6 H, the map G 3 g >-> Ug Ag is defined by means of left translation, i.e., l
2
l (AggiSI ^)(g) cf>(g; V^£ )(g) 4>(g;g), g), V l 2(G), ( G ) , h,9 e G, G, , Wj i €. jeG (A *)00 ==(Kg^g),
(1.8) (1.8)
Claim 1.5.1 The left regular representationt A of G is a continuous unitary representation of G on L2(G).
1.5. GROUP REPRESENTATIONS
33
Proof: In our proof we follow [7, pp. 135-136]. Clearly, everyi' A A 9s is is a linear operator. Furthermore, l 1 _1 _1 : [A ^]}( 91 [Aw92 ^]}(5) == [h [A }(9i 4>(< 4>{9?9?9) 0[(SiS2) tf)fo), {A„[A {A [< fl}(flr) M9i 9) l9)== =K9m) ](9?9) (A(A 0)(ff), i>(9 *s)= [A992S2 flKff) 2 ) " 1 ff] ^ ff]= =(A„„0)(ff), S1 92 Slfl2 Sl 9l52 2 *9i 9)
i.e., A9ln A and Ae = 1. 92 = A 9]92 Aj; "-S3 S2 :
Therefore, the map (1.8) defines a representation of G in L2(G). Furthermore, since dg is left invariant we have
h
_1 r_1 (A x) = {A = I/ 0(fl ,X), (<£,x)> ^9ggX) 5i)x(ff~ gg4>,A (,X), Ag>
hence, A9 is isometric, and since the range of A9 is all of L2(G), every A9 is unitary. Strong continuity is established as follows, let x € C0(G), where C0(G) is the set of all continuous functions with compact support on G, since every compactly supported continuous function on a Lie group G is uniformly continuous on its support (cf. [7, Proposition 2.2.4]), we have x X(5i)l<e', sup |? sip \x(a~1I9i) < e', sup|x(5 lx(
for g 6 V, where V is a neighborhood of the identity, e, of G. Moreover, since X 6 C0{G) there exists a fixed compact set K € G , supporting x and Agx for g sufficiently close to e, such that I|A,*-Xll I|A l|A --x l l I,2a2 9 X-Xll 9X
-1"I 11/2 \ f 1 /2 2 2 l 1 l doi 9\)-x{9\)\ dg\ / \x(g9i)-x(9i)\ dgi xGT fUG /lx( JG J l- 1 < sup \x(9~ 9i)-x{9i)\M <Me', V ) X(ffi)l^ < Me', < sup |x( |x(tf ffi)<
= =
9i€K Si6A' Bl&K
wheree M Now since C0{G) is dense in L2(G) there exists for each K dg\. Nc M = JJ JJ^dgi2 2 C L {G) C0{G) that ||0 \\4> - X[l:- <x||£'. 4> 6 i6-,2 (G) C0{G) 2
| | A ,^*---00*|| || | 22 = |l|A | A 99((0-x) 0 - x ) +- (A ( A3 X93 X- -X)x )■- ((04 >- x- x))ll l 22 l|A,0
34
CHAPTER 1. MATHEMATICAL
PRELUDE
< < I|A < ll|A | Aff9 (^-X)|| (tf-x)ll ( ^ - x ) | |l222 +I- l|A ||A l|A„x-x|| + Il0-x|| |ll||l*0* -- x | |22,, ||A9f9l x-xl|la2 -t+ using the fact that the Haar measure is invariant under left translations, we find + M)c' M) '< ||A,0 - - \\ 0|| < 2
22
f
99 9
X 22 2
f
Hence, ||A < <££2 < fore gforgeV. e V. geV. \\A93tf>-|| - 2\\ Therefore, we conclude that G 3 g7 H» in e. To show that H» A99 is ccontinuous < H» G 3 j •-» Aj is continuous in every point we have to show that l|A -A l|A9lSl - A £||2 2 < c/J^i c/22 ^V i £e V. tf OT ^9292< l l^|| ^ ££ for ff Using again the left invariance of the Haar measure we conclude ll I|A l ||AA9 1l^^^-A -- A9P2 0||22 = | A s9--. , S9i ^-^ll 0 - \\ fl|22 :<2 £< £forfor,92 € V. 9i5 lgi e V. 2 0|| 92 a -g? = |l|A 2 H-» Therefore, re, G 3 g >-* H-> yAA 9 9is is continuous in every point. Hence, A is a contin
2 uous unitary representation of G on Z, L2{G). (G). O a
1.5.1
Equivalence of Representations
Let f/1 and £/2 be two unitary representations of the same locally compact Lie group G on the Hilbert spaces Hi and H 2 , respectively. Then tf U11 is unitarily equivalent to U2 if there exists a unitary operator T : Hj — ► H 2 such that ffor every TU T ^ 1lg = = £/ U22TT every gff€G. g£ 6 G. G. Tttf: gT for
(1.9) (1.9)
v If1 U1 is unitarily equivalent to»t/ U2 th( then we writeit/ U11 S = U2. A A bounded operator
T from Hi to H 2 is called an intertwining operator for U1 tand1 U22iiTU] if TUl =
1.5. GROUP REPRESENTATIONS
35
UgT for all g € G. The set of all intertwining operators forms a linear space, which is denoted by TZ(UX,U2). Hence, two unitary representations U1 and U2 are unitarily equivalent if H(UX ,U2) contains a unitary operator from Hi onto H 2 . Observe that for U1 - U2, H(Ul,Ul)
is the commutant of the *-algebra
A(Ul) generated by the representation G 3 J H
Ug. Hence, 7£([/' ,Ul) is a
von Neumann algebra.
1.5.2
Irreducibility of Representations
A subspace Hj C H is called invariant under a unitary representation g >-* Ug if and only if UgH\ C Hi for all g € G. A unitary representation g h-> Ug of a locally compact group is called irreducible if U has no invariant subspaces other than Hi = H and Hi = {0}. A unitary representation, which has proper invariant subspaces is called reducible. Let Hi be a closed subspace of H, then one calls the restriction of U to Hi a unitary subrepresentation of U. A unitary representation is called completely reducible if it can be expressed as a direct sum of irreducible unitary subrepresentations. One can show that every finite-dimensional unitary representation of any Lie group is completely reducible (cf. [7, Corollary 5.3.2]). The following theorem is of fundamental importance in the theory of group representations:
36
CHAPTER
1. MATHEMATICAL
PRELUDE
1 T h e o r e m 1.5.1 (Schur's Lemma- unitary case) Let U J2 ,be irreducible, J1 aand U
unitary representations
of a Lie group G on H i and H 2 , respectively.
If T G
£ ( H ! , H 2 ) is such that
TV] =-U^T U22TgT .for all'igeG, jeG, TU]
then, either T is an unitary
operator from H j onto H2 (i.e.
Ul = U2), or
T = 0.
(For a proof see [7, pp. 143-144].) Hence, the set of all irreducible unitary representations of a Lie group G can be partitioned into equivalence classes. We denote by G the set of all equivalence classes of irreducible, unitary representations of a Lie group G. Schur's Lemma implies the following criterion of irreducibility for unitary representations:
Corollary 1.5.2 For a unitary representation
U of a Lie group G on a sep-
arable Hilbert space H to be irreducible it is necessary and sufficient
that the
only operators that commute with all the Ug are scalar multiples of the identity operator.
One can use Corollary 1.5.2 to give a new definition of irreducibility. A unitary representation U is called irreducible if the only operators that commute with all the Ug are scalar multiples of the identity operator. One refers to this formulation of irreducibility as operator irreducibility of U.
1.6. REDUCIBLE
1.6
REPRESENTATIONS
37
Reducible Representations
One can classify reducible unitary representations U according to the properties of the of the center Z of the von Neumann algebra TZ(U,U) of intertwining operators. Let us start with the case when 7l(U, U) is a factor.
Definition 1.6.1 A unitary representation
U of a Lie group G is said to be a
factor representation if and only ifTZ(U,U)
is a factor.
Representations
of
this type are called primary representations. Clearly, by Corollary 1.5.2 every irreducible unitary representation is a factor representation.
P r o p o s i t i o n 1.6.2 A representation
U is a factor representation
of type I if
and only if it is the discrete orthogonal sum of finitely or countably many equivalent irreducible representations,
i.e., if U is the multiple of some
irreducible
representation. (For a proof see [79, Proposition V.l.l].) Another interesting class of unitary representations is obtained if the von Neumann Algebra 1Z(U, U) is abelian, i.e., if the center Z of H(U, U) coincides
with n(U, U). Definition 1.6.3 A unitary representation if and only ifTZ(U,U)
U is said to be multiplicity free,
is abelian.
Observe that if U is both a factor representation and multiplicity free, then TZ(U,U) = {al}, i.e., U is irreducible by the virtue of Corollary 1.5.2. If U is
38
CHAPTER
1. MATHEMATICAL
PRELUDE
completely reducible and multiplicity free, then
u=
e) ^ CO oo
i
C=iI
where all £/"■ are mutually inequivalent and irreducible. Multiplicity free uni tary representations can be decomposed in an essentially reducible unitary subrepresentations.
unique way into ir
If G is a type I group then one can
decompose every unitary representation of G in an essentially
unique way into
irreducible unitary subrepresentations, as the following theorem shows:
T h e o r e m 1.6.4 Let (U, H) be a unitary type I representation
of a Lie group
G. Then there exists a standard Borel measure v and a function
h(-) such that
one has the decomposition
of the space H into a direct integral
L
L
U
JG
(For a proof see [79, Theorem V.2.8].) For the definition of a standard Borel measure see Appendix A.3. Note that one can show that any unitary representation can be written as a direct integral of irreducible representations, however only for unitary type I representations is the decomposition essentially example Mackey [75, pp.54-63].
unique.
For a discussion of this point see for
Chapter 2
PHYSICAL PRELUDE 2.1
Introduction
In non-relativistic quantum mechanics the states of a quantum mechanical system are given by unit vectors, such as T/>, , or 77, in some complex, separable Hilbert space H. For a single canonical degree of freedom problem the basic kinematical variables are represented on H by two unbounded self-adjoint operators P, the momentum, and Q, the position, with a common dense invariant domain D. These operators satisfy the Canonical (Heisenberg) Commutation Relation (CCR)
IQ,P] = U, [Q,P] H, where h — = 1. Let 7i(P,Q) H(P,Q) be the Hamilton operator of a quantum system, then the time evolution of this quantum system in the state tp £ D(H) is given by the time-dependent Schrodinger equation
id idtrl,(t) = n(P,QMt). n(P,QMt). tm = Since only self-adjoint operators may be exponentiated to give one-parameter unitary groups which give the dynamics of a quantum system, it will always be 19
40
CHAPTER
2. PHYSICAL
PRELUDE
assumed that the Hamilton operator is essentially self-adjoint, i.e. its closure is self-adjoint. If the Hamilton operator is not explicitly time dependent then a solution to Schrodinger's equation is given in terms of the strongly continuous one-parameter unitary Schrodinger group, U(t) — exp(—itTi),
by
Ht") 4>(t")-- =u{t" u(t"---t'y U(t" t')i>(f). OiKOOlKOFor a single particle system one traditionally chooses for the Hilbert space H the spaces L2(Si, dq), or L2{M, dp). On these Hilbert spaces the basic kinematical variables P and Q arc realized by the following two unbounded symmetric operators —id/dq and q, or p and id/dp,
respectively.
These operators are
essentially self-adjoint on the dense subspace S(JR) of L2{IR),
the space of in-
finitely often differentiable functions that together with their derivatives fall off faster than the inverse of any polynomial. Furthermore, since these operators leave S(M)
invariant, one can choose S(IR)
C L2(M)
as the common dense
invariant domain for these operators. One calls these operators together with their their common common dense dense invariant invariant domain domain S(1R) S(1R) the the Schrodinger Schrodinger representation representation on on q-space, or or on on p-space, p-space, which which is is denoted denoted by by 4>{q) 4>{q) €€ S{M) S{M) q-space,
or i>(p) i>(p) ££ S(M.), S(M), or
respectively respectively (cf. (cf. [11, [11, Appendix Appendix V]). V]). However, we would like to emphasize that there is nothing sacred about this choice of representation other than the time-honored custom of doing so. One can also choose one of the so-called continuous representations based on canonical coherent states (cf. [61]). In this representation the states are given by certain bounded, continuous, square integrable functions of two real parameters p and q. We denote these functions by i>n{p,q)- The functions ip (p q) 2 (10l\ span a subspace %V L2(R )
2
2
r,( L2 {R of (R2),),v,hwhere the subscript r\ denotes a unit fidu-
2.1.
INTRODUCTION
41
cial vector in the Hilbert space H on which the canonical coherent states are based. Let rj(x) € L2(IR) be a fixed normalized function and 4>{x) £ L2(M)
be
arbitrary, then an explicit representation of the functions ipv(p,q) can be given as follows:
I v^ ' I
q)dx. =rT]{x)t -j=r]{x)exp(-ipx)ip(x UP>l) )exp(—ipx)ip b(x + q)dx. UP>V) = = I/ -y=v{x)exp(-ipx)ip(x
(2.1)
From this form of the representation it may be seen that one obtains the Schrodinger representation in q-space or in p-space in appropriate limits. In particular, one obtains the Schrodinger representation on q-space by suitably scaling the i>v(p,q) so that the scaled fiducial vector approaches in the limit a delta function, and the Schrodinger representation on p-space as the fiducial vector approaches the indicator function of M times the constant e , p ' (see [62]). With each of these various representations one can associate a propagator: in q-space by
J
;q',t'Mq',t')dq>, i>(q",t") /J(.q",t";q\t'mq',t'W, W,t") ==J J(q",t"-q',t') J(.q",t";q\t'mq',t'W, in p-space by
J
i(p",t") ==JfJL(p",t";p',t')4,(p',t')dp\ L(p",t";p>,t')i>(p',f)dp', i>(p",t") L(p",t";p>,t')i>(p',f)dp', v'.t')dv\ HP"X): [p",t";p',t') and finally in the continuous representation based on canonical coherent states by n{[p ,q ( P ' , < 7 ' i', , WW- ? ' ,?'. ,t *"; ;? P\ til
II ill.
„
-/-
Of course each one of these propagators generally depends on the representation one has chosen. Physically, these propagators represent the probability amplitude for the quantum system under discussion to undergo a transition from an initial configuration to some final configuration, and they contain all the
42
CHAPTER 2. PHYSICAL
PRELUDE
relevant dynamical information for the quantum system. Let us ask whether it is possible to find a single propagator atorA-(p", K(p",q",t";p',q',t') o",q"X;p',
J
K(p\q",i"-p',q\t')^{p',q\t')dp'dq\ dp'dq', l";p',q',t')i> ^W,q"j") ( p ",",q",t" , q", t") = J K(p", q", t";p', q», i')^(p', ?', t')dp'dq', n(p ^(p',9',0 ■{p",q'\i" i>ni holds for >r an arbitrary fiducial fiducial vector. Stated otherwise, is there a prop propagator that is independent of the chosen continuous representation, but which, 2 L*(XP) nonetheless, propagates the elements of any representation space L (]R2) in
such a wayy that tthey stay in the representation space Li(m L2(M2)? )? •The answer is yes. We now outline the construction of this propagator for the Heisenberg Weyl group; for an alternative construction of this propagator see Klauder [66].
2.2
T h e Fiducial Vector Independent P r o p a g a t o r for t h e Heisenberg Weyl Group
Let P, Q, and / be an irreducible, self-adjoint representation of the Heisenberg algebra, [Q,P] = il, h = 1, [Q,I] = [P,I] = 0 on H. Then for an arbitrary normalized vector n 6 H define the following set of states v(P,q) V(P,<])= v(p,q) =-E=V(p,q)n, vv{p,q)v, q) = 7-E=V(p,q)r], y in
V2^
where V(p,q) = exp(-iqP)exp(ipQ). vQY In fact these states are the familiar canonical coherent states which form a strongly continuous, overcomplete family of states for a fixed, normalized fiducial vector 77 6 H and they admit the following resolution of identity:
-J
IH= ^H v(p,q){v(p,q),)dpdq. IH = /I V{P, (p,q){v(p,°),g (V(P,Q )dpdq.
2.2. FIDUCIAL
VECTOR
INDEPENDENT
43
PROPAGATOR
2 The map>C C„:H » L2{i (M (M22,,dpdq dpdq), defined for any H by: iy tpt/>66 H by Vv: : H[ --+
1 [C = V>„(p,9) == {■q{p,q),i>) (-j=r),V'{p,q)i}), [C^](p,q)v*( [Cvvvrp](p,q) rp](p,q)-i> {p,q)r) = (v(P,l),i>) ("/=»?> V*(P> = (vh (v(p,q),Tp Miv(p,q) {p,q),v) = (^T),V*(p,q)4>), P, 9)0), 0),
W
yields a representation of the Hilbert space H by bounded, continuous, square 2 2 integrable functions on a proper closed subspace L2(IR 2R ) )
resolution of identity one finds ,(p',q')dp'dq', p'dq', ^i>v(p>q) ( p , ? ?) ) = ICv(p,q;p',q')il> v{p',q')dp'dq', ^{v,q\p',q'y = /fiCvk i;p',q')i>v(p'>q'] / where ■;P',Q') = = {(v >Cv(p^,p',q') = n(p,q),v(p\q')) p,q),v{p\q')) == >,q;p,q) ^n{p^\p',q') {v{p,q),v(p',q'))
1 IW(P',. ; ^y/2*n'•(p,qT ^ )^'T/^^^^^'^'^T/^^ i (PW)^.*^V2n 2
is the reproducing kernel, which is the kernel of a projection operator from L2{JR2) onto the reproducing kernel Hilbert space L2(M2R).2 ). Le Let D be the common dense invariant domain of P and Q that is also invariant underrV(p,q), V(p,q), then one can easily show that the following relations hold on D : -id ~id idqqV'(p,q) V'(p,q ,9) qV*(p,q)
V*(, = V'(p,q)P, V'(p,q)P,
(2.2) (2.2)
)V'(v, id (q + idp)V'(p,q) p)V'(p,q)
= V'(p,q)Q. V(p,q)Q. V(p,q)Q. V(p,q)Q
(2.3) (2.3)
Notice, that the operator >r V*(p, V'(p, q) intertwines the representation of the Heisenberg-algebra on the Hilbert space H , with the representation of the Heisenbergalgebra by right invariant operators on any one of the reproducing . U I . U I I differential uiiii'irim. 2 2 kernel Hilbert spaces :es L2(R (IR2). Furthermore, these differential operators are S). ). Fui
essentially self-adjoint and an appropriate core for these operators is given by the continuous representation of of D , £>„ D,, = = C„(D). ILet n(P,Q) H{P,Q)
be the t
essentially self-adjoint Hamilton operator of a quantum system on H , then
44
CHAPTER
2. PHYSICAL
PRELUDE
using the intertwining relations (2.2) and (2.3) one finds for the time evolution 22 2 of an arbitrary element D,,, C L^(R nt1 il> ip D^ n{p,q,t) r,{p,q,t) ^L (Vi(R R)")) the the following v(p,q,t)
• K[p,q,t;p',q',t')^ri(,p'. t')^(p',q',t')dp'dq' 0<*P'niP, 0 t)==■=// J' A\,(p, #n(P. 9, 9,i; *i p', p«i \P'g', ?'. > ?'.OV»„(p', OV>„(p'> OV>„(p'>9'. 5'.?', i ' )0^ ^' ^9' V(P,g,
'
where A A ^ '('-np,(p,Qj:p',o',t') 7(.(pp, 9?,<;?', ,gg,, ,ii*;; pp; p',,9'9,9',*') 9' ,'t,'0)
j,g),exp[(r,[p, ?g)),,e«xcp[-t'(t q')) i't')H(P, <77(p, pp[[--it ( it -- t')w(i t')W(P, Q)]J7(P', «')) t ' M PP, «QMp', Q ) MPP' ,' 9, '«')> 2)MPW )M )> (»?(Pi?).
= =
1
V V pp 9p9 9eex 1 , ' 44= = V:V*(p,g)exp[-z| **( (*('' )') )x «Pl-*'(* PP'' - 1 ^t-t)H{P,QW{l t')H(P,Q)}V(p', = ^tl> = 4= - -<')H(P «')H(PQ)]V(p', Q)] V(p'q')^==v) 9, ,)4='?> 9V27T )4='?> \J2-K V2TT yi»> ZlT
1
exp[-i(< e x p [ - t ( t - t')H(-id t'yn q,°q,
=
W« \/ZW y/ZW
V27T
where the closure of the Hamilton operator has been denoted by the same symbol.
This construction holds for any r\ € H , therefore, one can choose
any complete orthonormal system {(f>j}'fL1 in H and write down the following propagator K(p,q,t;p',q',t') Kip,q,t;p',q\t') Kip,q,t;p',q',t') <;p'V,n oo OO
' ^ ( p , ?>';?'. ?'.'')') = =^ E^^-^p'^',*') ^2 tj(P>Q> ''P'> E^ K
t
t
3=1
OO3
°°
1
]£<■(h,'v^F 1:V(Z( P , ? ) ^ ' , ? ' ) -
e x p [ - i ( i - t')Hi-id„ -idq, q + idp)] + id,)
1
b*>'
= =
exp[-i((-0W exp[-i(t-t')n{-id + i 5 p ) ] ^ ; ( id ^ ,p)]^2( , 9 ) V ( p ' > O - ^ = ^ )j),, q,q ( - i 9 „ 9 + exp[—i(t <' H ( - «q,q t')H(-id q)V(p',q')] e x p [ - i ( <— - O A}i>q .9 + + «'9 )V(p' idp)h P )]^-tr[K*(p > 9 ) 9 ')] ZTT
=
eexp[-i(t-t')H{-id xp[-i(<- OH .9 + + «'9 P )]^-tr[K*(p idp)]±-ti[V'(p,q)V( q,q( -t « > 9 )V(p' P',q')] ) 9 ')] rA Letet {*,(*)}£i, and '*(p,g)V(p',g') ZlT
Let us now evaluate=
£
>t(x)>r = 1 1be
2 2 Let us now evaluate ^ti[V'(p,q)V(p',q')].i1 L L {R). Let { ^ ( x ) } ^ and {rpk(x)}^=1
two complete orthonormal systems in L2(1R).
Then using the representation
00
in1 (2.1) we can formally write 11 °° j -t tr r[[K7 (*p( p, g, ) Knp',
4>l) .. e-'' e --^'e''» e^'»"e V*>( 'V*>( ^ )^(Xe e' ' rl>k,J ^'>k,e ^^;>J , .ee'' ^'4>i) «^ ^, ) E«( ^^,e-''"Oe p0 ,
PPp
be
, VP P
Qy Q ppipQ
(^.e-'^e'^^Xe'^^.e^^,)
2.2. FIDUCIAL
1 °° 2~i ^ 2^r"
VECTOR
INDEPENDENT
PROPAGATOR
xx J - 22I // (z z + g)dz g)cfz q)dx / Vfc( ip + (z V/t( V/fc( f c(fc k(x' ' +
=
J
k,l=l J k,l=l=1 -
1
=
q')exp(ip'x')i(x')dx' i exp(ip'x')(j)i(x')dx' q')exp(ip'x')cj)i(x
*
oo
f x (p'x' 'x' — q')tpk( q')ipk{x ++ 9) q)exp[i(p'x'-px)]dxdx' - px)]dxdx' I )M ^ , (')^k(x'+ I)i> ')S ( ?+T+ ? q')M )*(I + 9) 'exp[i(p'x' (x> 111/ l{ ^(z)^l kV x
7j- / , 2TT
45
k,l=l
x
x
J
r
=
/ — / e x p [ --ii((pp - pp')x]{ '/)*]{ ) { 16(x /16(x S(x x')8[(x' x')6[(x' x) -- (g -- q')]dx'}dx «5(z (z -- x')6[{x' x')6[(x'x')£[(ar' --x)-(q:x) z) )]dx'}da 9 q')]dx'}dx 27. 2^/eX1
==
p')z]c(z ')x]dx S(q - ?99') <5(9 ''))^^: / e/ xe xp p[[-i(p [-- i ( p0 — - p')a]ofj! p')x]dx «(9
=
i{p-p')8{q-q'), «(p - PO
the fourth line follows from using the completeness relations for the 4>i(x) and tor K(n. a,,t-p',q',t')v t; p', ipk{x)- Hence, the propagator K(p,q,t;p',q',t')
is given by
K(p,q,t ,t;p',q',t') K(p, A'(p, g, q,tp',q',t') t-p\ t; p', q>, t') == = eexp[-i(t xp[-i(t t')H(-id„q + ididid p')S(q p')6(q ---9q>) q>). (2.4) (2.4) x p [ - --- t')H(-id t')H( g,q ,,q + p)}S(p p)]S(p p)]6(p---■p')S(q-, 'q>). ) As shown in [66] this propagator propagates the elements of any reproducing kernel Hilbert space L2(M2)
correctly, i.e.,
, ,{p',q',t P '(P, p', 9', O W , 7',t')dp'dq', 9', W ,, ^^ (( P )MP,I', p ,, 9«»»0*: ) =/ / A'<(p,q,t;p',q',t')il' '(P' *9,> *;\p',q',t')ip ?''"?'' *>„(?', ?'.)dp'dq', 00 << W
(2.5; (2.5)
The propagator in (2.4) is clearly independent of the chosen fiducial vector. A sufficiently large set of test functions for this propagator is given by C(St2) 2
2
L (M. ),
2
where C(IR )
every element of L2(IR2)
2
is the set of all continuous functions on M .
n
Hence,
is an allowed test function for this propagator. From
(2.4) it is easily seen that the fiducial vector independent propagator is a weak solution to Schrodinger's equation, i.e., n tr!p\ q\ n' l'\ idtK(p,q,t;p\q',t') = 'H(-id ,q+idp)K(p,q,t;p',q\t') K{p,q,t;p',q',t') a.t:v'a'.t') -id,, q,t;p\ q', t') K(p, 9, t; p', q', t') = qH(-id + id^.\K(n q, t; q, pq)K(p,
Taking in (2.4) the limit t —> >t' t'o one finds the following initial value problem a ,t;p',q',t') i-W V<'.,*:/ " > iddttK(p,q,t;p',q',t') K(p,q,t;p',q',f)
;=
n(-id -ididq,q,q n !nq q
+ ididp)K(p, ,q,t:p',q',t') q, t; p', q', t') p)K idp)K(p,q,t;p',q\t')
46
CHAPTER lin lhnK(p,q,t;p',q',t') lim l i r n, K(p, ,K(p A ' ( ,q,t;p',q',t') , lt;p', ; p ' , ?q',t') ',(') Mq,
2. PHYSICAL
PRELUDE (2.6) (2-6)
-= = S(p-p')S(q-q'). 6(p-p')S(q-q'). 6{p - p')< p') 8{q - q>).
We now interpret (2.6) as a Schrodinger equation appropriate t o two separate and independent
canonical degrees of freedom. Hence, p and q are viewed as
"coordinates," and we are looking at the irreducible Schrodinger representation of a special class of two-variable Hamilton operators, ones where t h e classical Hamiltonian is restricted to have the form H(k,q — x) instead of the most general form H(k,x,q,p). 3>P)- In fact the operators given by equation (2.2) and (2.3) are elements of the right invariant enveloping algebra of a two dimensional Schrodinger representation.
Based on this interpretation following standard
procedures (cf. [64]) one can give the fiducial vector independent propagator for the Heisenberg Weyl group the following regularized standard phase space iv,i . i n . I I V I ^ . ^ U E , • ■v.j' &
lattice prescription: lattice prescription: K{p,q,t\p',q',t') v. a. tw'.a'.t') A ' ( p , g , i ; p \ ?',*')
f
( ■ ■
=
f
N N
a: e !Xp{t x p -~Pj) + fc> /2(<}j+l lim 7 " 7/ e> 9;) 2(qj+l --~1j) ^ IZ^J+VzCPi+i Pi) ++ kj+l/ j+i/2(9i+i lim J / • • J• / exp{i ^ [ [z Jj-i-i/2(Pj+i -1}) q3 ) + i / 2 ( p J + i - pj) + kJ+l/2{qj+1 _ J J ]=0 i=o *°°' ]=0
N N
N
dl/2dxj+l/2 ' dk <% ■kj + + l/2
n
j=0 ;=0
where (pN+1,qN+iiN) + l) = (p,q),(p0,qo,
(
)
+ 1). (N+l),
Observe
that the Hamiltonian has been used in the special form dictated by the differential operators in equations (2.2) and (2.3) and that Weyl ordering has been adopted. After a change of variables (see [66]) the fiducial vector independent propagator Weyl Group becomes lupagfiuui for IU1 the LUC Heisenberg lldS Ka.Uv'.a'.t') .»'\ K(p,q,t;p',g'>t') ==
N N
f exp{; 1 i lim f eX x pp {{ z? 9;J(Pi+l Pj) p^ (( f( t?9'J^++ 1l1I ++q99;)(Pi+l P) i) N5OO N NZO ™oo JJ/ " 7 X E [^^2M N j)( ^pp^i+ i+ ' l+l 1 --~- pPi) PP;) i ^) __--~xxX;;^Zj ;+1/2(P;+1 ++ii++//2l/2(Pj 2((PP;; ++++iill----Pi) AT—OC Pi J J J J j=o j=0 1=03 j=0
J J
i(,
2.2. FIDUCIAL
VECTOR
INDEPENDENT
47
PROPAGATOR
+ eH(k / ,x ,2,x )]} ,2)]} )}} +■k^J+ - - qj) - (■eH(kj+1 eU(k ,xJ+1 +i/i /2(qj+i 2 ( 9 i +f i1 -qj)J+ j+l/2 ]+1/2 ]+1/2 j 1 *2/ ill — * v v n / »2 j+l »j+l/2 ~J+1/2
N N N a,i , ,,,,/ c a ,a : v- TT A*. A„ A„ TT■ a / c j +dk i /J+1 2 a/a2:dx i/2 xxY[dj 1 11 dpjdq, d dq, 1 1 ~— dpjdqj 11 — 52j +J+1/2 ,, V] r 2 (2*f i2w ( 2 ?> ) (2-)
j=o i=o i=0i
i=i 3=1
'
Taking an improper limit by interchanging the limit with respect to TV with the integrals one finds the- following formal standard phase space path integrals i W M w » » I J I o i w i m e w i 3 t u ( t u u i u y i i t t J t ojya.\-t; pen-it i i u c c K
>M
K(p,q,t;p',q',t') UV'.Q'J')
== .M M / eexpli exp {xp + kqx)]dt\ \ VpVqVkVx VpVqVkVx x p . i\ ti j [xp kq - H(k, H(k,q q - x))dt x))dt\ VpVqVkVx,
/'
and >.:W t;p',q',t') K{p,q,t;p',q',t') K(p, q, t; p', q',n't') t') = Mif,/ expP \ i jI [qp - xp VpVqVkVx, [qp-xp+kq•p -xp+kq— xp ++ kq kq -—H(k, "H(k,x))dt x)]dt x) \ VpVqVkVx,
here "fc" and "z" denote "momenta" conjugate to the "coordinates" uq" and "p", respectively. Despite the fact that the fiducial vector independent propagator has been constructed as a propagator appropriate to two (canonical) degrees of freedom, it is nonetheless true that its classical limit refers to a single (canonical) degree of freedom (cf. [66]). 2.2.1
E x a m p l e s of t h e F i d u c i a l V e c t o r I n d e p e n d e n t Propagator
Vanishing H a m i l t o n i a n We now look at two examples of the fiducial vector independent propagator. Our )ur first example is that of the vanishing Hamiltonian which leads to f: n' l)n' tr\ K(p,q,t;p',q',t K(p,q,t;p',q',t')
/ exp h
=
M / e exp x p i1 / (qp +- kqkq— - xp)dtdt M
=
Af // / exp exp (i exp f f ii j//qpdt) qpdt)^6{q}6{p}VpVq 6{q} 8{q} 6{q} 6{p}VpVq 6{p}VpVq b}VpVq 1 Af (i
=
6(p -p')8(q6(p-p')8(q-q>). 6(q-q').
VpVi VpVqVkVx VqVkVx
48
CHAPTER 2. PHYSICAL
PRELUDE
This is of course a trivial example; however, it shows that the fiducial vector independent propagator fulfills the correct initial condition as is expected from equation (2.6).
2 22 222 W = ((PP222+w + uw ,
The second example we consider is that of the Hamiltonian H(k,x)
= (k2 +
w 2 x 2 )/2. Here the fiducial vector independent propagator takes the following form: n 1-W K{p,q,t;p',q',f) K(p,q,t\p',q',t') P ' ,n'9 ' , 0t'\
=
M /lexp l exp?y<j ii MJ
2 22 2 22 2 > )/2]dt\ VpVqVkVx ;p-(k + )/2 \ 'pVqVkVx f[qp [qp+ - (k -ftuL0 x x)/2]dt [qp ++ kq kc - xp
"I
2 2 2 22 2 qp+(q )/2}dt\vpV<, )/2]dt \VpVq = Af I[exp / eexp x p11<j i:ij f[qp VpVq [qp ++ (q (q' + p) p/u/u)/2]dt\
csc{uT/2) csc((. 4wi Airi 4
')2 + u,( - ') 2 ]}), }) ,
2 2 2 X exp ( t j i ( 3 + q')(p - p ' )) + -cot(u> J-cot(wr/2) c o t ( Wrlr Y2)[-(p/ 2 ) ±(p-p') + u,( q')(p-p') q-q') g 9 «(9P ') + X ex 4 u)
.1 fr I . : . • _ __ - _ *.ii - _ i1 •:„ ._**. _ ^ - rt* .»u_ ~ where T = tA — itt'. TThis is an unusualI resulti i rfor or the propagator of the harmonic
oscillator. This result has the appearance of a propagator for a two-dimensional free particle in a uniform magnetic field (cf. [37, p. 64]). However, when one brings up an element of any of the reproducing kernel Hilbert spaces L2(M2) then this propagator acts like the conventional propagator for the harmonic oscillator. Moreover, in the appropriate limits one can recover the usual propagators in the Schrodinger representation (see [66]).
Chapter 3
A REVIEW OF SOME MEANS TO DEFINE PATH INTEGRALS ON GROUP AND SYMMETRIC SPACES This chapter is an introduction to some of the ways of constructing path integrals on group and symmetric spaces. Our arguments will be largely heuristic, but we will confront the issue again with rigor in chapter 5. In § 3.1 the construction of the Feynman path integral on K , group, and symmetric spaces is discussed. § 3.2 is devoted to a preliminary study of group coherent states, we take this subject up in more detail in chapter 5. The remaining part of § 3.2 is devoted to the construction of coherent state path integrals based on group coherent states.
3.1 3.1.1
Feynman Path Integrals Introduction
The year 1925 can be seen as the beginning of modern quantum mechanics marked by the two almost simultaneously published papers of Heisenberg [53]
4r
50
CHAPTER
3. PATH
INTEGRALS
and Schrodinger [92]. The former proposes the formalism of matrix mechanics, while the latter proposes the formalism of wave mechanics. Schrodinger [93] first showed that the two formulations are physically equivalent. Both of these approaches where combined heuristically by Dirac [24] into a more general formulation of quantum mechanics. The mathematically rigorous development of this general formulation of quantum mechanics was subsequently carried out by von Neumann [106]. This general formulation of quantum mechanics is based on an analogy with the Hamilton formalism of classical mechanics. It is well known that the Lagrangian formalism of classical mechanics has almost no place in this general formulation of quantum mechanics, except in the suggestive derivation of Schrodinger's wave equation from the Hamilton-Jacobi equation by the substitution, S — —ih\n(4>), where S denotes the Hamilton principal function. The first hint of the possible importance of the Lagrangian in quantum mechanics was given by Dirac [23]; he remarked that the quantum transformation (qt\qt0) corresponds to the classical quantity exp[(i//i) /, Ldt\. It was this remark by Dirac that led Feynman in 1941, then a student at Princeton, to a new formulation of quantum mechanics (see the account in [47, 126-129]). This new approach did certainly not break any barriers that could not be overcome from the operator or Hamiltonian point of view. Nevertheless, one might have gained in two ways from Feynman's work [35] and the ensuing work of other authors [21, 20, 22, 40, 59, 61, 62, 81, 98]. From a practical point of view, as pointed out by Feynman [35], this approach to quantum mechanics allows one to reduce a problem that involves the interaction of system A with system B to a problem, let us say, involving system A alone. This is clearly useful if
3.1.
FEYNMAN
PATH INTEGRALS
51
one wants to restrict oneself to questions concerning only one system. Another way one has benefitted from Feynman's approach to quantum mechanics is in the conceptual understanding of quantum mechanics, specifically in the understanding of the connection of quantum mechanics and classical mechanics (cf. [22, 61, 62]). There are several books and review articles on the subject of path integrals. The selection presented is not meant to be comprehensive but is rather reflective of the author's taste. Feynman and Hibbs [37] give a heuristic introduction to the subject, whereas Schulman [96], gives a more rigorous introduction to the Feynman path integral on configuration space and considers a number of applications of the method in different fields of physics. For a good and thorough introduction to the subject of phase-space path integrals Klauder's Bern Lecture Notes [59] and his Lectures [64] are an excellent choice. The review articles by Berry and Mount [9] and Marinov [76] also deserve to be mentioned. In addition, Kleinert's [72] recently published book contains many applications of the path integral method to problems in quantum mechanics, statistical, and polymer physics. Moreover, Inomata et al. [55] discuss various techniques of path integration not covered in the aforementioned monographs.
3.1.2
The Feynman Path Integral oni RRd
We will now describe a simple derivation of Feynman's path integral on the basis of the canonical formalism of quantum mechanics which was first published by Tobocman [98]. The idea is to find an appropriate approximation for the time evolution operator, U(t" — t') = exp[-(i/h)(t"-t')H] (introduced in chapter 2), {t"-i')H] <)H](intrc (h at small times and then to construct step by step the time evolution operator
52
CHAPTER
3. PATH
INTEGRALS
at finite times. We start from the identity
D)f+\
/ V ++1 TT/t" U(t" f(t" - t') t') [U((t" !{{t" -- t')/(N t')J(? 1))]l))f >(N+ + !))]> \, ') = = [U(
which holds for any N > 0. Let us now consider the case of large N, then the step size e = (t" - t')/(N
+ 1) is small and we have the following approximate
identity to first order in e il uu)» uu) U(e) «iii -- l-m. -m. e ) «« n h To ensure that the quantized Hamilton operator H(P, Q) To ensure that the quantized Hamilton operator H(P, Q) in order to avoid operator ordering problems, we consider in order to avoid operator ordering problems, we consider rlamutoman; Hamiltonian; 11 2 2 Hc(p,q)=\p V(q), lcl( Sd(p,q) + V(q): .) -=2 2 1P H+ tfc/(p,q)=^p + V(q),
l
-m.
is unambiguous, i.e., is unambiguous, i.e., the following simple the following simple (3.1) (3.1)
where q = (
^W)k) (pp-^w: (Pi -(p\(I-pH)\q)
= =
pp| q| q) ) [111[ l- -■^^ffff(( pv , q ))]]<
«
--eH(p,q) exp ——el ~ -fre€JH(P' T ( pc , q ) (p|q) (p|q), -JcJ(p,q)](p|q), h
/i
'
valid to first order in e. Here, / / ( p ,,qq)) is is defined denned as as ((((pppp|||H * |M t t((((P P,,,,Q Q))))|| |qq|qq)))) jw H PP QQ // // (( PP , qq)) == ' (Plq) • ((p|q) p|q)'
(3.2)
3.1.
FEYNMAN
PATH INTEGRALS
53
For the simple Hamilton operator H ( P , Q ) =: ((l1/ /22))PP 22 + V l ( Q ) we are consid ering H(p,q)
coincides with the classical Hamiltonian Hci(p,q).
Note that for
more complicated Hamilton operators this has no longer to be true (see below). Using fact that (p|i (p|q) = (27r)~dexexp[-(i/ft)pq] sing (3.2) and the f; ~(i/h)pq] \we find that (p|q) =--(2ir)J{q",t"-q\t') Jr(q",t";q\t') ( q V*"• ' ; nq '\*"» 0
= =
\U(t"-t')\q') (q"\U(t"-t')\q')
N—*oo X)
N-* 00
= ^
V . N
/
lim°°
J J
N
dq] ^n^+ii^ ( q J + 1 | [U(e)\q / (f)iq;> e ) | q 3j ))Y[d ] n •- • f•/f[{
)U * jJ=l :
]=0 =0 3=0 3=0
3j=\ =1
N
p N " « , / • • 7 e x pjj|>;+i/2| ^ SEi ^ ,+[ Pi J/+f2I /-22( -q•((qj+i jq +j +i i --- q■q »* )) = ^lim / • • • / n^+i w )iq,-> n ** =
N
-^(P
/ ,q;)]}fpqi 3i)]> n *U ii dt
^ c / (i+1 p j ++ ii/2/22, q, jq, j ) ] } n d q 3=1 ji=i =i
N
J
ft ~
rr ftn
aPj+1/2 —-£>
(3.3)
12=0 (( 22 77 rr )) i+i/2= i+i/2=o
where qo = q' and q;v+i = q". It follows from (3.3) that the q-space propagator 2,q,)]}n^ - S 1 ^ . (3-3) J(q",t";q',t'),*') satisfies the -^c following initial condition: n ; (p, +1/ i=l
; + l/2=0
1 "ff,« < "ff•;q',0 « ' <"l = lim j(qV';q',0 Jim *(q''-q'), = *( *(q q"-q'), 7(q ("—1
(27r)
where qo = q' and q;v+i = q"- It follows from (3.3) that the q-space propagator as it shouldi by its very definition. Observe in the phase-space path integral J(q",t";q',t') satisfies the following initialthat condition: Hm/t7(qV';q',0 = 5(q"-q'), representation (3.3) there is always one more integral over the p than there is as it should by its very definition. Observe that in the phase-space path integral over the q. The sum in the exponent has a natural interpretation: it is the representation (3.3) there is always one more integral over the p than there is finite sum approximation of the classical action functional along a path in over the q. The sum in the exponent has a natural interpretation: it is the phase space with fixed endpoints q" and q'. Now taking an improper limit finite sum approximation of the classical action functional along a path in by interchanging the limit with respect to TV with the integrals we find the phase space with fixed endpoints q" and q'. Now taking an improper limit following formal standard phase-space path integral: by interchanging the limit with respect to TV with the integrals we find the
J
V = Jj exp !XP path J(q", t') [i/Jdd(p(0, lVp, following formalJ(q",t";q',t') standard q Vt"; ; qq', ' , phase-space 0= (integral: p ( 0 . qq(0) ( * ) )I X>qPp, J(q", t"; q', t') = j exp [ i / d ( p ( 0 , q ( 0 ) ^ P ,
[s"
(3.4) (3-4)
54
CHAPTER 3. PATH
INTEGRALS
where, /"".
i(p,q)}dt. cd{\ hid '/fc/(p(0;q( cc //((Pp(i); ( 0);q(0) ;q(0) q ( 00) == / ' [pq [pq--ffc/(p,q)]«tt. [pq Hd(p, q)]*. l-J- H Jt' Jv
(3.5) (3-5)
J(q",t";q',t>) q",t";q>,t>) This forma] phase-space path integral for the q-space propagator J(q" ,t"; q', 0 V) was first written by Feynman [36, Appendix B], and then was subsequently rediscovered by other authors (see for instance Davies [20] and Garrod [40]). The integration ranges over all paths in 2d-dimensional phase-space which are pinned at q = q' and q = q", while the integration over the momenta is unrestricted. The Lagrangian form of the path integral, as originally proposed by Feynman [35], can be obtained from (3.3) by integrating out the momenta. That this can be done follows from the fact that the momenta enter quadraticly. Hence, if we carry out the TV + 1 Fourier transformations in (3.3) which are of the form: rf e 2 l 1 dPi Pj + l/2 / e exxp, p {{^ (q, + i1 --- qj) q,) lp> P +1/2] } - | ± p q; -- 2gPj {-^ [ [Ppii++i1//222 "••( (qj+1 ^ / a j ]J ((2n) 2. d
/2 ( 1 V / 2 C „ \** [ i ( q j -iH--q- qq, i))22 '" ^'(qj+i 6 X:P p \2-Kihi) \2itihe) e X [P[ — 27k 2^r
V
Ve)
h
(3.6)
then we find the following result: 2 " j " ~ ' ^/' , 0<. ) = ~I/Tdcc(xi(t)) Vq ,,/(q J{q",t"-q',t') / ( ql ",t, r ";;q ; q ',< =-- JV N /V J . .-.../J/eexexp x p » T^^/ ,(q(0) ( q ( 0))) 2?q, Vq,
n
4<(q«) / c /(q(0)
ft" n exp - / c (q) di, where e r e qq q(i') ( i , ) = =q q', 2^ "q". J 2 q 2 ~ V ( q ) dt> W hhere 'q' ' q ^q(<") " ) = =qq"-
■r
=
-V(q) £
(3.7)
This is the formal Feynman path integral over paths in configuration space pinned at q' and q". Before leaving this subsection we would like to make a
3.1.
FEYNMAN
PATH INTEGRALS
55
number of remarks concerning the just presented derivation of the Feynman path integral. (i) Canonical transformations.
Since the measure in (3.3) is a prod-
uct of Liouville measures dpdq one may be tempted to assume that the phase-space integral is invariant under general canonical transformations. However, this is not the case. As shown by Klauder [64, section II] the regularized lattice phase-space prescription (3.3) for the q-space propagator is only invariant, or better covariant, under the subset of point transformations among all canonical transformations. (ii) O p e r a t o r ordering.
If the Hamiltonian is no longer of the simple
form we have considered in (3.1) but has a more complicated ( p , q ) dependence, then one has to confront the issue of operator ordering in the Hamiltonian. For example, this is the case for a free particle moving on a Riemannian manifold, for which l « ( # c //((pp,,qq)) = -g' ^93t3t33(
56
CHAPTER
3. PATH
INTEGRALS
problem might lead to additional correction terms proportional to h in the action functional (cf. [96]). (iii) Integral over configuration-space trajectories.
The Feynman
path integral in (3.7) was obtained for the Hamiltonian (3.1).
If the
p-dependence of the Hamiltonian is no longer simply quadratic but more complicated, then the integral in (3.6) is no longer a simple Gaussian integral and does not result in the classical Lagrangian of the free particle. In this case the Feynman path integral (3.7) can not be used as a starting point for quantum theory.
3.1.3
T h e F e y n m a n P a t h Integral on G r o u p Spaces
The quantization of a free particle moving on a group manifold has been considered in a number of works [12-14, 28, 29, 48-50, 57, 71, 76-78, 95]. Schulman [95] introduced, starting from the known semiclassical approximation, a propagator for a free particle moving on the group manifolds of SO(3) SU(2).
and
However, Schulman did not present a simple path integral solution for
the problem, (cf. the remarks in [72], chapter 8). Dowker [28, 29] extended Schulman's approach to simple Lie groups, considering explicitly the motion of a free particle on the group manifold of SU(n).
It is shown in [28] that the
semiclassical approximation is only exact for the motion of a free particle on the group manifold of a semisimple Lie group and that it can in general not be expected that the semiclassical approximation is exact for all symmetric spaces, since it is not exact for the n-sphere, ^ n Sm ^ i SO(n+l)/SO(n), 5 0 ( 7 1 + 1 ) SO(n), 1), ni > 3.
3.1.
FEYNMAN
57
PATH INTEGRALS
The question as to what is the largest class of spaces for which the semiclassical approximation is exact seems still to be an open one. The beauty of the above result, as Dowker points out, is that in the cases for which the semiclassical approximation is exact, the propagator is obtained by summing only over classical paths. A Feynman path integral treatment of the motion of a free particle on compact simple Lie groups and spheres of arbitrary dimension has been proposed by Marinov and Terentyev [77, 78]. Before we consider their proposal we briefly outline the construction of path integrals on Riemannian manifolds. DeWitt [22] observed that for a free physical system moving on a d-dimensional unbounded Riemannian manifold with constant scalar curvature R and metric tensor [fft'j(q)] the propagator for infinitesimal time is given by the semiclassical sical aapproximation: d ill. q', „ ' t>) ti\ == A(q", Jjclt„" (q", A(q", t";q', q',*') *') expUrUl", t / cT(cl((q",t";q',t') qq " , t"; t') A(q",t"-q',t')e. t"■ ')exp exp (q ,tt"; ;q,t) ' t";;q', qq',, t') tt')} ) ,, ;q",i";q',0 J (q ,t ,q,t)
•[i*
,0
(3.8) (3.8)
where d 1 / 2/22 2 ")D( 1/2 2 '\ ")D(q",t";q',t')g A(q", t"; q', f) = (27Ttft)~ (2Kih)%-^(q")D(q", t"; q', )g-,1/2t')g-^(q')f A{q\t"-q',t') 0 = (2irih)%-V\ t')g-^(q')f q l)-d%' 2 ( q ' )^] 1 / 2.,,, [qVW.O l9(' q(q") J D(q",^";q^0ff-
and d2Ic, Zl±J : r ,, ll 1 k kk dq"dq" \ \\ ' [L dqdq'>dq" dq'
D(q",t";q\t') det q ,t ;q,t) == det\ Z?(q",*";q',0 KH
'
H
'
ft"
_ l 9iffijq'q^di —= \ if ff(q) = =: |I|det[ff,-j(q)]| det[ff,j(q)]| and and IIccci, = :< is van Vleck's determinant. Here g(q) det[
is the classical action functional. As observed by Marinov [76], this simple form of the semiclassical approximation is only valid for unbounded Riemannian manifolds, since the proof of the theorem that two points q" and q' at fixed small t" — t' may be connected by only one classical path (cf. [110, pp. 58-64]) uses the unboundedness of the manifold in an essential way. On the other hand,
58
CHAPTER
3. PATH
INTEGRALS
as is pointed out in [29] and [76], if one is dealing with bounded Riemannian manifolds there might exist a number of classical paths connecting two points on the manifold, each of these paths then enters into (3.8) possibly with a phase; see also in this respect the review article by Berry and Mount [9]. As an example we mention the case when the bounded manifold is multiply connected, the classical paths connecting two points on the manifold then divide into distinct homotopy classes; see the example below of a free particle moving on a circle and Schulman [96, 197-205] for a discussion of this point. For this case the semiclassical takes the following form: emiclassical approximation appro UUUW1I JUII:
*[*#
l ' I jl\ / c / / „ » 4». Jl, „„l' 4> 4I\ " ",t";q',t'[ ii\ q II ,t,11. )• exp p ^Q(q",t"-q\t')Jd(q",.'I t"; q', t>) = £ AAmm(q",t";q',t')e: (q", t";;q,t q', i')exp t>) f r^ rnW , i"■«"i„-Ky q',m>)-\* 0 - \*1n\ ""7m ,, , J (qi ,t ; q ;q,tj ,< ' d ( q 11 ; q , ' ) m
m
(3.9)
where the sum is over all classical paths connecting q" and q', I™ is the clas sical action functional along the mth path, and -ym is an integer that depends in general on all the arguments. Also note, as is remarked in [76], that the semiclassical approximation is only applicable if the action functional lc\ is generically dominated by a term proportional to i - 1 as t goes to zero, so that Ici becomes large in this limit and puts one into the semiclassical regime. This is of course the case for the free particle. For the case of an unbounded Rie mannian manifold the propagator at finite times is constructed by folding TV + 1 propagators of the form (3.8) >rs oi m e iorm (,i.a
Jn , N
N
oil' i=l
; q ,t == / H ' JC'(c[ '((
k=0 k=Q k=0
(3.10)
k-1
i=l
d J3 dqkk = = ,/g y/gl[H1T iq{. oneobtains a functional k.k. 'Taking wheree dq f ]} i == 11dq dq the limit N — -»►ooooone VaUU
integral over all intermediate coordinates that can be interpreted as the path integral. The final result is a path integral of the form (3.7); however
the
3.1. FEYNMAN PATH INTEGRALS
59
Lagrangian needs to be modified by a term proportional to h2R. DeWitt [22] found this term to be h2(R/12); this modification of the Lagrangian was also discussed by McLaughlin and Schulmann [74]. In the context of curvlinear coordinates the reason for modifying the path integral has been discussed by Arthurs [2, 3], Edwards and Gulyaev [32], and in the context of quantization of non-linear field theories by Gervais and Jevicki [42] and Salomonson [91]. If on the other hand, one applies this approach to a bounded Riemannian manifold, as is the case for compact Lie groups, then the resulting path integral, as pointed out in [76], is far from simple. Since one then has to use the semiclassical approximation presented in (3.9) and in addition to integrating over all intermediate coordinates, one also has to sum over all the different classical paths connecting q" and q'. Nevertheless, if the bounded manifold AA in question is isomorphic to a quotient space Af/T, i.e. M = Af/T, where T is a transformation group acting on Af and Af is an unbounded Riemannian manifold, then one can, as proposed by Marinov and Terentyev [78], construct a propagator on M by summing over the group T: JJ •q •,t q",;q,t) (q", t"-^q', Y, W , *"; 7 • q', 0==]T^(7V/W,0, £ Ml l"\ q', (3.11) 0(3.H) , V(qV;7-q',0 ^MM(q",t";q',t') (:(qV";q',0 qV 0 0===£^(qV^7V/) ;.V(7-qV;q',0, ;.w,*";7-q',o-•yer -yer -yer provided the propagator on Af is known. As pointed out above, since Af is an unbounded Riemannian manifold, by assumption a path integral representation for Jjj can be constructed and one sums over T in the last step to obtain JMIt has been shown by Marinov and Terentyev [78] that this approach is valid for any compact Lie group. In their work, see [78], Marinov and Terentyev take for Af the Lie algebra that is associated with the Lie group they wish to
60
CHAPTER
3. PATH
INTEGRALS
consider and for F the characteristic lattice of the group. As an application of this general formalism of Marinov and Terentyev we now consider the free motion of a particle on a circle. We will revisit this problem in chapter 5 where we present an exact path integral treatment of this problem without reliance on the semiclassical approximation. Let us consider a particle of mass m constrained to move on a circle of radius p. If we choose the arclength the particle has traveled as our generalized coordinate, the Lagrangian is given by 1l
L{,4>)= Lfai)-L(4>,i) ) = =-I4>\ \H\\H\ * ) ■ ■
Here / — mp2 denotes the moment of inertia of the particle and the angular
variable ranges from 0 < 4> < 2w, where we identify the points <j> = 0 and 4> = 2it. The solution of the equations of motion is found to be 4> == +ut, +ut, 0+ 4> 0 ut, where 4>0 and u> are arbitrary integration constants.
Considering a motion
starting at " ' - -~ 2-Knf ",t";<j>',t') / _J_ ">""- -4>' ' 222-KTI) ™ SSci{<j>",t";'j') (", t"; 4>',) 0=■=22{t" -'~ ™) 2f2,,, cl ( 2{ f / ) (tl)0 cl(4>",t";4>',t') - -t'y t y
(3.12) (3.12)
where n = 0, ± 1 , ± 2 , . . . . So we find that the classical action functional does not only depend on the initial and final position but also on the so-called winding number n, the number of times the particle moves counterclockwise minus the number of times it moves clockwise past the point <j>'. Hence the paths break up into distinct classes labeled by re; these classes are the so called homotopy classes. Any two paths in the same class with the same beginning and end point can be continuously deformed into each other (see [96, 197-2051).
3.1.
61
FEYNMAN PATH INTEGRALS
The canonical quantization for this example is straightforward, since p$ = diL = I<j>, we find for the Hamilton operator:
H-= ■H
t
h h2 2
-d \dl
2A 21
which has the following normalized eigenfunctions and eigenvalues: 1 . 1 = exp(imcf>), E m)2. 2,2, i'mW ^m{4>) -7=exp{im4>), E —(hm) fc*m mm== —(hm) 1>m(.4>) )= =- -^=exp(imcj>), 21 IT V21F where m = 0, ±1, ± 2 , . . . . The propagator is given in terms of the eigenfunctions tpm{4>) by the following spectral expansion: r
-f-OO -f-00
E*
"• ")MF)e*P f,t";n(cl>>)exp J(")M W;M nn=—oo =—00 n=—00 o 1 °°
=
1i
1 ^ 2TT
°°
r r
1 1
ii
eexp x :xp i(" <£') -- iii^h(t t,)n2 ~2 ex i( --— " *> 4>')n °° ^ "--~tt> V2 ■•■ P W 0> ^(h{t t t"" E ? w - 2i
-O =— 0O 0 2Jr nn=—oo ^
n=—o
The sum over n is related to the Jacobi theta function, + 00O 0
= E*
exp(intnn222 -48 = y y~]j exp(z7rtn + 2inz). * 33(z,t) (*,*) = nn = n = —oo —00 = —00
Therefore, with the following identifications we can write the propagator in closed form. Let
_ -h(t" -1') - t') f_-Ht"-n tt ==
then we find
1^1 _ 2nl 2*1 2wl
'
4>" - -' ' = " = 2 ~ 2 '
Zz
w.™o=£*(£.£). w,™o=£*(£.£). J(4>",t";',t') <j>",t":d>',t 1 == -!<03 ( { 2' 2TT
2-KI
,
\0 = = <j>" <£" - - <j>'1 and r = t" t"-=-e. A(j wheree A$ T t"f.- t'. Using the following property of 93, which follows from the Poisson summation formula (see [19, pp. 63-65]),
("£)w*(H)-0'
l / 21/a , P *(,,0 = M-) - 1/a e* (-.•£) * «3(f-j)' (f-7)' '2
fl3(«,*) == = fl,(,,0 («r expexp(-^) (»'*)" P 7rt y
2
62
CHAPTER
3. PATH
INTEGRALS
the propagator can also be written as a sum over classical paths, i.e., as a semiclassical series, + 0OO 0
£
J{4>",f-A',t') >",t";4>\t') --.= j{4>",f-A\t>)
<j>",t";<j>',t') Y, J{4?,i?'\4>\*) •/(*".*";*', 0 Jiy,t";*',0 n= —00 00 nn= = — —00
J(4>",t";4>\t') J{4>",t"\4>',t') >",t";
(
:
y/ y /22 exp
/1
I
2irih{t"
2-Knf " iHU 11{4>" . 'j? -4,' -4>' -- 2-KTlf h 2(t" - t') f) h'
PY
-t')J
(3.13) Observe that each of the propagators J in the series (3.13) is of the same form as the propagator of a free particle moving on the real line IR and that the series as a whole is a function of period 2w. The series (3.13) is a particular example of the general principle (3.11). Folding the propagator in (3.13) N + 1-times leads to the following path integral representation: J{4>",t"-A',t') t>",t":d)',t') s',0 z-2-. r-22*2»*
2* rr27r
Urn / N-ooJ0JO N->™Jo
J
-JO '0u
n::E G J N N W
0 H +++00 ° °0
/,
T
i=in^oo\27nheJ\2ixiKi) jj==OOn n, =, = - o
2 ! \\ 1 / 1/2
TfJ, 11A ^^ ' + i - ~A,A0jj+i FP
[ftft
22
2 72r 7wr w \2~'2
i
ft
j )j )
2e 2f 2e 2f
N
xxY[d4>j, Y\ d ;=i
where <J>N+I = ", 4>o = '', and £ = T/{N
+ 1). If we now shift the integratio
variable at each step, we can extend the N intermediate integrals to the whole real line, i.e.
+ °° 00 + + °°
££
n,=-oo
/•2(n + l)ir y/•2(7i 2 ( n jJJ + l ) "T l)ir
/
j2n n
J
'
y+00 +oo r+oo
'<% Hi-*-^ J//I GO d*iJ
-°°
3,1.
FEYNMAN
63
PATH INTEGRALS
As a final result, we find J {",t"-<j>',t') 4>",t";<J>\f) W,t";4>',t')
:=
r
/v 1
2
rT ■ -n( |(W N+ ++ l ))// 2
lim W,t";4>',t') = yv—oo lim» [2wihc -L- ( N-00 I2mhc\
+00 +°° +< +°°
.. ,.+00 r+00 r+00
r +I-CO 00 1r+oo
_ S/ } _ -)■r
W / £ *-
^ ^ n=—00 Tl=-
.../ .../
x
:=-ooJ-°° I v-^ / ( $»i+i , J+ ! -- $* ,;)) 2 2 -i-r
0 DO 0
J—00 J
-
f[K X e x[ piE'l f ; ^2e 2^; ^ 1n*>. n,^, N
X exp
]=0 i=o 3=0
where $N+I
= 4>" and $0 = ' + 27rn.
J j=\
(3-14) (3-14)
3=1
Note that the circle is the group
manifold of the simplest compact Lie group U(l) whose faithful irreducible representations are given by Dl(<j>) = exp(i<#) and D~1(). It is well known that the one-dimensional abelian translation group T\ of the real line JR is the universal covering group of U{1). Furthermore, the translations by 2irn, n — 0 , ± 1 , ± 2 , . . . form the cyclic subgroup (27r) of T\, which is the kernel of the homomorphism Ti B x —* f{x)
— exp(ix)
£ U(l).
Therefore,
by the Fundamental Homomorphism Theorem we have that U{1) = Ti/{27r). Hence, as stated in [78], the path integral representation (3.14) is a particular example of the general statement (3.11). For more complicated cases one might expect not to be able to obtain such a simple representation as (3.14) for the path integral, but one which involves the summation over the lattice group at each infinitesimal step. Nevertheless, Marinov and Terentyev have shown that in the case of the motion of a free particle on the group manifold of a compact simple Lie group the resulting path integral representation is of the from (3.14), with the only difference that the Lagrangian has to be modified to include a 'quantum' potential proportional to %2. One might ask if the approach of Marinov and Terentyev could be extended to more general systems than the free particle? The answer is no, since, as we have mentioned above,
64
CHAPTER
3. PATH
INTEGRALS
the semiclassical approximation is only exact for the case of the free particle moving on the group manifold of a semisimple Lie group.
3.1.4
T h e F e y n m a n P a t h Integral on S y m m e t r i c S p a c e s
More recently Bohm and Junker have used zonal spherical functions to construct path integral representations for a free particle moving on the group manifolds of compact and non-compact rotation groups (Bohm and Junker [12, 14]), the Euclidian group (Bohm and Junker [13]), and on symmetric spaces 1 (Junker [57]). However, a careful analysis of the construction presented in [57] reveals that it applies only to the case of a compact transformation group G acting on a compact symmetric space of the form G/H,
where H is a massive 2
subgroup of G. We will extend this construction below to a general unimodular transformation group G acting on a symmetric space M = G/H,
where
H is a massive compact subgroup of G. This will complete the argument of Junker [57] and achieve his proposed unification of the work in [12]—[14]. Path integrals on compact and non-compact rotation groups have also been discussed 'Let (S, T) and {T,U) be two topological spaces. A continuous one-to-one map f of S onto T is called a h o m e o m o r p h i s m if / is continuous. A topological space (S, T) is called h o m o g e n e o u s if for any pair u, v 6 S there exists a homeomorphism / of (S, T) onto itself such that / ( " ) = «. Let G be a connected Lie group and let a be an involutive automorphism of G, i.e. a2 = 1 and a / 1. Denote by G , the closed subgroup of G consisting of all elements G that are fixed points of a, i.e. o(g) = g, and by G'a the identity component of G„. Let H be a closed subgroup of G such that G!„ C H C G„, then one calls the quotient space G/H a s y m m e t r i c ( h o m o g e n e o u s space (defined by cr). The n-sphere S" is an example of a symmetric space. 2 Let T s be an irreducible representation of the group G on the space It. An element a £ It is called invariant relative to the closed subgroup H if for all h € H one has T/,a = a. A representation is called a r e p r e s e n t a t i o n of class 1 relative to H if its representation space contains elements that are invariant relative to H and if the restriction of Tg to H is unitary, i.e. (Th<j>,Tu-4>) = (0,i/>) V ft 6 H. One calls H massive, if there is only one normalized invariant element a 6 11 in the representation space li. of any representation of class 1 relative to H.
3.1.
FEYNMAN PATH INTEGRALS
65
by other authors; see for instance, Grosche and Steiner [48, 49], Grosche [50], and Kleinert [71, 72]. We will now close our account of the construction of Feynman path integrals on group spaces with the above announced extension of the method used by Junker to construct a path integral representation for a free particle moving on a symmetric space M. = G/H, where G is a unimodular Lie group acting on M and if is a compact massive subgroup of G. There is no loss in generality by assuming that M is of the form G/H since every homogeneous space can be represented in this form (cf. [7, Theorem 4.1.1]). Let us consider a free particle on a symmetric space M evolving from an initial state |x') at time t' to a final state |x") at time t". The coordinate representation of the Hamilton operator H of this system is given by // -A, 2m n *-£*■ here A denotes the Laplace-Beltrami operator on M &--= A = 9r
l/2
a0 0 ,9l/2x9y'V c dxp> eXag-^d x„ u
where ga^(x) is the inverse of the metric tensor3 on the space M and g = \det[gap]\. The path integral representation of the propagator describing this process is found to be ,.. ||x'> (x"| exp e x p -[l-{t" -i^(t"-t')H " - t')H*] |X'> J(x",t";x',t') ./(x",t";x',0 x",«";x',0 = (x"|exp
r f i lim (x"|fexpf(x
MN+1
t)\
= ^?j^h(-^)J jx W—oo
= =
JJ
^°° ~°°
3
/'nn^( ^> ' J ./••••/n
t m f/ ii m jlim / J
ft
N
N
J
j=0 3=0
, 7=0 = C
+j ix x
N
N
3 15 • • • / n ( i+ii i') n) n J ' J' ((3.15) - ) J
j
|x') |x/) 0
;xe £ d x dx 3, ==11
Since G is a Lie group and H is a compact s u b g r o u p of G an invariant m e t r i c exits on M (cf. [7, T h e o r e m 4.2.1].
66
CHAPTER
where x w + i1 = x " , x x', xo0 = x ' , £e = (t" (*" - t')/(N
3. PATH
INTEGRALS
+ 1),
(-K
J ( xxjJj++i1,,xxx>Ji ;i €e£ ) = ^)fW]|xi), xxpP (f - ^" «*-eft W « )j |xy) (x (*7ft)*W]|*i)> +11i | eeex] Xi) « w <x {lJ i++nl[l-(*'/B) 11|[l -- (»7fi)<W]l*i>. = (( xxji -+
(3.16) (3-16)
valid to first order in e, and dxj denotes the invariant measure on M.
Note
that for the cases we are considering, where G is a unimodular Lie group and H is a massive compact subgroup, an invariant measure always exists (cf. [7, Corollary 4.3.1]). In what follows we ask that the short time propagator (3.16) be invariant under the transformation group G, i.e., that ••/(^Xj+i^Xjse) / ( ^ X j - n ^ X j j e ) =: JI •/(0Xj+i,0Xj;e) .J /(^(X(xXJj+j++Ii,l,X.xXJi ;j JeCc)))
V G G,,, VVjsjeeeG
(3.17)
for j = 0 , 1 , . . . , N. As we will see below, this is a crucial assumption since it implies that A is an invariant elliptic 4 operator in the enveloping algebra on 'Denote by or = (<*i,..., am) a multiindex consisting of m non-negative integers. the length of a by
l°l = For every x e IRm let IC
a
Define
rri m
m
E
*
!
■
, * > ■
un > m m
=
Xj3
Let P be a polynomial of m variables of degree r, which has the form
E
a ?(*) = ^ Cw^ "Q ', ?(*) PI')' = E
|| oo || < < rr
where cQ are arbitrary complex numbers and cQ ^ 0 for at least one a with | a | = r. Then we denote the formal differential operator generated by P by:
v Q wi o ( -- , V ) °) : =E C»('° n ?{-&)=E n^ ^S/.1. = EcC«(-' E*cc «a ( -H. ,))'° I
P(-iV)V) =
||a|
|| oo||<
j=l
where V = (dti, . . . , dIm). The formal differential operator P(-iV)*) is called elliptic, if there exists a C > 0 such that 2 r/2
1li + P'(*)!>< + ||p(i)l>c(i ( ir ) | >> C ('(l l1 + + |r| | r | 22 ))rr // 22'
vVViregJztnr. fR f™ i . m™.
3.1.
FEYNMAN
PATH INTEGRALS
67
G.s
This can be seen by using the form (3.16) of the short time propagator
valid to first order in e in (3.17). From which it follows that the Hamilton operator H has to be an invariant operator for G if (3.17) is to hold, this in turn implies the above statement. Let a be a fixed point of M whose stability group is H, i.e. one has ha = a for all h £ H. Since G acts transitively on M we can write each Xj £ M as Xj = gj& gja for some g3 £ G. G.
(3.18)
Hence, using this construction one can view the short time propagator as a function on the group G: Jixj+uXj-^) J(x.j J{gi+Ugj\i). +1,x.f,e) ^(xj+i.XjSO J(9j+u9 };();0 ==-- -/(pj+uffjiOUsing the translation invariance of the short time propagator it follows that the short time propagator can only be a function of gj ffj+i, hence, J{9j+u93;e);0 ( f fu / 1^V!j19j+l\e)+ 0 -J(9j+i,9]\e) - J(99j+i,99j;e)nO :=: /J{9j J{9j+u9i;e) 0 = j +i ;i l O J(9j
(3.19) (3-19)
Since ha = a for any h £ H, we see that (3.18) is invariant with respect to right multiplication with elements of the stability group H. This implies that the short time propagator is invariant with respect to right multiplication with elements of H.
From (3.19) we see that the short time propagator is also
invariant with respect to left multiplication with elements of H.
Hence, we
conclude that the short time propagator J{g\ e) is a constant function on the two sided cosets HgH with respect to the subgroup H, i.e., J(h ;()i;0 = J(9\() J(h\gh J{g;e), xgh2;e) ;*), 5
ffor any hx,h2
e H.
If M. is a rank one symmetric space, then every invariant differential operator C is a polynomial in the second order Casimir operator of G which in a proper coordinate system on M is proportional to the Laplace-Beltrami operator (cf. [7, Theorem 15.1.1]).
68
CHAPTER
3. PATH
INTEGRALS
Let U^ be an unitary irreducible representation of class 1 on the Hilbert space lit.
i to>£5
Let us choose any complete orthonormal system {4>]}j^o 7=0
~
in L
^RV ' t*len
we can associate with U'- the following matrix elements
,vfa)
D((ii(g)=(4>i,ul
(3.20) (3-20)
These matrix elements play a special role in the theory of representations of groups. Unfortunately they are explicitly known for only a handful of cases (see [7, chapter 7]). Nevertheless, it can be shown that for simple Lie groups the matrix elements DJAg) are the regular eigenfunctions of a maximal set of commuting operators in the enveloping algebra, if this maximal set of commuting operators contains an elliptic operator (cf. [7, Proposition 14.2.2]). This property is often the starting point for an explicit calculation of the D^Ag), in § 5.3 we consider such a calculation in some detail for the case of SU(2).
If the
maximal set of commuting operators does not contain an elliptic operator then the matrix elements D^Ag) are generalized functions, i.e. distributions (cf. [7, Theorem 14.2.1]). In chapter 5 we will consider the construction of path inte-
4'
grals for the cases in which the matrix elements D{- are either not explicitly known, or are generalized functions. This construction will make no explicit use of the functions DtAg) but will only use the facts that they exist and form
■■ Dfcg)
a complete orthonormal set. Note that for the cases considered in this chapter the set of maximal commuting operators always contains the Laplace-Beltrami operator A, which, as we have remarked above, is an elliptic operator in the center of the enveloping algebra of G. Hence, the matrix elements D-(g) are 3 D%{g) ai regular functions on G. This shows that the assumption that the short time
3.1.
FEYNMAN
PATH INTEGRALS
69
propagator should be invariant under the transformation group G is crucial and can not be relaxed. Since H is a massive subgroup of G, there exists a unique normalized vector a 6 Hf- t h a t is invariant relative to H. Using the Gram-Schmidt orthogonalization procedure we can choose our complete orthonormal basis in such a way that o = a. Our interest now focuses on the (OO)-matrix elements
i4(s)=(4>o,^o>. 0D$Coo(9) o ( s ) =- < 0(a,V< o , £ t f^0oo>}0)..
(3.21)
One can easily convince oneself that this function is constant on the two-sided cosets HgH with respect to H. The function defined in (3.21) is called the zonal spherical function of the irreducible representation U^ relative to H. If we take G = SO(3), the group of rotations of iR 3 , and H = 5 0 ( 2 ) , the group of rotations of the plane, then M = S2, the two sphere and the zonal spherical functions are given by the Legendre polynomials
P((cos9).
Let us denote by G the set of all inequivalent irreducible unitary class 1 representations of G relative to H.
Then it is known, since H is a massive
subgroup of the unimodular group G that any function f(g) that is constant on the two-sided cosets HgH with respect to H, can be decomposed in zonal spherical functions £>o0(<7), £ £ G, of unitary irreducible representations of class 1 (see [105, pp.50-55]): /(
=
oo(s ttd( ( :DUa),
(3.22)
DiD0({g)f{g)dg. [ DUa)f(g (g)f{9)dg. 00(a)f(a)dg.
(3.23)
d c D
factDfag),
n
C6G G c
< = <
is
JG
JG
Heree V for the discrete or continuous orthogonal sum of all inequivalent ^ ^s stands t CEG CeG
irreducible unitary representations of class 1 of G with respect to H.
The
70
CHAPTER
3. PATH
INTEGRALS
constant d( appearing in (3.22) is given by
/I Di' Dg(g)DU9)dg d-l6((:,C), [g)dg==■-df6(C,0, =dfS(CC), dfS(U'), Qg)DL(g 0(g)DUg)dg O&Wla
(3.24) (3.24)
JG JG
where in suitable coordinates . ,. _ IJ he S^i *«,?) = (<■' C J ~ j ((- _- £/) S<5(C ^ ' *" ' ~ X \ *(C-C) C)
s discrete, if G G i is QGiissis ccontinuous. i fif fG continuous.
For the case of compact groups the constant d^ is the dimension of the repre sentation space VS of the unitary irreducible representation 17*, see also in this respect remark 5.1.1. We have now collected all the tools we need to construct the path integral representation for a free particle moving on M.
We have seen above that
the short time propagator is a constant function on the two-sided cosets HgH with respect to H, hence using (3.22) we can decompose it in zonal spherical functions: 1l l l J(g~ gJ+5i;<) J(9 J{9j9j j 9j+i;£) igj+i;<) +i;<
=
ip
C C 1 xC ^d
(3.25)
G
1U9T(g^gj+^Jig^gj+i-^)' {g] gj+\\t)dg Sj+i)J{gJ U9j gi+i) gi+i\e)dgj. (g^gj+i^)dgj-
C6C7 c
cc(«) c(0 =
I'
D
/
l
D
l l
lJ
r
(3.26)
JG JG
Moreover, let / e Ll(G),
where LX{G) is the space of all integrable functions
on G, then one has (cf. [7, Corrolary 4.3.1])
L
9 / f(g)dg f(g)dg --= JaJ^ JG
u
[ [ f(gh)dhdx, f(gh)dhdx,
JM JM
JH
which reduces for /f e6 L\H\G/H) L\H\G/H)1/H) to to
!{g)dg I(g)dg = jI >I(g)dx !{g)dx (I dh= I L/I' /GO f{g)dx
f{g)dg
JG JaG
JM JM JM
( dh lh =
JH JH
JM JM JM
f(f(g)dx, f(g)o g)dx,
(3.27) (3.27)
3.1. FEYNMAN PATH INTEGRALS
71
since f(h f(high = f(g)0 V/ii./ij dh = 1, Vhi,/»a 6G .// and where we have chosen JHHdh 1gh22)) = because H is compact. Using (3.25) and (3.27) in (3.15) one finds
I- i.nn
JJ(x",i";x',0 ( vx "".,
5
I ++I, ^ ^d(j+lHj+1 ) ^ +oo ( 50)(^jVj+l r^.+1) ^d c+ 1 ( e + <;+i 1 % (e)Dg^(g^ gj+1 G.J + ] (0^oo $ LJ + i6G
n AT
f j f[d *,,, 9j. (3.28) (3.28)
Using the orthogonality relations for the functions y/d^DJ. and the left invariance of dg one can easily show that the following relation holds
7
£
+
D&'igfg^D&igJ^gAdgj \9]) 9j ) 9j 9Jl9j+i)D*>&{Bj±i9i) &V \ / % A lJjG D^igr'g^D^gJ^dg,
JdiHi iHid(,
iG
Using (3.29) the Using (3.29) the and one finds as and one finds as
= I*(Ci,0-+i)i * ( C i ^ +0+i)£oo(s~-W )-. £(G,C, = ^i )otof(& s J0_£WWl1))-
d d
(3.29) (3.29) N intermediate integrations in (3.28) can easily be performed N intermediate integrations in (3.28) can easily be performed a final result that a final result that
1 ir } U j' s CeG
m [c J (( xx>"V = ii {{ l i Hm m [ ecc(c)] ( e )JV+1 ] M}d (^ C / fV c,"t",";i;"xx: 'x' ,,\ttt''')) == [cc(0]' JJ(x",t";x',t') } ^dcZ? cD$o(/ f ' - ^) " ). oGrVo L
3
(3.30) (3.30)
J
Let us now evaluate the limit N —> oo in the above expression, for large N one can write c^(t) as (e) = = Cc ccc(0) (0) ++ ccc
valid to first order in c. The value of c^(0) can be found from (3.26), using the fact that the short time propagator satisfies the following initial condition: l1 1'ij+ijf) S lim J(s J(o i9j+l) +i) J{gj'7 8e(ff, e(9j jxgj+i;e) e(gJ^9j+i) :V +ni ; 0 = <5 J J
e—0 e—»0 e—»0
J
J
Hence, one finds' C((0) = DQ 0(e) = 1. Therefore, one can write the limit in c=c(°) OJoto C(0) = ^oo(e) (3.30) as
AN+1 r+1 2" _ 11AT+l t" -f t' ii m m exp[( n 11+++ ^ p[(f" -- i')cc(O)]. o')cc(O)]. * .c(0)}. ^c(0) C c ( T JJlim ^ T 7 ^ ° ) == exp[(t" exp[(i" T^° t')c ♦ OO ivTT /V-»oo
N—too N—too L L
JV -\+ 1 1 1\
72
CHAPTER 3. PATH
INTEGRALS
E< =:=ih.f-.AC. ihc Or if we sett E(_ ihc^(0), ({0), we find
Mor
1 \unJcc(e)]#N++*=exp(-t(t"-t')E \hnJc lim *=exp(-^t"-t')E exp | ccy(y' " --t')E '£<)
N—oo
(
>
It is shown in [57] that the c^(Q) are the eigenvalues of the Laplace-Beltrami operator A o n M . Since in the proof of this statement no use is made of the compactness of M. it applies to the present situation as well. Finally using the l c group propertyf Dfog'-'g") D^ DUO'-'Q") Dl we can write (3.30) in the Z£kkfDl ckDi 0(g')Di 0(g')D 0(g") k0(g") v, 0(g')Dl 0(g") W0(g'~ g") 9 ) == ' --E
more familiar form
i
if e x r" *"■ v ' t'\ = t')lEE< Y1ckW Y(x')Y J(x",t"-x',t') ) = P [f-^ 4 f(\{t"-t')E, " - *')£<] (x"), J(x",t";x\t') = ^ E exp " I" *) c]; Y^d)Y W *a(x") "), a(x')Y ceG *
(3.31)
<€G
where Y(kk(g) Y< (g) Y< ■■= k(g)
yfcDi^g) Olo(s) \foL
One calls the matrix elements• DDio(9 m0(g) the associated spherical functions and they are the eigenfunctions of the Laplace-Beltrami operator on M. For l the case that M is the two sphere S 2 the functions y/diD VdiD'm0 m0{(8,cp) are the
classical spherical harmonics Yim(8,4>), which, as is well known, are the eigenfunctions of the Laplace-Beltrami operator on the two sphere S2. Hence, (3.31) is the well known spectral expansion of the propagator in terms of normalized eigenfunctions of the Hamilton operator. For specific examples we refer the interested reader to [57]. Let us close this section with two remarks, the first is that this approach can also be applied to Lie groups if we choose H as the closed subgroup consisting of the identity element, e, i.e. H = {e}. Then instead of using the zonal spherical functions one has to use the matrix elements DJAg). However, it should be clear from the remarks after (3.20) that one can construct path integrals this way
3.2.
COHERENT
STATE
PATH INTEGRALS
73
only for a handful of groups. In chapter 5 we will overcome the reliance on the matrix elements D^(g),
this will allow us to consider more general Hamilton
operators than just the Laplace-Beltrami operator. The second remark is of a more general nature and concerns the question: Why should one study quantum dynamics on group manifolds?
Clearly as we have seen in this chapter and
will see in chapter 5 the study of quantum dynamics on group manifolds uses interesting and deep mathematics. It is therefore, of considerable mathematical interest. Nevertheless, there are also physical reasons why the study of quantum dynamics on group manifolds is of interest, for instance the dynamics on a group manifold is of interest in some modern quantum field theories such as cr-models and non-abelian lattice gauge field theories.
3.2 3.2.1
Coherent State Path Integrals Introduction
The origin of coherent states can be traced back to the beginning of modern quantum mechanics. Schrodinger [94] introduced a set of non-orthogonal wave functions to describe non-spreading wave packets for quantum oscillators. In 1932 von Neumann [106] used a subset of these wave functions to study the position and momentum measurement process in quantum theory. It was not until thirty four years later that the detailed study of coherent states began ([6, 8, 59]). Klauder [59] introduced boson and fermion coherent states and used them both in the construction of path integrals for boson and spinor fields, respectively, whose action functional in each case is given by the familiar classical c-number expression. In 1963 Glauber [44, 45, 46] named the set of
74
CHAPTER
3. PATH
INTEGRALS
wave functions introduced by Schrodinger "coherent states" and used them in the field of quantum optics [68, 84] for the quantum theoretical description of a coherent laser beam. At about the same time Klauder published two papers [60, 61] dealing with the formulation of continuous representation theory, t h a t contain the seminal ideas for the construction of coherent states on general Lie groups. Coherent states for the non-compact affine group or ax + b group and the continuous representation theory using the affine group where introduced by Aslaksen and Klauder [4, 5] in 1968. Radcliff [87] constructed coherent states for the compact group SU(2) in 1970. In 1971 Perelomov [85] gave a general construction for coherent states for both compact and suitable non-compact Lie groups. Several books and review articles consider the definition and properties of coherent states ([17, 34, 69, 86, 111]). Klauder and Skagerstam [69] provide an introduction to the subject of coherent states in the form of a primer and offer a comprehensive overview of the literature until 1985 in the form of reprinted relevant articles dealing with the subject of coherent states. Perelomov [86] considers the usefulness of coherent states in the study of unitary representations of Lie groups and considers a number of applications. The review article by Zhang et at. [ I l l ] and the recently published proceedings of the International Symposium on Coherent States [34] also deserve to be mentioned.
3.2.2
Coherent States: Minimum Requirements
Let us denote by H a complex separable Hilbert space, and by £ a topological space, whose finite dimensional subspaces are locally euclidian. For a family of vectors {|0}/e£
on
H to be a set of coherent states it must fulfill the following
3.2.
COHERENT
STATE
PATH INTEGRALS
75
two conditions. The first condition is: C o n t i n u i t y : The vector \l) is a strongly continuous function
of the label I.
That is for all e > 0 there exists a 6 > 0 such that
£w I1110H O>-IOII<< - 1I O 0 1I 1I < «£ for alli VV € GL with
\l'-l\<8. <S.6. \l' - l\ <
Here, || • || denotes the norm on H induced by the inner product on H , i.e. || • || = (•,•)'.
Or stated differently, the family of vectors {|/ )} ( 6 £ on H form a
continuous (usually connected) submanifold of H. We assume that (/|/) > 0 for all / G L. In the applications we are considering the continuity property is always fulfilled. The second condition a set of coherent states has to fulfill is: C o m p l e t e n e s s ( R e s o l u t i o n of t h e I d e n t i t y ) : There exists a
sigma-finite
positive measure dfi(l) on L such that the identity operator / J J admits the following resolution of identity
L
)dpl(l) 7H == J\l)(l\-)drf) IK J\\W\-) \l)(l\-)drf) In In general, as pointed out in [69], p.
(3.32) (3.32)
5, "one has to interpret this formal
resolution of identity in the sense of weak convergence, namely, that arbitrary matrix elements of the indicated expression converge as desired."
3.2.3
Group Coherent States
To avoid unnecessary mathematical complication at this point we restrict our discussion to compact Lie groups. However, we would like to point out to the reader t h a t the discussion applies to a general Lie group, as defined in chapter 5. Let us denote by G a compact d-dimensional Lie group. It is well known
76
CHAPTER
3. PATH
INTEGRALS
that for compact groups all representations of the group are bounded and that all irreducible representations are finite dimensional. Moreover, one can always choose a scalar product on the representation space in such a way t h a t every representation of G is unitary, (cf. [7, Theorem 7.1.1]). Therefore, without loss in generality we assume that we are dealing with a finite dimensional strongly continuous irreducible unitary representation U of G on a (^-dimensional representation space Hf. Let us denote by {Xk}k= -l the set of finite dimensional }t.l1 self-adjoint generators of the representation U. The Xk,
k = l,...,d, .,d,
form
an irreducible representation of the Lie algebra L associated with G, whose commutation relations are given by [X [Xi,Xj] itXi]
= - i
dd k Cij xkkXk, iy~]cjj fc=i
E«
where cy* denote the structure constants. The physical operators are defined by Xk = hXk- For definiteness it is assumed that there exists a parameterization for G such that Ug{i) =- exp( e x p,(-il ((--ilX i/Z}1)^1 r^i ) . .,.exp(..exp(-il .exp(-il .-UkXk),hkXk), Ug(l) up to some ordering and where Z G £ . Here £ denotes the compact parameter space for G. For all Z G £ and a fixed normalized fiducial vector TJ £ E^ we define the following set of vectors on H,;: r,(l) vV) \f7(0 = = \Zd~cU {l)Vga{i)Ti. \Zd~
(3.33)
It follows from the strong continuity of Ugy) that the set of vectors defined in (3.33) forms a family of strongly continuous vectors on H<;. Furthermore, let us consider the operator
L
o 0= = JJv(nW),-)dg{l'), mivv(0,-)d()dg(0, cV(l')W),-)dg(l'),
(3.34) (3.34)
3.2.
COHERENT
STATE
PATH INTEGRALS
77
where dg(l) denotes the normalized, invariant measure on G. It is not hard to show, using the invariance of dg, that the operator 0 commutes with all
Ug^,
I £ L. Since Ug^ is a unitary irreducible representation one has by Schur's Lemma that 0 = A/u . Taking the trace of both sides of (3.34) we learn that , Xd tr(A/ =■ Xd A
2 Xd I[dg(l') dg(l') Ad< c\\r,fJdi :\\v\\ k( = = ddc\\ A =
1.
Hence, the family of vectors defined in (3.33) gives rise to the following resolution of identity:
{l)ii)W),-\ lK = JJcV(l)W),-)dg(l). IK m,-)dg{l) ?(')• cv(l)W),-)dg(l). IH(-((
b
(3.35)
Therefore, we find that the family of vectors defined in (3.33) satisfies the requirements set forth in §§ 3.2.2 for a set of vectors to be a set of coherent states. So we conclude that the vectors defined in (3.33) form a set of coherent states for the compact Lie group G, corresponding to the irreducible unitary representation Ug(iy
3.2.4
Continuous Representation
Analogously to standard quantum mechanics one can use the set of coherent states defined in (3.33) to give a functional representation of the space H<;. Let us define the map C„:H< C „v:::H< H< C 1p
-» 1L\G,dg) L\G,dg) L\G,dg) I-f—>
[CnW) =^(l) 1 = (r,(l)^).
78
CHAPTER
3. PATH
INTEGRALS
This yields a representation of the space H<; by bounded, continuous, square integrable functions 6 on some closed subspace L*(G) of L2(G).
Let us denote
by B any bounded operator on H<;, then using the map C„ and the resolution of identity we find that (3.35)
W)M worn')-,
(3.36)
(n(l),BrP) = J(V(l), Jw),Br,(l>))(r,(l'),iP)dg(l') W), ),B1>) BiP) = Br,(l')}(r,(l% ^)dg(l') (n(i),B4>) il')^)dg(l')
/<
holds. Choosing B = 7H C
we
R^A
')^(l')dg(l'), 1>nV) = J«JC„(/;l'M W w l')i>n{l')dg{l'), l')i>n{l')dg{l'), 1>nV) . /= Wl un =
(3.37) (3.37)
where IC £„(/;/') v(l;l')") = =
M0MO>M0,' U0>-
One calls (3.37) the reproducing property. Furthermore, as shown in Appendix A.2, the kernel £,,(/';/) is an element of L^{G) for fixed / G £ .
Therefore,
the kernel £,,(/';/) is a reproducing kernel and L*{G) is a reproducing kernel Hilbert space (cf.
Appendix A.2).
Note that a reproducing kernel Hilbert
space can never have more than one reproducing kernel (cf.
Claim A.2.1).
Therefore, since L^(G) is a space of continuous functions, ICv(l'; I) is unique. Moreover, since the coherent states are strongly continuous the reproducing kernel K.v(l'; 1) is a jointly continuous function, nonzero for I = I', and therefore, nonzero in a neighborhood of/ = I'. This means that (3.37) is a real restriction on the admissible functions in the continuous representation of H^. Of course a similar equation holds for the Schrodinger representation, however there one has (q\q'} = 8(q-q')
which poses no restriction on the allowed functions. In fact,
the reproducing kernel /Cv(l'\ I) is the integral kernel of a projection operator 66c
See Appendix A.l.
3.2.
COHERENT
STATE
PATH INTEGRALS
79
from L2(G) onto the reproducing kernel Hilbert space L^(G) (cf. Claim A.2.2). This ends our discussion of the kinematics (framework) and brings us to the subject of dynamics.
3.2.5
T h e C o h e r e n t S t a t e P r o p a g a t o r for G r o u p C o h e r e n t States
Let ip 6 H<
an
d denote by TL{X\,...
,Xj)
the bounded Hamilton operator of
the quantum system under discussion, then the Schrodinger equation on H<; is given by ihdti> -= ihdttjj ihdti>
7i{X ...,X )^ nH{XH{Xu---,Xd)i> l,...,Xd)ijju
d
U
since 7i is assumed to be self-adjoint and does not explicitly depend on time, a solution to Schrodinger's equation is given by
m'[->-* m-
^ ( i " ) == exp --%-{f-i>)H -t')K 1,{t") e t')H 0f(( i ' ) . Now making use of (3.36) we find
"J'd'A') b (l',t?)dg(L'l i>n(l", •t") (l", t"; I', t')^{l', t')dg(l'), ui",n-> =« JK /J Knv{l",t";l',t')U n
n
where i Kr)v(l",t"-l',t') (l",t";l',t')-'),expP -^t"~t')7i V(nexp ,(?",*";?',*') = (W'U "ft
-I(f-f)7<]„( n] 77(1'))77(0). ,n
Note that the coherent state propagator Kv'(l",t";l',t') „ ( f W , f ) 8 asatisfies the following initial condition: K (l",t";l',t') l",t";l',t>)
l i mi/ /I^v ( / " , i " ; / ' , * ' ) ( lim t"-*t'
= = £^ ,((//"";;//'')) -.
Hence, as t" —> t' we obtain the reproducing kernel ICv(l"; /'), which, as we have remarked above, is the integral kernel of a projection operator from L2(G) onto
80
CHAPTER
L*(G).
3. PATH
INTEGRALS
Moreover, since ICn(l"; I') is unique, we see that if we change the fiducial
vector from 77 to 77', save for a change of phase, then the resulting coherent state propagator is no longer a propagator for the elements of the reproducing kernel Hilbert space L*(G),
but is a propagator for the elements of the reproducing
kernel Hilbert space L"i,(G). Hence, we see that the coherent state propagator
Kvv(l",t";l',t') (l'\t";l',t')de ;/',t')d. ? depends strongly on the fiducial vector 77. Followine stanc Following standard methods in [64] and [69] we now derive a coherent state path integral representation for the coherent state propagator. We start as in § 3.1.2 from the basic idea
" - ')H 0 1i ==
i exp ;-(*•exp --(t"-t')H exp --(t"-t')H
=
N+1
/ \ -i/V+l 11 N+1 exp expf-JeWj
rT
expf-JeWj
whereS( = (t"-t')/(N (t" - t')/(N + 1l),, therefore, we find where e = (t" - t')/(N + 1), therefore, we find i Kn{l",t"-l',t') Kr,(l",t"-l',t') (l",t"-l',t')',*') = <7?(Z"),exp (Z"),exp -%-{t" - t')Hw] 77(0) 7?(0)
ft
Kv(l",t";l',t')
<7?(Z"),exp -l-{t"-t')H V+l 77(1')) i (/"), exp .-«*)] 7 V
=
= ( (l"), [exp ~h (-J«w)] \ ( 0 > = (V(l"), [exp (-J«w)] \ ( 0 > .
Inserting the resolution of identity (3.35) TV-times this becomes
o;
Inserting the resolution of identity (3.35) TV-times this becomes . N
J J (...
i Kv(l'\t"-l\t') ,(i",t";i',t')-(l", t";l',t')t') = = / • • • / n<77(/ t) J +-l),exp 1 ),exp (( - j e t w) in(h+i) 7J
Jy
j=0 J=O
v
h
J
N
)>n^(
.?&)) v(lj))f[dg(l^ Wi)> j=l
" and and lZ0 0 = = V. /'. This expression holds for any N, and therefore, it wheree /yv+i IN+I =- I" holds as well in the limit. N iV —»* 00 or it — —* > 0,0,i.e., i.e.,
exp
r /vN d" f"l':/'.*'' f'\ = mn lim» / / ■. ■. . / l ] ( » | i(Zj+i),ex Knn(l",t";l',t') ( / j + i ) , e x p- (jctt) -k) J £—0 J 3=0 \ n J J = 00
n<
7= 0
H
A'
d 7^■)>n 7 v(h)}]Jfc(W9(h)i=1
i=3
(3.38) (3.38)
fa(J/+i), >(-^W)r7(Z,))f< *tf0rtJ;)) for small e. For small £ Hence, one has to evaluate (77(^+1),exp {-^eH)f](lj)) one can make the approximation
3.2.
COHERENT
STATE
r
*HH <
(77(/ i+1 ),e> , exp f -
81
PATH INTEGRALS
■H
»
(v(h++ x ) . ( l - ;
-
WJ+IJ.WJJ)
,) «?(*;))
fCv,{,& H) + I ; 'h) J) ss
v(h)) i
i
Mh+i), ,Wi))l
~Ti~TTi ^—FTT\~ L »ft fo(** (ICJ+O^CJ)) . L " (ICJ+O^CJ)) . ii i i -- ^ ^,( ' «j + i i ' jj )
-r
i ^-TtH / e//^(/i+i; /v^(l(j+l ££,(*,-+!; „ ( / ,,-+i;Zj)exp + ! ; //j)exp ; ) exp !Xp --^eH J+ v/ ;l s)1 ; / J ) ~h
)].,
(3.39)
where Wj+ilHrtlj i),HT,(lj)) j+ „(l . n _ (y{l W:+i),nWi)) IK'3 +W: lJi +i),HWj)) "-'IV], 7(0)5 H HM v(h+i'>h) = l Wj+l),V{Ah) l))(0)) Inserting (3.39) into (3.38) yields .
-. N N
r
.
)n Nr -, , N N
1 i ... .../ H _ e-eH i ii;lj) ' [H[ <*
(l",t";l',t') Kvr)r)(l",t";l',t')-T,t"-l',t') l',t') == Km 0 lim .
/
■
^IP
(3.40) This is the form of the coherent state path integral one typically encounters in the literature. It is worth reemphasing that the coherent state path integral representation of the coherent state propagator (3.40) depends strongly on the fiducial vector. subsubsectionFormal Coherent State Path Integral
Even though there exists no mathematical justification whatsoever we now take in analogy to what we have done in § 3.1 and 3.2 an improper limit of (3.40) by interchanging the operation of integration with the limit e —<■ 0. As pointed out in [69, p. 63] one can imagine as e —> 0 that the set of points lj, j — 1,...,N, .,N, ,defines in the limit a (possibly generalized) function t' < t < t".
l(t),
Following [69, pp. 63-64] we now derive an expression for the
integrand in (3.40) valid for continuous and differentiable paths l(t). Note that the set of coherent states r)(l) we have defined in (3.33) is not normalized, but
82
CHAPTER
3. PATH
INTEGRALS
2
is o of constantt norm g given by■ dl' dl'2.. We now rewrite the reproducing kernel IS l m t n e £v{h+i'>h) following KvOi+uh) = (n(h+i)>v( j)) 'n U(h)) \h+iih) = - Wj+i)>V(h))
way:
r (vQi) = (V(h+i) )) -- W (v(h ),vik)) 7('i+i).Wi •?&)) h+i)> i ?('i+0 - rt< :+ (v(ii+i),v0i)) h+i)iv(ij-«i+l))
= d(cf[lc [ l-- ddc - 1 ( r ? ( / J+l)> + 1 )ddl-d-'W^),^^)-^))} 17r?('; ? ( / ;+ f+ i1 )) -- 7f (? (OZ)J )) )]>]] w S3
l 11 exp[-(f^ (77(f d^exp[dcfexv[-d~ -
1 cc
this approximation is valid whenever ||7y(/ 0,...,N. l|r?(f,+i) »?(/j)|| 0 the approximation becomes increasingly better since the 77(f) form a continuous family of vectors. Therefore, one finds ie imus 1
d - ^(c1 *(7?(f Z; + 1 ) - 7 T?^))], ^r,(f; +i;f;) « d^exp[—]. 7( Z j))]'
3 41 ((3.41) - )
for \\v(h+i) - V(h)\\ < 1, j = 0,...,7V. Using (3.41) in (3.40) and taking the limit e —> 0 the integrand in (3.40) takes for continuous and differentiable paths the following form:
dcexp dcexp
.; /•'"H l dt W) ddTl{ f'1 W) (77(0, dl)] l)) di,(0) ~TJVV ''^ >-id I B'WW* ^ ^ '' ~TJ - ~ ik~[, 1
where Hv(l(t))
=
(r,(l),H(X1,...,Xk)r,(l)) MO.w(*,
and where we have introduced the coherent state differential dV(l) = 7?(f + dl) - 77(f). dri(l) Hence, we find the following formal coherent state path integral expression for the coherent state propagator: IU1 . (l'\t";l',t>) . K
[<"
[hi Jf
(l), un] « «I dt\vVgV),
ih(v(i), iHv(l),~v(l)}-H,(l) >(0, ^ ( 0) r - Hv(
g
(3.42)
3.2. COHERENT STATE PATH INTEGRALS
83
where N
rn 7?(0 (l) = =UU V g{l) g{l) v(i) = a(') 1
and and
N+l N+l ■ W+l Vg(l) lim limi (d(d TT dg{l 0 == Um T>g{l) 33). k)(()()'N+l TJ Wi)TV—»oo TV—»oo N - o X) co
* * ■ ■**■
i=i
A discussion of what is right and what is wrong with (3.42) can be found in [69, pp. 64-66] we only remark here that (3.42) depends strongly on the choice of the fiducial vector and on the choice of the irreducible unitary representa tion of G. Hence, one has to reformulate the path integral representation for the coherent state propagator every time one changes the fiducial vector and keeps the irreducible representation the same, or if one changes the irreducible unitary representation of G. Now in many applications it is often convenient to choose the fiducial vector as the ground state of the Hamilton operator 7i of the quantum system one considers; see for instance Troung [102, 103]. Hence, one has to face the problem of various fiducial vectors. In chapter 5 we de velop a representation independent propagator, which nevertheless, propagates the elements of any reproducing kernel Hilbert space L2(G) associated with any irreducible, square integrable unitary representation of G. Hence, we can overcome the above limitation. Also note that coherent state path integrals afford an alternative way of constructing path integrals for quantum systems moving on group manifolds and on homogeneous spaces. For instance Klauder has used in [63] the coherent state path integral to describe the motion of a quantum system with spin s moving on the two sphere S2 and in [65] to describe the motion of a quantum system on the Lobachevsky plane. Klauder has also discussed a quantization procedure for physical systems moving on group manifolds and homogeneous spaces using the action functional in (3.42), see [61] and [62], and has therefore,
84
CHAPTER 3. PATH
INTEGRALS
provided an alternative method of quantization to the quantization methods discussed in §§ 3.1.3 and 3.1.4.
Chapter 4
NOTATIONS AND PRELIMINARIES
4.1
Notations
In this chapter, G is a real, separable, connected and simply connected, locally compact Lie group1 with fixed left invariant Haar measure dg, i.e. d(hg) = dg. Let A(g) be the modular function for the group G; i.e. d{gh) = A(h)dg. If A(g) = 1 then the group G is called unimodular. It is known that the following Lie groups are unimodular (cf. [39, p. 250] and [54, chapter X, §1]): (i) Every compact Lie group. (ii) Every semisimple Lie group. (iii) Every connected nilpotent Lie group. The affine group, which we will consider in chapter 5, is an example of a nonunimodular Lie group. Let V(G) be the space of regular Bruhat functions with 'For a summary of the basic facts of the theory of linear operators, Lie algebras, Lie groups, and the representation theory of Lie groups see Chapterl. 85
86
CHAPTER
compact support on G (cf.
4. NOTATIONS
AND
PRELIMINARIES
[15] and [79, pp. 68-69]). Let T be a closeable
operator on some Hilbert space H , then we denote its closure by T. Let L be the Lie algebra corresponding to G with basis X i , , . . , i j . Then we denote by X\ = U(xx),...
,Xd = U(xd) a representation of the basis of the Lie algebra L
by symmetric operators on some Hilbert space H with common dense invariant domain D . The commutation relations take the form [Xi,Xj]
;£ti
= i Sfc=i Ylk=i cijJ AXk<
A vector ip S H is called an analytic v e c t o r for a symmetric operator X acting on some dense domain in H if for some s > 0, the seriess X£^~ o KoKi s ) n / n \ ] X n i pnip is
defined andi Er= X )^^0l[0s[* [ns/nn/ !n !] ]l l l^^nnV V''|| | < °o . We say that the representation U of the 0
Lie algebra L satisfies Hypothesis (A) if and only if U is a representation of the Lie algebra L on a dense invariant domain D of vectors that are analytic for all symmetric representatives Xk = U{ U(xk) of a basis x-^,... x^,... Xk =
,x ,Xi.d. If Hypothesis (A)
is satisfied then by Theorem 3 of Flato et al. [38] the representation X\,..., X\,...,
Xi
of the Lie algebra L on H is integrable to a unique unitary representation of the corresponding connected and simply connected Lie group G on H . We will always assume that a representation of L by symmetric operators satisfies Hypothesis (A). Therefore, the representation of L by symmetric operators is integrable to a unique global unitary representation of the associated connected and simply connected Lie group G on H . Let there exist a parameterization of G such that the unitary representation U of G can be written in terms of the X Xkck as
= n n° d
U U U
9(i) 9(i) :9(')
xxp(—il F ) -iPXj) l 1 l )= = e) ex =xp(-i/ ■i^Xi)..^ . . deXxd), pP{-il ({-il - dXdd)Xd), J l\ l\ccM-^X M-^X exp{-il exp{-il Xl)...exp{-il Xl)...ex J J) i=i
(4.1) (4.1)
dj
U
U;'m {1)
1 xp(il = = exp(il eexp(il exp xJ>(il*Xd) p i dXddX).. = ..exp( . exp{il x p ( zlX/1xXx), r il)),, U.^P(^X X X d).d)...exp{il = ]l UexexpiiPX,) F rI)1 Xj) ])
J=l
( 4(4.2) .2)
4.1.
NOTATIONS
87
for some ordering, where / is an element of a d-dimensional parameter space Q. The parameter space Q is all of IRd if the group is non-compact and a subset of IR if the group is compact or has a compact subgroup.
Remark 4.1,1: Note that one obtains in this way a representation of all elements of G that are connected to the identity element. Since we are considering a connected and simply connected Lie group we have that Ugn\ is a representation of G. Since G is a manifold one needs in general a collection of proper coordinate charts that cover G, we will relabel the coordinates in each of these charts by the
00
f x,=- 2 2~( ~ ( \{ X ll =
Q
1 \
1 0 j ''
1 - "22\
X222: = XX
0 i
_ i 1l 0( T -t ),*3 = 0 ) ' ^ 33 =~ - 22 (( 00 ---11; ) ,)■ ) '' 2 V C
-f
rc\rr and satisfy the following well-knowni commutation relations:
[X„X } =[Xi,Xj] }*iJ where, djk}k
i": i +1 -1 0 -1 0
ieijkXk, itijkXk, Ijkl k,
for (ijk) an even permutation of (123), for (ijk) an odd permutation of (123), otherwise. for (ijk) an odd permutation of (123), otherwise.
CHAPTER 4. NOTATIONS AND PRELIMINARIES One possible parameterization of SU(2) is given in terms of the Euler angles by
where 0 5 4 < 2n, 0 < B
< n, -2n 5 [ < 2n.
Note that the points 0 = 0 and B = n have to be excluded since at these points only
4 + [ and 4 - [ are determined, respectively. The Euler angles are
analogous to geographical coordinates on the sphere S2in lR3. Just as, on S2, geographical coordinates are not uniquely determined at the north and south pole, the parameters
+,[
are not uniquely determined at the singular points
B = 0 and B = n. Hence, at these singular points of the parameter space, and
5 no longer
4
define a unimodular unitary matrix uniquely. Therefore, the
set of matrices for which the parameterization introduced above is unique is a proper subset of SU(2). However, as far as integration over the group SU(2) is concerned the above parameterization is adequate since the set of singular points forms a set of measure zero. As pointed out by Wybourne [108, pp. 38-39], a suitable choice of parameterization for the Lie group G one considers can generally be made as follows: One first obtains a matrix representation of the Lie algebra (Ado's Theorem [7, Theorem 1.2.11) associated with the Lie group one considers; these matrices are then taken as the group generators. Just as in the case of SU(2), considered above, one then determines a suitable parameterization and generates the group elements connected to the identity element in parameterized form by exponen-
4.1.
89
NOTATIONS
tiating the group generators. Note that the choice of parameterization of G can be made in many ways and should ideally be made such that singularities in Q about the identity element are avoided. Finally, a representation of the form (4.1) is obtained by exponentiating the self-adjoint representatives
{Xk}
of the basis {xk} of the Lie algebra associated with G using the parameters one has determined in the representation of the group elements connected to the identity element. O Let (U,H) and (U',H')
be unitary representations of G. A densely defined
closed operator S from H to H ' is called semi-invariant w i t h weight a if Vo5 € G G. V
U' SU; ==a(g)S, v(g)s, u' ssu; = g
In what follows we shall need a common dense invariant domain for X\,...,
Xd
that is also invariant under the one-parameter groups exp(itXk),h), k* = 1 , . . . ,d. Define D as the intersection of the domains of all monomials Xil ... Xjk for all 1 < i j , . . . , * * < d. By definition D contains D , hence is dense in H . Then by Lemma 3 of [38] the restriction of X\,...,
Xd to D is a representation of L and
by Lemma 4 of [38] D is invariant under all one-parameter groupss expl exp(itXk), k= c = l , . l,...,d. Let Xmk(g(l)) and Pm*(ff(0) be functions such that on D the following l(l)) and relations hold:1: dd
i
d
n
a ] J eXexp(il Xm IY[I a) m Xm I I exp(zl I aJiXXa)X
u■
a=m+l a=m+l m - 1l
n«
d b h exp(-il exp(—il ■il Xb)Xbi)) exp(-i/
b=m+l 66 = =m m+ + lli m --- 11l1
d
(g(l))X , , *, J(g(i))x = £EX^
a = lI a=l
=i 6=1 6=1
m
k
k
(4.3) (4.3) (4-
k=l k=l dd
£X>> *J(g(Wk( <s (70( )0*) f*cfc. -. (4.4) ((4.4) (4.- ) = XV
a a exp(z7 [U6 XXf cb.) ) = l[exp(-il [ exp(-i X-ila)X Xma)' a) Xma ]Jexp(z7 = ]I JT exp(i7 I ex]
b
k
=
fc mm
k
4 4
k=l
Note that, the parameterization of the Lie group G is chosen in such a way
m ^*0, respectively.
that det[A m fc (5(/))] '(0)1 / * 0 and det[pmk(g(l))]
Therefore, the
90
CHAPTER
inverse matrices [X~1mk{g(.l))]
4. NOTATIONS
and [p~lmk:{g{l))}
AND
PRELIMINARIES
exist. Furthermore, let U(l)
k
be the d x d matrix whose mfc-element is Um {l) such that on D d
kk k 7 U'{l)g{l) = u; xXmmXUg{l) umg{l)U9(1)g{l) = 52u {t)x xkk,k,,k, mm{t)X =- 52u m(l)X E<
uUg(i) xXmrnU u;9(1) g{l)XmU (l)) g(l) g(l
(4.5) (4.5)
k=\ k=l k=l k=\ dd
E<
l = YjU-'JWk, = VWk, = 22,U~ (l)Xk, m
(4.6) (4.6)
k=i k=\
holds. One can easily check that U(l) is given by exponentiating the adjoint representation of L,
n d
k
U(l)=JJex U(l) U(l) exp(l (l ckfc,k),cckc),k), ) = = J ] Pexp(/ k
fc=i
here ck denotes the matrix formed from the structure constants such that ck —
~ck(S). 4.2
Preliminaries
T h e o r e m 4.2.1 On the common
dense invariant
r
domain D of xX\,... u.
,Xd, Xd,
the following relations hold: d
\J{g{l))dl™X « " * * ,, a*(
dUdU (i) For For all all I G Q, Q, U; dUg{l) (i) {l) g{l) Z= --ii E '> UU; g(l){l) 9(l)
k
ik,m ,m = l
kk k and Z Lg[lo) X (g(l)) (g(l)) = Amm*(ff(/ \*( = (Z)). g{lo) m m m m fl(g(l)). , )A * «m . *(j(Z)) *($(■ fl(/o)
d k k mn (ii) For all I G - t E ! Pm' Pm Pm (g(l))dl W))dl Xk, ' m A-*, g{l} g{l) G Q, 0. dU ^ ( g{l} 0 U' ^ (g{l) 0 == -i »*GKO;
k,m = l
kk and (g(l)). J(Q(D). nd Rg(lo)Pm g{lo)Pm Pm Pm (g{l)). Wk(g(l)) C ^ O ) = />„*(«('))•
Proof.
(i) Let ip g D be arbitrary, then since Ug^
leaves D invariant, i.e.
t/ s (;)D C D , we define the differential of Ugm as follows: d
-E
, ) l ~ Ug(H ,...,ld)ll)~ t^/Jg[l',...J">+Al'",...,l<>)' A if lm" jrr i V ^ T Jim r (g l( 1 mI m+ + W^l m m m iELmVlim 'l mm^^ --^^l l( i ", . . ")^1 dl .. cd/ m ^ s(0^ ( O V=-^ VE [ )l?l — "-' -±jk ' dl
rft/
BV)r
Z^ m m= = ll
AA / - -">- 00o
A 7TTJ A // A
'
(4.7) (4.7)
4.2.
PRELIMINARIES
91
Now since Ug^ is the product of one-parameter unitary groups one finds for the differential of Ugm d( m mm m
du dC/ dU
dd
dr. 9(^=E= E£ ]n ex p(- i/a ^)(- iX -)m.) nI I eXP(-i/6x6)^r.
e; b -ilbXb)i>dF I I exp( —i exp(~il Xb)Hlm.
a (-U°Z exp( laXa)(-iX,■iXm) ]\zM-il Xa){-iXm)
: £ 9(/)^ = 9(l)^
m m ==l1\ aa ==ll m m ==l\ aa = l
Therefore, Therefore,
6=m ++ ll b i>=m 6=m + l
(d i d d d d
J U'^dU^ U'g{l]dUg{l)iP = U'g{l]dUg{l)iP =
a eexp(il x IIII exp P (ll /X/ "a *^.)) X XX, m m ta= m + ll exp(tZ°XB) Xm a =U
E II*
k
-* E -i mY,= l
a=m+l m =dl a = m + l a
d
E *( (Z))E m,ifc=l
=
-i
mm
dd d d
II] II\ exp(-t, c x p (-il- iX/b6)il)dl Y t )rl>dl tfdr ] \ eM-UbXb)Wlm 6 = m + l 6 =m+l =m+l b
m
b=m+l b=m+1
m
ff
Z^.
m,ifc=l
Since ip € D was arbitrary, one finds that on D c H , the following relation holds
dd
d
( 0 )drx E A^AM(s(O5(Z))
u du^d t / O == ~* «% V = -— J2 E h( 0m s C 0) = £
fc fc: m m 5
mm 1
(4.8) (4-8)
f efe
k,m-\
To establish the the second part of (i) let ^ ib 6 g D be arbitrary then
u U
m
9(l) 9(l) 9(0^ 9(^ W,(0tf *%> dU dU
rV «( o' -) sW [lo)9(l) 9- i-HioMV^*-'{'ohvY -> {UW) ( l „ .){si)dV '^( O (7)V>
dU U U U U dU dU J g{l)i> ('o) 9* =Z U9(l) hUU9(1) 3(lo) 9(lo) s{lo) 9('o) sit)^ 9(l)Tl ) 9(lo) k'o n^ = = S(>° UU
dU
Ut /U 9
1d (
dVdU
1
{dm
m Therefore, using (4.8) and the fact that both {Xk}i {Xk}dk=l =l *are lintk=i dand {dl }^}i=i
early independent families one finds fc kk k :k fc = A X (gL3{h) m(g(l)) (g(l f f - 1 (/ ( / O )5(0) ) 5 ( 00) ) = Amm*( )*m A *((ff(/)). 7(0)) = = A mm\'m((ffg{lo g{lo)*m 5 (0)™ (g(l))
rUMO)
L
fo(0)-
(ii) The first part of (ii) is similar to the first part of (i). To prove the second part of (ii) one can proceed as follows, let ij> £ D be arbitrary, then
dU
u du 'u9(io)U' ,u;{l^ i> = 9(nU' ;V (i^ ===: duMrtMiioWuMi)* 9(>) u' ;J(iMU)^9U)9(lo) *g( m^t oW) i)+• = du ^w (t mOi Qs )')C )>^. m 9(lo) 9^)u*g(l sdg{u)) u;
dU <*0j,(j) g^ ( O 1 9(nU' 5(0 g(i^
J
U
Q
U
{l)
dU
U
a
Therefore, by the same reasoning as above k: kk (i)). [n R {g{l)) i)) = (g(l)g(lo)) {g{i)g{io))!»))< = J(g(l))r Pmk(g(i (iu)Pm (9(l)) \g(l)g(lo)) = PPJ(g(l)),pJ(g(i)) = pmmPm\g Rg(lo)Pm 99{ o)Pm .0)- aD
CHAPTER
92 Since the Xmk(g(l))
4. NOTATIONS
AND
PRELIMINARIES
are left invariant functions on the Lie group G, the
relation (i) can be viewed as an operator version of the generalized MaurerCartan form on G, (cf. [18, p. 92]).
k Corollary 4.2.2 The functions ons X \mmk(g(/)) (g(l))
at and pmk{g{l))
are related as follows:
d d
E
= E>m -'^(0)^(0 W ) )0^(0 tfe*(0 =
A (; m Am*UK0) »*UK0) Proof.
ce=l =l
>t ? Let V/> e€ Dn be arbitrary then by Theorem 4.2.1 (ii),
dU dU uu <*C,<0 m h)+ mJ Ch)+ )^/)^
d
c dlmmX U U c: ( (/)). E p-pm (5(0)^ ^ci/ " (l)^g(l)u; (l) 9 (of/; ( ,)^ll = —t = = ~
U
g
cc ,, m m= = ll
Since Ug(i) leavess D in invariant, set 3=i/; U',,stp then multiplying the result ( / ) V €6DD, ,then ing relation from the left by V*n\ yields,
'
%
ad
E
^V(l) c g(l)
c c u; p mPm ( ff{g{i))dru; (/))drf/; Odr^; w(/)JXcU y s(0 ^ i)4>.CC((ff(0)rf u;'.V)du du
== --i£i J£]; Pm xT c(ug(i) 4>. u Ul(i)dU in gU(i)4> gin 9(l)(
in
gin
cc ,, m C,TTl m== =-1lll
Using Theorem 4.2.1 (i) and the definition of uj(i) Umk(l) tthe Corollary easily fol lows. □ Corollary 4 . 2 . 3 The functions
sat pmk(g{l))0)
equations: lah'ons: d '1
dd
E(
=E*
aa ( i ) ^ { ca 9, n, n[ p[ p- 1- /1(/ f(ff(f (0)]p" /( )/ ) ]) ]pp- -' k1' k"/(s(0)-a "( (3(Q)-ft-tP"V(9(»))]/ }==^^CCjjtt ^ - > / ( 9 (9(0), (0), 3( (; ;) ) )--aa, ,„„[[pp--\\ ((! !77((//))))]]pp--''//((99((//))))} 1=1 n=i
z=i = ii // =
d
d d
-E«.
1 1 1 1 1 o _ ^"[A-'/(!7(/)): = (ii) ^{a«"[A/( ! ?W (/))]A( s (i))]A-/(s(/))} /((<))}=X> 'A- 1 / o ( f f (()), VM O ■(0)-MA M V(j(i))] / i ''( s (0)-a,..[Ai(s(0)]Af l (/))}=^c Jk k 'A
/1=1 =1 /= = 1i
n1 = = 11l d
E^
dd
^E^
: X _1_ a,Oa<»[/> m [A" = £]pm*(s(J))3l»[p \V))]))]• ii) £J2 Z)^'(9('))ai-tp .V))](iii) A »"„'(9(/))9, '(J(J))M A - , / ( »SW ) ]l = E^'W)) -^ _1 -*(»('))]• [A-'/( (o)] E 3J = 1l
33 = == 111
where Cj^ are the structure constants for f o r GG..
4.2. PRELIMINARIES
93
Proof, (i) Let V € D be arbitrary then one easily finds using (4.7) that dimi d | m"8l"Ug < 9 / n i /J9{')^ dtndlmUg^lp, s(,)V> = (/)V>, =- din dim h»U tW
holds. Now picking out the terms di^Ug^ and di^Ug^ in Theorem 4.2.1 (ii) one finds dif acf
ifd
v>==aE EM=l
E* E
a aa a a 1 —zp m (g(i))x ua g{l) J > dim ™ [[-i }<3(l (9(l))X }ug{l)) rp [-ipm"^ n ((fi 5 (/))X )a)U*ag{l) Pn)n(9(l))X i> )aU W9ina,f/ Q ]
aa == ll = ll aa =
0aa ===1ll
- E^" d
E
f » ° a( s ( i' ))P)mP "(6.j(»(0)^.y» ( /( )s) (x' n): ) ^ ^[j ^' 9#0 * = \ ^-»a."-[Pna(9(0)]^<. ( 0) ^* = P" w))p< •'° -- EJ2J2EPP»"(»('))Pm l3|"« J^-i3,m [ P n[Pn"( °(S(i))]XaLo=l .0=1
,,
k
I
a,6=1 a,6=1
J
d
=
a , ( (0 ) P n 6 ( ( / )))XaX6 )X l| C/ --'*»[Pm i 3 i » [ p m "a(ff(0)]^a (j(/) 5 £/,(<)*• 9(,; J E a,6=1 - E ^"> Pm'(»(0)P-*(y('))^.^ ?•«*• (s(
l aa==ll
J
1.6=1 a.6=1 a.6=1
Since Ugm leaves DLa=l invariant, one can set = Ug^tp and rearranging the 1 invarian
terms yields
d dd /f f b6 E { -{W --i*.[ft. ' "-^W [ P(n97 (( s ( 0 ) ] } * / * = =E E PP»"(9(l))p n " ( 3 ( / ) )*»*(»(/))[*..*» Pm ( s ( / ) ) [ X o,X , Xbb6]4>. ]0. E 1(g(l))]+id,. + iidi-{p ]4>. m (9(l))iXa,X aa,6=1 .,,66==11 // == 111 a.6=1
E(
m}xj= E P»°(ff(0W fcW(s(0)]}-» = EX>°
^U'
Now making use of the commutation relations: [X X),} — X f this [Xa> = ii Ylj=i S / = i ccab* a,Xb] af/-X equation becomes if
Si'
a
6b f -f ( / ) ) ])]-d,n[pJ(g(l J6 / J\J **/<£ x/ f<'4> 4£ ===0. £ ] IISft-WOKO)] ^9(m[p --[Pn b n (nf(9(l))}-d, -- ft-^CiffO)] ft-On/foW)] ++ E EE Pn°(ff(0K n ° [g(i))Pm"(g{ ( f f ( 0 K ((5(0)Ca6 5(g(i))c ( 0 ) C aab 00. . (9(1))) + + } PPSW E< E
/
6
/=1 [
/
a,l 1,1=1 a,f>=l a,6=l a.6=l
J
Finally using the fact that the operators {Xk)i=i
^orm
a
^ a s ' s f° r 9 a n d
tnat
4> £ D is arbitrary one concludes
m-dmh
J ({dlAPn S / m ^ (9(l))}-- -^WOKO)]} »[Pm}(g(l d,n[pj(g(l))]} {S|»W(ff(0)] i(l))]} = ==[Pn'OKO)]
dif
s I (4.9) (9'(ff(O)c (g(l)l))Pm -E ft>'(?('W(^W. EE^Pn Pn(9(l))pm\9{l))c.t . (4-9) !))C l
K 5 ,\( = 1
.-,t=i
94
CHAPTER
4. NOTATIONS
AND
PRELIMINARIES
Now contracting both sides of (4.9) withhp~\ p~l 7jaa(g(l)) yields (5('))yi di
E
ll a a a0)]pm'<'(.9(0)} jV w : t jyJ(ff(i E d,Ap-'j{9{i))]pJw))} (9mpmf(g(i))} (ff(O)-^-b E {d,™[p(5;m[p ^ ^ v■w i?(0)]Pn W c(g(i)) i ) -- dAp-'i } (g(i))}Pn
d
d
5 xx aa (?(')K rMi))*** (< pp-y(
EE
// = = i ll s , s,t=l :,(ti == l!
where, d
d
52 E< ^p™ (9mp-\ w)) =-E*
d k n -1l c *C ,"(5(0)]P E ^9/c[p M / > [Pm*(s(0) mm* ( s ( 0))KV(
Finally contract both sides with p _1 jt' n (
to
obtain the desired relation,
ia
i_ n
l a l n 1 1 n {^[p•^[p-'/CffC ij (9(i))}p?(0)]P *kV( (gu)) ft"[p *"(ff(0)]pV(s(< 9(0)]P /GKO)} i i( (j(0)}= = "(5(0) ft" EEE( {^[p(s(0)]p(5(0) ---ft-[pfaip-wgmp-yw))} EW -1 n
71 = = 11 71
d
/ 1a !{g{l))p]k = EE Eycc^* ^C ^W)P~ / \g(')) (5 = ^ :o)p-1/°(ff(0)/=1
(ii) The proof of (ii) is similar to the proof of (i). (iii) Using Corollary 4.2.2 (iii) can be rewritten as dd
ad
/(s(0)5 /[/ V-\\l)p-\ {g(J))d \p 'W))]pl)K\6/(9(0)(s( . W)). ^~ t (ff(0)^ipr»'(s(0)]p~ '[pm*(ff('))]P '(9(0)= -E E trVtfKVfo' ^*'(0/'*
d h\t)p-\\s> <-\U-\\l)p-\\g{l))] \ V ) ) ] == -" E Y <-\u-\\i)p-\\g{i))] J IM^-Vw Eft Eftf1 = = 1]
J
1 /
s
(
ll
m
1 lb
/,5,t=l
<=i
/,»,< = !
This equation can be simplified as follows: d
i i i 7-\\l)p-\\ E pn = /(
h,j,t
= \
s
d
d-Y.d^p^W))] _x _I (g(l))]p-X>[ )]pP-\\g{i)) .V)) =■Eft-E4Pm '-^'W))]p .*(ff(0) $
5= 1
»=i
J2 u-^'m-wnmusv)] d„[(*(0)1. £E"C//. (03|m[p^(s(0)C/;''(0] m (0] === ft-lftn'(*(/))]. Pm'(g(i))}. / U-\JI(l)d,». di^PnHgiWj
h
Differentiating the product and rearranging the terms yields d
f
f J d,APn nf(9(l))} (g(l))} -~ Sin di»[pj(g(l))} dl™[Pn (i di«[pJ(g(l))} (g(i))} = =
J h } J - EPE PP» PnWi-luSH). (9(mAUj'■)}u~\\> (l)}U~\f(0(l). G7(0W-[tf/(O]tf~V(/). nJ(g(m J,h=l
4.2.
PRELIMINARIES
95
h n nh h ~Vi\V) := p»»%(0 Next using di)\" (l), lmU j {l) m'(g(l))cJs U Jsn U n (l), '(0, wwhich is proved along the same
lines as Theorem 4.2.1 (ii), we find f f ddi~WW))] tm[p>n n , (g(0)} )] - <9,n[p d,«[pj(g(l))] d,«[ J(g(I))] m PJ(g(i))}
d
E
f ]3 3 55 =-J2 Pn (g(i))Pm W))Pm (g(l))c (g(i))c 'nPn {g{i))p Pm l))c3d', = m' (g(i)) jd , -- Y,p-
i,s=\
which is equation (4.9), and therefore, establishes (iii). O One could ask if it is really necessary to use unbounded symmetric operators in the representation theory of Lie algebras or stated differently, can one develop a representation theory of Lie algebras using only bounded symmetric operators. We could then discard almost all the technical difficulties we have encountered in this chapter. This interesting problem has been considered by Doebner and Melsheimer [27] who have shown that T h e o r e m 4.2.4 (Doebner and Melsheimer [27]) A nontrivial
representation
of a non-compact
at least one
unbounded
Lie algebra by symmetric
operators contains
operator.
Since we are interested in quantum physics, we have to represent our basic kinematical variables by self-adjoint operators. Hence, we have to choose a representation of the basis of a given Lie algebra by symmetric operators satisfying Hypothesis (A), so that the closures of these operators are self-adjoint and the representation is integrable to a unique unitary representation of the associated Lie group. Therefore, in light of the above result, we can not avoid the use of unbounded symmetric operators when we are dealing with non-compact Lie algebras and Lie groups.
Chapter 5
THE REPRESENTATION INDEPENDENT PROPAGATOR FOR A GENERAL LIE GROUP In this chapter we present the detailed construction of the representation independent propagator for a general Lie group. In § 5.1 the construction of coherent states for a general Lie group is discussed. In order to keep the mathematics as simple as possible we first construct in § 5.2 the representation independent propagator for a compact Lie group and then introduce in § 5.3 the representation independent propagator for the special unitary group SU(2). In § 5.4 the construction of the representation independent propagator is then extended to the case of a general Lie group. The last section § 5.5 is devoted to the detailed construction of the representation independent propagator for the afflne group.
97
98
5.1
CHAPTER 5. REPRESENTATION CHAPTER REPRESENTATION
INDEPENDENT INDEPENDENT
PROPAGATOR PROPAGATOR
Coherent States for General Lie Groups
Let U be a fixed continuous, unitary, irreducible representation of a d-dimensional real, separable, locally compact, connected and simply connected Lie group G on the Hilbert space H . Let 0, £ € H then the function G 9 g -» (Ug^,(p) is called a coefficient of the representation
Definition 5.1.1 A continuous,
U.
unitary irreducible representation
U is called
square integrable if it has a nonzero square integrable coefficient, i.e. if there exist vectors , £ £ H such that {Ug(,,4>) ^ 0 and
b /I
JG JG
22 \{U \{U dg<<x. 9Z,4>)\ dg<<x. 9Z,4>)\
By a general Lie group G we mean in the following a real, separable, locally compact, connected and simply connected Lie group G with continuous, irreducible, square integrable, unitary representations.
For continuous, irre-
ducible, square integrable, unitary representations one can prove the following Theorem:
T h e o r e m 5.1.2 (Duflo and Moore [30]) Let U be a continuous, square integrable, unitary representation erator K in H, self-adjoint, satisfying
the following
irreducible,
of G, then there exists a unique op-
positive, semi-invariant
with weight A - 1 (g), and
conditions:
(i) Let Let ^<j>,£ ,( E 6 H JH, / 0. / Then 0. Then {Ugi,4>) {Ugi,4>) is square is square integrable integrable if andif only and ifonly if 1 2 {£ ee D D(A'( A ' - 1 //2 ))..
5.1.
COHERENT
STATES
FOR GENERAL
LIE GROUPS
99
(ii) Let X,X' € H , and £,£' e D ( / ( - 1 / 2 ) . Tfeen one /ms
J
l 2 /J(x,u (X,gU x')dg = = (x,x')(K(X, x')(K-l/21/2 t',1/2 O(u (x,x')(Kt',K< K~^i). (;)at)(U,?, (',K-^0ge,x')dg
(5.1) (5.1)
(For the proof see [30], Theorem 3.) Remark 5.1.1 Condition (i) shows that if a representation is square integrable then there exists a dense set S of vectors in H such that for £ 6 S the factor (Ug^,
n dd
d U um x p ( - i / A X AA ) = eexpi-il x p ( - 1i^).. / 1 T 1 ) . ...zxp(-il .exp(-il Xd)\ddXXdd); m .exp(-il m = I I eexp(-i/ I i k=\
where / G Q. Now let r] € D(A' 1 / ' 2 ); then we define the set of coherent states for G, corresponding to the fixed continuous, irreducible, square integrable, unitary
100
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
representation Ugm as: l/ = VmKUl'\, 17(f) = sil)K \,
1 2 ^r, D G fDA( 'A ''1 / 2) )
and and
Nl --
||T?|| = 1. 1.
(5.2) (5.2)
It follows directly from (5.1) that these states give rise to a resolution of identity of the form
-L
/ H = /I v(i)(v(i),-)dg(i), IH= V(I)W), ■ )dg(l), Jg Jg
(5.3) (5.3)
where dg(l) is the left invariant Haar measure of G given in the chosen param eterization by dg(l) =
dd k 1{l)^{dl ,
m
4/(0 = 7 ( 0 1 1 ^ ' k-i k-i
5 4 ((5.4) -)
k where w h e r e 7 ( 0 s | d j(l)=\det[\ e t [ A m * ( mf l(g(l))}\. (0)]|.
Remark 5.1.2 It follows from the strong continuity of Ug^ that the family of states defined in (5.2) is strongly continuous. Moreover, these states give rise to the resolution of identity (5.3). Hence, the family of states defined in (5.2) satisfies the requirements set forth in §§ 3.2.2 for a family of states to be a family of coherent states. O The map Cv : H —► L2(G),
defined for any ip £ H by:
l 2 l 21 2 l VK ,i>), [c„VK0 -= 1M0 iMO = WU) W), i>)=={V(u ' v,' ^), v>>, [C,VK0 (U^K mK gU)
(5.5) (5.5)
yields a representation of the Hilbert space H by bounded, continuous, square integrable functions on a proper closed subspace L2(G) of L2(G);
see Ap
pendix A . l . Using the resolution of identity one finds l^(/) M0 = n (l-J')i> v (l')dg(l'), = //IC |/C,(/;/')^(n^(0>
(5.6) (5.6)
5.1. COHERENT STATES FOR GENERAL LIE GROUPS
101
where 2 fCvv(l;l') W)MO) ^ = /2(/ a _Ug , ) /n/2 ) (v,K^U -,(l)g(v) IO/ tC (l;l') = (v(l)MO) == (r?,A'1{n,K^ K^n) 1( ,-, 7?n) g)9(( mil)
and KiUg-i^g^ijK* K*Ug-iti\s{ir\K*
denotes the closure of the operator
K^Ug-iingimK^. K*Ug-\n\gni\Ki.
One calls (5.6) the reproducing property. Furthermore, as shown in Appendix A.2, the kernel ICv(l'; I) is an element of L2(G) for fixed / £ Q. Therefore, the kernel fCv(l'',') is a reproducing kernel and L2(G) is a reproducing kernel Hilbert space; see Appendix A.2. One easily verifies (see Appendix A.l) that the map Cv is an isometric isomorphism from H to L2(G).
Now let the map
A : G 9 g >-* Ag be defined by left translation, i.e. 1 2 (A {g^(i)a{i% W ^ eI L\G), Vg(l),g(l') £e G. 9( ,)0)( fl (O) =
(5.7)
It is straightforward to show that the map defined in (5.7) is a continuous, unitary representation of G on L2(G).
This representation is called the left
regular representation of G. Lemma 5.1.3 The isometric isomorphism Cv intertwines the representation Ugin on H with a subrepresentation of the left regular representation A9(j) on L2V(G). Proof. Let ip £ H be arbitrary, then we have
[cvnuug{ngmW)m) == [c
wiu^) Wiu^)
= {ug-m<) (Ug-i{nUgil)Kl/2n,iP)
=
1 Mg-\i')9(i)) M9-
=
Ass(/(/')^(ff(0) «)^(ff(0)
A[C,V](0g{n[C^}(l). = A9(n
102
CHAPTER
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INDEPENDENT
PROPAGATOR
Since, ^ £ H was arbitrary and Cv is bounded we conclude that
C UUg(l') C vn g(l')
~
A ^g{i') g(l')CC^v~'
hence, Cv intertwines the representation U with a subrepresentation of the left regular representation A on L2(G).
□
Therefore, (U, H ) is unitarily equivalent to a subrepresentation of the left regular representation (A,
L2{G)).
L e m m a 5.1.4 The unitary representation
Ugm intertwines
sentation
{ X m } ^ = 1 of L on H , with the representation
invariant
differential
spaces L2(G)
the operator repre
of L by right and left
operators on any one of the reproducing kernel Hilbert
C L2(G).
In fact setting V; = (dp, ■ ■. ■ ,d[d) • ,djd) the thefollowing following relarela
tions hold: dd 1 m l id \(-idtm), (i) Letxkx(-iV,,l) k(~iV,,l)i = J2P~ k (9( ))(- t"')>
k = 1 , . . . , d, then k=l,...,d,
m=l m=l
xk(-iv,,i)u;{1) ip = v; u;m(l)xxkki>, i>, w> e D. {l)i>
d
{iV,,l) (ii) Let£k{iV,,l)=
X m1 m l = Y, ^k (9( 5^A(j(0)(*'ft») J k k=l,...,d, = l,...,d, then f c Wd^),
m = ll
ik(^i,l)U ^ ip ~x g{l)s{t) k(iVl,l)U
A common dense invariant of the L2(G) D„ s= C,(D).
C L2(G)
= Ug{l)Xki>,
Vt/. e D .
domain for these differential
is given by the continuous
operators on any one
representation
of D , i.e.
5.1.
COHERENT
Proof,
STATES
FOR GENERAL
LIE GROUPS
103
(i) Let V € D be arbitrary, then using the fact that dimUg^U*(l)i()
-Ug(i)dimU*,,^
-
it follows from Theorem 4.2.1 (ii) that d
E
ce -ifaV;,tfl> Y0p mrma*(g{!))U:.M, -id ( s ( 0 ) t f ; ((/) tf, -idlmlmU;u;(l)mi>i> == E^p (ff(/))t/; xccv>, 0*
m= rn
! ,i,...,d. ...,* l,...,d.
cc = = ll
After contracting both sides with p _ 1 J t m (^(Z)) one finds d
E
1 m Y,P~ (9(l))(-i^)U; ^ x^, {l)rp=U;= V)X T,p~1kkm(s(i))(-id u;k[l) t^)u;^
m m= = ll
, .i,...,d, ...d, k4 ==1k=l,...,d,
hence, ik(-i^iJ)U*^tp= ( / ) V: ifc(-lV|,Z)i/*
U*^X ip v; mxkki>
k=
l,...,d.
Using Corollary 4.2.3 (i) one obtains,
[5,-(-iVj,0.5iHV/,0]
-
E m,n=l
1mm l 1n [P" - 1 , m .( (s(l))(-id^),pf (ff(0)(-^),pf ( / ) ) ( - ^ ) , p - 1 jsn/(5(0)(-^n)] (5(0)(-^n)] \P~\ (gO)){-id^)]
771,71=1
1n ,1Bm = m,n=l i*EE EE {^[^ (ff(0)]p-1,"(5(0): {^[^1,1m,m(5(0)]p(5(0)]p-1i"(ff(0) (n[/J-i j (ff(0)]/»-Y(*(J))] i (ff(0) -- 5$-b1 _ — 1 77 77 11 = 71 = 11 7 1= =1 1
xX((-- i299;/m m )) m.= l n=l
\ m («7(0)(-«^) = 'Ecc«/ E^A^ox-*^) Jt=l «-/ LmE^A^ox-*^) =l = »E d X(-29/m)
= =
Jt=l *:=! dd Jt=l
Lm = l Lm Lm = l
i V, ,j I, i)).. !t^' d^CC, i/ ji*t i( i-(l-V !fc=i /fc=i c^= C l , / i t ( - i V , , 0 . fc=l Therefore, the differential operators { i i ( - i V / , / )/)}*=! } ^ = 1 with rators {xfc(-iVj, Therefore, the differential operators { i i ( - i V / , / ) } ^ = 1 with variant domain D,, form a representation of L on any one variant domain D,, form a representation of L on any one kernel Hilbert spaces L^(G). kernel Hilbert spaces L^(G). (ii) The proof of (ii) is similar to proof of (i). □ (ii) The proof of (ii) is similar to proof of (i). □
common dense in common dense in of the reproducing of the reproducing
104
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
d vv Corollary 5.1.5 The differential operators•s{x {ik{-iVi,l)} ({M* ({ifc(* <>0}5t=i) <>0}5t=i) k=1 k{-iVul)}ti({h(iV hl)}Ui)
are essentially
self-adjoint
on any one of the reproducing kernel Hilbert spaces
L*(G) and can be identified with the generators »{A(Xk)}f=i {A(^))Li {{P(X ( { P ( ^k)}*k=l ) }) L I )°f a subrepresentation Proof.
of the left (right) regular representation
of G on
L^(G).
Let ip £ D then it follows from Lemma 5.1.4 that xf fc (-tV,, Z)V>„(0 i)lM0 = = [C„XjtV](0. [CvXk^l), k{-iV,,
On the other hand, let Ugk tt\ = exp(-itX exp(-itXk), k), ak^
kk = =
l,...,d. l,...,d.
k — 1 , . . . , d, be one-parameter
subgroups of G. Then [CVXkkip](l) ip](l) can also be written as [C [C„X*V]W nXkiP](l)
c
clim ^ l "%](/) = [c, [C,*^P^]« = * ^ f - %^] («/ ) = =
Usk(
U }^Tt^ 9-k\t)9(tf^) fe^rew')^
U ~ 9(i)V^)} ~ (^w^)]
,.lim U9;\t)9(l))-U9(l)) U9k\tW))-*Mi)) Lim ;
= =
Lim f(—0 ~0 <~0 <~o n
=
; it
it
A
gt(0 -
l i m - itit^ it
(—0 t—0 •o
=
it zi V, (I)
A(XA(X k = k=l,..,,d l,...,d k)ipv(l) , k)^(l), Ag.(t\
—I
where the A(A"/t) = s—lim M , k = 1,...,<£, are the generators of a where the A(Xk) = s—lim — , k = l,...,d, are the generators of a subrepresentation of the left regular representation of G on lA(G). Hence, one subrepresentation of the left regular representation of G on lA(G). Hence, one can identify ik{ — iVi,l) with A{X k) on D,,, i.e., can identify xk{ — iVi,l) with A{Xk) on D,,, i.e.,
xkk{~iV (~iVhl)hl)
= A(X A(X k),k),
k=l,...,d. k=l,...,d.
•s iifc(-tV Clearly, the operators k(-iVi,l),( ,Z), ik — 1 , . . .,d, are symmetric, since the operators operators X Xfc, A: = 1 , . . . , d, are symmetric on H and since Cv is an isometric kt k isomorphism from H onto
L^(G).
5.1.
COHERENT
STATES
FOR GENERAL
LIE GROUPS
105
The essential self-adjointness of the operators x^ (^ -- i V ; , /), k = 1 , . . . , d, on L^(G)
X^, k = 1 , . . . , d, is established as follows. Since each of the operators Xk,
has a dense set D C H of analytic vectors, see § 4.1, we have by Lemma 5.1 in [83] that each Xk, k = \,...,d,
is self-adjoint.
Hence, the restriction of
Xk-, k — 1 , . . . , d, to D is essentially self-adjoint. Since, Cv is an isometric each Xk, isomorphism from H onto L^(G) we have that the closure of each £*.(—z'Vj,/), £/t(—z'Vj,/), k = 1 , . .. .. ,,d, d, contains a dense set of analytic vectors, namely, CV(T>), hence, is xjt(-iV;,I), by Lemma 5.1 in [83] self-adjoint. In particular, each xjt(-iV;, I), k = 1 , .l,...,d, . . , d, is essentially self-adjoint on D,,. Similarly one can prove that the operators {& k(iV i, l)}f=1 are essentially self-adjoint and that they can be identified with the generators {P(Xk)}f {P(-X"t)}^ =1 =1 of a subrepresentation of the right regular representation of G on L^(G). □ Corollary 5.1.6 The family
of right invariant
linear differential
operators
{{ik(-iVi,l)} {xi(-z'V/,/)}£ x i ( - i V ; , /dk=1 ) } ^ = 1 commutes with the family of left invariant linear
differential
operators {i^iVi, 0}/t=r {h(Wi,t)}Lt< Proof. Let £,•(—iV/,/) and x 3 (iV/,/) be arbitrary, then [xi(-iVi,l),Sj(iV [x (iVi,l)} t(-iV,,l),x:0,*i(»Vf,0] hl)]
=
1 m E b" . (5(0)(-^),A-11iJ"(n(f fff(Z))(ta (/))(ia/ n)J - 1 /"(fl(/))(-i5/"),A/n )] m,n=l
, - i . ^ m i a . [A-ywo)] \\-i. (n(i\W = E f Er {p-vorw)*{p-Tigim[A"VW0)1 m
n
n=l \ m = l
A i
/
1 n
m / ))^[p- 1 , nn( (0)]}j^ AA" i r^ --E w(0)ft-L»«>(0)J H&5 (0)]}j^ --EEX>-S U« " r^i (s(0)Mp~\ ))^[p- , ((s(0)]J 5
m =l m=J
J/
i n nn == vE ( E /»" rcs(0) {^/(9(i)w [p-\ w))} E lYp-'rigii)) y . U-'/igom, \ -\ (i))} P E= 1l (\ mmE /»-V"c*(0)/ ,E {A-VGKO^/ lp-\(9w))} «n = s=l ==l l «=1 \ m = l
/,s=l
106
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
, (ff(0)])W dm h4p'\ (9(i))U 9m -£ A- , (fl(0)9/™[pd
xpm s (s(0)} (s(/))} - £
n, nn
1m
m=l
d i d
\g{im, [p-\n(g(l))} = E{E A "V(J(0W/I^-VC»(0)] n=l n=l
^( / = 1 (/=1 d
-E ^
0. = o,
m = = ll\ m=
>_1
l
W4P~\ W))} \g(0)d,4p
m
Ul"
J)
where we have used Corollary 4.2.3 (iii) in the fourth line. Therefore, [xi(-iV,,l),2j(iV l>*j(*Vi,Z)] == 0, hl)] and since, x,-(-iV/,/) x , ( - i V ; , / ) and Xj(iVi,l) iij(iVi,l)
were arbitrary, this establishes the Corol
lary. □
5.2
T h e R e p r e s e n t a t i o n I n d e p e n d e n t Propagator for C o m p a c t Lie Groups
A representation independent propagator has first been introduced in [99, 100]. Let G be a d-dimensional, connected and simply connected, real compact Lie group G. For compact Lie groups all irreducible representations are finitely dimensional (cf.[7, Theorem 7.1.3]). Hence, let us denote the finite dimensional irreducible representations of G by U1- and their finite dimensional representa tion spaces by H ( . We denote the dimension of the representation space H<; by d(. One calls d^ the degree of the representation U^. Let X\,...,Xd
be an
irreducible representation of the basis of L by bounded symmetric operators on H<. Then Hypothesis (A) is trivially fulfilled for this family of operators since
5.2. CAMPACT LIE GROUPS
107
all vectors in H^ are analytic vectors for these operators, hence this representation of the Lie algebra L is integrable to a unique unitary representation of G on H<;. Let there exist a parameterization of G such that dd
U U9d) ==
m
11*
k ntl^P(-H ^ P H ' Xk) ***) =
1 1X )...ex (-ilddX ), exp(-il 1 V d exp(-« Xi)---exv(-U Xd),
k=l
k=\
where / S Q. Since G is compact, the parameter space Q is a bounded set, therefore, all irreducible representations are trivially square integrable. The positive self-adjoint operator K is given by K = d^I, hence, we can choose any normalized vector r/ £ H< and the coherent states for a compact Lie group G corresponding to a fixed irreducible unitary representation become: n(l) v(l)===-Jd~c VyfcV^n, ^V' see equation (5.2). As we have seen in chapter 3, the resolution of identity has the form
-L
IH IH I HCC,== = [
JG
r,(l)( r,(l)( y(l)(v(l),-)dg(l), V(l),-)dg(l), V(l),-)dg(l),
here dg(l) is given by
w^IK. Since all operators Xk, k — 1,.. .,d are bounded we have by Lemma 5.1.4 for any ^ £ H^, using the continuous representation Cn : H( —» L2(G), that i«fc(-iVj,J)M](0 * ( - i V , , 0 [ C „ ^ ( 0 = [C [C,X tf](Z), k = l,..,d, nXfc kmi Note that this relation holds independently of n. Since G is compact the center of the von Neumann algebra .4(A) generated by the left regular representation A of G contains a compact self-adjoint operator whose eigenspaces are A-invariant (cf. [79, Lemma IV.3.1]). Hence, A can
108
CHAPTER 5. REPRESENTATION
INDEPENDENT
PROPAGATOR
be decomposed into a direct sum of irreducible representations. In fact A is completely reducible into a direct sum of all irreducible unitary representations of G, where each t/1- occurs with multiplicity d(_ (see [7, Theorem 7.1.4]), i.e.,
= 0'
A = 0dc[/<, Ce6 CGG where G denotes the dual space of G; G is the set of equivalence classes of all continuous, irreducible unitary representations of G. Denote by H(Xk) the self-adjoint Hamilton operator of a quantum mechan ical system on H(. Then for V in the continuous representation of the solution to Schrodinger's equation,, tp(t) V(0 = exp[-z(£ ex P H(* - t')H{X t')H(X).)]rp{t'), k)]iP{t'), wheree h = 1, is given on L2n{G) by
J
V„(M) Kv(l,t;l',t')^(l',t')dg(l'), ^,(/,t) = Jj Kn(l,t;l',t'W v(l',t')dg(l'), where K Kvv(l,t;l',t') (i,t;i',t') =
(T)(l),exv[~i(t-t')H(X )]Ti(l')) fa(0,«pH(t-Owpr fck)MO>
= [C„ exp[= [Cnnexp[-i(t-t)7i(Xk)W)](l) t')[l vW)}(i) = U[t -u(t-t')[c = U(t-t')d w(*-O<*cto,P#0^(t,)»7), where U{t - t') . W(* i') = = exp [-i(t - t')H(x t')H(xkk(~iV (~iV,,l))] hl))] fc(-»V,,0)]In this construction r] was arbitrary, hence it holds for any 77 6 Hr. Therefore ( (ONB) {(f>j} {*,■}*, one can choose any orthonormal basis (ONB) =l in H<; and write down
5.2. CAMP ACT LIE GROUPS
109
the following generalized propagator <<<
= £E l iJr ^* (i (/ /,' *t;;/', W(t K {l,t;l\i') ^ *0/ ) ==W ( t ~- f f) ) d edetr[Uj* t r [ U j * 0 Dj ^ ( J(J00 ] *H (M;J',0 = H(C 3J ==1iI j=
_1 = U{t-t')d W(t-i')^Xc(5 (0ff(O). a^g-\l)g{l')),
(5.8) (5.8)
where xcGrHOffG')) s t r ^ * ,^%^[l^ ^,,,]. Lemma 5.2.1 The propagator A'JJ (/, t\ /', t') given in (5.8) correctly propagates all elements of any reproducing kernel Hilbert space l£{G), associated with the irreducible unitary representation U ,» of the compact Lie group G. Proof. Let T] € H( be arbitrary, then for ipv{l',i') g L^(G) one has (l,t;l',t')^(l',t')dg(l') (l,t;l',t')^(l',t')dg(l') ,{l',t')dg{l') jj/[' H , K KKH( niH{l,t;l\t')i>S'^)d9i}') 1 = /ju{t= |<w ( t - 0*x<(*-t')dad9-\iW))Ui',t'y9{i') (»M0)iW.*W)
c , = £d<w(< -1') J2<w J2cU{i w --*') -t')o//Ito, /(( to> ;3{l), u< f tu.U^U^JW), % ^ «{ll)u^u^Mvnmw) '
^
"4
d {V>
SUJ
j,n=\
T
£
( U(t = W(t ZY(t - *'t')(y/d^U )< (MtnMt')) g(l) Y, {n,r))4>n,W))
n =l n=l n=l [Cvexp[~i(t-t>)H(X [C,exp[-»(t-i')?t il v k)WW)
= [c,exP[-i(*-OwTOMO](0 =
Mht). = Mht)Therefore, V>„(U) = =■ J/ KH((l, t; I1, *>„(*'. t')dg(l') V77 e H c ,
110
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
i.e., the propagator Kj$ (ltt; l',t') propagates the elements of any reproducing kernel Hilbert space L^(G) correctly. □ Hence, we have succeeded in constructing for the irreducible representation U ns a propagator A'u that correctly propagates each element of an arbitrary reproducing kernel Hilbert space L^(G), i.e., we have succeeded in constructing a fiducial v e c t o r i n d e p e n d e n t propagator for a fixed irreducible unitary representation of G. Using the fact that the set { ^ } , ' l j is an ONB one can rewrite the group character x<(<7 - 1 (05(''))
m
terms of the matrix elements
D|.(/) = ( 0 „ Ujm4>j) of U< as follows:
xd9-\W)) E E 4(o4(0xds-\iw))=-E^ = 4(o4(0-
(5.9)
i,J=I
Therefore, A ' H can be written alternatively as h
>E
KK({l,t;t',t') =U{t - 0 £ d c 4 ( / ) 4 ( / ' ) .
(5.10)
1,3 = 1
In this construction the unitary irreducible representation U ,.. was arbi trary, hence one can introduce such a propagator for each inequivalent unitary representation of G, i.e., one can write down the following propagator for the left regular representation Affm of G on L2(G)
rf K(l, t-1', o0 E K(i, = £Y, AKHH{((1, (/,«;t;/',/',f)0==w(t U{t - o-EE o J2 E J2rfc4(o4(0c4(04(O')E C6G
(EG
Now it is well known from the Peter-Weyl Theorem that the functions y/d^D^l), ^D^(i), v7^
c ee G, d(,c, C G, 1i << ii,,jj << rf
5.2. CAMPACT LIE GROUPS
111
form a complete orthonormal system (ONS) in L2{G) (cf. [7, Theorem 7.2.1]. The completeness relation of this ONS is given by h
(g- (i)g E E dc4«4(0 ;4«4(0 =6MJ-'CMO), <eG'.J=i EE e
l
CeG"
_1 where the sum holds as a weak sum and <5e(<7 (/)g(/')) is defined as where the sum holds as a weak sum and <5e(p_1 (/)#(/')) is defined as
i
k k 6e{g-\l)g{l'))=^j-f[S(l l k -l' k ),
Se(g- (l)g(l')) =7(0 ^f[S(l : = 1 -l' ),
(5.11)
Therefore, we find as our final result K(l, t; I', 0t') = (-iVh = exp [-i(t - t')H(x t')H{xkk{-iVi,
/))] Se(g-\l)g(l')) I))} 6e{g-\l)g{l'))
(5.12)
This propagator, which is a tempered distribution, is clearly independent of the fiducial vector and the representation chosen for the basic kinematical variables {Xk}f—i- A sufficiently large set of test functions for this propagator is given by C{G), the set of all continuous functions on G. Hence, we have shown the first part of the following Theorem: Theorem 5.2.2 The propagator K(l,t;l',t')
in (5.12) is a propagator for the
left regular representation of the compact Lie group G on L2{G), which correctly propagates all elements of any reproducing kernel Hilbert space L^(G), associated with an arbitrary irreducible unitary representation t/^,„ of the compact Lie group G, ( £ G. Proof. To prove the second part of Theorem 5.2.2, let I/\j> and rj e H,;/ be arbitrary, then for any rpv(l) in some L^(G), associated with £/;,;, one clearly has that i/)v(l) S C{G). Hence, one can write
J[ K{l,t;l',t!)4, (r,t')dg(l') K(l,t;l',t')ipv(l',t')dg(l') v
y/d(i =E £ KKtiWWiyfcufwvMi'MgV [ K^l^l'My/d^uf^^))^) C6G
/
112
CHAPTER
5. REPRESENTATION
-J
PROPAGATOR
KnK(lH(l (l,t\l',t')MW)dg(0 (l,t;l',t')Ml',t')d9(0
= jJ =
INDEPENDENT
^ C ,(M).
The second equality holds since the elements of different representation spaces are mutually orthogonal, hence, only the £'-term remains.
In the last step
Lemma 5.2.1 has been used. D Hence, for any compact Lie group G we have constructed a propagator that is independent
of the chosen irreducible unitary representation of G. We call
this propagator a representation i n d e p e n d e n t p r o p a g a t o r for G. We will see in § 5.4 that the representation independent propagator for compact Lie groups can be given the following d-dimensional lattice phasespace path integral representation (see Proposition 5.4.4 or [99]): K(l",t";l',t') K(l", t";
\G\ |G| I', t') = = . = limlim I... i...ly ly
/
VWW)
J
iv)
ID
•I
N
x exp I iiY^lPj+i/2 ^b;+i/2 ■ • Cj+i ) - ~eHiZkiPj+i/illj+iJj))] Ci+i- - 0h) eMixkiPj+i/illj+iJj))]W)] >> ,=o N
s x J[dl}...dlt,
(5.13)
xjl' )"4 where lN+l = I", l0 = I' and e = (t"~t')/(N
+ l). The sum1 £ { p } appearing ^{P}
in the above expression is defined as .
E ^ K= E ^ i
=
A7
{p}
/2 PN + 1/2 PN+l/2
1
E •■]{r E 1^J(1^> '-*K KE'E '
K E
P PN-l/2 PN-1/2
1 .
' A7-
P3/2 /2
Pl/2 Pl/2
the sums are over the spectrum of the operator — iV;, denned in § 5.4.2, and where K is the appropriate normalization constant such that g| , |\G\ KVWW)P1 KJWW)
Y ' Pi
Pit
k l exp V - l'kk)) -= =6Jg-6e(g-\l")g(l% (l")o(l')) rPJkpdl" (z"*-/'*) *p «*E =i . jt=i
5.3. EXAMPLE:
THE SPECIAL
UNITARY
GROUP SU(2)
113
The arguments of the Hamiltonian in (5.13) are given by the following functions:
xdv ,•/■ ik(p]+i/2;h+i'h) I
d
M
=£
- f ^,p-\' *{g{lm]+l))^ )±UTT_F ) ++ < ' Vp-\W{g{l)])))_ , "P m
m
2 ^
/clPj + l/2>'j + l , ' j j ~ 2 _ , m=\ m=l
mJ
+ l / 2 >,
P m J + i/2>
^
-, fck=
« =
! , . . . , rf <*.
1, . . . , 0 .
Remark 5,2.1 Observe, that the lattice expression for the representation inde pendent propagator exhibits the correct time reversal symmetry, which means that K{l",t";l',t') K(l",t";l'J')
= K(l', K(l',t';l",f). t'; I", t").
Also note that in the construction of the representation independent propagator for compact Lie groups and its path integral representation no explicit use is made of the ONS s/kD\.(l), ^/d^Dctj(l),
(C £6 G and i,j
= l,...,dc,
in L2{G)
whose
existence is guaranteed by the Peter-Weyl Theorem, but merely the facts that it exists and is complete are used. Moreover, we have made no assumptions about the nature of the physical systems we are considering, other than that its Hamilton operator be self-adjoint. Therefore, the path integral representation (5.13) can be used to describe the motion of a general physical system, not just that of a free particle, on the group manifold of any compact Lie group and it does not matter if the D^-(l) are explicitly known or not. Hence, (5.13) represents a clear improvement over the path integral formulations describing the motion of a free particle on a group manifold presented in chapter 3. O
5.3
Example: The Representation Independent Propagator for SU(2) € G ^ 4 ( 0 , ((eG
While the Peter-Weyl Theorem assures that the ONS y/d^D^(l),
and
i,j = 1 , . . .,df exists and is complete, the construction of such a set is frequently
114
CHAPTER
5. REPRESENTATION
INDEPENDENT
a difficult task. The functions ■J^D^Al)
PROPAGATOR
are known only for a limited class
of groups and will now be constructed for SU(2). It turns out that this is an exercise in harmonic analysis. We will now explicitly describe the maximal set of commuting operators in L2{SU{2)).
We will take the set of infinitely
often differentiable functions, C°°(SU(2)),
as their common dense invariant
domain. Since SU(2) is a rank one group, there exists one two-sided invariant operator C\ in the center of the enveloping algebra £ of SU(2).
Moreover,
since SU(2) is compact the maximal set of commuting right (left) invariant differential operators in the right (left) invariant enveloping algebra £R
(£L),
can be associated with the Casimir operator of the maximal subgroup U(l) of SU(2). Let Si, S2, and S3 be an arbitrary irreducible representation of the Lie algebra su(2) by self-adjoint operators satisfying the commutation relations 3
[5,-,5j] i [Si, Si]:=- i)
^ijkSk-
t(ijkSk-
k=i
Since the Casimir operator of SU(2),
3
2
= £T; = i Sj, 9 commutes with all the C\ =
generators of the Lie algebra su(2), its eigenspaces are invariant under the Lie algebra, and all vectors of an irreducible representation must be eigenvectors of C\ with a fixed eigenvalue.
We denote by H ( the (2£ (2( + l)-dimensional
complex eigenspace corresponding to the eigenvalue C(C + 1), C = 0> 1 / 2 , 1 , . . . . We denote the pairwise non-equivalent irreducible representations of SU(2) on any of the H<; by £/<. One can show that every irreducible representation U of SU(2) is equivalent to one of the representations U^, C f = 0 , 1 / 2 , 1 , . . . , (cf. [105, Theorem III.5.1]).
5.3. EXAMPLE: THE SPECIAL UNITARY GROUP SU(2)
115
For SU(2) in the Euler angle parameterization an arbitrary unitary irreducible representation of SU(2) is given by C U = ex J7g(8,*,t) exp(-iS exp(-iC53 ), P(-^- s , 3)exp(-tfl52)exp(-iC5 3) exp(-i9S2) 3 ),
where the domain of the parameters 8, <2w, O<0<2?r,
-2TT -2-K<£<2K. < f < 2TT.
With this choice of parameterization of SU(2) the operators {5/t}^=1 defined in Lemma 5.1.4 (i) are given by: si(—ide,—id,f,,—id£,8,<j),£) = i sin 4>dg + i cot 8 cos 4>d^ — i cos 4> csc 9d^, si(—idg,—id,f,,—id£,9,<j),£) si{—idg,—id^,—id^,6,4>,^) — — S2(—idg,—id^,—id^,9,<j),^) —icos^dg icos<j>dg + +icotOs'irKpd^-isiiKficscOd^, icot^sin^S^ - ism,(,)
= = -id,/,. -ify.
(5.14)
By Corollary 5.1.5 these operators can be identified with the generators of a subrepresentation of the left regular representation of SU(2), (i.e., belong to the right invariant Lie algebra of SU(2)). Similarly the operators {sk}l-l
defined
in Lemma 5.1.4 (ii) are given by: si(idg,id,£) = i sin fdg — i csc 9 cos (,3$ + cot 8 cos £8^, Si(idg,id,£) S2{idg,idj,,id£,8,4>,£)
= icos£dg + icsc8sin£dj, -
S3(idg,id$,id$,B,el>,£)
= id(_, i%
icot8s\n£d£, (5.15)
and can be identified with the generators of a subrepresentation of the right regular representation, (i.e., belong to the left invariant Lie algebra of SU(2)). From (5.14) and (5.15) we easily identify the Casimir operators of U(l) as Ax = id* Ai id^,
Bi B\ = id/:. id{.
(5.16)
116
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
For the Casimir operator of SU(2) one finds
1 dC, = = -- (( 11 -- zz22))dd22 + + 22<9 2zdz 2-- Y1 -^2(2^d l --
2 & ^ ++ 9a|), « )•
2zd
5J7 ((5.17) )
where z = cos 8 and the identity - sin 6dcoso = dg has been used. Since C\ commutes with all elements of the enveloping algebra £ and Ugig^t)
1S
irreducible,
C\ is a multiple of the identity on any one of the reproducing kernel Hilbert spaces L2(SU(2))
associated with the irreducible representationn U ,g , ,,, i.e., ^(SAO' C i ==CC(C+ ( C + i)J l1)4^ ) % s{SU{2)) y(2)) C\ C(C
(5.18) (5-18)
Let = _/ be an orthonormal basis in H ( , then we can associate with Let {V'n}„ {VUL-< each irreducible representation U^,e.,\, where £ = 0 , 1 / 2 , 1 , . . . , the following matrix elements:
£ >nL(0,4>,Z)= ( M , 0 = (i>m, (tm.U^^n), l7j (Mi< )^n), Di
-( < < m,n rn,n < < Q. (. "C
We shall now determine the matrix elements Z)ron(#, ,£) as the common eigenfunctions of the operators A\, Bi, C\. Equations (5.16) and (5.17) suggest that the common eigenfunctions of the operators A\, D\, and C\ are of the form
Z& e-i(m^+n() ^ *e-'^+^Pi + ^ j r t B ( cno(cose). s«). nia{e,,0 n (0,*,O = = Using this form of: Dmn{0,4>, 6) m (5.18) one finds £4»(M.0i
_(1 - ( 1 - z2) 2 2 ) - 2^ - + + 2z 2 2+— +
m 2 ++ n2 _- 2mn2) - ^~z idzdz r\1h—- {2^z-{m " 2mnz)] pi£„M ™(z) ==c(cC(C++1i)J*,(*). ) p -( 2 )dd'-z 2 2
2
l
The functions Pmn(z),J, which as the Wigner functions, are given by W1UU11 are c U t known H.UUW
;C-m)!(C /(C - m)!(C + m)!, m)! _ 2J ^m2—n g Z /(C-m)!(C+^)! (m -„. m+n) / z.. :^7 — rp{m-n,m+n) m -a-*: 1( 1 j U(l++ *) j) F C-m 2 yV (C(C-n)!(C ;C-n)!(C + + n)! n ) ! * m Tnn\*'J *„W- 2m )!(C + n ) ! * <-™ (*)>\ )i B
_ 1 PPC - {z) (z)
5.3. EXAMPLE: THE SPECIAL UNITARY GROUP SU(2) where P^™~n'm
117
(z) are Jacobi polynomials, (see [105, p. 125]). Also observe
that Pmn(z) - (~l)n~mP^ Pmn(z)n(z).
Therefore, one finds for the matrix elements
of the irreducible representation U ,g , f> the following: n — g—«'(ra^+ 0 Din(e,4>,£) = e-'^^Pi n(cos6),
as pointed out above these functions form a complete ONS on L2(SU(2)). The completeness relation for this ONS takes the form, see (5.11),
E E
w
l)D(mn(6",f,Z") ,« )0L(O'.e')
(2C +
<e<3 7i, n = — (
16TT2 ,
sin 8"'&(e"-e')6("-^')6(C-n
By equation (5.12) the representation independent propagator for SU(2) is then found to be: =
K(l",t";l',t')
2 16TT2 exp[-i(t" 0, S 2 (-iVj, I), 5 3 (-tV/, /))] 167r expH(i"-- t')H(h(-iVh t')H(h(-iVl,OM-iVl,l)M-Wul))]
X e W Wr -W - #)6{4>" -
?),
where / = (#,,£) and V; = (de,d^,,d^). By Equation 5.13 the regularized lattice phase-space path integral representation for the representation indepen dent propagator for SU(2) is given by A'(0",tf>",£",i";0',^',O K(e",4>",e'J" ;e',4>',e,t') = =
2 1fi 167T2
! tff ■/ ••••■ i&L •••/ /£/ i6" sin ■ 8'aiai11m i&L / w-oc/
N N N
a 1 EK+1/2^+1 *;) /■ P^ EK+i/'t^ - -**) 'sin {c,/3,-y} J=0 Vsin 8" sin 0' N-><x> j J ,„ , ~L +^+1/2(^+1 - 4>i) + lj+l/2(Cj+l lj+i/2(Cj+i ~ Zj) tftiskiPj+ifrh+i'h)}} tftiskiPj+ifrh+i'h)]} • • J — ■
1
N
x TT[ d9jd4>jd£j, d9jd4>jd(ij, 3=1
■E E
»Ei
exp{i ex
118
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
where, *i(Pi+i/2)'(j+i»'j)
=
) aQr/ii+i/2 - n2 ( s i n 0^ ji i+i + i ++ sssin0 iinn0^Jj-) ++ 1i / 22 --- -(cot0 - ( c o t 00]JJJ++++ 111l COSj)PJ+3+ COS 4>j+\+ COtfljC0Sj)P COSj)P i/2 COSj)Pj+\/2 : J +i/ -(cot0 COSj c s c 0 J ++ i1 + -I- COS cos 4>j csc0j)"/j+i/2, I j+ii+i CSC0J+1 + -(coSj csc CSC0j)"/j+i/2, +
h(P}+i/2',lj+uh) '■}) = h(Pj+i/2'Jj+iJj) =
22 (( cC o0 Ss ^^ +i +1i // 22 ++ Cc0oS s^^) )a aJ i++11//22 -- - ( c o t 0Oj+i <j>j)Pj+i/2 ]+i sin j+i + cot Bj sin <j>j)Pj+i/2 + csc0j0jsin sin 4>j)lj+i/2, 4>j)lj. + -(csc 0j+i 0 j++i , sin 4>j+i 0j j+\ ++csc 4>j)lj+i/2,
h(j>j+l/2'ilj+lJj) h(Pj+i/2',h+iJj)
= Pj+1/2Pj+1/2-
As an example let us calculate this propagator for the following two Hamilton operators (i) 0 , 3 aa((-tV,,0a3(-tV,,0)=5jS3. -«V,,0a3(-»V,,0) 3 (i) H{h(-tV,, W(5,(-«V,,0,5
2/
(ii) H ( 5 1 ( - t V ;l,o.s2(-^i.O»S3(-*'v*,0) / ) , s 2 ( - l V i , / ) > 5 3 ( - » V / ! 0 ) == (u) w(gi(-iv,
5.3.1
^(sl + s3 1? ++ 5i)si)^(Si
T h e H a m i l t o n O p e r a t o r• W ( s i , S 22,,S § 33)) = ^^jig W(si,S S^
As announced in chapter 3 we now revisit the free particle moving on a circle and present its exact path integral treatment. The Hamiltonian W(ih., ib, %) = P2/2I
describes a free particle moving on a circle with fixed axis, like a bead
on a hoop. We analyze this problem in two steps. First we proceed naively, assuming that the Hamilton operator is self-adjoint. In particular we assume that 53 = -id$ is self-adjoint on L 2 ([0,27r)) and has a spectrum of the form P = n,n = 0 , ± 1 , . . . . Then in a second step we reexamine this assumption and show that 53 = -id^, self-adjoint with spectrum P = n is only one particular choice of
5.3. EXAMPLE:
THE SPECIAL
UNITARY
GROUP SU(2)
119
uncountably many. With this choice of Hamilton operator the representation independent propagator takes the form:
K(6",»,e',t";6',\e,t') ",iM";6',<j>\e,t') 16TT 2 .
/
*-f ./J/ { E ~} e*—f a,/?,7}
■
V sin 5" 6" sin 5'oaN-oo J y Vsm6"sm6'N-, J
ex
Q il/2(0j 2 +l p{ i*—' Et f^0 j+ / (
- 8j)
i=°
/32
Mfj+i + /3i+l/2(^/'i+i +Pj+iMh+i to) M+7i ++ 7;7. 2(o l/2(6 f,o fi) - - 4^^^~]11}) nI I <^^ < *^ « i+l -" to) +1/+ +i + l --" fi) 3=1
+l
^C ^ »-_I^^) 5-(O _ n! -UL rn [...[(±-Y = i^ f»J = IY*' sin6»" ^ ^ i V - WI - K 7 V27ry V
;
;
°°
oo
xx
00 oo
E XI ••• ••• •• •E EE
/3w+^
= - o00 o
C N AT
e
(_ j ' = 0
*
[
ex e xexp p
/Pi ? ^=—00 =-oo
i 1 NN
1
d [ft+i/aWi+i- -to)*i)~ -2 /2/4n/a] 4 n / a ] f fI II dI^ '^'p VVE E [ft+i/aWi+i J i=l
This last integral al can be evaluated evalu as follows:
m
[
f (
1 \\ WNN +++ l1l
■Kir
jN™ao v l J o' o" J/ - (iir) i l d.-)J N->ca \Zlt ^°°-/
V
y
JV n TT
e o2
^ + l /72]} 2]} f ~ 21jPj+1/2 /
-I \N+\ \AT+1
N->oo\Zir/
E
ex ex i A/ i PP WN+I/2^" •' •••E•EE/ //•'-•7-• //~ p{W ^ /+W-^P1 W T—^ /
a ftv 1/2 l/2 0NV+++1/2 PN
a JJ ft/2 ft/2 ft/2 01/2
/
, ., „
/ _ , *>(/W V /2 - ft-i/a) Mj-1/2; - ^ -~ E * j>^ i(+/lw -
2 /
i3=1 =i 3=1
i &O 22aT X) T T E*'" ^ ^°° N N
6XP 6XP
ftv+1/2 ftv+1/2
i3=0 =0 3-0
I. (.
i 2! j8(0"-0 - ^ ] ]}. = = Is E-p{ E ^ •l{i'k"-^-^ «*"-*''-§" } - }• 27T
' 0 /8== - o o
exp<
k
L
,i
N
i == 1i J 3=1
1 ii^+ ^+11/2 /2<<^" ^" --ft/a*' Pil^' ~- ^£r E E ^i+1/2] ^i+i/J ff
/^HP * InI ^ >spj+wpi-v2 ,
n
/^Vj+llllS 1 ^2+1/2]) 1III E #H/2» I d"**i^^
X
1
,//
JJ
N
ft/2 01)2
jJ==00
/■
E-E/-7E
«—^
1
1 22 p^ p{*E^;+i/ ^;+i-£ P {T, E '+1-^) T ^+ / ^
,i
N
= =
N
e ex xe x
„ ftv+i/2 == = -~°° -oo /3, / 2== -0O ftv+i/2 o o ft/2 oo 0N+l/2 ft/2 =-~0O
/•
N+l I \
= ^lim^ ( — J
00 oo
E ^X "•„ E E X) ••• ^—' *—'
i-
= ^{i) /
00 oo
3=° J=°
J)
120
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
Hence we find for the representation independent propagator -6(9" - o>) 9') 6(£" K(e",",e\t";e>,) = -^6(0" sie - - ^')o sin 9" 8TT
K{e"A\Ot";e'A'^,t')
ntr9" sin
T
oo
2 \ e x p j i n(4>" - ')- —n XE f; expjzL^"-^')-^ Yin ]}. n = —oo n= —c
^
2
'■
The sum over n is related to the Jacobi theta function, oo OO
exp(i7rin E exp(i7r£n
93(z,t)
— \] =
2
+
2inz).
n —oo n= = —oc
Therefore, with the following identifications we can write the representation independent propagator in closed form. Let
t=
-T 2TT/ 2^7
'
Z
__ (4>" (4>" --- f) f) -
2
'
then our final result for the representation independent propagator becomes
, K(8", 4>", f, t";;W,M *', 0', e, 0 == - ^ 7- 5(0" W" - 9') O 6({" *(e" W,r,?',< " sin 0" 8TT
sin v"
[(£-#) T 0" ~ 4>') - -T] OO:h \{-^-^-, Oh ~ [ 22 '2^7. 2TT/ •. i
I
Znl
This result agrees with the one found by Schulman [95] expect for an arbitrary phase factor. We now follow our analysis in [101, section III.c]. It is well known that the symmetric operator -id* on L 2 ([0,27r)) with domain f2?
(W22 ++ |V>f)d0 |^'|2)d0 < < oo, 0 0 , ^V(0) (0) = V(2TT)} D(-id<,) = {i> = 00 == T/>(2TT)} D(-tty) {V> : / \\M
'JoJo
has deficiency indices (1,1), hence, it can be extended to a self-adjoint operator but not uniquely. In fact there exists a one-parameter family of self-adjoint extensions which we denote by -id^ -idt = = -id* -id* with the domain r ' 2 7T
D (-t9|) = = {V D(-t9|) {V :
/ \\ip\ (|V|22 + + WfW | ^ T ) # << oo, V(2TT) = e'V(O)}, Jo 7o
5.3. EXAMPLE:
THE SPECIAL
UNITARY
121
GROUP SU(2)
where 6 € [-■*,*) (see [88, pp. 257-259]). Note that the choice 6 = 0 corresponds to the case of periodic boundary conditions, which we have assumed above. The spectrum of each — -idiid\ is straightforwardly found as follows, let A 6 R then s
-id M) =A^#) -idii>{4>) this implies that the eigenfunctions are given by ib(d>) M<j>)= =eiXe*.iX+,
Fitting the boundary conditions ip(2ir) = e l S ^(0), yields the following set of eigenvalues,
K)
Tl A„= ^J K = ( +2^
nn £ = {...,-2,-l,0,+l,+2,...} {...,-2,-l,0,+l,+2,...} Z = 6 Z
Therefore, the spectrum of — id^ is given by
2* ("£)
{
= jlA\ : :A\== s p eecc((--i di ^^ ) =
*}
(n + — j , 6 5 £ 6 [-%, [ —7T,7r) ir) and nn ec Zz i .
Hence, the choice of periodic boundary conditions is only one of uncountable many possibilities.
If we choose instead the boundary conditions ip(2ir) =
e%sip(0), \where 6 £ [—7T,7r) is arbitrary, our expression for the representation e*V(0), e'V( independent propagator becomes K(0"d>"f"i":0',d>',t:'j') K(e",",e\t";9',
sin 0" sin a"
8{9" - 0') «5(£" - e') 6\2' T_W - ~ (n 4- — ^ ~ 21 JJ^J
x£-J«[(.+ ±) w ^-|:(. + ±)1} V
exo exp
n=-oo n = — oo n=-oo 0~
( n + — ^ f<6» - ^
^V LL
= sinJ^^-o^'-nexpfifc^-^l 0" I 2n siv'I V. L
XiMA" _ AA
,T^2T J ) 8TT2/.
oo
n W ex] ) I i < » x- £• f;exp(.-{ - t f - ^ - 27" B [(^-0')-^]-^}) nn=-oo = —oo
122
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
Therefore, with the following identifications we can write the representation independent propagator again in closed form. Let T -T '' 2*1 2x1
tf===
T6 _ {£ - 4>') T6 = ixl 2 ' 4nl' ~ 2 4x1'
Zz
'
then our final result for the representation independent propagator with arbi trary 6 becomes 8TT
a K(e", '\ e, t"; e', 4?, e, sin 0 6"-8(6"-0'), = -^w9') s(e -*')- *«* K(e",",e,t";e',',e,t')
j^-mt xel
o
■*H_ix£_] .
(6" --4>') \(" -d>')40 T8T, - T " \e 63u3 — [—2 S7'277. • [2
9!
4%I'2TCI 4TT
'
This result exhibits the same 0-dependence as does the one found Schulman [95], which also encompasses all spins.
5.3.2
T h e H a m i l t o n O p e r a t o r W ( s i , s 2 , s 3 ) = ^j(sf 2l(Sl + + ss^ 2 ++ s i3f))
Our second example is that of the Hamilton operator 7i(s-i, 52, S3) = C\(9,4>,£), where C\(9,4>,(.) is the Casimir operator of SU(2) given in (5.17). Note that the Hamilton operator 7i(5i,S2,S3) = C\(9,4>,£) is essentially self-adjoint since C\{0,,^) is a symmetric and elliptic central element of the enveloping algebra £ of 517(2), (cf. [79, Corollary VI.3.1]). This Hamiltonian describes the motion of a free particle on the group manifold of SU(2).
With this choice of the
Hamilton operator the representation independent propagator becomes:
K{9",4>",e,t"^A',i',t') = =
1 116TT 6 7 rV2 e£- £c 'C(>f lC" "• *■*"•«")" • « " ) ^ 7-6(9" * ' ( 0 " -- 9')6(4>" e')6{<j>" - - $>)§{(" o 4>'W -- O ;in 8' Cc , 1c 9 2 »,H ^ ' 2 ; >< "-*"-<"> l6 -'SCi(«".*".f") £ £ m + + i)fli«(«",^,f)i?L(^^,0 l)Din(6»,ci>»,e>)Din($'\ct>>,ti') 167rV'& T C )/&„(*",*",£") (2C CeG"--
E
5.3. EXAMPLE: = =
THE SPECIAL
2 167T 1 6 r 2 ^3(2C+l)exp ( 2 C + l)exp
7
UNITARY
GROUP SU(2)
123
' .T 1 C+D [-*^JC«(C l)\xd91)1 X c(9",4>",t"W,4>',e)), to^CV.fMO'.O).
where T = t" - t' and equations (5.8) and (5.18) have been used. Here x<(s) denotes the character for the representation b'(.,,,,
i.e., if one denotes the
t g-\o"\r> e'w,',e)byby(0,<£,() (M,Oone
Euler angles of the element g-1(0",",{")g{6',4>',Z')
xMl*,t)) = X<(ff(M,f))=
finds:
ci £
0}Pim(cos6) eexxPp[-im(4> [-.m(^ + + 0]P4m(costf)
m=-<
Observe that, the character of the group can be expressed as a function of a single variable as follows. It is well known that the character as a function of the group is constant on conjugacy classes, i.e., for any two elements g and <7i one has OiOO.l) M = xd9\ggi') xd9\ggi = xdg)
Therefore, to show that X((ff) is a function of one variable, it is sufficient to show that the conjugacy classes of SU(2) can be labeled by a single parameter. As is well known from linear algebra any unitary unimodular 2 x 2 matrix g can be written as g — <7i7<7i_1, where 51 £ SU(2) and 7 is of the following diagonal matrix:
{ nr/2) _ (/ • eeeW W 7 = V 1 ~ \ 00
00 e-fPV'J e-fPV'J
e ->(r/2)
\ ^) )
Furthermore, among all matrices equivalent to g there exists only one other diagonal matrix 7' obtained from 7 by complex conjugation. Therefore, each conjugacy class of elements of SU(2) is labeled by one parameter F, ranging from -2K
< T < 2TT and where T and -F give the same class. Hence, the
characters X ( ( P )
can
be regarded as functions of one variable F that varies
between 0 and 27r. The geometrical meaning of the parameter F is that it is
124
CHAPTER 5. REPRESENTATION
INDEPENDENT
PROPAGATOR
equal to the angle of rotation corresponding to the matrix g. In terms of the Euler angles (6", &',£") and (0', <£',£') T is given by T T
= =
a r c c o s [ c o s ( 0- " B') - 0 ' )cos(<£" c o s ( 0 "-- <')sin(£" sin(£"- - (')] £*)]
One can derive an explicit formula for X((d) the matrix U / 0 rrj)
t n a t corres
as a
(5.19) (5.19)
function of I\ Note that
P o n d s to 7 6 SU(2) is given by the diagonal
matrix of rank 2( + 1 having diagonal elements e~ ! a r , —( < a < £, Now let 9 = 5i75i~\
then
c X<(ff) xc(ff) == [^ [^(o,r,o)l ( o,r,o)] = E
„ c„i
t.[r/<
1
tr tr
V"; e„-«r e-»r
E a=-C a=-<
=
ssin(c+ i"nI S( CT-+ •■! li/2)r / 2<■) )r ■ ssin i n r T/2 /2 '
Hence, the group character can be written as sin(C+l/2)r
2)r l xd9-\e"A",i")9{e'A',0) '' x((s(«''f»nK«,^'f)) = sin(CsinV/o r/2 T/2 sin 111
where T is given in (5.19). Therefore, one finds for the representation indepen dent propagator
K^W.fW^'.M
K(0",<j>",e',t";6',4>',e,t') 7 -1 c
~
16^
2
r
r
E
N^
Et
1. „ = .lim / . . . / Y ^ exp{i exp{iy^[Q,+i/2(0,+1 6" «-o /J }.*-?. ^ ' J+i/Ji W Vsin 6»" sin sin=6" 6" t—0 / {a,P,l} Vsin 0" sin 6" « j=0 + PJ ^ ++11//22{i) + j)++ 7j + l / 2 ( G + l N - eC x (0 i+1 , 0 i+1+ i,0, i+fc+1; *,-, ^ , fc)]} Yl eCi($j i,Ci+i;0j,
9,) J; &)
ddjd^d^ d0jd
5b(c+ ' (t" f i " -- i'l0 1 sin( sin(C+ 2)r + l /i/2)l ] ( 2 CC++1) l ) e x p„ [-^Q«c - i6»» S(2 + D1 .ffft 21 '} sinT/2 16TT 2
n
C€G.
^J
s
sinr/2
5.4. GENERAL
LIE GROUPS
125
This result agrees with the one found by Schulman [95] which was obtained by the methods mentioned in chapter 3.
5.4 5.4.1
The Representation Independent for General Lie Groups
Propagator
Construction of t h e Representation Independent Propagator
Now let G be a general Lie group. Let us again denote by U an arbitrary, fixed, square integrable unitary representation of G. Then it is a direct consequence of Lemma 5.1.4 (i) that for any ip G D
5fc(-iVf,0M](Q = [c„x lCvXkW), s*(-tv,,/)[c,,tf](0 k =l,...,d. \,...,d. ki>](i), k = holds independently
of n. Therefore, the isometric isomorphism Cv intertwines
the representation of the Lie algebra L on H , with a subrepresentation of L by right-invariant, essentially self-adjoint differential operators on any one of the reproducing kernel Hilbert spaces L^(G).
To summarize, we found in § 5.1
that any square integrable representation U of G is unitarily equivalent to a subrepresentation of the left regular representation A on L^(G).
Furthermore,
the generators of G are represented by right invariant, essentially self-adjoint differential operators on
L^(G).
Let ( T , H ) be a representation of G, then we denote by A(n) the von Neumann algebra generated by the operators itg, g G G. By Proposition 5.6.4 in [25] there exists a projection operator P ; in the center of the von Neumann algebra .4(A) such that the restriction A/ of A to the closed subspace
P[[L2(G)}
of L2(G) is of type I, and such that the restriction of A to the orthogonal
126
CHAPTER 5. REPRESENTATION
INDEPENDENT
PROPAGATOR
complement of Pj[L2(G)] has no type I part. Since G is separable and locally compact there exists by Theorem 5.1 in [30] a standard Borel measure i> on G, the set of all inequivalent irreducible unitary representations of G, and a ^-measurable field (P^,Hf),e(A, of unitary representations of G, such that the type I part of A, A/, can be decomposed into a direct integral,
L
A/== [u<®I dv(Q, A/ G( (dv(0,
i fG G xx GGoon n H^ ncc® where U^ ® /,; is a representation of ® He. H(. Denote by H(X k) the essentially self-adjoint Hamilton operator of a quantum system on H^. Then the continuous representation of the solution to exV[-i(t-t')H(X )}iP(t')1 takes, on L2{G), the Schrodinger's equation,,iP(t) ip(t) = exp[—i(t — t')H(Xkk)]i>(t'), following form:
-I
Mht) == J KKnn{l,t;l\t')%{l',t')dg(l') {l,t;l',t')i>n(l',t>)d9(l') Mht) where Kv(l,t;l',t')
= = =
(r)(l),ex-p[-i{t-i')n(Xk)]rj(l')) (iK0,™pH(t-OW(*OMO>
= [Cv exp [C{C nex vexp{-i(t-t')n(X ?[-i(t-t')H(Xk)]T,(l'))(l) k)Hl')}(l) = U(t-t>)[C - t') = U{t u(t OM nr,(l>)](l) OK0 1/22 = U(t - f) U(t-t')(v,K U m-t'){riJOl UQg-Hl)a(n 10l^) v) i(l)g(v)IOn
= U(t~t')IC U(i~t')fC n(l;l'), n{l;l'),
(5.20)
where U[t - t') - t>)7i(xk(-iV,,l))] n = ex exp[-i(<-OW(£fe(-iV,,/))] ?[-i(t and A'J[/s-i(;)s((,)A'2 denotes the closure of the operator
K?U.-in\gn,\K%.
Note that for non-compact Lie groups it is not true that every symmetric Hamilton operator is also essentially self-adjoint, as was the case for compact
5.4. GENERAL
LIE GROUPS
127
Lie groups. To illustrate this important fact, we consider the following two examples: E x a m p l e 5 . 4 . 1 : Let G be the non-compact two parameter group of transformations x' — p~xx — q, 0 < p < oo, —oo < q < oo of the real line M. and let H = L2(IRt),
where JR^ = (0, oo). An irreducible unitary representation of G
on H is given by the formula: (Ug{p = p-Vh-^ftp-'k), p -- 1^/ 2Ve '- ^' ' p^ (- p1 -*^)) ,, gM,W) (u q)4>)(k) =
€ H. 4>en.
The generators of the one-parameter unitary subgroups are given by i U(X2) = "2
U(Xx) = k, We choose the set S(flit)
as
(kdki+ dk.ikd_k) •
the common dense invariant domain for these
operators. As our first example we consider the operator
T,
=
U{Xl + X2) --
D(Tj)
=
S(Rt). S(S(R+).
i
d
k «+\ 2 [k"dk dk +' dk dk ) '
Clearly the operator T\ is symmetric. To show that T\ is essentially self-adjoint it is necessary and sufficient to show that the kernel of the operator T* + il, ker(Tj* + il), consists only of the zero vector, i.e., ker(T'1* + il) = {0}. In other words we have to show that the equation: T*<j>±(k) = ±i(j>±(k), ±i±(k) has no solutions in H other than 4>±(k) = 0. One finds the following solution
128
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
for the above equation:
(H
2 k1/2expfc^expQfc * " ), ■
*+(*) ■■= Mk)
4>-{k) = «-(*) =
A:-
3/2
-p(jf).
2 ex]k'^exp^k * s ^.
Both of these functions are not in H , since they are not square integrable. The function \4>+(k)\2 diverges at infinity and the function \4>-{k)\2 diverges at the origin. Therefore, we conclude that T\ is essentially self-adjoint. As our second example we consider the operator T22
== U{X2X2<2 + + =
xX22xX22)
k
+ t + 1k = a*)* dk,/ - 'K'a+s^K'a 'K'sdk +ass* K(45 +*is') ' d
:
=
D(T») D(T 2 )
k2
d_ '
+
■(*:
[ \ dkj
t
2\
2
,('i) ' " I2\ k*
[\ dkj
+ -*te) (•£ '(«£)< (4)*
=
+ 5(iR. ). S(IRt)-
This operator is clearly symmetric, and one determines the following solutions for the equation T2<j>±(k) = ±i±(k):
i-^)>
* T 3 / 2 efc_3/2eX xp *+(*) *+(*) = 4-^)' V p(4*a ;' 4>-(k)
k-3'2e exp 2 ) • \4k V4*V'
Clearly 4>-{k) is not integrable since it has a non-removable singularity at the origin, however +(k) is square integrable, and hence, belongs to H . Therefore, even though the operator T2 is symmetric it is not essentially self-adjoint and can also not be extended to a self-adjoint operator since it has deficiency indices (1,0). O
129
5.4. GENERAL LIE GROUPS
We now proceed with our construction of the representation independent propagator. Let Q,/3 6 27(G), then put U(a) = U{a)
Ja(g(l))Udg(l), dg(l), 1°a(g(l))U
1 a*(g(l)) a*(g(l)) = = A( f f -A(g-i(l))a(g-\l)), (/)) a ( 5 - 1 (0)» a n d
gU) g{l)
define t h e ma
P V(G) x V{G) ^ (",/?) ->
a * /? € 27(G) aas follows: a*P
J
(a*P)(g(l)) a(g(l'))P(g-\l')g(l))dg(l'). (Q*/3)(5(0) = = J/ a(g(l'))(3(g-\l')g(l))dg(l'). With these definitions we find that IC £n(a,jfl) n(a,/3) =
f
ICv(l;l')a(g(l))P(g(l'))dg(l)dg(l') 1 1 IC /C,(i; l')a{g{l))/3{g(l'))dg(l)dg(l') v(l;l')a(g(l))P(g(l'))dg(l)dg(l')
(77, (A'i/2C/ ,)IO/ /^/22r,)a(g(l))P(g(l'))dg(. 1(0s(( r?)Q(5(/))/5(ff(/'))(iff(/)rfff(/') =-- jjJ jjJf (r,,KWU (r,,K^U IOnr,)a(g(l))P(g(l'))dg(l)dg(l') = VJO^U g9-_ i{l)g{n g-> mg-imil) 2 2 2 = y jJf (r,,K^U (r?, IOl U JO/^) IOl yj J[alg(l))p(g(l)g(l'))dg(l)\dg(l') a(g(l))(3(g(l)g(l'))dg(l) a(g(l))P(g(l)g(r dg(l') (v,IO/ UK^ IO/h g{ll) g(v) V ) V) oin = (r,,KVZU KV*v) [J a(g(l))P(g(l)g(l'))a gm
= ==
2 J[f It,, {77,lOnU^IOl^ia' K^U KVr,){a* (77, IOl Uma(i ,,IOl2ri)(a
** /J)(s(/'))«fo(f) /3Ml'))dg(l') 0)(g(l'))dg(l')
(■q,IOl {■q,IOl (rj,IO/22U{a**P)K U{a'*P)lOl U(af*P)lO/ll22r]). r1). )
Note that K,n(a,(3) is a bilinear, separately continuous form on 27(G) x 27(G). We call the bilinear separately continuous forms on 27(G) x 27(G) kernels on G. Also observe that ICv(a,P) is a left invariant kernel, i.e., IC(LBa,Lg0)
= fCv(a,(3), for every geG,a,0e g£G,a,Pe P 27(G).
Therefore, we can write (5.20) as Kr!{a,t;/3,t,)
- t'y n(a,/3). = u{tK{t-t')IC
130
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
In the above construction n £ D D(A ( A-1^^22)) was arbitrary, furthermore as shown x x2 2 l 2x 2 U{a)K [30, Corollary 2] 2] for a £ V(G) the operator ■KK l lU(a)K l l elsewhere [30,
isi trace
class. Therefore, we can choose any ONS {<}> {j}jeN inD(A' D(A'1 /1' /2')2 ) and andwrite write :}j£ivin 000 0
1 2
1 2
/ ( / ( a ' * / 3 ) A /l/2]}-. ICH(<*,0) = ] [ >£,*>>, ,// ?3 ) = ICH(<*,0)-=: t r [ A tr[lO/2U(a**P)K J = I
Note that /Cn(a,f3)
is a left invariant kernel on G, since each / C ^ ( a ; / J ) is a left
invariant kernel on G. Therefore, by Proposition VI.6.5 in [79] there exists a unique distribution 5 in T>'(G) such that fCu(a,0) 1 22
1
= S(a' */3). In fact we see
12 2
r [ A i / £/ [ /ss-i- (1,() W i / / ]].. Therefore, we find the following that S(ff(0eC')) = ttr[A' j ( / ), A ) A' ?(}. {l)g{ll)IO/ \. mil)K^}. AHH(l,( Mt; -I',/ ,00 =
(5.21)
Remark 5.4.1 This propagator is clearly independent of 77 the fiducial vec tor that fixes a coherent state representation. However, this propagator is in general no longer a continuous function but a linear functional acting on T>{G). We will see below that the elements of any reproducing kernel Hilbert space he in the set of test functions for this propagator. O L e m m a 5.4.1 The propagator K K'n(l,t;l',t') H(l,t-l',t') ,0 igiven in (5.21) correctly propagates all elements
of any reproducing kernel Hilbert space L2{G),
with the irreducible, square integrable unitary representation
associated
U„n\ of the gen-
eral Lie group G. Proof. Let n e D( A 1 / 2 ) be arbitrary, then for ^ ( / ' , t') e L 2 ( G ) one can write
I
J f KB
Kn(l,Pj',t')i> KH{l,t;l',t')W,t')dg(l') v(l',i>)dg(l')
5.4. GENERAL LIE GROUPS
131
I'
= Ju{t,)Ki/^n(l', t')dg(l') U{t - t')tt[KU2Ug_imit'MK^U^v^IO/^l'j'W) OO
=
E
I
I/2 C /, K1'2T1,i>(t')}dg(, /2 -Hl)g YtWU{t -- *') 0 / (4> ( ^ , ^3JO/>U ^ - . (gO P (n,' '))KV^ ^ 1 / 2J)(U ^ ) ( g{l 5) ( ' ' ) ^ ] ^ . V ' ( < ' ) > ^ ( O J
i=i
oo 00
== u{tU{t-t'){K" - t')(K"2U2ug(lg(l) J^tiWhW)) > 7 , nto, m) 1=1
=-=
[C [C vzxv[-i{t-t')H(X vexp[-i(t-t')H(X k))4>{t')]{l) k)}4,(t')}(l) i>r,(!,t),
where the fourth equality holds by Theorem 5.1.2. Therefore, 1 2
12 rh,(l, t)== Jh
■I
i.e., the propagator propagates the elements of any L^(G) correctly. D In the above construction the unitary irreducible representation Ug^ was arbitrary, hence we can introduce such a propagator for each inequivalent unitary representation of G. By Corollary 5.1 in [30] there exits a positive, cr-finite standard Borel measure u on G, a immeasurable decomposition (U f .,,Hf) f g A of A/, and a measurable field (/{'<;),_g of nonzero, positive, self-adjoint operators such that K
in H^ for
j/-almost all C £ G such that for a,/3 e Pi[V(G)}
I
l ,2 l2 (5)K\ \dv{(), S 6 (a**(3) * / ? )= = / ti[Kl tT[K\,2l2U<(a' U<{a- * p)K [a**P) e(a* c ]dv(Q, M
JG
is well defined; see Appendix A.3. Here,
JJ
6e(a'*P) 5e(a'*P) == J J 6See(g-\l)g(l'))a(g(l))P(g(l'))dg(l)dg(l') (g-\l)g(l'))a(g(l))P(g(l'))dg(l)dg(l'), and 8e(g~l(l)g(l')) is given in the chosen parameterization by d
1 k k k lk 6ee(g-\l)g(l'))-(g-\l)g(l'))=:±rfl6(lS(l -l>- ).l ). 6 7( 7(0 - '
k=l
(5.22)
132
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
Hence, we can write down the following propagator for A; of G on L
K(a,t;P,t') K(a,t;0,t')
=
(G),
(a,t;(3,t')d»(0 )«MO If JKjr («,*;/M H( He
,
«L
= U{t - t') ^/c = U(t-t')J KH< {a;0)dv(Q Hi(a;/3)<M0 =
U(t - t') 0 /[ K[K\ ti[Klcl/2*U<{a* U<(a* Jt JG JG
l , 2l7 *0)K *0)K\ ]dv(Q ( ]dv(Q
= U{t-t')L U{t - U(t-t')6 t')Se(a* * P). e(a**P).
Therefore, we find the following propagator for the type I part of the left regular representation A/: K(l,t;l',t') = «exP?[-i(t if(/,t;l',O = H ( « - t-' M * * ( - » V / t')H(x . 0 ) ] k^(-iV ( i fhl))Mg-\l)g(l')). -1(/)5(n).
(5.23) (5-23)
Remark 5.4,2 Observe, that this propagator is clearly independent of the fiducial vector and the irreducible, square integrable unitary representation one has chosen for G. A sufficiently large set of test functions for this propagator is given by C(G) n L2(G),
where C{G) is the set of all continuous functions
on G. Hence, the elements of any reproducing kernel Hilbert space L2(G)
are
allowed test functions for the propagator given by (5.23), and therefore, for the propagator given by (5.21). Therefore, we have shown the first part of the following Theorem: T h e o r e m 5.4.2 The propagator K(l,t;l',t')
in (5.23) is a propagator for the
type I part of the left regular representation
of the general Lie group G which
correctly propagates all elements of any reproducing kernel Hilbert space L2 (G) associated with an arbitrary irreducible, square integrable unitary U< ofG,CeG. u^ofccec. g{l)
representation
5.4. GENERAL
LIE GROUPS
133
Proof. To prove the second part of Theorem 5.4.2, let U m and T}(, 6
D(K\{2)
be arbitrary. For any ipV(,(l) € L* ,(
L
JG
K(l,t;l',t'W
= =
¥vl
(l',t')dg(l')
u(t-- <')^e(ff(/M/'))(^ 4 V> MO)f U(t e
1
K) 1/2/
U{t-t')[C v^c{t')){l) )[Cvtfc"
mx )\i
=
[ C v exp[~i(i - f )W(X*)]^ t')H{X c ,(O](0 k k)}^,{t')){l)
=
V> V (M).
Therefore,
■I K(l,t,r,t')1> M',t')dg(l'),)dg((%
i>n(,{l,t) = 1/2
n
D(Kf)1/2 for all r]^ G T)(KJ ) and any (' 6 G, i.e. this propagator propagates all 77,;/ G £ T)(KJ for all r]^
) and any ( ' 6 G, i.e. this propagator propagates all
elements of any reproducing kernel Hilbert space L^ ,(G) associated with an %{G) elements of any reproducing kernel Hilbert space L^ ,((?) associated with an u f;% correctly. □ arbitrary irreducible representation U
m
arbitrary arbitrary irreducible irreducible representation representation U U ff;;% % correctly. correctly. Hence, we have succeeded in constructing a Hence, we have succeeded in constructing a propagator for a general Lie group. propagator for a general Lie group. 5.4.2
□ □ representation representation
independent independent
P a t h I n t e g r a l F o r m u l a t i o n of t h e R e p r e s e n t a t i o n Independent Propagator
From (5.23) it is easily seen that the representation independent propagator is a weak solution to Schrodinger's equation, i.e., K{l,t-l',t') K(l,t;l',t') idttK(l,t;l',t') K(l,t;l',t')
= 'H(x (-iV1{-iV,,l),...,x l,l),...,xdn(x l,l))K(l,t;l',t'), W(il(-iV d(-iVhl))K(l,t;l',t'), 1 (-iV,,/),...,i l / (-tV,,/))A'(/,t;/',0,
(5.24)
Taking in (5.23) the limit t —* t' yields the following initial value problem: id K(l,t;l',t') ttK(l,f }l',t') idkK(l,t;l',t') K(l,t;l\t')
= =
H(xh1,Xd(~ (-iV (-iylh,l),...,x l),...,x (-iV d(-iV,,l))K(l,f hl))K(l,t;l',t'), ll',t'), H(x1(-iV l),...,x W(*i(-iV,,0.. d(-iV hl))K(l,t;r,t'),
134
CHAPTER
5. REPRESENTATION
\imK(l,t;l',t') lim Urn , K{l,t;l',t')
=
INDEPENDENT
PROPAGATOR
6e(g~\l)g(l')). a'1
(5.25)
Remark 5.4.3 Observe that the coherent state propagator given in (5.20) is also a weak solution to the Schrodinger equation (5.24). However, it satisfies the initial value problem idtKn(l,
t; I', t')
]imK bmfKvn{l,t;l',t') lim {l,t;lf,t')
=
W (f S oC i (--ttV V/, , I),..., / ) , . . . , xd(-iV,,
=
KnvW). IC (l;l').
l))Kn{l,
tt; /', I', t'), (5.26)
Therefore, we can write idtK#(l,
t; I', f) = Wtt(£i(-iV (£i(-iV (! ( ! 0 , -., xd(-rV,,
l))K#{l,t; l))K#(l, t; /', *'), *'),
(5.27)
where Kj. denotes either Kn or K. Note that the initial conditions, i.e., either (5.25) or (5.26) determine which function is under consideration. O We now interpret the Schrodinger equation (5.27) with the initial condition (5.25) as a Schrodinger equation appropriate to d separate and dent canonical degrees of freedom.
indepen-
Hence, / 1 ,...,/ < i are viewed as d "coordi-
nates", and we are looking at the irreducible Schrodinger representation of a special class of d-variable Hamilton operators, ones where the classical Hamiltonian is restricted to have the form H(±i(p,l),...,x H(ii (p, / ) ,d.(p,l)), . . , xd(p, /)), instead of the most 1d d general form n( H(pi,...,Pd,l ). ). d). ,l\...J Pl,...,Pd,i\...,i,...,l
In fact the differential operators given in
Lemma 5.1.4(i) are elements of the right invariant enveloping algebra of the ddimensional Schrodinger representation on L2{G).
Based on this interpretation
one can give the representation independent propagator the following standard formal phase-space path integral formulation in which the integrand assumes
5.4. GENERAL LIE GROUPS
135
the form appropriate to continuous and differentiable paths K(l",t";l',t') K(l",t";l>,t>)
n
d
=
m M ,/),M
'.•!•
n=l
It i1 J ] dl{t)dp{t), dl(t)dp(t), te[t',t"]
(5.28) where "pi",...,"pd" denote "momenta" conjugate to the "coordinates" "Z 1 ",..., "ld".
Note that we have used the special form of Hamiltonian and that its
arguments are given by d
*k(pj) 2k(p,l)
E
_1 m
m m = ^2 P _1 )t (5(0)Pm, kk {g{i))Pm
k = l,...,d. l,...,d.
m m= = ll
The integration over the "coordinates" is restricted to the parameter space Q. If part of Q is compact the momenta conjugate to the restricted range or periodic "coordinates" of this part of the group are discrete variables. For this class of momenta the notation J FJ dp(t) is then properly to be understood as sums rather than integrals. Before we can turn to a regularized lattice prescription for the representation independent propagator, we first have to spell out what we mean by a Schrodinger representation on L2{G).
Let {Ck,PCk}f=1
be a
family of symmetric operators with a common dense invariant domain D on some Hilbert space H satisfying: 1. The following canonical commutation relations (CCR): a b [C =o 0 [ca,c,Cb}} =
;;
[F£»,P£6] = = Q 0 [Pc«,Pc»}
;i
[£aa,Pcb i6abaI, [C cb}} = i6 bI,
i , 6 = a,b=l,...,d. \,...,d.
2. The operator A = £ L i [(-Pc*)2 + (£*) 2 ] is essentially self-adjoint. Z,/t=i
136
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
is a amilv Note that condition 2 above ensures that {Ck, P of essentially essentially Pc*)t=\ £ t}jt=i is a ffamily of
self-adjoint operators (cf. [83]). Let * be a dense set of analytic vectors for A and the family {Ck, PCk}d=l,
then we denote by # the closure of <& in the
nuclear topology on * (e.g. [11], Chapter 1.3). In fact, one can show that n the families of operators {Pc {-Pk£)i=i *}£=1 aand d {{Ck} A } Ad=1= I form two separate complete
commuting systems of operators, respectively (cf. [11], Chapter 1.3). Let us denote by <J>' the dual space of # , that is the space of all 7$- continuous linear functionals acting on . Then by the Nuclear Spectral Theorem there exist common generalized eigenvectors, (ll,...,ld\
6 ' and ( p i , . . . , p , / | 6 4>',
respectively, such that
A
{l\...,l \£^ ! |£ a t (l\...,ldd\C^
=
l"(l\...,ldd\,\, r(i\...,i
] {pi,...,Pd\Pc* {P\,---,Pd\P Vd\Pl c«
= =
Pa(Pi,---,Pd\, P*(pi,'--,Pd\,
Ila" ees pspec(£°), e c c(Z>), (Z=),
aa = = l1, ,.... . , d ,
pa G spec(P£a), spec(F£«), Pa spec(F £ a),
a = 1a=l,...,d, , . . . ,d
normalized such that d d d ,d {i"\...,i" (l"\...,l" \l'\...,l' \i'\...,i ) )
=
6e(g-\l")g(l')), 6Jg-\l")g(l')).
n< d
(p'{,...,M,---,Pd) (p'{,...,p"d\p'i,--.,Pd)
=
;*(?*.f[M,A), p*),
where u a
i
i \ — \ ^T>'iv'k ^(p'k~p'k)
*0>M = 1
n e sspectrum If of PPCCkk is is discrete discrete ^ tthe P e c t r u m of If If the the spectrum spectrum of of PP££** is is continuous continuous
and giving rise to the resolutions of identity /
_
\l)(l\dg(l) _ . \l)(l\dg(l)
I
_
_
f
=
/,
\p)(p\dp \p)(p\dp = =
I, r,
■)
d ' s p e c ( £ JJ ) X ...xspec{C ^Jsp ...xspec(C))
-,'ssppcccc((PP££ii))xx......xxssppeecc((PP/£; d )
=
/ ,
5.4. GENERAL LIE GROUPS
137
where \l) = \l\...,l*) \l\...,ld) ') aand \p) = \Pl,...\pu...,p ,pd). ), Remark 5.4.4 If the spectrum of PCk is discrete then dpk denotes a pure point measure such that the integration over pk reduces to summation over spec(P^"). [PCK O On L2(G) these operators can be represented as a Ca = la and Pc£°. = = -idi* = -w> -i[di* ++^T\ni)i (l)},
(5.29)
where D5 = S(G), the set of functions of rapid decrease on G, is chosen as the common dense invariant domain of these operators. Here T a (/) is defined as Ta(7) = 9;aln7(/) and where 7(/) is given in (5.2). It is easily seen that these operators satisfy the CCR, are symmetric on L2(G), and that — tVj has the following generalized eigenfunctions: d
£
k 1/2 (l\p) -^2(l)ex V(iY/Pkl Y (1\P) ===7- 1(0exi > (*&*n' k=\
where V; = (dfl (d/i,..., dtd) We normalize these functions so that ,... ,d,d) -
_ 11 , "
J
d
!T
m
Ky/l{l")i{V)
-i
fc )]dp, ...dp . ..dpd = = expfiVp, // exp[i £ >> (( // "' ** -- l'/'k)]dp, d
k=\
(g-\l")g(l') 6See(g-\l")g(l%
where K denotes the normalization constant. Therefore, we find for the normahzed generalized eigenfunctions of — iVf. (1\P)
1
d
= exp(t2 Pklk)-
V*7(0
is
(5.30)
fc=l
We call (5.29) a d-dimensional Schrodinger representation on
L2(G).
Moreover, the differential operators {zi( —iV/,/)}jLj can be written as follows:
138
CHAPTER
5. REPRESENTATION
INDEPENDENT
L e m m a 5.4.3 Using the differential operators {-id[*}d=1 right invariant
differential
operators {ik(—iVi,
l)}f=i
PROPAGATOR given in (5.29) the
defined in Lemma
5.1-4
(i) can be written as: di
*
E
i1
„ ~
= l,...,d, ^ 2[P~\ 2b-1*mm(»(0)(-tf/m) K "\ ( j (*(0)], S*(-iV,,0 = E W)){-idlm) + (H- i$^») A ( 0 ) ] . kA = l,...,d,
£k(-iViJ)
=
m=l m=
(5.31) (5.31)
where V ( = ( d ; 1 , . . . , dp). where V ( — ( d ; 1 , . . . , dp). Proof.
Since -idp
+ (i/2)Ta(l),
- -idp
a = 1 , . . . , d, the differential operators
{xk( —*V/,f)}Jt=1 become after substitution of this expression 1 ift(-«» k(-idli+(i/2)r . 7 2 ) ^l<( 0 , . . . , - ^ ++ P + ((i),...,-id
1 (i/2)r f(i/2)r (0,i(i),i\...,i ,...,O) d
d
d
= E< = EL>-V"c*< ^ mm(s(0)H&» + ^ ( 0 ] m == l1 771
-
■(a(l))(-idpn)] E ^ ^ T ( 5( ((/))(0 ) ( - ^^)) + />pp-- ,"(g(i))(-i^)} ( ( 0 ) ( - ^ ) ] ++ + E m
771 =
5
l 1m m 5 kt
l
lp-\ lp-\mm(g(l))T (9(i))Tm(l). (i).
1
m mm g{l)),-ldlm] \ ( Using [[pp-1_ A: \ m m( 5(s(0)> ( / ) ) , -i5,m] id, -*#/"•] = 0/mp idimmp-\ P~1i(g(l)) c (9(0)
x£kf(-id c ( - in5 „ + (i/2)T\l),..., d d<<
E
-idp
m yield ^ tthe h e definition of T m(l) (/) yields
aand n(
+ (i/2)Td(l),
l\...,ld)
ii
l^ m 2 ^ "m1(9{l)){~id^) ^ ( ^ / ))) _( - ^ )) + (-idlm-)p-\ = E 4p"\ + _(-id,-n)p-\ W))] (
j5/m
+
(
t5/m)p
(5(Z)
m m=l= l
d al
■^> W)
ftSk»[p-1*nar(0)7(03.
Since the operators xk(—*V;,/)
are essentially self-adjoint on any reproduc-
ing kernel Hilbert-space L^(G)
(cf. Corollary 5.1.5) and since 7(/) j£ 0 one
concludes that d
,...,d, Eld,4 ^ b ^-\ r(g(i)h(i)} c ^ M ^ ] = =o ,o,,fc k == li,...,d, E m
P
m=l
5.4. GENERAL
LIE GROUPS
139
and therefore id i f c ( -iiV V/,,,//)) = E
.i i D o ^ ' ^ ^ W X - ^ m ) + (-ia,-.)p- 1 f c mff(y(/))],, *t == 1L, ., . ...,,dd.. □
^[p-'r^ox-^)+(-^)p- r( (0)],
m=l '
Remark 5.4.5 The above Lemma shows that the linear differential operators {xfc(-iV/,/)}^ = 1 are elements of the right invariant enveloping algebra of the (^(-iV,,0}ti d-dimensional Schrodinger representation on L2(G). O Adapting methods used in [64] and [69] we can give the representation independent propagator the following regularized lattice prescription.
eH Proposition 5.4.4 LetH LefHet =-HI{I = H/(I + c1i tH22)) be a sequence of regularized Hamil ton operators on H, where (e = = (t" (*" - t')/(N t')/{N + 1). Then provided the indicated integrals exist (see below) the representation can be given the following d-dimensional
independent
propagator in (5.23)
lattice phase-space path integral rep
resentation: K(l",t";l',t') K(l",t";l',t') = =
l
= Urn lim / . . . / . ' ooj / >/7('")7('') *
/v
■I'D
xx eexp x p O ' ^zJ>j+i/2• b j + i / 2 • (O+i (0+i -h)~ - 0 ) - f W (0)))? 2;0+i.0))]j
■)
=0
(V
n
r xj[dl}...dl< J c f / ) . . . ^ J] ;= i
jJ++ l / 2 = 0
dp aPu-n/a ■ ■■"PjitW' ^,2Kddp^'\
#'
where i/v + i = I", 'o = V an(t the arguments of the Hamiltonian following
(5.32) are given by the
functions:
E
+ Po-'rWi)), ■f , M= V P'P-'fWiOi ~V"(g(/J+')) + ~ 1 * m ( g t t— ) ) PP m , + l / 2,,, fc-i d k= l,...,d. iXk{Pj * ( P j++\/2
140
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
Remark 5,4,6 If part of the parameter space Q is compact then we denote by 72 the class of momenta conjugate to the restricted range or periodic "coordinates".
If pi 6 72. then dpk denotes a pure point measure such that the
integration over p^+1/2 reduces to summation over the discrete spectrum of PCk. />£.*.
For the case of a compact parameter space Q (5.32) reduces to (5.13). O
Proof.
Since the Hamilton operator H is in general arj unbounded operator,
we introduce the following sequence of regularized Hamilton operators on H
n6 =
~
H I+6H2'
6 > 0.
Then it is straightforward to show, by using the Spectral Theorem and the Monotone Convergence Theorem, that for all ip € D(7i) C H one has s-lim 7i{ Kg = = ~H, «, and that on all of H one has
[i-iaufv
s-lim [/ - ieHt}N+l N-* ao
where e = [t" - t')/(N
=--exp[-i(t" exp[-i(t"
-
t')H],
+ 1). Now in order to obtain the lattice phase-space
path integral in (5.32) one can proceed as follows. Let { 0 , } ° ^ be an arbitrary ONS in $ C H , then K(l",t";l',t') K(l",t";l',t')
aa -exp[-i{t"-t'Mx t'y, =: exp[-i(i" exp[-i{t"-t'Mx ))}(l"\l') k(-id l*,l))}(l"\l') k(-id l*,l
= =
(l"\exp[-i(t" t')H(xk(P «,Caa))W) )W) k(P \' I' p[-i(t" -- t'p t')K(x Cco,C
=
E
OO OO
a a E (l"\ - OW(^(/ 3 £°,£ ))]0;>(^lO C"l^*)(^.exp[-t(t" - OW(**(ft«»£ ))]^){0ilO k)(4>k,ex?[-i(t"
J,k=\=
1
0 00 0
E
Ar+1 imc{xk{pc^ca))],N+l = jj™, E (Htf*X&.[i - *We(Sfc(P£.,£ ))r4> tfi)tolO 1){4> J\i')
i,fc=i
5.4. GENERAL
141
LIE GROUPS
Urn lim Urn (/" { / * | [ l - « H e ( i t ( P £ . , £ « ) ) ]r "Y + 11 | 0
= =
TV—oo /V —*oo
im lim
r
/
■
■
/
;
N
N
= ri oo / ••• / ,
n
Il(tl ^i+i\[l-itKt(£k(Pc<>;lj+i,lj)Wj) + i l t 1 - »w«(**(^-;^+1,0))]l'i>
z d/ ■y(lj)dl}.,.dl*, IIT( i) i - - • d / j .
;=0
where /" = //v+i, I' = 'o, and (• | •) denotes the generalized inner product. Note that the third line holds true since each 4> € $ gives rise to a linear functional acting on <£ in the following manner L^(ip) = (<j>\ip) = (4>,ip) for all ifr £ «£. Hence, one has thatt (j) t')K],) = = <&|exp[-t(t" - t')7i\\4> C)W]|^) L*^k((expl-i(t" 3) = L ^ ,,eexxpp[[ -- it ( t " - *t')H}4>j)■ (& ' ) W t o ) . The fourth line follows from the fact that <& C H and that the approximation we are using holds for all elements of H (see above). That we can interchange the limit with the infinite sum at the third step above follows from Moore's Interchange of Limits Theorem (see [31, Lemma 1.7.6]). Hence, we find the following expression for
K{l",t";l',t') K{l>',t";l',t>)
=
lim lim
J
N—co j
r
N N
( P £ . ; Z i , h))]\W J l[(ln i\{li | [ l --« W ( i *i0U(*k{Pc';lj+uh)Ws) f
...
K(l'\t";l\t'):
j+ J +
e
i+
N
xn[ 7 ( Z , V / ! . . . ^ .
(5.33)
x'[[l(li)dllj...dl*.
(5.33)
Therefore, we have to evaluate (/ J+1 |[l - i«Wt]|/j). This can be done as follows: Therefore, we have to evaluate (/ J + 1 |[l - i«W t ]|/j). This can be done as follows: i l l 1 c- (xk(Pc«;lj+uh))]\h) (l]+1\[l-iM P£-iWj))]|*i)
(lm\[l-u-Ht(xk(Pc^lj+i,li))}\li) | [ l fl/2 (Pj+i/2l[l - -ic?ic(xieHi(xk(Pc°4Jj+i,li))]\h}dPj+i/2 < + l|Pj + l/2>(Pj + l/2l[l ~ //<'j -irtU(xk(PC>l}+l,lj))]\lj)dP3+l/l =
l/2)(h\Pj ++\/2) \/2) [1[1 -ilKt(ik(P] -«'^((ii(Pj-H/2;'i+l,'j))]< l/2, {h + + l\Pj \\Pj + + l/2){h\Pj + l/2;lj+l'l}))\dPj : 'Pj ++ l/2, / I<(h
where d
Xk{P]+i/2,h+uh)= SfcfPj+i/a.fj+i.'j)
1
^ - ^ ( f C i t O l + p - ' t " ^ ' , ) )) „ ... k=l,...,d. l_s 22 l,...,d. - Pm'+m> Pmj+i/J. k =
m=l
771=1
142
C H A P T E R 5. REPRESENTATION
INDEPENDENT
PROPAGATOR
Substituting the right-hand side of (5.30) into the above expression yields
{lj+1 \l-kH
=
/ f
/
1
g'Pj + l / 2 ( ' j + l _ ' j )
W M
f^n-^-h)
%/7(',-+l)7(',-)» -/
c 4.1/2 dp; .JPj+l/2 J1I J xx[l-ien [ l - z e€(x ? ikE(p(mf2i x / tj+ll ( plJj))} ]- K + 1 / 2^; /-j + i , / j ) ) )]
(5.34) (5-34)
Now inserting (5.34) into (5.33) yields K(l",t";l',t') (l",t";l',t') --
(
1
ex
■I
= lim 11
~7WW)' ) N —>ooo J/
N
^ b j + i / 2 • (*i+i - h)\ h) 3=0 ({ 3=0
NV
e W EE ( (£fc(Pi+i/2^i+i^i))] xfc(pi+1/2;/i+1,;i))] X ~][1 J|[l- - i ieW 3=0
n N
xftdZ]...^ 33=1 =1
ft
3 ++ 1/2=0 1/2=0
-dp j + l/2 d^PlJ-fl/2 Pl /2 - ■ P "' K/ ,/d ^d 1 / %
(5.35) (5-35)
Equation (5.35) represents a valid lattice phase-space path integral represen tation of the propagator K(l",t";l',t'). i(Tic(£k)
One can now interpret the term 1 -
as the first order approximation of exp[—WHc{xk)\ for small c. Hence,
provided the indicated integrals (or sums as necessary) exist one may replace (5.35) by the more suggestive expression: 1 A'(Z", *";/',*') = K(l",t";l',t') =
= hm
7 nwhm vW'M'') V^OTP n i n)) N^a 1
r
I
■I
/.../
zv
X exp { ^bi+1/2 • ('3+1 - '3) - e K ( * j t ( j P i + i / 3 ; i i + i ^ i ) ) ] | x exp <*']T[Pi+i/2 • Cj+i - h) ~ ^(^kiPj+iZ-i-Jj+ulj))]) 3=0 N
dZj . . . dlj
n ft N
■)
dpv+i/2
.. .dp^.u/2
«n«fl}..^ **>»»-***»»,' 3=1 3 + l/2=( j=i
which is the desired expression. D □
ITSrl f
3+1/2=0
Remark 5.4.7 Observe that even though the group manifold is a curved manifold the regularized lattice expression for the representation independent propagator
5.5. EXAMPLE:
THE AFFINE
GROUP
- save for the prefactor 1/ y/f(l")'y(l')
143
- has the conventional form of a lattice
phase-space path integral on a d-dimensional flat manifold. Also note that the lattice expression for the representation independent propagator exhibits the correct time reversal symmetry. Furthermore, we have made no assumptions about the nature of the physical systems we are considering, other than that their Hamilton operators be essentially self-adjoint. Hence, one can use (5.32) in principle to describe the motion of a general physical system, not just that of a free particle, on the group manifold of a general Lie group G. In addition, there are no ft2 corrections present in the Lagrangian. Therefore, we have arrived at an extremely natural path integral formulation for the motion of a general physical
system
on the group manifold of a general Lie group that is (a) more general than, (b) exact, and (c) free from the limitations present in the path integral formulations for the motion of a free physical system on the group manifold of a unimodular general group discussed in chapter 3. O
5.5
Example: The Representation Independent Propagator for the Affine Group
We now introduce a representation independent propagator for the affine group. The affine group is the group of linear transformations without reflections on the real line, M 9 x —» p~lx
— q, where 0 < p < oo and — oo < q < oo.
This group has been used by Klauder [65] for the coherent state path integral quantization of one-dimensional systems for which the canonical momentum p is restricted to be positive for all times. For further applications of the affine group in quantum physics the reader is referred to [65] and references there
144
CHAPTER 5. REPRESENTATION
INDEPENDENT
PROPAGATOR
in. The affine group is also an example of a locally compact, non-unimodular Lie group, its modular function in the adopted parameterization is given by A(#(p, 9)) = p _ 1 and its left invariant Haar measure is given by dg(p,q) = dpdq.
5.5.1
Affine Coherent States
Let us denote by X\ and X2 a representation of the basis of the Lie algebra associated with the affine group by self-adjoint operators with common dense invariant domain D on some Hilbert space H. Since X\ and X2 are a representation of the basis of the Lie algebra associated with the affine group, it follows that these operators satisfy the commutation relations [X1,X1} = 0,
[X2,X2} = 0, and [X1,X2] =
-iXx.
From these commutation relations it is easily seen that the Lie algebra associated with the affine group is solvable, therefore, the affine group is a solvable Lie group. Since Xi and X2 are chosen to be self-adjoint they can be exponentiated to one-parameter unitary subgroups of the affine group, cf. example 5.4.1. Since the affine group is a connected solvable Lie group every group element can be written as the product of these one-parameter unitary subgroups (cf. [7, Theorem 3.5.1]). With the above parameterization the map: r r g{p,q) -> f/9(Pi,) = exp(-29A exp(-iqX 1 )exp(ilnpA 1)exp(i\npX2 )
provides a unitary representation of the affine group on H, for all (p, q) g P+, where P + = {(p,g) : 0 < p < 00, -00 < 9 < 00}. The unitary representations of the affine group have been studied by Aslaksen and Klauder [4] and Gel'fand and Neumark [41] and it is known that there exist only two (faithful)
5.5. EXAMPLE:
THE AFFINE
145
GROUP
inequivalent irreducible unitary representations for this group, one for which X\ is a positive self-adjoint operator and one for which X\ is a negative selfadjoint operator. We denote the irreducible unitary representation of the affine group corresponding to Xx positive by Ub
% and to Xx negative by U2,
>,
respectively. The continuous representation theory using the affine group has been investigated by Aslaksen and Klauder [5] where it was shown that for f, 4> £ H , ■£ 0 the factor ([/;,p , ) £ , 4>),(, = 1,2, is square integrable if and only if £ 6 D ( C _ 1 / ' 2 ) , where the operator C is given by C = 4tXi
and X\ is restricted to be pos-
itive. Hence, the irreducible unitary representations of the affine group are square integrable for a dense set of vectors in H . Moreover, in [5] the following orthogonality relations have been established for the irreducible unitary representations of the affine group: 1 2 l/2e,C-^20, J(x^^OiU^'^^dpdq ^ (X',x)(C\x^)0(U^J\x^dpdq=^)X)(C-^e,Cf^ /
1,2 C= 1,2
C=
where x , x ' £ H , and £,£' 6 D ( C - 1 / 2 ) . Hence, each of the irreducible unitary representations can be used to define a set of coherent states:
v(p,i) u<0M°cl'\7?' V{P,V == S(P.9j where r\ £ D(C1^2)
2, c =l ,1,2,
C =
and \\T]\\ = 1. These states give rise to a resolution of
identity and a continuous representation of the Hilbert space H on any one of the reproducing kernel Hilbert spaces L 2 ( P + ) C Z 2 ( P + ) . 5.5.2
The Representation Independent Propagator
Using Theorem 4.2.1(ii) we find: idU fl(P.?)i)
s(p.?) SIP.?)
^id? +
p P
-
1, -X,)dp,
— —J
P V
C=l,2
146
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
from which we identify the following 2 x 2 coefficient matrix:[/>m*(ff(p,9))]: [pmk(g(p,
(i J).
[Pmkh(g(p,q))) (g(p,q))}= = ( [pm
a
\ p
_i
)•
p /
Inverting this 2 x 2 matrix we find: 1
■ ■ ( J; -_; J) )■{p-lm\g(p,q))}-[pm %(P,?))]=( With these coefficients we find by Lemma 5.1.4 for the differential operators that describe the action of the affine operators X\ and X2 on any reproducing kernel Hilbert space L*(P+)
the following: ixxl
=
x2
=
Thus, if we denote by %{X\,X%)
-idqq, ipdp - iqdq.
the essentially self-adjoint Hamilton opera-
tor of a quantum mechanical system on H , cf. example 5.4.1, then by Theorem 5.4.2 the representation independent propagator for the affine group is given by: K(p", q", t"; p p',1, q', f) = exp[-i(t" e x p [ - i(t" - t')H{xx,
x2)] 6(p" - p')6(q" - q').
By Proposition 5.4.4 we can give the representation independent propagator for the affine group the following regularized lattice phase-space path integral representation: K(p",q",t"-p',q',t') A'(p'',
lim NZOJ.
/
/V—*oo u
■
x p \xi+\/2{P]+i -~ Pi) Pi) + + *i+i/a(«+i kj+l/2(qj+1 qA • / eaxp r £[■a i+x/2(Pi+i -- qj) t 3=0 !
• ■ / •
-cH■nufIfej-n/2, -m + PPjj )} *7+l/2> ^i+i/stej+i glwM&'+J + + qj) 9;) -- xxJ+1/2 {pJ+1 ) } \J 1j ] + 1/2(p ]+1
')]}
5.5. EXAMPLE:
THE AFFINE
N N
N
147
GROUP
.,
,
ax +1/2 +j+i/2 ~ TT J J TT dkj aK j+i/2 7 i/2dXj [ dpjdq. X{[dpsd , qj[[ (2*)' i=i j=o ^ ^
where ( P N + I , 9 W + I ) = (p",q"),
(po,9o) = (p',g')> and e = (f" - *')/(JV + 1).
In this expression one can preform the following three consecutive variable changes. For all j , one first lets xj+1/2
-> xj+1/2
+ (qj+1+qj)kj+l/2/(pj+l '(Pi+i+Pj), +Pj),
followed by the substitution z ; + i / 2 —► - 2 X J + 1 / 2 •7(Pj+i /(PJ+I + Pi), and finally one + Pi) lets kj+lj2
—* 2 (Pj+i + Pj)kj+i/2-
Then the resulting regularized phase-space
path integral is given, by K(p",q",t";p\q',t') K(p",q",t";p',q',t') ';p',g',0 N
lim
/
/■
/
/ exp■(■
]| ■E<
2*i+i/2[(9j+i + Qj){p}+i ~ Pj)
3=0 2 2(?j+i Pj) , / , w M (Pj+i ~ Pi) + (Pi+l + Pi)(9j+i - &)] - ^ i + i 12 /3: (Pj+i VPj+1 + TP Pj\) j)
N
«,)}) n"T
- e ^ f Tjfai+ii ++PPi i-) f c j + i / 2 , ^ + i / :
dk3+i/2dXj+l/2
(2*)'
;'=o
Therefore, taking an improper limit by interchanging the operation of integra tion with the limit with respect to TV we find the following formal phase-space path integral representation for the representation independent propagator for the affine group:
K(p",q",t";p',q',t') MJexv{iJ[k(ip)-x(h^)-H(pk,x)]dt\ - z(ln p) - 7i(pk, x)]dt dt\ \ « ( p , 9 , * ; P . ? . *) = = 7W / exp < f[k{qp) /
{';
xVpVqVkVx, XVpVqVkVx, where ( ~ ) denotes d/dt( ■ ). This expression agrees with the one found in [70] 1 l up to a numerical factor M , given by' M TI), v) which is used in the M == (TJ,X^ (v,x; normalization of the resolution of identity in the definition of coherent states
148
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
for the affine group due to Aslaksen and Klauder [5]. We now formally evaluate the representation independent propagator for two soluble examples. For the exact lattice calculation of these two examples see Appendix B.
T h e Free Particle X2/(2m).
Our first example is that of the free particle where 7 i ( X i , X 2 ) =
Since X\ is self-adjoint and g(x) = x2 is a real-valued Borel function on M, g(X1)
= X2 is self-adjoint on D g == {4> \2(,pXl Px*(d\)4>) it'- : J^ JZo^(^ (dX^)
< co} (cf.
[88, Theorem VIII.6]). In this case the representation independent propagator becomes
is( 1 ft A ( p 11, * ,t \p ,q ,t) ;p,q,t) A (p ,
It
II
4JII "
I
I
I\ J4 ' \
dt \
J dtJ\
VpVqVkVx VpVqVkVx • dp{t)
- ^ / - { . • / [ ^ " - © H Mn I , ^ L
V P /
AT/e: = = AfjexpliJ AfjexpliJ
P(0
j(q+q-J y (
dt U{p}D«j£>p
Carrying out the remaining two integrations we obtain as our final result, Carrying out the remaining two integrations we obtain as our final result,
ff K(p",q",t";p',q',t)
I
rh
= — 6(p" - p') exp <»"«"•'"■'''•''«>-i/is^*"-"')"[jpztF-tf ■■ \J 2ni(t" - V) 'X (t"~t'y 7-M" i -) i'f q
1 2
Observe that, up to the presence of the delta function 6(p"—p'), this result is in perfect agreement with the usual result for the free particle, even though we only consider the positive or negative half of phase-space, i.e., p is constrained to be either positive or negative.
149
5.5. EXAMPLE: THE AFFINE GROUP wX T h e Hamilton Operator ■H(X ) 2 ) == ^2j kXXf l + u>X W ( 1X,Xl 2t X 22
The second example we consider is that of the Hamilton operator H{X\, X?) = X\)1m
+ wXi. We have seen in example 5.4.1 that this Hamilton operator is
essentially self-adjoint. The representation independent propagator takes the following form K(p" ,q \p,q,t) ,t) ) . V ,t > ' ;p,q yF ->H ■>•■ =■ M jexpli j expli
j
J
Jo
k(pq) — - x(hTp) x(\np - ~ - ~ux dt\ 2m J J ■irn
VpVqVkVx
+
^/-{■r[f(f)'-'(f ")]+',n^- ^{«jn?(«^)'-^ ?R)' + -)]*}-. > n i ^= Af
exp < i
( "u
■»(H)"_. x r
L
rp
^•
'
'' JJ
JJ
te[t',t te[t',t»]
rv
'
•\ 2
, P \ -x(~/ P +i w) dt ivq J — Af .A/- / exp < i J { Jo I* V + gV) - x(-PP +u -jJ JJ (€[('" ^^PW p/ 2 P m / p\ = Affexplif Aff / exp expli< i f j(q + q-j dti dt 1 6{p <5{p + wp}VqVp up}VqVp = AT Af
+ ( .up}VqVp L»Tiy, / p; dt f . I /6{prm _- wuqf] T r 2T 2 _ e -.T/2 r 2 A ^ g ) 2 i ddt} t | p gVq, > [|(<j = 56 ((f6(e" W eexPx{i£ " -- e-^p')/ p') AfJex e -'y/ 'pp"where T = t" — t'. The final path integral we have to solve is a Lagrangian path integral for a quadratic Lagrangian which can be done using extremal methods; ve have to solve is a La see [96]. The action for this Lagrangian path integral is given by r
m
m
fT
hi c = 2" '0 ' 2" / variation of which yields the equation variation of which yields the equation
\2 ; (q — u>q) dt 9 Uq ' ~ ' of motion of motion I ■
q = u22q, q = u q, which has the general solution which has the general solution q(t) q(t) = = Asinh(wi) Asinh(wf) + + Scosh(wi)Bcosh(ujt).
150
CHAPTER
5. REPRESENTATION
INDEPENDENT
PROPAGATOR
After imposing the proper boundary conditions, one finds the evaluated classical action to be mu 2 2 5Sd = q ] mu-K?") (q'f}cosh(uT)-2 "q'}« =2sinh(u>T) , T Tr){[(q") ^ ' ++ (9')2] cosh(wT) - 2qq" ' ~ ^{{q"- )2 wn - {q'n So that our final result for the representation independent propagator with this Lagrangian becomes: K(p", ",t";p', K{p", ',i') n\p qq> 9 i i qq',t>) ;P > ? >i) raw
/Y) S ( > r / 2 p » _ e-" e - TT '2p') 27nsinh(wT) V 27risinh(wT) yy 27rzsinh(wT) ziri smn{u>i j VV \ // X exp9
/
imui
-A[(q") V2sinh(wT)
2
+ (
imw
- ^ V-I(9") ) 2 - (2 s- '(?') ) 2 ]2]j) .•
Observe that the evaluated action functional in the exponent of this propagator, save for the term -(mu>/2)[(q")
— (q1) ], agrees with the evaluated action
functional one obtains for the propagator of the harmonic oscillator in imaginary time formulation although it is not of the same "physical origin".
Chapter 6
CLASSICAL LIMIT OF THE REPRESENTATION INDEPENDENT PROPAGATOR Even though the regularized lattice phase-space path integral representation for the representation independent propagator has been constructed by interpreting the appropriate Schrodinger equation (5.27) as a Schrodinger equation for d separate and independent canonical degrees of freedom, it should, nevertheless, be true that the classical limit for the representation independent propagator refers to the degree(s) of freedom associated with the Lie group G. In particular we will show that this is true for a general Lie group since the classical equations of motion obtained from the action functional for the representation independent propagator imply the classical equations of motion obtained from the most general classical action functional of the coherent state propagator for G. We first discuss the classical limit for compact real Lie groups and then turn our attention to the classical limit for general non-compact real Lie groups.
151
152
6.1
CHAPTER
6. CLASSICAL
LIMIT
Classical Limit for Compact Lie Groups
It is known that any compact Lie group is the direct product of its connected center 1 and a finite number of simple subgroups (cf. [7, Theorem 3.8.2]) and that all irreducible unitary representations of compact Lie groups are finite dimensional (cf. [79, Lemma IV.3.2]). Subsequently we consider the classical limit of semisimple compact Lie groups, i.e. compact Lie groups which have a discrete center. This includes the physically important examples of SU(2) and SU(3),
whose centers are given by C = { - e , e } and C = { - e , e x p ( 2 f i ) e , e } ,
respectively. The problem of taking the classical limit of semisimple compact Lie groups has been previously considered by Gilmore [43] and Simon [97]. In Simon [97] the classical limit of quantum partition functions is discussed, extending previous work by Lieb [73] to any semisimple compact Lie group. The classical limit is taken by using coherent states built up from a maximal weight vector in an irreducible representation. While in Gilmore [43] and Simon [97] the general problem of taking the classical limit of operators belonging to a semisimple compact Lie algebra is considered, we, on the other hand, consider the problem of taking the classical limit of the most general action functional appropriate to the coherent state propagator for a semisimple compact Lie group. Let us denote by Xj — hX}, j = 1 , . . . , d, the physical operators. Then the most general action functional appropriate to the d-dimensional semisimple compact Lie group G is given by (see Chapter 3, Eq. 3.42): d_
/ = J[ih( / [ihW), {i)} -- ((VV(l), (i),n(x H(XuU...,x ..., d)r,(i))]dt. Xd)V(l))]dt. V(l)) v(i), -jtV /
(e.i) (6.1)
TBy the center of a group G we mean the set of elements of G which commute with every element of G, that is, C = {a £ G : ax = xa V x 6 G}.
6.1. COMPACT
153
LIE GROUPS
Let us assume that the semisimple compact Lie group G we are considering has rank n < d, i.e., there exist n self-commuting operators HT, r = 1 , . . . , n , that form the Cartan subalgebra H of the Lie algebra L associated with the Lie group G. Moreover, let us denote by m = ( m i , . . . , mn) the highest weight of the finite dimensional irreducible unitary representation (U, H ) of G. Using the non-degenerate Cartan metric tensor gis = J2j k=i cik:'csjk we construct the Casimir operator d
s
Ei g' x,xs,
c2 =
(,.9=1 l,s=l
which satisfies
[C22,,Xk] Xk]
=
dd U - Y,~]g9l'([Xi,X {[Xi,Xk]X k]X ts
-Si
+ +
X,[X Xi[Xst,Xk}) ])
l,s=l /,s=l dd
i
= -Ei - Y 9 //,s=l ,s=l (,s=l dd
U
(P d
dd
\
3j=l =1 j=l
JJ
£
1j Y^k'XtXs YYcc*k skeik'XtX) X iXJ +Y XiXj £ ctJXtX, + +
\t=l \t=l
d
dd
d
c j s ■ik'XtX l Csk'X'X, = sk X X, = -- Y Y cc^XtX' ^XtX ++ YY CsJX'Xi 1
l,t =l i,t=i l,t=l l,t=l
j,s=l j,s=l
d
- £
s ll = -- J£(«**.+ cu) XX*X XSX' X = = 0, 0, > j * . + cu) l,s=l /l,s=l ,s=l
since cTSt — J2i=i crsl9it is totally antisymmetric under any interchange of its indices. Since the Cartan metric tensor is symmetric and is non-degenerate for semisimple compact Lie algebras it can be diagonalized, i.e., may therefore be taken in the form gu — —6u- Hence, without loss of generality we can assume that the Casimir operator is given by d
£
c, ?■ c2 = Y;x*?1=1 (=i
154
CHAPTER
6. CLASSICAL
LIMIT
The operator C 2 can be written in the standard Cartan-Weyl basis of the Lie algebra L as follows: 71
E
E
2
cC =V Y, ^^r ^K ++ X X h E^EaEC 222 == aE-aa,, r=l 7= l
(6.2) (6.2)
aa
where J^ Q denotes the sum over the nonzero roots of the Lie algebra L. When this operator acts on the highest weight vector w m of the irreducible unitary representation U, one obtains, because of the condition Eaum
= 0 for positive
roots, 2 2 a 2 2j YlK ; ,EWm m C uj l'£ K ml rr$m? + + ]\um C = \^h |X)A + XY'n ( B[Eo 'a£[E,E. - aaU]\u +x 2um = 2
m
2
f
2
ll rr = = ll
22
a
o>0 a>0 o>0
71
n
m2
a
))
w
== ^ h n X" Xm ty , r a' ;+++XX EVXE* EF°,°J^^^i <W*»-r = l\ T n
a>0 j=1
J
a TQrmTTl ,T Y ^arTTlr == hh ) ^22 S^XEI mm? Ir ++ XE« 2
r
.. rr = = ll
a Q >> 0 a>0
\
r
W mm. W
'
/ .
It is well known that every irreducible unitary representation is characterized by the components of the highest weight m. By Schur's lemma every invariant operator in the carrier space of the irreducible unitary representation U is proportional to the identity operator, i.e. C2 = A/, where A is given in terms of the components of the highest weight m, in particular we have that n
c22: = h2 C C2 =
2 m +2grmr)I, 2 T 2 r h Y,(m T+2g mr)I, r= l
where where 9T =
1 2 >
E
a>0 a>0
r
-
We We now now consider consider the the classical classical limit limit of of the the action action functional functional given given in in (6.1). (6.1). Since deal with with general general fiducial fiducial vectors vectors we have to Since we we want want to to deal we have to consider consider , k= k= {
l,...,d,
6.1. COMPACT LIE GROUPS
155
where the x/t,,, k = 1,.. .,d, are real numbers given by Xk„ = {Vi^kV)- We insist on vanishing dispersion as h —► 0 and m —<• oo (i.e., mT — ► oo, for each r), namely, that d d
lim ; ( 7 7 , ( l f c - ( 7 7 l ^ 7 ? ) )22 7 ? ) = 0 , hm Y,(V, (Xk ~ (V,XkV)) V) = 0, A-.0
E<
771—*00 771—*:=i fc=l
(6.3) (6.3)
where the limit h —»0 0anc and m —» oo is taken in such a way that the product /x = hm stays finite. We denote the set of fiducial vectors that satisfy (6.3) by T.
If we choose for the fiducial vector the highest weight vector of the
irreducible representation U then we find dd
lim y^(u) (Um,(X , (X k
£<
m
k
A-.0 A —0 ■*—* m—>oo
(u> ,X Lj ))2oj )
-
(U>m , m X A :kW m m) )20Jm m }
-
d
=
lim (((w ((cjmm,C 22w mm) -- ^(w ,X/twm )22)) ■D (wmm,X/tw
A—0 m—*oo m—>oo
j tk=l =i
n 71
= =
Um Urn [[h22 ^2(m22 + 2grrmTT) - h lim
ft—0 m—>oo 771—»00
=
£(
rr = = ll n
nn
22
2 ^ mmV\] r=l
lim S ^ 22 V 225s r m r = 0.
A—0 A—o m—*oc m—*oo
E
•"-—'
7r== 1l
Hence, the highest weight vector satisfies (6.3), and therefore, the set T contains at least one vector. Since for fixed / £ C, Ug^ is a unitary operator on H there exists a /,, such that V=
u9 Ug(l )"Tnn
Therefore, we find d
Xkr, {uTn,U* ) ) = ^ tlU ,XmtU, mX) t w m ) . Xkn = (V,XkV) {V,XkV} = K , t / * ({lri) , / X^kjV(g[ln) l , )UJ Umm Vk(%)(u ' 7 , ) (mw t=l
Only the terms for which (ojm,Xtu)m) Only the terms for which (ojm,Xtu)m)
^ 0 contribute to this sum, hence we ^ 0 contribute to this sum, hence we
156
CHAPTER
6. CLASSICAL
LIMIT
find
E Uk (l ) i(r)> r
Xk„ =
n
m
where / = {r £ { 1 , . . . , d} : XT 6 H) and m,( r ) denotes the component of the highest weight m for which XT — H,. For finite h the term that represents the classical Hamiltonian in the coherent state propagator for the Lie group G is given by m Hnn(l)(l)==(V(r (l),n(X = (V= ,H(v,'H\ (£ ] U H 1/^(0^ JIJ), IJ), 1,...,Xd)r,(l)} k (l)Xm] }(l),'H(Xu---,XdWl))
\m=l
where we assume that H{X\,.-., operators {Xk}i=i-
Xd) is an arbitrary polynomial of the physical
Therefore, if we now take the limit h —» 0 and m —> oo in
the above mentioned sense, then the classical Hamiltonian is given by lim Urn
^—o m—»oo m—oo
Uk lh U \l)v , = 7 ^(0 H r,(l)= = nlJ2 k''( >i>) \6=1 \6=1
/
where
-E
i = lim = lim "fc= jim (77,Xfc7?) = Urn ?%„ = Er ^t */ /' ^ ^ )) ^ ^ ), ,' A—0 h—0
m—*oo 77i—*oo
m—+00 m —»oo
k =fc=1i,...,d. >--->rf-
rrg£//
Hence, the classical limit of the action functional given in (6.1) becomes Hence, the classical limit of the action functional given in (6.1) becomes
/c/ = um r^nw),Jt-n{i)) - W),n{xu...,Xd)v{i)))dt Id
=
lim / m—*oo m—*oo A
[rt{7?(/),|7?(I)}-{r?(0>H(Xlj...,X(f)7?(0)]^
h
m
/A
6
J d
6
\
/ E ^ (gm vk -H(±V - H ( E ^i (o^, • • •, E ^ (o«6) A, [*,m=l ,m = l [*,m=l
\6=1 \6=1 \6=1
6=1 6=1 6=1
//
(6.4) Extremal variation of this action functional, with respect to the independent labels lb, holding the end points fixed, yields the equations of motion d
E
dd
E
s s s E ]v»«{di^>, (9(l)) - dlb\cbs. (g(l))} lb = Y, w'W.'COK. -Had,c[Uj(l)]vj, s{dic\b (g(l))-d lb\c (g(l))}i
6,5=1l 6,5=1
a,/=l
(6.5)
157
6.2. NON-COMPACT LIE GROUPS
where Ha denotes the partial derivative of H with respect to the a-th argument a=
l,...,d.
6.2
Classical Limit of N o n - C o m p a c t Lie groups
Our discussion of the classical limit of compact semisimple Lie groups cannot be generalized to non-compact semisimple Lie groups, since they do not admit faithful unitary finite-dimensional representations (cf.
[7, Corollary 8.1.4]).
Hence, we must follow a different route to achieve a well defined classical limit of the most general action functional appropriate to a general non-compact Lie group G, given by
7'
d_ \\K^\\-222 J[th(ai), [ih(ai),-m)-(mnxu---,xMiwt, i/=HA-mr == \\K^\\|«o> - («o, nxu- ■ ■ - Xd)wwt, ' dt where £(/) = Ug{i)Kll2£ and it is assumed that A 1 / 2 ^ 6 D. Without loss in generality we can set 77 = A'1/,2f/||A'1/,2f||, then our most general action functional becomes d
/I = = J]ih{t,{l), ~i,(i)) (r,{l),H(Xu. / [ttMO, dt ^(O) - W),n(x xdMi))}dt, V{ d)v(lWt, u..., ■ .,X />
(6.6) (6.6)
where r](l) — Ug^rj and where it is assumed that 77 6 D n D(A r _ 1 / 2 ). In our discussion of the classical limit of the action functional given in (6.6) we use an abstract formalism for taking the h —» 0 limit developed by Yaffe [109]. Yaffe [109] considers a family of quantum theories characterized by some parameter x, such as Ti, and studies the limit of these theories as x approaches zero. It is assumed that each theory is defined on some Hilbert space H x with some Hamilton operator Hx. Furthermore, it is assumed that
158
CHAPTER
6. CLASSICAL
LIMIT
there exists a Lie group G, with associated Lie algebra L, that has on each Hilbert space H x a unitary representation Ux. We assume for definiteness that Ux is parameterized as
(v'***) ^(0 =nT*p 6 ^(T«>) ( 7 ^ )•' d
S(')
k=i
up to some ordering. Then the first assumption, which restricts the choice of the group, is: A s s u m p t i o n 1.
Each unitary representation
of G on H x is irreducible.
Hence, on each Hilbert space H x one can define a set of coherent states nx(l) = Ux,l-.nx. For any operator 0 acting on H x , we define the u p p e r s y m b o l
0Vx(l)
by Or,x(l) = (vx(l),0 (l),On Vx(l)} x(l)}
for all I € C,
i.e., the upper symbol is a set of coherent state expectation values. The second assumption restricts the possible fiducial vectors nx one can choose. For each value of x we require:
A s s u m p t i o n 2. Zero is the only observable whose upper symbol
identically
vanishes. By an observable we mean a family of self-adjoint operators consisting of one self-adjoint operator acting in each Hilbert space H x . An example in which assumption 2 is not valid is given by SU(2) coherent states based on a fiducial vector that is not the highest (or lowest) weight vector, in this case a unique specification of any observable by its upper symbol may not be possible (cf. [69, p. 34]). Note that assumption 2 implies that two different operators cannot
6.2. NON-COMPACT
LIE GROUPS
159
have the same upper symbol. Hence, one can uniquely recover any operator from its symbol. As pointed out in Yaffe [109, p. 411] "this means that it is sufficient to study the behavior of the symbols of various operators in order to characterize the theory completely". Observe that the x -* 0 limit of an arbitrary observable does not have to exist. In order to have some control over the x —* 0 limit one introduces the concept of a classical observable.
According to Yaffe [109] an observable
0 is called a classical observable if the limits of its coherent state matrix elements exist, ( i xx(l),Oy ( ' ) . Q ' hx(n) (O) lim, (y x-»c
o (vx(i),vx(n)'
and are finite for all /,/' € C by Oc.
The set of all classical observables is denoted
Clearly the set Oc is a subset of all possible observables, hence it is
possible that measurements using only observables of Oc may fail to distinguish between different coherent states. Therefore, two different coherent states, nx(l) and rjx(l') are called classically equivalent if for all O € Oc one has ]im(r, \im( x(l),Or, x(l)) Vx(l),0 Vx(l))
x -.o
l ' ) ,x(l'),0 OnVxx{l')). = UrnM ]lm(ri (l')). x-»o
The third assumption states that classically inequivalent coherent states become orthogonal in the x —* 0 limit. In particular: A s s u m p t i o n 3. The limit SfaxCOi^x^')] - ~ ^ m x-*oX m ( 7 ?x(0! 7 ?x('')) for all 1,1' G L and $[yx(l),yx(l')]
satisfies the
(i) 3?e{$[7/ x (/),77 x (/')]} > 0 if nx(l) and nx(l')
exists
conditions: are classically
inequivalent.
160
CHAPTER
6. CLASSICAL
LIMIT
(ii) Ke{$[7? Ke{$[77x (/),77 x (/')]} = 0 ifr) if r)x(l) x(l) and r]x(l') are classically equivalent,
and
idt{[r, t5,{*fo x ( ?' )).,«e xp p( y( ^^tix7)x^((07]' )}]l}«W = o = 0 VX g L. x(l),exp^txjr,x(l)} x (0] - * [h% x (/),exp(-^A-)i; As shown in Yaffe [109] assumption 3 implies that classical observables cannot "move" the coherent states. Hence, any fixed [7*,n cannot be a classical observable except t/* = In
. However, as shown in Yaffe [109] assumption
3 implies that any X e L is an acceptable classical observable. Moreover, as pointed out in Yaffe [109], assumption 3 implies that if T]X(1) and rjx(l') are classically equivalent then
.
(v (i) ovAn)
As shown in As shown in establish the establish the
Um x ; =■■lim0 (/) f of ro ar U OOe ae a . lim lnO i mVx(l) aU x - oo (VX(1),VX(1')) 1 w x - o (vx(l),Vx( ')) x-o " * Yaffe [109], this fact together with assumption 3 allows Yaffe [109], this fact together with assumption 3 allows following factorization for any pair of classical observables following factorization for any pair of classical observables
one to one to 0 and O and
0': Hm[
(6.7)
With these three assumptions one gains some control over the x ~* 0 limit, however the quantum dynamics is left completely unrestricted. In order to gain complete control over the x —» 0 limit one has to require A s s u m p t i o n 4.
Tix is a classical
observable.
As shown in Yaffe [109] this set of assumptions is sufficient to show that a quantum theory reduces to a classical theory as x ~> 0. We now discuss the classical limit of the action functional in (6.6). Since we are working with Lie groups that have irreducible square integrable representations assumption 1 is automatically satisfied. We assume that we have selected
6.3. REPRESENTATION
INDEPENDENT
PROPAGATOR
161
the fiducial vector r\ such that assumption 2 is satisfied and we also assume that assumption 3 is satisfied. To satisfy assumption 4 we restrict ourselves to d Hamilton operators that are arbitrary polynomials of the generators {x{Xk}f. . k} k=1.=1
Then the most general classical action functional appropriate to the coherent state propagator for G is given by
»A tt\ \ = // s £
lhia == KmJ[ih(71{l),j^mJ[ih(r,(l),^r,(l))-(r {l))]dt )(l),n(X 1,...,X tr1{l))-{T1{l),H{X l,...J d)T1d)r,(l))}dt d
" xA „k ' ( j ( ! ) m) ! \ - ? < ( ^ , ( l ) )
■*
= /I ^
|_i,m=l
££
(6.8)
&
m (g(i))i vk-n{xkri(i))
dt
(6.8)
XJWWvk-H A m i ( K ! ) ) ! > -[Y w J(U^x^\l)v ( Ii,...,Y K , . .jVMd\l) p /%(\l h ) \6=1 \6=1
ik,m , m ==]l
6=1 6=1
dt,
/
; linn-.of'Ji^t'))) Mm.K^o(n,X where we have used (6.7) and and(4.5). (4.5). The ThevVk = limft_,o(7?,^/t?7), k, k — — kn), k =
l,...,d, l,...,d,
are constants. are real real constants. Extremal variation of this action functional, with respect to the independent labels / 6 , holding the end points fixed, yields the equations of motion d
d
£
=£
s 6 n d,c[Uj(l)}vff,, cXb (g(l))-d lb\c (g(l))}i = £ Vs{d, Md,cA = ££ Hadl'[Uj(l)]v 6 (
s
6,s=l
6,s=i
b
a
(6.9) (6.9)
o,/ = l
o,y=i
where Haa denotes the partial derivative of H with respect to the a-th argument; where H denotes the partial derivative of H with respect to the a-th argument;
a= a=
l,...,d. l,...,d.
Remark 6.2.1 Generally the constants V\, ...,vd are nonzero and are the vestiges of the coherent state representation
induced by n that remain even after the limit
h —> 0 has been taken. A similar statement also applies to the case when G is compact. O
162
CHAPTER 6. CLASSICAL LIMIT
6.3
Classical Limit of the Representation Independent Propagator
In the case of the representation independent propagator one identifies the classical action functional as (see Proposition 5.4.4) J Id d
rf
dd
(6.10) (6-10)
- (H(ii{p,/),..., p ,)0])A ]^ / C P ]PjP i'J-'W * i ( P , 0 . " - . MzPd .( O
=
3J=I =1
J
t /
\d V=
l))Pi), P;' ; -WE^ 1 i i W))ft,-,E' , " 1 ^sW)ft
(§• V =1
x
JJ=I =1
A dt. dt.
/.
Varying this action functional holding the end points fixed yields the following set of equations of motion: d
£
a = ^H = p-\b(g(l)), "wv-^Y^o),
/» ib
(6.11)
aa = = ll
d
£
a pPc = - J2 ■H dlc[p-\\g(l))]Piw°3ic[/rVaK0)]p;c
(6-12)
a,j=l s nto Substitution of H Hss -= « T}*_i Pf"(g(l)W (6.12), and contraction of both S /==*ii P/ (ff(0)' /7 iiinto (6-12), P/'OKO)' _1 C sides of the resulting relation with A 7/ l( 5(5(0) ( 0 ) yields: d
d
d
1i . (ff(0)p>. ( (0)ft.(p/(j(0)]p" . (j(0)py, =££ £E2//A <»~1A-"c(s(0)^[p/(ff(0)]p-
£ A>-Y(ff(/))Pc -Y(KO)PC = cc = = ll
/
c,s CS == l\ //,,;; == 1!
1 B h fl
1 i
where d
d
3= 1
5S == 1l
>/'(ff(0)&<[p-VGK0)] == E£p/G?(0)Mp-V(s(0)] £-0/<[p/W))]p-7(s(O) E >0,«[p/'($(O)]p-7(s(O) has been used.
(6i3) (6-13)
6.3. REPERESENTATION Claim:
INDEPENDENT
PROPAGATOR
The following relation holds:
d
E £
163
d
E
ft-[A-l^(ff(0)]rpi. £ I S,~[A"V(;•
//7
, e c , , . (j(/))ft-=' A>A (•?('))?; = k ( S((0)ft«[p / '(ff(0)]pff (0)9,< [p/( ff (0)K 7
(6.14) (6.14)
Proof. To establish equation (6.14) it is sufficient to show that Proof. To establish equation (6.14) it is sufficient to show that d
A-V($(0K/(s(0)fy[p-VGK0)] £Y X-\ (g(l))p '(g(l))d [p-\ (g(l))}
dimX-\\g(l))
d
dtm\-\\g{l))=
i
/
m
(6.15)
b
lf
(6.15)
holds. Equation (6.15) can be rewritten as
1
b
E;v(ff(0)*-[A-V(5-(0)j v w o i M ^ ^ ) ) ] == '£p^■CafCOJft-lp(g(im . ^))]. £< m*(g(t))dt~ip-\ 1 4
5 ==11
5s =:1 1l
which is the expression given in Corollary 4.2.3 (iii), and therefore, establishes (6.14). ). D a If one inserts (6.14) into (6.13) one finds
i r*
= 0. ja!/ XZ^hO/MPi > - V 0 / ( O ))Pi > i =0-
(6.16) (6-16)
Therefore, we can choose a set of integration constants, cc\,j , ...,c
(6.17)
m=l
Substitution of this form of pj into (6.11) and (6.12), yields the following set of 2d equations d
lh :
t d
a EZn (jZuk'(i)cAp-\\g(i)), o=l
d
0=1
d
Ea>[Ac'(ff(0)K ([A/(ff(/))K = - E E
£* j= l J=I
/
VJ=1
d
-££
a=l ;,m=l a=l ;,m=l
wo
/ d
\
\J = 1
/
I/ , / (EE^ *'((/)c.j3/.^)'=.)ai.^-11flflJJ'(ff(0)]A,'(ff(0)]A,-mm(ff(0)cm. (ff(0)cm.
(S
164
CHAPTER 6. CLASSICAL
LIMIT
After differentiation with respect to time these equations take the form d
P =
E? X>Vx,\ff(Q). a= l
(6.18)
a= l
d
d
3,»[Ae'Gr(0)]/**. 4,3 6 , s = l1
d
l i
mm
h.[p-Jw))]*j Jwmi W))<:m(*w)*» - - E E n'd,.[p-
(619)
a = l ; ' , m = li a = l j',m = l
EL
Next contract (6.18) with £ * = 1, ddie find [\hss{g(l))]c {g{l))]csss and find tc [\b d
E
&-[Vfo(0)]J*C ft-[V(j(0)]J*«.
=
6,j == l
fc
(/))]/ ^[A '(ff(/))]/'c, E^^[A/(
b,j = l
&,»=!
(6.20) (6.20)
,i = = 1l ao == li l6,3
d
E
E E^ W OJWVbWMc, EE" « VVV.WW))a-[vwi))]c„
e fl
Cj
==
E EE ]
- E
a mm i W ^(p- , a JI'(5(0)]A, W°a,.[p(»(0)cmm.. (6.21) o (s(0)]Ay(i?(/))c
a = l j',m =— l1 a = l _;',m = l
Subtracting (6.21) from (6.20) yields d
£
d1
£
c a adic[Uj{l))c , {^A ^ A /c'( ( S5 (Z))} ( 0 ) } /l6b = E 2 H-H E :C *S{$« V6(Ss((50( )0 )- - fyA di.[Uj{l)]cf, }
6,s=li
6,5 = 1
(6.22)
a,/=l
where Corollary 4.2.2 has been used. Among all possible allowed values of Ci,..., c& are those that coincide with » ] , . . . , vj, for an arbitrary fiducial vector. Hence, for this choice of Cj,..,,Cd the above equations coincide with the equations of motion obtained from the most general classical action functional for the coherent propagator for G [see Eq.(6.5) and Eq.(6.9)]. Therefore, the set of classical equations of motion obtained from the classical action functional of the representation independent propagator implies the set of classical equations of motion obtained from the most general classical action functional of the coherent state propagator for G. Thus, we find that the set of solutions of
6.3. REPERESENTATION
INDEPENDENT
PROPAGATOR
165
the representation independent classical equations of motion appropriate to the representation independent propagator for a general Lie group G with square integrable, irreducible representations includes every possible solution of the classical equations of motion appropriate to the most general coherent state propagator for G. We summarize all this in the following Proposition: P r o p o s i t i o n 6.3.1 Let G be a real, separable, locally compact, connected and simply connected Lie group whose unitary irreducible representations
are square
integrable. If the fiducial vector satisfies (6.3) when G is compact and tion 2 when G is non-compact, action functional
Assump-
then the equations of motion obtained from the
of the representation
independent propagator imply the equa-
tions of motion obtained from the most general classical action functional the coherent state propagator for G.
for
Chapter 7
CONCLUSION AND OUTLOOK In this chapter, as before, we mean by a general Lie group a real, separable, locally compact, connected and simply connected Lie group with irreducible, square integrable unitary representations, unless we explicitly state otherwise.
7.1
Extension to Groups t h a t do not possess square integrable, irreducible Representations
We have focused our attention in this book on general Lie groups with square integrable, irreducible, unitary representations, since for this case the existence of a resolution of identity is guarantied in general by Theorem 5.1.2 and we were able to construct a representation independent propagator rigorously. It would be interesting to see if the construction of the representation independent propagator presented in §§ 5.4.1 can be extended to Lie groups that do not posses square integrable, irreducible representations, such as the Euclidean group. The obstacle one has to overcome when one considers such groups is the introduction of a resolution of identity. This problem has recently been 167
168
CHAPTER
7. CONCLUSION
AND
OUTLOOK
solved by Isham and Klauder [56] for the n-dimensional Euclidean group
E(n).
In Isham and Klauder [56] attention is focused on reducible, square integrable representations of E{n).
In this case it becomes possible to introduce a set of
coherent states, i.e., to establish a resolution of identity, and to introduce a coherent state propagator. Therefore, if the problem of introducing coherent states for these groups can be solved, then one can use the following argument to introduce a fidu cial vector independent
propagator for these groups. Denote by U a generic,
continuous, unitary representation of a Lie group G on some Hilbert space H , which does not need to be a square integrable, irreducible representation. For definiteness let us assume that we have parameterized the Lie group G such that the representation U is given by:
Ug(l) =■ exp(—il=exp(-il exp(-il1Xl)...exp(-ildXd), X\,..., for some ordering, where the X\,...,Xj
Xj, form an integrable, representation
of the associated Lie algebra L of G by essentially self-adjoint operators on some common dense invariant domain D € H and where / is an element of a rf-dimensional parameter space Q. Let us denote by r)(l) the coherent states associated with the representation Ug^ of G, where r\ 6 H is the fixed, normalized fiducial vector. Let us furthermore assume, that these states give rise to a resolution of identity 'ih H=
i
/ I ij(0W),-)<W9, ?(0M0.->«M0.
Jg
where dji(l) denotes the normalized, left invariant group measure given by where dfi(l) denotes the normalized, left invariant group measure given by {
*tii) : -M ), W\
7.1. POSSIBLE EXTENSIONS
169
where l/\G\ denotes the normalization; for the definition of dg(l) see (5.4). We can now use this set of coherent states to give a continuous representation of H. We define the map C„:H
-♦
LL22(G,dfi{l)) (G,dn(l))
4> -> M ] ( i ) = A,m = w / U ) , which as we know yields a representation of H by bounded, continuous, square integrable functions on the closed subspace L2(G,dfi(l)) of L2(G,dfi(l)). We now introduce the fiducial vector independent propagator Kn(l", t"; I', t') as follows, it is a single, (possibly generalized) function that is independent of any particular choice of the fiducial vector, which, nevertheless, propagates the ipv correctly, i.e.,
■i
KH(l,t;l',t')iP i> (l,t) = If KH(l,t;l',t')iP v(l',t')d»(l') Mln>t)= v(l',t')dn(0 Jg5 If equation (7.1) is to hold for arbitrary n, we must require that 1 Jim \imKKHH(l,t;l',t')=\G\6 (l,t;l',t') =e(g\G\6 (l)g(l')), e(g-\t)g(l%
where ^ f a " 1 (/)»(/'))
is
(7.1)
(7.2)
defined in (5.22).
An analysis of our results presented in chapter 5 shows that Lemma 5.1.4, Corollary 5.1.5 hold for reducible, square integrable representations.
Even
though we have stated Lemma 5.1.4 and Corollary 5.1.5 for irreducible rep resentations, this property of the representation is not used in the proofs of these results, hence these results also apply to the case when one considers reducible representations. Therefore, it is a direct consequence of Lemma 5.1.4 (i) that for any ip £ D, xik(-iVhl)[C k(-iVl,l)[CvviP](l) m)
= [C„X [CvfcXVK0. kWl),
k =
l,...,d,
170
CHAPTER 7. CONCLUSION AND OUTLOOK
holds independently of rj. Hence, we find that Cv intertwines the representation of the Lie algebra L associated with G on H, with a subrepresentation of L by right invariant, essentially self-adjoint differential operators on any one of the reproducing kernel Hilbert spaces
L^{G,d^{l)).
Denote by Ti{Xk) the essentially self-adjoint Hamilton operator of a quantum system on H. Then the continuous representation of Schrodinger's equation on H, idti>{t) = Hrp{t), takes on L*(G,dp.(l)), the following form: idttU^t) V„(M)
= =
[CnnH(X H(Xkk)i>{t)]{l) )iP(t)](l) \C
{-ivuhi))^{i,t). = «(**! n{x n(xk(-iV l))^(l,t). Using (7.1) we find that the fiducial vector independent propagator Kn is a solution to this Schrodinger equation, i.e., i
=
/ H(xh(-iVt,l))K H{l,t;l',t'), W(i t (-»V,,J))jr ), H (M;J',t
l = \G\S = \G\6 {l)g(?)). a(g~\l)g(l')). a(g-
(7.3)
It was such an initial value problem that we have taken as our starting point for the path integral formulation of the representation independent propagator. Hence, we find that one can, using Proposition 5.4.4, introduce path integral representations for general Lie groups that have reducible square integrable representations. However, observe that we can only introduce & fiducial vector independent propagator in this case. This program has been explicitly carried out for the case of E{2) by Tulsian and Klauder [104] .
7.2. WHAT HAS BEEN ACCOMPLISHED
171
Observe that if one considers groups with reducible, square integrable representations one has to proceed on a case by case basis since a general theory in this case is lacking. It would therefore, be of some interest to see if the theory developed by Duflo and Moore [30] for locally compact groups with irreducible, square integrable, unitary representations can be extended to locally compact groups with reducible, square integrable, unitary representations.
7.2
W h a t has been Accomplished
We have seen in chapter 3 that the quantization of physical systems moving on group and symmetric spaces has been an area of active and on-going research over the past three decades; see for instance [12-14, 28, 29, 48-50, 57, 71, 7678, 95]. In particular we have reviewed in §§ 3.1.3 in detail the approach of Marinov and Terentyev [77, 78] to the construction of path integral representations for a free particle moving on group manifolds of compact simple Lie groups and spheres of arbitrary dimension. As we have pointed out, in this approach the Lagrangian needs to be modified to include a 'quantum' potential proportional to h2. However, the need to include a correction term of order h2 into the Lagrangian has also been found to be necessary by other investigators who, have starting from the semiclassical approximation, constructed path integral representations of the free particle moving on unbounded Riemanian manifolds; see for instance [22] and [74]. More importantly, this approach cannot be extended to more general physical systems than the free particle since the semiclassical approximation, which has been used in an essential way by Marinov and Terentyev, is exact only for the case of a free particle moving on the group manifold of a semisimple Lie group.
172
CHAPTER 7. CONCLUSION AND OUTLOOK We extended in §§ 3.1.4 a method used by Junker [57] to construct path
integral representations for a free particle moving on a compact symmetric space to symmetric spaces of the form M = G/H, where G is a general, not necessarily compact, unimodular transformation group acting on M and H is a massive compact subgroup of G. Again we found that this method could not be extended to more general physical systems moving on M. than the free particle, since we were asking that the short time propagator be invariant under the transformation group G. We found that this assumption implied that the Laplace Beltrami operator, was an element of the enveloping algebra of the transformation group G. This in turn implied that the matrix elements D^Ag) were regular functions on G. This showed that the assumption that the short time propagator should be invariant under the transformation group G was crucial and could not be relaxed. Moreover, as we have remarked at the end of §§ 3.1.4, one can construct path integrals this way only for a handful of groups, since one needs to know the explicit form of the spherical zonal functions D^Q{g) in order to carry out the construction; see also in this respect the remarks after equation 3.20. In §§ 3.1.5 we discussed the construction of coherent state path integrals. We found that the final path integral representation for the coherent state propagator exhibited a strong dependence on the choice of the fiducial vector and on the choice of the square integrable, irreducible, unitary representation of the general Lie group G under consideration. Hence, one has to reformulate the path integral representation for the coherent state propagator every time one changes the fiducial vector and keeps the irreducible representation the same, or if one changes the irreducible representation of G. As we have pointed out,
7.2.
WHAT HAS BEEN ACCOMPLISHED
173
in many applications it is often convenient to choose the fiducial vector as the ground state of the Hamilton operator Ti of the quantum system one considers; see for instance [102], and [103]. Hence, one has to face the problem of various fiducial vectors. In chapter 5 we have constructed the representation independent propagator. In § 5.1 we have defined coherent states for a general Lie group G and have proved Lemma 5.1.4 and the Corollary 5.1.5 which we have applied in the construction of the representation independent propagator and the construction of regularized lattice phase-space path integral representations of the representation independent propagator. Prior to the construction of the representation independent propagator for a general Lie group we have constructed in § 5.2 the representation independent propagator for any compact Lie group G. We showed that the representation independent propagator for any compact group correctly propagates the elements of any reproducing kernel Hilbert space associated with an arbitrary irreducible unitary representation of G. Hence, it solves the problem of various fiducial vectors. We observed that in the construction of the representation independent propagator for compact Lie groups and its path integral representation no explicit use has been made of the ONS y/d^D^Al), i,j
= l,...,d(,
in L2(G)
£ £ G and
whose existence is guaranteed by the Peter-Weyl
Theorem, but merely the facts that it exists and is complete have been used. Moreover, we have made no assumptions about the nature of the physical systems we were considering, other than that its Hamilton operator be self-adjoint. Therefore, the path integral representation (5.13) can be used in principle to describe the motion of a general physical system, not just that of a free parti-
174
CHAPTER
7. CONCLUSION
AND
OUTLOOK
cle, on the group manifold of any compact Lie group and it does not m a t t e r if the matrix elements D~(l) are explicitly known or not. Hence, we found that the path integral quantization (5.13) represented a clear improvement over the path integral quantizations describing the motion of a free particle on a compact group manifold presented in chapter 3. As an example we have constructed the representation independent propagator for SU(2) and presented the exact path integral treatment of a free particle moving on a circle and on the group manifold of 5/7(2). In § 5.3 this construction was suitably extended to a general Lie group and we showed that the results obtained for compact Lie groups also hold for a general Lie group. We then established t h a t it is possible to construct regularized phase-space path integrals for a general Lie group G. Even though the group space generally is a multidimensional curved manifold, we have shown that the resulting phase-space path integral has the form of a lattice phasespace path integral on a multidimensional flat manifold.
Furthermore, since
we have made no assumptions about the nature of the physical systems we were considering, other than that their Hamilton operators be essentially selfadjoint, we found that our path integral representation can be used in principle to describe the motion of a general physical system, not just t h a t of a free particle, on the group manifold of G. In addition, we found t h a t there were no h2 corrections present in the Lagrangian. Note that conventional path integral expressions for propagators are representation dependent. Rather than having to adapt each propagator to the representation in question we have succeeded in introducing for a general Lie group a propagator that is representation independent. For a given set of kine-
7.2.
WHAT HAS BEEN ACCOMPLISHED
175
matical variables this propagator is a single generalized function independent of any particular choice of fiducial vector and irreducible representation of the general Lie group, which nonetheless, correctly propagates each element of the coherent state representation associated with these kinematical variables. In our opinion the significance of our work lies in the fact that it affords a solution to the construction of path integrals for general physical systems on a wide class of curved spaces, i.e. path integrals for quantum systems whose operator quantum mechanics is not given by the usual canonical commutation relations but by more complicated group commutation relations. This work offers to the quantum mechanics of such systems a global universal solution which allows uncluttered concentration on the fundamentals of a problem. In fact, we have arrived at a novel, extremely natural phase-space path integral quantization for the motion of a general physical system on the group manifold of a general Lie group that is (a) more general than, (b) exact, and (c) free from the limitations present in the path integral quantizations for the motion of a free physical system on the group manifold of a general unimodular
Lie
group discussed in chapter 3. It is worth emphasizing that the representation independent propagator evolves any state in a way that leaves the choice of rj invariant.
In as much as
the choice of n corresponds to the choice of polarization in the sense of geometric quantization it follows that the representation independent propagator evolves the system while at the same time preserving the polarization. Since this fact holds for a general Hamiltonian we see that the representation independent propagator for a general Lie group provides an acceptable solution to the longstanding problem posed by geometric quantization.
176
CHAPTER
7. CONCLUSION
AND
OUTLOOK
In chapter 6 we discussed the classical limit of the representation independent propagator of a general Lie group G and have shown that its classical limit refers indeed to the degrees of freedom associated with G. In § 6.1 and § 6.2 we have discussed in detail the classical limit of the coherent state propagator for compact Lie groups and non-compact Lie groups. In § 6.3 we proved that the equations of motion obtained from the action functional of the representation independent propagator for a general Lie group indeed imply the equations of motion obtained from the most general action functional of the coherent state propagator for a general Lie group.
7.3
Possible Further Directions
In our opinion another interesting avenue to achieve the path integral quantization of the form (5.32) for general Lie groups that do not have square integrable, irreducible representations would be to start from the classical mechanics associated with the particular Lie group one considers and to try to derive the form of the action functional we have arrived at in Proposition 5.4.4.
The
quantization would then be achieved by postulating (5.32) as the path integral quantization for these kinds of Lie groups. Furthermore, we believe that the representation independent propagator holds considerable interest for quantum field theory.
We have used in this
book the word representation independent in a dual meaning, its first meaning pertained to the fact that the representation independent propagator is independent of the choice of the fiducial vector and its second meaning to the fact that this propagator is also independent of the choice of the unitary, irreducible representation of the Lie group G. In the case of quantum field theory these
7.3. POSSIBLE FURTHER DIRECTIONS
177
two meanings of the word representation independent are inextricably related, since the dynamics chooses a representation for the basic kinematical variables, see for instance Haag [51, pp. 56-57] and Klauder and Skagerstam [69, pp. 8283]. We therefore, believe that it would be a worthwhile task to extend our concept of a representation independent propagator into the realm of quantum field theory.
Appendix A
CONTINUOUS REPRESENTATION THEORY A.l
Continuous Representation
In this Appendix we confine our attention to real, locally compact, separable, connected and simply connected Lie groups G that have irreducible, square integrable unitary representations.
Let us denote by U a fixed continuous,
irreducible, square integrable unitary representations of G on the Hilbert space H. Let Xi, ...,Xd be an irreducible representation of the basis of the Lie algebra L corresponding to G, by symmetric operators on H satisfying Hypothesis (A), see § 4.1, then L is integrable to a unique unitary representation of G on H. Suppose there exists a parameterization of G such that d
ug[g[i) = Y[exPP(-iikkxkk) = expi-u'x,)..
.exPP(-uddxdd);
where / is an element of a d-dimensional parameter space Q. 179
180
APPENDIX
A. CONTINUOUS
REPRESENTATION
THEORY
We defined in chapter 5 the set of coherent states for G, corresponding to a fixed square integrable unitary representation Ugy) as: V(l)
l K '\ =U Ua(l) K^\ g{l)
2 T)eeB(K^ T>(K ) 1'2) and V
n\ = h
\\n\\ = 1,
where K is the unique self-adjoint, positive, semi-invariant operator with weight A~l(g(l))
given in Theorem 5.1.2.
Claim A . 1 . 1 The map Cv : H -+ L2{G),
defined for anyip £ H by:
[C^m) M ( ' ) == Ul) W )==W\i>) W0i) maps the elements
of H into complex, bounded, continuous,
(A.l) square
integrable
functions. Proof. Let ip £ H be arbitrary, then
WOI 1^(01
= \W),i>)\ =
\(U \{umK Kll'\i>)\ iSM
2 < \\U < WU^K^VWUW g(l)K'l n\\U\
<
1/a ||0-, | | ^ Ww| |||||jr | | J f 1 / a i!7||IMI 7||||^||
< M, M, v leg, where we have used the Cauchy-Schwarz inequality in the third step and where M is given by M = ||ij' 1 '' 2 7?||||^||. Hence, the functions ipn(l) are bounded. That the functions ipv(l) are continuous follows from the fact that the unitary representation Ug^ is strongly continuous, and therefore, weakly continuous. To see that the functions ^,,(/) are square integrable, let ip e H be arbitrary, then since rj 6 D(A - 1 / 2 ) it follows that Kxl2n
e D ( A ' - 1 / 2 ) , therefore, by The-
orem 5.1.2 (i) the function ipv(l) = (Ug^Kl/2n,tp)
is square integrable. Since
A.l.
CONTINUOUS
REPRESENTATION
181
V> 6 H was arbitrary it follows that the functions rf>^(l) are square integrable (cf. remark 5.1.1). □ We denote by L*(G) the space spanned by the bounded, continuous, square integrable functions ipv(l). The space L2{G) is clearly a subspace of L2(G),
and
is therefore, an inner product space. The inner product on L"l(G) is given by the restriction of the inner product on L2(G) to L2(G).
We denote the inner
product and the norm on I»(G) L*(G) by by (•»•)_ (.,.)„ and a a d|| |• [|| . |=| ,( (3 •(, (• )„) v }1 ^/2 , respec tively.
Claim A. 1.2 The map Cv defined in (A.l)
is an isometric
the Hilbert space H onto the inner product space
isomorphism
from
L^G).
Proof. To show t h a t the map Cv is an isomorphism from H to L^G)
we have
to show that Cv is linear, one to one, onto, and continuous (cf. [31, II.3.17]). By definition the map Cv is clearly a linear operator from H onto
L^G).
To see that Cv is one to one we have to show that [CV4>](1) = [Cv(j>](l) V / 6 5 implies ip =
4>)==oo Vie Viea.a. [c„(v - *)](/) = MO.tf - +)
(A.2) (A.2)
Since the unitary representation J/Sm is irreducible the set T = {r](l) : I € S} is a total set in H (cf. [7, Proposition 5.4.2]). Therefore, (A.2) implies that ip = 4>. Hence, the map Cn is one to one. We show next that the linear operator Cv from H onto L^G)
is continuous.
To show that Cv is continuous it is necessary and sufficient to show that Cn is
182
APPENDIX
A.
CONTINUOUS
REPRESENTATION
THEORY
bounded (cf. [88, Theorem 1.6]). Let tjj 6 H be arbitrary then -,1/2
/ \Ui)\a
=
w UQ r /■
J
/2 "Ii 11/2
V(t))(V(l),i})dg(l) ) / (i,, (iM(0)M0,VW) UQ J
= IIV'II, = Ml where we have used (5.3).
Hence, Cv is a continuous linear operator.
therefore conclude that Cv is an isomorphism from H onto
We
L^(G).
We now show that Cn is isometric. Let , TJJ € H be arbitrary then
([C0](O, [Cnm\ ([c nmuc,m%
L
= / t(,vV)Mi),1>)dg(i) (*, 9(0)M0.tf),1>), {<£, V>),
(A.3)
where we have used (5.3) in the last step. Equation (A.3) shows that Cn is isometric. Therefore, we conclude that Cv is an isometric isomorphism from H onto L*{G).
□
Claim A.1.2 shows that the Hilbert space H is isometrically isomorphic to the inner product space L*(G) and it follows from this that L*(G) is a Hilbert space (cf. [107, Theorem 4.9]). We call L^(G) the c o n t i n u o u s representation of H.
A.2
Reproducing Kernel Hilbert Spaces
A reproducing kernel is abstractly defined as follows (cf. [80, pp.42-43]): Definition A . 2 . 1 Let £ be a topological space and R be a Hilbert space whose elements are functions the function only if
from £ to C, the set of complex numbers.
We say that
K : £ X £ -> C, (/', I) i-> /C(/'; /) is a r e p r o d u c i n g kernel, if and
A.2. REPRODUCING KERNEL HUBERT SPACES
183
(i) K,(l'\ I) belongs to R as a function of I' for all I. (ii) K.(l'\ I) has the reproducing property: for all R, 4,(1) 4>(l)==(£(/'; (£(/';I),I),0(Z')),, 4>(l'))v 4> inin R,
(A.4)
We say that R is a reproducing kernel Hilbert space if there is a function K. satisfying conditions (i) and (ii), i.e., if it has a reproducing kernel. It follows from this definition that the reproducing kernel has the following properties: £(/;/) > 0, £(/;/) >0, \IC(l';lf 2 \IC(l';l)\
K.(l'\l) = lC(^V) K(l\V) < IC(l';l')IC(l,l). IC(l';l')IC(l,l).
(A.5) (A.6)
To see that the inequality (A.5) holds we use the reproducing kernel property (A.4); since for fixed / we have that /C(/'; ! ) ? R one finds
IC(l;l)==(W;i)Mr-,i)h /c(/;0 {/C(r;0,/C(/';J))j<==ll^';')ll WW'Jt2 >>o.0. From l (£(/"; (/c(/"; /'),£(/"; /'),ic(i"\ /)),„ i)h ==£(/'; W i)0 (£(/"; W";1),£(/"; 0.W\ /')),« %»==K(f; «(f;0 0
and (ii) in Definition 1.2.1 the second statement in (A.5) easily follows. (A.6) is established as follows: |/C(/';0| 2 |/c(/';0l
,
IC(l';l)IC(l';l) = = /»(/c(/";/'),'C('";0>/» ;/').«('"; 22 = !(£(*";/'),£(!"; 0M = |(/c(/";O.'C('";0) I»l 2
2
||^(;';/")II 2II^(M")II < ||^(;';/")II II^(M")II2 <
)C(l';l'))C(l;l), JC{l';l')IC(l;l),
184
APPENDIX A. CONTINUOUS REPRESENTATION
THEORY
where we have used the Cauchy Schwarz inequality in the third step. One can generalize the inequality (A.5) as follows; let A„, v - 1,2,3,... ,n, be arbitrary complex numbers then n
WC(/„,y>o E ] £ A A^/C(/ ,/ )>0 —1
v
v M
(i„,LeC). (i^L eC).
Using (A.4) and the linearity of the inner product we find Using (A.4) and nthe linearity ofn the inner product nwe find
E<'A„/C(7 ;/„), n
n
j
AM£(/';/M));,
E n
Aj/A^/C^i!/;
l^).
o < ( £ *„£(/'; k),EW'';W)i< = E ^WrfUHence, the reproducing kernel is a positive definite function. Claim A.2.1 A reproducing kernel Hilbert space can never have more than one reproducing kernel. Proof. In our proof we follow [80, p. 43] suppose there exists another function /C'(/';/) which has the reproducing property (A.4), then ||/C(/';/)-£'(/'; OH OH'2 =
(/C —(K-K.',K-K.') IC j IC — /C V)[/
— (/C,/C — = — /C K,)^fp — —(/C \K,,/C ,/C——/C/C)^/);<
=
/C(M)-/C'(/;/)-/C(/;0 + £'(/;/) IC(l;l)-IC'(l;l)-K(l;l)
=
0,
since both kernels have the reproducing property. □ We now show that the function £,,(/'; /) defined in chapters 3 and 5 as
^a';o = W).^)>, ICv(l';l)=(r,(l'),V(l)},
is a reproducing kernel. Repeating the proof of claim A.1.1 word by word, sub stituting ICn(l';l) - (*?('')> ^(O) w ' t n ' arbitrary but fixed for ipv(l) everywhere
A.2. REPRODUCING KERNEL HUBERT SPACES
185
we see that ICn(l'\ I) is a continuous, bounded, square integrable function of /'. Therefore, /C„(/'; /) belongs to L2(G) as a function of V for all I. From (5.6) we see that the function ICv(l'; I) satisfies the reproducing property. Hence, Kn(l'; I) is a reproducing kernel and therefore, by Definition A.2.1, lJl(G), which is a Hilbert space of continuous functions, is a reproducing kernel Hilbert space. Claim A.2.2 // /C(/'; /) is the reproducing kernel of a proper closed subspace R' of a Hilbert space H, then (l%; is the projection of 6 H onto R'. Proof. In our Proof we follow [80, pp. 47-48] by the Projection Theorem, see [88, Theorem II.3] every element of H can be written uniquely as
0(O = ^'(O + X(O. x(O. where, a11 4>' S R', and {4>',x)r (<j>',x)i> all = °= f0o r for 'cj>' € R g' R'.
Since IC(l'; I) belongs to R' as a function of /', we have
/' (0}/< == 0, o, (/C(r and hence, we see that (IC(I';lU(l%, x{f))v (£(/'; l)Ml%> == (IC(?iI),*V') Wil),W) ++ xW))v =
{IC{l';l)A\l'))v
= 4>V)- Da Since, Kv(l'; 1) is the reproducing kernel of the proper closed subspace L2(G) of L2{G) it is the kernel of a projection operator from L2{G) onto X^(G).
186
APPENDIX
A.3
A.
CONTINUOUS
REPRESENTATION
THEORY
Proof t h a t Equation (5.22) is Well D e n n e d
All measures considered in this section are positive and
of measurable sets in the
a-algebra B such that OO oo
-u[jx
X X==
Xnn
n=l
and v(Xn)
< oo. The Lebesgue measure dx on ( - 0 0 , 0 0 ) is an example of a
(7-finite measure. Definition A . 3 . 1 A measure v on G is a standard B o r e l m e a s u r e if there exists a Borel set S such that v(S) dard Borel space (it is isomorphic
= 0, and the subspace G — S is a stan-
with a Borel subset of some complete
metric
space). By Corollary 5.1 in [30] there exists a standard Borel measure 1/ on G, a //-measurable decomposition (U^H^),
^ of the type I part of the left regular
representation of G, and a measurable field (A'^)../, of nonzero, positive, self1) adjoint operators such that K{ is a semi-invariant operator of weight AGr A(<7 -1 )
in H<; for ^-almost all C € G such that (i) For Q £ Pj[V(G)] the operator ICJ U(-{a)K^2 (ii) For a,P £ Pi[V(G)},
is trace class.
one has
■L
1/2 <5 (a**/3) <5ee(Q* * / ? ) = / tt{Kl ti[Kl/2m(a* m(a**(3)K^]d * /3)A ]^(C)v(C). C
JG
Claim A . 3 . 2 The equation (A.7) is well defined.
(A.7)
A.3.
187
PROOF THAT EQUATION (5.22) IS WELL DEFINED
Proof. Suppose there exists another standard Borel measure v1 and measurable fields (U'g ,H^)-e(A and (Ki),~g with the same properties as above, then by Theorem 5.2 in [30] the measures v and v' are equivalent, and one has, for v- almost all C G G
1
(A8)
M^fW -
K'c =
\di/(Qj
)v
where Vj is for all £ 6 G an operator intertwining Ug and U'g , i.e. V(Ug = U'g V<; and V( is bounded forall £ £ G . Therefore, using (A.8) one can write I/ tT[K' U'<{a' * *P)K' P)K'1/li2\di/{C) }dv'(() tt[K'll/2/2U'((a* *P)K'\
JG
=
« */3)]if/(C) [/ tr[X' c lA'C(A-i/a tTlK'cU'^b-Wa'KWdi/iQ A~i/2 a0**/?)]A/(C)
JG
N
<M0 l = V-*U«(A-^a'*P)} = ^It t rtr[V V f JcK f (cV ^ | )) d»>(( «fr'(C) c- ^(A-i/»a«*/3)] ( | ^ = Jf66tT{V = ti[VlccK *fi)V?) ((~n (K((U<(A-V*a**0)V (U<(A-V*a' v c = = = =
( | 4^£ L) ) du'(C) & /(0 dv'{()
MO/ / .yt .r [t A C (' A -/ ., / 3 ) ] ( ^ § ) M 0 r [^ A c c / 2Ha A - ^ a . ^ ^ ^)^ MC
JG
//
'■■
"
'
J
V M O ;
/2 /2 /2/2 ti[Kl V<(a'*/3)Kl ]dv(0, [_tT[Kl ti[Kl/2U<(a'*0)Kl V<(a'*/3)Kl/2}dv(O, ]dv(0,
JJ G JG G
where the fourth equality holds by tr(A5) = tr(5A), for A trace class and B bounded and the last equality holds by Theorem 32.B in [52], since v and v' are equivalent and a-finite. Hence, we have shown that
L
-L
/a 2 / ti[K>\ tx[K'\f2/2U"-{a* ]di/'(0 ti[Kl/22U<{a* U<{a* U'((a* **/?)Jir'J P)K'V ]dv\Q = /f ti[K\'
JG
*P)Kl,2,2]dv((), }dv{(),
JG
for any two decompositions of the type I part of the left regular representation of G. Therefore, equation (A.7) is well defined. D
This page is intentionally left blank
Appendix B
EXACT LATTICE CALCULATIONS In this Appendix we present the exact lattice calculation of the representation independent propagator for the affine group for the following two Hamilton operators:
(i) W ( 1 H(X , J 2uX ) 2=)=£.Xl ^,2, 2 (ii) W(X H(X1,X 1,A' 1 + W X 23.. i) 2) = ^A:&X?+uX
B.l
The T h e Free Particle
Our first example is that of the Hamiltonian H(pk,x) H(pk,x)
2
2 {pk) /2m. Here p = (pk) /2m.
is restricted to be positive and the variables k,x are unrestricted.
Let pj =
(Pj Apj = ppj+iJ + i -- pj, = (
lim
//
N — oo
,0
. f yv yv p • / e x P v ' ] C k: + i/i(iiAPj J
{ f?o L
189
+ P J A 9 J ) - *i+ 1/2-r1 p
>
190
APPENDIX B. EXACT LATTICE
=
€€
(P;fcj + 1/2)22"]1 (P;fcj+i/2) l'[ £j T^-
2m
T dkj + l/2dxj + i/7 r^r^ rffcj+1/2^+1/2
)j I I dqdq>dVdV> 11
J
CALCULATIONS
> ; = 1
<2*T (SO 3 (2TT)2 ^ '
,=0
\2
1 / . ■ . /exp j i f [ - £ ^ ± tM ^2^l ! + +fcj+lM&Ap, kj+wtoApi+pjbqt) + P; A 9; ) N—»oo / / I *—* Zm m J J L
ton
m
^
-
^
{ ;=0
/ -
f*.Aii, J
W N
m
m
X?
A
«.A/».1Z_L.
AT
,,
/ -
9 Apv 1 * A \2 ij Apj + p) A?,-)' - Xi+Wfy f
( i-i A n . _1_ « ■ A /» • 1 * — *r • . . -,.
,
TT W„ ,/„ TT ^ ; + l/2<^;+l/2 xx[[d Yl dp, ^ Pldqi[[ v
M3 = 1I ^ ^ U — ( 2 ^ —
2
=
r I2 ( J N f - t p_22 T m /r/. . . .r /exp A A9;)l ?. A + K m e x n l i V j ^ L ? lim J" *, +i/2 2^?(9i W +P> JL oo/-/ ( g { ^ r [ & * / * -r 5 ^ ^ * « >
N-ooJ
\ ;=0 ^
+ j|m{qAp, Ap + pA?j) A
2^ > >+^ ;
+ fe
2 9j)
L
J
J
Ap
^ ^ TT ^i++ 1 / 2 ^i +41 / -~Ji +, +1i//22^^> }}\ ^)J TT n/Pi^n/' g nd«d*> n ((W)2
Now for all j we let fc,k]+1+/i/2 fcj+1/2 + j^i^j^Pi j^ilj^Pi 2 = fcj+1/2
Pj&Qj) + Pj^Qj)
an
d carrying out
the integrations over fy+1/2 fcj+1/2 we find find
K(p" tf(P",>q",t";p', ,",*"; qP',f) ',
+ ^]} ^]} 7-/ eexpx p fs[s( AA9 i+++^. 2 ij+i/2 d *W» 4M
m == A&(*?)' oo(—j
.. n j . J . TT
++1)/ 1)/
t y7-7 ■••y exp {|'g[£( §[&l A *«+^"; ^)'-' ^) -
dl
;+i/2
As a next step we carry out the integrations over the Xj+1/2 Xj+l/2 and find /'<(P < ( p ",9, 9 ",t, i "\p ; P ' ,q , 9 ' , f,t) )
-■ Jtc?r7 b.&TJ-h -/-{'§[£(-*»i) >t§ [£(**+IN'])
•
B.l.
THE FREE PARTICLE
191
-n^CfOn**, On* ;=0
•'
" "
'
j=l
a ++ W -„-(^r,>7/-{'S[s( " W]) I ;=oL S(> P;
JV JV N
N«
Jj3=0 =0
j3=1 =l
>n< \("+: lim f • ts.(&r"i~iJ
J
xTT6(APj)TT
JV—00 V
N N
x
L
N N N
£ kbit J=0 3=0 j=0
n s(p>+ >+11 -~ppi) i) nn ^ft'^'j ^ft'^'j ••
jJ3=0 == 00
;; = =1 1
Making use of the identity
J
/ <5(x 6(x - y)6(y - z)dyy = = S(x6(x S(x - zz),
and carrying out the integrations over the pj successively we find K(P",q",t"-p',q',t') Sip" - p')') line lim ( ~ )
'7-/
K(N+1)/2
'
N
IN
2 !?i+i -?;") P iiJZ^ili+i-Qj) J/ " l j=0 ;=0
J
X
[T%-
J ;=1
This last integral can be straightforwardly evaluated using the following definite integral:
A
a
v£
/
ab
ab
p rh a• a(cl( x - u ) 2] W - ex5xp[-6(u-y) exp — ; ( x - y ) 2 •2 p[. ^ / ^— ] < f u= = /-co i/\ /^—e xexp[—a(a -«) ]\/r xPH(uv)) 22]du= ]du \ / ^ 6 ) ye Xe Fx p a[-£-b(*- y) • J —oo 1
2
2
o)
L +'
(B.l)
for Ke(a) > 0,Re(6) > 0 extended to &e(a) = 0 = 3?e(6) as an improper integral. This identity is easily proved by completing the square in the exponential.
192
APPENDIX
B. EXACT
LATTICE
CALCULATIONS
Using ( B . l ) the integral over qi is given by
/ m \ f e x \im, .,1 e x rim ,,1 \im, ,2I '*»«/ ^2 ' m /. \2l d JJ, e xP Trill-9°) ^92 - 9i [■w-r) P IT— (\2wit) — ) JJ/ exp ^It- ( 9 i --9 o9o) ) a«i xp 27(93-91) v^lirii/ It- ((??23-- ?? Il ) ■ I- 27(91 a9i / m im "' / /\2 eXP eXP3 = VM27) = V 2^(27) 2^(20 eXP [2(20 [2(27; ( « " ' ) . ■
where we have included two of the (m/2nie)1/2
factors and have identified q0
with g'. The effect of the 91-integration is to change e to 2e, both in the ex ponential and in the square root, and to replace the exponentials by a single exponential which depends only on
has been inte
grated out. Proceeding in this manner carrying out all the ^-integrations we find as our final result:
/i(p",?",<";P',9',0
'fevW ~q). % i{p x p9 (9 ?,)2 ■6(p"\ji^-t') " ) e')exp [W^)^''- ~q).'' =z v^^ "^' [^^ :
6(p"-p<) Jim c\/27ri(Ar+l)fexp . P
where we have made use of the following identities QN+I - q" and (t" - t') = (N + l)e. Hence, our final result is given by
I m im im f 11 /x2 2 K(p",q",t";p',q',t') K(p", q", t"; P>, q>, t>)== sf^Kv" - ^ < 5 ( p " - ~p 'P') ) e x ex f P [ g( ( g " - 9 ' ) " , \j 2wiT 2T p 2f{g -qf , where T = t" - t'. Observe that this result agrees with the usual result for the free particle except that here p is restricted to be positive, i.e. p > 0.
B.2.
B.2
= A"S+-x.
THE HAMILTON OPERATOR W(Xi, X 2 ) = ^ X f + uX2 2M
193
T h e Hamilton Operator 7i{X =^ Xf+ « ( XuiX, X 2^x2 2) 2 ) = u;X22 o;X = (pk)2/2m
Let us now consider the Hamiltonian H(pk,x)
+ wx. With the
notations of the previous section, the representation independent propagator for the affine group takes the following form:
,, >",q",t";p\q',t')= K{p",q",t"-p',q K{p",q",t",t') = Jim lim^J. lim I ... . ./exp ex i i £ [[fcy kj+lf2 +pjAqj) pjAqj) V',q ,t') + 1 /{qjApj 2 (?jAp J -+ N-coJ J ( ;=o 2 22 A (pjk Ap> (pjk U TT TT JJ J J TT TT dkj+i/-2(ix (Pjkj+i/2) 7 + l+/i/-2dx 2 ^ j +j+1/7 l/2 Pj j+1/7 Ap> ^P] (pjk WKjj+l/2 +))h U dkj j+l/2 -Xj + i/2-r--( ;1m 3+1/2 m ' ' pj 2m J j1 v(2*)' y " - 3jf=i = 1 3=0
Vj
= =
J
J
3=1
3=1
fe t /l)i exp JJ< iJ2 iJ2 [_ € (Pi^±lZll! fc(^.AAp iPiP;-j ++ jA ? ;qi9i))) lim im y...|expiig // ....... / exp i > [lim exp '2 fm + +P pj, A + Jb +1/2(?,PjA >+1/2 j *—'L f JJ L L N-coJ t ;=o ^ ;=0 w-»y J ^ [I \&i ;=0 m/ - . A .N?2 - . / , -- • . - A \2 2 AA AAA +PjAg AA P3+ 2^ ^ 2 2? ( ?(«i (;? i PPP Ji +i+ +P PPJ J i 9??JJi))) + ^ 2^2 (?i,Ap PiAA
N—co j
m
m
- ^ 2 ( « i A P i + PJ A ? J )
+ ^ 2 ( ? J : APi + PiA«i)
f+i/2
)l}n*3^3 /J
v"pr
J
J
•
dkj+i/2dXj+i/2
3=1
■^.(^+-)]}n**fi*^ A -1 2 2
eXP eXP
lim /f . . . y e x p , = fc / - 7// ^'g|lS**+*/ ^i=o§ I1 ^ L [W-5& ^ 7^83) '3
^lim / . - / e x p ^ E J ^ ^ + 1 / 2 - ^(?; A P;+Pi A
/ -
A
%2
A
^
^
< % n —7^2 n <*>^ «iII (£j5 ^^ — •• TT
3; ==1 1
,■ d.,
///222 T T d«*i TT a**ji + lil // 22 a"a xx*;ii++Hiil/2 V V
3=0
;;
Just as in the case of the previous section integration over fcj+1/2 yields K(p",q",t";p',q',t') A'(P", ? V ; P ' , ?'.*') =
A + +^« -A »m,'teV" r ^+,,r'a/.. /,"w"|,fbU, S'M s *" -^'' f^' /n \ (N+11/2 / Z7rm \ lim — —
N—00 \
If
/
r
r
(
N r
/
A +
A
2
n
Pi
J
194
APPENDIX
B.
EXACT
LATTICE
CALCULATIONS
-*i+i/a(^+«-)]} )]} (*+-)]} -*j+l/2
/v N r g^j + 1/2 [Jc/pjdg, 2 =i i-oiv ( 2 T ) P J '
Next we carry out the integrations over the i ; + i/2 and find K(p",q",t";p', *(*>", ?",*";q',i') P', q'.t') (N+l)/2 ■, (N+D/2 f2nm\ '27rm\C' lim oo \ ie
/,
,.
/
/ . . ./ /«.P
>{■£;[=(*«+>f ")]} /.^[m/
A
Qj H
Ap; _ \ 21 1 Pi
-<(IH.OnW,V, =- &(&) . im ^v N+i)/2 /.../ex / z f h+»]} feu,+^-,Vll P;§[s(/-/-{'S[£^^)1} N N
1
rr L 2^—6
i ^
;=0
=
lim I N—oo 2iric)
'
[j_rfPj-rf9j-
j = l
N+1J/2
exp <
P;
N
; 6 ( A P ; + £(jp7) [
>II
;' = '
r / mi *v(JV+i)/a ) / 2 lim ( / . / e x p 1 \27TIf/
tf]) = ^(^r""/ ••/-{'gs^-*)']) ^(Ag,
-euq,f] >
=0
N
i+
* n« [('+?) [( T)»+I --(■-=)»] (i - y)»] I[dpjdqj. T** >,]n< -
Pi+i
j=0
1
;=0
;=1
Now using t h e i d e n t i t y
(ax -— by)6(ay by)iS(ay -— b. / <5(ax S(ax by)S(ay 62)^2/ bz)dy = <5(a 6(a2x2x -- 6 2bz2z), ),
/'
a n d c a r r y i n g o u t t h e P j - i n t e g r a t i o n s we find
/ < ( ? " , ? " , «« "" ;;o',q',t') /<(?",?", P P '' , ? ? '' , 0
\ Af + l lim 6 ( w—00 w-oo [V\
AT + i ;
'
7-/•
v(JV+i)/: x lim ( — /V~oo V27T V27rie/
^ f ef—V i
, \N+1 JV + 1iy w N -+
V' - (V
N
•y/ « epxxpp 'iJ2 E *—^ -( A- «9 ; --« fa;> • ) .
;=0
L J=0
N
n<% ;=i
J /=!
B.2. THE HAMILTON OPERATOR W(Xj, X 2 ) = ^ X ? + wX 2 = =
195
T :i 7 ' p"-eu-«' 7TV'2V) p') 6(e"'6(e» VV'-e~
,. /i m \(w+i)/2 AN m , \("+i)/2 /•f f/■ x hm / • • / exp — (Aq} N-oo\27r:<:/ JJ JJ N-00 \2iri(/ Lj=0 = 6(e"T'2p"-e-"T'?p')
,1 A
f n^
-tuiqj)
J •'= J = 1i
x W I- .f/ e xexp xM p if
{q - uq) uqf d\ dt Vq, Vq, ^{q-wqfdt ■[■!>
where T = t" — t'. The final path-integral we have to solve is a Lagrangian path integral for a quadratic Lagrangian. This final Lagrangian path integral can be done using extremal techniques which are exact in the case of a quadratic Lagrangian, see [96]. The action for this Lagrangian path integral is given by /c,= Icl =1
777
f
rT
22~y {q-uqfdt ti-"tfdt 'I
variation of which yields the equation of motion 2 q
which has the general solution q(t) — Asinh(wt) + B cosh(ut). Imposing the boundary conditions q(Q) = q' and q(T) = q" yields 1
q(t): { -{9'sinh[w(T - t)] + q"smh(ut)}. «(*) = • u nWsinhKT 9 "sinh(wi)}sinh(wT) sinn(wi j Therefore, using (B.2) one finds the evaluated classical action to be Scl = 5d
{[(q"f . " " 7F) -{[(q"f 2sinh(wT 2sinh(w7)
(B.2)
2 (q'f}coW + ( 9 ') 2 ]cosh( T) JT)-2q"q'}- 2q"q'} - ^^[(q") [ ( 9 "2) 2 - (q') }Sh(o (q'n I
So that our final result for the representation independent propagator with this Lagrangian becomes
196
APPENDIX
B. EXACT
LATTICE
CALCULATIONS
Kip",q"t";jJ,q',t') K(p",q",t";jJ,q',t') I
/
2-KTTIU)
y 2Sinh(u)i) isinh(wT)
5(e^/2p"_e-"T/2p') \ /
irruj , 22 < < ) 22 + ((?') cosh(.T) - 2 99V } - ^ M " [1(' 9 T" ) 2 -' ( ?W?)) ' ) 2 ] j■. x exp I( ^ - ^ { [ ( ?^{[(?") 'V: g ') ] ]cosh(o;T)-2 iT 2sinh(uT)" (B.3)
Bibliography [1] N.I. Akhieser and I.M. Glazman, "Theory of Linear Operators in Hilbert Space I," Ungar Publ. Co., New York, 1961. [2] A.M. Arthurs, Path integrals in polar coordinates, Proc. Roy. Soc. Lond., A 3 1 3 (1969), 445-452. [3] A.M. Arthurs, Path integrals in curvlinear coordinates, Proc. Roy. Soc. Lond., A 3 1 8 (1970), 523-529. [4] E.W. Aslaksen and J.R. Klauder, Unitary representations of the affine group, J. Math. Phys., 9 (1968), 206-211. [5] E.W. Aslaksen and J.R. Klauder, Continuous representation theory using the affine group, J. Math. Phys., 10 (1969), 2267-2275. [6] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Commun.
Pure and Appl. Math., 14 (1961), 187-214.
[7] A.O. Barut and R. Raczka, "Theory of Group Representations and Applications," second revised edition, World Scientific, Singapore, 1986. [8] F.A. Berezin, "The Method of Second Quantization," Academic Press, San Diego, 1966. 197
198
BIBLIOGRAPHY
[9] M.V. Berry and K.E. Mount, Semiclassical approximations in wave mechanics, Rep. Prog. Phys., 35 (1972), 315-397. [10] A. Bohm, Associative algebra in the problem of mass formulas, J. Math. Phys., 8 (1967), 1551-1558. [11] A. Bohm, "The Rigged Hilbert Space and Quantum Mechanics", Springer Verlag, Berlin 1978. [12] M. Bohm and G. Junker, Path integration over compact and noncompact rotation groups, J. Math. Phys., 28 (1987), 1978-1994. [13] M. Bohm and G. Junker, Path integration over the n-dimensional Euclidean group, J. Math. Phys., 30 (1989), 1195-1197. [14] M. Bohm and G. Junker, "Group theoretical approach to path integration on spheres." In: S. Lundquist et al. (eds.), Path Summation
Achievements
and Goals, Proceedings, Trieste, Italy, 1987, 469-480, World Scientific, Singapore, 1988. [15] F. Bruhat, Distributions sur un groupe localement compact et applications a l'etude des representations des groupes p-adiques, Bull. Soc. Math. France, 89 (1961), 43-75. [16] A.L. Carey, Square-integrable representations of non-unimodular groups, Bull. Austral. Math. Soc, 15 (1976), 1-12. [17] P. Carruthers and M.M. Nieto, Phase and angle variables in quantum mechanics, Rev. Mod. Phys., 40 (1968), 411-440.
BIBLIOGRAPHY
199
[18] P.M. Cohn, "Lie Groups," Cambridge University Press, London, 1965. [19] R. Courant and D. Hilbert, "Methoden der Mathematischen Physik I," Dritte Auflage, Heidelberger Taschenbiicher 30, Springer Verlag, Berlin, 1968. [20] H. Davies, Hamilton approach to the method of summation over Feynman histories, Proc. Camb. Phil. Soc, 59 (1963), 147-155. [21] B. Davison, On Feynman's integral over all paths', Proc. Roy. Soc. Lond., A 225 (1954), 252-263. [22] B.S. DeWitt, Dynamical theory in curved spaces. I. A review of the classical and quantum action principles, Rev. Mod. Phys., 29 (1957), 377-397. [23] P.A.M. Dirac, The Lagrangian in quantum mechanics, Phys. Ztschr. der Sowjetunion, 3 (1933), 64-72. [24] P.A.M. Dirac, "The Principles of Quantum Mechanics," fourth revised edition, Oxford Univ. Press (Clarendon), Oxford, 1967. [25] J. Dixmier,"C*-Algebras," North Holland, Amsterdam, 1977. [26] J. Dixmier, "Von Neumann Algebras," North Holland, Amsterdam, 1981. [27] H.D. Doebner and 0 . Melsheimer, Limitable dynamical groups in quantum mechanics. I, J. Math. Phys, 9 (1968), 1638-1656. [28] J.S. Dowker, When is the 'sum over classical paths' exact?, J. Phys. (London), A3 (1970), 451-461.
200
BIBLIOGRAPHY
[29] J.S. Dowker, Quantum mechanics on group space and Huygens' Principle, Ann. Phys., 62 (1971), 361-382. [30] M. Duflo and C.C. Moore, On the regular representation of a nonunimodular locally compact group, J. Fund. Anal., 21 (1976), 209-243. [31] N. Dunford and J. Schwartz, "Linear Operators," Part 1, Wiley, New York, 1958. [32] S.F. Edwards and Y.V. Gulyaev, Path integral in polar coordinates, Proc. Roy. Soc. Lond., A 2 7 9 (1964), 229-235. [33] G.G. Emch, "Algebraic Methods in Statistical Mechanics and Quantum Field Theory," Wiley-Interscience, New York, 1972. [34] D.H. Feng, J.R. Klauder, and M.R. Strayer (eds.), "Coherent States: Past, Present, and Future," Proceedings, Oak Ridge, Tennessee, 1993, World Scientific, Singapore, 1994. [35] R.P. Feynman, Space time approach to non-relativistic quantum mechanics, Rev. Mod. Phys., 20 (1948), 367-387. [36] R.P. Feynman, An operator calculus having applications in quantum electrodynamics, Phys. Rev., 84 (1951), 108-128. [37] R.P. Feynman and A.R. Hibbs, "Quantum Mechanics and Path Integrals," McGraw-Hill, New York, 1965. [38] M. Flato, J. Simon, H. Snellman, and D. Sternheimer, Simple facts about analytic vectors and integrablility, Ann. scient. Ec. Norm. Sup., 5 (1972), 423-434.
BIBLIOGRAPHY
201
[39] S. Gaal, "Linear Analysis and Representation Theory," Springer Verlag, Berlin, 1973. [40] C. Garrod, Hamiltonian path integral methods, Rev. Mod. Phys., 38 (1966), 483-494. [41] I.M. Gel'fand and M.A. Neumark, Unitary representations of the group of linear transformations of the straight line [russ.], Doklady Akad. Nauk SSSR, 55 (1947), 571-574. [42] J.-L. Gervais and A. Jevicki, Point canonical transformations in path integral, Nucl. Phys., B110 (1976), 93-112. [43] R. Gilmore, The classical limit of quantum nonspin systems, J. Math. Phys., 20 (1979), 891-893. [44] R.J. Glauber, Photon correlations, Phys. Rev. Lett, 10 (1963), 84-86. [45] R.J. Glauber, The quantum theory of optical coherence, Phys. Rev., 130 (1963), 2529-2539. [46] R.J. Glauber, Coherent and incoherent states of the radiation field, Phys. Rev., 131 (1963), 2799-2788. [47] J. Gleick, "Genius," Vintage Books, New York, 1993. [48] C. Grosche and F. Steiner, Path integrals on curved manifolds, Ztschr. Physik, C36 (1987), 699-714. [49] C. Grosche and F. Steiner, The path integral on the pseudosphere, Ann. Phys. (N. Y.), 182 (1988), 120-156 .
202
BIBLIOGRAPHY
[50] C. Grosche, "Path Integrals, Hyperbolic Spaces, and Selberg Trace Formula," World Scientific, Singapore, 1996. [51] R. Haag, "Local Quantum Physics: Fields, Particles, Algebras," Springer Verlag, Berlin, 1992. [52] P.R. Halmos, "Measure Theory," Springer Verlag, Berlin, 1974. [53] W. Heisenberg, Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Ztschr. Physik, 33 (1925), 879-893. [54] S. Helgason, "Differential Geometry and Symmetric Spaces," Academic Press, Orlando, FL, 1962. [55] A. Inomata, H. Kuratsuji, and C.C. Gerry, "Path Integrals and Coherent States of SU(2) and 5f/(l,l)," World Scientific, Singapore, 1992. [56] C.J. Isham and J.R. Klauder, Coherent states for n-dimensional Euclidean groups E(n) and their application, J. Math. Phys., 32 (1991), 607-620. [57] G. Junker, "Path integration on homogeneous spaces." In: V. Sa-yakanit et al. (eds.), Path Integrals from meV to MeV, Proceedings, Bangkok, Thailand, 1989, 217-237, World Scientific, Singapore, 1989. [58] A.A. Kirillov, On unitary representations of nilpotent Lie groups, Soviet Math. Doklady, 1 (1960), 108-110. [59] J.R. Klauder, The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers, Ann. Phys. (N. Y.), 11 (I960), 123168.
BIBLIOGRAPHY
203
[60] J.R. Klauder, Continuous-representation theory. I. Postulates of continuous representation theory, J. Math. Phys., 4 (1963), 1055-1073. [61] J.R. Klauder, Continuous-representation theory. II. Generalized relations between quantum and classical dynamics, J. Math. Phys., 4 (1963), 10581073. [62] J.R. Klauder, Continuous-representation theory. III. On functional quantization of classical systems, J. Math. Phys., 5 (1964), 177-187. [63] J.R. Klauder, Path integrals and stationary-phase approximations, Phys. Rev., D19 (1979), 2349-2356. [64] J.R. Klauder, Path integrals, Acta Phys. Austriaca Suppi, XXII (1980), 3-49. [65] J.R. Klauder, "Path Integrals for Affine Variables." In: J.P. Antoine and E. Tirapequi (eds.), Functional Integration Theory and Applications, Proceedings, 101-119, Plenum, New York, 1980. [66] J.R. Klauder, "The universal propagator." In: D. Han, Y.S. Kim, and W.W. Zachary (eds.), Workshop on Harmonic Oscillators, Proceedings, Maryland, USA, 1992, 19-28, NASA Publication 3197, 1993. [67] J.R. Klauder, "The Sum over Histories: Formalism and some Applications," Lecture Notes, Universitat Bern, Bern, 1962, (unpublished). [68] J.R. Klauder and E.C.G. Sudarshan, "Fundamentals of Quantum Optics," W.A. Benjamin, New York, 1968.
204
BIBLIOGRAPHY
[69] J.R. Klauder and B-S. Skagerstam, "Coherent States: Applications in Physics and Mathematical Physics," World Scientific, Singapore, 1985. [70] J.R. Klauder and W.A. Tome, The universal propagator for affine [or SU(1,1)] coherent states, J. Math. Phys., 33 (1992), 3700-3709. [71] H. Kleinert, Path integral on spherical surfaces in d-dimensions and on group spaces, Physics Letters, B 2 3 6 (1990), 315-320. [72] H. Kleinert, "Pfadintegrale: In der Quantenmechanik, Statistik und Polymerphysik," Bl-Wissenschaftsverlag, Mannheim, 1993. [73] E.H. Lieb, The classical limit of quantum spin systems, Commun.
Math.
Phys., 31 (1973), 327-340. [74] D.W. McLaughlin and L.S. Schulman, Path integrals in curved spaces, J. Math. Phys., 12 (1970), 2520-2524. [75] G.W. Mackey, "Unitary Group Repesentations in Physics, Probability, and Number Theory," Advanced Book Classics, Addison-Wesley, Reading, Massachusetts, 1989. [76] M.S. Marinov, Path integrals in quantum theory: An outlook of basic concepts, Phys. Rep., 60 (1980), 1-57. [77] M.S. Marinov and M.V. Terentyev, A functional integral on unitary groups, Sov. J. Nucl. Phys., 28 (1978), 729-737. [78] M.S. Marinov and M.V. Terentyev, Dynamics on the group manifold and path integral, Fortschr. Physik, 27 (1979), 511-549.
BIBLIOGRAPHY
205
[79] K. Maurin, "General Eigenfunction Expansions and Unitary Representations of Topological Groups," PWN-Polish Scientific Publishers, Warsaw, 1968. [80] H. Meschkowski, "Hilbertsche Raume mit Kernfunktion," Springer Verlag, Berlin, 1962. [81] C. Morette, On the definition and approximation of Feynman's path integrals, Phys. Rev., 81 (1951), 848-852. [82] J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Camb. Phil. Soc.,45 (1949), 97-124. [83] E. Nelson, Analytic vectors, Ann. of Math., 70 (1959), 572-615. [84] H.M. Nussenzweig, "Introduction to Quantum Optics," Gordon and Breach, New York, 1973. [85] A.M. Perelomov, Coherent states for arbitrary Lie group, Commun. Math. Phys., 26 (1972), 222-236. [86] A.M. Perelomov, "Generalized Coherent States and their Applications," Springer Verlag, Berlin, 1986. [87] J.M. Radcliffe, Some properties of coherent spin states, J. Phys. A: Gen. Phys., 4 (1971), 313-323. [88] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, Vol I: Functional Analysis," revised and enlarged edition, Academic Press, Orlando, FL, 1980.
206
BIBLIOGRAPHY
[89] F. Riesz and B. Sz.-Nagy, "Functional Analysis," Dover, New York, 1990. [90] J.E. Roberts, Rigged Hilbert spaces in quantum mechanics, Commun. Math. Phys., 3 (1966), 98-119. [91] P. Salomonson, When does a non-linear point transformation generate an extra 0(h2) Potential in the effective Lagrangian?, Nucl. Phys., B121 (1977), 433-444. [92] E. Schrodinger, Quantisierung als Eigenwert Problem, Ann. Physik, 79 (1926), 361-376, 489-527. [93] E. Schrodinger, Uber das Verhaltnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen von Erwin Schrodinger, Ann. Physik, 79 (1926), 734-756. [94] E. Schrodinger, Der stetige Ubergang von der Mikro- zur Makromechanik, Die Naturwissenschaften, 14 (1926), 664-666. [95] L.S. Schulman, A path integral for spin, Phys. Rev., 176 (1968), 15581569. [96] L.S. Schulman,"Techniques and Applications of Path Integration," Wiley, New York, 1981. [97] B. Simon, The classical limit of quantum partition functions, Commun. Math. Phys., 71 (1980), 247-276. [98] W. Tobocman, Transition amplitudes as sums over histories, Nuovo Cimento, 3 (1956), 1213-1229.
BIBLIOGRAPHY
207
[99] W. Tome, A representation independent propagator. I. Compact Lie groups, Ann. Inst. Henri Poincare, Theor. Phys., 63 (1995), 1-40. [100] W. Tome, A representation independent propagator. II. Lie groups with square integrable representations, Ann. Inst. Henri Poincare, Theor. Phys., 65 (1996), 175- 222. [101] W.A. Tome and J.R. Klauder, The universal propagator for spin [or SU(2)] coherent states, J. Math. Phys., 34 (1993), 2893-2913. [102] T.T. Troung, New integral equations for the quartic anharmonic oscillator, Nuovo Cimento Lett., 9 (1974), 533-536. [103] T.T. Troung, Weyl quantization of anharmonic oscillators, J. Math. Phys., 16 (1975), 1034-1043. [104] G. Tulsian and J.R. Klauder, The universal propagator for E(2) coherent states, Comm.un.in Theor. Phys., 4 (1995), 63-90. [105] N.J. Vilenkin, "Special Functions and the Theory of Group Representations," American Mathematical Society, Providence, RI, 1968. [106] J. von Neumann, "Die mathematischen Grundlagen der Quantenmechanik," Springer, Berlin, 1932. [107] J. Weidmann, "Lineare Operatoren in Hilbertraumen," Teubner Verlag, Stuttgart, Germany, 1976. [108] B.G. Wybourne, "Classical Groups for Physicists," Wiley, New York, 1973.
208
BIBLIOGRAPHY
[109] L.G. Yaffe, Large N limits as classical mechanics, Rev. Mod. Phys., 54 (1982), 407-435. [110] L.C. Young, "Lectures on the Calculus of Variations and Optimal Control Theory," W.B. Saunders Co., Philadelphia, 1969. [Ill] W.-M. Zhang, D.H. Feng, and R. Gilmore, Coherent states: Theory and some applications, Rev. Mod. Phys., 62 (1990), 867-928.
Index Abelian Lie Algebra 30
Classical Observable 159
Adjoint Operator 4
Closeable Operator 7
Affine Group 143
Closed Operator 6
Analytic Vector 86
Closed Topological Space 18
Associated Spherical Function 72
Closure of an Operator 7
Associative Algebra 1
Coherent States 74
Atlas 27
Coherent State Path Integrals 81 Coherent State Propagator 79
Bicommutant 15
Commutant 15
Bounded Operator 4
Commuting Operators 25
Bra Vector 23
Compact Operator 5
Canonical Coherent States 42
Compact Topological Space 28
Canonical Commutation
Complete 3
Relations 39
Complete Commuting System
Canonical Transformation 55
of Operators 25
Cartan Metric Tensor 153
Completely Reducible Representa-
Cauchy Sequence 3
tion 35
Center of a Lie Algebra 30
Complex Lie Algebra 29
Center of a von Neumann
Complex Manifold 28
Algebra 16
Connected Topological Space 29 209
210
INDEX
Continuous Representation 32
Fiducial Vector Independent
Continuous Representation of a Hilbert Space 182
Propagator 110 Fixed Points 64
Continuously Diagonal Operator 13
Formal Degree 99
Coordinate Chart 27
Fundamental Family of Vector Fields 10
Countably Hilbert Space 20 Countably Inner Product Space 20 Countably Countably Normed Normed Space Space 20 20 Cyclic 25 Cyclic Vector Vector 25
G 36 Gel'fand Gel'fand Triplet Triplet 24 24 General General Lie Lie Group Group 98 98
Decomposeable Operator 14
Generalized Eigenvector 25
Differentiable 27 Differentiable Manifold Manifold 27
Generalized Maurer Cartan
Direct Integral 11
Form Form 92 92
Direct Orthogonal Sum 12 _ , _. . Domain ofr an Operator 4 Domain ot an Operator 4 Elliptic 66 Elliptic 66 Enveloping Algebra 32 Enveloping Algebra 32 Equivalent Functions 12 Equivalent Functions 12 Equivalent Vector Fields 11 Equivalent Vector Fields 11 Essentially Self-Adjoint Essentially Self-Adjoint Operator 9 Operator 9 Extension of an Operator 4 Extension of an Operator 4
Haar Measure 85 Hamilton Principal Function 50 Hamilton Principal Function 50 Harmonic „ Oscillator 48 Harmonic Oscillator 48 Hausdorff 27 Hausdorff 27 Heisenberg Algebra 42 Heisenberg Algebra 42 Hilbert Schmidt 6 Hilbert Schmidt 6 Hilbert Space 3 Hilbert Space 3 Homeomorphisim 64 Homeomorphisim 64 Homogeneous 64
Factor 16
Homogeneous 64 Space 64 Homogeneous
Factor 37 Factor Representation 16
Homotopy Classes Homogeneous Space6064
Feynman Path Integral3754 Factor Representation
Hypothesis Classes (A) 86 60 Homotopy
Fcynman Path Integral 54
Hypothesis (A) 86
INDEX
211
Ideal 29
Massive Subgroup 64
Inner Product 3
Matrix Mechanics 50
Inner Product Space 3
Measurable Operator Field 13
Intertwinig Operator 34
Measureable Vector Field 10
Invariant Element 64
Minimal Projection 16
Invariant Subspace 35
Modular Function 85
Involutive Automorphism 64
Multiplicity Free Represen-
Irreducible Unitary Representation 35 Isomorphic 5 " Jacobi Polynomial 117 117 Jacobi Theta Function 61, 61, 120 120
tation 37 Mutually Orthogonal 5 Nelson Operator 20 Nilpotent Lie Algebra 30 Nilpotent Lie Group 30 Non-closeable Operator 7 Non-closeable Operator 7
Kernels on a Lie Group 129 Kernels on a Lie Group 129
Nuclear Space 22 Nuclear Space 22
Ket Vector 20 Ket Vector 20
Nuclear Spectral Theorem 26 Nuclear Spectral Theorem 26
% k, nw on Am*(«7(0) 89
O P e n Set 18
Lagrangian 50 _ . . _ „, Laplace-Beltrami 65 Laplace-Beltrami Operator Operator 65 Left Invariant Kernel 129 129 Left Regular Representation 32 Lie Algebra 29 Lie Group 28 Linear Functional 23 Locally Compact 28
Operator 4 Operator Field 13 Operator Irreducibility 36 Operator Ordering 55 Orthogonal Projection 5 Orthogonality Relations 99 Parameterization 86
INDEX
212 Path Integral Integral on an unbounded Riemannian Riemannia Manifold 57
Reproducing Property 101, 183 Resolution of Identity 75 Rigged Hilbert Space 24
on a bounded Riemannian Manifold Manifold 59 59 . . c _„ c on a Symmetric Space 72 on a Symmetric Space 72 , nGroup n on a General on a General Group Manifold 139 Manifold 139 Phase-Space Path Integral 53 Phase-Space Path Integral 53 Primary 37 Primary Representation Representation 37 Propagator 41 Propagator 41 Proper Coordinate Chart 27 . .. T , , , -,,, n Projection-Valued Measure l10 Projection-Valued Measure 10 D
Tl(U\U22)35 TZ(U\U ) 35 Pmkk(9V)) 89 Pm (g(l)) 89 Range of an Operator 4 Range of an Operator 4 Real Lie Algebra 29 Real Lie Algebra 29 Reducible Representation 35 Reducible Representation 35 Representation of Class 1 64 Representation of Class 1 64 Representation Independent Representation Independent Propagator 112 Propagator 112 Reproducing Kernel 183 Reproducing Kernel 183 Reproducing Kernel Hilbert Reproducing Kernel Hilbert Space 183 Space 183
♦-Algebra 55 *-Algebra Schrodinger Equation 39 e> ■» Schrodinger Group 40 c ° Schrodinger Represen Schrodinger Representation of L2(G) 137 tation of L 137 *W Self-Adjoint Operator 5,8 Self-Adjoint Operator 5,8 Semi-invariant Operator with with Semi-invariant Operator
weight a 89 ° 31 Semisimple Lie Algebra 31 r t> Semisimple Lie Group 31 Semisimple Lie Group 31 Separable Hilbert Space 3 Separable Hilbert Space 3 Separable Topological Space 28 Separable Topological Space 28 Separated Sets 29 Separated Sets 29 Simple Lie Algebra 30 Simple Lie Algebra 30 Simple Lie Group 31 Simple Lie Group 31 Simply Connected Topological Simply Connected Topological Space 29 Space 29 Solvable Lie Algebra 30 Solvable Lie Algebra 30 Solvable Lie Group 30 Solvable Lie Group 30 Special Unitary Group 5/7(2) 114 Special Unitary Group SU{2) 114 Spectral Expansion 61 Spectral Expansion 61
INDEX
213
Spectral Theorem 9
Unitarily Equivalent Represen-
Square Integrable Representation 98 Square Integrable Vector Field 11 Standard Borel Measure 186 Standard Standard Borel Borel Space Space 186 186 Strong Operator Convergence 6 Strong Operator Convergence 6 Stronger Topology 19 Stronger Topology 19 Subalgebra 29 Subalgebra 29 Subrepresentation 35 Subrepresentation 35 Symmertic Operator 8 Symmertic Operator 8 Symmetric Space 64 Symmetric Space 64 System of Generators 2 System of Generators 2 7$-continuous 7$-continuous 23 23 Topological Topological Space Space 18 18 Topology Topology 18 18 Trace Trace Class Class Operator Operator 66 Trivial Trivial Representation Representation 32 32 Type Type II 16 16 Type Type II Representation Representation 37 37 U(l) 90 Unbounded Operator 4 Unimodular 85
tations 34 Unitary Operator 5 Unitary Representation 32 Upper Symbol 158 Van Determinant 57 57 Van Vleck's Vleck's Determinant Vector Field 10 Vector Field 10 von Neumann Algebra 15 von Neumann Algebra 15 Wave Mechanics 50 Wave
Mechanics Weaker Topology5019
Weaker Wigner Topology Functions 19 116 Wigner 116 WindingFunctions Number 60 Winding Number 60 „ . 0Spherical , . T Function „ Zonal 69 Zonal Spherical function 69
