Quantization and representation independent propagators

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Quantization and representation independent propagators
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Includes bibliographical references (leaves 161-167).
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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 Marie-Jacqueline and Anne-Sophie































"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 Marie-Jacqueline, 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


on-going 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 phase-space 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 non-relativistic 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 self-adjoint operators P


, the momentum,


and Q


the position,


with a common dense invariant domain D


. These operators


satisfy the Canonical (Heisenberg) Commutation Relation (CCR)


[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


time-dependent Schrddinger equation


iat(t)


= Wt(P, Q).(t)


Since only self-adjoint operators may be exponentiated to give one-parameter uni-


groups which


give the dynamics of a quantum system


it will always be as-


sumed that the Hamilton operator is essentially self-adjoint, 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 one-parameter


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 self-adjoint


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 q-space,


or on p-space,


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 time-honored custom of doing so.


One can also


choose one of the so-called 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 q-space or in p-space in appropriate limits.


In particular, one obtains


the Schrbdinger representation on q-space 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 q-space by


"; q', t') (q'


,t')dq'


in p-space 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, self-adjoint 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
V-T


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 self-adjoint 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


two-variable 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 dp-dqj 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 self-contained.

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 Maurer-Cartan 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 phase-space 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 phase-space 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


phase-space path integral has the form of a lattice phase-space path integral on a mul-


tidimensional flat manifold.


Hence


, we obtain a novel and very natural phase-space


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 non-compact 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 Hamilton-Jacobi 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


126-129])


. This new approach did certainly not


break any barriers that could not be overcome from the operator or Hamiltonian point


of view.


Nevertheless, one might have gained in two ways from Feynman's work [35


and the ensuing work of other authors [21, 20,


, 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 .... A-t 11


U(e)


+ V(q),


. .., pd).








- -H(p, q)](p|q)


- (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
N-too


" 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 q-space propagator


q', t') satisfies the following initial condition:


q', t')


= 6(q"


as it should


ts very definition.


Observe that in the phase-space 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)~


(p|7H(P, Q)lq)


(Pl|q)


J(q",


J(q"


- q'),


lim J(q"
t"- t'







with respect to N with the integrals we find the following formal standard phase-space

path integral


J(q"


I- (p(t), q(t))
ti


DqDp,


where,


- Ha(p, q)]dt.


This formal phase-space path integral for the q-space 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 2d-dimensional phase-space 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 phase-space prescrip-


tion (2.3) for the q-space 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 configuration-space trajectories.

(2.7) was obtained for the Hamiltonian (2.1).


[95]).


The Feynman path integral in

If the p-dependence 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 n-sphere,


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 d-dimensional unbounded Riemannian manifold with constant scalar


nn+..... ...., oi m+,- toncnr fi-nl


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. 58-64]) 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,


197-205] 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 non-linear 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 so-called 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=5--oo


- -E (tt


- i-h(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.


63-65]),


= (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 + 1-times 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 D-1( )


-=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 non-compact 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 one-to-one map f of S onto T is
called a homeomorphism if f-1 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 n-sDhere 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 non-negative 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 g-gj+', 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 Laplace-Beltrami 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 Gram-Schmidt 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 two-sided 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 two-sided 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.50-55]):


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 two-sided 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
N-oo 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
N-d+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 Laplace-Beltrami 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 Laplace-Beltrami 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 Laplace-Beltrami


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 a-models and non-abelian 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 non-orthogonal wave functions


to describe non-spreading 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 c-number 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 non-compact 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 sigma-finite 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 d-dimensional 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 dc-dimensional representation space H


Let us denote by {Xk}dI=l


the set of finite dimensional self-adjoint 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 self-adjoint 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) N-times 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.


63-64,


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 --gA-f *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,


64-66


, 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.


68-69])


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 d-dimensional parameter space C


. The


parameter space Q is all of jd if the group is non-compact 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 d-tuple 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 self-adjoint 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 semi-invariant 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 one-parameter 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 one-parameter 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(-


b-X,


k((l))Xk,


E^Am


(3.3)


b=m+l


m-1


m-1


-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 mk-element 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 one-parameter 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))dlm-Xk'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 Maurer-Cartan 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


U-T*


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 p-m (g(l))p


desired relation,
d


{ [P-ViO(()]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[U-1


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


non-compact 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 non-compact 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 d-dimensional


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,


self-adjoint, positive,


then there exists a unique operator K


in H


if there exist vectors 6,


semi-invariant 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 Heisenberg-Weyl 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)


, K-1/2 ).


-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


p-l/"(g(1))(


[pm(g())(
m,n=l


({. [P-'m(g(1))]p


' (g ())


- Dl[p-i}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 self-adjoint 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 one-parameter


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 self-adjoint. Hence, the restriction of each Xk,


= 1,...,d,


to D is essentially self-adjoint.


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 self-adjoint.


particular, each tk(


-iV


..., d, is essentially self-adjoint 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


p-1"((1))


jl(g(1))] -


-if (g(l))8,,[p


A-11m(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


A-imW())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 d-dimensional,


connected and simply connected


real compact Lie group G.


For compact Lie groups


all irreducible representations are finitely dimensional (cf.[7


Theorem 7.1.3]).


Hence,


let us denote the finite dimensional irreducible representations of G by 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 self-adjoint


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 self-adjoint operator whose


eigenspaces are A-invariant (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 self-adjoint 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 Peter-Weyl 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 Peter-Weyl


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 two-sided 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 self-adjoint 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


-1-z


-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(m-n,m+n)
(-mi


where p(_m-n'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') l-l(-iV


(-iV, /),


1
sin 0


- )


where 1


= (0, ,E) and


= (8e, 9 ,,-).


Equation 4.13 the regularized lattice


phase-space 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 4-1 + 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 self-adjoint.


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 self-adjoint 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
N--oo


...


S
a,'v


exp{i


- 6,) + %-A-i


( 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
N--too


.. .


n+l/i2=p-oo


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^ )+j-1/2) 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 j-1/
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 self-adjoint 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' )

-i-C1 ("
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 self-adjoint 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 v-measurable 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 self-adjoint 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 non-compact Lie groups it is not true that every symmetric Hamilton


operator is also essentially self-adjoint,
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 non-compact 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-/2e-iqk?(p


E H.


The generators of the one-parameter 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 self-adjoint 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 self-adjoint.


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 non-removable 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 self-adjoint and can also not be
extended to a self-adjoint 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 Ug-l()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 ciivr-h 7 nrnnTsQrnr fAr ono r Tnonr111Tlonf 11n +TTlnr r7onre_


E D(K/2),


}(t'))


', t') (l'








a semi-invariant operator of weight A (g


-1) in Hc for v-almost 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),


1-f ln t nn 4-b 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 d-variable 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 d-dimensional Schrodinger representation on L2(G)


. Based


on this interpretation one can give the representation independent propagator the fol-

lowing standard formal phase-space 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)


S-IPl,.


Remark 4.4.4


If the spectrum of P-k 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 d-dimensional 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


r-m(1)]


-1'k (g())(


El1/
m=1


-^jm


-ia'm


Using [p-'


-itfm


= im p-km( ((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

Hilbert-space L(G) (cf.


-iV


1, 1) are essentially self-adjoint 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 d-dimensional 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 d-dimensional lattice phase-space 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+)) + P-kmt))


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,)


(l1-l








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


s-lim 916
6--0


-=


and that on all of H one has


s-lim [I
N--oo


- ie91,


N+1 = exp[-i(t"


where E


- (t"


- t')/(N + 1)


. Now in order to obtain the lattice phase-space 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
N--oo


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).


-i-t'


-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
') N-oo


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 phase-space 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 phase-space path


integral on a d-dimensional flat manifold.


Also note that the lattice expression for the


representation independent propagator exhibits the correct time reversal symmetry.


Furthermore


, we have made no assumptions about the nature of the physical


systems we are considering,


other than that their Hamilton operators be essentially


self-adjoint. Hence, one can use (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"







one-dimensional systems for which


the canonical momentum


p is restricted to be


positive for all times.


For further applications of the affine group in quantum physics


the reader is referred to Ref.

example of a locally compact,


adopted parameterization is given


64 and references there in.

non-unimodular 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 self-adjoint 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 self-adjoint they can be exponentiated to one-parameter


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 one-parameter 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 self-adjoint 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(C-1/2


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'


,C-1/2 ,


,x)(C


I v TT1


I