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PATH INTEGRALS AND COHERENT STATES OF SU(2) AND SU(1,1)

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PATH INTEGRALS AND COHERENT STATES OF SU(2) AND SU(1,1)

A. Inomata H. Kuratsuji C. C. Gerry

V f e World Scientific

«

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661

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UK office: 73 Lynton Mead, Totteridge, London N20 8DH

PATH INTEGRALS AND COHERENT STATES OF SU(2) AND SU(1,1) Copyright © 1992 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 orany information storage and retrieval system now known or to be invented, without written permission from the Publisher.

ISBN 981-02-0656-9

Printed in Singapore by Utopia Press.

v

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PREFACE

Most of the exactly soluble systems in quantum mechanics may be classified by their group structures into two types: one is the SU(2) type and the other is the SU(1, 1) type. The same can be said of path-integrable systems. Those problems which are exactly soluble by conventional methods or algebraic methods are also often soluble by path integration. The group structure in fact helps path integration. To those which are not exactly soluble, we would have to explore appro­ priate approximation methods. Feynman's path integral, which is difficult to evaluate explicitly, has long been used as a semiclassical tool even for exactly soluble problems. However, it is not simple to incorporate other approxima­ tion schemes to the path integral. The coherent state formulation could provide a framework in which the path integral may be treated by non-serniclassical approximation. Since Feynman's path integral is based on the c-number Lagrangian, the concept of spin, whose classical counterpart does not exist, cannot trivially be accommodated in it. The coherent state formulation supplies a practical basis for handling spin. In this regard, taking up path integrals and coherent states, based on the groups, SU(2) and SU(1, 1), is not a specialization of interest but an important enterprise to extend the limit of Feynman's path integral and to develop its applications. This volume consists of three parts written separately by three authors. Part I provides a general introduction to path integrals in quantum mechanics and group theoretical preliminaries for path integrals on SU(2) and SU(1, 1). Various techniques for path integration, applications of polar coordinate path integrals, and harmonic analysis based on SU(2) and SU(1, 1) are discussed by using exactly soluble examples. Part II deals with the SU(2) coherent states and their applications. Construction and generalization of the SU(2) coherent states, formulation of coherent path integrals for spin and unitary spin, and semiclassical quantiza­ tion in the coherent state formulation are presented. Applications are made to the study of quantum fluctuation, the nonlinear field model and phase holonomy. Part III presents the theory of the SU(1, 1) coherent states and their applications. The SU(1, 1) coherent states and their path integrals are con­ structed. The conserved Noether current for dynamical groups, the most general coherence preserving Hamiltonian, and the Large-N approximation

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vi

are discussed. Applications in quantum optics include such topics as squeezed states, the phase operator formalism, and Berry's phase in the degenerate parametric amplifier. When this book was initially planned by the first author some five years ago, there were only two books on path integrals. They are those of Feynman and Hibbs (1965) and of Schulman (1981). Then Wiegel's (1986) and Kleinert's (1991) appeared. They are all excellent and unique as textbooks as well as reference books. However, in recent years, there have been considerable developments of path integration techniques and their applications. There are a number of new interesting topics that deserve attention but are not included in these books. This volume contains some of the new topics on path inte­ grals, particularly those which are related to the dynamical groups of SU(2) and SU(1, 1) and those which are based on the coherent states of SU(2) and SU(1, 1). In preparing materials for a general introductory book, the first author requested the help of the second and the third authors who are pioneers and experts of the coherent state path integrals of SU(2) and SU(1, 1). Their responses were enthusiastic and their contributions turned out to be substan­ tial. With their well-prepared manuscripts, he found it rather difficult to merge them organically and coherently into the planned book. What seemed more reasonable than diluting the originality of each author by mixing was to include the works of three authors as independent parts of one volume. While their original manuscripts already written two years ago were modified with more pedagogical sections, the first author had to shift his aim from an introductory level to a slightly advanced level. Since each of the three parts is self-contained and has a sufficiently pedagogical introduction, the reader may be able, with­ out difficulty, to get a general view of the fields. At the same time, the reader may be entertained by some crude ideas contained in each part.

A.I. January 1992

VII

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CONTENTS

Preface

v

Part I Path Integrals for SU(2) and S U ( 1 , 1) Akira Inomata

1

1 Introduction

3

2 Techniques for Path Integration

8

2.1 2.2 2.3 2.4 2.5 2.6

Rules of Thumb in Path Integral Calculus Change of Variables in a Path Integral Local Time Rescaling Techniques Path Integrals in Polar Coordinates Asymptotic Recombination Techniques Dimensional Extension Techniques

3 Path Integrals on SU(2) 3.1 3.2 3.3 3.4

The Group SU(2) Path Integrals on SU(2) The Poschl-Teller Oscillator The Hulthen Potential

4 Path Integrals for SU(1, 1) 4.1 The Group SU(1, 1) 4.2 Continuous Basis for Representations of SU(1, 1) 4.3 The Modified Poschl-Teller System 4.4 Harmonic Oscillators Revisited 4.5 The Coulomb Problem 4.6 The Morse Oscillator 5 Exactly Path-Integrable Examples 5.1 5.2

Basic Soluble Examples Other Soluble Examples

Appendix. Representations of SU(2) and S U ( 1 , 1)

8 12 16 22 28 32 40 40 56 61 68 76 76 85 92 98 106 112 115 115 117 121

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VIII

Part II Path Integrals in the SU(2) Coherent State Representation and Related Topics Hiroshi Kuratsuji

139

1 Introduction

141

2 Grauber States Revisited

145

3 SU(2) Coherent States and their Generalization

149

3.1 3.2

Elementary Construction by Spin Algebra Lie Group Construction of the GCS

149 157

4 Path Integrals for Spin and Unitary Spin

164

4.1 4.2

Path Integrals for Spin Path Integrals for Unitary Spin

5 Semiclassical Quantization Theory 5.1 5.2 5.3

Formula without Quantum Fluctuations Formula with Quantum Fluctuations Topological Aspect of Semiclassical Quantization

6 Applications 6.1 6.2

Topological Invariants in Many-Particle Systems Geometric Characteristics of the Canonical Term

7 Nonlinear Field Model 7.1 7.2

General Formulation PX(C) Model (Continuous Spin)

8 Phase Holonomy

164 169 174 174 178 184 188 188 191 194 194 197 202

8.1

General Remarks

202

8.2

Coherent State Path Integrals for Adiabatic Motion

205

Appendix A. The Kernel Function

210

Appendix B. Invariant M e a s u r e of t h e Grassniann Manifold

211

Appendix C. T h e Poisson Bracket in Generalized P h a s e Space

213

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ix

Part III S U ( 1 , 1) Coherent States a n d Path Integrals for S U ( 1 , 1) Christopher C. Gerry

219

1 Introduction

221

2 S U ( 1 , 1) and t h e Perelemov C o h e r e n t S t a t e s

224

2.1 2.2 2.3

SU(1, 1) and Its UIRs Examples of SU(1, 1) as an SGA SU(1, 1) Coherent States

3 Path I n t e g r a l a n d Classical Dynamics 3.1 Path Integrals 3.2 Conserved Noether Currents for Dynamical Groups 3.3 Most General Coherent Preserving Hamiltonian 3.4 Time Dependent Invariants 3.5 Direct Path Integral Approach to Time Evolution 4 Applications in Q u a n t u m Mechanics 4.1 Radial Isotropic Oscillator 4.2 Radial Coulomb Problem 4.3 The Morse Oscillator 4.4 Application to the Large-N Approximation 5 Application t o Q u a n t u m Optics 5.1 Introduction 5.2 Squeezed States and SU(1, 1) CS 5.3 Interactions of the Squeezed Vacuum States 5.4 Phase Operator Formalism for SU(1, 1) 5.5 Berry's Phase in the Degenerate Parametric Amplifier 5.6 Related Matters A d d e n d u m - Further Developments

224 227 229 233 234 238 239 244 246 251 251 257 266 271 281 281 284 289 296 301 308 309

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1

Part I

PATH INTEGRALS FOR SU(2) A N D SU(1,1)

Akira Inomata

Department of Physics State University of New York at Albany Albany, New York 12222, USA

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3

1

Introduction

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" The formulation is mathematically equivalent to the more usual formulations.

There are,

therefore, no fundamentally new results. However, there is a pleasure in recognizing old things in a new point of view In addition, there is always the hope that the new point of view will inspire an idea for the modification of present theories, ...." R.P. Feynman, Rev. Mod. Phys. 20,367(1948)

In 1948, Feynman proposed the path integral approach to quantum mechan­ ics as an alternative to Heisenberg's matrix mechanics and Schrodinger's wave mechanics. While Heisenberg's approach and Schrodinger's approach are both based on the Hamiltonian H of the canonical formalism, Feynman's proposal is unique in that its formulation depends on the Lagrangian L. He asserted that the propagator tf(x", x'; t", f) = (x" | exp[-irH/%

| x') = £ C ( x ' ) Vn(x") e~"B^

(1.1)

n

defined for r > 0 can be given by a path integral,

K(x",x'; t",t') = lim / ft *{*h*i-U Tj) II

C1-2)

4 where

in/2

I

27rifi Tj

ri

exp KS>

(1.3)

with the short time action,

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Sj=ftj

L(x,x)dt.

(1.4)

In the above, we have used the notations, Xj = x(tj),

t' = t0, t" = tjqy r = t" — 1 \

and Tj = tj — tj_j. Suggesting the trapezoidal rule (or the mid-point prescription) by which the short time action on cartesian coordinates is approximated by S^TjL^tXj), r i X ( ^ ' * ' ) '

(1.5)

where AXJ = Xj — Xj_! and Xj = (x^ 4- Xj_x)/2, and deriving the Schrodinger equation in cartesian variables, Feynman claimed that the path integral approach is equivalent to the canonical formulation. However, it is a curious fact that the p a t h integral cannot be calculated for systems other than a free particle, a har­ monic oscillator and more general quadratic systems. For instance, the hydrogen atom problem whose solution once symbolized the success of Schrodinger's wave mechanics cannot be solved by Feynman's approach. Since the path integral has a close proximity to classical mechanics, it has been taken as a powerful tool in semiclassical calculation. Therefore, Feynman's p a t h integral has been nearly synonymous with a semiclassical approximation method. Is Feynman's path integral only a method in quantum mechanics or a formulation equivalent to other quantization schemes? Is the path integral in­ herently incapable of producing an exact solution for such a standard problem as the hydrogen atom problem? Or is the failure of the path integral approach due to the lack of techniques? Is Feynman's framework too restrictive to accommo­ date a wider class of solutions? If so, how can it be extended? Answers to these questions do not come from the study of approximate solutions. It is necessary to pursue ways to solve the path integral exactly. In fact, in recent years, there has been remarkable progress in developing techniques of path integration. It has now become clear that Feynman's path in­ tegral is capable of solving problems other than quadratic systems. For instance,

5 Feynman's path integral can explicitly be calculated for the Poschl-Teller oscil­ lator by the dimensional extension technique and yield exact results. It has also been recognized that the framework of Feynman's path integral has to be slightly extended in order to solve many of the standard problems.

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When we say a problem is solved for a quantum system, we usually mean that the energy spectrum and the energy eigenfunctions of the system are found. Thus it is sufficient to find the propagator which contains information of the spectrum and the energy eigenfunctions.

However, the propagator is not the

only object that contains the information. The energy Green function, G ( x " , x ' ; £ ) = (x" | (E - H)^

I x') = £

^WU*")

n

& -

^

(1 6)

&n

can also supply the same information. If it is difficult to find the propagator, then we can try to calculate the energy Green function which is the Fourier transform of the propagator. A question is how the Green function can be accommodated in Feynman's formulation. Although the Green function cannot be expressed as a path integral, we can extend Feynman's assertion, defining an entity called the promo tor, P ( x " , x ' ; r ) = (x" | e x p [ - t r ( f f - E)/K\ | x'),

(1.7)

and observing that the promotor can be given as well in a path integral form, P ( x " , x ' ; *",*') = lim / I I P^j^j-uTj) where M P(Xj,Xj-l\Tj)

=

f[

dnxh

(1.8)

n/2

2m%r3}

exp

vnw>.

(1.9)

with the short time action modified by the energy term, Wj = f'

[I(x, x ) + E]dt = Si + ETJ.

(1.10)

The energy Green function is then evaluated by 1 f°° G(x", x'; E) = P ( x " , x'; r ) dr. in Jo

(1.11)

When the path integral for the propagator is difficult to calculate, the p a t h in­ tegral for the promotor is also difficult to evaluate. However, the Green function

6

has a remarkable property in that it is invariant under a certain time transfor­ mation of the promotor. The property is based on the following proposition; Proposition 1: The promotor (1.7) if ImE promotor defined by Path Integrals and Coherent States of SU(2) and SU(1,1) Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 12/31/14. For personal use only.

P(x",x';r//V') = (x"|exp

> 0 is equivalent to a modified

L^fME-mgU

|x'>

(1.12)

under the integration over r € [0, oo), where f = /(x') and g" = g{x") are c-numbers, and f(x) and g(x) are positive-definite q-number functions of the position operator x. The proof of this proposition is straightforward. Since the promotor (1.7) is a special case of the modified promotor (1.12) with / ( x ) = g(x) = 1, it is sufficient to show that the r-integration of (1.12) is independent of / ( x ) and g(x). The rintegration of (1.12) can be computed as follows; for Im E > 0, j T P(x», x'; T/f'g")dr = / o V ' | exp ^j^f(y.)(E

- ff ) ff (x)j \x!)dr

= (tft)/y'(x"|j-1(£-j)-1/-1|x')

(i-is)

= (ift){xl(£-20-1IA The result is indeed independent of / ( x ) and g{x). As a consequence of the above equivalence proposition, we obtain G(x", x'; E) = ^ f°° P(x", x'; a)(dr/da) da (1.14) in Jo where we have set a = T/f'g" = r/[f(xf)g(x")]. This implies that the Green function can be evaluated by choosing an appropriate generalized promotor. The long-standing hydrogen atom problem has been solved by path integration in this extended framework of Feynman's path integral. A majority of exactly soluble problems can be classified by their solutions into three types: those of the hypergeometric type, those of the confluent hypergeometric type, and others. The systems of the first type have usually the SU(2) group structure, whereas those of the second type have the 517(1,1) group structure. Very few examples belong to the last class. In Part I of this volume, we shall discuss how to solve by path integration those systems which have the SU(2) or 5I7'(1,1) group structure. The discussions

7 on the path integrals of the coherent state bases for SU(2) and 517(1,1) will be left to the authors of Part II and Part III. Chapter 2 of Part I is devoted to various path integration techniques which have been developed in the last ten years. In Chapter 3, the group SU(2) is re­

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viewed in a way pertinent to path integral calculations and its unitary irreducible representations, and is utilized for constructing a path integral for a free particle on the SU(2) manifold. Then, as examples, the Poschl-Teller oscillator and the Hulthen system are solved by path integration on the SU(2) manifold. Chapter 4 begins with a review of the group 51/(1,1) and employs the unitary irreducible representations of 5Z7(1,1) on a continuous basis to analyze polar coordinate path integrals and to solve the harmonic oscillator, the Coulomb problem and the Morse oscillator. The last chapter lists some exactly path integrable systems, their group properties and the techniques needed to solve them. The Appendix presents a unified way to determine unitary irreducible representations of SU(2) and 5 ^ ( 1 , 1 ) . For more detailed information on the representations of the groups

SU(2)

and 517(1,1), we refer the reader to Vilenkin (1968) and Sugiura (1975). For a more elementary introduction to path integrals and more detailed discussions of path integration techniques, the reader may consult a forthcoming monograph, Path Integrals: Methods and Applications, by the present author. It also contains such topics as harmonic analysis of path integrals; topologically constrained path integrals as applied to the Aharonov-Bohm effect, anyon physics, and entangled strings; time-dependent conformal transformations with or without magnetic fields; the pulsed oscillator; a path integral treatment of the Dirac equation for the Coulomb problem; quantization of the Kaluza-Klein monopole; and others. The citations (GR:....) for mathematical formulas used in the text indicate the formula numbers of Gradshteyn and Ryzhik (1980).

8

2

Techniques for Path Integration

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The rules in standard calculus are mostly applicable to path integration. However, there are also rules unique to path integral calculus. In performing explicit path integration, the subtle difference in rules sometimes results in sig­ nificant errors. To attack a non-quadratic path integral, there are a variety of tools which have been developed for path integration. The purpose of this chap­ ter is to provide an overview of the rules of thumb in path integral calculation and various techniques available for explicit path integration.

2.1

Rules of Thumb in Path Integral Calculus

Once the path integral is written in the product form (1.2), most of the rules in standard calculus apply. In performing path integration explicitly, however, we have to recognize certain basic premises and restrictions made in defining the product form, and understand unique properties of the integrand denned only for an infinitesimal period of time. In what follows, we shall gather all practical rules of thumb for path integration. To define the path integral in the product form (1.2), we have partitioned the finite time interval r = t" — t' into N subintervals in such a way that t — to ^ ^1 5: ^2 ^ * ' * ^ tN-1 ^ ^AT

and for all j

==:

t

(j = 1 , 2 , . . . , N) , Tj = tj — tj_i —> 0

as

N —► oo.

For the short time action (1.5) or (1.10), therefore, we are only concerned with terms of zeroth order and first order in Tj. It has been our premise that the

9 contributions from all other higher order terms are unimportant and may be ignored. Suppose the short time action (1.4) is explicitly given in the form,

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Sj = ^-Axj

• M • Ax,- -

Tj

where M is a diagonal mass matrix and M = Ml

V(Xi),

(2.1)

in a Euclidean space. Then

Feynman's path integral in the product form is a collection of the integrals of Fresnel type which are generally oscillatory. 1

A trick suggested by Feynman

was to add a negative imaginary part to Planck's constant. This converts the oscillatory integrals into the Gaussian integrals and makes the path integral convergent. The trick known as Wick's time rotation, t —> —it, often used in field theory in converting a Feynman path integral into a Euclidean path integral, results in a similar effect on path integration. However, such a universal change of a parameter does not give a desirable result in a space with an indefinite metric. A more generally applicable trick is to assume that each element of the diagonal mass matrix M has a positive imaginary part. Under this assumption, the path integral can be convergent independent of the metric of the space. After integration, we may arbitrarily let the imaginary parts vanish. Which trick should be chosen for the convergence is not a real issue here. What is important is to use some reasonable trick to make the path integral convergent. For the usual non-relativistic quantum mechanical setup, any of the tricks mentioned above is acceptable. In order to cover the path integral on the manifold of the noncompact group 517(1,1), we employ the imaginary mass trick as our choice. This trick has an added advantage in making the asymptotic recombination technique workable as will be discussed in Section 2.5. From the Gaussian integral formula follows the asymptotic relation valid for e very small, f exp[-(a/e):r 2 + a'x2 + fix4 + fi'x4 + jx6} x2n dx ~ rc = j exp[-(a/e)x2 + a'eAn + fix4 + fi'e2Bn + Q(e 3 )] x2n dx,

J c

with ^ee, e.g., J. Rzewuski (1969) p.236.

(2.2)

10 An = (2n + l ) / ( 2 a ) ,

Bn = (2n + l)(2n + 3 ) / ( 2 a ) 2 .

In the above, c is a real constant, and a (Re a > 0), a', /?, /?', and 7 are constants. The asymptotic formula (2.2) is a simplified version of that obtained by MacLaughlin and Schulman (1968), but it is a version slightly modified for Path Integrals and Coherent States of SU(2) and SU(1,1) Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 12/31/14. For personal use only.

practical convenience, particularly for angular integrations for which c = 7r or 2TT.

Furthermore, when we employ (2.2) as an equivalence relation in path inte­ gration, we must keep in mind, even though we do not have to be explicit, that for any coordinate variable g, ///(»,»-i)x(t),

(2.17)

into a new one, (x" | e-""'* | x') = jexp

\± J'" L(x,i]

a) da Vx(a),

(2.18)

with £(x,x;s) = X(x,x/t;

t(a))t

where r = t" - *', a = a" - a' = a(t") - s(t'), and ° = d/da.

(2.19)

The original

and new propagators are identical objects expressed in terms of two different time parameters. The global time transformation technique has not been much help in solving nonquadratic path integrals, but has been useful in generating a new propagator from an old one (Cai, Inomata and Wang 1982; Junker and

17 Inomata 1985). The time-dependent conformal transformations considered by Alfaro, Fubini, and Furlan (1976) and by Jackiw (1980) is an example of a global time transformation combined with a coordinate transformation.

Since their

benefit is rather limited in path integration, we shall focus our attention only to

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the non-integrable case in the present volume. If (2.16) is not integrable, it is a position- and time-dependent scaling of an infinitesimal time interval. The value of s may be integrated along a chosen path, but it varies depending on the choice of the path. Since all possible paths must be taken into account in path integration, the path-dependent value of s cannot be taken as a global "time" parameter for the propagator. It is futile to apply a local time transformation by expecting some physically meaningful timedependent propagator. In fact, the local time transformation technique has been successful only when it is applied for evaluating the time-independent energy Green function. The equivalence proposition prepared in Chapter 1 provides a ground for the application of a local time transformation. The promotor (1.7) can be put in the infinite product form,

P ( x " , x'; r ) = jUm / ft to | exp[-(i/h)Tj(H

- E)\ | *,•_,) " j f dxj.

(2.20)

Similarly, the modified promotor (1.12) may be expressed in an infinite product form. To do this, we divide the new interval a = r/f'g"

into iV subintervals Gj

(j = 1,2,3,..., N) in such a way that CFJ > 0 for all j and Cj tends to zero as N goes to infinity, and write (1.12) as N-l

P ( x " , x V ) = j h n / n > i | e x p [ - ( » 7 » t o / ( x ) ( J J - E)g(x)} | x ^ ) -

*°°

J

II i=i

j=i

(2.21) with a = ]CjLi ; r) P,(x" • x')

(2.55)

25 where the radial path integral, N

N-l

Kt(r", r'; r ) = lim^ j J[ Ke{rh ~*°°

r>_i;

T,-) J[ r) drh

i=i

(2.56)

i=i

still remains to be evaluated with the short time propagators (2.52), contingent Path Integrals and Coherent States of SU(2) and SU(1,1) Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 12/31/14. For personal use only.

on specification of the potential. 2.4.2

H a r m o n i c Oscillator in E3

A harmonic oscillator in E3 can be solved in terms of cartesian variables because the potential is separable. However, if there is an additional central potential, the separation in cartesian coordinates is not possible. Here, we wish to evaluate Feynman's path integral for the three-dimensional isotropic harmonic oscillator in polar coordinates. Since any spherically symmetric potential enters only to the radial propagator in (2.55), we shall calculate the radial path integral (2.56) for the harmonic oscillator potential, V{r) = -Mu2r2.

(2.57)

The short time radial propagator for this system is

x exp

[£(>-W)«+*.>]M&«-')- («•>

In the above, we have adopted the local average value of the potential, V- = j M W 2 ( r , ? + '-i-i)-

(2.59)

Since the potential term in the exponential contains the short time interval Tj = tj — tj-i linearly, the local potential Vj chosen above is equivalent in path integration (according to the rules of thumb) to the end point approximation V(rj), or the mid-point approximation V(fj)

.

Now, we introduce a new parameter

0.

(2.60)

26

Since we have cos(/?j = cos[arcsin(u>Tj)] = 1 — -(^2rf + O(r^),

(2.61)

the short time propagator (2.60) may be written in the form

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rs

x

(

. _1/2

Mu in

x exp

iMw 2%

cot

(Mu

^i(ri+rj-i)

>

(2.62)

At this point, we define 0 < 77, 77' < 00, 0 <

—1 the following function, vv{r), 77'; ip) = —z csc (f exp[i(?7 + 77') cot 0 and Re v > —1, then we can easily verify the following simple composition formula, J

[°° v„0A m v"K0?, */; v>) ^ = ^(*A i\ - tz {* " i)] —

JA[(a + *>]

(2.80)

where A = J(a + b)v2/a — 6/(4a). The asymptotic recombination techniques freely exploit these replacement rules to convert a modified Bessel function into an exponential function and also reversely an exponential function into a modified Bessel function within a path integral. There are some skeptics about the validity of the Edwards-Gulyaev formula in a path integral. However, no serious counter examples have been proposed so far. 2.5.1

Introduction of an Inverse Square Potential

The asymptotic recombination technique works well in introducing an inverse square potential into the radial path integral (2.56). Suppose the following short time radial propagator is given, v t

\

M

iriTjrj

| ^ ) - , > - " < - , *

28

M f )• ' ^-l^' Mujy/r'r" iMw n (r + ihsm.(u>T) exp "** L 2%

dt)Vr

r"2)zot(uT)

r u

t Mur'r" \ \ifism.(u)r))

(2.92)

where Re v > 0 and v = V v* Finally we wish to point out that the Gaussian path integrals on the left hand side of (2.91) and (2.92) are not directly calculable. Explicit path integration can be performed only by converting them into the Besselian path integrals with the help of the Edwards-Gulyaev formula. Remember that the results on the right hand side have been obtained by using the recurrence property of the modified Bessel function.

2.6

Dimensional Extension Techniques

The dimensional extension techniques may also be used to simplify some nontrivial path integrals. The main idea of the techniques is to convert a complicated

33 lower dimensional path integral into a simpler higher dimensional p a t h integral. The conventional way to increase the dimensionality of a path integral is to make use of the Gaussian integral formula, which adds extra cartesian variables. An unconventional approach is to add angular variables to a path integral based on

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an asymptotic integral formula of the modified Bessel function. For instance, the one-dimensional path integral for the symmetric Poschl-Teller oscillator may be mapped by the latter into the path integral for a free particle moving on a unit sphere S2. 2.6.1

Conventional Dimensional Extension

The dimensional extension can be performed rather trivially in cartesian co­ ordinates. From the Gaussian integral formula, M 1 / 2 / ~ exp [-(y - y0)2/a} dy = 1

(2.93)

where Re a > 0 and y0 is a constant real number, we can easily derive the useful identity, roo

[7ra1]1/2[7ra2]1/2 /

exp[-(yx

- y)2/at] exp[-(y

-

y2)2/a2]dy

J—oo

= K a i + a 2 )] 1 / 2 exp [ - ( y i - y 2 ) 2 /(«i + ^)] ,

(2.94)

where Re ai > 0 and Re a2 > 0. By induction, it is also easy to show that (2.94) can be extended to the following multiple integral formula, roo

I

N

II[™;]

M-1 1/2e

~

(A2/j)2/aj

J

-°° i=i

I I dVj = M 1 / 2 exp \-(yN - y0)2/a] ,

(2.95)

i=i

where N i=i

This formula is indeed the one used by Feynman to calculate explicitly the path integral for the free propagator. Integrating both sides of (2.95) over the entire range of y^ with the aid of (2.93), we get an idem factor, r J—oo

I I K - ] 1 / 2 e x p H A ^ f / a , ] dVi = 1, - i

(2.97)

34 which is the basic tool for the conventional dimensional extension in cartesian coordinates. As a simple example, let us examine the one-dimensional path integral for a free particle,

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N

/

N-l

N

[(iM^hrjXAxj)2]

J[exp i=i

J[ [MfaifiTj]1'2

J[

j=i

i=i

dxj.

(2.98) If the left hand side of the idem factor (2.97) is inserted into the right hand side of (2.98) by setting ctj = 2fiTj/iM, then we obtain oo

/

K(x",y";x',y';T)dy"

(2.99)

-OO

where K(x",y";x',y';T)

+

(2.103)

where p £ Z. The present formula, a key to the unconventional dimensional extension, is valid only in a short time propagator as it is subjected to the Edwards-Gulyaev formula. The dimensional extension technique may be best explained by applying it to a specific example. For this purpose, we take the symmetric Poschl-Teller oscillator whose Lagrangian is . where VQ = ft2/Ma2,

1 2 -Mx

■2V0X[X-V

(2.104)

sin ( 2 x / a ) '

A > 1, a is a constant and x £ [0,7ra/2]. The path integral

we calculate is one-dimensional: N

jr(^,*';r) = j B m / n o p [ ^ ]

M jR 2«'ftn

1/2

N-l

Y[dxj 3=1

(2.105)

36 with the short time action, Sj =

,3

^L(Axj)2-2VoTj-r

A(A-l) sm(2xj/a)sm(2xj-i/a)

(2.106)

The path integral (2.105) with (2.106) is certainly not trivial. It cannot directly

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be path-integrated.

The unconventional dimensional extension technique was

indeed devised originally for solving this (Inomata and Kayed 1985a) and the Rosen-Morse problem (Inomata and Kayed 1985b; Junker and Inomata 1986). The first step to simplify the integral is to change the x-variable into an angular variable 0 G [0,7r] by letting 0j = 2xj/a.

(2.107)

Then, the short time action becomes 51

=

^ ! (VA J/ ^ - 2 VUo r i ^ A - - 1 ) 8TJ sin 0J sin0j_i

(2.108)

Next, we use the following effective equality in path integration, i(A^)2 = [l-cos(A^)] + l ( A ^ ) 4 ,

(2.109)

and replace the fourth order term by an equivalent term. Then we express (2.108) in the form, Ma2 . / A ^ x, Sj =-rr 4*3 [l-cos L- — - v(A6j)} —,,j

*2Ma2 , , - » 4 , .

A


J(Q

= [(J

m){J ±m + l)]1'3^1^)

T

J

D (J3)W(C)

(3.49)

= m^y(C).

(3.50)

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The set of the orthonormalized functions { ^ 7 ( 0 } (~J < m < J) is indeed the socalled canonical basis for the representations DJ(g) of SU(2).

Every irreducible

unitary representation of SU(2) is equivalent to one of these representations DJ(g)

with J = 0,1,2,... 7 - 1 3 5

m = 0, ± 1 , ± 2 , . . . ± J - l l l 3 l 5

m

(3.51)

+ /

The relations (3.49) and (3.50) may formally be expressed as J± | Jm) = [(J T rn)(J ± m + l)] 1 / 2 | J m ± 1)

(3.52)

J3 I Jm) = ?7i I J m ) .

(3.53)

The ^-representation in which

(c\j±\c) = {63)

(3.64)

51

[

dJmn{6) ' and A — " - '. Comparing (3.77) and (3.78), we obtain the useful expansion formula, C\J (cos " ) =

£

e ' - C - * ) e"") and and the the orthogonality orthogonality relation relation (3.63), (3.63), we we get get

/

3.1.8

XJ(p.-1ff»)xJ"(ffr1ffc)dM»6) = ^-^Sjj^ig-Jgc).

(3.82)

Fourier E x p a n s i o n s

Any function f(g) square-integrable on SU(2) can be expanded in a Fourier series of t h e functions D^n(g)

as

co

J

/(?)=£

£

J

E ( 2 ^ + l)«L^n(i/)

(3-83)

2J=0 m=-J n=-J

and the Fourier expansion coefficients aJmn are given by

aJmn = Jf(g)DJm:(g)dii(g).

(3.84)

If the group elements are parameterized by means of Euler angles, we have CO

J

J

E

E (2J + 1) a£»e''("*+,*)W)e"{m*+n*)dL((>)s™0deddil>.

(3.86)

In particular, if f(g) does not depend on the Euler angle t/>, that is, if f(g) is a function square-integrable on 5 2 , then CO

m

) - E

J

E

J=0m=-J

( 2 J + 1) «me*'m* i T ( c o e 0)

(3.87)

54

where Pf(cos0) = ( - l ) m ^ / ( J + m)\/(J - m)\ dJm0(6), and the expansion coeffi­

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cients are < = y^T^'v r f /(*» «A)e-'m*Pr(cos 9) sin 9 d9 d. (3.88) 47r(y + m)\ Jo Jo Finally, we consider the case where the function /(#), square-integrable on 5(7(2), is central, that is, /(hgh'1) = f(g) for any h € 5(7(2). As we have seen from (3.67), the character functions XJ(Q) a r e a^so central, and form an orthogonal set satisfying the orthogonality relation (3.81). Therefore, any central function square-integrable on 5(7(2) can be expanded in terms of the character functions: oo

/(