Path Integrals on
Group Manifolds
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The Representation Independent Propagator for General Lie Groups
Path Integrals on
Group Manifolds
Wolfgang Tome University of Florida
World Scientific Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co Pte. Ltd Lid P O Box 128, Farrer Road, Singapore 912805 USA office Suite IB, 1060 Main Street, River Edge. NJ 07661 UK office- 57 Shelton Street. Street, Covent Garden. London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
PATH INTEGRALS ON GROUP MANIFOLDS — THE REPRESENTATION INDEPENDENT PROPAGATOR FOR GENERAL LIE GROUPS Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-02-3355-8
This book is printed on acid-free paper
Printed in Singapore by Uto-Piint UtoPiint
For Marie-Jacqueline and Anne-Sophie
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"The highest reward for a man's toil is not what he gets for it, but what he becomes by it." -John Ruskin
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PREFACE The quantization of physical systems moving on group and symmetric spaces has been an area of active and on-going research over the past three decades. It 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 variables, which nonetheless, correctly propagates each element of a continuous representation based on the coherent states associated with these kinematical variables. We begin our discussion with a short chapter on some of the standard results from the fields of Algebra, Functional Analysis, and Representation Theory. The results discussed are used freely in the remainder of the book. The reader unfamiliar with these results should refer to chapter 1. In chapter 2 we discuss the relevance and physical interpretation of representation independent propagators. We have chosen the simplest case of a single canonical degree of freedom as a vehicle to introduce the reader to the concept of representation independent propagators. IX
X
In chapter 3 we discuss the construction of path integrals on group and symmetric spaces. In § 3.1 we review the Feynman path integral on flat, group, and symmetric spaces. § 3.2 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 4 we introduce the notations and basic definitions used throughout the volume. The main result of this chapter is Theorem 4.2.1, in which we derive an operator version of the generalized Maurer-Cartan form. Chapter 5 contains the construction of the representation independent propagator for a real, separable, locally compact, connected and simply connected Lie group with irreducible, square integrable representations. We refer hereafter to a real, separable, locally compact, connected and simply connected Lie group with irreducible square integrable representations 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 § 5.1 we define coherent states for a general Lie group and prove Lemma 5.1.4 and the Corollary 5.1.5 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 § 5.2 the representation independent propagator for any real compact Lie group. It is shown in Theorem 5.2.2 that the representation independent propagator for any compact group correctly propagates
XI
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 § 5.3 this construction is then suitably extended to a general Lie group and we show in Theorem 5.4.2 that the result obtained in Theorem 5.2.2 holds for a general Lie group. In Proposition 5.4.4 we establish that it is possible to construct regularized phase-space path integrals for a general Lie group. Even though generally the group space is a multidimensional curved manifold, it is shown that the resulting phase-space path integral has the form of a lattice phase-space path integral on a multidimensional flat manifold. Hence, we obtain a novel and very natural phase-space path integral quantization for systems whose kinematical variables are the generators of a general 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 the representation independent propagator for the affine group is constructed. In chapter 6 we discuss the classical limit of the representation independent propagator of a general Lie group and show that its classical limit refers indeed to the degrees of freedom associated with the general Lie group. § 6.1 and 6.2 contain a detailed discussion of the classical limit of the coherent state propagator for compact Lie groups and non-compact Lie groups. In § 6.3 we prove that the equations of motion obtained from the action functional of the representation independent propagator for a general Lie group
xii imply the equations of motion obtained from the most general action functional of the coherent state propagator for a general Lie group (cf. Proposition 6.3.1). Chapter 7 is an outlook on further directions that can be pursued in the implementation and investigation of representation independent propagators. Gainesville, Florida, September 1997
Wolfgang Tome
ACKNOWLEDGEMENT I would like to thank John Klauder for his friendship, advice, and constructive criticism without which this book would never have been written. I wish to thank Stanley Gudder for introducing me to the foundations of quantum theory and thus making this book possible. I am grateful to Hajo Leschke for his helpful remarks on the universal propagator for affine coherent states. His remarks proved to be very valuable in the further course of my investigations. I am also grateful to Max Brocker from the Studienstiftung des deutschen Volkes, whose valued assistance and support have made it possible for me to attend various meetings from which I have benefited a great deal. Most importantly I wish to express my deepest gratidude to my wife MarieJacqueline for making me feel complete and for always being an unfailing companion throughout my adult life and to my daughter Anne-Sophie for willingly sharing her joy for life with us. Last but not least, I would like to thank all my teachers for their encouragement and having taught me well.
xii
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Contents 1 MATHEMATICAL PRELUDE
1
1.1
Algebra
1
1.2
Functional Analysis
3
1.2.1
Operators on Hilbert Space
3
1.2.2
Direct Integrals
10
The Nuclear Spectral Theorem
17
1.3.1
Some Topological Notions
18
1.3.2
Nuclear Space
19
1.3.3
Linear Functionals
23
1.3.4
Generalized Eigenvectors and the Nuclear Spectral The-
1.3
orem 1.4
Lie Groups and Lie Algebras 1.4.1
1.4.2 1.5
25 27
Nilpotent, Solvable, Semisimple, and Simple Lie Algebras and Lie Groups
30
The Enveloping Algebra of a Lie Algebra
31
Some Basic Notions of the Theory of Group Representation . .
32
1.5.1
Equivalence of Representations
34
1.5.2
Irreducibility of Representations
35
XV
xvi
CONTENTS 1.6
2
3
37
PHYSICAL PRELUDE
39
2.1
Introduction
39
2.2
The Fiducial Vector Independent Propagator for the Heisenberg Weyl Group
42
2.2.1
47
Examples of the Fiducial Vector Independent Propagator
A R E V I E W OF SOME M E A N S TO D E F I N E PATH INTEGRALS ON GROUP A N D SYMMETRIC SPACES
49
3.1
Feynman Path Integrals
49
3.1.1
Introduction
49
3.1.2
The Feynman Path Integral on Md
51
3.1.3
The Feynman Path Integral on Group Spaces
56
3.1.4
The Feynman Path Integral on Symmetric Spaces
3.2
4
Reducible Representations
...
64
Coherent State Path Integrals
73
3.2.1
Introduction
73
3.2.2
Coherent States: Minimum Requirements
74
3.2.3
Group Coherent States
75
3.2.4
Continuous Representation
77
3.2.5
The Coherent State Propagator for Group Coherent States 79
NOTATIONS A N D PRELIMINARIES
85
4.1
Notations
g§
4.2
Preliminaries
go
CONTENTS
xvii
5 THE REPRESENTATION INDEPENDENT PROPAGATOR FOR A GENERAL LIE GROUP
97
5.1
Coherent States for General Lie Groups
98
5.2
The Representation Independent Propagator for Compact Lie Groups
5.3
5.4
106
Example: The Representation IndependentPropagator for SU(2) 113 5.3.1
The Hamilton Operator H(si,S2,S3) = JJS^
118
5.3.2
The Hamilton Operator H(si,S2,S3) — 2/(^1 + ^2 + ^3)
122
The Representation Independent Propagator for General Lie Groups 5.4.1
125 Construction of the Representation Independent Propagator
5.4.2
125
Path Integral Formulation of the Representation Independent Propagator
5.5
6
133
Example: The Representation Independent Propagator for the Affine Group
143
5.5.1
Affine Coherent States
144
5.5.2
The Representation Independent Propagator
145
CLASSICAL LIMIT OF THE REPRESENTATION INDEP E N D E N T PROPAGATOR
151
6.1
Classical Limit for Compact Lie Groups
152
6.2
Classical Limit of Non-Compact Lie groups
157
6.3
Classical Limit of the Representation Independent Propagator . 162
xviii 7
CONTENTS
CONCLUSION AND OUTLOOK 7.1
167
Extension to Groups that do not possess square integrable, irreducible Representations
167
7.2
What has been Accomplished
171
7.3
Possible Further Directions
176
A CONTINUOUS REPRESENTATION THEORY
179
A.l Continuous Representation
179
A.2 Reproducing Kernel Hilbert Spaces
182
A.3 Proof that Equation (5.22) is Well Denned
186
B EXACT LATTICE CALCULATIONS
189
B.l The Free Particle B.2 The Hamilton Operator H(X (Xi,X UX22)]
189 = ^Xf r?-+ uX >X22
■
193
BIBLIOGRAPHY
197
INDEX
209
Chapter 1
MATHEMATICAL PRELUDE In this chapter we have collected some of the standard results from the fields of Algebra, Functional Analysis, and Representation Theory which are well known to the specialist working in those fields, but which may not be necessarily assumed to be common knowledge for physicists.
1.1
Algebra
Let A / {0} be a vector space over the complex numbers C. A is called an associative algebra with unity over C, if a product A x A —► A,(A,B)
t-* AB
is defined on A such that (AB)C
=
A{BC),
A(B + C)
=
AB + AC,
(A + B)C
=
AC + BC,
{aA)B
=
A(aB)
= aAB,
for
a£C,
and if there exists an element / G A such that IA = AI = A for all A £ A. 1
CHAPTER
2
1. MATHEMATICAL
PRELUDE
A set Q of elements of A is called a system of generators of A if the smallest closed subalgebra with unity containing Q coincides with A. The unity / is not included in the system of generators. Let us assume that the associative 1 4 1 1 V+a', a°I+J2 ' X,> ++ £Yl a'tf'XiXj + ...• 1X J +
!j=l =1
a°,a\ai ll,...eC. a° j\-
3
i,j=l »,J
••
(1.1)
ec.
We restrict the discussion to the case of a finite number of generators and we will not discuss topologies of A, i.e. it is assumed that the above sums for every A are arbitrarily large but finite. Defining algebraic relations are relations among the generators P(X1,...,X ,...,Xdd)) ,xd) = = = 0, where P(x\,.. Xi,.
(1.2)
.,Xd) Xd) is a polynomial of d variables with complex coefficients.
Let B G A be represented by dd
d
£
6°/ VXiU + Y, »,;. B = b°I+Y,b% b°I+J2b% £ VfXiXj V'XiXj x + ....... i = li
•°,
/ . bi} b°,b\b»,...6C. b°,b\b'\...ec.
ti ,,Jj ==1i >*,]'
J +
(1.3)
ec
If one can bring (1.3) into the same form as (1.1) with the same coefficients a0, a',... a° »V
by using the denning algebraic relations (1.2), then B is equal to A.
1.2.
FUNCTIONAL
1.2 1.2.1
ANALYSIS
3
Functional Analysis Operators on Hilbert Space
We list here some definitions and properties of operators on Hilbert spaces which are used in the main body of the text. As in the previous section we denote by C the set of complex numbers.
Definition 1.2.1 A complex vector space H is called an inner product space if there exists a complex valued function (•, •) on H x H satisfying for all <j>, ip, n and a, b 6 C: (i) (^) (xi) (4>^)===(TM>, J^A), J^A), > > >0 0and (Hi) {<j>, (p) cj>) and(4>, (4>,<j>) <j>)—— 0 0if ifand andonly onlyif if4>4>= =0.0. The function
(•, •) is called an inner product.
One can show that every inner product space is a normed space with norm ||. || = (. ( . ) 1 ' 2 . A sequence {4>n} in H is called a Cauchy sequence if for every e > 0 there exists a N(e), such that > N(e). \\ N(e). Hn: € :< i n,m An inner product space H is called c o m p l e t e , if every Cauchy sequence in H converges (that is has a limit in H). A c o m p l e t e inner product s p a c e is called a H i l b e r t s p a c e . A Hilbert space is called s e p a r a b l e if it contains a countable dense subset. One can show that a Hilbert space is separable if and only if it has a c o u n t a b l e orthonormal bases.
CHAPTER
4
1. MATHEMATICAL
PRELUDE
Let H and H ' be Hilbert spaces and let T be a map from a linear subspace D ( T ) C H to H ' , such that
&vo
T(a(f> + T(a
, ip 6 D ( T ) and all a,b £ C, then T is called an o p e r a t o r from H to H ' . The linear subspace D ( T ) is called the d o m a i n of T. The set R ( T ) = T ( D ( T ) ) is called the range of T. An operator T : H —> H ' is called b o u n d e d if there exists some constant C > 0 such thatt ||T(/)||JJ, II^Hi < C|||| 0 e£ H , -IMIHH, for all
n} C H , the sequence {T<j>n} C H ' has a subsequence that converges in H ' . Let us note the following facts about nonzero self-adjoint compact operators in £ ( H ) . (i) Every nonzero self-adjoint compact operator has at least one eigenvector cj>\ which belongs to a nonzero eigenvalue A. (see [1, pp. 124-126]). (ii) From the Rellich-Hilbert-Schmidt Theorem [79, p. 42] we have the following spectral resolution for nonzero compact self-adjoint operators: CO 0 0
E
T T= = Y,*kPk, Y,*kPk, M>Pk, k=l k=l
(1.4) (1-4)
6
CHAPTER
1. MATHEMATICAL
PRELUDE
where the Pk are mutually orthogonal projections on the finite dimensional eigenspaces H* = PP*H kH and \Xk\ -» 0 as k -> oo. Moreover, one has that H = ©^° =1 H :*©k f c ® ker(T), where ker(T) = { = 0} is the kernel of T. Hence, the Hilbert space H decomposes into a direct orthogonal sum of mutually orthogonal finite dimensional subspaces. If T is one-to-one, then ker(T) consists only of the zero element. An operator T is called trace class if and only if t r ( vVT^*TT ) < oo. If T is trace class and B 6 £ ( H ) , then TB and BT are trace class, furthermore, one has trfTjB) =: t r (ti(BT). i An operator is called H i l b e r t - S c h m i d t if and only if tr(T , *T) < oo. One can show that if T is trace class or Hilbert-Schmidt then T is compact. We will now discuss unbounded operators on H . An operator T on H is called closed if the relations llim i m , lim l i m "0n T<j>n = ^, r/,, Pn n = 0,
n—*oo n—*oo
n—+00 n—+oo n—+00
{<j>n} C D(T) D(T) {0.}
imply 4> £ D ( T ) and T<j> = tp. Here, the notation limn-,00 ??„ = 77 means strong convergence and is shorthand for: For every c > 0 there exists a N(() that [|T7„ — t)\\ < £ for all n > N(e).
such
Closedness is a weaker condition than
continuity. If T is a continuous operator on H , then h m n _ 0 0 n = / implies that the sequence {Tcf>n} converges. On the other hand if T is only closed then the convergence of the sequence {4>n} C D ( T ) does not imply the convergence of the sequence { T 0 n } . Nevertheless, if T is closed and the sequences {<j>n}, {V'n} C D ( T ) have the same limit, then the sequences {Tc/>n} and
{Tipn}
cannot have different limits. It is worth emphasizing that an operator is continuous if and only if it is bounded. Therefore, unbounded operators on H can
1.2. FUNCTIONAL ANALYSIS
7
not be continuous, however they can be closed. If T is not closed, one can sometimes find a closed extension of T. If a closed extension of T exists then T is called closeable, the smallest closed extension of T is called its closure, which is denoted by T. An operator T : H -» H' is closeable if and only if the following holds: If {<j>n} is a sequence in D(T) with lim^oo n — 0 and {T4>n} C H' is convergent, then limn_oo T } in D(T)1 with l i m n -lim o c n ^oo n u , = n
n
such that■{T4> {T4>n} n} is convergent}, T<j> = limlimn T<j>n for (T). 6 D(T). D(f). n—»oo n—*oo n—»oo
There exists a simple relationship between the notions of adjoint and closure (For a proof of these relationships see for instance [88, p. 253]). Let T be a densely defined operator on H. Then: (i) T* T" is closed.
T)
(ii) T is closeable, if and only if D(T*) is dense in H in which case r T = rpmm T""\ [T)' = T*. (iii) If T is closeable, then (T)* T'.
Example of a non-closeable operator: Let H = *£22,, the space of all absolutely square summable sequences, i.e. £22 := {{{a Y*n=\ l a «|22± has no solutions in H other than 4>± — 0. For self-adjoint operators one has the following spectral decomposition [88, p. 263]:
T h e o r e m 1.2.2 (Spectral Theorem) There is a one-to-one correspondence between self-adjoint
operators T and the projection valued measures PT( •) on H .
This correspondence
is given by +oo
XPpT{d {dX), dX),). T == / . -oo ' - // -oo
10
CHAPTER
1. MATHEMATICAL
PRELUDE
where a p r o j e c t i o n - v a l u e d - m e a s u r e is a map from the Borel measurable of R into the set of all orthogonal projections
TT(H) satisfying 7r(H) sai
the
sets
following
conditions: r T (i) F P1 ( 0 ) = 0 and P (R)M) =-- I.
sequ (it) If {£i},e/v., ^V. { 1 , 2 ), .*, ..*. •}} ,) is a sequence of mutually .}i6/V !V„ = {1,
disjoint
real
T T/I I D jp ' .)Z-.eTv. ^ Pr (T£(Ei)Borel measurable sets, thennPf T( (Ute7V. Jiefv. &*) ■ ) =- X^eJV. T T PT{ ■(F) = = PP T (P£T{EnF). IF). (Hi) P (E)PT)P (F)(F
1.2.2
Direct Integrals
All Hilbert spaces in this section are separable.
! t E bbe a locally compact Let
. there separable space and let vi/ be a positive measure 6E EE1 let sure on on E. z.. For every (Ce: exist a Hilbert space H (c with inner product (•, •),..
rw
i/i(.) is a :map from E to l i c e n c e ' such that: E 99 (( >-> A v e c t o r fieldi ipt.) —> 4>( V^e£ H Hee . A countable family of vector fields {4>', J■,}
• »'€JV.
r
isis cal called a f u n d a m e n t a l
in
family if the following two conditions are fulfilled: (i) All functions ES 93 C( ' t-» (ipljipi}^ i^}( are //-measurable for i,j 6 IN.. ■*w\,ii (ii) For all ( € E, the family of vectors {V>->
\A(rp^ipi),, #o
i £ IN., are (/-measurable. Let us note the following facts about measurable vector fields (cf. [79, Lemma 1.6.5])
rwH,:.
(i) The measurable vector fields form a linear subspace of FIi-eH^f:ce:- k
1.2.
FUNCTIONAL
ANALYSIS
11
i s a f-measurable function. (ii) If i>(.) V>(.) is ||! V»(.)lk V(-)He is is a measurable vector field, then en ||V>(.
(iii) If ^(.j then is a measurt h {ip(_,4>()( tor fields, fiel (.) are a ^-measurable vector i>(. andi 0(.) (V'Ci^)c *s a m e ; able function. tion. Using the Gram-Schmidt othogonalization procedure, one can construct a complete orthonormal set of vector fields, i.e., a sequence 1, 2,... of measurable vector fields such that:
(%,fyc
] H ( = oo, then the set {}^j w bwhere (i) If dimH,; = l,
T T?4>1 IN,. 's ' i iG6I V - I Hence, C H-» {t^\,T^4>i)(_ T{4>i isis measurable for every 6c>c = ~( ^(^
* cC: *- is T Tjj-dense in H , since * is already 7fj-den Tfj-dense in H . H -de; :J Let us consider a.. TLie algebra L of symmetric operators on a Hilbert space
H which have a common dense invariant domain D . Let X\,...,Xd Xd be an
LU
operator basis for L such that the Nelson operator A = Yli=i XfCfh is essentially
1.3.
THE NUCLEAR
SPECTRAL
self-adjoint. It then follows that Xi,..
THEOREM
21
.,Xd are essentially self-adjoint (cf. [83,
Lemmas 5.2 & 6.2]). Since the Xk, k = l , . . . , d , are symmetric A is a positive definite operator. Furthermore, since A is essentially self-adjoint there exists on H by Theorem 5 in [83] a unique unitary representation U of the simply connected locally compact Lie group G which has L as its Lie algebra such that for all X in L, U(X) U(X) = X. INow let us denote by A y the dense set of analytic ;atio U of G. It is shown in [7, pp. 364-365] that the vectors for the representation dense set A y C H of analytic vectors forms a common dense invariant domain for the basis X\,..
.,Xd of L and its enveloping algebra £(G)1'.
Therefore,
every element of A y is in the set of analytic vectors forr A. Let ty be the dense set of analytic vectors Ay for V, then we can as outlined above construct a countably Hilbert space . We now show that the elements of the enveloping algebra £ ( G ) are ccontinuous with respect to 7 $ and are S(G) therefore, uniquely defined on the whole whc space 6 * one has P+
IfX {4>,Xi{A < k(,{A ), {4>, Xi(A IfX^) k(4>, V), (A ++IfX^) s(A(A+++/I)I)p+1p+V), Xri)>n} be a sequence converging to zero in the 74,-topology, i.e., l i m n _ 0 0 ||n)--
71—*OQ
n—*oo
'For the definition of an enveloping algebra of a Lie group G see 1.4.2 '1 'For the definition of an enveloping algebra of a Lie group G see 1.4.2
(1.6) (1.6)
CHAPTER
22
1. MATHEMATICAL
PRELUDE
Let Xi be arbitrary, then to show that X{ is a continuous operator it is sufficient to show that (1.7)
fc every q, lim1 ||X,0 |]X,-0JL \\Xi.. n |L = 0 for (in\\q
X) n—»oo n—*oo
i.e., that X , n0) n ) - / )I)'q' .X,$ A +r /iyXtfn) lim (<j> + lim {Xi {Xl(f)n n, ((A ■•■(A n, X t{A ) 9%A: == lim (4> ,X n—»ooi
n—»oo n—»oo
= 0 for e v e r y q,
(cf. [88, T h e o r e m 1.6]). By (1.5) o n e h a s
0'*
A ++ lyXiin) \(( 007n l, n,X,Xi(A ^>i((A
o ,+1
*:(„, * | | * n l | J + il , < k{4> A H+ I)q+l4>n) n,, (I( A > » ) < fcll*nll?+l.
however, by (1.6) the right-hand side converges to zero as n —* oo for every q, and therefore, the left-hand side also converges to zero for every q, this establishes (1.7). Since X{ was arbitrary this shows that all generators are continuous operators. Since 'I' is a 7 $ dense linear subspace of 4» we can, using the B.L.T. Theorem (cf. [88, Theorem 1.7]), uniquely extend the linear operators Xi, i = l,...,d
on ty to operators on the whole space . Note
that since the operators X,-, i = 1 , . .. ..,d ,,dd aare ar continuous on they are defined everywhere on , hence, domain questions do not arise. We are now ready to give the definition of a nuclear space.
Definition 1.3.3 t-> L(4>) = (L\) £ C and +++L(x/>)i L(a L(a
) aL{4>) aL{4>) bL(ip) bL(ip) for 4>,ip <j>,ip 1 anda,b a,b£C. a,b £C.£C. L(c rfor , ij) £S £ £S I Sand ,(a
