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QUANTIZATION AND REPRESENTATION INDEPENDENT PROPAGATORS By WOLFGANG TOME A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY For MarieJacqueline and AnneSophie "The highest reward for a man's toil is not what he gets for it, but what he becomes by it." John Ruskin ACKNOWLEDGEMENTS I would like to thank Prof. Dr. John R. Klauder for being my thesis advisor for the past 4 years. His ideas, guidance, constructive criticism, advice, encouragement, support, trust and confidence in me are greatly appreciated. I wish to thank Prof. Dr. Stanley P. Gudder, from the University of Denver, for introducing me to the foundations of quantum theory and for serving as my master's thesis advisor. I also would like to thank Prof. Dr. Alwyn van der Merwe for making my stay at the Department of Physics at the University of Denver possible. Prof. Dr. James couragement, Dufty is cordially thanked for his advice, support, and en without which this work might never have been completed. I also would like to thank Prof. Dr. Gerard G. Emch and Prof. Dr. Stephen J. Summers for their constructive criticism at an early stage of this work and Prof. Dr. Bernard F Whiting for helpful discussions on the classical limit of the represen station independent propagator. I also wish to thank Prof. Dr. Khandkar Muttalib for his service on my committee. I am grateful to Prof. Dr. Hajo Leschke from the University of Erlangen, Ger many for his helpful remarks on the universal propagator for affine coherent states, during the Conference on Path Integrals in Physics, held in Bangkok, the Winter of 1993. my investigations. Thailand, in His remarks proved to be very valuable in the further course of I am also grateful to Dr. Max Brocker from the Studienstiftung Finally, I wish to express my gratitude to my wife MarieJacqueline, who has al ways been a supportive and understanding companion in the at times very demanding life of a physicist. Financial support for the work presented here has been provided in part by a doctoral fellowship from the Studienstiftung des deutschen Volkes and by a graduate research award from the Division of Sponsored Research at the University of Florida. TABLE OF CONTENTS ACKNOWLEDGEMENTS S S S S * mi ABSTRACT CHAPTERS INTRODUCTION * 8 1 The Fiducial Vector Independent Propagator for the Heisenberg Weyl Group 1.1.1 Examples of the Fiducial Vector Independent Propagator General Overview of the Thesis A REVIEW OF SOME MEANS TO DEFINE THE FEYNMAN PATH INTEGRAL ON GROUP The Feynman Path Integral on Rd 2.1.1 2.1.2 AND SYMMETRIC SPACES , Group, and Symmetric Spaces Introduction . . . The Feynman Path Integral on Ri 2.1.3 The Feynman Path Integral on Group Spaces . 2.1.4 The Feynman Path Integral on Symmetric Spaces Coherent States and Coherent State Path Integrals . 2.2.1 Introduction . . . . . Coherent States: Minimum Requirements. 2.2.4 2.2.5 Group Coherent States . Continuous Representation * S * S 5 0 8 The Coherent State Propagator for Group Coherent States NOTATIONS AND PRELIMINARIES Notations . 42 . 46 Preliminaries THE REPRESENTATION INDEPENDENT PROPAGATOR FOR A GENERAL LIE GROUP Coherent States for General Lie Groups. . . . . The Representation Independent Propagator for Compact Lie Groups __ **__* , 1 in 1 T i f/^t T rt L'r Page Missing or Unavailable Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy QUANTIZATION AND REPRESENTATION INDEPENDENT PROPAGATORS By WOLFGANG TOME August 1995 Chairman: Dr. John R. Klauder Major Department: Physics The quantization of physical systems moving on group and symmetric spaces been an area of active and ongoing research over the past three decades. is shown in this work that it is possible to introduce a representation independent propagator for a real, separable, connected and simply connected Lie group with irreducible, square integrable representations. For a given set of kinematical variables this propagator is a single generalized function independent of any particular choice of fiducial vector and the irreducible representations of the Lie group generated by these kinematical variable es, which nonetheless, correctly propagates each element of a continuous representation based on the coherent states associated with these kinematical variables. Furthermore, it is shown that it is possible to construct regularized lattice phase space path integrals for a real, separable, connected and simply connected Lie group with irreducible, square integrable representations, and although the configuration path integral quantization is obtained for general physical systems whose kinematical variables are the generators of a connected and simply connected Lie group. This novel phasespace path integral quantization is (a) more general than, (b) exact, and (c) free from the limitations of the previously considered path integral quantizations of free physical systems moving on group manifolds. To illustrate the general theory, a representation independent propagator is plicitly constructed for SU( ) and the affine group. CHAPTER 1 INTRODUCTION In nonrelativistic quantum mechanics the states of a quantum mechanical system are given by unit vectors, such as b 4, or 77, in some complex, separable Hilbert space For a single canonical degree of freedom problem the basic kinematical variables are represented on H by two unbounded selfadjoint operators P , the momentum, and Q the position, with a common dense invariant domain D . These operators satisfy the Canonical (Heisenberg) Commutation Relation (CCR) [Q, P] = iI, where h . Let Wi(P, Q) be the Hamilton operator of a quantum system, then the time evolution of this quantum system in the state 4 E D(Q) is given by the timedependent Schrddinger equation iat(t) = Wt(P, Q).(t) Since only selfadjoint operators may be exponentiated to give oneparameter uni groups which give the dynamics of a quantum system it will always be as sumed that the Hamilton operator is essentially selfadjoint, i.e. adjoint. its closure is self If the Hamilton operator is not explicitly time dependent then a solution to Schridinger's equation is given in terms of the strongly continuous oneparameter unitary Schridinger group, U(t) = exp(itW), 2 variables P and Q are realized by the following two unbounded symmetric operators id/dq and q, or p and id/dp, respectively. These operators are essentially selfadjoint on the dense subspace S(1R) of L2(IR) the space of infinitely often differentiable functions that together with their derivatives fall off faster than the inverse of any polynomial. Furthermore, since these operators leave S(R) invariant, one can choose S(R) C L2(R) as the common dense invariant domain for these operators. calls these operators together with their common dense invariant domain (11) the Schrbdinger representation on qspace, or on pspace, which is denoted by b(q) (R) or (p) However, (R), respectively (cf. we would like Appendix V]). to emphasize that there is nothing sacred about this choice of representation other than the timehonored custom of doing so. One can also choose one of the socalled continuous representations based on canonical coherent states [60]). In this representation the states are given by certain bounded, continuous, square integrable functions of two real parameters p and q. these functions by ',(p, q). The functions i,(p, q) span a subspace L(1R2 We denote ) of L(R2), where the subscript i7 denotes a unit fiducial vector in the Hilbert space H on which the canonical coherent states are based. Let r)(x) E L2(JR) be a fixed normalized function S(z) L (JR) be arbitrary, then an explicit representation of the functions n(p, q) can be given as follows v%(P, q) r(x) exp( ipxz)t (x + q)dx. (1.1) FYom this form of the representation it may be seen that one obtains the Schrbdinger representation in qspace or in pspace in appropriate limits. In particular, one obtains the Schrbdinger representation on qspace by suitably scaling the 4n(p, q) so that the 1 a 1 a I f ...! a 1 _ l J a a U 1a J27 ,1 L0  3 With each of these various representations one can associate a propagator: in qspace by "; q', t') (q' ,t')dq' in pspace by '(p"' ,t')dp' and finally in the continuous representation based on canonical coherent states by ,P(p" K,(p" q,t ',( ',t')dp'dq'. 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, for the quantum system. LU and they contain all the relevant dynamical information et us ask whether it is possible to find a single propagator t') such that P1 0(p11 I", ") holds for an arbitrary fiducial vector. It'), ,(p' ,q',t')dp'dq', Stated otherwise, is there a propagator that is independent of the chosen continuous representation but which , nonetheless, propa gates the elements of any representation space Lr(R2) in such a way that they stay in the representation space L2(nR)? 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 [65]. The Fiducial Vector Independent Propagator for the Heisenberg Weyl Group Let P and I be an irreducible, selfadjoint representation of the Heisenberg r rl T nf D Ti Sa TT II I .. I* I j I l . Than few an arbitrary normalized r n . J(q" S(q" L(p" , q",t" K (p" K(p" nrn  1 1 1 where V(p, q) E exp( iqP) exp(ipQ). In fact these states are the familiar canonical coherent states which form a strongly continuous, a fixed, normalized fiducial vector 7n overcomplete family of states for E H and they admit the following resolution of identity: > )dpdq. The map C, 4 L(IR2 , dpdq) defined for any EH by: [C,p](p,q) 1 V/27? = ((p, q) yields a representation of the Hilbert space H by bounded, continuous, square inte grable functions on a proper closed subspace L2(R2) of L2(R2). Using the resolution of identity one finds K,(p, q;p' q') dp'dq' where, K, (p, q; p' ',q'))  (,(p, q),S (p =<( , 1 i2n; V'(p, q)V(p' is the reproducing kernel which is the kernel of a projection operator from L (1R2) onto the reproducing kernel Hilbert space L2(R2). Let D be the common dense invariant domain of P and Q that is also invariant under V(p, q), then one can easily show that the following relations hold on D: iaV* (p, q) V (p, q)P, (1.2) (q + ip)V* (p, q) V*(p, q)Q. (1.3) Notice, that the operator V*(p, q) intertwines the representation of the Heisenberg 1 VT r(p,q) (l(p,q), = C0(P,q) , V* (p,q) ) ,') ,(p' )(P, q) 5 and an appropriate core for these operators is given by the continuous representation of D = C,(D). Let 'H(P, Q) be the essentially selfadjoint Hamilton operator of a quantum system on H, then using the intertwining relations (1.2) and (1.3) one finds for the time evolution of an arbitrary element '(p, q,t) of D, C L(R2) the following K, (, q, t; p ', t')dp'dq' where, K,(, q, t; P' (r(p, q), exp[i(t 1 V/27  t')(P, Q)1p ')) V*(p, q) exp[ 1 V/27 t')(P, Q)}V(p'  t')N( i8,q + iQp)] (, 1 J2v v (p,q)V(p' where the closure of the Hamilton operator has been denoted by the same symbol. This construction holds for any 7r EH therefore one can choose any complete or thonormal system {(j>&}=1 in H and write down the following propagator K(p, q, t; p' K, (p, q, t; p', i(t t')n( iq, q + ip)] Z j 1 V/27T: V*(p,q)V(p ioq, q + i8p) tr[V* (p, q) V (p' Let us now evaluate etr[V*(p,q)V(p', q' complete orthonormal systems in L2 (R). {a(x) }=1i {k (z) }=i be two Then using the representation in (1.1) we 1 V^TT 1 J2 ,D, ,t'), (p' t(p, q, t)  t') (  tr[V(p, q)V(p' ipQe)qP (e. P\, eip'Q4 ) k, = kc,/=l ' #1 (x)exp( ipx)lk(X + q)dx + q')lk(x / k(x' + q') exp(ip'a')fl(x')dx' + q) exp[i(p'x'  px)]dxdx' Sexp[  x')S[(x'  p')z]{  q')]dx'}dx x) q')  p')6(q  p')x]dx  q'), the fourth line follows from using the completeness relations for the '(sx) and k (x). Hence , the propagator K(p, q, t; p' , t') is given by: K(p, g, t; p'  exp[ ia,, q + i8p)]S(p q'). (1.4) As shown in [65] this propagator propagates the elements of any reproducing kernel Hilbert space L1(R2) correctly, i.e. ,p(p,q,t) K(p,q,t; p ', ', t')In(p' ,t')dp'dq', (1.5) The propagator in (1.4) is clearly independent of the chosen fiducial vector. A suf ficiently large set of test functions for this propagator is given by C(R2) n L2(i2), where C(o2) is the set of all continuous functions on JRZ Hence every element of L()4R2) is an allowed test function for this propagator. From (1.4) it is easily seen that the fiducial vector independent propagator is a weak solution to Schrbdinger equation, i9tK(p,q,t; p' =w ig, q + i.p)K(p, q,t;p'  t . (q ( ,e 6i() 1(x}^WCk (x'  )t'( 7 We now interpret (1.6) as a Schrbdinger equation appropriate to two separate and in dependent 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 twovariable Hamilton operators, ones where the classical Hamiltonian is restricted to have the form 7U(k, q  x) instead of the most general form U(k, q,p). In fact the operators given by equation (1.2) and (1.3) are elements of the right invariant enveloping algebra of a two dimensional Schrbdinger representation. interpretation following standard procedures (cf. Based on this [63]) one can give the fiducial vec tor independent propagator for the Heisenberg Weyl group the following regularized standard phase space lattice prescription: K(p,q,t;p' lim N+oo . .*/ exp{i (qe+1 qj) (qj+1 + qj)/  xj+1 )]} J dpdqj f dkj+i/2dCj+/, 2 where (pVN+1, qN+l) _= (P, ) (Po, qo) = (p' and e  (t  t')/(N + 1). Observe that the Hamiltonian has been used in the special form dictated by the differential operators in equations (1.2) an After a change of variables (see (1.3) and that Weyl ordering has been adopted. [65]) the fiducial vector independent propagator for the Heisenberg Weyl Group becomes K(p, q,t;p' lim N+o S./ N1 exp{it (qj+l + qj)(pj+i j=o  pi)  xj+l/2(pj+l  p) + k+1l (qj+i qj)  eW(k,+i/2, )] Jdpdqj /2dZ+1/2 i=1 i=o (2.r Tankiner an imnrnnpr limit hv intprhanatrinaT thp limit. with rsnnPrtt tn N with thP  eH(Cj+i pj) + k+1 (pj+l K(p, q, t; p' here = M denote /exp i momentt" Jqp  zp + k4 conjugate to the 7/(k, x)]dt p:qDkDx, "coordinates" respectively. Despite the fact that the fiducial vector independent propagator has been con structed 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 [65]). 1.1.1 Examples of the Fiducial Vector Independent Propagator 1.1.1.1 Vanishing Hamiltonian We now look at two examples of the fiducial vector independent propagator. first example is that of the vanishing Hamiltonian which leads to K(p, q, t;p' exp /(qp+ k4  xp)dt DpDqDkDx exp (i  p') (q This is of course a trivial example; however Sqpdt 6{4} 6{p}VpTdq q'). r, it shows that the fiducial vector inde pendent propagator fulfills the correct initial condition as is expected from equation (1.6). 1.1.1.2 The Hamiltonian I7 = (P2 + w2Q2)/2 The second example we consider is that of the Hamiltonian l(k, z) = (k2 w2z2)/2. Here the fiducial vector independent propagator takes the following form: 1( n t *tn n' ' csc(wT/2) exp i (q + q'(p 1 p')+ cot(wT/2) 4 1 (p W p ') +w(q where T = t t' . This is an unusual result for the propagator of the harmonic os cillator. This result has the appearance of a propagator for a two dimensional free particle in a uniform magnetic field (cf. 64]). However, when one brings up an element of any of the reproducing kernel Hilbert spaces L (R2) then this prop agator acts like the conventional propagator for the harmonic oscillator in the appropriate limits one can recover the Moreover, usual propagators in the Schrbdinger representation (see [65]). General Overview of the Thesis This thesis is organized into six chapters and three appendices. is this introduction and the last chapter is a conclusion. The first chapter The results of our research are contained in chapters 3, and 5. The three appendices have been added to make this thesis reasonably selfcontained. In chapter 2 we discuss the construction of path integrals on group and symmetric spaces. In section 2.1 we review the Feynman path integral on flat, group, and sym metric spaces. Section is devoted to the study of group coherent states associated with a compact group and the construction of coherent state path integrals based on group coherent states associated with a compact group. In chapter 3 we introduce the notations and basic definitions used throughout the thesis. The main result of this chapter is Theorem 3.2.1, in which we derive an operator version of the generalized MaurerCartan form. Chapter 4 contains the construction of the representation independent propagator  q') locally compact, connected and simply connected Lie group1 with irreducible square integrable representations2 as a general Lie group. For a given set of kinematical variables this propagator is a single generalized function independent of any particular choice of fiducial vector and the irreducible representation of the general Lie group generated by these kinematical variables. In section 4.1 we define coherent states for a general Lie group and prove Lemma 4.1.4 and the Corollary 4.1.5 which we apply in the construction of the representation independent propagator and the construction of regularized lattice phasespace path integral representations of the representation independent propagator. Prior to constructing the representation independent propagator for a general Lie group, we construct in section 4.2 the representation independent propagator for any real compact Lie group. It is shown in Theorem 4.2.2 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. As an example the representation independent propagator for SU(2) is constructed. In section 4.3 this construction is then suitably extended to a general Lie group and we show in Theorem 4.4.2 that the result obtained in Theorem 4.2.2 holds for a general Lie group. In Proposition 4.4.4 we establish that it is possible to construct regularized phasespace path integrals for a general Lie group. the group space is a multidimensional curved manifold, it is s Even though generally hown that the resulting phasespace path integral has the form of a lattice phasespace path integral on a mul tidimensional flat manifold. Hence , we obtain a novel and very natural phasespace path integral quantization for systems whose kinematical variables are the generators ar\ Tll.njlnn 'r^i ra+l kn nrc 1 +br'i.?^ +rbo rnnnran4 n+Srv an n Aanan nFf ft WAmtAft T /i mr"i In chapter 5 we discuss the classical limit of the representation independent prop agator of a general Lie group and show that its classical limit refers indeed to the degrees of freedom associated with the general Lie group. Sections 5.1 and 5.2 con tain a detailed discussion of the classical limit of the coherent state propagator for compact Lie groups and noncompact Lie groups. In section 5.3 we prove that the equations of motion obtained from the action functional of the representation independent propagator for a general Lie group imply the equations of motion obtained from the most general action functional of the coherent state propagator for a general Lie group (cf. Proposition 5.3.1). CHAPTER 2 A REVIEW OF SOME MEANS INTEGRAL ON GROUP TO DEFINE THE FEYNMAN PATH AND SYMMETRIC SPACES This chapter is somewhat independent of the rest of this thesis and serves as an introduction to some of the ways of constructing path integrals on group and issue again with rigor in chapter 4. path integral on JRd, gr< to a preliminary study of group coherent states, but we will confront the Section 2.2 is devoted detail in chapter 4. The remaining part of section 2.2 is devoted to the construction of coherent state path integrals based on group coherent states. The Feynman Path Integral on Rd year 1925 can be seen as the beginning modern quantum mechanics marked by the two almost simultaneously published papers of Heisenberg [52] and The former proposes the formalism of matrix mechanics, the latter proposes the formalism of wave mechanics. Schridinger f92 while 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 [104]. This general formulation of quantum mechanics is based on an analogy with the symmetric spaces. Our arguments will be largely heuristic, In section oup, and symmetric spaces is discussed. the construction of the Feynman we take this subject up in more Introduction , Group, and Symmetric Spaces Schridinger quantum mechanics, except in the suggestive derivation of Schridinger's wave equa tion from the HamiltonJacobi equation by the substitution, ihln('0), 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 qto) corre sponds to the classical quantity exp[(i/h) ft Ldt]. led Feynman in 1941, then a student at Princeton, It was this remark by Dirac that to a new formulation of quantum mechanics (see the account in [47 126129]) . 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, , 58, 60, 61, 80, 97] . 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 use ful if one wants to restrict oneself to questions concerning only one system. one has benefitted from Feynman's approach Another 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. ,60, 61]). There are several books and review articles on the subject of path integrals. 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 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 contains many applications of the path integral method to problems in quantum mechanics, statistical, and polymer physics. Moreover Inomata et al. [54] discuss various techniques of path integration not covered in the aforementioned monographs. 2.1.2 The Feynman Path Integral on Rtd 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 To bocman [97]. The idea is to find an appropriate approximation for the time evolution operator, U(t"  t') = exp[ (i/h)(t"  t')9i] (introduced in chapter 1), at small times and then to construct step by step the time evolution operator at finite times. start from the identity U(t" which holds for any  t') = [U((t"  t')/(N + 1))1N+ 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 t S1  .1Cfc To ensure that the quantized Hamilton operator 1{(P, Q) is unambiguous, i.e. in order to avoid operator ordering problems, we consider the following simple Hamiltonian Ha (p, q) = 1 (2.1) where q = (qi, ..., qd) and p  (pi Furthermore, we use the mixed (p, q) matrix element of the time evolution operator U(t): (q"JU(t"  t')lq') /(q"IP') (p U(t"  t')lq'>dp', j_ L i __ _____ ... _t i TI* *!I .... At 11 U(e) + V(q), . .., pd).  H(p, q)](pq)  (pq)] ft (Pq) , (2.2) valid to first order in E. Here , (p, q) is defined 1H(p, q) For the simple Hamilton operator '1(P, Q) = (1/ + V(Q) we are considering A(p, q) coincides with the classical Hamiltonian Hd(p, q). Note that for more corn plicated Hamilton operators this has no longer to be true (see below) . Using ( and the fact that (p q) =(27r) d exp (i/h)pq] we find that t"; q', t') (q" IU(t"  t')lq') lim (qf(U (C)]+ N*oo lim N oo f l .+iIU(e) J d C C lim Ntoo " exp [Pj+i (q +1 q)  eHd(pj+i N j=1 N j+1/2=0 (2.3) dpj+l/ (271) where qo = q' and qN+1 = q". It follows from ( .3) that the qspace propagator q', t') satisfies the following initial condition: q', t') = 6(q" as it should ts very definition. Observe that in the phasespace path integral representation (2.3) there is always one more integral over the p than there is over 1ml ,1 a I J r r A..!  A. S.. n:J . qj)~ (p7H(P, Q)lq) (Plq) J(q", J(q"  q'), lim J(q" t" t' with respect to N with the integrals we find the following formal standard phasespace path integral J(q" I (p(t), q(t)) ti DqDp, where,  Ha(p, q)]dt. This formal phasespace path integral for the qspace propagator J(q" first written by Feynman [36, Appendix B.], q',t') was and then was subsequently rediscovered other authors (see for instance Davies (20] and Garrod [40]). The integration ranges over all paths in 2ddimensional phasespace which are pinned at q Sq," ' and , while the integration over the moment is unrestricted. The Lagrangian form of the path integral as originally proposed by Feynman can be obtained form (2.3) by integrating out the moment. done follows from the fact that the moment enter quadraticly. That this can be Hence, if we carry out the N + 1 Fourier transformations in (2.3) which are of the form: i . exppy z ' S dij+/2 2'rihc~ 2 1  q) 2Pj++/2J (q j+ (27)d (2.6) then we find the following result: J(q" ... exp Id(q(t)) hi c (q(t)) V(q) , where q(t') = q" ', q(t") (2.7) 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 number of remarks  S 1  ]l(p(t);q(t)) dpj+l/2 i(q+i qj) invariant under general canonical transformations. However, this is not the case. As shown by Klauder [63, section II] the regularized lattice phasespace prescrip tion (2.3) for the qspace propagator is only invariant, or better covariant, the subset of point transformations among all canonical transformations. Operator ordering. under If the Hamiltonian is no longer of the simple form we have considered in (2.1) but has a more complicated (p, q)dependence, then one has to confront the issue of operator ordering in the Hamiltonian. is the case for a free particle moving on a Riemannian manifold, Hd(p, q) For example, for which gi(q)pipj. The basic principles one uses for the resolution of the operator ordering prob lem are (a) the Hamilton operator has to be symmetric and, (b) if the classical system has a symmetry group, the corresponding quantum theory must have this symmetry. As Marinov remarks , "the first of these conditions is evident while the second is more arbitrary and not always constructive" [75, In particular, condition (a) cal Hamiltonian Hda(p, q) = implies that we should associate with the classi SF(q)p the following quantized Hamilton operator 9i(P,Q) = (1/2)[PF(Q) + F(Q)P]. Using the principles (a) and (b) in the resolution of the operator ordering problem might lead to additional correction terms proportional to h2 in the action functional (cf. Integral over configurationspace trajectories. (2.7) was obtained for the Hamiltonian (2.1). [95]). The Feynman path integral in If the pdependence of the Hamil tonian is no longer simply quadratic but more complicated, then the integral in (2.6) is no longer a simple Gaussian integral and does not result in the classical 2.1.3 The Feynman Path Integral on Group Spaces The quantization of a free particle moving on a group manifold has been con sidered in a number of works , 48, , 56, , 76, Schulman [94 introduced starting from the known semiclassical approximation, propagator for a free particle moving on the group manifolds of SO(3) and SU(2). However, Schulman did not present a simple path integral solution for the problem, (cf. the remarks in Ref. 71, 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 Ref. 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 nsphere, SO(n + 1)/SO(n), n> 3. The question as to what is the largest class of spaces for which approximation is exact seems still to be an open one. the semiclassical 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 Marinov and Terentyev , 77]. Before we consider their proposal we briefly outline the construction of path inte grals on Riemannian manifolds. DeWitt [22] observed that for a free physical system moving on a ddimensional unbounded Riemannian manifold with constant scalar nn+..... ...., oi m+, toncnr finl l th nrnnacratnr for infinitesimal time is riven ,94]. where q', t') = (2rih) (q")D(q" q', t')g D(q" q',t') = det 21ia agOq'iqk is van Vleck' determinant. Here g(q) and Id  , 2 JC gij*1'qdt is the classical action functional. As observed by Marinov [75], this simple form of the semiclassical approximation is only valid for unbounded Riemannian manifolds, since the proof of the theorem that two points q" connected by only one classical path (cf. [108, and q' at fixed small t"  t' may be pp. 5864]) uses the unboundedness of the manifold in an essential way. On the other hand, as is pointed out in Refs. 29 and 75, 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 (2.8) possibly with a phase; Berry and Mount [9]. is multiply connected, see also in this respect the review article by As an example we mention the case when the bounded manifold 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 [95, 197205] for a discussion of this point. For this case the semiclassical approximation takes the following form: J (q," q', t') Am (q" _i ,,t., q', t') exp q', t') (2.9) where the sum is over all classical paths connecting q" and q' If is the classical action functional along the mth path and 7m is an integer that depends in general on all the arguments. Also note, as is remarked in Ref. that the semiclassical 9) 11'2 d/2 [g A(q" det[gj (q)] 1 I particle. For the case of an unbounded Riemannian manifold the propagator at finite times is constructed by folding N + 1 propagators of the form (2.8) q', t') N k= tk+1; qk,tk) Jdqk, (2.10) where dqk = V dgi. Taking the limit N  oo one obtains a functional integral over all intermediate coordinates that can be interpreted as the path integral. final result is a path integral of the form (2.7) however, the Lagrangian needs to be modified a term proportional to h2R. DeWitt found this term to be h2 (R/12); this modification of the Lagrangian was also discussed by McLaughlin and Schulmann [73]. In the context of curvlinear coordinates the reason for modifying the path integral has been discussed by Arthurs [2, Edward and Gulyaev [32], and in the context of quantization of nonlinear field theories by Gervais and Jevicki and Salomonson [90]. If on the other hand , one applies this approach to a bounded Riemannian mani fold, as is the case for compact Lie groups, then the resulting path integral, as pointed out in Ref. , is far from simple. Since one then has to use the semiclassical ap proximation presented in ( "coordinates 9) and in addition to integrating over all intermediate , one also has to sum over all the different classical paths connecting q" and q'. Nevertheless, if the bounded manifold M in question is isomorphic to a quotient KN/lr space /r/F is an unbounded Riemannian manifold where I is a transformation group acting on NA then one can and A , as proposed by Marinov and Terentyev [77], construct a propagator on M by summing over the group F JM (q" q', t') J (q" JNr7 J(q" r, t'>} shown by Marinov and Terentyev [77] that this approach is valid for any compact Lie group. In their work, see Ref. , Marinov and Terentyev take for AN the Lie algebra that is associated with the Lie group they wish to consider and for r 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 4 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, Lagrangian is given by L(q, S) 1  I Here I = mp2 denotes the moment of inertia of the particle and the angular variable 4 ranges form 0 , where we identify the points 4 = 0 and 4 = 27r . The solution of the equations of motion is found to be A <> + 4t, where 4o and w are arbitrary integration constants. 4' and ending at #"' Considering a motion starting at we find for the classical action functional: ', t') S(t')(  27rn) (2.12) 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 socalled winding number n, the number of times the particle moves counterclockwise minus the number of times it moves clockwise past the point 6' Hence, the paths break up into distinct classes Sd (# The canonical quantization for this example is straightforward, since p7 = 8L 16, we find for the Hamilton operator h2 2I "' which has the following normalized eigenfunctions and eigenvalues: 1 S2W lbm(4) exp(im), = (hm) where m = 0, 1, 2,.... The propagator is given in terms of the eigenfunctions bm(4) by the following spectral expansion: (") (') exp n=oo n=5oo  E (tt  ih(t" 2I t)n The sum over n is related to the Jacobi theta function, +00oo exp(iirtn2 + 2inz). n= oo Therefore , with the following identifications we can write the propagator in closed form. t') then we find 03 2 hnT 24Il where A4) =6 " 'and T = t" '. Using the following property of 03, which follows from the Poisson summation formula (see ,pp. 6365]), = (t1/2 exp( z2 z it (73  nft t the propagator can also be written as a sum over classical paths, series, i.e. as a semiclassical  ix 1 ti  t') J("  ')n 03 (Z, h(t" J(" 03 (Z, 23 Observe that each of the propagators J in the series (2.13) is of the same form as the propagator of a free particle moving on the real line R and that the series as a whole is a function of period 2,r . The series ( 13) is a particular example of the general principle (2.11). Folding the propagator in (2.13) N + 1times leads to the following path integral representation .2w lim o N+o0 o "."0 0o 2= N j=o I irihe nj =0 i i(j+l   2rny) j=1 where N +1 = t", to = , ande = T/(N+ 1). If we now shift the integration variable at each step, we can extend the N intermediate integrals to the whole real line, i.e. ni=o0 2(nj+l)ir r +oo0 00 d4,j. As a final result we find J( " lim N*oo 2Xihe (N+1)/2 n=oo +00 00 +00 . .. 'O0  U) N d1 dt , (2.14) where ON+1 = and 4o = t' + 2rn. Note that the circle is the group manifold of the simplest compact Lie group U(1) whose faithful irreducible representations are given by = exp(i4) and D1( ) =exp( It is well known that the one dimensional abelian translation group Ti of the real line JR is the universal covering group of U(1). Furthermore, the translations by 27rn, = El, +2, form the cyclic subgroup (27r) of T1 which is the kernel of the homomorphism Ti x + f(x) exp(ix) e U(1). a , Therefore, by the Fundamental Homomorphism Theorem we have J(w" I( +1 D() i>). 24 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 (2.14), difference that the Lagrangian has to be modified to include a 'quantum' proportional to h2 with the only potential . One might ask if the approach of Marinov and Terentyev could be extended to more general systems than the free particle as we have mentioned above The answer is no, since, , the semiclassical approximation is only exact for the case of the free particle moving on the group manifold of a semisimple Lie group. 2.1.4 The Feynman Path Integral on Symmetric Spaces More recently Bbhm and Junker have used zonal spherical functions to construct path integral representations for a free particle moving on the group manifolds of compact and noncompact rotation groups (B6hm and Junker [12, 14]), the Euclidian group (BShm and Junker [13]), and on symmetric spaces1 (Junker [56]). However, a careful analysis of the construction presented in [56] 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 massive2 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 56] and achieve his proposed unification of the work 1Let (S, 7) and (T, U) be two topological spaces. A continuous onetoone map f of S onto T is called a homeomorphism if f1 is continuous. A topological space (S,7) is called homogeneous if for any pair u, v S there exists a homeomorphism f of (S, 7) onto itself such that f(u) G be a connected Lie group and let a be an involutive automorphism of G, i.e. a2 = v. Let = 1 and a #l. Denote by G, the closed subgroup of G consisting of all elements G that are fixed points of o, i.e. = g, and by GC the identity component of G,. SHC GC,, Let H be a closed subgroup of G such that then one calls the quotient space G/H a symmetric (homogeneous) space (defined The nsDhere S" is an example of a symmetric space. by a). Page Missing or Unavailable 26 valid to first order in e, and dxj denotes the invariant measure on M. the cases we are considering, Note that for where G is a unimodular Lie group and H is a massive compact subgroup, an invariant measure always exists (cf. what follows we ask that the short time propagator (2.16) transformation group G Corollary 4.3.1]). be invariant under the that J(gxj+l, gxj; e) = J(xj+l ,x,; e) gEG, (2.17) for j =0.1 As we will . , see below this is a crucial assumption since it implies that A is an invariant elliptic4 operator in the enveloping algebra5 on G.6 This can be seen by using the form (2.16) of the short time propagator valid to first order in e in (2.17) From which it follows that the Hamilton operator 71 has to be an invariant operator for G if (2. 17) is to hold this in turn implies the above statement. 'Denote by a length of a by = (1, ... ,am a multiindex consisting of m nonnegative integers Define the For every z S"m let =nI Let P be a polynomial of m variables of degree r, which has the form where ca are arbitrary complex numbers and ca # 0 for at least one a with a = r. Then we denote the formal differential operator generated by P by: P(iV) = where V = (O,, , ** ,0,m). The formal differential operator P(iV) is called elliptic, if there exists 0 such that  e r (i)ola' J=l =5:  k 1 A i j __ i ilr 't^ ^ I*' r i i r ^ a Let a be a fixed point of M whose stability group is H, i.e. one has ha = a for all h Since G acts transitively on M we can write each xj EM as for some gj eG. (2.18) Hence, using this construction one can view the short time propagator as a function on the group G: J(xi, xj;e) = J(gj+l, g; ). Using the translation invariance of the short time propagator it follows that the short time propagator can only be a function of ggj+', hence, J (g,+, gj; e) = J(ggj+1, ggj; ) = J(g 1j+l; e). (2.19) Since ha = a for any , we see that (2.18) is invariant with respect to right multiplication with elements of the stability group H time propagator is invariant with respect to right m . This implies that the short ultiplication with elements of From (2.19) 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; c) is a constant function on the two sided costs HgH with respect to the subgroup H J(high2; ) = J(g; e), for any hi, h2 EH. Let UC be an unitary irreducible representation of class 1 on the Hilbert space RC. Let us choose any complete orthonormal system {(}mo 1 in RC then we can associate with UC the following matrix elements D((g) = (<, UU q). (2.20) xj = gja (2.20) 28 Dt(g) 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. Proposition 14.2.2]). This property is often the starting point for an explicit calculation of the Dt (g), in some detail for the case of SU(2) in section 4.3 we consider such a calculation If the maximal set of commuting operators does not contain an elliptic operator then the matrix elements D (g) are generalized functions, i.e. distributions (cf. [7, Theorem 14.2.13). In chapter 4 we will consider the construction of path integrals for the cases in which the matrix elements Dt are either not explicitly known, or are generalized functions. This construction will make no explicit use of the functions D (g) but will only use the facts that they exist and form a complete orthonormal set. Note that for the cases considered in this chapter the set of maximal commuting operators always contains the LaplaceBeltrami operator which as we have remarked above enveloping algebra of G. Hence, , is an elliptic operator in the center of the the matrix elements Dt(g) are regular functions on G. This shows that the assumption that the short time propagator should be invariant under the transformation group G is crucial and can not be relaxed. Since H is a massive subgroup of G, a E 7& that is invariant relative to H. U there exists a unique normalized vector sing the GramSchmidt orthogonalization procedure we can choose our complete orthonormal basis in such a way that = a. Our interest now focuses on the (00)matrix elements = (t0, (2.21) One can easily convince oneself that this function is constant on the twosided costs HgH with respect to H . The function defined in (2.21) is called the zonal spherical function of the irreducible representation U relative to H If we take G = SO(3), Doo (g) US4o). Let us denote by G the set of all inequivalent irreducible unitary class 1 represen stations 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 twosided costs HgH with respect to H can be decomposed in zonal spherical functions Doo(g), , of unitary irreducible representations of class 1 (see [103, pp.5055]): dccDoo(g9), Do (g) f (g) dg. Here stands for the discrete or continuous orthogonal sum of all inequivalent irreducible unitary representations of class 1 of G with respect to H . The constant de appearing in (2.22) is given by = d c(C (2.24) where in suitable coordinates if G is discrete, if G is continuous. For the case of compact groups the constant de is the dimension of the representa tion space 7R of the unitary irreducible representation U , see also in this respect remark 4.1.1. We have now collected all the tools we need to construct the path integral repre sentation for a free particle moving on M. We have seen above that the short time propagator is a constant function on the twosided costs HgH with respect to H hence using (2.22) we can decompose it in zonal spherical functions: D (g) Doo (g)dg Moreover ,let f e LI(G), where L1(G) is the space of all integrable functions on G, then one has (cf. Corrolary 4.3.1]) f(g)dg f(gh)dhdx, JM H which reduces for f e Li(H\G/H) to f(g)dg f(g)dx f(g)dx, (2.27) since f(high2) = /f () Vhl E H and where we have chosen fi dh because H is compact. Using (2.25) and (2.27) in ( .15) one finds J(x" x', t') lim Noo G .G. . J dgj. (2.28) Using the orthogonality relations for the functions /d'D\ and the left invariance of dg one can easily show that the following relation holds 00 Sr'gj+l)Do(gj1gj)dgj = 6({, C+i)D)o(g~i gy.j+) (2.29) Using ( 29) the N intermediate integration in ( .28) can easily be performed and one finds as a final result that J(x" x', t') { lim [c(e)]N+ dD~ (g'1 Nd+Do Let us now evaluate the limit N + oo in the above expression, for large N one can write c((c) as = cc(0) + valid to first order in e. The value of c(0) can be found from (2.26), using the fact that the short time propagator satisfies the following initial condition him J(g7 5j+i; e) 9j+i) = Se(gj c6,+ ()Doo+ (gI gj+l) dt+1 d"+i cC ( ) 7 Or if we set E = ih (0), we find lim N.. c (4)N+l = exp(  (t" It is shown in Ref. 56 that the (0) are the eigenvalues of the LaplaceBeltrami oper ator A on 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 group property = E kDio(g')Do(g1") we can write (2.30) in the more familiar form J(x" X', tl) (2.31) where YCk(9) = dDio(g) One calls the matrix elements D50o(g) the associated spherical functions and they are the eigenfunctions of the LaplaceBeltrami operator on M. the two sphere For the case that M is the functions 1/dD, o(O, ) are the classical spherical harmonics which, as is well known operator on the two sphere , are the eigenfunctions of the LaplaceBeltrami Hence, ( 31) is the well known spectral expansion of the propagator in terms of normalized eigenfunctions of the Hamilton operator. specific examples we refer the interested reader to Ref. 56. 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 Dt(g). However, it should be clear from the remarks after (2.20) that one can construct path integrals this way only for a handful of groups. In chapter 4 we will overcome the reliance on the matrix elements D (g), t')E, D,(g 'l g) '(t" h^ YCk(x')Ya,(x ), Yie(0), as we have seen in this chapter and will see in chapter 4 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 amodels and nonabelian lattice gauge field theories. Coherent States and Coherent State Path Integrals 2.2.1 Introduction The origin of coherent states can be traced back to the beginning of modern quan turn mechanics. SchrSdinger [93] introduced a set of nonorthogonal wave functions to describe nonspreading wave packets for quantum oscillators. mann In 1932 von Neu 104] 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, ,58]) Klauder 58] 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 cnumber expression. In 1963 Glauber [44, , 46] named the set of wave functions introduced by Schrodinger "coherent states" in the field of quantum optics [67 coherent laser beam. and used them 83] for the quantum theoretical description of a At about the same time Klauder published two papers [59, 60] dealing with the formulation of continuous representation theory, that contain the seminal ideas for the construction of coherent states on general Lie groups. Coherent states for the noncompact affine group or ax + b group and the continuous represen tation theory using the affine group where introduced by Aslaksen and Klauder [4, 5] 33 Several books and review articles consider the definition and properties of coherent states ([17 34, 68, 85, 109]). Klauder and Skagerstam [68] 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 [85] 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 al. [109] and the recently published proceedings of the International Symposium on Coherent States [34 also deserve to be mentioned. 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 )}LEC on H to be a set of coherent states it must fulfill the following two conditions. The first condition is: Continuity: That is forall e The vector 1) is a strongly continuous function of the label 1. > 0 there exists a 6 > 0 such that Sfor all 2' Here, ., *)1/2 E C with * I denotes the norm on H induced by the inner product on H, i.e. II  = . Or stated differently, the family of vectors {I )}I on H form a continuous (usually connected) submanifold of H. We assume that (Il ) > 0 for all I E C. In the applications we are considering the continuity property is always fulfilled. The second condition a set of coherent states has to fulfill is: Completeness (Resolution of the Identity): There exists a sigmafinite posi resolution of identity ) d (1) (2.32) In general, as pointed out in Ref. 68, p. , "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." 2.2.3 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 that the discussion applies to a general Lie group, as defined in chapter Let us denote by G a compact ddimensional Lie group. It is well known that for compact groups all representations of the group are bounded and that all irreducible repre sentations are finite dimensional. Moreover, one can always choose a scalar product on the representation space in such a way that every representation of G is unitary, Theorem 7.1.1]). Therefore , without loss in generality we assume that we are dealing with a finite dimensional strongly continuous irreducible unitary representa tion U of G on a dcdimensional representation space H Let us denote by {Xk}dI=l the set of finite dimensional selfadjoint generators of the representation U The Xk, ..., d, form an irreducible representation of the Lie algebra L associated with , whose commutation relations are given by d [Xi, Xj =i where cj denote the structure constants. The physical operators are defined by = iXk. For definiteness it is assumed that there exists a parameterization for G such that cekX, following set of vectors on He 17(1) = VdUz(ln?. (2.33) It follows from the strong continuity of Ug(I) that the set of vectors defined in (2.33) forms a family of strongly continuous vectors on He. Furthermore, let us consider the operator 0= (2.34) where dg(1) denotes the normalized, invariant measure on G. It is not hard to show, using the invariance of dg, that the operator O commutes with all Ug9(, I LC. Since Ug() is a unitary irreducible representation one has by Schur's Lemma that 0 = AIH . Taking the trace of both sides of ( .34) we learn that tr(AIH) = Ad trt[r(') (r)(/'), > dg>(') dc i77 dg(l') Hence, the family of vectors defined in (2.33) gives rise to the following resolution of identity: )>dg(l). Therefore (2.35) , we find that the family of vectors defined in (2.33) satisfies the requirements set forth in subsection 2.2.2 for a set of vectors to be a set of coherent states. we conclude that the vectors defined in (2.33) form a set of coherent states for the compact Lie group G, corresponding to the irreducible unitary representation Ug(. 2.2.4 Continuous Representation *)dg ('), rl(')(() r7(1)(}7(1), L (G, dg) [c,& ] (0) This yields a representation of the space H by bounded, continuous, square integrable functions8 on some closed subspace L((G) of L2(G). operator on H Let us denote by B any bounded then using the map C, and the resolution of identity we find that (2.35) (t (Z) Bn(1')) (r( (2.36) holds. Choosing B =/H we find ,(1) (2.37) where K, (; ') = 4 One calls (2.37) the reproducing property. (Furthermore, as shown in ),Appendix B.(')) Furthermore, as shown in Appendix B.2, the kernel K(l' 1) is an element of L2(G) for fixed I Therefore the kernel 1) is a reproducing kernel and L (G) is a reproducing kernel Hilbert space (cf. Appendix B.2). Note that a reproducing kernel Hilbert space can never have more than one reproducing kernel of continuous functions, K(l' Claim B 1) is unique. Therefore, since L (G) is a space Moreover since the coherent states are strongly continuous the reproducing kernel ,(l' 1) is a jointly continuous function, nonzero for I = 1', and therefore, nonzero in a neighborhood of I This means that (2.37) is a real restriction on the admissible functions in the continuous representation Of course a similar equation holds for the SchrSdinger representation, however KM'( E C. =(q W(), = ), () '), ) dg (') (W(1), B ) of He. from L2(G) onto the reproducing kernel Hilbert space L2(G) (cf. Claim B.2.2). This ends our discussion of the kinematics (framework) and brings us to the subject of dynamics. The Coherent State Propagator for Group Coherent States Let 'i E Hc and denote by 7(X1,..., Xd) the bounded Hamilton operator of the quantum system under discussion, then the Schridinger equation on He is given by ihdt h = (X ..1 d) since 7t is assumed to be selfadjoint and does not explicitly depend on time, a solution to Schridinger's equation is given by: b(t") = exp Now making use of (2.36) we find 14 (I" K,(l" ,t";1' ,t')dg(l'), where K,(Z" ,t') = (7(l"),exp  t')7t Note that the coherent state propagator K,(l" , t"; 1' , t') satisfies the following initial condition lim K,(1" t' . tt' ,t') = (l" Hence, as t"  t' we obtain the reproducing kernel C,(l"" which, as we have remarked above, is the integral kernel of a projection operator from L2 (G) onto L2 (G). Moreover, since /,(l";1') is unique, we see that if we change the fiducial vector from v1 to 7', save for a change of phase, then the resulting coherent state propagator is no 1,,,,, .. L 1. .. .  .1 .. 1.. .. 1 T..IT:1 L. ... T.2 rf2 r Ii Ptl  t')( .(t'). ,t') ,(l' ,t") (t"  i(f11 q(l')). 38 Following standard methods in Refs. 63 and 68 we now derive a coherent state path integral representation for the coherent state propagator. We start as in section 2.1.2 from the basic idea t Ii(t" i r a where e = (t" therefore we find K, (" ((/"), exp (t" ?r t') l7(')) (i1(I") i  ) exp ( Inserting the resolution of identity ( .35) Ntimes this becomes N ( 3 =0 e inl) .t N / ** where 1N+1 = l" and 1 = ' 4., This expression holds for any and therefore it holds as well in the limit N  oo or e = lim fso. N i. N h 7 (l)> jdg( ). Hence, one has to evaluate (7(lj+1), exp (  e)ri(lj)) for small e. For small e one can make the approximation i  'l 7 )) P~tn ,_ ()(lj+}), th (n7(Ij+l)  (l Pt K ,(Ij+i; l) exp  H(l+; lj) (2.39) "'a')) K (2" t"; 1' 1, (l)) 3, '(l))  t')  t')/(N + 1) K,(1" (77(lj+1), exp( (<(2+l) 7lj+i(lj), ,(l+1; l) xp Inserting (2.39) into (2.38) yields I', t') =lim O Nj= *...JJ KK(li+i j=0 lj) exp cH,(ly+,. A N fldg(l) . (2.40) 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 (2.40) depends strongly on the fiducial vector. 2.2.5.1 Formal Coherent State Path Integral Even though there exists no mathematical justification whatsoever we now take in analogy to what we have done in sections 1 and 2.2 an improper limit of (2.40) by interchanging the operation of integration with the limit e + 0. As pointed out in Ref. 68, p. , one can imagine as e + 0 that the set of points ij, j . .. defines in the limit a (possibly generalized) function l(t), t' Following Ref. 68, pp. 6364, we now derive an expression for the integrand in (2.40) valid for continuous and differentiable paths *(t). in (2.33) is not normalized, the reproducing kernel K, (/Z+i; l) Note that the set of coherent states rl(1) we have defined but is of constant norm given by d'2. We now rewrite = ((lb+i), r(lj)) in the following way (bj+i)  '(j)) dc[1 deex  d((l+), (l(j+) dr' ((ij+i), r(1+1 this approximation is valid whenever W(lj+1)  ()(l) ... Hence, as e + 0 the approximation becomes increasingly better since the rl(1) form a continuous family of vectors. Therefore one finds a ' 9 t 1 I It \ ,. \'t1 iA ja' Sf f Ik a j n  I gAf *I U I f U NE S U  )  (l))] ) (1)], g(l" (,7(1j+i) rl(lj)) (W7(j+i) p following form: I, 11   dc vo (7l(1), dr)(l) i ft H, (1(t))dt where Hr,(l(t)) = () 7(X1, X...) and where we have introduced the coherent state differential dr(1) 7(1 + dl) f(l). Hence, we find the following formal coherent state path integral expression for the coherent state propagator: exp x t K, (" ih(,(l) (2.42) where = U(1)7 VDg(l) lim N+oo (d)N+l' fdg(l). j=1 A discussion of what is right and what is wrong with (2.42) can be found in Ref 68, 6466 , we only remark here that (2.42) depends strongly on the choice of the fiducial vector and on the choice of the irreducible unitary representation of G. Hence 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 Ii of the quantum system one considers; see for instance Troung [100, 101]. Hence, one has to face the problem of various fiducial vectors. chaDter 4 we develoD a reDresentation indeDendent DroDafator. which nevertheless. d dt gg(1), l',t') H, () Also note that coherent state path integrals afford an alternative way of con structing path integrals for quantum systems moving on group manifolds and on homogeneous spaces. For instance Klauder has used in Ref. 62 the coherent state path integral to describe the motion of a quantum system with spin moving on the two sphere and in Ref. 64 to describe the motion of a quantum system on the Lobachevsky plane. Klauder has also discussed a quantization procedure for phys ical systems moving on group manifolds and homogeneous spaces using the action functional in (2.42), see Refs. 60 and 61, and has therefore, provided an alternative method of quantization to the quantization methods discussed in subsections 2.1.3 and 2.1.4. CHAPTER 3 NOTATIONS AND PRELIMINARIES 3.1 Notations In this chapter, G is a real, separable, connected and simply connected, locally compact Lie group' with fixed left invariant Haar measure dg, i.e. d(hg) = dg A(g) be the modular function for the group G; i.e. d(gh) = A(h)dg. If A(g) then the group G is called unimodular. It is known that the following Lie groups are unimodular (cf. 39, p. 50] and [53, chapter , 1]): Every compact Lie group. Every semisimple Lie group. Every connected nilpotent Lie group. affine group, which we will consider in chapter is an example of a non unimodular Lie group. compact support on G (cf. D(G) be the space of regular [15] and (78, pp. 6869]) Bruhat functions with Let T be a closeable operator on some Hilbert space H, spending to G with basis xl, then we denote its closure by T .. Z .Xd. Let L be the Lie algebra corre Then we denote by XI = U(Zd) 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]  i =1Ci jkXk. A vector 4 E H is called an analytic vector = U(x1), . .., Xd _ __ I I I LI ___ 43 the representation U of the 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(xk) of a basis X1,..., Xd. Hypothesis (A) is satisfied then by Theorem 3 of Flato et al. the representation , Xd 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 Hypoth esis (A). Therefore, the representation of L by symmetric operators is integrable to a unique global unitary representation of the associated connected and simply con nected 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 H exp( i TXj)  exp( ilX1) ...exp( ilXd), (3.1) f exp(ilVXj) = exp(ildXd) ...exp(ilXt), for some ordering, where I is an element of a ddimensional parameter space C . The parameter space Q is all of jd if the group is noncompact and a subset of Id$ if the group is compact or has a compact subgroup. Remark 3.1.1 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 Ug( is a representation of G. Since G is a manifold one needs in general a collection of proper coordinate charts that cover G (see Appendix A.4), by the dtuple 1. we will relabel the coordinates in each of these charts Nevertheless, in practice it is often possible to work with a single proper coordinate chart (parameterization) as the following example shows. However, , . half times the Pauli matrices and satisfy the following well known commutation relations Xi, Xj = icijkXk, where, for (ijk) an even permutation of (123), for (ijk) an odd permutation of (123), otherwise. One possible parameterization of SU( ) is given in terms of the Euler angles by u(O, ,) exp( 4X3) exp( iOX2) exp( cos(0/ sin(0/2) where, Note that the points 0 = 0 and 9 = w have to be excluded since at these points only + and  are determined, respectively. The Euler angles are analogous to geographical coordinates on the sphere in JR3 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 0 = 0 and 0 . Hence, at these singular points of the parameter space, and unitary matrix uniquely. no longer define a unimodular Therefore, the set of matrices for which the parameterization introduced above is unique is a proper subset of SU( However , as far as integration over the group ) is concerned the above parameterization is adequate since the 0 S 0t = ei( 0)/2 sin(0/2) 27 < < 27r = 7" i(d i(4+ ei(+()/2 COS(/ < 2x7. 1.2.1]) associated with the Lie group one considers; these matrices are then taken as the group generators. 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 exponentiating the group genera tors. 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 (3.1) is obtained by exponentiating the selfadjoint representatives { k} 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. Let (U, H) and (U', H') be unitary representations of G. operator A densely defined closed S from H to H' is called semiinvariant with weight a if gsvU = (g) SG. In what follows we shall need a common dense invariant domain for X1, S.., Xd that is also invariant under the oneparameter groups exp(itXk),  1, Define D ... * as the intersection of the domains of all monomials . ., ik .. Xi. for all 1 By definition D contains D, hence is dense in H. Then by Lemma 3 of Ref. 38 the restriction of X1,. * S d to D is a representation of L and by Lemma 4 of Ref. 38 D is invariant under all oneparameter groups exp(itXk),  1, Let Am k((g()) and pm k (g()) be functions such that on D the following relations hold: exp(ila a=m+l exp( bX, k((l))Xk, E^Am (3.3) b=m+l m1 m1 F r . r SC U . * m ces [A 1mk(f(1))] and [p . k (g(I))] exist. Furthermore, let U(1) be the d x d matrix whose mkelement is Um k(l) such that on D Us XmUg(l) U9Q)X mUg* holds. k()X&k, (3.5) (3.6) ,k(l) One can easily check that U(1) is given by exponentiating the adjoint repre sentation of L, = exp(lCk), here ck denotes the matrix formed from the structure constants such that Ck Preliminaries Theorem 3.2.1 On the common dense invariant domain D of X1, the fol lowing relations hold, For all I E U~() dUgQc) k (g())dlm =i k,m=l k(g(0)). and Lg(lo)Am k(g()) For all I E G, dU, Z Pm k(g(1))dlmXk, and Rg(Lo)pm t(g(t)) k,m=  Pm Proof. (i) Let '1' E D be arbitrary, then since Ug() leaves D invariant, i.e. u9(L)D we define the differential of Ug,() as follows:  Ug(l,...,ltd) lim A ,O (3.7) Now since Ug() is the product of oneparameter unitary groups one finds for the cG(i) SAm m=l S...,Xd, CD, ,...,vld) dUg(I) Therefore, U9() dUg(l)9 exp(ilZ exp(il b)mdlZ m=l a=m+l b=m+1 k(g(1))dlmXk'L Since 4 e D was arbitrary, one finds that on D , the following relation holds U*(i)dUg(1 k To establish the second part of (i) let 4' k(g() )drXk, E D be arbitrary then $gll)dUg(l)' = U (n o Ug(lo dUg(1)  Ul(lo)(10)(1dU9 Therefore , using ( the fact that both {dlZ m=1 are linearly independent families one finds Lg)(o)Am k(g(0))  m = Am k g(Z)) (ii) The first part of (ii) is similar to the first part of (i) (ii) one can proceed as follows, let 14 To prove the second part of E D be arbitrary, then dUg(l)U9(1) / = dU,( U,(10o)U(,*o wU9 Therefore , by the same reasoning as above Pmk 9) = pmk (g(l)g(o)) Since the Amk(g(1)) 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 (, cf. p. 92]). 1(lo)W(O) = Pm m,kc=l 1(lo)9() = dUg(l)g(lo)U (l)g( k)) Ro(zo Proof. Let v E D be arbitrary then by Theorem 3. 1 (ii), dU,() U,Ui pm (g (1) )dlmX c U,() U( c,m=1 Since Ugc, leaves D invariant, set UT* E D , then multiplying the resulting relation from the left by UZ*, yields, i r c,mn=1l pmc(g(l))dlmU;(, Using Theorem 3. .1 (i) and the definition of Um k(1) the Corollary easily follows. Corollary 3 functions Pm k(g(1)) and Am k(g(0)) satisfy the following equa tions  kn((/))  in 1\ 9(} = cjk1P 1 (9(1), k (g(9))  On  1 ((0)} =z Cjkf A' (g(1))dm [A 's (g( z )}O [p where Cjk1 are the structure constants for G. Proof. (i) Let b E D be arbitrary then one easily finds using (3.7) that am Dgn Ugci}1P a  Ojm Ugq)tb, holds. Now picking out the terms Ojm Ug,() and 9pn Ug() in Theorem 3. 1 (ii) one finds ipnW(g()) _;a. ^ aI,(1'tl1 aUg(I)] 4 n n i [ ipm g"())X g] g  .Irr ....   1( 1 a( cUg(l) . dUgn) Ika(g( (9(1))]p ls b(g())]. Pmn(g( _ . one can set 6 + i, I[Pm = Ug(1) and rearranging the terms yields p= PnC(g(1))Pm(g(Q))E,'Xa"b]". Now making use of the commutation relations [Xa, Xe] becomes = i El=, c~sf X this equation Om [Pno(g(l))  d.n (()] + pnitg (())pmb(g(1))Ca6 Finally using the fact that the operators { is arbitrary one concludes {Ol [p, (g(l)  ln  3 PnS(g1))Pmt(g(())cjf! f(g()]} (3.9) Now contracting both sides of (3.9) with p 1a (gQ()) yields ,I (g(O)]pa(g(Q))  9n 11a(gQ())]Pm (g(1) )pg(l) )c, p (g)), where, 8ac [PP k (g(()) k(g(0), has been used. Finally contract both sides with pm (g(l))p desired relation, d { [PViO(()]p k(g())  8n[p' (g())]p cjk/(g())p fa(g()). (ii) The nroof of (ii) is similar to the nroof of (i. Since Ug() leaves D invariant, Sk} =1 form a basis for g and that 4) 1 "(g(1)) to obtain the 1(n) E { iagm[p, (g(1)) (g(1)) E(8". (g(O)} 8a[p k (g(l))Pm k (g()1P 50 This equation can be simplified as follows d Pn (g(Zl))Uvjh()Om[U1 t(l)p ian [Pm $(g(l))]p hj,t=l h (1)atm[pln(g(l))Ujh(1)]) din [Pm hj=l Differentiating the product and rearranging the terms yields: [PnT(g())] n [Pm Pn (g(1)),m [Uj"(1)]U1 Next using 6,mUjh(1) = Pm W(g())cs "U^(1), which is proved along the same lines as Theorem 3.2.1 (ii), we find ,m[p (g())] an [pm (g(1))] = Pngp(1))Pm '(g(l))cjd which is equation (3.9) and therefore, establishes (iii). E 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 rep presentation 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 Theorem 3.2.4 (Doebner Melsheimer [27]) A nontrivial representation noncompact Lie algebra by symmetric operators contains at least one unbounded op erator. Since we are interested in quantum ph ,,.,,. .. ^ 1 .. .,l :.. ..1 .. i .. .1 ,I,,., _  we have to represent our basic kine LI j w nr .n a a. r a a a in an a h n an a at 4 *a I: n 1 ( ) 'tb(9(1)) f(g(l))]. S (1). (g(l))] = j,h=l 51 light of the above result, we can not avoid the use of unbounded symmetric operators when we are dealing with noncompact Lie algebras and Lie groups. CHAPTER 4 THE REPRESENTATION INDEPENDENT PROPAGATOR FOR A GENERAL LIE GROUP Coherent States for General Lie Groups Let U be a fixed continuous, unitary, irreducible representation of a ddimensional real, separable, locally compact, connected and simply connected Lie group G on the Hilbert space H. Let , e H then the function G 9 + (<, >) is called a coefficient of the representation U. Definition 4.1.1 A continuous, unitary irreducible representation U is called square integrable if it has a nonzero square integrable coefficient, E H such that (Uag 0 and 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, irreducible, square integrable, unitary representations one can prove the following Theorem: Theorem 4.1.2 (Duflo and Moore 30]) Let U be a continuous, irreducible, square integrable, unitary representation of G, selfadjoint, positive, then there exists a unique operator K in H if there exist vectors 6, semiinvariant with weight A and satisfying the following l(g), Let x' EH , and ,' ED(K Then one has (x, U , x')dg (x, x')(K (4.1) (For the proof see Ref. Remark 4.1.1 Theorem 3.) Condition (i) shows that if a representation is square integrable then there exists a dense set S of vectors in H such that for E E S the factor (Ug,, ) is square integrable for all 4 EH. One refers to condition (ii) as the orthogonality relations for U A result similar to (ii) has been obtained by Carey [16, Theorem 4.3] by realizing the square integrable representation U in a reproducing kernel Hilbert space. For the HeisenbergWeyl group these othogonality relations have first been proved by Moyal [81]. The operator K is called the formal degree of the representation . When G is unimodular, K is a scalar multiple of the identity operator which is the usual formal degree. Let X1, ..., Xd be an irreducible representation of the basis of the Lie algebra L corresponding to G, by symmetric operators on H satisfying Hypothesis (A), then L is integrable to a unique unitary representation of G on H. Let there exist a parameterization of G such that, d flexp( il'Xk) = exp( ilXi) il Xd); where I E. Now let n} E D(K1 then we define the set of coherent states for G, correspond ing to the fixed continuous, irreducible, square integrable, unitary representation Ug = Ug()K'/2r; t E D(K1 1. (4.2) , K1/2 ). 1/2C' )(U,(' ... exp( 54 where dg(l) is the left invariant Haar measure of G given in the chosen parameteri zation by dg(l) = 7() dlk (4.4) where 7(l) det[Am Remark 4.1.2 It follows from the strong continuity of Ug() that the family of states de fined in (4.2) is strongly continuous. Moreover, these states give rise to the resolution of identity (4.3). set forth in subse Hence, the family of states defined in (4.2) satisfies the requirements action 2 for a family of states to be a family of coherent states. The map C, 4 L(G) defined for any 4 EHby: [c,](1)  ((1) =(U (4.5) yields a representation of the Hilbert space H by bounded, continuous , square in tegrable functions on a proper closed subspace L~(G) of L2(G); see Appendix B.1. Using the resolution of identity one finds ()(0 l')t,(l')dg(l') (4.6) where ,(t; 2')  ((0) = (q, K' /2U, (l)g(l)K1/217) and K1 1OsC')K denotes the closure of the operator K'/2Ug 1(1)(1)KK/2 .One calls (4.6) the reproducing property. Furthermore, as shown in Appendix B.2, kernel KP,(l' 1) is an element of Lg(G) for fixed I E G. Therefore, the kernel K,(l'; 1) is a reproducing kernel and L (G) is a reproducing kernel Hilbert space; see Appendix B.2. One easily verifies (see Appendix B.1) that the map C, is an isometric isomorphism from H to LT(G). Now let the map A 9 g '* A0 be defined by left translation, k(5(0) r, i), = (1) r(l')) It is straightforward to show (see Appendix A.5 continuous, unitary representation of G on L2(G ) that the map defined in (4.7) is a ). This representation is called the left regular representation of Lemma 4.1.3 The isometric isomorphism C, intertwines the representation Ug,( H with a subrepresentation of the left regular representation Ag() on L((G). Proof: Let E H be arbitrary, then we have Since, , E H was arbitrary and C, is bounded we conclude that = Ag(n,)C ; hence , C, intertwines the representation U with a subrepresentation of the left regular representation A on L~(G). Therefore, (U, H) is unitarily equivalent to a subrepresentation of the left regular representation (A, L2(G)). Lemma 4.1.4 The unitary representation U,(L) intertwines the operator representa tion Fnn of L on H, with the representation of L by right and left invariant dif ferential operators on any one of the reproducing kernel Hilbert spaces L2(G) L2(G). In fact setting V = (8 1 ...,O/d) the following relations hold: (< (1), 1 (fl')g(1)) [CWKOU] (1) Ug. (a ) ) UmgK'I2h, , (g c, u,g, k (iV km(9(l))(i81m ) S1, aI d. then: m=l SXk , k(iV l)Ug(O)  s ED. A common dense invariant domain for these differential operators on any one of the L2(G) C L2(G) is given by the continuous representation of D, a C,(D). Proof. arbitrary, then using the fact that O9mUg(l)U;*(O Ug() ,pmU*;)1 it follows from Theorem 3 1 (ii) that i8. U*( m After contracting both sides with p Pm,(g(1))vU,* = 1,..., d. km(g()) one finds p k(m9))( iaim = U;()Xk, =1, hence, iV = U';( = 1, .* ., d. Using Corollary 3.2.3 (i) one obtains, iV i~am pl/"(g(1))( [pm(g())( m,n=l ({. [P'm(g(1))]p ' (g ())  Dl[pi}m(}g())lp (g))} (i,) m=l n=l p km(g(l))( Cij k( iV iL. : a1.F.+;n 01 aarmofa r o I T r 1d with cmmon dense invariant m=l . ., d, S, I,)U 1), i( iagn ialm i rrti .^  *I i l / I . Corollary 4.1.5 The differential operators {xk(i sentially selfadjoint on any one of the reproducing kernel Hilbert spaces L2(G) can be identified with the generators ({P( =,) of a subrepresentation of the Proof. left (right) regular representation of G Let 4 = 1,..., d. on L(G) the other hand, Ugsk(t) = exp( itX  1,.. ,d, be oneparameter subgroups of G. Then [C,iXk]](I) can also be written Cr,X ](l) C lim I (+ 1 (,(gk Agk(t) 4(1) where the A(Xk) Agk(t) = 1,. .., d, are the generators of a subrepre sensation of the left regular representation of G on L (G) . Hence , one can identify ',l) with A( k) on D,, i.e. Clearly, the operators S1,. ,l), k = 1,..., d, are symmetric, since the operators nd since C, is an isometric isomorphism from H onto L!(G) )({ ,I)L =1)are E D then it follows from Lemma 4.1.4 that )r,ip) . 7 = 1, .d. .., d, are symmetric on H a iVI, 1), (1)  S =A( k ,] (1) = [C, Ugk(t) S(O^(r ) (t)g(l))  ,(s(1)) fc); (Q> . each Xk, = 1,..., d, is selfadjoint. Hence, the restriction of each Xk, = 1,...,d, to D is essentially selfadjoint. Since, C, is an isometric isomorphism from H onto Lg(G) we have that the closure of each Xk( iV 1, a^ .., d, contains a dense set of analytic vectors, namely, C,(D), hence , is by Lemma 5.1 in Ref. 82 selfadjoint. particular, each tk( iV ..., d, is essentially selfadjoint on D1n" Similarly one can prove that the operators {ik(iV /)k=1l adjoint and that they can be identified with the generators {P( presentation of the right regular representation of G on L"(G). E are essentially self k)}k=i of a subrep Corollary 4.1.6 commutes with the The family of right invariant differential operators {xk( family of left invariant differential operators {i(iVVi, iV )}L=1 Proof: Let i{( 1) and i,l) be arbitrary, then (ii' ilmn m,n=1 d d E E d d E L n=l \m=1 l im(g(l))8im p1"((1)) jl(g(1))]  if (g(l))8,,[p A11m(g())Om(p 1)] 1"(g)] Pm(g())} 1"(g())]) d d En=1 =1 n=l L/=1 ~lm(g(l))Ogm(p m=l where we have used Corollary 3 3 (iii) in the fourth line. Therefore, OIkd=r1 1 (g()) A gn(g(1))( i8 AimW())8m  "(g())]) } . A'/(g(1))8f;[p' n(g(1))] 59 The Representation Independent Propagator for Compact Lie Groups In this section we follow our presentation in Ref. Let G be a ddimensional, 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 Uc and their finite dimensional representation spaces by H We denote the dimension of the rep presentation space H by de. One calls d, the degree of the representation U' . Let ...,Xd be an irreducible representation of the basis of L by bounded symmetric operators on H(. since all vectors ir Then Hypothesis (A) is trivially fulfilled for this family of operators iH are analytic vectors for these operators, hence this representa tion of the Lie algebra L is integrable to a unique unitary representation of G on He. Let there exist a parameterization of G such that, d = Texp( ilkXk)  exp( ilXi) ildXd), where I EG. Since G is compact, the parameter space C is a bounded set, therefore, all irreducible representations are trivially square integrable. The positive selfadjoint operator K is given by K = dcl hence , we can choose any normalized vector i E He and the coherent states for a compact Lie group G corresponding to a fixed irreducible unitary representation become: = d U' see equation (4 As we have seen in chapter 2, the resolution of identity has the form =~ ...exp( r(1) ((1) , *)dg(l), Since all operators Xk, k E He, = 1, ..., d are bounded we have by Lemma 4.1.4 for any using the continuous representation C,  L2(G), that iV = [CXk](), Note that this relation holds independently of r7. Since G is compact the center of the von Neumann algebra A(A) generated by the left regular representation A of G contains a compact selfadjoint operator whose eigenspaces are Ainvariant (cf. Lemma IV.3.1]). Hence, A can 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 U7 occurs with multiplicity d, (see [7 Theorem 7.1.4]), i.e. GdcU CEO where G denotes the dual space of G G is the set of equivalence classes of all con tinuous, irreducible unitary representations of G. Denote W(Xk) the selfadjoint Hamilton operator of a quantum mechani cal system on H . Then for UC the continuous representation of the solution to Schrbdinger's equation, = exp  where ht , is given on L (G) by i (1, t) K,(2, t; I' where, K,Q(, t; l', t') (<7(1), exp  t') C(Xt)X}(L)> [C, exp  t')') , t')dg(l'),  t') (Xk)( ) I,1)[C,](1) , Mt'),(' In this construction r was arbitrary, hence it holds for any i E H(. Therefore, one can choose any orthonormal basis (ONB) {q, } xinH and write down the following generalized propagator KH (, t; l' = (t  t') dtr[U9(c) (4.8) where  tr[ Lemma 4.2.1 The propagator KHt (l,t; ',t') given in (4.8) correctly propagates al elements of any reproducing kernel Hilbert space L (G) associated with the irreducible unitary representation U9 of the compact Lie group G. Proof. Let r/ E He be arbitrary, then for ,(l'' L~(G) one has KH, (I, t; 1' ,t')dg(1') =  t')dcx dcU(t U(* U g(J* ) ( (O u( U9(1) U9 j,n=l 1)4n. [Cn exp  t')91(X)]'(bQ')](1) Therefore, a  t')d ,t')dg(l') t')( U, ; ', t') 1(1)g(l')) '(1)g(l')) ,t').(1' u(t 1(l)g(l'')) '), 0(t'))dg(l') )< ) n (t')) )d (l') dd(, n)U(t )(t')) (1, t). Hence, we have succeeded in constructing for the irreducible representation UC a propagator KH, that correctly propagates each element of an arbitrary reproducing kernel Hilbert space L((G), i.e., we have succeeded in constructing a fiduc independent propagator for a fixed irreducible unitary representation of G. fact that the set {} i1 is an ONB one can rewrite the group character xc(g in terms of the matrix elements D (1) zial vector Using the 1(1)g(l')) E (4,U Ug>) of UC as follows, D{()Dt (l'). (4.9) Therefore, KHe can be written alternatively as KHC (l, t; 1' ,t') =(t t') (4.10) this construction unitary irreducible representation was arbitrary, hence one can introduce such a propagator for each inequivalent unitary represen station of G, one can write down the following propagator for the left regular representation Ag(, of G on L2(G) K(l, t; ' t')  KH (l, t; l' ,t') =u(t cet ij=l Now it is well known from the PeterWeyl Theorem that the functions Vo/ D o(1), form a complete orthonormal system EG, (ONS) in L2(G) Theorem 7.2.1]. completeness relation of this ONS is given by DQ(l)D(l') = 6e(g ()g(l')) = x9(g deDi,(1)D (,(l').  M(1)g(1')), /L ffl ^ Therefore we find as our final result K(l, t; 1', t') = exp [i(t  t')7t(k(iV Q)y e(9 1 (9 )) (4.12) This propagator, which is a tempered distribution, is clearly independent of the fidu cial vector and the representation chosen for the basic kinematical variables {Xk}.=. 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 4.2.2 propagator K(1, t; ' (4.12) is a propagator for the left regular representation of the compact Lie group G on LZ(G), which correctly propa gates all elements of any reproducing kernel Hilbert space L (G), associated with an arbitrary irreducible unitary representation Ug(1) of the compact Lie group G Proof. To prove the second part of Theorem 4.2 , let U , and 77 EH be arbitrary, then for any ,(l) in some L (G), associated with U,) one clearly has that {,(l) C(G). Hence one can write K (, t; I' , t')dg (') KHQ (1, t; h' (2, t; 1' ,t') ,(' , t')dg(') = ,,(2, t). The second equality holds since the elements of different representation spaces are mutually orthogonal, hence, only the C'term remains. In the last step Lemma 4.2.1 has been used. Hence, for any compact Lie group G we have constructed a propagator that is independent of the chosen irreducible unitary renresentation of (. We call this ,t') ( d ,t')W( 1h, (t'))dg(l1') 64 integral representation (see Proposition 4.4.4 or [98]): G I lim JvyF)^N* ^oc .../ * ii x exp i *(j+1 lj) kC(Pj+l/2; 1 j+1, j)) x fdli (4.13) where lN+1 = ", and = (t"f  t')/(N + 1). The sum appearing in the above expression is defined PN+1/2 1 PN.1/2 Z 1 K P3/2 the sums are over the spectrum of the operator itV defined in section 4.4. where K is the appropriate normalization constant such that K Xy(/")7(l') p Pk (lik  rk) =e (g9 The arguments of the Hamiltonian in (4.13) are given by the following functions: + p Remark 4.2.1 Observe, that the Pmi+i/2i, attice expression for the representation independent propagator exhibits the correct time reversal symmetry, which means that K(l",t" l',t') = K(l' 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 v/32D,(L), CE and i,  1,..., de, in L2(G) whose existence is guaranteed by ,t"; ', t') Xk (Pj+l m=l t"). *** d K(l"  7(t Pj+l ... dlf, 1 n ),  w(g(l)) 65 path integral representation (4.13) 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 Dj(1) are explicitly known or not. Hence, (4.13) represents a clear improvement over the path integral formulations describing the motion of a free particle on a group manifold presented in chapter 2. O Example: The Representation Independent Propagator for SU(2) While the PeterWeyl Theorem assures that the ONS EA = 1, ... ,de exists and is complete, the construction of such a set is frequently a difficult task. The functions J/dqDj(1) are known only for a limited class of groups and will now be constructed for SU(2) monic analysis. ators in L2(SU( It turns out that this is an exercise in har We will now explicitly describe the maximal set of commuting oper We will take the set of infinitely often differentiable functions, Co( SU(2)) as their common dense invariant domain. Since SU(2) is a rank one group, there exists one twosided invariant operator C1 in the center of the enveloping algebra E of SU(2). right (left) i gebra R (E Moreover, since SU(2) is compact the maximal set of commuting nvariant differential operators in the right (left) invariant enveloping al L), can be associated with the Casimir operator of the maximal subgroup U(1) of SU(2). Let S1 and S3 be an arbitrary irreducible representation of the Lie algebra su(2) by selfadjoint operators satisfying the commutation relations 3 Ss,. Since the Casimir operator of SU(2) of the Lie algebra su(2), 4ijkSk* commutes with all the generators = its eigenspaces are invariant under the Lie algebra, and all VdD,4(1), irreducible representations of SU(2) on any of the H by Ul . One can show that every irreducible representation U of SU(2) is equivalent to one of the representations U( =0,1/ [103, Theorem III.5.1]). . S.. For SU(2) in the Euler angle parameterization an arbitrary unitary irreducible representation of SU(2) is given by exp( iS3) exp i0S2) exp( iCS3 where the domain of the parameters 0, 4, and C is given by < 2r , 27w <27 With choice parameterization the operators {(k=L1 defined Lemma 4.1.4 (i) are given by: i90, i0,c 0,4, i sin 40e + i cot 0 cos 4 i cos t csc e00, iO,, isa i cos 10e + i cot 0 sin 480  i sin csc 0D8, ia, 0, 4', i9,. (4.14) By Corollary 4.1.5 these operators can be identified with the generators of a subrepre sentation of the left regular representation of SU(2), belong to the right invariant Lie algebra of SU(2)) . Similarly the operators { k k=1 defined in Lemma 4.1.4 (ii) are given by: i(ioe, i8A, i04 i sin aeo csc 6 cos (08 + cot 0 cos C8, (i9e, i cos gde + i csc 0 sin (a9  i cot 0 sin t(f, 3(iOe, i9,, i,0, 4, C) (4.15) and can be identified with the generators of a subrepresentation of the right regular O,4,0) = i~e, ie0, i8e, i4,, i i9 i ,0,, , For the Casimir operator of SU(2) one finds (1 z)8 1z 2z 9 c + i), (4.17) where = cos 0 and the identity  sin O9cos G = 09 has been used. Since C1 comriutes with all elements of the enveloping algebra is irreducible, multiple of the identity on any one of the reproducing kernel Hilbert spaces L4(SU( associated with the irreducible representation U,, 1 = C Let {b,} + 1)IL2(SU(2)) be an orthonormal basis in H(, (4.18) then we can associate with each irreducible representation U^(8,^,), where = 0,1/2,1, the following matrix ele S.*.. ments Dn,(0, t, = (m, m, n We shall now determine the matrix elements D&,(6, B, () as the common eigen functions of the operators A1, B1, C1. common eigenfunctions of the operators D (0, ) Equations (4.16) and (4.17) suggest that the and C1 are of the form i(m+n) p (cos 0). Using this form of DL, (0, , C) in (4.18) one finds: z2 d z P4n(z) + dz  2mnz) z22 = C(C + 1)Pn (z). The functions Pijn(z), which are known as the Wigner functions, are given by 2m (C m)!(C 4+ m)! p(mn,m+n) (mi where p(_mn'm+n)( where. m z) are Jacobi polynomials, (see [103, 125]). Also observe that a ~I I as ~ aI a a C UIr(,,Ct) ) p ( ' dPz mnv z)+ p ( rn\ ( )('!(1 ( n)!(+n)! i I I as pointed out above these functions form a complete ONS on L2(SU(2)). The com pleteness relation for this ONS takes the form, see (4.11), +1)D ,(0" m,n= , ")oD (e' 16wr2 s 6(0" sin 9 By equation (4.12) the representation independent propagator for SU(2) is then found to be: l',t') 167r2 exp[i(t"  t') ll(iV (iV, /), 1 sin 0  ) where 1 = (0, ,E) and = (8e, 9 ,,). Equation 4.13 the regularized lattice phasespace path integral representation for the representation independent propaga tor for SU(2) is given by 0' , 167w2 r sin sin lim o'sin 9" sin O' moo +/3j+1/2( j+1 Cj) + . . exp{i [OLj+1/2(j+1 j) j=0 N 7j+l/2(Ej+i j) t(sk(Pj+1/2; lj+1, Zj)3} J7J djdjd , j=' where, 1 2(sin &+1 + sin gj)&j+1/2 2  (cot 0%j^ cos 41 + cot Oj cos Cj)/ j+1/2 (cos 4jy+ CSC 0+l + Ccos (j csc Oj)7Yj+l/2, 1 j+1/2 + cos Cj)tj+1/2 1  (cot 6jQl sin j+1 + cot 0j sin j) 3j+1/2 2 0+il sin j+l + csc Oj sin #j)7y+l/2, JI ,)= S3(iV 0')6("  ') " ('. 1,))]> K(l"  e')(" ')(('" K(0" (Pj+1/2; j+1 ,j ) (Pj+1/2; j+l j) 2),2C( iV iV iV iV iV 2I 4.3.1 The Hamilton Operator 7i(&1 St, S3) = 1 21 As announced in chapter 2 we now revisit the free particle moving on a circle and present its exact path integral treatment. The Hamiltonian t( i, S2, = 32/21 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 selfadjoint. In particular we assume that 83 adjoint on L2([0, 27r)) and has a spectrum of the form f/ =nq1n = 0, 1, = io8 is self Then in a second step we reexamine this assumption and show that i98 selfadjoint with spectrum /3 = n is only one particular choice of uncountably many. With this choice of Hamilton operator the representation independent propagator takes the form , ", ," I ',(' ,t') 16r2 /sin O" sin 0' lim Noo ... S a,'v exp{i  6,) + %Ai ( k+1 + 7j+i 167r2 sin 6(" (j+1 j)   6') 6C(" Pj+1/2 21 ) lim N*oo =00 E exp i =0o j*+1 (41+1  4j) _ 2/ a~ N ,j=1 This last integral can be evaluated as follows N+1 lim Ntoo .. . n+l/i2=poo 01/2==o N E 1 11 >} de0jd3 dy .. ** 1j 2ir exp{i 5[tij+i (j+l  j) 1)w3 31(~i + S23). K(e"  *) (0441 j+il Su+4 j=1 1= 1 NN (4^ )+j1/2) 2 +1/2} j j=o0 j=1 1/2 exp i[ N+1/2" 2I NJ+1/2 J+1/2] j+1/2 j1/ j=1 exp i T  #) 2 2I Hence we find for the representation independent propagator .1 9, ."6("0" smin" 8"  0') 6((" exp i n= oo T  ') n 2I The sum over n is related to the Jacobi theta function, 00 exp(iirtn2 + 2inz). n=oo Therefore, with the following identifications we can write the representation indepen dent propagator in closed form. qS') then our final result for the representation independent propagator becomes 8fr t') = s 6(0"" ' sin 0"  0') 6("  ()03 T '27IrI This result agrees with the one found by Schulman [94] expect for an arbitrary phase factor. We now follow our analysis in Ref. 99, section III.c. It is well known that the symmetric operator i9 on L2([0, 27r)) with domain A "T lim N>oo Q=oo , ", "  f1/2 ' w(" K(6" n(4" 03(z,t) = ') K(0" which we denote by i9 iOa with the domain 2w * L S oo, i(27r) = e'"(0)}, where 8 r, 7) (see [87 257 Note that the choice 6 = 0 corresponds to the case of periodic boundary conditions, spectrum of each which we have assumed above. il, is straightforwardly found as follows, let A ER then this implies that the eigenfunctions are given by '() = iA Fitting the boundary conditions rf(27r) = e'i(0) yields the following set of eigen values, = n n+ Therefore, the spectrum of nEZ S. , i96 is given by spec( 6 n +  "2^ = A E [7, 7r) nE Z . Hence, the choice of periodic boundary conditions is only one of uncountable many possibilities. If we choose instead the boundary conditions 4(27r) where = e (0), E [i, x) is arbitrary, our expression for the representation independent propagator becomes , ', ' 87sin f(  6') 6(s"  ') exp i n=oo 6 2v} T n+ 21 2ir ={ D( 59]). ia~iW~ = A() ,0, +1, K(0" n+ ') A 1 72 Therefore, with the following identifications we can write the representation indepen dent propagator again in closed form. 4,') then our final result for the representation independent propagator with arbitrary 6 becomes ,t", ,e 87r sin "6( " sin 0  0') 6(C" C) iS(" iT62 87i 2  ') T6 This result exhibits the same dependence which also encompasses all spins. as does the one found Schulman [94 4.3.2 The Hamilton Operator ?7( 1, '2, =7( Our second example is that of the Hamilton operator 7%( 1, 2, where C1(0, 4, () is the Casimir operator of SU(2) given in (4.17). Note that the Hamilton operator WI( 1,2, 3) 4) is essentially selfadjoint since C1(0, 4, 4) = C1(, , is a symmetric and elliptic central element of the enveloping algebra E of SU(2), 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: , ", ( A'ElC1 O', >' (' t' ) iC1 (" 21 ", ,(") ^ C)" 1 sin 8,(0" sime 0  e')6("'  ')  ') T i c (e" 21 4, 4,n E m,i= (2( + 1)DS4(0" an x .T af\ \ I .i I I  k^ll ,. I [ I II l R I I J, t. r r / a4\ I 1 n.l1 i1l A.1\ lid 9 T1 '274r ,")D6,(' , ',(' K(0"  ') + j) K(0" I lir' \ the element g 1(0", , ') by (9, ) one finds: Xc(9(0,, )]Pm(cos 0) m=Observe that, the character of the group can be expressed as a function of a single 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 gl one has (919911 Therefore, = Xc (9) to show that xc (g) 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. known from linear algebra any unitary unimodular As is well x 2 matrix g can be written as , where gl SSU( ) and 7 is of the following diagonal matrix i(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( ) is labeled by one parameter r , ranging from 2r and where r and r give the same class. Hence, the characters xc(g) can be regarded as functions of one variable r that varies between 0 and 27r. The geometrical meaning of the parameter r is that it is equal to the angle of rotation corresponding to the matrix g. In terms of the Euler angles (9" ,C") and (0U , #,, ') r is given by = arccos[cos(0"  O') cos(4"  ') cos((" C)  cos(0" + 9') sin(t"  ') sin(('" (4.19)] (4.19) One can derive an explicit formula for xc(g) as a function of r U'(,r,o) that corresponds to 7 Note that the matrix E SU(2) is given by the diagonal matrix of rank 2C +1 = 9g17g1 ,(")g(e' im(( + ei(r/2) 74 Hence, the group character can be written as 1(0" where r is given in (4.19). ,',C)) Therefore sin(C + 1/2)r sin r/2 , one finds for the representation independent propagator , E", ,t"; 0', 16trr vi. sn 0, flim Jsm 0 sn + exp{i ... (0,+1 0) + /j+1 Cj) {(,O,}  cC1(0j+i + 1)exp (t" /  t') +1) +1/2)r sin r/2 This result agrees with the one found by Schulman [94] which was obtained by the methods mentioned in chapter The Representation Independent Propagator for General Lie Groups 4.4.1 Construction of the Representation Independent Propagator Now let G be a general Lie group. square integrable unitary representative Let us again denote by U an arbitrary, fixed, n of G. Then it is a direct consequence of Lemma 4.1.4 (i) that for any 4 = [C1, kl]()0, = 1, ..., d. holds independently of ir. Therefore , the isometric isomorphism C, intertwines the representation of the Lie algebra L on H, invariant with a subrepresentation of L by right , essentially selfadjoint differential operators on any one of the reproducing kernel Hilbert spaces L'(G}. To summarize, we found in section 4.1 that any square xd ( 2E( , ", ") g(0' K(0" [aj+i (5+1 4, () ]}} dejd~jdf> + 7i+i I,)[c,]( Let (X , H) be a representation of G then we denote by A(4r) the von Neumann algebra generated by the operators 79 g G (cf. Appendix A.2). By Proposition 5.6.4 in Ref. 25 there exists a projection operator PI in the center of the von Neumann algebra A(A) such that the restriction A1 of A to the closed subspace PI[L2(G)] of L2(G) is of type I, and such that the restriction of A to the orthogonal complement of P[L2(G)] has no type I part. Since G is separable and locally compact there exists by Theorem 5.1 in Ref. 30 a standard Borel measure v on G , the set of all inequivalent irreducible unitary representations of G, and a vmeasurable field unitary representations of G such that the type I part of A, can be decomposed into a direct integral, where UC Ic is a representation of G xG on H Denote by system on He. 7L(Xk) the essentially selfadjoint Hamilton operator of a quantum Then the continuous representation of the solution to Schrbdinger's equation, = exp  t')1(Xk(t') takes , on L(G) the following form K,,(Q, t; I' , t') (1' ,t')dg( where,  [C exp[ k )3flC')3(2)  t')[) t')(QhK (4.20)  t'); '), ;', t') ,Hep d ((), ,(,t) 76 Note that for noncompact Lie groups it is not true that every symmetric Hamilton operator is also essentially selfadjoint, illustrate this important fact, we consid Example 4.4.1: as was the case for compact Lie groups. ler the following two examples: Let G be the noncompact two parameter group of transformations p < oo, 00 oo of the real line JR and let H = L (Rj), where 1R" = (0, An irreducible unitary representation of G on H is given by the formula: (Ug, ) )(k) = p/2eiqk?(p E H. The generators of the oneparameter unitary subgroups are given by U(X,) U(X2) i d 2 dk d dk We choose the set (Ri) as the common dense invariant domain for these operators. As our first example we consider the operator i d d 2 dk dk D(T1) (IR). Clearly the operator T1 is symmetric. To show that Ti is essentially selfadjoint it is necessary and sufficient to show that the kernel of the operator Tj + iI ker(Tj* + ii), consists only of the zero vector, ker(Tt + iI) = {0} . In other words we have to show that the equation: has no solutions in H other than + (k) One finds the following solution for the above equation 1 it 1 T{q(k) = tfi+(k), =k, = kf2 U(X2 + .,,, / i . Both of these functions are not in H , since they are not square integrable. function M+(k) diverges at infinity and the function diverges at the origin. Therefore, we conclude that T1 is essentially selfadjoint. As our second example we consider the operator r d ki + dk i d d i d d  k+ + k + k k 2 dk dk 2 dk dk \ dIf dd dk dk c D(T2) (Rfl). This operator is clearly symmetric, and one determines the following solutions for the equation T'*4+(k) = +ig((k): exp + (k) _(k) 1 4k 4kc2 1 ? exp Clearly (k) is not integrable since it has a nonremovable singularity at the origin, however 4+(k) is square integrable, and hence, belongs to H. Therefore, even though the operator T2 is symmetric it is not essentially selfadjoint and can also not be extended to a selfadjoint operator since it has deficiency indices (1,0). O We now proceed with our construction of the representation independent propa gator. Let a, f E D(G), then put U(a) a(g(l)) Ug( a'(g(1)) SA(g 1(1)), and define the map DT(G) x D(G) 3 (a, 3) + a*/0 e 1 .\  u(X2x dk dk r +X dg(l, '(1))a(g With these definitions we find that: K,(a,) )= /2r7)a(g(l))f( g(l'))dg l)dg(l') a(g(1))/(g(1)g(l'))dg(1) r>)(a"* 3)(g(l'))dg(l') U(a* Note that Kf(a, 1) is a bilinear, separately continuous form on 1D(G) the bilinear separately continuous forms on 2D(G) that K/(a, 3) is a left invariant kernel, that is C, (Lga, Lgf0) x (G). x 1D(G) kernels on G. for every g We call Also observe E D (G). Therefore , we can write (4.20) K,(a,t; t')  t')}a, 8). In the above construction in E D(K1 ) was arbitrary, furthermore as shown else where [30, Corollary ] for a E V(G) the operator is trace class. Therefore , we can choose any ONS {j})jev in D(K1/2) and write KH(a, 3) = K (a, /) = trt[K/2U(a* fl)K1/2] Note that KH(Ca, /) is a left invariant kernel on G, since each K1j (a; /3) is a left invariant kernel on G. Therefore, Proposition VI.6.5 in Ref. 78 there exists a unique distribution S in )'(G) such that KH(a, /) = S(Q* /). In fact we see that dg(') K,(l; l')a(g(l))W(g(l'))dg(l)dg(l') (7,K K()g( (O, KI U7(zKI ({h,Ki2Ug( )K1 (r, K1 * ))K1 = K, (a, 3), , a, u (t K'/2U(a)K1/2 79 Remark 4.4.1 This propagator is clearly independent of ij the fiducial vector that fixes a coherent state representation. However, this propagator is in general no longer a continuous function but a linear functional acting on {D(G) . We will see below that the elements of any reproducing kernel Hilbert space lie in the set of test functions for this propagator. Lemma 4.4.1 The propagator KH(1, given in (4.21) correctly propagates all elements any reproducing kernel Hilbert space associated irre ducible, square integrable unitary representation Ug(1) of the general Lie group G. Proof. Let i E D(K1/2) be arbitrary, then for 4,(1' E L2(G) one can write KH (, t; 2' = u(t ,t')dg(l') = ,t') (l'  t')tr[KU/2 Ugl()g(l')K/2]),(l' ,t')dg(l') U(t t') U(t t')(K1/2U( (., K1/2 U9g ()9(l,)K1/2 j) (UlgI')K1/2 (t') ) dg (') (fjr }7>j, [C, exp[i(t t'))](t')]( (Jt ), where the fourth equality holds by Theorem 4.1.2. Therefore, (l,it) = KH(1, t; 1 ,t')dg(l'), the propagator propagates the elements of any L2 (G) correctly. In the above construction the unitary irreducible representation Ug() was arbi tirTTnT hono Tx r r on n+TrAln/aO ciivrh 7 nrnnTsQrnr fAr ono r Tnonr111Tlonf 11n +TTlnr r7onre_ E D(K/2), }(t')) ', t') (l' a semiinvariant operator of weight A (g 1) in Hc for valmost all E G such that for P [D(G)] 6e(a* *) = tr[K/2U( )K1/2 trtK~~ V ~ /5 (4.22) is well defined; see Appendix B.3. Here, s(a* p) = and 6e(g (1)g(l')) is given in the chosen parameterization by (1)g(l')) = y 6(lk v(1) =1 Hence, we can write down the following propagator for A1 of G on L2(G), K(a, t; f, t') u(t t') U(t t') tr[KI/2UC(a* 1)K1/2 ]dv(C) u(t t')S* *fl). Therefore, we find the following propagator for the type part of the left regular representation AI: K(1, t; I' ,t') = exp[i(t t')7H((k(iVl, l))]6e(g  (I)g('). (4.23) Remark 4.4.2 Observe, that this propagator is clearly independent of the fiducial vec tor 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), 1f ln t nn 4b n nr n nCl  ()g(') )a(g(t) )3(g(l') )dg(1)dg(Z'), _ 'k) ]dv(C), KH (a, t; t') dv( () ICH((cu; )dv wrrtl nrn v^^\ fC 'f /n Cnrnr iC n9 f^ +l^ f^^ /+ r ll ^/^t^t' /\Il^4i t/^ ~ \^ / Theorem 4.4.2 The propagator K(l, t; 1' , t') in (4.23) is a propagator for the type I part of the left regular representation of the general Lie group G which correctly prop agates all elements of any reproducing kernel Hilbert space L4(G) associated with an arbitrary irreducible, quare integrable unitary representation U1) of G Proof. To prove the second part of Theorem 4.4.2, let U and 7K' g(1) ED( arbitrary. For any ic, (1) E L2 , associated with U', we can write JGK(l, t; ' G = [c,,  t')6e(g ~ 9 )U 'K. 1/2 l(1)gl ))(Uri,)K, tiC, (t'fl(2)  = (lt) Therefore S(, t) for all rk, E D(K,/2) and any C' , i.e. this propagator propagates all elements of any reproducing kernel Hilbert space L (G) associated with an arbitrary irreducible representation UC) correctly. Hence, we have succeeded in constructing a representation independent propagator for a general Lie group. 4.4.2 Path Integral Formulation of the Representation Independent Propagator From (4.23) it is easily seen that the representation independent propagator is a weak solution to Schrbdinger's equation, , Xd( = H(\. iV L))K(, t; 1' (4.24) ;1',t t')dg ('), e 4 t'),{(l' t')dg(l') n(t'))dg(l')  '[C,,(t](  U(t  t')n( i8,K((, t; 1' _ Remark 4.4.3 Observe that the coherent state propagator given in (4.20) is also a weak solution to the Schridinger equation (4.24). However it satisfies the initial value problem im K,I( C,(2 (4.26) Therefore we can write iOtK#(, t; ' iV =i(1 ., Zd( iV l))IK (, t; 2' (4.27) where K# denotes either K, or K Note that the initial conditions, i.e. either (4.25) or (4.26) determine which function is under consideration. We now interpret the SchrSdinger equation (4.27) with the initial condition (4.25) as a Schrbdinger equation appropriate to d separate and independent canonical de grees of freedom. Hence, I1 Id are viewed ,..., as d "coordinates" , and we are looking at the irreducible Schrbdinger representation of a special class of dvariable Hamil ton operators, ones where the classical Hamiltonian is restricted to have the form '(Ui(p, ), ...,Xd(p, 1)) instead of the most general form fl(pl, ..., pd11 , ...,d). In fact the differential operators given in Lemma 4.1.4(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 fol lowing standard formal phasespace path integral formulation in which the integrand assumes the form appropriate to continuous and differentiable paths K(l" exp i pmlm ( i(p,l) m= " ,sd(p,))dt t',t tEt (',tlf dl(t)dp(t), a i8 K,,( ...,Xd( ; ', t') , t"; l',) t') iV,,1))K(,t; ' N(Wl Page Missing or Unavailable normalized such that , *.* l"d l, Sp Pp'd) , . Sd) I 6(p ,pjk) where 6(P, pk) 6 ,, / Fkrk If the spectrum of Pq& is discrete If the spectrum of PT& is continuous and giving rise to the resolutions of identity dg(l) spec ...xspec(Cd Ip) (pdp spec(Pi )x ...xspec(P d where ..., d) and ]p) SIPl,. Remark 4.4.4 If the spectrum of Pk is discrete then dpk denotes a pure point measure such that the integration over pk reduces to summation over spec(Pc). On L2(G) these operators can be represented = ialo. 1 (4.29) = i[o.a where D the set of functions of rapid decrease on G, dense invariant domain of these operators. is chosen as the common Here ra(0) is defined as r(0) = al In y7() and where 7(l) is given in (4 It is easily seen that these operators satisfy the CCR, are symmetric on L2(G), and that iV t has the following generalized eigenfunctions (l)exp (i where S= (8^1 ..., ad) We normalize these functions so that K y'7tr("'') exp[i pk "kU  r)ldpl ... dpd =6e (g I 1 i 1 _ .. a C  mx. ra~n Pk k 1 n ),  p'k) .,Pd)"  r ')y), C.. 1 L, rl,, n Arm nl: nn~ ']r We call (4.29) a ddimensional Schridinger representation on L2(G). Moreover differential operators {k Lemma 4.4.3 iV )}=1 can be written as follows: given in (4. 9) the right invariant differential operators {xk iV defined in Lemma 4.1.4 (i) can be written as: iaim i1im)p 1 (g(1))], . , (4.31) where Proof. Since iBda iao. + (i  1, . ..,, the differential operators {Xk( iVt, l)},= become after substitution of this expression  + (i/2)r(), ..., iOld + (if )rd(, 11 ij'm rm(1)] 1'k (g())( El1/ m=1 ^jm ia'm Using [p' itfm = im pkm( ((1)) and the definition of r"(l) yields ..., i^Od + (i/ 1km 27(1) m m=1 km(g(l)) m[p k (()]. Since the operators 4r Hilbertspace L(G) (cf. iV 1, 1) are essentially selfadjoint on any reproducing kernel Corollary 4.1.5) and since 7(l) 0 one concludes that '1km(g())7()] = 0, = 1, Using the differential operators {it^o}=1 iV 'k (1())( m=l S= ( .., .. ,By P'1 mg()) m=l d 1 E9 * .. d. d I)}L p)r(t) 'k"( (l))( 1km m) )+p km (f( ), i~p )rd( , .. d) )r'(1), +(i/ 8^lm th )+( iim Remark 4.4.5 This Lemma shows that the differential operators {hk are elements of the right invariant enveloping algebra of the ddimensional Schrbdinger representation on L2(G). Adapting methods used in Refs. 63 and 68 we can give the representation inde pendent propagator the following regularized lattice prescription. Proposition 4.4.4 operators on H, Let 7, where be a sequence of regularized Hamilton Then provided the indicated integrals = (t" exist (see below) the representation independent propagator in (4 following ddimensional lattice phasespace path integral representation: lim N*oo x exp i [pj+I Il)  e1,((k(pj+1 4+I, j)) x Jdl j=1 dpu+1/2 ... dl ...dpCj+i/2 (4.32) j+1/2=0 where lN+1 = l"I = 1' and the arguments of the Hamiltonian are given by the following functions: p km(g(lj+)) + Pkmt)) Xk (Pj+ j+l1 Pm+ /2, m=l Remark 4.4.6 If part of the parameter space C is compact then we denote by the class of moment conjugate to the restricted range or periodic "coordinates". If pk E R then dpk denotes a pure point measure such that the integration over phj+1/2 reduces to summation over the discrete spectrum of Pr. For the case of a compact parameter space C (4.32) reduces to (4.13). ) can be given the .. .S . * = 7(I + e?12)  t')/(N + 1) , t"; ', t') K(I" (l,,)y ( t,) (l1l Then it is straightforward to show, by using the Spectral Theorem and the Monotone Convergence Theorem, that for all 4 E D(N) C H one has slim 916 60 = and that on all of H one has slim [I Noo  ie91, N+1 = exp[i(t" where E  (t"  t')/(N + 1) . Now in order to obtain the lattice phasespace path integral in (4.32) one can proceed as follows. {&} be an arbitrary ONS in CH then K(l" = exp ,5a))r (a" exp Sk(Pc" ,C))]I>r <*>) (k, Xk(PcG lim Noo lim (1" lim / N*oo (Irl, k)(k,  i'H k(Pc"  iE (k (Pco ca))N+ N i,} n ^w" where 1" = IN+1, = o10, and ( *) denotes the generalized inner product. Note that the third line holds true since each E 4 gives rise to a linear functional acting on * in the following manner Lb() = (41 ) = (<, for all . Hence one has that (kl exp[ = L4k (exp[  t')] j) = k, exp The fourth line follows from the fact that 4" C H and that the approximation we are r9nn hnlAlre ew all olomaont nf I (Co alhnvpV That we rran interchanre the limit  t') ], a))N+">1 kI) ... d E 4  t')w(  t')( ,c)^io  t')W( 1 i ,(ik(Pcl t )W). it' t') Limits Theorem (see [31, Lemma I.7.6]) Hence we find the following expression for r, t'): , t"; l', t') lim N oo N ieH,( Xk(aPc; lj+l ,j)) lij) 7(lj)dl2 (4.33) Therefore, we have to evaluate (lj+< I[1  ill( This can be done as follows:  i ", k(PcO ie ,( /2) (Pj+l ) (ljpj+1/2 Xk (PCG 1 itC,(~k (pj+1 where P km(gj+))+ p lj+1, 1 k(mC)) ~,7p+1I2, S1, ... d. m=l Substituting the right hand side of (4.30) into the above expression yields (lj+ ll  icU e pj+1/2  i ", ( k (pj+1l S)) dPj+l/2 j (4.34) Now inserting (4.34) into (4.33) yields K (" , t"; 2', t') = lim ') Noo x exp i tpj+i S(1j+1 N  li) 1 5 =0  iE',(kO(pj+1/2; j+ x n dl ... dpS+*/2 ...dld (4.35) j+1/2=0 Eauation (4.35M represents a valid lattice phasespace path integral representation of l+1, Zk (p+1 1 f7(li+W) ( > K(l" ...dl K(l" (/j+lI[1 lj )} Ijldp hWPj +1 lj+1 dplj+l/2 expression: x exp i [pj+I * (+1 lj) fd,(xkj+1/2; j+1' j))I N 1 d3 x j=..1dl j=l N j+1/2=0 dplp+1/ . .. dpdg+l/2 which is the desired expression. Remark 4.4.7 Observe that even though the group manifold is a curved manifold the regularized lattice expression for the representation independent propagator  save for the prefactor 1/Xy(l") ')  has the conventional form of a lattice phasespace path integral on a ddimensional 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 selfadjoint. Hence, one can use (4.32) 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 h2 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 that is (a) more general than, (b) exact, the group manifold of a general Lie group 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 2. O 4.5 Example: A Representation Independent Propagator for the Affine Group We now introduce a representation independent propagator for the affine group. ,t "; ', t') ..9. K(I" onedimensional 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 Ref. example of a locally compact, adopted parameterization is given 64 and references there in. nonunimodular Lie group, A(g(p, q)) The affine group is also an its modular function in the and its left invariant Haar measure is given by dg(p, q) 4.5.1 Affine Coherent States us denote by and X2 a representation of the basis of the Lie algebra associated with the affine group by selfadjoint operators with common dense invariant domain D on some Hilbert space H. Since X1 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 [X2, XZ iX1. 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 X1 and X2 are chosen to be selfadjoint they can be exponentiated to oneparameter unitary subgroups of the affine group, example 4.4.1. Since the affine group is a connected solvable Lie group every group element can be written as the product of these oneparameter unitary subgroups (cf. Theorem 3.5.1]). With the above parameterization the map: g(P,q)  Ug(pg)  exp( iqX1) exp(iln pX2) provides a unitary representation of the affine group on H for all (p, q) P+ where = {(P,q) 00 < < 00}  . S The unitary representations of the affine = dpdq. and one for which X1 is a negative selfadjoint operator. We denote the irreducible unitary representation of the affine group corresponding to X1 positive by U7,q) and to X1 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 (, SH 0 the factor ,2, is square integrable if and only if E D(C1/2 where the operator C is given by C = 27T and X1 is restricted to be positive. Hence irreducible unitary representations of the affine group are square integrable for a dense set of vectors in H. Moreover, in Ref. 5 the following orthogonality relations have been established for the irreducible unitary representations of the affine group: , x')dpdq = (X' = 1.2 where X' eH eD(C Hence , each of the irreducible unitary repre sentations can be used to define a set of coherent states: r (p, q) = UC ,qC  j^ r1) where ,? E D(C' ) and 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 (P+) 4.5.2 C L2(P+). The Representation Independent Propagator Using Theorem 3.2.1(ii) we find: idUU g(p,q) g(p,q) Xdq + (qXx  p from which we identify the following 1 X)dp p 1.2 coefficient matrix [Pm k(g(p, q))]: (a(f v.a I)) , < # = 1 I  U(U)(, (x, U )(U' ,C1/2 , ,x)(C I v TT1 I 