From Field Theory to Quantum Groups: Birthday Volume dedicated to Jerzy Lukierski

+ 1 ®
(5.2)

8Notice that, wcrc it not by the q-bracket [x]q = (qX - q-X)!(q - q-l), these redefinitions would be equivalent to those in (4.3); this exhibits once more the non-commutative nature of many contraction! deformation diagrams. 9The above oscillator algebra, where N is treated as an independent gcnerator, has a non-trivial central element, z q-N+l([N]q - a+a) and many irreducible representations (for 0 < q < 1)[34] unequivalent to the Fock space ones with vacuum state and number operator N, [N]q = a+a, for which z = o.

=

Contractions, Bopf Algebra Extensions The duals

fJ

15

and ex of ex and !~ are found to be

~(x) =

x ()\: cos 'P

+ y C9 sin
(3(y) = -x

ri(x C9

Q)

sin

(9

cos
(5.3)

n(y C9

,

The compatibility conditions (8' .IO)-(B' .14) are satisfied, and (B' .15), (B' .16), (8' .17) and (B' .18) determine the Bopf structure of Funw( £(2)), [x,y]=-wy,

[x,
Ll

[y,
Llx = 1 Q) x + x ® cos

,

Lly=lQ9y-x!X:sin
, ,

(5.1)

t(
S(y) = - sin
,

(3) A discussion of the Galilei case will be presented elsewhere. (4) Consider now the case of the deformed 1/eisenberg- Weyl 'group' Funp(HW), (see [39]) dual of the algebra U p ( 1/W) as given in (4.6) (i.e., (2.6) for w= 2p). It was shown in Sec. 3 (4b) that Up(HW) could be obtained as the cocycle bicrossproduct [11,12] (Appendix B) Up(HW) = H~ A of the undeforrned algebras H = U(T1'(2)) and A = U(u(I)) by using the non-trivial ~ and ~ given (3.8), (3.9). Thus, the deformed Ileisenberg- Weyl group algebra Funp(IlW) may be found as the cocycle bicrossproduct of 1/ = 1'1'(2) and A = U( 1) using the duals n : A C9 H ---> Hand '0 : A ---> H (9 1/ of ~ and ~ respectively. Using (Yl' Y2; X) for the parameters of 1'1'(2) and U(I), < Y;, Yj >= Dij , < X >= 1 the dualization of ~ immediately leads to

-=,

n(X,Yi) == Xf>Yi

= -2PYi

or

l-ex J;4 P

[X,YiJ

= -2pYi

,

(5.5)

= 1,2. Let us now dualize ~ = '=). What was really needed in (3.9) to compute [Y1 , Y2] was the difference ~(Yl, Y2) - ~(Y2, Y1 ); the ambiguity in ~(YI' Y2) is related to the cohoundary ambiguity. A suitable election produces p

i

-

1

V'(X) = 2"(Yl C9 Y2 - Y2 is!

ytl ,

from which Ll(X) is easily found using (8'.34) since Funp(HW) is determined by fYi, Yj] = 0 LlYi

,

[X, Yi] = -2pYi

= Yi ~ 1 + 1 C·) Yi

S(Yi) = -Yi

,

{3 is trivial (0: is trivial). In all,

,

,

S(X) = -X

(5.6)

(5.7) ,

E(Yi, X) = 0

.

16

J.A. de Azcarraga and J.C. Perez Bueno

The coproduct mimics tlw familiar H W group law, and the non-commutativity is just reflected in the non-zero [X, Yi] commutator. (a) Extended Euclidean group Fun p (E(2)). The dual Funp(E(2)) of the algebra (a) given byegs. (4.6) (and (4.4)) is generated by the clements (YhY2,X,'P) « li,Yj >= 5ij , < N,'P >= 1, < :=,X >= I) for which

Ll'P = 'P (8 1 + I Q<; 'P LlY2 = I 19 Y2

,

LlYI == I ® YI

+ Y2 19 cOS'P -

YI ® sin 'P

+ YI

® cos'P

+ Y2 19 sin 'P

,

+ YI

;

,

LlX=I®X+X®1 I + 2"[YI 19 cos 'PY2

+ Y2 ::>9 sin 'PY2 -

Y2 19 cos 'PYI

Q9 sin 'Pyd

(5.8) It is not difficult to check directly that Fun p (E(2)) [(5.8)] is a Hopf algebra; we shall now obtain (5.8) by dualization in two different ways. For the dual in the basis (4.15), see

[24].

al) Fun p( ];(2)) is the bicrossproduct FunU(1) .,:lFunp(HW), where U( I) is generated by 'P and Funp(HW) is given in (5.7). To see this, it is sufficient to dualize the right action
(j(yt) = YI ::>9 cos 'P {3(X) = 1 Q9 X

+ Y2 ® sin 'P

,

fJ(Y2) = -YI Q9 sin 'P

+ Y2 19 cos 'P

,

,

(5.9) for which the coproducts and antipodes in (5.8) are obtained from (8'.16) and (8'.18). Clearly [Yi,'P] = 0 = [X,'P] since ii (~) is dual to {3, which is trivial. a2) lo'ullAE(2)) has also a cocycle bicrossproduct structure. To see this, we take A as the Hopf algebra generated by X with primitive coproduct and H as the (un deformed) Euclidean group Hopf algebra of generators (Yll Y2, 'P) with LlYi ,Ll'P , S(Yi, 'P) and t(Yi, 'P) as in (5.8). Then, since 0' and 1/J were trivial

Contractions, Hopf Algebra Extensions

17

in Sec. 4 a2), fJ and 1, are trivial (fJ(a) = a @ Ill, (a,b) = f(a)f(b)IH) and - t ff @ H , (i : A @ /I - t /I may be found from (4.8) to be

;j; : A

1;;(x) =

~[Yl iZ: cos r.pY2 + Y2 @ sin r.pY2 -

Y2

@

cos r.pYI + Yl @ sin r.pYd

,

(5.10) Thc relations (fl' .19)-(B' .28) arc fulfilled 10 and the cocycle bicrossproduct structurc of Fun p (E(2)) follows from (B'.29) (which for 1, trivial and a with primitivc coproduct leads to [a, h] = af;h) and (B'.34).

(b) Extended Euclidean group Fun p(E(2)). This is the dual Fun p(E(2)) of the lIopf algebra Up(E(2)) (scc cqs. (1.11) and (4.13)). It is gcncrated by the elcmcnts (x,y,r.p,X) « Px,x >=< Py,Y >=< J,r.p >=< ~,X >= 1) with relations [x,r.p] = [y,r.p] = [X,r.p] = [x,y]=O LX,x]=-2psinr.p

,

Llr.p = r.p 0) 1 + 1 Q9 r.p

,

LX,y]=2p(l-cosr.p) ,

Lly = 1 Q9 y + y 09 cos r.p - x @ sin r.p

S'(x)==--x

,

y Q9 sin r.py - y 09 cos r.px + x 0 sin r.px] ,

S(x)==--cosr.px+sinr.py

f(r.p,X,y,X)==-O

,

,

~[x (j) cos r.py +

LlX = 1 :i9 X + X @ 1 + S(r.p)==--r.p

+ x Q9 cos r.p + Y :i9 sin r.p

Llx = 1 Q9 x

,

S(y)=-cosr.py-sinr.px

,

,

(5.11) which define a Bopf algebra as it may be checked. Now, we take A as the Hopf group algebra gcnerated by Xand 1I as the dual un deformed Euclidcan group lIopf algcbra Fun( E(2)) (eqs. (5.4) for w ==- 0) of generators (x, y, r.p). If we now define fJ, 1, to be trivial plus

J}(X) =

~[x Q:; cosr.py +

y!Xl sinr.py - [email protected]+ [email protected]]

, (5.12)

Xe-x=-2psill:p

,

Xf;y=2p(l-cosr.p)

,

X"r.p=O

,

lOThe only non-trivial properties are (8'.23) and (B'.25). The first one is the dual cocycle condition, verified because the dual cocycle ,p is the undeformed one, and the second one is due to the compatibility between the coproduct and the commutators.

18

.LA. de Azca.rraga and J.C. Perez Bueno

from ~ and f3 in eq. (4.14), the Hopf algebra (5.11) is recovered using (8'.29) and (13'.34), which exhibits the cocycle bicrossproduct structure of (5.11). Due to the commutators [X, xl, [X, y], there is no Hopf Funp(HW) subalgebra here and no bicrossproduct structure in contrast with the previous al) case.

6.

Differential calculus on the Euclidean and Galilean planes

We shall now introduce a covariant differential calculus [40] (sec Appendix A) on the different homogeneous spaces which can be constructed. Clearly, to have a proper action on the 'homogeneous' part, a bicrossproduct structure is needed. Let us consider now a few different cases. (1) Due to the lack of a bicrossproduct structure, the inhomogeneous part of the Uq ( [(2)) algebra (PI, P2 ) docs not constitute a Hopf subalgebra, and the construction of the space algebra as the dual of (PI, P2) cannot be performed. (2) The Euclidean plane E~ is introduced as the dual « Pi, Xj >= bij) of the translation Hopf subalgebra Uw(T r(2)) of Uw([(2)) generated by Pi (eq. (3.2)). Since Uw(Tr(2)) is commutative but not cocommutative, we obtain (eqs. (5.1))

dx = x 1291 + 1 QSix

dy=yQSiI+IQSiy

[x,y] = -wy

(6.1 )

for the E~-plane algebra associated with Uw ([(2)). Let us construct a bicovariant differential calculus on E~ which is consistent (i.e. covariant) under the action of J. The (left) action of J on E~ is defined by duality, < Px <1 J, x >=< Px, J i> X > etc., from which it follows that

Ji>X = Y

J

i>

(6.2)

Y =-x

To define a first order (J - )covariant differential calculus we have to determine all commutators [Xi, dXj] in a way which is consistent with the action (6.2) (which for instance, implies Ji>xdy = (J(1) i>x)d(J(2) i>Y)) and with the Jacobi identity. Although it is not difficult to check that the set of covariance equations (like J i> (XidxJ) - J i> (dxjx;) = J i> [Xi, dXj]) has a unique solution given by

[x, y] = -wy

[:c, dx] = 0 = [x, dy]

[y, dx] = wdy

[y, dy] = -wdx

(6.3) the above COI1lmutators do not satisfy the Jacobi identity and thus fail to provide a consistent differential calculus. This situation is not new, and has already appeared for the differential calculus on other spacetime algebras [28,29]. We now show that the solution proposed there, and which involves an enlargement of the algebra which has been found to be associated with a Hopf algebra cocycle extension [29], also applies here. We stress that this problem is associated to the deformed character of (3.2) as expressed by w, being of course absent for the undeformed Euclidean Hopf algebra U([2).

Contractions, Hopf Algebra Extensions

19

Consider the trivial extension Uw (t:(2» xU(u(1» mentioned in Sec. 4, obtained by adding the primitive Hopf algebra generated by := to Uw ( [(2)). The previous procedure applied to (Px , Py ,:=) leads now to an enlarged Euclidean algebra Ew generated by (x,y,X) « :=,X >= I) and to the additional relations

[x,x]=O=[x,y]

,

J

LlX=XI8l1+1I81X

t>

X= 0

(6.4)

.

Proceeding as before, we find that there is a unique solution for the rotation covariant differential calculus on the above enlarged Euclidean 'space' specified by (d. (6.3»

[x, y]

= -wy

[y,X]

=0

, [x, X] [y,dx]

,

=0 ,

= wdy

[x, dy]

[X,dx] = wdx

[x, dx]

= wdX

, [x, dy]

=0 ,

[x, dX]

[y,dy] = -w(dx - dX)

= wdy

[X,dX]

= wdx

[y,dX]

,

= wdy

= wdX

(6.5) and satisfying Jacobi identity. (3) We define the two-dimensional Galilean plane Gi as the dual « X, x >= 1=< X"t >, < X,t >= 0 =< X"x » ofUw(Tr(2». The commutativity (noncocommutativity) of Uw(Tr(2» implies the relations

[x, t]

= 2wx

;

Llx

= x 181 1 + 1 181 x

Llt

= t 181 1 + 1 181 t

(6.6)

for the Gi algebra. Following the same pattern of case (2) we construct a bicovariant differential calculus (covariant under the action of the 'boost' V) that satisfies Leibniz's rule and Jacobi identity. The (left) action of V on Gi is given by

V

t> X

= -t ,

V

t>

t=0 .

(6.7)

Using (6.7), we find that the covariance requirement implies the system of equations

V

t>

[x, dx] = -[t, dx]- [x, dt]

V

t>

[t, dx] = -[t, dt]

V

t>

[x,dt] = -[t,dt]

V

t>

[t, dt] = 0

+ 2wdt (6.8)

.

The unique solution linear in dx, dt that satisfies (6.8), Leibniz's rule and Jacobi identity is l l

[x, dx]

=0

,

[x, dt]

= wdx

,

[t, dx]

= -wdx

[t, dt]

= wdt

(6.9)

11 Even if there is no deformation (w = 0) there exists a non-trivial solution (see [29]) given by [x, dx] = pdt and all other commutators equal to zero.

20

J.A. de Azcarraga and J.e. Perez Bueno

Thus, this case is different from the Euclidean case E~. On Gi there is a covariant differential calculus without any additional one-form. l2 (a) It was seen (eq. (6.5)) that to define a }-covariant differential calculus on E~ it was necessary to enlarge it to Ew . Let us now show that two N -covariant calculi may be similarly constructed on Funp(HW) (eqs. (5.7)) as the dual of the Up(HW) subalgebra of Up ( £(2)) (Sec. 4( a)). The left action t> of N on (Yb Y2, X) is obtained from (4.7) and given by N

t>

Yl

= Y2

N

,

t>

Y2

= -YI

(6.10)

Proceeding as before, we find the commutators [Vi, Yj] = 0

fYi,

,

[x, dy;] = )"dy,

xl =

2PYi

[Vi, dxl = ()..

+ 2p)dYi

[X, dX] = W1X

(6.11 )

The Jacobi identity requires)., = -2p or )., - fL = -2p. The bicovariance requirement now determines two bicovariant differential calculi over Funp(HW) (on the E~ plane the coproduct of the generators was primitive, hence the differentials are bi-invariant by (A.4) and the bicovariance is trivial). We first find, using (5.7) and (A.4),

+! i1 R dX = dX ® 1 +!

i1L dX = 1 ® dX

(YI ® dY2 - Y2 ® dyIl

(6.12)

(dYI ® Y2 - dY2 ® YI)

it is easy to show that the coact ions (6.12) satisfy (A.2). If we use now (A.I) to calculate i1dx, dxl = fLi1LdX we find fL = 2).,; the same condition is obtained Ilsing i1 R • Then, (6.11) leads to ()., = -2p, fL = -4p) [Vi, Yjl

=0

,

[Vi,

[x, dy;] = - 2pdYi

and ()..

xl = 2pYi [Yi,dXl = 0

,

[X,dX] = -4p dX

(6.13)

= 2p,fL = 4p) [Vi, Yj] = 0

,

[X, dy;] = 2pdYi

[Vi, xl = 2pYi [Vi, dxl = 4pdYi

[X, dxl

= 4pdX

(6.14)

12For the differential calculus on the deformed Newtonian spacetime associated with the (I version of the deformed Galilci algebra 9. see [29].

+ 3)

Contractions, Hopf Algebra Extensions

21

Since (A.3) is satisfied, eqs. (6.13), (6.14) determine two first order N-covariant differential calculi over Ep • Acknowledgments This paper has been partially supported by the CICYT grant AEN93-187. One of us (JCPB) wishes to acknowledge a FPI grant from the Spanish Ministry of Education and Science and the CSIC. Both authors wish to thank M. del Olmo for very helpful discussions.

Appendix A:

Bicovariant differential calculus

Let A be a Hopf algebra and let Ll and E be its coproduct order bicovariaul differential calculus over A is defined [40] by d : A -> r is a linear mapping satisfying Leibniz's rule and r bimodule (r, LlL, Ll R ) i.e., the linear mappings LlL : r -> A 0 r and the exterior derivative d satisfy

and counit. A first a pair (r, d) where is a bicovariant A, Ll R : r -> r 0 A

LlR(aw) = Ll(a)LlR(w) Lldwa)

= Lldw)Ll(a)

,

LlR(wa)

= LlR(w)Ll(a)

(A.I)

(A.2)

(f 0 id)LlL = id

(A.3)

(A A)

r

where the left (right) equations in (A.2) express that is a left (right) A-comodule, (A.3) is the result of bicovariance (commutation of the left and right coaetions), and (AA) expresses the compatibility of the exterior derivative d with Ll and LlL,R. Eqs. (A.I), (A.2) and (A.3) characterize (r,LlL,Ll R) as a hicovariant bimodule over A; the addition of (AA) determines a first order hicovariant differential calculus (r, d). An element w E r is called left (right) invariant if Lldw) = I 0 w (Ll R ( w) = w Q:) I). As in the undeformed (Lie) case, the basis elements of the vector space Iinv C r of the left-invariant elements generate as a left free module.

r

Appendix B:

Bicrossproduct of Hopf algebras and co cycles

We list here for conveuience the basic formulae of Majid's bicrossproduct and cocycle bicrossproduet constructions and refer to [i 1,12] (see also [14]) for details.

22

.l.A. de Azcarraga and J.e. Perez Bueno

TIlE' expressions which characterize 'H~ A (used in Secs. 3,1) involve the mappings a: A®'H -> A (right 'H-module action), (3: 'H -> A®'H (left A-comodule coaction), 'H Q9 'H -> A (two-cocycle) and 1/J : 'H ---+ A Q9 A (hence the more detailed notation 'H{3''''~Q,~ A, see [11]). Those of the dual case ( 1I",( ~t3,,z; A when all ingredients are indicated) involve the respective dual operations; they were used in Sec. ,J. We may think of 'Hr>~ A as emphasizing the 'algebra-like' aspects and of IJ .<4 A may be found in the original papers [11,12] (or in the Appendix of [29] with the same numbers they are referred to in the main text, also corresponding to the dual formulae for IJ ~A below). Let H and A be Hopf algebras and let

e:

e,

a) H be a left A-module algebra (H >:JA) b) A be a right H-comodule coalgebra (H lOC A) i.e., there exist linear mappings

a:AC5iH---+H

® h)

o:(a j3(a)

== at>h

(WI)

a E A, hE H

= a(1) 0 a(2)

A,

a(1) E

a(2) E

IJ

(B'.2)

such that the properties of al) 0: being a left A-module action i>:

(B',3) a't>(a~h) = a'ai>h

(B'A)

a2) H being a left A-module algebra:

(B'.5) bl)

13

being a right IJ -comodule coaction: a(1)

a(l)(l)

\1 fll(a(2)) = a ® IH

== a [(id 0 f)

\1 a(I)(2) (1 a(2) = a(l) \1 a(U) 0 a%

[((3 \1 id)

013

013=

=

id]

(id ® L1)

(B'.6) 0

13]

(B'.7)

13We use the bicrassproduct notation ~ or ~ rather tban the (right, left) crassproduct (e><:, >:I) or the (left, right) cross coproduct ()oIl,~) even if the caactians /3, /3 or the actions n, ii are trivial, and omit explicit reference to them (or to {, '" etc.)

Contractions, Hopf Algebra Extensions

23

b2) A being a right II-comodule coalgebra:

(B'.8) a [(,1 09

id)

0

rj

(I) (I)

.

(I)" (2)::>9

0) a

= (id ® id (2)

(2)

a

(I)

(1)

(2)

(2)

= a(1) (9 a(2) ® a(l) a(2)

mJl) 0

(id

® id)

({3 ® (3)

,1

== (l3c29{3)

,1] ,

(9

T

T

is the twist mapping, are fulfilled.

0

0

0

(8'.9) where l/IH is the lllultiplication in II and Then, if the compatibility conditions

(WID) ,1(ai>h) == (ai>h)(I) <X: (a[>h)(2) = (a(l;li>h(1))

is)

a(W(a(2)i>h(2»)

ij(lA) == I~) is: I~) = I A

is)

III

(J(ab) == (ab)(1)

(9

(ab)(2) = adl)b(1)

is)

,

(8'.11 ) (8'.12)

,

a(g)(a(2)i>b(2))

a(g) 0 (a(1)i>h)a(g) = a(g) 0 a(g)(a(2)t>h)

,

,

(8'.13) (13'.14)

r

are satisfied l 4, there is a Hopf algebra structure on [11 f{ == H Q9 A called the (left-right) bicrossproduct III'> ~IJ A (II ~A for short) defined by

(h®a)(g(\\:b)=h(a(1)[>g)®a(2)b, ,1/.,:(h Q9 a) qt·

h,gEII;a,bEA

= h(l) Q9 a(g) ® h(2)a(g) is) a(2)

= (Jl is) fA

,

1/\

= If{ Q9 I A

S(h 0 a) = (If{ @SA(a(1)))(Sll(ha(2))

IA)

(13'.15) (B'.16)

,

(8'.17)

,

(9

,

.

(13'.18)

In f( = II ® A, h == h IX: I A and a == If{ Q9 a; thus, ah = a(l)i>h @ a(2). There are two cases of special interest [15] (see also [II]). When i3 = I Q9 If{ i.e. (J(a) = a Q9 I Jl (trivial coaction) and A is cocommutative, J( is the scmidirect product of Hopf algcbras since then ,1n(h 0 a) = (a(1) 0 h(1)) Q9 (b(2) 0 g(2»)' When 0' is trivial, 0: = (A @ IJl (a[>h = hfA(a)) and H is commutative, J( is the semidircct coproduct of lIopf algebras since (h ® a)(g Q9 b) = hg 0 abo When 0: is trivial, (3(ab) = i3(a)(J(b) (algebra homomorphism). As for Hr>.ttl A, the above construction may be extcnded to accommodatc cocycles [11,12]. Let A and H two lIopf algebras and 0- and (J as in (B'.I), (B'.2). Then H is a left A-module cocycle algebra if (B'.3), (B'.5) are fulfilled and thcre is a linear (two-cocycle) map ~ : A ® A -+ f{ such that

(13'.19) 141f A is cocommutative and II commutative, condition (B'.14) is automatically satisfied.

J.A. de Azca.rraga and J.e. Percz Bueno

24

a(1)t>'(b(1) Q0 c(1))'(a(2) 181 b(2)c(2)) = ~(a(1) 181 b(1))'(a(2)b(2) 181 c) , Va, b, c E A, (B'.20) (cocycle condition) and (B'.4) is replaccd by

which for' trivial rcproduces (B'.4). Similarly, A is a right lJ-comodule coalgebra cocyclc if (B' .6), (B' .8), (B' .9) are fulfilled, and there is a linear map 1/J : A -+ H 181 H, 7/J(a) = ~(a)(1) 0 7jJ(a)(2), such that

t(1/J(a)(1))~(a)(2) = 1t(a)

= ~(a)(I)t(1/J(a)(2)),

.::1~(a(I))(1){>(a(W) Q9 ~(a(1))(2)a(g)

[(( 181 id)

0

~

= (id 181 () o~] ,

= ~(a(1))(1) Q9 .::11/J(a(1))(2)~(a(2))'

(B'.22) Va E A , (B'.23)

(dual cocycle condition) and (B'.7) is replaced by (1

Q9

{>( a(I)))( (,8 Q0 id)

0

;1( a(2»))

= a(g) Q9 .::1a(g)~( a(2))

(B'.24) = ((id 181 .::1),8(a(I)))(l Q9l}(a(2)))

Then, if the compatibility conditions (B'.10), (B'.12), (B'.14) and .::1(a(1)c>h){>(a(2)) = ~(a(1))[a(g)i>h(1) Q9 a(~~)(a(3)i>h(2))1 (1181 ~(a(1) 09 b(1))),8(a(2)b(2))

(B'.25)

,

= a(i;lb(g) 181 a(g\a(2)i>b(g)),(a(3) Q9 b(2))

,

(B'.26)

(which replace (B'.11)15 (H'.13)), together with .::1'(a(1) ® b(I))lNa(2)b(2)) = 1/J(a(l)) [(a(~;)i>1/J(b(1))(1))~(a(~;) 181 b(~;))181 a(W( a(3)i>1/J( b(1))(2))a(~~\ a(5)i>b(W)'( a(6) Q9 b(3»)] , (B'.27) tWa 1& b)) = ((a)t(b) , ~(1A) = III 1&111 (B'.28) hold, (A, H, ix,,8, " ~) determine a cocycle left-right bicrossproduct bialgebra Hi. ~<J,J; A. In it, the C0l11lit and unit are defincd by (B'.17) and the product and coproduct (B'.15), (H'.16) arc replaced by (h 00 a)(g ® b) = h(a(1)t>g)'(a(2) ®

b(I)) Q9

u(3)b(2)

.::1(h ® a) = h(l)1/J(a(l))(l) 181 a(g) Q9 h(2)~(a(1))(2)a(g) 181 a(3) 15With ~(u(l)

@

b(l))C 1 (U(2)

(8:

(B'.29)

, .

b(2)) = {(a){(b) (convolution invertible [11]), eq.

(B'.30) (B.26) gives

l3(ab) = a(Wb(W ® C 1 (a(l) (8: b(l»)a(g)(a(J)t>b(~~)){(a(4) @ b(J»). If A is Abelian, as is always the case in the cocycle bicrossproduct structures in the main text, this formula reduces to (B'.13).

Contractions, Hopf Algebra Extensions

For

25

It is convenient to have the explicit expression of (8'.27) in the more simple cases. 1f; trivial it reads

"b) d~ ( a ®

:=

(2») -( ~t( a p(1) ) Q9 b(I)(I») Q9 ap(2)( ) a(2)l>b(l) ~ a(3) Q9 b) (2)

(B'.31)

For t> trivial, it gives

.I.( )[.I.(b )(l)i( (I) b (I») .I'(b )(2) a(2) (2)b (2) (2)(( b) 'f/ ap) 'f/ (I) ~ a(2) ® (2) Q9 'f/ (1) ~ a(3) 181 (3)

(B'.32) For

1f;

and t> trivial, it reduces to -(

""(

(I)

"

d~ a 181 b) = ~ a(l) Q9

b(I)(I») ® a (2)) b(l)(2)"~ ( a(2) iI9 b(2) ) p

(B'.:J3)

For (trivial [((a ® b) = c(a)t(b)IHl (B'.21) reduces to (B'.4), (B'.26) to (B'.13) and (13'.29) to (B'.15). For 1f; trivial [1f;(a) = IH il91l/c(a)], (B'.24) reduces to (B'.7), (B'.25) to (B'.II) and (B'.30) to (B'.16). For (3(a):= a 181 I trivial, (B'.30) gives for the elements of A with original primitive coproduct the cocycle extension expression

d(l Q9 a) = I ® a ® I iI9 I + I 181 11811181 a + ~(a)(I) Q911811f;(a)(2) 181 1

(B'.34)

which in f{ simply reads d(a) := 1181 a + a 181 1 + 1f;(a). This was used for (5.7) [(5.6), (5.8) [(5.10)] and (5.11) [(5.12)].

REFERENCES [1] E. iniinii and E.P. Wigner, Proc. Nat. Acad. Sci. 39 (1953) 510. [2] E. iniinii, "Contractions of Lie groups and their representations" in Group theor. concepts in elem. part. physics, F. Giirsey cd., Gordon and Breach 1964, p. 391. [:1] V. Aldaya and .LA. de Azcarraga, Int. J. of Theor. Phys. 24 (1985) 141.

[4] J. A. de Azcarraga and J. M. Izquierdo, Lie algebras, Lie groups cohomology and some applications in physics, Camb. Univ. Press 1995. [5] E. J. Saletan J. Math. Phys. 2 (1961) 1. [6J V. G. Drinfel'd. in Pmc. of the 1986 Int. Congr. of Math., MSRI Berkeley, vol I, 798 (1987) (A. Gleason, ed.). [7] M. .limbo Lett. Math. Phys. 10 (1985) 63; ibid 11 (1986) 247. [8] 1"D. Faddeev, N. Yu. Reshetikhin and 1. A. Takhtajan, Alg. i Anal. 1 (1989) 178 (Leningrad Math. J. 1 (1990) 193). [9J E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini J. Math. Phys. 31 (1990) 2548; ibid 32 (1991) 1155, 1159.

26

.LA. de Azcarraga and J.e. Perez Bueno

[10J E. Celeghini, R. Giachetti, E. Sorace, and M. Tarlini, "Contractions of quantmn groups", in Lee. Notes Math. 1510, (1992) p. 221. [IIJ S. Majid J. Algebm130 (1990) 17; Israel J. Math. 72 (1990) 133. [12J S. Majid and Va. S. Soibelman J. Algebra 163 (1994) 68. [13J W. Singer J. Algebra 21 (1972) I. [14] R. J. Blattner, M. Cohen and S. Montgomery, Trans. Am. Math. Society 298 (1986) 671; R. J. Blattner and S. Montgomery, Pac. J. Math. 137 (1989):n. [15] R. Molnar J. Algebra 47 (1977) 29. [16] J. Lukierski, A. Nowicki, H. Ruegg and V.~. Tolstoy Phys. Lett. B264 (1991) 331; J. Lukierski, H. Ruegg, and V.N. Tolstoy, "K-quantum Poincare 1994", in Quantum groups: formalism and applications, J. Lukierski, Z. Popowicz and J. Sobczyk eds, PWN, Warsaw 1994, p. 3.19. [17] S. Majid and 11. Ruegg Phys. Lett. B334 (1994) 348. [18] S. L. Woronowicz ComrmlT!. Math. Phys. 149 (1992) 637. [19] P. Schupp, P. Watts and B. Zumino Lett. Math. Phys. 24 (1992) 141. [20J A. Ballesteros, E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini J. Phys. A26 (199:3) 7495. [21] A. Ballesteros, E. Celeghini, F. .I. Herranz, M. A. del Olmo and M. Santander J. Phys. A27 (1994) L369. [22] N. A. Gromov and V. I. Man'ko J. Math. Phys. 33 (1992) 1374. [23] D. Ellinas and J. Sobczyk J. Math. Phys. 36 (1995) 1404. [24J W. K. Baskerville and S. Majid J. Math. Phys. 34 (1993) 3588. [25J F. Bonechi, R. Giachetti, E. Sorace and M. Tarlini Commun. Math. Phys. 169 (199.1) 627. [26] L. L. Vaksman and 1,. I. Korogodskii, Sov. Math. Dokl. 39 (l9R9) 173. [27J J. A. de Azcarraga, M. del Olmo, J. C. Perez Bueno, in preparation. [2SJ A. Sitarz Phys. Lett. B349 (1995) 42. [29] J. A. de Azcarraga and J. C. Perez Bueno J. Math. Phys. 36 (1995) 6879. [30J S. Giller, P. Kosinski, M. Majewski, P. Maslanka and J. Kunz Phys. Lett. B286 (1992) 57. [31] A. J. Macfarlane J. Phy". A22 (1989) 4581.

Contractions, Hopf Algebra Extensions [:J2] L. C. Biedenharn J. Phys. A22 (1989) L873. [3:J] M. Arik and D. D. Coon J. Math. Phys. 17 (1976) 524. [:J4] P. P. Kulish, Theor. Math. Phys. 86 (1991) 108. [35] C. Il. Oh and K. Singh J. Phys. A27 (1994) 5907. [36J C. Quesne and]\,. Vansteenkiste, q-algj9510001. [:J7] P. Maslanka J. Math. Phys. 35 (1994) 76. [38] J. Sobczyk, Czech. J. Phys. 46 (1996) 265. [:J9] V. Hussin, A. Lauzon and G. Rideau Lett. Math. Phys. 31 (1994) 159. [40] S. L. Woronowicz Commun. Math. Phys. 122 (1989) 125.

27

From Fidd TllCory to Quantum Groups

THE LINEAR DIFFERENCE DERIVATIVES AND SOME Q-SPECIAL FUNCTIONS MALGORZATA KLIMEK

Institute of Mathematics and Computer Science, Technical University of CZljstochowa ul. Di}browskiego 73, 42-200 CZljstochowa, Poland E-mail; klimek at matinf.pcz.czest.pl

Abstract: We introduce the simplest generalization of difference and q-derivatives the linear difference derivatives and investigate their properties. They are used in explicit construction of Q-deformed independent oscillators. The applications to deformation of quantum oscillator, Legendre and Hermite equations are studied and the eigenproblem for discrete part of their spectra is solved.

1.

Introduction

The q-special functions and their application have been widely studied in the literature (see for example [1,2]). In the paper we propose the generalization of the q-derivatives and difference derivatives, used before in deformation of classical problems. We shall show that Q-commutation relation can be realized by means of the operator which we call the linear difference derivative. The creation and annihilation operators, constructed using this derivative, will then be applied to construction of the set of independent Q-oscillators. In the section 3 we consider also some classical equations deformed using the linear difference derivative. The preliminary results include the eigenstates of deformed quantum oscillator and polynomial solutions of the Legendre and Hermite equations, connected with the discrete part of their spectrum. The interesting feature of this deformation of quantum oscillator is the finitness of dicrete spectrum (33,40). The number of possible eigenvalues depends on parameters, describing the initial linear difference derivative used in the deformation.

30

2.

Malgorzata Klimek

Generalized Difference Derivatives and Their Properties

Let us remind the non-symmetric generalized difference derivative acting as follows on the real functions of real variable [3,4]:

(I) where 1; denotes the transformation of the set of real numbers 1; : JR -> JR. In the paper, we shall also consider the symmetric case given by the following formula

In the above definitions, we have used the transformation operator ( which acts on functions in the following way:

C f(x)

(+ f(x) = f(¢(x)),

= f(1;-I(X)) ,

Acting with the (-operator on the product of functions we obtain:

([J. g(x)] = U(x). (g(x). Finally, the important feature of the calculus of new derivatives is the fact that they do not commute with the transformation operator

(a~f(x) = cP2~~~)_~~x)a~u(x),

(3)

([)~J(x)

(4 )

=

cP(X~(-t-I(X)[)~U(X). 1;

X

-

X

Let us notice that when we choose ¢ as the translation (J(x) = J(x the known difference derivatives: I

fhJ(x) = Tz[J(x

+ h) -

J(x)],

I

DhJ(x) = 2h [J(x

+ Il)

+ h)

we obtain

- f(x - h)]

(5)

and far the transformation of dilatation U(x) = f(qx) we end up with q-derivatives

a J(x) = q

J(qx) - J(x) , (q - I)x

[) J(x) = J(qx) - J(q-I X) . q (q_q-l)X

(6)

Both types of derivatives are known from the theory of iterative functional equations and are also used in realizations of the quantum algebras generators [5-8]. We propose the generalization of the operators given by (5,6) (first introduced [3,4]) and we would like to present in the paper its application to deformation of

The Linear Difference Derivatives and Some Q-Special Functions

31

quantum oscillator and to construction of certain Q-deformed special functions. First we derive the realization of the Q-dcformed commutation relations. It is clear that not all transformations ¢ yield the algebraic relations of this type. We will show, however, that we need not restrict the calculus to the known difference and q-derivatives. We start from the Leibniz rule for genemlized difference derivatives: (7)

Comparing with the Leibniz rule for differential derivative, we notice the modification that the transformation operators appear on the right-hand side of both formulae. This is transferred onto the commutation rule for the derivatives and coordinate variables which we obtain from the formulae (7) taking f( x) = x and 9 an arbitrary function (8) The commutation relations can be reformulated using the properties of the transformation operators in order to compare them with the standard formula for the coordinate- and momentum operator from quantum mechanic:

(9) As we see on the left-hand side, we have now the isotopic commutation relations [9] for the redefined coordinate operator (+x, but on the right-hand side the isotopic identity (in this case the (+ -operator) does not appear. Thus the relations need to be transformed in order to obtain closed Q-isotopic commutators. This procedure can be realiwd only when the transformation operator and the generalized difference derivatives Q-commute (3,1). This condition implies the following iterative functional equations for the transformation ¢:

a",¢(x)

= const = Q

fJ",¢J = const = Q

(10)

It is easy to check that apart from dilatation and translation transformations also the linear transformation fulfills the above equation. We shall investigate from this point the properties and applications of this simplest generalization of the operators used in the literature. We call them lincur difference derivatives. They commute with the linear transformation operator in the following way:

where the ( operators describe now the linear transformation

U(x) = f(Qx

+ h)

Malgorzata Klimek

32

Taking this into account we obtain after redefinition of the momentum and coordinate operators :

(i) the isotopic Q-commutation formula for the non-symmetric case [10]: PTX - QXTP =

r- 1

(11)

with the momentum, coordinate and isotopic operators looking as follows:

T = (-8-t where we used the non-symmetric linear difference derivative which acts on functions according to general formula (I):

fjJ( )=J(Qx+h)-J(x). q, x (Q-I)x+h' (ii) the isotopic Q2-commutation relation for the symmetric case: (12)

with the following realization of the momentum, coordinate and isotopic operators

t

= (-8-t+1

where the symmetric linear difference derivative acts following the formula (2):

fJ J( ) = J(Qx + h) - J(Q-I X 1>

x

- Q-Ih) (Q_Q-I)X+(I_Q-I)h

We see that in both cases we deal with a two-parametric realization depending on the integer numbers.'> and t.

In order to restrict these algebras to the standard (that means non-isotopic) relations we choose correspondingly: for the non-symmetric case for the symmetric case

s+ t

= 0 => T = 1

s + t - 1 = 0 => T

=1

(13)

(14)

When these conditions are fulfilled we obtain the Q- and Q2-dcformed commutation relations: (15) PX-QXP=I,

Px - Q2 XP = 1

( 16)

The Linear Difference Derivatives and Some Q-Special Functions

33

with the redefined momenta and coordinate-operators looking as follows: X

= (s+JxC s =

X=

("+I X

(Qs+l x +[s+I]Qh)(+,

c s+1 =

(QS+1 X

+ [s + I]Qh) (2

(17) (18)

where

In both cases the momenta are represented by the linear difference derivatives multiplied with transformation operator. The representation of the coordinate operator depends strongly on the choice of the free integer parameter s. We obtain the simplest formulae for s = 0:

3.

Applications

3.1.

Linear Difference Derivative and Independent Deformed Oscillators

The proposed realization of the deformed commutation relations can be easily extended to the multidimensional case, where the momenta are realized using the partial linear difference derivatives and partial transformation operators acting on functions of many variables:

(19)

od(x) := [((tx;) - xit 1[(t - IlJ(x) ,

(20)

(jd(x) := [((tx;) - (C X i)tl[(t - C]f(x).

(21 )

When we take in all these partial transformation operators the same parameters

QJ = Q2 = ... = Q" we obtain two sets of commutation relations (22) (23) which are known from the model of independent deformed oscillators [5,6, II] used in the realizations of generators of quantum algebras as functions of oscillators.

3.2.

The Deformation of Quantum Oscillator

Let us consider the following iterative functional equation (24)

34

Malgorz ata Klimek

with the momen tum and coordin ate operato rs realized accordin g to formula (17). To calculat e the spectru m En we apply the techniq ue known in the theory of differential equatio ns as models with shape-in variant potentia ls [12]. It was generali zed to Hamilto nians which allow decomp osition into creation and annihila tion operato rs dependi ng on parame ters [3,13-15]. In our case it is enough to use one-par ameter decomp osition which looks as follows:

(2.5) with the explicit formulae for the creation and annihila tion operato rs

(26) It is easy to check that the following commu tation relation is fulfilled (27) where the function F depends on the transfor med parame ter A 2

A'

F(A' ) = ~+­ N Q'

A'

= f(A) = QA.

It was proven in the literatu re that when the formula (27) is valid the spectru m of the Hamilto nian At(A)A(A) can be constru cted using iteration s of transfor mation of parame ter A At(A)A( A)P n = unP n n

Un

=

L: F[f(A)] = 1=]

w

(28)

1

± m( -[n]l + Q[n]Q) v"'t Q Q

where the initial value of A = ±Qf w is determi ned by conditio n (25). The eigensta tes P n are constru cted using the creation operato rs At

(29) provide d that the n-th vaccum

cPn fulfilling the equatio n of first order

(30) docs exist. To obtain this function we use the Q-comm utation relation for monom ials:

Applyin g the above relation , we can constru ct the n-th vaccum as the Q-expo nential function :

The Linear Difference Derivatives and Some Q-Special Functions

35

where I 0 > is arbitrary constant function and [2k]Q!! := [2]Q[4]Q ... [2k]Q. This construction is correct only when the series representing the function is convergent. Using the d'Alembert criterion we conclude that this is the case for

Q< I

(33)

Let us notice that the number of eigentates depends strongly on the value of characteristic parameters defining the creation and annihilation operators (17,26). For h =J 0 we obtain finite number of possible vaccums what results in finite number of eigenstates. Putting h = 0 we deal with standard Q-dcformation of quantum oscillator with infinite number of eigenstates. When both the above conditions (3:3) are fulfilled the function given by (32) is defined properly for arbitrary x E R. Like in the classical model we choose the branch of solution with the" -" sign (corresponding to the initial value of A = -w,fiJ) to obtain the function with proper asymptotic behaviour. The result of our construction is the following spectrum and eigenstates of deformed quantum oscillator constructed using the linear difference derivative: 2 2 W 1[ 1 [ ]Q (P 2 -w,l' )<1>n=- ,fiJ((jn&+Qn

<1>11 =

-

Q )<1>n,

At(A)At(f(A)) ... At(fn-l(A))cPn.

In this case the deformation consists of the modification of spectrum as well as of the entirely different construction of eigenstates. The transition from one-vaccum to many-vacua eigenstates is characteristic feature of Hamiltonians with splitting (25) and commutation relation (27) depending on transforming parameters. All the presented formulae are valid also for deformation via symmetric linear difference derivative. Let us report briefly the results. As before, the Hamiltonian is splitted into product of annihilation and creation operators depending on parameter:

p2 _ W

At (A)

2

X 2 = At(A)A(A)

= P + AX ,

+ const, w2

..

A(A)

=P

-

(35) _

TX

.

The commutation relation for these new operators is similar to the previous one (27) with changed transformation of parameter A

(36) where

w2

F(A')

,\'

= l' + Q2 '

36

Malgorzata Klimek

It implies the following spectrum and eigenstates of Hamiltonian (:JS)

(37)

The n-th vaccum,
Applying the d'Alembert criterion to these function we conclude that it is correctly defined for x E R when the corresponding conditions are fulfilled:

Q < 1,

(40)

As before, these conditions yield the finite number of eigenstates for h for h = O.

3.3.

#- 0 and infinite

The Deformed Legendre and Hermite Equations

We shall study two examples of deformation of second-order differential equations with polynomial solutions [16]. We replace the differential derivative in the operator of equation with the operator P as well as the standard variable with X given by (17). We start with the Legendre equation:

(41 ) The problem of finding the spectrum Un can be solved by the technique applied in the previous section. In this case, however, we would deal with many-parameter decomposition (2.5). The simpler way is to follow the analogy with the case of differential equation and assume that the solution is the polynomial of n-th order: n

L" =

L

ak,yk

I0 >

(42)

k=O

where the monomials X k I 0

X k I 0 >= (Qs+1 X

> look as follows

+ [s + l]h)(Qs+2 x + [5 + 2]h) ... (Qs+k x + [s + k]h) 10>

(43)

The Linear Difference Derivatives and Some Q-Special Functions

37

Throughout this section we shall denote [n]Q = [n]. In the calculation, we use the commutation relation between the operator of Legendre equation and the polynomial:

2 [P(l - X )P,

n

n-2

n

k~O

k=O

k=O

'E akXkj = 'E ak+2[k + 2][k + l]X k - 'E ak[k][k + I]X k .

(44)

It yields the following conditions for coefficients of polynomial L n :

[k][k + I] - (1n ak+2 = ak [k + 2][k + I]

0:S;k:S;n-2,

(45)

an-l ([n][n - I] - (1n) = 0, a,,([n][n

+ I]

- (1n) = O.

Solving this set of equations we obtain the explicit expression for eigenvalues

(1n = [n][n

+ I]

(16)

which are connected with the polynomials determined by coefficients given by the formulae: for n even:

ak+2

= ao

([k

+ I][k]- (1n)([k -

l][k - 2]- (1n)"'([O] - (1n) [k + 2]!

(47)

where k is an even integer number fulfilling 0 :S; k :S; n - 2; -- for n odd

ak+2

= al

([k

+ l][k] -

(1n)([k - l][k - 2]- (1n) ...([2] - (1n) [k + 2]!

(48)

with k an odd integer number fulfilling 0 :S; k :S; n - 2. Thus the deformation of Legendre equation constructed using linear difference derivative looks as follows:

(49) with the following explicit form of lowest order polynomial solutions:

Lo = ao,

L 2 = ao(1 - [3]X 2 ) I 0 > , L. = a (1 3

1

+ [2] -

[3][1]X 3 ) 1 0 > [2][3]

The next example we study is the Hermite equation written down using the linear difference derivative P and new variable X given by (17 )

(50)

Malgorzata Klimek

38

The assumption that the solution is the polynomial of n-th order yields now also the equations for its coefficients

ak+2 = ak[k

2[k]- 17n + 2][k + 1]'

an _l(2[n - 1]-

17 n )

a n (2[n] -

0~k~n-2,

(51 )

= 0,

17n )

= 0

which imply the explicit formula for eigenvalues 17 n

(52)

= 2[n]

as well as expressions for coefficients of polynomials H n for n even (2[k]- 17 n )(2[k - 2] - 17n ) ... (2[0]ak+2 = ao [k + 2]! where k is an even integer number fulfilling 0 ~ k ~ n - 2; - for n odd (2[k] - 17n )(2[k - 2]- 17n ) ... (2[1]ak+2 = aj [k + 2]!

17n )

17n )

(53)

(54)

with k an odd integer number fulfilling 0 ~ k ~ n - 2 . The polynomials determined by the above equalities solve the eigenproblem for the deformed Hermite equation: (P 2

_

2XP

+ 2[n])I1n

~ 0

(55)

with the lowest order solutions of the form:

Ho = ao,

Let us notice that the calculations for the deformation with symmetric derivative are very similar, so we have omitted them. is the orthogonality of solutions is the open problem. We should construct the scalar product and check the properties of solutions. As the definite integral for generalized difference is known [3,4], we expect to be able to use it in the construction for redefined variables X. 4.

Final remarks

We have derived in the paper the deformed commutation relations for linear difference derivatives and redefined variables using the properties of generalized difference

The Linear Difference Derivatives and Some Q-Special Functions

39

deri vat ives. It appears that the linear difference derivative is the simplest generalization of the difference allCl q-derivatives which allows us to realize the set of deformed independent oscillators. In this context it would be interesting to deform field-theoretic equations using linear difference derivative and construct the Hopf algebras of their symmetry operators in the way it was done for q-derivatives [17,18] and for difference derivatives [7,8,19] . We applied the linear difference derivatives to deformations of quantum oscillator as well as to the Legendre and Hermite equations. Using different techniques we have constructed the discrete part of their spectra and eigenfunctions. It seems that the introduced derivatives would be also useful in deformed realization of the general hypergeometric equation. This problem as well as the question of orthogonality of solutions needs further investigations.

REFERENCES [I] H. Exton, q-J/ypergeometric Functions and Applications Ellis Horwood, Chichester, 19H:l. [2J R. Askey, S.K. Suslov, Lett. Math. Phys. 29 (199:l) 12:l. and references therein. [:lJ M. Klimek, Noncanonical Extensions of Dynamical Systems Ph.D. Thesis, University of Wrodaw, 1992 (in Polish). [4] M. Klimek, J. Phys. A: Math. Gen. 26 (199:l) 955. [5] A. MacFarlane, J. Phys. A: Math. Gen. 22 (1989) 4581. [6] L.C. Biedenharn, J. Phys. A: Math. Gen. 22 (1989) L87:l. [7J .J. Lukierski, A.

~owicki,

H. Ruegg, Phys. Lett. B293 (1992) :l44.

[8J J. Lukierski, II. Ruegg, W. Riihl, Phys. Lett. B313 (1993) 357. [9J R.M. Santilli, Hadr. J. 1 (1978) 223. [10] M. Klimek, in Proceedings of the XXX Winter School of Physics: Quantum Groups Formalism and Applications, ed. J. Lukierski, J. Sobczyk, Z. Popowicz, Polish Scientific Publishers PWN, Warszawa, 1995. [IIJ T. Hayashi, Commun. Math. Phys. 127 (1990) 129. [12J D.L. de Lange, A. Welter, J. Phys. A: Math. Gen. 25 (1992) 5753 and references therein. [13] L.E. Gendenshtein, JETP Lett. 38 (1983),299

40

Malgorzata Klimek

[14] A. Arai, "Exatly Solvable Supersymmetric Quantum Mechanics", Hokkaido University Preprint in Mathematics 62/1989 [15] M.Klimek, J. Phys. A: Math. Gen. 25 (1992) Lll. [16] G.M. Murphy, Ordinary Differential Equations and Their Solutions, D. Van Nostrand Company, New York 1960. [17] R. Floreanini, L. Vinet, Quantum Symmetries of q-DifJerence Equations, Preprint Universite de Montreal, CRM-2167/1994. [18] R. Floreanini, L. Vinet, Lett. Mat. Phys. 32 (1994) 37. [19] A. Nowicki, E. Sarace, M. Tarlini, Phys. Lett. B302 (1993) 419.

From Field Theory to Quantum Groups

THE K-WEYL GROUP AND ITS ALGEBRA

Department of Theoretical Physics University of Ladt ul. Pomorska 149/159, 90-236 Mdt, Poland

Department of Functional Analysis University of Ladt ul. Ranacha 22, 90-238 Ladt, Poland

Abstract: The K-Poincare group and its algebra in an arbitrary basis are constructed. The K-deformation of the Weyl group and its algebra in any dimensions and in the reference frame in which 900 = 0 are discussed.

1.

Introduction

It is our great pleasure to contribute to this volume. In last years we had an opportunity to collaborate with Prof. Jerzy Lukierski. Our common topic has been the deformed symmetries of space-time, mainly the so called K-Poincare algebra invented by Lukierski, Nowicki and Ruegg [1]. Apart from investigating the formal properties of K-Poincare algebra and looking for its possible physical applications one of the main ideas of Prof. Lukierski is to extend the notion of K-deformation to larger groups of space-time symmetries. This idea resulted in series of papers [2-6] devoted to the K-deformation of SUSY extensions of the Poincare symmetry. The next step to be done is to look for K-deformed conformal group / algebra. This problem has not been fully solved yet but some preliminary steps were already undertaken [7,8]. Inspired by these papers and numerous discussions with Prof. Lukierski we attempt here to make it small step toward the solution of this problem. lSupported by KBN grant 2P30221706p02

42

P. Kosinski and P. MaSlanka

Classically, the conformal group in four dimensions is nothing but SO( 4,2). However, the standard (matrix) parametrization of S'O( 4,2) is not used, when SO( 4,2) is viewed as conformal group. On the contrary, the conformal group is obtained from the action of SO( 4,2) on light cone in sixdimensional space-time. But the light-cone coordinates are related in rather complicated way to Minkowski coordinates in four dimensions. Consequently, the standard parametrization of SO(4,2) is related to the "conformal" one by a complicated (even somewhere singular) change of group parameters. This poses no problem on the "classical" level. However, if we are passing to the "quantum" (i.e. deformed) case we are faced with typical ordering problems of quantization procedure. This gives some flavor of difficulties one meets trying to deform the conformal group. In the recent paper [8], Lukierski, Minnaert and Mozrzymas considered a new class of classical r-matrices on conformal algebras in three and four dimensions, which obey the classical Yang-Baxter equation and depend on dimensional parameter. An important observation concerning the d = 4 case was that the classical r-matrices obtained by them depend only on generators belonging to Poincare subalgebra of conformal algebra. Due 1.0 the fact that they obey the classical Yang-Baxter equation (and not modified one) they provide r-matrices for any algebra, containing Poincare algebra as subalgebra. One of the r-matrices considered in [8] leads to the so called [lIlll-plane deformation of Poincare algebra found, by different methods, in Ref. [9]. This deformation is similar to the standard K-deformation. The only difference is in the choice of undeformed subalgebra which is the stability algebra of light-like fourvector instead of time-like one. However, this difference is significant: in the standard case the Schouten bracket is ad-invariant but does not vanish. Therefore, the relevant r'-matrix does not provide automatically the r-matrix for any extension of Poincare algebra. Actually, the invariance is broken already after adjoining the dilatation generator D. Our aim here is to put the results of Ref. [8J in more general setting. In Sec. 2. we review the properties of Poincare group for arbitrary chosen metric and discuss the Poisson structure on it. In Sec. 3. the quantization of this classical structure is performed. The bicrossproduct form of resulting quantum group allows us to find, by duality, the relevant algebra. The Weyl group and its algebra are constructed in Sec. 4.. Finally, Sec. 5. is devoted to some conclusions.

2.

Relativity theory in an arbitrary coordinate system

Let us consider the n-dimensional linear metric space M with metric tensor 9 v (1"v=O,I, ... ,n-I) given by an arbitrary non degenerate symmetric n x n matrix (not " necessary diagonal). Poincare group P is the group of inhomogeneous transformations of the space M:

The

43

Weyl group and its algebra

K-

where the matrices A"v (Lorentz group) satisfy the condition:

It is easy to see that the Poincare algebra

P reads:

0,

[P",Pv] [M"v,P,,] [M"v,M",,]

i(9v"P" - 9,,"Pv) , i(9""M v" - 9v"M""

+ 9v"M"" -

9""M v,,)

where

(M",{3)"v = i(li"",9v{3 -li"{39v",). Now consider r E

1\ 2 P given as follows, [12]: . _

l

l--Mov AP

II _

-r

K

1J.1I,O

M"v

A

P",

(2.1)

whcre

r"v,,,, = ..!-(li"o9V '" -li vo9"C»

2K

and

K is a real deformation parameter. A calculation of Schoutcn bracket of I' with itself yields

[ r,

_

r] -

1900 - 2 M",{3

'"

{3

A PAP ,

(2.2)

K

It is not difficult to see that [I', r] is invariant, hence r is defines a structure of a Poisson Lie group on P, by the formula:

(2.3) where Xl;, XI~ arc the right- and left-invariant vector fields. It is easy to find thc following exprcssions for the invariant vcctor fields:

XC> L

X'" R

P. Kosinski and P. MaSlanka

44

In the Lie algebra basis corresponding to the above vector fields we have the following relation between the generators of the Poincare algebra and the invariant vector fields:

This enables us to calculate the Poisson brackets of the coordinate functions P:

{aU, a"}

-~((A"o K

- 8"o)A U,IJ

~(8Uoa" -

8"0 aU)'

K

{A",IJ' 3.

J1i'J

+ (Ao,IJ -

gO,IJ)g"U) ,

O.

(2.4)

The K-Poincare group and K-Poincare algebra in an arbitrary basis

If we perform the standard quantizations of the Poisson brackets of the coordinate functions on P by replacing { , } ---+ 1obtain the following set of commutation relations:

H'

-~((A"o - 8"o)A U,IJ + (Ao,IJ - gO,IJ)g"U) , K

[aU, a"]

~(8Uoa" - 8"0 aU)' K

[A"{J,Al'vJ

O.

(3.1)

This standard quantization procedure is unambiguous: there is no ordering ambiguity when quantizing the right-hand side of Eg.(2.4) due to the commutativity of A's. Since the composition law is compatible with Poisson brackets, the above commutation rules are compatible with the following coproduct:

AI'" 0A"v' Al'v (9 aV + al'

(9

I.

(3.2)

The antipode and the cOlltlit are given by:

8(Al'vl S(al')

c:(Al'vl c:(al')

o.

(3.3)

If we define the *-operation in such a way that Al'v and al' are selfadjoint clements, we conclude that the relations Eq. (3.1), (3.2), (3.3) define a Hopf *-algebra - the

The

K-

Weyl group and its algebra

45

K-Poincare group Pt<. It follows frolIl the Eg. (3.1), (3.2), (3.3) that the form of the K-Poincare group does not depend on the choice of the metric tensor g"v. The differences between the various K-Poincare groups are related to the fact that AOfi appearing in the first commutation relation of Eg. (3.1) are not the independent variables but are the linear combinations of the independent ones: AO/l = 90"A"/I' It should be stressed that the K-Poincare group can be defined as a right-left bicrossproduct [13,14]: P" = T*t><4C(L). To see this it is sufficient to define the structure maps:

A"v

(X) XV ,

-':((A"o - 8"0)Ae v + (A ov - gov)g"e). K ~oreover, while C( L) is the standard algebra of function defined over Lorentz group, T* is dcfined by the following relations:

[a",a V ] .:1 (a")

S(a")

_alJ. ,

c:(a")

o.

The bicrossproduct structure of the K-Poincare group allows us to define the dual object, the K-Poincart~ algebra P" as a left-right bicrossproduct:

where T is dual to T* and U(L) is the universal enveloping algebra of the Lorentz algebra. The duality 'j'* {=} T is defined by:

The duality betwccn thc Lorentz group and algebra is defined in the standard way:

< A"v' M°{J >= i(gO"8 fiv - gfi"8"'vl· The structure maps are defined by the following duality relations:

< t, M ofi < A l> t, M°{J >

< fJ(t), M",{J (>9 P.., >, < A (>9 t, 8(M"'/I) >,

here t is arbitrary product of a's while A is an arbitrary product of A's. Finally, using the method described in [1.1] we arrive at the following explicit form of the K-Poincare algebra:

P. Kosinski and P. MaSlanka

46

a) the commutation rules:

[Mij, Pol

0,

[M'J,Pk ] [Mia, Po]

iK(O\gOi - O\gOj)(1 - e-lf) + iWkg i,' - o\gjS)P., iKgi0(l - e-~) + igik pk, . K 00 ri (1 - 2.!:2. ) lv . ri AS l' - .!:2. -z2 kg se +, 9 vk - C

[MiO,pk]

K

-

K

. Oi i i rs p p.--9 i is p.p , +zg Pk(C- ~ -I)+-okg k K

T

2K

K

0,

[1'", Pv ]

i(g"a M V'\ _ gva M"'\

[M"v,M'\a]

+ 9 v,\ M"a

_ g"'\ M va ) ;

b) the coproducts:

+ Po I8i I ,

L1Po

I

L1Pk

Pk I8i eMij I8i I

L1M

ij

L1M iO

Q9

Po

.!:2. K

+ I I8i pk ,

+ 118i Mij , I ® Mia + Mia ® c-lf

_ .!.Mij ® p K

J

where i,j,k = 1,2,:l, ... ,n - I. Let us note that the K-Poincare algebra and group in any dimensions and for the diagonal metric tensor were obtained in 14 and 15. However the duality between the K- Poincare algebra and group was not discussed in these papers. Let us remark in the end of this section that the classical r-matrix Eq. (2.1), r = M Oi 11 pi, does not modify the coproducts for the generators Mij, forming undeformed Lie subalgebra as well as the component Po of the fourmomentum. The algebra with the generators Mij, Po describes the classical subalgebra of our K-Poincare algebra.

4.

The

K- Weyl

group and algebra

The classical Weyl group W consists of the triples (a, A, eb ), where a is an-vector, A is the matrix of the Lorentz group in n-dirnensions and b E R, with the composition law: (a", A"v' cb) * (a 'V , A'~, eb') = (A"veba'v + a!', A"vA'~, ebc b'). Its Lie algebra, the Weyl algebra

[p",Pv ] [M"v,l',\] [M"v,M,\a] [M"v,D] [p",Dj

W, reads:

0,

i(gv,\p" - 9,,'\pv ), i(g"lTMv'\ - gVlTMI'>' 0,

-iP" ;

+ gv,\M"a

- g,,>.Mvu ),

The

K-

Weyl group and its algebra

47

here M,w, 1'JI. are the generators of the Poincare algebra and D is the dilatation generator. We would like to obtain the K-deformation of the Weyl group and its Lie algebra. It is clear that the classical r-matrix for the Poincare algebra satisfying the classical Yang-Baxter equation (CYSE) is also a classical r-matrix for the Weyl algebra. It follows from the Eq. (2.2) that our classical r-matrix Eq. (2.1) satisfies the CYBE iff goo = O. This means that our r-matrix defines a structure of a Poisson Lie group on the Weyl group only in the basis in which the metric tensor takes such a from that goo = O. We shall consider only these types of matrices. In order to obtain the K-deformation of the Weyl group, we firstly find the invariant fields:

A'''' D AJI.{i 8 8AJI./3 8AJI." '

XL"

c

bAJI." a 8aJl.'

8

b

c aeb '

X (i/3 H

A{i 8 A" 8 /3 a ,,8 /l v -8A -il v8A +a--;----) - a - a ' il"v /1/3v ( a" a/3

a

xrt

aa" '

XH

a 8aJl.

JI.

a

+e

b

a

aeb'

Then, using the Eq. (2.3) we calculate the Poisson brackets of the coordinate functions on Wand perform the standard quantizations of the Poisson brackets, by replacing { , } ---+ Finally, we obtain the following set of commutation relations:

H' ].

_~((Cb A"o - 8"0)A e{i + (A o/3 - ebgo(J)g"e) , K

[a Q , a"]

~(8Qoa" - 8"oa e ) , K

0,

[A"/3' AJl.vl [AJl.v' b]

0,

[a'" b)

o.

This standard quantization procedure is unambiguous: there is no ordering ambiguity. Since the composition law is compatible with the Poisson brackets, the above commutation rules arc compatible with the following coproduct:

L1A"v L1aJl.

AJI." (gi A"v , eb A"v is: a + aJl. is: I, V

b® I

+ I ® b.

(4.1 )

48

P. KosiJlski and P. MaSlanka

The antipode and the cOllnit are given by:

S(AI'") S(al')

=:

S(b) e(AI'J

-b, =:

e(al') e(b)

A/, _e- bA/a v ,

81'v'

0, =:

O.

(4.2)

We conclude that the Eq. (4.1), (4.2) define the Hopf algebra - -- K- Weyl group WK' If we forget for a moment about our general theory, w.e can chcck by explicit calculation that our structure is self consistent (Jacobi identities, the relations [L1a, L1bJ = L1[a, bJ) iff goo =: 0 01' b = O. For exam pic:

It is easy to sec that the

W~

has a right-left bicrossproduct structure:

wherc etA) is the standard algebra of functions defined ovcr group A. Thc group A consists of the pairs (A, eb ), wherc A is a matrix of the Lorcntz group and b E R, and with the composition law:

T* is defincd by the rclations:

S(a")

_all ,

e(a")

O.

To sce this it is sufficicut to define the structurc maps as follows:

fJ(al')

b

e JlI'" Q9 a" , b

_!..( (e A"o - bl'o)Jle" K

+ (A o" -

b

e go" )g"e) ,

O. This right-left bicrossproduct structure of the K- Weyl group allows us to define the Weyl algcbra WK as a left-right bicrossproduct structurc:

K-

ThE'

K-

49

Weyl group and its algebra

where T is dual to T* and U(A) is the universal enveloping algebra of the Lie algebra A of the group A. The duality T* ¢=? T is defined by:

< aI', Pv >= i8l'v. The duality between the group A and the algebra

A is defined in

the standard way:

i(g"1'8 13v - g131'8"vl ,

< Al'v' M"13 > < b,M"f3 >

0,

< A"v' D > < bn , D >

0,

i8n ,l.

The structure maps are defined by the following duality relations:

< I, M"f3


< f3(I), M"13 ® P-, >,

P-, >

<1,D I,M"13> < f r> I,D>

, ;

[Mij,PoJ =

[Mij,Pk] = [MiO,PoJ

[MiO,PkJ

=

[D,PoJ

[D, P;] b) the coproducts:

+ I ex> D -

~) - -gOi 1 M ik 1/9 gOi M iO ® ( 1 - e-·

11D

D· I6i I

11 Po

11Pk 11M i }

+ Po ® /, + I 1/9 Pk , Mij 1/9 I + I 1/9 Mij ,

11M iO

II/9M' +M' l/ge-· --M'}I/9PJ'

K

/0 Po

Pk Q9 e-!! ·0

·0

~

1

K

f)k,

50 5.

P. Kosinski and P. MaSlanka Conclusions

We have constructed the K-Poincare group resulting from Poincare group formulated in an arbitrary basis. The quantization is unambiguous due to the absence of ordering problems. The resulting quantum group has a bicrossproduct structure. Using this and the methods developed in [I5] we were able to construct the relevant K-Poincare algebra. The Schouten bracket of the classicalr-matrix we have used appeared to be proportional to the 900 component of the metric tensor. Therefore in the reference frame chosen in such a way that 900 = 0 the relevant Poisson structure can be extended to any group containing Poincare group as a subgroup. This was used to define the Poisson structure on the Weyl group, which allowed us to construct K-deformation of this group. Again we obtained a bicrossproduct structure which allowed us to construct the relevant K- Weyl algebra. The above construction seems to us to be a proper introductory step toward the definition of K-deformed conformal group. One can attempt to quantize the Poisson structure on conformal group resulting from the same r-matrix we used in the case of the Weyl group, hoping that the ordering problems could be overcome in some way (as for example in the case of K-Poincare supergroup [2-6]). As a next step one tries to construct the relevant algebra. This might be more difficult as the bicrossproduct structure is lacking in the case of conformal group. An alternating way of attacking the problem would be to try to incorporate on the quantum level the property of conformal group that it can be obtained from Weyl group by adding (in a special way) the operation of inversion.

REFERENCES [I] .1. Lukierski, A. Xowicki, H. Ruegg, Phys. Lett. B293 (1993) 419. [2] 1'. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, J. Phys. A27 (1994) 6827. [3] P. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, J. Phys. A28 (1995) 2255. [4] P. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, Mod. Phys. Lett. AIO (1995) 2599. [5] P. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, preprint 1FT UWr887/95, q-alg. 95-01010, to appear in J. Math. Phys. [0] P. Kosillski, J. Lukierski, P. Maslanka, J. Sobczyk, "Quantum deformation of the The Poincare supergroup", in: "Quantum Groups. Formalism and applications". Proceedings of XXX Karpacz Winter School of Theoretical Physics, eds. J. Lukierski, Z. Popowicz, J. Sobczyk, PWN 1995, p. 353. [7]

~1.

Klimek, J. Lukierski, Acta Phys. Pol. B26 (1995) 1209.

[8] J. Lukierski, P. Minnaert, M. Mozrzymas, preprint CPTMB/PT/95-0. [9] A. I.lallesteros, FoOl. Ilerranz, M.A. del Olmo, M. Santander, preprint March 1995.

The

K- Weyl

group and its algebra

[10] S. Zakrzewski, J. Phys. A27 (1994) 2075. [11] P. Zaugg, preprint MIT-CTP, (1994). [12] S. Majid, H. Ruegg, Phys. Lett. B334 (1994) 348. [1:1] 1'. Kosinski, P. :vIaslanka, preprint IMUL3/94. [14] J. Lukierski, H. Ruegg, Phys. Lett. B329 (1994) 189. [15] 1'. Maslanka, J. Phys. A26 (1993) L1251.

51

From Field Theory to Quantum Groups ----_._- ------- - - - - - - - -

RATIONAL SOLUTIONS OF YANG-BAXTER EQUATION AND DEFORMATION OF YANGIANS S.M. KHOROSHKIN

Institute of Theoretical and Experimental Physics 1.17259 Moscow, Russia (e-mail: khoroshkinatvxitep.itep.ru)

A.A.

STOLIN

Department of Mathematics, Royal Institute of Technology S-10044 Stockholm, Sweden (e-mail: astolinatmath.kth.se) AND V.N. TOLSTOY

Institute of Nuclear Physics, Moscow State University 119899 Moscow, Russia (e-mail: tolstoyatanna19.npi.msu.su)

Abstract: A quantization scheme of a certain class of nonstandard rational solutions of CYBE for any simple Lie algebra 9 is given explicitly. We obtain this quantization using a twisting of the usual Yangian Y(g). Explicit computations are performed for the case of S12' The corresponding quantum object (deformed Yangian Y~,«SI2)) is a two-parametric deformation of the universal enveloping algebra U(SI2[UJ) of the positive current algebra sI 2[u]. We consider the quantum double DY~,«sI2)' its the universal R-matrix and also the RTT-realization of Y~,«SI2)'

Introduction This paper is a survey of results obtained by the authors in [1] and [2]. It is devoted to the problem of a quantization of a non-standard rational solution of classical YangBaxter equation (CYUE) for any simple Lie algebra g. We propose here a general scheme of deformation of the Yangians as a solution of this problem for a certain class of such non-standard rational solutions. Yangian of Y(g) of a simple Lie algebra 9 was introduced by V. Drinfeld [3,4] as a deformation of the universal enveloping algebra U(g[u]) of the current algebra g[u]. It

54

S.Y1. KhoroshkiiJ, A.A. Stolin and V.N. Tolstoy

has a lot of applications in the mathematical physics and in the representation theory ([5]-[15]). In particular, the Yangian symmetry Y(si n) was shown for the following one-dimensional N-body integrable models: the Hubbard model [5,6], the classical sin Euler-Calogero-Moser model confined in an external harmonic potential [7], the quantum sin Calogero model confined in the harmonic potential [8], and the quantum Sutherland model [9,10]. The Yangians are employed in [13,14] for an explicit description of the center of the universal enveloping algebra U(g), where 9 is a simple Lie algebra of A-, B-, C-, D-series. A connection between the Yangian Y(gln) and the classical construction of the Gelfand-Zctlin basis for the Lie algebra gin was established in [15]. Tensor products of finite dimensional representations of the Yangian Y(g) produce rational solutions of the quantum Yang-Baxter equation (QYRE). For instance, for 9 = sin these solutions can be obtained by the fusion procedure applied to the Yang solution R( u) = 1 + plu, where p is the permutation of factors in en (9 en. However, there exist other rational solutions of the classical Yang-Baxter equation (CYRE). These solutions were studied in [17]. Every rational solution of CYBE provides a bialgebra structure on g[u]. These structures have not been quantized yet except the case l' = cdu, when the Yangian is exactly the quantization. We present here a quantization scheme for rational solutions of the form cdu + constant. We obtain this quantization using a twisting of the comultiplication in the usual Yangian Y(g). The existence of such element was proved by V. Drinfcld [3]. In Appendix we reproduce from ([18]) another general construction of the twisting element. We realize explicitly the twisting for the simplest non-standard rational r-matrix for S12, namely r = c21u + h", 1\ C"" Here C±a, h", is the standard Chevalley basis for S12' We note that the additional term h", 1\ c'" leads to a deformation of the co-algebra structure of Y( sI2 ). We perform this deformation by means of the twisting of U(SI2) 129 U(SI2) by some special two-tensor of U(SI2) (9 U(SI2) which (appeared in [19]. Moreover, this two-tensor enables us to write down a quantum Rmatrix corresponding to the classical r-matrix r = c2Iu+h",l\e_",. On the other hand, writing down this R-matrix explicitly in the fundamental representation, we develop RTT-formalism (see [20]) to get another presentation of the deformed Yangian. We discuss also properties of the corresponding quantum determinant, and the realization of the deformed Yangian Y,I.{( S12) in terms of generating functions ("field" realization). 1.

Rational solutions of classical Yang-Baxter equation

Let 9 be a simple Lie algebra over C, {Ii} be an orthogonal basis of 9 with respect to the Killing form. Let C2

P(u,v)= --+r(u,v) u-v be a function from C 2 to 9 0 g, where definitions.

C2

= Li Ii (9 l;.

(I.l ) We introduce the following

Rational solutions of Yang-Baxter equation

55

Definition 1.1 We say that P( u, v) is a rational solutions of the classical YangBaxter equation (CYBE) if the function P( u, v) satisfies the following conditions: 12

21

(i) it is skew-symmetric, i.e. P (u,v) = - P (v,u), (ii) the function 1'( u, v) is a polynomial in u, v, i.e. it has the form 1'( u, v) = Lij Pij (u, v) Ii 161 Ij , where Pij (u, v) are usual polynomials of two variables u and v. (iii) the function P( u, v) satisfies the CYBE

13

Here l' (U1' U3) means 13

P (U1' U3) =

L (hAUl, U3){Ii 010 Ij ) ,

( 1.3)

i,j

if P(u,v) has the form P(u, v) = Lij rPij(U, v)Ii0Ij , where rPij(U, v) are usual functions , 12 23 of two variables u and v (rPij : C 2 -> C). The clements P (Ul,U2) and P (U2,U3) are defined analogously. The commutators in the left side of (1.2) are considered as commutators in U(g)8 3 •

Definition 1.2 We say that two rational solution P1(u,v) and P2(U,V) are gauge equivalent if there exists an automorphism 7>. of algebra g[u] such that (7). 0 7>.)PI (u,v) = P2(U,V). It turns out that the degree of the polynomial part r(u,v) for rational solutions of CYBE can be estimated. More exactly, the following result was proved in [17]: Theorem 1.1 Let P( u, v) = ~ + 1'( u, v) be a rational solution. Then there exists a rational solution p( u, v) being gauge equivalent to 1'( u, v) and of the form -

C2

P(u, v) = - u-v where roo,

1'10, 1'01, I'll

+ 1'00 + rlOu + 1'01V + rlluv

,

(1.4 )

E g 119 g.

We will be dealing in the present paper with the case C2

P(u,v) = - u-v

+ 1'0,

(1.5)

where 1'0 := 1'00 E 9 (>9 g. In what follows, we need the following result. Let g((u- 1)) := g 0 C((u- I )). One can define the non-degenerate ad-invariant inner product on g( (u- 1 )): (x, y) = Resu=o tr( ad x . ad y). The following theorem is valid.

56

S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy

Theorem 1.2 1) There is a 1-1 correspondence between the set of rational solutions of CYBE of the form (l.5) and subalgebras W C g[[u-llJsuch that: (i) u- 2 g[[U- l ]] eWe g[[u- l ]], (ii) w.L = W with respect to the form (.,.) introduced above, (iii) WCDg[u]=g((u- l )). 2) Any W satisfying conditions (i-iii) above defines a subalgebra Leg and a nondegenerate 2-cocycle B on L. In other words the element B is skew-symmetric and satisfies the equation

B([x, y], z)

+ B([z, x], y) + B([y, z]x) = 0

( 1.6)

for any x,y,z E £. Moreover, the element r·o E L 1\ f and it is non-degenerate, l = B E C 1\ C. 3) Conversely, any pair (f, B) such that f is a subalgebra of 9 and B is a nondegenerate 2-cocycle on f, defines a rational solution of the form (l. 5).

ra

r'o

ro

l l Here the symbol means the following. Since ro E f ® f '2:' Hom(C, f), E Hom(f, C) '2:' C (l) C. It should be noted that a Lie algebra with a non-degenerate 2-cocycle is called quasi-Frobenius Lie algebra. Proof (see also [18], [17]). Let P(u,v) be a rational solution of the form (1.5), where ro is given by the following general expression ro = Li,j rij Ii 129 I j. Let us define a linear functional on u-lg[[U-llJ by the formula

pta) =

'L rij(1i,a)Ij

(1.7)

i,j

for any a E u-1g[[U- I ]] and let us define a subspace W C g[[u-llJ consisting of the elements of the form a + pta), where a E u-lg[[U-1lJ. Since P(u,v) is a rational solution, therefore we have that rij = -rji. It is easy to see that this is equivalent to the following condition: W = W.L, which proves property (ii). Now we prove that W is a subalgebra of g[[u-llJ. Taking into account that (ii) is already proved, it is sufficient to prove that ([x,y],z) = 0 for all x,y,z E W. Since the elements x, y, z have the form x = a + pta), y = b + p(b), z = e + pte) for some a, b, e E u-lg[[U-llJ, we have to consider ([a

+ pta), b + p(b)], e + pte))

= ([a, p(b)], pte))

+ ([p(a), b], pte)) + ([p(a), p(b)], e)

Since the inner product is ad-invariant, we have:

([a, p( b)], p( e))

= (a, [p( b), p( e)]) = 'L

rijrkle'ji(1i, b)( h, e)(1m, a) ,

ijklm

where we take into account that [Ii, Ij] ="Lk e7/k' On the other hand

.

Rational solutions of Yang-Baxter equation

57

and the fact that 1'0 satisfies CYBE (this is the same as P( u, v) satisfies CYBE) is equivalent to that ([x,y],z) = 0 and hence to that W is a subalgebra. It is clear that the properties (i) and (iii) are fulfilled because of the construction of Wand therefore every rational solution of the form (1.5) determines W, Conversely, let W satisfy conditions (i) - (iii). Since we have another decomposition u-lg[[U-llJEBg[u] = g[[u- I ]] we infer that W = {a+p(a) : a E u-lg[[U- I ]]}, where p: u-lg[[U-llJ ---> 9 is a linear map. Let fin := Iiu- n- I be a basis of u-lg[[U- 1]] and fin := Iiv n be a basis of g[v]. Then the element L:in(ein is) (!in + p(fin))) is a rational solution of CYBE of the form (1.5). The first statement is completely proved. Let us prove the second statement. Define a projection 1r : g[[u-llJ ---> 9 as follows: 1r( aD + al u- I + ...) = aD. Clearly 1r(W) = L is a subalgebra of g. Moreover, W c .c EB u-lg[[U-1lJ C g[[u- 1 ]]. Then the condition (ii) implies that u-1.c.L EB u- 2g[[u- 1lJ C W where .c.L egis the orthogonal complement to .c with respect to the Klling form. It is easily seen that the subalgebra N = u-1.c.L EBu- 2g[[u- 1 lJ is an ideal in the algebra M = .cEBu-1g[[U- I ]]. The quotient algebra MIN is isomorphic to the semidirect product of .c and commutative algebra £*, where £* is the dual space to .c on which .c acts by the coadjoint representation. We can write MIN = .c + (£*, where (2 = O. Then MIN is equipped with the non-degenerate ad-invariant symmetric inner product coming from the inner product on Meg is) C((u- 1 )). One can check that (a,b) = 0 if a, b E .c: (ex,ey) = 0 and (a,ex) = x(a). Denote by ¢ the natural projection M ---> N. Then ¢(W) = S is a subalgebra of MIN, which coincides with its orthogonal complement. Thus, dim(S) = dirn(.c). Recalling the construction of .c we deduce that S is projecting onto .c under the natural projection .c + e£* --->.c. Therefore S is of the form S = {I + (f(l): I E.c} for some f E Hom(.c,£*). Let x: Hom(.c,£*) ---> £* is) £* be the natural isomorphism. Then we have that B = x(f). One can check that the equality S = S.L implies that B is skew-symmetric and the fact that S is a subalgebra implies that B is a 2-cocycle. The non-degeneracy of B comes from the property (iii), which implies that S n .c = (0) in MIN. The second statement is completely proved. Clearly, all the considerations above can be converted, which proves the third statement. The theorem is proved completely. It should be noted that in the case of 9 = sIn all the rational solutions can be described in a similar way [i 7]. Let

dk = diag( 1, ... ,1 t, ... ,t) E GL(n,C((CI))).

---.,.........k

(1.8)

Then every rational solution of CYRE defines some Lagrangian subalgebra W contained in d};l . sl( n, C[[t- 1]]) • dk for some k. The corresponding combinatorial data are: (i) subalgebra .c c sl(n, C) such that .c + Pk = sl(n, C), where P k is the maximal parabolic subalgcbra of sl(n, C) not containing the root vector ea. of the simple root Ok;

,'58

S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy

(ii) 2-cocycle B OIl L which is non-degenerate on L n P k . In the case of slz, one has just two non-standard rational r-matrices (up to the gauge equivalence): (1.9) and

cz

Pz(u, v) = - u-v

+ h", iZI e_",u -

e_", iZI h",v .

(1.10)

The corresponding Lagrangian subalgebras are

and

Wz = C:l(slz[[t- 1 ) EB C(e",t- 1 - h",t) ED C{e_",C1)ED Ell C(h",C 1 )

(IJ

C{ C"'C Z )

e C{e_",C Z ) Ell C{h",C Z + 2c",) .

( 1.12)

Now we are going to quantize the rational r-matrix (1.5).

2.

Deformation of Yangians

In this section we describe a procedure for quantization of any rational solution of the form (l.Ei). The output is a deformed Yangian. The Yangiall V(g) as a quantum deformation of the universal enveloping algebra U(g[t]), where g[t] is an algebra of polynomial currents over a simple Lie algebra g, was introduced by Drinfeld [3] firstly in terms of generators which actually are not associate to choice of a concrete basis in g. Later Drinfeld gave in [3,4,21] another realization of the Yangians in terms of generators connected with Cartan- Weyl basis in g. In this paper we use this second Drinfeld realization. We remind the definition of this realization and the basic properties of the Yangian. Let 9 be a simple Lie algebra with a standard Cartan matrix A = {aij li,j=o, a system of simple roots // := {Ol" .. ,02} and a system of positive roots .1+(9). Let ei := e a " hi := ha,. fi := fe>, := C_"'" (i = 1•... ,1'), be Chevalley generators and {e.."f..,}. hE .1+(g), be a basis Cartan-Weyl in g, normalized so that (ea,fa) = I.

Definition 2.1 The Yangian Y := Y(g) associated to g is the Hopf algebra over C generated (as an associative algebra) by the elements eik .- e""k, h ik := h aik , fik := f""k, (i = 1, ... , r; k = 0,1,2, ... ), with the relations:

(2.1) [hiO.ejd = (Oi.Oj)Cj/,

[hio.h] = -(ui,uj)fjl. 1

[hi,k+l,eJtl- [hik,cj,l+d = 1]2"(oi,oj){hik ,Cjd ,

(2.2)

(2.3)

Rational solutions of Yang-Baxter equation

59

1 [hi,k+l,!Jd - [hik,!j,l+d = -1)2(Oi, Qj){h ik , !jt} ,

(2.4)

1

(2.5)

[ei,k+l, eJd - [eik' ei,l+ll = 1)2(0i, OJ){ eik, Cjd , [!i,k+l,!il] - [Jik,fJ,I+1]

1

= -1)"2(Oi,Oj){Jik,!jd

Sym{k} [ei,k, [ei,k 2 . , . [Ci,k n ,) , Cjd ' .. J] = 0 { Sym{k} [fi,kl [Ji,k2 ... [Ji,k n .) , lid·· .J] = 0

,

(2.6)

lor i -=f- j ,

(2.7)

where {a, b} := ab+ ba, nij := 1 - A ij , the symbol "Sym"{k} denotes a symmetrization on k 1 , k 2 , • •• , k;J' The co-multiplication map of Y is given for basic generators eik, hik , lik, (k = 0, I), by Ll(x)=xlSil+llSix,

(2.8)

x E9 ,

L

Ll (eid = Cil (8)1 + 1 lSi eit + hiO lSi eiO - 1) .

I'Y lSi [e"" e'Y]

,

(2.9)

'YELl +(g) Ll (fid = fit lSi 1 + 1 (8) fil + fio lSi hiO +.,.,.

L

[f"" l'Yl lSi

c-y ,

(2.10)

-YELl +(g) Ll(hitl=hit09I+llSihil+hiO®hiO-1)'

L

(Oi,')!'Y®C'Y'

(2.11)

'YELl +(g) We explicitly introduce in the relations (2.3)-(2.6) and (2.9)-(2.11) the Yangian deformation parameter 1) E C\ {O} 1 and therefore we shall use further the notation Y~(g) for the Yangian Y(g). Remarks. (i) The universal enveloping algebra U(g) is generated by the elements eiO, hiO, Jio and it is embedded in Y~(g): U(g) ~ Y~(g). (ii) One can show that Y~(g) is generated only by the elements eiO, hiO, liO, fil, (i = I, .. . ,1'), and therefore we can obtain the co-multiplication map for all generators cik, fib h ik , !ik, (i = 1, ... ,7'; k E Z). (iii) If we replace the right parts of (2.3)-(2.6) by zeros, we obtain the algebra isomorphic to U(g[t]). For any ,\ E C define an automorphism 7>. of Y~(g) by the formulas:

xEg,

(2.12) (2.13)

This "translation" automorphism is compatible with the comultiplication, i.e. (2.14) 1 All Yangians with different that TJ = 1.

11

are isomorphic one to another and therefore one usually assumes

S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy

60

The Yangian is a pseudotriangular Hopf algebra [3]. It means that there exists an unique element R(>.) = 1 + Lk:1 >.-kRk, where Rk E Y~(g) 0 Y~(g), such that

(7), 0

~)R(II)

= R(1I + >. -

(7), 0 id)L1'(x)

p.) , 13

12

23

(,10 id)R(>') =R (A) R (A) ,

= R(A)((J;, 0 21

R (>') R (->.)

id)L1(x))R(A)-1 , (2.15)

= 10 1 ,

(2.16)

where 7>, is the translation automorphism and, as usually, we denote by ,1' the opposite comultiplication. Now we return to the rational solution of CYBE in the form (1.5). We recall that this solution P( u, v) can be characterized by availability of the following combinatorial data: (i) subalgebra £ of g; (ii) non-degenerate 2-cocycle B on .c. Our approach to quantization of a rational solution of CYBE is based on the following result borrowed from [3]. Theorem 2.1 There exists an element FE (U(£)[[7]]])18I 2 of the form f' =

1

1 + "27]ro

~ 7] kf'k, + L.

(2.18)

k=2

o

where r 1 = B (in sense of the Theorem 1.2), and this element satisfies the following condition: 12

23

F (,1 0 id) F = F (id 0 ,1 ) F ,

(2.19)

where ,1 is the trivial (eocommutative) comultiplieation in U(g). Remark. Another construction for F is described in [18] and we show in Appendix an approach, which enables one to consider both cases simultaneously. Let Y~(g) = Y(g)[[7]Jl. For any F satisfying the conditions of the Theorem 2.1 we define a new algebra y~(F)(g), which has the same multiplication as Y~(g) but its comultiplication is defined as follows (2.20) The following theorem is valid. Theorem 2.2 The algebra Y~(g) is the pseudotriangular Hopf algebra with ,1 (F) and R(F)(u) =F R(u)F- 1.

Rational solutions of Yang-Baxter equation

61

Proof. First of all it is not difficult to verify that the comultiplication (2.19) is a coassociative operation. Now we prove that the element R(F) (u) satisfies the relations (2.15)-(2.17) with .1 (F). The first relation from (2.15) and the second relation from (2.16) are obvious for .1 (F) and R(F)(u). It is easy to see that the following chain of relations takes place

(T" is! id).1 (F)' (x) = (T" Q9 id)

= (j.!

F.1'(x)r =F R(>.)((T" is! id).1 (x))(R(>.))-1 (}\)-I l

R(A)P-I)((T" is! id).1(F)(x)) (

FR(>.)r't

l

=

R(F)(>.) ((T" 0 id).1 (F)(x))(R(F)(Ajt' . This proves the second relation from (2.15). It remains to prove the first relation of (2.16) and (2.17). The first of these relation is a consequence of (2.15) and the following equalities (which are equivalent to (2.19)): 12

21

13

F (.1 is! id) F=F 0-(.1 is! id)1" , l

(U(.1 ® id)1") ((id is! .1 )Ft =

d;~)-I ~ ,

where u(a 0 b 121 c) = b 0 c 121 a. At last, We deduce (2.17) using (2.15) and (2.16). The proof is complete.

3.

Non-Standard Quantization of U(51 2 )

Let e±c" he. be the Chevalley basis for the universal enveloping algebra U(51 2 ) of the Lie algebra 51 2 with the standard defining relations: (3.1 ) We put here and anywhere (0:,0:) = 2. Let U( L) E U(51 2 ) be the universal enveloping algebra of the Borel subalgebra L of 51 2 , generated by the elements he. and E-e.. Let us introduce the following two-tensor (a formal series) F of some extension of U(b_) 121 U(b_):

where ~ E C is some parameter. We have borrowed the element F from [19]. It is not difficult to verify that the following series

is an inverse element to F, i.e. 1"1"-1 valid.

=

F- 1F

=

1. The following proposition is

62

S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy

Proposition 3.1 The element F (as formal series) satisfies the following relation (3.4 )

where L1 is the usual eomultiplication in U(b_), i.e. L1(a) = a 0 1 + 1 0 a for any a E b_. Proof By direct calculations. 0 As consequence of this proposition we have Corollary 3.1 Let A be an arbitrary !fopf algebra containing the l/opf algebra U(b_) and let A~Fl be the algebra A[[~lJ (i.e. the algebra A over C[[~lJ) with co-multiplication map L1(Fl given by the formula

(3.5) Then Ar) is a l/opf algebra. Proof Coassociativity of L1(Fl follows from the formula (3.4). Existence of the antipode is proved in [21].0 We introduce now the following notations for some elements of utJ(L): 1'.:, := 1 -

~e_o

,

(3.6)

Proposition 3.2 The elements ho , 1':1 satisfy the following relations:

1'cJ,,-1

= 7';:11'0

, [h o , To]

L1(Fl(h o ) = h o 091';;1+10h o ,

= 2(1

- To) ,[h o , I~-I]

L1(FJ(1'.:,)

8(h o ) = -hoI'.:, ,

= 1'.:,01'.:"

= 2(1';:1

L1(Fl(1~-I)

= 1;1 , 8(1';:1) = 1'.:, e(I'.:,) = e(1';;I) = 1 .

5(1'.:,)

e(h o ) = 0,

_1~~2) ,

= 1';;1 09 1';;\

,

The algebra Utl(b_) generated by ho' 1':1 is a Hopf subalgebra of Dtl(L). Proof By direct calculations with (3.1), (3.4) and (3.5).0 The Hopf algebra Ut\b_) is triangular with the universal R-rnatrix R

=

p21

F- 1 = 1 + ~e_o 1\ h o

+ ... =

(3.7)

(3.8) (3.9) (3.10)

Rational solutions of Yang-Baxter equation

63 (3.12)

This algebra is a quantization of the Lie bialgebra b_ defined by the classical r-matrix h",. According to Corollary 3.1 we can extend the twisting by F to 0t)(.gI2)' Then we have

r = e_" 1\

Proposition 3.3 Let ut)(sI2) be an algebra generated the elements h"" e" and 1';;1 with the defining relations

l' 1'-1 0' Q

[h", e,,] L1(F)(h",)

= 1'-11' n

Q,

= 2e"

,

[h a, l' J = 2(1 - 1') [h 0', 1'-1] Q

Q

[1'", e",J

= h"091';I+I09h,,,

= 2~h",

,

Q

= 2(1'-1 a

- 1'-2) Q'

(3.13)

= -2~T;1 haT; I , (3.14) ,1(1")(7:') = 1',,0'1:', L1(F)(T;I) = 1';1 0 1';1, ,

[1';1, e,,]

(3.15)

L1(F)(e,,)

= e,,09T;1 + 10ge"-~h"@T;lh" 5(h,,)

= -h,,7:'

,

5(1',,)

5(e,,) = -e"T" c:(h,,)

ih",(h" -2)@T;I-i h,,(h,,+2)@1';2 ,

= T;l

,

5(1';1)

~h"(h,, + 2)7:'(1'", -

= c(e,,) = 0,

c:(T,,)

= 1'"

2) ,

= c:(T;I) = 1 .

,

(3.16) (3.17) (3.18) (3.19)

Then ut)(sI2) is a Hopf subalgebra of ut)(sI2) and it is a triangular deformation of U(sI2) in the direction of the classical r-matrix r = L " 1\ h". Proof By direct calculations with (3.1) and (3.5).0 Remark 1. In every finite dimensional representation of sl2 the element Tu = 1 - 2~L", is always invertible since L" is nilpotent. Therefore, the theory of finite dimensional representations of Ut l (sI2) is the same as the theory for .gI2' Remark 2. Similar computations for sl2 with another twisting element F were carried out in [24]. However, using Theorem 2 from [22] one can prove the following result. Theorem 3.1 There exists an invertible element l' E U(gI2)[[~ll such that F (7' 09 7')P-1 ,1(7'-1), c(7') = 1 and PI(T) = 1 E GL 2, where PI is the two-dimensional representation of g12' In other words /lopf algebras obtained from U(sI2)[[~ll by twisting by F- I and F are isomorphic as /lopf algebras and the isomorphism is given by conjugation by T.

t

Proof The matrix (PI @ p.)((p 2I l p) was computed in [24J. We can calculate the matrix (PI @ Pl)( p 21 F- I ) and it turns out that we obtain the same matrix (see Lemma 6.1 further). Then Theorem 2 from [22J implies all the statements.O

S.M. Khoroshkin, A.A. Stalin and V.N. Tolstoy

64

4.

Twisting of Yangian Y(sI2)

One can show that the Yangian Y,,(slz) (as a Hopf algebra) can be defined by Chevalley generators h"" e±,,, co-" with the defining relations [16]:

[e""e_,,] == h" , [h,,,c±,,] == ±2e±" ,

(4.1)

[h", eo-oj == eo-" , [e_o, eo-oj == TJe_" ,

(4.2)

[c"" [e", [eo, eo-,,]]] == 6TJe; , [[[e"" eo-",], eo-a], eo-oj == 6TJeL", .

(4.3)

,1( ho )

== h" ® 1 + 1 0 ho ,

,1( e±o)

== e±o ® 1 + 1 0 e±o ,

,1(e o_,,) == eo_", 0:9 1 + 1 0 eo-", 8(h",) == -h" ,

S(e±o) == -e±ci ,

+ Tie o 0

ho

(4.4)

(4.5)

,

8(c o_,,) == -e o- o - TJc",h" ,

c(h o ) == e(e±o) == e(e o-,,) == 0,

(4.6)

e(l) == 1 .

(4.7)

Since Yry(slz) contains U(slz) as a Hopf subalgebra the Corollary 3.1 implies that the algebra

Yry(2

isomorphic to Yry[[~JJ with the cOIIlultiplication (3.5) is a Hopf algebra.

Proposition 4.1 The elcments ho ,e" ,1~±1 (sec (3.6)) and e6-", satisfy the relations:

1'0 T,,-I == 1;;IT", ,

[h,,, 7'c,] == 2(1 - T",) , [T"" e,,] == 2~h" ,

[h", e",] == 2c", ,

[1'c"

eo_",]

== - 2: (T; - 21::' + 1),

[e"" [c"" [e"" e6-,,]]] == 6TJe; , ,1(F)(h,,) == h",0T;;I+I®h""

rho, T;;I] == 2(7~-1 - T;;Z) , [T;;l, e",] == -2~T",-1 hS;;1 ,

[1:;-1, eo-a] == - 2: (T;;Z - 21:;-1 [[[e", e6-",], e6-",], eo_",] == 6TJeL", .

,1(F)(T",) == T",®T", ,

,1(F)(1~-I)

(4.8) (4.9)

+ I) , (4.10) (4.11)

== T,,-1 0 T;;I, (4.12)

,1 (F) ( e,,)

== e" 0 7;;1 + 10e" -~h" 0T;;1 h", -

,1(F)(eo_",) ==

e",

0 7::'

+ 1 ® CO-'" + ~h",

5(h",) == -h",1'c, ,

~h"(h,,, ® T;;I

8(To ) == T;;I ,

5(e",) == -e",T", -

2) 01:;-1- ~ho(ho + 2) 01~-z ,

+ 2:h", 0

(I -7::') ,

5(T;;I) == 1'c, ,

~h",(h", + 2)T",(1'c, -

1) ,

5(co- o ) == -eo_",T;:1 - {h",To + TJ.h",T;;1 - TJ. h", , ~

TJ

e(h",) == c(c",) == e(e6_",) == 0,

2~

2~

c(T",) == e(1:;-I) == 1 .

(4.13) (4.14)

(4.15) (4.16 )

(4.17)

(4.18)

The algebra Yry ..dsI2) generated by the elements h"" e"" 7;=1, e6_", is a Hopf subalgebra - (F) of Yry,e (sI 2)'

Rational solutions of Yang-Baxter equation

65

Proof By direct calculations.D One can see that the Hopf algebra Yry(2(sI2) is a quantization of the Lie bialgebra sI2[u] corresponding to the rational solution (1.5). 5.

Deformed Yangian double DYry(sI2)

Since Yry(.~12) C Yry,~(sI2) C Yry(2(sI2) (as associative algebras) and since T>. acts identically on U(sI2)[[~]] and since Yry,~(sI2) differs from Yry(sI2) by elements from U(sI2)[[~]] therefore the automorphisms T>. extended to Yry<2(sI2) preserve Yry,~(sI2)' Thus we have the following result. Proposition 5.1 The Hopf algebra Yry,~(sI2) is pseudotriangular with R(F)P..) :=

1"21 R(),,)F- 1. In particular, R(F)(>\) is a rational solution of QYBE.

Let us recall that the Yangian double DYry(sI2) (see [16]) is a quasi triangular Hopf algebra with an universal R-matrix R which lies in some extension of DYry(sI2) ® DYry(sI2)' Since U(sI2) C DYry(sI2) we can twist the Yangian double DYry(.~12) by F. Using formal algebraic arguments similar to that of Section 2 we get as result the following proposition. Proposition 5.2 The deformed Yangian double DYry,dsI2) is a quasitriangular Hop!

algebra with the universal R-matrix R(Fl = F 21 RF-l.

In what follows we need the realization of the Yangian double DYry(sI2) given in [16]. In this realization the Yangian double DYry(sI2) is generated by the elements h ko , CkH", (k E Z)2, which are composed into generating functions h+( u) = 1 + r"'Lk>O hkou- k- I , (h o := h,,), 4,,( u) = Lk>O CkH" U- k- 1 , and h- (u) = 1 - TJ Lk
(5.2) (5.3)

(5.4) (5.5) (5.6) 2These notations of the generators are connected with the notations in [16] as follows: hkb := h k , !k (k E Z).

Ckb+" := Ck, Ckb-n :=

S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy

66

(5.7) where {a,b} = ab+ba. It turns out that Yry(s12) is generated by h+(u), etc,(u), while the dual to Yry(s12) algebra YryO(s12) is generated by h-(u), e;;;,,(u). The universal R-matrix R found in [16] can be factorized as follows: (5.8) where

-

-

k?O

k?O

R+ = IIexp(-I/C(k+l)b-"c;;:ekS+")' R_ = II exp(-17e_(k+l)S+"° ekb-,,) , (5.9) Ro =

II exp(17Resu=v(17-1In h-(v + 2n + I) 1/9 dd 17-

1

In h+(u))) .

(5.10)

u

n?O

Here Resu=v(f(u) I8ig(v)) = Lkfk I8ig-k-1 if f(u) = LfkU-k-l, g(v) = Lgkv - k- I .

Corollary 5.1 The element R(F) = FRF- 1 satisfies QYBE, where F is the same as in (3.2) and R is defined by (5.8)-(5.10). 6.

RTT-realization of the deformed Yangian Yry.dsI2)

We develop in this section so called RTT-formalism (see [20]), i.e. we obtain the RTT-realization or, in other words, the realization in terms of the L-operator. Let p(1) be the two-dimensional representation of s12 in C 2 with the basis II) and 1- I). It is well-known that p(1) is extended to a representation p~1) (u E C) of the Yangian v,/( 051 2 ) by means of

pS1)(h±(w)) II) = (1

pS1)(h±(w)) 1- I)

+ _17_) II) , w-u

pSl)(e~(w)) II) pSI)(e:,,(w))

II)

p!/)(e~(w)) 1- I)

0,

=

= _1_

w-u

1_ I)

,

=

= (I -

_17_)

w-u

_I_II) , w-u

pS1)(e:,,(w))

1- I)

1- I) . (6.1)

= 0,

We find with the help of these formulas

(pSI)

(9

pSI))(R)

=
v)( I

+ ~)=
v)R (u - v),

(6.2)

where

2

2

1

R(u - v) L (u) L (v) =L (v) L (u)R(u - v) , I

2

(6.3)

where L (u) = L( 11) 09 id, L (v) = id 09 L( v). The matrix L( u) is a generating function for }'~(gI2) and Y(s12) ~ Yry(gI2)j(qdetL(u) - I). More exactly, we can formulate the following result.

Rational solutions of Yang-Baxter equation

67

Proposition 6.1 Let L( u) be a 2 x 2-matrix with non-commuting entries, such that 1

2

2

1

(i) R(u - v) 10 (u) 10 (v) =L (v) 10 (u)R(u - v) , [)O) L( i) IP) (l!..) L( U ) = I + --;;+ --;;2 + ... --;;;.+ ... , (iii) qdet L(u) = 11l(u)122(u-l)-lzl(u)llz(u-l) = lzz(u)ln(u-l)-itz(u)lzl(u-l) = I. Then the matrix coefficients of L(u) generate a Hopf algebra isomorphic to The comultiplication L1 and the antipode 8 are given by the formulas

Y~(sI2)'

(6.4)

L1(lij{U)) = 'Llki(u) 01jk(u) , k

8(L(u))

= L-1(u).

(6.5)

The deformed Yangian Y,J.dslz) admits a similar representation. We start with the lemma. Lemma 6.1 In the representation p~i)Q9p~i) the universal R-matrix 'RP') = p 21 nF- 1 has the form R~.((u - v) := (p~1) 0 p~l))(n(F)) =

= (1 +~p~I)(e_,,)0P~1)(h,,))(I-TJuP~2J(1 ~~p~I)(h,,)0P~i)(e_,,)) 1

=

(

-=-~~v ~ _~~v ~

--~

e

~ )

1

~~v

~

0

1-

=

(6.6)

u~v

Proof. By direct calculation.D Let us consider an algebra A of matrix elements of 1.,( u) satisfying the relation 1

2

Z

1

R~.((u - v) L (u) L (v) =L (v) I., (u)R'J.du - v) .

(6.7)

It follows that L(11) = (p~i) 1/9 id)n(F) satisfies (6.7). Algebra A together with the comultiplication (6.4) and antipode (6.5) constitutes a Hopf algebra. The following lemma takes place. Lemma 6.2 (i) The matrix R~.((TJ) is the project07' onto the one-dimensional subspace C(ll) G 1- 1) -1- 1) ® 11) - ~I- 1) 01- 1)) up to a scalar factor. (ii) The following relations hold

S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy

68 qdetry,~

= 111(u)122(u -1)) -121(u)ln(u - "1) - ~111(u)112(u In(u)ll1(u - "1) -112(u)121(u -1)) + ~112(u)ll1(u - "1) ,

L(u) =

"1)

= (6.9)

t1(F)(qdetry,~ L(u)) = qdetry,~ L(u) 119 qdetry,~ L(u) ,

where the quantum detenT/inant

qdetry,~L(u)

(6.10)

is an element of the Hopf algebra A.

Proof The part (i) is verified by direct calculations. The proof of the second part is standard (see [13]).0 qdetry,~

Lemma 6.3 The quantum determinant

L( u) is a central element of the alge-

bra A. Proof The formula for the quantum determinant qdetry,~ and the quantum YangBaxter equation for Rry,~( u) provide the following equality:

R;~~("1)R~~~(u)R~~~(u + "1)R;~~("1) 1(v) (qdetry,~ L(u))R;~~("1)

= (qdet'I,~ L(u))R'I,~("1) 23

1

23

12

13

L (v)Rry,~(Tl)Rry,~(u)Rry.~(u

23

=

+ "1)Rry,~("1)

,

1

where we use the standard notations: L (u) := L( u) 119 I 119 1, and R~~~ = Li ai 119 bi 119 I and so on if Rry,~ = Li ai 119 bi . Direct calculations show that

( )RI2( )R 13 ( )R23() R 23 ry,~ "1 ry,~ u ry,~ u + "1 ry,~ "1

23 = 2(u-"1)R "1 ry,~

.

Therefore R;~~("1)L 1 (v)( qdetry,~ L( u) )R;~~("1)

and

qdet'l~

= R;~~("1)( qdetry,~ L( u))L 1 (v )R;~~("1))

,

L( u) commutes with L( v), i.e. [qdetry,~

L(u), L(v)] = O.

The proof is complete.O At last we have the following theorem. Theorem 6.1 Let A be a Hopf algebra generated by matrix elements of the generating function L( u) satisfying the following conditions: R'I.~(U

1

2

2

1

- v)· L (u) L (v) =L (v) L (u)·

L(u) =

'1'1 ( ~h"l:f

0) + -

L(o)

I

I

'I';;'

qdetry,~

t1(l;j{u)) =

u

Rry,~(u -

L(l) + -, + 2

v) ,

(6.11 ) (6.12)

u

L( u) = I

L lki(u) 119ljk(u) ,

(6.13) (6.14)

k

S(L(u)) = L -I(U) , where T,,:= I -

2~e_".

Then A is isomorphic to Yry,dsI2)'

(6.15)

69

Rational solutions of Yang-Baxter equation

Proof The proof is analogous to that of Proposition 6.1. Taking into account that

we see that there exists a homomorphism from A to Y~,{(sI2)' To notice that this is an epimorphism, one can use the formula (7.2). To prove that this homomorphism is a monomorphism one can use the same arguments as for non-deformed Yangians (see for instance [23]).0 Remark. To find the constant term in the decomposition of L( u) into the series in u- 1 , one should find

(Z-I 0) z '

o) 1

where Z

= 1 + ~e_"

3!!

+ -21. ~

It is not difficult to see that

Z2

2 2

e_"

+ ... +

(2n - 1)!!

,

n.

~

n

0

n

e_"

+ ...

(6.16)

(6.17)

= 1';;1, hence (6.18)

7.

Realization of the deformed Yangian functions

Y~.{(sI2)

in terms of generating

The realization of the usual Yangian Y~(sI2) in terms of the generating functions ("fields" realization) ht(u) and 4,,(u) (see Section 5) can be obtained from the Gauss decomposition of the L-operator:

L(u) == (p(1)(u) 0 id)(R+Ro1L)

== (

1 0 ) ( kl(u) -et(u) 1 0

0 ) (1 k 2 (u) 0

-e~,,(u))

(7.1 )

1

where k.(u)k 2(u - 1) = 1, k 2(u)k j l(U) = h+(u) (see [16] and [23]). The same procedure can be applied to R(F). We have

L(u) = (p(1)(u)

==

Q9

id)(F 2I R+RoR_r l ) ==

(~~" ~) (-e~(u) ~) (kl~U) k2~U)) (~ -e~t(U))

(1'!

T~~) == (7.2)

=(

1

~h" - et(u)

0 ) ( kl (u)71 1 0

1) (10

0 k 2 (u)T;>

-T;;~e~,,(U)T;t) . 1

(7.3)

S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy

70

Remark. Strictly speaking, the dccomposition (7.3) is valid in the algebra A defined by the relation (6.7), but simultaneously the formula (7.3) shows that all the generators {h", c,,, 1;'1, co_a} of the deformed Yangian y,1.{($!2) can be expressed in terms of the I.-operator (6.:J), what proves that the homomorphism constructed in Theorem 6.1 is an epimorphism. The Gauss decomposition (7.3) provides the following choice of generators for Yry.{("!2): ht(u)

= 1;,-t ht(u)T,-;t

. et(u)

= et(u) - ~hcx , e~cx(u) = T:te~cx(u)1~t.

(7.4)

Using relations (5.1 )-( 5. 7) one can obtain thc following relations betwecn -

±I

hcx(u), e±cx(u) and 1;'

2:

T:th-:;(u)T: t = 1;;l et(u)T: 1 = et(u) =F 217~

(I ±17~e~a(u)r1i!t(u)(1

±17~e~cx(U)rJ,

(Vi)

± 217~(1 ± 17~e~cx(u)tl ht(u)(1 ± 17~e~cx(u)rl,

(7.6)

T:te_cx(u)T: t = e_cx(u)(1 ±17~e_cx(u)r1.

(7.7)

With the help of the relations (5.1)-(5.7) and (7.5)-(7.7), we can prove the following thcorem. Theorem 7.1 Defining relations for the "field,," i!t(u), e!,,(u) have the form: Hdu)h~(v) = H1(V)i!~(u) ,

(7.8)

r 217~(et(

[et(u), et( v)] = 1/ (et(v) - et(u) + u) - et( v)) , v-u (u - v + 17)91(u)e~cx(v) - (u - v -17)gl(v)e~cx(u) ==

(7.9)

17(gl(u)e~cx(u)) + gl(v)e~cx(v)) ,

(7.10)

(U-1J-17)Hl(u)e~cx(v)-(u-V+1/)gl(V)ht(u)

== 17Hl(v)e~cx(u)+17g1(U)i!cx(v),

( e~(u) - 217~ + 2TI~H2(U))92(V) - 92(v)et(u)

= -

1/(H2(u) - H 2(v)) u-v

(7.11)

+ 217~92(v)

,

(7.12)

(u - v

+ 17)(e~(v) =

217~ + 217~H2(v))H2(u) - (u - V -17)H2(u)

17H2(v)e~(u)

where H 1(u) =

=

+ 17(e~(u) - 21/~ + 2TI~H2(U))H2(U)

,

(7.13)

(I - 217~e~cx(u)rlht(u)(1 - 217~e~,,(u)rl

,

(7.14)

H 2(u) = (I -17~e~a(U)r\t(u)(1 -17~e~,,(U)rl , 91(U)

= (I - 21IU~a(u)r1e~,,(u)

•

92(U)

= (I -17~e~cx(u)rl e~cx(u).

(7.15)

(7.16)

Rational solutions of Yang-Baxter equation

71

Proof. Let us prove for instance the formula (7.10). From (5.6) we have

Substituting c_et(u)

1]~C~,,(u)rl (see 8.

=

1

1

1

I

TJc+et(u)TJ and using that T;;2e~et(u)TJ

=

C~et(u)(l -

(7.7)), we obtain the formula (7.10).0

Appendix. An integral associative formula and quantization of Poisson brackets

Let M be a smooth manifold equipped with a Poisson bracket. To quantize this bracket means to find a family of new operations * on C=(M) depending on a formal parameter h ("Planck constant") such that: (i) f * 9 = f 9 + ~ih{f, g} + L~2 hkFkU, g), where Fk are bidifferential operators on C=(M); (ii) the operation * is associative; (iii) it is a distributive operation with respect to addition; (iv) for any a constant real function a E R is valid the relation a * f = f * a = af. We are interested in the following case: M = £*, where £* is the dual space to a real Lie algebra £, and M is equipped with a Poisson bracket {".} B, where B is a 2-cocycle on £. Let us defiue the following bracket {".} B:

{f,g}B(X)

= (J,g](x) + BU,g)

,

(8.1 )

where f and 9 are two linear functions on £* and we extend the definition using the Leibnitz rule. We do not claim that B is non-degenerate. The following lemma is valid.

Lemma 8.1 The bracket (8.1) is a Poisson bracket.

Proof. The Leibnitz rule {fg, h}H = f{g, h}B + g{f, h}B follows from definitions immediately. Further, it is sufficient to check the .Jacobi identity for linear functions. The latter is straightforward since BU, g) = const and hence {BU, g), h} B = O. Thus, the ,Jacobi identity for {.,.} B follows from the Jacobi identity for £ and the 2-cocycle property for B. Let Ch(x, y) be the Campbell-Hausdorf series, uniquely defined by the relation chx . chy = ehCh(x,y), where x and yare elements of the Lie algebra £. The function Ch(x,y) has a form Cd:r, y) = x

h

h2

+ y + "2[x, y] + 12([~' [C 1]]] + [1], [1],~]]) + ...

(8.2)

and satisfit's tlw following associativity condition (8.3)

S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy

72

It is well- known that G" (x, y) is a commutator series, namely

where [XI'" Xn] == ~[ .•• [Xl, X2]X3,'" xn]. Define a form B" (x, y) with values in R as follows

h2

h

B,,(x,y)== 2B(x,y)+ 12(B(x,[x,y])+B(y,[y,x]))+...

(8.5)

In other words, we substitute ~ B([[Xh X2] ... Xn-I], xn) instead of [XI, ... ,x n] in the Campbell-Hausdorf series. The following proposition takes place. Proposition 8.1 The form (8.5) satisfies the relation

B,,(C,,(x,y),z)

+ B,,(x,y) == B,,(x,C,,(y,z)) + B,,(y,z)

(8.6)

for any x, y, z E L. Proof Consider the central extension LB of the Lie algebra.c. As a vector space 2" L CD R and admits the following commutator

LB

(8.7)

[(a,A), (b,JI)] == ([a,b],B(a,b)).

It is easy to check that (8.8) Then

A :==

(exp"(x,O) . exp"(Y,O))

. exp"(z, 0) ==

exp("C\(x,y),O) . exp"(z,O)

. exp(O,Rh(X,Y))

(8.9)

,

i.e., exp(O,Bh(a,b)) is a central element of the group G(LB) and hence the clement A has the form A == (exp("Ch(Ch(X,y),z), expBh(Ch(X,y),z)+Bh(X,y)) . (8.10) The statement of the Proposition follows from the associativity in the group As a consequence of this Proposition we have the following result.

G(LB ).0

Corollary 8.1 Let f,9 E COOte), then the following convolution

f *9

==

Jj(Og(

1]) exp'<X,Ch(C~»

.

exp'Bh(e,~) d~d1]

(8.11 )

is an associative operation on COOte), which quantizes the bracket {', .} B. Here is the Fourier transform of the function f.

i(O

Rational solutions of Yang-Baxter equation

73

Proof It is not difficult to see that the operation (8.11) is an associative operation. Indeed, we have

Further It is obvious that

Therefore, we have

(f * g) * h

=

J

j().)g(Jl )h( 1])' expi<X,Ch(Ch(A,/l),ij)> . expiBh(Ch(A,/l),ij) . expiBh(A,/l) d>.dJId1] (8.14)

and Proposition 8.1 implies associativity. Further, if we decompose f * 9 in the series in h, the constant term is

Jj(<)g(1])expi<x,~+ij> d~d1]

=

f(x)' g(x)

(8.15)

and the coefficient at the linear term is equal to

~

J

j(<)g(1])( < x, =

z

[~, 1]] > +B(~, 1])) expi<x,e+ij> d~d1] =

2"( < x, [\7 f, \7g] > +B(\7 f, \7g) .

(8.16)

Clearly, it is nothing but ~([J,g](x) + B(f,g)) if f,g are linear functions. The proof is completed.D Example. Let.c be arbitrary and B = O. Then this formula provides a quantization of the Lie-Kirillov-/<,:ostant bracket (see [IB}). Now we define an action of C(.c) on C, which depends on B satisfying the cocycle condition of Theorem 1.2. Consider £B as in Proposition 8.1. Then £*a contains a hyperplane .ci = {h E £*a : h(O,I) = I}. Clearly.ci is parallel to .c~ = {h E £*a : h(O,I) = O} and .c~ is nothing but C. It is easy to see that .c'B = .c~ EB R and let p be the projection from .c~ onto .ci along R. Then define PB(g) E Diff(.c~) as follows:

Here Diff(.c o) is the set of diffeomorphisms of .co; 9 = (9, A) E G(£B), where A is an arbitrary element of R', "y E G(.c) and the action docs not depend on A. Moreover, PB is well-defined since (0,1) E Center(£B)'

Lemma 8.2 (8.17)

74

S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy

The Proof is straightforward. Remark. The action PR is non-linear unless B the coadjoint representation of G(£).

= O.

If B

= 0,

po is nothing but

Theorem 8.1 ([22]). (i) Let B be non-degenerate. Then PH acts locally transitively at the point (0,0) E and hence, defines an equivariant local diffeomorphism between G(£) and £'0 (equivariancy should be undel'stood with respect to right shifts on G(£) and PB on £0)' This local diffeomorphism enables one to carry the *-operation from £0 ~ L* to G(£). (ii) Such obtained *-operation on C00( G(£)) uniquely defines

Co

PI E (U(£)[[h]])0 2

and f*g = m(Fd-I(f®g), where f,g E COO(G(L)), 12 23 (iii) (.::1 09 I)(Fd- 1 . (F)-1 = (I QS:.::1 )(Fd- 1(F)-I.

m is the usual multiplication.

l'\ow we assume that £ is a Frobenius Lie algebra, which means that there exists such that 1[( x, y]) is a non-degenerate 2-form on £.

J E L*

Theorem 8.2 ([18]). Let £ be a Frobenius Lie algebra with J E L* and B = O. Then Po acts locally transitively at the point J E L* and uniquely defines r2 satisfying conditions 2) and 3) oj Theorem 8.1. Remark. In both cases F was computed (partially)

Acknowledgments

The authors are thankful to the Swedish Academy of Science for the support of the visit of the first and third authors to the Royal Institute of Technology (Stockholm), during which visit the paper was completed. The first and third authors would like also to thank the Russian Foundation for Fundamental Research, grant 95-01-00814A, and the ISF, grant MH1300, for the financial support.

Rational solutions of Yang-Baxter equation

75

REFERENCES [1] A. Stolin, TRITA-MATH 1994 0042 preprint, Stockholm, 1994. [2J S.M. Khoroshkin, A. Stolin and V.N. Tolstoy, TRITA-MATH 1995-MA-17 preprint, Stockholm, 1994; TRlTA-MATH 1994 0042 preprint, Stockholm, 1994; Lett. Math. Phys. (1995) (to appear). [3] V.G. Drinfeld, Soviet Math. Dokl. 32 (1985) 254. [4] V.G. Drinfeld, Soviet Math. Dokl. 36 (1985) 212-218. [5J D. Uglov and V. Korepin, Phys. Lett. A 190 (1994) 238. [6] D. Uglov, preprint of State University of New York at Stony Brook, (Sept. 1994). [7] K. Hikami, Phys. Lett. A 197 (1995) 393. [8] K. Hikami, J. Phys. A: Math. Gen. 28 (1995) L131. [9] F.D.M. Haldane, Z.N.C. Ha, J.C. Talstra, D. Bernard and V. Pasquier, Phys. Rev. Lett. 69 (1992) 2021. [10J D. Bernard, M. Gaudin, F.D.M. Haldane, and Y. Pasquier, J. Phys. A: Math. Gen. 26 (1993) 5219. [11] T. Nakanishi, Nucl. Phys. B439 (1995) 441. [12] A. LeClair and F. Smirnov, Int. J. Mod. Phys. A7 (1992) 2997. [13J A. Molev, M. Nazarov and G. Olshanski, preprint of Australian National University, Canberra, 1994. [\4] A. Molev, J. Math. Phys. 36 (1995) 923. [15J M. Nazarov and V. Tarasov, Pub/. RIMS. Kyoto Univ. 30 (1994) 459. [16] S.M. Khoroshkin and V.N. Tolstoy, Lett. Math. Phys. (1995) (to appear). [17] A. Stolin, Math. Scand. 69 (1991) 57. [18J M. Gekhtman, A.Stolin Proceedings of the workshop in algebra "Moscow-Tainan" (1996) (to appear). [19] M. Gestenhaber, A. Giantino and S.D. Schack, in Lecture Notes in Math. on Quantum Symmetry, 1510 (1992) 9. [20J L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan, AIgebm Analysis, 1 (1989) 178. [21] V.G. Drinfeld, Leningmd Math. J. 1 (1989) 1459. [22] V.G. Drinfeld, Soviet Math. Dokl. 28 (1983) 667. [23] J. Ding and LB. Frenkel, Commun. Math. Phys. 156 (1993) 277. [24J Ch. Ohn, Let. Math. Phys. 25 (1992) 85.

Chapter II. QUANTUM GROUPS: APPLICATIONS

From Field Theory to Quantum Groups

LARGE N MATRIX MODELS AND Q-DEFORMED QUANTUM FIELD THEORIES LYA.AREF'EVA

Steklov Mathematical Institute, Vavi/ov 42, eSP-I, 117966, Moscow, Russia e-mail: arefevaatarevol.mian.su

Abstract: Recently it was shown that an asymptotic behaviour of SU(N) gauge theory for large N is described by q-dcformed quantum field. The master fields for large N theories satisfy standard equations

of relativistic field theory but fields satisfy q-deformed commutation relations with q = O. These commutation relations are realized in the Boltzmannian Fock space. The master field for gauge theory does not take values in a finite-dimensional Lie algebra however there is a nonAbelian gauge symmetry. The gauge master field for a subclass of planar diagrams, so called half-planar diagrams, is also considered. A recursive set of master fields summing up a matreoshka of 2-particles reducible planar diagrams is briefly described.

1.

Introduction

With great pleasure I dedicate my paper to Jurek Lukierski. He made a great contribution in quantum field theory. I hope that this paper will be interesting for him sincc his last interests are relatcd with q-deformed algebras [1,2]. In thc last years there were considerations of models of quantum field theory based on quantum groups and q-deformed commutation relations [1]-[20]. The main problem with these approaches was how to find a physical justification for q-dcformed quantum field theory. Recently, it was shown that there is a remarkable physical justification for q-deformcd quantum field thcory at least for q = O. In [21] it was shown that such theory describes an asymptotic behaviour of SU(N) gauge theory for N ---> oc.

80

LYa. Arcf'eva

The large N limit in QeD where N is the number of colours enables us to understand qualitatively certain striking phenomenological features of strong interactions [22]-[25]. To perform an analytical investigation one needs to compute the sum of all planar diagrams. Summation of planar diagrams has been performed only in low dimensional space-time [26,27]. It was suggested [24] that there exists a master field which dominates the large N limit. There was an old problem in quantum field theory how to construct the master field for the large N limit in QCD. This problem has been discussed in many works, see for example [29]-[35]. More recently the problem has been reconsidered [36][46] by using methods of non-commutative (quantum) probability theory [41,47,48]. Gopakumar and Gross [36] and Douglas [37] have described the master field using a knowledge of all correlation functions of a model. Finally the problem of construction of the master field has been solved in [21]. It was shown that the master field satisfies to standard equations of relativistic field theory but it is quantized according to q-deformed (q = 0) relations

a(k)a*(k l ) = 8(k - kl ), where a(k) and a*(k) are annihilation and creation operators. These operators have a realization in the free (Boltzmannian) Fock space. Therefore, to study the large N limit of QCD it seems reasonable to develop methods of treatment of theory in the Boltzmann space. Let us note that the fact that the master field satisfies the same equations as usual relativistic fields makes actual a development of non-perturbative methods of investigation of these equations. Quantum field theory in Boltzmannian Fock space has been considered in [43][46]. Some special form of this theory realizes the master field for a subset of planar diagrams, for the so called half-planar (HP) diagrams and gives an analytical summations of HP diagrams [4:l]-[46]. In this paper a realization for the master field for lIP diagrams of gauge theories will be given. Then some recursive set of master fields summing up a more rich subclass of planar diagrams will be also sketched. This subclass of diagrams contains a matreoshka of 2-particles reducible planar diagrams. In Section 2 consider the master field for matrix models including the master field for gauge field will be considered. In Section 3 Boltzmann quantum field theory and the lIP master field for scalar and gauge theories will be considered.

2. 2.1.

Master Field for Planar Diagrams Zero-dimensional Master Field

The master field <1> for the Gaussian matrix model in zero dimensional space-time is defined by the relation (2.1 )

Large N Matrix Models for k = 1,2... , where the action S(M) = ~ tr M matrix. The operator !/J = a + a'

2

81

and M is an Hermitian N x N

acts in the free or Holtzmannian Fock space over C with vacuum vector aiD

>= D

10 > (2.2)

and creation and annihilation operators satisfying the relation aa' = 1

(2.3)

Let us recall that the free (or Boltzmannian) Fock space F(ll) over the Hilbert space H is the tensor algebra over H

Creation and annihilation operators are defined as

u'(f)II Q9 ... Q9 In = I Q9 II

(9 ...

(0 In

a(f)II 0 ... 0 In =< I, II > 0h 0 ... 0 In where < 1,9 > is the inner product in H. One has

a(f)a'(9) =< 1,9> . Here we consider the simplest case H = C. The relation (2.1) has been obtained in physical [29] and mathematical [47] works. It can be interpreted as a central limit theorem in non-commutative (quantum) probability theory, for a review see [41]. The basic notion of non-commutative probability theory is an algebraic probability space, i.e a pair (A, h) where A is an algebra and h is a positive linear functional on A. An example of the algebraic probability space is given by the algebra of random matrix with (2.1) being non-commutative central limit theorem. As another example one can consider quantum groups. Theory of quantum groups have received in the last years a lot of attention [49]-[52]. In this case A is the Hopf algebra of functions on the quantum group and h is the quantum I1aar measure. The relations of theory of the master field in the Boltzmannian Fock space with quantum groups was discussed in [42].· There we defined the master field algebra and showed that this algebra is isomorphic to the algebra of functions on the quantum semigroup SUq (2) for q = D. In fact the master field algebra coincides with the algebra of the so called central elements of the quantum group Hopf algebra. Let us

82

I.Ya. Aref'eva

repeat the main steps of these observations. In the Boltzmannian Fock space F( C) we have (2.4) a'a == 1 -10 >< 01. Let us define an operator

F == c'IO >< 01, where r/J is an arbitrary real number. following relations:

(2 ..5)

Then from (2.2), (2.4), (2.3) one has the

aF == 0, aF' == 0, F F' aa'=I, a'a+FP'=1.

= p' F,

(2.6)

We call the algebra (2.6) the master field algebra. From equations (2.6) we get

and the operator P F' is an orthogonal projector. Now let us recall the definition of algebra of functions A q = Fun(SlJq (2)) on the quantum group ,c..;Uq (2). The algebra Aq is the Hopf algebra with generators a, a', c, c' satisfying the relations

°

ac' = qc'a, ac = qca, cc' = c'c, a'a + cc' = 1, aa' + q2cc' = 1

(2.7)

°

where < Iql < 1. Taking q = in (2.7) one gets the relations (2.6) if F == c. Therefore the master field algebra (2.6) is isomorphic to the algebra A o of functions on the quantum (semi) group SUq (2) for q 0. A q is a Hopf algebra with the standard coproduct,

=

,1 :

Aq

->

Aq

~

Aq

,1(gj) = L gl ® gJ. k=O,l

The Boltzman field 1> is a central element of the Hopf algebra A g • The bosonization of the quantum group SUq (2) [20] gives bosonization for the master field. If band b' are the standard creation and annihilation operators in the Bosonic Fock space, [bY] == I, blO >= 0, then 1a

==

q2(N+l)

N

+1

b, c =

satisfies the relations (2.7). Here N = b'b, r/J is a real number. If q (2.8)

a=

1

~b,

vN+ 1

c= e

i,p

(2.8)

e'qN

10 >< 01.

---->

°one gets from (2.9)

Large N Matrix Models

83

Therefore the master field takes the form

1> = b'

1

+

1

IN+T IN+T

b.

(2.10)

The operator N can be also written in terms of creation and annihilation operators a+, a, N = L::;;"=l(a+)k(al.

2.2.

Master Field as a Classical Matrix

Let us consider UrN) invariant correlation functions for a model of selfinteracting Hermitian scalar matrix field M(x) = (Mij(x)), i,j = 1, ... , N in the D-dimensional Euclidean space-time (2.11 ) where S(M) is the action

and M is an Hermitian N x N matrix, (2.12) Witten suggested [24] that there exists a master field which dominates in the large

N limit of invariant correlation functions of a matrix field, i.e.

where M is some 00 x 00 matrix. Since 00 x 00 matrix can be considered as an operator acting in an infinite dimension space one can interpret the RHS of (2.13) as an expectation value of the product of some operators 1>(x;) (2.14) This interpretation gives an alternative definition of the master field 1>(x) as a scalar operator which realizes the following relation (2.15) where < . > means some expectation value. Therefore the problem is in constructing of a scalar field 1> acting in some space so that the expectation value of this field reproduces the large N a'iymptotic of UrN) invariant correlation functions of given matrix field.

I. Va. Aref'eva

84 2.3.

Free Master Field

To construct master field for the free matrix field let us calculate the expectation value for free matrix field in the Euclidean space-time (2.16)

where the action

We have 1

N3 < ir (M(xIlM(X2)M(X3)M(X4)) D(xI - x4)D(X2 - X3)

D(XI - X2)[)(X3 - X4)+

>(0)=

1

+ ND(xI

(2.17)

- x3)D(X2 - X4),

Here we use

< Mij(x)Mj'i'(Y)

>(0)=

bii,bjj,D(x - y),

(2.18)

D is an Euclidean propagator, dD k

D(x - y) =

eik(x-y)

J(21l')D k2 +

m2

(2.19)

Let (2.20) be the Holzmann field with creation and annihilation operators satisfying the relations rp-(x)¢i+(y) = D(x - y),

(2.21 )

It is easy to check that

where [0 > is a vacuum ,p-(x)IO >= 0 =< OI4>+(x). The similar relation is true for an arbitrary n-point correlation function. This consideration proves that the Euclidean Holtzmann field is a maste~ field for the Euclidean free matrix model. Moreover, if we assume the relations (2.23) then we get (2.24)

Large N Matrix Models

D(XI - X2)f)(.T3 - X4)

+ D(x]

- x4)D(X2 - X3)

85

+ qD(xI

- x3)D(X2 - X4),

i.e. relation (2.24) reproduces (2.17) if we identify 1

(2.25)

N = q.

We can also consider the Minkowski space time. To avoid misunderstanding we use a notation M(in) for the free Minkowski matrix field. One has (2.26) where

/F(J:) =

('2~)3

J

o eikxO( _k )b(k 2

-

2

m )dk.

onsider the free scalar Boltzmannian field .p(in)(x) given by

.p(in)(x) = _1_

(21r )3/2 where w( k) = y'P

+ m 2•

J~(a'(k)eikX + J2W( k)

a(k)e- ikx ),

(2.27)

It satisfies Klein-Gordon equation

(0

+ m 2 ).p(in)(x) =

0

and it is an operator in the Boltzmannian Fock space with relations

a(k)a*(k') = 8(3)(k - k')

(2.28)

and vacuum 1.00), a( k) 1.00) = O. A systematical consideration of the Wightman formalism for Boltzmannian fields is presented in [44]. a(k) and a*(k) act in the Boltzmannian Fock space T(H) over 11 == U(R3 ) and one uses notations such as aU) == J a( k )f( k )dk. An n-particle state is created from the vacuum lila) == 1 by the usual formula

but it is not symmetric under permutation of ki . The the following basic relation takes place lim _1_. < 01/1' ((M(in)(y!lJPl ... (M(in)(Yr)yTJIO > NI+,

(2.29)

N-oo

= (Jlol( .p(in) (yd )PI ... (.p(in)( X T lJPT lila)

where k = PI +... + pro To prove (2.29) one uses the Wick theorem for the Wightman functions and 't Hooft's graphs with double lines. According to the Wick theorem we represent the vacuum expectation value in the L.II.S. of (2.29) as a sum of't lIooft's graphs with the propagators ('2.26). Then in the limit N --+ 00 only non-crossing (rainbow) graphs are nonvanished. We get the same expression if we compute the R.H.S. of (2.29) by USiIlg the relations (2.28), i.e. by using the Boltzmannian Wick theorem.

I. Ya. Aref'eva

86

2.4.

Master Field for Interacting Matrix Scalar Field

To construct the master field for interacting qualltum field theory [21] we have to work in Minkowski space-time and use the Yang-Feldman formalism [53]-[56]. Let us consider a model of an Hermitian scalar matrix field M (x) == (M i ) (x)), i, j == 1, ... , N in the 4-dimensional Minkowski space-time with the field equations

(0

+ m 2 )M(x) == J(x)

(2.30)

We take the current J(x) equal to (2.31) where 9 is the coupling constant but one can take a more general polynomial over M(:r). One integrates eq (2.30) to get the Yang-Feldman equation [5:1,54]

M(x) = M(in)(x)

+

J

f)ret(x - y)J(y)dy

(2.32)

where Drct(x) is the retarded Green function for the Klein-Gordon equation,

and MUnl(x) is a free Bose field. The U(N)-invariant Wightman functions are defined as

W(XI,···,Xk) =

1

-k

NJ+'j

< Oltr(M(xd···M(Xk))IO >

(2.:13)

where 10 > is the Fock vacuum for the free field M{in) (x). We will show that the limit of functions (2.33) when N -+ 00 can be expressed in terms of a quantum field ¢>(x) (the master field) which is a solution of the equation

¢>(x) == ¢>(in)(x)

+

J

nret(x - y)j(y)dy

(2.34)

where (2.35) The master field 4>( x) does not have matri x indexes. The following theorem is true. Theorem 1. At every order of perturbation theory in the coupling constant one has the following relation (2.36)

where thf field M(J;) is dffined by (2.:12) and ¢>(x) is defined by (2 . .'14). The proof of the theorem can be found in [21].

Large N Matrix Models 2.5.

87

Gauge field

In this section we construct the master field for gauge field theory. Let us consider the Lagrangian I 2 I 2' L = l7' { -';/"V - 2a(o"A,,) +co"V'"c} (2.37) where A" is the gauge field for the SU(N) group, c and c are the Faddeev-Popov ghost fields and Q is a gauge fixing parameter. The fields A", c and c take values in the adjoint representation. Here

9 is the coupling constant. Equations of motion have the form

(2.38)

One writes these equations in the form

(2.:m) Dc

where

= J,

Dc

= J,

J V = -~o,,[AI" A v]- ~[AJ.l' }'"v] - ~Ovcc - ~efM.,

N2

N2

J =

-~o"[A,,,c], J N2

N2

=

N2

-~[A",o"c] N2

From (2.:39) one gets the Yang-Feldman equations (2.40)

c(x) = c(in1(x)

+

J

Dret(x - y)J(y)dy, c(x) = c(in)(x)

where

D::t(x) = (g"v - (I _

+

J

Q)o~V)lyet(x),

and 9"v is the Minkowski metric. Free in-fields satisfy

Dret(x - y)J(y)dy,

I. Ya. Aref'eva

88

and they are quantized in the Fock space with vacuum 10 >. The vector field A~n) is a Bose field and the ghost fields e(in), e(in) are Fermi fields. Actually one assumes a gauge Q = 1. In a different gauge one has to introduce additional ghost fields. We introduce the notation 1j.'i = (A", c, c) for the multiplet of gauge and ghost fields. The U(N)-invariant Wightman functions are defined as (2.41) We will show that the limit of functions (2.41) when N --> OC! can be expressed in terms of the master fields. The master field for the gauge field A,,(x) we denote BJ1.(x) and the master fields for the ghost fields c(x),c(x) will be denoted ry(x),ry(x) . The master fields satisfy to equations

(2.42) where (2.43) These equations have the form of the Yang-Mills equations (2.39) however the master fields B", ry, ry do not have matrix indexes and they do not take values in a finite dimensional Lie algebra. The gauge group for the field B" is an infinite dimensional group of unitary operators in the I30ltzmannian Fock space. Equations (2.42) in terms of currents read (2.44)

Dry=j,Di/=j, where

jv = -ga"[B,,, Bv]- g[13", F"v]- gavi/ry - gryovi/, j

= -go"[B,,, ry], J = -g[B", O"i/].

We define the master fields by using the Yang-Feldman equations

B,,(x) = T/(:r)

= ry(in)(x) +

J

13~m)(x) + D::t(x - y)jv(y)dy,

JDTet(x - y)j(y)dy,

ry(x)

= ii(in)(x) +

(2.45)

JDTet(x - Y)J(y)dy,

The in-master fields are quantized in the I30ltzmannian Fock space. For the master gauge field wp have (2.46 )

Large N Matrix Models

89

where f~.\l( k) are polarization vectors and annihilation and creation operators satisfy (2.47) The expression (2.46) for the field BI'(x) looks like an expression for the photon field. However because of relations (2.47) the commutator [HI'(x), B,,(x)] does not vanish and it permits us to develop a gauge theory for the field BI'(x) with a non-Abelian gauge symmetry. We quantize the master ghost fields in the Boltzmannian Pock space with indefinite metric 3 (inl( ) _ 1 d k ( *(k) ikx 4(k) -ikx) (2.48) 11 x - (271-)3/2 e + fJ e , 7](in l (x)

= _1_

(271" )3/2

JJ2TkT' J~((3*(k)eikX + J2TkT

,(k)e- ikx ),

where creation and annihilation operators satisfy

,(kh*(k')

= 5(3 l (k -

IJ(k)(3*(k') =

_t5(3 l (k

k'), - k').

(2.49)

We also assume that the prodnct of any annihilation operator with a creation operator of a different type always is equal to zero, i.e.

,(k){r(k') a(.\l(k){r(k')

= (3(kh*(k') = a(.\)(kh*(k) = 0, = ,(k)a(.\)*(k') = (3(k)a(.\)*(k') = 0.

(2.50)

The Boltzmannian Pock vacuum satisfies

(2.51) Let us denote Xi = (HI" 1], Ti) the multiplet of the master fields. The following theorem is true. Theorem 2. At evcry order of perturbation theory in the coupling constant one has the following relation

whlTc the fields AI'(x),c(x) and c(x) are defined by (2.40) and HI'(x),1](x) and Ti(x) are defined by (2.45). The proof of Theorem 2 is analogous to the proof of Theorem 1. We get relations (2.49) for master fields by taking into account the opposite statistics of the ghost fields.

I. Va. Aref'eva

90

Master Field for HP Diagrams

3. 3.1.

Half-Planar Approximation for the One Matrix Model

A free n-point Grccn's function is defincd as the vacuum expectation of n-th power of mastcr field

(3.1) As it is well-known, the Green's function (2.4) is given by a n-th moment of Wigner's distribution [26,47]

dO)

=

2"

j2 d)' ),2"~ = n!(n+I)!' (2n)! -2211'

This rcpresentation can be also obtaincd as a solution of the Schwinger-Dyson equations ,(O) GJ2n

-

-

~

L...J

G(O)

2m-2

G(O)

2n-2m'

m=l

Interacting Boltzmann correlation functions are defincd by the formula [43] (3.2) In contrast to the ordinary quantum field theory where onc dcals with the exponcntial function of an interaction, here we deal with thc rational function of an interaction. In [45] it was shown that under natural assumptions the form (5.1) is unique one which admits Schwinger- Dyson-like equations. For the case of quartic interaction Si"t = 9<jJ4 the Boltzmannian Schwinger-Dyson equations have the form k-l

G

n

G " = """G(O) ~ k-/-l ..I/+n-k-l + /=1

-g[G n -

k G k+ 2

""" G(O) G ~ /-k-l ..I n +k-I-I

(3.3)

/=k+1

+

G,,-k+l Ch+l

+ Gn - k+ 2 G k + G"-k+3(h-d·

For 2- and 4-point correlation functions we have

(3.4) 3.2.

Boltzmann Correlation Functions for D-DimensionaI Space-Time

Hcre we present the Schwinger-Dyson equations for Boltzmann correlation functions in f)-dimcnsional Euclidean space. To avoid problems with tadpoles let us following [46] Consider the two-field formulation. We adopt the following notations.

Large N Matrix Models

91

Let V,(x) = Jj>+(x)+V'-(J:), ¢(x) =¢+(x)+¢-(x) be the Holzmann fields with creation and annihilation operators satisfying the relations

where D( x, y) is V-dimensional Euclidean propagator. The n-point Green's function is defined by

We define an one-particle irreducible (IPI) 4-point function J'4(x,y,z,l) as

where :F4 is a connected part of F4

r4(x,y,Z,t) = :F4 (x,y,z,t)

+ f 2(x,t)D(y,z).

Kote that in the contrast to the usual case in the RHS of (:J.6) we multiply :F4 only on two full 2-point Green functions while in the usual case to get an I PI Green function one multiplies an n-point Green function on n full 2-point functions. Let us write down the Schwinger-Dyson equations for the two- and four-point correlation functions. We have

r 4 (p, k, r) = -g - 9 .1'2(P) = 9

J

dk'/'2(P + k - k')D(k')/'4(p + k - k', k', 7')

JdkdqFi(k)D(q)D(p - k - q)r (p, k, q). 4

(3.7)

(3.8)

where ,

h

1

=

2

P

j~ +m 2 + ~2

Equation (3.7) is the Bethe-Salpeter-like equation with the kernel which contains an unknown function Fl.. Equation (3.8) is similar to the usual relation between the self-energy function 1.,'2 and the 4-point vertex function for
I. Va. Aref'eva

92

3.3.

A Matreoshka of 2-particles Reducible Diagrams

In [43] has been shown that equations sum up HI' diagrams of planar theory. Let us remind the definition of HI' diagrams. Sometimes they are called the rainbow diagrams. The free rainbow diagrams are dual to tree diagrams and they have been summed up in the zero dimensions [57]. The half-planar diagrams for < ir' (M(x n ), ... M(x n )) > are defined as a part of planar non-vacuum diagrams which are topologicaly equivalent to the graphs with all vertexes lying on some plane line in the left of generalized vertex represented tr (M(x n ), ... M(x n )) and all propagators lying in the half plane, We can use F"2 and F4 to construct correlations functions which correspond the sum of more complicate diagrams. Let us consider the following correlations functions (3.9)

J

i::::4

(1

+ II dDx;l4(x2' X3, X4, XI )lj;1 (xd

: r/J(X2)r/J(X3) : lj;1( X4)tIIO),

t=)

here

(3.10) The Schwinger-Dyson equations for the 2- and 4-point correlation functions rjl) and FP) satisfy t.o equations similar to equations (3.7) and (3.8). The obtained FJ!) and 1,~1) may be used to define the next approximation to planar diagrams. One can see that such procedure sums up special type of 2-particles reducible diagrams. These diagrams are specified by the property that they contain two lines so that after removing these lines from the given diagrams one reminds with two disconnected parts and each of these disconnected part is itself a connected 2-particle reducible diagram. It is natural to call this set as an matreoshka of 2-particle reducible diagrams.

3.4.

Master Field for HP Gauge Theory

In the case of gauge theory the set of the HI' master field is given by the field satisfying the following relations (3.11 )

1 c-(x)c+(y) = (27r)D

1 c-(x)c+ (Y)=-(21f)D A;(x)c+(y)

J J

dDk PI cxpik(x - y),

1 . y ), dD kpexpzk(x-

= A;(x)c+(y) =

°

(3.12)

(3.13) (3.14)

Large N Matrix Models

93

A unrcnormalized interacting Lagrangian which assumed to enter in the correlation functions as is (3.5) (a generalization to two-fields formalism is evident) has the form

4g2{2AvAI'AvAI' - AvAI'AI'Av - AI'AI'AvA v } - g{ Ol'cAl'c

+ cOl'cAI' + Al'col'c -

ol'ccAI' - Al'ol'cc - cAl'ol'c}

(a.15)

Divcrgences in onc-loop correlation functions may be removed by the following renorlIlalizations

2

+~ z1/f'(Zf P

r {2AvAI'AvAI' 1

AvAI'AI'Av - A,.AI'AvA v }-

fLZllP(ZllP)-I OI'C-A I'C - C-A I' 0I'C-) 8 1 2 (3.16) where 2

ZllP = I 1

Z.

HI'

2

ZHP 1

= 1

+ -g-(-O') In A 2 161r

g2 13 +(- 161r2 3

O')ln A

= 1' 2 ZHP = 1 + L(~ 161r2 2

- ~)lnA 2

(3.17) (3.18)

(3.19)

Z2 factors enter in the nonstandard way since only two legs (the first and the last) may bring the wave function renormalization. So we have a modification in the definition of beta function (3.20) Therefore we have

(3.21) Recall that the IIsual beta function is /3 = -~¥-. Note that we get good results: beta remains negative and in this approximation it does not depend on the gauge fixing parameter 0', i.e. it is gauge invariant.

94 4.

LYa. Aref'eva Concluding remarks

In conclusion, models of quantum field theory with interaction in the Holtzmannian Fock space have been considered. To define the master field for large N matrix models we used the Yang-Feldman equation with a free field quantized in the Boltzmannian Fock space. The master field for gauge theory does not take values in a finite-dimensional Lie algebra however there is a non-Abelian gaugf' symmetry. For the construction of the master field it was essential to work ill Minkowski space-time and to use the Wightman correlation functions. The fact that the master field satisfies the same equations as usual relativistic fields push us to develop a non-perturbative methods of investigation of these equations. :\'ote in this context that in all previous attempts of approximated treatment of planar theory were used some non-perturbative approximation [59 61]. To sum up a part of planar diagrams we have used the new interaction representation with a rational function of the interaction Lagrangian instead of the exponential function in the standard interaction representation. The Schwinger-Dyson equations for the 2- amI -i-point correlation functions for this theory form a closed system of equations. The solntions of thf'se equations may be used to sum up a more rich class of planar diagrams. This is a subject of further investigations. Acknowledgments The author is grateful to P. Medvedev, L Volovich and A. Zubarev for useful discussions.

REFERENCES

[1J

Y. Frishman, J. Lukierski, W.J. Zakrzewski, J. Phys. A: Math. Gen. 26 (199:n 30l.

[2] P. Kosinski, .1. Lukierski, P. Maslanka, and J. Sobczyk

K. "q-Dcformation of Poincare Superalgebra with Classical Lorentz Subalgebra and its Graded Bicrossproduct Structure" J. Phy... A28 (1995) 2255; M. Klimek and .1. Lukierski "K.-deformed realisation of D=4 conformal algebra" Acta Phy... Polan. B26 (199.1) 1209.

[3] I. Va. Aref'eva and I. V. Volovich, Mod. Phy... Lett. A6 (1991) 893.

[4J A.P. lsaev and Z. Popovicz, Phys. Lett. B281 (1992) 27l. [.5J 1. Castellani, Phys. Lett. B292 (1992) 93. [6] V.P. Akulov, V.D. Gershun and A.I. Gumenchuk, Pis'ma Zh. t'ksp. Them'. Fi::. 56 (1992) 177. [7] K. Wu and R.J. Zhang, Comm. Thcor. Phys. 17 (1992) 175-182. [8] M. Hiroyama, P1'Og. Theor. Phys. 88, !'Io I (1992).

Large N Matrix Models

95

[9] S. Watamura, "Hicovariant Differential Calculus and q-deformation of Gauge Theory", preprint II D-Til EP-92-45. [10] T. Brzezinski and S. Majid, Phy". Lett. B298 (1993) 339. [Ii] D.D. Coon, S. Yu and M.M. Backer, Phy". Rev. D5 (1972)1429. [12J V.V. Kurishkin, Ann. Found. L. de Broglie 5 (1980) 111. [1:3J 1. Va. Aref'eva and 1. V. Volovich, Phys. Lett. B264 (1991) 62; L Aref'eva, "Quantum Group Gauge theories", in Proc. oJ the First Sakharov Memorial ConJ. (1991); L Aref'eva and L Volovich, Phy". Lett. B286 (1991) 179; LV. Volovich, "Quantum Group Shear', hep-th/9404110. [14] O.W. Greenberg, Phys. Rev. D43 (1991)4111. [15] J. Rembielillski and K.A. Smolinski, Mod. Phys. Lett. A8(1993) 3335; T. Brzezinski, J. Rembielinski, K.A. Smolinski, Mod. Phys. Lett. A8 (1993) 409; J. Remhielinski, preprint, lIEP-TH/9302019 [16] M. Chriaco, Mod. Phys. Lett. A8 (1993) 2213. [17] LYa. Aref'eva and G.L Arlltyunov, Geometry and Physics, 11 (1993) 409-423. [18] A.P. Isaev and Z. Popowicz, Phys. Lett. B307 (1993) 353. [19] A.P. Isaev and P.N. Pyatov, Phys. Lett. A179 (1993) 81. [20] LYa. Aref'eva, K. Parthasaraty, K. Viswanathan and LV. Volovich, Mod. Phys. Lett. A9 (1994) 689. [21] L Aref'eva and L Volovich, hep-th/9510210 (to be published in Nuel. Phys. B). [22] G. t'Hooft, Nuel. Phys. B72 (1974) 461. [23] G. Veneziano, Nuel. Phys. B160 (1979) 247. [24] E. Witten, in: Recent Developments in Gauge Theories, eds. G. t'Hooft et a!., Plenum Press, New York and London 1980. [25] LYa. Aref'eva and A.A. Slavnov, Lectures in the XIV International School oj Young Scicntists, DlIbna, 1980; A.A. Slavnov, Acta Physiea Austriaca, Suppl. 25 (198:3) :3.')7. [26J E. Brezin, C. Itzykson, G. Parisi and J.-B. Zuber, Comm. Math. Phys. 59 (1978) 35. [27J A. Migdal, .n'Tp, 69 (197.')810; B. Rusakov, Mod. Phys. Lett. A5 (1990)695. [28] YII. Makeenko and A.A. Migdal, Nucl. Phys. B188 (1981) 269. [29] O. Haan, Z. Physik C6 (1980) :345. [30] M.B. Halpern and C. Schwartz, Phys. Rev. D24 (1981) 2146.

96

I. Va. Aref'eva

[31] P. Cvetanovic, Phys. Lett. 99B (1981) 49; P. Cvetanovic, P.G. Lauwers and P.N. Scharbach, Nucl. Phys. B203 (1982) 385. [32] LVa. Aref'eva, Phys. Lett. 104B (1981) 453. [33] .1. Greensite and M.B. Halpern. Nucl. Phys. B211 (1983) 343. [34] L.G. Yaffe, Rev. Mod. Phys. 54 (1982) 407. [35] A. Migdal, "Second Quantization of Wilson Loop", hep-th/9411100. [36] R. Gopakumar and D. Gross, Nucl. Phys. B451 (1995) 379. [37] M.R. Douglas, "Stochastic master field", RU-94-81, hep-th/9411025; [38] L Singer, Talk at the Congress of Mathematical Physics, Paris (1994). [39] M.B. Halpern and C. Schwartz, Phys. Rev. D24 (1981) 2146. [40J M. Douglas and M. Li, Phys. Lett. 348B (1995) 360. [41] L. Accardi, Y.Lu, L Volovich, hep-th/9412246 [42] L. Accardi, LYa. Aref'eva, S.V. Kozyrev and LV. Volovich, Mod. Phys. Lett. A10 (1995) 2323. [43] L. Accardi, LYa. Aref'eva and LV. Volovich, preprint CVV-201-95, SMI-05-95, hepth/9502092. [44] L. Accardi, L Aref'eva and L Volovich, "The Master Field in Euclidean and Minkowski Formulation and Free Non-Commutative Random Variables", in preparation. [45] L Aref'eva and A. Zubarev, preprint SMI-35-95. [46] L Aref'eva and A. Zubarev, preprint SMI-36-95, hep-th/9512166. [47] D. Voiculescu, K..J. Dykema and A. Nica, "Free random variables", CRM Monograph Series, Vol. 1, American Math. Soc. (1992); D.V. Voiculescu, Invent. Math., 104 (1991) 201. [48] 1. Accardi, Rev. in Math. Phys. 2 (1990) 127. [49J V. Drinfeld, in Proc. ICM Bcr'keley CA, Providence, RI: AMS, ed. A.M. Gleason (1986) 798. [50] 1.D. Faddeev, N. Reshetikhin and L.A. Tachtadjan, Alg. Anal. 1 (1988) 129. [51] S. Woronowicz, Commun. Math. Phys. 111 (1986) 613; Publ. RIMS: Kyoto Univ. 23 (1986) 117. [52] M. Jimbo, Lett. Math. Phys. 11 (1986) 242. [53] C.:". Yang and D. Feldman, Phys. Rev., 79 (19,')0) 972.

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97

[.54] J.D. Rjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill, New York, 196.5. [.55] FoOl. Dyson, Phys. Rev. 82 (19,')1) 428. [,')6] G. Kallen, Arkivf. Phys.2 (19.51) 371. [,')7] D.V. BouJatov, V.A. Kazakov, I.K. Kostov and A.A. Migdal, Nucl. Phys. B275 (1986) 641. [58] K. Rothe, Nucl. Phys. B104 (1976) 344. [,')9] A.A. Slavnov, Phys. Lett., 112B (1982) 1.54; Theor. Math. Phys., 54 (1983) 46. [60] I.Ya. Aref'eva, Phys. Lett. 124B (1983) 221; Theor. Math. Phys., 54 (1983) 1.54. [61] G. Ferretti, Nucl. Phys. B450 (199.5) 713.

FroIll Field TlJeory to QuantuIll Groups

QUANTUM GROUP COVARIANT SYSTEMS M.

CHAICIIIANt AND

P.P.

KULISHt

t IIigh

Energy Physics Laboratory, Department of Physics, Research Institute for High Encrgy Physics P.O. Box 9 (Siltavuorcnpenger 20C), University of Helsinki Helsinki SF-OOOI4, Finland e-mail: chaichianatphcu.helsinki.fi +St.Petersburg Branch of Steklov Mathematical Institutc Fontanka 27, St.Petersburg, 191011, Russia e-mail; kulislJ atpdmi.ras.ru

Abstract: The meaning of quantum group transformation properties is discussed when instead of the Lie grouJl the corresponding quantum group is used while the algebraic (matrix) form of the transformation looks the same. Different algebras are considered which are covariant with respect to the quantum (super) groups 811q('2), SUq(l, I), SlJq(ljI), SUq(n), SUq(mln), OSpq(112) as well as deformed Minkowski space-time algebras. 1.

Introduction

The transformation properties of physical systems related to the Lie groups are of great importance for the understanding of Nature. As a result, applications of the Lie group theory take place in quite different branches of physics and the corresponding formalism is very well developed. The quantum groups and quantum algebras extracted from the quantum inverse scattering method (QISM) happen to be quite similar or even richf'r mathematical objects as compared to Lie groups and Lie algebras. In this paper we shall point out some peculiarities of the quantum group interpretations when the formal transformations of the physical quantities winside with the usual ones whi Ie the coefficients (elements) of these transformations started to be lIOn-commutative quantities belonging to a quantum group (QG) or a quantum algebra.

M. Chaichian and 1'.1'. Kulish

100

These objects (QG and q-algebras) are described using the language of Hopf algebras. In the general situation of Lie group theory one has the Lie algebra Lie( G) (or better to say the corresponding universal enveloping algebra) with non-commutative multiplication and symmetric coproduct .:1, and the commutative algebra of functions F on the Lie group manifold G with non-symmetric coproduct .:1 : F ---+ F @ F. After a q-deformation (or" quantization") the corresponding objects Lic q ( G) and Fq start to be much more similar in between as the Hopf algebras with non-commutative multiplications and non-symmetric coproducts in both cases. Hence, it looks natural to have the same physical interpretation for transformations including both of them. Putting aside the complicated integrable models solved by the quantum inverse scattering method and the quantum conformal field theory we consider systems with finite degrees of freedom such as a set of q-oscillators A covariant under the coaction 'P of the quantum (super-) group Fq • When the coaction 'P : A ---+ Fq @ A does not preserved the physical observables such as Hamiltonian, momenta, etc, the standard problem of the tensor product decomposition of 11.£0' (g/HA emerges, where 11.£0' and HA are the state spaces of the corresponding algebras. If instead of f~ we have the quantUIIl algebra Lie q ( G) then its representations are almost identical to the undeformed algebra (let us omit plenty of technicalities related with the case when q is a root of unity). However, the deformed algebra of functions F q is a new algebraic object with its own representation theory and the corresponding decomposition problem of 11.£0' !XI HA or 11.£0' ® 11.£0' is new as well.

2.

Covariant systems

2.1. A q as su q (2)-covariant algebra. Let us start with a simple covariant system. The q-oscillator algebra A q has three generators with commutation relations (for q real one has A = qN/2a = qNa )

(1 ) (2) (3) The third set can be obtained with fixed q from the quantum algebra contraction procedureS) (,\ = q _ q-l) 0= lirn)..1/2X+/qs,

S

---+ 00,

N

=s-

s1L q (2)

by a

J.

From this contraction one could find that although there is no natural coproduct for A q , the formulas for the su q (2) coproduct that survive under the contraction procedure could be interpreted as covariance of the algebra A q with respect to the

Quantum Group Covariant Systems

101

quantum algebra 8u q (2) (with generators X+, X _, J and well known commutation relations and coproduct .1). The corresponding map is 1/J : A q -> A q 181 8u q (2), such that

1/J(N) = N - J,

(4)

1/J(at)

= atq-J

+ /).q-N X_.

It is easy to check the following consistency properties for this coaction: (1/Jl8lid)o1/J = (id 181.1) o1/J and (id 181 c) 0 1/J = id as well as that 1/J preserves the defining relations of A q . However, the central element z of the algebra A q

(5) is not invariant under this coaction: 1/J( z) # z. If we choose the Hamiltonian of the q-oscillator as H = at 0' and restrict ourselves to the irreducible representation 'H F of A q with the vacuum state (for q E (0,1) there are other irreps5,8)) then z = and

°

(6) The spectrum of If and its eigenstates are obvious

8pee H

= {[n;q-2J,

n = 0,1,2,., .},

After the coaction the changed Hamiltonian describes an "interaction" of the qoscillator with the q-spin

It acts iu the space of physical states 'H ph = 'HF 181 Vj, where Vj is an irreducible finite dimensional representation of the su q (2) of dimension 2j + 1. This space is decomposed into the direct sum of 2j + 1 irreducible representations of A q : 'Hph = L\ 'H~) with corresponding vacuum states 10 >k, k = 0, 1, ... ,2j k

10 >k=

L

1m> I8Ilj - k + m;j > e(Tn, k) ,

m=O

where 1m >E 'HF , I/;j >E Vj , JI/;j >= lll;j >,

(8)

102

M. Chaichian and P.P. Kulish

m-l

c(m, k) = (_v'1qj-k+l)m([m;q-2]!rl/2

IT

N+(j - k + I, j) .

1=0

The spectrum of HI coincidcs with that of H up to the multiplicative factor q2(J-k) in cach subspace 1t~), but it has the multiplicity 2j + I

spec II -- { q2(j-k) [n,. q-2] ,n -- 0, I , 2,... }. The central element w(z) has 2j + I eigenvalues -[k _ jj q-2]. It is interesting to point out that this coaction has no classical (non-deformed) countcrpart in the limit q --> I in the quantum thcory. Such a limit exists in the Poisson-Lie theory (see e.g. [22]). The connection of the q-oscillator algebra A q with su q (2) through the contraction proccdure gives rise also to a more complicated coaction of thc quantum group SUq (2) on A q (scc Subsec. 2.5 and 2.6).

2.2. A q as SUq(l, I)-covariant algebra. Lct us consider the second set (2) of the q-oscillator algebra A q generators with relation

AAt = qAtA

+ 1.

redenoting q2 by q and putting q E (0, I). This relation reminds of the quantum plane with xy = qyx a central extcnsion. Using the two componcnt column Xl = (A, At) it can be rewrittcn in the R-matrix form 7.10)

(9) whcre if = P R is the R-matrix of su q (2) and J is thc four component column j! = (0, I, -q, 0) obviously relatcd to the well-known q-metric 2 x 2 matrix (q. This relation is prcserved under the transformation W: X ---> w(X) = T X, with I' the 2 x 2 matrix of the quantum group SU q(l, 1) gencrators

1'=(ab* a*b) which satisfies the FRT-relation fiT ® T = l' ® TR [1]. The invariance of the inhomogeneous term is just another form of the q-metric relation

T(qT t

= (qdeiqT,

T I8i T J

= 11l 2) = delqT J

,

(10)

provided that the quantum dcterminant of Tis 1: deiqT = aa* -qbb* = a*a-b*b/q = 1 (a defining condition for ,'iUq ( 1,1)). (One can consider central extension of the real

Quantum Group Covariant Systems

103

quantum plane as well with Iql = I covariant with respect to SL q (2, R). Then the reality condition will fix the phase of the constant term.) The map ljJ : A q --> A q 0 SUq(l, I) or in terms of the generators

(11 ) satisfies all properties of a coaction. Its form is reminicent of the famous Bogoliubov transformation. However, now the" coefficients" are non-commuting. The q-oscillator Hamiltonian acting in the same space 'H F as (6) /I

= AtA = [N; q] = (qN - I)j(q - I)

( 12)

(it differs from the previous one by qN factor and renotation of q2) is also not invariant under the coaction ljJ

(13) The commutation relations of the SUq(1, 1) generators (as well as those of SUq(2))

abO = qb*a,

ab = qba

[a, a*] = >"b*b,

bb* = bOb ,

>..=(q-ljq),

themselves rellli nd of the q-oscillator algebra (a "- at, a* "- a, b "- b* "- q-N) with the additional condition due to delq'l' = 1

aa* =

1+ qb*b,

a*a =

1+ b*bjq.

The deformation parameter q being less than 1 forces us to consider the irreducible representation of SUq(1, I) in the Hilbert space 12(Z) with basis 1m >,m = ... , - 2, -1,0, 1,2.... which consists of the eigenstates of commuting operators b, b* for which a acts as a creation (shift) operator

c~

=

1+ q-2m-l.

lienee the transformed Hamiltonian is defined in the space 'HF (9 12( Z). It has the same spectrum {[n; q], n = 0, 1, ... } with infinite multiplicity. The corresponding vacuum states are

10 >~~)=

f) -1)i xi I2j > 01k - j >,

i=O

(14)

104

M. Chaichian and P.P. Kulish

where the second vector in the tensor product belongs to the SUq(l, I) irrep space /2(Z). As in the case of the representation theory the invariant subspaces of the QG F (coaction) corepresentation V can be defined as W C V such that c/J : W -> F @ W . The invariant elements of the F-corepresentation V do not change at all: c/J( v) = v (or better to write IF ® v for one has the possibility in the corresponding representation of the dual Hopf algebra (F)* to contract a dual element X E (F)* with IF to get a number X(IF)). The extensions of the previous examples to higher rank quantum groups give rise to covariant algebras corresponding to different quantum homogeneous spaces 23 ), systems of (super) q-oscillators 4-6,15) and examples of non-commutative geometry.

2.3. The covariant super-q-oscillator algebra s-A q [4] refers to the quantum supergroup S'Uq(lII), with the T-matrix of the gencrators

and the commutation relations

a(i dfJ fJ, = -,fJ,

= qfJa, = qfJd,

a, d,

(i2 =,2 = 0,

= na, = nd, [a, d] = (q - l/qJifJ.

Fixing the central element (super-determinant) sdetqT = (a - fJh)/d = I one gets a simple relation betwccn the even generators d and a

d = a - fJ,/qd

=a-

fJ,/qa

due to the nil potency of the odd generators. The involution (*-operation) is introduced as follows: d = I/a*, , = dfJ*d = afJ*a . This involution leads to TTt = I and it is consistent with the Z2-grading. One has for the generators u, a*, fJ, (3*

u"fJ = (3a*/q,

[a,a*] = (I-l)fJ*fJ,

aa" = I

+ (3(3*,

fJ(3* = _q2fJ"fJ,

a"a = I - (3* (3 .

Introducing a' = a(1 + (i*fJ/2) = (I - (3fJ"/2)a, one gets a'a'" = a'"a' = I and the factorization of the T-matrix (a' = I/a'" )

Quantum Group Covariant Systems

T

= (a' o

105

0 ) ( (I -/3"/3/2) /3/qa' ) /3" /a'" (I + /3"/3/2)

I/a'"

with unit super-determinant. One concludes that the q-deformation (quantization) of the SU(III) super-group is realized by the unitary scaling operator A, A" = A-I acting on the Grassmann variablies "I and "I" , which are not quantized /3 = ATf ,

A/3

=

q/3A,

A/3"

=

q/3*A,

Al = I

and a' == cxp( i

AAt- q2 AtA=I, AB = qBA,

BBt+StS=I+(l-l)AtA, ABt = qBt A,

B 2 = Bt 2 = 0 .

Using the R-matrix formalism similar to (9) it is not difficult to show that these relations as well as the Hamiltonian H = AtA+BtB are invariant w.r.t. the coaction

However, one can consider Hamiltonians which are not invariant w.r.t. the QG transformation. The latter one will extend the initial system after the transformation. In particular, one can consider different versions of the q-dcformed N = 2-SUSYalgebra taken as the super-charges

where I and It are free fermions I = q-N B commuting with A , At [4]. In all the cases the coaction does not extend the space of states for the representation theory of SUq ( 111) is rather poor and only additional Grassmann parameters appear.

2.4 SlJq(n)- and SUq(mln)-covariant (super) algebras. Let I1S introduce 2n generating elements of the SUq(n)-covariant oscillator algebra A q ( n )5), writing them as n-component column and row

Their commutation relations in the R-matrix form (a spectral parameter independent Zamolodchikov - Faddeev algebrafl

M. Chaichian and 1'.1'. Kulish

106

demonstrate easily that, due to the FRT-relation [R,TI T 2 ] = 0 and TTt = 1 the coaction ¢(A) = l' A, ¢(At) = AtTt satisfies all the requirements. One can rewrite these relations in the form (14) using the 2n-component vector X = (AI, ... , An; At, ..., and the corresponding 2n x2n matrix R, which happens to be the R-matrix of the quantum group Spq(2n). Then the inhomogeneous term (2ncomponent vector J) is expressed using the invariant matrix G of Spq(2n): TIGT =

AD

GI,IO).

The invariant Hamiltonian w.r.t. the SUq(n)-coaction is

which in the Fock space can be written in terms of the mode number operators N k ,k = 1,2, ...

It was already pointed out that the SUq (l, 1) quantum group is related to the qoscillator algebra. The same is true for the SUq(2): its defining relations coincide with (3) up to notations and some factors. A realization of the SUq(n) requires n(n - 1)/2 q-oscillators and n - 1 phase factors I4 ). Hence, transforming the algebra A q(n) -+ S Uq(n) 09 A q(n) one jumps from the n degrees of freedom to the n( n + 1) /2 ones. The situation for the quantum super-group SUq(mln) is similar. The covariant super-algebra s-Aq(mln) has Tn boson and n fermion mutually non-commuting qoscillators 4 ) or vice versa. The realization of the SUq(rnln) requires m(m - 1)/2 + n( n - 1) /2 q-oscillators and rn x n Grassmann paramclers as well as some phase factors. 2.5 The next example of a covariant system is related to the reflection equation algebra K (or the q-Minkowski space-time algebra, or the quantum sphere algebra) (see e.g. [18]). Its quantum group covariance depends on the set of R-matrices in the defining equation (a reflection equation) R(2)/' R (I)/' 12 "I 12 "2

=

I ' 1",(3)/' R(4) "2 "12

"1

(15)

12'

with the coaction cp(]() = ](' = TKS where RU), j = 1, ... ,4 define the commutation relations of the I' and S entries I8). For the simple SUq (2) covariant case one has

R\1) = R\~) = R R\;) = R\~) = R matrix Rand l' = st = S-I. The algebra K 12 ,

with the slq(2)

21

has four generators:

Quantum Group Covariant Systems

0', 13, "

{i

107

with relations

0'/3 = q-2/30', [0, /3] = q-I >'0'13, = q2,O', [0,,] = _q-I>.,O', [,8,,] = q-I >'(0 - 0')0', [0',0] = 0,

a,

and two central elements

One has the covariance of JC with respect to the quantum group SUq (2) with the coaction <.p : JC ---+ SlJq (2) 0 JC which is easy to write using the matrix form

<.p(J<) where

J(

= f(' = UJ
and U = (Ut)-I are the following 2 x 2 matrices of generators

J<_(O' ,

(3) {i ,

a

qb) .

U = ( -b t at

Due to the fact that the q-determinant is equal to one

thc quantum group SlJq ('2) has cssentially one unitary irreduciblc rcpresentation H F 23 ) with vacuum state 10 >: ala >= 0, bla >= 1,

The algebra JC with the *-opcration J< = J
a

/(' = ( -bt

qb) (0' ,t) (at -b) at, 0 qbt a '

hence e.g. Let us consider the very simple (onc-dimensional) irreducible reprcscntation of thc algebra JC : 0' = 0 = 0, , E R. The factor HI is one-dimensional and H F has to be decomposcd into the irreducible representations of )C. To reach this aim one has to find eigenvalues>. and the corrcsponding eigenvectors I>. > of 0" = <.p(O') = qf(ba t + ab t ) in HF. Those of them

M. Chaichian and P.P. Kulish

108

related by IAn+1 >~ 'P(TlIA n >, A,,+I = q2 An give rise to the invariant subspace of H F w.r.t. Je'. The Hermitian operator (bat + ab t ) is a Jacobian matrix with the entries qn c" on the sub-diagonal. Hence the problem of the non-trivial deficiency indecies could take place I6 ). 2.6 A more complicated covariant system is related to the quantum super group

OSPq( 112)2). According to the general arguments of the Introduction, the coaction map gives rise to the extension of the dynamical system and to the representation of the covariantly transformed system in the tensor product with one of the factor being an irreducible representation of the corresponding QG. To find an irreducible unitary representation of the quantum super-group OSPq( 112) one has to introduce a *-operation and to analyse the commutation relations among the generators Tij , i, j = 1,2,3. The matrix T of the OSpq{l12) generators is even and has the dimension 3 in the fundamental representation and the grading (0, 1, 0). The compact form of the quadratic relations among the generators is given by the Z2-graded FRT-relation ( Z2-graded tensor product [2, 3])

ilTiSl1'=T®TR. The osp( 112)- ii-matrix

R has the spectral

R=

qPs - q-I p.,

decomposition [2]

-

q-2 PI ,

where the projector indices refer to the dimension 4s + 1 of the subspaces corresponding to spin 1, 1/2, (see their explicit expresions in [9]). Due to the structure of the R-matrix and the orthosymplectic condition Ts1Cq T = IC q [2], there are only three independent generators among nine entries of T. One can easily see this from the Gauss decompositionl 7) of the matrix l'

°

where TL , Tu are lower and upper triangular matrices with unities on their diagonal and the diagonal factor 1'0 = diag(A, B, C). Among the Gauss decomposition generators one finds three independent ones: A, (1Lb and (Tv h2' while the element B is centraP7). Introducing the nine elements of the Gauss decomposition: Tf) = diag(A,B,C),(liJ21,31,32 = (x,y,z) and (l'uh2,13,23 = (u,v,w) one finds from the FRT-relation 2l : A = U = T12 / qA,

Tn,

x = T2I/A,

v = u 2/w,

y = x 2/w,

w = _ql/2U,

z = X/ql/2,

B = 1'22 - 721 (Tn )-1 TI2 ,

where the elements B = AC = CA are central and w = ql/2 _ q-I/2. Due to the commutativity of 1'13 and T31 which are conjugated to each other 1':l1 = -1'/3 /q according to the *-operation from 2), these elements are diagonal in the

Quantum Group Covariant Systems

109

°

Zrgraded Fock representation rtF with the vacuum: T21 10 >= and the element 1'12 as a creation operator. From the structure of the quadratic relations among the generators Ti/l it follows that the four clements 1'\2, 1'32, 1'\3, 1'31 form a subalgebra of the OSPq( 112)

Hence, the irreducible representation in the Fock space rtF is given by 1'\2 as creation operator and 1':12 as annihilation operator, while

with 7'13 and 7 31 diagonal in the basis In >c::= (1'd n lO >. Let us now define the quantum OSp-plane, which is an associative super algebra A with three generators a, ~, b and the Z2-grading p(a) = p(b) = 0, p(~) = 1. Taking into account the similarity of the quantum super-group OSpq(112) to the symplectic group case [I, 7] the defining relations of A can be written in the R-matrix form with a central extension (16)

where

it is the ospq(112)

R-matrix, X = (a,~,b)t and the nine component vector

J

= (0, _q-I/2,0,0, 1,0,0,ql/2,0)

is the rewritten invariant matrix Cq 2). relations

One has for the generators the quadratic

( 17) where /1 = ql/2 + q-I/2 = >../w. The vector J is the eigenvector of the rank one projector PI which gives rise to the centrality of the element

This central element is invariant under the OSPq( 112) coaction: X ---t T X. The algebra A was identified in [9] as a twisted q-super-oscillator. Although A has the same irreps as (3) with b = at the coaction is more complicated with respect to (11) of the Subsec.2.2 for it includes the number or scaling operator ~ = TJqN as well.

110 3.

M. Chaichian and P.P. Kulish

Conclusion

The problem of the quantum group coaction interpretation and the corresponding tensor product decomposition are especially interesting in the framework of the Poincare group deformation ll - 13). The corresponding quantum group has many generators and rather complicated quadratic relations among them. Even in the very simple (trivial ?) case when the deformation of the Poincare group is given by the twisting 12 ) there are two Weyl generators defining the representation. The Hamiltonian of the relativistic system being only covariant under the group transformation law will get extra degrees of freedom after the quantum group coaction 13). Another kinematical group: the q-Galilei algebra G q , was connected with the X X Z-model dispersion relation due to the eqivalence of the trigonometric function addition law and a non-commutative coproduct 20 ). Realizing the generators of G q in terms of the local spin operators onc can obtain by the duality the quantum group coaction. Further interesting possibilities for the representation theory refer to the case when coproduct or coaction maps the original algebra into a tensor product with non-commutative factors (see e.g. [18,21]). Acknowledgments The authors thank P. Presnajder and M. Scheunert for useful discussions. PPK appreciate the influence and support of the :"Jon-perturbative quantum field theory workshop of the Australian National University and Laboratoire de Physique Theorique et Haute Energie associe au C.i\ .R.S. REFERENCES [I] L. D. Faddecv, ~. Yu. Reshetikhin and L. A. Takhtajan, Alg. Analiz 1 (1989) 178; (Leningrad Math. J. 1 (1990) 193). [2] P. P. Kulish and N. Yu. Reshetikhin, Lett. Math. Phys. 18 (1989) 143.

[3] M. Chaichian and 1'. P. Kulish, Phys. Lett. B233 (1990) 72. [4] M. Chaichian, P. P. Kulish and J. Lukierski, Phys. Lett. B262 (1991) 43.

[5] W. Pusz and S. L. Worollowicz, Rep. Math. Phys. 27 (1989) 231. [6] V. Rittenberg and M. Scheunert, J. Math. Phys. 33 (1992) 436. [7] P. P. Kulish, Phys. Lett. A161 (1991) .50. [8] P. P. Kulish, Theor. Math. Phys. 85 ( 1991) 157; ibidem 94 (1993) 193.

[9] F. Thuillier and J. C. Wallet, Phys. Lett. B323 (1994) 153. [10] P. P. Kulish, Proc. of Varenna School on Quantum Groups: Theory and Applications, to be published by Plenum Press, 1995.

Quantum Group Covariant Systems

III

[II] J. Lukierski, A. Nowicki, 11. Ruegg and V. Tolstoy, Phys. Lett. B264 (1991) 331. [12] M. Chaichian and A. Demichev, J. Math. Phys. 36 (1995) 39R. [13] J. A. de Azcarraga, P. P. Kulish and F. R6denas, Phys. Lett. B351 (1995) 123; hepth/9405161,Fortschr. Phys. (1995). [14] M. Arik, in: Symmetries in science VI, Plenum Press, 199:1, p. 47. [15] M. Chaichian, H. Grosse and P. Presnajder, J. Phys. A27 (1994) 2045. [16] I. M. Burban and A. C. Klimyk, Lett. Math. Phys. 29 (1993) 13. [17] 1::. V. Damaskinsky, P. P. Kulish and M. A. Sokolov, Zap. Nauch. Semin. POMI, 224 (1995) 155; £SI-95-217; q-alg/ 9505001. [18] P. P. Kulish and R. Sasaki, Prog. Theor. Phys. 89 (1993) 741. [19] P. P. Kulish, Alg. Analiz, 6 (1994) 195. [20] F. Boneehi, 1::. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, J. Phys. A25 (1992) 1.939; Phys. Rev. B46 (1992) .5727. [21] T. 11. Koornwinder, in: Orthogonal polynomials: theory and practice, ed. P.1'\evai, :'\ATO AS Serie 1!J91, p. 257. [22] O. Babelon and D. Bernard, Commun. Math. Phys. 149 (1992) 279. [23] L. Vaksman and Va. Soibelman, Func. Analiz Pril. 22:3 (1988) 1; Va. Soibelman, Int. J. Mod. Phys. 7 Supp. 18 (1992) 859.

From Field Theory to Quantum Groups

THE RENORMALIZATION GROUP METHOD AND QUANTUM GROUPS: THE POSTMAN ALWAYS RINGS TWICE MIGl:EL

A. MARTIN-DELGADOt AND GERMAN SIERRA

t

t Departamento de Fisica Te6rica I Universidad Complutense. 28040-Madrid, Spain +lnstituto de Matermiticas y Ffsica Fundamental. C.S.I.C. Serrano 12.1, 28006-Madrid, Spain

Abstract: We review some of our recent results concerning the relationship between the Real-Space Renormalization Group method and Quantum Groups. We show this relation by applying real-space RG methods to study two quantum group invariant Hamiltonians, that of the XXZ model and the Ising model in a transverse field (ITF) defined in an open chain with appropriate boundary terms. The quantum group symmetry is preserved under the RG transformation except for the appearance of a quantum group anomalous term which vanishes in the classical case. This is called the quantum group anomaly. We derive the new qRG equations for the XXZ model and show that the RG-f1ow diagram obtained in this fashion exhibits the correct line of critical points that the exact model has. In the ITF model the qRGflow equations coincide with the tensor product decomposition of cyclic irreps of SUq (2) with q4 = 1. 1.

Introduction

The Renormalization Group method has become one of the basic concepts in Physics, ranging from areas such as Quantum Field Theory and Statistical Mechanics to Condensed Matter Physics. The many interesting and relevant models encountered in these fields are usually not exactly solvable except for some privileged cases in one dimension. It is then when we resort to the RG method to retrieve the essential features of those systems in order to have a qualitative understanding of what the physics of the model is all about. This understanding is usually recast in the form

114

M.A.

~artin-Oelgado

and G. Sierra

of a RG-flow diagram were the different possible behaviours of the model leap to the eyes. Many authors in the past have contributed significantly to the idea of renormalization and it is out of the scope of this paper to give a detailed account on this issue here. We shall be dealing with the the version of the RG as introduced by Wilson [1] and Anderson [2] in their treatment of the Kondo problem, and subsequent developments of these ideas carried out by Orell et al. at the SLAC group [3] and 1'feuty et al. [1]. It was Wilson in the late sixties and early seventies who set up the framework of the method in its more thorough and complete version. According to his own words, he did so in his search for a better understanding of what a quantum field theory is. To this end he also introduced another tool, the Operator Product Expansion (OPE) for the field operators. Both the RG method and the OPE have become two cornerstones in modern Quantum Field Theory. It has been long known that the physics of (1 + 1)-dimensional quantum many-body systems and 21) statistical field theories were very special in many aspects when compared to their higher dimensional generalizations. It was after the seminal work by Belavin, Polyakov and Zamolodchikov (B1'Z) [5] when the special role of conformal symmetry in two dimensions was brought about in connection to those many special properties exhibited by 20 systems such as, criticality, integrability, etc ... The basic tool employed by BPZ to develop their conformal program was precisely one of the tools introduced by Wilson, the OPE, which with the help of two-dimensional conformal symmetry is powerful enough so as to classify the local fields forming the local algebra according to the irreducible representations of the Virasoro algebra and to determine the correlation functions of those fields. Among all the Conformal Field Theories studied by BPZ they singled out what they called Minimal Models as those for which the conformal program is more successful in determining their properties the best. The minimal models, also called Rational Conformal Field Theories (RCFT), are those CFT which contain finite number of primary fields in the operator algebra. For these models the anomalous dimensions of the operators, or equivalently the critical exponents, are known exactly and moreover, their correlations functions can be computed as solutions of special systems of linear differential equations. The underlying phenomenon responsible for such remarkable features is the truncation of the operator algebra, that is, the primary fields form a closed operator algebra. After the introduction of Quantum Groups by Orinfeld [6] in the mid-eighties, some CFT theorists realized the relationship between these new algebraic structures and those appearing in the RCFT. It was shown in reference [7] that the representation theory of SL(2, q) with q a root of unity provides solutions to the polynomial equations of RCFT, as well as a rather efficient way to compute the duality matrices. The rationality condition is met by requiring q to be a root of unity. In this case

The Renormalization Group Method

115

the representation theory of the quantum group is significantly different from the classical one. The first surprise is that when qN = I there are only N - 1 distinct finite dimensional irreps with spins j = 0, 1/2, I, ... ,( N - I )/2, this last one irrep being singular. This truncation of the representation theory of quantum groups was put in correspondence [7] with the previous truncation of the operator algebra found by BPZ in the minimal models. With this historical perspective in mind, what we have suggested by introducing what we call the Renormalization Quantum Group Method (qRG for short) [8] is to establish the connection of the truncation of states characteristic of the Real-Space RG methods (the other tool introduced by Wilson) with the special features of the (1+I)-dimensional systems exemplified by the RCFT. The connection we have found can be cast into the following squematical form, qRG-truncation ...... RCFT whose precise content will be explained in the following sections with several examples. Physicists working in field theory and condensed matter generalized the Real Space Renormalization Group methods introduced by Wilson [I] to other problems by using the Kadanoff's concept of block [3,4]. The Block method (BRG) has the advantage of being conceptually and technically simple, but it lacks of numerical accuracy or may even produce wrong results. For this reason the analytical BRG methods were largely abandoned in the 80's in favor of numerical methods such as the Quantum Monte Carlo approaches. In the last few years there has been new developments in the numerical RG methods motivated by a better understanding of the errors introduced by the splitting of the lattice into disconnected blocks. A first step was put forward in [9] where a combination of different periodic boundary conditions applied to every block lead to the correct energy levels of a simple tightbinding model. This method however has not been generalized to models describing interactions. Recently, we have clarified the role played by the boundary conditions in the real-space renormalization group method [10] by constructing a new analytical BRG-method which is able to give the exact ground state of the model and the correct 1/N 2 -law for the energy of the first excited state in the large N(size)-limit. A further step in the generalization to interacting models was undertaken by White in [II] where a Density Matrix algorithm (DMRG) is developed. The main idea is to take into account the connection of every block with the rest of the system when choosing the states which survive the truncation procedure. The standard prescription is to choose the lowest energy states of the block Hamiltonian. Instead, in the DMRG method one replaces the block Hamiltonian by a block density matrix and chooses the eigenstates of this matrix with the highest eigenvalues. The density matrix is constructed out of the ground state of a superblock which contains the desired block. In these notes we want to explain another RG method, the qRG method, which

116

M.A. Martin-Delgado and G. Sierra

uses the concept of quantum groups. This mathematical notion emerged in the study of integrable systems and it has been applied to conformal field theory, invariants of knots and manifolds, etc. [6,12]. The new application of quantum groups that we envisage has been partially motivated by the aforementioned work of White, Noack and collaborators and it is probably related to it. This relation is suggested by the fact that quantum groups describe symmetries in the presence of non-trivial boundary conditions. The typical example to understand this property of quantum groups is given by the lD Heisenberg-Ising model with anisotropic parameter,1. The isotropic model ,1 = ± 1 is invariant under the rotation group SU(2), but as long as /,11 i- 1 this symmetry is broken down to the rotation group U(1) around the z-axis. One can "restore" this full rotation symmetry by adding appropriate boundary operators to the Hamiltonian of the open chain. The classical group SU(2) becomes then the quantum group SUq (2), where the quantum parameter is related to the anisotropy by ,1 = q+r' [J:I,14]. The "restoration" of a classical symmetry into a q-symmetry is achieved at the price of deforming the algebra and the corresponding addition rule of angular momentum. The q-sum rule, which is called the comultiplication, becomes non local and violates parity. The total raising (lowering) operators acting on the whole chain are a sum of the raising ( lowering) operators acting at every single site times a non-local term involving all the remaining sites which appear in an asymmctric way: sites located to the left or to the right at a given site contribute differently. These features of q-groups made them specially well-suited to implement a RG method which takes into account the correlation betwecn neighboring blocks. In the forthcoming sections we show how this can be done explicitly in two examples in 1D: the Heisenberg-Ising model and the Ising model in a transverse ficld (ITF). This paper is organized as follows. In Sect.2 we present a brief introduction to the Block Renormalization Group methods based upon the concept of the intertwiner operator T. This will allow us to see the truncation procedure inherent to the BRG method as tensoring representations of the Hamiltonian symmetry algebra, the intertwiner operator T being a Clebsch-Gordan operator. In Sect.3 we use the HeisenbcrgIsing model to show how the truncation process pertaining to the rcal-space RG is nothing but a tensor product decomposition of irreps of the symmetry algebra. The properties of the model are qualitative and quantitative well described by the BRG equations in the massive region ,1 > 1, while in the massless region one predicts the massless spectrum but not criticality at each value oj ,1 > 1. This latter fact is rather subtle and elusive. In Sect. 4 we set up the foundations of the qRG method using the Heisenberg-Ising model as an example. We derive the new qRG equations and show that the RG-f1ow diagram obtained in this fashion exhibits the correct line of critical points that the exact model has. Moreover, the qRG equations for the renormalized spin operators show the appearance of a novel feature: the quantum group anomaly. In Sect. 5 we apply the qRG method to another model, the Ising

The Renormalization Group Method

117

model in a transverse field (ITF model). Here, the qRG-How equations coincide with the tensor product decomposition of cyclic irreps of SUq (2) with q4 = 1. Seet.6 is devoted to conclusions and perspectives.

2.

A Brief Review of Block Renormalization Group Methods (BRG)

In this sectiou the block renormalization group method is revisited and we preseut a new and unified reformulation of it based on the idea of the intertwiner operator T to be discussed below. For a more extensive account on this method we refer to [15] and chapter 11 of reference [16] and references therein. The block RG-method is a real-space RG-method introduced and developed by the SLAC group [:1]. Let us recall that Wilson developed his numerical real-space renormalization group procedure to solve the Kondo problem [1]. It was clear from the beginning that one could not hope to achieve the accuracy Wilson obtained for the Kondo problem wheu dealiug with more complicated many-body quantum Hamiltonians such as Heisenberg, Hubbard, etc. ... The key difference is that in the Kondo model there exists a recursion relation for Hamiltonians at each step of the RG-elimination of degrees of freedom. The existence of such recursion relation facilitates enormously the work, but as it happens it is specific of impurity problems. From the numerical poiut of view, the Block Renormalization Group procedure proved to be uot fully reliable in the past particularly in comparison with other numerical approaches, such as the Quantum MonteCarlo method which were being developed at the same time. This was one of the reasons why the BRG methods remained undeveloped during the '80's until the beginning of the '90's when they are making a comeback as one of the most powerful numerical tools when dealing with zero temperature properties of many-body systems. Let us first summarize the maiu features of the real-space RG. The problem that one faces generically is that of diagonalizing a quantum lattice Hamiltonian H, i.e.,

HI1/J >= EI1/J >

(2.1 )

where 11/J > is a state in the Hilbert space H. If the lattice has N sites and there are k possible states per site then the dimension of H is simply (2.2) As a matter of illustratiou we cite the following examples: k = '1 (Hubbard model), k =:l (t-.) model), k = 2 (Heisenberg model) etc. When N is large enough the eigenvalue problem (2.1) is out of the capability of any human or computer means unless the model turns out to be integrable which only happens in some instances in d = 1. These facts open the door to a variety of approximate methods among which the RG-approach is one of the most relevant. The main idea of the RG-method is the

M.A. Martin-Delgado and G. Sierra

118

mode elimination or thinning of the degrees of freedom followed by an iteration which reduces the number of variables step by step until a more manageable situation is reached. These intuitive ideas give rise to a well defined mathematical description of the RG-approach to the low lying spectrum of quantum lattice hamiltonians. To carry out the RG-program it will be useful to introduce the following objects:

• H : Hilbert space of the original problem.

• H': Hilbert space of the effective degrees of freedom. • H: Hamiltonian acting in H.

• ll': Hamiltonian acting in H' (effective Hamiltonian). • T : embedding operator: H' ---; H

• Tt :truncation operator: H ---; H' The problem now is to relate H, H' and T. The criterium to accomplish this task is that If and H' have in common their low lying spectrum. An exact implementation of this is given by the following equation: (2.3)

HT= I'll'

which imply that if IJi E, is an eigenstate of H' then TlJii, is an eigenstate of fJ with the same eigenvalue (unkss it belongs to the kernel of T: TlJii, = 0), indeed, IfTIJi~, = TH'IJi~., = E'TIJi~,

To avoid the possibility that TIJi ' = condition,

(2.4)

°with IJiI i- 0, we shall impose on I' the (2.5)

such that (2.6) Condition (2.5) thus stablishes a one to one relation between H' and Im(T) in H. Observe that Eq. (2.3) is nothing but the commutativity of the following diagram:

H' /1'

1

.2..

H

1

7-{'.2..H

1I

The Renormalization Group Method

119

Eqs. (2.3) and (2.5) characterize what may be called exact renormalization group method (ERG) in the sense that the whole spectrum of H' is mapped onto a part (usually the bottom part) of the spectrum of H. In practical cases though the exact solution of Eqs. (2.3) and (2.5) is not possible so that one has to resort to approximations (see later on). Considering Eqs. (2.3) and (2.5) we can set up the effective Hamiltonian II' as:

II' = Tt HI'

(2.7)

This equation does not imply that the eigenvectors of H' are mapped onto eigenvectors of II. Notice that Eq.(2.7) together with (2.5) does not imply Eq. (2.3). This happens because the converse of Eq.(2.5), namely TTt =1= 17-( is not true, since otherwise this equation together with (2.5) would imply that the Hilbert spaces 11. and 11.' are isomorphic while on the other hand the truncation inherent to the RG method assumes that dimH' < dimH. The fact that Ttl' =1= hi reflects nothing but the irreversibility of the RGtransformation. Indeed, we go from 11. to 11.' as prescribed by eq. (2.7) but we cannot reverse that equation. What Eq.(2.7) really implies is that the mean energy of H' for the states p' of 11.' coincides with the mean energy of II for those states of 11. obtained through the embedding T, namely,

< p'1H'1tfJ' >=< TP'lHITp' >

(2.8)

In other words Tp' is used as a variational state for the eigenstates of the Hamiltonian H. In particular T should be chosen in such a way that the states truncated in 11. , which go down to 11.', are the ones expected to contribute the most to the ground state of IJ. Thus Eq. (2.7) is the basis of the so called variational renormalization group method (VRG). As a matter of fact, the VRG method was the first one to be proposed. The ERG came afterwards as a perturbative extension of the former (see later on). More generally, any operator 0 acting in 11. can be "pushed down" or renormalized to a new operator 0' which acts in 11.' defined by the formula,

(2.9) ;\fotice that Eq.(2.7) is a particular case of this equation if choose 0 to be the Hamiltonian II. In so far we have not made use of the all important concept of the block, but a practical implementation of the VRG or ERG methods does require it. The central role played by this concept makes all the real-space RG-methods to be block methods. Once we have established the main features of the RG-program, there is quite freedom to implement specifically these fundamentals. We may classify this freedom

120

M.A. Martin-Delgado and G. Sierra

in two aspects: • The choice of how to reduce the size of the lattice. • The choice of how many states to be retained in the truncation procedure. We shall address the first aspect now. There are mainly two procedures to reduce the size of the lattice: • by dividing the lattice into blocks with n s sites each. This method introduced by Kadanoff to treat spin lattice systems.

IS

the blocking

• by retrieving site by site of the lattice at each step of the RG-program. This is the procedure used by Wilson in his RG-treatment of the Kondo problem. This method is clearly more suitable when the lattice is one-dimensional. We shall be dealing with the Kadanoff block methods mainly because they are well suited to perform analytical computations and because they are conceptually easy to be extended to higher dimensions. On the contrary, the DMRG method introduced by White [11] works with the Wilsonian numerical RG-procedure what makes it intrinsically one-dimensional and difficult to be generalized to more dimensions. The first step of the BRG method consists in assembling the set of lattice points into disconnected blocks of n s sites each. In this fashion there are a total of N' = N Ins blocks in the whole chain. This partition of the lattice into blocks induces a decomposition of the Hamiltonian (2.1) into an interblock Hamiltonian H B and a interblock Hamiltonian H BB : (2.10)

where>' is a coupling constant which is already present in H or else it can be introduced as a parameter characterizing the interblock coupling and in this latter case one can set it to one at the end of the discussion. Observe that the block Hamiltonian H B is a sum of commuting Hamiltonians each acting on every block. The diagonalization of lis can thus be achieved for small n s either analytically or numerically. Eq. (2.10) suggests that we should search for solutions of the intertwiner equation (2.:3) in the form of a perturbative expansion in the interblock coupling constant parameter >., namely, (2.11 ) }/' = }/~

To zeroth order in

+ >.H; + >.2 H~ +...

>. Eq. (2.3) becomes

(2.12)

The Renormalization Group Method

121

(2.13) Since HB is a sum of disconnected block Hamiltonians defined through the relation

h;f l , j' =

1, ... , N' implicitly

N'

L h;f)

HB =

(2.14)

j'==1

one can search for a solution of To in a factorized form N'

(2.15)

To = IITo,j' j'=1

and an effective Hamiltonian

Hb

which acts only at the site j' of the new chain, N'

' -- '" H0 L..J his') j' - H'8'

(2.16)

j'=1

Observe that H;, is nothing but a site-Hamiltonian for the new chain. Eg. (2.13) becomes for each block: (B)

(s')

•

h j , 7 0 ,j' = To,j,h j ,

(2.17)

The diagonalization of h;~l for j' = I, ... , N' will allow us to write

h;~)

k ns -k'

k'

=

L Ii))'

{i

j,(il

+

i=1

where Ii >j' for j

=

L

la)j' (" j,(al

(2.18)

,,=1

1, ... , k' are the kl-Iowest energy states of h;~). Moreover, we

suppose that h;f l is the same Hamiltonian for each block so that {i does not depend on the block. The truncated Hamiltonian h;~) and the intertwiner operator l'o,j' are then given by: k' " I')' hj'(s') == 'LJ 1- j'

('I'

(i j' Z

(2.19)

Ii)], j,(il'

(2.20)

i=]

k'

TO,j' =

L i=]

Later on we shall show examples of these relations. The final outcome of this analysis is that the effective Hamiltonian HI has a similar structure to the one we started with, namely 11. The operators involved in Ii.:, and H~,s' may by all means differ from those of lis and lIss> but in some cases the only difference shows up as a change in the coupling constants. This is known as

122

M.A. Martin-Delgado and G. Sierra

the T'eno1'Tnalization of the bare coupling constants. When this is the case, one may easily iterate the HG-transformation and study the RG-flows.

3.

Block RG-Approach to the Heisenberg-Ising Model

To exemplify the standard BRG-method we shall study a Id-lattice Hamiltonian, the Heisenberg-Ising model. Prior to considering in detail the Antiferromagnetic (AF) Heisenberg model we shall make some general considerations concerning Hamiltonians which commute with a symmetry group g. Notice that for the AF Heisenberg model 9 is nothing but the rotation group SlJ(2). Let us call 9 an element of the group 9 and JrH(g) a representation of 9 acting on the Hilbert space 'H. We say that 9 is a symmetry group of the Hamiltonian H if

\lg E 9

(3.1 )

Similarly we want the effective Hamiltonian H' to be invariant under the action of 9 acting now on the Hilbert space 'H',

\lg E 9

(3.2)

For this to be the case, the RG-transformation must preserve the symmetry of the original hamiltonian. This call be simply achieved if one chooses T as the intertwiner or Clebchs-Gordan operator. Indeed we may recall that if a representation say fl v, is contained in the tensor product fl v, ® Il v2 then one can define the CG-operator as,

C>12 3

• .

(3.3)

which in fact satisfies the intertwiner condition:

(3.4) This equation expresses the commutativity of the following diagram,

p.5)

Thus from a theoretical point of view, the truncation process pertaining to the realspace HG is nothing but a tensor product decomposition of representations of the group g. To make this point more explicit let us suppose that 'H and 'H' are given as

tensor products as: (3.6)

The Renormalization Group Method

H' = 0~'V'

123

(3.7)

where V and V' are irreducible representation spaces of g. Then the block method of the previous section applied to this case is equivalent to the tensor product decomposition:

V:2I .n.,. :81 V -----+ V'

(3.8)

In Eq. (:l.8) one is establishing that the irrep fl v ' is contained in the tensor product of N copies of the irrep flv. Obviously, the tensor product decomposition usually contains different irreps. The criterion to choose a particular irrep, or a collection of irreps is the one of minimum energy. All the states of a given irrep V' will have the same energy. The summary of the discussion so far is that the intertwiner operator' To can be identified with the Clcbsch-Gordan operator,

(3.9) Let us illustrate this ideas with the AF Heisenberg-Ising model whose Hamiltonian is given by: N-l

/IN

=J

:L (5j8j+l + 5J 8J+l + L15i8i+1)

(3.10)

j=l

where L1 2': 0 is the anisotropic parameter and J > 0 for the antiferromagnetic case. I[ L1 = lone has the AF-Heisenberg model which was solved by Bethe in 1931. I[ L1 = 0 one has the XX-model which can be trivially solved using a Jordan- Wigner transformation which maps it onto a free fermion model. For the remaining values of L1 the model is also solvable by Bcthe ansatz and it is the 10 relative of the 20 statistical mechanical model known as the 6-vertex or XXZ-model. The region L1 > 'I is massive with a doubly degenerate ground state in the thermodynamic limit N -+ 00 characterized by the non-zero value of the staggered magnetization, mst =

(~ :L Si(-I)j).

(3.11 )

J

The region 0 S L1 S I is massless and the ground state is non-degenerate with a zero staggered magnetization. The phase transition between the two phases has an essential singularity. 'INc would like next to show which of these features arc captured by a real-space ltG-analysis. The rule of thumb for the ltG-approach to half-integer spin model or fermion model is to consider blocks with an odd number of sites. This allows in principle, although not necessarily, to obtain effective Hamiltonians with the same

M.A. Martin-Delgado and G. Sierra

124

form as the original ones. Choosing for (3.10) blocks of 3 sites we obtain the block Hamiltonian:

=

~ {['~1 + 52 + '~1]2 -

(51

+ '~3)2 -

3/4}

+ ((5: 5; + 5;5~),

(3.12)

t=:.d-l. If ( = 0 the block Hamiltonian liB is invariant under the 5U(2) group and according to the introduction to this section, we should consider the tensor product decomposition: (3.13)

The particular way of writing H B given in Eg. (3.12) suggests to compose first '~1 and '?3 and then, the resulting spin with .C;2. The result of this compositions is given as follows:

3 3

'2' 2) 3 1

=

I iTT), EB

= J/2,

1

12, 2) = y'3(1 Hi) + llii) + I TTL)), 1 1

'2' 2)1 1 1

12'2)0=

1 = y'2(1

(3.14) EB

Hi) -I liT)), EB

1 J6(2 Iili )-11H)-ITT1)),

= J/2,

(3.15)

= 0,

(3.16)

ER=-J.

(3.17)

lIence for ( = 0 we could choose the spin 1/2 irrep. with basis vectors I~, ~)o and ,~, -~)o in order to define the intertwiner operator To. However, if (-I- 0 the states (3.14) -(3.17) are not eigenstates of (3.12). The full rotation group is broken down to the rotation around the z-axis. The states 1~, &) and I~, ~)o are mixed in the new ground state which is given by:

(:3.18) where x = _ _2-.:(_.d_----,=1) == 8 + .d + 3v'.d 2 + 8

and its energy is

(3.19)

The Renormalization Group Method

J

E B = --[,1 1

+ J LV + 8]

125

(3.20)

along with its 1- ~) partner. This are now the two states retained in the RG method. To be more explicit, wc havc I

1+ -2)

1

=

1- -2) = -

I J6(1+2x

2

1 J6(1+2x

[(2x + 2)1 ili) + (2x - 1)1 iil) + (2x - 1)llij)],

(3.21 )

)

2)

[(2x + 2)lli1) + (2x -1)llLj) + (2x -1)1 ilL)]·

(3.22)

Thc intcrtwincr operator To reads then,

1'0 = I + ~)(i I' + 1- ~)(ll' 2

2

(3.23)

J

where I i)' and 1)' form a basis for the space V' = C2 • The RG-cquations for the spin operators Si (i = 1,3) are thcn givcn by

t

~

~x

(3.24)

ToSfTo=CS'i i= 1,3,

(3.25)

t S~zl' = T.0 10

(Z

c:,z . =

~~)ll

1, (3

(3.26)

where C, etc. are the renormalization factors which depend upon the anisotropy parameter by (X

<,

=

~

y = 2( 1 + x)(1 - 2x) 3( 1 + 2x 2 ) z _

2(1 +X)2 +2x 2 ) '

~ = 3(1

,

(3.27)

(3.28)

Observc thc symmetry bctwccn thc sitcs i = 1 and 3 which is a consequencc of the even parity of the states (3.21) -(3.22). The renormalized Hamiltonian can be easily obtained using Eqs.(3.21)-(3.28) and (3.10), and apart from and additive constant it has the same form as II, namely [17J, (3.29) wherc

M.A. Martin-Delgado and G. Sierra

126

(3.30)

(3.31 ) Iterating these equations we generate a family of Hamiltonians H}:;;~m(.I(m), ,1(m)). The energy density of the ground state of llN in the limit N ---> 00 is thcn given by lim Eo

N

-00

"

i'V

= e BRG = L.. ~ 00

_I_ e (J(m) ,1(m)) 3m +1 B ,

(3.:32)

m=O \

whcrc initially .l(O) = .I, ,1(0) = ,1 and Egs.(3.30) -(3.31) provide the flow of the coupling constants. The analysis of Eq.(3.:11) shows that there are 3 fixed points corresponding to the values ,1 = 0 (isotropic XX-model), ,1 = I (isotropic Heisenbcrg model) and ,1 = 00 (Ising model). The computation of C:;'RG in this case is facilitated by the fact that (3.32) becomes a geometric series at the fixed point. The exact results concerning thc models ,1 = 0 and ,1 = I are extracted from references [18] and [19]. The case with ,1 ---> 00 is exact because the states I ± given in (3.21) - (3.22) tend in that limit to the exact ground state I i 1i) and 11 i 1) of the Ising model. As a matter of fact,

k)

The region 0 < ,1 < I which flows under the RG-transformation to the XX-model is massless since both .1(m) and ,1(m) go to zero. We showed at the beginning of this section that all this region is critical (a line of fixed points) and therefore massless. The RG-equations (3.30) -(3.31) are not able to detect this criticality except at the point ,1 = o. Only the massless ness property is detected. The region ,1 > I which flows to the Ising model is massive and this follows frolIl the fact that thE' product .l(m),1("') gocs in the limit In ---> 00 to a constant quantity .1(=),1(00) which can be computed from Egs. (:UO) -(:l.:ll) and (:1.23) .l(=),1(oo) =

IT ~ (I + X )4 m

m=O

9 (I

+ 2x;,Y

(3.33)

where x'" is given by (:3.19) with ,1 replaced by ,1(m). This quantity gives essentially the mass gap above thc ground state and also the end-to-end or LRO order (Long Rangc Order) given by the expectation valuc [(S(1). k~(N))[ in the limit N ---> 00.

The Renormalization Group Method

127

In summary, the properties of the Heisenberg-Ising model are qualitatively and quantitatively well described in the massive region Ll > 1 while in the massless region o < Ll < lone predicts the massless spectrum but no aiticality at each value of Ll. This latter fact is rather subtle and elusive. One would like to construct a RGformalism such that the Hamiltonian HN(Ll) would be a fixed point Hamiltonian for every value of Ll in the range from -I to 1. Hence we postpone this discussion to next section. The phase transition between the two regimes is correctly predicted to happen at the value Ll = 1. This is a consequence of the rotational symmetry, namely at Ll = 1 the system is SU(2) invariant and the RG transformation has been defined as to preserve this symmetry. When Ll =1= I the SU(2) symmetry is broken and this is reflected later on in the RG-ftow of the coupling constant Ll. The region 0 < Ll < I corresponds to a central charge c = 1, namely, it is realize by a boson compactified in a circle which radius depends on Ll. We may wonder whether the criticality of the region ILlI ::; 1 is due to some non-trivial symmetry underlying the anisotropic Hamiltonian fl.

4.

Quantum Groups and the Block Renormalization Group Method for the Heisenberg-Ising Model

We present in this section a novel treatment of the Block Renormalization Group method for one dimensional quantum Hamiltonians based on the introduction of a quantum group. Our aim is to address the important questions left open in the previous section and to clarify the peculiar role played by the Renormalization group in the one dimensional physics [8]. Let us consider the following open spin chain Hamiltonian,

(4.1 )

where q is an arbitrary quantuIll parameter. This Hamiltonian is known to be integrable [13,20,23,24]. In an interesting paper Pasquier and Saleur [14] established the q-group invariance of (4.1) which served to get a better understanding of the interplay between q-groups and CFT at a discrete or lattice level. We have already talked about the relation between q-groups and CFT in the introduction when reference [7] was mentioned, but this concerned the models in the continuum. What we would like to show now is that the real-space RG method applied to (4.1) may be perhaps the way to link both the discrete and continuous approaches between the relation of q-groups and eFT. As a matter of fact, (4.1) is invariant under the following quantum group generators [14]: S+,8- and SZ

128

Yl.A. Martin-Delgado and G. Sierra

C'Z

"

=

1~ z 2" L.. (jj ,

(4.2)

J=I

N

S± =

L

q-t(u:+"'U;_l)(j;qt
(4.3)

j=1

which satisfies the quantum group algebra:

(4.4 ) (4.5) In the limit q - t lone recovers from Eqs.(4.2) -(4.5) the usual algebra and addition rules of su(2). For q i= 1 this algebra is the quantum universal enveloping algebra Uq (su(2)), or simply the quantum su(2) group denoted by SUq (2). The important property is that the generators (4.2)-(4.3) commute with the Hamiltonian (4.1):

(4.6)

If we set q + q-I

Ll = - 2 -

(4.7)

we observe that the bulk terms of Eq.(4.1) and the one in Eq.(:UO) coincide. The only difference appears in the boundary term of Eq.(4.1) which is essential for the existence of the quantum group symmetry. There are two important cases which we can consider:

q : real and positive

Iql =

1

=> Ll 2:: 1,

=> ILlI s:: 1.

(4.8)

(4.9)

If q is real then HN(q) is Hermitean while if q is a phase then HN(q) is not Hermitean but nevertheless its spectrum is real for H N( q) and HN( q-l) are related by a similarity transformation. This case is the most interesting one. In particular if we write q as a root of unity q = e;~l, then the Hamiltonian (4.1) is a critical Hamiltonian with a Virasoro central algebra given by q=

..

e;;+T

6

=>

c = 1 - j.l(j.l

+ 1)

(4.10)

However for the time being let us keep q as an arbitrary parameter characterizing the anisotropy of the model (4.7).

The Renormalization Group Method

129

Let us try to apply the Block Renorrnalization Group Method to the analysis of the Hamiltonian (4.1). To this end we have to write (4.1) in the following form: N-I HN(q) =

L

hj ,j+l(q,

J),

(4.11 )

j=l

Observe that each boundary term in hj,j+1 cancels one another leaving only those at the end of open chain (i.e. j = 1 and N). The nice feature of the site-site Hamiltonians (1.12) is that all of them commute independently with the q-group generators S± and 8 z ; [hj,j+I,S±]

= [hj,j+t,SZ] = 0

Vj

= 1, ... ,N -1.

(4.13)

Using hj,j+1 we can construct q-group invariant block Hamiltonians. If the block has for example 3 sites then we will have, (4.14) In the isotropic case (i.e. q = 1) we employ the Clebsch-Gordan decomposition (:1.13) in order to find the eigenstates of the block Hamiltonian H B (E = 0) given in (3.12). For quantuIll groups, we can also perform q-CG decompositions. The new feature is that now the q-CG coefficients depend on the value of q. For generic values of !J the analogue of Eqs.(:3.l4) -(3.17) are given by: (4.15)

eB

+

q q-I = .1--- ,

4

(1.16)

q

eB

+ q-I

= .1--4 •

130

M.A. Martin-Delgado and G. Sierra

eB =

J

2 _ q _ q-I 4 '

where [:3]q denotes the q-number with the usual definition

[n]q =

q" - q-" 1 n EN. q - q-

(4.20)

The normalization of the states in Eqs. (4.15) - (4.19) is as if q were always a real number, a bra vector (I means transposing a ket vector. If q goes to 1 the vectors in Eqs. (4.15) - (4.19) go over to the vectors in Eqs. (3.11) - (3.17). On the other hand, for q i- 1 there is a certain similarity between the states It,±~)o and the states I ±~) given in Eqs. (3.21) - (3.22). However the main difference is that I~, ±t)o are not parity invariant (1 <--> 3), (2 <-> 2), while states (3.21) - (3.22) are. If we let q go to zero then Ll goes to +00. Let us recall that in this limit the states I ± ~) (3.21) - (3.22) go to the exact ground state of the Hamiltonian (3.12). However in the case of the states I~, ±~)o one does not recover the exact eigenstates in this limit. This shows that the states (1.15) - (1.19) are not appropriate for a discussion of the AF region Ll > 1 (or q real). lIence we shall confine ourselves to the critical region ILlI < 1 where q is a pure phase. As we did already for the isotropic case we shall truncate the basis (1.1.,)) - (1.19) to the states I~, ±t)o and therefore the intertwincr operator To is given by

To =

I~, ~)o(i I' + I~, -~)o(ll'

which satisfies the normalization condition

(4.21)

The Rellormalization Group Method

T~To

=1

131

(4.22)

where the superscript t stands for the transpose of the operator (instead of the adjoint ). This means that we can get the effective Hamiltonian H' through the formula:

H' = T~IITo. The RG-equations for the spin operators t....

...x

ToSiTo = ~S'

.5;

(4.23)

(i = 1,3) are then given by

= 1,3,

(4.24 )

t uV1' - . c:,v. 1, 3 , T,O"j 0 - ~" 1 =

(4.25)

i

(4.26) where ~ is the renormalization factor which depends upon the anisotropy parameter through the q-parameter in the following fashion,

~=

q

+ q-l + 2

2(q+q-l

+ 1)

(4.27)

and

""1=-r/3:=""= 4(q+q-l+l)'

(4.28)

Observe that there are quite a few remarkable differences between this "quantum" renormalization prescription Eqs. (4.24) - (4.27) with respect to the ordinary renormalization expressed in Eqs. (3.24) - (3.28). To begin with, the renormalization constant ~ is common to all the spin operators regardless of its spatial component. This is a reflection of the SUq (2) preservation of the RG method adopted. And last but not least, observe the presence in Eq.(4.26) of an extra term proportional to the identity operator. We may call this term a quantum group anomaly. Using these equations we can compute the renormalization of the block-block Hamiltonian hBB which turns out to be of the same form as the original site-site Hamiltonian (4.12) with the same value of q, namely (4.29) -1

- q -2q with

(O"~z - O"~z+l)

+ eBB( q, J)]

132

M.A. Martin-Delgado and G. Sierra

e

+ 3q-l + 4) . 32(q+q-l+1F

J) = J (q - q-l )2(3q

(

BBq,

(4.30)

Combining Eqs.(4.), (4.:30) and (4.15) - (4.19) we finally arrive at quantum group RG-equations (4.31) with (4.32)

q' = q,

(4.33)

(4.:34) IIcuce we obtain a quite remarkable result we were searching of, namely, that the coupling constaut Ll or alternatively q does not flow under the HG-transformation, while J(m) goes to zero in the limit when m ---+ 00, which in turn implies that theory is massless. The computation of the ground state energy of liN for any value of N is very simple since one has only to compute a geometrical series. The result is

Eo(N) = N

1 - (e/3)M 3_ (eB

e

+ eBB) -

1 -eM 1_ eBB

e

(4.35)

where AI is the number of RG-steps we have to make in order to resolve the chain of N = :3 M sites. As a check of the validity of this expression and because of its own interest as well, we shall consider the case q = e i1f / 3 in (4.35) which yields

E'0 ( N, q = ei1f/3) =

-83 N +"83

(4.36)

This expression coiucides with the exact result obtaiued through Bethe ansatz in reference [20]. Recall that the finite-size corrections to the free-energy for conformally invariant two-dimensional systems behaves as [21,22],

E = eN + C s

7rC

-

1

(4.37)

24 N

where e is the bulk energy per unit length and Cs is the surface energy (which vanishes for periodic boundary conditions). Since the term proportional to 1/ N is absent in (4.36) one observes that this value of q corresponds to a central extension c = 0 of the Virasoro algebra, in agreement with equation (4.10) (p 2).

=

The Renormalization Group Method

133

At first sight it looks surprising that an approximation method such as the Block Renormalization Group yields the cxact rcsult at least in thc case q = ei~/3. The peculiarity of this value of q, and in general when q is a root of unity, has been noticed in various contexts [12]: conformal field theory, quantum groups and in fact they are intimately related. The first thing to be noticed is that at q = ei~/3 the two denominators of the states I~, t) and It, t)l vanishes reflecting the fact that the "norm" of these states is zero, i.e., they are null states and therefore they must be dropped out in a consistent theory. It has been shown in [14] that because of (S±)3 = 0 the 6 states I~, rn) and It, mh do not fortn two irreps of dimensions 4 and 2 but rather a simple indecomposable but not irreducible representation of SUq (2). All these facts motivates that the tensor product decomposition (3.13) for the case q = ei1f / 3 should be really be taken as (4.38) where the irrep 1/2 on the right hand side denotes the one generated by It,m)o. This truncation is mathematically consistent and coincides precisely with the truncation we havc adopted in our Block Renormalization Group approach to the q-group invariant Hamiltonian (4.1). From a physical point of view, the truncation (4.38) means that the states I~, and It, th are not "good excited states" above the "local ground state" given by It, m)o. In other words, above the ground state there are not well behaved excited states. This is why thc central extension is c = 0 which means that the unique state in the theory is actually the ground state. What the BRG method does is to pick up that piece of the ground state which projects onto a given block! In the case of q = e i1f / 3 we have therefore construct for chains with N = 3 M sites the exact ground state of the model through the BRG method. It is worthwhile to point out that this derivation is independent of the Bethe ansatz construction and relics completely on the quantum group symmetry. Another interesting example is provided by q = e i r./4 which corresponds to the critical Ising model (c = 1/2). The RIIS for this q in eg. (4.38) contains two spin-l/2 irreps. According to thc qRG method the truncation of the spin-3/2 irrep shoulcl be a legitimate operation involving no approximation at all. In references [7] the representation thcory of q-groups was put in one-to-one correspondcnce with that of Rational Conformal Field Theories (RCFT). There it was observed that the truncation inhcrent in the construction of the RCFT's has a parallel in the truncation of thc representation theory of q-groups with q a root of unity. The result we have obtaincd in this letter suggests that q-group truncations can be carried over a RG analysis of q-group invariant chains. In other words, using q-groups we can safely truncate states in the block RG method. We may summarize this discussion squematically as we have mentioned in the introduction.

t)

134

5.

M.A. Martin-Delgado and G. Sierra

qRG Treatment of the ITF Model

This simple model has been widely used to test the validity of BRG methods [3,4]. The Hamiltonian of an open chain is given by II = L::7=1 1 hj •j + J where (5.1)

r

The standard choice is p = pi = 12, in which case (5.1) has 4 different eigenvalues. The BRG method with a block with two sites chooses just the 2 lowest ones. However if (p, p') = (r, 0) (or (O,!')) the Hamiltonian (5.1) has two doubly degenerate eigenvalues ±CB (ea = J.J2 + f2). This choice is not parity invariant but it implements the self-duality property of the ITF model, yielding the exact value of the critical fixed point of the lTF which appears at (1'IJ)c = 1 [25]. In the following we shall make the choice (p, pi) = (r, 0). This degeneracy of the spectrum of (5.1) has a q-group or·igin. The relevant quantum group is again SUq (2) with q4 = 1. However the representations involved are not a q-dcformation of the spin 112 irrep. as in the previous example, but rather a new class of irreps. which only exist when q is a root of unity. They are called cyclic irreps. and neither are highest weight nor lowest weight representations as the more familiar regular irreps. If we call E, F and I< the generators of SUq (2), which correspond essentially to S+, S- and q2S in the notation of the previous example, then a cyclic irrep. acting at a single site of the chain is given by: Z

Ej =

Fj

aCTi,

= bCT},

J()

= ACT;

(5.2)

where a = ~~, b = -~~. The parameter..\ is the label of the cyclic irrep. Strictly speaking, we have a particular kind of cyclic irreps. Indeed, the ones which allow the existence of an intertwiner for their tensor product. Using (5.2) and the addition rule of SUq (2) we can get the representation of E, F and I< acting on the whole chain:

E

a

LN

..\j-1 CT z .,. CTj_ICTj z x , 1

(5.3)

j=1 N

F

b

L ..\j-N CT·CT· Y )

z z )+1 ... CTN ,

(5.4)

j=1 N

J(

..\N

II CTJ.

(5.5)

j=1

Now it is a simple exercise to check that these operators commute with (5.1), [h).J+I, I~'] = [hj,)+J,

F]

= [hj,j+l, J(] = 0,

Vj,

(5.6)

The Renormalization Group Method

135

assuming that we choose (5.7)

A = 1'/.1 .

The last of the equalities in (5.6) expresses the well-known Z2-symmetry of the ITFmodel which allows one to split the spectrum of the Hamiltonian into an even and odd subsectors. The other two symmetries are new and explain the degeneracy of the spectrum of hj.j +l . By all means the whole Hamiltonian H = L:j hj.j+1 is also invariant under (5.3). l'\otice that Il differs from the standard ITF simply in a term at one of the ends of the chain. This is the same mechanism as for the XXZ Hamiltonian: one needs properly chosen operators at the boundary in order to achieve quantum group invariance. Similarly as for the XXZ model the RG-analysis of the ITF becomes a problem in representation of quantum groups: blocking is equivalent to tensoring representations. What is the tensor product of cyclic irreps.? Here it is important to realize that all cyclic irreps. of SUq (2) have dimension 2, what distinguishes them is the value of A. The tensor product decomposition of two cyclic irrep. Al and A2 of the type given in (5.2) is given by:

(5.8) where the 2 means that AIA2 appears twice in the tensor product. If we perform a blocking of two sites we will get two cyclic irreps. corresponding to A2 • Then we expect from If-group representation theory that the new effective Hamiltonian hj,j+1 will have the same form as (5.1) but with new renormalized coupling constants .1' and r' satisfying:

A' = T' .1'

= (!-)2 = .1

A2

(5.9)

This is indeed the result obtained in [25]. We arrive therefore at the conclusion that the HG-flow of the ITF Hamiltonian (5.1) is equivalent to the tensor product decomposition of cyclic irT'Cps of SUq (2). This q-group interpretation of the RG-flow is independent of the size of the blocks: for a n-site block the RG-flow would be A ----> An. The fixed point A = I of (5.9) describes the critical regime of the 11'10' Hamiltonian and it corresponds to a singular point in the manifold of cyclic irreps. [27]. At A = I the operators (5.3) are still symmetries of the Hamiltonian (a, b taking any non-zero value) and they recall the Jordan-Wigner map between Pauli matrices and Id-Iattice fermions. Of thc two cquivalent irreps A2 appearing in the tensor product A I8l A we only pick up onc of them. which is the lowest cncrgy. Observe that thc loss of information implied by the R(; method is in the ITF case somehow redundant information from the q-group point of view. The RG-flow in A is in some sense exact and not affected by the R(;-procedure.

136

M.A. Martin-Delgado and G. Sierra

The results we have obtained in the ITF model may perhaps be realized in IIlore complicated models. Namely, that the space of coupling constants or equivalently, the space of Hamiltonians of a given theory which is where the RG takes place, is the Spec manifold of an underlying quantum algebra, so that the RG-ftow is given by the tensor product decomposition of the algebra. For this to work we have to consider quantum groups with a rich and "exotic" representation theory, which is indeed the case when q is a root of unity. In these cases we know from references [26] and [27] that the Spec of these quantum groups is indeed very rich. We have used here the simplest situation. As we said above, it would be interesting to know whether there are more complicated realizations of these ideas. The chiral Potts model is a potential candidate for this realization due to its well known connection to S'Uq (2) with qN = 1.

6.

Conclusions

We have presented in this paper a brief description of the Renormalization Quantum Group Method (qRG) which is specially well-suited to treat 1D quantum lattice Hamiltonians. We have applied real-space RG methods to study two quantum group invariant Hamiltonians, namely, the Heisenberg-Ising model and the Ising model in a transverse field (ITF model). They are defined in an open chain with appropriate boundary terms. The defining feature of this qRG method is that the quantum group symmetry is preserved under the RG transformations except for the appearance of a quantum group anomalous term which vanishes in the classical case. We have called it the quantum group anomaly. As for the Heisenberg-Ising model, with the aide of the qRG equations we have shown that the q parameter describing the anisotropy coupling constant L1 does not flow under the RG-transformation when q is of modulus one, while the coupling constant J goes to zero implying that the theory is massless. In this fashion, the RG-ftow diagram obtained with the qRG method gives the correct line of critical points exhibited by the exact model. In the ITF IIlodel, we have shown that the qRG-flow coincides with the tensor product decomposition of cyclic irreps of SUq (2) with q4 = 1. Cyclic irreps. were used in [28] to derive the Boltzmann weights of the Zwchiral Potts model [29]. In [28] the labels of the cyclic irreps. have the meaning of rapidities rather than coupling constants as in our realization. The Z2 CP-model is nothing but the ITF model. We may wonder whether the general Zwchiral Potts model admits a q-group RG treatment along the lines of this work. This problem will be considered elsewhere. A model that admits a qRG analysis is the XY model with a magnetic field h (XYh). The results will be presented in [30]. It suffices to say here that the q-group underlying the model is SUq (2) with q4 = I and the representation used are the so called nilpotent irrcps. [12], which are also described by a parameter .\ analogue to that in (5.2) and related to the magnetic field h. The XYh model is equivalent to a free fermion with chemical potential.

The Renormalization Group Method

137

The results we are obtaining can be translated into a q-group symmetry between fermions, either free as in the XY or ITF models or interacting as the XXZ model. Another interesting model of interacting fermions is the Hubbard model, which has been studied using RG-methods in [31] . The integrability of the I D Hubbard model [32] suggests that it might be studied using our qRG techniques. All the Hamiltonians analysed in this letter are one dimensional, so the quantum groups are of the type that we know. Despite the fact that the Yang-Baxter equation (the precursor of q-groups) has a higher dimensional analogue called the Zamolodchikov or tetrahedron equation [33], the corresponding high dimensional analogue of quantum groups is not known. This fact represents a barrier to a qRG analysis of Hamiltonians defined in dimensions higher than one. Another possibility, which is suggested by our results, would be to define quantum groups as those which contain symmetries which are anomalous under RG transformations. This definition is independent of the space dimensionality. The quantum anomalous term in equation (4.26), and an analogous term also present in our qRG treatment of the ITF model, gives a discrete realization of this idea. A continuum analogue of this anomaly is given by the Feigin-Fuchs current, which has an anomalous operator product expansion with the energy-momentum tensor [34]. At this point it may be worth to recalling the continuous version of quantum groups in CFT of refcrence [35], which uses the Feigin-Fuchs or free field realization of the latter. Putting all these arguments together, we arrive at the conclusion that quantum groups are indeed defined by symmetries anomalous under RG transformations. This point of view about quantum groups may set up the pathway to new developments in the ficld. Finally, it is somcwhat amusing the way in which the word group enters the title of these notes. It refers both to the Renormalization Group method and to Quantum Groups, but neither of thcm are really groups! Acknowledgments Work partially supported 111 part by CICYT under contracts AEN93-0776 (M.A.M.-D.) and PB92-I092, European Community Grant ERBCHRXCT920069 (G.S.).

REFERENCES [I] K.R. Wilson, Rev. Mod. Phys. 47 (1975) 773. [2] P.W. Anderson, J. Phys. C3 (1970) 2436. [3] S.D. Drell, :vi. Weinstein, S. Yankielowicz, Phys. Rev. D 16 (1977) 1769. [4] R. Juillen, P. Pfeuty, J.N. Fields, S. Doniach, Phys. Rev. B 18 (1978) 3568. [5] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. 8241 ( 1984) 333.

M.A. Martin-Delgado and G. Sierra [6] V.G. Drinfeld, "Quantum Groups" in Proceedings of the 1986 International Congress of Mathematics, cd. A.M. Gleason (Am. Math. Soc., Berkeley); M. .limbo, Lett. Math. Phy••. 10 (19R5) 63, Lett. Math. Phys. 11 (1986) 247. [7] L. Alvarez-Gaume, C. Gomez, G. Sierra, Phys. Lett. 8220 (1989) 142; Nucl. Phys. 8330 (1990) 347. [8J M.A. Martin-Delgado and G. Sierra, UCM-CSIC preprint, 1995, to be published. [9] S.R. White, R.M. Noack, Phys. Rev. Lett. 68 (1992) 3487. [10] M.A. Martin-Delgado and G. Sierra, L"CM-CSIC preprint, 1995, to be published in

Phys. Lett. B. [11] S.R. White, Phys. Rev. Lett. 69 (1992) 286:l; Phys. Rev. B 48 (1993) 10345.

[12] C. Gomez, M. Ruiz-Altaba, G. Sierra, Quantum Groups in Two-Dimensional Physics, Cambridge University Press (to be published). [1:l] E.K. Sklyanin, J. Phys. A 21 (19RR) 2375. [14] V. Pasquier, H. Saleur, Nucl. Phys. B 330 (1990) 523. [15] P. Pfeuty, R . .IulIien and K.A. Penson, in Real-Space Renormalization, editors T.W. Burkhardt and .I.M ..I. van Leeuwen, Series Topics in Current Physics 30, SpringerVerlag 1982. [16] .I. Gonzalez, M.A. Martin-Delgado, G. Sierra, A.II. Vozmediano, Quantum Electron Liquids and Hight-To Superconductivity, Lecture Notes in Physics, Monographs vol. 38, Springer- Verlag 1995. [17] .I.M. Rabin, Phys. Rev. IJ 21 (1980) 2027. [18] E. Lieb, T. Schultz and D. Mattis Ann. Phys. 16 (1961) 407. [19] R. Orbach, Phys. Rev. 112 (19.58) 309. [20] F.C. Alcaraz, M.N. Barber, M.T. Batchelor, R..I. Raxter, G.R.W. Quispel, J. Phys. A 20 (1987) 6397. [21] H.W. Hlote, .I.L. Cardy, M.P. Nightingale, Phys. Rev. Lett. 56 (1986) 742. [22] L Affleck, Phys. Rev. Lett. 56 (1986) 746. [23] M. Gaudin, Phys. Rev. A 4 (1971) 386. [24J LV. Cherednik, LV., Theor. Math. I'hys. 61 (1984) 977. [25] A. Fernandez-Pacheco, Phys. Rev. D 19 (1979) 3173. [26] D. Arnaudon, A. Chakrabarti, Comm. Math. Phys. 139 (1991) 461. [27] C. de Concini, V.G. Kac, P7'Og. Math. 92 (1990) 471.

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139

[28] E. Date, M. .limbo, K. Y1iki, T. Miwa, Comm. Math. Phys. 137 (1991) 133; Publ. Res. Inst. Math. Sci. 27 (1991) 639. [29] 11. Au-Yang, H.M. McCoy, .I.II.H. Perk, S. Tan, M.L. Yan, Phys. Lett. A 123 (1987) 219; 11. Au-Yang, R..I. Baxter, .I.H.H. Perk, Phys. Lett. A128 (1988) 138.

[:30] M.A. Martin-Delgado, G. Sierra, paper in preparation. [31J .I.E. Hirsch, Phys. Rev. 822 (1980) 52.59; C. Dasgupta, P. Pfeuty, J. Phys. C 14 (1981) 717; C. Vanderzande, Phys. Lett. A18 (1985) 889;.1. Perez-Conde, P. Pfeuty, Phys. Rev. B 47 (1993) 8.56. [32] E. Lieb, F. Wu, Phys. Rev. Lett. 20 (1968) 1445; B.S. Shastry, Phys. Rev. Lett. 56 (1986) 2453. [33] A.B. Zamolodchikov, Comm. Math. Phys. 79 (1981) 489. [34] V.S. Dotsenko, V.A. Fateev, Nucl. Phys. B (1995). [35J C. Gomez, G. Sierra, Phys. Lett. 8240 (1990) 149; Nucl. Phys. 8352 (1991) 791.

From Field Theory to Quantum Groups

REMARKS ON THE QUASI-BIALGEBRA STRUCTURE IN QUANTUM MECHANICS ANATOL NOWICKI 1

Dipartimcnto di Fisica, Universita di Firenze and INFN - Firenze, Italy

Abstract: It is shown that the description of the time evolution for many-particle quantum mechanical states can be embedded into the quasi-bialgebra structure where the time parameter plays the role of a deformation parameter. Space rotations and translations are also briefly discussed using the notion of the noncoassociative comultiplicatioIl.

1.

Introduction

Recently, several applications of Hopf algebras to physics as quantum groups and deformed quantum algebras are investigated. The deformed space-time symmetry groups and algebras are extensively considered with a hope that these notions turn out to be suitable in the description of high energy phenomena. However, till now the fundamental problems related to a physical interpretation of the Hopf algebra structure are not solved, for instance the notion of the deformed momentum coproduct is not clear [1]. It is hard to find the applications of the Hopf algebra formalism even in the case of simple quantum mechanical systems. Perhaps, this strong mathematical structure can not be realized in quantum mechanics at all. In this note, we relax the notion of Hopf algebra to the notion of quasi-bialgebra introduced by Drinfeld [2], and we present some simple quantum mechanical examples which can be described using a quasi-bialgebra structure. In particular, we deal with the concept of noncoassociative comultiplication obtained by the twisting procedure

[a,4].

142 2.

A. Nowicki

Noncoassociative coproduct and quasi-bialgebra structure

Let us recall the basic notions related with noncoassociative comultiplication and quasi-bialgebras introduced by Drinfeld in [2]. We denote by

A

= (A, L1, E) an algebra with comultiplication L1 and counit

E.

Definition 1. The algebra A = (A, L1, E) is quasi-bialgebra if and only if there exist invertible clement 4> E A ® A ® A satisfying 3-cocycle condition: (104)) ((10 L1 ® 1)4» (4) 0 I) = ((10 1 ® L1)4» ((L10 1 (1)4» and invertible elements

f. 9 E

(I

Q9

(2.1)

A such that for all a E A the following relations hold:

L1)L1(a) = 4> ((L1 ® 1)L1(a)) 4>-1 ,

(2.2a)

(I ® E)L1(a)

jar l

,

(2.2b)

(E 0 1)L1(a)

gag-I,

(2.2c)

(IQ\:E01)c1> = j&;g-I.

(2.2d)

The relation (2.2) tells us that in the case c1> -I- id (c1> is called the Drinfeld associator) we deal with nonassociative coproduct L1(a). This fact is the fundamental difference between quasi-bialgebras and bialgebras for which the coproduct is coassociative.

It is well known, that for a given quasi-bialgebra by the twisting procedure we obtain a new quasi-bialgebra.

Definition 2. The quasi-bialgebras A = (A,L1,E,4» and A 1" = (A,L1 F ,E,4>F) are related by twisting if there exists an invertible clement FE A Q9 A such that (2.6) for all a E A. In this case the following relation holds

(2.7) Element F satisfying (2.6) is called twist junction. In fact, this general definition of a twist function F in the case of quasi-bialgebra gives us the similarity transformation of the coalgebra sector. If one compares this definition of the function F with that one for the Hopf algebras [3], it is seen that the counital and 2-cocycle conditions are relaxed.

Proposition 1. [2] If the algebra A F = (A, L1 1" , E, 4>1") is related to the quasi-bialgebra

Remarks on the Quasi-Bialgebra Structure in Quantum Mechanics

A

= (A, Ll,E, 1'/»

143

by the twist function F then A F is a quasi-bialgebra also.

Let us notice that if A is a bialgebra (i.e. with coassociative coproduct) then A F is not (in general) a bialgebra. The properties of the quasi-bialgebra A F depend on the explicit form of the twist function F. It is easy to prove the following statement:

Proposition 2. If A is a bialgebra and the twist function F is of the form F = J Q9 where 11(f) = J (<: J (i.e. group-like element) then the following relation holds:

J

(2.8) for all a E A.

In this case, the relation (2.8) allows us to replace the noncoassociativity condition (2.2) for the comultiplication I1 F by this one. It appears that we can build coassociative tensor product realization of the quasi-bialgebra A F using the relation (2.8). We shall see in the next Chapter that the conditions of the Proposition 2a. are satisfied in quantum mechanical description of the time evolution and space rotations for noninteracting physical systems. Let A = (A,I1,c:,P = id) be a bialgebra with trivial, non deformed coproduct Ll( a) = a Q9 I + I Q9 a then the following holds:

J Q9 9

Proposition 3. If a twist function F is of the form F =

11(/) =

J (X) J,

EA

(X)

A where:

l1(g) = 9 ® g,

(2.9)

then

FLl(a)p-l = (far l ) Q9/ + I Q9 (gag-I), l I I'/> = (X) Jgrlg- Q9 9

r

(2.10) (2.11 )

and the quasi-bialgebra structure is given by:

(I Cy I1 F )Ll F(a) = I'/>

((d,' ® I)LlF(a)) p- I ,

(I Q9c:)I1 F (a) F

(c: Q9I)I1 (a) (I

r;)

c: Q9 1)1'/> =

(2.12)

Jar l ,

(2.13)

gag-I,

(2.14)

J r;) g-I

.

(2.15)

We finish with our general considerations of the quasi-bialgebra structure the following:

144

A. Nowicki

Remark. If the twist function F is of the form F = f ® f where f is a group-like element then this twist generates an authomorphism (i.e. linear homomorphism) of the dual algebra. In the other cases we obtain a nonlinear homomorphism of the dual algebra.

3.

Quasi-bialgebra structures in quantum mechanics

We shall consider well known simple examples from quantum mechanics where the quasi-bialgebra structure is explicitly realized. Let us consider the standard undeformed Poincare algebra with the commutation relations in the form:

0(3) rotation sector (i,j,m = 1,2,3) [M k , Md = ifkl",M",

(3.1 )

- boost sector: (3.2)

.. four-momentum sector (Il, v

= 0, 1,2,3):

[P", P v ] = 0

[Mi, Pol

0,

[Mk, Pd

iCklm Pm,

(3.3)

with trivial, ulldeformed coproduct:

L1(M k ) = Mk®I L1(.c k ) L1(P,,)

+ I®M k

+ I ® .c k P" ® I + I ® P" .c k ® I

(3.4) (3.5)

(3.6)

Example 1. Time evolution of many-particle states. First, we llotice that the element f = U(t) = eitPo is group-like because of the trivial coproduct (:t6). Therefore, the twist function F(t) = f ® f = U(t) 09 U(t) satisfies the conditions of Proposition 2. and the formula:

L1 t (a) = FL1(a)F- 1 =

U(t)aU+(t) 09 I =

+ I ® U(t)aU+(t)

a(t)09I+I®a(t)

(3.7)

Remarks on the Quasi-Bialgebra Structure in Quantum Mechanics

145

defines the observable a in time t for the two-particle states. It is obvious that .:1 t (a) give us a noncoassociative cornultiplication i.e.

(I

C;;;

.:1 t ) .:1 t (a) =

!p

((.:1 t Q';/).:1 t (a))

(3.8)

!P-l

wit.h the Drinfeld associator of the form (2.11): !P

= U+(t) (9 1(9 U(t)

(3.9)

using the relation (2.8) we can define the observable a in time t for three-particle statc as follows:

which is the standard quantum-mechanical formula for three-particle observable a in time t. This procedure can be immediately generalized for many-particle statcs. Therefore, we see that the time evolution of many-particle states can be put in thc quasi-bialgebra schemc with time t treated as a deformation parameter. One can repeat this procedure for the K-Poincare algebra [5] because in this case the clement f = eitPo is also group-like one and the conditions of Proposition 2. are fulfilled.

Example 2. Three-dimensional rotations of coordinate system

In this case one can choose the twist function F in the form: (3.11 )

where ii describes O(3)-rotations and iiM = (}kMk. It follows from (3.4), that f(ii) is a group-like elemcnt i.e.

f( ii)

Q';

f( ii)

(3.12)

and we obtain the twisted coproduct:

.:1 F (o) (M k)

R k/(ii).1(Md,

(3.13)

.:1f"(O)(Pd

R kl(ii).:1(P1) ,

(3.14)

146

A. Nowicki

L1 F (a)(£k) = Rkl (ci)L1(£t} , L1 F (a)(po) = L1(Po)

(3.15)

(:1.16)

where Rkl(ci) is the three-dimensional rotation matrix. The noncoassociativity of L1F(5) is described by ep (2.11) in the form: (3.17)

Similarly, one can find the quasi-bialgebra structure for the boost transformations with (3.18)

In the case of K-Poincare algebra [5], Proposition 2 holds only for threedimensional rotations because they are not deformed. It does not hold for the boost transformations.

Example S. Translations in space and dual algebra

In this example we express the twist function F by the transformation of shift on a vector r : (3.19)

;-.Ioncoassociative twisted coproduct is given by :

L1 F (M) = L1 F (i) = L1 F (P) = L1 F (po) =

L1(M) + (r x P) 01 L1(i) + rL1(Po) , P 0 1+ 10 P , Po 0 +10 Po

+ 10 (r x

P), (3.20)

where (x denotes vector product). We obtain a quasi-bialgebra structure with the Drinfcld associator:

(3.21) On the other hand, the usual associator has the form:

(10 L1F (1 C9 L1 F (1 C9 L1 F

L1 F «) l)L1 F (M) = 1 Ii) 1 ® (i x P) - (i x P) d'li) l)L1 F (i) = r(1 @ 1 rg; Po - Po Ii) 1 ® l), L1 F (><) l)L1 F (P!'l = 0

on the linear basis of Poincare algebra.

0-)

1 C9 1, (3.22)

Remarks on the Quasi-Rialgebra Structure in Quantum Mechanics

4.

147

Final remarks

The aim of our contribution was to point out the fact that basic mathematical formalism describing timc evolution and spacc symmctry transformations of quantum systcms can bc expresscd by the notion of noncoassociative coproduct in quasibialgebra framework. Thc next stcp is to consider the examples of interacting quantum systems in terms of quasi-bialgebra. This problem is under investigation. Acknowledgments I would like to thank R. Giachetti. E. Sorace, M. Tarlini for interesting discussions on this subject and Dipartimcnto di Fisica and INF'\! in Fircnze for financial support. I would likc to express my gratitude to prof. J. Lukierski - who introduced me to the world of theoretical physics.

REFERENCES [I] J. Lukierski, ll. Ruegg, V.N. Tolstoy, "Quantum ,,-Poincare 1994" in Proceedings of xxx Karpacz Winter School of Theoretical Physics, ed. J. Lukierski, Z. Popowicz, J. Sobczyk, PWN, Warszawa 1995. [2] V.G. Drinfeld, Algebra Anal. 1 (1989) 114 (in Russian); V.G. Drinfeld, Leningr'ad Math. Joum. 1 (1990) 1419. [3J N. Rcshetikhin, Lett. Math. Phys. 20 (1990) 331; B. Bnriquez, Lett. Math. Phys. 25, (1992) Ill. [4] J. Lukierski, ll. Ruegg, V.!\'. Tolstoy and A. Nowicki, J. Phys., A27, (1994) 2389; J. Lukierski, A. Nowicki, V.N. Tolstoy, "Twisted Poincare Algebras" in Proceedings of xxx Karpacz Winter School of Theoretical Physics, ed. J. Lukierski, Z. Popowicz, J. Sobczyk, PW:'\, Warszawa 1995; A. Nowicki, "!\'oncoassociative Twisting; Its Application to ,,-Poincare Algebra" ill Pr'oceedings of the 4th International Colloquium "Quantum Groups and Integrable Systems", Praguf.:, 22-24 June, 1995. [5] J. Lukierski, A. :':owicki, ll. Ruegg and V.N. Tolstoy, Phys. Lett. B 264, (1991) 331; J. Lukierski, A. :':owicki and H. Ruegg, Phys. Lett. B 293, (1992) 344.

Chapter III. SUPERSYMMETRY

From Field Theory to Quantum Groups

LAGRANGIAN MODELS OF PARTICLES WITH SPIN: THE FIRST SEVENTY YEARS ANDRZEJ FRYDRYSZAK

Institute of Theoretical Physics, University of Wroclaw, 50-204 Wroclaw, Pi. M. Borna 9, Poland

Abstract: We briefly review models of relativistic particles with spin. Departing from the oldest attempts to describe the spin within the lagrangian framework we pass throught various non supersymmetric models. Then the component and superfield formulations of the spinning particle and superparticle models are reviewed. Our focus is mainly on the classical side of the problem, but some quantization questions are mentioned as well. 1.

Introduction

The aim of the present brief review is to indicate some essential aspects of the theory of relativistic point particle with spin. Selected models are presented mostly historically as they were appearing, to show the development of ideas in the period of 70 years - starting from the very begining. The first published work concerning the lagrangian description of the relativistic particle with spin was the paper by Frenkel [i] which appeared in 1926. In that time main considerations go towards the derivation of the equations of spin precession in the external electromagnetic field. Then to its relativistic generalization. The last one was achieved by Bargmann, Michael and Telegdi [3] 33 years after the Frenkel model was constructed. However only the work of Frenkel contains the Lagrangian defining the model of a particle, not only a considerations on the equations of motion level [4]. Then there is a forty years long gap in the activity in constructing such a models. However, in the meantime - in the fifties, the idea of anticommuting coordinates emerges in works of Martin [20], Matthews and Salam [21], and Tobocman [22]. Later it strongly influenced particle models [30,48]. The silence was broken with the work of 13arut [5]. Then in early seventies wider interest in the subject begins with the works of Hanson, Regge [6]; Grassberger [8];

Andrzej Frydryszak

152

Casalbuoni [14]; Rerezin, Marinov [18]. Some of these models involve anticommuting coordinates. Further growth of the interest was stimulated by dynamically developing research in the supersymmetry and supergravity and then superstrings theory with the wide use of Z2-graded structures. This period lasts from the eighties to the present decade with such a new models: Brink-Schwarz [28], Brink-diVecchia-Howe [26], de Azca,rraga-Lukierski [24], Siegel [35,36], Volkov-Soroka-Tkach [71]. Above metioned models fall in principe into different categories. The classification can be made due to the such attributes as mass, algebraical (conventional or anti-commuting) and geometrical character of the internal degrees of freeedom (vectorial, spinorial, twistorial). In the sequel we shall adopt the following naming conventions. Models involving only conventional coordinates will be called the classical models. The models involving anti-commuting coordinates are generally called here pseudoclassical. These with the anti-commuting vectorial degrees of freedom are called the spinning particles and these with the spinorial anti-commuting degrees of freedom are called superparticles. The type of extension of the configuration space of the relativistic particle by the commuting or anti-commuting coordinates determines the symmetry and the behaviour of the model in the external field and upon quantization. Starting demand is to have object which is at least Poincare invariant. Classical vectorial particles and spinnig particles couple properly to the external fields, however only the latter ones can be correctly quantized and do not give undesirable classical selfacceleration and effect of Zitterb('wegung type. Spining particles are in some sense the classical limit of the Dirac particle. After the first quantization these new anti-commuting variables are mapped into the Dirac matrices and they disappear from the theory. This is a general feature of the spinning particle models. On the other hand the extension of the configuration space by the anti-commuting spinorial variables yields the models which are super-Poincare invariant. In contrast to the spinning particle models their first quantization gives theory which still involves the anti-commuting variables. As a result of quantization we get rather not a single quantum particle with spin but a (minimal) supermultiplet. The organization of the review follows the models classification sketched above. We begin with the two principal categories of the classical models and the so called arbit.rary spin particles. Then the pseudoclassical group of models is presented including spinning particles, snper-particles, twistorial and harmonic particles, arbitrary superspin models. f\ext we comment the double supersymmetric models with the spinning superparticle in the component and superficld form. We conclude this brief review recapitulating some new developments including first attempts of q-deformation the relativistic model of the spinnig particle and the K-relativistic model of a particle (exceptionally without spin). 'I'll(' literatur(' on the particle with spin is vast. We include here, only the very selective list of references. Among them are some of the works of Professor Lukierski,

t

LAGRA~GIAN

MODELS OF PARTICLES

153

who has been very active in this area. He has introduced (together with J.A de AZCiirraga) one of the nontrivial models of the superparticle. I am very happy to dedicate this brief review to .Jurek on the occasion of his sixtieth birthday.

2.

Classical models

In this section I shall briefly present the classical models of the relativistic particle with spin. The adjective classical means here not only that a model is not quantum but also that it is described by means of the commuting variables only.

2.1.

Vectorial models

Historically the first model has been introduced by Frenkel. The spin of the particle in his model is described directly by a tensor of spin S"v, which is assumed to be proportional to the tensor of internal magnetic moments M"v' It enters the lagrangian via the "transversality condition"

(I) to reduce the number of independent degrees of freedom. We shall call it the Frenkel condition. Explicit form of the action is as follows

(2) It yields the equations of the motion of the form

S"v - (x"Svp - xvS"p)a P (AX" + S"paP)'

° °

(3) (4)

Some developments of this model were done 33 years later by Barut. To describe internal degrees of freedom he introduces the frame of four fourvectors q(;) i = 0, 1,2,3; such that q(O) is proportional to (x") and the rest is orthogonal to (x"). Csing implicit form of the action (5) s = drL(x",q(;),q(;))

J

and the following definition of the tensor of spin

(6) he gets the following form of the equations of motion

o o

(7) (8)

154

Andrzej Frydryszak

Moreover the Fr<'nkel condition is valid. Historically the next classical model was of different kind. It was an exemplification of the idea that particle is an irreducible representation of the Poincare group. Namely in the model of Hanson and Regge the configuration space has coordinates (x,,, A,w), where A E Ll (Ll - orthochronous Lorentz group). It turns out that dependence of the lagrangian function on the x, A, Ii should be restricted to L(x",O""V), where o",'v = ,1.\" Ii~. Now the tensor of spin is given by the formula

(9) and the equations of motion take the same form as in the Barut model. The demand, that in the non-relativistic limit the particle has only three spin degrees of freedom is realized by the condition

(10) It was originally introduced by Dixon [2.5]. In this model it should be included into the action. Let us note that the Frenkel condition and Dixon condition yield the essential differences in possible motions of particles, even in the free case. Explicit realization of such an action takes the form

(11 ) (12)

o

( 13)

o

(11 )

The last constraint follows from the reparametrization invariance of the action. The mass of the particle is renormalized here by the square of the S"v. This model does not give after the first quantization the Dirac particle. Along similar lines is constructed the BMSS model proposed in Ref. [10]. Here again the particle with spin is directly tied up to the irreducible representations of the Poincare group To this end, as a configuration space one takes (zl', A) E The matrix is decomposed into the momentum and spin tensor, where

pl.

pl.

mAllo,

s

mO

(15)

i,XAO"12 A - 1 .,X E ~

(16)

-i8~~

(17)

This means that

(18)

LAGRANGIAN

~ODELS

155

OF PARTICLES

and by the construction (19)

(20) Therefore the model written on such a space has to be of Dixon category. lagrangian finally defining the model is taken in the following form

L = p"i"

i>..

.

+ "2Tr(a12;1-1;1)

The

(21)

The resulting equations of motion are the same as in the Hanson-Regge model. The four-momenta and four-velocities are related in the standard way. This type of model has been recently reformulated and a correspondence to the psudoclassical model was proposed [11]. ]\'ow let us corne back to the vectorial models. In 1978 Grassberger proposed the description [8] in which the Minkowski space is extended by the two four-vectorial internal degrees of freedom (x,,) >--+ (x",a",b,,). The Poincare invariant action is defined by means of the lagrangian I .2

x ) + x,,((Jb" -

L = 2n!(1 -

au")

I .

+ 2(b"a" -

CL"b")

(22)

The Lagrange multipliers m, a, {3 are introduced to provide the necessary constraints

x·2

I

a"x" b"x."

0 0

(23) (24) (25)

The tensor of spin obtained from the above lagrangian is composed of the new vectorial internal co-ordinates i.e.

(26) and it obeys the Frenkel condition with S"vS"v = const. Now there are some new features in the equations of motion of this model, namely

d

.

-(mx" + 7]")

dT

.S'"v + P[vXv]

o

(27) (28)

This means that if such a particle is coupled to the external electromagnetic field, or has only passed through the bounded area with non-vanishing field, due to the presence of Tlv term the center of mass and the center of the charge need not coincide. Five years later Cognola, Soldati, Vanzo and Zerbini [9] proposed another vectorial

156

Andrzej Frydryszak

model. Its configuration space is the same as for the Grassberger model, but the new lagrangian takes the form

I - -m 2(pJ.l.vX." X.v)l2 j

-

-

a. v bV

(29)

with S'!'v given by eq.(26) and (30)

The constraints are now of the form

(31)

o

(32)

o

(33)

what obviously means that the Dixon condition is fulfilled. The internal degrees of freedom are here othogonal to the momenta, moreover the direction of fourvelocity and fourmomenta can be different. Equations of motion take the standard form. Another branch of the models of the classical spinnig particle is connected with the Souriau's notion of the space of motions [56J and the coadjoint orbit method. As a sample we indicate here the Refs. [56,58,59J and recently [57]. The classical models sketched in this section have one property in common, they do not give after the first quantization the accepted quantum relativistic Dirac particle. On the other side, coupled to the external electromagnetic field, in the limit of the weak homogenous field, they yield the Hargmann-Michel-Tclegdi equations. The models fulfilling the Frenkel condition have helisoidal curves as the solutions of the equations of motion. This can be interpreted as a counterpart of the Zitterbewegung solution for the Dirac particle, however from the classical point of view such a trajectory for a free particle can be hardly accepted. Despite the technical subtleties of different models of this kind the behaviour of the particular type of the particle with spin depends mainly on the type of the "othogonality condition" for the internal degrees of freedom i.e. the Frenkel or the Dixon condition. The latter one seems to be more natural. 2.2.

Spinorial models

Finally let us comment classical models of the particle with spin described by the spinorial coordinates. The presence of the commuting spinoI' not necessarily means that model describes the particle with spin [61,62]. In the twistor-like approach the massless point particle for example, has the action of the form (34)

LAGRANGIAN MODELS OF PARTICLES

157

where Pm is the particle momentum and A" is a commuting spinorial variable, needed to ensure the mass shell condition (another interesting spinorial model has been discussed in Refs.[68] and [69] with the action of the form S = J dT),'mA:i;m). The bove action is a good starting point to supersymmetric generalizations. The model of a point particle with spin described by commuting spinors as dynamical variable has been proposed originally by Barut and Duru [31, :32]. The free part of the lagrangian is given in the form

(35) A is a Dirac spinor, p is a constant. An interesting property -of this model is that it exhibits the classical analog of the Zitterbewegung, which can be seen directly from the solutions of the equations of motion [31]

+ ~",PI' sin TnT)A(O) m

A(T)

(cos mT

).(T)

A(O)( cos mT - -,I'P" sin mT)

-

(36)

z

m

where

i,I'PI'A

(38)

-i)"l'pl'

(39) (40)

A,I' A

XI'

The deep analysis of this model can be found in [:31,32] and some recent comments in [33]. In thE' last decade there were considered other models [63-67]. As an ilustration let us consider two of them. The lagrangian of the first model [63J is closely related to the spinning supcrparicle model [77], and is given in the form

L

= ~(C-li:2 + em~) -

h:i;· j

+ 2iji"

(41 )

where h, rna E ~+ and T! is the Majorana spinor. The current ja = ij,aT! has vanishing square. The resulting equation of motion have the form Ii

hj)

0

(42)

2

0

h:i;,T! -2-;'

0

(43) (44)

-(e-1:i; -

dt

:i;2

+ rnGc

Obviously the conserved angular momentum tensor has a contribution from the "internal" degrees of freedom

(45)

158

Andrzej Frydryszak

where Pa is defined by expression in the first equation of motion given above. The Dirac quantization of this particle, after solving the second class constraints (and hence with the breaking of the Lorentz covariance in the spinorial sector of the phase space) gives the condition on states which singles out arbitrary spin and relates the mass and the spin mJ± =

J+- 1- + ±h 2

J

h 2 (J

1 + mo + -)2 2

(46)

The limit for the massless case can be considered as well and gives the description of particles with arbitrary helicity. The second example, the arbitrary spin particle model comes [64] from the geometrical construction of the model on the six dimensional product space of the Minkowski space M and the two-dimensional sphere 8 2 • The family of Lagrangians of the model involves the joint interval in M and 8 2 , with the metric on 8 2 depending explicitly on fourvelocities. It is parametrized by the mass and spin (m, s) and has the following form

(47) where;; is a complex coordinate on the 8 2 , (z") = (1, z) and ~a = (O"a)""z" z", eJ, 2 e2 are einbein fields associated with the reparametrizations in M and 8 . The.d is an additional ("spherical") mass .d = hmcJs(s momentum tensor has the form AA IVl

ab = J:"J!b - XbP"

+

(

+ I)

(s - spin). The conserved

( ) ,,/3;; Ii J!z ) z-0 (-O"ab ) "iJ Z. iJ pz - Z" O"ab

(48)

with the "internal" part defined by complex spherical coordinates. The Dirac quantization of this model can be performed in the covariant way and gives irreducible representations for arbitrary choice of spin.! Let liS note that analogously to the Barut-Duru particle this model exhibits the Zitterbewegung effect as well. Let us recall that this phenomenon is typical for the vectorial models.

Pseudo classical models

3.

In this section we pass to the models with the internal degrees of freedom"described by the anti-commuting co-ordinates. The origins of the pseudomechanics should be dated back to the 1956, to the work of Martin [20]. However, anti-commuting variables appear firstly in the context of the functional integral for fermions in the works of Matthews, Salam [21] and Tobocman [22]. The extension of a configuration space to superspace enlarges the underlying symmetry group. Depending on the type of I

The author thanks S. L. Lyakhovich for comments to this paragraph

159

LAGRANGIAN Y10DELS OF PARTICLES

the model the extension of the Poincare algebra yields the super-Poincare algebra or some super algebra of the other kind. There are two types of such a models: vectorial and spinorial. In models of vectorial type (the spinning particle models) the extension gives a untypical vectorial superalgebra, with the odd generators having vectorial index. Characteristic feature of such models is conventional character of the first quantized theory. Namely, the odd variables upon quantization are mapped into Dirac matrices and disappear on the quantum level. Such particle can be considered as a pseudoclassical limit of the conventional Dirac quantum particle. In fact, this was the non-achieved goal of the vectorial classical models presented in the previous section. The spinorial models (the superparticle models) have the super-Poincare algebra (or its extension) as a symmetry generators. This models are connected more closely to the relativistic supersymmetry and the superparticles can be viewed upon, as a minima!. irreducible representations of the super-Poiucarc group. However, such an objects contains the whole multiplet of fields with different spin but not only the spin one half component. On the first quantized level one still deals with the anti-commuting variables and instead of the wave functions the the wave super-functions have to be considered. The Dirac equation is not used literary but finds its superspace counterpart. It is worth noting that there exists an equivalence betveen some pseudoclassical and classical models of particles with spin which allows to generalize the notion of Zitterbewegung to the pseudoclassical case [16] (d. as well Ref. [17]). 3.1.

Spinning particles

Twenty years after the anti-commuting variables were introduced into the physical literature for the first time, there was proposed the spinning particle model by Berezin, Marinov [is] and Barducci, Casalbuoni, Lusanna [19]. The configuration space for this model is described by the set of co-ordinates (xl" 01" Os), where the O-variables are anticommuting between themselves; 01' beeing fourvector and Os a scalar. Proposed lagrangians were of the form 1·

I·

-m

1·

1

.

(X" - ;;:OIJ.Os)(x" - ;;:0,,(5) - 2°1'0" - ;,/50S

-rrl\/-x

2

+~

(0,,01'

+ 0505 -

(J~x201J. + Os)'\)

(49)

(50)

Let us focus on the model given by the first of above lagrangians. It is invariant under the supertranslations X JL

0" Os

, !---4' X J1.

0'" O's

xI' -

0" Os

{IJ.A0 5 + {sHO"

+ t" + (s

(51 ) (52)

(53)

160

Andrzej Frydryszak

where A, B arc numerical constants. The algebra of generators of these transformations is defined by the following relations

{Q", Qv}

(54)

{Qs, Qs}

b

(55)

{Q", Qs}

(H-C)P"

(56)

The Q", Q., commutes with P". Performing canonical analysis of the model one gets the first class constraints

P2 -m 2

o

(57)

p"O" - mOs

o

(.'i8)

After the first quantization one obtains precisely the Klein-Gordon and Dirac equations. For the O-sector of the phase space the anti commutation relations

[OI',Ovt

-hgl'v

(59)

[Os, Os] +

n

(60)

Ost

0

(61 )

[0

1"

show that the classical variables originating from the Grassmann algebra arc mapped after quantization to the elements of the appropriate Clifford algebra, here 01, I-> /'f"'lI''''IS, OS I-> /'f"'lS (where II" "'Is - Dirac matrices). Above model can be generalized taking into account the repararnetrization invariance, which yields the supergravity in d= 1 [19] (cf. also J. van Holten's contribution to this volume). For the relativistic point particle one can explicitly achieve time reparametrization invariance by means of einbein field e( T). In the case of the spinning particle it is necessary to introduce its supersymmetric partner 1jJ(T). Resulting lagrangian takes the form

L = e- 1 :i: 2

+ em 2 + i(Oj)1' + OsOs) + i(mOs -

e-1:i:1'01')1jJ

(62)

The action given by this lagrangian is invariant under Poincare transfo\,mations, reparametrizations and the local supersymmetry transformations, what justifies the associacion with the J) = 1 supergravity. Namely,

/jT

-O(T)

(63)

/jxl'

0:( T):i:1'

+ it( T)01' a(T)OI' + (2e)-lf(T)(2:i:1' -

(64)

/jOI' 60 s

6t {)1./J

a(T)Os - mC(T) d T(O:(T)e + i((T)V') (T d dT (O(T)ljJ + 2t(T))

i1jJOI')

(65) (66) (67) (68)

LAGRANGIAN MODELS OF PARTICLES

161

The Euler- Lagrange equations for the e and 'IjJ are of algebraic character and this fields can be easily eliminated what yields the other version of the lagrangian which was considered in [29, :30]. The most general form of the action for the spinning particle with the supergravity multiplet can be given by the lagrangian of the form [39]

L

ige- I ;j;2 - ibcm 2 + goj)1' + &0 505 + mb05'IjJ (ge- I + 2e- 3;j;2 g')'IjJ(;j;I' BI') + 2e- 2g'(;j;I' BI')2

(69) (70)

All the spinning particle models have the property that they are the classical limits of the Dirac field theory and the anticommuting variables are present only in the classical description. The coupling of such models to the external electromagnetic or Yang-Mills fields yields vectorial superspace versions of the Bargmann-Michel-Telegdi or Wong equations [19]. Therefore in some sense the spinning particle models are improved versions of the conventional vectorial models, now with the proper quantum picture. Let us finish this section with the superfield formulation of the spinning particle proposed by Ikernori [49,50]. The first step consists in considering instead of single conventional time parameter a generalized super-time as a (111) dimensional superspace with coordinates (t, 1/), where 1/ is the new anticommuting variable. This means that trajectories of a system will take values in the superspace too. Kamely,

X(t,7]) = x(t)

+ i7]O(t),

X E C OO (t)[7]]o

(71)

Above superfield unifies in one object the even and odd coordinates of the spinning particle. The supersymmetry present in the super-time space is called the little SUSY: (t, 7]) - - t (t + T, aT/, 1/ + Cl'), where T, is an even and Cl' an odd infinitesimal parameter. The algebra of supercharges and covariant derivatives is of the form

017 D = 017 [Q, Q]+

-

7]Ot,

= 20t

o

=

07]'

(72)

(73) (74)

[D, D]+ = -20t

(75)

[Q, IJ]+ = 0

(76)

To introduce the local invariance the d = 1 supergravity multiplet is needed. It enters the super-zweibein field (EX), where aM = (at, a,,) and \7 A = EX aM. The action takes the form

(77)

162

Andrzej Frydryszak

The customary choice of the gauge for the super-zweibein field is the following

FM

-

(

( JA ) -

£ -1

_e-1,1, )

-e-l17

e-1£

-

'"

'

E = e + T/JjJ

(78)

Recently the model of the spinning particle with arbitrary number of supersymmetries on the world-line has been constrncted [51,55]. Such an lV-extended little SCSY in the massive model of the spinning particle, after the field redefinitions in the equations of motion, yields the supersymmetric Lax equation. Moreover it can be used in the study of hyperbolic Kac-Moody algebras.

3.2.

Superparticle models

The extension of the Minkowski space to the superspace with the additional spinorial coordinate is the basic structure of the supersymmetric field theories [18]. The super-Poincare group becomes the fundamental symmetry of the theory. Now, the superparticles are generalizations of the relativistic point particle from the Minkowski space to such a superspace, with still the same requirement at the background - to describe" correctly" the spin. Because they incorporate super-Poincare invariance there is a close connection between superparticles and representation of supersymmetry. Some models provide a natural examples of actions which yield, after the first quantization, the minimal irreducible representations of the given superPoincare superalgebra. The superparticles have reach symmetry, as local as rigid [a1]. In many respects the quantization procedure is diflicult because of the complicated structure of the phase space constraints. This aspect of the superparticle models makes that they are instructive toy models used to understand the superstrings and the variety of their quantization procedures. There are important differences between massless and massive models, ho~ever we will not stress them, aiming only to ilustrate generally the historical development in the construction of the models. The first pseudoclassical relativistic particle model with the spinorial grassmanian coordinates was proposed by Casalbuoni in 1976 [14]. On the configuration superspace (xl" (Jo" iF') the he defined the lagrangian of the form L = -rnJw!,w!'

(79)

dw" = dx" - i(dOu!,(J - (Ju"dO)

(80)

where is the super one-form, invariant under supertranslations

,

X"f-7x!,

x" - i(tu,,(J - Ou"t)

0",

0", Oex

0"

f-7 f-7

0' a

B~

(81 )

+ fa

(S2)

+ (0.

(Sa)

163

LAGRANGIAN MODELS OF PARTICLES

The lagrangian is too poor to give after the first quantization the Dirac equations and some interesting supersymmetric multiplets. The chiral supermultiplet content of the Casalbuoni's G4 model was analysed by Almond [15]. To improve this model the first order fermionic kinetic terms are needed. But they cannot be introduced in a strightforward way, because of the relation

o"dT ~()" = ~(O 0") dT"

(84)

Four years later Volkov and Pashnev [23] tried to cure this drawback using more general super one-form, invariant under super-Poincare transformations. Namely, 2

d8 = dw"dw"

+ adO"'d()" -

a*dO"dO o ,

aEC

(85)

and then the action of the form S=

_rnJ

T

_mJ Jw"w" + aO"O" - a·eo0" T

2,Jd;;i. Tl

=

2

T

(86)

TJ

Now the fermionic kinetic term is present and gives the first class constraints. However, not the one playing upon quantization the role of the Dirac equation. Nevertheless the content of the model is more reach, since the first quantized theory contains some multiplets (two scalar multiplets and one vector multiplet of states with the negative norm). The Brink-Schwarz action for a superparticle of mass Tn in d dimensions uses again invariant super oncform w.The reparametrization invariance is provided by the einbein field what enables to consider a massless superparticle as well. In 1981 they proposed an action of the form [28]

(87) where

w"=:i;n+ior"o,

n=I,2, ... ,d-l

(88)

Specialy massless case is intersting here, because there is an additional invariance present (Siegel [:37]) b"O" O,.:l'H

bKt'

r;:;"fJ KfJ,

r:::;"'fJ =

Wn

(l''')''fJ

(89)

-ibKOr"()

(90)

4ieOK

(91)

whre K is an anticommuting spinoral parameter. This symmetry allows to reduce some of the () - degrees of freedom (here half of them, in general at most half)

[36-38]. The first massive superparticle model which exhibits

K -

symmetry was introduced

164

Andrzej Frydryszak

in 1982 by de Azcarraga and Lukierski [24]. In their model this symmetry was firstly observed but the role of such a gauge invariance in reduction of the degrees of freedom was first pointed out by Siegel [37] and he introduced modified action. To finally overcome the problem with the fermionic kinetic term present in the Casalbuoni's model one has to enlarge the superspace. In the de Azca.rraga-Lukierski model it is done by considering the N-extended Y1inkowski superspace (xl',Oj",Of), i = 1,2, ... , N and introducing central charges to the superalgebra. I1ence the resulting underlying rigid symmetry gets enlarged to N -extended super- Poincare superalgebra. The new "isotopic" structure allows to use internal symplectic metric A ij = -Aji and the expression of the form Oi" A ij 1;oj" now is not a total time derivative and can contribute nontrivially to the action. After obvious modification in the super one form (92) the lagrangian function can be written as

~ + Z'(Oi0 A ij O'jo 1 = -mVwJlwl' J

Oi A ij O'-F') +"

(93)

The fermionic kinetic term is in fact of the Wess-Zumino type and changes under supersymmetry transformations by a total time derivative. Indeed, let Z/J be a symmetric, Lorentz invariant matrix (where I, J could be multi-indices e.g. 1= (Q, i) and Z/J = f"BA ij ), then the simple example of the WZ-term for a supersymmetric particle is of the form (94) What means that one starts from the closed super-twoform h = idOZdO. It;s exact; with b = idOZO one can write h = db. From the invariance of of h under supertranslations it follows that d(b,b) = 0 and at least locally b,b = df for some superfunction f. For the AL-action it means that f-variation yields the total time derivative change in the lagrangian. The AL-model after the first quantization yields the irreducible representations of the N -extended super- Poincare superalgebra. The whole spectrum of supersymmetric multiplcts was found as a result of the first quantization [43,81] not only for the massive case but also for the massless [44]. In the quantization of this model there was firstly applied the supersymmetric generalization of the Gupta- Bleuler [40,41] quantization method [42,43], which later was used in quantization of various systems exhibiting the similar structure of the second class constraints (i.e. hermitean splitting of the set of the second class constraints into the subsets of conjugated, relatively first class constraints). The coupling of this model to the external fields gives interesting results. Comparing to the traditional equations of the spin precession in the external electromagnetic field we obtain that the superspace generalization of the Bargmann-Michel-Telegdi

LAGRANGIAN MODELS OF PARTICLES

165

equations takes the form [45] , L ',,". e, x"

+ 9 s,spin PA "" FPA

(95)

U

9 F,," W "

+ (- e.-

L'"PW W. 9 ). W", p ,,-

(96)

2m e (W' - 2m "w" - W· "w" )F"P'zp,

(97)

-- i2 E: ""PAP P('O AO-) - _i_ h W,,were 2m 2 E: ""PAP "S'PA • Spin an d .JcSpin "" l kO' k. F'or th' c ex t erna I YangMills field we obtain, within the minimal coupling, the generalized Wong equations [45] >

p" i"

91

F"" I"'x" + 92 SSpin 0" FPA I" a PA a

ra A b I C X. " 91} bc " a W"I

92 F ""

a

2m

92) bc

Fb

91 + ( -2m -

91 (W· vWIJ.

--

ra

-

''"

ICS'''" Spin

). F"PW W. la 92 W" a p "

W·) LWp' la J.lW V I'a Zp ,

(98) (99)

(100) (101 )

However, the AL-model is supersymmetric therefore the fully supcrsymmetric coupling to the supersymmetric field is of greater interest. It can be found in Ref. [60]. In the case of the supersymmetric Yang-Mills and supergravity theories it gives in a natural way the conventional sets of constraints for these fields. 3.3.

Twistorial models

The supersymmetric particle models using the twistor-like variables were developed in the eighties, firstly in the component formulation then in the superfield one [61,62,65,66,71,72]. The very important result obtained within this formulation consists in re-expressing upon use of the equations of motion the local world-line supersymmetryas K-transformation [71,72]. General feature of this kind of models is a possibility of manifestly covariant quantization. To merely signal the existence of very reach developments let us recall the superfield version of the model beeing generalization of the following component action [71J

(102) Namely, SI = -i

J

dTd7] Pm (DX m + i81m De),

(103)

where D = a~ + i7]OT and Pm = Pm + i7]Pm, X m = X m + i7]Xm, eo = Oalpha + 7]>'",. In the component version this action contains additional to the S an auxiliary term of the form

(104)

Andrzej Frydryszak

166

The mechanism of trading the twistorial superparticle's K-symmet.ry for world-line supersymmetry is analysed in series of papers [71-73] and recently in Ref. [74]. Relation between the different forms of the superparticle dynamics, involving spinorial coordinates is analysed in Ref. [75]. The possibility of manifestly covariant quantization of the massless particle model was the motivation of development of the model of harmonic superparticle [76]. The action of this model is a generalization of the Siegel model with some new (harmonic) bosonic variables which are parametrising a suitably choosen coset spaces.

3.4.

Arbitrary superspin models

The model of the classical arbitrary spin particle [64] discussed in Sec.2.2. can be generalized to the pseudoclassical model with the I\'-extended super-Poincare symmetry [70]. After the Dirac quantization this model gives the on-shell massive chiral superfields (the central charges can be introduced as well). The extension of the configuration space M x 8 2 is done in the Minkowski sector, it is changed into the N-extended superYrinkowski superspace with coordinates (x",O"IJN) I = 1,2, ... , N, a = 0,1,2,3. On the new configuration space M 4 14N x 8 2 there is defined the Lagrangian of the form

( lOti) where

n"

." '0 10" " 0'-1 - O· I 0" "0 I x+z ( 0" )

'z"ziJ

" ,,(I

mJV(V

+ 1)

(106) ( 107) (108)

The V is a superspin parameter. The central charges analogous to those of the Azcarraga-Lukierski model can be considered here as well [70].

4.

Doubly supersymmetric models

The doubly supersymmetric models were considered firstly by Gates and ;'IIishino [51]. To extend the ~SR string theory they proposed a new class of superstring models which possess both spacetime and world-sheet supersymmetries. Then within this scheme the particle model was considered [52,53]. There are also two other approaches to such particle models: the twistor-like superfield models (commented in previous section) and the spinning particle model (invented firstly in the component form [77, 78]). The spinning particle models are revieved in Ref. [82]. here we shall restrict ourselves to the brief ilustration of the superfield realization [79,80]. In the (supersymmetry)2

LAGRAKGIA;-.J MODELS OF PARTICLES

167

particle models one introduces the supertime space (t, 1/) and the superMinkowski superspace. Therefore the trajectories of a point object are the mappings

(109) where

X"'(t,ll) eA(t,ll)

::= ::=

x"'(t) + 1/A"'(t) OA(t)+1I'pA

(110) (III)

Introducing the covariant object

(112) where V is the covariant superderivative given by VA == E';!iJM (cL eg. (71)) one can write the supersymmetric invariant action in the form

s

~

Jdtd1/8det(E,;!)V7}Y'" . Y", ==

2"1Jdt( e - [ ,w2 _e('P/ m p)2

- 2C -1,/.''''' 'f/A W",

+ A~)

-

2'('''' 1 W

(11:1) -

,1.,,,,)'f/A 'P/mp-

(114) (115)

The superfidd covariant phase space description of this model in the rigid supersymmetry case was given in Ref. [81]. One can say that developments in the spinning and superparticle models has been resumed in their superfield formulation which appeared in the second half of the eighties. It has turned out that all types of the pseudomechanical description can be put together and organized in a joint superfield model (Jet us note that there exists the superfield formulation of a spinning particle alone, but not of the superparticle, which has to coexist in the superfield formulation with the spinning particle).

5.

Recent developments: q-deformed spinning particle and K-relativistic model

Finally let us mention the brand new aspect of the relativistic particle models, namely their deformations. Actually there are not well established deformed models. However, without entering into the question why things have to be (or not to be) deformed we shall recall two examples: the q-dcformed spinning particle and t.he Krelativistic part.icle. The example of t.he q-deformed relativistic spinning particle was considered by Malik [83]. With the use of the first order Lagrangian of the spinning particle and the q-deformed graded commutation relations for (x m , 1/;"', p"', e) in the phase space he introduces" deformed" G Lq (2)-invariant Lagrangian

(116)

168

Andrzej Frydryszak

The model is under investigation and its Dirac" deformed" quantization is still to be performed. The K-rclativistic particle is more "physical". It lives in the K-deformed Minkowski space [86,87] with the mass shell condition modified to the following form

(2K sinh po )2 _ 2K

jJ

= m2

( 117)

This model can bl' described within the formalism with commuting as well as noncommuting space-time coordinates. The interesting properties of the object of this kind are discussed in Refs. [84,85]. However an inclusion of spin to this model is still an open question. Acknowledgements This work is supported in part by KHN Grant

#

2 1'302 087 06.

REFERENCES [I] Frenkel .1., Z. fiir Physik 37 (1926),243 [2] Thomas L.H., I'hil.Magazine 3 (1927), 1 [3] Ilargmann V., Michell,., Telegdi V.L., Phys. Rev. Lett. 2 (1959),435 [4] Corben H.C., "Classical and Quantum Theories of Spinning Particles", Holden-Day, San Francisco 1968 [il] Barut A.a., "Electrodynamics and Classical Theory of Fields and Particles", MacMillan, :\ew York 1964 [6] Hanson A.J., Rcggc T., Ann.Phys. 87 (1974),498 [7] Hanson A.J., Regge T., Teitelboim C., Constrained Hamiltonian Systems", Accad.Naz.dei Lincei, Rome 1976 [8] Grassberger 1'., J.Phys. A: Math Gen. 11 (1978), 1221 [9] Cognola G., Soldati R., Vanzo 1,., Zerbini 5., Phys. Lett. 104 B (1981),67 [10] Balachandran A.P., Marmo G., Stern A., Skagerstam Bo-S., Phys. Lett. 89 B (1980), 199 [II] Cho J-H., Hyun S., Kim J-K., "A Covariant Formulation of Classical Spinning Particle", prcprint YU:vIS-93-09, Seoul 1993 [12] Cog[]ola G., Soldati R., Zerbini 5., preprint CTFn, Univ. di Trento, 1982 [13] Stern A., Skagerstam Bo-S., Physica Scripta 24 (1981),193 [14] Casalbuoni R., Nuovo Cim. 33 A (1976), 389

LI\GRA:\'GIAN MODELS OF PARTICLES

169

[15] Almond P., "The Suprsymmetry Extended Weyl Algebra and Casalbuoni's G4 Model", preprint QMC/02-81, London 1981 [16] Barut A.O.,Pavsic ~1., Phys. Lett. B 216 (1989), 297 [17] Cho J-H., Hyun 5., Kim J-K., "Relation between Classieal and Pseudo-classical Spinnig Particle", preprint YCYlS-93-8, Seoul 1993 [18J Berezin F.A., Y1arinov M.S., Ann. Phys. 104 (1977),

3;~6

[19] Harducci A., Casalbuoni R., Lusanna 1,., Nuovo Cim. 35 A (1976), :H7 [20] Martin J., Proc. Roy. Soc. of London A 251 (1959),536 [21] Matthews P., Salam A., Nuovo Cim. 2 (1955), 120 [22] Tobocman W., :-,'uovo Cim. 3 (1956), 134 [23] Volkov D.V., Pashnev A.T., Teor. Mat. Fiz. 44 (1980),321 [24] de Awirraga J .A., J.ukierski J., Phys. Lett. 113 B (1982), 170 [25] Dixon W.G., Nuovo. Cim. 34 (1964),317 [26] Brink L., Di Vecchia P., Howe P., Nud. Phys. B 118 (1977),76 [27] Brink L., Oeser 5., Zumino B., Di Vecchia 1'., Howe P., Phys. Lett. 64 B (1976),437 [28] Brink L., Schwarz J.H., Phys. Lett. 100 B (1981),310 [29] Galvao C.A.P., Teitelboim C., J. Math. Phys. 21 (1980), 1863 [30] Sundermeyer K., "Constmined Dynamics" Springer, Berlin 1982 [31J Harut A. 0., Duru 1.11., Phys.Rev.Lett. v.52, 23 (1984),2009 [32] Barut A. 0., Duru 1.11., Phys.Rev.Lett. v.53, 25 (1984),2355 [33] Leon J., Martin J.M., "Introducing spin to classical phase space", LAEFF preprint #26/1995, Madrid [34] Townsend P.K., "Spacetime supel'syrnrnetl'ic particles and strings in background fields" in proceedings of the first Torino meeting on Superunification and Extra Dimensions,1985 [35] Siegel W., Class. Quantum Grav. 2 (1985). L95 [36] Siegel W., Phys. Lett. 203 B (1988),79 [37] Siegel W., "Introduction to string field theory", World Scientific, Singapore 1988 [;~8]

Evans J.M., ;\"ucl. Phys. B 331 (1990),711

[;~9]

Harducci 1\., Giachetti R., Gomis J., Sorace E., J. Phys A: Math. Gen. 17 (1984), 3277

170

Andrzej Frydryszak

[40] Gupta S., Proc. Roy. Soc. of London 63 A (1950), 6R1 [41] Bleuler K., Helv. Phys. Acta. 23 (1950),567 [42] Lusanna 1.., in "Supcrsymmetry and Supergravity" proceedings of XIX Karpacz Winter School ofThpor. Phys. World Scientific, Singapore 1983 [43] Frydryszak A., Phys. Rev. D 30 (19R4), 2172 [44] Frydryszak A., Phys. Rev. D 35 (1987),2432 [45] Frydryszak A., "Supersymmetric particle with internal symmetries in external electromagnetic and Yang-Mills fields" 1FT/UWr 1982 [46] de Azc:irraga J.A., Lukierski .1., Phys. Rev. D 28 (1982), 1337 [47] Frydryszak A., Lukierski .I., Phys. Lett. 117B (19R2), .51 [48] Gates Jr. S.J .. Grisaru :\1.'1'., Rocek M., Gates W., "Superspace" Benjamin/Cummings, London 198:3 [49J Ikemori 11., "Superjicld formulation of superpar·ticle" preprint, 19RR

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DPNU-88-03, :'lagoya Univ.

[50] Ikemori H., Z. fUr Physik C: Particles and Fields 44 (1989),625 [51] Gates S.J. Jr., Nishino II., Class. Quantum. Grav. 3 (1986),745 [52] Gates S.J. Jr., Majumdar P., Mod. Phys. Lett. 4A (1989),339 [53] Gates S../ . .I r., in "Functional integration, geometry and strings", proceedings of XXV Karpacz Winter School of Theor. Phys., Hirkhiiuser, Basel 1989 [54] Gates S.J. Jr., Rana L., "A Theory of Spinning Particles for Large N-extended Supersymmmeil'y " hep-th/9504025 [55] Gates S.J. Jr., Rana , "A Theory of Spinning Particles for Large N-extended Supersymmmetry (II)" hep-th/9510151 [56] Souriau .I.-M., "Structure des systemes dynamiques", Dunod, Paris 1970 [57] Zakrzewski S., "Extended phase space for a spinning particle ", hep-th/9412100 [58] Duval Ch., Horvathy P., Ann. Phy,. (:"Y) 142 (1982),10 [59] Duval Ch., Ann. lnst. 11. Poincare A XXV (1976), :l45 [60] Lusanna 1.., Milewski B., :"ucl. Phys. B 247 (1984),396 [61] Bengtsson A.h.II., Bengtsson 1., Cederwall M., Linden N., Phys. Rev. D 36 (1987), 1766 [62] Penrosp R., Mac Callum M.A.H., Phys. Rep. 6 (19n), 109

LAGRANGIAN MODELS OF PARTICLES

171

[6aJ Hasiewicz Z., Siemioll P., Defever F., Int. J. Mod. Phys. A 17 (1992), :J979 [61] Kuzenko S.M., Lyakhovich S.L., Segal A.Yu., Int. J. Mod. Phys. A 10 (1995), 1529 [65] Penrose R., Rindler W., "Spinal's and space-time", Cambridge Cniv. Press, Cambridge 1986 [66J Gershun V.D., Tkach V.I., JETI' Lett. 29 (1979),320 [67] Howe P.S., Penati S., Pernici M., Townsend 1'., Phys. Lett. B 215 (1988),255 [68] Ferber A., t"ucl. Phys. B 132 (1977),55 [69] Shirafuji T., Prog. Theor. Phys. 70 (1983), 18 [70] Kuzenko S.M., Lyakhovich S.L., Segal A.Yu., Phys. Lett. B 348 (1995),421 [71] Sorokin D.P., Tkach V.I., Volkov D.V., Mod. Phys. Lett. A4 (1989),901 [72] Sorokin D.P., Tkach V.I., Volkov D.V., Zheltukhin A.A., Phys. Lett. 216B (1989),302

[ra] Bandos LA., t"urmagambetov A., Sorokin D.P., Vo1kov D.V., Class. Quantum. Grav. 12 (1995), 1881 [74] Galperin A.S., Howe P.S., Stelle K.S., ~ucl. Phys. B 368 (1992),248 [75] Cederwall M., "A note on the Relation between Different Forms of Superpartiele Dynamics", preprint ITP-9a-33, Goteborg 199a (hep-th/9:310177) [76] Sokatchev Eo, Class. Quantum. Grav. 4 (1987),237 [77] Aoyama S., Kowalski-Glikman .1., Lukierski J., van Holten J .W., Phys. Lett. 217B (1989),95 [78J Kowalski-Glikman J., Lukierski J., Mod. Phys. Lett. A4 (1989), 2437 [79] Kavalov A., Mkrtchyan ILL., "Spinning superparticle", preprint Yer I'hI/1068(31)-88, Yerevan, 1988 (unpublished) [80] Frydryszak A., "Superjield Spinning Superpartiele Model and (Supersymmetryj2 ", preprint ITP UWr 726/89, Wroclaw, 1989 (unpublished) [81J de Azca.rraga J.A., Frydryszak A., Lukierski J., Phys. Lett. B 247 (1990),289 [82] Frydryszak A., Lukierski .1., "Spinning Superparticle Models - Recent Developments", preprint ITP /UWr 752/90, WrocJaw 1990 [83] Malik R.P., "On q-deformed spinning t'elativistic pat·ticle", hep-th/950302 [84] Lukierski J., Ruegg H., Zakrzewski W.J., in "Quantum Groups. Formalism and Applications" proceedings of XXX Karpacz School of Theor. Phys., PWN, Wroclaw 1995 [85J Lukierski J., Ruegg II., Zakrzewski W.J., Ann. Phys. 243 (1995),90

172

Andrzej Frydryszak

[86J Lnkierski .J., :\'owicki A., Ruegg H. Phys. Lett. B 264 (1991), 3:11 [87] Lukierski J., :\'owicki A., Ruegg 11. Phys. Lett. B 313 (1993), 357

From Fidd Theory to Quantum Groups

D = I SUPERGRAVITY AND SPINNING PARTICLES J.W.

VAN HOLTEI\"

NIKHEF!FOM, P.O. Box 41882 1009 DB Amsterdam NL

Abstract: In this paper I review the multiplet calculus of N = I, I local supersymmetry with applications to the construction of models for spinning particles in background fields, and models with space-time supersymmetry. New features include a non-linear realization of the local supersymmetry algebra and the coupling to antisYIllmetric tensor fields of both odd and even rank. The non-linear realization allows the construction of a D = 1 cosmological-constant term, which provides a mass term in the equations of motion. D

1.

=

Worldline supersymmetry

Supersymmctry, as a symmetry between bosons and fermions, was discovered almost 25 years ago [1,3]. Apart from mathematical elegance, supersymmetry has the quality of improving the short-distance behaviour of quantum theories and has therefore been proposed as an ingredient of many models of physical phenomena, most often but not exclusively in the domain of particle physics and quantum gravity. Many of these applications are presently speculative, but it was realized already early that supersymmetric extensions of relativistic particle mechanics describe ordinary Dirac fermions [4,7]. Supersymmetric theories of this type are known as spinning particle models. They are useflll providing low-energy descriptions for fermions in external fields [7,12] and path-integral expressions for perturbative amplitudes in quantum field theories [13,20]. They are also useful in studying aspects of higher-dimensional supersymmetric field theories and superstring models [21]. Like in string theory, in the discussion of supersymmetric point-particle models one has to distinguish between worldline supersymmetry, where the supersymmetry concerns transformatiolls of the worldline parameters of the physical variables (proper Dedicated to Jurek Lukierski on the occasion of his 60th anniversary.

174

J.W. van Holten

time), and space-time supersylllmetry which refers to supersymmetry in the target space of the physical variables. Both types of supersymmctry are encountered in the literature. In fact, in [22J a model was constructed possessin~ both types of supersymmetry simultaneously. In the present paper I review the construction of pseudo-classical spinning particle models with worldline supersymmetry. To this end I present the multiplet calculus for local worldline supersymmetry (N = 1 supergravity in one dimension) and construct general lagrangians for D = 1 supersymmetric linear and non-linear (j-modcls with potentials; many elements of this formalism were developed in [24,25J. The coupling to all kinds of background fields, including scalars, abelian and non-abelian vectors fields, gravity and anti-symmetric tensors is discussed in some detail. I finish with the construction of a model which exhibits space-time supersymmetry as well [22].

2.

D = 1 supermultipIets

Supergravity models in a+-I space-time dimensions describe spinning particles. Indeed, the local supersymmetry and reparametrization invariance generate first-class constraints which after quantization can be identified with the Dirac and Klein-Gordon equations. IIence the quantum states of the model are spinorial wave functions for a fermion in a d-dimensional target space-time. The models I construct below are general N = 1 supergravity actions in f) = 1 with at most 2 proper-time derivatives, and the correponding quantum theories. IIigher-N models have been considered for example in [26,27]. Extended target-space supersymrnetry has been studied in [28,29]. In one dimension supersymmetry is realized off-shelP by a number of different sets of variables, the supermultiplets (superfields): 1. The gauge multiplet (e, X) consisting of the einbein e and its superpartner X, the gravitino; under infinitesimal local worldline super-reparametrizations, generated by the parameter-valued operator D(~,e:) where ~(T) is the commuting parameter of translations and e:( T) the anti-commuting parameter of supersymmetry, the multiplet transforms as

De

d(~e) . ---2ze: v dT '"

d(~X)

de:

~+-dT'

(1)

2. Scalar lIlultiplets (x, 1/'), used to describe the position and spin co-ordinates of particles. The transformation rules are lThe term off shell implies that the supersyrnmetry algebra is realized without using dynamical constraints like the equations of motion.

[) =

1 Sllpergravity and Spinning Particles

175

dx

8x = ~ dr - icV',

(2)

where the sllpercovariant derivative is constructed with the gravitino as the connection:

dx

+ ix1/;·

dr

(3)

3. Fermionic multiplds (TJ,j) with Grassmann-odd TJ and even f. The fcomponent is most often used as an auxiliary variable, without dynamics of itself. The transformation properties under local super-reparametrizations are


U

df r

. 1 '1"l e

(4)

= ~-d - lc-vTTJ·

The supercovariant derivative is formed using the same recipe as before:

(5)

4. A non-linear multiplet consisting of a single fermionic component transformation rules {j 17 =

d17 . 1 '1"l (-+c-zc-17v17 T , r e

~d

17

with the

(6)

with the supercovariant derivative

(7)

On any componcnt of any multiplet thc commutator of two infinitesimal variations with parameters (6.2,101,2) results in an infinitesimal transformation with parameters (6,103) given by

(8)

J.W. van Holten

176 For the non-linear representation variation

(T

the proof requires use of the supersymmctry

(9) Then a simple result is obtained:

(10) It follows that each of these multiplets is a representation of the same local supersymmetry algebra, and this algebra closes off-shell. However, the parameters of the resulting transformation depend on the components of the gauge multiplet (e, X), indicating that the algebra of infinitesimal transformations is a soft commutator algebra, rather than an ordinary super Lie-algebra. Among the representations discussed, the gauge multiplet and the non-linear multiplet have mauifestly non-linear transformation rules. The variations of the other two multiplets are linear in the components of these rnultiplets. For this reasou the scalar and fermionic rnultiplets arc called linear representations of local supersyrnmetry, although some of the coefficients depend on the gauge variables (c, X).

3.

Multiplet calculus

The linear representations (scalar and fermionic) satisfy some simple addition and multiplicatiou rules; this tensor calculus has been developed iu [25]. The rules can also be formulated in tenus of D = I superfields [24]. As concerns addition, any linear multiplets of the same type can be added component by component with arbitrary real or complex coefficients. The linearity of the transformation rules then guarantees the sum to be a multiplet of the same type. The multiplication rules are also simple. There are 3 different product formula's: I. The product of two scalar multiplets E = (x,1/J), E' = (x',1//) is a scalar multiplet

E x~' = 1-'" = (J:.r',J:li,'

+ x'0).

(II)

This rule can be extended to arbitrary powers of scalar multiplets, for example:

(12)

In this way one can define functions of scalar rnultiplets by power series expanSlOns.

D = I Supergravity and Spinning Particles

=

2. The product of a scalar lTlultiplet E (TI, J) is a fermionic multiplet

177

(x, lj;) and a fermionic multiplet P

Ex p = pi = (xTJ,xf - ilj;TJ).

=

(13)

3. The product of two fermionic multiplets is a scalar multiplet:

- ip

X

pi = E ' = (-iTJTJI,fTJ' - fry).

( 14)

:"iext I introduce the operation of derivation of scalar and fermionic multiplets; the super-derivative on linear multiplets is a Grassmann-odd linear operator turning a scalar multiplet into a fermionic multiplet, and vice-versa, with the following components: 'D2.' = pi =

(01,0/, l'D e

T

x) .1

(15) On product multiplcts this super-derivative satisfies the Leibniz rule, with in particular the result 'DE" = n'DE x 2.,,,-1.

(16)

The super-derivative satisfies an operator algebra similar to the supersymmetry algebra:

(17) where 'D T is the supercovariant proper-time derivative on componcnts, encountered before in eqs.(3) and (5).

4.

Invariant actions

1. Invariant actions can bc constructed for the lincar as well as the non-linear multiplcts. As there exists no intrinsic curvature in D = I, there is no invariant action for the gauge multiplet involving the einbein, but there is a very simple action for the gravitino, namely

J

dTiAX,

( 18)

178

J.W. van Holten

where A is a constant. The equation of motion for this action by itself is not consistent (it requires A to vanish); but this is changed if one adds other terms to the action, like the ones discussed below. Also, one can replace X in the action by dxldT, but then the action becomes a total derivative. A cosmological-constant like action can be constructed with the help of the nonlinear multiplet; it reads

(19) This is also the kinetic action for the fermionic r5 variable, which in view of the anticommuting nature of 17 can be only linear in proper-time derivatives. :"Jote, that the non-linear nature of the multiplet allows one to rescale the variable r5 and thereby change the relative co-efficients between the various terms in the action. A rescaling of 17 by a factor lie gives the action

5 nl (r5;e)

=

2' ' JdT (e-"':"'X(J-..!....(J{;-) , 2 c

C

(20)

where I have introduced the dot notation for ordinary proper-time derivatives. Of course, the rescaling also changes the non-linear transformation rule for 17 under supersymmetry to (21) Combining the actions S'A and Snl in such a way as to get standard normalization of the fermion kinetic term for 17 leads to the action (with the dimension of h)

SgTaV

=

m

J

dT (ieAX -

~ e + iCX(J + ~ (J{;-) ,

(22)

where m is a parameter with the dimension of mass and c has the dimension of a velocity, The Euler- Lagrange equations for the fermions X and (J' then give 17

= A,

I d(J X = - - = O. C

dT

(23)

Thus the constant A is like a vacuum expectation value of the fermionic variable 17. However, the variation (21) of (J' is such that for non-zero C it can be gauged away completely by a supersymmctry transformation. Therefore it does not represent a true physical degree of freedom. Of course, this seems to contradict the equation of motion (2:1), but we observe that also the equation for the einbein is inconsistent, requiring the constant c to vanish. Again, these problems are solved by adding further terms to the action. In applications c usually represents the velocity of light, which

D = 1 Supergravity and Spinning Particles

179

can convcnicntly ht> taken as unity (c = I). 2. ;"ext we turn to a formula for the construction of invariant actions for linear multiplets. Given a fcnnionic multiplet !P = (1),!), an action invariant under local supersymmetry transformations is Slin

=

J

dT

(ef - iXT/).

(24)

Note, that in eqs.(19) and (24) the integrand itself is not invariant, but transforms into a total proper-time derivative:

J

5SIin =

dT ddT

(-in/) ,

(25)

and the same for (iSnl with T/ -> u. Eq.(25) shows, that (is vanishes for variations which are zero on the endpoints. Eq.(21) can be applied to the construction of actions for scalar multiplets if one applies an odd number of super-derivatives so as to obtain a composite fermionic multiplet, sometimes called the (fermionic) prepotential. A simple example of this construction is the free kinetic action for a scalar multiplet, constructed from the composite fermionic multiplet

!P bn =

~V21-'

x Vl-',

(26)

Inserting the components of this multiplet into the action formula (21) gives

, J (1. 2e

Skin =

dT

-

X

2

i ·+i- x1/Jx.) . + -1/J1/J 2 e

(27)

If one extends this formula to d free multipJets E" = (x", 1/J"), J1 = 1, ... , d, then using the appropriate minkowski metric it becomes the action for a free spinning particle in d-dimensional space-time. This is a special case of the most gcneral action involving only scalar multiplets and quadratic in proper-time derivatives of the bosonic co-ordinates x": the D = 1 (non- )linear u-model in a d-dimensional target space, constructed from a fermionic multiplet

(28) Here g"v( E) is a symmetric tensor in the space of the scalar multiplets, which can be interpreted as a metric on the target manifold. Using !P[g] in the action formula (21) gives the component expression

(29)

J.W. van Holten

180 when~

D denotes a target-space covariant derivative (30)

with [~/'(x) the Riemann-Christoffel connection. The action Ski,,[g] is manifestly covariant in the target space. The symmetries of this action have been investigated in detail in [30,31], with applications to special target manifolds like Schwarzschild space-time and Taub-:WT in [12,31,32]. The simplest action for scalar multiplets involves only one super-derivative. It starts from the general fermionic multiplet (super I-form)

P [A] = AI'(E) x

P~'I',

(31)

with AI'(E) an abelian vector field on the target space. Inserting the components of

P[A] in the action formula (24) gives the result Svec[A] =

JdT (AI'(x)xl'-IFl'v(x)1/JI'1/Jv),

(32)

with Fl'v the field strength tensor of the abelian vector field AI'(x), A similar construction can be carried out using arbitrary odd super p-forms [24]. For example p = 3 gives

P [H]

~ HI'VA(E) x P~'" x PE v X PEA.

(33)

This gives an action involving the anti-symmetric 3-tensor Ill'v>-' (x):

Soda [Il] =

~Hl'v>-,(X)1/JI'1/JV (x>-. + ~i x1/J>-') + ~()"Hl'v>-.(x)1/J"1/JI'1/Jv1/J>-'.

(34)

The inclusion of even p-forms is also possible, but requires one or more fermionic multiplets; details are given below. In the sallie spirit one can find odd p-form extensions of the kinetic term

P [G] = Gl'vl ...vp(E) x p 2 EI'

X

P~Wl ..•

X

EVP.

(35)

For a discussion of these unconventional actions I refer to [24]. Finally, actions with higher powers of p 2 and/or with P" (n 2': 3) lead to higher-derivative component lagrangians. I do not consider them here. Combining the results for scalar fields, within the restrictions we have imposed the general action for scalar multiplets is of the form

me2 .)' [L;] = mS'kin [g] + qSvcc [A] + aSadd [Il] + meSA - -2-5,,1,

(36)

This action describes a spinning particle in background electro-magnetic and gravitational fields, with the possible inclusion of torsion for ai-D. The first-class constraints obtained from the equation of motion for the einbein and gravitino are

D = 1 Supergravity and Spinning Particles

lng llv x·,"V 1jJ

W H 'lVA JjJIlJjJVJjJA -:3

--

mee

(11 -

(J

181

)•

Hcre f'"v(11) and F"" VA (H) arc the field strenghts of the vector All and 3-form H IlVA ' rcspectively. Eqs.(37) are the pseudo-classical cquivalcnts of the Klein-Gordon and Dirac equations. :\Tote that local supersymmetry can be used to chose a gauge (J = 11 in which the expressions on both sides of the second equation vanish. If e = 0 (absence of Snl and SA) the particle is massless. With the inclusion of Snl (e f- 0) the particle acquires a non-zero mass. If the kinetic terms arc normalized in the standard way, thc relativc co-cfficicnt q between the first two terms represents the electric charge of the spinning particle, as defined by the generalized Lorentz-force [II]. Then the anti-symmetric tensor D"V = QV,1l 1V represents the electric and magnetic dipole momcnts. The terms involving the ;l-form Hllv.\(x) combine to form an anti-symmetric contribution to thc Riemann-Christoffel connection, representing torsion indeed. S. Finally we turn to the construction of actions involving elementary fermionic multiplets. To begin with, there is the simple action formula (24) lincar in the components of a single fermionic multiplet. It involves no proper-time derivatives, and therefore it can constribute only to potential terms. A natural and straightforward generalization of this action involving r fermionic llluitiplets
with the Ui(E) a set of scalar-multiplet valued potentials. The component action then is

(39) As the equation of motion for fi requires all Ui(x) to vanish, this action by itself is useful only to impose constraints on the target manifold. This conclusion is modificd when additional (kinetic) terms are added to the action. More complicated actions obtained using the multiplet calculus with both ferlllionic and scalar multiplets must havc an odd total number of fermionic multiplets and super-derivatives. Therefore the next complicated type of action involves thc product of two fermionic multiplets including a super-derivative. The general form of the fermionic prepotential is

J.W. van Holten

182

(40) with J(;j(E) a scalar-multiplet valued symmetric matrix. The component action for this prepotential is

It contains kinetic terms for the fermionic variables r/, but the variables P only appear without derivatives and are auxiliary degrees of freedom. In combination with the potential term SpodVj its elimination turns the constraints Vi into a true potential, allowing the bosonic variables to fluctuate around the solutions of the constraints

Vi(X) =0. Other actions can be constructed by replacing some of the super-derivatives 'DEJ1. in the odd p-form prepotentials like rJ>[Il] (33) by fermionic multiplets. I give the details for the case p = 3. First consider a prepotentiallinear in fermionic multiplets: (42) The potentials SiJ1.v(X) define l' anti-symmetric tensors (2-forms) on the target space of the scalars. Thus this construction and its higher-rank generalizations allows the inclusion of even p-forms in the action. Substitution in the linear-multiplet action 5Ii", eq.(24), gives

Seven

[il]

iCJiB (),I J1.,I.V iJ1.V X 'I' 'f/ J dT ( -'2

-

. is' (),I.J1. • v TTJ ;J1.V X 'f/ X

1 isip.v (),I.J1.,I.V + '2XTJ X 'f/ 'f/

(43) Next consider the case of a quadratic expression in fermionic multiplets. The prepotential is (44) The vector field V;j,,(x), anti-symmetric in [ij], takes values in a gauge group G <;:; 50 (1') for l' even (in the quantum theory this is always the case [19]). Thus this action describcs the coupling to Yang-Mills fields. After quantization the fermionic variables TJi generatc a Clifford-algebra represcntation of the group G cmbedded in 50(1') on the particle wavc-functions. The explicit cxpression for the pseudo-classical action is

[) = 1 Supergravity and Spinning Particles

183

S'YM[Vj = Jdr (~2 TI i 1] j\l;'J" ( x )'''+=x 4 1] i 1] jp(Oj ij"v ( X ),I,",I,V+' 'f' 'I-' ze fi\l;'J" ( x )1] j,I,") 'I-'

'

(45)

Here FJ~) represents the abelian (linear) part of the field-strength for the vector field \1", The non-abelian part can be obtained by a proper choice of J
t


[1'] = ~ 7i J k(E)

X


X

3. The action constructed from this prepotential becomes