodynamics (the unified electro-weak interactions) with the SC/(2) x U(l) of Glashow, Weinberg, and Salam. Indeed, for quantum chromodynamics, it would be easy to imagine a gedanken triple-EPR experiment, based on deep inelastic e N scattering processes inside a nucleon N, with apparent causally unrelated but quantum-correlated measurements of the color-S(7(3) of the three quarks (constrained by the color scalar property of the nucleon). The answer is not clear for a variable such as isospin or flavor-SL7(3) (unitary spin). At the phenomenological level, these are also gauge theories, with p, to , K*, and
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able, but the constraints on the geometric structure of the fiber bundle do transmit the relevant information over the entire base space. This is represented by the curvature (or the gauge field). At point B, the information is acknowledged, thus reproducing EPR classically. All points of the base manifold thus "know" of the alteration at x, even though it cannot be known at x itself. Aharonov has conjectured that, in quantum mechanics, it is the uncertainty principle that represents an analogous local partial unobservability of A at x, with (only) B at x' being automatically affected by that measurement or alteration at A. Possibly, an n-constituents quantum EPR could be represented by a discrete or lattice analog of our classical bundle. In this article, I have not touched on the other puzzling feature of quantum mechanics—i.e., the "collapse of the statevector. " I conjecture that the nonquantum features of the apparatus can be represented by a phase-transition description. In the gravitationally induced transformation of a "normal" (classically describable) star into a white dwarf or a neutron star— states in which the collapse is stopped by quantum degeneracy—we observe the transition from classical to quantum as a critical phenomenon. The same happens in the cooling down of a superfluid or a superconductor. It is thus plausible that an appropriate generalization might work for macroscopic bodies with the characteristics of the Copenhagen school apparatus. I would like to thank F. Dyson, S. Nussinov, and Y. Aharonov for several useful discussions. I am grateful to Drs. H. Woolf, J. Bahcall, and R. Dashen for the hospitality of the Institute for Advanced Study, where this paper was conceived. 1. Aharonov, Y. & Bohm, D. (1959) Phys. Rev. 115, 485-491. 2. Dirac, P. A. M. (1931) Proc. R. Soc. London Ser. A 133, 60-72. 3. 't Hooft, G. (1974) Nucl. Phys. B 79, 276-284.
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4. Yang, C. N. & Mills, R. (1954) Phys. Rev. 95, 631. 5. Yang, C. N. & Mills, R. (1954) Phys. Rev. 96, 191-195. 6. Polyakov, A. M. (1975) Phys. Lett. B 59, 82-84. 7. Belavin, A. A., Polyakov, A. M., Schwartz, A. S. &Tyupkin, Yu. S. (1975) Phys. Lett. B 59, 85-87. 8. Einstein, A., Podolsky, B. & Rosen, N. (1935) Phys. Rev. 47, 777780. 9. Chambers, R. G. (1960) Phys. Rev. Lett. 5, 3-5. 10. Yang, C. N. (1974) Phys. Rev. Lett. 33, 445-447. 11. Lubkin, E. (1963) Ann. Phys. (N. Y.) 23, 233-283. 12. Wu, T. T. & Yang, C. N. (1975) Phys. Rev. D 12, 3845-3857. 13. Kobayashi, S. & Nomizu, K. (1963) Foundations of Differential Geometry (Wiley Interscience, New York), Vol. 1. 14. Sternberg, S. (1964) Lectures on Differential Geometry (Prentice-Hall, Englewood Cliffs, NJ). 15. Choquet-Bruhat, Y., de Witt-Morette, C. & Dillard-Bleick, M. (1977) Analysis Manifolds and Physics (North-Holland, Amsterdam). 16. Yang, C. N. (1977) Ann. N.Y. Acad. Sci. 294, 86-97. 17. Drechsler, W. & Mayer, M. E. (1977) Differential Geometry and Gauge Theories, Springer-Verlag Lecture Notes on Physics (Springer, New York), Vol. 67. 18. Atiyah, M. F. (1979) Geometry of Yang-Mills Fields (Acad. Naz. dei Lincei Pisa). 19. Ne'eman, Y. (1979) Symetriesjauges et Varietes de Groupe (Univ. of Montreal Press, Montreal). 20. Bohm, D. & Aharonov, Y. (1957) Phys. Rev. 108, 1070-1077. 21. Bell, J. S. (1966) Rev. Mod. Phys. 38, 447-452. 22. Clauser, J. F„ Home, M. A., Shimony, A. & Holt, R. A. (1969) Phys. Rev. Lett. 23, 880-884. 23. Clauser, J. F. & Shimony, A. (1978) Rep. Prog. Phys. 41, 18811927. 24. Aspect, A., Dalibard, J. & Roger, G. (1982) Phys. Rev. Lett. 49, 1804-1807. 25. Weyl, H. (1929) Z. Phys. 56, 330-352. 26. Ne'eman, Y. & Thierry-Mieg, J. (1980) Proc. Natl. Acad. Sci. USA 77, 720-723.
Foundations of Physics, Vol. 16, No. 4, 1986
The Problems in Quantum Foundations in the Light of Gauge Theories1 Yuval Ne'eman23 Received Mav 1, 1985 We review the issues of nonsepat-ability and seemingly acausal propagation of information in EPR, as displayed by experiments and the failure of Bell's inequalities. We show that global effects are in the very nature of the geometric structure of modern physical theories, occurring even at the classical level. The Aharonov-Bohm effect, magnetic monopoles, instantons, etc. result from the topology and homotopy features of the fiber bundle manifolds of gauge theories. The conservation of probabilities, a supposedly highly quantum effect, is also achieved through global geometry equations. The EPR observables all fit in such geometries, and space-time is a truncated representation and is not the correct arena for their understanding. Relativistic quantum field theory represents the global action of the measurement operators as the zero-momentum (and therefore spatially infinitely spread) limit of their wave functions (form factors). We also analyze the collapse of the state vector as a case of spontaneous symmetry breakdown in the apparatus-observed slate interaction.
1. INTRODUCTION John Archibald Wheeler's contributions have spanned almost every area of twentieth century physics, and yet it is in the two fields of general relativity and the foundations of quantum mechanics that his influence has reached beyond the formalism and phenomena, down into the philosophical layer—the metaphysics, in the original sense. In his work on the general theory of relativity, it was Wheeler who emphasized the importance of the 1
Work supported in part by U.S. DOE Grant DE-FG05-85ER4020O. Center for Particle Theory, The University of Texas at Austin, Austin, Texas 78712, and Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. 3 Wolfson Chair Extraordinary in Theoretical Physics, TAUP N-161-85. 2
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geometrical interpretation. His interest in "wormholes" and "black holes" first pointed to physical effects whose origin is purely topological—an idea that has matured into our present "instantons," etc. Coining the term "geometrodynamics," he kept physicists aware of the challenge of doing to electromagnetism and the other interactions that which Einstein had achieved for gravitation. The ideal of a fully geometrical theory of physics was always there is his teaching. Nowadays, with the verified unified weakelectromagnetic theory (quantum asthenodynamics or QAD) and with the somewhat less manifestly validated quantum chromodynamics (QCD) for the strong nuclear interactions, that ideal is partially realized by the geometrical character of these theories. As gauge theories, they are already geometrically realized as fiber bundle manifolds (FBM). And yet, beyond the present stage, we have glimpses of further extensive geometrization— supermanifolds in the unification with gravity through supergravity, spontaneous Kaluza-Klein compactification for the internal symmetries, and geometrical lattice-spanned manifolds for superstrings, the present candidate universal theories. In the foundations of quantum mechanics (QM), Wheeler has again played a leading role. In the fifties he guided and sponsored the introduction of Everett's relative-state ("many-worlds") interpretation." 1 Since then he has mostly pressed for a more positivistic approach, insisting on the essential role of the (physical) observer in mapping out the limits of quantum physics' residual reality and its contents. (2) Here again, as in general relativity, he has put his conviction, imagination, and eloquence at the service of the philosophical foundations. In the present work, we would like to connect the two areas. We shall utilize the great advances in the geometrization program to throw new light on the problems and prescriptions of quantum-measurement theory, paradoxical as they might appear. We show that the discrete (2-point) nonlocal features of the Einstein-Podolsky-Rosen (EPR) quantum situations are entirely similar to the infinite and continuous nonlocal correlations of gauge theories. Moreover, it seems very natural that such features should appear in measurement theory, based as that theory should be on paralleltransportable observables. We also present a symmetry interpretation of the nature of the measurement-interaction inducing the collapse of the state vector. Our picture is inspired by the notion of spontaneous symmetry breakdown (SSB) with its geometrical-topological features.
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2. EPR-TYPE NONLOCAL CORRELATIONS As background for our geometrical discussion, we summarize current views of the EPR' 1 ' "paradox." The EPR setup has been generalized in various suggested experiments,(4) especially processes involving the detection of spin correlations (these are more conclusive, due to their discrete properties). The key tool has been J. S. Bell's theorem,{S) providing testable inequalities'6' derived from the local-type deterministic alternatives to QM. The results' 78 ' have provided an overwhelming refutation of the localdeterministic alternatives and also, at least to a very large extent, a confirmation of QM. The latter somewhat cautious phrase we use is only due to the remaining deterministic alternative of a nonlocal "hidden variables" theory, following de Broglie and Bohm.' 9 ' To this author, however, it appears somewhat futile to replace QM by another (and perhaps more complicated) nonlocal nonlinear theory, seemingly with predictions everywhere identical to QM. Such a theory should thus anyhow be regarded at this stage as an interpretation, like the "relative-state interpretation,"" 10 ' unless a Popperian falsifying experiment is suggested, capable of distinguishing between these ideas and "orthodox" QM. The refuted local-deterministic theories make two postulates"" which do not appear in the axiomatics of QM but were originally assumed by EPR. (a) The existence of a committed "underlying reality," committed in the sense that it is immutably "there," whether or not it is observed; any probabilistic description provided by QM thus results only from our lack of information—rather than from a true undetermined state of affairs, lasting up to the instant of measurement, as claimed by orthodox QM. (b) The separability of mechanically and spatially (or causally) isolated systems. This feature was especially tested in a recent experiment' 8 ' in which the inexistence of a causal link between the measurements at different points was carefully ensured. We eschew the discussion of those deeper issues relating to the uncommitted nature of the underlying reality, if any. J. A. Wheeler12' has given us a beautiful description of the manner in which the elements of that reality are "created" through their interaction with instruments and recording observers, the "Abrahamic" view (see footnote 36 in Ref. 2). The recording which shapes and commits reality is Bohr's "closure by irreversible amplification." (Note that this does not entail "conscious" registration, as is sometimes suggested. Any such suggestion is yet another interpretation.) All that is implied by QM is a material interaction serving as a
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"measurement," something irreversibly recorded by the mere presence of a background of macroscopic matter in the universe. Returning to the EPR setup, we quote B. d'Espagnat's resume" 2 ' of the proben status of nonseparabihty, the denial of EPR's second assumption: "When two quantum systems that originally could be considered as each having a complete set of definite properties of its own...." Again, "... in spite of the possible separation of the corresponding wavepackets [moving shutters may be introduced (see the optical switch in the experiment of Aspect et a/. ( 8 , )j, they are—at least as regards spin—to be considered as forming only one system...." For the sake of clarity, we assume the EPR experiment to involve the emission at the origin of two particles with spins / = 1 or 7 = \ from an original J=0 state, and the simultaneous measurement of spin components or polarizations of the emitted states A and B at +.v respectively (or at least measurements representing space-like separated events). The simplest such experiments exploit the decay of n° -»2y, or an atomic transition in Ca 40 with a similar emission of two gamma rays. The two states are found to have correlated helicities J3[A( -Kv)] = ±{, 7 3 [ S ( - x ) ] = + 5 . However, the violation of Bell's inequalities contradicts the possibility that this be the result of the definite helicities being there already prior to the measurement (as a committed "underlying reality"); and neither does the experimental setup allow the information about the measurement at A to arrive in B before the simultaneous measurement there!
3. GAUGE THEORIES AS FIBER BUNDLE MANIFOLDS QED, the more recent QAD unifying it with the weak interactions, and the (as yet separate) QCD strong interaction theory—all of these are gauge Theories. H. Weyl" 31 suggested this feature for QED, and C.N. Yang and R. Mills" 41 generalized the concept to non-Abelian gauge groups. It was then shown" 51 that such gauge theories should be regarded as specific geometric constructs, namely fiber bundle manifolds (FBM). A principal fiber bundle" 61 &>(.-$, G, n, •) is a manifold defined by a base space Jt (space-time or some subspace of it, in gauge theories), a structure group G [1/(1) in QED, SU(2)x U(\) in QAD, SU(3) in Q C D ] , a "vertical" projection n,
n: p(x, r) -*x,
pe»,
xeJt,
TeG
Locally, this is just a direct product of the "fiber" G with x
p(x, r) = x®T
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but if extended, that would trivialize /f. The fourth defining element is the right action (•) of G on .^,
vr, r,, r2 e0\ Pe // n{p-n
= Mp),
p-r=p'eS
p-(r,r2) =
(la)
ip-rx)-r.
The right action is thus a mapping / from g, the abstract Lie algebra of G, with the commutator as Lie bracket ("LB") /'=exp/(y, g) = exp/#
(lb)
into J^, the tangent-manifold of,'/, i.e. the basis of "vector-field" differential operators realizing g through Poisson brackers (PB). Fiber bundles such as -^ are then endowed with a specific tool, the connection one form (i>, essential to our discussion. This is an (almost) inverse map from ^ onto g (it cannot be a true inverse, as the dimensionality of . ^ is larger than that of G)
ML, Leg,
t(L) = L'e.^
\_L, L{]LB=
[L ,
(2)
L\jPB
The connection w is Lie-algebra valued, u> = u>aLu,
io» = WMd:",
a
K = l"jZxi
(o"Md="Jl^=dNM5«h
<3a» (3b)
so that we indeed get a mapping
= L'a J w = L'a J iohLh = L„
(3c)
As a result of the two mappings- the "dot" right action and the connection, the principal fiber bundle ?/ is a strongly correlated manifold. The group G and its Lie algebra acquire global characteristics: although there is a fiber G(x) at every point .ve,#, all local algebraic generators L'Jx) are assured to be correlated at all different points .v, x' B.H. This is guaranteed
761
Ne'eman
366
by the connection to, which reproduces the same abstract (and thus global) La at all points. The original definition of a "vertical" direction is inherent in FBM, as it is defined by the projection n. Now it is the connection that defines the subset H of horizontal vector fields: those that yield nothing upon contraction with co, the kernel H at a point pe:^. < # ) = * ' J » , = 0, ^r = Hn+ V„,
A'e//„c^, H
r = kerK),
H„.K=Hp-g
(4)
V„ = Im,(g)'
co defines horizontal displacements via "covariant derivatives" — parallel displacement vector fields (/ is a 1-form) D„ = dp-oj«t{x)Lu, D = dx"Dtl,
(5)
DX = dX - [co, A]
The curvature 2-form is given by R = dco — \\_w, co]
(6)
This is the field strength in a physical theory, while co is the potential. The curvature results from the noncommutativity of the covariant derivatives, ID,D-\ = R
(7)
It thus represents the changes when "moving" from A to B in a parallelogram ACBD either via C or via D. This curvature is then guaranteed to be horizontal and fulfills the Bianchi identity L'JR
=0
DR = 0
(8) (9)
The FBM thus supports an observable (the Lie algebra g) which is guaranteed to be well defined everywhere over ,M. It is thus omnipresent everywhere in M, capable of "materializing" locally at any x. All of this does not involve special relativity or causality. The omnipresence and correlations are generated by the specific geometry of the FBM. To rephrase these facts in a language reminiscent of EPR: two measurements at x and x', at spacelike distances, and even over a curved space, have the same correlated definitions of observables such as spin and can define parallel directions ("vertical" in their character here) at both places, without any signals being transmitted.
762
Quantum Foundations and Gauge Theories
367
4. GAUGE THEORIES AND QUANTUM PROBABILITIES In these definitions, the FBM is a "classical" theory, and yet it has these EPR-type correlations." 7 ' That "classical" theories should possess the nonseparability features generally associated with QM appears to us to clear the latter theory from some of the accusations often levelled at it. Alternatively, we can regard gauge theories—and for all we know these may be the only physical theories relevant to the physical world—as already containing some of the features characterizing the quantum picture. When electromagnetism or a Yang-Mills interaction were first described geometrically, it seemed that only the nature of the fiber variable (the phase, in QED) was a reflection of QM, the picture being otherwise classical. It then turned out that the guaranteeing of the conservation of probability (unitarity), a highly quantum feature, is also already present in the geometry. (,8) The RQFT systematics in Z)-dimensional space generate two unphysical components in the massless Atl photon (D-vector potential) or for each of the a = 1,..., N internal dimensions of the Yang-Mills potential A" (p. = 0,..., 3 in space-time). This is because the physical Hilbert space carries the photon or YM quanta as massless irreducible representations of the Poincare group. These correspond to the representations of the subgroup SO(D — 2) of the D —2 transverse directions for a D-dimensional base manifold. In physical Minkowski space, D = 4 and the group is SO(2). (We have used the more general definition, current in 1985 when the conjecture of a larger primordial dimensionality, a la Kaluza-Kiein, is favored among particle physicists.) The Yang-Mills interaction thus suffers from the unphysical propagation of two inexistent modes, and "ghost fields" are introduced artificially so as to correct these unphysical effects. However, it turns out" 8 ' that there is no need for any artificial intrusion—the "ghost" fields are already present there automatically in the geometry! Moreover, the Curci-Ferrari equationst,9) constraining the ghost fields and guaranteeing unitarity are just the Cartan-Maurer structural equations of the manifold. The shock of this revelation may explain the attempts on the part of some theoreticians to search (hopelessly it seems) for alternative interpretations of these facts. The geometrical identification has, however, been extremely useful, for instance in the study of "anomalies," yet another RQFT feature of these geometrical gauge interactions. The ghost fields x" constitute in essence the "vertical" component of the connection, which can be expanded locally once we select a section, (o = Al dx" + x" dy' = A^dx" + x" or, on a FBM where G is doubled (for right and left action) Q) = A"dx" + x'' + x"
763
(10)
368
Ne'eman
and the x" and x" fulfill the relevant structural equations, guaranteering the horizontality of R of (9). As a two-form over the entire FBM dimensionality, R involves differentials in dx'' A dx' (the physical field strength or Maxwell tensor F"n,), in dx1' A dy', dx1' A dy', dy' A dy', dy' A dy', dy' A dy'. The structural equations are just the statement guaranteeing the vanishing of all but the Maxwell tensor F"n,. (dx
A
dy)
sA = Dx;
(dy
A
dy)
•IX = - i l l , * ] ;
(dy A dy):
sj> = ~~\[X, X]
(dy
A
dy)
•** = l>\
(dy
sx = -b - [*, * ]
(dx A dy):
sA = Dx (11
A
.v, .fare vertical exterior derivatives, ^
= rf + .v + j ,
.y2 = .r = 0 (12)
.s\v + .w = 0,
«/+<& = (),
sd+ds
=0
b is an auxiliary field. These arc precisely the Curci-Ferrari unitarity e q u a t i o n s . " 9 ' The FBM geometry of the gauge theories (and these may be the only theories used by nature) thus "knows" about the probabilistic aspect of QM and guarantees the conservation of unitarity. Note that even though we used a localization in (10), the ghost fields are also really global objects, definable over the entire FBM. It thus appears to us that the key to the understanding of EPR is in the recognition of the global geometrical nature of the act of observation and measurement. If gauge theories and their F B M geometry require nonlocalized operators to define the conserved observable, it is natural that a measurement at A should in effect have measured B as well. The "commitment" at B happens when we measure. These are nonlocal features that correspond to the same approximation as the entire definition of the system to be measured. In his "delayed choice experiment,"' 2 ' Wheeler has even constructed an example on a cosmic scale. It all reflects a "reality" that is highly global in its geometry. What makes EPR appear unreal is the fact that (with Einstein, Podolsky, and Rosen) we tend to visualize the experiment in the sectional setting of Galilean 3-space or even Riemannian 4-space, whereas the true arena is one in which the latter are just the base manifolds. It is the additional dimensionality of the fibers—without which our measured observables would be meaningless—it is in this geometry that we should consider the measurement. This is a geometry with essentially global features, and they are reflected in QM.
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Quantum Foundations and Gauge Theories
369
5. THE AHARONOV-BOHM EFFECT There are other nonlocal QM effects which are again the result of global geometric features in FBM. The simplest is the Aharonov-Bohm effect.(20) Fringes indeed appear on a screen' 21 ' in the interference between two electron beams passing in an F^ = 0 region on either side of a solenoid containing a magnetic flux (note that this effect too was greeted with disbelief when it was first predictd). It is highly instructive to follow the derivation of the effect by considering the FBM geometry.' 22 ' The cause is that the base manifold (coordinates r, 6) .WAB=M2-J2
(13)
is the plane with a finite disk extracted (even if the radius rA -> 0, i.e., only a point missing), a multiply connected region. Using the path integral formalism,'23' one sees that if a nonzero magnetic flux
nel
(14)
The (nonrelativistic) path integral' 23 ' from the electron gun at rG to the screen at r 9 . (if is the classical Lagrangian function, H the Hamiltonian) K(rs, r c ; 0 :=
|r G >
= |<Jr(/)exp|/J o 'rfrJ?(r(T),f(T);T)J
(15)
has a kernel resulting from a summation over all paths n e Z; n < 0 refers to \n\ clockwise windings; « = 0 just stays to the right of A; n= 1 to its left; n > 1 implies n — 1 counterclockwise windings. KAB(rs,rG;t)=
£
e x p [ i e ( 0 s - 0 e + 2/nr)] #„(r s , r G ; t)
(16a)
with a phase shift
765
(16b)
370
Ne'eman
for the free-particle kernels Kn(rs,0s,rc,0a;t) y)
1/2
j
r
dk dk 2^exp{Jfc(0 s -0 c + 2 « j r ) } Z | i | ( r s , r G ; O
(17a)
where the reduced kernel l\k\(rs,rG;t)=
\3r(x)expU 1 /?
2
= j(rsr(;)"
exp 2/
I dx
'
=
(4 + 4
*
<
>
2wr 2 (r).
•
*
'
-
'
-/)'*'./^(/M/WO
(17b)
J\k\(pr) the appropriate Bessel function. The calculation is that of a path integral over the universal covering space, the FBM of (11), using appropriate techniques.' 24 ' Summing up, the nonlocal QM fringes can be calculated directly from the FBM geometry, a sum over all paths on 3PAB and not on .JiAB. The effect results from a global feature of S'AB. Wu and Yang<15) have in fact used these global features of the Aharonov-Bohm effect to define a gauge theory. Working relativistically, it is the phase factor (integrating over a closed loop) or path integral expj —j
/*„(*) dx'4 = exp
0
(18a)
that should, on the one hand, cause fringes to appear at Q, but should also display the same fringes for two magnetic fluxes (18a), (18b) related by he d& =
^ neZ
(18b)
This necessitates a gauge invariance of the left-hand side of (18a) for a Au and potential {Afl)a have to be gauge transformable into \jjh and {A J,, without affecting (18a). This is solved by _
«h = s-V„,
„-/oc(.v)
S=e
he (A/J)/, = (A,)a-i—SdllS-1
he = (A/l)u + — dfla(x)
766
(19a)
Quantum Foundations and Gauge Theories
371
S must be single-valued, but a need not be, provided J*=^-j
[ M „ ) A - M „ ) J dx" = YC ( 0 " " *«>
(19b)
is the gain for every winding. It is thus through the global feature that the electromagnetic gauge invariance and the entire electromagnetic interaction are defined. Yang125' has generalized this idea as a definition for any Yang-Mills gauge, thus emphasizing the significance of the global features. In contradistinction to FBM-structured gauge theories, gravity was generally studied locally rather than with a global emphasis. However, Aharonov Bohm effects have been pointed out in gravity recently,<26) and the global structure is sure to be even richer than in an FBM. The local structure is now understood to be that of a "soft" group manifold (SGM) ,27) with spontaneous fibration(28) and factorization,' 27 ' and these provide for large global variety.
6. MONOPOLES AND INSTANTONS Another example of a global effect is the Dirac monopole. As reintroduced by Wu and Yang," 5 ' this corresponds to a base manifold .//n = U}~A}
(20)
3-space with a sphere (rA ->0) at the origin removed. The Dirac monopole with strength g is placed there, at the origin. Taking an uncontractable sphere S2 in ,'UD, they show first that it is impossible to have a singularityfree A over the entire sphere. A loop integral Q = j>Alld,x"
(21)
taken on a horizontal (r, 9) circle will, by Gauss' theorem, correspond to the flux through that circle Q(r, 9) = 2ng{\-cos9) We start with a small 9 at the North Pole (NP) Q(r, 0) = 0 and keep sliding to the equator Q(r, n/2) = 2ng
767
(22)
372
Ne'eman
At 9 = n — c we have Q(r, 7i — f,)->4ng
as
e-*0
but with zero cross section Q(r,n) = 0 i.e., the local description has to fail, at least at the South Pole (SP). This is typical of an FBM, in which indeed one only invokes regional coordinate charts Ux, Up. Taking Ux from the NP "down" all the way beyond the equator, and Up from the SP "up" to the equator and beyond, a band with 9 = n/2 + S just S of the equator can be described in both overlapping coordinate charts, in the region {n/2)-6^9^(n/2)
+ S,
S
Taking for Afl at / = 0 the spherical components (axially symmetric) Ua:A, = Ar = Ae = 0,
A^ =
-^--(\-cosO) r sin 0 (23a)
U„:At = Ar = Ae = 0,
^ = T- ^gr s d + c o s 0 ) r sin 9
we find that the gauge transformation, defined for an FBM as the transformation relating the U3 and Up descriptions in the overlap region, is given by 5=(/j = e x p ( - / a ) = e x p ( — — J
(23b)
which has to be single-valued, thus yielding Dirac's quantization condition, 2ve -j- = n, tic
nel
(24)
Sxp = exp(/«^)
(23c)
Therefore
The phase factor (21) will have a unique value. All of that, of course, provided we stay out of A3 itself. In fact, homotopy tells us that when *,(£/(! )) = Z
768
(25)
373
Quantum Foundations and Gauge Theories
we shall have quantized monopoles, corresponding to the first Chern class of &{MD, U(\), 7t, •). For a Yang-Mills theory with SU(2) as structure group, there are only two classes of monopoles, depending on whether one is on the sheet connected to + 1 in SU(2). 't Hooft and Polyakov' 29 ' have shown how a Yang-Mills theory generates magnetic monopoles under spontaneous symmetry breakdown (SSB). The SSB is achieved through the action of a Nambu-Goldstone SSB field, with a nonvanishing vacuum expectation value (VEV) according to Higgs and Kibble. Such a field x behaves as a representation &{G) of the gauge group G. The nonzero VEV <0|x"|0>*0
(26)
breaks G'-symmetry, leaving a stability subgroup H. For # \ the third component of an 5(9(3) vector, when G = SO(3), H = SO(2). The SSB is thus described by S2 = SO(3)/SO(2). The vacuum in Minkowski space (i.e., the base manifold .# 3 ,) is given by a sphere S2 at r -> oo, so that SSB is given by the second homotopy group, which reduces according to a theorem, n2(G/H) = n[(H) = Z
(27)
Similarly, if we use the supergroup' 301 G = SU(2/\) as the simple unification of the Salam-Weinberg SU(2)x U{\) QAD, and take the SSB as given by the sixth component of an SU(2j\) octet, / / = U(\) and (27) holds. (Note that it does not hold for G = SU(2) x U(\) and the usual choice of a Higgs-Kibble meson.) Instantons' 3 " are yet another type of global objects, solutions to the Yang-Mills equations in a classical Euclidean space-time. It has been shown that tunnelling transitions between any two such solutions correspond to global quantum solutions.<32) In the Euclidean base space Jd,, the vacuum corresponds to the asymptotic equatorial 3-space S3, and the instantons are given by the third homotopy group 7t3(S£/(2)) = Z
(28)
For SU(3), the count is more complicated, with seven zero modes, but we shall not discuss this result here. We only mention that it is now believed that instantons (or inter-instanton transitions = QCD vacua) with their topological charges play an important role in QCD and quark confinement. We have gone into some detail in relating the Aharonov-Bohm and monopole cases, less for the more complicated instantons. There are additional global effects relating to "Gribov ambiguities," etc. Summing up,
769
374
Ne'eman
we emphasize that the geometrical-topological "classical" structure of our present field theories—the gauge models—is highly global and materializes in very nonlocal effects.
7. APPARENT ACAUSALITY AND RQFT Nonrelativistic QM cannot pretend to causal propagation of information, as it assumes Galilean covariance, with c-* oo. To check on this aspect, we are required to use the relativistic theory in which QM is embedded, RQFT. This theory led to difficulties for a while in the 1960's, when the successes of QED seemed hard to generalize. After the successes of 1971 (renormalization of Yang-Mills theories, including SSB) we seem to have an excellent theory. Supersymmetry, the embedding of the Poincare group in a supergroup, has even yielded finite four-dimensional field theories, such as 'W = 4 super Yang-Mills," with no ultraviolet divergences altogether. In RQFT, the Yang-Mills theories have the gauge group algebra Lu€ g of (lb), (2) realized through the Noether theorem, L = d r
» \ ° Wjn^{LMY
(29)
where a'' is a spacelike 3-hyperplane in Jf3A, if the Lagrangian 4-density, and 3>(La) the unitary representation of L„e g carried by the field components <j>r{x). The generator is thus a global feature, integrated over the entire space. To understand its global action in terms of QM, we note that (29) represents the charge in a Coulomb-like sense, for the Yang-Mills interaction. Dynamically, we know that the non-Abelian group results in the gauge-field potential itself carrying charge, so that the long-range behavior is ~r rather than ~r x. However, the Coulomb charge can be regarded as the zero-momentum limit of the form factor, y(k) = j > x e x p ( / k - x ) / V )
(30)
where f{x) is the time component of the current density, the integrand of (29). With k -+0, the uncertainty principle spreads (30) over an ever larger region of space. The global action of the Lie algebra of the observables in the FBM is thus guaranteed by RQFT to be spread out over the entire space! Indeed, in the sixties, .S-matrix theory, entirely represented in momentum space, was the favored theory. It had originally been advanced by Heisenberg to avoid future difficulties with space quantization. We observe
770
Quantum Foundations and Gauge Theories
375
that within RQFT, it is k -> 0 momentum space that plays the most important conceptual role for QM. Summing up, we have shown that although the measurement apparatus in EPR may be localized at A as a single-body wave function, its action as a measurement device is just the localization L'(xA) [Eq. (2)] of a globally defined operator L whose region of definition (resulting from Eqs. (29) and (30)) extends all the way to xB; otherwise there would be no meaning in preserving the same definition of an observable at A and B, a precondition in checking whether or not that measured variable (spin, etc.) maintains its correlation and how it is done. The only known alternative to RQFT as a relativistic generalization of QM is the method of path integrals, and we have seen in Sects. 5 and 6 that this also involves globally defined operations and is even more explicitly a topologically global theory.
8. THE COLLAPSE OF THE STATE VECTOR VIEWED AS SSB The great remaining "mystery" of QM is the state-vector collapse, viewed as a physical process. This is the above-mentioned "closure by irreversible amplification"'2' of N. Bohr. It would appear that this process is very much in the class of a spontaneous symmetry breakdown. Prior to the masurement, we have a quasisymmetric state vector: a^+A^1
(31)
for the two superposed spin-up and spin-down eigenstates, for instance. The measurement causes only one of the above to subsist (in our world, at least, if we take the relative-state interpretation). This is entirely similar to a Goldstone-Nambu or Higgs-Kibble situation, in which the stable solution is not the symmetric one. Similarly, if we should measure our electron's position, we would break a certain translation quasi-invariance in which all x-space is allowed, though constrained by the values of i//(x). In all cases, of course, the uncertainty principle compensates with a new quasi-invariance ( = indeterminacy) in the conjugate variable. Modern physics both in the particle domain and in condensed matter is very much dominated by SSB and corresponding phase transitions. In that sense, we should not be surprised at the nature of the measurement process. It just means that the measuring apparatus is by definition a quantum-mechanical array with a Goldstone-Higgs-like interaction as its main required feature.
771
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Ne'eman
This article has been in the nature of an effort at demystification. It seems to us that just as Newton's mechanics gained in insight from developments in the next century—from the work of d'Alembert, Laplace, Lagrange, Euler, Hamilton, and Jacobi—so can QM be better understood today. Gauge theories and spontaneous symmetry breakdown have acquired a key role in the actual physics—and have also clarified much of what is mysterious in QM. It could be that this does not bring the quantum picture of "reality" any closer to our everyday intuition, but it shows that it is in harmony with geometrical concepts that we seem to have accepted independently. It also exorcises any apparent contradiction between the quantum scene and the concepts which have evolved in classical (relativistic) field theory.
REFERENCES 1. J. A. Wheeler and W. H. Zurek, Quantum Theory of Measurement (Princeton University Press, Princeton, New Jersey, 1983). 2. J. A. Wheeler, "Delayed-Choice Experiments and the Bohr-Einstein Dialog" (American Philosophical Society Publication, 1981). 3. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). 4. M. Schwarz, Phys. Rev. Lett. 6, 556 (1961); B. d'Espagnat, Nuovo Cimenlo 20, 1217 (1961); R. Armenteros el at.. Proceedings, 1962 International Conference on High-Energy Physics (CERN), J. Prentki, ed. (CERN Publication, Geneva, 1962); D. Bohm and Y. Aharonov, Phys. Rev. 108, 1070 (1957). 5. J. S. Bell, Rev. Mod. Phys. 38, 447 (1966); Physics 1, 195 (1964). 6. J. F. Clauser, M. A. Home, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969). 7. J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978); A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49, 91 (1982). 8. A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982). 9. H. Everett III, Rev. Mod. Phys. 29, 454 (1957); J. A. Wheeler, Rev. Mod. Phys. 29, 463 (1957); N. Graham, Ph.D. Thesis, University of North Carolina at Chapel Hill, 1970; B. S. DeWitt, Phys. Today 23, 30 (1970). 10. D. Bohm, Phys. Rev. 85, 166, 180 (1952); D. Bohm and B. Hiley, Found Phys. 14, 270 (1984); L. de Broglie, Tentative d'Interpretation Causale et Nonlineaire de la Mechanique Ondulatoire (Gauthier-Villars, Paris, 1956); J. S. Bell, Found. Phys. 12, 989 (1982). 11. B. d'Espagnat, Sci. Am. 241/5, 158 (1979). 12. B. d'Espagnat, Conceptual Foundations of Quantum Mechanics (W. A. Benjamin, Menlo Park, California, 1971), Chap. Ill, Sects. 7 and 9. 13. H. Weyl, Z. Phys. 56, 330 (1929). 14. C. N. Yang and R. Mills, Phys. Rev. 95, 631 (1954); 96, 191 (1954). 15. E. Lubkin, Ann. Phys. (N.Y.) 23, 233 (1963); J. Math. Phys. 5, 1603 (1964); A. Trautman, Rep. Math. Phys. 1, 29 (1970); H. G. Loos, Phys. Rev. D 10, 4032 (1974); T. T. Wu and C. N. Yang, Phys. Rev. D 12, 3845 (1975). 16. W. Drechsler and M. E. Mayer, Differential Geometry and Gauge Theories, SpringerVerlag Lecture Notes in Physics, Vol. 67 (Springer-Verlag, New York, 1977); S. Sternberg, Lectures on Differential Geometry (Prentice-Hall, Englewood Cliffs, New Jersey, 1964);
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17. 18.
19. 20.
21. 22. 23. 24. 25. 26. 27.
28. 29. 30.
31.
Y. Ne'eman, Symetries Jauges el Varietes de Groupe (University of Montreal Press, Montreal, 1979). Y. Ne'eman, Proc. Natl. Acad. Sci. U.S.A. 80, 7051 (1983). J. Thierry-Mieg, J. Math. Phys. 21, 2834 (1980); Nuovo Cimento A 56, 396 (1980); Y. Ne'eman and J. Thierry-Mieg, Proc. Natl. Acad. Sci. U.S.A. 77, 720 (1980); Nuovo Cimento A 71, 104 (1982); L. Beaulieu and J. Thierry-Mieg, Nucl. Phys. B 197, 477 (1982). G. Curci and R. Ferrari, Nuovo Cimento A, 35, 1, 273 (1978). Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959); 123, 1511 (1961); 125, 2192 (1962); 130, 1625 (1963); W. Ehrenberg and R. E. Siday, Proc. Phvs. Soc. London B 62, 8 (1949). R. G. Chambers, Phys. Rev. Lett. 5, 3 (1960). G. Morandi and E. Menossi, Eur. J. Phys. 5, 49 (1984). R. P. Feynman, Rev. Mod. Phys. 29, 337 (1948); R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). C. Morette-DeWitt, A. Maheshwari, and B. Nerlin, Phys. Rep. 50, 255 (1979); M.S. Marinov, Phys. Rep. 60, 1 (1980). C. N. Yang, Phys. Rev. Lett. 33, 455 (1974). L. H. Ford and A . Vilenkin, J. Phys. A: Math. Gen. 14, 2353 (1981). Y. Ne'eman and T. Regge, Phys. Lett. B 74, 54 (1978); Riv. Nuovo Cimento 1, # 5 (series 3), 1 (1978); Y. Ne'eman, E. Takasugi, and J. Thierry-Mieg, Phvs. Rev. D 22, 2371 (1980). J. Thierry-Mieg and Y. Ne'eman, Ann. Phys. (N.Y.) 123, 247 (1979). G. 't Hooft, Nucl. Phys. B 79, 276 (1974); A. Polyakov, JETP Lett. 20, 194 (1974); J. Arafune, P. G. D. Freund, and C. J. Goebel, J. Math. Phys. 16, 433 (1975). Y. Ne'eman, Phys. Lett. B 81, 190 (1979); D. B. Fairlie, Phys. Lett. B 82, 97 (1979); Y. Ne'eman and J. Thierry-Mieg, in Differential Geometric Methods in Mathematical Physics, P. L. Garcia, A. Perez-Rendon, and J. M. Souriau, eds., Springer-Verlag Lecture Notes in Math., Vol. 836 (Springer-Verlag, New York, 1980), pp. 318-348; Proc. Natl. Acad. Sci. U.S.A. 79, 7068 (1982). A. Polyakov, Phys. Lett. B 59, 82 (1975); A. Belavin, A. Polyakov, A.Schwartz, and Y. Tyupkin, Phys. Lett. B 59, 85 (1975).
773
CAN EPR NON-LOCALITY BE GEOMETRICAL ?
Yuval Ne'eman* # Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University, Tel-Aviv, Israel 69978 and Alonso Botero Center for Particle Physics Physics Department University of Texas, Austin, Texas 78712, USA
1.
Introduction
The presence in Quantum Mechanics of non-local correlations is one of the two fundamentally non-intuitive features of that theory; the other revolves around the (so-called) collapse of the state vector, i.e. the need to assign special non-Schrodinger dynamics to the (generalized) "measurement" process. The non-local correlations themselves fall into two classes: "EPR"1 and "Geometrical". The latter include the Aharonov-Bohm2 and Berry (or Geometrical) 3 phases plus a variety of global solutions to quantum gauge theories, namely Maxwell's (a la Weyl), Yang-Mills theories with or without spontaneous symmetry breakdown, and Quantum Gravity state-of-theart (i.e. simplified) models. This category includes magnetic monopoles (in U(l) gauge theories4 and in some spontaneously broken Yang- Mills theories 5 ), instantons 6 , merons, etc. The non-local characteristics of the "geometrical" type are well- understood and are not suspected of possibly generating acausal features, such as faster-than-light propagation of information. This has especially become true since the emergence of a geometrical treatment for the * Wolfson Distinguished Chair in Theoretical Physics # also Center for Particle Physics, University of Texas, Austin, Texas 78712, USA
42
774
relevant gauge theories, i.e. Fibre Bundle geometry, in which the quantum non-localities are seen to correspond to pure homotopy considerations. We review this aspect in section 2. Contrary-wise, from its very conception, the EPR situation was felt to be "paradoxical" 0 . It has been suggested7 that the non-local features of EPR might also derive from geometrical considerations, like all other non-local characteristics of QM. In [7], one of us was able to point out several plausibility arguments for this thesis, emphasizing in particular similarities between the non-local correlations provided by any gauge field theory (such as parallel-transport and the existence of a connection) and those required by the preservation of the quantum numbers of the original (pre-disintegration) EPR state-vector, throughout its (post-disintegration) spatiallyextended mode. The derivation was, however, somewhat incomplete, especially because of the apparent difference between, on the one hand, the closed spatial loops arising in the analysis of the geometrical non-localities, from Aharonov-Bohm and Berry phases to magnetic monopoles and instantons, and on the other hand, in the EPR case, the open line drawn by the positions of the two moving decay products of the disintegrating particle. In what follows, we endeavor to remove this obstacle and show that as in all other QM non-localities, EPR is similarly related to closed loops, "almost" involving homotopy considerations. We shall develop this view in section 3. Before presenting our "resolution" of the EPR "paradox", we should state our reading of the actual answers provided by experiment. This is necessary since some schools in the Foundations of Quantum Mechanics have not yet accepted the finality of these answers (pointing at possible "outs" which would have to be checked before a final verdict); in addition, there is the alternative de Broglie - Bohm interpretation which chooses to preserve the deterministic features, at the expense of having actual action-at-a-distance. EPR were assuming (in the hope of preserving the intuitive view of locality, as inherited from classical physics) that the uncertainty relations of QM are due to statistical or information-wise considerations (e.g. involving hidden variables), whereas there does exist nevertheless an underlying determinisitic reality in which the two components of the disintegrating particle have already acquired their new quantum numbers at the moment of disintegration, even though these still remain hidden from us - until a measurement has been performed on one of them. Bell's inequalities8 made it possible to test this thesis and our understanding is that the Aspect experiments9 indeed falsified it - with the possible alternative we mentioned, namely the non-locally-acting hidden variables stressed by the late David Bohm. ° One advantage of this situation is the fact that after sixty years, there is still need for further clarification - thus also providing an excellent opportunity for a public celebration of yet another anniversary of our good friend Nathan Rosen.
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In our present context of conjuring away the non-local features, however, Bohm's theory appears somewhat purposeless. 2.
Global Effects from H o m o t o p y in F i b r e B u n d l e G e o m e t r i e s
A Principal Fibre Bundle V(M,G,ir, x) 1 0 is given by a base-manifold M, the structuregroup Q, the 'vertical' projection ir and the multiphcation x. The Fibre Bundle trivializes locally into a direct product; the projection ir maps an entire fibre over one point in the base manifold; and the " x " product has the group Q mapping the bundle manifold onto itself, while preserving the group's associativity: V(p &V,
x e M,
aeO),
p = x ® a, TT(J> x a) =
V xg ->V, (pxa)xa'
w(p),
(2.1)
=pxaa'
The " x " action is achieved through the group Q's generating Lie algebra 7, (o(g) = exp^g1^)), as represented by vector-fields (of the mathematical terminology) on the bundle's tangent manifold 7 C V„ acting on V; the Lie bracket is realized through the usual vector-field construction (i.e. the differential operators acting mutually as derivatives on each other's coefficient functions). The " x " action can thus be reinterpreted as a mapping from the abstract Lie algebra r o j j onto a submanifold of V,, (x) = r o t , -» V., 7 -» 7 e P . , V 7 € Tab,. The dimensionalities obey dim(V*) = dim(r o j,) + dim(M.*). kernel -M,, is performed by the connection w.
(2.2)
The "inverse" mapping 11 , with
w : V, -» r o 6 j ) V 7 6 r o 6 „ w(7) = 7-
(2-3)
with the "abstract" (or matrix commutator) Lie bracket relating to the vector-field realization through 17) 7 Imatrizcom.
17) 7
jvectorfield
(2.4) The connection is a one-form, whose action in the above map is realized through an inner product or contraction with the vector-fields of the Lie algebra's realization in V,, U\jj)
= 6)
(2.5)
where the coefficient functions of dp~ in the o> one- form and of 6** in the vector-field 7 are quantum-fields with arguments p~, p' , respectively, and where the right-hand side of (2.5) is supplemented, within the inner product, by the appropriate Dirac delta function 6(p — p') in the corresponding integration.
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The Fibre Bundle (FB) is trivial, if a cross-section can be drawn on it globally. Taking the example of the Mobius strip, what makes it non-trivial is the fold or twist. Should we select some point p and draw a cross-section through it, the twist can always be pushed further away, as long as we do not close a loop. As a result, (I) any open line (such as is drawn in the EPR particle's disintegration) will follow the geometrical constraints of the direct product M x Q. If M is Minkowski spacetime, Poincare invariance is thus guaranteed. To the extent that we shall show that EPR occurs within a FB manifold, this is then why it will not involve faster-than-light communication. The corolary is also relevant to our physical issue: (II) Closed loops - and only closed loops - probe the connectedness characteristics of a FB Manifold. Thus, only homotopy can reflect global topological features. This is why all the global quantum effects we cited (Aharonov-Bohm 2 and Berry 3 phases, monopole*'5 and instanton 6 YM solutions, etc) do involve closed loops, when they display their non-local characteristics. The emergence of these effects in the formalism of plain quantum mechanics follows. The wave-functions, given as sections on the bundle, are represented over the fibre bundle Projective Hilbert Space representations. The connectedness features revealed by homotopy are induced from the bundle over these representations. It is not possible to use one coordinate system without encountering singularities. On the bundle, instead, one utilizes coordinate patches, clean of singularities, each in its own sector; continuity is ensured by requiring a smooth transition beteen different patches in the overlap region 12 ' 13 . Such a requirement, coupled with the constraint of obtaining single-valued wave functions there, defines the gauge transformations. The reader, if unfamiliar with this description, is encouraged to read the example treated in ref. [14], where M is the 2-sphere with a "hole" at the center (containing a magnetic monopole) [S2 — (0)]; at least two patches are required (one for the "northern" and one for the "southern" hemispheres), with the overlap region covering a broad equatorial belt. Q emerges as a 17(1) gauge group, as a result of the single-valuedness requirement in the overlap transition function. This is the passive approach to gauge transformations and to fibre bundles. 3.
Fibre Bundle Embedding of the E P R Processes
In the usual treatment of the EPR processes, non-relativistic quantum mechanics serves as the arena. This in itself makes one wonder why would the EPR authors or anybody else expect the portrayed experiments to display a relativistic behavior, i.e. exclude faster-than-light transmission of information. From a non-relativistic treatment, one might have expected Galilean rather than Lorentz invariance. In the FB approach, M obeys special relativity because of the local triviality of the FB (reducing to a direct product, with the Poincare group as the local
45
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isometry of M, one of the factors in the direct product). £From I-II we also know that this also automatically relegates non- local effects to closed loops. The need for the FB embedding here can best be understood in terms of the need for active gauge transformations and for a connection. What is at issue is parallel transport: How else could we know that the symmetry group, used to describe the pre-disintegration compound in terms of its components (e.g. spin angular momentum, in the singlet state S = 0), is unbroken and that the eigenvalues of its Lie algebra retain their value, scale and meaning all along the paths of the decay products? In a FB, this is guaranteed by the existence of the connection. Any measurement performed at the disintegration point x0 € M can be parallel-transported to any points along this section of the FB (the decay path), using a covariant derivative D,,9m(x) = l v (S™dn — wJl(x)(7i)JJ )*' (2).. The connection regenerates the Lie algebra at any point along the path, following (2.3). What about the non-local features? They have to relate to closed paths and holonomy. We do have a closed path, since the punchline in EPR - namely checking that when particle A is measured to have its spin 'up', particle B indeed turns out to have its spin 'down' - requires closure of the path with 'messages' from A and B back to the origin I, or, more precisely, to F, where we have meanwhile arrived from I. In a space-time diagram, in which space is reduced to one dimension, we have a diamond shaped path (fig. 3.1) F
A
B
\
I \
I \
I I
Fig. 3.1: Closing the Loop in EPR
The "paradox" in EPR consists in the existence of non-local correlations between A and B, i.e. in the survival of the original amplitude as long as the measurement has not been performed, either at A or at B. This is a natural geometrical result in the FB.
46
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Note the complementarity between the two types of non-local quantum effects: the explicitly homotopical situations require the loop to be uninterrupted, i.e. having no localized interaction along the paths between J and F, whereas EPR by definition requires a "measurement" at A or B. It might be possible to invent an EPR experiment in which the carriers of the information from A or B to F would consist of the same type of particle as those resulting from the disintegration at I (and thus moving along IA and IB). However, this would still not lead to a "combined" effect, because the interaction at A or B would automatically exclude the explicit global result. 4.
Discussion
This FB realization of the EPR situation posits that (non- relativistic) QM is indeed "incomplete", but only in the following sense: (1) there would be no justification, anyhow, to expect a non- relativistic theory to display "Einstein locality" or "separability" (modern "local causality" in axiomatic relativistic quantum field theory); (2) the marriage of QM with Special Relativity has been long (since 1948) known to require Quantum Field Theory, but we now also know (from our acquaintance with the Standard Model of Particle Physics) that these have to be Gauge Field Theories. Such theories are geometrical in their nature and are realized by Fibre Bundle geometries. All EPR situations can therefore be embedded in some bundle geometry deriving from an appropriate combination of the basic geometries of the Standard Model gauges and/or Gravity. (3) Thus, QM is a truncation of the 7?s spatial submanifold of a FB's Minkowski base manifold. As such, it carries a structure allowing it to fit properly into the complete relativistic FB and excluding faster-than-light effects. QM has no formal 'knowledge' of relativity; however, having emerged from studies of electromagnetism, it is endowed with several such "pre-coded" interfaces with Special Relativity. Example: E = hu and p = A/A will yield E2 — (cp)2 = 0. Similarly, the non-local structure does not represent a violation of SR because it is a part of the FB 'heritage', its active aspect being limited to global homotopy. This is the structure behind that situation which has been termed "peaceful coexistence" by Shimony14. In an indirect way, this picture is related to the fact that we have a complex Hilbert space, involving phases, the "basic element" of all gauge groups Q. The complex Hilbert space itself derives directly from the Heisenberg algebra [x,p] = ih, once we require z and p to have real eigenvalues, i.e. to be hermitean. Intuitively, one might have indeed expected quantum nonlocality to reflect the blurring of spacetime, due to the uncertainty relations. Something of the sort is happening in the FB approach, but only in a rather loose sense.
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References [1] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 (1935) 777. [2] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [3] M.V. Berry, Proc. Roy. Soc. London A 392 (1984) 45. [4] P.A.IvI. Dirac, Proc. Roy. Soc. London, Ser. A 133 (1931) 60. [5] G. 't Hooft, Nucl. Phys. B 79 (1974) 276; A.M. Polyakov, JETP Lett. 20 (1974) 430; see also J. Arafune, P.G.O. Freund and C.J. Goebel, J. Math. Phys. 16 (1975) 433. [6] A.M. Polyakov, Phys. Lett. B 59 (1975) 82; A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Phys. Lett. B 59 (1975) 85. [7] Y. Ne'eman, Proc. Natl. Acad. Sci. USA 80 (1983) 7053. [8] J.S. Bell, Rev. Mod. Phys. 38 (1966) 447. [9] A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 47 (1981) 460 and 48 (1982) 91; A. Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett 49 (1982) 1804. T.E. Kiess, Y.H. Shih, A.V. Sergienko and C O . Alley, Phys. Rev. Lett. 71 (1993) 3893. [10] see for example S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, J. Wiley Interscience, N.Y. (1963), vol. I; S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ (1964); Y. Choquet-Bruhat, C. de Witt-Morette and M. Dillard-Bleick, Analysis, Manifolds and Physics, North Holland Pub., Amsterdam (1977). [11] Y. Ne'eman and J. Thierry-Mieg, Proc. Natl. Acad. Sci. USA 77 (1980) 720. [12] C.N. Yang, Ann. N. Y. Acad. Sci. 294 86. [13] T.T. Wu and C.N. Yang, Phys. Rev. D 12 (1975) 3845. [14] A. Shimony, Intern. Philos. Quarterly 18 (1978) 3.
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7 March 1994 PHYSICS LETTERS A ELSEVIER
Physics Letters A 186 (1994) 5-7
Localizability and the Planck mass Yuval Ne'eman Beverly and Raymond Stickler Faculty ofExact Sciences', Tel Aviv University, Tel Aviv 69978, Israel, Center for Particle Physics', University of Texas, Austin, TX78712, USA Received 22 June 1993; revised manuscript received 5 October 1993; accepted for publication 10 January 1994 Communicated by J.P. Vigier
Abstract We combine the assumption of environmental decoherence, as the mechanism generating the classical (i.e no quantum interferences) nature of spacetime, with the limit on its other classical feature, point-like continuity, namely Planck length. As a result, quantum extended objects with masses larger than the Planck mass have to derive their quantum behaviour from longrange correlations; objects with masses smaller than the Planck mass cannot display classical behaviour.
As an effective procedure, the Copenhagen prescriptions for the application of quantum mechanics have been overwhelmingly successful. Weighing this record, when considering the experimental verification of the Bell inequalities [ 1 ] by Aspect et al. [2], it seems hard to escape the conclusion that as nonintuitive as this be, and EPR [3] notwithstanding, the only "reality" in this world is the "quantum reality", the "reality" of complex quantum amplitudes. This is a reality represented by Hilbert space, rather than by spacetime. Moreover, the probabilistic nature of quantum mechanics is proven not to be of the "subjective" type. Rather than having fuzziness in a quantum variable just representing a lack of knowledge about its value, that fuzziness turns out to be "objective", i.e. the variable "really" does not possess a definite value, in that quantum reality, until this is settled in a measurement. This includes, first and foremost, the system's location, namely its spacetime coordinates. 1 2
In the last twenty years, the fact that we nevertheless appear to exist in seemingly "classical" surroundings is generally understood (though this is still far from unanimity) to arise dynamically, through the decohering action of the environment [4-9]. Starting from this paradigm, Joos has also shown [10] that the effects of decoherence include the emergence of a classical spacetime. The derivation uses a cube of volume L3 situated in a homogeneous gravitational field gM,. Assuming the (quantum) state to be in a superposition of two values of thefield(or of the corresponding acceleration y) >=a\yl>+b\y2>
Objects travelling through this cube (including dust particles) effectively become an "apparatus", "measuring" the field; their state |*> evolves according to the Schr&dinger dynamics for the product l*(0)>|*(0)> ^{a\yl>\xr3(t)>
Wolfson Chair Extraordinary in Theoretical Physics. Supported in part by US DOE Grant DE-FG05-85-ER40200.
(1)
+
b\y2>\xr\T)>]
As a result, the density matrix
0375-9601/94/S07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI0375-9601 (94)00035-N
781
(2)
6
Yu. Ne'eman IPhysics Letters A 186 (1994) 5-7
P(yl,y2,r)=p(f,y2,0)
<xr'(r)\xrl(t)y
the embedding gravitational field and the resulting decoherence process. Like other "measurement" processes in QM, the decoherence-generating measurement of the field should involve a "macroscopic" apparatus (the dust particles) acting irreversibly (the "pointer"). We can combine Joos' results with the strong-field horizon limitation and formulate
(3)
involves (the last factor in Eq. (3)) the overlap of the two evolved combined states (of that natural "apparatus" and the interacting field), which is approximately proportional to <5(3>1-y2). This gradually destroys the contribution of the interference between the two values of y. We refer the reader to Ref. [10] for details and orders of magnitude in this example. In this study, we assume with Joos that this is indeed the type of mechanism which would thereby attenuate the contribution of quantum phases and interferences and leave us with impression of a classical spacetime in that sense. However, Joos' derivation assumes another classical attribute of spacetime: the interaction between the gravitational field and matter obeys Einstein's equation *,»(*)-ifc»(*)*(*)=-KT, w (*)
Conclusion 1. Spacetime is fully classical, i.e. pointlike, continuous and devoid of quantum interferences down to dimensions /P. In addition, we have a criterion for "macroscopicity": objects whose Compton wavelengths have no physically definable spatial spread, i.e. M> mv, but which nevertheless interact with the gravitational field over more than one distinguishable point:
(4)
Conclusion 2. Macroscopic objects, in the sense of the apparatus in the Copenhagen interpretation, have M> w P and /> /P.
and is thereby a local interaction in the sense of continuity (we have explicited the x arguments to make this obvious). The quantum wavefunctions are delocalized, but the x variable is continuous. Localizability in both senses is an evident requirement for our overall physical description of canonical spacetime. We would therefore now like to derive some additional information with respect to the notion of localizability. First, what is then the limit of localizability? Treatments of quantum gravity appear to have formulated an answer years ago: the "Planck length" is assumed to represent the ultimate "definition" of spacetime (the term being used in the optical sense). This is an accepted notion. Although Planck's introduction of this length was based on dimensional arguments, we nowadays define "Planck dimensions" by equating the Compton wavelength Ac=h/mc to the Schwarzschildradius, rs=2Gm/c2, yielding i
lP=y/2Gh/c ,
mp^y/hc/lG.
We now have, however, another corollary relating to Planck dimensions. With our new definition of macroscopicity, since the Compton wavelength XM< /P, the "extension" in this context can only have a "classical" implication. What then of macroscopic quantum systems? Conclusion 3. Quantum behaviour in macroscopic objects involves long-distance correlations solely. Examples include Cooper pair coherence lengths, white-dwarf electron degeneracy coherence lengths, etc. Another corollary relates to "microscopic" masses M<mP. Conclusion 4. Microscopic objects (M<mP) are not fully localizable (i.e down to / P ) and display quantum behaviour.
(5)
Using the decoherence paradigm, we can now formulate an additional interpretation of the logic behind this notion: a particle with mass mP would generate its own strong gravitational field, rather than coupling mildly to our background spacetime field as in (4). It would develop a horizon, of radius /P, a black hole, thus also further inhibiting its interaction with
Examples: buckyballs (M~10 3 GeV), DNA {M~ 108 GeV), etc. Developing criteria to distinguish between quantum and classical objects has been an aim of many studies in the foundations of quantum mechanics. The above considerations appear to us to point to the
782
Yu. Ne'eman /Physics Letters A 186 (1994) 5- 7
notion of Planck dimensions as a related notion, thus providing at least one important criterion relating to localizability, perhaps the only quantifiable one - and a direct result of the decoherence mechanism philosophy.
References [ 1 ] J.S. Bell, Rev. Mod. Phys. 38 (1966) 447; Physics 1 (1964) 195. [2] A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 49 (1982)91;
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7
A. Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett. 49 (1982) 1804. [ 3 ] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 (1935) 777. [4] H.D. Zen, Found. Phys. 1 (1970) 69. [ 5 ] M. Simonius, Phys. Rev. Lett. 40 (1978 ) 980. [6] W.H. Zurek, Phys.Rev. D24 (1981) 1516; 26 (182) 1862. [7] E. Joos, Phys. Rev. D 29 (1984) 1626. [8] E. Joos and H.D. Zeh, Z. Phys. B 59 (1985) 223. [9JJ.B. Hartle, in: Proc. Jerusalem Winter School for Theoretical Physics, Quantum cosmology and baby universes (1989), eds. S. Coleman, J.B. Hartle, T. Piian and S. Weinberg (World Scientific, Singapore, 1991) pp. 67158. [ 10] E. Joos, Phys. Lett. A 116 (1986) 6.
DECOHERENCE PLUS SPONTANEOUS SYMMETRY BREAKDOWN GENERATE THE "OHMIC" VIEW OF THE STATE-VECTOR COLLAPSE Yuval Ne'eman Beverly and Raymond Sackler Faculty of Exact Sciences* Tel-Aviv University, Tel-Aviv, Israel 69978 and Center for Particle PhysicsH University of Texas, Austin, Texas 78712
ABSTRACT The collapse of the state-vector is described as a phase transition due to three features. First, there is the atrophying of indeterminacy for macroscopic objects including the measurement apparatus. Secondly, there is the environmental decohering mechanism, as described by Zeh, Joos and others - dominant in macroscopic objects. As a result, the classical background, an input in the Copenhagen prescriptions, is generated as an "effective" picture, similar to the "effective" introduction of Ohmic resistance or of thermodynamical variables, when going from the micro to the macro-scopic; in this case, the collectivized substrate is provided by the multiplicity of photon scatterings, etc., on top of the effect of the large number of particles in macroscopic objects. Thirdly, there is the Everett "branching", i.e. the materialization of one of the now decoherent states, accompanied by the destruction of the other branches. By definition, quantum indeterminacy represents a symmetry; in a measurement, or in a branching, this symmetry is broken "spontaneously", involving a Ginzburg-Landau type potential with asymmetric minima, thus concretizing the quantum "dice" without the burden of "many worlds". We review and systematize the various phase transitions relating quantum to classical phenomena.
* Wolfson Chair Extraordinary in Theoretical Physics 1 Supported in part by US DOE Grant DE-FG05-85-ER40200.
784
290 Their 1 s not to reason why. Their's not to make reply. Their's but to do with psl. (paraphrasing Tennyson, "The Charge of the Light Brigade", 1855)
1.
THE COLLAPSE OF THE STATE-VECTOR As an effective procedure, the Copenhagen prescriptions for the
application of Quantum Mechanics have been overwhelmingly successful. Their experimental validation through the verification of the I-lJ Bell inequalities by " ] Aspect et al. was also a reconfirmation of some of the least intuitive features in the Copenhagen story. The probabilistic nature of physics at its fundamental level (Einstein's "God does not play dice") might be philosophically unpalatable to somebody reared in a deterministic age
but it does not hurt
the inner consistency of the theory. Paradoxes are however generated by two other aspects: the non-local features and the "collapse of the state-vector" extraordinary dynamics (Involving a classical breakground as input). The non-local features are either global-geometrical 1n origin (e.g. the Aharonov-Bohm effect, the Berry phase, QCD Instantons, etc.) or derive from the preservation of quantum correlations over macroscopic distances (EPR-type effects), a result whose paradoxical features Involve the state-vector collapse as well.
I have conjectured
3 5
elsewhere ^ ~ ^ that such EPR-type non-local1t1es are also related to geometrical features.
* This is less so 1n an age 1n which much of classical physics has also turned out to be non-determin1st1c, through "chaotic" behaviour.
785
291 The collapse of the state-vector is paradoxical 1n that it postulates a separate dynamical process for the description of measurements ("process 1" In Everett's notation £ 6 -b, whereas everything else 1s described by the Schroedinger equation ("process 2") for the case of non-relativistic quantum mechanics (or by the appropriate Dirac or Klein-Gordon, etc., for relativistic situations)
. Everett's work
launched a program in which the fundamental physics Is entirely quantal and does not require the Copenhagen classical framework.
Process 1
then becomes a special case of process 2, in a treatment involving "relative states", emphasizing the fact that any amplitude represents a subsystem of the "world function of the universe".
Zeh and Joos C 7 - 1 0 ^
have improved on this program by showing how the classical framework input of the Copenhagen prescriptions emerges as a result of the interaction with the environment. This picture has recently been given a path-integral formulation by Hartle and Gell-Mann t 1 1 ] .
what is still not resolved in this
approach is the dice-throwing: the concretization of one of the decoherent components in the resulting universal amplitude, while the other branches appear to live on. This is the feature which causes the Everett scheme to be described as involving a "many-world" interpretation - the worlds in which the other branches become concrete instead. I have suggested [4,12] j . ^ ^ n ^ s i n v 0 ] v e breakdown mechanism (SSB).
a
spontaneous symmetry
The measurement (or other branchings')
Hamiltonians would then contain a potential with asymmetric minima. This formulation does away with the "many world" implications of the unrealized branches - thus answering the criticism expressed in the "Note added in proof" on p. 320 of ref. £6], without negating the Popperian falsification requirements for scientific theories, a weak ** The choice is unfortunate, since "process 2" is the basic modus operandi of Quantum Mechanics. We have however preserved the notation, to reduce confusion.
786
292 point in Everett's answer.1' An SSB solution has also been put forward by Ghirardi, Rimini and Weber tl4] j w1|.|1 a potential which localizes the macroscopic wave-function (apparatus plus system) and annihilates the unrealized branches (corresponding to other locations).
A solution
involving a thermodynamical version has also been suggested t 15 J.
The SSB solution has to be superimposed on the Joos-Zeh hypothesis E
7-10
3 , in which the relevant Hamiltonian describes scattering by a
very large number of photons, etc., and in which the local nature of spacetime itself £9] stems from the continuous interaction of the gravitational field with macroscopic bodies.
I have also suggested a
general recasting of the classical/quantum transitions in a formulation based on the concept of phase transitions £12].
2. DECOHEREMCE IN MACROSCOPIC OBJECTS Incorporating the collapse of the state-vector with the related "measurement" (or branching), coupled to a weaning away of the other branches, in the amplitude-determining Schroedinger equation, includes three notions: (a) a macroscopic device involving irreversibility, registering the measurement; (b) the decoherence of this device as a result of multiple scatterings - thus yielding probabilities; (c) dicing - i.e. the realization of one possibility out of several and the vanishing of the other branches in the original amplitude. The effective classical characteristics of the measurement apparatus are due to its essentially macroscopic nature. Quoting Wheeler Clf>J, paraphrasing Niels Bohr,
1
Assuming that the theory indeed makes it unphysical for different
branches to Interact. Some science fiction treatments have explored the alternative option, opening up the possibility that different branches could become connected - see for example ref.[13].
787
293 "a phenomenon is not yet a phenomenon until it has been brought to a close by an Irreversible act of amplification, such as the blackening of a grain of silver bromide emulsion or the triggering of a photodetector" thus generalizing the apparatus' dial. In a macroscopic object, the (Ax.Ap) s h uncertainty is shared by N particles, thus reducing the uncertainty per particle.
In the apparatus' amplitude (s is the action
per particle, S the total action) iS/ti X = c e
iNs/li
,„ % (2.1)
= c e
the N -» - limit yields the same effective results as -n" •+ 0. The now effectively classical combined object, apparatus together with measured quantum system, is then exposed to decohering multiple scatterings. For an apparatus |x(0)> and a system |*(0)> = c1|q(0)> + c2|q'(0)>, the Schroedinger common evolution runs according to |#(0)>|x(0)> •* c 1 |q(t)>| X q (t)> + c2|q'(t)>|xq'(t)>
(2.2)
The density matrix p(q,q',t) is multiplied by the overlap of the two resulting states of the apparatus, p(q.q'.O) - p(q,q',t) = p(q,q',0)<Xq'(t)|xq(t)>
(2.3)
with interference terms becoming unobservable if <X q '|x q > = S(q-q')
(2.4)
In the language of sums-over-histories this defines a decoherence matrix (for this fine-grained case)
788
294 D[q'(t),q(t)] = S(q-q') exp 2iri {(S[q'(t)]-S[q(t)])/1i} p(q',q,0)
(2.5)
which yields for coarse-grained histories, with the two paths q(t) and q'(t) allowed respective variation ranges A(t), A'(t), D[q'(t):[A'(t)].q(t):[A(t)]] = /6q' CA']
/6q
(2.6)
S[q'(t f )-q(t f )] exp 2irl{(S[q'(t)]-S[q(t)])/h} p(q'.q.O)
[A]
The integration is over all that is unspecified: here, over the range of values of the q(t) and q'(t), with their final values coinciding. As a result of (2.4), when the off-diagonal elements of (2.6) will be sufficiently small, the amplitude (2.2) decoheres and the two states it describes become approximately orthogonal, with |c l d |' and |c2 d | 2 (the
coefficients of the post-decoherence orthogonal states) providing
the respective materialization probabilities.
3. SPONTANEOUS SYMMETRY BREAKDOWN FOR DICING We now have to represent dicing, namely the realization of one state and the destruction of the amplitude for the other.
Note that we
have no such problem with classically derived probabilities, since they are a-priori derived as statistical probabilities. The quantum probability derives from the quantum postulates for the amplitude, with the definition strongly tied to process 1 (the state-vector collapse). With Everett, we assume that the quantum amplitude is the only existing "underlying reality", as resulting from the Bell inequalities and the Aspect experiment; note that we are using the EPR terminology, even though we imply its opposite, since EPR were assuming a deterministic causal and local underlying reality.
Having adopted the view in which
physics is indeed a-priori quantal, we now have to complete the "liberation" process of the emerging classical result by ridding it of the residual unrealized piece of the quantum amplitude.
789
295 Let us take, as a concrete example, the usual Stern-Gerlach apparatus and measure the s z component of the spin for a beam of spin J=l/2 particles.
Assume (as in ref. L"173) that we have first measured
the s x component, so that the s z states are mixed and overlap.
Through
decoherence, when measuring the s z component this time, we have the possibility of identifying whether this particle is in a |ft(q*)> or in a |*(q+)> state; in most situations (if we have not "prepared" this beam) the probabilities will be 1/2 for each state. To describe the apparatus' dial, with the classical variable o, i.e. « o, we have to include 1n the action S[q(t)] a potential V = -u 2 q 2 + Aq»
(3.1)
By Ehrenfest's theorem, the values of q resulting from the Schroedinger evolution (2.3) will be the same as we would get for o with a classical potential V c = -u 2 o* + Ao*. i.e. o = ± (2uJ/A)1/2
(3.2)
We couple the apparatus to the system so that the final decohered wave function will contain the factor, 6[o - | ( 2 u 2 A ) 1 / 2 l ] P[*(qt)] + S[a + | (2u a /A) 1 / 2 ' ] P[*(q+)] P represents the two projection operators respectively.
(3.3)
Thus, when the
"solution" (3.2) will correspond to one elgenstate, the normalization 1s for that single state; the combination (3.3) in the Hamiltonian (or in the action) 1s an either/or statement, not a superposition amplitude. The potential V in (3.1) thus achieves the task of dicing, in the absence of a statistical (hidden variables) substratum in quantum mechanics.
790
296 We note that the "loss of uncertainty + decohering + dicing" combined process (=the collapse) occurs both in measurement and in all similar branchings. Of course, we have just replaced the many worlds incongruity by the adhoc insertion of an SSB potential, every time there is a process involving a macroscopic body. The aim should be to derive the SSB potential from the environment-generated decoherence process itself. We have dealt elsewhere with criteria of "macroscopicity" [18], Suffice it to say that these are based on the superposing of Joos' derivation of some classical features observed in spacetime L9J (namely the lack of phases and thus of quantum interference) from decoherence processes caused by the environment (e.g. dust particles "falling" in a gravitational field) with the other quantum restriction on spacetime - namely the emergence of the Planck length - providing a limit on pointlike smoothness and continuity. Our result indicates that Planck mass is also the lower bound for macroscopic objects with spatial extension exceeding their DeBroglie-Compton wavelengths; it is the upper bound for quantum behaviour not involving long-range correlation.
4. CLASSICAL TO QUANTUM AND QUANTUM TO CLASSICAL In this context, it is worth reviewing the class of phase transitions relating a classical to a quantal phase. Very roughly, we have the "recognized1 transitions, a) the low temperature classical to quantum transitions (superconductivity, superfluidity) b) the classical to quantum transitions induced by gravitational collapse (white dwarfs, neutron stars, black holes) to which we have just added the set,
791
297 Cj) the large N (number of particles) quantum to classical transition t 12 J. c 2 ) the scattering-induced quantum-coherent to decoherent transition (both Cj and C£ were described in section 2) C3) the dicing, quantum-decoherent to classical transition of section 3. To read the entropy-symmetry aspects, we write the partition function Z = J dM(x) e
(4.1)
where M(x) is the order parameter, T the temperature, L the entropy and the bracket in the numerator of the exponent is the free energy, F[M(x)] = E[M(X)] - T L[M(x)]
(4.2)
In this picture, the "order" term E[M(x)] wins over the entropy when T decreases beneath some critical temperature. For the transitions of class (a), the quantum phase is thus an ordered phase (with broken symmetry)*; the same holds in class (b), with nuclear energy supplying the temperature in the star. When the nuclear fuel is exhausted, the temperature falls, order wins and the star becomes quantum coherent. E then includes gravity, which causes the star to contract until electron (or nucleon) degeneracy is reached. We now turn to the state-vector-collapse transitions. The effective classical characteristics of the measurement apparatus are due to the combined action of (cj) and (C2-C3). The macroscopic nature of the
* We do not address here the more complex SSB struture relating to the electric charge local gauge invariance in BCS theory.
792
298 generalized apparatus decreases its indeterminacy, with the N •* - limit yielding similar results to those of h •* 0.
The now effectively
classical object is then exposed to the decohering multiple scatterings, followed by the dicing transition. In this sequence, it is the quantum phase which is intrinsically symmetric, due to quantum indeterminacy. This symmetry is destroyed together with the phases, with dicing completing the transition to a less-symmetric (and more ordered) classical phase.
As discussed in ref.E 1 2 ^, the symmetry-breaking arrow
points here in a direction opposite to that of the classical to quantum macroscopic transitions (a-b).
The order parameter in (c) is the
measured variable q(t). The number of particles N and the number of decohering scatterings N'(N) determine the abstract temperature, which is their inverse (see (2.1)), very roughly
T(c) s l/N
or 1/N1
(4.3)
so that the ordered phase, using (4.1), is indeed the classical phase. Note that it is in the dicing transition we described in sec.3 that q(t) and its classical parallel o play the role of M(x).
5. Conclusions This treatment of (c), the canonical quantum to classical transition, thus provides a sound qualitative and formal basis for a physics in which the quantum description is the only one at the fundamental level.
It would be interesting to check whether this phase
transition aspect - if pursued further in a more evolved treatment might yield new relations and some experimentally testable results, perhaps in combination with our gravity generated criterion of macroscopicity L 1 8 J.
The combined phase transition (c) thus yields an effective picture fitting the Copenhagen prescriptions. In this presentation, these prescriptions - including the necessary classical apparatus - are now
793
299 of a "collective" nature, in a generalized sense. There are in fact two types of reductions of this sort in physics: 1)
In thermodynamics, a micro substrate has been abstracted into the
thermodynamical variables, through statistical mechanics.
Something
similar happens in the physics of particles and fields, where it was shown by 't Hooft E 1 9 ] that, provided one imposes some constraints relating to the chiral anomalies, one has the option of replacing a treatment in terms of the (fundamental?) level of quark and gluon fields (QCD) by the composite level of baryon
and mesons (chiral and
gauged flavour-SU(3). This is "complete reduction". 2) Alternatively, there is the modern rederivation of Aristotle's physics.
Aristotle takes his models from "real life" (e.g. people pushing
a boat) in which friction (or air resistance, for falling bodies) is always present. Aristotle thus postulates a linear law of force i.e. force is proportional to velocity.
F = v,
It needed Galileo and Newton's
abstract (and physically almost unachievable) picture of a (perfect) vacuum, to get F • ma. Aristotle's law Is then recovered by Introducing friction, for instance 1n evaluating the final velocity of a stone falling in water, or of a paratrooper falling in air. Similarly, classical electrodynamics involves concepts such as electrical resistance, yielding Ohm's linear law V = IR, (whereas with the evolution of electronics we have come across non-linear behaviour, demonstrating the peculiar nature of the macroscopic behaviour represented by Ohm's law.
We suggest calling this type of partial reduction
"Ohmic", honouring Georg Simon Ohm (1787-1854) who produced his physics here in Cologne, as a highschool teacher. I am told the University was then closed.. Still, I am reminded of a Hasldic story.
For some reason
or other, a famous rabbi had exchanged places with his coachman, on a trip to a neighbouring town.
When they arrived, the people addressed
the supposed rabbi (i.e. the coachman, sitting in the rabbi's place) with some hard scholastic questions. The fake rabbi answered "for such simple questions you can talk to my coachman" - who indeed gave the correct answers. "If the coachman is such an expert, imagine what the
794
300 rabbi can do!" was the reaction.. If Ohm as a highschool teacher could do what he did, imagine what should be the required standards, here in Cologne.. This may explain why I have had such a fruitful collaboration with Friedrich Hehl and his group.. We thus contend that with the transitions (c), the Copenhagen systematics receive an Ohmic interpretation; improving this derivation as suggested could be described as the Cologne program (1993). Its completion would allow us to postulate an overall quantum foundation for physics, ruled by orthodox quantum equations ("process 2") and with no need for "prescriptions".
795
301 (i. Bibliography 1.
Bell, J.S., "On the Problem of Hidden Variables in Quantum Mechanics", Rev.Mod.Phys. 38 (1966) 447-452; Physics 1 (1964) 195.
2.
Aspect, A., Grangier, P., and Roger, G., "Experimental Realization of Einstein-Podolsky-Rosen-Bohm 'Gedankenexperiment': A New Violation of Bell's Inequalities", Phys.Rev.Lett. 49, No. 2 (1982) 91-94; Aspect, A., Dalibard, J., and Roger, G., "Experimental Test of Bell's Inequalities Using Time-Varying Analyzers", Phys.Rev.Lett. 49 (1982) — 1804.
3.
Ne'eman, Y., "Classical Geometric Resolution of the Einstein-Podolsky-Rosen Paradox", Proc.Natl.Acad.Sc1.USA 80 (1983) 7051. ~
4.
Ne'eman, Y., "The Problems in Quantum Foundations in the Light of Gauge Theories", Found.of Phys. 16 (1986) 361-377.
5.
Ne'eman, Y., "EPR Non-Separability and Global Aspects of Quantum Mechanics" in Symposium on the Foundations of Modern Physics, P. Lahti and P. Mittelstaedt editors (Proc. Joensuu 1985 Symp.), World Scientific, Singapore (1985), pp. 481-495.
6.
Everett, H.III, "'Relative State' Formulation of Quantum Mechanics", Rev.Mod.Phys. 29 (1957) 454-462.
7.
Zen, H.D., "On the Interpretation of Measurement in Quantum Theory", Found.of Phys. 1_ (1970) 69-76.
8.
Joos, E. and Zeh, H.D., "The Emergence of Classical Properties Through Interaction with the Environment", Zeit.fiir Physik B59 (1985) 223-243.
9.
Joos, E., "Why Do We Observe a Classical Spacetlme?", Phys.Lett.A116 (1986) 6.
10. Zeh, H.D., "There are no quantum jumps, nor are there particles!", Phys.Lett.A172 (1993) 189. 11. See for example J.B. Hartle, in Quantum Cosmology and Baby Universes, S. Coleman, J.B. Hartle, T. Piran and S. Weinberg editors (Proc. Jerusalem Winter School for Theoretical Physics, Xmas 1989), World Scientific Pub., Singapore (1991), pp. 67-158. 12. Ne'eman, Y., "Classical to Quantum: A Generalized Phase Transition", in Microphysical Reality and Quantum Formalism, A. van der Merwe et al. editors (Proc. Urbino Symp., 1988), pp. 145-151.
796
302 13. See for example Frederick Brown's "What Mad Universe", with General Eisenhower conducting the war against Arcturus.. 14. Ghirardi, G.C., Rimini, A. and Weber, T., "Unified dynamics for microscopic and macroscopic systems", Phys. Rev. D34 (1986) 470-491. 15. Haake, F.f "Decay of Unstable States", Phys. Rev. Lett., 25 (1978) 1685. 16. Wheeler, J.A., in Proc. Am. Phil. Soc. and Roy. Soc. Joint Meeting, 5 June 1980, Am. Phil. Soc. Pub. 1981, pp. 9-40. 17. Peierls, in Symposium on the Foundations of Modern Physics, P. Lahti and P. Mittelstaedt editors (Proc. Joensuu 1985 Symp.), World Scientific, Singapore (1985), pp. 187-196. 18. Ne'eman, Y.N., "Localizability and the Planck Mass", Phys.Lett.A, to be publ. 19. 't Hooft, G., NATO Adv. Study Inst, on Recent Developments in Gauge Theories, Series B, 59 (1980) 117.
797
CLASSICAL TO QUANTUM: A GENERALIZED PHASE TRANSITION
Yuval Ne'eman ' 3 Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel
ABSTRACT We present a new interpretation of the classical-to-quantum transition, as a generalized phase transition. It replaces the inadequate present identification of macroscopic = classical and microscopic = quantum, which fails for low-temperature and Paulidegenerate stellar systems. We also apply the method of phase transitions to the quantum measurement process. This explains the anomalous features of observation and the "collapse" of the state vector. 1.
THE REVOLUTION IN ONE'S INTUITIVE NOTIONS
The revolutionary aspect of twentieth century physics is caused by the profound chasm between one's intuitive picture of the physical world and the teachings of the new physics. In the special theory of relativity, it was mainly the nature of time that clashed with one's ordinary notions, deriving as they do from everyday communication between people, i.e., reference frames connected by Lorentz transformations that are very close to the identity. It is possible that when, perhaps in several thousand years, children will grow up in an environment in which spaceships will be as common as present-day automobiles, the old ("Galilean") concept of time will vanish altogether. Today, physicists or electronic engineers learn to live with the new relativistic concepts of space and time and gradually develop the appropriate intuition. The old view is rejected, and there is no residual feelings of mystery. It is only when the need arises to explain or communicate a relativistic situation to the uninitiated that one realizes that one's Galilean sense of absolute time has become atrophied, but that this is not yet the norm. In fact, the total number of people who understand relativistic time, even after eighty years since the advent of special relativity, is still much smaller than the number of people who believe in horoscopes.
145 A. van der Merwe et al. (eds.), Microphysical Reality and Quantum Formalism, 145-151. © 1988 by Kluwer Academic Publishers.
798
146 2.
EPR
I n quantum mechanics t h e chasm i s much w i d e r . I t i s our e n t i r e s e n s e of l o c a l i z a t i o n t h a t h a s t o be d i s c a r d e d . EPR made i t o b v i o u s , and t h e e x p e r i m e n t a l l y v e r i f i e d v i o l a t i o n of B e l l ' s i n e q u a l i t i e s i m p l i e s non l o c a l b e h a v i o u r . No wonder t h a t many amongst the p h y s i c s community i t s e l f c o n t i n u e E i n s t e i n ' s s e a r c h f o r a more p a l a t a b l e a l t e r n a t i v e ; and y e t non l o c a l i t y now p e r v a d e s e v e r y t h i n g . I have l i s t e d e l s e w h e r e ( l » 2 ) m any o t h e r examples of such non l o c a l e f f e c t s . Most of us quantum p h y s i c i s t s have l e a r n e d to beware of our i n g r a i n e d s e n s e of l o c a l i z a b i l i t y , and y e t i t i s h a r d to r e p l a c e i t by s o m e t h i n g e l s e t h a t would b e as i n t u i t i v e . There i s always t h e f e a r of a c t i o n a t a d i s t a n c e w i t h a c a u s a l b e h a v i o u r o r s u p e r l u n i n a l p r o p a g a t i o n of i n f o r m a t i o n — a n d y e t we a l s o know t h a t t h i s r e a l l y n e v e r a p p e a r s . Y a k i r Aharonov(3) h a s p r e a c h e d t h e o n l y a l t e r n a t i v e way of t h i n k i n g I am aware of, w i t h a s i m p l e i n t u i t i v e g r a s p of t h e quantum w o r l d . The f a c t t h a t , f u n d a m e n t a l l y , we e x i s t i n H i l b e r t s p a c e r a t h e r t h a n s p a c e - t i m e , t h i s f a c t i s t h e o r i g i n a l cause of a l l t h e s e " p a r a d o x e s " ; h o w e v e r , one can s t a y w i t h t h e s p a c e - t i m e p i c t u r e provided one accepts full non local connectivity as t h e b a s i c t e n e t of quantum m e c h a n i c s , and provided one also preserves the axiom of local causality. The uncertainty principle t h e n becomes the mechanism that will intervene every time the two notions disagree. Wherever we t r y t o use "EPR n o n - l o c a l i t y " o r i t s l i k e s as a way of s e n d i n g messages f a s t e r t h a n l i g h t , we f i n d t h a t t h i s i s i m p o s s i b l e due t o the u n c e r t a i n t y r e l a t i o n s . P r o p a g a t i o n over a s p a c e - l i k e i n t e r v a l i s always l i a b l e t o l e a d t o t h e n e g a t i o n of c a u s a l i t y , as two d i f f e r e n t r e f e r e n c e frames may have o p p o s i t e t i m e - o r d e r i n g s f o r b o t h ends of t h e p r o p a g a t i n g l i n e . T h i s i s why a d e t a i l e d a n a l y s i s w i l l always show t h a t t h e u n c e r t a i n t y r e l a t i o n s make i t i m p o s s i b l e to cause such a v i o l a t i o n . On t h e o t h e r h a n d , when t h e non l o c a l e f f e c t s r e l a t e to a c l o s e d p a t h ( a s i n t h e Aharonov-Bohm and o t h e r " t o p o l o g i c a l " e f f e c t s ) t h e r e i s no way of making t h e L o r e n t z group i n t e r f e r e , and non l o c a l i t y can be e x p l i c i t . We have d e a l t w i t h t h e non l o c a l f e a t u r e s e l s e w h e r e . ( 1 > 2 , 4 )
3.
THE COLLAPSE OF THE STATE VECTOR
The o t h e r s o u r c e of uneasy f e e l i n g s i n quantum mechanics i s t h e " c o l l a p s e " of t h e s t a t e v e c t o r ( o r t h e p a r a d o x of S c h r o d i n g e r ' s c a t ) . Here i t i s n o t a c l a s h w i t h o n e ' s everyday i n t u i t i o n , i t i s more l i k e an i n t e r n a l i n c o n s i s t e n c y of t h e t h e o r y i t s e l f : The measurement ( o r o b s e r v a t i o n a l ) p r o c e s s does n o t f o l l o w t h e quantum r u l e s as s e t by S c h r o d i n g e r ' s e q u a t i o n . The t h e o r y i s f o r c e d t o p r o v i d e a d i f f e r e n t p o s t u l a t e f o r measurement p r o c e s s e s . I n t h e p r e s e n t a r t i c l e we s h a l l m o s t l y touch upon t h i s i s s u e and r e l a t e i t t o a n o t h e r q u e s t i o n , t h a t of t h e i n t e r p r e t a t i o n of t h e c l a s s i c a l t o quantum t r a n s i t i o n . To s t a r t w i t h , we p o i n t o u t t h a t the "collapse" represents a breakdown of symmetry. For a s p i n - h a l f p a r t i c l e , t h e wave f u n c t i o n i s a p r i o r i symmetric between t h e two h e l i c i t y s t a t e s , and t h e r e i s a 50% chance i t w i l l be i n e i t h e r s t a t e . We measure t h e h e l i c i t y and we
799
147
o b t a i n " u p " or "down" as a unique a n s w e r . This i s t h e breakdown of S 2 , the symmetric t w o - e l e m e n t g r o u p . I f we have a non l o c a l i z e d p a r t i c l e and f i x i t s p r e c i s e p o s i t i o n , we have a breakdown of t r a n s l a t i o n i n v a r i a n c e . F i x i n g t h e momentum amounts t o the same t h i n g i n t h e momentum r e p r e s e n t a t i o n , e t c . Summing u p , t h e measurement p r o c e s s t r e a t s t h e quantum s y s t e m a s symmetric ( d i s o r d e r e d ) and t h e p o s t - o b s e r v a t i o n s y s t e m as asymmetric (ordered).
4.
THE CLASSICAL/QUANTUM LIMIT
C o n v e n t i o n a l l y , one assumes t h a t t h e t r a n s i t i o n from quantum to c l a s s i c a l o c c u r i n g f o r m a c r o s c o p i c b o d i e s i s r e l a t e d to t h e l a r g e number N of p a r t i c l e s i n v o l v e d . The l i m i t N->°° y i e l d s c l a s s i c a l systems(5) i, = c e i A /tf = c e ± N a / K ,
(4.1)
where t h e a c t i o n i s A = Na, i n terms of t h e a c t i o n f o r i n d i v i d u a l p a r t i c l e s , and N-*" y i e l d s r e s u l t s s i m i l a r t o t h e VL -> o . I n quantum m e a s u r e m e n t s , as n o t i c e d , t h e apparatus is postulated to be classical ( t o a c h i e v e t h e n o n - S c h r b d i n g e r b e h a v i o u r we m e n t i o n e d ) . This i s r e g a r d e d as n a t u r a l , t h e a p p a r a t u s i n g e n e r a l c o n s i s t i n g of a m a c r o s c o p i c s y s t e m . I n f a c t , i n W h e e l e r ' s p a r a p h r a s e ^ ) of B o h r ' s d e f i n i t i o n of an o b s e r v a t i o n ( i n t h e c o n t e x t of EPR " p a r a d o x e s " ) , "a phenomenon i s n o t y e t a phenomenon u n t i l i t h a s been b r o u g h t to a c l o s e by an irreversible act of amplification such a s the b l a c k e n i n g of a g r a i n of s i l v e r bromide emulsion o r t h e t r i g g e r i n g of a p h o t o d e t e c t o r , " the i r r e v e r s i b i l i t y r e q u i r e m e n t d e n o t e s such c l a s s i c a l b e h a v i o u r , presumably due t o l a r g e N. However, t h e r e seems t o be a flaw i n t h i s a r g u m e n t . We know of s e v e r a l c a t e g o r i e s of m a c r o s c o p i c b o d i e s w i t h d e f i n i t e l y q u a n t u m - t y p e behaviour. These a r e g e n e r a l l y u n d e r s t o o d as l o w - t e m p e r a t u r e phenomena, i n which t h e z e r o - p o i n t energy E i s of t h e same o r d e r as t h e t e m p e r a t u r e T. I n o t h e r words,^'-' t h e p a r t i t i o n f u n c t i o n (M(x) i s t h e o r d e r p a r a m e t e r and S t h e e n t r o p y ) Z = J
DM(x)
involves a free
exp {-|[E
- T S]}
(4.2)
energy
F(M(x)) = E(M(x)) - T S<M(x))
(4.3)
in which the "order" term E may win over disorder as represented by S, due to T -> 0.
800
148 I n d e e d , we know of s u p e r c o n d u c t o r s ( t h e o r d e r p a r a m e t e r i s t h e e l e c t r o n C o o p e r - p a i r a m p l i t u d e ) , s u p e r f l u i d s ( t h e He^ a m p l i t u d e ) , white-dwarf s t a r s (the electron-degeneracy amplitude), neutron s t a r s (neutron-degeneracy amplitude), e t c . In the a s t r o p h y s i c a l examples, i t i s the g r a v i t a t i o n a l p o t e n t i a l t h a t keeps decreasing the entropy term u n t i l t h e e q u i v a l e n t of low T i s r e a c h e d . 5.
MEASUREMENT AS A SPONTANEOUS SYMMETRY BREAKDOWN
The f r e e energy ( 4 . 3 ) can be expanded i n even powers of t h e o r d e r parameter M(x), F(Mx)) - F Q = b j £ M 2 (x) + b2(l i=l i=l
M2(x))2 i (5.1)
+
n
d
kl
I
3M. „
(-^) 2 + ...;
i = l y=l
3x^
i = l,...,n is the dimensionality of the order parameter representation, V = l,...,d that of the system (we have left out the case of a non euclidean metric, etc.). Taking as an example n = 1 and k = o, we maximize the partition function Z in (4.2) by minimizing F in (4.3) and (5.1). The conditions for the minimum are
0
F
>0
(5 2)
"' "I" - "=5 -
'
3M We find, since F' = 2M(b1 + 2 b 2 M 2 ) ,
F" = 2 b± + 12 b 2 M 2 ,
two solutions: (1)
the symmetric solution M = o, b,> o;
(2)
(5.3)
the broken symmetry solution M = ±(- b l/2b 2 )\ b±<
o,b2>o;
(5.4)
the b,
801
2
and t h u s have two r e g i m e s : 2 (1) T>T , b..>o, M = o (symmetric solution); c
(2)
(5.6)
JL
T
2
(asymmetric s o l u t i o n ) .
(5.7)
T h i s asymmetric s o l u t i o n i s t h e Ginzburg-Landau method f o r p h a s e t r a n s i t i o n s , which h a s b e e n used i n t h e p h y s i c s of p a r t i c l e s and f i e l d s as t h e H i g g s - K i b b l e method f o r s p o n t a n e o u s symmetry breakdown. We now r e t u r n to o u r o b s e r v a t i o n i n S e c . 3 , where we n o t e d t h a t a quantum measurement r e p r e s e n t s a symmetry b r e a k d o w n . Using t h e above f o r m a l i s m , we may t a k e t h e unmeasured wave f u n c t i o n J , "up" +j^ "down"
(5.8)
a s t h e symmetric s o l u t i o n , as i n ( 5 . 6 ) . A f t e r measurement we h a v e e i t h e r " u p " or "down," j u s t l i k e t h e two s o l u t i o n s i n ( 5 . 7 ) . We c o n c l u d e t h a t a quantum measurement can be described as yet another phase transition. The non linear aspects of measurements are thus related to the type of structure we have in (5.1).
6.
A GENERALIZED PHASE TRANSITION SEQUENCE
I n S e c . 4 , we d i s c u s s e d s e v e r a l s y s t e m s where a phase t r a n s i t i o n i s a c t u a l l y known t o o c c u r i n t h e c l a s s i c a l / q u a n t u m c h a n g e . Moreover, t h e l a r g e - N c h a r a c t e r i z a t i o n f o r c l a s s i c a l systems o b v i o u s l y f a i l s t h e r e , w i t h m a c r o s c o p i c and even a s t r o n o m i c a l o b j e c t s f o l l o w i n g t h e quantum r e g i m e . On the o t h e r h a n d , we saw t h e r e t h a t the l a r g e - N c h a r a c t e r i z a t i o n i s a l s o used f o r t h e quantum measurement a p p a r a t u s . However, i n t h e l o w - t e m p e r a t u r e and a s t r o p h y s i c a l systems we m e n t i o n e d , t h e a p p a r a t u s i s p h y s i c a l l y much s m a l l e r t h a n t h e o b s e r v e d quantum s y s t e m s ! We therefore suggest that the classical/quantum transition be interpreted as a phase transition for all situations, not j u s t the e x p l i c i t l o w - t e m p e r a t u r e c a s e s . . Any quantum s y s t e m i s i n t h a t s e n s e i n a l o w - t e m p e r a t u r e r e g i m e . Any quantum measurement a p p a r a t u s i s i n t h e c l a s s i c a l p h a s e , above c r i t i c a l t e m p e r a t u r e . Our a n a l y s i s i n S e c . 5 shows t h a t t h e c o n c e p t of a phase t r a n s i t i o n can be a p p l i e d t o quantum measurements and r e s o l v e s two d i f f i c u l t i e s : (1) (2)
the n o n - S c h r o d i n g e r n a t u r e of measurement p r o c e s s e s ; t h e " c o l l a p s e " of t h e s t a t e - v e c t o r .
At t h e same time we now have an i n t e r p r e t a t i v e p r i n c i p l e f o r t h e c l a s s i c a l / q u a n t u m c h a n g e , b a s e d on a g e n e r a l i z e d p h a s e t r a n s i t i o n . This r e s o l v e s two o t h e r i s s u e s : (3)
t h e f a i l u r e of a "small-N c h a r a c t e r i z a t i o n " f o r t h e
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150 quantum transition in low T systems; the failure of the "large-N characterization" for the measurement apparatus in these systems. It is not impossible that further development of our new classical +-> quantum transition postulate might lead to previously unsuspected physical effects. In addition, there ought to be a way of connecting the two ideas relating to phase transitions: that of the classical/quantum change and that of the quantum measurements discussed in Sec. 5. Using disorder as our criterion, we indeed present the following sequence, involving two (generalized) phase transitions: (4)
(a) (b)
the classical system has the largest entropy; at T-K), we enter the (less-disordered) quantum regime, whose entropy is only due to indeterminacy; (c) observation (or measurement) of the quantum system removes that uncertainty and yields the most ordered system. The growth of order, from the classical to the observed quantum, is thus a useful organizing principle. 7.
ACKNOWLEDGEMENTS
I would like to thank Y. Aharonov and Y. Bendov for several discussions. 8.
NOTES
1. 2.
Wolfson D i s t i n g u i s h e d C h a i r i n T h e o r e t i c a l P h y s i c s : TAUP N171-85 . Also U n i v e r s i t y of T e x a s , A u s t i n , s u p p o r t e d i n p a r t by U . S . DOE Grant DE-FG05-85ER40200. Supported i n p a r t by t h e U . S . - I s r a e l B i n a t i o n a l S c i e n c e F o u n d a t i o n .
3.
REFERENCES 1. 2.
3. 4. 5.
Y. Ne'eman, "The Problems i n Quantum F o u n d a t i o n s i n t h e L i g h t of Gauge T h e o r i e s , " Found. Phys., A p r i l 1986. Y. Ne'eman, "EPR N o n - S e p a r a b i l i t y and Global Aspect of Quantum Mechanics" i n Foundations of Modern Physics ( J o e n s u u 1986 S y m p . ) , P . L a h t i and P. M i t t e l s t a d t , e d s . (World S c i e n t i f i c , S i n g a p o r e , 1 9 8 5 ) . For an a d d i t i o n a l i n t e r e s t i n g example, s e e T. D. L e e s , Columbia U n i v e r s i t y r e p o r t CU-TP-305. Y. Aharonov, p r i v a t e communication. Y. Ne'eman, Proa. Nat. Aoad. Sai. USA 80, 7051 ( 1 9 8 3 ) . S e e , f o r example, L. D. Landau and E. M. L i f s h i t z , Quantum Mechanics ( N o n - R e l a t i v i s t i c Theory) (Pergamon P r e s s , London 1 9 5 9 ) , p. 20.
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151 J . A. W h e e l e r , "Delayed-Choice E x p e r i m e n t s and t h e B o h r - E i n s t e i n D i a l o g , " i n Proceedings of the American Philosophical Society and Royal Society Joint Meeting, 5 June 1980 (Am. P h i l o s . S o c . P u b . , 1 9 8 1 ) , p p . 9 - 4 0 . See a l s o N. Bohr, Atomic Physics and Human Knowledge, (Wiley, New York, 19 5 8 ) , p p . 7 3 , 8 8 . S e e , f o r example: G. Toulouse and P . P f e u t y , Groupe de Renormalization et ses Applications ( P r e s s e s U n i v e r s i t e de G r e n o b l e , 1 9 7 5 ) ; H. E. S t a n l e y , Phase, Transitions and Critical Phenomena (Oxford U n i v e r s i t y P r e s s 1 9 7 1 ) ; S. K. Ma, Modern Theory of Critical Phenomena (W. A. Benjamin, R e a d i n g , 1 9 7 6 ) .
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C H A P T E R 11: PHILOSOPHY A N D SOCIOLOGY O F SCIENCE: EVOLUTION A N D HISTORY JOE ROSEN Retired from Tel Aviv University, Tel Aviv, Israel. Present affiliation: Department of Physics, The Catholic University of America Washington DC, USA Yuval's interest in the philosophy and history of science has been manifest throughout his career as a physicist and even before he became a physicist. His work in the foundations of physics, the subject of the previous chapter, necessarily involved philosophical considerations. Yet this specific interest found increased expression as Yuval spent more time thinking about matters of philosophy and history in the 1990's and on into the 21st century. Indeed, five of the six articles that were chosen for the present chapter are from this period. But the first article of this chapter, "Concrete Versus Abstract Theoretical Models," serves as evidence of earlier thinking in the philosophy and history of science: it was presented at a symposium in 1971. Here Yuval discusses the progress of theoretical science in terms of concrete and abstract models. The former models are based on an accepted physical picture, while the latter are expressed as mathematical formalism, devoid of a clear physical interpretation. Science fruitfully develops through the transformation of abstract models into concrete ones, as new abstract models are proposed to deal with the latest data. These, in turn, later become concretized, and so on. Development can be delayed by premature demands for concrete formulations. Kepler's laws explained by Newton serve as an archetypal example of fruitful development. To illustrate his ideas, Yuval presents two detailed examples: (a) Einstein's theory of gravitation and the Mach principle and (b) unitary symmetry and hadron structure (in which field Yuval was playing an active role at the time). Yuval's interest in the sociology of science is revealed in his analysis of the negative influence that Leninist ideology had on particle physics in Japan. In the next article, "Symmetry, Entropy and Complexity," published in 1991, Yuval starts with a brief historical review of symmetry and symmetry breaking in physics in general and in the physics of particles and fields in particular. Then he presents an idea, based on modern approaches to entropy and complexity, about using symmetry as a measure of information content. The third article, "Cosmological Surrealism: More than 'Eternal Reality' Is Needed," published in 1994, presents an implication of inflationary cosmology in which baby universes come into existence within our universe. Yuval points out that if we assume that such baby universes are born in the same way that our universe is assumed to have been born, then an observer in such a baby universe will experience time that is "outside" our time. Thus we need to recognize that our space-time reality does not comprise all of reality.
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Indeed this conclusion may be obtained already within general relativity proper and single universe cosmology. It is however eliminated by black hole evaporation with the advent of Bekenstein entropy and the consequent Hawking radiation. "Multiverse" relativity remains to be explored. The next article, "Pythagoreanism in Atomic, Nuclear and Particle Physics," published in 2000, is an example showing Yuval's interest in the history of science. Pythagoreanism claims that nature must be describable in terms of arithmetical relationships, in particular in terms of sequences of integers. Yuval describes the history of this idea, from its inception in the era of ancient Greek science, through its deep freeze as dogma during the dark ages, then through its disrepute as "music of the spheres" during the age of enlightenment and reason, and finally to its revival and vindication in the 20th century. Modern pythagoreanism takes the form of the whole-number relationships that are given to us by quantum physics, most recently through the tool of group theory. Many and various historical facts are presented along the way. The last two articles of this chapter, "Paradigm Completion for Generalized Evolutionary Theory with Application to Epistemology" and "Evolutionary Epistemology and Invalidation", both published in 2004, are closely related. In them Yuval extends the classical idea of evolution-involving random variation and selective retention-to include the effect of the environment, thus allowing for such as dinosaur extinctions. He generalizes from biological evolution to the evolution of systems as diverse as human societies and the universe, with special emphasis on the evolution of our understanding of nature. In the latter Yuval points out that Popper's falsification, which Yuval prefers to call invalidation, plays the role of extinctor. The present chapter brings to a close this collection of selected papers by Yuval Ne'eman. In much of his published work in the physics areas covered by chapters 1-9, one can discern the underlying common thread of Yuval's inclination toward the foundations of physics, represented explicitly in chapter 10. And in his articles on the foundations of physics per se, Yuval's interest in and knowledge of the philosophy and history of science often make themselves evident. Thus the present chapter fittingly concludes this book.
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REPRINTED PAPERS OF CHAPTER 11: PHILOSOPHY AND SOCIOLOGY OF SCIENCE: EVOLUTION AND HISTORY
11.1
Y. Ne'eman, "Concrete Versus Abstract Theoretical Models", in Interaction Between Science and Philosophy, Proc. Jerusalem Sambursky Symp. 1971, Y. Elkana, ed. (Humanities Press, Atlantic Heights, 1974), pp. 1-25.
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Japanese version in Shizen 71-12 (1971) pp. 94-105. 11.2
11.3
11.4
11.5
11.6
Y. Ne'eman, "Symmetry, Entropy and Complexity", in Differential Geometry, Group Representations and Quantization (Festschrift Honouring M. D. Doebner), J. Hennig, W. Lucke and J. Tolar, eds. Lecture Notes in Physics 379, (Springer Verlag, 1991), pp. 257-264.
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Y. Ne'eman, "Cosmological Surrealism: More than "Eternal Reality" is Needed", Found. Phys. Lett. 7 (1994) pp. 483-488.
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Y. Ne'eman, "Pythagoreanism in Atomic, Nuclear and Particle Physics", in Proc. Wenner-Gren Center Int. Symp., Stockholm, September 2000, pp. 265-278.
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Y. Ne'eman, "Paradigm Completion for Generalized Evolutionary Theory with Application to Epistemology", in Origins, J. Seckbach, ed. (Kluwer Academic Publishers, 2004), pp. 251-260.
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Y. Ne'eman, "Evolutionary Epistemology and Invalidation", in Evolutionary Theory and Processes: Modern Horizons, S. P. Wasser, ed. (Eviatar Nevo festschrift, Kluwer Academic Publishers, 2004) pp. 109-112.
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CONCRETE VERSUS ABSTRACT THEORETICAL MODELS VUVAL NE'EMAN
Department of Physics and Astronomy and Institute for the History and Philosophy of Science, Tel-Aviv University, Tel-Aviv and Center for Particle Theory, Physics Department, University of Texas, Austin A.
INTRODUCTION: CONCRETE AND ABSTRACT IN KEPLER'S CONTRIBUTION
This paper represents an attempt to abstract a lesson in the method of science—in particular theoretical science—from some of the more recent developments in physics. The main examples are taken from the study of Gravitation, and at greater length from the Physics of Particles and Fields. Presumably, these examples only emphasize well-known principles. However, it still appears that the lessons to be learnt have not been really absorbed in the conventional methodology. This is why they seem worth stressing. Generally, the term "concrete" as used in reference to science should not be restricted to what appears concrete to a physicist in the second half of the twentieth century. To the ancients, concreteness implied the appearance of familiar objects in the structure of the universe. The sky had to be a giant tent or vault with little holes. Planets were later made to move on wheels, the wheels on other wheels or circles. To Kepler it appeared necessary at one time to postulate the existence of perfect solids embedded in each other—another type of clockwork. Since, however, my major examples relate to recent developments, the term "concrete" will designate in their case conceptions where a so-called "good physical picture" appears in the background, namely a picture in terms of received relativistic and quantum effects: the causal propagation of an interaction, the inexistence of preferred reference frames, particle realization of fields satisfying the orthodox spin-statistics correlations, geometrized space-time. The "abstract" counterparts will appear in the form of mathematical formalism lacking such a clear "phys1
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ical" interpretation. In time, this formalism will itself become "concretized" through the perception of some new and appropriate physical picture. This generation's abstract becomes the next generation's concrete, except that at the present rate of development, a "generation" may last as little as ten years! My basic thesis in this paper is that this kind of progress in which the abstract anticipates the concrete is commonly part of fruitful developments in science, while a premature demand for concrete formulations in accordance with some predetermined scheme may easily delay such developments. For instance, Kepler's real contribution consisted of his three laws of planetary motion, all three of them unrelated at the time to any concrete model. That the planets should choose to move in elliptical rather than circular trajectories fitted no clockwork and even violated Kepler's aesthetic bias. His intuitive ideas—we would say nowadays "his physical intuition"—yielded nothing. Instead, it was only when he transcended his sixteenth century models that he finally produced a major advance in physics. After Newton, ellipses and velocity-area ratios ceased to be the abstract phenomenological regularities they had been to Kepler. They merged into the concrete physical picture now evoked in any mind trained in "classical"' physics. The remainder of this paper will discuss two further examples in more detail, one of which is already in large part historical, and one in which I am still myself involved. B.
EINSTEIN'S THEORY OF GRAVITATION AND MACH'S PRINCIPLE
1. The Direct Experimental Predictions This is a case which has already been discussed in part in the past, for instance by R. H. Dicke.1 I shall review his analysis and emphasize some results which have been derived more recently. As enunciated by Mach2 himself, "Mach's Principle" embodies the idea of relativity, and carries it over to the interpretation of inertial forces, iR. H. Dicke, T h e Many Faces of Mach" in H-Y Chin and W. F. Hoffman eds., Gravitation and Relativity, W. A. Benjamin, New York and Amsterdam, 1964, p. 121. 2E. Mach, The Science of Mechanics, 5th English edition, La Salle, III., 1942, Ch. 1, as quoted in ref. 1.
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"For me only relative motions exist. . . . When a body rotates relatively to the fixed stars, centrifugal forces are produced. . . ." This idea, which dates back to Bishop Berkeley, certainly appeals to our physical intuition, once the ether has been removed and wc have no absolute space to stick to. The water in the rotating bucket climbs at the periphery because of an interaction with the distant masses of the universe, since there is no absolute space. It was thus natural that Einstein should feel so strongly about Mach's Principle. Indeed, he considered the fact that his new theory of gravitation reflected this principle as its main advantage, and named it accordingly the General Theory of Relativity. Quoting Einstein,8 . . . "the theory of relativity makes it appear probable that Mach was on the right road in his thought, that inertia depends upon a mutual action of matter. . . ." Einstein went on to list three effects "to be expected" (i.e. by "concrete" thinking) if Mach's principle was valid: 1) a body must experience an accelerating force when neighbouring masses are accelerated, in the direction of that acceleration, for instance: a body inside a massive hollow sphere should experience an acceleration when we accelerate the sphere; 2) similarly, if we rotate the hollow sphere, we should produce Coriolis and centrifugal forces on the body inside it; 3) a body's inertia should augment (though to a very small extent) when additional masses are added in its neighborhood. General relativity indeed predicts the first two effects. Einstein thought that it predicted all three, but he seems to have been mistaken about the third.4 The correction to Einstein's theory suggested by Brans and Dicke1 does add such an effect. However, I think most workers in the field would be willing to take at least equal bets about the question of whether or not experiments will vindicate Einstein's version of the 8
A. Einstein, The Meaning of Relativity, Princeton University Press, Princeton, NJ., 1955, p. 100. *C. Brans, Phys. Rev., 125 (1962), 2194.
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theory, rather than the Brans-Dicke modification.5 Point 3 is then a first point in our list here, where the actual mathematical-physical theory differs from the intuitive thinking which led to it, and may even be the correct answer. That "concrete" thinking which led to the above mentioned third point may turn out to have been just "hand-waving." (I use this expression as a convenient way of referring to superficial reasoning which gives a plausible impression that a logical difficulty has been removed, or that an explanation has been supplied in terms of accepted principles—but which fails the test of a thorough and more formal and quantitative analysis.) It is indeed difficult to distinguish between handwaving and deductive thinking when the foundation is an "intuitive" idea rather than a mathematical theory. The issue will of course be settled by experiment—and the mathematical theory adopted accordingly. Then, a new "concrete" picture will follow as an interpretation. 2. GodeVs Universe We now go on to "Godel's universe." This is a solution to Einstein's equation, discovered8 by K. Godel in 1949. It was the first model to display esoteric effects due to space-time structure in the large. Many of these effects run counter to our "intuitive" views, e.g. a closed timecoordinate where the future connects back to the past. In addition, the model showed that in an infinite space, the matter of the universe can be made to rotate absolutely, i.e. not with respect to any distant masses (since they themselves are rotating). Here is a result which is certainly in violation of Mach's Principle; however, it derives from the equations of a theory which was supposed to embody that idea. Again, I think most physicists would change nothing of Einstein's equations. Clearly, we might add some boundary conditions (e.g. no such rotation!) which will preserve the principle. Nevertheless, even this ad-hoc insertion should wait until we first check the actual behavior of the universe phenomenologically. It might even be rotating! a
In 1967, Dickc published the results of an experiment showing that the sun is oblate. Assuming this to be correct, he found fault with the value of the precession of Mercury, as given by Einsteinian gravitation theory. In 1970, a J.P.L. measurement of the bending of radio signals from Mariners 6 and 7, when the spacecraft were passing behind the sun, has just brought at least equally strong evidence in favor of Einstein. «K. Godel, Proc. 1952 Inter. Congr. Maths (Cambridge, Mass.), Vol. 1, p. 175.
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3. Inertia in an Empty Universe After Gbdcl, this entire field of non-Friedmannian models has flourished.7 The most relevant model for our discussion is the OzsvathSchucking "Anti-Mach" metric.8 This is a universe with no matter in it, but it does have a non-vanishing Riemann tensor, i.e. it will display inertial features! Isn't that terrible for the original intuition leading to Mach's principle? However, the physical picture can now perhaps be saved through a "broader" interpretation: this is a world filled with gravitational radiation, entirely self-generated. To particle physicists who think in terms of gravitons, this isn't very strange, since the gravitons couple to anything carrying energy, including gravitons. It is just like a YangMills Isospin gauge field,9 which is coupled to itself since it carries isospin. Indeed, the replacement of current-commutators by the corresponding Yang-Mills field commutators10 is an exact analog, since it represents a hadron system in which only the Yang-Mills field is left. Returning to Mach, we now feel that Mach's original idea was at least somewhat ambiguous! 4. The Actual World Obeys Einstein's Theory and Is not Relativistic in the General Sense Our fourth step in this survey of the non-Machian aspects of Einstein's theory of gravitation relates to space-time itself. Clearly, if absolute rotation is allowed by the theory, we should be prepared to discover additional anti-relativistic aspects in this supposed General Theory of Relativity! Indeed, this has led Synge11 to rewrite the theory, emphasizing his view (or Minkowski's, according to Synge) which considers Einstein's theory of gravitation as a theory based on an absolute space-time, T
See for example, O. Hackmann and E. Schiicking, "Relativistic Cosmology" in L. Witfen ed., Gravitation, J. Wiley and Sons, New York and London, 1962, Ch. 11, p. 438. 8 Ozsvath and E. Schiicking, "An Anti-Mach Metric," in Recent Developments in General Relativity (The Infeld Festschrift), Pergamon Press, New York—Oxford-Paris-Frankfurt a.M. and PWN (Polish Scientific Publishers), 1962, p. 339. °C. N. Yang and R. L, Mills, Phys. Rev., 96 (1954), 191. 10 T. D. Lee, S. Weinberg and B. Zumino, Phys. Rev. Letters, 18 (1967), 1029. 1J J. L. Synge, Relativity: the General Theory, North-Holland, Amsterdam, 1964, preface p. 9.
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. . . "However, we need not bother about the name (Relativity), for the word 'relativity' now means primary 'Einstein's theory,' and only secondarily the obscure philosophy which may have suggested it originally." N. Rosen has recently inspected the inertial systems realized in an Expanding Universe.12 This is now a plain non-exotic Conformally-flat Universe; still, Rosen finds that even though local gravitation is entirely "relativistic," the Universe is not. Anisotropy experiments, and even a Michelson-Morley experiment, will detect motion with respect to that "cosmic" frame. Indeed, the times 7\ and Tt for the return of the two light-rays travelling parallel and perpendicular to the earth's motion will differ by a very small amount, depending entirely upon the expansion rate of the universe. For paths of the order of the distance to the nearest quasar, 3C 273 (about 3 billion parsecs) it amounts to some 2 hours! Experiments are indeed being done now to detect our "proper motion," through the anisotropy it should create in the 3°K "background" radiation filling the universe. The fact that we are now discussing the whole, rather than the part, goes beyond Einstein's original philosophy. As long as we do not introduce intrinsic structure in space-time via the choice of a Godel type cosmology, this is indeed the General Theory of Relativity, when applied to a localized problem. However, the same equations enabled Einstein to make a further conceptual jump, and to invent Cosmology. It seems that it is at this stage that he "unrelativitized" his theory. Mathematically, we are only feeding in the "particular" data corresponding to this problem. However, there is only one universe, and it seems improper to regard it as just one particular set of data. Rosen shows that it does define a preferred frame of reference; of course, this is the one corresponding to the matter at large, which is just what was needed to make the local inertial effects become relativistic. I think that at this stage we can admit that Mach's Principle, though an excellent trigger to Einstein's creativity, has by now been overtaken by the resulting theory, and left behind conceptually. Intuitive pictures are essential to a theoretician's progress, but his mathematically forI2 N. Rosen, "Inertial Systems in an Expanding Universe," Proceedings of the Israel Academy of Sciences and Humanities (Section of Sciences), 12 (1968).
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mulated creations transcend that kind of thinking and bring in new pictures, ever closer to the physical world he describes. C.
UNITARY SYMMETRY AND THE STRUCTURE OF HADRONS
1. Unitary Symmetry I shall now try to analyze objectively (I hope this is possible, even though I have been involved at the personal level) the sequence of ideas which led to the introduction of Unitary Symmetry in hadron physics. Hadrons are particles which react to the Strong Interaction (i.e. the strong nuclear force). We now have experimental proof that they can be classified18-u according to the unitary representations of the group U(3), for each spin, each parity and charge-parity assignment. Hence, since classification implies a clustering in the energy levels, one was led to believe in an approximate symmetry of the strong Hamiltonian (or 18
C/(3) is the group of unitary 3-dimensional matrices; its simple subgroup is 5(7(3), the group of unitary unimodular 3-dimensional matrices. The observed hadrons all belong to representations of a subgroup £C/(3)/Z(3) of SC/(3), in which Z(3) is the center, i.e. the subgroup commuting with the entire group. Z(3) contains the 3 elements (exp (27r'/3), exp ( 4 ^ / 3 ) , 1). The generator algebra of SC/(3) and SUO)/Z{3) is Cartan's A7, the traceless matrices in 3-dimensions over a complex field. In 1/(3) there is in addition an identity-generator corresponding to baryon-number. For further details, see: M. Gell-Mann and Y. Ne'eman, The Eight/old Way, W. A. Benjamin, N.Y., 1964; Y. Ne'eman, Algebraic Theory of Particle Physics, W. A. Benjamin, N.Y., 1967. "The SU(3) classification was suggested as a formal realization of the Sakata model by: M. Ikeda, S. Ogawa and Y. Ohnuki, Prog. Theoret Phys., 22 (1959), 715; Y. Yamaguchi, Prog. Theoret Phys. Suppl., 11 (1959), 1; O. Klein, Arkiv Fysik, 16 (1959), 191; W. Thirring, Nucl. Phys., 10 (1959), 97; J. E. Wess, Nuovo Cimento, 15 (I960), 52; It was independently developed on the basis of the (later observationally confirmed) octet model for baryons (the "Eightfold Way)" by: Y. Ne'eman, Nucl. Phys., 26 (1961), 222; M. Gell-Mann, Cal Tech report CTSL 20 (1961), unpublished at the time. The original draft of my above mentioned paper was an entirely independent identification of the role of SU(3) in hadrons. I was not aware of the Japanese work at the time. It was upon submitting my results to A. Salam that I heard from him about Ohnuki's talk at the 1960 Rochester conference and was given the Ikeda et al. preprints. I then cut out of my paper the entire mathematical introduction (thus making it almost unreadable . . .) and reexpressed my simple matrices as linear combinations of the rather more complicated matrices of the Nagoya group.
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5-matrix) under the isomorphisms of £/(3). All this emerged in 19591961; at the present time we have indeed direct evidence of the symmetry itself, in the form of symmetric couplings, i.e. a symmetric law of force. Moreover, by Noether's theorem we can regard Unitary Symmetry as indicative of the conservation of Unitary Spin (the Generator Algebra of S£/(3)), an eightfold complex, in addition to the well-known conservation of baryon number. 2. Emergence of Strong Interactions and the First Hadron Model The story of the discovery of the strong interaction goes back to Chadwick's discovery of the neutron in 1932 and the subsequent realization of the need for a new force to bind it (and the protons) in nuclei. This imaginative jump was taken by Yukawa and Stuckelberg in 193S, in the form of generalization of the idea of the electromagnetic potential: the nuclear force was assumed to be generated by a short-range potential 0(r)
- - G C Z 0 L 4TT r The parameter/x which fixes the range (l//x) corresponds to a mass. In a corpuscular picture, this is then the mass of the "exchanged" particle. With the subsequent discovery of the pion by Lattes, Occhialini and Powell in 1947 (after the ir-fi confusion to which we shall later return) we also observe the emergence115 of the first hadron "model," suggested by Fermi and Yang in 1949. This hypothesizes that the pions are bound states of the nucleon-antinucleon system, and attempts to calculate the parameters of the necessary binding force. Such a calculation is still problematic today, even though our knowledge of strong interaction dynamics has increased tremendously. It was certainly very speculative in 1949. However, all quantum numbers other than the mass (or energy level) could be reproduced by this "model." 3. Nagoya Dialectics I shall describe in succeeding paragraphs a certain sectarian approach which developed in Japan after World War II. It is interesting that in
«E. Fermi and C. N. Yang, Phys. Rec, 76 (1949), 1739.
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the over-enthusiasm of a politically-motivated telling of history, the Fermi-Yang model has been totally disregarded* (seep. 23 of ref. 16 ). Instead, Fujimoto told the Lenin Symposium that the 1955 introduction of the Sakata model" was a "revolutionary development" because it suggested a "composite model of elementary particles." As we shall see, the Sakata model only differs from the Fermi-Yang idea by the addition of the A hyperon as the third sub-particle, a necessary extension after the discovery of strange particles. Quoting Fujimoto, however, Sakata's "revolutionary" step was . . . "a step forward with the dialectic-philosophical view of Nature —the strata-structure of Nature—and destroyed an old belief that the elementary particles were the ultimate element of matter, and he considered the existence of the fundamental particles as the substance of the stratum existing deeper beneath the stratum of elementary particles." In the next sentence, Fujimoto relates this view to his (and Sakata's) basic dogma "One may say that he (Sakata) re-discovered in modern physics the inexhaustibility of an electron expressed by a famous phrase of Lenin's." 4. Strange Particles Let us return to the historical sequence, before discussing this socalled "Nagoya" philosophy. At about the time that the pion was discovered, Rochester and Butler18 observed "V events." Various hyperons and mesons were grad*I have since come across the following illustrations of this point "A large difference should be pointed out between the Sakata theory and the theory of Fermi and Yang. Main concern of Fermi and Yang's theory was on calculation of a bound state of a nucleon pair which could correspond to a -n-meson, while the Sakata theory was proposed to disclose internal structure of elementary particles in terms of their composite nature" (Y. Fujimoto, appendix to Takctani's letter to the Nobel Committee, published in Soryushiron Kenkyu (Study of Elementary Particles) Nov. 1970, p. 262). This seems to me like empty gobblcdygook. 16 Y. Fujimoto: Presentation of S. Sakata's "Theory of Elementary Particles and Philosophy" to the Lenin Symposium, 1970. (English Version published by the Department of Physics of Nagoya University.) i»S. Sakata, Prog. Theoret. Phys., 16 (1956), 686. « G . D. Rochester and V. V. Butler, Nature, 160 (1947), 855. A previous event had been observed in 1944 by Leprince-Ringuet.
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ually identified, but one was faced with the puzzle of their rapid production and slow decays. This was solved in two steps between 1952-53: associated production19 and strangeness.20 Attempts had just been made at an explanation which would be based upon "known" features only, such as centrifugal barriers. However, a phenomcnological analysis and the various evidence about spins etc. seemed to indicate that the symmetries of space-time weren't involved. Realizing that in a new field of exploration it is legitimate and necessary to generate new concepts as you progress, Nakano and Nishijima and independently Gell-Mann introduced a new "internal" quantum number, strangeness or hypercharge. "Internal" is a misnomer; it simply implies a quantum number which doesn't involve the structure of space-time, such as electric charge (at the present stage of the development of physics, at least). Here was a Keplerian advance, the observation of a regularity and its mathematical description, without first attempting to suggest a "good physical explanation." Such suggestions did follow—for instance, Feshbach's generation of strangeness as an effective selection rule due to para-statistical behavior of the strange particles (a hypothesis which has been disproved experimentally); none worked, and strangeness is still and abstract concept to date. It proved—and still is—extremely useful, though it hasn't yet found its Newton, taking the analogy with Kepler's laws of planetary motion. 5. The Sakata and Octet Models Now back to the question of a hadron "model." With strange particles around, the Fermi-Yang model had to be extended. Goldhaber and others21 replaced it by the sets (NK) or (A,K+K°), i.e. three or four basic states rather than two as in the nucleon-pion case. Sakata17 propounded the more elegant solution of (p,n,A). Since the dynamical theory of the binding was obscure, the whole approach made very little progress until it received in 1959-60 an algebraic formulation in the form of Unitary Symmetry.14 The fact that 3 basic complex fermion states could reproduce the quantum numbers predicted by the Gell-Mann Nis19
A. Pais, Phys. Rev., 86 (1952), 663. A similar though less formalized suggestion was made by Nambu, Yamaguchi, Nishijima and Oneda in 1951. 2°T. Nakano and K. Nishijima, Prog. Theoret. Phys., 10 (1953), 581; M. Gell-Mann, Phys. Rev., 92 (1953), 833. 21 M. Goldhaber, Phys. Rev., 101 (1956), 433. Similar suggestions were made by G. Gyorgyi, R. Christy, G. Derdi, M. A. Markov and Y. B. Zel'dovich.
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hijima formula—and that the basic states were close in their energy levels —this fact could be abstracted in a generalized higher internal symmetry based on the Lie group U(3). This also answered a quest for a "global" symmetry, started by Gell-Mann and Schwinger in 1956, a quest for a law of force which would relate IT and K couplings. However, the Sakata model was based on the assignment of a special role to the A hyperon, relatively to the 2 or S. There seemed to be no experimental justification for such a choice. Indeed, the model assigned the X to a multiplet where it had to appear with another nucleon-likc state and a Z +, positive strangeness hyperon. For the S, it had to assign it to the same multiplet as the famous Fermi / = 3/2, J = 3/2 resonance of the N-TT system. It thus predicted that its spin would be / = 3/2, like that ( A ) resonance. One is reminded of Newlands' "model of the octaves" in pre-Mendeleev chemistry, where gold was put in a column with chlorine, disregarding the experimental situation. The interest in a materialist lower stratum as predicted by Dialectical Materialism had indeed been useful in triggering the introduction of the f/(3) group. However, the motivation was so overwhelming that it overshadowed the experimental facts. Independently of these developments which were based on the Sakata model, and of which I was entirely unaware, I was trying in the second half of 1960 to find that "global symmetry." In the context of a methodical search through the classification of Lie algebras, I hit upon the possibility of identifying (N,A,2,H) as an SU(3) octet; this still enabled one to assign the mesons to a similar octet, since the product of baryon and antibaryon octets contained octets,
8 X£=j$-r- K)+i£* +l + ?2+i (just as in the Sakata case, where 3 X 3 = 8 + 1). Thus both models predicted the existence of the eighth meson" (the""? at 560 MeV, discovered in 1961) but the octet (also suggested simultaneously by M. Gell-Mann in his unpublished "Eightfold Way") predicted J = yz+ for the B, and even 2-A relative parity, and put Fermi's A resonance in a 10 or a 27. The latter choice was subsequently excluded by the observation of the Goldhaber gap, i.e. the inexistence of positive-strangeness resonances between 1000-1800 MeV, so that Gell-Mann and I
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both settled on the 10 and predicted the existence of the Omega Minus.23 As a lesson in the psychology of research, it is worth noting that nobody knew the ins and outs of SU(3) in 1960 as well as the Nagoya group. Still, they did not observe the seemingly obvious better fit provided for the baryons by that same octet representation that they were using for mesons! Yamaguchi, who was at that time at CERN, tells mc he did indeed think of that possibility though he never published it. Even in 1962 he still considered it so implausible that as chairman of the Symmetries session of the CERN ("Rochester") conference, he did not think that the octet model deserved discussion at the meeting, and I was denied the possibility of presenting the first strong evidence of the failure (in pp annihilation into 2 mesons) of the Sakata model (a selection rule discovered by Lipkin et al.) and of the good fit provided by the octet.23 The observation of die Or hyperon at the predicted energy (with strangeness - 3 ) provided a final spectacular confirmation of the octet model2* even though it had practically been confirmed adequately by that time through a variety of other predictions. 6. More about Nagoya Dogmas I cannot pretend to be an expert on the history of that unusual group in theoretical physics centered around S. Sakata and M. Taketani. Let me first state that both these physicists made important contributions to the development of nuclear and particle physics. S. Sakata, who died in 1970, predicted the existence of ir° in 1937; in 1946, he published24 with T. Inoue a solution to the riddle posed by the muon's properties (which had proved very different from those expected from Yukawa's 22
M. Gell-Mann made the prediction in a remark from the floor at a final plenary session (Proceedings 1962 International Conference on High-Energy Physics at CERN, p. 80S). 1 had submitted a similar suggestion in a written communication presented to G. Goldhaber earlier during the meeting. 23 P. T. Matthews, M. Rashid, A. Salam and H. J. Lipkin, C. A. Levinson and S. Meshkov, Phys. Letters, / (1962), 125. " V . E. Barnes et al., Phys. Rev. Letters, 12 (1964), 204. Since then about 30 Q events have been observed. At the time of the writing of this article, G. Goldhaber et al. have just announced the observation of a Q in a SLAC bubblechamber picture. 25S. Sakata and T. Inoue, Prog. Theoret. Phys., / (1946), 143; Y. Tanikawa, Prog. Theoret. Phys., 1 (1946), 200; M. Taketani, S. Nakamura, K. Ono and M. Sasaki, Phys. Rev., 76 (1969), 60; R. E. Marshak and H. A. Bethe, Phys. Rev., 72 (1947), 506
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meson, with which it had wrongly been identified). This was the "two meson theory," suggested independently and somewhat later by Marshak and Bethe in the U.S. Sakata also made contributions to the renormalization problem (his was a "regulator" type solution) etc. As to Takctani, besides important contributions to the study of nuclear forces, be has also authored a theory of the methodology of science. His theory identifies a recurring cycle made of three stages—phenomcnological, substantial and essential (see pp. 4-9 of ref. I f l ). Thus my emphasis upon patterns would relate to a detail in their phenomenological stage, while Sakata and Takctani emphasize the next step in which one looks for structure. Indeed they really jump from experiment to structure and point to cases where this happened, such.as in Yukawa's meson idea. The Nagoya dialectics school suffers from a strong bias against the U.S. and the West in general. This may in part be a residue of the Second World War, but there are also personal grudges. The "comments" by Taketani in the 1965 Yukawa conference20 are enlightening. Let us quote "comment 2" in full; "Some English speaking people talk ten words when we talk one word, and do hardly take care of one word which we talk. This point is one of our complaints in any international conference. We should like to ask English speaking people to hear about our talks with the special care. Otherwise our attendance to the international conferences would lose its true meaning, and we are led to consider that we are not welcomed as a matter of fact. To my present lecture the above statement will be applied. The disadvantageous conditions which arc imposed upon us will not be essentially improved even if a certain person would take trouble to invite some of us as a result of personal good will." "Comment 1" refers to the two-meson theory. We quote the essential points; "The theory of two mesons and two neutrinos proposed by Sakata and his co-workers should be noticed in its remarkable perspectives and completeness by itself. But the theory has been intentionally neglected by some of the foreign physicists (see, for instance, A. S. Wightman, Britannica, 1957, 343B) and also suffered from the unjust critics at its unimportant points, which I think is entirely an unfair matter. I shall, therefore, repeat here the essential points of the theory. ^Proceedings of the International Conference on Elementary Particles (Kyoto 1965), p. 170.
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"The Sakata theory was published in 1943 in Japanese. The translation of it in English was made public in 1946. Until 1940, it became clear that the cosmic meson just did not possess the properties of the Yukawa particle which had been introduced on the basis of the nuclear force. The difficulties of the meson theory, which were a puzzle at that time, were pointed out in the discrepancy between theory and experiment, for instance, on the cross sections of scattering and absorption of meson in matter, the energy loss of meson due to the electromagnetic interactions, and the decay lifetime of meson." "Here they had to introduce the existence of the neutral meson (muon) n besides the charged ones m ± . About the possible properties of the neutral muon they remarked in the following way: 'neutral meson which is assumed in the following discussions to have a negligible mass, and consequently may be regarded as equivalent with the neutrino.' "This statement clearly leads us to the two-neutrino theory, in which v (a partner of e) is distinguished from n (a partner of m) and n is assumed to have the mass of the negligible magnitude and, therefore, can be equivalent with the neutrino. "Is there anything to be added to their statement in order to give the correct theory for the two-meson problem? On the basis of the above argument, we proposed that n should be called Sahatorino while v should be called Paulino." "We believe that the paper of Sakata and Inoue were received by the workers in U.S. in the year 1947. In 1948, the other paper on the two-meson theory by Taketani, Nakamura, Sasaki and Ono was sent to U.S. It is regrettable for us to find some workers in the major country insisting that the article which they did not read could have no contribution to the progress of science. We know that the works done by Sakata and his co-workers made the important contribution at least to the progress of physics in Japan. No one will deny that the achievements made by Japanese workers played an important role in the international developments of the meson theory." For completeness* sake, we also quote from Marshak's answer:37 "I have been involved in 'this question of priority with some of my Japanese colleagues. Now Profs. Sakata and Inoue without question sug"R. E. Marshak, on p. 180 of ref. 26.
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gested the two-meson theory several years before I did in my paper with Prof. Bethe. But due to the war their paper did not reach to U.S. until 1948, which was at least 6 months after I presented my theory at the first Shelter Island Conference. In their paper, Sakata and Inoue, working within the framework of die M0ller-Rosenfeld model, considered two mesons, a heavy and a light one, we call them tr and /x. now, and they deduced a lifetime of 10 -21 sec for the decay of re. I don't blame them for deducing such a short lifetime, which is too fast by a factor 10 13 , because they were basing their work on the scattering experiments. I had the benefit of the Conversi, Pancini and Piccioni experiment and I deduced a lifetime of 1CH sec which turned out to be correct. On the other hand, Sakata and Inoue assumed the spin of v to be 0 and the spin of /x to be Vi while in my paper I assumed the reverse in making an illustrative calculation of the lifetime. Ever since then, Sakata claims that he had the correct two-meson theory. I say that we were both wrong. I mean, he had the wrong lifetime, I had the wrong spin. Or perhaps a more flattering way of saying it is that we both made contributions and we were both equally thrilled by Powell's discovery of the pion." It is thus a combination of an anti-West bias, suspicion of American physicists' motives, some just resentment over important contributions which were disregarded for a time in the West—all these mingle with genuine belief in the ideas of Marx, Engels, Lenin as relating to atomic physics. It is interesting that the latter belief is an extreme orthodoxy which can only be compared with a fundamentalist's attachment to the biblical story of Genesis, or the resistance to the study of the theory of evolution in some southern states in the U.S. The dogma itself is simple enough, as can be seen in this quotation from Sakata's 1965 paper: 28 "One may quote the following two points as remarkable features of the physics of the present century. The first is the recognition of the strata-structure of Nature, in particular the discovery of a series of new strata of the microscopic world, namely, molecules—atoms—atomic nuclei—elementary particles. The second is the recognition of a limit of validity of the physical laws, in particular the discovery that the Newtonian mechanics is not the eternal truth of perfection. As a result 28
Published in Japanese in Kagaku, on the occasion of the thirtieth anniversary of Yukawa's theory. English translation included in ref. 16; see pp. 3-4 of that source.
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it established the following point of view for Nature: there exist in Nature an infinite number of strata with different qualities amongst each other, including the nebulae and the esolar system as examples of the large scale, and the molecules, the atoms and the elementary particles as examples of the small scale. Each of those strata is governed by its respective and proper laws of physics, and all of the strata are always in the middle of creation and annihilation, and they compose Nature as the one and whole unified existence through their correlation and mutual dependence among themselves. This point of view is called the dialecticphilosophical view of Nature, and it was already put forward by Engels at the end of the nineteenth century. One may say as a conclusion that the atomic physics of the twentieth century re-discovered the dialecticphilosophical view of Nature." 7. The Doctrine of Inexhaustibility and the Bootstrap The doctrine would be tame, were it not for the assertion of the existence of an infinite sequence of strata. It may of course be true: if nature is not an a-priori mathematical construct, it is possible that our physical analysis is just some kind of series expansion, with an infinity of terms. However, considering for example that we know nothing about happenings under 1(H 8 cm, any such statement is entirely non-scientific. Indeed, there are even indications that it may well be wrong: on the one hand, if we extend the quantization procedure to gravitons, space-time itself becomes quantized at 1(H 3 cm. This would then set a lower limit On the other hand, we are now witnessing at Berkeley and elsewhere another attempt at the description of hadron matter, pursued with an almost equally dogmatic single-mindedness, and in which the basic motivation stems from the belief that we have already reached the end of the way. This is the "bootstrap" movement, in which all hadrons are believed to be dynamical constructs satisfying self-consistency conditions. This is like Aristotle's hyle, in contradiction to the "atomistic" (or strata) approach. Quoting Chew:28 "The revolutionary character of nuclear particle democracy is best appreciated by contrasting the aristocratic structure of atomic physics 2,I G. F. Chew, on pp. 105-106 of M. Jacob and G. F. Chew, Strong Interaction Physics, W. A. Benjamin, New York, 1964.
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as governed by quantum electrodynamics. No attempt is made there to explain the existence and properties of the electron and the photon; one has always accepted their masses, spins, etc., together with the finestructure constant, as given parameters. There exist composite atomic particles, such as positronium, whose properties are calculable from the forces holding them together, but so far one does not see a plausible basis, even in principle, for computing the properties of photon and electron as we compute those of positronium. In particular the zero photon mass and the small magnitude of the fine-structure constant appear unlikely to emerge purely from dynamics. Among strongly interacting particles, on the other hand, we have yet to see very small masses or other properties that cannot plausibly be attributed to a dynamical origin. "The bootstrap concept is tightly bound up with the notion of a democracy governed by dynamics. Each nuclear particle is conjectured to be a bound state of those S-matrix channels with which it communicates, arising from forces associated with the exchange of particles that communicate with 'crossed' channels. (The principle of crossing is reviewed in Chapter 1.) Each of these latter particles in turn owes its existence to a set of forces to which the original particle makes a contribution. In other words, each particle helps to generate other particles which in turn generate it. In this circular and violently nonlinear situation we shall see that quite plausibly no free parameters appear, the only self-consistent set of strongly interacting particles being the one we find in nature. "If the system is in fact self-determining perhaps the special stronginteraction symmetries are not arbitrarily to be imposed. No convincing explanation has yet been given for the origin of isotopic spin, strangeness, or the newly discovered eightfold way, but many physicists believe that the secret will emerge from requirements of self-consistency in a democracy. Hopefully the origin of these symmetries will be understood at the same moment we understand the pattern of masses and spins for strongly interacting particles—both aspects of the system emerging from the dynamics of the bootstrap." It is interesting to observe that each party likes to think of its approach as "revolutionary." Actually, the drive provided by the bootstrap approach has indeed been extremely useful for the phcnomcnolog-
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ical charting of the high-energy domain, through the introduction of concepts such as poles in the complex angular-momentum plane (these represent dynamical metastable particles with lifetime sometimes shorter than 10 -28 sec). Much has also been learned with respect to the dynamics of strong interactions, the structure of the S-matrix and its analytical properties, etc. It is thus not excluded that for hadrons, the sequence of strata may have reached its endl However, this is at present also very far from certain, and seems contradicted by the quark hypothesis, which we shall discuss in coming paragraphs. Moreover, since the basic idea of the bootstrap is that self-consistency will allow the calculation of all parameters, quoting Salam:"0 "We may yet find that we are living (with Voltaire) not only in the best of all possible worlds, but indeed in the only possible world." Now since any calculation is bound to leave out some feature in order to be manageable, its failure can be considered here as a success, as pointed out humoristically (on various occasions) by Salam and by GellMann, and sometimes naively by the "bootstrapists" themselves. Actually, either the self-consistent system should break up into a number of such independently self-consistent systems, or else it can only be useful for the description of some particular general features, and can never become a physical theory, even if it is true in some absolute sense. Returning to the doctrine of inexhaustibility of the strata, it is now clear that its exact negation has turned out to be just as useful a motivation in the study of hadrons I The term "inexhaustibility" has been taken from a quotation often encountered in the writings of the Nagoya school: "Even an electron is an inexhaustible as an atom," (V. I. Lenin, in Materialism and EmpiricoCriticism.) Two points should be made with respect to that quotation. First, it is often encountered in general articles in physics published in Communist countries—completely unbiased articles in which the actual scientific treatment is superbly free of dogmatic thinking. In such cases it just represents either a bow to a greater National or International leader, or a protective gesture in times when physicists have had reason to fear hostility from the regime. However, in the case of the Nagoya school, it is taken literally to represent revelation. It is somewhat in»°A. Salam, on p. 34 of Contemporary Physics, Vol. II (Trieste Symposium 1968), LA.E.A. Vienna 1969.
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congruous when Sakata, Taketani or Fujimoto refer to bootstrap physics as being religiously oriented, without ^noticing their own dogmaticism:31 "In so far as the current view is'adopted, various properties of elementary particles, which are introduced ad hoc into the theory, for example, masses, spins, symmetries, strengths and types of interactions and so forth, should be regarded as to be given by the Providence of God. Although there are recently some attempts to derive these elements by a self-supporting mechanism such as a bootstrap approach, the situation remains to be unaltered. Because such attempts, combined with the current view, will lead us to the philosophy of Leibniz that universe is in a pre-established harmony. Thus the current view will always introduce religious elements into the science and stop the scientific thinking at that stage." Sakata held the same opinion about die phenomenological approach, if it did not bow to the inexhaustibility dogma:32 "Here, a current abstract method of the group-theoretical approach will be useful only in preventing fixation of a certain concrete model gained at a certain stage of the experimental progress. Once one will forget this remark and will fall into a way of abstraction without any precaution, one will spread an inverted viewpoint of believing the ultimate aim to be a discovery of the symmetry properties as the 'providence of God,' and then the physics will fall down into one of the theologies." The first quotation is taken from a paper which, after more of the same, goes on to present some extremely pertinent remarks on quarks and the like, It led F, Bopp, who was chairing the session, to make the following comment,88 when opening the discussion which followed that paper presented by Sakata: "Thank you very much Prof. Sakata for your talk. On an occasion of New Year's day in 1611, Kepler had written a letter to a friend on the sexangular snow. In this letter he tried to explain the structure of the snow crystals according to the view that it must be what we call today cubic package of spheres and by good observation and sharp reasoning, he came to the result that this was impossible and that he must 8iZ. Maki, Y. Ohnuki and S. Sakata, p. 109 of ref. 26. 82S. Sakata, on p. 18 of ref. 16. sap. Bopp, p. 119 of ref. 26.
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replace the cubic package of spheres by a most dense one as we know it today. But this was impossible for him to believe and so he said that the atomic view must be from the Devil. I am feeling that we all have our leading ideas and we are all in danger to say that other ideas are from the Devil. But we are feeling from this letter of Kepler that by good observation and hard arguments we are coming to good results. So I propose to open the discussion with hard arguments. Thank you. 8. Effects of the Doctrine on Theory and Experiment An important weakness which emerges from papers where the dogmatic elements preponderate is that they tend to encourage "handwaving" at the expense of real theory. The ancient Greeks invented atomicism; nevertheless it only became a theory when Dalton made it quantitative and related to observation and measurement The Nagoya school now has a favorite doctrine of "B matter." This is something which turns a lepton (the electron, the muon and the two corresponding neutrinos) into a baryon. It was suggested by the Sakata triplet (p,n,A), which seemed "obtainable" from (v,e-,/^) the supposed lepton triplet (prior to the discovery of the two neutrinos in 1962). The Sakata triplet can now be replaced by the hypothetical fundamental triplet suggested by Unitary Symmetry as the basis of hadron structure ("quarks"). However, there are four leptons! One can invent a variety of amendments, but since the entire idea contains nothing as far as actual dynamical computation, its only advantage was in the suggestive correlation of 3 hadrons to 3 leptons. Nevertheless, it still comes up as an example of "correct thinking" and a victory of the inexhaustibility dogma. There is no doubt that some future Dalton will indeed find the manner in which hadrons relate to leptons, and the inexhaustibilists will then claim the credit for their doctrine. . . . Our next comment about that school relates to their attitude towards what they call84 the "mist of positivism." "The mist of positivism has been thick around all fields of science since the beginning of this century. It is a famous story that there were repeated fruitless discussions of skepticism about the objectivity of the atom among physicists, including Ostwald and Mach, even on the last night before the day when the internal structure of the atom was dis3
*S. Sakata, on p. 2 of ref. 16.
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closed. The moment of proposal of Yukawa's theory was in the middle of a revolutionary development of nuclear physics, in which the positivistic philosophy of the Copenhagen school headed by Bohr was not able to have a correct perspective of this revolution." There is no hesitation about quoting high energy experiments when they happen to lead to the validation of their own theories. However, they strongly resisted the building of a powerful accelerator in Japan in 1965, claiming that they could guess at all the answers anyhow, from their doctrine. A similar situation occurred in Europe in 1968, when Heisenberg opposed the building of the European 500 GeV machine. His main objection was based upon his "knowledge" that all the answers were already present anyhow in his nonlinear theory, another idea of the bootstrap type (the original one in fact) which being over-ambitious (an equation which should yield all particles, Ieptons and hadrons) has gone very little beyond the hand-waving stage in 15 years. These examples display some of the real dangers facing this part of physics whenever a doctrinaire approach wins over the combination of healthy unbiased experimental discovery and uncommitted abstract charting of new territory. One last point relating to the "inexhaustibility" of the electron itself, as stated in Lenin's much quoted passage. There is nothing wrong, and perhaps even a certain perceptiveness, in a non-scientist such as Lenin guessing at further structure. It is however a sad development when a scientist falls back into the medieval way of preferring dogmas to actual physical theory. Fujimoto18 is scandalized at Gell-Mann: "Repeatedly I want to mention that the Sakata theory has its essence in the philosophical method of discovering the 'logic of matter'—the fundamental particles—as a 'causa formalis' of the phenomcnologjcal regularities of the elementary particles, such as the symmetry property. In this point, there exists a fundamental difference between his point of view and views of the positivists. I might quote a statement of Gell-Mann as an example of the latter view. He stated in Tokyo in 1964 that he thinks not necessary to reject the point model of the elementary particle." Well, it is a fact that present physical theory treats the electron as a point particle, and is entirely validated by experiments. It may well happen that indeed somewhere beyond present available transfer-momenta we shall discover structure in the electron, and Lenin will be vindicated. It hasn't happened yet, and the point picture is in fact the
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only available proper theory. To date we have no experimental indication of space-structure; neither do we have an alternative theory. 9. Quarks and SU(6) We now return to developments in particle physics since 1962, with several more lessons pertaining to the irrelevance of concrete models. The Sakata model, based upon a concrete set of "fundamental particles" making up the "elementary particles" had failed. The more abstract octet model provided a map of the world of particles; however, an octet is already a mixed tensor of the second order in SU(3) (i.e. a piece of the 3 X 3* product of basic covariant and contravariant triplets). Alternatively, it is an unmixed tensor of the third order in SU(3). (i.e. a piece of the 3 X 3 X 3 product of 3 contravariant—or 3 covariant-triplets). There was thus little hope of regarding the octet itself as a building block. Moreover, there were various indications of structure even in protons and neutrons, such as that revealed by the scattering of electrons on nucleons. In a study performed with Haim Goldberg (Auphir) in 1962 at the Israel AEC Soreq Research Laboratory,85 we suggested a Neo-Sakata like view, in which the fundamental field (or "model") would be represented by a triplet with baryon number B = V6, so that a nucleon would be made of 3 such objects. This article, which appeared in n Nuovo Cimento only on 1.1.63 (having been lost for a time by an editor in some drawer) went by almost unnoticed. This was due to the general preponderance of the Sakata model at the time; it was also due to bad writing, since it was very formal and did not point at experimental conclusions. Indeed, we did not know how seriously we should take our own suggestion—would these B = 16 fields actually materialize as particles (with fractional chargesl), or would they just stay as a mnemonic device? Alternatively, a theory might develop in which they wouldn't appear as single particles, but they would still play a fundamental physical role. Some time in 1963, Gell-Mann arrived at the same idea. He published i t " in 1964 in Physics Letters. By the time it appeared, the Omega S5
H. Goldberg and Y. Ne'eman, Nuovo Cimento, 27 (1963), 1; and report IAEC 725 (Feb. 1962). "M. Gell-Mann, Phys. Letters, 8 (1964), 214; G. Zweig. unpublished CERN reports 8182/TH.401 and 8419/TH.412.
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Minus experiment had vindicated the octet, and the world of physics became interested in these triplets, which he named "quarks." GellMann's paper was also extremely readable, and carefully pointed to the possible existence of these fractional charge states: "It is fun to speculate about the way quarks would behave if they were physical particles of finite mass (instead of purely mathematical entities as they would be in the limit of infinite mass). Since charge and baryon number are exactly conserved, one of the quarks (presumably «% or d-V*) would be absolutely stable, while the other member of the doublet would go into the first member very slowly by /3-decay or Kcapture. The isotopic singlet quark would presumably decay into the doublet by weak interactions, much as A goes into N. Ordinary matter near the earth's surface would be contaminated by stable quarks as a result of high energy cosmic ray events throughout the earth's history, but the contamination is estimated to be so small that it would never have been detected. A search for stable quarks of charge —1/3 or + 2 / 3 and/or stable di-quarks of charge —2/3 or -f- 1/3 or + 4 / 3 at the highest energy accelerators would help to reassure us of the non-existence of real quarks." A similar suggestion was made by G. Zweig, then at CERN.80 In terms of our thread, here then was a Sakata-like model, unsuspected to start with, derived from the abstract identification of the octet, unbiased by a concreteness complex. To date, we do not know what its final role will be; it is certainly much more sophisticated than the (p n A) choice. Funnily enough, however, an almost identical lesson was to follow, still relating to quarks. With the experimental confirmation of 5C/(3) in the octet version, many workers felt that this was the time to tie up this "internal" symmetry with the "external" ones of space-time, i.e. the Poincare group. I shall later relate the story of my own attempt in this vein and the identical methodological error which made me miss the point. Let us first mention the work of Giirsey and Radicati, "7 of Zweig88 and of Sakita." All of these authors conceived the idea of combining Unitary Spin (SU 87
F. Giirsey and L. A. Radicati, Phys. Rev. Letters, 13 (1964), 173. G. Zweig, Proc. of the 1964 Intern. School of Physics "Ettore Majorana," Academic Press, New York 1965. »°B. Sakita, Phys. Rev., 136 (1964), B1756. 8a
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(3)) and space-spin (an SU(2) subgroup of the Poincare group) for particles at rest, by considering the six states of a quark (spin-up and spin-down for each quark) as a basis for the group SU(6). The question then arose of the SU(6) assignment for the physical baryons. Considering the quarks as fermions, the physical baryons should correspond to the totally-antisymmetric (in spin / and SU(3) indices) product of 3 quarks. This has 20 components, including 16 for the baryon octet (with / = 1/2) and a unitary singlet with / = 3/2. B. Sakita did make this assignment, which was the direct choice if you believed in concrete quarks. The other authors tried other representations as well as the 20. They noticed that the totally-symmetric product with 56 components fitted perfectly the baryon / = 1/2 octet and / = 3/2 decimet (containing the ft-). They could see several interesting applications to this marriage, and thus assigned the baryons to 56. This forced the quarks to have para-statistics!<0 A short time after publication of the 56 assignment, spectacular results started to appear, all of them pertaining to this choice. It turned out that it predicted a ratio of —3/2 between the magnetic moments of the proton and the neutron; the experimental figure is —1.46! Since then, the 56 assignment has been accepted everywhere. Again, it was the freedom of picking an abstract representation which produced the right result, rather than the "good physical picture" of 3 quarks. And again, one could now readapt that "physical picture," by replacing the quarks by paraquarks, corresponding to a type of statistics as yet unseen. Why not? In the words of Sakata:28 "As soon as scientific research penetrates into a new and unknown stratum of Nature, physical concept and laws established in the old strata lose quite often their validity." Now to my own mistake. With J. Rosen, we were trying in 1963-64 to combine SU (3) and space-time aspects.41 To ensure that we would not make the mistake (common to various other attempts in 1963-65) of generating SU(3) transformations which wouldn't commute with the Lorentz group, we simply adjoined the two metrics. We added to the Minkowski metric another six real dimensions in which we could represent 5t/(3). This led us to a geometrical model involving general rela40
O. W. Greenbcrg, Phys. Letters, 13 (1964), 598. *»Y. Ne'eman and J. Rosen, Ann. Phys.,57 (1965), 391.
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tivity and cosmology. Engrossed in this "good physical picture," I never checked the enveloping symmetry group (which of course contained SU{6)) and its representations. Instead of trying to keep aloof from a concrete model I fell into that same "concrete" pitfall. CONCLUSION
Let me end this story of trial and error with two remarks. First, about physics in Japan, which has taken up so much space in these comments. The Japanese contribution to Particle Physics has been of the first rank. Taking the period since the mid-fifties, they have produced several of the main leaders in the field: Y. Nambu, K. Nishijima, B. Sakita, J. J. Sakurai, K. Igy, M. Suzuki, H. Sugawara, Y. Hara etc. In axiomatic field theory, H. Araki is a central figure. Nambu's role is second to none. It should be noted however that most of that work was done in the U.S. In Japan proper, only Tokyo was relatively free of the doctrinaire atmosphere. It is only now that a gradual normalization is taking place. So much for the ill effects of dogmas. The interaction of science and philosophy—or rather the influence of philosophy on science—may be useful at the level of the individual scientist, in triggering ideas. The results may or may not be relevant to the original philosophy. It can become disastrous if the link between philosophy and science is enforced by the intellectual establishment. DEDICATION
My last remark is a dedication. I "discovered" the beauty of physics in 1940, at the age of fifteen, while reading Jeans and Eddington. This led me to attend evening courses given by Prof. Sambursky in a TelAviv school; they were inspiring and augmented my interest, which was to be given a final boost through the teachings of Prof. Ollendorff at the Technion in 1944/5.1 am thus repaying a debt in this meeting.
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Symmetry, Entropy and Complexity Yuval Ne'eman* Raymond and Beverley Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel 69978 and Center for Particle Theory" University of Texas, Austin, Texas 78712, USA
Abstract: We review the role of symmetry in Physics and its interrelationship with order and with information, in the light of modern approaches to the concepts of entropy and of complexity versus disorder.
1 Introduction It is with great pleasure and deep appreciation that I dedicate this essay to the 60th anniversary of Professor H.D. Doebner. Throughout the last fifteen years I have gained much insight from following his work in the intersection of differential geometry and physics. I have also greatly enjoyed the topical conferences he has organized in this field - not the least for the opportunity it provided for discussions with Doebner himself, enhanced by the hospitality of the Doebner household and conversation with Mrs Doebner and their daughter. Through his initiative, Clausthal has become a Mecca for workers in geometry and group theory as applied to physics. In some ways, it revives the memories of nearby Gottingen in the hallowed days of Gaus.s and Hilbert. Heine's Harz Mountains travelogue missed an important intellectual site by coming 150 years too early. My debt to Professor Doebner goes even further. He has introduced many students to research in physics; considering his interest in groups and geometry, it is thus not surprising that I should have come across several of them in my work. Beyond that, however, is the fact that I have developed extensive collaborations with two of them. In my work on the extension of the idea of symmetry to spectrum generating groups, I have collaborated since 1968 with Arno Bohm, who independently conceived ideas similar to mine in 1965, when we introduced SGG * Wolfson Chair Extraordinary in Theoretical Physics » Supported in part by Grant DE-FGO5-85ER40200
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as algebraic systems connecting all solutions of a quantum problem, i.e. all energy levels. My work on world-spinors and on the gauge approach to quantum gravity was triggered in 1977 when reading articles by Friedrich W . Hehl on gravity with torsion, which he succeeded in putting on the same footing as other gauge theories. This was the start of an extremely pleasant and fruitful ongoing collaboration. I have thus greatly benefited indirectly as well from Prof. Doebner's efforts as a teacher.
2 Symmetry Implies Abstraction and Loss of Information
Physics is an experimental and observational science and thus deals with the "real world". Its method, however, uses abstraction. The aim is to achieve a unified and coherent presentation of all natural phenomena. To treat different phenomena in a single formulation, physics has to strip away the circumstantial details and identify the essentials and discover the common denominators and their constrained behaviour - the laws of physics. This then implies sweeping generalizations a n d a loss of information about the individual systems. The more phenomena are encompassed by a law - the more it has to become simple and rely on less specification. The information about the individual systems is left to the boundary conditions, if at all. Symmetry laws are in that category. They represent negative statements embodying powerful generalizations. They are "Postulates of Impotence"' [20], though highly potent ones. IMPOTENCE, because they state that it is impossible to prefer one frame over the rest. If a crystal is hexagonal, it has a symmetry under rotations by 360/6 = 60deg.. and it is impossible to select one face out of the six as a "preferred'" face. To the extent that we wanted to preserve the identity of one of the faces - it is lost in the symmetry. Notice the closeness to the classical concept of entropy in such a S3*mmetry law; as to leaving the information to the boundary conditions - the modern theory of Chaos tells us t h a t this is often also a way of loosing information, since many dynamical systems lead to entirely different evolutions even though the initial conditions may be so close as to be indistinguishable. The Principle of Covariance in Einstein's General Theory of Relativity states that it is impossible to select a preferred reference frame - i.e. the laws of gravity do not depend on the selection of a particular reference frame, all reference frames are equivalent. The French saying goes "la nuit, tous les chats sont gris" - at night, all cats are grey - i.e. it is impossible to distinguish or specify a preferred cat. There is then a symmetry between cats, they all look the same.
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3 Broken Symmetries - Imposed or Spontaneous Symmetry can sometimes be in the laws of physics and then has a great range of applications: in the example of Einstein's theory of gravity, for instance, whatever the gravitational problem, the laws will still have to be stated "covariantly" (i.e. independently of the selection of a reference frame, of a coordinate system). Sometimes, however, there is a symmetry that relates to the boundary conditions. In the Kepler problem (sun and planets) for instance, there is an a priori spherical symmetry in the givens themselves: the sun is assumed to be spherical, and therefore there will be no preferred direction for its gravitational pull - in the way that would happen in a description of gravity in this room, where we would be forced to assume a preference for the downwards direction in the action of gravity. It so happens that the laws themselves also contain no preferred direction and are spherically symmetric, even for this room, even though the boundary conditions are less symmetric. In fact, the symmetry of the laws is generally greater than that of the givens; in the case of Einstein's theory, for instance, the laws are also locally Lorentzinvariant, which includes, aside from insensitivity to rotations of the system in space, an invariance under accelerating boosts ("special Lorentz transformations"). Sometimes, we are surprised by the amount of symmetry sustained by the boundary conditions. In Cosmology, for example, there is no known a priori reason for the boundary conditions to be very symmetric. They could have been as complicated and asymmetric as we wish - and yet in reality, the observations show that the cosmological boundary conditions are highly spherically symmetric. In modern treatments, there is a delicate interplay between laws and boundary conditions. We shall see that symmetry has to be broken at some stage, when we deal with the real world. In the words of Francis Bacon, "there is no excellent beauty that hath not some strangeness in the proportion". Rather than break the symmetry of the Laws, it is more convenient - and useful - to find formulations in which the Laws are entirely symmetric, and the symmetry breakdown is "blamed" on some boundary conditions. In Quantum Mechanics, the "real world" is given by the Hilbert Space. We now return to the breaking of symmetry. This can be explicit in the dynamics: the Hamiltonian or Lagrangian will have a contribution breaking the symmetry in a given direction. We use this approach for Unitary Symmetry [17,7] where the breaking of SU(3) (now known as "flavor" SU(3)) is inserted by postulating a higher mass for the "s" quark. The assumption is that this is due to some new and different interaction. I called it the "Fifth Interaction" when I first suggested [18] that the Strong Interactions are really SU(3) invariant and that the symmetry breakdown was due to another perturbative interaction - as against the non-perturbative features of the "true" Strong Interactions. The Fifth Interaction can now be generalized to cover the force responsible for the "generations" structure displayed by quarks and leptons. In my 1964 papers, I had already suggested that it could also generate the mass of the muon, i.e. be responsible for the apparition of a second generation of quarks and leptons. The present "standard model"
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in which the interquark Strong Interaction is described by Quantum Chromodynamics ("QCD") indeed postulates an SU(3)-flavor invariant Strong Interaction, because the color-SU(3) gauged by QCD commutes with flavor-SU(3). The alternative way in which a symmetry can be broken is "Spontaneous" symmetry breakdown. It corresponds to cases in which it is possible to "blame" the symmetry breakdown on the boundary conditions. T h e Laws are assumed t o continue to obey the full symmetry, but the basic state in the Hilbert space, the "vacuum state'* does have a preferred direction. If, for instance we are dealing with a type of "charge" (that is not explicitly conserved because the symmetry is broken) we already endow the vacuum with a certain amount of that charge, and the particles built on this vacuum will also have that feature. In this manner, we continue to have a preferred direction imposed by the boundary conditions of the problem, in this case the Hilbert space. "Spontaneous" is sometimes taken to mean more t h a n that. Since the "states" of the system are solutions of the dynamical equations, we search for equations whose solutions will indeed carry quantum numbers breaking the symmetry. T h e present work on the STRING, for instance, attempts to find such equations (String Field Theory, or other non-perturbative techniques) that will yield solutions with broken supersymmetry etc. and looking like the real world. This approach was first introduced in the study of superconductivity in the physics of condensed matter. In that discipline, the method was invented [10] to explain phase transitions, such as the transition in a material between a paramagnetic and a ferromagnetic state when it is cooled down to the critical temperature - or the transition to the superconducting state at very low temperatures (since 19S5, the temperatures are no more that low). In a more structural theory of superconductivity [2] we can understand the asymmetric behaviour of the vacuum from the dynamics. In that problem, a "false vacuum" state is created, when the overall interaction between the electrons and the atomic lattice in the metal produces a "pairing" between electrons: two noncontiguous electrons start acting as if they were bound. This then becomes the lowest-energy "ground" state and acts as a vacuum for that particular situation; but this vacuum is not really a "neutral" empty vacuum, and thus contains characteristics that break the symmetry of the equations. The method was successfully generalized to the physics of particles and fields [15,16,12]. Here, the assumption of a "directed" vacuum requires the existence of massless particles - massless in the approximation in which all other effects are removed. The massless particles are needed to complete the vacuum's multiplet. To summarize, all symmetry breakdowns are dynamically caused - almost by definition of what physics is all about. However, explicit symmetry breaking is due to an extraneous force, whereas the spontaneous breakdown is caused by that same force that obeys the symmetry, and corresponds to the mathematical feature that a solution can have less symmetry than the equation. In an unbroken symmetry, the vacuum is invariant, i.e. if we apply to it the symmetry"s transformations, it does not change. In other words, the symmetric vacuum is a "scalar", forming a single-state multiplet. This is the algebraic char-
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acterization. But when the vacuum has a direction, applying the symmetry operations to that state should rotate it into some other state. What would that state be like in the case of the vacuum in spontaneous zvmxaetry breakdown? It turns out that a particle with zero mass could serve ad a "partner" for our non-single vacuum. The idea was very successfully applied to the understanding of the Yukawa force. This is the force responsible for the attraction between nucleons (protons and neutrons) in any atomic nucleus. It involves the exchange of pions (the "meson" postulated by Yukawa and Stueckelberg in 1934) between nucleons, like volley balls in that game. The force obeys a certain symmetry called SU(3)xSU(3)-"chiral", because the relevant conserved currents are characterized - on top of the "unitarysymmetry" charges they carry - by left or right "handedness". The two SU(3) in the name of the symmetry correspond to two currents, one an SU(3)-left and the other an SU(3)-right. Note that Parity is conserved because both chiralities are present; it is only when the left-chiral current of SU(3)-left comes by itself - in Fermi's Weak Interaction - that Parity is thereby broken. The doubling of the SU(3) currents and symmetry is quite analogous to what we observe in the case of angular momentum. In very low energy atomic physics we can have a separate conservation of spin and orbital angular momentum, i.e. two SU(2) currents of angular momentum. However, once we increase the energies involved, the spin and orbital angular momenta mix, and only total angular momentum is conserved. The same happens with the unitary symmetry chiral currents. Once the symmetry is broken, only the sum of SU(3)-left + SU(3)-right subsists as a conserved quantity. This sum is plain SU(3), and in a certain approximation it is even locally conserved. Its currents then couple universally to an octet of spin 1 vector-mesons. Chiral unitary symmetry together with this "*SU(3) gauge" provide a good phenomenological working theory for the physics of hadrons - the hundreds of different particles that feel the "strong" nuclear interaction and that we now consider as consisting of bound systems of either three quarks or a quark and an antiquark. The theory is sometimes described as "'current algebra". It fuses two theoretical discoveries of 1959-64: unitary symmetry ("SU(3)") and spontaneous symmetry breakdown using techniques [8] inspired by Heisenberg's version of Quantum Mechanics, the "Matrix Mechanics" [19,1].
4 Symmetry, Order and Information We already noted the negative correlation between symmetry and information. Symmetry represents a lack of information, an impossibility to specify, to provide identification, which is an important type of information. Lack of information in large ensembles is traditionally connected with entropy, disorder. However, this statement is not precise enough. Missing information may be connected with disorder, in the sense that it becomes too difficult to specify that information because it relates to myriads of turbulent molecules, for example. In
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computer language, it would involve myriads of information bits. This type of lack of knowledge is described as "subjective" because it is due to our own limitations. But in Quantum Mechanics, on the other hand, missing information just corresponds to its inexistence - the physical state has not yet been generated, as long as a measurement has not been performed (a "measurement" in the sense of an irreversible interaction with a macroscopic system). At this stage, all there is is just a wave-function, with a probabilistic interpretation. We know from the m a n y experiments that have realized the E P R idea [6] and applied the test provided by Bell's [3] inequalities that there is no physically concrete "underlying reality" other than the wave-function. This lack of knowledge is then an "objective" lack of information, information that does not yet exist. In the case of the grey cats of the French proverb, the lack of information is due to darkness - not to inexistence - i.e. to a difficulty in the acquisition of the information, resembling the case of disorder. It is subjective. Very recently, an advance in the study of "chaotic systems" has revealed the existence of objective entropy in non-quantum situations. There are problems in which an infinitesimal difference in the initial conditions will lead to totally different evolutions of the systems. These are then "unstable" initial conditions, generated in collective states by the internal interactions between the constituents. T h e phenomenon of turbulence in a liquid or in a gas is one such situation. The entropy of a symmetry is the magnitude of the "Whittaker impotence" it represents. This can be given a quantitative definition by taking, for instance, the volume of the Lie group - or some quantity related to the group dimensionality. SU(3) invariance is related to an 8-dimensional manifold. However, SU(3) is a broken symmetry. It is broken through the "c" quark being about 30 times heavier than the "a" and "6" quarks. This therefore reduces the overall symmetry, leaving a subgroup U(2) as the residual invariance. U(2) has a 4-dimensional group manifold with a smaller volume and is therefore a smaller symmetry and represents less entropy. The study of entropy in relation with the need to describe complexity has produced in recent years completely different approaches to the objectivisation of entropy. The aim is to have a description that would represent, for instance, the complexity of a living cell or of an organism. One such measure was "algorithmic complexity" [13,5]. The quantity characterizing the state is the length of the shortest computer program that can describe the state. It will represent the information content of that state, a kind of inverse of the state entropy. A crystal can be described by a much shorter list of instructions than a living being (who's DNA is probably the relevant program). This means that the crystal embodies less information and has a higher intrinsic entropy t h a n a living system. On the other hand, a gas with quintillions of quintillions of molecules could only be described by a program listing them all with their locations or momenta (or both, classically), i.e. the state and the design program are of the same magnitude. This corresponds to algorithmic incompressibility. This would either imply t h a t the gas contains a very large amount of information - and little entropy in the usual
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definition in which information is "subjective" - which is not what we would like to understand by entropy, which should be an objective notion. Should we then define entropy as proportional to the program's length? This seems OK in the comparison between a crystal and a gas, but it would assign a large entropy to a living being or DNA - again a paradoxical result. The missing feature in this analysis is the notion of randomness as against complexity. Both notions require longer programs. The characterization should account for the fact that the tremendous amount of information relating to the initial condition of a gas has little meaning because it is random. Any small change has no effect on the physics. In the case of DNA the list is also enormous, but a tiny change will produce a new and different being. Complexity is not disorder. This issue is resolved in a proposal due to Bennett [4]. He measures order the opposite of entropy - by the "logical depth" of the system. It represents the logical length of the program for the realization of the state, once the data is fed. To construct a living cell one would require an extremely long set of instructions. For a crystal, a limited number of steps would suffice. For a gas of molecules, the INITIAL DATA would be of an enormous magnitude, but the instructions program would consist in a trivial "copy that data". This definition therefore does fit the concept of objective entropy. It has since been further developed [14]. We can adapt these concepts to symmetry. Instead of the dimensionality or volume of the group, we could measure the information content of the vacuum, i.e. of the multiplet containing the Nambu-Goldstone boson. One way of measuring this quantity could draw from the structure of the Young tableau for that representation of the group, which is similar to a computer program for its construction. This does not appear interesting in finite-dimensional Lie groups, but something similar might be possible and helpful in infinite cases such as the presently fashionable group of conformal transformations (transformations preserving angles) in two dimensions - a symmetry of the theory of the Quantum Superstring, a "great hope" at present, as a candidate "Theory of Everything". The subject calls for further investigation. In fact, the present search for an equation or a method that would yield the physical vacuum - one out of billions of allowed ways to go from 10 to 4 dimensions - is precisely the type of case that would fit the above discussion and relate directly the notion of volume in phase space with the volume of a symmetry group.
References 1. 2. 3. 4. 5. 6. 7.
S.L. Adler, R.F. Dashen: Current Algebras (W.A. Benjamin Inc., New York, 1968) J. Bardeen, L.N. Cooper, J.R. Schrieffer: Phys. Rev. 126 162 (1957) J. Bell: Rev. Mod. Phys. 38 447 (1966) C.H. Bennett: Found. Phys. 16 585 (1986) G. Chaitin (1965): see his book Algorithmic Information Theory (Cambridge University Press, Cambridge, 1987) A. Einstein. B. Podolsky, N. Rosen: Phys. Rev. 47 777 (1935) M. Gell-Mann: "The Eightfold Way", Caltech report CTSL 20 (1961), unpub.
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8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
M. Gell-Mann: Phys. Rev. 125 1067 (1962) M. Gell-Mann: Phys. Lett. 8 214 (1964) V.L. Ginzburg, L.D. Landau: JETP 20 1064 (1950) H. Goldberg, Y. Ne'eman: Nuov. Cim. 27 1 (1963) J. Goldstone: Nuov. Cim. 19 154 (1961) A.N. Kolmogoroff: Probl. Peredachi Inf. 1 1 (1965) S. Lloyd, H. Pagels: Ann. Phys. (N.Y.) 188 186 (1988) Y. Nambu: Phys. Rev. Lett. 4 380 (1960) Y. Nambu, G. Jona-Lasinio: Phys. Rev. 122 345 (1961), Phys. Rev. 124 246 (1961) Y. Ne'eman: Nucl. Phys. 26 222 (1961) Y. Ne'eman: Phys. Rev. B1S4 1355 (1964) Y. Ne'eman: Algebraic Theory of Particle Physics (W.A. Benjamin Pub., New York, 1967, 334 pp.) E. Whittaker: From Euclid to Eddington (Cambridge University Press, Cambridge, 1949)
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of Physics Letters, Vol. 7, No. 5, 1994
COSMOLOGICAL SURREALISM: MORE T H A N " E T E R N A L R E A L I T Y " IS N E E D E D
Yuval N e ' e m a n 1 ' 2 , 3 Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University, Tel-Aviv, Israel Received May 27, 1994 Inflationary cosmology makes the universe "eternal" and provides for recurrent universe creation, ad infinitum - making it also plausible to assume that "our" Big Bang was also preceeded by others, etc.. However, GR tells us that in the "parent" universe's reference frame, the newborn universe's expansion will never start. Our picture of "reality" in spacetime has to be enlarged. Key words: inflationary cosmology, eternally recurring inflation, black hole, horizons, Schwarzschild radius, underlying reality. One is used to associate the foundations of quantum mechanics with fundamental metaphysical issues, such as EPR's "is there an underlying reality?" [1]. I would like to suggest that, as a result of the recent advances in cosmological studies, in the context of the inflationary model [2,3], physics has effectively undergone yet another, perhaps its most profound, revolution. This is conceptually comparable to the 1905 rejection of absolute time (the negation of absolute space was conceptually natural, as noted by Bishop Berkeley, by Newton himself, and especially by Mach; the revolutionary aspect was limited in that case to the rejection of the Newtonian formalism). Yet another comparable conceptual transformation happened with Aspect's (experimentally derived [4]) negative answer to EPR's above-mentioned querry, as explicited by Bell's inequalities [5]. Here, one's intuitively perceived "objective" material reality is now replaced by just potentially materializable, but otherwise "ethereal"
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amplitudes. In fact, the latest revolution, which we are noting and delineating in this letter, does bear some resemblance to DeWitt's version [6]) of Everett's [7] "many-worlds" interpretation of quantum mechanics; except that the latter example should not - in our view be considered as final, since a literal interpretation might yet be dispensed with, should a proper mechanism for "dicing" be developed (following Einstein's "God doesn't play dice"). We have conjectured that a mechanism in the nature of a spontaneous symmetry breakdown [8,9] might bring about the materialization of one component of the state-vector and the cancellation of the rest, thus avoiding the need for the "many worlds" to exist. The present observation is the following: (a) In the creation of black holes through gravitational collapse (a topic first investigated after the discovery of quasars and the conjecture that their energy is supplied by gravitational collapse of very massive stars [10]), the collapsing matter will never reach its Schwarzschild radius [11], in the reference frame of a distant outside observer A [12]. However, in the reference frame of an observer B, sitting on the collapsing star and falling into the black hole, the Schwarzschild radius is reached and crossed within hours or minutes from the collapse's start; the unfortunate B is "eaten up" by the r = 0 singularity after a comparable stretch of (his) time. This is best studied in Kruskal-Szekeres coordinates [13]. Since the principle of covariance denies the existence of any "preferred" reference frame, the "post-future" (i.e., that which comes after A's future, which is also "our's") "last trip" of B already contains the seeds of our announced metaphysical revolution: where (and "when") indeed will A (or the outside) "be", when B is half-way between the Schwarzschild radius and r = 0? Or alternatively, how can B be allowed his (or her) reference frame, in the equalitarian regime of covariance, if we can claim in all finality that B will never cross that Schwarzschild radius, in our spacetime reality?. Before the emergence of inflationary cosmology, however, B could be dismissed as some kind of thin "fringe" on the borders of reality - "an extra half-hour" added to eternity, perhaps an oddity of our description of spacetime. And yet, t h a t half-hour somehow does not overlap with our reality? Are there perhaps other "realities"? Can we accept more than one reality, just as there are any number of reference frames? (b) A similar situation arises in "eternal" inflationary cosmology [3]. New universes can be created (e.g., [14,15]) through a mechanism (inflation) which emulates the de Sitter model [16] in the first 1 0 - 3 5 sec, then "exits" this mode and settles in a flat k = 0 Friedmannian quasi-linear expansion. The first (inflationary) phase can be induced whenever a vacuum fluctuation, or some other mechanism, e.g., a collision between two 10 14 GeV cosmic rays [17], might generate, in a very tiny spatial region, an energy-density larger t h a n 1075g/cm3, i.e., about 10 14 - lQ15GeV, contained in a volume whose
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linear dimensions are of the order of the corresponding Compton wavelength. The tiny system might then "settle" for a while as a "false vacuum" in that state: in an unstable (symmetric?) solution of the (otherwise) spontaneously-broken-symmetry mechanism of a GUT, provided it would have gotten there through supercooling, for instance, or some alternative non-turbulent phase; this would correspond to having a region with a cosmological constant A, the classical GRG representation of the quantum vacuum energy. It would then trigger a de Sitter exponential expansion S = exp(Ht), with Hubble constant H — y ^ ^ 2 . Outside observers A will just note the creation of a tiny black hole, a Schwarzschild solution as in (a) above, with only the very beginnings of an expansion, lasting in this state "forever", i.e. with t —> oo. One way of visualizing this phase is to remember that the exponential growth of the tiny region is like a very fast "unfurling" of huge amounts of new space, i.e., the larger parts of the original de Sitter new universe are infinitely red-shifted with respect to A. Our entire universe is in an A-type frame and will never see the transformation of that tiny false vacuum region into anything else. However, for an inside frame of reference B, we have the birth of a de Sitter universe, a Big Bang, followed by the exit phase, then evolving into a new Friedmann (fiat) universe - and perhaps, some 10 10 years later, physicists discussing concepts of reality. The B picture is best studied in Gibbons-Hawking coordinates [18]. The new universe might have involved a singularity (a time-like half-line) due to the Penrose theorem - except that quantum tunneling makes it possible, for that new universe, to avoid the singularity. In one such solution [14], the new cosmos starts with a total mass smaller than some critical value. Classically, it would then recollapse without inflation and would reach its singularity in the future. Instead, however, it quantum-tunnels into the exponentially inflating solution (occuring only for masses larger than the critical value, classically) whose classical singularity would have lain in the past and it then goes on to make a universe, having thus managed to skip the singular stage in both world lines. As a result, the new universe carries no singularity blemish and is no different from its parent, "our" present universe. Presumably, this is also how the universe we live in came into being, with an eternal lifetime and with no singularities, neither in its past nor in its future. We should thus extend the principle of covariance to all such universes. They are all eternal - except that this is meaningless within our present conceptual framework: The new universe will never exist in our frame A, in all our time; and yet it is as good as our own universe, will have (in its B frame) galaxies and suns and perhaps physicists. So, where and when does it exist? Never, says A. Forever, says B. Note that the two did overlap before the
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"happy event" which triggered the birth of a universe, out of a given false vacuum in a region of "our" universe. They then separate, B going it by itself, observing A fading flashing out its eternity in the infinitely redshifted environment of the new Big Bang.. Clearly, "eternity", as mentioned in inflationary cosmology [3] is not an adequate answer - it just relates to A, to the eternity of "our" reality. There is, however, (perhaps) a countable infinity of such "eternities", branching out from each other, then separating, with the offspring "hibernating" and never being born, in the parent universe's reality, to "the end of time" = our eternity. And yet, beyond this eternity, there is another full-fledged universe, the offspring, flourishing and "realizing itself. Clearly, this new picture calls for our conceptual framework to admit "surrealism", i.e., "existence" beyond space and time as we know them. In the direction of the past there is just one world - line tying up together all past eternities. The theoretical basis for this conceptual jump has been around since the earliest beginnings of general relativity, since all it involves is the Schwarzschild solution [11] and the de Sitter model [16], perhaps also the Einstein-Rosen bridge [19]. R. Penrose and S. Hawking have clarified the role of horizons extensively. Interestingly enough, we came close to such a picture in our lagging-core hypothesis for the quasars [20], except that that quasar interpretation required all these de Sitter solution quasars to emerge into the same universe - no trivial requirement. The issue does exist for collapsing black holes, but these could be disregarded as far as their B picture was concerned, by regarding them as "odd" pieces of our reality, exceptional covariant frames never realizing their full physical content. This position can no more be justified in an ever multiplying inflationary cosmology, in which one of the main points is the physical non-uniqueness of universe creation, yet another sur-grandiose Copernican rejection of "our" centrality. We thus have to learn to enlarge our conception of what "is" beyond our space and time. This is sur-history and surreality.. I would like to thank Prof. D. Lynden-Bell and the Institute of Astronomy at the University of Cambridge for the Institute's hospitality during the fall trimester of 1993; it was in the inspiring creative atmosphere of the Institute, that these ideas first started forming. I would also like to thank Prof. C. Isham for the hospitality of Imperial College and for interesting discussions relating to quantum gravity.
REFERENCES 1. Einstein A., Podolsky B., and Rosen N., 1935, Phys. Rev. 4 8 , 777.
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2. Guth A.H., 1981, Phys. Rev. D 23, 347; Linde A., 1982, Phys. Lett. B 108, 389; Albrecht A., and Steinhardt P.J., (1982), Phys. Rev. Lett. 48, 1220; La D. and Steinhardt P.J., 1989 Phys. Rev. Lett 62, 376. 3. Recent reviews: Guth A.H., 1993 Proc. Nat. Acad. Sci. USA 90 4871; Linde A., 1991, Gravitation and Modern Cosmology 1991, Zichichi A., ed. (New York: Plenum); Steinhardt P.J., 1993, Class. Quantum Grav. 10, S33. 4. Aspect A., Grangier P., and Roger, G., 1982, Phys. Rev. Lett. 49, 91; Aspect A., Dalibard J. and Roger G., idem 49, 1804. 5. Bell J.S., 1966, Rev. Mod. Phys. 38, 447. 6. DeWitt B.S., 1968 Batelle RencontresI, C. DeWitt and J.A. Wheeler, eds. (New York: Benjamin) 7. Everett H. Ill, 1957, Rev. Mod. Phys. 29, 454. 8. Ne'eman Y., 1988, Microphysical Reality and Quantum Formalism, A. van der Merwe et al., eds. (Dordrecht: Kluwer), p. 141. Ne'eman Y., "Decoherence plus spontaneous symmetry breakdown generate the ohmic view of the state-vector collapse", appear in Symposium on the Foundations of Modern Physics 1993 (proceedings of symposium, Cologne, Germany, 1993). 9. Ghirardi G.C., Rimini A., and Weber T., 1986, Phys. Rev. D 34, 470; 36, 3287. 10. Hoyle F., Fowler W.A., Burbidge G.R., and Burbidge E.M., 1964, Ay. J. 139, 909. 11. Schwarzschild K., 1916 Sitzber. Deut. Akad. Wiss. Berlin, Kl. Math- Phys. Tech. 189. 12. A beautiful illustration of this state of affairs is provided in Frederick Pohl's novel, Beyond the Blue Horizon. The hero suffers for thirty years from a depression caused by his realization that, throughout his entire lifetime, his fiancee is suffering and shocked by his own behavior. She is in a spaceship falling into a black hole, which they had been exploring together, each in his own spaceship. A fatal mistake on his part caused her ship to be shoved into the black hole, while his ship thereby recoiled and made it to safety. His entire lifetime therefore coincides with one second of her time, just that second in which she is wondering why he has abandoned her and perhaps even suspects his motives. She is finally extracted from the hole and is now very much younger than her lover of the previous second. 13. Kruskal M.D., 1960 Phys. Rev. 119, 1743; Szekeres G., 1960 Pub. Math. Debrecen 7, 285. 14. Farhi E., Guth A.H., and Guven J., 1990, Nucl. Phys. B 339, 417. 15. Fischler W., Morgan D., and Polchinski J., 1990, Phys. Rev. D 42, 4042. 16. de Sitter W., 1917, Proc. Kon. Ned. Akad. Wetensch. 19, 1217.
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Hut P. and Rees M., 1983, Nature 302, 508. Gibbons G.W. and Hawking S., 1977, Phys. Rev. D 15, 2738. Einstein A., and Rosen N., 1935, Phys. Rev. 48, 73. Novikov I.D., 1964, Astr. Zh. 41, 1075; Ne'eman Y., 1965, Ap. J. 141, 1303; Ne'eman Y. and Tauber G., 1967, Ap. J. 150 755.
NOTES 1. Wolfson Distinguished Chair in Theoretical Physics. 2. Also on leave from the University of Texas, Center for Particle Physics, Austin, Texas. 3. Royal Society - Israel National Academy of Sciences Visiting Professor, Institute of Astronomy at the University of Cambridge, Cambridge, United Kingdom, and Physics Department, Imperial College of Science, Technology and Medicine, London, United Kingdom.
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Pythagoreanism in atomic, nuclear and particle physics Yuval Ne'eman 1 School of Physics and Astronomy.Tel Aviv University, Ramat Aviv,Tel Aviv 69978, Israel, and Center for Particle Physics, University ofTexas.Austin.TX, U.SA
Introduction: Pythagoreanism as part of the Greek scientific world view — and the three questions I will tackle Our interest is in the physics of the present and in the regularities which ushered it in. Surprisingly, such regularities had been postulated by Pythagoras of Samos and his school, some 26 centuries ago. In this context, the American Heritage Dictionary of the English Language [1] defines Pythagoreanism as: "the syncretistic philosophy, expounded by Pythagoras, chiefly distinguished by its description of reality in terms of arithmetical relationships". In particular, the Pythagoreans expected Nature to be describable in terms of tetracytes — dimensionless ratios between ordered integers, such as 1:2:3:4. Pythagoreanism and tetracytes were the result of the Pythagoreans' study and understanding of the physics of music. They used a monochord [2], but whatever the instrument, the same systematics amounted to the laws of wave motion in an artificially quantized system. The ends of the cord — or the air column in a woodwind instrument — are fixed and do not oscillate, and the only waves allowed are those whose wavelength is equal to the distance between the two holds, or to half that distance (so as to accommodate two wavelengths between these holds), or to one third that distance, etc. The Pythagoreans also discovered the role of harmonics, i.e. a conceptual introduction to harmonic analysis. They were strongly impressed by the role played by discrete quantities; in modern parlance, their basic 'quantum number' spanned Z, i.e. the integers. We shall name this feature 'musical' quantization, as this describes how they came by it. Somehow, they were so impressed that they conjectured that any other area of physics should end up being describable by similar constructions, a conjecture that later became known as the music of the spheres. Until the 20th century, the music of the spheres appeared to belong in the same class as the Philosophers' Stone, the Fountain of Youth and a few other medieval myths. The programme as such in fact appeared to have failed almost immediately — although this was disregarded in light of what had been learned in the attempt. Pythagoras had announced and proved his famous theorem, namely that the sum of the areas of the squares built on the orthogonal sides of a rightangle triangle is equal to the area of the square built on the hypotenuse. It was natural that this should almost directly lead to look at the case of an isosceles right-angle triangle and to the resulting question of the value of \/2. Pythagoras tried a 'Pythagorean' solution, namely y/2=p/q, with^ and q being any integers, 'Correspondence should be sent to the School of Physics and Astronomy, Tel Aviv University (e-mail [email protected]).
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i.e. pfq is a rational number, in modern parlance. Squaring both sides yields 2q2—p1, which, Pythagoras realized, contains a contradiction, since the left-hand side contains the factor 2 to an odd power, while the right-hand side can only have an even (or zero) power of this factor. The lesson was thus that arithmetic itself contains non-pythagorean elements, namely the irrationals. The importance of the discovery overshadowed the failure of Pythagoras' musical conjecture, as applied to arithmetic. Note that yet another important theorem in arithmetic is apparently due to Pythagoras, namely the elegant proof that there can be no largest prime. Note that within Greek science in general (lasting roughly from 600 B.C. to 400 A.D.), Pythagoras' school contributed an important element at the formatting stage, namely the emphasis on both geometry and on mathematics in general as the logical engine of the scientific drive. It also contributed to the direct development of these disciplines, culminating in the work of figures such as Euclid, Appolonios of Perga and Archimedes. It is that same drive which in its 'second coming' (c. 1500 A.D. to the present) has changed our world. True, scientific observations had started much earlier in Mesopotamia, Egypt, India and China, but they were not taken as making up a welt-anschau, i.e. a world view meant to describe and explain Nature in terms of a system of laws of Nature. The development of astronomy and astronomical models (in Greece, going beyond the descriptive treatments in the Middle East) and much more so the evolution of geometry, led to a gradual replacement of logical hermeneutics by mathematical derivation. There was no a priori reason guaranteeing that Nature would indeed obey mathematical reasoning, a fact that has been described in our age as a surprising feature by Einstein and others. The new approach was later stressed by Plato, but we are especially indebted to the Pythagoreans for the emergence of this new language of science as represented by mathematics. Greek science made tremendous progress during its 1000 year history [3,4], whether in astronomy, measuring the radius of the Earth (Erathostenes, c. 250 B.C.) and the distance to the Moon (Hipparchos, 150 B.C.) with less than 1 % error in either measurement, conceiving the Earth's daily spinning motion (Heraclides, c. 320 B.C.) and the heliocentric model for the solar system (Aristarchus, c. 250 B.C.), in both solid and fluid mechanics (Archimedes, c. 250 B.C.) and especially in geometry, now our model structure for all science. So, we are bound to ask why then did (Greek) science stop and what finally gave science a new lease after such a long time? This, touches upon the entire problem of the Middle, or Dark, Ages, which cannot be dealt with in the present work. Instead, I shall only tackle three pieces of this puzzle: • (i) In the general freeze (400-1500 A.D.), during which science was replaced by Aristotelian dogma, was there a specific turning point that opened the way for a scientific revival and without which it might not have occurred — a fact that would help explain the long wait? • (ii) Was there some limitation in the observational reach of physics prior to the 20th century, because of which Pythagoreanisms could not have been observed earlier, and will the present display continue? • (iii) Was there, until the last century, something missing in the intellectual toolkit for the identification of such arithmetical relations?
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Even though my first question touches upon a much broader issue, I believe the aspects I raise have not been given their proper weight to date and deserve further study.
Point 1: the impact of Gersonides and Crescas, two scientific anti-Aristotelian rebels I have discussed elsewhere [5-8] the gradual decay, followed by the forced closure of the Greek schools, ending with the murder and martyrdom of Hypatia of Alexandria [9] in 415 A.D. and the closing of the Athens Academy in 529 A.D. lam seriously worried about the possibility of a second freeze, i.e. of an end to modern science. I can point to many signals, such as the 'post-modern' description of science as wholly subjective [10], perhaps caused by confused ideas held by some of the main leaders in the quantum revolution; books like John Horgan's The End of Science [11], declarations such as that of Claude Allegre, France's Minister of Scientific Research, who in March 2000 called on the French to stop studying mathematics because it can all be done by computers [12] (we have had an identical call in Israel in 1997, against which I wrote [13]), apocalyptic titles (or beliefs) by scientists, such as Stephen Hawking's inaugural professorial lecture The End of Physics [14], the mounting price of telescopes and accelerators and the precedent created by the U.S. Congress' decision to close the superconductiong super-collider in Dallas, TX [15]. Perhaps the worst signal is the flight of students from physics and mathematics to business administration and the like. So much for the prospects of a future end to science. In Greece, during the decay period, the (so-called) Neo-Platonist academies had turned into centres of Aristotelian teaching. The closure of the School of Athens completed the eradication of the entire system of schools of Greek philosophy, but it also represented a hope, as the nine Athenian teachers were allowed to move to Sassanid, Persia, with their books, and thereby saved the essential results of 1000 creative years. That first school in Persia soon developed extensions and by the time of the Muslim conquest they were flourishing. The Abbassid caliphs in Baghdad opened an extension in their palace, and soon their Ommayad rivals in Spain opened theirs in Cordova. From there, Greek scientific material gradually seeped back into Western Europe and helped boost the development of the early universities, Bologna, Montpellier, Oxford, the Sorbonne, etc. in the 12th and 13th centuries. All this is well known, but one question is unanswered, namely, why was it only in the 16th century that new physical theories appeared on the scene — Copernicus' in particular? The immobilization was due to Aristotle's dogmatization and to the readers' scholastic attitude. One exception was Rabbi Levi ben Gerson (1288-1344, alias Gersonides or Leo Hebraeus), 'mathematicus' to the exiled Popes in Avignon, the author of 118 chapters on astronomy and inventor of the sextant [16,17]. Having once measured an angle between two stars and later remeasured it, finding a small difference, he wrote that this was probably the result of atmospheric aberration, whereby George of Trebizond, mathematicus to the Rome antipope, then wrote a tract in which he represented a millennium of
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dogmatic Aristotelian orthodoxy and declared that it was Gersonides' own mental aberration in repeatedly measuring things rather than just believing in Ptolemy and in the grand master, Aristotle. It is generally not realized outside of the circle of experts that this canonization of Aristotle was shared by the entire Judeo-Christian-Muslim culture from the 7th century onwards. All Muslim mathematicians and astronomers, from Al-Khwarizmi (whose name gave us the words algorithm and logarithm), to Al-Battani (Albategnius), Ibn Sina (Avicenna) and Ibn Rushd (Averroes), were all orthodox Aristotelians, who even improved and made Aristotle's presentation more systematic. Another important Aristotelian was Maimonides, the great Jewish medical researcher and religious philosopher, and the same is true of the succession of Jewish astronomers in Spain, starting with R. Abraham (Bar-Hiyya of Barcelona) and his pupil Abraham Ibn Ezra, then the group who compiled the Alphonsine tables, and later Zacut of Salamanca and of Sagres — they too were Aristotelians. As Aristotle was supposed to have established that motion had to be in circles or in straight lines, and as The Bible was interpreted as asserting the Earth's immobility, they were all led to accept Ptolemy's epicycle model. Note that the only resistance to Aristotelian teachings came from the opposite side, namely circles related to the Church who felt that Aristotelian philosophy might lead to a return to idolatry. There was nobody to raise scientific criticisms until two scholars dared rebel; Gersonides, the astronomer and experimentalist discussed above and a theorist, Hasdai Crescas (1350-1412), rabbi of Saragossa and high official in the Kingdom of Aragon. Their criticisms of Aristotelianism were well quoted and disseminated by Giovanni Pico della Mirandola (1463-1494) and Giordano Bruno (1548-1600), who was sentenced to death partly because of his criticisms. The criticisms are also quoted at various stages by Regiomontanus (1436-1476), Kepler, Galileo and others. First, there was the outstanding Popper-type 'falsification' experiment in which Gersonides [16,17] put an end to the original Ptolemaic model for the planetary motions. He had improved the camera obscura and had developed a system of measuring the apparent brightness of stars, using a set of tiny holes of different sizes. He then conceived the idea of checking on the correlation between the epicycles' geometrical interpretation of a planet's projected orbit and the changes in its apparent brightness. He focused on Mars and followed it for 5 years, finally reaching the conclusion that die correlation does not exist and duly publishing his negative result, namely that the Ptolemaic epicycles do not correlate with the changes in apparent brightness. Note that as epicycles represent a mathematical Fourier-like expansion, one could have corrected by bringing in higher orders to fit the complicated curve of Mars' orbit as observed from an Earth-bound system, but this too would have meant dropping the canonical Ptolemean model. Let me give you some additional information about Gersonides' stature. He is the only astronomer prior to the 17th century, when Copernicus' solarcentred model provided a basis for distant parallax measurements, who put the nearest stars at their correct distances (several light years) rather than just behind the Moon. He also remarked that "given a theory [or dogma] and measurements that do
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not agree with the theory, I trust measurements" [18]. He wrote in Hebrew and his writings were translated into Latin by Mordechai Finzi, mathematicus to the Duke of Mantova. Gersonides also discussed religious themes and cosmology; he developed the idea of continuous creation, contradicting the simplest fundamentalist reading of Genesis chapter 1. In Jewish tradition he is respected as a philosopher of religion but strongly criticized for having disagreed with Maimonides. Now to Hasdai Crescas, theorist [19] and deep thinker, whose life was strewn with sufferings. His son was killed in a massacre and he himself was forced several times into religious 'disputations' in which the losing side — and this was always the Jewish side, as it was the King or Queen who served as umpires — risked forced conversion and sometimes even death. Plato was a mathematician and had distinguished between space itself, known when empty as the vacuum, and its material contents. We know from the work of Galileo and Newton that the vacuum is a very useful concept; essential, for instance, for the definition of inertia, and for Newton's second law. Aristotle, on the other hand, was a 'realist' as a physicist, claiming that "Nature does not tolerate a vacuum", always including friction-type forces. He thereby ruled that F=kv, i.e. force (F) is proportional to velocity (v; this is the 'final velocity' in a viscous medium), as opposed to Newton's proportionality to acceleration (a; F=ma). Note that although it is easy to derive the 'final velocity' result for a viscous medium starting from Newton's second law, imposing it as in Aristotle's approach leads nowhere. Crescas wrote a scathing attack on Aristotle's error in ignoring the vacuum as a concept and defended Plato's view with imaginative arguments, allowing Galileo, 200 years hence, to disregard air resistance when throwing a stone and a sheet of paper from the Tower of Pisa and claim that they would fall at the same velocity in a vacuum. Both Gersonides and Crescas were criticized by the Jewish intellectual leadership (although they were not excommunicated, unlike Spinoza). They started the ground shaking under Aristotelianism everywhere and paved the way for Copernicus' and Galileo's innovative ideas, which inaugurated the new age.
Point 2: Kepler's spheres to Bohr's orbits — Pythagoreanisms at last! Macroscopic physics in general does not involve quantization, and the quest for the music of the spheres in the distribution of planetary orbits in the solar system involving the five perfect polyhedra [20] got nowhere — Kepler's excitement notwithstanding when he wrongly thought he had found the geometrical model for the solar system. On the other hand, Kepler's abstracted laws of planetary motion, which to him were definitely less exciting than the polyhedra, nevertheless do represent the closest thing to Nature's music of the spheres of which Kepler was dreaming. Discovering the regularities in an unknown domain both supplies hints with respect to the dynamics and serves as a test for candidate theories, precisely as one reproduces Kepler's three laws from Newton's dynamics. And yet, the expected Pythagoreanisms, that is, frequencies involving ordered low integers, finally appeared in physics in 1885, when Johann Jakob
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Balmer (1825-1898), a school teacher in Basel, Switzerland, discovered an empirical ad hoc algebraic formula fitting a series of lines in the spectrum of hydrogen. In a generalized version it said: X.-1 = [l/(91.14«w)][(l/« 2 )-(l/^ 2 )] where n and k are integers, with k>n. Balmer's formula [21] had «=2, but by 1925 similar series were found with n = \ [22] in the ultraviolet, andw=3 [23], rc=4 [24] and n=5 [25] all in the infrared. The (91.14»m) -1 in these formulae is J.R. Rydberg's (1854-1919) constant. These formulae were among the main indications which ushered in quantum mechanics [26]. Note that between 1913 and 1916, when Niels Bohr [27,28] introduced his ad hoc atomic quantized model, his basic assumption was the quantization of angular momentum L, namely L = mvr=nh (where m and v are the orbiting electron's mass and velocity, r is the radius of the orbit and h is Planck's constant). After de Broglie's introduction of matter waves in 1924 [29], this was replaced by a "generalized musical quantization" condition, namely n\=n(h/p)=2irr. Applying the musical condition to a closed orbit implies that only those diameters yielding circumferences accomodating precisely an integer number of wavelengths survive, other wavelengths vanishing by self-interference. This musically inspired picture is still the closest semi-classical approximation to quantum mechanics and is often quoted in textbooks.
Point 3: Aristotle to Maupertuis, Emmy Noether, Schwinger Examining the reformulation of the musical condition in quantum mechanics proper, however, we encounter yet another conceptual element of Greek descendance, namely the idea of a variational calculus, of invariance and of extremization. This variational approach starts with Aristotle, who was a strong proponent of the anti-abstraction, effective approach, as we saw in the vacuum issue. And yet his 'deep freeze' of physics was mostly due to a later dogmatic reading and canonization, rather than to his own assertions. What then did Aristotle state? The question was investigated by two colleagues in high-energy physics, W. Yourgrau and S. Mandelstam [30], who reached the following conclusion. Aristotle noted that motion seemed to be mostly either in straight lines or in circles — not such a bad first approximation — and searched for a reason. His suggestion was that in Nature there must be some principle of economy: the straight line is the shortest route between two given points, while the circle is the shortest closed perimeter, given the enclosed area. Applying Aristotle's variational approach to optics, Heron of Alexandria (c. 100 A.D.), also the inventor of the steam engine, proved that the angle of reflection from a mirror has to be equal to the angle of incidence by requiring the ray of light to travel over the shortest route (or in the shortest time). We then jump to Pierre de Fermat (1601-1665), who similarly applied the minimal-route or minimal-time idea to prove the law proposed by W.v.R. Snell (1591-1626) for refraction (in the transition of light between two media of different densities). C. Huygens (1629-1695) then showed that this is consistent with light being interpreted as wave motion. The economy
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principle worked in these simple cases but what was missing was a generalized mathematical technique, a variational calculus. The next installment was indeed mathematical and was supplied by Gottfried Leibniz (1646-1716), co-inventor (with Newton) of the infinitesimal calculus. Whereas Newton's version was motivated by his needs in defining velocity, acceleration, force, etc., all involving time-derivatives, Leibniz was motivated by his variational interest, as reflected in the title of his relevant book Nova Methodus pro Maximis et Minimis. Leibniz passed on the problem of a calculus of variations to the brothers Jacques I (1654-1705) and Jean I (1667-1748) Bernoulli, who passed it on to Leonhard Euler (1707-1783). Meanwhile, PierreLouis de Maupertuis (1698-1759), who had read Fermat's proof of Snell's law, independently adopted the 'shortest-route' methodology — partly influenced by religious ideas — and embarked upon an attempt to improve on Fermat, ending up with his principle of least action. He assumed that the function to be minimized should depend on mass, m, velocity, v and distance, s, and tried, as the simplest possible candidate expression for his function, the product mvs, which is equivalent to the product of linear momentum p by distance/« and thus has the same dimensions as angular momentum. I regard this choice of dimensionality in itself as a fundamental element in modern science. This was much improved and completed as to the mathematical construction by Leonhard Euler, who was in friendly contact wih Maupertuis throughout that period, and then also by Euler's successor as Director of the Prussian Academy of Sciences, Count Joseph Louis de Lagrange (1736-1813), who reformatted an 'extended' version of Newtonian mechanics in a variational mould. I have related elsewhere the story of Voltaire's hypocrisy in his two interventions, first ridiculing Leibniz in Candide for his assumption that "this is the best of all possible worlds" (Leibniz was trying to define a function which would be maximized) and later, after Maupertuis' definition of the action function, issuing against him a deadly pamphlet, accusing him of plagiarizing Leibnitz' discovery and hounding him out of Frederick's academy. Maupertuis, who had meanwhile become its President, was forced to resign and go. I should mention that Euler stood by Maupertuis in this crisis. The results at that stage represented a method yielding useful equations, the Euler-Lagrange equations of motion, which indeed effectively express the vanishing of the functional variational derivative, 8 XS(X, 3^=0, i.e. the condition for the extremization of the action S, itself a function of the variable X and its time derivative 3 ^ . When the action is written as the integral of the Lagrangian function 3 over a time interval from t, to t2, this becomes the condition for the integral to be stationary. The Lagrangian function 3 is assumed to involve the dynamic 'coordinates' X, their time dervatives d^X and the time t itself, so
h 5=J"3(X, d,X, t)dt
I have denoted the 'co-ordinate' X rather than x in order to be able to preserve the notation without confusion in the modern relativistic case [X(x) is then the field, t
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becomes the four-dimensional Minkowski space-time co-ordinate x"-, the fourdimensional integration interval is now between two three-dimensional space-like hyperplanes A" and the Lagrangian function is a four-density], either classically or in quantum field theory, where the X are also field operators acting on Hilbert space states, e.g. in the Heisenberg representation. Note that for the above SxS=0 to yield the Euler-Lagrange equations of motion 8X3 = 0, we have first to cross over into the integral with the variational derivative. Aside from the measure, which should be treated according to Jacobi's prescriptions, we expand 53 into the sum of the X-induced variation, 53/8X, plus the one induced by d^X, namely ( 8 3 / 8 3 ^ 8 5 ^ . Focusing on the latter term, we note that under some conditions, i.e. either for 'global' transformations or, if for a 'local' one in the relativistic case, after replacing djt by the covariant derivative D X (see the sequel for both these conditions), this second and last term on the right-hand side lends itself to a commutation between 8 and 3 (or its covariant extension), resulting in a term that is integrable by parts and yields a surface term [B3/d (dr.X)](8.X) . For the Euler-Lagrange equations in 3 to derive uniquely from the vanishing of the variational derivative of the action, it is also necessary to ensure the vanishing of this surface term. In the classical case, one just did not vary the variables on the boundaries. This surface term has since become the key to the success of Pythagoreanism in particle physics. An alternative version to the Euler-Lagrange analytical mechanics was later developed by William R. Hamilton (1805-1865) and K.G.J. Jacobi (1809-1851). It was the application of the action principle, using the latter treatment, which enabled E. Schroedinger in 1925 to devise his equation for the quantum mechanical wave function, i.e. the 20th century reincarnation of Pythagoras' musical quantization condition [26], now found to dominate the fundamental level. Since 1925 we know indeed that physical phenomenology appears quantized at the molecular, atomic and subatomic levels, and that under some special conditions this holds macroscopically (e.g. white dwarfs and neutron stars, superconductors and superfluids, etc.). Moreover, at the fundamental conceptual level, Pythagoreanism is fully vindicated, i.e. Pythagoras' intuitive generalization from the physics of music, namely that physical problems should obey quantized conditions, has turned out to be true at the fundamental level! This is so because the modern assumption is that everything basically consists in quantum amplitudes, while the reason that there is a classical world which appears unquantized is that for macroscopic bodies the interaction with the environment generates a decohering effect, losing the quantum properties in the limit of a very large number of such interactions. Note that while Schroedinger was solving his equation for the case of the hydrogen atom, reproducing the spectroscopic series, Pauli obtained the same result using Born-Heisenberg matrix mechanics methods involving Lie group theory, developed in the 19th century. This was an element that had been missing in the exploitation of the variational approach, namely a methodology for the description of the effects of a set of organized transformations (a group), whether discrete or continuous, on the action function. The action has to be invariant, to constrain the variation to vanish on the boundary. Thus the missing methodology was group theory and it was launched in the early 19th century. This was the Romantic Era and one couldn't find two more
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romantic heroes than the two inventors of group theory, Niels Henrik Abel (1802-1829) and Evariste Galois (1811-1832). I shall not repeat here the details of their tragic lives and deaths, except for mentioning that Abel died of hunger (or of eating the straw of his matress due to hunger) 1 week before the arrival of a letter appointing him as professor at a prestigious chair in Berlin. As to Galois, the story about the duel in which he was killed has now been modified. Recent research has found proof that the duel was just a trap set by the secret police, who considered Galois to be a dangerous revolutionary [31]. Yet another related feature has been very useful in this century, namely the so-called Erlangen program launched in 1872 by F. Klein (1849-1925) and S. Lie (1842-1899) after adventures almost as 'romantic' as in the Abel-Galois story, although with a happy ending [32]. The main contribution of the Erlangen programme was in its linkage of algebraics (group theory) to geometry. At this point, a helpful picture is provided by a solid but thin cardboard cylinder (e.g. like the ones used for toilet or facsimile paper). It is invariant under rotations around its axis, provided that we turn it by the same amount everywhere. This is a global symmetry. However, should the cylinder have been made of rubber, we could also have applied to it rotations by different angles along the cylinder's axis, yet without causing visible changes. Still, that cylinder's rubber body would have to absorb strains due to curvature caused by those differences betweeen rotation angles at any two places. Geometrically, this is a fibre bundle, with the one-dimensional cylindrical axis as base manifold and the circular crosssections along the axis forming the fibre over any point in the base manifold. The fibre bundle's 'structure group', our invariance under rotations of the fibre, is SO(2), the orthogonal group in two dimensions, with the group U(l) as its infinite covering and the circle as a representation. In the global case, the (rigid) bundle itself is trivial. Modern 'gauge theories' are fibre bundles with four-dimensional space-time as base manifold, and some unitary representation of the gauge Lie group, as realized by the particles' Hilbert space states, making up the fibre. The topological solutions to the Yang-Mills equation (instantons, etc.) have been used in geometry to study four-manifolds (the three- and four-dimensional ones are the only ones as yet untamed) and have provided an entirely new perspective in this sector of the Erlangen programme. The marriage of group theory with the variational approach was achieved by an Erlangen University graduate, Emmy Noether, in 1918, in her two Goettingen theorems [33], Her finding had the character of a treasure trove, hidden within the 'parasitic' surface term of Maupertuis' principle of least action, now renamed the action principle. That surface term represented the contribution at the boundaries and had to vanish, achieving it trivially by keeping the variables unchanged on the boundary in the transformation. In the relativistic picture, the integration domain is a four-dimensional region and the boundary is made of two space-like three-volumes X at times tx and t2, plus the 'cylinder' at spatial infinity, which is made to have no contribution by making the variables vanish at spatial infinity. The overall cancellation of our surface term now occurs when the values of the spatial integral over the two hyperplanes cancel, i.e. when for
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we have \fd3xjo\-\S
A(g
A(t.)
i.e. the space integrals constitute conserved total charges. In her first theorem, Noether thus went on to show that in the case of a continuous and global symmetry transformation (a Lie group) one obtains one conservation law per dimension of the Lie algebra, and vice versa, i.e. to each conservation law there is a corresponding subgroup of the symmetry group generated by a one-dimensional subspace of the Lie algebra, taken as a manifold. The above integrals of the relevant conserved charge-density over spatial sections of space-time are identical with the relevant Lie algebra generator. The main applications relate toflavoursand are described in the last paragraphs of this chapter. Noether's second theorem treated the case of a local symmetry, which then imposes an interaction mediated by a local connection field (or space-time tensor or four-vector potential), whose presence is required for the sake of parallel transport. It enters the expression for a covariant derivative, D =d —y(xY\a, where X.^ is an operator in the basis of the Lie algebra's generators. The connection y(x)* transforms non-linearly so as to cancel the unwanted gradient term in the group parameter d <x*(x) created by the action of the partial derivative when the parameter is not a constant, due to its local nature. The case of a local 'internal symmetry' has been studied and presented in detail by H. Weyl [34] for Abelian symmetry groups and by C.N. Yang and R. Mills [35] for the non-Abelian case. This is a formalism which has been applied for the local gauge [36] of S[£/(2)°°£/(3)], known as the 'standard model', the grand synthesis of all basic interactions other than gravity, i.e. the link up between the Weinberg-Salam electroweak unification's [37,38] spontaneously broken U(2), and the quantum chromodynamics (QCD) [39,40] of the strong interactions. The formalism's latest advance has been related to the mathematical theories of non-commutatie geometry [41,42] and the superconnection [43]. I discovered [44] a supersymmetry scheme based on the supergroup SU(2/l)=OSp(2/2) (and D. Fairlie found it independently and almost simultaneously [45]). It had various puzzling features, mainly through its Z(2) grading's identification with chirality and no change of statistics within the multiplets under the action of odd-graded generators. All of this was before the mathematical advances. The puzzles posed by this symmetry were eventually all explained and settled, after it was shown to derive from these mathematical innovations. To illustrate the applications of SU(2/1) we observe that the mass of the Higgs meson [responsible for the spontaneous breakdown of U(2), leaving only the electromagnetic U(l) unbroken], left unconstrained by the Weinberg-Salam theory, appears in the SU(2/1) mesonic mass relation [46], which should hold at around 5 GeV: m2[H]:OT2[Z]:m2[W] = 4:3:l, a rather neat Pythagoreanism. The Higgs' mass will be determined around 2005, when the L H C (large hadron collider) new collider at the European Centre for Nuclear Research (CERN) in Geneva, Switzerland will be working. Note that Einstein's theory of general relativity, as yet an unquantized (i.e. a classical) theory, is also built around local gauge symmetries, as a matter of fact around three such gauge groups, namely (i) the passive diffeomorphisms over
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.R4, a purely passive symmetry represented non-linearly over its SL(4,R) subgroup [47], (ii) the Lorentz group with its double-covering SL(2,C)=SO(l,3) as an active gauge symmetry on the local frames and (iii) 'translations', except that these now have to be covariant, becoming Lie derivatives, which causes the algebra's commutation relations to go on ad infinitum. This is indeed also the status of the supersymmetric translations in 'supergravity' theories. Another comment about the formalism: it had to be adapted further to the particle world and its Hilbert space picture. Schwinger [48] adapted the action principle to a quantum system, connecting the action with the S-matrix. Feynman developed a quantum version of the path-integral [49-51]. Now, back to results. The drama in both nuclei and particles has been in the interplay between structural and phenomenological methods. In the first case, one relies on a basic structural guess or extrapolation, from which it becomes possible to derive the system's spectrum of excitations according to that hypothesis, to be compared in the next stage with the observations. In the second approach, one first maps what is available in the system's spectra and then attempts to identify the algebraic structure characterizing it. Once this stage has been reached and validated, one analyses the algebra and starts a search for a structure that might result in its emergence in this problem. Finally, one designs experiments that could check and either invalidate or confirm that structure. In nuclei, the structural approach first appeared unreliable because the forces involved were known to be short-ranged. Various physicists, looking at the nuclear spectra and their magic numbers had the impression of a repeat of the atomic model, and yet this was strongly criticized by Bohr and others, stressing the absence of a central potential in nuclear dynamics. With the further accumulation of such central-potential-like spectral data, however, and several other arguments, mathematical treatments were shown in which the set of inner-positioned nucleons do produce an effective central potential. It was only at that stage that the nuclear shell model [52] was accepted by the consensus. In time, the collective model [53] was added and took care of other configurations, and finally the interacting boson model (IBM) [54] with a hierarchy of symmetry breaking, including the use of SU(3) {as suggested by Elliott [55] — an SU(3) whose maximal subalgebra SO(3) corresponds to angular momentum and the other five generators make up a / = 2 quadrupole) and some other subsystems connected with quadrupolar excitations [56]. The nuclear case was thus one in which one could have used the structural hint provided by observations, but one did not, out of a feeling of inappropriateness of the hydrogen-atom-like model when dealing with nucleons. In the particle world, the meanderings were similar but sometimes also more fundamental, for example when rotator or vibrator models were tried as a higher symmetry, the aim being to reproduce the SU(2) of isospin, i.e. an algebra in which the raising and lowering operators also change the electric charge. Around 1960, the Institut Henri Poincare in Paris was a centre of dialectical materialism. A group of physicists from this institution, headed by Prince Louis de Broglie, plus some allies in Japan, tried to develop such a 'spinning top' model [57], which would even have explained electric charge by a mechanical model. They failed, and neither could they show that angular momentum and isospin could add up as a single entity. Another line, also mechanical and dialectically materialistic, picked up the Sakata model [58] and developed the corresponding
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SU(3) symmetry. Here the idea was that one was ideologically convinced that there had to be a materialistic 'fundamental brick'. Perhaps the best example of a global symmetry at the fundamental level is the 'flavour' SU(3) of the hadrons, which we discovered (or identified) in the late fall of 1960 ([59], M. Gell-Mann, unpublished work). I had joined the quest, taking the phenomenological line. I mapped the spectrum, then compared the algebras as listed in Elie Cartan's 1899 classification of all semi-simple Lie algebras. Applying a set of algebraic criteria that I had abstracted from the phenomenology, 1 proceeded through a step-by-step elimination, until it yielded SU(3) with the spinJ=V2 baryons (the proton, neutron and six 'hyperons' — similar particles carrying a 'charge' known as 'strangeness') fitting nicely in a (composite) 'octet' family (although the proton and neutron had until then been considered as 'elementary'). I could similarly classify in various such families all other known particles and predict the missing ones within their respective 'multiplets'. The physics consensus adopted this scheme early in 1964, after an experiment had clearly validated one such prediction, involving the ft" hyperon, with very unusual properties, unique to this classification (among the two or three surviving and competing schemes). Meanwhile I had been thinking about the proton and neutron as 'composites' and was led, with H. Goldberg early in 1962, to define a new set of 'elementary' objects, 'building blocks' with which one could 'construct' everything. This was a triplet, with fractional electric charges [60], soon to be baptized 'quarks' by Murray Gell-Mann [61], who arrived at the same elementary bricks, as did George Zweig (G. Zweig, unpublished work) some 2 years later, and sharpened the model — just when our classification was proved to be correct — whereas nobody had noticed my suggestion in 1962, when very few considered the classification itself as a serious proposition. Once the pattern had yielded, group theory provided Pythagoreanisms in the hundreds [62], whether relating to the strong force, such as the ratios between relative probabilities for baryons with spin/=V 2 in the 10-member multiple! ([A ++ , A + , A0, A-];[Y* + , Y*°, Y*-];[E*°, S*-];[ft-]), decaying into a spin/='/ 2 baryon, sitting in the octet ({p+, tf°];[X+, X°, 2"];[A°];[H°, E~]) or a 'meson' with spin / = 0 , a member of another octet family {[K+, X°];[TT + , IT 0 , TT""];[T|0];[.K~, K?~\) and with relative orbital angular momentum L = l. In this listing, the square brackets denote closer subfamilies with common isospin and strangeness charges. We denote by ANTT the probability that A will decay into N+TT, etc. and recover the following Pythagoreanism. 3(A7Vir)=12(Y*NK)=8(Y*ATT)=12(y*2ir)
=12(a*ZK)=\2(E*ZK)=6(n.aK) =3(A2X)=12(y*EA)=8(K*ET,) = 12(E*ET))=12(E*EIT)
Or, counting quarks, the 1965 Levin-Frankfurt relation [63] between total crosssectionsCTwhen scattering a meson beam (say any of the three ir mesons), versus a nucleon beam (denoted N, consisting of either protons or neutrons), off the same target X. The meson is a compound made of one quark plus one antiquark (particles and antiparticles have the same cross-sections at very high energies) whereas the nucleon is made of three quarks, so that o-(ir, X):a{N, X)—2\i which
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fits the observations very well. I recall that in 1963, after I had published a popularized article in a (now defunct, I believe) magazine, International Science and Technology, with results of that type, I received a letter from Sweden. It was signed by Rydberg's daughter, who described the excitement of the days of atomic spectroscopy and considered the situation very similar (she was, of course, right in principle). For our ending, here is one more example, namely the Giirsey-Pais-Radicati relation [64] between magnetic moments, obtained by counting quarks and their helicities in a certain static approximation, u.(p):u,(») = —(3:2), also good within 1%. Wouldn't Pythagoras have been happy? References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.
Morris, W. (ed.) (1976) American Heritage Dictionary of the English Language, Houghton Mifflin Co., Boston Helm, E.E. (1967) Sci. Am. 217,92-103 Heath, T.L. (1991) Greek Astronomy, Dover Publications, New York Heath, T.L. (1981) A History of Greek Mathematics (2 vol.), Dover Publications, New York Ne'eman, Y. (1998) Pub. IHES, 145 Ne'eman, Y. (2000) in Visions in Mathematics 2000 (Milman, V., et al., eds), pp. 383-405, GAFA, Birkhauser-Verlag, Basel Ne'eman, Y. (1998), Func. Diff. Equations 5(3-4), 19-34 Ne'eman, Y. (2000) in Nuclear Matter, Hot and Cold (Alster, J. and Ashery, D., eds), p. 12-22, Tel Aviv University, Tel Aviv Dzielska, M. (1995) Hypatia of Alexandria, Harvard University Press, Cambridge, MA Weinberg, S. (1996), New York Rev. Books, 19 March H o r g a n J . (1996) The End of Science, Addison-Wesley, Reading, MA Allegre, G. (2000) Le Figaro, 19 March Ne'eman, Y. (1996) Letter to the Editor, Ha'aretz Magazine, 19 January Hawking, S. (1979) The End of Physics, Inaugural lecture, Cambridge University Ne'eman, Y. and Kirsh, Y. (1996) in The Particle Hunters, 2nd edn, section 11.3, pp. 281-284, Cambridge University Press, Cambridge Goldstein, B. (1985) The Astronomy of Levi ben Gerson, Springer-Verlag, New York Goldstein, B. (1969) Proc. Israel Nat. Acad. Sci. Hum. 3,239-254 Ben Gerson, L. Paris manuscript 724, fol. 75r Koestler, A. (1959) in The Sleepwalkers, part IV, Hutchinson, London Balmer, J.J. (1885) Verhondlungen der Naturforschenden in Basel, 7,548-560 Lyman, T. (1914) Phys. Rev. 3,504-505 Paschen, F. (1908) Annalen der Physik 27,537-570 Bracken, F. (1922) Nature (London) 109,209 Pfund, A.H. (1924) J. Opt. Soc. Am. 9,193-196 Thornton, S.T. and Rex, A. (1993) in Modern Physics, chapter 5, Saunders Coll., San Diego Bohr, N. (1913) Phil. Mag. 28,1 Bohr, N . (1915) Phil. Mag. 30,394 De Broglie, L. (1923) Comptes-Rendus 177,507-510 Yourgrau, W. and Mandelstam, S. (1968) Dynamics and Quantum Theory, Dover Publications, New York Toti Rigatelli, L. (1996) Evariste Galois, 1811-1832, Birkhauser-Verlag, Basel Kastrup, H.A. (1987) in Symmetries in Physics (1600-1980) (Doncel, M.G., et al., eds), pp. 113-164, University of Barcelona, Barcelona Noether, E.A. (1918) Nach. d. Kgl. Ges. d. Wiss., Math.-phys., Klasse2,17 Weyl, H. (1929) Z. f. Physik 56,330-352 Yang, C.N. and Mills, R.L. (1954) Phys. Rev. 96,191-195 Weinberg, S. (1975) Phys. Rev. D l l , 3583-3598 Weinberg, S. (1967) Phys. Rev. Lett. 19,1264-1266 Salam, A. (1968) Elementary Physics (Svartholm, N , ed.), Almqvist and Wiksells, Stockholm Fritzsch, H., Gell-Mann, M. and Leutwyler, H. (1973) Phys. Lett. B47,365-368 Weinberg, S. (1973) Phys. Rev. Lett. 31,494-497 Connes, A. (1994) Noncommutative Geometry, Academic Press, San Diego Madore, J. (1995) An Introduction to Noncommutative Differential Geometry and its Physical Applications, Cambridge Univeristy Press, Cambridge Quillen, D. (1985) Topology 24,89-95
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44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64.
Ne'eman, Y. (1979) Phys. Lett. B81,190-194 Fairlie, D.B. (1979) Phys. Lett. B82,97-100 Ne'eman, Y. (1986) Phys. Lett. B181,308-310 Sijacki, Dj. and Ne'eman, Y. (1985) J. Math. Phys. 26,2457-2464 Schwinger, J. (1951) Proc. Nat. Acad. Sci. U.S.A. 37,452-459 Feynman, R.P. (1948) Phys. Rev. 74,939-946 and 1430-1438 Feynman, R.P. (1949) Phys. Rev. 76,749-759 and 769-789 Feynman, R.P. (1950) Phys. Rev. 80,440-457 Goeppert-Mayer, M. and Jensen, J.H.D. (1955) Elementary Theory of Nuclear Shell Structure, Wiley, New York Bohr, A. and Mottelson, B.R. (1975) Nuclear Structure, vol. 2, W.A. Benjamin, Reading, MA Arima, A. and Iachello, R (1975) Phys. Rev. Lett. 35,1069-1072 Elliott,J.P. (1959) Proc. Roy. Soc. Lond. A 245,128-145 and 562-581 Draayer, J.P. (1985) Proc. IXth Oaxtepec Symp. Nucl. Phys., Notas de Fisica 8,97 de Broglie, L., Bohm, D., Hillion, P., Halbwachs, P., Takabayasi, T. and Vigier, J.P. (1963) Phys. Rev. 129,438-450 and 451^*66 Sakata, S. (1956) Prog. Theor. Phys. 16,686-688 Ne'eman, Y. (1961) Nucl. Phys. 26,222-229 Goldberg, H. and Ne'eman, Y. (1963) Nuovo Cim. 27,1-5 Gell-Mann, M. (1964) Phys. Lett. 8,214-215 Ne'eman, Y. (1967) Algebraic Theory of Particle Physics, W.A. Benjamin, Reading, MA Levin, E.M. and Frankfurt, L.L. (1965) Zhur. Eksp. I Teor. Fiz., Ptzma v. Red. 2,105 Gursey, R, Pais, A. and Radicati, LA. (1964) Phys. Rev. Lett. 13,299-301
861
PARADIGM COMPLETION FOR GENERALIZED EVOLUTIONARY THEORY WITH APPLICATION TO EPISTEMOLOGY
YUVAL NE'EMAN School of Physics and Astronomy Tel-Aviv University, Tel-Aviv, Israel 69978
1. Evolution Fully Generalized Evolution was discovered and named in the XlXth Century within the (Darwinian) biological setting, but in the XXth, it spilled over in all directions. A solid* evolutionary theory of nucleosynthesis [Bethe (1967); Burbidge et al. (1987); Pagel (1997)], coupled together with astrophysical evolution, now explains the making of the material content of the observed universe, both as to the "cooking" of the chemical elements of which it is composed and as to the formation of the astronomical features (galaxies, stars, planets etc) in which this matter is organised. Further on in the same past direction, we encounter the Big Bang and now even have several (more or less speculative) theories of the evolutionary making of universes. [Carter (1993), Barrow and Tipler (1986), Guth (1983), Linde (1990), Harrison (1995), Smolin (1992)]. In the opposite (more complex) direction, biological evolution, after reaching man, has been continued by the evolution of human societies [Ne'eman (1980)] or socioanthropological evolution; in this sector, the evolutionary levels are characterized by technologies (from the Paleolithic to the Age of Information Technology) as befits " M the toolmaker". This lead M. Bradie [Bradie (1986)] and Ruse [Ruse (1986)] to extend the application of the method to Epistemology, i.e. to the making of science, the latter being taken as the extension of man's tools (literal Evolutionary E.) . D.T. Campbell and K. Popper [Popper (1972), Campbell (1974)] then realized that what was emerging was the universality of the evolutionary mechanism, whether the evolving population be universes, stars, living beings or even ideas\ (analogical E.E.) - it all occurs via blind variation and selective retension, the essence of evolution. Note that some doubts about the "blindness" of the variations should have been assuaged after our reallocation of that role to serendipity [Kantorovich and Ne'eman (1989); Ne'eman (1993); Ne'eman (1999); Kantorovich (1993), and Kantorowitz (2000)].
* I use the adjective "solid" to imply that it has been checked in the laboratory.
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252 2. Entropy: Gravity as Model Earlier, Schroedinger [Schroedinger (1967)] had considered the overlap with thermodynamics and pointed out that evolution creates order, or negative entropy (which might have appeared paradoxical). However, as it involves dissipative systems, the environment's rise in entropy more than compensates for this local loss, thus maintaining the Second Law. We now have proof that a similar "compensation" mechanism occurs at the fundamental level, with the generation of order in (fermionic) matter being compensated by disorder in the (bosonic) field which supplies and replaces the dissipating energy. Growing a crystal by deposition in a saturated solution is a classical example, with gravity (and some atomic contributions to surface tension) supplying the ordering energy with a related generation of compensating (positive) entropy, "paying the price" of order. Classical thermodynamics covers a physical region in which the short-ranged (nuclear) interactions (Strong and Weak) have been integrated out, together with the atomic and molecular electromagnetic bindings, leaving us with just kinetic energy, plus - perhaps — some weak chemical potentials. All other parts of the fundamental Hamiltonian are included indirectly through the masses, angular momenta and various structural parameters at the molecular level. It is under such an assumption of nointeraction conditions that we present the elementary illustration demonstrating the relationship between entropy and the arrow of time: two pictures of a group of molecules, one [S] showing them spread out over a large volume, the other [D] showing them all in one relatively dense bunch. [D] is the relatively more ordered set up, [S] is the less ordered one. With chance as the only intervening factor, and as the probability of many molecules accidentally converging towards and arriving at the same point is negligible, we conclude that [D] is the earlier take, [S] the later one, reached naturally as a result of the molecules' random motion. It is instructive, at this point, to review the findings with respect to the only other identified contribution to irreversibility, the Bekenstein component [Bekenstein (1973)], corresponding to the action of gravity in its strongest attractive phase, namely in the formation of black holes, and thus in a region of phase space which is far from thermodynamics. At first sight, it would seem that here too, order is generated, with all masses converging as in [D] onto the singularity at the center of the black hole (if it is a spherical one). Thus, as for [D] in our previous discussion, a black hole would represent the generation of order, i.e. negative entropy, a la Schroedinger. This approach, however, is wrong in that it follows the history of matter solely (as represented by the energy—momentum tensor current density) coupled to the gravitational field in the Hamiltonian and does not consider radiation and the gravitational field itself, with the tensions it exerts in its strong binding. I have shown elsewhere [Ne'eman (2000)] the special-relativistic approximation to Black-Hole physics, in which we assume that a black-hole is a stellar object whose gravitational binding's (negative) self-energy has managed to "eat up" the entire invariant mass energy (k a numerical factor of order 1) Mc2 - (k GN M2)/r = 0, yielding the GR result for the Schwarzschild radius when k ~ Vi. Considering the matter content solely would thus be a grave error. Let us consider this question using what has been learned about Black Hole Physics.
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253 First, there is J. Bekenstein's intuitive formula [Bekenstein (1973)], namely positive entropy, proportional to the area of the black hole's envelope. This result was tested and confirmed by S. Hawking, using thermodynamical considerations and quantum barrier-penetration [Hawking (1975)]. What is this entropy? Full insight indeed came after Bekenstein's formula was rederived by C. Vafa and A. Strominger within Quantum Gravity, as described by String Theory [Strominger and Vafa (1996)]. This revealed where the disorder was: in the organisation of the quanta of the gravitational field ! The states counted by Strominger and Vafa are solitons and topological realizations of the gravitational field in its binding action! Thus, there is indeed a negative contribution to the total entropy coming from the "improvement" in the orderliness of the nuclei and electrons now imprisoned within the black hole, or even better, now stuck on its envelope (in view of the holographic interpretation of the conservation of quantum information [Susskind (1995)]. As against this negative increment, there is a (larger) positive contribution originating in the tensions created within the black hole's gravitational field quanta. Note that in Cosmology, with yet another arrow-of-time (linked with the cosmological expansion), it was assumed that the dense state [D] being the most ordered, with the lowest entropy, a contracting universe would produce negative entropy, in violation of the Second Law. Prior to Bekenstein's identification of the black hole's contribution [Bekenstein (1973)], the conclusion used to be that a collapsing universe would have an inverted time-arrow and would then become an expanding one [Ne'eman (1970); Aharony and Ne'eman (1970)]. This picture ignored the contribution of the main actor in cosmology, namely gravity.
3. Evolution and Entropy: Measures of Complexity Refocusing on Evolution, we are clearly again in regions in which the Hamiltonian contributes through binding components. These range from the role of Quantum Chromodynamics in nucleosynthesis to the biophysical contributions (mostly electromagnetic) making up nature's own genetic engineering. As with the area of a black hole's surface in the Bekenstein formula, we have to identify a structural complexity function, a time-arrowed quantity characterizing the action of the evolutionary drive. Moreover, as against Schroedinger's negative entropy which relates to matter, the complexity function will represent the positive entropy produced by the tension within the binding fields. Two approaches have been used to date, the more abstract [Li and Vitanyi (2000); Chaitin (1975); Bennett (1988); Fogelman (1991); Goertzel (1992); Szwast et al. (2002); Becker (2002)], inspired by Kolmogorov's treatment of information, and given by the length of the shortest program describing the system - and a pragmatic one, used in the biological domain, inspired by genetic studies. Here, effective measures of complexity have been abstracted from experimental requirements, e.g. in cases involving two species deriving from the same ancestry, estimating the time elapsed since that branching. This is done by counting the number of mutations which are not common to the two species, a linear procedure. In nucleosynthesis, it appears obvious that parametrizing the growth of complexity will involve the advance in the Atomic Mass and Charge Numbers reached, perhaps the path
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254 in the Segre diagram. As to the sign, here again it probably indicates positive entropy originating in the binding fields. Once a useful and informative level of complexity has been properly defined, it should be appended to the Second Law, in an extension of thermodynamics, or better to a related formulation in Shannon's Information Theory.
4. Extinctions and a Balanced Evolutionary Paradigm The importance and dimensions of the massive extinctions [Becker (2002)] occurring on the border between two geological eras has become clear in recent years. Survival of the fittest is replaced by survival of the luckiest. More specifically, it would appear that man's presence on earth is no less due to such opportunities created for the ("scalawag"**) small mammals by the catastrophic extinction of the unlucky dinosaurs — than to the normal evolutionary survival of the fittest. Darwinian evolutionists have essentially been biologists and have thus relegated the catastrophic changes in the environment to the boundary conditions. Both mutations and environmental catastrophic extinctions, however, are tychic*** interventions, except that mutations occur in a cybernetic program in very small steps (through errors entering in the routine procedure of copying the DNA molecules.) whereas catastrophes are single one-time rare events. Other examples of generalized evolution also point to the importance of the extinctions, as we shall see. We reformulate the evolutionary paradigm accordingly. The components are (1) a population of N individual systems S ~ with (2) each S controlled by its cybernetic program P{S} , with S existing in (3) an environment V, i.e. P{S}@V. P undergoes (4) a routine Rp which exposes it to random errors, so does (5) the environment V, which undergoes R v . Chance T thus enters through two gates, namely M: T# (RP{S}) -> P'{S'} (a mutation in the system; "type M" for short) and E: T#(RV) -> V (a "passive" mutation, a change in the environment, possibly a potential extinction; a "type E" mutation, for clarity). The new state of affairs is S'@V or S @ V which may or may not be as good and stable as the original S@V, depending on (6) the selection criteria C, acting like a sieve. Thus C [S'@V] -> 0 describes a bad type M mutation, while C [S'@V] -> S' is a good or indifferent one, C [S@V] -> 0 is a bad type E mutation, an extinction. The characteristics of evolutionary processes are (a) the creation of order, (b) increasing complexity, (c) dissipation, (d) teleonomy (or an apparent teleology), (e) tinkering. We have discussed the inter-relationship between the first two. Dissipation implies being far from equilibrium, thus requiring a steady replacement procedure for the matter and energy exiting the system. A relatively simple example would be represented by a tornado, which is also a good example of the (rotational) order generated and of the external over-compensation in entropy (as experienced by the inhabitants of Japan or of the Caribbeans).
** scalawag: undersized and "worthless" animal profitting from a catastrophe which hit the dominating predators. In the American civil war, used of white republican southerners during Reconstruction. *** tychic: "blind", random, derived from Tyche, Greek Goddess of Chance
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255 Evolution always gives the impression of working towards an aim, of an intention behind any sequence. Surveying the previous steps S(n), S(n-l), S(n-2), .. ,S(n-m) while advancing from S(n) to the next, i.e. S(n+1), one comes to think of the sequence of m+1 steps, as a machine fed with S(n--m) and now producing S(n+1). Kantorovich introduced the term "tinkering" to emphasize the "unengineered" character of the production. Flying animals (birds) were not designed and produced - as airplanes were. Instead, the existence of feathers, which evolved as a warming device on some reptiles led to wings, etc. In a sense, tinkering also explains the difference between teleology (where there is an a-priori intention to produce the final state) and teleonomy (the real picture, in which the final state is selected by the small steps along the chain).
5. The Evolution of Human Society - the Age of Information as example. Here the evolutionary levels are characterized by technologies — from the Paleolithic, etc., the Ages of Bronze and of Iron etc.. to the present Age of Information Technology - and I have shown elsewhere that the tychic elements enter via scientific discovery [Ne'eman (1980); Kantorovich and Ne'eman (1989); Ne'eman (1993); Ne'eman (1999)]. Let us analyze one such example. In 1905, mathematical logicians were exposed to ridicule everywhere. Impressed by advances in logic (such as Boolean algebra) and by Cantor's Set Theory with its provision of a method of handling infinities, Russell and Whitehead had initiated an ambitious program [Crossley et al. (1972)] of axiomatization for the entirety of Mathematics - but the whole edifice suddenly seemed to collapse when they hit the "Russell-Whitehead Paradox", namely ("a includes b" means "b is a member of a") does the set of all sets which do not include themselves include itself? # The unhappy logicians were forced back to square one, checking all their steps; they ended up in 1921 with the Zermelo-Fraenkel set of axioms and some very precise open questions. K. Goedel's incompleteness theorems threw additional light on the issues; further improvements were introduced by two doctoral students, J. v. Neumann (Budapest), who improved on the axioms, and A. Turing (Cambridge). The latter was asked, in this context, to check the concept of "computable functions" used by Goedel in his proof. Turing solved the problem by conceiving the Turing Machine, a programmable computer, which he later developed into a universal Turing machine basically characterizing all present computers. At the same time, Turing thereby also "solved" a famous problem in the Foundations of Mathematics, namely Hilbert's Entscheidungsproblem - by demonstrating it to be unsolvable (a conclusion independently reached by A. Church). The Second World War had meanwhile started and Turing was mobilized and assigned with other mathematicians and physicists to the Blechley Park center, where the Allies were trying to break German codes. On the other hand, von Neumann, being Jewish, had fled the Nazis and was now in the USA, involved in helping the Armed forces first in calculating counter-battery fire and later in the context of the Manhattan Project.
# An easier analog would be represented by a city contracting with a barber to have him shave all male citizen who do not shave themselves. Should the barber shave himself? The reader should try to check the barber's options as limited by his contract.
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256 The need for much calculation, whether in cryptography or in artillery-control, was met by the construction of primitive computers — the "Bombe" in the UK and ENIAC in the USA. Within a few years computers were everywhere, and especially when they became transistorized, after some twenty years of vacuum tubes. The Age of Information had dawned, and this was due to a chain of events triggered by the RussellWhiteheadparadox[Ne'eman(1995), Davis (1994), Aspray (1994)]. Note that in the early XlXth Century, Charles Babbadge, helped by Lady Lovelace had launched (with the financial support of he British gonernment) the building of a calculating machine and later conceived the idea of a programmable machine (but never got to build this "Analytical Engine" project) and yet it fizzled out one reason being that the status of technology was inappropriate, heavy mechanical gears instead of the electronics of the XXth century. Similar reasons explain the dead ends reached by such smart developers as Descartes, Pascal or Leibniz (the latter developed the binary number system for this purpose). A hundred years after Babbadge, his project was revived at Harvard by H. Aiken and completed around 1940, but only as a calculator, not as a "logic machine ".
6. High-Energy Physics and the World Wide Web The European Centre for Nuclear Research was founded after World War II as a scientific venture supported and exploited by 14 European countries. Scientifically, it managed throughout the Cold War to keep multinational nonfederated Europe advancing neck to neck with the two main world-powers, the USA (working mainly from Brookhaven, Fermi Lab., Stanford and Berkeley) and the USSR (with Dubna and Serpukhov-Protvino), with Japan and China joining the race from time to time. One great difficulty was due to the multinational composition itself, whether linguistically or at the technical level - each country with its preferred computerware or devices. Having to work together in large mixed experimental collaborations starting from the Seventies, a unified and elegant information system evolved and by the early Eighties various technical communications were exchanged between labs and accelerator facilities, finally crystallizing in a system now universal and known as "e-mail". Other features were later borrowed by a USA government Agency, ARPA and its ARPANET. With some interest displayed by the USA presidency (a formal initiative by Vice President Al. Gore), this has evolved into the World Wide Web, perhaps the most characteristical feature of an Information-shaped modern world. Again, a research program which was totally unrelated to a world-information issue ended up resolving it [White (1998); Berger (1996); Kouzes et al (1996)].
7. Twentieth Century Epistemology has Strong (de facto) Evolutionary Elements Reviewing epistemology as it was understood and formulated in the XXth Century shows that several of the features characterizing an evolutionary theory were identified. Four key ideas were launched:
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257 [a] Thomas Kuhn (1922-1996) stressed the role of normal science ( [RP(S)] in our characterization) and of the paradigm (which we identify with the theoretical environment [V]) [Kuhn (1962)], to be replaced sometime (T#RV -> V ) by revolutionary science R v and its resulting in a change of paradigm [ V ] . [b] Sir Karl R. Popper (1902-1994) emphasized falsification*, which consists in testing the theory in a region of its parameter space in which it has not yet been tested [Popper (1935)]. This is a change of environment (in parameter space) possibly leading to extinction. [c] Imre Lakatos (1922-1974) organized the history of science [Lakatos (1991)] according to research "programs". This is very much teleonomy, the viewing in retrospective. Example: the title of Gordon Frazer's book "The Particle Century" [Fraser (1998)] is fitting, as the electron was identified in 1897 and the top quark in 1996, with nothing earlier and nothing yet since - but it represents retrospective viewing and is very much a matter of teleonomy. [d] Paul Feyeraband (b. 1924) has emphasized chance and summarized with the phrase "anything goes" [Feyeraband (1977)]. The tychic interventions are indeed the key to evolutionary processes.
8. The discoveries towards the beginning of the XXth Century Any textbook on Modern Physics will cover these experiments, and we list some examples in our bibliography [Thornton and Rex (1993); Serway et al. (1997); Segre (1980)]. Several of these experiments involved cathode rays and we start with these. [a] Johann Hittorf (1824-1914) discovers (1875) that cathode rays consist of negatively charged particles. This was a repeat of the experiments of Julius Pluecker (1801-1868) with better vacuum technology [RP(S)]. However, an object which was left inside by mistake [TM] threw a "shadow" on the anode, showing that the rays originate in the cathode, i.e. they are electrically negatively charged [S']. [electrons] [b] Wilhelm C.Roentgen's (1845-1923) discovery of X-rays (1895). The routine [Rp (S)] involved further study of cathode rays, using an evacuated Hittorf tube, placed inside a black cardboard box. Roentgen had also prepared a set of screens made of paper with a layer of barium-platinum cyanide, a [RP(S)] phosphorescent material which he intended to use later. Suddenly, however, he saw one such screen, laying by mistake near the box [TM], starting to phosphoresce. He reached with his hand and saw his bones. [c] Radioactivity (1896) Henri Becquerel hears Poincare reporting Roentgen's discovery to the Academie des Sciences and conjectures that X-rays are linked to cathode rays in the same manner that luminescent radiation is excited by exposing the relevant mineral to sunlight [roughly correct!]. He starts (routine Rp ) research on luminescence, selecting out of his father's collection a sample of pechblende (a mineral containing uranium [TM] ) as luminescent mineral, exposing it daily to the sun, with a well-protected photographic film underneath [Rp]. The film was guaranteed to be fully protected from the Sun for an entire day - and yet [TM] one could observe the mineral's silhouette on the film. After several days, the weather changed and Becquerel postponed further exposures, leaving the mineral and film in a drawer together [TM]. * invalidation
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258 When the weather improved, Becquerel took mineral and film out of the drawer - and observed to his surprise that the film had been exposed, indicating the presence of a new type of (unstimulated) hard radiation [S'j\ Note that the conjecture (linking the production of X-rays by cathode rays to the mechanism behind luminescence) was correct, though irrelevant to the new discovery. (d) The Michelson — Morley experiment (1887) The aether was introduced as the carrier of electromagnetic waves, whose existence had just been experimentally verified by H. Hertz and were now believed to include light as one sector. A.A. Michelson's idea was to measure (R v P(S)@V) the earth's velocity in the aether, based on Galilean-Newtonian Kinematics (the conceptual environment V). However, this stretched the use of such kinematics (e.g. addition of velocities) very much beyond any previous check (R v ) and resulted in an extinction, (TE #V -> 0) namely the extinction of "unlucky" Newtonian Mechanics as a universally valid theory and its replacement in 1905 by a "scalawag" - namely Special Relativity. (e) Max Planck's analysis of the spectrum of Black-Body radiation (1900) Here too, there was a classical body of theory, a conceptual environment V with predictions ("the ultra-violet catastrophe") in the untested region [R v ] of ever shorter wavelengths and ever higher frequencies. The result was TE #V -> 0, the extinction of classical thermodynamics and its replacement by ("scalawag") Quantum Mechanics.
9. Summary and Conclusions We have discussed the scope of evolution, from the making of universes to the growth of ideas, and functions such as complexity and entropy, characterizing evolutionary processes. We have then modified the basic paradigm of evolutionary theory so as to include the massive extinctions in transition layers between geological eras in the evolutionary history of life on earth. We then applied the improved paradigm to case studies in Evolutionary Socio-anthropology and Evolutionary Epistemology. The improvement in the paradigm is especially important in the latter. In the example of the discoveries at the turn of the Century, the first three cases, namely the discoveries of cathode rays, X rays and of radioactivity are type M mutations in which an accident in the performance of a set program reveals new unknown phenomena. The last two, namely the eather-drift experiment and the black body radiation spectrum, are type E (extinctions), i.e. a body of theory (the conceptual environment of the moment) is suddenly demoted and becomes limited in its applicability to a restricted region in parameter space.
10. References Aharony, Y. and Ne'eman, Y. (1970) Time Reversal Symmetry and the Oscillating Universe, International Journal of Theoretical Physics 3, 437-441. Aspray, W. (1994) The Mathematical Reception of the Modern Computer: John von Neumann and the IAS Computer, In: R. Herken (ed.) The Universal Turing Machine, a Half-Century Survey II, Springer Verlag, New York, p. 166. Barrow, J.D. and Tipler, F.J. (1986) The Anthhropic Cosmological Principle, Clarendon Press, Oxford and New York.
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259 Bekenstein, J.D. (1973) Black Holes and Entropy, Physical Review D5, 2333-2346. Bennett, C. (1988) Logical Depth and Physical Complexity, In: R. Herken, (ed.) The Universal Turing Machine: a Half-century Survey. Oxford University Press, London, United Kingdom, pp. 227-258. Berger, E. (1996) Birthplace of the Web, Fermi News, 19, #16, 7. Bethe, H. (1972) Nobel Lecture, in: Nobel Lectures in Physics 1963-1970, Elsevier Publishing, Amsterdam, The Netherlands, 209-236. Bradie, M. (1986) Assessing Evolutionary Epistemology, Biology and Philosophy 1, 401-459. Becker, L. (2002) Repeated Blows, Scientific American 286, #3, 62-69. Burbidge E.M., Burbidge, G.R. Fowler, W.A. and Hoyle F. (1987) Synthesis of the Elements in Stars, Reviews of Modern Physics 29, 547. Campbell, D.T. (1974) Evolutionary Epistemology, In: P.A. Schilpp (ed.) The Philosophy of Karl Popper, 1 La Salle: Open Court Publishers, 413-463. Carter, B. (1993) The Anthropic Selection Principle and the Ultra-Darwinian Synthesis, In: F. Bertola and U. Curi (eds.) The Anthropic Principle, Cambridge University Press, United Kingdom, 33-66. Chaitin, G.J. (1975) A Theory of Program Size Formally Identical to Informaiton Theory, Journal of the Association for Computing Machinery 22, 329-340. Crossley, J.N. et al (1972) What is mathematical logic! Oxford Uuniversity Press, United Kingdom. Davis, M. (1994) Mathematical Logic and the Origin of Modern Computers, In: R. Herken (ed.) The Universal Turing Machine, a Half-Century Survey II, Springer Verlag, New York, p. 137. Feyeraband, P. (1977) Against Method, Schocken Books, New York. Fraser G. (ed.) (1998) The Particle Century, Institute of Physics Publishers, United Kingdom. Fogelman, F. (1991) Les Theories de la Complexite (Toeuvre d'Henri Atlari) Ed. Seuil, Paris. Goertzel, B. (1992) Self-Organizating Evolution, Journal of Social and Evolutionary Systems 15, 7-53. Guth, A.H. (1983) Inflationary universe: A possible solution to the horizon and flatness problems, Physical Review D23, 347; (1987) in The Inflationary Universe, Addison-Wesley Publishing, Reading, Massachusetts. Harrison, E. (1995) The natural selection of universes containing intellegent life, Quarterly Journal of the Royal Astronomy Society 36, 193-203. Hawking, S. (1975) Particle Creation by Black Holes, Communications in Mathematical Physics 43, 199220. Kantorovich, A. and Ne'eman, Y. (1989) Serendipity as a Source of Evolutionary Progress in Science, Studies in the History and Philosophy of Science 20, 505-529. Kantorovich, A. (1993) Scientific Discovery - Logic and Tinkering, State University New York Press, Albany, New York. Kantorowitz, A. (2000) From the Amoeba to Einstein, University of Haifa and Zamora-Beitan Publishing, Haifa, Israel. Kouzes, R.T, Myers, J.D., Wulf, W.A. (1996) Collaboratories: Doing Science on the Internet, Computer 29 (8), 40 Kuhn, T. (1962) The Structure of Scientific Revolutions, University of Chicago Press, Chicago, Illinois. Lakatos, I. (1991) Falsification and the Methodology of Scientific Research, In: I. Lakatos and A. Musgrave (eds.) Criticism and the Growth of Knowledge, Cambridge University Press, United Kingdom, 91-196. Li, M. and Vitanyi, P. (2002) An Introduction to Kolmogorov Complexity and its Applications. Springer Verlag [translation from Chinese]. Linde, A.D. (1990) Inflation and Quantum Cosmology, Academic Press, Boston, Massachusetts. Ne'eman, Y. (1980) Science as Evolution and Transcendence, Acta Cien. Venez. 31, 1-3. Ne'eman, Y. (1993) Serendipity, science and society - an evolutionary view, in: Proc. Kon. Ned. Akad. v. Wetensch. 96, 433-448. Ne'eman, Y. (1999) Order out of Randomness — science and human society in a generalized theory of evolution (in Hebrew), Van Leer Jerusalem Institute and Hakibbutz Hame'uchad Publishers. Ne'eman, Y. (2000) Heuristic Methodology for Horizons in GR and Cosmology, Gravitation and Cosmology 6, Supplement, 30-33. Ne'eman, Y. (1970) CP and CPT Symmetry Violations, Entropy and the Expanding Universe, International Journal of Theoretical Physics 3, 1-6. Ne'eman, Y. (1995) The sophism which ushered in the Age of Information Technology, In: R. Stavy and D. Tirosh (eds.) Theory and Practice in Mathematics, Science and Technology Education (in Hebrew), TelAviv University, School of Education Publishing, Tel-Aviv, Israel, pp. 15-20. Pagel B.E.J. (1997) Nucleosynthesis and Chemical Evolution of Galaxies, Cambridge University Press. Popper, K. (1972) Objective Knowledge, Oxford Clarendon Press.
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260 Popper, K. (1935) Logik der Forschung. Ruse, M. (1986) Taking Darwin Seriously, Oxford-Blackwell Publishers. Segre, E. (1980) From X-rays to Quarks, W.H. Freeman and Co. Publishers, New York. Schroedinger, E. (1967) What is life, Cambridge University Press, United Kingdom. Smolin, L. (1992) Did the Universe Evolve, Classical Quantum Gravity 9 (1), 173-191; (1997) The Life of the Cosmos, Weidenfeld and Nicholson Publishers, London. Strominger A. and Vafa C. (1996) Microscopic origin of the Bekenstein-Hawking Entropy, Physics Letters B379, 99-104. Susskind L. (1995) The world as a hologram, Journal of Mathematical Physics 36, 6377- 6396. Szwast Z., Sieniutycz S. and Shiner J.S. (2002) Complexity principle of extremality in evolution of living organisms by information-theoretic entropy, Chaos, Solitons and Fractals 13, 1871-1888. Thornton, S.T. and Rex A. (1993) Modern Physics for Scientists and Engineers, Saunders College and Harcourt, Brace Jovanovich Publishers; also Serway, R.A., Moses C. and Moyer C.A. (1997) Modern Physics, same publishers. White, B. (1998) The World Wide Web and High Energy Physics, Physics Today, November, 30-36.
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EVOLUTIONARY EPISTEMOLOGY AND INVALIDATION
Yuval Ne'eman School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Abstract:
1.
I show that the two most important conceptual advances in epistemology made by Karl Popper, namely, invalidation (1935) and evolutionary epistemology (1972), though apparently unrelated, fit very precisely together if one applies an improved generalized evolutionary paradigm in which the extinctions are included as chance "mutations" of the environment.
INTRODUCTION
It is a pleasure for me to contribute to this Festschrift honoring Eviatar Nevo upon his 75th birthday. As one always interested in evolution - while working mostly in other fields such as Particle Physics, Cosmology, and Philosophy of Science, I was happy to have had the chance to be in contact with Eviatar throughout the years. The topic I have selected relates to evolutionary epistemology, a philosophical doctrine launched by Karl Popper (1972) and developed by Donald Campbell (1974), with significant contributions by my friend and former student Aharon Kantorovich (1993) including Kantorovich and Ne'eman (1989) and Ne'eman (1999). I do believe in Evolutionary Epistemology. Popper is the most quoted modern philosopher of science and is well known for his work on the concept of invalidation (or falsification, the term used by Karl Popper - which I believe was a rather S.P. H'asser (eel.). Evolutionary Theory and Processes: Modern Horizons, Papers in Honour of Eviatar Nevo Y. Ne'eman. Evolutionaty Epistemology and Invalidation, 109-112. © 2004 Khmer Academic Publishers. Printed in the Netherlands
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unfortunate choice). I use "invalidation", which fits the concept more precisely, in view of its intended meaning, namely, trying to show that some theory or definition, which had been validated for a certain range of values of some parameters, no longer holds within a range of these parameters previously untested, but is now experimentally within reach. Falsification implies trying to show that it is false over the new range. But the term has a different and rather ugly meaning in everyday life (such as falsifying a document), whereas invalidation is precisely what we mean. Taking evolutionary epistemology as the correct theory and looking in retrospect at what Popper said about invalidation (Popper, 1935), one tends to wonder: if ideas develop according to the evolutionary paradigm, how does "invalidation" - a most important concept in the history and philosophy of science - fit? What is its evolutionary role? I hope to show that it indeed fits beautifully, provided we first update our evolutionary paradigm.
2.
EXTINCTIONS AND A NEW EVOLUTIONARY PARADIGM
Looking at the history of life on Earth we come across the primordial importance of the extinctions (Becker, 2002) at the transition layers between any two geological eras, as first noted and studied by L. and W. Alvarez in the 1960s. We now know several other cases of massive extinctions which have also given a boost to some otherwise un-evolving species. It is scientifically wrong to leave out these catastrophic developments from evolutionary studies. In my amended paradigm we shall see how it becomes possible for extinctions to be considered part of the evolutionary processes. Here is a modified schema of evolutionary processes: A system [S], governed by a program P[S] exists in an Environment [E]. It undergoes some routine R[P] which exposes it to the impact of chance, ("tychic" intervention [T] of random results), causing mutations M(S). Dynamical constraints "filter" the mutations; if it is an improvement, it prospers. Up to here we have had only "active" mutations of the system's program. What is the role of the catastrophes which caused the extinctions (such as a hit by a comet, etc.)? These are tychic interventions in a routine exposure of the Environment R[E] , or "passive" mutations. The routine in this case is the motion of the Sun and Earth through different debris with the passage of time. Here, too, there is a dynamic selection: the Tunguska meteor only set fire to a large area in Siberia - whereas the one that killed the dinosaurs also
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gave an unexpected chance for the smaller mammals to take over. I have used the term "scalawag" for this kind of improvement by survival.
3.
EVOLUTIONARY EPISTEMOLOGY - ACTIVE MUTATIONS
Let me immediately place the various actors in their roles in this case. The "system" is a conjecture or theory which motivated the present stage. The "environment" is the existing body of theory, the paradigm that it has to fit. The mutation occurs when the system's program is exposed to random developments. The intention was to study X - this is the routine R(P) - and something has happened by chance and either nothing important develops, or B is discovered. Examples abound, such as (Segre, 1980; Thorton et al., 1993) the discovery of x-rays by Roentgen in 1896: he was studying cathode rays when a screen, left on a table nearby from a previous experiment, started to glow. He put out his hand and saw the bones through the flesh. Another such example is Cade's discovery of the psychochemical properties of lithium (Lickey and Gordon, 1983). Cade was a physician associated with an asylum. He planned to test uric acid, reputed as producing active personalities, and tried it as an energizer for the sufferers of depression. Uric acid was obtained as its lithium salt. Having administered it to all inmates, he discovered that the schizophrenics calmed down. Let us quote some other examples in other fields out of the multitude, i.e., in experimental physics, the discovery of radioactivity by Becquerel or in theoretical physics the emergence of string theory as a quantum and the theory of gravity; in mathematics, both pure and applied. We have the emergence of the modern computer from the after-shocks of the Russell-Whitehead paradox (Ne'eman, 1998).
4.
EVOLUTIONARY EPISTEMOLOGY: INVALIDATION AS AN EXTINCTION
We now come to experiments that have broken an existing paradigm. An example is the Michelson-Morley experiment (1897) (Segre, 1980; Thorton and Rex, 1993), with the resulting extinction of Galilean symmetry and the emergence of relativity. The experiment assumed that the addition of velocities always follows the simple rules of Galilean symmetry. This had been verified over a range of velocities up to 0.1% of the velocity of light. The Michelson-Morley experiment tested it de facto at the velocity of light itself - and it destroyed the basic assumption. Galilean physics became the 874
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low-velocity approximation of a new theoretical environment, namely, relativity. Popperian invalidation is thus a routine "motion" within the environment by changing one parameter (in this case velocity), thereby entering a previously unexplored sector and opening the possibility of finding that this is not at all the theoretical environment we started from. As to the "willed action" aspect, in Michelson's de facto selection of the new theoretical kinematics environment by his (unconscious) choice of the value of the velocities appearing in the addition equation, the only relevant criterion is his total ignorance of what the kinematical paradigm (the environment) will be like - whether it will be the same Galilean symmetry (i.e., no need to change the current paradigm) or not; neither did he know what it could change into. Research is a "blind" activity by definition. Otherwise it becomes development. As to the evolutionary role in epistemology of the researcher who, unlike Michelson, consciously initiates a falsification/invalidation experiment, he is the direct analogue to that of the biologist who modified the biological environment by irradiating Drosophila flies. Both are in one class with a new predator appearing in a given region, or a dam built on the river home of a species. The falsification/invalidation scientist hitchhikes upon an evolutionary feature and becomes part of it.
5.
REFERENCES
Becker L. 2002. Repeated Blows. Sci Amer. 286, 62-69. Campbell D.T. 1974. Evolutionary Epistemology, in The Philosophy of Karl Popper, P.A. Schilpp, ed., La Salle: Open Court, v.l, pp. 413-463. Kantorovich A. and Ne'eman Y. 1989. Serendipity as a Source of Evolutionary Progress in Science. Studies in the History and Philosophy of Science 20, pp. 505-529. Kantorovich A. 1993. Scientific Discovery - Logic and Tinkering. SUNY Press, Albany, pp. 281 Lickey M.E. and Gordon B. 1983. Drugs for Mental Illness. W.H. Freeman, New York. Ne'eman Y. 1998. The sophism which ushered in the Age of Information Technology. In Theory and Practice in Mathematics, Science and Technology Education (Hebrew), R. Stavy and D. Tirosh, (Eds.), Tel Aviv University School of Education, Tel Aviv, pp. 1520. Ne'eman Y. 1999. Order out of Randomness: Science and Human Society in a Generalized Theory of Evolution (Hebrew). Van Leer Jerusalem Institute and Hakibbutz Hame'uchad, Tel Aviv, 112 pp. Popper K. 1935. Logik der Forschung. English version, 1959. Popper K. 1972. Objective Knowledge. Clarendon Press, Oxford. Segre E. 1980. From X-rays to Quarks. W.H. Freeman, New York. Serway R.A., Moses C. and Moyer C.A. 1997. Modern Physics. Saunders College and Harcourt, Brace Jovanovich, New York. Thornton S.T. and Rex A. 1993. Modem Physics for Scientists and Engineers. Saunders College and Harcourt, Brace Jovanovich, New York.
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