Field theory : a path integral approach

A PATH INTEGRAL APPROACH

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World Scientific Lecture Notes in Physics -Vol. 52

A PATH INTEGRAL APPROACH

Ashok Das University of Rochester

Y£ World Scientific m

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

FIELD THEORY: A PATH INTEGRAL APPROACH Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orby 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., 27 Congress Street, Salem, MA 01970, USA. ISBN 981-02-1396-4 ISBN 981-02-1397-2 (pbk)

Printed in Singapore.

To Lakshmi and Gouri

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Introduction Traditionally, field theory had its main thrust of development in high energy physics. Consequently, the conventional field theory courses are taught with a heavy emphasis on high energy physics. Over the years, however, it has become quite clear that the methods and techniques of field theory are widely applicable in many areas of physics. The canonical quantization methods, which is how conventional field theory courses are taught, do not bring out this feature of field theory. A path integral description of field theory is the appropriate setting for this. It is with this goal in mind, namely, to make graduate students aware of the applicability of the field theoretic methods to various areas, that the Department of Physics and Astronomy at the University of Rochester introduced a new one semester course on field theory in Fall 1991. This course was aimed at second year graduate students who had already taken a one year course on nonrelativistic quantum mechanics but had not necessarily specialized into any area of physics and these lecture notes grew out of this course which I taught. In fact, the lecture notes are identical to what was covered in the class. Even in the published form, I have endeavored to keep as much of the detailed derivations of various results as I could - the idea being that a reader can then concentrate on the logical development of concepts without worrying about the technical details. Most of the vii

concepts were developed within the context of quantum mechanics - which the students were expected to be familiar with - and subsequently these concepts were applied to various branches of physics. In writing these lecture notes, I have added some references at the end of every chapter. They are only intended to be suggestive. There is so much literature that is available in this subject that it would have been impossible to include all of them. The references are not meant to be complete and I apologize to many whose works I have not cited in the references. Since this was developed as a course for general students, the many interesting topics of gauge theories are also not covered in these lectures. It simply would have been impossible to do justice to these topics within a one semester course. There are many who were responsible for these lecture notes. I would like to thank our chairman, Paul Slattery, for asking me to teach and design a syllabus for this course. The students deserve the most credit for keeping all the derivations complete and raising many issues which I, otherwise, would have taken for granted. I am grateful to my students Paulo Bedaque and Wen-Jui Huang as well as to Dr. Zhu Yang for straightening out many little details which were essential in presenting the material in a coherent and consistent way. I would also like to thank Michael Begel for helping out in numerous ways, in particular, in computer-generating all the figures in the book. The support of many colleagues was also vital for the completion of these lecture notes. Judy Mack, as always, has done a superb job as viii

far as the appearance of the book is concerned and I sincerely thank her. Finally, I am grateful to Ammani for being there. Ashok Das, Rochester

IX

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Contents

1

2

3

Introduction

1

1.1

Particles and Fields

1

1.2

Metric and Other Notations

2

1.3

Functionals

3

1.4

Review of Quantum Mechanics

9

1.5

References

13

P a t h Integrals and Quantum Mechanics

15

2.1

Basis states

15

2.2

Operator Ordering

18

2.3

The Classical Limit

29

2.4

Equivalence with Schrodinger Equation

32

2.5

Free Particle

36

2.6

References

42

Harmonic Oscillator

43

3.1

43

Path Integral for the Harmonic Oscillator xi

xii

4

5

6

CONTENTS

3.2

Method of Fourier Transform

47

3.3

Matrix Method

51

3.4

The Classical Action

62

3.5

References

70

Generating Functional

71

4.1

Euclidean Rotation

71

4.2

Time Ordered Correlation Functions

79

4.3

Correlation Functions In Definite States

82

4.4

Vacuum Functional

87

4.5

Anharmonic Oscillator

97

4.6

References

100

P a t h Integrals for Fermions

101

5.1

Fermionic Oscillator

101

5.2

Grassmann Variables

106

5.3

Generating Functional

113

5.4

Feynman Propagator

118

5.5

The Fermion Determinant

126

5.6

References

132

Supersymmetry

133

6.1

Supersymmetric Oscillator

133

6.2

Supersymmetric Quantum Mechanics

141

6.3

Shape Invariance

145

CONTENTS

xiii

6.4

Example

151

6.5

References

153

7

8

9

Semi-Classical M e t h o d s

155

7.1

WKB Approximation

155

7.2

Saddle Point Method

7.3

Semi-Classical Methods in Path Integrals

168

7.4

Double Well Potential

174

7.5

References

185

(

164

P a t h Integral for t h e Double Well

187

8.1

Instantons

187

8.2

Zero Modes

196

8.3

The Instanton Integral

202

8.4

Evaluating the Determinant

207

8.5

Multi-Instanton Contributions

215

8.6

References

219

P a t h Integral for Relativistic Theories

221

9.1

Systems with Many Degrees of Freedom

221

9.2

Relativistic Scalar Field Theory

226

9.3

Feynman Rules

241

9.4

Connected Diagrams

244

9.5

References

248

xiv

CONTENTS

10 Effective A c t i o n

249

10.1 The Classical Field

249

10.2 Effective Action

258

10.3 Loop Expansion

268

10.4 Effective Potential at One Loop

272

10.5 References

279

11 Invariances and Their Consequences

281

11.1 Symmetries of the Action

281

11.2 Noether's Theorem

286

11.2.1 Example

291

11.3 Complex Scalar Field

295

11.4 Ward Identities

300

11.5 Spontaneous Symmetry Breaking

306

11.6 Goldstone Theorem

318

11.7 References

321

12 S y s t e m s at Finite Temperature

323

12.1 Statistical Mechanics

323

12.2 Critical Exponents

331

12.3 Harmonic Oscillator

337

12.4 Fermionic Oscillator

345

12.5 References

349

13 Ising Model 13.1 One Dimensional Ising Model

351 351

CONTENTS

xv

13.2 The Partition Function

358

13.3 Two Dimensional Ising Model

366

13.4 Duality

368

13.5 High and Low Temperature Expansions

374

13.6 Quantum Mechanical Model

382

13.7 Duality in the Quantum System

392

13.8 References

395

Index

396

Chapter 1 Introduction 1.1

Particles and Fields

Classically, there are two kinds of dynamical systems that we encounter. First, there is the motion of a particle or a rigid body (with a finite number of degrees of freedom) which can be described by a finite number of coordinates. And then, there are physical systems where the number of degrees of freedom is nondenumerably (noncountably) infinite. Such systems are described by fields. Familiar examples of classical fields are the electromagnetic fields described by E(x,t)

and B(x,t)

or equivalently by the potentials ((x, t), A(x, t)).

Similarly, the motion of a one-dimensional string is also described by a field (f>(x,t), namely, the displacement field. Thus, while the coor1

2

CHAPTER 1. INTRODUCTION

dinates of a particle depend only on time, fields depend continuously on some space variables as well. Therefore, a theory described by fields is usually known as a D + l dimensional field theory where D represents the number of spatial dimensions on which the field variables depend. For example, a theory describing the displacements of the one-dimensional string would constitute a 1+1 dimensional field theory whereas the more familiar Maxwell's equations (in four dimensions) can be regarded as a 3+1 dimensional field theory. In this language, then, it is clear that a theory describing the motion of a particle can be regarded as a special case, namely, we can think of such a theory as a 0+1 dimensional field theory.

1.2

Metric and Other Notations In these lectures, we will discuss both nonrelativistic as

well as relativistic theories. For the relativistic case, we will use the Bjorken-Drell convention. Namely, the contravariant coordinates are assumed to be aj" = (t,s)

^ = 0,1,2,3

(1.1)

while the covariant coordinates have the form x^ = •qlll/xu = (t, -x)

(1.2)

3

1.3. FUNCTIONALS

Here we have assumed the speed of light to be unity ( c = l ) .

The

covariant metric can, therefore, be obtained to be diagonal with the signatures ^

= (+,-,-,-)

(1.3)

The inverse or the contravariant metric clearly also has the same form, namely, iT = (+,-,-,-)

(1.4)

The invariant length is given by Jb

—

JU

*Lr #j - —

ft

JU ul**-'l/

—

ILLIS

—

V

/

The gradients are similarly obtained from Eqs. (1.1) and (1.2) to be

s

> = -L=

<">

so that the D'Alembertian takes the form -2

n = tf^ = iT0,A = j ^ - V"

1.3

(1-8)

Functionals In any case, it is evident that in dealing with dynamical

systems, we are dealing with functions of continuous variables. In

4

CHAPTER 1. INTRODUCTION

fact, most of the times, we are really dealing with functions of functions which are otherwise known as functionals. If we are considering the motion of a particle in one dimension in a potential, then the Lagrangian is given by L(x,x)

= -mx2 -V(x)

(1.9)

Li

where x(t) and x(t) denote the coordinate and the velocity of the particle and the simplest functional we can think of is the action functional defined as S[x] = ftfdtL(x,x)

(1.10)

Note that unlike a function whose value depends on a particular point in the coordinate space, the value of the action depends on the entire trajectory along which the integration is carried out. Thus, a functional has the generic form

F[f] = jdxF(f(x))

(1.11)

where, for example, we may have F(f(x))

= (f(x)r

Sometimes, one loosely also says that F(f[x))

(1-12) is a functional. The

notion of a derivative can be extended to the case of functionals in a natural way through the notion of generalized functions. Thus, one

5

1.3. FUNCTIONALS

defines the functional derivative or the Gateaux derivative from the linear functional

Equivalently, from the working point of view, this simply corresponds to defining

**•(/(*)) _ lhn nn*)+e&i* - y)) - nm) 6f(y)

<™

^

e

n 14. >

It now follows from Eq. (1.14) that

sf{y)

= S(x-y)

(1.15)

The functional derivative satisfies all the properties of a derivative, namely, it is linear and associative,

8f(Xy

llJi

6

6f(xy

'""

mmif}) llJi

'""

6f(x)

6f(x)

6 = 6JMF2[f] + F1[f] ^ 1Wi

6f(x)

'">

Sf(x)

(1.16)

It also satisfies the chain rule of differentiation. Furthermore, we now see that given a functional F[f], we can Taylor expand it in the form

F[f] = JdxP0(x) + J dxxdx2 + J dxidx2dx3

P2(xux2,

P1(x1,x2)

f(x2)

x3) f(x2)f(xz)

-\

(1.17)

CHAPTER 1. INTRODUCTION

wh ere Po(x) =

P1(x1,x2)

P2(x1,x2,x3)

=

F(f(x))\f(x)=0 SF(f(Xl)) Sf(x2)

(1.18) /(x)=0

1 S2F(f{Xl)) 2\Sf(x2)6f(x3) f(x)=0

and so on. As simple examples, let us calculate a few particular functional derivatives. i) Let

F[f] = jdy F(f(y)) = J dy (f(y))n

(1.19)

where n denotes a positive integer. Then, SF(f(y)) Sf(x)

lim

F(f(y)+e8(y-x))-F(f(y))

0

Um

lim

(/(y) + eg(y ~ x)f (/(»))" + ne(f(y)r^S(y

(f(y)r - x) + 0(e 2 ) - (/(„))»


=

n(f(y)y-l6(y-x)

(1.20)

7

1.3. FUNCTIONALS

Therefore, we obtain

Jay

Sf(x)

6 f{z) jdyn{f{y))n-18{y-x)

=

= nM*))-1

(1.21)

ii) Let us next consider the one-dimensional action in Eq. (1.10)

S[x}= /*'df

L(x(t'),x(t'))

(1.22)

with L(x(t),x(t))

l

=

-m{x{t)f-V{x{t))

= T(x(t))-V(x(t))

(1.23)

In a straightforward manner, we obtain SV(x(t')) Sx(t)

=

^ V(x(t') + e6(t' - t)) *->o e

= V'(x^))S(l/-t) where we have defined V'(x(t')) =

dV(x(t')) dx(t')

V(x(t')) (1.24)

8

CHAPTER 1. INTRODUCTION

and

mm) = ^ww+e&v-tv-TizV)) 8x(t)

e-+o

€

= mx(t')—6(t'-t)

(1.25)

It is clear now that SL(x(t'),i(t')) 6x(t)

=

6(T(x(t'))

- V(x(t'))) 6x(t)

= mxW-^Stf

-t)-

Consequently, in this case, we obtain for U
=

ft,

V'(x{t'))6(t' - t) (1.26)
6L(x(t'),x(t')) Sx(t)

k

= ft' dt' (mi(t')i(t' - *) - V'(x(t'))6(t' - t)) =

-mx(t)-V'(x(t)) ddL(x(t),x(t)) dt dx(t)

dL(x(t),x(t)) dx(t)

{

'

The right hand side is, of course, reminiscent of the Euler-Lagrange equation. In fact, we note that 6S[x] 6x(t)

d dL dtdx(t)'t

6L dx(t)

K

>

1.4. REVIEW OF Q UANTUM MECHANICS

9

gives the Euler-Lagrange equation as a functional extremum of the action. This is nothing other than the principle of least action expressed in a compact notation in the language of functionals.

1.4

Review of Q u a n t u m Mechanics In this section, we will describe very briefly the essential

features of quantum mechanics assuming that the readers are familiar with the subject. The conventional approach to quantum mechanics starts with the Hamiltonian formulation of classical mechanics and promotes observables to noncommuting operators. The dynamics, in this case, is given by the time-dependent Schrodinger equation

« 3 f

= HW»

d-29)

where H denotes the Hamiltonian operator of the system. Equivalently, in the one dimensional case, the wave function of a particle satisfies

iR^lfi

=

tf(s)#M)

10

CHAPTER 1. INTRODUCTION

where we have identified V>(*,i) = <*#(<)>

(1-31)

with \x) denoting the coordinate basis states. This, t h e n , defines the time evolution of the system. T h e m a i n purpose behind solving t h e Schrodinger equation Ues in determining the time evolution operator which generates t h e time translation of t h e system. Namely, t h e time evolution operator transforms t h e q u a n t u m mechanical state at an earlier time, t2, to a future time, t 1 } as hKti)> =

tf(*i>*2M*2))

(i.32)

Clearly, for a time independent Hamiltonian, we see from Eq. (1.29) (the Schrodinger equation) t h a t for ti > t2, U(t1,t2) = e-^-^H

(1.33)

More explicitly, we can write U{tu t2) = 8(U - t2)e-i^-^H

(1.34)

It is obvious t h a t the time evolution operator is nothing other t h a n the Greens function for t h e time dependent Schrodinger equation a n d satisfies {ih—

- H)U(tut2)

= ihSih

- t2)

(1.35)

1.4. REVIEW OF QUANTUM MECHANICS

11

Determining this operator is equivalent to finding its matrix elements in a given basis. Thus, for example, in the coordinate basis defined by A » = a;|a:)-

(1.36)

we can write (xi\U(th

t2)\x2) = U(h, xn t2, x2)

(1-37)

If we know the function U(t\,xi; t2, x2) completely, then the time evolution of the wave function can be written as 4>(xi,h) = Jdx2 U(ti,xi;t2,X2)i>{x2,t2)

(1-38)

It is interesting to note that the dependence on the intermediate times drops out in the above equation as can be easily checked. Our discussion has been within the framework of the Schrodinger picture so far where the quantum states \ip(t)) carry time dependence while the operators are time independent. On the other hand, in the Heisenberg picture, where the quantum states are time independent, we can identify using Eq. (1.32)

H)H = m = o))s = m = o)) = e*tH\i>(t))=eitH\iP(t))s

(1.39)

In this picture, the operators carry all the time dependence.

For

example, the coordinate operator in the Heisenberg picture is related

12

CHAPTER 1. INTRODUCTION

to the coordinate operator in the Schrodinger picture through the relation XH{t)

= e*tHXe-VH

(1.40)

The eigenstates of this operator satisfying XH(t)\x,t)H

= x\x,t)H

(1.41)

are then easily seen to be related to the coordinate basis in the Schrodinger picture through \x,t)B

= eitH\x)

(1.42)

It is clear now that for t\ > t2 j(xi,*i|»2,t2>ir = = =

(xx\e-^Re&E\x2) {xx\e-^-^H\x2) {xi\U(t1:t2)\x2)

= U{tux^t2,x2)

(1.43)

Thus, we see that the matrix elements of the time evolution operator are nothing other than the time ordered transition amplitudes between the coordinate basis states in the Heisenberg picture. Finally, there is the interaction picture where both the quantum states as well as the operators carry partial time dependence. Without going into any technical detail, let us simply note

1.5. REFERENCES

13

here that the interaction picture is quite useful in the study of nontrivially interacting theories. In any case, the goal of the study of quantum mechanics in any of these pictures is to construct the matrix elements of the time evolution operator which as we have seen can be identified with transition amphtudes between the coordinate basis states in the Heisenberg picture.

1.5

References

Dirac, P.A.M., "Principles of Quantum Mechanics", Oxford Univ. Press. Schiff, L.I., "Quantum Mechanics", McGraw-Hill Publishing.

Chapter 2 P a t h Integrals and Quantum Mechanics 2.1

Basis states Before going into the derivation of the path integral rep-

resentation for U(tf, xf, ti, Xi) or the transition ampUtude, let us recapitulate some of the basic formulae of quantum mechanics. Consider, for simplicity, a one dimensional quantum mechanical system. The eigenstates of the coordinate operator, as we have seen in Eq. (1.36), satisfy X\x) = x\x)

(2.1)

These eigenstates define an orthonormal basis. Namely, they satisfy

15

16

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

{x\x')

— 8{x — x1)

J dx\x)(x\ = I

(2.2)

Similarly, the eigenstates of the momentum operator satisfying P\p)=p\p)

(2.3)

also define an orthonormal basis. Namely, the momentum eigenstates satisfy

W)

=

8(p-p')

Jdp\p)(p\ = I

(2.4)

The inner product of the coordinate and the momentum basis states gives the matrix elements of the transformation operator between the two basis. In fact, one can readily determine that

(p\x) = T = ^ ~ * P * = W

(2-5)

These are the defining relations for the Fourier transforms. Namely, using the completeness relations of the basis states, the Fourier transforms of functions can be defined as

2.1. BASIS STATES

17

/(*) = (x\f) = /dp (x\p)(p\f)

= ±=jdkeik*f{k) /27T

f(k)

=

(2.6)

y/%f{p)

= -l=Jdxe-ikxf(x) .

(2.7)

2TT

These simply take a function from a given space to its conjugate space or the dual space. Here k = | can be thought of as the wave number in the case of a quantum mechanical particle. (Some other authors may define Fourier transform with alternate normalizations. Here, the definition is symmetrical.) As we have seen in Eq. (1.42), the Heisenberg states are related to the Schrodinger states in a simple way. For the coordinate basis states, for example, we will have \x,t)H

= e*m\x)

It follows now that the coordinate basis states in the Heisenberg pic-

18

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

ture satisfy H{x,t\x',t)H

=

(x\e-*tHe*tH\x')

= (x\x') = 6(x - x')

(2.8)

and

J dx\x,t)H

H(x,t\

= Jdx e*tH\x){x\e = e*tB J dx =

*'

,~M# \x)(x\e^^

eitHIe-itH

= I

(2.9)

It is worth noting here that the orthonormaUty as well as the completeness relations hold for the Heisenberg states only at equal times.

2.2

Operator Ordering In the Hamiltonian formalism, the transition from clas-

sical mechanics to quantum mechanics is achieved by promoting observables to operators which are not necessarily commuting. Consequently, the Hamiltonian of the classical system is supposed to go over to the quantum operator H(x,p)

-•

H(xop:pop)

(2.10)

2.2. OPERATOR ORDERING

19

This, however, does not specify what should be done when products of x and p (which are noncommuting as operators) are involved. For example, classically we know that xp = px Therefore, the order of these terms does not matter in the classical Hamiltonian. Quantum mechanically, however, the order of the operators is quite crucial and a priori it is not clear what such a term ought to correspond to in the quantum theory. This is the operator ordering problem and, unfortunately, there is no well defined principle which specifies the order of operators in the passage from classical to quantum mechanics. There are, however, a few prescriptions which one uses conventionally.

In normal ordering, one orders the prod-

ucts of x's and p's such that the momenta stand to the left of the coordinates. Thus, N.O.

xp

— • px N.O.

px 2

— • px N.O.

2

x p — • px N.O.

xpx

• px

2

In i i \

(^-H)

and so on. However, the prescription that is much more widely used and is much more satisfactory from various other points of view is the

20

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

Weyl ordering. Here one symmetrizes the product of operators in all possible combinations with equal weight. Thus, xp —^ -(xp + px) px

w.o. 1, . A — • -{xp + px) Li

2

W.O. 1 / 2

i

,

2\

x p —> -(a; p + xpx + px ) xpx

—^ - ( x 2 p + xpa;+ px )

(2-12)

and so on. For normal ordering, it is easy to see that for any quantum Hamiltonian,

H(x,p), {x'\HN-°-\x)

= J

dp{x'\p)(p\HN-°-\x)

= f£;e-i«-«)H{x,p)

(2.13)

Here we have used the completeness relations of the momentum basis states given in Eq. (2.4) as well as the denning relations in Eqs. (2.1), (2.3) and (2.5). To understand Weyl ordering, on the other hand, let us note that the expansion of (ax + (5p)N

2.2. OPERATOR ORDERING

21

generates the Weyl ordering of products of the form xnpm naturally if we treat x and p as noncommuting operators. In fact, we can easily show that

(ax + (3p)N=

-^p- aym(xnpm)w-°-

£ n+m=N

(2.14)

fl>.m.

The expansion of the exponential operator e(ax

+ /3p)

would, of course, generate all such powers and by analyzing the matrix elements of this exponential operator, we will learn about the matrix elements of Weyl ordered Hamiltonians. From the fact that the commutator of x and p is a constant, we obtain using the BakerCampbell-Hausdorff formula

.ax.

,ax.

,ax.

.„

ax

,(ax + f3p)

Using this relation, it can now be easily shown that

iha3.

(2.15)

22

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

(x'le^

+ PP^x)

=

ax^ ,cnx, (x'|e ( 2 K ^ 2 )\x)

,OLX. . ax = fdp{x'\e'~2~'ePP\p){p\e<~2''\x)

i ,

IN

,a(x

+x>)

(2.16) n s

Once again, we have used here the completeness properties given in Eq. (2.4) as well as the defining relations in Eqs. (2.1), (2.3) and (2.5). It follows from this that for a Weyl ordered quantum Hamiltonian, we will have

W-M-Jm*'*1"**

X

~¥'p)

(2 17)

'

As we see, the matrix elements of the Weyl ordered Hamiltonian leads to what is known as the mid-point prescription and this is what we will use in all of our discussions. We are now ready to calculate the transition amplitude. Let us recall that in the Heisenberg picture, for tf > t{, we have U(tf, xfi tu Xi) =

H(xf,

tf\xu

ti)H

Let us divide the time interval between the initial and the final time into N equal segments of infinitesimal length e. Namely, let

2.2. OPERATOR ORDERING

23

Thus, for simplicity, we discretize the time interval and in the end, we are interested in taking the continuum limit e —> 0 and N —* oo such that Eq. (2.18) holds true. We can now label the intermediate times as, say, tn = U + ne

n = l,2---,(N-l)

(2.19)

Introducing complete sets of coordinate basis states for every intermediate time point (see Eq. (2.9)), we obtain U(tf,Xf,ti,Xi) =

lim

e—>0 •>

=

H(xf,tf\xi,ti)H [dx1---dxN_1H{xf,tf\xN-i,tN_1)n:

N—>co

H{xN-l,tN-l\xN-2,tN-2)H

' ' 'H (xi,t\\Xi,

U) H

(2.20)

In writing this, we have clearly assumed an inherent time ordering from left to right. Let us also note here that while there are N inner products in the above expression, there are only (N — 1) intermediate points of integration. Furthermore, we note that any intermediate inner product in Eq. (2.20) has the form

24

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

I

ir(a:Tl,tn|a:n_1,in_i)ff

_ ~~ /

=

(x„|e

=

(^n|e

=

(xn\e

r dp^ 2TT^

i

ft

ft

eft

|z n -i)

ft

l^n-l)

|xn_!)

(2.21)

^Pn{Xn-xn-1)--eH(

,p n )

e

Here we have used the mid-point prescription of Eq. (2.17) corresponding to Weyl ordering. Substituting this form of the inner product into the transition amplitude, we obtain U(tf,xf,ti,Xi) =

lim / dx\ • • • CLXN-I^T-Z • • • e->0 J 27Tft

N—too

dpN

* E (Pn(«n - * n - l ) ~ e f f ( "

J^-enn=i

" 2

1

,pn))

(2.22)

In writing this, we have identified XQ = Xi

xjf = Xf

(2.23)

25

2.2. OPERATOR ORDERING

This is the crudest form of Feynman's path integral and is defined in the phase space of the system. It is worth emphasizing here that the number of intermediate coordinate integrations differs from the number of momentum integrations and has profound consequences in the study of the symmetry properties of the transition amplitudes. Note that in the continuum Hmit, namely, for e —»• 0, we can write the phase factor of Eq. (2.22) as

l i m r E (Pn(xn ~ «n-i) - eH(

"~ ,p n ))

iV—too

=

u

m-e£(p„( *->° n „ = i

)-H( e

,pn)) I

N—>oo

= ^1' dt(px - H(x,p)) = l- f' dtL

(2.24)

Namely, it is proportional to the action in the mixed variables. To obtain the more familiar form of the path integral involving the Lagrangian in the configuration space, let us specialize to the class of Hamiltonians which are quadratic in the momentum variables. Namely, let us choose H(x,p) = ^-

+ V(x)

(2.25)

26

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

In such a case, we have from Eq. (2.22)

U(tf,xf,ti,Xi) lim / dxi • • -dxpi-i €->0 J

e

h S(Pn(

dpi 2irh

dp N 2irh

}

- 2^ -

e

Vi

)}

2

(2.26)

The momentum integrals are Gaussian and, therefore, can be done readily. We note that it,P2n

Pn(xn-Xn-i),

J 2nh ie {Pn

=

2

2mpn(xn - x n -i) A

f^e-2^n

}

;

J 2nk =

d - ^2mh I t l(Pn f^!LPn e

J

"

) - (

"

)J

2TT" 11716 ,Xn

1

Xn—j

( 2 ^ ) 1 ^

2irfi

-2

)

ie Z7Y16 ,Xn

m

i

e

,I

=W

- /""Tl

2R

Xn^.\

.2

"Tl-l \^

£

(2 27)

-

Substituting this back into the transition amplitude in Eq. (2.26), we obtain

27

2.2. OPERATOR ORDERING

U(tf,xfitilxi)

= 5m(2^)T 26 —^ ,777- ,Xn n

Idxi---dxN^e

v

= AJVxehk

= AJVxe

"=i 2

S[x]

Xn—\ . 2

z

e

v

T

^ / «^n "T «^n—1

)2 - ^ (

:

L

))

" (2.28)

where A is a constant independent of the dynamics of the system and S[x] is the action for the system given in Eq. (1.10). This is Feynman's path integral for the transition amplitude in quantum mechanics.

To understand the meaning of this, let us try to understand the meaning of the path integral measure Vx. In this integration, the end points are held fixed and only the intermediate points are integrated over the entire space. Any spatial configuration of the

28

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

intermediate points, of course, gives rise to a trajectory between the initial and the final points. Thus, integrating over all such configurations (that is precisely what the integrations over the intermediate points are supposed to do) is equivalent to summing over all the paths connecting the initial and the final points. Therefore, Feynman's path integral simply says that the transition amplitude between an initial and a final state is the sum over all paths, connecting the two points, of the weight factor es^W. We know from the study of quantum mechanics that if a process can take place in several distinct ways, then the transition amplitude is the sum of the individual amplitudes corresponding to every possible way the process can happen. The sum over the paths is, therefore, quite expected. However, it is the weight factor es5txl that is quite crucial and unexpected. Classically, we know that it is the classical action that determines the classical dynamics. Quantum mechanically, what we see, however, is that all the paths contribute to the transition amplitude. It is also worth pointing out here that even though we derived the path integral representation for the transition amplitude for a special class of Hamiltonians, the expression holds in general. For Hamiltonians which are not quadratic in the momenta, one should simply be careful in defining the path integral measure Vx.

29

2.3. THE CLASSICAL LIMIT

2.3

The Classical Limit As we have seen in Eq. (2.28), the transition amplitude

can be written as a sum over paths and for the case of a one dimensional Hamiltonian which is quadratic in the momentum, it is represented as U{tf, xf; U,Xi) = AJ Vx e*s[x] = Um AN fdxx • • • dxN^e c-tO

(2.29) ^ n=i 2

e

2

J

where AN =

\2She) 2

Even though one can be more quantitative in the discussion of the behavior of the transition amplitude, let us try to be qualitative in the following. We note that for paths where %n ^

-En— 1

the first term in the exponential would be quite large, particularly since e is innnitesimally small. Therefore such paths will lead to a very large phase and consequently, the weight factor can easily be positive or negative. In other words, for every such xn, there would be a nearby xn differing only slightly which would have a cancelling effect. Thus, in the path integral, all such contributions will average out to zero.

30

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

Let us, therefore, concentrate only on paths connecting the initial and the final points that differ from one another only slightly. For simplicity, we only look at continuous paths which are differentiable. (A more careful analysis shows that the paths which contribute nontrivially are the continuous paths which are not necessarily differentiable. But for simplicity of argument, we will ignore this technical point.) The question that we would like to understand is how among all the paths which can contribute to the transition amplitude, it is only the classical path that is singled out in the classical limit, namely, when h — • 0. We note here that the weight factor in the path integral, namely, e * 5 ^ , is a phase multipHed by a large quantity when h — • 0. Mathematically, therefore, it is clear that the dominant contribution to the path integral would arise from paths near the one which extremizes the phase factor. In other words, only the trajectories close to the ones satisfying

SS[x] Sx(t)

(2.30)

2.3. THE CLASSICAL LIMIT

31

would contribute significantly to the transition amplitude in the classical limit. But, from the principle of least action, we know that these are precisely the trajectories which a classical particle would follow, namely, the classical trajectories. Once again, we can see this more intuitively in the following way. Suppose, we are considering a path, say # 3 , which is quite far away from the classical trajectory. Then, because h is small, the phase along this trajectory will be quite large. For every such path, there will be a nearby path, infinitesimally close, say # 2 , where the action would differ by a small amount, but since it is multiplied by a large constant would produce a large phase. All such paths, clearly, will average out to zero in the sum. Near the classical trajectory, however, the action is stationary. Consequently, if we choose a path infinitesimally close to the classical path, the action will not change. Therefore, all such paths will add up coherently and give the dominant contribution as h — • 0. It is in this way that the classical trajectory is singled out in the classical limit, not because it contributes the most, but rather because there are paths infinitesimally close to it which add coherently. One can, of course, make various estimates as to how far away a path can be from the classical trajectory before its contribution becomes unimportant. But let us not go into these details here.

32

2.4

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

Equivalence with Schrodinger Equation At this point one may wonder about the Schrodinger

equation in the path integral formalism.

Namely, it is not clear

how we can recover the time dependent Schrodinger equation (see Eq. (1.30)) from the path integral representation of the transition amplitude. Let us recall that the Schrodinger equation is a differential equation. Therefore, it determines infinitesimal changes in the wave function. Consequently, to derive the Schrodinger equation, we merely have to examine the infinitesimal form of the transition amplitude or the path integral. From the explicit form of the transition amplitude in Eq. (2.29), we obtain for infinitesimal e U(tf = €,xf;ti

— 0,Xi)

= (JO-)* eh{r2{—T-) ~

V {

- ^ -

) )

(2.31)

We also know from Eq. (1.38) that the transition amplitude is the propagator which gives the propagation of the wave function in the following way, OO

/

dx' U(e,x;0,x'U(x',0)

(2.32)

-OO

Therefore, substituting the form of the transition amplitude namely, Eq. (2.31) into Eq. (2.32), we obtain

2.4. EQUIVALENCE WITH SCHRODINGER EQUATION

33

(2.33) Let us next change variables to ri = x'-x

(2.34)

so that we can write 2771

26

^ ' 6 ) = (2^) ^ /Idr? ^ ^ " * ^

7?

+

^ W + »/, 0) (2.35)

It is obvious that because e is infinitesimal, if 77 is large, then the first term in the exponent would lead to rapid oscillations and all such contributions will average out to zero. The dominant contribution will, therefore, come from the region of integration

0< M<(—J

(2.36)

where the change in the first exponent is of the order of unity. Thus, we can Taylor expand the integrand and since we are interested in the infinitesimal behavior, we can keep terms consistently up to order e. Therefore, we obtain

34

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

i T n

l

2

(V(z, 0) + ^ ' ( x , 0) + | V ( * , 0) + ^(T?3)) i

^

2

= ® t * > * ' «..0)-^(«tf(.,0) +ni,'(x, o) + ^ v > , o) + 0(n\
/

im drj T) e2%e

i

2

\

m J

2

-oo

im 7^~V / drjr)2e2he '-oo

ihe {2irihe\ ? = — m \ m J

(2.38

Note t h a t these integrals contain oscillatory integrands and t h e simplest way of evaluating t h e m is t h r o u g h a regularization, namely,

35

2.4. EQUIVALENCE WITH SCHRODINGER EQUATION

im

im

~z

oo

/ -oo

=

drj e2ne

hm /

dn e 2%e

£-+0+ S_+n+ J—oo /-oo

(

\ IT

hm

\{8-2heh 2 2irihe\ (2mhe\

_ \

m

(2.39)

)

a n d so on. Substituting these back into Eq. (2.37), we obtain tp(x,e)

=

m

\5

-^-J

2-Kihe

U(x,0)--V(x)4>(x,0)

ihe (2ivi%€\ 2 +T^ J 2m \ m =

V (*, 0) + ~ r 2m

or,V'(a;,e) — ip (x,0)

i>"(x,0) + O{e2)

(x, 0) - ^V h

h2 d2

ie

M 2m9^

(x) i, (x, 0) + O (e2) +V

^)^x'0)

+

O

^ (2.40)

In t h e limit e —• 0, therefore, we obtain t h e time dependent Schrodinger equation (Eq. (1.30)) dip(x,t)

h2

d2

2m dx2

+

V(x)\iP(x,t)

36

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

T h e p a t h integral representation, therefore, contains t h e Schrodinger equation a n d is equivalent t o it.

2.5

Free Particle We recognize that the path integral is a functional inte-

gral. Namely, the integrand which is the phase factor is a functional of the trajectory between the initial and the final points. Since we do not have a feeling for such quantities, let us evaluate some of these integrals associated with simple systems. The free particle is probably the simplest of quantum mechanical systems. For a free particle in one dimension, the Lagrangian has the form

L = Knx2

(2.41)

Therefore, from our definition of t h e transition amplitude in Eq. (2.28) or (2.29), we obtain U(tf,xf;U,Xi) fcc

(

m

\ 2i /• — / dxi • • • dxN^e 2irih€/ J

ira = Hm —) e^o \2mheJ N->oo

lhe

dXl---dxN^e n=\ J

_—^

fit/

I ™-"[\

^n—1

p. 2—i zo a "=i **

,2 (2.42) '

v

37

2.5. FREE PARTICLE

Denning / ra \ 2

» = ^

I *.

(2.43)

we have m

\ 2 (2he\

tffr, «,*,*,) = Km (^Y

2

(~)

N—*oo N

, iH{Vny dyi • • • dyN^e *=i

2/n-l) (2.44)

This is a Gaussian integral which can be evaluated in many different ways. However, the simplest method probably is to work out a few lower order ones and derive a pattern. We note that / dyx e*[(j/i ~ Vo)2 + (2/2 ~ 2/i)2]

= y d y i e i [ 2 ( y i -^ ) 2 + ^2-yo)2] |)%^

( y 2

-y°

) 2

If we had two intermediate integrations, then we will have

(2.45)

38

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

e z '[(^ - W>)2 + fa - yi) 2 + (?/3 - 2/2)2]

J dyidy2

i7r

\ * /" j 72 e^ 2 o ^ 2 ~ y°) 2 + (^3 - 2/2)2]

-T/*

w , . [2^ ( y 2 - ^3 ^ ) 2 +3^(y3-yo)2] Y) /d%c o (2/3 - y o ) 2

^TT\2 /2i7ry

'(«r) 2 V ^ ( y 3 - y o ) 2 1

(2.46)

eo

A p a t t e r n is now obvious a n d using this we can write U(tf,xf,ti,Xi)

lim (

m

f

) I—)**

m

\ 2 /2irihe\

JV-1\ \

1

l

I.. .. \2 (yjv - 2/0)

(^

iV->oo

lim +0 \2ivihe)

K

,«

\

.*.

m

-iv-i 2

)

„

1 y/N

^ ^ e 27iiVe

/

(XJV

\2

- x0)

N—>oo 1

=

im

2

2 {Xf Xi) lim -—-—— ~ e2hN-e e^o \2itihNe

N-KX

1 m

2nih (tf - t{)

~\h

im(xf

- Xj)

2(tf-ti)

(2.47)

2.5. FREE PARTICLE

39

Thus, we see that for a free particle, the transition amplitude can be explicitly evaluated. It has the right behavior in the sense that, we see as tf — • £;, U(tf, xf; tu x{) — • 8(xf - x{)

(2.48)

which is nothing other than the orthonormality relation in the Heisenberg picture given in Eq. (2.8). Second, all the potentially dangerous singular terms involving e have disappeared. Furthermore this is exactly what one would obtain by solving the Schrodinger equation. It expresses the well known fact that even a well localized wave packet spreads with time. That is, even the simplest of equations has only dispersive solutions. Let us note here that since S[x] = /

-mx2

dt

the Euler-Lagrange equations give (see Eq. (1.28)) 6

S[x]

..

n

m =0

4r *

(2.49)

This gives as solutions %cl(t) = v = constant Thus, for the classical trajectory, we have

(2.50)

40

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

S[xcl] = I*' dt -mx2cl = -mv\tf

- U)

(2.51)

On the other hand, since v is a constant, we can write Xf -Xi = v(tf - U) or,

v = ^ — ^

(2.52)

Substituting this back into Eq. (2.51), we obtain

^ M I ^ ' - ^ T ^

<253)

-

We recognize, therefore, that we can also write the quantum transition amplitude, in this case, simply as

g(«/,»/i'.,-.)=(

m * 2>ft(t/ t )

2

_ , J «t,M

P.54)

This is a particular characteristic of some quantum systems which can be exactly solved. Namely, for these systems, the transition amplitude can be written in the form U(tf,xf;ti,xi) = Ae*s^

where A is a constant.

(2.55)

2.5. FREE PARTICLE

41

Finally, let us note from the explicit form of the transition ampHtude in Eq. (2.47) that dU dtf

_

U 2(tf - U)

im fxf — x^ 2h \tfU

dU dxf

im fxf — Xi . h \tf — t

d2U dx2f

im U h tf — ti

(im\2 (xf - xA \ h J \tf — ti

2m I U "~ftM~* 2 (tf -ti) 2m

+

m I Xf — X{ j[tf-ti

f.^dU

2

(2 56)

= -w 1% ,•

'

Therefore, it follows that

*§L. _£*£ dtf

2m dx)

(2v .57)

'

which is equivalent to saying that the transition amplitude obtained from Feynman's path integral, indeed, solves the Schrodinger equation for a free particle (compare with Eq. (1.35)).

42

2.6

CHAPTER 2. PATH INTEGRALS AND QUANTUM MECHANICS

References

Feynman, R.P. and A.R. Hibbs, "Quantum Mechanics and Path Integrals", McGraw-Hill publishing. Sakita, B., "Quantum Theory of Many Variable Systems and Fields", World Scientific. Schulman, L.S., "Techniques and Applications of Path Integration", John Wiley publishing.

Chapter 3 Harmonic Oscillator

3.1

Path Integral for the Harmonic Oscillator As a second example of the path integrals, let us consider

the one dimensional harmonic oscillator which we know can be solved exactly. In fact, let us consider the oscillator interacting with an external source described by the Lagrangian L = -mx2 — -mui2x2 + Jx

(3-1)

with the action given by S = jdtL

(3.2)

Here, for example, we can think of the time dependent external source 43

44

CHAPTER

3. HARMONIC

OSCILLATOR

J(t) as an electric field if the oscillator is supposed to carry an electric charge. The well known results for the free harmonic oscillator can be obtained from this system in the limit J(t) — • 0. Furthermore, we know that if the external source were time independent, then the problem can also be solved exactly simply because in this case we can write the Lagrangian of Eq. (3.1) as 1 -2 1 T -mx mti) 2x 2 +i Jx 2 2 l , l J y2 JT2 2 / -mx — 2-moo \ x moo2) + 2muj2 2

1

= -mx 2

JL2

1

2—2 .

moj x -\ 2

J

/,3\

^

2moj2

(o.o) '

v

where we have defined J x = x—

rauj2

(3.4)

In other words, in such a case, the classical equilibrium position of the oscillator is shifted by a constant amount, namely, the system behaves like a spring suspended freely under the effect of gravity. The system described by Eq. (3.1) is, therefore, of considerable interest because we can obtain various known special cases in various limits. The Euler-Lagrange equation for the action in Eq. (3.2) gives the classical trajectory and takes the form

3.1. PATH INTEGRAL FOR THE HARMONIC OSCILLATOR

45

SS[x] = 0 6x(t) or, mxci

+ muj2xci — J = 0

(3.5)

and the general form of the transition amplitude, as we have seen in Eq. (2.28), is given by U(tf,Xfiti,Xi)

(3.6)

= A J Vx eW*

To evaluate this functional integral, let us note that the action is at most quadratic in the dynamical variables x(t). Therefore, if we define x(t) = xcl(t) + V(t)

(3.7)

then, we can Taylor expand the action about the classical path as 6S[x] S[x] = S[xcl + V] = S[xcl] + fdt rj(t) . . . J oxyt)

X=Xci

6x(t1)6x(t2)

(3.8)

We note from Eq. (3.5) that the action is an extremum for the classical trajectory. Therefore, we have SS[x] Sx(t)

= 0

(3.9)

46

CHAPTER 3. HARMONIC OSCILLATOR

Consequently, we can also write Eq. (3.8) as (3.10) X—Xcl

If we evaluate the functional derivatives in Eq. (3.10) for the action in Eq. (3.2), we can also rewrite the action as S[x] = S[xcl] + I [tf dt (mf/2 - mu2r]2)

(3.11)

The variable rj(t) represents the quantum fluctuations around the classical path, namely, it measures the deviation of a trajectory from the classical trajectory. Since the end points of the trajectories are fixed, the fluctuations satisfy the boundary conditions r,(U) = r,(tf) = 0

(3.12)

It is clear that summing over all the paths is equivalent to summing over all possible fluctuations subject to the constraint in Eq. (3.12). Consequently, we can rewrite the transition amplitude as U(tf,xf,U,Xi) * f^

= AjVr\e\ft

KS[XC1} +

— (' dt{mtf

2n^

-rS\xcA t — I = A eh l cts JVr]e2hJti

-

mu2rf)

dt (mri2 — mu2n2) v ' '>

(3.13)

47

3.2. METHOD OF FOURIER TRANSFORM

This is an integral where the exponent is quadratic in the variables and such an integral can be done in several ways. Since the harmonic oscillator is a fundamental system in any branch of physics, we will evaluate this integral in three different ways so as to develop a feeling for the path integrals.

3.2

M e t h o d of Fourier Transform First of all, we note that the integrand in the exponent of

the functional integral does not depend on time explicitly. Therefore, we can redefine the variable of integration as

t —>t-U

(3.14)

in which case, we can write the transition amplitude as

U(tf,xf;ti,Xi)

= Ae^Xd]

jVTje2hJo

V

U

^

^

where we have identified the time interval with T = tf-U

(3.16)

The variable 77(f) satisfies the boundary conditions (see Eq. (3.12)) 77(0) = V(T) = 0

(3.17)

48

CHAPTER 3. HARMONIC OSCILLATOR

Consequently, the value of the fluctuation at any point on the trajectory can be represented as a Fourier series of the form r)(t) = £> n sm (-jr)

n - integer

(3.18)

Substituting this back, we find that fT i, -2 dtr

Jo

x-^ tT i.

fmt\ fm'K\ dtanam

> = £/o

/nnt\ cos

/rmrt\ cos

{T){-Y) [-r) [-l^) <3 19)

= 2 ? ( T ) "«

-

where we have used the orthonormality properties of the cosine functions. Similarly, we also obtain /•T 2 J0 dtV(t)

rT = gyQ

= ?£«£

. (nixt\ . frmrt\ d*a„amsiii^—jan^—j

(3-20)

Z n

Furthermore, we note that integrating over all possible configurations of rj{t) or all possible quantum fluctuations is equivalent to integrating over all possible values of the coefficients of expansion an. We also note that since we have chosen to divide the trajectory into TV intervals, namely, since there are (N — 1) intermediate time points, there can only be (N — 1) independent coefficients, a n , in the Fourier expansion in Eq. (3.18). Thus we can write the transition amplitude also as

3.2. METHOD OF FOURIER TRANSFORM

49

U(tf,xf,ti,Xi) lim A'e*s^ e—»0 e->0 N—»oo

I dax • • • daN_i e' J

imTN-1

= lim A'e*s^

Ucn-'-daN^e

4h

I(mr\2

'iffl-")<

« = 1 VV J >

I

(3.21)

Here we note that any possible factor arising from the Jacobian in the change of variables from r\ to the coefficients, an, has been lumped into A' whose form we will determine shortly. We note here that the transition amplitude, in this case, is a product of a set of decoupled integrals each of which has the form of a Gaussian integral which can be easily evaluated. In fact, the individual integrals have the values (see Eq. (2.39)) imT ((nit\2 / dan e

,\

,

-srU-r)-"^

= 0'((T)2-;

-(^)'(?r(-oV **> Substituting this form of the individual integrals into the expression for the transition amplitude in Eq. (3.21), we obtain

50

CHAPTER 3. HARMONIC OSCILLATOR

U(thxr,U,Xi) = lim A'^i^'^Jl

n=1

jfcSo

(l - (—) ) '

V

U7ry

(3.23)

/

If we now use the identity, ,.

x-1 /

/wT\2\

fen !-(-)

sinwT

,„„,.

= ^jr-

0.24)

we obts obtain U(tf, xf;tl,

Xi)

= Mm A " C H - 1 ( 5 ^ ) ~ 5

(3.25)

N—>oo

We can determine the constant A" by simply noting that when to = 0, the harmonic oscillator reduces to a free particle for which we have already evaluated the transition amplitude. In fact, recalling from Eq. (2.54) that

U,,(t„V,t,^)=(2^^ti))'ei^

(3.26)

and comparing with Eq. (3.25), we obtain

lim A" = ( — ^ - V 6^o N->oo

\2mhTJ

(3.27) v

'

Therefore, we determine the complete form of the transition amphtude for the harmonic oscillator to be

3.3. MATRIX

METHOD

51

l

1 1

^-^-)=(^) m" ^ f

J*e* 5t *' ]

T"

(3-28) v

\27riftsinw27

7

It is quite straightforward to see that this expression reduces to the transition amplitude for the free particle in the limit of UJ —• 0.

3.3

Matrix Method If the evaluation of the path integral by the method of

Fourier transforms appears less satisfactory, then let us evaluate the integral in the conventional manner by discretizing the time interval. Let us parameterize the time on a trajectory as tn — ti + ne

n =

0,l,---,N

Correspondingly, let us define the values of the fluctuations at these points as ViQ = Vn

(3.29)

Then, we can write the transition amplitude in Eq. (3.15) in the explicit form

52

CHAPTER 3. HARMONIC OSCILLATOR

U(tf,xf;ti,Xi) =

i fit j Vr} e2h Ju

A e*s^

v

>

N

=

Mm \27rzn,e; (—fc J e-»o

e^'^Jdin---dvN-i

JV-KX)

ie " (( it SU

(i -Vn-lV [Vn

MLrv

muj

2 (Vn +

Vn-l''2^

(3.30)

In this expression, we are supposed to identify Vo = VN = 0

(3.31)

corresponding t o t h e b o u n d a r y conditions in Eq. (3.12), namely,

v(U) = v(tf) = o To simplify the integral, let us rescale t h e variables as

,„^(£)V

(3.32)

T h e transition amplitude, in this case, will take the form U(tf,xf;ti,Xi)

e—o \2nineJ

N-

E

e n=i

lI

\m

J

I \2

((Vn ~ Vn-lY

2 2/Vn

~ ?U (

i

Vn-1\2\

) )

^

(3.33)

53

3.3. MATRIX METHOD

If we think of the 77„'s (there are (TV - 1) of them) as forming a column matrix, namely,

T) =

V2

(3.34)

\ VN-I

)

then, we can also write the transition amplitude in terms of matrices as JV , „ .

,

K=l

¥

W»*r,u.«)= £s ( ^ ) ( * ) ' ."*/*,<*'»'

(3-35)

N—>oo

Here rjT represents the transpose of the column matrix in Eq. (3.34) and the (N — 1) x (N — 1) matrix B has the form

t

2 -\ -1

B =

0

2-1

0-1 \

'

•

•

0 0

2-1 '

•

'

•

2 10

eW

12 0 12

0

\

10 1

'•

This is a symmetric matrix and, therefore, we can write it as

(3.36)

54

CHAPTER 3. HARMONIC OSCILLATOR

( x y 0 0 y x y 0

B

(3.37)

0 y x y

where we have defined x = 2

1-

y = -

i +

eW eW

(3.38)

The matrix B is clearly Hermitian (both x and y are real) and, therefore, can be diagonalized by a unitary matrix (more precisely by an orthogonal matrix) which we denote by U. In other words,

( bi 0 BT

0

0 b2 0 0 \ \

= UBU]

(3.39)

0 b3 \

\

Therefore, defining ( = UV we obtain

(3.40)

55

3.3. MATRIX METHOD

Jdrje^Bv

JdCeiCTBDC

=

N-l

=n

Vun

n=l

=

(iir)^

(det Byl

(3.41)

Here we have used the familiar fact that the Jacobian for a change of variable by a unitary matrix is unity. Using this result in Eq. (3.35), therefore, we determine the form of the transition amplitude for the harmonic oscillator to be U(tf,xf;thXi)

=

Hm p M

( ^ ^ ( v^ ^ ( dv e t ^ - e ^ M

t^o \2niheJ

e^o \2mheJ =

f

\ m )

v

Mm L • t m , J V e^o V27rtaedetB;

'

'

' S

M

v(3.42) ;

JV—too

It is clear from this analysis that the transition ampMtude can be defined only if the matrix B does not have any vanishing eigenvalue.

56

CHAPTER

3. HARMONIC

OSCILLATOR

We note here that the main quantity to calculate in order to evaluate the transition amplitude is lim e det B

e-»0 JV->oo

Let us note from the special structure of B in Eqs. (3.37) and (3.38), that if we denote the determinant of the sub n x n matrices of B as /„, then it is easy to check that they satisfy the recursion relation In+l = xIn-y2In_x

n = 0,1,2,-••

(3.43)

where we restrict J_! = 0

Jo = 1

(3.44)

This recursion relation can be checked trivially for low orders of the matrix determinants. Substituting the form of x and y, we obtain /

eW\

(

T

In+i = 2[l-—jln or, In+1 - 2In + In-i In+1 - 2J n + J n _! or,

2

e

eW\2

- (l + — j =

T

/-!

e 2 u, 2 / . e2uj2 \ — I /„ + In-i -\ ~j^~In-i I

w2/ , e W T ) = —— 2 In + In-i + —— i n _i

/,«) (3.45)

We are, of course, interested in the continuum limit. In order to do so, let us define a function {tn - U) = (ne) = eln

(3.46)

3.3. MATRIX METHOD

57

In the continuum limit, we can think of this as a continuous function of t. In other words, we can identify t = ne as a continuous variable as e —• 0. We note then, that Urn e det B = Urn e/ JV _ 1 = (j>(tf - U) = (T)

e—»0 JV—»oo

(3.47)

£—»0 JV—•oo

We also note from Eq. (3.46) that, in the continuum limit, 0(0) = lim el0 = Urn e = 0

(3.48)

and similarly,

*°> - £ » e ( ^ ) = Ss (2" ¥ ~x) =J

<3 49)

'

Furthermore, from the recursion relation for the J n 's in Eq. (3.45), we conclude that in the limit e —• 0, the function (t) satisfies the differential equation dt2

= -u2<j)(i)

(3.50)

We recognize this to be the harmonic oscillator equation and the solution subject to the initial conditions (Eqs. (3.48) and (3.49)) is clearly

, . . sin u)t {t) = UJ

It now follows from this that

(3.51)

58

CHAPTER 3. HARMONIC OSCILLATOR

Urn e d e t £ = Urn e/jv-i = {T) = N—too

^

^

(3.52)

N—*oo

Consequently, for t h e harmonic oscillator, we obtain the transition amplitude in Eq. (3.42) to be U(tf,xf;ti,Xi) m

= Mm e^o

s x

ei

i ci]

\2irihedetB

N—»oo VTUJJ

2wih sin UJT

is[xci] e*-

(3.53)

This is, of course, w h a t we h a d already derived in Eq. (3.28) using t h e m e t h o d of Fourier transforms. Let us next describe an alternate way to determine det B which is quite useful in studying some specific problems. Let us recall from Eq. (3.43) t h a t t h e determinant of the nxn

matrices, In, satisfy

t h e recursion relations J„+i = xln -

y2In-i

We note here t h a t we can write these recursion relations also in the simple m a t r i x form as L

n+\

( x

2 W In -yl In-l

(3.54)

3.3. MATRIX METHOD

59

Iterating this (N — 2) times, for n = N — 2, we obtain (

IN-\ IN-2

2

x

-yl

V1

°

(N-2)factors

\

(if-i)

x

-y

X

1

0

1

(3.55)

We can determine the eigenvalues of the fundamental 2 x 2 matrix in Eq. (3.55) in a straightforward manner. From det

x-X

-y2

1

-A

= 0

we obtain x ± \fx2 — Ay2

(3.56)

Furthermore, the 2 x 2 matrix can be trivially diagonalized by a similarity transformation. In fact, if we define (

S =

c\+

dA_ ^

(3.57)

with

s-' =

( cI _±\ c K

V

d

d

)

(3.58)

60

CHAPTER

3. HARMONIC

OSCILLATOR

where c and d are arbitrary parameters, then it is easy to check that x

-y

1

2\

/

=s

0

A+

0

0

A_

(3.59)

Using this in Eq. (3.55), we obtain IN-I

^

(

= S

IN-2

A*" 2 0

\

0

x

A"-

2

(3.60)

1

We recall from Eq. (3.56) that x = A+ + A_ Using this as well as the forms for <S and S

(3.61) 1

in Eqs. (3.57) and (3.58),

we obtain from Eq. (3.60) IK-!

= j

^

-

(A? - A-)

(3.62)

We can easily check now that =

0

h

=

1

h

=

A+ - A _ • w

=

A+ + A_

/-l

1

=

X

(3.63)

3.3. MATRIX METHOD

61

which is consistent with our earlier observations in Eqs. (3.43) and (3.44). Let us next note from Eqs. (3.56) and (3.38) that A+-A_-(x2-4y2)1

4(l-^f) 2 -4(l + ^f) 2 ( - 4 e V ) * = 2ieu> a; + y/x2 — 4y2

A? =

2(1

= XN

JV

JV

e2^2

—) + 2ieu>

(1 + ieu + O (e 2 ))" ~ (1 + tew)* x — y/x2

=

2(1

—• 4 y 2 N

e2w2

JV

) — lieuj

\

N

\ =

(l-ieu>

+ 0 (e2))N ~ (1 - iew)*

(3.64)

Consequently, substituting these relations into Eq. (3.62), we obtain

62

CHAPTER 3. HARMONIC OSCILLATOR

lim eIN^

=

<=->0

N—too

Urn e , e->0

N—too

=

1 A

A+ - A_

)im e-^—((l

(A* - A*)y V + iecj)N-(l-iecu)N)

+

iV-too

JV-.00

= J - (e^ - e - n = ^ ^

(3.65)

v

2zo; ' u> Here T is the time interval between the initial and the final times and this is, of course, what we had obtained earher in Eq. (3.52), namely, that ii r. sinwT hm e det B — JV-»co

In any case, we obtain the transition amplitude for the harmonic oscillator to be

u{t

>'*>^ = (*£L*f

eisM

3.4

(366)

'

T h e Classical Action Once again we see from Eq. (3.28) or (3.66) that the

transition amplitude has a generic form similar to the one found for the case of the free particle. Namely, it is proportional to e ^ 5 ^ .

3.4. THE CLASSICAL ACTION

63

A complete determination of the transition amplitude, in this case, therefore, would require us to evaluate the classical action for the system. This can be done simply in the following way. We recall that the Euler-Lagrange equations for the present system are given by (see Eq. (3.5)) mxci + muj2xci — J = 0 In other words, the classical trajectory is a solution of the equation

The solution, obviously, consists of a homogeneous and an inhomogeneous part and can be written as xd(t) = xH(t) + xi(t)

(3.68)

where the homogeneous solution is of the form xH{t) = Aeiujt

+ Be~iu,t

(3.69)

with A and B arbitrary constants. To determine the inhomogeneous solution, we use the method of Greens function.

Here the Greens

function for Eq. (3.67) is defined by the equation [~

+ ^

G(t - t) = -6(t - t')

(3.70)

It is clear that if we know the Greens function, G{t — £'), then, the inhomogeneous solution can be written as

64

CHAPTER 3. HARMONIC OSCILLATOR

Xl(t)

= - £' dt' G(t - * ' ) —

(3.71)

The Greens function can be easily determined by transforming to the Fourier space. Thus, defining

G(t-t>) = S(t-t')

[^Le-ik(t-t')G(k)

J

v27r

= f^e'iHt-t')

(372)

where G(k) is the Fourier transform of G(t — t'), and substituting these into Eq. (3.70) we obtain (Note from Eq. (2.6) that the Fourier transform in time is defined with an opposite phase.) (™+^)G((-(')

=

-S(t-t')

"•Gw-jzwh>

(3 73)

-

Consequently, the Greens function takes the form

G(t-t')

[^Le-ik(t-t')G{k)

= J

\l/.ir

1

,

P-ik(t

- t')

3.4. THE CLASSICAL

ACTION

65

A quick inspection shows t h a t t h e integrand has poles at k = ±u>. Therefore, we must specify a contour in the complex fc-plane in order to evaluate t h e integral. Normally, in classical mechanics, the Greens functions t h a t are of fundamental interest are t h e retarded and t h e advanced Greens functions. B u t a Greens function t h a t is of fundamental significance in q u a n t u m theories is t h e F e y n m a n Greens function a n d corresponds to choosing a contour as shown below. Im k

Im k

-UJ +

—UJ

X

UJ

Re k

i6 *— UJ

— i8

Re k

Equivalently, it corresponds to defining (see Eq. (3.73))

GF(k)

=

1 1 lim e-»o+ y/^K k2 - UJ2 + ie

=

1 1 1 lim s-+o+ ^/2TT k + UJ — i8 k — UJ + i6

where we have defined

(3.75)

66

CHAPTER 3. HARMONIC OSCILLATOR

In other words, we can think of the Greens function in Eq. (3.75) as the Fourier transform of the function which satisfies the differential equation d2 Mm ( — + u>2 - ie)GF(t - t') = -6(t - *')

(3.76)

We note here, for completeness, that the retarded and the advanced Greens functions, in this language, correspond respectively to choosing the Fourier transforms as 1 GR'A(k) = Urn 6-o+ V^TT (k ± ie)2 - LJ2 with the respective contours Im k

Im k

-CO + l€ x

Re k

X

-u} — ie

UJ

*—

u) + ie Re k

— ie

With such a choice of contour for the Feynman Greens function, enclosing the contour in the lower half plane for t — t' > 0, we obtain

3.4. THE CLASSICAL ACTION

G^Ht-t')

67

-ik(t - t')

= lim — fdk 6-*o+ 2TT J

1 / „ .x = —(-2m) 2TT V

e

(k + ui - iS)(k — u + iS)

-IUJ (t

'

- t')

2u

= — e-M*-**)

(3.77)

On the other h a n d , for t — t' < 0, enclosing the contour in t h e upper half plane, we find -i

H

,

.-ikit - t')

G ( t - f ) = lim-Wdife7

6-^o+ 27T-/ =

1

27T2

e*

2TT

^—

(fc + u; - « 6 ) ( f c - u ; + z<5) w

(*-0

- 2 W

= _L e M*-*')

(3.78)

2iu) T h u s , t h e F e y n m a n Greens function has t h e form p~iu(t

- t')

aF{, -1) = .(< - f)—^-

Jult

- t')

+ «(«• - O ^ - s —

0.79)

68

CHAPTER 3. HARMONIC OSCILLATOR

Using this Greens function in Eq. (3.71), we can now obtain the inhomogeneous solution as

„m

xi(t) = - [ ' dt'GF(t - t')

m

-i ,, „-iui(t — t') ~( dfe-4-.—-j(t')+ — ^ ( / ' dte-™^

. „iu)(t — t') dfe l j(t>))

tf

~ ^JM+jt'

dt'e^i1

~ *') J ( f ) ) (3.80)

Thus, substituting Eqs. (3.69) and (3.80) into Eq. (3.68) we can write the classical trajectory as xcl(t) = xH(t) + xr(t) = Aeiu}t +

Be-iujt

_ _J_(J*di>e-iu(t

~ 0 J(*') + J'' dt'e^V

~ *')/(*')) (3.81)

Imposing the boundary conditions

= Aeiujti

+ Be~iuti

- — L _ f' dt'eiu}& 2imuj hi

~

^Jit')

xci(tf) = xf = Aeiujtf

+ Be~iujtf

— /*' dt'e'^f 2imu Jti

~ 0 m') (3.82) \ J\

J

3.4. THE CLASSICAL ACTION

69

we see that we can solve for A and B in terms of the initial and the final coordinates of the trajectory as A =

-^=[(xfe~i^-xie-iutf)

2i sin ail

-iu>t f tt

I' Msmuj(t'

-U)J(t')\

moj

\(xieiujtf K 2isinwr

B =

-

}

xJ^) '

J"U fI** ' dt' *, sin u{tf-t')

J•(«')]

mu) Substituting these relations into Eq. (3.81), we determine the classical trajectory to be Xci(t) = —

—[xfsinuj(t - U) + Xismvttf

+—*— ftf dt'J(t')(e-iujT

- t)

cosu;(< - t') - cosu(t f + U - t - t'))}

_ _ ! _ (J* dt'J{t')e-iu)^ ~ *') + £' dt'Jity^

~ 0)

(3.83)

We can now derive the classical action from Eqs. (3.2) and (3.1)in a straightforward manner to be

70

CHAPTER 3. HARMONIC OSCILLATOR

S x

[ d] = ir~.

if, [ixl, + x)) costuT -

2x{xf]

& s i n UJ j . X{

[*' dtJU) sinuUf - t) + Xf [*' dtJ(t) smu(t - tA sin uT JU sin u)T JU 1 smu(tf — t)s'maj(t' — ti)J(t') (3.84) muj sin

UJT JU

This, therefore, completes the derivation of the transition amplitude for the harmonic oscillator interacting with a time dependent external source.

3.5

References

Feynman, R.P. and A . R . Hibbs, "Quantum Mechanics and Path Integrals", McGraw-Hill publishing. Kleinert, H., "Path Integrals", World Scientific publishing. Schulman, L.S., "Techniques and Applications of Path Integration" , John Wiley publishing.

Chapter 4 Generating Functional 4.1

Euclidean Rotation We have seen in Eq. (2.39) that the standard Gaussian

integral (where the exponent is quadratic), namely,

[~ dx e™2 = (±)* J-oo \a / generahzes in the case of a n XTCmatrix as (see Eq. (3.41))

provided A is a Hermitian matrix. In fact, we will now see explicitly that this result holds true even when we replace the matrix A by a 71

72

CHAPTER 4. GENERATING FUNCTIONAL

Hermitian operator. In other words, we will see that we can write r

i ftf dt

J VV e Ju

•n(t)0(t)n(t) / w W /W

= N [det 0 ( O P

(4.2)

where 0{t) is a Hermitian operator and N a normalization constant whose explicit form is irrelevant. To establish the identification, let us go back to the harmonic oscillator and note that the quantity of fundamental importance in this case was the integral (see Eq. (3.13))

= /t,.-ii **K*+^

(4.3)

with the boundary conditions r,(U) = V(tf) = 0

(4.4)

The value of this integral was determined earHer (see Eq. (3.25)) to be

4.1. EUCLIDEAN ROTATION

73

where T = tf — t{ represents the total time interval. We evaluated this integral earlier by carefully discretizing the time interval and calculating the determinant of a matrix (see Eqs. (3.30) and (3.35)) whose matrix elements were nothing other than the discrete form of the matrix elements of the operator in the exponent in Eq. (4.3). This would already justify our claim. But let us, in fact, calculate the determinant of the operator in the exponent of Eq. (4.3) explicitly and compare with the result obtained earlier. The first problem that we face in evaluating the functional integral is that the exponential in the integrand is oscillatory and, therefore, we have to define the integral in some manner. One can, of course, use the same trick as we employed in defining ordinary oscillatory Gaussian integrals (see Eq. (2.39)). Namely, let us define i m

fvneihJu

ftf

J* I -2

2 2\

dt

(v-"v)

im

dt

rtf ,

/

. . d2

+ 2

,

= ^Jv,e-mf, «Wl* " -'^

. .

,v

(45)

This provides proper damping to the integrand and, as we will see, leads to the Feynman Greens functions for the theory. In fact, as

74

CHAPTER 4. GENERATING FUNCTIONAL

we have already seen in Eq. (3.76), the inverse of the operator in the exponent in Eq. (4.5) gives the Feynman Greens function which plays the role of the causal propagator in the quantum theory. It is in this sense that one says that the path integral naturally incorporates the causal boundary conditions. There is an alternate but equivalent way of defining the path integral which is quite pleasing and which gives some sense of rigor to all the manipulations involving the path integral. Very simply, it corresponds to analytically continuing all the integrals to imaginary times in the complex £-plane. More explicitly, we let

t —> t' = —IT

T real

(4.6)

Ira t

J

Re t

With this analytic continuation, then, the integral in Eq. (4.3) becomes

4.1. EUCLIDEAN ROTATION

75

(4.7)

Here we have scaled the variables in t h e last step a n d N' represents t h e Jacobian for t h e change of variables. Furthermore, we have to evaluate the integral in Eq. (4.7) subject to the b o u n d a r y conditions

v(n) = v(rf) = o

(4.8)

T h e right h a n d side of E q . (4.7) is now a well defined quantity since the integrand is exponentially d a m p e d . (The analytically continued o p e r a t o r h a s a positive definite spectrum.)

We can now evaluate

this integral a n d at t h e end of our calculations, we are supposed to analytically continue back to real time by letting r - • T1 = it

t

real

(4.9)

From Eq. (4.2) we see t h a t the quantity which we are interested in is d e t ( — J ^ + to2).

Furthermore, this determinant has

to b e evaluated in t h e space of functions which satisfy t h e b o u n d a r y

76

CHAPTER 4. GENERATING FUNCTIONAL

conditions V(Ti) = V(Tf) = 0 Therefore, we are basically interested in solving the eigenvalue equation d2 i-J^

+ W2)lpn = Ki>n

(4.10)

subject to the boundary conditions in Eq. (4.8). The normalized eigenfunctions of Eq. (4.10) are easily obtained to be WK{T — Ti)

2

V'n(r) = , .

r sin^

y-

,

(4.11)

with n a positive integer. The corresponding eigenvalues are

Hi^))2^

(4 i2)

-

Thus, we see that the determinant of the operator in Eq. (4.7) or (4.10) has the form d2

det(-jL+c ) aT

°°

2

=

n A„ n=l

n-K

= B

Sinh

"(r/ - ^

,

2

(4.13)

77

4.1. EUCLIDEAN ROTATION

where we have used a relation similar to the one given in Eq. (3.24) and B is a constant representing the first product whose value can be absorbed intothe normalization of the path integral measure. Analytically continuing this back to real time, we obtain

det(-^W)

-det(-gW) w(tf-ti)

OJT

y

'

Here, as before, we have identified T = tf — U with the total time interval. This is, of course, related to the value of the path integral which we had obtained earlier in Eq. (3.28) through a careful evaluation. Therefore, we conclude that

(4i5)

^•(^r^K^r -

Later, we will generalize this result to field theories or systems with an infinite number of degrees of freedom. Let us next discuss in some detail the analytic continuation to the imaginary time. Consider a Minkowski space with coordinates x» = (t, x)

78

CHAPTER 4. GENERATING FUNCTIONAL

where we leave the dimensionality of space-time arbitrary. Then, under the analytic continuation, x^ = (t,x) x2 = (t2-x2)

—• (—ir,x) -

-(T2 + X2)

(4.16)

Therefore, we note that r is nothing other than a Euclidean time. The analytic continuation, consequently, corresponds to a rotation to EucMdean space. (Although, in one dimension, it does not make sense to talk about a Euclidean space, in higher dimensional field theories it is quite meaningful.) The sense of the rotation is completely fixed by the singularity structure of the theory. Let us note that an analytic continuation is meaningful only if no singularity is crossed in the process. We know from our study of the Greens function in the last chapter (see Eq. (3.75)) that in the complex energy plane, the singularities occur at

—u> + id X

\ u) — iS

It is clear, therefore, that an analytic continuation from Re k° to Im k°

79

4.2. TIME ORDERED CORRELATION FUNCTIONS

is meaningful only if the rotation is anticlockwise, namely, only if we let k° - • jfe'° = iK

K real

(4.17)

Since we can represent dt it follows now that in the complex £-plane, the consistent rotation will be t -> t' = -ir

4.2

T real

(4.18)

Time Ordered Correlation Functions Let us recapitulate quickly what we have done so far. We

have obtained the transition amplitude in the form of a path integral as H(xf,tf\xi,ti)H

= N JVxeh

[Xl

(4.19)

Let us next consider a product of operators of the form

xH{h)xH{t2) and evaluate the matrix element H{x},tf\XH{tl)XH{t2)\xhti)H

tf>t1>t2>

U

80

CHAPTER 4. GENERATING FUNCTIONAL

Since t\ > t2, then we can insert complete sets of coordinate basis states and write HiXfitflXBit^Xgfa^Xi, = J

ti)H

dxidx2H(xf,tf\XH(ti)\xi,t1}H H{xuti\XH{h)\x2,

= J dxxdx2 x1x2H(xf,

t2)H H(x2, t2\xi, ti)H

tf\x!, h)H E{xi,ti\x2,

t2)H

(4.20)

H{x2,t2\xu

ti)H

Here we have used the relation in Eq. (1.41). We note that each inner product in the integrand represents a transition amplitude and, therefore, can be written as a path integral. Combining the products, we can write (for t\ > t2)

E{xf,tf\Xs{h)XB{t2)\xi,ti)E

= N fVx

z(ti)x(t 2 ) eh

*

(4.21)

Here we have used the identification xi = x(h)

x2 = x(t2)

Similarly, we note that for t2 > tt, we can write

(4.22)

4.2. TIME ORDERED CORRELATION FUNCTIONS

H{Xf,

=

tf\XH(t2)XH(tl)\Xi, Jdxxdx2

81

ti)H H(xf,tf\XH(t2)\x2,t2)H

H{X2, t2\XH(tl)\xi,

tijH H(xi,tl\Xi,

ti)H

—
= N j Vx x(U)x{t2)

eh

m

(4.23)

In the last step, we have used the fact that factors in the integrand such as x(t\) and x(t2) are classical quantities and, therefore, their product is commutative. Thus, we see from Eqs. (4.21) and (4.23) that the path integral naturally gives the time ordered correlation functions as the moments —
= NjVx

z(
where the time ordering can be explicitly represented as TiXHitjXnfa)) = 0{tx - t2)XH(t1)XB(t2)

+ B(t2 - t^Xg^XHih)

(4.25)

CHAPTER 4. GENERATING FUNCTIONAL

82

In fact, it is obvious now t h a t t h e time ordered p r o d u c t of any set of operators leads to correlation functions in the p a t h integral formalism as

H(xf,

tflTiO^XsiU))

• • • On{XH{tn)))\xi,

U)H —
= NjVx01(x(t1))---On{x(tn))eh

m

(4.26)

F u r t h e r m o r e , t h e b e a u t y of the p a t h integrals Mes in the fact t h a t all the factors on the right h a n d side are c-numbers (classical quantities). There are no operators any more.

4.3

Correlation Functions In Definite States So far, we have calculated the transition amplitude be-

tween two coordinate s t a t e s . In physical applications, however, we are often interested in transitions between physical states. Namely, we would like t o know t h e probability amplitude for a system in a state |i/>;)ff a t time £; to m a k e a transition to a state \ipf)H a t time tf. This is w h a t t h e S-matrix elements are supposed to give. Let us note t h a t by definition, this transition amplitude is given by

83

4.3. CORRELATION FUNCTIONS IN DEFINITE STATES

= J dxfdxi

H(ipf\xf,

tf)H

H(xf,tf\xi,

U)H H(xi, ti\i/ji)H i

= N J'dxfdxityix^tfWiixiiU)

j'Vxeh

[Xi

(4.27)

Here we have used the usual definition of the wavefunction. Namely, H{x,t\lp)H = lf>(x,t) Following our discussion earlier (see Eq. (4.26)), we see that the time ordered correlation functions between such physical states can also be written as ir(V/|r(Oi(Xjr(ti)) • • • On(XH(tn)))\i>i)H

(4.28)

-S\x] Nfdxfdxity(xf,tf)il>i(xi,ti)JVx01{x(t1))---On(x(tn))en

=

In dealing with physical systems, we are often interested in calculating expectation values. This is simply obtained by noting that BWTiO^Xnfr))

• ••On(XH(tn)))\iPt)H /•

(4.29) — S[x]

/

dxfdxi ip*(xf, tf)ipi(xi, ti) y X>xOi(a;(
On(x(tn))eh

84

CHAPTER 4. GENERATING FUNCTIONAL

Since the states need not necessarily be normalized, we obtain the expectation value to be (TiO^XHih))

••

-On{XH(tn))))

=

HWiinO^XHJt!))

• • • On(*g(«n)))|^)g

_

fdxfdxiip*(xf,tf)il>i(xi,ti)SVxOi(x(ti)) S dxfdxi ipi(xf, tf)tpi(xi, U)

(4.30)

••

^S[x] •On{x{tn))eri

~S[x) jVxeri

Note that the normalization constant N has cancelled out in the ratio and it is for this reason that we do not often worry about the explicit form of the normalization constant. In most field theoretic questions one is primarily interested in calculating the expectation values of time ordered products in the ground state and, consequently, one tends to be sloppy about this factor in such cases. From now on, let us suppress, for convenience, the subscript H signifying the description in the Heisenberg picture. Let us next note that we can generate the various correlation functions in a simple way in the path integral formalism by adding appropriate external sources. Thus, if we define a modified action of the form S[x, J] = S[x] + f' dt x(t)J(t) then, clearly,

(4.31)

4.3. CORRELATION FUNCTIONS IN DEFINITE STATES

85

S[x,0] = S[x] where S[x] defines the dynamics of the system. Let us further define

{il>i\tl>i}j = Nj

dxfdxi

tf(xf,

-S\x tf)1>i(xi, U)jVxeh

J] '

(4.32)

Clearly, then, i

r

r

(•4>i\4>i)j=o = N J dxfdxi^*(xf,tf)rl)i(xi,ti)

J

t5W

Vxen

= WiWi)

(4-33)

It is clear now from Eqs. (4.31) and (4.32) that (tf > t\ > ti)

S{ipi\ipi)j SJ(ti)

= N J dxfdx^*(xf,tf)M*»ti)

f^i6-^^^1^

= N f dxfdxitfixf^ftyifati)

{X

It follows, therefore, that

JVx^xitJeh

'

(4.34)

86

CHAPTER 4. GENERATING FUNCTIONAL

styM. sj(h)

J=0

N J dxfdxi tp*(xf, tf)ipi(xi, U) J Vx -x{tx)eri

=

(4.35)

-(iPiixin)^)

where we have used the relation in Eq. (4.29). Similarly, we have for tf > £l>^2 > ti

s2{^mi/J 6J{h)6J{t2) = NJ

J=0

dxfdxixpi(xf,tf)tpi(xi,ti)

/-a)'

i\2 SS[x,J]8S[x,J] 6J(ti) 6J(t2)

r r = NJdxfdxiiPl(xs,tf)^i(xi,U)jVx[-\ (|)2

i

^S[x,J] j=o

(i\2

WTiX^XfrWi)

TS[X\

xit^x^eh (4.36)

In general, it is quite straightforward to show that

«J(ti)---6J(t„)

= (^)n(i,i\T(X(t1)...X(tn))\1pi)

(4.37)

87

4.4. VACUUM FUNCTIONAL

Consequently, we can write (T(X(tl)...X(tn))) _

(ipi\T(X(t1)...X(tn))\il>i)

_ {-ih)n (Vi|Vi)j

6n{^\iPi)j 6J{t1)---8J{tn)

(4.38) j=o

It is for this reason that (4>i\ipi)j is also known as the generating functional for the time ordered correlation functions.

4.4

Vacuum Functional An object of great interest in quantum theories is the

vacuum to vacuum transition ampUtude in the presence of an external source. The simplest way to obtain this is to go back to the transition amplitude in the coordinate space. i

{xf,tf\xi,ti)j

r i:S[x,J] = N JVxeh r TS[X\ + T f1 dtJ(t)x(t) u w = NJVxeh L J hk

(4.39)

It is clear from our earlier discussion in Eq. (4.26) that we can also think of this quantity as the matrix element of the operator

CHAPTER 4. GENERATING FUNCTIONAL

(xf,tf\xuti)j

=

NJVxeh

Ii

fa

f'dtX(t)J(t) W

= {xf,tf\T(ehJu

y)

)\xl:tt)

(4.40)

Let us next take the limit ti —• —oo

tf —» oo

That is, let us calculate the amplitude for the system to make a transition from the coordinate state in the infinite past labelled by the coordinate X{ to the coordinate state in the infinite future labelled by Xf in the presence of an external source which switches on adiabatically. We will consider this limit by assuming that the external source is nonzero within a large but finite interval of time. That is, let us assume that J(t) = 0

for |t| > r

(4.41)

and we will take the limit r —> oo at the end. In such a case, we can write l

Urn {xf,tf\xi,ti)j if—too

= N

t

rco

— / VxehJ-oo

dt( x) + Jx) v L(x, v '

(4.42)

4.4. VACUUM FUNCTIONAL

89

Alternately, we can write from Eq. (4.40)

lim

{xf,tf\xi,ti)j =

fe.Um

'ti—*—oo

-I (*/,*/|r(eft J-r

ti—>—oo if—too

dtJX )\xi,ti) (4.43)

Let us further assume that the ground state energy of our Hamiltonian is normalized to zero so that H\0)

= 0

H\n)

= En\n)

En > 0

(4.44)

(We wish to point out here that in a relativistic field theory, Lorentz invariance requires P„|0) = 0 which leads to a vanishing ground state energy. In quantum mechanics, however, the ground state energy does not vanish in general and in such a case, the asymptotic limits are not well defined and the derivation becomes involved. We, therefore, choose a derivation parallel to that of a relativistic quantum field theory and assume that the ground state energy is zero.) Although for simplicity of discussion we have assumed the energy eigenstates to be discrete, it is not essential for our arguments. Introducing complete sets of energy eigenstates into the transition amplitude, we obtain

90

CHAPTER 4. GENERATING FUNCTIONAL

lim

{xf,tf\xhti)j

ti—y — oo if—too

i

=

fe,1"11

T,{xf,tf\n)(n\T{enJ-r

tT

- I

dtJX

)\m)(m\xhti)

tf—KX>

= ^6,hm

H

Ew^

\n)

tf —»00

l

- r dt JX {n\T{ehJ-r )\m)(m\eh i

%

-Eti

\xt)

i

— — £ / n t f -f- —hjmt{

=

lim lim ^ en.

n

{xf\n)

tf—tOO i

[T

• r dt JX (n\T(ehJ-r )\m)(m\xi)

(4.45)

where we have used Eqs. (1.42) and (4.44). In the limit t{ —> —oo and tf —• oo , the exponentials oscillate out to zero except for the ground state. One can alternately see this also by analytically continuing to the imaginary time axis (Euclidean space in the case of field theories). Thus, in this asymptotic limit, we obtain

4.4. VACUUM FUNCTIONAL

lim

91

(xf,tf\xi,ti)j

ti—>—oo

i rr - I dtJX ]im(xf\0)(0\T(ehJ-r )\0)(0\xt)

=

J—oo = {xf\0)(0\xi)(0\T(ehJ-co~"

)|0)

(4.46)

Consequently, we can write %

XU|.V...

/"OO oo

/

-°°dUX)\0)=

Um

{

^t

n

\

U

\

J

(4-47)

fy—>oo

The left hand side is independent of the end points and, therefore, the right hand side must also be independent of the end points. Furthermore, the right hand side has the structure of a path integral and we can write Eq. (4.47) also as - I dtJX t {0\T{ehJ-oc )\0) = {0\0)j = NjVxeh without the end point constraints and with

-S[x, J] l

J

(4.48)

92

CHAPTER 4. GENERATING FUNCTIONAL

OO

/

dt (L(x,x)

+ Jx)

Let us note that if we define Z[J] = (0\0)j = NJVxeh

[X

'

J

(4.49)

then, it follows from Eq. (4.38) that {-ih)n Z[J]

SnZ[J] 6J(t1)---8J(tn)

= {T(X(t1)---X(tn)))

(4.50)

Namely, Z[J] generates time ordered correlation functions or the Greens functions in the vacuum. If one knows all the vacuum Greens functions, one can construct the S-matrix of the theory and, therefore, solve the theory. In quantum field theory, therefore, these correlation functions or the vacuum Greens functions play a central role. Z[J] is correspondingly known as the vacuum functional or the generating functional for vacuum Greens functions. In quantum mechanics, we are often interested in various statistical deviations from the mean values. This can be obtained in the path integral formalism in the following way. Let us define

z[

,]

or, W[J]

=

>

[ J 1

= -ih In Z[J]

(4.51)

4.4. VACUUM FUNCTIONAL

93

We have already seen in the case of the free particle as well as the harmonic oscillator that the path integral for the transition amplitude is proportional to the exponential of the classical action (see Eqs. (2.54) and (3.28)). It is for this reason that W[J] is also called an effective action. Let us note that by definition 8W[J] SJfa)

J = 0

-

(

%h

> Z[J]

SJfa) j=o

= (xfa))

(4.52)

where (• • •) stands for the vacuum expectation value from now on. Next, note that {-ih)

=

62W[J] 8Jfa)8Jfa)

j=o

62Z[J] (-;«)= (\Z[J) 6J(t!)SJ(t2)

6Z[J] 6Z[J] \ Z [J] 6 J fa) 6 J fa)) j=o 2

=

{T(Xfa)Xfa)))-(Xfa))(Xfa))

= (T ((Xfa)

- (Xfa)))(Xfa)

- (Xfa)))))

(4.53)

We recognize this to be the second order deviation from the mean and we note that we can similarly, obtain

94

CHAPTER 4. GENERATING FUNCTIONAL

)

^ =

8 J {^83^)8 (-^)

3

J (h)

S3Z[J] Z[J] 6J(i1)<5J(<2)5J(<3)

1 S2Z[J] Z2[J] 8J{t1)8J{t2)

1 S2Z[J] SZ[J] 2 ~ Z [J]8J(t3)8J(tx)8J{t2) ~ +

2 6Z[J] 6Z[J] SZ[J] \ ~Z*8J{t1)8J(t2)8J{t3))

= (r(x(<x)x(<2)x(<3))) -(T(X(t3)X(tl)))(X(t2)) +2(X(t1))(X(t2))(X(t3))

1

SZ[J] SJ(t3)

82Z[J] 6Z[J] 2 Z [J]8J{t2)8J{h)8J(tl)

j=o

(TixwxMMXiu)) - Wi)) (4.54)

= (T ((xin) - (x(tl)))(x(t2) - (x(t2)))(x(t3) - (x{u))))) We can go on and the expressions start to take a more complicated form starting with the fourth functional derivative of the effective action W[J]. However, W[J] can still be shown to generate various statistical deviations and their moments. In quantum field theory, W[J] is known as the generating functional for the connected vacuum Greens functions. Let us next go back to the example of the harmonic oscillator which we have studied in some detail. In this case, we have

4.4. VACUUM FUNCTIONAL

95

:S[x,J] jVxehL

Z[J] = N where Six, J] = I

J-oo

dt(-mx2

— -mu)2x2 + Jx)

2

(4.55)

2

Obviously, in this case, we have (X(tl))

=

(-ih)

1 8Z[J\ Z[J\6J{h)

j=o

±S[x]

N

SVxx(tx)eh N

JVxefi

(4.56)

S[x]

This vanishes because the integrand in the numerator is odd. Therefore, for the harmonic oscillator, we obtain from Eqs. (4.50) and (4.53) that (T(X(tl)X(t2)))

=

=

(-ih)2

(-ih)

S2Z[J] Z[J]6J(«i)5J(« 2 ) 62W[J) 6J{h)8J{t2)

j=o

(4.57) J-0

Let us also note that because the action for the harmonic oscillator is quadratic in the variables, we can write

96

CHAPTER

i

J*ll

f°°

r ~z I Z[J] = NjVxehJ-oo

2 2 .

FUNCTIONAL

T \

— -mu> x + Jx) 2 '

(4.58)

at z

hm JV / £>a:e 2a •'-°° €~>0+ ;->0+

1

-2

dtl-mx 4

4. GENERATING

m. m

J

Let us recall that (see Eq. (3.76)) d2

lim ( ^ + ^ - *)<M* - *') = S(t - t') Using this, we can define

x(t) = x(t) + — [°° dt'GF(t - t')J(t')

(4.59)

777, J — OO

and the generating functional will then take the form im r<x> Z[J] = L J

Urn N 6—0+

d

9

. . _, .

fVxe J

dtdt J G

' ^ ^-^J^

xe-2^nJL =

Hm N d e t ( -

+

^-*e)

e->0+

- - ^ - ir

dtdt'j(t)GF(t - t')j(t')

X e 2?lm •'•'-oo = Z{0]e

- - ? - /7°° dt dt' J(t)G F(t - t')J(t') W M y V y 2hmJJ-™

(4.60)

4.5. ANHARMONIC OSCILLATOR

97

We now obtain in a straightforward manner 62Z[J) 6J(ti)6J(t 2 )

= -^-GF{h

~ t2)Z[0]

(4.61)

Consequently, for the harmonic oscillator, we have (see Eq. (4.57))

(T(X(tl)X(t2)))

= (-ihY

1

Z[J]

62Z[J]

SJ^SJih)

= (-ih)2 ( - ^ ) GF{h - t2) = ^GF(h

j=o

- t2)

(4.62)

In other words, the two-point time ordered vacuum correlation function, in the present case, gives the Feynman Greens function. This is a general feature of all quantum mechanical theories, namely, that the two-point connected vacuum Greens function is nothing other than the Feynman propagator of the theory.

4.5

Anharmonic Oscillator Just as there are a handful of quantum mechanical prob-

lems which can be solved analytically, similarly, there are only a few path integrals that can be exactly evaluated. (Fortunately, there is a one to one correspondence between the quantum mechanical problems that can be analytically solved and the path integrals that can

98

CHAPTER 4. GENERATING FUNCTIONAL

be exactly evaluated.) T h e Gaussian (recall the free particle and the harmonic oscillator) is the simplest of t h e p a t h integrals which can be exactly evaluated. However, we also know t h a t if we p e r t u r b the harmonic oscillator even slightly by an additional potential, say a quartic potential, t h e problem cannot be analytically solved. In other words, t h e q u a n t u m mechanical system corresponding to the Lagrangian L = -mi2 2

- -mw2x2 2

- -x4 4

A> 0

(4.63) '

v

is impossible to solve exactly even when A
r •,

t

Z[J] = NJVxeh =

TS[X, J] L

J

i roo \ , —/ dti-mx v NjVxehJ-™ 2

„

1 2

A

2 2

mu> x

4

4

x +

. Jx)

(4.64)

If we write x4

S[x, J] = S0[x, J] - [°° dt^ J — OO

4

(4.65)

99

4.5. ANHARMONIC OSCILLATOR

where S0[x, J] = J™ dt ( - m i 2 - -mu2x2

+ Jx)

(4.66)

is the action for the harmonic oscillator in the presence of a source, then we note that

In other words, operationally we can identify

im ~~x{t)

(4 68)

'

when this acts on So[x, J]. Now, we can use this identification to write

Z[J\

—7Z I dtxA ^ —Sofa;, Jl A%J-™ )eft L J

t = NJVx(e

i\

r°° „ ,

..

S u

i

d

{-TnL

= = (e

4hJ

~°°

6J

(t)

^m))Njvxe-hsM )Z0[J]

(4.69)

where ZQ[J] is the vacuum functional for the harmonic oscillator in-

100

CHAPTER 4. GENERATING FUNCTIONAL

teracting with an external source. We have already seen in Eq. (4.60) that it has the form Z0[J] = Z0[0]e

JJ^ dt' dt" J(t')GF(t' - t")J(t")

2hmJJ-oo

^ ' ^

' ^ '

(4.70)

Substituting this back then, we obtain iX too , ,

Z[J] = Z 0 [0][e

4hJ

~°°

$

,

J)

]

6J<

jfjT dt' dt" J{t')GF(t' - t")J(t") It is clear that if A is small, i.e., for weak coupling, we can Taylor expand the first exponential and we will be able to obtain the vacuum functional as a power series in A. Consequently, all the vacuum Greens functions can also be calculated perturbatively. This is perturbation theory in the framework of Feynman's path integral.

4.6

References

Coleman, S., "Secret Symmetry", Erice Lectures (1973). Huang, K., "Quarks, Leptons and Gauge Fields", World Scientific publishing. Schwinger, J., Phys. Rev. 82, 914 (1951); ibid 91, 713 (1953).

Chapter 5 P a t h Integrals for Fermions

5.1

Fermionic Oscillator

As we know, there are two kinds of particles in nature, namely, bosons and fermions. They are described by quantum mechanical operators with very different properties. The operators describing bosons, for example, obey commutation relations whereas the fermionic operators (i.e., operators describing fermions) satisfy anti-commutation relations. As a preparation for such systems, let us study a prototype example, namely, the fermionic oscillator. There are many ways to introduce the fermionic oscillator. Let us discuss one that is the most intuitive. Let us recall that the bosonic harmonic oscillator in one dimension with a natural fre101

102

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

quency u> has a Hamiltonian which, written in terms of creation and annihilation operators, takes the form

HB = ~{aBaB + a-BaB)

(5.1)

Here, for simplicity, we are assuming that h = 1. The creation and the annihilation operators are supposed to satisfy the commutation relations [aB,aB} = l

(5.2)

with all others vanishing. The symmetric structure of the Hamiltonian, in this case, is a reflection of the fact that we are dealing with Bose particles and, consequently, the states must have a symmetric form. Fermionic systems, on the other hand, have an inherent antisymmetry. Therefore, let us try a Hamiltonian for a fermionic oscillator with frequency u of the form

HF = iziapap - aFaF)

(5.3)

If ap and aF were to satisfy commutation relations like the Bose oscillator, namely, if we had

5.1. FERMIONIC OSCILLATOR

103

[aF,aF] = l

(5.4)

with all others vanishing, then using this, we can rewrite the fermionic Hamiltonian in Eq. (5.3) to be HF = ~(aFaF

- (aFaF + 1)) = —-

(5.5)

In other words, in such a case, there would be no dynamics associated with the Hamiltonian.

Let us assume, therefore, that the

fermionic operators aF and aF satisfy, instead, anti-commutation relations. Namely, let [aF, aF}+

= aF + a2F = 2a2F = 0

[ 4 , 4 ] + = (aF)2 + (aF)2 = 2(aF)2 = 0 [aj?,a^] + = aFaF+

aFaF = 1 = [aF,aF]+

(5.6)

In contrast to the commutators, therefore, the anti-commutators are by definition symmetric. An immediate consequence of the anti-commutation relations in Eq. (5.6) is that in such a system, the particles must obey Fermi-Dirac statistics. To see this, let us note that if we identify the operators aF and aF with the annihilation and creation operators for such a system, then we can define a number operator as usual as

104

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

NF = aFaF

(5.7)

From the anti-commutation relations in Eq. (5.6) we note that NF

=

aFaFaFap

= aF(l — aFaF)aF — aFap — NF or, NF (NF - 1) = 0

(5.8)

Therefore, the eigenvalues of the number operator can only be zero or one. This is the reflection of the PauH principle or the Fermi statistics, namely, that we can at the most have one fermion in a given quantum state. Thus, we see that the anti-commutation relations are the natural choice for a fermionic system. Given this, then, let us rewrite the Hamiltonian for the fermionic oscillator in Eq. (5.3) as

HF

= -^{aFaF - (1 = u(a}FaF-±)

apaF))

= u>(NF-±)

(5.9)

Furthermore, the commutation relations between aF, aF and NF can now be calculated in a straightforward manner.

5.1. FERMIONIC OSCILLATOR

105

[aF,NF]

= [aF,apap] = [ap,ap}+aF = aF

[aF,Np]

=

[aF,aFap} =—aF[aF,ap}+=-aF

(5.10)

Consequently, if we assume an eigenstate of Np to be denoted by \np), we have Np\nF) = nF\nF)

(5-H)

with np = 0,1. The ground state with no quantum is denoted by |0) and satisfies

NF\0)

= 0

HF\0)

=

W

(^_I)|0) = -||0)

(5.12)

Similarly, the state with one quantum is denoted by |1) and satisfies

NF\1)

= |1>

HF\1)

= w(JV>-i)|l> = | | l >

(5.13)

The ground state is annihilated by ap and we have

aF\0)

= 0

4|0> =

|1)

(5.14)

106

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

It is clear from the anti-commutation relations in Eq. (5.6) t h a t

4|1) = 4.4,|0) = 0

(5.15)

Therefore, t h e Hilbert space, in this case, is two dimensional a n d we note here t h a t t h e ground s t a t e energy has the opposite sign from the ground s t a t e energy of a bosonic oscillator.

5.2

G r a s s m a n n Variables Since fermions have no classical analogue, we cannot di-

rectly write down a Lagrangian for the fermionic oscillator with t h e usual notions of coordinates and m o m e n t a . Obviously, we need the notion of anti-commuting classical variables. Such variables have been well studied in m a t h e m a t i c s a n d go under the n a m e of G r a s s m a n n variables.

As one can readily imagine, they have very u n c o m m o n

properties a n d let us note only some of these properties which we will need for our discussions. For example, if 6i,i = 1,2, • • • , n, defines a set of G r a s s m a n n variables (classical), then they satisfy

eiej + ejei = o

»,j = i,2,.--,n

This, in particular, implies t h a t for any given i,

(5.16)

5.2. GRASSMANN VARIABLES

Q\ = 0

107

i not summed

(5-17)

In other words, the Grassmann variables are nilpotent. This has the immediate consequence that if f{6) is a function of only one Grassmann variable, then it has the simple Taylor expansion

f(6) = a + be

(5.18)

Since 0;'s are anti-commuting, the derivatives have to be defined carefully in the sense that the direction in which the derivatives operate must be specified. Thus, for example, a right derivative for Grassmann variables would give

whereas a left derivative would give

Thus, the sense of the derivative is crucial and in all our discussions, we will use left derivatives. Let us note that like the Grassmann variables, the derivatives with respect to these variables also anti-commute. Namely,

108

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

d_d_

d d

These derivatives, in fact, behave quite Hke the exterior derivatives in differential geometry. We note in particular that for a fixed i

d \2

80 J = °

(5 22)

-

In other words, the derivatives, in this case, are nilpotent just hke the variables themselves. Furthermore, the conventional commutation relation between dervatives and coordinates now takes the form

iw^

=

^+e'wr6"

(5 23)

'

The notion of integration can also be generalized to Grassmann variables. Denoting by D the operation of differentiation with respect to one Grassmann variable and by / the operation of integration, we note that these must satisfy the relations

ID

= 0

DI

= 0

(5.24)

Namely, the integral of a total derivative must vanish if we ignore surface terms and furthermore, an integral, being independent of the

5.2. GRASSMANN VARIABLES

109

variable must give zero upon differentiation.

Note that since dif-

ferentiation with respect to a Grassmannn variable is nilpotent (see Eq. (5.22)), it satisfies the above properties and hence for Grassmann variables integration can be naturally identified with differentiation. Namely, in this case, we have I = D

(5.25)

Thus, for a function of a single Grassmann variable, we have

Jdem = ^p-

(5-26)

This immediately leads to the fundamental result that for Grassmann variables

fde = o J dee = 1

(5.27)

This is an essential difference between ordinary variables and Grassmann variables and has far reaching consequences. An immediate consequence of the definition of the integral in Eq. (5.26) is that if we redefine the variable of integration as e' = ae

a^O

(5.28)

110

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

then, we obtain

IM

W

=

9

^

1

= «-a#-

= ajdff /(£)

(5.29)

But this is precisely the opposite of what happens for ordinary variables. Namely, we note that the Jacobian in the case of redefinition of Grassmann variables is the inverse of what one would naively expect for ordinary variables. This result can be easily generalized to integrations involving many Grassmann variables and it can be shown that if Q\ = aijOj

(5.30)

with det a,ij ^ 0 and repeated indices being summed, then

/ ft d6i f(6i) = (det aij) J ft dffJiarft) i=l

(5.31)

j=l

We can also define a delta function in the space of Grassmann variables as 6(0) = 9

(5.32)

That this satisfies all the properties of the delta function can be easily seen by noting that

111

5.2. GRASSMANN VARIABLES

JdB6(9)

= fd0 0 = l

(5.33)

which follows from Eq. (5.27). In addition, if

f(0) = a + be then,

Jd68(0)f{e) = jdeef{e) = jdee(a + be) = Jddea = ^

= a = f(0)

(5.34)

Here we have used the nilpotency of the Grassmann variables. Furthermore, consistent with the rule for change of variables for the Grassmann variables, if g{0) = aO then, we obtain 8{g(9)) =a6 = a6(6) = ^~~m

(5-35)

where g{0) is assumed to be Grassmann odd. An integral representation for the delta function can be obtained simply by noting that if ( is also a Grassmann variable, then

112

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

JdCeW = Jd((l+i(6) = —(l+i(9)

= i9 = i6(0)

(5.36)

Let us next evaluate the basic Gaussian integral for Grassmann variables. Let us consider two sets of independent Grassmann variables, namely, (#i, #2> • • •, 9n) and (0j,0%, • • •, 9%)

an

d analyze the

integral

I = JIId9* d0j e-(diMiiei

+ c*i°i + di°i)

(5.37)

where we are assuming that Q and c* are independent Grassmann variables. Furthermore, the convention for summation over repeated indices is always assumed. Note that if we make the change of variables (we are assuming that M _ 1 exists) e'i = MijOj + a or,

Oi = Mrfty-cj)

(5.38)

Of = 0t + c*jM~1

(5.39)

and

then, we obtain using Eqs. (5.31), (5.38) and (5.39) that

113

5.3. GENERATING FUNCTIONAL

i = J iide; M; e-WiWi+

«) + &*)

= det M, / n dffl de> e-VM + <M?W ~ ci))

= det M , / n <w; cw;. c - ( W + c ^ ^ -

<M^

ij

= det M„ / II dfff dff, e-6'6* +

<M^

= N detMijeciMi~Jlci

(5.40)

Here N is a constant and we note that the Gaussian integral in the case of Grassmann variables has the same form as the integral for ordinary variables except for the positive power of the determinant. This leads to an essential difference between quantum mechanical bosonic and fermionic theories.

5.3

Generating Functional With all this background on Grassmann variables, we can

now ask whether it is possible to write a Lagrangian for the fermionic oscillator. Indeed, let us consider the Lagrangian

L= ^ - ^ ) - | h M

(5.41)

114

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

where ip and ip are two independent Grassmann variables. Quite often one eliminates a total derivative to write an equivalent Lagrangian also as L = iW-'%$,1>]

(5-42)

We will, however, continue with the first form of the Lagrangian, namely, Eq. (5.41). This is a first order Lagrangian and one can define canonical conjugate momenta associated with the Grassmann variables ip and ip as usual dL

TT

n, = |

ij

= -i,

(5,3)

With the convention of left derivatives, the proper definition of the Hamiltonian (which is only a function of coordinates and momenta and which also leads to the correct dynamical equations of motion) is

H = ipu^ + ipn^-L

2

— •

= f[<M

—

%

— *

—

UJ —

(5-44)

5.3. GENERATING FUNCTIONAL

115

It is clear, therefore, that this simple Lagrangian will yield the Hamiltonian of Eq. (5.3) for the fermionic oscillator if we identify

ip = ap

i> = 4

( 5 - 45 )

With this identification of tp and V> with the annihilation and the creation operators respectively, the Hermiticity properties for these variables now follow. Namely, we note that

ft

= if)

ft = ^

(5.46)

We also note that with this convention, the number operator defined as NF = a)FaF = i>ip

(5-47)

is Hermitian since NF = (^V) f - ft ft = W

(5.48)

Since Grassmann variables have no classical analogue, even when we are dealing with the reality questions of ordinary Grassmann variables

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

116

(not operators), we follow the above prescription in defining complex conjugation. Namely, we define for any pair of Grassmann variables 7? and x, (VX)* = x V

(5-49)

In other words, even classically, we continue to treat Grassmann variables like operators. This is the only way a consistent transition is possible from a classical to a quantum Lagrangian involving fermions. With this prescription, let us note that the Lagrangian for the fermionic oscillator given in Eq. (5.41) is Hermitian (real). L\

= ^(^_^)_|[^^

i -•

-

u) -

= L

(5.50)

With this, then, we can write the vacuum functional for the fermionic oscillator as z — Z[r,,f}] = (0\0)v,n = Njvj>Vi>eh ' ' ' S[ 4 ,Av ll]

(5.51)

117

5.3. GENERATING FUNCTIONAL

where we have denoted the sources for ip and ip by rj and 77 respectively. The complete action for the oscillator, in this case, has the form

Sty,$,v,v] = Sty,$] + j^dtiftf

+

tiri)

(5.52)

with

Sty,® = jdtL = jdt(]i(U

- h ) - |[VsV<])

(5-53)

Once again, we will assume the Hermiticity conditions for the sources similar to the ones given in Eq. (5.46), namely,

T7f

=

fj

rf = 77

(5.54)

in order that the complete action in Eq. (5.52) is Hermitian. Just as an ordinary derivative with respect to a Grassmann variable is directional, similarly, there are right and left functional derivatives with respect to fermionic variables. The definition of the functional derivative is still the same as given in Eq. (1.14), namely, 8F(rl>(t))=Y

8^{t<)

So

F(iP(t) +

e6(t-t'))-F(iP(t))

e

(5.55)

118

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

However, since e is now a G r a s s m a n n variable, t h e position of e _ 1 in the expression defines the direction of the derivative. one can think of e

_1

simply as ^ .

(Incidentally,

Secondly, we note here t h a t for

polynomial functionals, t h e limit e —• 0 is r e d u n d a n t since e2 = 0 and t h e highest power of e in t h e expansion of t h e functional is linear.) T h u s , a left functional derivative corresponds to defining

6 F S

^

] )

= Hm e- 1 [F y,(t)

+ eS(t - f ) ) - F (^(t))]

(5.56)

whereas a right functional derivative would be defined as

S F 6

^

]

= lim [F ( m

+ eS(t - t')) - F ((*))] ^

(5.57)

As we have mentioned earlier, we will always work with left derivatives even when dealing with functionals involving fermionic variables.

5.4

Feynman Propagator Let us next go back to t h e Lagrangian in Eq. (5.41) for

the fermionic oscillator a n d note t h a t when fermions are involved, we essentially have a m a t r i x s t r u c t u r e . This is another reflection of the fact t h a t t h e G r a s s m a n n variables inherently behave like operators. Let us define t h e following two component matrices.

5.4. FEYNMAN PROPAGATOR

*

—

*

=

119

U"

U, &(T3

= 1?

-i>

0 = (5>V3

= »/

-v

0 = (5.58)

Here (73 denotes t h e PauH m a t r i x , namely,

&3

1

0

0

-1

=

We note, t h e n , t h a t

f

d

T

at

yjfi$

( .d dt —if) 0

oW..\

^

=

tp

iip 1

=

t\)

— ip

=

i [iptp — V'1/')

=

ijj

-ip

-i

1>

-u;

dt J i (•?/>?/> + -0-^

120

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

t $0

= xj) =

-i\)

V

\

(5.59)

(rpr) + f}ip) = 0 *

where we have used the anticommuting properties of the Grassmann variables. It is now straightforward to show that the action for the fermionic oscillator in Eq. (5.52) can be written as oo / -oo

i — d 3 7 dt

U

K

* * + 0*)

= /_~ d '(^*( i f f 3^-w)¥ + e*)

(5.60)

We note here that, using Eq. (5.59), we could have written the source term also as oo

/ -oo

_

dt # 0

In other words, 0 and 0 are not really independent and, therefore, we can write the vacuum functional as (h=l)

121

5.4. FEYNMAN PROPAGATOR

Z[Q] = N , = N J We

jV*eiS i f°° dt(-y(ia3-J ~~ V2 v sdt

- w)# + 0 # ) ' '

(5.61)

We know that we can evaluate the generating functional if we know the Greens function associated with the operator in the quadratic term in the exponent (see Eq. (5.40)). Let us, therefore, study the equation (icr34- - u)G(t - t') = 6{t - t') at

(5.62)

This is clearly a matrix equation and it can be solved easily in the momentum space. Thus, we define

G(t-f)

6{t-t')

=

f^G(k)e-ik(t-t')

= J-ldjfee-^C-'')

(5.63)

where G(k) is a matrix in the Fourier transformed space. Substituting these expressions back into Eq. (5.62), we obtain

122

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

/L(<73* - U)G(k) = i /Z7T

or, G(k) =

Z7T

'2ir a3k — u> 1

a3k + u 2

/2TTfc

-

(5.64)

W2

Consequently, the Greens function in Eq. (5.62) has the form G{t-H)

f-^G(k)e-ik(t-t')

=

\f /.IT

J

i / . B ^ e - a C - ' ) 2-K J kz — LJ2

(5.65)

The singularity structure of the integrand is obvious and the Feynman prescription, in this case, will lead to the propagator (see Eq. (3.75))

GF(t-t) = Km i / a J* XI- e " it( '' t ' ) 7)-»0+ 27T •/

= lim

kZ - UZ + IT]

l

a3k

-Uk

17-^0+ 27T

1.2

/

+U Z?

e~^-0(5.66) ? \2

This can also be written in the alternate form GF(t - t1) = Urn — I dk — — ^ e-ik<J v ; e^o+ 2TT J a3k - u + ie

~ *')

(5.67) '

K

5.4. FEYNMAN PROPAGATOR

123

and satisfies the equation (see Eq. (3.76))

Urn (ia3— -u> + ie)GF(t - t') = S(t - t') €—>o+

(5.68)

at

This defines the Feynman propagator in the present case. Going back to the vacuum functional, we note that we can write

, =

KmN e-.0+

r 1d i / dt(--$(ia3— - u) + ie)$ + 9 * ) ! VVe J V 2 k dt '

J

Therefore, the 1-point function can be obtained to be

6Z[S] SQ(h) 0=0=0 =

, ]im+Njv$(M(ti))e2J

% r — iJL d - I dt *(z + i e ) * dt '

= 0

(5.70)

This is because the integrand in Eq. (5.70) is odd under

* -• - *

* -• - *

(5.71)

124

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

Similarly, we can obtain from Eq. (5.69)

6Z[G] = 0 60^) 0=0=0

(5.72)

Thus, we see that if we write

ftl = JW[Q] Z[Q]

(5.73)

then, in this case,

62W[B]

(-0 <50(t1)56(f2) 1 (

z)

0=0=0

62Z[@]

z[e] 6B(h)6e(t2)

0=0=0

= -(r(*(t 1 )*(t a ))>

(5.74)

(Compare this with Eq. (4.57) for h = 1.) Incidentally, time ordering, in the case of fermionic variables, is defined as T ( * ( t 0 * ( t 2 ) ) = 0(ti - t 2 )*(*i)*(<2) - 0{h - *i)*(* 2 )*(*i)

(5.75)

The relative negative sign between the two terms in Eq. (5.75) arises from the change in the order of the fermionic variables, which anticommute, in the second term. Going back to the vacuum functional, we

5.4. FEYNMAN PROPAGATOR

125

note that since the exponent is quadratic in the variables (namely, it is a Gaussian integral), it can be explicitly evaluated using Eq. (5.40) to be Z[&] r =

V^fe

limJV

J

V

12 v

d dt

£-•0+

= JVe 2

J J dt1dt2@(t1)GF{t1 - t2)e(t2)

= zroi e~ytdtldt2 ® (fl)GF{tl ~ *2)0(*2)

(5.76)

where Z[0] represents the value of the functional in the absence of sources. It is obvious now that 82Z[G]

seitjsefa) 0=0=0

= iGF{h - t2) Z[0]

(5.77)

Therefore, we have, using Eq. (5.74)

-H)' =

1 Z[Q]

62Z[@] SQitJSQfa)

0=0=0

" H)2 zioi iG ^ i_i2)z[0]

= iGp(ti

— t2)

(5.78)

126

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

This again shows (see Eq. (4.62)) that the time ordered two-point correlation function in the vacuum gives the Feynman Greens function. As we have argued earlier, this is a general feature of all quantum theories.

5.5

The Fermion Determinant The fermion action following from Eq. (5.41) or Eq. (5.42)

is quadratic in the dynamical variables just like the action of the bosonic oscillator in Eq. (3.2). Therefore, the generating functional can be easily evaluated. In this section, we will evaluate the generating functional for the fermions in the absence of any sources. For simplicity, let us take the dynamical Lagrangian of Eq. (5.42). Then, we can write the generating functional, in the absence of sources, to be

Z[0] = N

jVi>V^eiS&^\

= NJV^V^e^

y

'

(5.79)

5.5. THE FERMION DETERMINANT

127

Here we have used the anti-commuting properties of the Grassmann variables to rewrite the commutator of the fermionic variables in a simpler form. The constant N, representing the normalization of the path integral measure is arbitrary at this point and would be appropriately chosen later. We can once again define tf-U

= T

as the time interval and translate the time coordinate to write the generating functional of Eq. (5.79) also as

Z[0] = Nf

Vm

J f

dt

^

~ ^

(5.80)

To evaluate the path integral, we should discretize the time interval as in Eq. (2.18). Thus, defining the intermediate time points to be tn = ne

n = 1,2, • • -,N — 1

where the infinitesimal interval is defined to be T

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

128

we can write the path integral to be

Z[0] = lim N / dtp! • • • dipN-xdipi • • • dip^-i N—*oo

« E (*V»n( x e n=i

"

) - W^n(

e

= i

)) (5.81)

Here we have used the mid-point prescription of Weyl ordering as discussed in Eqs. (2.17) and (2.21) alongwith the earlier observation that the variable ip represents the momentum conjugate to ip. The exponent in Eq. (5.81) can be written out in detail as

N—l

ifu)

_

if to —

E (1 + -irWrdn + (1 + -^WN^N .71=1

- E

^

(5.82)

L

(1 - -g-^nV'n-l - C1 - -^-)V'lV,0 - (1 " ~Y)i>Ni>N-\

Thus, defining (iV — 1) component matrices

5.5. THE FERMION DETERMINANT

129

*l>2

v> =

y V'iV-l /

' 4, =

&

^

V>2

\ V'JV-I /

J

=

0

-d-^)

\° / I 0 ^ J =

-d-f)

(5.83)

0

we can write the path integral of Eq. (5.81) also as l€UI>

ZM - lim N [*#

e ' ^

+

+

*** ^

+ (1 +

-

T>*»*'>

N->oo

(5.84)

130

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

where we have defined a (iV — 1) x (iV — 1) matrix B as ( x 0 0 0 ••• \ y x 0 0 •••

B=

(5.85)

0 y x 0 ••

V

/

with

y =

(5.86)

-(i-^-)

The path integral in Eq. (5.84) can now be easily evaluated using Eq. (5.40) and the result is

ZO

=

Mm i V d e t £ e V

V

2

^NyN>

JV— oo

=

Hm N l«Be{J"-lB"-"Jl

'

(

' + T^ls.87)

JV-.00

The matrix B has a very simple structure and we can easily evaluate the detrminant as well as the appropriate element of the inverse matrix which have the following forms.

131

5.5. THE FERMION DETERMINANT

detU = xN~l = (1 + 1

BZ ,,

.Jf-2

= (-1)**—=.

~f-x (1 - — ) * ~

2

2 (1 + ^ " ) i V " 1

(5.88)

In the continuum limit of e —» 0 and JV —> oo such that iVe = T, the path integral, therefore, has the form iuT Z[0] = N e~2~

e

e(

~1WT^^° ~

^ ^ )

= N e ~ T e ( e " i w T ^/V'i - ^/V>/)

(5.89)

Here we have identified

^o = A

i>N = 4>f

(5.90)

We choose, for simplicity and for future use, the normalization of the path integral measure to be N = 1 so that the free fermion path integral takes the form

Z[0] = e~T

iwT

e{e-

4>fipi ~ 4>fi>f)

(5. 9 1 )

132

5.6

CHAPTER 5. PATH INTEGRALS FOR FERMIONS

References

Berezin, F., "The Method of Second Quantization", Academic Press. D e W i t t , B., "Supermanifolds", Cambridge Univ. Press.

Chapter 6 Super s y m m e t r y

6.1

Super symmetric Oscillator We have seen in Chapters 3 and 5 that a bosonic oscilla-

tor in one dimension with a natural frequency UJ is described by the Hamiltonian HB = -x {aBaB + aBaB) = u(aBaB

+ -)

(6.1)

while a fermionic oscillator with a natural frequency ui is described by the Hamiltonian itJ

t

i

4.

1

HF = -z {aFaF - aFaF) = w(aFaF - - )

(6.2)

Here we are assuming that % — 1. (See Eqs. (5.2) and (5.3).) The cre133

134

CHAPTER

6.

SUPERSYMMETRY

ation a n d t h e annihilation operators for t h e bosonic oscillator satisfy

[a*,4] = l

(6.3)

with all others vanishing. For t h e fermionic oscillator, on t h e other h a n d , t h e creation and the annihilation operators satisfy the anticomm u t a t i o n relations (see Eq. (5.6))

[aF,aF]+=

0 = [a] ? ,aj r ] +

[aF,aF]+=

1

(6.4)

Let us note here (as we have pointed out earlier in chapter 5) t h a t t h e ground s t a t e energy for t h e bosonic oscillator is ^ whereas t h a t for the fermionic oscillator is — TT. Let us next consider a system consisting of a bosonic and a fermionic oscillator with the same n a t u r a l frequency u>. This is known as the super symmetric oscillator.

T h e Hamiltonian for this

system follows from Eqs. (6.1) and (6.2) to be

H = HB + HF

=

—{aBaB

+ aBaB

+ aFaF

=

/ t ui(aBaB

I t + - + aFaF

=

uj(aBaB

+ aFaF)

-

aFaF)

1

-

^ -) (6.5)

6.1. SUPERSYMMETRIC

OSCILLATOR

135

We note from Eq. (6.5) that the constant term in the Hamiltonian for this system has cancelled out. If we define the number operators for the bosonic and the fermionic oscillators as NB

=

aBaB

NF

= aFap

(6.6)

then, we can write the Hamiltonian for the system also as H = u(NB + NF)

(6.7)

It is clear from Eq. (6.7) that the energy eigenstates of the system will be the eigenstates of the number operators NB and NF. Consequently, let us define \nB,nF) = \nB) ® \nF)

(6.8)

where NB\nB)

= nB\nB)

nB = 0 , 1 , 2 , . . .

nplnp)

— np\np)

n^ = 0,l

(6-9)

Here we are using our earlier result in Eq. (5.11) that the eigenvalues for the fermionic number operator are 0 or 1 consistent with the Pauli

136

CHAPTER 6.

SUPERSYMMETRY

principle while the eigenvalues for the bosonic number operator can take any positive semidefinite integer value. From Eqs. (6.7), (6.8) and (6.9) we note that the energy eigenvalues for the supersymmetric oscillator are given by HB\nB,nF)

= EnBtnF\nB,nF)

= uj(nB + nF)\nB,nF)

(6.10)

with nB — 0,1,2 . . . and nF = 0,1. We also note from Eq. (6.10) that the ground state energy of the supersymmetric oscillator vanishes, namely, £o,o = 0

(6.11)

Incidentally, the ground state is assumed to satisfy a B |0) = 0 = 0^0)

(6.12)

The vanishing of the ground state energy is a general feature of supersymmetric theories and as we will see shortly it is a consequence of the supersymmetry of the system. We also observe from Eq. (6.10) that except for the ground state, all other energy eigenstates of the system are doubly degenerate. Namely, for nB ^ 0, the states \nB, 1) and \nB + 1,0) have the same energy. The degeneracy in the energy value for a bosonic and a fermionic state, as we will see, is again a consequence of the supersymmetry of the system.

6.1. SUPERSYMMETRIC

137

OSCILLATOR

Let us next consider the following two fermionic operators in the theory. Q =

aBaF

Q = aFaB

(6.13)

We can show using the commutation relations in Eqs. (6.3) and (6.4) that (The bosonic operators commute with the fermionic ones.) [Q,H] = [aBaF,u(aBaB = uj(aB[aB,aB]aF

+ aFaF)] +

aB[aF,aF]+aF)

= u)(—aBaF + aBaF) = 0

(6.14)

and similarly, [Q,H] = [aFaB,u(aBaB

+ aFaF)] = 0

(6.15)

The operators, Q and Q, therefore, define conserved quantities of this system (charges) and would correspond to the generators of symmetries in the theory. We also note that

138

CHAPTER 6.

[Q,Q]

SUPERSYMMETRY

[aBaF,aFaB]+ aB[aF,aF]+aB dBaB +

-

aF[aB,aB]aF

aFaF

- H u>

(6.16)

T h u s , we see from E q s . (6.14), (6.15) a n d (6.16) t h a t t h e operators Q, Q a n d H define an algebra which involves b o t h commut a t o r s a n d a n t i c o m m u t a t o r s . Such an algebra is known as a graded Lie algebra a n d defines t h e infinitesimal form of the supersymmetric transformations.

An immediate consequence of the s u p e r s y m m e t r y

algebra is t h a t if t h e ground state is invariant under s u p e r s y m m e t r y transformations, namely, if (see Eq. (6.12)) Q|0) = 0 = Q|0)

(6.17)

t h e n it follows from Eq. (6.16) t h a t (0|fT|0) = (0\QQ + QQ\0) = 0

(6.18)

Namely, t h e ground state energy in a supersymmetric theory vanishes. F u r t h e r m o r e , we note from Eq. (6.13) t h a t Q is really the Hermitian conjugate of Q.

Consequently, it follows from Eq. (6.16) t h a t in a

6.1. SUPERSYMMETRIC

OSCILLATOR

139

supersymmetric theory H is really a positive semidennite operator and, therefore, its expectation value in any state must be positive semidennite. Let us next analyze the effect of Q and Q on the energy eigenstates of the system. We note from the commutation rules of the theory that [Q,NB]

=

[aBaF,aBaB]

=

aB[aB,aB}aF

= —aBaF = —Q [Q,NF]

=

[aBaF,aFaF]

=

aB[aFiaF\+aF

= aBaF = Q

(6.19)

and similarly

[Q,NB]

=

Q

[Q,NF]

= -Q

(6.20)

In other words, we can think of Q as raising the bosonic number, nB, by one unit while lowering the fermionic number, nF, whereas Q does the opposite. It now follows that for

CHAPTER 6.

140

SUPERSYMMETRY

(a) \nB

\nB^F) = ^>(aF)^\0)

(6.21)

where we recognize nF — 0,1 and nB = 0,1, 2 , . . . , we have

Q\nB,nF)

=

s/nB + l\nB + 1,n F - 1) if nF ^ 0 0

if nF = 0

n\ \ v f c l T C 5 - 1 , ^ + 1) i f n B ^ 0 o r n F ^ l Q\nB,nF) = \ ^B (6.22) 0 if nB = 0 or nF = 1 Namely, we note that acting on any state other than the ground state, the operators Q and Q take a bosonic state (with nF = 0) to a fermionic state (with nF = 1) or vice versa. This is the manifestation of supersymmetry on the states in the Hilbert space of the Hamiltonian, i.e., the bosonic and the fermionic states are paired. Furthermore, since Q and Q commute with the Hamiltonian of the system (see Eqs (6.14) and (6.15)), it now follows that such paired states will be degenerate in energy. Namely H(Q\nB,nF))

= Q(H\nB,nF))

=

EnB:nF(Q\nB,nF))

H{Q\nB,nF))

= Q(H\nB,nF))

= EnBtnF(Q\nB,nF))

(6.23)

The supersymmetric oscillator is the simplest example of supersymmetric theories. The concept of supersymmetry and graded

6.2. SUPERSYMMETRIC QUANTUM MECHANICS

141

Lie algebras generalizes to other cases as well and there exist many useful realizations of these algebras in the context of field theories.

6.2

Supersymmetric Quantum Mechanics Let us next study a general supersymmetric, quantum

mechanical theory. From our discussion in the last section, we note that supersymmetry necessarily involves both bosons and fermions and, therefore, let us consider a Lagrangian of the form

L = \i2~ \f(x) + iU ~ f{x)W

(6.24)

Here, for consistency with earlier discussions we have set m = 1 for the bosonic part of the Lagrangian. We also note here that f(x) can be any chosen monomial of x at this point. It is clear from Eq. (6.24) that when f(x) = LJX

(6.25)

the Lagrangian of Eq. (6.24) reduces to that of a supersymmetric oscillator discussed in the last section. In general, we note that under the infinitesimal transformations

142

CHAPTER 6.

SUPERSYMMETRY

*<* = 72^ 6J =

--^ie-±f(x)e

6e$ = 0

(6.26)

and

slX = - L ^ 6rf> = 0 8rf = -^xe-~f(x)e

(6.27)

where e and e are infinitesimal Grassmann parameters, the action for the Lagrangian can be seen to remain unchanged. In other words, the transformations in Eqs. (6.26) and (6.27) define symmetries of the system. (See chapter 11 for a detailed discussion of symmetries.) These symmetry transformations mix up the bosonic and the fermionic variables of the theory and, therefore, are reminiscent of the supersymmetry transformations which we discussed earlier. In fact, one can explicitly show that the two sets of transformations in Eqs. (6.26) and (6.27) are generated respectively by the two supersymmetric charges Q and Q in the theory. We also note here, without going into detail, that while the Lagrangian in Eq. (6.24) is supersymmetric for any

6.2. SUPERSYMMETRIC QUANTUM MECHANICS

143

monomial f(x), in the case of even monomials the presence of instantons breaks supersymmetry. (Instantons are discussed in chapter 8.) Consequently, let us consider monomials only of the form. f(x)~x2n+1

n = 0,1,2,...

(6.28)

With these preparations, let us next look at the generating functional for supersymmetric quantum mechanical theory in Eq. (6.24) (h = 1) Z = N J V^V^Vx

eiSix> ^ $\

(6.29)

As we have seen in the last section, the spectrum of a supersymmetric theory has many interesting features. Correspondingly, the generating functional for such a theory is also quite interesting. In particular, let us note from Eq. (6.24) that since the Lagrangian is quadratic in the fermionic variables, the functional integral for these variables can be done easily using our results in chapter 5. (See Eqs. (5.40) and (5.87)). Thus, we can write

= N'det(i^--f(x)) Substituting Eq. (6.30) into Eq. (6.29), we then obtain

(6.30)

144

CHAPTER 6.

eiSM>$\

Z = N JV^V^VX

* /dt(\x 2 -I fix)) , _

, = NJVxe

- t

J

SUPERSYMMETRY

V

2

d

2J

= NJVxdet(ij-f'(x))eJ

v

i I dtUii--

J

" JVxpVipe

V V

dt

f ix)U)

J y

"*'

/• 1 1 i dti-x2 -fix)) V

2

2Jy"

(6.31)

Let us next note that if we define a new bosonic variable through the relation p = ix- f(x)

(6.32)

then the Jacobian for this change of variables in Eq. (6.31) will be given by j = [ d e t ( i A _/'(*))]-!

(6.33)

This is precisely the inverse of the determinant in Eq. (6.31). Furthermore, we note that

J dtp2 = J dt(ix - f(x))2 = Jdt(-x2-2ixf(x) + f(x)) =

-Jdt(x2-f(x))-2if™oodxf(x)

= - j dt(x2 - f(x))

(6.34)

Here we have used the fact that for f(x) of the form in Eq. (6.28), the

6.3. SHAPE INVARIANCE

145

last integral vanishes. (For even monomials, on the other hand, this does not vanish giving the contribution due to the instantons which breaks supersymmetry.) Substituting Eqs. (6.32), (6.33) and (6.34) into the generating functional in Eq. (6.31), we find - , Z = NjVpe2J

[dtp2 r

(6.35)

In other words, we see that the generating functional for a supersymmetric theory can be redefined to have the form of a free bosonic generating functional.

This is known as the Nicolai map (namely,

Eq. (6.32)) and generalizes to field theories in higher dimensions as well.

6.3

Shape Invariance As we have noted earlier, there are only a handful of

quantum mechanical systems which can be solved analytically. The solubility of such systems now appears to be related to a specific symmetry of the systems known as shape invariance. This symmetry is also quite useful in the evaluation of the path integrals for such systems. Let us consider a one dimensional quantum mechanical system described by the Hamiltonian

CHAPTER 6.

146

SUPERSYMMETRY

H

- = J + U-(x)

(6-36)

If we assume the ground state of the system to have vanishing energy, then we can write the Hamiltonian in Eq. (6.36) also in the factorized form H_ = QQ

(6.37)

Q = J-(p V + iW(x))

(6.38)

U.(x) = \{W2(x) - W'(x))

(6.39)

where

V2

and we identify

We note that Q and Q are Hermitian conjugates of each other and that given these two operators, we can construct a second Hermitian Hamiltonian of the form H+ = QQ = £ + U+(x) = ^ + \{W\x) + W'(x))

(6.40)

It now follows that if \if>) is any eigenstate of the Hamiltonian ff_ other than the ground state, namely, if

6.3. SHAPE INVARIANCE

147

H-\i/>) = QQ\j>) = \\if>)

A^O

(6.41)

then it follows that QH-\r/>) = A(Q|V))

or, QO(QIV)) =

KQW))

or, H+(Q\1>)) = KQW)

(6-42)

Namely, we note that the two Hamiltonians, H_ and H+, are almost isospectral in the sense that they share the same energy spectrum except for the ground state energy of H_. The potential, of course, depends on some parameters such as the coupling constants. If the potential of the theory is such that we can write U+(x, o0) = U-(x, a^ + R(ai)

(6.43)

with R(ai) a constant and the parameters ao and ai satisfying a known functional relationship ai = /(a„)

(6.44)

then we say that the potential is shape invariant. In such a case, we can write using Eq. (6.43)

CHAPTER 6.

148

SUPERSYMMETRY

P2

H_(a0) = Q(a0)Q(a0)

= — + U-(x,a0)

H+(a0) = Q(a0)Q(a0)

P2 = — + U+(x,a0) V2

= j + U_(x,ai) = jy_(ai) +

+ R(ai)

R(ai)

= Q(a 1 )Q(a 1 ) + i?(a 1 )

(6.45)

Since we have assumed that the ground state energy of H_ vanishes and since we know that H+(ao) and H-(ao) are almost isospectral, it follows now from Eq. (6.45) that the energy value for the first excited state of H-(ao) must be #i = i ? ^ )

(6.46)

It is also easy to see now that for a shape invariant potential, we can construct a sequence of Hamiltonians such as

#«

= H+{ao) = H-(ai) + R{ai)

R-M = # + K - i ) + £ k=i

= H_{as) + t k=i

R(ak)

R(ak)

(6.47)

6.3. SHAPE INVARIANCE

149

Here we have identified as = /'(a„) = / ( / • • • ( / ( a 0 ) ) . . . )

(6.48)

All the Hamiltonians in Eq. (6.47) will be almost isospectral and from this, with a little bit of analysis, we can determine the energy levels of H-_(a0) to be En = £ R(ak)

(6.49)

k=i

Given the sequence of Hamiltonians in Eq. (6.47) we can write down the relation Q{as)Q{as) or, Q(as)H^

= Q(as+i)Q(as+1)

+ R(as+1)

= H^s+1^Q(as)

(6.50)

This defines a recursion relation between the sequence of Hamiltonians. Furthermore, for t > 0, defining the time evolution operator for a particular Hamiltonian, H^s\ in the sequence to be u(s)

= e-itHM

(6>51)

we note that Eq. (6.50) gives Q(as)U^

=

Q{as)eitH{s)

= e-itH(S+1)Q(as)

= U^+1^Q(as)

(6.52)

150

CHAPTER 6.

SUPERSYMMETRY

Similarly, by taking the time derivative of Eq. (6.51), we obtain

at = -i(Q(a.)Q(a.) = -if:

+ f

R(ak)U{s) -

R(ak))e-itH{s) iQ{as)U(s+VQ(as)

k=i

= -iQ(as)U^+1^Q(as)

or, (^ + ifR(ak))U^ ut

(6.53)

ib=i

The relations in Eqs. (6.52) and (6.53) define recursion relations for the time evolution operator and have the coordinate representation of the form

(jx+W(x,as))U(s\x,y-t) =

-(-^-W(y,as))U^(x,y;t)

(^- + =

i±R(ak))U(°\x,y;t)

-^(^-^(x,as))(^-W(y,as))C/(s+1)(x,y;0

(6.54)

It is clear from this discussion that for a shape invariant potential if one of the Hamiltonians in the sequence coincides with a system which we can solve exactly, then using the recursion relations

151

6.4. EXAMPLE

in Eq. (6.54), we can solve for the time evolution operator of the original system. This will determine the path integral for the system.

6.4

Example Let us consider a quantum mechanical system with W(x,ao)

= a0 tanha;

a0 = 1

(6.55)

From Eqs. (6.39) and (6.40), we find U4x,a0)

l(W2(xta0)-W'{x,a0))

=

sech2x

= 2 U+(x,a0)

= ^(W2(x,a0)

+

W'(x,a0))

= \

(6-56)

In this case, therefore, we can identify o0 = 1

;

ai = a0 - 1 = 0

;

R(ax) = 0

The Hamiltonian for the system, in this case, is

(6.57)

152

CHAPTER

H{0) = H-(a0) = ^r-

6.

SUPERSYMMETRY

sech2x + \

(6.58)

Zi

Zi

and the next Hamiltonian in the sequence is given by (see Eq. (6.47))

H™ =

H+(a0)

= H_{ax) + R(ai)

= ^ +\

(6-59)

This is, of course, the free particle Hamiltonian for which we know the transition amplitude to be (see Eq. (2.47))

UW(x,y;t)

=-7==

Ax-yf e 2t {

_ it
(6.60)

\J2-Kit

Substituting Eq. (6.60) into Eq. (6.54), it is easy to see that U^°'(x,y;t) = -secha; sechy Li

it 1 roo „ (ik - tanhx)(ifc + tanhy) (ik(x - y) - — (k2 + 1)) — — / dk-±— e I 2TT •/-OO

1 + k2

This determines the transition ampUtude for t h e original system.

6.5. REFERENCES

6.5

References

Das, A. and W . J. Huang, Phys. Rev. D 4 1 , 3241 (1990). Gendenshtein, L., J E T P Lett. 38, 356 (1983). Nicolai, H., Nucl. Phys. B170, 419 (1980). W i t t e n , E., Nucl. Phys. B202, 253 (1982).

153

Chapter 7 Semi-Classical M e t h o d s

7.1

W K B Approximation

As we know, most quantum systems cannot be solved analytically. In such a case, of course, we use perturbation theory and perturbation theory brings out many interesting properties of the system. However, by definition, perturbation theory cannot provide information about nonperturbative aspects of the theory. For example, the Born approximations used in scattering theory give more accurate estimates of the scattering amplitudes as we go to higher orders of perturbation, but we cannot obtain information on the bound states of the system from this analysis. Similarly, even though we may be able to obtain the energy levels and the eigenstates for the motion 155

156

CHAPTER 7. SEMI-CLASSICAL METHODS

of a particle in a potential well by using p e r t u r b a t i o n theory, we will never learn a b o u t barrier p e n e t r a t i o n from such an analysis. These are inherently nonperturbative p h e n o m e n a . It is, therefore, useful to develop an approximation scheme which brings out some of these n o n p e r t u r b a t i v e characteristics. W K B is such an approximation scheme. T h e basic idea behind this is quite simple. Let us assume t h a t we have a particle moving in a complicated potential V(x).

T h e n , the stationary states of t h e system will satisfy

t h e time-independent Schrodinger equation given by

Here E is a constant representing t h e energy of t h e s t a t e . We know t h a t if t h e potential were a constant, namely, if

V(x)

= V = constant

t h e n , t h e solutions of Eq. (7.1) will be plane waves (for E >

V).

Namely,

i>(x) = A e

±— px h

where p = y/2m(E

-

V)

(7.2)

7.1. WKB APPROXIMATION

157

When the potential changes with the coordinate, but changes slowly, then it is easy to convince ourselves that within a region where the potential does not change appreciably, the solutions of Eq. (7.1) can still be written as plane waves of the form of Eq. (7.2) with p{x) = sj2m{E - V(x))

(7.3)

It is clear, therefore, that we can try a general solution to the timeindependent Schrodinger equation of the form i>{x) = NA(x)eiB(x)

(7.4)

where N is a normalization constant and furthermore, noting that we can write (for nonnegative A(x))

A(x) =

elnA(x)

we conclude that the general solution of Eq. (7.1) can be represented as a phase where the phase, in general, is complex.

Since

the Schrodinger operator depends on h, the phase clearly will be a function of h. With all these information, let us write the general solution of the time-independent Schrodinger equation to have the form

i>(x) = Netiy

'

(7.5)

158

CHAPTER 7. SEMI-CLASSICAL METHODS

If we substitute this wave function back into the Schrodinger equation (Eq. (7.1)), we obtain d2tp(x)

2m + -gr(E-V(x))il>(x)

=0

or, (-^'(*)) 2 + ^ » + jr(E - v(x))) **m or, [-^'{x)f

+l-"{x)+ *g(E

=o

- V{x))} = 0

(7.6)

So far, everything has been exact. Let us next assume a power series expansion for <j>(x) of the form (x) = Mx)

+ H\{x)

+ h2(j)2{x) + •••

(7.7)

It is clear, then, that (j)o(x) will correspond to the classical phase since that is what will survive in the limit h —• 0. Other terms in the series, therefore, represent quantum corrections to the classical phase. Substituting the power series back into Eq. (7.6), we obtain 1

[-(<#,(*) + H\{x)

+ • • -)2 + 2m(E -

+±(
V(x))]

+ ...) = 0

or, ^ ( - ( < # , ( * ) ) 2 + 2 m ( £ - V ( a O ) ) + 1 (i[(x)) + O(h0) = 0

(7.8)

7.1. WKB APPROXIMATION

159

For this to be true, the coefficients of the individual terms in Eq. (7.8) must be zero. Equating the coefficient of the A term to zero, we obtain

-(^(x))2

+ 2m(E-V(x))

or, (ct>l0(x))2 = 2m(E-V(x)) or,

= 0 p2(x)

=

<j>'0(x) = ±p(x) or,

4>Q(x) = ± f dx'p(x')

(7.9)

JXo

Here p(x) is the momentum of the particle at the point x, defined by Eq. (7.3), corresponding to motion with energy E. Furthermore, let us note that even though both the signs of the solution are allowed in Eq. (7.9), consistency of the subsequent relations will pick out only the positive sign. This, therefore, determines the classical phase to be

Mx)

= [X dx'p(x')

(7.10)

JXo

If we keep only the leading order term in the expansion of <j)(x), then the wave function would have the form

—4>a(x) —/ •4>(x) = Neh^y ' = NehJx0

dx'p(x') vy

'

(7.n)

160

CHAPTER 7. SEMI-CLASSICAL METHODS

The time-dependent stationary wave function, in this case, would be given by --Et = e n

ip(x,t)

ip(x)

^Et + UXdx'p(x')

= N e ft.h %

=

Neh

hJx JXn0

ft.

et

Jo dt1 (px - E)

= NeKS[Xd]

(7.12)

This is exactly what we would have expected in the classical limit. Let us next include the first order correction to the phase. Setting the coefficient of the | term in Eq. (7.8) to zero, we obtain

i^(x)

- 2#,( a J )#(a J ) = 0 i(

t>o(x)

OI

or,

'

X)=

^

2Ux)

*n_j.i,_\\i

=

2(ln^(:C))

0 1 (x) = - b i 0 o ( x ) = - l n p ( x )

(7.13)

The constant of integration in Eq. (7.13) can be absorbed into N. We also note here that this selects out the positive root for <j>o(x) in Eq. (7.9).

7.1. WKB APPROXIMATION

161

Thus, keeping up to the first order correction to the classical phase, we can write the wave function to be

.. .

AT

ip(x) ~ =

TMx) + hi(x))

Nen Uf Neh

dx1 p(x') +

%

—\np(x))

NeyxdX'P(X') \/P(x) This is known as the WKB approximation for the quantum mechanical wave function of the system.

This approximation, clearly, breaks

down for small p(x) and in particular when

p(x) = 0

(7.15)

Namely, at the classical turning points, the classical momentum vanishes and consequently, in these regions, the WKB approximation breaks down and we must examine the Schrodinger equation and its solutions more carefully. From the form of the WKB wave function in Eq. (7.14), we note that

162

CHAPTER 7. SEMI-CLASSICAL METHODS

i/>*(x)1>(x) oc - L p(x)

(7.16)

In other words, the probability density, in this case, is inversely proportional to the momentum or the velocity. This is, of course, what we would expect from classical considerations alone. Namely, classically, we would expect qualitatively that the system is more likely to be found at points where its velocity is smaller. Thus, the WKB approximation gives us a quantum wave function which retains some of the classical properties. It is for this reason that the WKB approximation is often also called the semi-classical approximation.

Xa

X})

X

Let us consider a particle moving in a potential of the form shown above. Then, the normalization constant for the WKB wave function can be determined approximately in the following way. Since the wave function damps outside the well, we can write the normalization condition to be approximately

7.1. WKB APPROXIMATION

163

fXb

OO

/

dxi/j*(x)il)(x)

~

/

dxifj*(x)ip(x) da; = 1

- I*I*JC

(7.17)

Recalling that the classical period of oscillation is given by

T = 2

-—=

rxb

2m|

dx

p(x)

(7.18)

we obtain \N\2—

or,

=

N = N* =

2m

(7.19)

Therefore, we can write the normalized WKB wave function to be

ip(x) ~

2m Tp(x) 2-K

T /

dx'p(x')

u)

-z I dx' p(x') FK eh J > •Kv(x)

(7.20)

where w = — denotes the classical angular frequency of motion.

164

7.2

CHAPTER 7. SEMI-CLASSICAL METHODS

Saddle Point Method Let us consider an integral of the form /•oo I=f

dxeay

—f(x) '

(7.21)

J—oo

where a is a very small constant. Furthermore, let us assume that the function f{x) has an extremum at x = xo which is a maximum. In other words, we are assuming that

/'(*)U 0 = o / » U

0

< 0

(7.22)

We can now expand the function around this extremum as f(x) = / ( x 0 ) + \{x - x 0 ) 2 /"(xo) + 0((x - x 0 ) 3 )

(7.23)

Substituting this back into the integral in Eq. (7.21), we obtain

165

7.2. SADDLE POINT METHOD

I =

too

["(/(*<>) + \{x - x0)2f"(x0) A axe a *>

+ 0((x - x0)3))]

J — oo

-/(*„) , » A {-Ux = e& axe ta

- x,f\f{x,)\

+ 0((x - x0)3))

J—oo

=

-/(xo) ,00 (-^2|/"(^o)| + 0 ( ^ efl / dyy/ae l

3

))

J—oo

~

-f(xo) ea

2TT

va

2-Ka

N \f"(x0)\

N l/"(*o)

1 ~f(xo) ea

(7.24)

It is easy to see that the terms we neglected are higher order in a and, therefore, this is the most dominant contribution to the integral.

jTntf

Note from Eq. (7.23) that if we consider the function in the complex x-plane, then along the imaginary axis it has a minimum at the extremal point. Therefore, the extremal point is really a saddle point in the complex x-plane.

Hence the name, the saddle point

CHAPTER 7. SEMI-CLASSICAL METHODS

166

method. Let us also note that the direction of our integration has been along the direction of steepest descent (along real axis) and, therefore, this method of evaluating the integral is also referred to as the method of steepest descent. It is worth emphasizing here that if the function in the exponent has several extrema, then the value of the integral will approximately equal the sum of the contributions around each of the extrema. This method is quite useful in obtaining approximate values of very complicated integrals. As an example, let us analyze the Gamma function for large values of the argument. The Gamma function has the integral representation given by j°°dxxne~x

r ( n + l) = n! =

= [°°dxe(-x Jo ,00

=

+

nl x

*)

n(lna;

dxe

)

n

(7.25)

Jo Let us next assume that n is very large. This, then, would correspond to ^

m

°ur previous discussion in Eq. (7.21). In the present case, fix)

= In a; n

/(,)

= I -i

(7.26)

7.2. SADDLE POINT METHOD

167

Requiring the first derivative to vanish gives XQ

/(x0)=lnx0-^

=

n

= (lnn-1)

(7.27)

Furthermore,

/"(*) = Ax2

(7-28)

and, therefore, we have /"(so) = ~

<0

(7.29)

Therefore, x0 — n is a maximum of the function and, in this case, is the only extremum. Thus, for large n, we can write

r ( n + l) -

e»/(*o)jrefcen<-^<X-*0>a>

enf(**)rdxe-k{x-Xo)2

= Jo

enf(**)rdxe-h{x-X°)2

* J—oo

= or, n<

V^i^en(lnn-1) \/27rn f - J

for large n

(7.30)

168

CHAPTER 7. SEMI-CLASSICAL METHODS

(Since XQ = n is large and positive, we have extended the integration to the negative axis in the intermediate step because the contribution from this region is negligible.) This is known as the Stirling's approximation and holds when n is large. Sometimes, this is also written in the form Inn! ~ n(lnn — 1)

7.3

for large n.

(7-31)

Semi-Classical M e t h o d s in P a t h Integrals The saddle point method or the method of steepest de-

scent can be applied to path integrals as well. Note that so far we have only evaluated path integrals which involve quadratic actions. But any realistic theory will involve interactions which are inherently nonlinear. The path integral cannot always be evaluated exactly for such systems and the method of steepest descent gives rise to a very useful approximation in such a case. Let us consider a general action S[x] which is not necessarily quadratic. The transition amplitude associated with this action, as we have seen, is given by (see Eq. (2.28))

(xf^flxiiU) = N JVxeh

(7.32)

7.3. SEMI-CLASSICAL METHODS IN PATH INTEGRALS

169

Since h is a small parameter, this integral is similar to the one in Eq. (7.21), but is not exactly in the same form . Namely, this integral is oscillatory. However, we know that, in such a case, we can rotate to the Euclidean space as we have discussed earlier and the integrand becomes well behaved. We will continue to use the real time description keeping in mind the fact that in all our discussions, we are assuming that the actual calculations are always done in the Euclidean space and then the results are rotated back to Minkowski space. Let us recall that the classical trajectory satisfies the equation (see Eq. (1.28) or (3.9)) 6S\x\ 6x(t) v /

x=xci

Therefore, it provides an extremum of the exponent in the path integral. Furthermore, the action is a minimum for the classical trajectory. Thus, we can expand the action around the classical trajectory. Namely, let us define x(t) = xd(t) + v(t) Then, we have S[x] = S[xj + T,}

(7.33)

170

CHAPTER 7. SEMI-CLASSICAL METHODS

Substituting this back into the transition amplitude in Eq. (7.32), we obtain (Xf,tf\xi,ti)

= NjvJ{S[Xc'] ^II^^^M^)^ Vr)e2n

~ Neft

M

°"

oxci{t1)bxci(t2)

N

1

+ ( 3)

TS[XC1]

&S[xd\

(7.35)

K

h8xcl(u)8xd{t2y

where we have used the result of Eq. (4.2). (We are also keeping the ft term explicitly to bring out the quantum nature of the calculations.) Clearly, the saddle point method breaks down if , det

62S[xcl] ; r— — r = 0 oxcl(ti)bxci{t2)

Normally, this happens when there is some symmetry or its spontaneous breakdown occuring in the theory. When this happens, the path integral has to be evaluated more carefully using the method of collective coordinates as we will discuss in the next chapter. We note here that the form of the transition amplitude

7.3. SEMI-CLASSICAL METHODS IN PATH INTEGRALS

171

in the saddle point approximation in Eq. (7.35) is surprisingly similar in form to the WKB wave function in Eq. (7.20). (Recall from Eq. (1.35) that the transition amplitude is a Schrodinger wave function for a delta function source.) In fact, the phases are identical (see Eq. (7.12)). It is the multiplying factors that we readily donot see to be comparable. We will describe below, without going into too much detail, how we can, in fact, relate the two multiplying factors as well. Let us note that a general action has the form S[x] = j ' ' dt (^mx2 - V(x)\

(7.36)

For simplicity, we will choose ti = — ^ and tf = ^ which will also be useful later. In this case, we note that 62S[xd]

Sxd^Sxdfa)

= - (m^-2

\

dt\

+ V"(xcl)) 6(tl - t2)

(7.37)

Therefore, we see that det(j-

f^ [ f c ' ] ,

J oc detfhm^

+ V"(xd)))

(7.38)

Thus, we are interested in evaluating determinants of the form det(^(m^ +

W(x)))

172

CHAPTER 7. SEMI-CLASSICAL METHODS

in the space of functions where ~q(^) = 0 = 7?(—f-)- We have already evaluated such determinants earlier in connection with t h e harmonic oscillator (see Eq. (4.10)).

Let us recall here some general results

which hold for determinants of operators containing b o u n d e d potentials. W i t h a little bit of analysis, it is possible to show t h a t , if

±{m^

+ W(x))1®

= \1,$>

with t h e initial value conditions V'vr( — f) 1 d2 d e t ( - ( m — + Wiix))

n where W\{x)

and W2(x)

=

0

an

- A)

(7.39)

d i>w{~f)

=

1> then

J,W(T)

at* are two b o u n d e d potentials. We note t h a t

for A coinciding with an eigenvalue of one of t h e operators, t h e left h a n d side will have a zero or a pole. B u t so will the right h a n d side for t h e same value of A because in such a case ip would correspond to an energy eigenfunction satisfying t h e b o u n d a r y conditions at the end points. Since b o t h the left a n d the right h a n d side of t h e above equation are entire functions of A with identical zeroes a n d poles, they must be equal. It follows from this result t h a t

det

& T O |s + Wl{x))) h

<*f±_

1*(f)

det(

=

^m^ + S


^!(f)

W

*W = constant (7.41

7.3. SEMI-CLASSICAL METHODS IN PATH INTEGRALS

173

That is, this ratio is independent of the particular form of the potential W(x) and can be used to define the normalization constant in the path integral. We will define a particular normalization later. But, for the moment, let us use this result to write the transition amplitude in Eq. (7.35) in the form

N

/ , i ,\ (x{,tf\xhti)

=

^et(^m^-2

N

lS^ +

V"(xcl)))

eri

i5M

(7.42)

i*kf) Let us note here that by definition (see Eq. (7.39)), ipv»(t) satisfies the equation (it is an eigenstate with zero eigenvalue)

(m~

+ V"(xcl))^l(t)

= 0

(7.43)

We note here that the classical equations (Euler-Lagrange equations) following from our action have the form (see Eq. (1.28))

It follows from this that

174

CHAPTER 7. SEMI-CLASSICAL METHODS

d2 ,dxci or

m

{

' dT> -dT or,

,dxcl

T/H/

) + VM

(rn^

lH

= °

+ VM)^

= 0

(7.45)

Comparing with Eq. (7.43), we readily identify that

V$(0«^«pOrcJ)

(7.46)

Consequently, we recognize that we can write the transition amplitude in Eq. (7.42) also in the form T T N (*f,j\*i,-±) = - r r

2

2

i

e

h

=S\xci] (7-47)

vw)

The correspondence with the WKB wave function in Eq. (7.20) is now complete and we recognize that the method of steepest descent merely gives the WKB approximation.

7.4

Double Well Potential As an application of the WKB method, let us consider a

particle moving in one-dimension in an anharmonic potential of the

7.4. DOUBLE WELL POTENTIAL

175

form V(x) = | ( * 2 - a 2 ) 2

(7.48)

where g and a are constants. Consequently, the action has the form S[x] = [dt (\mx2

- ^-(x2 - a2)2)

(7.49)

This potential, when plotted, has the shape of a double well.

a

—a

x-

This is a very interesting potential and shows up in all branches of physics in different forms. Note that it is an even potential with a local maximum at the origin. The two minima of the potential are symmetrically located at x = ±a The height of the potential at the origin is given by

(7.50)

176

CHAPTER 7. SEMI-CLASSICAL METHODS

V(x = 0) = i£

(7.51)

Let us also define here for later use V"(x = ±a) = g2a2 = W

(7.52)

where we can identify u with the natural angular frequency of harmonic oscillations near the minima. We note from Eq. (7.51) that for infinitely large coupling, the potential separates into two symmetrical wells with an infinite barrier. The motion of the particle is easy to analyze in this case. Each well has quantized levels of energy and if the particle is in one well, then it stays there forever.

Namely,

there will be no tunneling from one well to the other. Furthermore, from the symmetry of the problem at hand (namely, x *-> — x ), we conclude that both the wells in the present case will have degenerate energy levels. Thus, if ipo{x) denotes the ground state wave function of the well in the positive cc-axis with energy EQ, then tpo(-x)

will

describe the ground state wave function of the left well with the same energy. In fact, any linear combination of the two wave functions and, in particular, the combinations

if>i(x) = -j=(il>o{x) + *l>o(-x))

M*) = ^(.M*)-M-*))

( 7 - 53 )

7.4. DOUBLE WELL POTENTIAL

177

will also be degenerate in energy. When the coupling constant g is finite, then the potential barrier will be finite. Consequently, the particle initially confined to one well can tunnel into the other well and the states of the two wells will mix. The symmetry of the system (Hamiltonian) still dictates that the eigenstates of the Hamiltonian can only be even and odd linear combinations as described above. They will, however, not be degenerate in energy any longer because of tunneling and we wish to calculate the splitting in the energy levels due to tunneling using the WKB approximation. Let us note that ipi(x) is a symmetric wave function whereas ip2(x) is anti-symmetric. Consequently, it is obvious that ^i(a;) would represent the ground state of the system (after taking tunneling into account). Let us write down the time-independent Schrodinger equations that various wave functions satisfy.

£ME>

+

^

+ ! ? ( £ , - F(*)W>,(s) = 0

(7.55)

+ £ < * - V M ) * M = 0

(7.56)

^

^

*to,-V(x))*

W

..0

(7.54)

178

CHAPTER 7. SEMI-CLASSICAL METHODS

Furthermore, let us assume, for simplicity, that the wave functions are all real. If ipo(%) denotes the wave function in the well I - that is, the well on the right - then, we can easily see that its value will be vanishingly small in the well II, or the well on the left. Thus, we can normalize the wave function as j°°dxil>l(x)~l

(7.57)

Let us note similarly that if ipo(-x) denotes the wave function in the well II, then it would have vanishingly small value in well I. Consequently, a product such as tp0(x)tjJo(—x) will be negligible everywhere. We see from Eq. (7.53), therefore, that M*)M*) * ^

ri(*)

(7-58)

so that /o°° dx M*)M*)

- -^ /0°° dx V-o(z) ^ -^

(7-59)

Let us next multiply Eq. (7.54) by i>\{x) and Eq. (7.55) by ^>o0*0

M x )

an(

f ^ l

i subtract one from the other. This gives

_ V o ( * ) ^ ^ + f ? ( £ o - 2 W i ( * ) * , ( * ) = 0 (7.60)

Integrating this equation and using Eq. (7.59), we obtain

7.4. DOUBLE WELL POTENTIAL

179

2m

+ - ^ ( £ 0 - £7x) /o°° dz V i ^ ) ^ ) = 0

or, ^(£0 - Ex)± = - (M*W*) ' M*W*))\? = ^i(0)^(0)-Vo(0)V'i(0) or,

£0 _ ^ =

* (V-i(O)^(O) - Vo W i ( 0 ) ) m\/2

(7.61)

Let us note that since

Mx) = -fiiM*) + M~x)) ^i(O) =

>/2^o(0)

V-i(O) = 0

(7.62)

Substituting this back into Eq. (7.61), we obtain

E0-E1

= - ^ = V^Vo(0)Vi(0) = —Vo(0)Vi(0)

(7-63)

It is here that we would like to use the WKB approximation. We recall from Eq. (7.20) that we can write the WKB wave function as

180

CHAPTER 7. SEMI-CLASSICAL METHODS

V-o(O) =

*(0) *

_^_ -fif^l^)!

\ ?™(o)

^)^(O)

(7.64)

Here, we note that

u(0) = ^ - ( ^ ( 0 ) - £o)

V(0) »

£„

(7.65)

Putting these back, we obtain

EQ

— E\

=

h2mv(Q) m h

= hv(o)

(Mo))2

UJ

f°dx lp(a

irv(0) (7.66)

This is the splitting in the energy level of the true ground state from the case of the infinite well. We can similarly show that 2 huj - - Jo dx |p(a E2 - E0 = — e h 7T

and, consequently,

(7.67)

181

7.4. DOUBLE WELL POTENTIAL

&-* = *£,-&

****)]

(768)

7T

which gives the splitting between the two degenerate levels in this approximation. The damping exponential in Eq. (7.68) reflects the effects of tunneling. In fact, note that because of the reflection symmetry in the problem, we can write 2i

ta

1

?&

- - / dx \p(x)\n - - / dx \p(x)\ e hJo '^ =e hJ-a ^K n

(7.69)

and this gives the coefficient for tunneling from the minimum at x = —a to the one at x = a. This, of course, assumes that the particle under consideration has vanishing energy. In reality, however, we know that the quantum mechanical ground state energy is nonzero in general. In fact, if we approximate the potential near each of the minima by a harmonic oscillator potential, then we can identify the ground state energy of the system with £o = ~

(7.70)

This would, then, imply that the turning points for motion in both the wells, in this case, are given by

182

CHAPTER 7. SEMI-CLASSICAL METHODS

1

,/

r-f

\2

- mu (x - a)

^

= E0 = —

or, x — a = ±<

or,

xi

h muj

h

= a±

muj

(7.71)

and, similarly, mtjj2{x + a ) 2 =

EQ =

or, a; + a = ±<

or, x / 7

ftu

mu

= -a±

ft mu

(7.72)

The tunneling from one well to the other, in this case, therefore, would correspond to tunneling from —a + J— to a

A. mu

7.4. DOUBLE WELL POTENTIAL

183

Correspondingly, for a more accurate estimation of the splitting in the energy levels, we should replace the exponential in Eq. (7.68) by

[ dx" \p(x)\ _^ - - /o"

mu

inr* i\p(x)\ r > i T* 11 dx

(7.73)

Furthermore, recalling that (see Eq. (7.3)) \p(x)\ = y/2m(V(x)

- EQ)

we can evaluate the exponent in a straightforward manner as

J0

m

"dx\p(x)\ haj x

=

~ _

r«->/i!i; , mu(a2 - x2) X •/o 2a r ^ Jo

dx (mL0{a2 ~ ^ 2a

dX[

{

4a2h i 2 2 2 muj(a -x y> ^—) {a2-x2)>

This integral can be trivially done and has the value

l

(7 74) ^4j

184

CHAPTER 7. SEMI-CLASSICAL METHODS

ra-v J0 ^

dx

\P(X)

muia2

h

3

2

+ _h,l n 2

h

ft

,

A muja

mii>aj

ls°+ll«\^+0^

;)+om (7.75)

Here we have denned 2muia2

(7.76)

Substituting E q s . (7.73) a n d (7.76) into Eq. (7.68), we obtain the splitting in t h e energy levels to be

E2-Ei

~

2hcj

4mwa 2 e 2 -^r^o \ z e Ti

4e /—- 3 -TSO — vmnujia e n

(7.77)

TV

In t h e next chapter, we will calculate this energy splitting using t h e p a t h integrals and compare t h e two results.

7.5. REFERENCES

7.5

185

References

Coleman, S., "The Uses of Instantons", Erice Lectures, 1977. Landau, L.D. and L.M. Lifshitz, "Nonrelativistic Quantum Mechanics", Pergamon Press. Migdal, A . B . and V . Krainov, "Approximation Methods in Quantum Mechanics", Benjamin Publishing. Sakita, B., "Quantum Theory of Many Variable Systems and Fields", World Scientific Publishing.

Chapter 8 P a t h Integral for the Double Well

8.1

Instantons Let us next try to do the path integral for the double well

potential. We recall from Eq. (2.28) that the transition amplitude is defined as T T -%-HT , {xf,-\xu--) = {xf\e h \Xi)=NjVxeh

%

-S\x\ (8.1)

where for a double well potential (see Eq. (7.49)) the action is given by S[x] =

(\dt(\mx2-V{x))

= j^dt(\m#-£{x*-a*)*) 187

(8.2)

188

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

As we have seen earlier in chapter 4, the best way to evaluate the path integral is to go to the Euclidean space. Thus, by rotating to imaginary time t —•

-it

we obtain using Eq. (1.42)

-THT (xf\e

h

,

-lsE[x]

\Xi)=NJVxe

h ^

J

(8.3)

where the Euclidean action has the form SE[x] = \ \ dt^mx2

+ V(x))

(8.4)

In the semi-classical method, we can evaluate the path integral by the saddle point method. The classical equation which is obtained from the extremum of the action in Eq. (8.4) has the form SSE[x] 6x(t) or, mx - V'(x)

=

= 0

(8.5)

V(x) = | V - a2)2

(8.6)

with

8.1.

INSTANTONS

189

The Euclidean equations, therefore, correspond to a particle moving in an inverted potential otherwise also known as a double humped potential. The Euclidean energy associated with such a motion is given by E = -mx2

- V(x)

—a

a

(8.7)

a:-*.

Two solutions to the Euclidean classical equation of motion in Eq. (8.5) with minimum energy are obvious. Namely, x(t) = ±a

(8.8)

satisfy the classical equation with E = 0. In other words, the particle stays at rest on top of one of the hills in such a case.

Quantum

mechanically, this would correspond to the case where the particle executes small oscillations at the bottom of either of the wells in the

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

190

Minkowski space and these small oscillations can be approximated by a harmonic oscillator motion. However, in the large T limit, (namely, when T —»• oo) in which we are ultimately interested in, there are nontrivial solutions to the Euclidean equation of motion in Eq. (8.5) which play a dominant role in evaluating the path integral. Let ~ c'

xd(t) = ±a tanh ^

(8.9)

Li

where tc is a constant and we have identified as in Eq. (7.52) mu2 = V"(±a) = g2a2

(8.10)

Then, we see that xcl

acu j ^ t ' — *c) = ±—sech

= i ^ r i 1 - TJ) = ^^\xci I

or

..

_

w

=

^p —

a

)

la 2V(xd)

^W-^N Xc\

-

(8.11)

m

. 1XC\XC\

la a u2

= -V'(xcl) m

la a2

(8.12)

8.1.

INSTANTONS

191

Consequently, it follows that mxcl - V\xcl) = 0

(8.13)

and we conclude that the solutions in Eq. (8.9) represent nontrivial solutions of the Euclidean equation of motion, namely, Eq. (8.5). We also note that, for these solutions,

xci{t —• — oo) = =pa xci(t -> oo) = ± a

a-

J

f

tc

(8.14)

-a \ t~~

—a

—a-

tc

v-

These solutions, therefore, correspond to the particle starting out on one of the hill tops at t —> — oo and then moving over to the other hill top at t —• oo. Let us also note (see Eq. (8.11)) that for such solutions 1 1 2 -mx2cl = - m - V(xcl) = V(xc}) z

46

in

(8.15)

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

192

Therefore, for such motions, the Euclidean energy defined in Eq. (8.7) has the value E = ^mx2cl-V(xc!)

= 0

(8.16)

In other words, these also correspond to minimum energy solutions like t h e trivial motion. T h e action corresponding to such a classical motion can b e easily calculated.

SE[xci] = S0

=

dt(-mx2cl

I J—oo oo

/ -oo

=

V(xcl))

„

/-±a

dt mx., = m la

T

1 6

dxc\ xc\

J^a

m J f a a dxci(T—(x2cl mui la

=

+

£

3

2 , 2m2uj3 - mua = 3 Zg2

-

a2))

±a

_ mw 4a 3

2 Ta

2a

3 (8.17)

where we have used E q . (8.11). We note t h a t this action h a s t h e same value as the 5o defined in Eq. (7.76) in connection with t h e W K B calculation of t h e splitting of the energy levels for t h e double well. T h e solutions in Eq. (8.9) are, therefore, finite action solutions in t h e Euclidean space a n d have t h e graphical form as shown above. T h e y are known respectively as t h e instanton a n d the anti-instanton

8.1.

INSTANTONS

193

solutions (classical solutions in Euchdean time). If we look at the Lagrangian for such a solution, then we find, using Eq. (8.11), that

LE = -^mx2cl + V(xci) = mx2cl = =

m ^ s e c ^ - ^ )

K

4

2

'

ma2u2 , 4uj(t — tc) sech — 4 2

J

(8.18)

\^ I tc

t~*

In other words, the Lagrangian is fairly localized around t

tc with

a size of about

u>

ga

(8.19)

It is in this sense that one says that instantons are localized solutions in time with a size of about - . The constant tc which signifies the time when the solution reaches the valley of the Euchdean potential

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

194

is really arbitrary. This is a direct reflection of the time translation invariance in the theory. Just as we can have a one instanton or one anti-instanton solution, we can also have multi-instanton solutions in such a theory. However, before going into this, let us calculate the contribution to the transition amplitude coming from the one instanton or anti-instanton trajectory.

From our earlier discussion in Eq. (7.35) of the saddle

point method in connection with path integrals, we conclude that (O.I. stands for One Instanton)

-\HT (a\e h =

r^i

| - a)o.i.

N JDxe Ne

h

—ZSE[X]

n

M

Jvv

-y«^i$$^

N

e-hSo

(8.20)

d e t ( | ( - m f + V"{xcl)))0.i Here we have defined x(t) = xd(t) + t](t)

(8.21)

and, for the one instanton case, as we have already seen in Eq. (8.9),

8.1.

195

INSTANTONS

xcl(t -tc)

= a tanh ^ — ^

(8.22)

Let us next analyze the determinant in Eq. (8.20), for the one instanton case, in a bit more detail. We know from Eq. (8.11) that dxj dt

=

au 2

Mt

- U) 2

( v

23)

'

satisfies the zero eigenvalue equation (see Eq. (7.45) rotated to imaginary time)

In fact, using Eq. (8.17), we can define the normalized zero eigenvalue solution of Eq. (8.24) as

As we had seen earlier in Eq. (7.41), the determinant in Eq. (8.20) can be obtained from this solution simply as det(i(-m ^

+ V"(xcl))) oc V o ( | )

T - oo

(8.26)

But from the form of the solution in Eq. (8.9) or (8.14), it is clear that

196

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

Hm V, 0 (|) _ > 0

(8.27)

In other words, in this case, the determinant identically vanishes. The reason for this is obvious, namely, ipo{t), in the present case, happens to be an exact eigenstate of the operator (—m-^ + V"(xci)) with zero eigenvalue. (This means that ^o(±|-) = 0 for T —• oo.) This is what one means in saying that there is a zero mode in this theory.

8.2

Zero Modes As we have argued before, a zero mode is present in the

theory whenever there is a symmetry operative in the system. To see this, let us recall that the determinant in Eq. (8.20) arose from integrating out the Gaussian fluctuations. Therefore, the term that we need to re-examine is

[Vrie

2hJJ

6xci(h)6xcl(t2)

( 8 2 8)

Note that, in this case, since tpo(t) represents a zero mode of the operator

Sx

1 % ^ \, if we make a change of the integration variable

as 6rj(t) = eV>o(0

(8.29)

8.2. ZERO MODES

197

where e is a constant parameter, then the Gaussian does not change. In other words, the transformation in Eq. (8.29) defines a symmetry of the quadratic action. Another way to visuaUze the trouble is to note that if we were to expand the fluctuations around the classical trajectory in a complete basis of the eigenstates of the operator

Sx

,t ^

,f >,

then we can write

v(t) = £ cnipn(t) n>0

VV(t)

=

I I dcn

(8.30)

n>0

Substituting this expansion into Eq. (8.28), we obtain

fvVe~^

!! dtldh

V{tl)

^M^)Vit2)

- ^ = / I I dcne J

V A c2 n>o

2n

n>0

r

t

~ Ofe ^

= JdcoJ II

dc

«e

m

">°

AnCn

(8.31)

n>0

where A„ denotes the eigenvalues corresponding to the eigenstates ij)n. Here we note that the zero mode drops out of the exponent and, consequently, there is no Gaussian damping for the dc0 integration.

198

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

In such a case, we have to evaluate the integral more carefully. To understand further the origin of the problem, let us examine a simple two dimensional integral. Let

&x\dxiea

I — // J J—oo

y

-(-X2

= 11°° dx1dx2ea 2

-

Q2(x2)2)

y v

''

(8.32)

Here a is a small parameter and we have defined f(x)

=

\S2-9\S2)2

x2 = x\ + x22

(8.33)

This example is, in fact, quite analogous to the instanton calculation. The classical equations, in this case, lead to the maximum

g = *(i-V*) = o or,

\xcl\ = - ^ 2ff

(8.34)

It is easy to see that the other solution, namely, the origin, in this case, corresponds to a local minimum. The most general classical solution following from Eq. (8.34) can, therefore, be written as

199

8.2. ZERO MODES

1 Xlcl

— — COS6>

2# X2d = 7T sinO 2g

(8.35)

where 6 is an arbitrary constant angular parameter.

This is very

much like the arbitrary parameter tc which arises in the case of the instantons. In the present case, the presence of the angular parameter, 9, is a consequence of the rotational invariance of the function f(x) in Eq. (8.33). Expanding around the classical solution, namely, choosing

xa = xaiCi + r]a

a = 1,2

(8.36)

we have

f(x) = f(xcl + ff) ~ f(xcl) + L ^ g ^ 2 dxddxd

+ 0(V3)

(8.37)

From the form of f(x) in Eq. (8.33), we note that

dxddxd = -8g2x*xd

(8.38)

200

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

In t h e m a t r i x form, therefore, we can write d2f(xcl)

= -2

dx%dxPd

cos 2 9

cos 6 sin 6

cos 8 sin 6

sin 2 6

(8.39)

T h u s , if we use t h e saddle point m e t h o d naively, we would obtain t h e value of t h e integral in Eq. (8.32) t o be !

1

I~ea

d2f(xcl) ox ox

I drjae

d cl

Let us note t h a t the m a t r i x

(8.40)

}*cl} in Eq. (8.39) has two

eigenvalues, A = 0, —2. As a result, t h e Gaussian integral in Eq. (8.40) does not exist. This is very much like t h e instanton calculation t h a t we did. In fact, it is easy to check from Eq. (8.39) t h a t the eigenstate with zero eigenvalue has t h e form

(8.41)

XQ

Consequently, under a transformation of the variables of integration of t h e form

ex0 XV2f

=

-2ge

(8.42)

201

8.2. ZERO MODES

which we recognize as an infinitesimal rotation, t h e quadratic exponent in Eq. (8.40) does not change. In fact, writing out the exponent completely, we have -f(Xcl) J~efl

--(77iCOS0 + 7? 2 sin0) 2 a

rrco // drjid^e

(8.43)

F u r t h e r m o r e , redefining t h e variables as jj! =

rji cos 6 + r\2 sin 8

rJ2 =

— Vi s m $ + V2 cos 6

(8.44)

we note t h a t we can write t h e integral in Eq. (8.43) also as

I

~

- / ( * c i ) /Y°° ,_ , . — m ea II dr)idri2e a

2

JJ — OO

=

ea

/ 7—oo

d^ 2 /

dr?ie

a

(8.45)

J—oo

T h e analogy with t h e instanton case is now complete. There is no damping for t h e dfJ2 integration.

T h a t is t h e origin of

the divergence a n d it is a consequence of rotational invariance in this case. In this simple example, t h e solution to the problem is obvious. Namely, since t h e function / ( # ) , in Eq. (8.33), is rotationally invaria n t , it is appropriate t o use circular (polar) coordinates. T h e angular

202

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

integral, in this case, can b e trivially performed giving a finite result after which t h e saddle point approximation can b e applied t o t h e radial integral which will have no zero m o d e . This m e t h o d generaUzes readily t o other systems with more degrees of freedom and is known as t h e m e t h o d of collective coordinates. This is w h a t we will t r y t o use in order t o evaluate t h e instanton integral.

8.3

The Instanton Integral In t h e case of t h e instanton, we have already seen in

Eq. (8.31) t h a t S2S[xcl)

/^e"^//AA^l)^S&,l('a) = J

which is divergent.

dc0

J

T[dcne

Zn

n>o

(8.46)

n>0

T h e divergence, in t h e present case, is a con-

sequence of time translation invariance, namely, t h e position of t h e center of t h e instanton can b e arbitrary. So, following our earlier discussion, we will like t o replace t h e dc0 integration by a n integration over t h e position of the center of the instanton. Let us discuss very briefly how this is done.

Let us recall t h a t expanding around t h e

8.3. THE INSTANTON

203

INTEGRAL

instanton trajectory yields x(t) or,

= xcl(t - tc) + rj(t - tc)

x(t + tc) = xcl{t) + t)(t) = xcl(t) + £ c„ V»„(t)

(8.47)

n>0

(Since the trajectory is independent of the center of the instanton trajectory, the fluctuations must balance out the tc dependence.) Multiplying Eq. (8.47) with ipo(t) and integrating over time, we obtain

t-

l'Tdtx(t

J

+ tc)tl>0(t)

—2

= f \ dt(xcl{t) + £ J

~i

M(£> V -« = c0

cnxjjn{t))^(t)

n>0

+ c0 T

(8.48)

The first term vanishes because it has the same value at both the limits. (It is worth emphasizing here that we are only interested in large T limits when all these results hold.) This simple analysis shows

204

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

that co = c0(tc)

(8.49)

Therefore, we can easily change the co-integration to an integration over tc. To obtain the Jacobian of this transformation to the leading order, let us consider an infinitesimal change in the path in Eq. (8.47) arising from a change in the coefficient of the zero mode. Namely, let

(8-5°)

6rj = 6coMt) where we assume that 6c0 is infinitesimal. In this case,

6x(t + tc) = SV = 6c0Mt)

= *co ( § )

2 d

^jp-

(8-51)

where we have used Eq. (8.25). However, we also note that this is precisely the change in the path that we would have obtained to leading order had we translated the center of the instanton as

(—9 \J ~\ Namely, in this case,

(8.52)

8.3. THE INSTANTON

INTEGRAL

205

Sx(t + tc) = x(t + tc + 8tc) - x(t + tc) .dx(t

a fco

+ tc) dt

.

(S0\~*dx(t \mj

is.yi

i^M

\m/

+ tc) dt

(8 . 53 )

dt

Thus, from Eq. (8.52), we note that to leading order, we can determine the Jacobian of the transformation from the integration variable c0 to tc to be dc0(tc) dtc

r^j

(SQY \m

(8.54)

A more direct way to arrive at this result is to note from Eq. (8.48) that since c0{tc)

i2Tdtx(t

=

+ tc)ip0(t)

J

~~2

dc0

-dZ

=

/•!

dx(t + tc)

dt

U —dT—Mt) dx(t + tc)

r% , ,dxei{t) •'-T at

„>0

dipn(t) at

(§) " j \ dttf(t)+ £ c„ j \ dt d^f> Mt) (8.55)

206

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

where we have used Eq. (8.25). The n = 0 term drops out in the second term because for n — 0, the integrand is a total derivative of V'oW which vanishes at both the hmits. It is clear, therefore, that to leading order (since ipo is normalized to unity), dc0(tc) dtr

(So\ (^j2+0(h)

(8.56)

Namely, we are using here the fact that the higher moments of a Gaussian of the kind that we are dealing with in Eq. (8.46) are higher orders in h. Thus, we are ready to do the determinant calculation now. We substitute Eq. (8.54) or (8.56) into Eq. (8.46) to obtain

i^-y^^^-^^w^ 1

\m)

-(2)

J—

" -

2

J n„>>00

/ l dt<

(8.57)

^ / d e t ' ( i ( - m f + V»(xcl)))

Here det stands for the value of the determinant of the operator without the zero mode. Let us also note here that even though the

207

8.4. EVALUATING THE DETERMINANT

dtc integral in Eq. (8.57) can be done trivially, we will leave it as it is for later purposes. Thus, from Eqs. (8.20) and (8.57) we obtain the form of the transition amplitude in the presence of an instanton to be 1

\HT

(a\e h

\-a)OJ.=

- ^hs 0 Nl^Ye V m ^

j\dte

(8.58)

/det'^-mJL + ^ M ) ) ^ 8.4

Evaluating t h e Determinant To evaluate det', let us define the quantity

A(E) (

=

'

m i - ^ t V M ) - t ) det(l(-mg + m ^ ) - £ )

,

n (E

" ;-E) 0) - E) n.(M

(8.59) '

X

where the determinant in the denominator corresponds to that of a free harmonic oscillator which we have already evaluated. It is easy to see again that both sides of Eq. (8.59) have the same analytic structure and, therefore must be equal. We note that A(E = 0) = 0

(8.60)

since there is a zero eigenvalue for the determinant in the numerator and further,

208

CHAPTER

8. PATH INTEGRAL

FOR THE DOUBLE

A(E = oo) = 1

WELL

(8.61)

If we eliminate the zero mode in Eq. (8.59) by dividing it out, then we obtain

dA(E) or,

=

lE

dE

=°

det'(l(-m^

+ V"(xcl)))

det(I(-mg + a;>))

(8.62)

Clearly, if we can evaluate the left hand side of Eq. (8.62), then we would have evaluated det' since we already know the value of the determinant for the harmonic oscillator. To evaluate this, let us consider the scattering problem for the Schrodinger equation \(-m^ or, (-mlL

+ V"(xel)W

=

Eif,

+ V"{xel))il> = hE?p

(8.63)

If we define the asymptotic solutions (Jost functions) as lim f+(t,E)

—

e~ikt

t—KX>

Urn f-(t,E) t—*•—oo

—+ eikt

(8.64)

209

8.4. EVALUATING THE DETERMINANT

where we identify hE

-u2

m

Km V"(xcl)

raw2

,2

—•

(8.65)

The Jost functions are two linearly independent solutions of the Schrodinger equation in Eq. (8.63) and consequently, any general solution can be written as a linear combination of the two. In particular, we can write Urn f+(t,E)

— • A+(E)eikt

+

B+(E)e-ikt

— • B(E)eikt

+ AJE)e-ikt

t—*—oo

Urn f-(t,E)

(8.66)

The linear independence of the Jost functions can be easily seen by calculating the Wronskian which has the value

W(f+(t,E),f-(t,E)) -

-

f (t na/-(*'E>

oMi^f

f+ E)

^ —~dt

= 2ik B+(E) = 2ik B-(E)

(f

F,

f {t E

df- - ' > (8.67)

where the equality in the last step results from evaluating the Wronskian at the two different time limits t —• ±oo. This, in fact, shows

210

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

that the two coefficients B+(E) and B-(E)

are identical. With a bit

more analysis, they can also be shown to be equal to A(JE'), namely, B+(E) = B-(E) = A(E)

(8.68)

Let us also note from Eq. (8.25) that the zero mode has the asymptotic form limiMi)

=

lim

( ^ r ^ s e c h wt

(££)

e^

2

^ f ^

= Ke^ut

(8.69)

where we have defined 2

K = 2auj{—)

(8.70)

Thus, from Eqs. (8.64), (8.65), (8.66) and (8.69), we note that we can identify (k(E = 0) =

-iu)

lim f±(t,E

= 0)=

\t\—>oo

Hm l ^ 0 ( i ) - ^

e^l*l

(8.71)

|<|—too A

Thus, comparing Eqs. (8.71) and (8.72) with the asymptotic form of the Jost functions in Eqs. (8.64) and (8.66), we conclude that A+(E

= 0) = 1

A_{E = 0) = 1

B+(E = 0) = 0

B_(E = 0) = 0

(8.72)

8.4. EVALUATING THE DETERMINANT

211

Consequently, we obtain A(E = 0) = B+(E = 0) = 0

(8.73)

a result which we already know. The asymptotic equations which the Jost functions satisfy (see Eqs. (8.63) and (8.65)) are -™d2fft2E)

~ V& - ™2)U(t,E) = 0

9 2 /

- (HE' - mu,2)f_(t, E') = 0

-m

^

g )

(8.74)

Multiplying the first of these equations by /_(£,£") and the second by f+(t,E)

and subtracting one from the other, we obtain

=

h(E-E')f+(t,E)f4t,E')

=

^(E-E')f+(t,E)f_(t,E>)

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

212

or, -W(f+(t,0),f_(t,E)) or

'

=

--Ef+(t,0)f„(t,E)

— U(t,o)f-(t,o)

^diW{f+{t>0)J-{t>E)) E=0

m

mK2

V-o(')

(8-75)

Integrating this equation between (—f,f) with T —> oo, we obtain

0

t=oo

7

J^oA^^' )' -^

mK2

(8.76)

On the other hand, from the asymptotic form of the Jost functions in Eq. (8.66), we see that

ul = (e-—ut (ikB.(E)etKl

ikt -

mi ikA_(E)-ikt\ e- )

+u>•,—wtt e-ul(B-(E)e%KlAkt -(ewtikeikt-ueut

+

—ikt\-lKl)) A_{E) e t= (8.77)

£*%=_„

from which we determine

jte^wr(/+('.°)./-('.jE0) =

2w

dB-(E) dE

= 2ui E=0

dA(E) dE

(8.78) E=0

213

8.4. EVALUATING THE DETERMINANT

Here we have used the identification in Eq. (8.68) as well as the results of Eq. (8.72). Comparing Eqs. (8.76) and (8.78), then, we obtain

2u

dE

E-0

dA(E) dE

or,

h mK2

dA(E)

h 2muK2

E=0

(8.79)

We, therefore, determine the ratio in Eq. (8.62) to be

drt'(JK-m£ + V^sc))) det(\(-m§ + u>2))

=

dA(E) dE

E=0

2TUUJK2

(8.80)

The one instanton contribution in Eq. (8.58) can, now, be explicitly determined and takes the form

(a\e

n

|-

a)o.i.

&*(*(-">& +"'))

N

^ / d e t ( I ( - m f + W»))N d e t ' ( | ( - m f +

s0y

-TSO

,f

V»(xcl)))

214

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

mu\

muj\

2 —

2

T

—

2mu

/5o 2

\


\~h

t

mu)\ 2

"Z^O

e

K e~hS°

n

/_! ^

j\_dtc (8.81)

dtr

where we have used Eq. (8.80) as well as the value of the p a t h integral for the harmonic oscillator given in Eq. (3.28) (rotated to E u c h d e a n space). We have also defined a new quantity, r, whose value using Eq. (8.70) is given by

2S0w \~h =

Ken

-T$o

1 2m -^So 23 2,A —— ui a e n \ n

(8.82)

T h e transition a m p h t u d e , in this case, separates into a product of two factors - one t h a t of a simple harmonic oscillator arising from the trivial solution of the Euclidean equation of motion a n d the second giving t h e t r u e contribution due to an instanton. We can, similarly, calculate t h e transition a m p h t u d e in the presence of an anti - instanton a n d it can be shown to be identical to the result obtained in Eq. (8.81).

8.5. MULTI-INSTANTON

8.5

CONTRIBUTIONS

215

Multi-Instanton Contributions As we h a d discussed earlier, a string of widely separated

instantons a n d anti-instantons also satisfies t h e Euclidean classical equation given in Eq. (8.5). T h e instanton density is small for weak coupling a n d in such a case these multi-instanton solutions will contribute to t h e transition amplitude as well and their contribution can be evaluated under an approximation commonly known as the dilute gas approximation. A typical example of a multi-instanton solution has the following form.


<2 — ^3

£4 — ^5

Let us consider a n-instanton solution with centers at 11,*2, ••• ,tn

satisfying

- -g < *n <
( 8 - 8 3)

In such a case, t h e integral over the centers of t h e instantons gives

216

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

/ A dh _zdt2.../ 2

d*„ = - r

2

(8.84)

I!

2

Furthermore, since the instantons and the anti-instantons are assumed to be noninteracting, their contributions to the transition amplitude will simply be multiplicative.

Thus, a n-"instanton"

solution will

contribute an amount (see Eqs. (8.81) and (8.84))

We have to recognize here that only an even number of instantons and anti-instantons can contribute to the transition amplitude of the form

-\HT {a\e h

-\HT \a)

or

(-a\e

h

\ - a)

(8.86)

Similarly, only a total of odd number of instantons and anti-instantons can contribute to transition amplitudes of the form 1 ~HT (a\e h

| - a) or

(-a\e

1 -THT ft \a)

(8.87)

Adding all such "instanton" contributions, we see from Eq. (8.85) that we will have

8.5. MULTI-INSTANTON

217

CONTRIBUTIONS

-\HT (-a\e

n

| - a)

V V vrft / =

f ^ V e ,

" coah(rr) LUJL

{muy

UaJ

2n!

—Y (

e

rp

e

rp

)

2

. ll™)\e<-')T 2 \

+e

TT^ /

+ e<

+ r)T

)

(8.88)

Similarly, we can show that for odd number of instanton contributions, we will obtain

(a\e~hHT\

= -(—j

- a) = {^f

(e

2

e~T~ sinh(rT)

_

c

2

)

(8.89)

If we identify the two low lying states of the Hamiltonian as |±), with energy eigenvalues E± respectively, then we note that by inserting a complete set of energy states we will obtain for large T

218

CHAPTER 8. PATH INTEGRAL FOR THE DOUBLE WELL

-\HT (—a\e n ~

{-a\e

=

e

| — a) ——TJT ——MT h | _ ) ( _ | _ a ) + (-o|e h | + ) ( + | - a)

ft " ( _ a | _ ) ( _ | _ a ) + e

+

ft

(_a|+)(+|-a)(8.90)

Comparing this with Eq. (8.88), then, we obtain E± = h(^±r)

(8.91)

Therefore, the spMtting between the two energy levels is obtained to be AE = E+ - £_ =

2ftr 2m

3

-T&o

= 2ft x %t -—(J2a e n \ ft 3

= W2mhujia

—rSo

e h

(8.92)

This spMtting of energy levels calculated in the path integral formaMsm can, then, be compared with the result obtained through the WKB approximation in Eq. (7.77).

8.6. REFERENCES

8.6

219

References

Coleman, S., "The Uses of Instantons", Erice Lectures, 1977. Sakita, B . , "Quantum Theory of Many Variable Systems and Fields", World Scientific Publishing. Shifman, M. et al, Sov. Phys. Usp. 25, 195 (1982). Zinn-Justin, J., "Quantum Field Theory and Critical Phenomena", Oxford Univ. Press.

Chapter 9 P a t h Integral for Relativistic Theories

9.1

Systems with M a n y Degrees of Freedom

Thus far, we have only discussed one particle systems. However, the method of path integrals generalizes readily to systems with many particles or systems with many degrees of freedom. Let us consider a system with n-degrees of freedom characterized by the coordinates xa(t),

a = 1,2, ••-,«.. These coordinates, for example,

can denote the coordinates of n-particles in one dimension or the coordinates of a single particle in re-dimensions. If S[x] denotes the appropriate action for the system (namely, if it describes the dynamics of the system), then the transition ampHtude in Eq. (2.28) can be 221

222

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

easily shown to generalize to

a

{xf,tf\xi,ti)=NjVx eh

—S\xa]

(9.1)

The action generically has the form S[xa} = ff

dtL(xa,xa)

(9.2)

and we are supposed to integrate over all paths starting at x" at t = ti and ending at xj at t = tj. We can also introduce appropriate sources, in this case, through the couplings S[xa, Ja] = S[xa] + J*' dt Ja(t)xa(t)

(9.3)

to define the transition amplitude in the presence of these sources as (see Eq. (4.40))

J

(xf,tf\xi,ti)

, = N JVxaeh

^S[xa,Ja] (9.4)

As we have seen earlier in Eqs. (4.38) and (4.50), this allows us to derive the various transition amplitudes or matrix elements in a simple manner. We can also define, as before, the vacuum to vacuum transition amplitude in the hmit of infinite time interval as (see section 4.4)

9.1. SYSTEMS WITH MANY DEGREES OF FREEDOM

Z[J] = (0\0)J = Nj

VxaehS^X

223

'J

]

(9.5)

where in the infinite time interval limit, the action in Eq. (9.3) has the form S[xa, Ja] = r

dt (L(xa, xa) + Ja(t)xa{t))

(9.6)

J—oo

and the integration over the paths in Eq. (9.5) has no end point restriction in the sense that the initial and the final coordinates of the paths can be chosen arbitrarily. The path integrals can also be extended to continuum field theories once we recognize that these theories describe physical systems with an infinite number of degrees of freedom. Thus, if <j>(x, t) is the basic variable of a 1 + 1 dimensional field theory, then the vacuum to vacuum transition amplitude in the presence of an external source can be written as Z[J] = (0|0) J = NJV(j>eh

Lr

'

J

(9.7)

where S[(f), J] = S[] + ff°° dt dx J(x, t)(f)(x, t)

(9.8)

(Incidentally, in all these discussions, we are going to assume that the

224

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

relation between the Lagrangian and the Hamiltonian of the system is the canonical one which would lead to path integrals of the form in Eq. (9.5) or (9.7). If this is not the case, then one should take as the starting point, the path integral in the phase space as obtained in Eq. (2.22).) Before going into the discussion about the functional integration in the present case, it is worth emphasizing what we have discussed earlier, namely, it is the time ordered Greens functions in the vacuum which play the most important role in a field theory because the scattering matrix or the S-matrix can be obtained from them. This is why it is the vacuum functional which is the quantity of fundamental significance in these studies. The second point to note is that we have left the specific form of S[] arbitrary. Depending on the particular form of the action, we will be dealing with different kinds of field theories - both nonrelativistic and relativistic. Returning now to the question of the functional integration, let us recall that in the 0 + 1 dimensional case, we defined the path integral by dividing up the time interval into infinitesimal steps (see Eqs. (2.18) and (2.20)). Here, in addition, we have to divide up the space interval into infinitesimal steps as well. Thus, let us assume that - § < * < !

(9.9)

with the understanding that we will take the limit L —* oo at the end.

9.1. SYSTEMS WITH MANY DEGREES OF FREEDOM

225

Let us further divide the length interval into N equal steps of length e such that Ne = L

(9.10)

I 111 I II I I I I II I II I I I I I _L

L 2

2

If we now label the position as well as the value of the field variable at any intermediate point on the trajectory as xm

-

L —^ + me

(xm) = m

0<m
(9.11)

then, just as in the case of quantum mechanics, we can define (see Eq. (2.29), for example) / l t y = lim

Mm

[ Ud4>m

(9.12)

Ne=L

Unlike the case of quantum mechanics, which we have extensively discussed, however, the path integral, in the present case, does not exist in the sense that the integrations defined in Eq. (9.12) are divergent in the continuum limit. However, if we absorb the divergence into the normalization constant, N, in Eq. (9.5) or (9.7), then the Greens functions can still be defined uniquely since they are defined as ratios for which the divergent constants simply drop out.

226

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

For field theories in higher dimensions where the basic variables are (f)(x,t), we can define the vacuum generating functional exactly in an analogous manner. Namely, we have

, -SU, J] Z[J] = (0\0)J = NJV4>eh Lr ' J

(9.13)

where

S[, J] = S[4>] + J dnx J(x, t)(x, t)

(9.14

and the integrations are over the entire space-time manifold in higher dimension with n denoting the dimensionality of the space-time manifold. The functional integral, in such a case, is defined by taking a hypercube divided into a lattice of infinitesimal spacing and then identifying the functional integral with a product of ordinary integrals of the field values at each of the lattice sites.

9.2

Relativistic Scalar Field Theory With all these preliminaries, let us take a specific form

of the action in Eq. (9.14). Namely, let us choose a relativistic scalar field theory in 3 + 1 dimensions described by the Lagrangian density

£(0, d,cf>) = \d,d»4> - ^-4>2 - ^

(9.15)

9.2. RELATIVISTIC SCALAR FIELD THEORY

227

with A > 0.(This condition merely corresponds to the fact that we would like the potential to be bounded from below so that the quantum theory will have a meaningful ground state.) so that

S[(/)] = J d i x C{, d„<j>)

(9.16)

and S[, J] = S[] + J d*x J(x, t)(j>(x, t)

(9.17)

It is worth noting here that the Lagrangian can be obtained from the Lagrangian density in Eq. (9.15) by integrating over the space variables as

L = Jd3x £(, d^)

(9.18)

This theory is quite similar (see Eq. (4.63)) to the anharmonic oscillator which we discussed earlier except that it is a relativistic field theory invariant under global Poincare transformations. This is a selfinteracting theory which can describe spin zero particles with mass m. The Euler-Lagrange equations following from the action in Eq. (9.17) take the form ^ y !

= d^cf> + m V + \

(9.19)

Here we have chosen to represent the space-time variables in a compact notation of x for simplicity. Commonly, this theory described by

228

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

the action in Eq. (9.17) or by the dynamical equations in Eq. (9.19) is also known as the ^ 4 -theory. In the absence of interaction, namely, when A = 0, the action in Eq. (9.17) is quadratic in the field variables and hence the generating functional can be evaluated in much the same way as in the case of quantum mechanical systems (see chapters 2, 3 and in particular, 4). However, let us first define the Feynman Greens function associated with this theory. The equation satisfied by the Greens function is (fyd" + m2) G(x - x') = -64(x - x')

(9.20)

Defining the Fourier transforms as

6\x-x') = J^S^e-ik-ix-x')

(9.21)

and substituting these back into the differential equation, Eq. (9.20), we obtain

"•

G{k)

=<4ra

(9 22)

'

9.2. RELATIVISTIC SCALAR FIELD THEORY

229

Here we are using the scalar product for the four vectors with the metrics introduced in Eqs. (1.3) and (1.4) and k2 represents the invariant length square of the conjugate four vector k11. The Feynman Greens function or the propagator is, then denned following Eq. (3.75) as 1

-—— e~ik ' (x ~ *')

GF(x - x') = lim / ~ v

'

4

2

6-o+ J (2TT) k - m2 + ie

(9.23) y

'

We can also think of the Feynman Greens function as satisfying the differential equation (see Eq. (3.76)) lim+ (dud" + m2- ie) GF(x - x') = -<54(x - x')

(9.24)

Going back to the generating functional in Eq. (9.14), we note that for the present case, if A = 0, then we can define r -zSnlx, J] Z0[J) = N J V eh 0 l J

= sfv*e-y*'il"S'*,

+m )

' *-'*)

(9.25)

with SQ[X, J] representing the free, quadratic part of the action. The field, (j)(x,t), is assumed to satisfy the asymptotic condition lim 4>(x, t) — • 0 |x|—too

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

230

Furthermore, the normalization constant, N, for the path integral is normally chosen such that

Z[0] = 1 Let us note here once again that the integral in Eq. (9.25) should be properly evaluated by rotating to Euclidean space as discussed in section 4.1. Alternately, we can also define

Z0[J]=

j

Urn N fve h e—>0+

+ m2-ie)- J)

jd*x(-{d^ 2

(9.26)

J

If we now redefine the variable of integration to be 4>(x) = <j>(x) + / d V GF(x -

x')J(x')

(9.27)

with GF defined in Eq. (9.23), then, we obtain Urn. j d*x ; U ( < V + m2 - ie)4> =

lim / d*x \Mx) €—*o+ •

'

+ / d V GF(x - x')J(x')){dad^

2

x((x) + jdix"GF(x =

+ m2 - ie)

J

-

x")J(x"))

lim j d 4 x L2 - 0 ( a ) ( ^ a * 4 + m2 - ie)(x) - J(x)<j>(x) -I

[ d V J(x)GF(x

2J

-

x')J(x')

(9.28)

9.2. RELATIVISTIC SCALAR FIELD THEORY

231

where we have used Eq. (9.24). Substituting Eq. (9.28) back into the generating functional in Eq. (9.26), we obtain (Note that the Jacobian for the change of variable in Eq. (9.27) is trivial.) Z0[J] =

lim N

r ^ ^ y d ^ W i d ^

+ m'-ieWx)

- J(x)<j>(x))

-^JJdixdix'J{x)GF(x-x')J(xl) e->0+

= i V [ d e t ( a ^ + m 2 )]"^ xe

~

[[dtxePx' 2nJJ

J(x)GF(x

r , -^ir II d4x d V J(x)G F(x K = Z 0 [0]e 2 f t " ' M

-

x')J(x') x')J(x') ' V '

(9.29)

Here we have used a generalization of the result in Eq. (4.2) for a field theory. As in the case of the harmonic oscillator, we note that when A = 0,

232

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

(0\
(-ih) sz0[j] Z0[J]

z0[j]

8J(x) v(~

n

j=o

J dV GF(x - x')J(x'))Z0[J] J=0

= 0 <0|T(^(«)^(y))|0)

=

(-ih)2

62Z0[J]

Z0[J]

SJ(x)8J(y)

j=o

j=o

ihGp(x

— y)

(9.30)

Namely, we obtain once again the result that the Feynman propagator is nothing other than the time ordered two point correlation function in the vacuum (see Eq. (4.62)). Just as the path integral for the anharmonic oscillator cannot be evaluated in a closed form, the 0 4 -theory does not also have a closed form expression for the generating functional. However, we can evaluate it perturbatively at least when the coupling is weak. We note that we can write (as we had also noted earlier in Eq. (4.68) in the case of the anharmonic oscillator)

4>{x)

8 8J(x)

(9.31)

9.2. RELATMSTIC

233

SCALAR FIELD THEORY

when acting on the free, quadratic action So[, J]. Therefore, we can rewrite the generating functional of Eq. (9.13), in the present case, also as

, l- f d*x (-d^d^ - —4>2 - ^U4 + Jet)) K Z[J) = NJV(j>ehJ 2 "v * 2Y IV ^ Y'

d, { ih

{-Ml

= = (e

m

'

* - sW/)NjvJs°^ SJ X

()

)Z0[J]

(9.32)

Once again, we note that this is very analogous to the result obtained in Eq. (4.69) for the anharmonic oscillator. A power series expansion in A for the generating functional in Eq. (9.32) follows by Taylor expanding the exponential involving the interaction terms. Thus, we have

234

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC

7\ 71 Z[J]

n ll

i\h3 fr J* ,A

lAn

= -^rJdx6j^)

4 S °

THEORIES

3

i\h \ 2 , L1 i, lAn + )

^-ir

xe~2h 11dixi

^*2 J(XI)GF(XI

~ x2)J(x2)

To obtain a feeling for how the actual calculations are carried out, let us derive some of the Greens functions to low orders in the coupling constant for the 0 4 -theory. Let us recall from Eq. (4.50) that, by definition,

(oirMa!)^)...^))^) (-ih)n Z[J]

6nZ[J] 8J(xl)8J{x2)---8J{xn)

(9.34) j=o

Furthermore, from the symmetry of Z[J] in Eqs. (9.29) and (9.32) (namely, from the fact that it is invariant under J <-» — J ) , we conclude that the vacuum expectation value of the time ordered product of an odd number of fields must vanish in this theory. In other words,

9.2. RELATIVISTIC SCALAR FIELD THEORY

(O|r(0(x!)

235

•#c2n+i))|0)

62n+1Z[J] 6J(Xl)---6J(x2n+1)

(-ih)2n+1 Z[J]

= 0

(9.35)

j=o

Consequently, only the even order Greens functions will be nontrivial in this theory and let us calculate only the 2-point and the 4-point functions upto order A. By definition, {o\T((Xl)<j>(x2))\o} =

{-ihf Z[J]

62Z[J] 8J{xl)8J{x2)

(9.36) j=o

Keeping terms upto order A, we note from Eq. (9.33) that Z[J] ~ Z0[0] (1

i\h3

i*

SJ^x)'

— jj dix1dix2J(x1)GF(x1

-

x2)J(x2)

x e To evaluate this, let us note that i 62 e - ' ^ JJ d4x1d4x2 J(x1)GF(x1 e 2h SJ2(x) ° 8

SJ(x)

L

K

h

x e 2h =

[~GF(0)

-

x2)J(x2)

[-kJd4x3GF(x-x3)J(x3)) Jj d4xidix2J(x1)GF(x1

— x2)J{x2)

(9.37)

236

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

^(J d4x3GF(x - x3)J(x3))(JdixiGF(x h i i i x e 2 J / d x1d x2J{xl)GF(x1

-

xi)J(xi))}

x2)J(x2)

With some algebraic manipulations, this, then, leads to the result

[d*x-;

<54

8J\x) t ,4

=

—rz JJ(J dixidix2J(xi)GF(xi e Zn

*2 , S2 '6J2(xySJ2(x)

-

x2)J(x2)

ffdix1d4x2J(x1)GF(x1-x2)J(x2)

~ i

f*x[-^GF(0)GF(0) 6z

+^GF(0)(J

d*x3GF(x - x3)J{x3))(J

+ Z ? ( / dix3GF(x x (J dix5GF(x

dixiGF{x

- x3)J(x3))(J - x5)J(x5))(J

dAXiGF{x

d^xsG^x

-

-

Xi)J(xt))

xi)J(x4))

- a: 6 )J(x 6 ))]

Putting this back into Eq. (9.37), we obtain the generating functional to linear power in A to be

237

9.2. RELATIVISTIC SCALAR FIELD THEORY

i\h

Z[J] = Zo[0) [1 + l— GF(0)GF(Q) J d4x J d*x (J dAx3GF(x

+jGF(0) i\ — Jd4x A\h

(J d4x3GF(x

x (J dix5GF(x

- x3)J(x3))(f

d4XiGF(x -

- x3)J(x3))(J

- x5)J(x5))(J

dix6GF(x

- r r Jj dixldix2J(xi)GF{x1

dAx4GF(x

-

-

x4)J(x4))

xi)J(xi))

- x6)J{x6)) } x2)J{x2)

(9.40)

xe It now follows that Z[0] = Z0[0] (1 +

i\h % —GF(0)GF(0) / d*x)

(9.41)

This is clearly divergent and as we have argued earlier, the divergence can be absorbed into the normalization constant. From Eq. (9.40) we can also calculate to linear order in A

62Z[J] 8J{x\)8J(x2)

j=o

l = Z0[0] (1 + -^GF(0)GF(0) J <***)(-£
+ ^GF{0)

[d4xGF{x

-

Xl)GF{x

- x2)

(9.42)

238

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

Therefore, to linear order in A, we obtain (Q\T((x1)4>(x2))\0) {-ihf Z[J]

62Z[J] SJ(Xl)6J(x2)

J=0

h2 ifGF(0)GF(0)Jd*x)

Z0[0](l +

xZ„[0] (1 + l-^GF(0)GP(0) J d*x)(-^GF(Xl - x2)) + O G F ( O ) [d*xGF(x 2

~ -til

~-GF(xi

^

- x2) + -GF(0)

-

Xl)GF(x

J dixGF(x

= ihGF(x1-x2)-—-GFxh (0)[dixGF(x-x1)GF(x-x2)

- x2) - xx)GF{x

- x2)

(9.43)

Here in the second term, we have only kept the leading order term coming from the expansion of the denominator since we are interested in the 2-point function upto order A. We note that the first term is, of course, the Feynman propagator for the free theory defined in Eq. (9.23). The second term, on the other hand, is a first order quantum correction. It is worth pointing it out here that GF(0) is a divergent quantity as we can readily check from the form of the propagator. Thus, we find that the first order correction to the propagator in this theory is divergent. This is, indeed, a general feature

9.2. RELATIVISTIC SCALAR FIELD THEORY

239

of quantum field theories, namely, the quantum corrections in a field theory lead to divergences which are then taken care of by what is commonly known as the process of renormalization. Next, let us calculate the 4-point function upto order A. To leading order, we note from Eq. (9.40) that PZ[J\ 8J(x\)8J(x2)8J(x3)8J{xi) = Z0[0] - i (K l + h' ~ '

j=o l

-^GF(0)GF(0) 8

+GF(xi - x3)GF(x2 2A

~7^GF(0)

- x2)GF{x3

- xA)

- xi) + Gp(xi - X4t)GF{x2 - x3))

t

j <£x {GF{xi - x2)GF(x

+GF(x1 - x3)GF(x +GF(X\

J d*x)(GF(Xl

- x3)GF(x

- x2)GF(x

- X4)

- Xi)GF(x - x2)GF{x

- x3)

- xA)

+GF(x2 - x3)GF{x — x\)GF(x — x4) +GF(x2 - Xi)GF{x - xi)GF(x — x3) +GF(x3 - Xi)GF{x - xi)GF(x iX — — / d4xGF(x

— xi)GF(x

— x2)GF(x

-

x2)} — x3)GF(x

— x4)

(9.44)

240

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

Substituting this back into the definition of the 4-point function in Eq. (9.34) and keeping terms only upto order A, we obtain (0\T((x1)4>(x2)(l>(x3)(xi))\0)

8iZ[J]

{-ihf Z[J]

8J{x1)8J{x2)8J(x3)8J{xl)

= -H2(GF(XI

- x2)GF(x3

j=o

- Xi) + GF(xi - x3)GF(x2

+GF(x! - xi)GF(x2

- x4)

- x3))

i\h3

4 2 -GF(0) J d x {Gp(xi — X2)GF(X

+GF(XI

- x3)GF(x

+GF(X\

— xA)Gp{x - x2)GF{x

+GF(x2 — X3)GF(X

-i\h3

- X2)GF(X

- x\)GF{x

— X3)GF(X - x4) - x3) - Xi)

+GF(x2 - Xi)GF(x - xi)Gp(x

- x3)

+GF(x3 - Xi)GF(x - xi)GF{x

- x2)}

J <&XGF{X — X\)GF(X

— X2)GF(X

— a;4)

— X3)GF(X

(9.45) — x4)

241

9.3. FEYNMAN RULES

9.3

Feynman Rules These lowest order calculations are enough to convince

any one interested in the subject that a systematic procedure needs to be developed to keep track of the perturbative expansion.

The

Feynman rules do precisely this. Let us note that the basic elements in our (/>4-theory are the Feynman propagator for the free theory and the interaction. Let us represent these diagrammatically as

Xi

X\^

x2

= ihGp(xy

— X2)

^X4

, - X ^r £, 3 £2

=V(x1,X2,X3,Xi) <54S[0] tl

(9.46)

S(f>(xi)S(f)(x2)Scj)(x3)S(f)(x4) ^=0

iX

/ dixS(x — xi)6(x — X2)8(x — x$)8(x — X4)

The interaction vertex is understood to be the part of the graph without the external lines or the propagators. It is clear that given these basic elements, we can construct various nontrivial graphs by joining the vertex to the propagators. Let us further use the rule that in evaluating such graphs, we must integrate over the intermediate points where a vertex connects with the propagators. With these rules, then, we can obtain the value for the following simple diagram to be

242

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

x\

2/i 2/2

x2

J diy1d*y2diyidiyiihGF(xl

=

x ihGF(y3 =

(ihf

- yi)ihGF(y2

-

x2)

- 2/4)^(2/1,2/2,3/3,2/4)

J d4yd4y1d4y2d4y3d4y4GF(x1

- y\)GF(y2

- x2)GF(y3

- 2/4)

iX x ( - y < % - 2 / i ) % - 2 / 2 ) % - # » ) % - 2/4)) -Xh2GF(0)Jd4yGF(x1-y)GF(y-x2)

=

(9.47)

T h e r e is one final rule. Namely, t h a t if t h e internal p a r t of a diagram has a symmetry, t h e n t h e true value of t h e diagram is obtained by dividing with this s y m m e t r y factor. For t h e present case, t h e internal bubble in t h e diagram for Eq. (9.47) is invariant under a rotation by 180°. T h e s y m m e t r y factor, in this case, is 2 1 = 2. (The s y m m e t r y factor for a F e y n m a n diagram is t h e most difl&cult to determine by naive inspection a n d should be obtained t h r o u g h a careful evaluationwhen necessary, going back t o t h e Wick expansion of field theory.) Dividing by this factor, we obtain the value of this diagram to be

Xl

-2-x

2

= ~~GF(0)

J d4x GF(Xl - x)GF(x

- x2)

(9.48)

which we recognize t o b e t h e first order (linear in A) correction to t h e propagator in Eq. (9.43).

9.3. FEYNMAN RULES

243

Each such diagram that can be constructed from the basic elements in Eq. (9.46) is known as a Feynman diagram of the theory and corresponds to a basic term in the perturbative expansion. Thus, for example, we now immediately recognize from Eqs. (9.43) and (9.45) that upto order A, we can write

(o|r(^( a;i )0( a!2 ))|o) — xi

x2 + xi

V

X2

(0\T((Xl)(x2)(x3)(x4))\0) x

X\

X2

X3

X 4 + X2

x

l

4 . x2 X2 + X\

x

3

x

l

i

Xi + Xi

x 3 +X\

V

x

Xi +X2

\)

X/| Xi_\J__X3 Xi X3 + X2 Xi + X3

V

x

i . x2 X3 + Xi

£3 V

\)

x

3 £4

Xn Xi

+

X2^><^X3

(9.49)

244

9.4

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

Connected Diagrams The Feynman diagrams, as can be seen from the diagrams

in Eq. (9.49), clearly consist of two classes of diagrams - one where each part of the diagram is connected to the rest of the diagram and another where parts of the diagram are disconnected. The Feynman diagrams, which consist of parts that are not connected to one another, are known as disconnected Feynman diagrams. It is clear from the above simple example that the generating functional Z[J] generates Greens functions which contain disconnected diagrams as well. As we have discussed earlier (see the discussion following Eq. (4.51)), the logarithm of Z[J] generates Greens functions which contain only the connected diagrams (otherwise known as the connected Greens functions) and these give rise to the physical scattering matrix elements. Namely, W[J] = -ih\nZ[J] generates connected Greens functions.

(9.50) We note from Eqs. (9.50),

(9.34) and (9.35) that

6W[J] ih 6Z[J] 6J{xx) 7=o " ~W\ 6J&)

= (Ol^xOlO) j-o

Similarly, for the two point function, we obtain

(9.51)

245

9.4. CONNECTED DIAGRAMS

—ih

82W[J] 8J(xi)8J(x2)

(-my

J=0

82Z[J] Z[J]8J{Xl)8J(x2)

l

8Z[J] 8Z[J] Z [J]8J(x1)8J(x2)i 2

=

(O|r(0(aiM*2))|O> -

=

(0\T(4>(x1)4>(x2))\0)c

J=0

(9.52)

In a similar manner, with some algebra, it can be shown that

i-iny

83W[J] 8J(xi)8J(x2)8J(x3)

j=o

= {OlTMziMxiWx^lO) -

- ( 0 1 ^ 0 1 0 ) (0|r(^(a !2 )^(x3))|0)

(0|^(x 2 )|0) (0\T((x1))\0) - {0\<j>(x3)\0) (0\T((x2))\0) +2<0|# B l )|0) (01^2)10) (0|
=

(O|T(0(x 1 )0(x 2 )0( a; 3)|O) c

(9.53)

and so on. We can, of course, check exphcitly that the connected Greens functions involve only connected Feynman diagrams as follows. Note that for the (/>4-theory, as we have discussed earlier in Eq. (9.35)

(o|<Mz)|o) = o

(9.54)

246

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

Consequently, upto order A, we note from Eq. (9.52) that (0\T(<j>(Xl)ct>(x2))\0) = (0|T(^( a J l )^(x 2 ))|0) c -

xx-

•x2 + Xl-&-x2

(9.55)

Similarly, for the 4-point function, we have {Q\T({xl)4>{x2)
(OFMxMfaMxsMxtVlO)

-(OlTMxM^miOlTMxsMxjm -(OlTi^x^ix^miOlT^xMx^lO)

_

Xi^^Xi

(9.56)

Thus, we see that W[J] generates connected Greens functions. Given this, we can write down the diagrammatic expansion of the connected 2-point Greens function upto order A2 in this theory simply as

247

9.4. CONNECTED DIAGRAMS

= X\

x2 + xi

V

+ a;i Q

a;2 + %i ^

x2 + #i

a;2

0 a?2 (9.57)

The organization of the perturbation series now becomes quite straightforward. Note that while W[J] generates connected diagrams, it contains diagrams that are reducible to two connected diagrams upon cutting an internal line. Thus, in the 2-point function represented above, the third graph is reducible upon cutting the internal propagator. Such diagrams are called I P (one particle) reducible. It is clear that the I P irreducible diagrams are in some sense more fundamental since we can construct all the connected diagrams from them. We will take up the study of the I P I (one particle irreducible) diagrams in the next chapter.

248

9.5

CHAPTER 9. PATH INTEGRAL FOR RELATIVISTIC THEORIES

References

Huang, K., "Quarks, Leptons and Gauge Fields", World Scientific publishing. Jona-Lasinio, J., Nuovo Cimento 34, 1790 (1964). Schwinger, J., Proc. Natl. Sci. USA 37, 452 (1951); ibid 37, 455 (1951).

Chapter 10 Effective Action

10.1

T h e Classical Field As we saw in the last chapter, the generating functional

for Greens functions in a scalar field theory is given by l

-W\J\

Z[J} = eh

,

li

=NjVcf>eh

is[(f>,J} Lr

'

J

(10.1)

where W[J] generates connected Greens functions. We note that the one point function in the presence of an external source is given by

For the (/>4-theory, we have seen in Eq. (9.35) that this vacuum expec249

250

CHAPTER 10. EFFECTIVE ACTION

tation value of the field operator vanishes in the absence of external sources (namely, when J = 0). In general, however, let us note that we can write —p . x p . (O|0(x)|O) = (0|eft 0(O)e h

x

|0)

(10.3)

where we have identified e~iPx with the generator of space-time translations. Assuming that the vacuum state in our Hilbert space is unique and that it is Poincare invariant, namely, that it satisfies

e h or,

|0) =

|0)

PM\0) = 0

(10.4)

we obtain from Eqs. (10.3) and (10.4) that {0\(x)\0) = (O|0(O)|O) = constant

(10.5)

Thus, from the symmetry arguments alone, we conclude that the one point function can, in general, be a constant, independent of spacetime coordinates. For the ^-theory, this constant coincides with zero, namely, we have (0|(/>(x)|0) = 0

(10.6)

10.1. THE CLASSICAL FIELD

251

The value of the one point function is quite important in the study of symmetries. As we will see later, a nonvanishing value of this quantity signals the spontaneous break down of some symmetry in the theory. In the presence of an external source, however, the vacuum expectation value of the field operator is a functional of the source and need not be zero. Let us denote this by

and note that it is indeed a functional of the external source. The field 4>c(x) which is only a classical variable is known as the classical field. To understand the meaning of the classical field as well as the reason for its name, let us analyze the generating functional in Eq. (10.1)

Z[J] = NJV4>ehS[4>'J] in some detail. Since Z[J] is independent of <j)(x), an arbitrary, infinitesimal change in (x) in the integrand on the right hand side will leave the generating functional invariant. Namely, under such a change,

252

CHAPTER 10. EFFECTIVE ACTION

*

J

8Z[J] = N [vUs[,J]efr ^ •>

n

= x Jv {Jd

xS

^-6^r)eh

= 0

(10.8)

Here we are assuming that the functional integration measure in Eq. (10.1) does not change under a redefinition of the field variable. (If it does change, then we will have an additional term coming from the change of the measure. Such a term plays an important role in the study of anomalies.) Since the relation in Eq. (10.8) must be true for any arbitrary variation 8(j){x) of the field variable, it now follows that we must have

J v/r*«SM,5 J

sl

*"' 1 = o

(io.9)

0(p\X)

This merely expresses what we already know from our study of quantum mechanics. Namely, that the Euler-Lagrange equations of a theory hold only as an expectation value equation (Ehrenfest's theorem) or more explicitly,

<0|

^f

|0)J =

°

(1 10)

°-

10.1. THE CLASSICAL FIELD

253

The Euler-Lagrange equation (the classical equation) has the generic form (see, for example, Eq. (9.8) or (9.17))

~ *H?

= F{ X))

^ ~

J{X)

=°

where the quantity F((j)(x)) depends on the specific dynamics of the system and for the 0 4 -theory, we note from, Eq. (9.15), that it has the particular form

F(0(X))

= +m2)(t){x)+ 3(x)

=-HI ^

^

(io i2)

-

Using Eq. (10.11) in Eq. (10.9), then, we obtain

-N jv<j)

6S[, J] ^S[,J] h 64>(x)

= NJV(j>(F{
l

=0

(10.13)

Let us recall that

«*!#T = 6J(x)

N

I V4>{x)ehS[^

which allows us to use the identification

^

^

~ihJJ{x)

J]

(10.14)

254

CHAPTER 10. EFFECTIVE ACTION

using which, we can write Eq. (10.13) also as

g or, [ir(_iR -TW\J\

or, e h

-W\J] ) - J(x)]eh =0

6 [

TW[J]

\F(-ihj-^)-Jix)]eh

or,

LJ

=0

F(0 c (a;) - i f e - ^ r ) - J(x) = 0 (10.15)

This is, of course, the full dynamical equation of the theory at the quantum level. It is quite different from the classical EulerLagrange equation in Eq. (10.11). But let us note that in the limit h —>• 0, the complete equation in Eq. (10.15) reduces to the form F(4>c{x)) - J{x) = 0

(10.16)

which is the familiar classical Euler-Lagrange equation in Eq. (10.11). It is for this reason that (/>c(x) is called the classical field. To get a better feeling for the quantum equation in Eq. (10.15) as well as Eq. (10.16) in the ti —> 0 Emit, let us consider specifically the 4theory.

In this case, as we have noted earlier in Eq. (10.12), the

255

10.1. THE CLASSICAL FIELD

Euler-Lagrange equation takes the form F((x)) - J(x) = ( < V + m2)<j>{x) + ^(x)

- J(x) = 0

Consequently, the quantum equation takes the form

% W - injj^)

- J(x) = ° m2)((f>c(x)-iTij^)

or, (d,dK +

4^-ihJ(x-/-J{x)=0 or, ( 5 ^ + m 2 )^ c (x) + ^ ( x ) - J ( a ; ) i\h

, , % W

Aft2

S2<j)c(x)

-2T+°W JJ{x) ~ -U6J{x)8J{x) = °

nn17^

(1

°- 1 7 )

In terms of W^[J], this can also be written as

i\h8W[J]82W[J] 2! 6J(x) 8J\x)

\h283W[J} 3! 8J3(x)

= K

'

We can think of Eq. (10.18) as the master equation governing the full dynamics of the quantum theory. By taking higher functional

256

CHAPTER 10. EFFECTIVE ACTION

derivatives, we can determine from this, the dynamical equations satisfied by various Greens functions of the theory. These equations are also known as the Schwinger-Dyson equations and the Bethe-Salpeter equations are a special case of these equations. We should note here that in quantum field theory, whenever there are products of field operators at the same space-time point, the expressions become ill defined and have to be regularized (defined) in some manner. The manifestation of this problem is quite clear in Eq. (10.18). The equation involves second and third functional derivatives at the same spacetime point which are not at all well defined. One needs to develop a systematic regularization procedure to handle these difficulties. The discussion of these topics lies outside the scope of these lectures. Let us note, on the other hand, that in the limit h —»• 0, all such ill defined terms vanish and we have from Eq. (10.17) ( c V + rn2)c(x) + ±
(10.19)

which is the classical Euler-Lagrange equation. Furthermore, let us note that we can solve this equation iteratively through the use of the propagator defined in Eqs. (9.23) and (9.24) (Greens function) as follows. First, we note that the solution for <j>c{x) can be written as an integral equation

10.1. THE CLASSICAL FIELD

257

4>c(x) = - / d V G ^ - x')(J(x') =

- / d4x'GF(X

- x')J(x')

A - ^CV))

(10.20)

+ ^ / d V GF(x - x')4>3c{x')

which can be solved iteratively. The iterative solution has the form 4>c(z) ~ - / d V GF(x +± Jd4x'GF(x — — J d4x'GF(x

x')J(x')

- x')(fd*x»GF(x' — x')J(x')

x GF(xi - x2)GF(xi

- x")(-J(x")

+ |^(x")))3

— — J dixidiX2d4x3d4X4GF(x

- x3)GF(x1

- xi)J(x2)J(x3)J(xi)

— xi) -\ (10.21)

We can diagrammaticaUy represent this if we introduce a vertex describing the interaction of the field with the external source as

* ss[4>,j] h 6(f)(x)

= -J(x)

(10.22)

In this case, the iterative solution for the classical field can be written diagrammaticaUy as c(X) = X 1 2!3! x

" + 3! *

-*+...

(10.23)

258

CHAPTER 10. EFFECTIVE ACTION

In other words, the classical field in the limit h —•> 0 generates all the tree diagrams or the Born diagrams. It is, therefore, also known as the Born functional or the generating functional for tree diagrams. The combinatoric factors cancel out when taking functional derivatives of 4>c with respect to sources and we obtain the n-point tree amplitudes.

10.2

Effective Action The classical field, as we have seen in Eq. (10.7), is defined

as

SW[J] 6J(x) = 4>M) This relation indicates that the variables J(x) and (f>c(x) are in some sense conjugate variables. As we have argued earlier, the classical field is a functional of the source J{x).

However, we can also invert the

defining relation for the classical field and solve for the source J{x) as a functional of (f>c(x) at least perturbatively. In fact, let us define a new functional through a Legendre transformation as r[&] = W[J] - j (Px J(x)c(x)

It is clear, then, that

(10.24)

259

10.2. EFFECTIVE ACTION

b
0(pc[x)

J

oc(x)

8W[J] 8J{V) - (d% [aydiv8Jy ^8WW ~ Jdyf>.T(v)fi bJx\ J 66Jx)6J(v) SJ(y)( 6c(x) 6cf>c(x) 6J(y)

6T& or 6<j>c(x) = -J(x)

J{ J(X)

>

(10.25)

In this derivation, we have used the chain rule for functional derivatives as well as the definition of the classical field.

We note that

Eq. (10.25), indeed, defines the source as a functional of the classical field and has the same structure as the classical Euler-Lagrange equation for a system in the presence of an external source. Let us recall from Eqs. (10.11) and (10.12) that it is given by

f f = -'<*>

<«>•»>

It is for this reason that r[<^>c] is also known as the effective action functional. Note that as we have discussed earlier in Eq. (10.5), when

J —> 0

<j>c{^) —• constant

In the framework of the effective action, we see from Eq. (10.25) that the value of this constant is determined from the equation

260

CHAPTER 10. EFFECTIVE ACTION

6T[c 6c(x)

= 0

(10.27)

0c(a:)=const.

This is an extremum equation and is much easier to analyze to determine whether a symmetry is spontaneously broken. To understand the meaning of this new functional, r[<^>c], let us note that if we treat 4>c(x) as our independent variable, then we can write 6 c(y) /J * " 6J(x) 6c(y)

4

6J{x)

s

6J(x)6J(y) 8c{y) Using this in Eq. (10.25), then, we obtain

8

:(S4) = -*(*-*)

8J(yy8c{x)'

Introducing the compact notation \y(n)

rW

=

SnW[J] SJ(x1)---SJ(xn)

= ^,s?['"i<

>

OQc^X!) • • • 0
<">•»>

10.2. EFFECTIVE ACTION

261

we can write Eq. (10.29) also in the compact form (We can view this as an operatorial equation where the appropriate coordinate dependences will arise by taking matrix elements in the coordinate basis.)

WWTW

_i

=

(10.31)

We recall from Eqs. (9.30) and (9.52) that the full propagator of the theory is defined to be WW\J=Q = -G

(10.32)

Furthermore, recalling that when J = 0, 4>c(x) = (j>c = constant, we have from Eq. (10.31)

^ ( 2 ) l,=or ( \ = - i or, Gr<2>|^ = 1

(10.33)

In other words, r ' 2 ' | , is the inverse of the propagator at every order of the perturbation theory. Thus, writing r (2)

|

=

r (2) +

s

=

G

-l

+

s

( 1 0 3 4 )

262

CHAPTER 10. EFFECTIVE ACTION

where S denotes the quantum corrections in ^2'\,

, we have from

Eq. (10.33)

1

G{G-t or, G —

1

+ £

Gp

=

1

1

1 =

Gp

Gp

Gp

E

1 '-rS Gp

Gp

Gp — GpYlGf ' + GFEG,?YiGp

+

(10.35)

Introducing the diagrammatic representation ^S = - O -

(10.36)

we have the diagrammatic relation for the propagator as

~&~

x

y

+xy

+

x +•••

(10-37)

It is clear from the above relation that S is nothing other than the I P irreducible (1PI) 2-point vertex function. It is also known as the proper self energy diagram.

263

10.2. EFFECTIVE ACTION

Given the relation in Eq. (10.29)

f *z J

62w

* 2 iU]

W

=

we can differentiate this with respect to

J

_«, _ ,

6J(y)6J(z)84>c(z)84>c(x)

{

V)

S SJ

,, to obtain

83W[J] 82r[c] 8J{u)8J(y)8J{z)8 o)6J(y)8J(z) 8 c{z)84> c{x) c{z)8(t> c{ 62W

f.^j\r J

6 T

W

8J{y)8J(z)

K

*M

'

'

62W[J]

8(j>c{z)8cf>c(x)84>c{a) 8J{a)8J(u)

Recalling that ^(2)r(2)

=

_X

we can also rewrite this equation as 8*W[J] 8J(x)8J(y)8J(z)

= JfdHMdH U

52W[J]

62W[J]

62W[J]

8J(x)8J{x')8J{y)8J{y')8J{z)8J{z')

83T[c] 6e(x')6c(y')6e{z')

(10.39)

In compact notation, we can write this equation also as W^

= W^W^W^T^

(10.40)

264

CHAPTER 10. EFFECTIVE ACTION

Furthermore, introducing the diagrammatic representations, x-^-y

= -MWW(x,y)\J=0

x

z = (-ihfW^ix^z)^ y

z = ^T^(x,y,z)l y '

(10.41)

n

we note that Eq. (10.39) or (10.40) can also be diagrammaticaUy represented as x (10.42)

This shows that T^\, gives the proper 3-point vertex function (in other words, it is 1PI). A similar calculation for the 4-point connected Greens function leads to the diagrammatic relation

265

10.2. EFFECTIVE ACTION

x v

sw

yW x v

+ permutations (10.43) z

Thus, we see that we can expand the effective action functional as 00

r[&]= £

r i

„=i

J

1

d x1-.-dixn-rW(xu---,Xn)\ n\

'^

&(*!)•••&(*„) (10.44)

where (we are assuming (f>c = 0 in the above expansion) r ( B ) (*i, •••,*»)!,. is the proper (IPI) n-point vertex function of the theory. It is for this reason that T[(f>c] is known as the I P I generating functional. Let us note here that the I P I vertex functions with suitable external wave functions lead to the scattering matrix of the theory. Since T[c] is an effective action, it can also be expanded alternately in powers of the derivative or momentum like the classical action. Thus, since the tree level action has the form

S[) =

j#x(ld^+-^
CHAPTER 10. EFFECTIVE ACTION

266

we expect to be able to write T[c] also in the form

r[&] = Jd4x {-veff{4>c{x)) + l-A{4>c{x))d^c{x)d^c{x) + •••) (10.46) where the terms neglected are higher order in the derivatives. Let us recall that when the sources are set equal to zero, then the classical field takes a constant value 4>c{x) = 4>c. In this limit all the derivative terms in the expansion in Eq. (10.46) vanish, leading to

T[c] = -Jdix

Veff{c) = -Veff((f>e) J d4x

(10.47)

In other words, in this limit, the effective action simply picks out the effective potential including quantum corrections to all orders. The quantity J d4x which represents the space-time volume is also conventionally written as J dAx =

(2TT) 4 5 4 (0)

(10.48)

Let us also note that the constant value of (f>c = ((f)) when the sources are turned off is obtained from the extremum equation in Eq. (10.27)

5r[0 c 8(j>c(x)4>c(<») = c

= 0

(10.49)

10.2. EFFECTIVE ACTION

267

In t e r m s of t h e effective potential, this condition becomes equivalent to

dVeff(c)

dc

= 0

(10.50)

={*)

which is a familiar extremization condition from t h e s t u d y of classical mechanics. Note here t h a t since (f>c is a n ordinary variable (namely, it does not depend on space-time coordinates), only ordinary partial derivatives are involved in the above e x t r e m u m condition.

In

this sense, this equation is much easier to analyze t h a n a functional equation. We also note t h a t t h e renormalized values of the masses a n d t h e coupling constants (including all q u a n t u m corrections) can be obtained (in this theory) from t h e effective potential simply as

d2VKeff

m\

dfr <94Keff

d:

=

\R

(10.51)

«=(*>

T h e s t u d y of the effective potential is, therefore, quite i m p o r t a n t . It is particularly useful in analyzing when t h e q u a n t u m corrections can change t h e qualitative tree level or classical behavior of a theory.

268

10.3

CHAPTER

10. EFFECTIVE

ACTION

Loop Expansion We have already described the Feynman rules in the co-

ordinate space. However, for most practical calculations, it is quite useful to work in the momentum space. The Feynman rules can be readily generalized to the momentum space given the rules in the coordinate space. (This simply involves taking Fourier transforms.) For the 0 4 - theory, for example, the momentum space Feynman rules take the form ih *z;— = ihGF(p) = lim— p

*w

p2Xp3=

e ^ 0 p2

. _m2

+

l€

-jHPi+P2+P3+P4) (10.52)

In evaluating a Feynman diagram, we should integrate over the intermediate momenta, namely, the momenta of the internal propagators. Thus, for example, let us evaluate the 1PI 2-point vertex function at order A. k

SI (

Pi

According to our rules, we obtain

P2

10.3. LOOP

269

EXPANSION

(10 53)

= 1 *<*-») I $pi?h?

'

Note that the factor of | in front of the integral in Eq. (10.53) simply corresponds to the symmetry factor of the diagram which we discussed earlier. In writing the propagator, we have not explicitly included the ie term although it should always be kept in mind in evaluating the integral. We will discuss the actual evaluation of the integrals later. For the present, let us simply note that the calculations indeed take a simpler form in the momentum space. Let us next recognize from the form of the exponent in the path integral that the quantity which determines the dynamics of the system is \c{4>, d,4>) = \(\d,&r4>

- ^

- ^

(10.54)

The Planck's constant which measures the quantum nature of an amplitude comes as a multiplicative factor in the exponent. As we have seen in Eqs. (9.46) and (10.52), the consequence of this is that each vertex has a factor of \ associated with it. On the other hand, the propagator which is the inverse of the operator in the quadratic part of the Lagrangian, comes multiplied with a factor of h. Thus, sup-

270

CHAPTER 10. EFFECTIVE ACTION

pose we are considering a proper vertex diagram (1PI diagram) with V vertices a n d I internal lines or propagators, then the total n u m b e r of h factors associated with such a diagram is given by

P = I-V

(10.55)

In other words, such a diagram will behave like ~ h . Let us also calculate the n u m b e r of independent moment u m integrations associated with such a diagram. First, let us note t h a t in a proper vertex diagram, there are no external propagators or legs. Second, all t h e m o m e n t a associated with the internal lines must be integrated. Since there are I internal lines, there must, therefore, be I m o m e n t u m integrations. Of course, not all such m o m e n t a will be independent since at each vertex there are m o m e n t u m conserving 6functions. Each such ^-function will reduce the number of m o m e n t u m integration by one. Since there are V vertices, there will be as m a n y m o m e n t u m conserving 6-functions. However, we will need to have an overall m o m e n t u m conserving <5-function for the amplitude.

Hence,

the <5-functions will effectively reduce the number of m o m e n t u m integration by V — 1. Therefore, the n u m b e r of independent internal m o m e n t u m integrations will be given by

L = I-(V-1)

= I-V

+ 1 = P+1

(10.56)

10.3. LOOP EXPANSION

271

But the number of independent momentum integrations precisely measures the number of loops in a diagram and from the above relation we note that the number of loops associated with a diagram is related to the power of h associated with a diagram. In fact, the number of loops exceeds the power of h by one. Therefore, expanding an amplitude in powers of h is also equivalent to an expansion in the number of loops. The loop expansion provides a valid perturbative expansion simply because h is a small quantity. This expansion is quite useful and is very different from expanding in powers of the coupling constant. This follows mainly from the fact that the expansion parameter, h, multiplies the entire Lagrangian. Consequently, it is insensitive to how we divide the Lagrangian into a free part and an interaction part. The loop expansion is, therefore, uneffected by any such separation. This is particularly useful if the theory exhibits spontaneous symmetry breakdown in which case, as we will see later, the vacuum expectation value c = (0) becomes dependent on the coupling constants of the theory. Shifting the fields around such a value complicates perturbation in powers of the coupling constants. However, as we have argued, the loop expansion is uneffected by such a shift.

272

10.4

CHAPTER 10. EFFECTIVE ACTION

Effective Potential at One Loop Let us next calculate the effective potential for the (/>*-

theory at one loop. In this case,

s[] = jd'x (\d,d^ - ^ > - ±
Furthermore, we note from Eq. (10.17) that the classical field, (f>c{%), satisfies the equation

If we now use the relations (see Eqs. (10.7) and (10.25))

6c(x)

=

-J(x)

8<j>c(x) _ 62W[J] = -G(x-y,c) 8J(y) ~ SJ(x)6J(y)

(10.58)

273

10.4. EFFECTIVE POTENTIAL AT ONE LOOP

and remember that we are interested only in one loop effects, then keeping terms up to linear power in h, we obtain from Eq. (10.57)

(d,d» + m2)«M*) + ^c(x)

__ ^ 0 c ( x ) ^ M + 0(%2) 6T[c) 84>c{x)

or

> -7^r\ 0(pc{x) + ^(x)G(x 2

- x, 4>c) + 0(h2) 6T[c] 6c(x)

or

5

> jJrr\ W J "

^)

=

-^^(X)G?(O,

4>c) + o(h2)

(10.59)

Now, if we expand the effective action as

r[<j>c] = s[c] + hSifc] + 0{h2)

(10.60)

then, Eq. (10.59) gives —^

(fc^faj) + 0(h2) = _!^ c ( x )G(0,«k) + 0(h2)

(10.61)

Therefore, to Unear order in h, we can consistently write

^

= -f^)G(0,«

dO.62)

274

CHAPTER 10. EFFECTIVE ACTION

From the structure of the action (see Eq. (10.46)) S1[c] = Jd!lx(-V1(c(x))

(10.63)

+ ---)

we obta obtain

SSM 6c(x)

(10.64)

dc

c{x)-

Thus, if we restrict to (f>c(x) = c — constant, then Eq. (10.62) takes the form - ^ -

=

(10.65)

y0cG(O,0c)

Let us note that although the Greens function G(x — y, (f>c) can itself have a power series expansion in h, consistency requires that we only use the lowest order expression for the Greens function in the above equation. Furthermore, noting from Eq. (10.29) that

*ar[6J

U*z

s2w[J] __ _

JdZ6Mx)SM^SJ(z)6J(y)~

4

[

_ V)

to the lowest order of the Greens function that we are interested in, this relation gives

1*'

S'Sfa}

6c(z)

or, / d'zdd^

(-G(z-y,c))

=

-6A(x-y)

A + m 2 + -ct>2c)8\x - z))G(z - y, <j>c) = -8\x

- y)

10.4. EFFECTIVE POTENTIAL AT ONE LOOP

275

or,

( « V + m 2 + ~cj>2c)G(x - y, <j>c) = -S\x

or,

(d^

+ m2eff)G(x-y,4>c)

- y)

= -6\x-y)

(10.66)

Here we have used the form of the action from Eq. (9.15) as well as Eq. (10.32) and have denned mlff = m 2 + ^

2

(10.67)

The Greens function can now be trivially determined and as we have seen before in Eq. (9.23) has the form

Substituting this back into Eq. (10.65), we obtain <9Vi(<£c) _

dcf>c

iA IA

i

W

iA IA

.

°'

4) =

ft


11

I^(^^m:*«//

or, W e ) = g f - « W | ^ , _ J _ ^

(10.69)

Here, we are assuming that V\(
276

CHAPTER 10. EFFECTIVE ACTION

to Euclidean space and doing the integral. First, we note that if we interchange the orders of integration in Eq. (10.69), we obtain

W c )

=

y / ( ^ / o ' r

a K

c

^ - ^

c

- ^ 2

r

4 2 J' (2TT) (2ir)*./OJo A^2 +

m2_fc2

dAk i f d4k , ,U2c + m2-k2 -2/(2^'"'' m . - f '

< 10 - 7 °)

Now, rotating to Euclidean space (see section 4.1), we obtain

W c )

i f id*kE (\4>l + m2 + k2E = -2i(2^ln( m* + k2E ) 1 r d3Q = 2/(2^

M

^

l n (

| ^ + m2 + fc| m' + fc* }

(10J1)

Since the integrand does not depend on the angular variables, the angular integration can be done trivially and has the value

Jd3n

= 2ir2

(10.72)

10.4. EFFECTIVE POTENTIAL AT ONE LOOP

277

so that we have

= ^/>^<^f)

dO.73)

x = k2E

(10.74)

Defining

we note that the effective potential at one loop takes the form W c ) = 3 ^ 2 /0°° dx x(\n(x + m2eff) - ln(x + m 2 ))

(10.75)

Clearly, the integrals are divergent and, therefore, we have to cut off the integral at some high momentum scale to obtain Vi(c) = ^ ^ l

m

0

eff,

dxx(ln(x

* 2 ,

m

e / /

+ m2eff) - ln(x + m 2 ))

n

2

l

- [ y ( l n A2 - \) + m 2 A 2 - ^ - l n A 2 + ^ - ( I n m 2 - ±)]}

278

CHAPTER 10. EFFECTIVE ACTION

1

r/ 2

2^A2 2

m

2

tff->

A-2

™4,

A2

3^[K / / -m )A -^ln- + - l n ™*ff„

1

rA

,,,,

rn2eff ^2 A ,,.

m 4 m 2 1, 2l % 2 2'

1 2j 9

A ,,,,

A2

^ (2l(1° ! ^ I,'*'' ± M l -_ ^;)] n<^ +. iVK ++ ^)

m 4 ., m 2 1.

(10.76)

Here we have introduced a n arbitrary mass scale, /J,, t o write the expression in a meaningful manner. Note t h a t the one loop potential, a s it s t a n d s , diverges in t h e limit A —> oo which is the physical limit for t h e true value. This brings out one of the essential features of q u a n t u m field theory. Namely, point-like interactions necessarily induce divergences simply because the Heisenberg uncertainty principle, in this case, allows for an infinite uncertainty in the moment u m being exchanged. This necessitates a systematic procedure for eliminating divergences in such theories. This is known as t h e renormalization theory which we will not go into. Let us simply note here t h a t u p to one loop, t h e n , we can write the effective potential of the (/>4-theory to be

V}}} = V + Vi

(10.77)

10.5. REFERENCES

10.5

279

References

Goldstone, J., A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962). Jona-Lasinio, J., Nuovo Cimento 34, 1790 (1964). N a m b u , Y., Phys. Lett. 26B, 626 (1966). Zinn-Justin, J., "Quantum Field Theory and Critical Phenomena", Oxford Univ. Press.

Chapter 11 Invariances and Their Consequences

11.1

Symmetries of t h e Action Let us continue with the 4-theory and note that, in this

case, we have

S[] = J d 4 x £(<j>, < 9 »

(11.1)

where the Lagrangian density has the form

c(4>,
- ^

(n.2)

We can write the action in Eq. (11.1) also in terms of the Lagrangian 281

282

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

in the form S[] = JdtL

(11.3)

where L = Jd'xC^d^)

(11.4)

Given this theory, where the basic variables are the fields 4>(x), we can define the momentum conjugate to the field variables in a straightforward manner as

n

^

= TSITT = ^ d(p(x)


This is the analogue of the relation between the momentum and velocity in classical mechanics, namely, p = x (for m = 1). In quantum field theory, in operator language, this then is the starting point for quantization. However, in the path integral formalism, we treat all variables classically. Therefore, let us analyze various concepts in the classical language. First, let us note that given the Lagrangian in Eq. (11.3), we can obtain the Hamiltonian through a Legender transformation as H = Jd*xU{x)<j)(x) - L In the present case, we can write this out in detail as

(11.6)

283

11.1. SYMMETRIES OF THE ACTION

H = jd3x(U(x)^x)-^2(x)

+ ^V(f>(x)-V(f>(x)

+^02(*) + ^ 4 (*)) = Jd3x(±n\x) + U/4> • v^(x) + ^4>2(x) + ^ 4 (x))(n.7) Sometimes, this is also written as H = J d3x {\j>\x) + \vcj>{x) • V4>(x) + ^ V ( x ) + ^(x))

(11.8)

Given a Lagrangian density, which depends only on <j>(x) and d^^x),

the Euler-Lagrange equation is obtained to be (This is

simply the generalization of Eq. (1.28) to the case of a field theory.)

^dd,j,(j){x)

d
which gives the dynamics of the system. Given the dynamical equations, we can ask how unique is the Lagrangian density for the system. The answer, not surprisingly, turns out to be that the Lagrangian density is unique only up to total derivatives. Namely, both £(<£,
and £(<j>, drf) + d^tf,

dx)

(11.10)

give the same Euler-Lagrange equation. We can, of course, check this directly. But a more intuitive way to understand this is to note that

284

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

with the usual assumptions about the asymptotic fall off of the field variables, we have SK = jdixdllK>i(4>,dx<j>) = Q

(11.11)

In other words, a total divergence in the Lagrangian density does not contribute to the action. Consequently, the variation of SJC cannot contribute to the dynamical equations. (We note here that even when the asymptotic fall off of the fields is not fast enough, this statement remains true.) We can, of course, check this for specific examples explicitly. Thus, choosing CK = d^K* = d^d^)

= d^d^

+ 0<W

(11.12)

the Euler-Lagrange equation gives dCK ^ dd,dA

a v

dC dd^+

ail

dC d

= 3 M <9^-2<9 M <9^ + ( 9 ^ 0 = 0

(11.13)

With this analysis, therefore, it is clear that a given system of dynamical equations will remain invariant under a set of infinitesimal transformations of the field variables of the form —• 4> + 6

(11.14)

11.1.

SYMMETRIES OF THE ACTION

285

if and only if the corresponding Lagrangian density changes, at the most, by a total divergence under the same transformations. Namely, if C —•4£ + dltK't

(11.15)

under a field transformation, then it defines an invariance of the dynamical equations. Note that in the special case when K^ = 0, then the Lagrangian density itself is invariant under the set of field transformations in Eq. (11.14) and, therefore, also defines a symmetry of the system. However, this is a very special case. In general, if under — • <j> + 8<j> S[] — + S[c/> + 6cj>} = S[4>]

(11.16)

then we say that the field transformations define an invariance or a symmetry of the system, namely, the dynamical equations. Continuous transformations, by definition, depend on a parameter of transformation continuously. This parameter can be a space-time independent parameter or it can depend on the coordinates of the field variables. In the first case, the transformations would change the field variables by the same amount at every space-time point. On the other hand, the change in the field variables, in the second case, will be different at different space-time points depending

286

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

on the value of the parameter. Accordingly, these transformations are called global and local transformations respectively. The basic symmetries in gauge theories are local symmetries.

11.2

Noether's Theorem Noether's theorem, very simply, says that for every con-

tinuous global symmetry of a system, there exists a current density which is conserved. More specifically, it says that for a system described by a Lagrangian density C{4>, <9M0), if the infinitesimal global transformations

(x) —> 4(x) + e^x)

(ii.i7)

where e is the constant parameter of transformation, define a symmetry of the system, in the sense that under these transformations

C^C

+ dnIfifadxfaSet)

(11.18)

then,

dd^x) defines a current density which is conserved.

(11.19)

287

11.2. NOETHER'S THEOREM

To see t h a t j£ is indeed conserved, let us note t h a t

9C

d(f)(x)

8e(x) + - ^ ddf,(x)

8c(d,4>(x)) - d»K»

(11.20)

Here, we have used the Euler-Lagrange equation in Eq. (11-9) as well as the fact t h a t d/j,8e(x) = 8e8IJ,(f){x). We note next t h a t t h e first two t e r m s in Eq. (11.20) simply give the change in the Lagrangian under t h e transformations. Therefore, we can also write using Eq. (11.18) 8^

= 8^-8^

= 0

(11.21)

This shows t h a t t h e current density given in Eq. (11.19) is indeed conserved. T h e current density defined in Eq. (11.19) depends on the p a r a m e t e r of transformation as well. A more fundamental quantity is t h e current without t h e p a r a m e t e r of t h e transformation a n d let us denote this symbolically as JT = if

(H-22)

We have to r e m e m b e r t h a t this is only a symbolic relation simply because t h e p a r a m e t e r e may, itself, have a tensorial structure in which

288

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

case t h e current without t h e p a r a m e t e r will have a more complicated tensor structure as we will see shortly. As in classical electrodynamics, we know t h a t given a conserved current density, we can define a charge which is a constant of motion as

Q = fd3xj°(x,t)

(11.23)

T h e fact t h a t this charge is a constant, independent of t i m e , can b e seen simply as follows.

§ - 5/*•**.'> =

fd3xdof(£,t)

= jd3x (d0j°(x,t) + V • j(x,t))

(11.24)

Here, we have added a t o t a l divergence which vanishes under our assumptions on t h e asymptotic behavior of t h e field variables. T h u s , we have

dt

= Jd3xd^f = 0

(11.25)

which follows from E q s . (11.21) a n d (11.22), namely, t h e conservation of t h e current density. This shows t h a t t h e charge is a constant of motion.

11.2. NOETHER'S THEOREM

289

Another way to understand this result is to note that this implies classically that the Poisson bracket of Q with H vanishes. Quantum mechanically, the commutator of the two operators must vanish. [Q,H] = 0

(11.26)

But this is precisely a symmetry condition in quantum mechanics. Namely, we know from our studies in quantum mechanics that a transformation is a symmetry if the generator of infinitesimal symmetry transformations commutes with the Hamiltonian. Conversely, any operator which commutes with the Hamiltonian is the generator of a symmetry transformation which leaves the system invariant. Thus, we recognize Q to be the generator of the infinitesimal symmetry transformations in the present case. This simply means that the infinitesimal change in any variable can be obtained from 8£4>=-i[eQ,4>)

(11.27)

(Classically, we should use appropriate Poisson bracket relations.) It is now clear that the vanishing of the commutator between Q and H simply corresponds to the Hamiltonian being invariant under the symmetry transformations-which we expect. In quantum field theory, the operator implementing finite symmetry transformations can be written in terms of the generator

290

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

of infinitesimal transformations as U(a) = e-iaQ

(11.28)

where a is t h e p a r a m e t e r of finite transformation.

A field variable,

under such a transformation, is supposed to change as <j){x) —> U{a)4>{x)U-\a)

= e~iaQ (x)eiaQ

(11.29)

And, furthermore, a true s y m m e t r y is supposed to leave t h e ground s t a t e or t h e v a c u u m invariant, namely, U(a)\0)

= e-iaQ\0)

= \0)

(11.30)

Equivalently, it follows from Eq. (11.30) t h a t Q|0} = 0

(11.31)

In other words, for a true s y m m e t r y , t h e conserved charge annihilates the vacuum. Therefore, in such a case, we note from Eq. (11.27) t h a t

( 0 | ^ ( x ) | 0 ) = -i(Q\[eQ,(x)]\0) = 0

(11.32)

where we have used Eq. (11.31). As we will see later, if there is a spontaneous breakdown of a symmetry, t h e n , t h e conserved charge,

11.2. NOETHER'S THEOREM

291

Q, does not annihilate the vacuum and that the vacuum expectation value of the change in some operator in the theory becomes nonzero.

11.2.1

Example As an example of Noether's theorem, let us study global

space-time translations as a symmetry of quantum field theories. Let us continue to use the 0 4 -theory for this discussion. Let us define the infinitesimal translations a^ + e" ^

or,

(11.33)

as the global transformations, where eM is the constant parameter of transformation. In such a case, 6e(x) = 4>(x + e) - (f>(x)

6ed^{x)

= dIM{8i4>{x)) = evdlldv(j>{x)

(11.34)

Given this, we can, of course, obtain the infinitesimal change in the Lagrangian density in Eq. (11.2)

292

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

C = l-d,ct>{x)d^{x) - ^ V ( z ) -

±
in a straightforward manner. A much simpler way to evaluate this, however, is to note that the Lagrangian density is effectively a function of x, namely, C = C(x). Thus, 8eC = C(x + e) - C(x) = e^d^x)

= d^K"

(11.35)

Therefore, we readily identify K» = e»£(x) = e^C(, d^)

(11.36)

On the other hand, we see from Eq. (11.2) that for this theory

dC dd,j.(f>(x)

= d»(t>{x)

(11.37)

As a result, we see from Eqs. (11.34), (11.36) and (11.19), that the Noether current defined in Eq. (11.19), in this case, follows to be

=

d»{x)(evdu4>{x)) - e»C

= e" (<9^(*R(*) - KQ = ev(d^(f>(x)&'
(11.38)

11.2. NOETHER'S THEOREM

293

This is, of course, the conserved current density and the current without the parameter of transformation has the form (see Eq. (11.22)) j?(x) = evT»v

(11.39)

where we see from Eq. (11.38) that T^ = d^{x)d,/(t>{x)

- rfvC

(11.40)

There are several comments in order here. First, let us note that the fundamental conserved quantity (in this case T^v) is not necessarily a vector. Its tensorial character depends completely on the parameter of transformation. Second, we note from Eq. (11.40) that the conserved quantity, in this case, is a symmetric second rank tensor, namely, Tia,

_ Tv»

(11.41)

This is known as the stress tensor of the theory. Let us also note from Eqs. (11.23) and (11.40) that the conserved charge, in this case, has a vectorial character and has the form

Ptl = Jd3xT0>"

(11.42)

294

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

To understand the meaning of the charges in Eq. (11.42) (there are, in fact, four of them), let us write them out explicitly. We note from Eqs. (11.40) and (11.42) that

P° =

fd3xT00

= f d3x {(fa))2 - C)

= J d3x {\j>\x) + I V 0 • V 0 + ^ V ( x ) + ±\x)) = H

Pl = J d3xT0i = Jd3x 4>d{4> or, P = - Jd3x<j>(x)V
(11.43)

We recognize the first quantity (namely, P°) as the Hamiltonian of the system obtained in Eq. (11.8) and from relativistic invariance we conclude that P must represent the total momentum of the system. Thus, we recover the familiar result that the space-time translations are generated by the energy-momentum operators of the theory.

295

11.3. COMPLEX SCALAR FIELD

11.3

Complex Scalar Field So far we have discussed the ^-theory where the basic

field variable is real. Such a theory, as we have mentioned before, can describe spin zero mesons which are charge neutral. Let us next consider a scalar field theory where the basic field variable is complex. Namely, in this case, {x)

(11.44)

One way to study such a theory is to expand the complex field in terms of two real fields as, say <£(*) = ^ ( 0 i ( z ) + i&(*))

(11.45)

However, let us continue with the complex field, (f)(x), as the basic variable. The real Lagrangian density describing quartic interactions can be generalized from Eq. (11.2) and written as £ ( 0 , *) = drf*^

- m24>*4> - ^{<j>*4>)2

(11.46)

with A > 0. We can treat and (f>* as independent dynamical vari-

296

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

ables. Correspondingly, the two Euler-Lagrange equations following from Eq. (11.46) are given by Ou„„ "7*

,. - -^—7 =

0

or, ( a ^ + m 2 ) 0 + ^ ( ^ » V ) 0 = 0

(11.47)

or ( c ^ + m V + ^

(11.48)

and

W

= 0

Thus, the two dynamical equations in Eqs. (11.47) and (11.48) correspond to two coupled scalar equations. We should have expected this since having a complex field doubles the number of degrees of freedom. Let us next note that if we make a phase transformation of the form 4>(x)

—*

e'ioL4>{x)

<$>*(x) —+ eia*(x)

(11.49)

where a is a real, constant (global) parameter of transformation or

11.3. COMPLEX SCALAR FIELD

297

equivalently, an infinitesimal transformation of the form (e is infinitesimal) 6€4>{x) 8e*{x)

= — ie{x) = ie<j>*(x)

(11.50)

then, we note that under such a transformation * —* eia*e-ia<j>

= <j>*4>

(11.51)

Equivalently, under the infinitesimal transformations of Eq. (11.50), we note that

«e(^V) = (ScPW + WJ) =

ie - ie*4> = 0

(11.52)

Namely, under the transformation in Eq. (11.49) or (11.50) (f>* remains unchanged. Similarly, we note that under the transformation of Eq. (11.49) d^Vt

— a M (e* a 0*)^(e-* a ^) = d^d^

(11.53)

298

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

Alternately, from the form of the infinitesimal transformations in Eq. (11.50), we obtain 6,(8^*8^)

= (8e(8licl>*))d^

+8^(8,(8^))

= 8,(8^)8^6+

8^(8^8^))

= ied^*d^4>-ied^4>*d,i(t> = 0

(11.54)

(It is important to recognize that the invariance in Eqs. (11.53) and (11.54) results because the parameter of transformation is assumed to be independent of space-time coordinates.) In this case, therefore, we see that the constant phase transformations define a symmetry of the theory, in the sense that, £ = 8^*8^

- m2cf>*(j> - ^{(f>*(f>)2 — • £

(11.55)

Equivalently, 8€C = 0

(11.56)

Such a symmetry is called an internal symmetry since the transformations do not change the space-time points. For such an invariance, we note that K* = 0

(11.57)

11.3.

COMPLEX SCALAR FIELD

299

Therefore, the conserved current constructed through the Noether procedure has the form (see Eq. (11.19)) .„

dC ddy.<j)* =

dC <9<9M0

2

<9 M 0(ie0*) + &*<(>* (-ie)

= ie(*d*-&**<(>) = e(z0* B1 0) = ef

(11.58)

where we have defined f = i<j>* & 4> = i^d^cj)

- <9M0* <j>)

(11.59)

The conserved current, j ^ , in this case, has a vectorial character very much like the electromagnetic current density. Therefore, it can be identified with the electromagnetic current associated with this system. This theory, therefore, can describe charged spin zero mesons. The conserved charge, for the present case, can be written as

Q =

fd3xj°

= J dzx i{4>*0-0* ) = i J d3x (0*0-0*0)

(11.60)

300

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

In quantum field theory, Q would represent the electric charge operator. As we have mentioned earlier, if the phase transformations in Eq. (11.49) or (11.50) define a true symmetry of the system, then the charge operator in Eq. (11.60) must annihilate the vacuum. In other words, we must have, in such a case, Q|0) = 0

11.4

(11.61)

Ward Identities Symmetries are quite important in the study of physical

theories for various reasons. First of all, they lead to conserved quantities and conserved quantum numbers. But more importantly, they give rise to relations between various Greens functions and, therefore, between the transition amplitudes. Thus, as an example, let us consider the generating functional for the complex scalar field. Z[J,J*} = eiW{J'J*}=NfV(l)V(t>*eiS&(l>*,J'J*}

(H.62)

Let us note here that we have now set h = 1 for simplicity and that we have defined

S[, 4>\ J, J*] = S[<j>, 4>*] + J d4x{ J*(j) + Jcf>*)

(11.63)

11.4. WARD IDENTITIES

301

with S[,*] representing the dynamical action for the system. Here, we note that J* is the source for the field whereas J corresponds to the source for *. Note also that even though the action S[,4>*\ is invariant under the global phase transformationsof Eq. (11.50), the complete action S[(f>, *, J, J*] is not unless we simultaneously change J and J* also. In fact, let us note that infinitesimally, 6eS[,<j>*,J,J*] = 8eS[4>,4>*]+8e{j'(Px(J* + J*))

= Jd*x(J*8e(j>+J6£(l)*) = -ie fd4x(J**)

(11.64)

Since the generating functional does not depend on the field variables (that is, the fields are all integrated out), making a field redefinition in the integrand of the path integral should not change the generating functional. In particular, if the redefinition corresponds to the infinitesimal phase transformations defined in Eq. (11.50), then we will have 6eZ[J, J*] = 0 = N J V(f>VV*(e J#x(J*-

^*> J> J 1

jp^j^P**'1*}

(11.65)

where we have used Eq. (11.64). In general, one should also worry about the change coming from the Jacobian under a field redefinition.

302

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

In the present case, it does not contribute. Let us recall that by definition,

6J(x) S

J

4¥^T

= *iV/P<W*#C)e^>V'J1

(11.66)

Using this then, Eq. (11.65) becomes

e

/ *' W - ^ > " J^~^

=°

<1L67>

This must hold for any arbitrary value of the parameter e and, therefore, we conclude that

r

A

f)W

8W

«• / <''(»)«^-'W«w> = °<"-68> This is the master equation for defining symmetry relations. By taking higher functional derivatives of Eq. (11.68), we can obtain relations between various connected Greens functions as a result of the symmetry in the problem. In this case, the symmetry

303

11.4. WARD IDENTITIES

relations are quite simple (simply because the symmetry transformations in Eq. (11.50) are simple), but in the case of more compMcated symmetries such as gauge symmetries, such relations are extremely useful and go under the name of Ward Identities of the theory (also known as Slavnov-Taylor identities particularly in the case of gauge symmetries). It is interesting to note that we could also have obtained the Ward identities from a combined set of transformations of the form 6e(j> = —iecf) 6eJ = -ieJ

6e(f>* = ie(j>* 6eJ* = ieJ*

(11.69)

In such a case, it is easy to see that the complete action in Eq. (11.63) is invariant. Namely, 6eS[, (j>*,J, J*] = 0

Therefore, from

Z[J, J*} = eiW^ we obtain

J

1 = iV J V<j)V(t>* eiS& ^*' J ' J*]

(11.70)

304

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

6£Z[J, J*] = Nj

VVcf>* {i8eS) eiS^

or, iStW[J, r\eiW\-J^ or,

^*> J ' J *l

J

*\ = 0

6eW[J,J*]

= 0

or

-• / ' " ' ^ - ' w ^ ) ' = ° (11-n) This is, of course, the same relation as in Eq. (11.68). Let us note that in the case of a complex scalar field, we will have a complex classical field defined by (see Eq. (10.7))

From the transformation properties of the fields 4>{x) and *(x) in Eq. (11.50), we can immediately determine the transformation properties of the vacuum expectation values in Eq. (11.72). Namely, we obtain (This corresponds to asking, by how much would c(x) change

11.4. WARD IDENTITIES

305

if we change 4>{x) according to Eq. (11.50).) 6ee(x) = (0\6e{x)\0)J'r S4t(x)

=

-ie{0\<j>(x) |0) J ' J * =

-ie^c(x)

= {0\6e*(x)\0)J>J' = ie(0\*(x)\Q)J'r = iecj>*c{x) (11.73)

where we have assumed the invariance of the ground state under such a transformation. From the transformation properties of the classical fields in Eq. (11.73), we can now work out the Ward Identities for the 1PI vertex functions. In the present case, we note that we have r [ & , # ] - W[J,J*] -Jd*x(J*e(x)

+ J(x)
(11.74)

from which it follows that « e r [ & , # ] = -jdix{J*6etl>c(x)

+ J{x)6t*e(x))

= ie J d*x (J*(x)<j>c(x) - J(x)*e(x))

= 0

(11.75)

Here in the last step, we have used the relation in Eq. (11.68). On the other hand, using Eq. (11.73), we obtain

306

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

= -i£/A(^w-^j^w) Therefore, following Eq. (11.75), we can set this to zero and noting that the parameter e is arbitrary, we obtain

/ d'x (drf^*) - i£rA*cW = ° J

0c(x)

(n-77)

0
This is the master equation from which we can derive relations between various I P I vertex functions, as a consequence of the symmetry in the theory, by taking higher order functional derivatives.

11.5

Spontaneous Symmetry Breaking Let us next consider the complex scalar field theory de-

fined by the following Lagrangian density. C = 8^*8^

+ m20V - - ( 0 »

2

A> 0

(11.78)

This is the same Lagrangian density as in Eq. (H-2) except for the sign in the mass term which is opposite. It is clear that this Lagrangian

11.5. SPONTANEO US SYMMETRY

307

BREAKING

density is also invariant under the global phase transformations in Eq. (11.49) or (11.50) since each of the terms is. Therefore, the phase transformations define a symmetry of this theory as well and according to Noether's theorem, there exists a conserved charge which is the same as given in Eq. (11.59). However, if we look at the potential of this theory, namely,

V{,<j>*) = - m W + ^ V )

2

(H.79)

then, we note that for constant field configurations, the extrema of the potential occur at

~

= (-m2

+

-0»0* =

O

(11.80)

The solutions of these extremum conditions are easily obtained to be

0 c = <j>: = o 2m 2

or,

4>*c(j>c = -—

However, it is quite easy to see that

(11.81)

308

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

d2V d4>*d

(11.82)

= —m i=<**=0

whereas d2V d(/)*d(j)

.2m' " A

= 5*"*

.im' " A

= m

(11.83)

Consequently, the extremum at
#
2m2

(11.84)

Note that since for constant field configuarations the derivative terms vanish, this also defines the true minimum of energy or the true ground state of this theory. To better understand what is involved here, let us rewrite the complex field in terms of two real scalar fields. Namely, let us write 1

= ~^(a

+ i

P)

(11.85)

where we assume that a and p are real (Hermitian) scalar fields. In terms of these variables, then, the minimum of the potential occurs at

11.5. SPONTANEOUS SYMMETRY

or,

BREAKING

309

ol + pl = ^f

(11.86)

It is clear that, in this case, there is an infinite number of degenerate minima lying on a circle in the a — p plane. For simplicity, let us choose pc = 0. Then, the minimum of the potential can be chosen to be at 9

or,

4m 2

2m ac = ±-j=

(11.87)

Let us, in fact choose the minimum to be at 2m
V\

pc = 0

(11.88)

In this case, therefore, we see that one of the fields develops a vacuum expectation value, namely, ac = (0\a(x)\0) = pc = (0|p(aj)|0) = 0

~ (11.89)

310

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

To understand further what is involved, let us plot the potential in Eq. (11.79) as a function of a and p for constant values of the fields. 171

V(*,p) = -^(*i

2\ , ^ + P>) +^

' 2 ,

/ 2

+

pif

A> 0

(11.90)

Thus, the potential, in the present case, is very much like the instanton potential in Eq. (7.45), but the minima are infinitely degenerate. Popularly, such a potential is also known as the Mexican hat potential.

Let us also note that since

we can deduce from the transformation 8(f) = —iecj) rule in Eq. (11.50)

11.5. SPONTANEOUS SYMMETRY

BREAKING

311

that -j=(6a + iSp)

=

or,

= e(p — ia)

6a + i6p

-ie-j=(a

+ ip) (11.91)

From this, we conclude t h a t under t h e global phase transformations, t h e real scalar fields transform as 6a

=

Sp =

ep -etr

(11.92)

In other words, the global phase transformations correspond to a rot a t i o n in t h e a - p plane. Let us also note from our earlier discussion in Eq. (11.27) t h a t t h e infinitesimal change in any operator can be expressed as a c o m m u t a t o r with t h e charge associated with the transformation as 6a = —ie[Q,a]

=

ep

6p = -ie[Q,p]

=

-ea

Therefore, since with our choice in Eq. (11.89) (0\a(x)\0)=ac

=

^

(11.93)

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

312

we conclude using Eq. (11.92) that 9m

(0\Sp\0) = -e(0\a\0) or,

= - e ^

2m -»e(0|[Q,d|0) = - e -^=

(11.94)

It is clear, therefore, that in the present case, we must have Q|0)^0

(11.95)

in order that the relation in Eq. (11.94) is consistent. In such a case, we say that the symmetry of the Hamiltonian (or the theory) is spontaneously broken. Since Q does not annihilate the vacuum of the present theory, let Q\0) = \X)

(11.96)

We know from Eq. (11.26) that the symmetry of the Hamiltonian implies that [Q,H] = 0 Assuming that the vacuum state has zero energy (i.e. H|0) = 0), we then obtain using Eq. (11.96)

11.5. SPONTANEOUS SYMMETRY

BREAKING

313

[Q,H]\0) = 0 or, {QH-HQ)\0)

= 0

or, HQ\0)

= 0

or, F | X )

= 0

(11.97)

In other words, the state |%) denned in Eq. (11.96) would appear to be degenerate with the vacuum in energy. We can, therefore, think of this as another vacuum. The problem with this interpretation is that this state is not normalizable. This can be easily seen from Eq. (11.96) and (11.23) as follows. (Q is seen from Eq. (11.60) to be hermitian.) (x\x)

= =

(0\QQ\0) (0\jd3xj°(x,t)Q\Q)

= Jd3x{0\eiP-xj°(0)e-iP-xQ\0)

(11.98)

We have already seen that the Hamiltonian commutes with Q expressing the fact that it is independent of time. Since Q does not depend on spatial coordinates, it follows that the momentum operator also commutes with Q. In fact, in general, we can write [P„Q} = 0

(11.99)

There are many ways of obtaining this result besides the argument

314

CHAPTER

11. INVARIANCES

AND THEIR

CONSEQUENCES

given above. The most intuitive way is to note that PM generates space-time translations whereas Q generates a phase transformation in the internal Hilbert space. Both these transformations are independent of each other and, therefore, their order should not matter which is equivalent to saying that the generators must commute. A consequence of their commutativity is that we have e-iPxQ

= Qe-iPx

(11.100)

Using this in Eq. (11.98), then, we obtain (X|X> =

Jd3x(0\eiP-xj°(0)Qe-iP-x\0)

=

fd3x(0\j\0)Q\0)

=

(0|j°(0)Q|0) Jd3x—>oo

(11.101)

where we have used the property of the ground state, namely, PM|0) = 0

(11.102)

In other words, the state \x) is not normalizeable and hence cannot be thought of as another vacuum. This analysis also shows that the finite transformation operator U(a) = e~iaQ

(11.103)

11.5. SPONTANEOUS SYMMETRY

BREAKING

315

does not act unitarily on t h e Hilbert space. In fact, it is straightforward t o show t h a t t h e charge Q does not exist when there is spontaneous breakdown of t h e symmetry. Let us note, however, t h a t even t h o u g h Q m a y not exist, c o m m u t a t o r s such as

are well defined in such a theory a n d as a result expressions such as

U{a)(j)(x)U-\a)

e-iaQ(x)eiaQ

=

are also well denned. Another way of saying this is to note t h a t while the operator U(a)

defines unitary transformations for t h e field vari-

ables, it does not act unitarily on the Hilbert space. This is another manifestation of spontaneous s y m m e t r y breaking. To analyze further the consequences of spontaneous symm e t r y breaking, let us note t h a t even classically, if the potential has a nontrivial minimum, t h e n a stable p e r t u r b a t i o n would require us to expand the theory a b o u t t h e stable minimum. T h u s , let us expand

a p —•

> {a)+a p

=

2m -y~+a (11.104)

T h e n , t h e Lagrangian density of t h e theory in Eq. (11.78) would be-

316

CHAPTER

11. INVARIANCES

AND THEIR

CONSEQUENCES

come

+ p2) - A(a2

= \a^a + \dd&p + ^ V

+^((- + ^ ) =

-d^d»a

2

+ P2)-^((- + ^ )

+ -d)ipd»p + —(a2 A .9

,

2

+ P2)2

+ p2 + -^a

4m

4m

+ p2f

+ —) 2

,

= ^ a v + ^ V + ^t^-"-2-^) 9.m

2

m 2 , ,2m 3

V ( T -T) +^

2m 3 ,

- ^

= ia„
+ p 2 )

_^

( ( r

2

+ /

,

V

( U 1 0 5 )

Thus, we see the interesting fact that while the field a

11.5. SPONTANEO US SYMMETRY

BREAKING

317

remains massive with the right sign for the mass term, the field p indeed has become massless. This is a general feature of spontaneous symmetry breaking, namely, whenever a continuous symmetry is spontaneously broken in a manifestly Lorentz invariant theory, there necessarily arise massless fields (particles). These are known as the Goldstone fields or Goldstone modes (particles). In the present case, we note that p corresponds to the Goldstone field and let us recall our earlier result, namely, 2TT?

(0|fip|0) = - e ^

(H-106)

This is also a general feature of theories with spontaneously broken symmetries. Namely, in such theories, the change in the Goldstone field under the symmetry transformation acquires a nonzero vacuum expectation value. In terms of the potential, it is easier to understand the Goldstone mode intuitively.

The minimum of the potential occurs

along a valley and the Goldstone mode simply reflects the motion along the valley of the potential. In particle physics, one does not know of elementary spin zero particles which are massless. The closest that comes to being massless is the pi-meson. The Goldstone particles were, therefore, not received well by the particle physics community. However, in the presence of gauge fields like the photon field, the Goldstone modes get absorbed into the longitudinal modes

318

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

of the gauge bosons effectively making t h e m massive. This is known as t h e Higgs mechanism and is widely used in t h e physical models of fundamental interactions. A massless field or a particle, of course, has associated with it an infinite characteristic length (Compton length). T h e most familiar massless field is the photon field and we know t h a t as a consequence of the p h o t o n being massless, the Coulomb force has an infinite range. In fact, we recognize t h a t t h e two point function in such a theory will have an infinite correlation. Namely, two particles at infinite separation will still feel t h e presence of each other.

Therefore, we

conclude t h a t when Goldstone modes are present, certain correlation lengths will become infinite.

11.6

Goldstone Theorem In a manifestly Lorentz invariant q u a n t u m theory with

a positive metric for the Hilbert space, t h e Goldstone t h e o r e m states t h a t if there is spontaneous breakdown of a continuous symmetry, t h e n there must exist massless particles (Goldstone particles) in t h e theory. To see a general proof of this theorem, let us assume t h a t we have a theory of n-scalar fields described by the Lagrangian density

319

11.6. GOLDSTONE THEOREM

C = C{<j>i,d,d>i)

« = l,2,.--,n

(11.107)

Furthermore, let us assume t h a t t h e global transformations

S^i = T^e)^

(11.108)

where we assume s u m m a t i o n over r e p e a t e d indices a n d where t h e global p a r a m e t e r of transformation, e, may itself have an index, define a s y m m e t r y of t h e Lagrangian density in Eq. (11.107). In this case, we can define the generating functional with appropriate sources as

Z[Ji] = eiWW

= N JvieiS^

J

^

(11.109)

Furthermore, t h e classical fields are defined to be

&=(*) = T ^ T = <0|<M*)|0) Jfc oJi(x)

(11.110)

T h e case of spontaneous symmetry, of course, corresponds t o having nontrivial iC's when t h e sources are t u r n e d off. Namely, even if for one of t h e values of i, 6W ic = ic{X)\jk=0 = SJj(x)

^0 Jk=0

(11.111)

320

CHAPTER 11. INVARIANCES AND THEIR CONSEQUENCES

then, we will have spontaneous breakdown of the symmetry. Let us note from Eqs. (11.73) and (11.108) that the classical fields would transform under the symmetry transformations as

8eic(x) = Tij{e)ie{x)

(11.112)

We also know from Eq. (10.25) that the 1PI vertex functional satisfies the defining relation 8ic(x)

= -Ji(x)

(11.113)

When the source is turned off, this defines an extremum equation whose solutions, 0j c , will have at least one nonzero value if the symmetry is spontaneously broken. Given the above relation, we can also obtain -SeJi(x)

=

f^y ' c 6ejc(y) J 8(f>ic{x)8(f>jc{y)

When we switch off the sources, consistency of Eq. (11.114) will lead to (in this case (f>ic(x) = <j>ic = constant)

321

11.7. REFERENCES

Jdy

8(f)ic(x)S(j)jc(y)

Tjk(e)(f>kc = 0

or, J' d4y (G-F1(x-y))..Tjk{e)<}>kc or, (GF\Pfi

= 0

= 0)).. Tjk(e)kc = 0

(11.115)

This system of equations will have a nontrivial solution (namely, there will be spontaneous breaking of the symmetry) only if d e t ( G ^ ( p M = 0)).. = 0

(11.116)

In other words, there must exist massless particles in the theory. This proves the Goldstone theorem.

11.7

References

Goldstone, J., Nuovo Cimento, 19, 154 (1961). Guralnik, G. S. et al, in "Advances in Particle Physics", Ed. R. L. Cool and R. E. Marshak. Hill, E. L., Rev. Mod. Phys., 23, 253 (1957). Itzykson, C. and J . - B . Zuber, "Quantum Field Theory", McGrawHill Publishing. N a m b u , Y . and G. Jona-Lasinio, Phys. Rev., 122, 345 (1961).

Chapter 12 Systems at Finite Temperature 12.1

Statistical Mechanics

Let us review very briefly various concepts from statistical mechanics. Let us consider not one q u a n t u m mechanical system, b u t a whole collection of identical q u a n t u m systems-an ensemble. T h u s , for example, it can b e a n ensemble of oscillators or any other physical system. Let us further assume for simpMcity t h a t t h e physical system under consideration has discrete eigenvalues of energy. Each system in this ensemble can, of course, be in any eigenstate of energy. T h u s , we can define pn to represent t h e probability of finding a system in the ensemble to be in an energy eigenstate \n).

This is, of course,

completely statistical in t h e sense t h a t pn can be identified with the 323

324

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

n u m b e r of physical systems in the state \n) divided by t h e total number of systems in t h e ensemble. Such a situation is quite physical as we know from our studies in statistical mechanics. Namely, we may have an ensemble of physical systems in t h e r m a l equilibrium with a heat b a t h . For a given ensemble, t h e value of any observable quantity averaged over t h e entire ensemble will take t h e form {A) = A = EPn(n\A\n)

= ^j>nin

n

(12.1)

n

where we are assuming t h a t the energy eigenstates are normalized a n d that An = (n\A\n)

(12.2)

denotes the expectation value of t h e operator in t h e q u a n t u m mechanical state | n ) . T h u s , there are two kinds of averaging involved here. First, we have t h e average in a q u a n t u m state (expectation value) a n d second, we have t h e averaging with respect to the probability distribution of systems in t h e ensemble. Being a probability, pn has to satisfy certain conditions. Namely, 1 > Pn > £Pn n

=

0 1

for all n (12.3)

12.1. STATISTICAL MECHANICS

325

It is in general very difficult to determine the probability distribution for an ensemble. However, if we are dealing with a thermodynamic ensemble, namely, an ensemble interacting with a large heat bath, and if we allow sufficient time to achieve thermal equilibrium, then we know that the probabihty distribution, in this case, is given by the Maxwell-Boltzmann distribution. Namely, in this case, we can write En Pn = ^e'kT

(12.4)

Here En is the energy of the nth quantum state, k is the Boltzmann constant and T the temperature of the system. The normalization factor Z can be determined from the relations for the probabilities in Eq. (12.3) as

X>n = 1 n

or, ^ £ e P Z n

En kT

En or, Z = Y.e'kT or,

= 1

=

Y.{n\e~^H\n)

Z(/3) = Tre-PH

(12.5)

326

CHAPTER 12.

SYSTEMS AT FINITE TEMPERATURE

where we have defined (12.6) Z((3) is known as the partition function of the system and plays the most fundamental role in deriving the thermodynamic properties of the system. For a statistical ensemble, it is easy to see that the thermodynamic average of any quantity defined in Eq. (12.1) will be given by (A)p

=

Y.Pn{n\A\n) n

=

mVe~PEn{n]A]n)

= mT,('~f"'A) -

(12.7)

Tr(e~PHA) Tre-(3H

In particular, the average energy associated with the system follows from Eq. (12.7) to be

327

12.1. STATISTICAL MECHANICS

Tr(e-PHH) Tre-PH

_

_ ~

a/3 A ' c Tre-f3H 1 . { Z((3)

dZ(/3) d(3 > (12.8)

T h e a m o u n t of order or the lack of it, for an ensemble, is denned t h r o u g h t h e entropy as S = -£Pnlnp

n

= -(lnp)

(12.9)

n

By definition, it is clear t h a t t h e entropy is always positive semidefinite since 0 < pn < 1. F u r t h e r m o r e , its value is zero for a pure ensemble for which

Pn = <5nm

for a fixed m

(12.10)

For such an ensemble, all the individual systems are in the same energy s t a t e and, therefore, it is an ordered ensemble. On the other h a n d , the larger the n u m b e r of states the physical system can be in, t h e more

328

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

disordered the ensemble becomes and the entropy increases. For a thermodynamic ensemble, as we have seen in Eq. (12.4),

Pn

Z([3)

Therefore, we can calculate the entropy of the ensemble to be

n

=

-Y,Pn(-pEn-hiZ(f3)) n

= pYl PnEn + In Z ( / 3 ) £ P n n

n

= (3U + In Z{/3) = -P-^\nZ((3)+ln

= -P2^^2^))

Z((3)

( 12 - n )

Here we have used Eq. (12.8) and (12.3) in the intermediate steps. Given the internal energy, U, and the entropy, 5 , the free energy for an ensemble can be written as

329

12.1. STATISTICAL MECHANICS

= -^lnZ(/3) + ^lnZ(/3)-ilnZ(/3) =

—InZOS)

(12.12)

In terms of the free energy, we can define the other thermodynamical quantities as U = -±UZW

= ±V>F) = F + f>%

S=-e%i\^m)=P%

(12.13)

It is also interesting to note from Eq. (12.12) that the partition function takes a particularly simple form when expressed in terms of the free energy. Namely, we can write Z(P) = e-PFW

(12.14)

We have gone over some of these concepts in some detail in order to bring out the essential similarities with the concepts of path integral that we have been discussing so far.

330

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

One of the major interests in the study of statistical mechanics is the question of phase transitions in such systems. Phase transitions are all too familiar to us from our studies of the different phases of water. Even in solids, such as the magnets, the hysteresis effect or the effect of spontaneous magnetization provides an example of a phase transition. Namely, we know that below the Curie temperature, Tc, if a magnetic material is subjected to an external magnetic field, then the material develops a residual magnetization even when the external field is switched off. The amount of residual magnetization decreases as the temperature of the system approaches the Curie temperature and vanishes at Tc. For T > Tc, the system exhibits no spontaneous magnetization. The temperature T = Tc is, therefore, a critical temperature separating the different phases of a magnetic material. The behavior of physical systems near the critical point is of great significance. This can be studied from the point of view of statistical mechanics quite well. They can also be studied with equal ease using the concepts of path integrals. However, before we discuss this, let us recapitulate how one uses statistical mechanics to study critical phenomena.

331

12.2. CRITICAL EXPONENTS

12.2

Critical Exponents To fix ideas clearly, let u s discuss the critical exponents

in t h e context of a specific model which explains t h e properties of magnetization quite well. This model goes under the n a m e of planar Ising model or t h e Ising model in two dimensions. T h e crucial feature of this model is t h a t it ascribes the magnetic properties of a material to its spin content. This should b e quite familiar from our studies of atomic systems where we know t h a t elementary particles with a nontrivial spin possess magnetic dipole m o m e n t s Let us consider a square lattice with equal spacing in b o t h x and y directions. Let us also assume t h a t a t each lattice site labelled by n = (TCI,ra2),there is a spin S(n) which can either point u p or down. Accordingly, we assume „/ x I S(n) = {

1 f°r

S

P m UP _ - 1 for spin down

(12.15)

F u r t h e r m o r e , let u s assume t h a t the spins interact as locally as is possible. I n fact, t h e Hamiltonian for the Ising model is taken t o b e H = -JY,S(n)S(n

+ ji)

(12.16)

n,/i

where we have assumed a simplified coupling for t h e problem. Here

332

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

jl s t a n d s for either of the two unit vectors on the lattice. In simple language, t h e n , the Ising model assumes nearest neighbor interaction for t h e spins which are supposed to be pointing only along t h e z-axis. T h e constant, J, measures t h e strength of the spin-spin interaction. It is clear t h a t if its value is positive, t h e n a m i n i m u m of t h e energy will be obtained when all t h e spins are pointing along the same direction-either u p or down. Accordingly, such a coupling is known as a ferromagnetic coupling.

Conversely, if J is negative, t h e n the

coupling is known as anti-ferromagnetic.

I t is worth pointing out

here t h a t t h e Hamiltonian for t h e Ising model has a discrete symmet r y in t h e sense t h a t if we flip all the spins of the system, t h e n the Hamiltonian does not change. Let us next subject this spin system to a constant external magnetic field B. In this case, t h e Hamiltonian becomes H=-JYiS(n)S(n

+ fi) + B'£S{n)

(12.17)

n

n,fl

T h e partition function defined i n Eqs. (12.5) a n d (12.14), for the present case, takes the form Z(f3, B) = e - W , B)

= Tre-(3H

=

^

e-(3H

(12

18)

config

T h e s u m m a t i o n , here, is over all possible spin configurations of the system. Let us note t h a t at every lattice site, t h e spin can take two

12.2. CRITICAL EXPONENTS

333

possible values. Consequently, if N denotes the total number of lattice points, then there are 2N possible spin configurations over which the summation in Eq. (12.18) has to be carried out. The true partition function, of course, has to be calculated in the thermodynamic limit when N —> oo. Let us note now from Eqs. (12.17) and (12.18) that, in this case, we have

», | f

= ^Tr(Y,S(„)e-lm)

= (T,S{n))l!

(12.19)

Using the translation invariance of the theory, we can write (S(n))p

= (5(0))^

(12.20)

= N(S(0))p

(12.21)

so that we obtain %

Thus, from Eq. (12.21), the mean magnetization per site can be obtained to be M(P, B) = jf(Z S(n))p = (S(0))p = 1 g

(12.22)

334

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

This is, of course, a function of both the temperature and the applied magnetic field and its value can be calculated once we know the free energy or the partition function. The magnetic susceptibility is proportional to the rate of change of magnetization with the applied field and is defined to be X

dM dB

=

J=0

1 N

d2F dB2

B=0

= 4(£<S(n)S(m))^-<£S(n)}§)B=0 iV

n,m

n

- iV 2 (5(0))|) B = o

= | : (N £ (S(n)S(0))p

(12.23)

where we have used Eqs. (12.19) and (12.20). Thus, we see that the magnetic susceptibility is related to the fluctuations in the spin. It is large at those temperatures where the correlation between the spins is large. Note that if the system has no net magnetization, i.e., no spontaneous magnetization, namely, if (S(0))p\

B=0

(12.24)

= 0

then, the magnetic susceptibility is completely determined by the spin-spin correlation function. Namely, in this case, we have

X=P'E{S{n)S(0))fl

(12.25) B=0

12.2. CRITICAL EXPONENTS

335

If, for some temperature j3 > (3C, we find in our spin system that (S(0))P\B=0

+ 0

(12.26)

then, the system shows spontaneous magnetization or residual magnetization. In this case, we note that the discrete symmetry of the system is spontaneously broken. The spontaneous magnetization vanishes as we approach the critical temperature and for (3 < f3c, the system will show no spontaneous magnetization simply because the thermal motion will dominate. The critical temperature and the behavior of spontaneous magnetization near the critical temperature, namely, how the magnetization vanishes as the temperature approaches the critical temperature M\B=0 ~ (T - Tcf

(12.27)

can be calculated once we know the partition function. Let us emphasize here that the parameter f3 in the exponent is not T4; which was defined earlier but represents a critical exponent. (The notation is unfortunate, but this is the convention.) Furthermore, the spontaneous magnetization defines an order parameter in the sense that its value separates the two different phases. We can similarly calculate the correlation length between the spins, £(T), at any temperature by analyzing the magnetic sus-

336

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

ceptibility. For very high temperatures, it is clear that the thermal motion will not allow any appreciable correlation between the spins. However, as the temperature of the system is lowered to the critical temperature, the system may develop long range correlations and the behavior of the correlation length near the critical temperature is parameterized by another critical exponent of the form

£(T) ~ ( r - Tc)~v

(12.28)

The magnetic susceptibility may similarly become large at this point and its behavior near the critical temperature is parameterized by yet another critical exponent as

X(T)

~ (T - T c )~ 7

(12.29)

Similarly, other thermodynamic quantities in the system such as the specific heat defined as C

T

d2F

= - gf-2

( 12 - 3 °)

may also display a singular behavior at the critical point and all these can be calculated once we know the partition function.

337

12.3. HARMONIC OSCILLATOR

12.3

Harmonic Oscillator The calculation of the partition function for the one di-

mensional quantum harmonic oscillator is quite straightforward. We know that for an oscillator with a natural frequency LJ, the energy levels are given by En = (n + l)u

n = 0,1,2,..-

(12.31)

where we have set h — 1. For this system then, the partition function can be derived using Eqs. (12.5) and (12.31) to be Tre-P11

Z(/3) = n

-(3(n + hu

= E

e

l

n

= e""2"(£ e-n^) n=0

{3u>

= -

2

(7-4^7) \-e-P"'

338

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

Since we know the partition function, we can calculate the thermodynamic properties of the system. Let us next see how we can calculate the partition function for the harmonic oscillator through the path integral method. Let us recall that we have already calculated the transition amplitude for the harmonic oscillator which has the form (see Eq. (3.66))

(xf,T\xi,0)

-

C'^W

= (xf\e-iHT\Xi)

=

NfVxeiSW

• ^ ) ' ^ l

(-33)

whereas we have noted earlier (see Eq. (3.84) with J = 0) S[xd] = T T ^ ; [(*? + */) cos LJT - 2xiXf\ 2 sin UJI

(12.34)

We note here that we have set h = 1, J = 0 in the above equations and T here denotes the time interval between the initial and the final points of the trajectory. From the definition of the partition function, Z(J3) =

Tre~PH

we note that the trace can be taken in any basis. In particular, if we choose the coordinate basis in the Schrodinger picture, then we can

12.3. HARMONIC OSCILLATOR

339

write Z(P) = Jdx (x\e-PH\x)

(12.35)

We now recognize the integrand in Eq. (12.35) merely as the transition amplitude (see Eq. (1.43)) for the harmonic oscillator with the identification T

=

-i/3

Xf

=

Xi = x

(12.36)

In other words, the integrand really is the transition amplitude between the same coordinate state in the Euclidean time with (3 = -^ (T is the temperature here) playing the role of the Euclidean time interval. Using this, then, we obtain from Eq. (12.35)

\")

x

= }

x e

^27rt(-tsiiih/3a;)J

(_, %T!a J2x2 cosh/3u, - 2x2)) 2(—ismh.(3u) muj

i

\2

r

.

[dxe 2ir sinh (3u)) '

_

~L„v. Vs

„

inh/3u;

(coshpu - l)x

, 8u> 2 i mu> \2 //• dxe - ( m w t a n h — 2 x) 2n sinh {3cu

)

340

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

TV

mu 2TT s i n h /3u>

/

flu

mw tanh -— .

Y /3u 2 sinh (3LJ tanh -— / \5

4 sinh

2

put

2 sinh

(12.37)

/3u>

This is, of course, the partition function which we had found by a direct calculation in Eq. (12.32). Let us note next that since

Z{p) = e~PF =

1 2 sinh

/3u

(12.38)

we obtain F == - \ I n Z = \ (In2 + In sinh ^ ) Consequently, we note from Eq. (12.13) that

(12.39)

341

12.3. HARMONIC OSCILLATOR

(H)0 = U =

<91nZ

m

h =•

a/3

(3u cosh — 2 3u 2

UJ

sinh — 2

PUJ

(3UJ UJ

e 2 + e ~~2~ PUJ

2 e 2 —e 1 + e-/ 3 " 2 1 _ e-P" i

UJ

uje -PUJ ^

UJ

2 or, <*>, = "

+l

= l + ^

_ e-poJ Y

(12-40)

This is exactly what we would have obtained from Planck's law (remember that h = 1). Among other things, it tells us that for low temperatures or large /3, we have (B)p = Uc^

(12.41)

Namely, in such a case, the oscillators remain in the ground state. On the other hand, for very high temperatures or small p, we get

342

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

(H)p = U^^

+ ^-^±

= kT

(12.42)

This is, of course, the expression for the equipartition of energy. (We expect the system to behave in a classical manner at very high temperature.) This analysis of the derivation of the partition function for the harmonic oscillator from the path integral is quite instructive in the sense that it shows that a (D+l)-dimensional Euclidean quantum field theory can be related to a D-dimensional quantum statistical system since the Euclidean time interval can be consistently identified with /3 = ^ as in Eq. (12.36). In fact, the relation between the two for a bosonic field theory can be simply obtained as

Zip)

= N f

V(j>e~SE

= N[

Vcf>e-JodtJd3xCE

(12.43)

Here we are assuming integration over the end points which is equivalent to taking the trace. Furthermore, we note that the field variables, in this case, are assumed to satisfy a periodic boundary condition 4>{t + l3) = 4>{t)

(12.44)

This, as is clear, arises from the trace in the definition of the partition

12.3. HARMONIC OSCILLATOR

343

function. A careful analysis for the fermions shows that the partition function, in such a case, can be written exactly in the same manner but with anti-periodic boundary conditions. Namely, for fermions, we have Z((3) = N

jVi>V^e-S^^

= Nfv&tye-iyj**1111^®

(12.45)

with the boundary conditions ^(0) = -(/3) $(0) = -${p)

(12.46)

These boundary conditions can be shown to be related to the question of quantum statistics associated with the different systems. This way of describing a quantum statistical system in equilibrium through a Euclidean path integral is known as the Matsubara formalism or the imaginary time formalism (since we rotate to imaginary time). Let us note here without going into details that there exist other formalisms which allow for the presence of both time and temperature in the theory simultaneously. These go under the name of real time formalisms. The Matsubara formalism also suggests that a (D + 1)dimensional bosonic Euclidean quantum field theory can be related

344

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

to a (D + l)-dimensional classical statistical system in the following way. Let us consider a quantum mechanical system described by the Lagrangian L = - mi2 - V(x)

(12.47)

Then, the generating functional for such a system will have the form

f

Z = NJVxeh

-zS\x\

L J

t

—

dt L

=NJVxehJo

(12.48)

Here we have put back Planck's constant for reasons which will be clear shortly. If we rotate to Euclidean time, the generating functional takes the form , Z = NjVxe

- - / dt(-mx2 K hJo 2

+ V(x)) y "

(12.49)

This has precisely the form of a classical partition function if we identify H = j^dt^-rnx2

+ V{x))

(12.50)

as governing the dynamics of the system and h = kT=^

(12.51)

12.4. FERMIONIC OSCILLATOR

345

Here we should note t h a t t h e variable, t, in this case should b e t r e a t e d as a space variable a n d n o t as a time. Planck's constant measures q u a n t u m

Let us also recall t h a t t h e

fluctuations

in a q u a n t u m me-

chanical system whereas t e m p e r a t u r e measures t h e r m a l

fluctuations

in a statistical system. T h e identification of t h e two above, therefore, relates the q u a n t u m with t h e t h e r m a l system.

fluctuations

fluctuations

in a q u a n t u m mechanical system

in a corresponding classical statistical

This connection can b e simply extended to a field theory

where t h e Euclidean action would act as t h e Hamiltonian for the corresponding classical statistical system.

12.4

Fermionic Oscillator Let us next calculate the partition function for a fermionic

oscillator with a n a t u r a l frequency u, b o t h using t h e p a t h integrals and t h e s t a n d a r d m e t h o d s . We note from our discussion in section 5.1 t h a t t h e Hilbert space for t h e fermionic oscillator is quite simple. In fact, from E q s . (5.12) and (5.13), we note t h a t it is Mke a two level system with energy eigenvalues

Ei

=

-

(12.52)

346

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

Once again, we have set h = 1 for simplicity here. It, then, follows from the definition of the partition function in Eq. (12.5) that for the fermionic oscillator we have using Eq. (12.52) Z{p)

=

Tre~PH

= e~2~ + e"~2~ /3u> = e~2~(l + e - ^ ) = 2 c o s h ^

(12.53)

The evaluation of the partition function for the fermionic oscillator follows from the form of the transition amplitude derived in Eq. (5.91). Following our discussion in the last section, we note that in the case of fermions, we have to impose anti-periodic boundary conditions (see Eq. (12.46)), namely, for the calculation of the partition function, we require

fy = -i>i 4>f

= -i>i

(12.54)

We can now calculate the partition function for the system with the identifications in Eqs. (12.36) and (12.54) as well as the result in Eq. (5.91) as

347

12.4. FERMIONIC OSCILLATOR

0u Z((3) = Jd$idA{e~2

ei-^Mi-Mi))

= e~2~ J d ^ # i ( l - (1 + e-Pu)$nl>i) (3u> = e~2~'(1 + e~Pw) = 2 cosh ^

(12.55)

Here we have used the nilpotency properties of Grassmann variables (see Eq. (5.17)) as well as the integration rules given in Eqs. (5.26) and (5.27). This is exactly the same result which we had obtained earlier in Eq. (12.53) for the partition function of the system. We note now from the definition in Eq. (12.12) that the free energy for the fermionic oscillator is given by F(f3) = - I lnZ(/3) = - i ( l n 2 + l n c o s h ^ )

(12.56)

The average energy for the ensemble can now be calculated from Eq. (12.13) to be

348

CHAPTER 12. SYSTEMS AT FINITE TEMPERATURE

(Hh = u = §p(pm) U)

2

sinh cosh

(3UJ

/3u

(3LO /3w e 2 u —e 2 /3u> 2 (to e 2 +e 2 U)

UJ

or, (H)0 = U = - - +i - ^ - ^ 2 ' ePu + 1

(12.57)

It now follows that for low temperatures or large /3 (/3 = ^ ) , we have

(H)f, = U~-^

(12.58)

Namely, the system Hkes to remain in the ground state for low temperatures whereas for high temperatures or small /3, we obtain

(*)> = »*-*£—&

(1259)

'

In this case, we see that the average energy of the system goes to zero inversely with the temperature which amounts to saying that the system tries to populate equally the two available energy states.

12.5. REFERENCES

12.5

349

References

Huang, K., "Statistical Mechanics", John Wiley. Kogut, J., Rev. Mod. Phys., 5 1 , 659 (1979). Matsubara, T., Prog. Theor. Phys., 14, 351 (1955). U m e z a w a , H . et al, "Thermofield Dynamics and Condensed States", North Holland Publishing.

Chapter 13 Ising Model 13.1

One Dimensional Ising Model Let us pursue the ideas of statistical mechanics, which we

have developed in the last chapter, with the example of the one dimensional Ising model. The Hamiltonian for spins interacting through nearest neighbors on a one dimensional lattice (chain) is given by H=-JJ2sisi+1 + B'£Si i=\

(13.1)

i=l

Here we have assumed that the total number of lattice sites is N and that the spin system is being subjected to an external magnetic field, B, which is a constant. The classical partition function for this system is, by definition, 351

352

CHAPTER 13. ISING MODEL

N

Z(P)=

E S;=±l

N

(PJ E SiSi+l ~ PB E Si) e -i -i

arr e~PH = E

(13.2)

S,=±l

To study this system, let us assume periodic b o u n d a r y condition on the lattice (cyclicity condition), namely, Si+N = Si

(13.3)

a n d ask whether there exists a q u a n t u m mechanical system whose Euclidean generating functional will give rise to t h e partition function for t h e one dimensional Ising model. Let us consider t h e q u a n t u m mechanical system described by t h e Hamiltonian Hq = -a<Ti + ya3

(13.4)

where <7\ a n d o$ are the two PauH matrices a n d a and 7 are two arbitrary constant p a r a m e t e r s at this point. Let \s) denote t h e two component eigenstates of a$ such t h a t a3\s) - s\s)

s = ±l

(13.5)

We can now calculate t h e Euclidean transition amplitude for t h e quant u m system described by Eq. (13.4) between two eigenstates of (73, which is defined to be

13.1.

ONE DIMENSIONAL

ISING MODEL

353

(sfin\e-THi\sin)

(13.6)

Dividing the time interval into N steps of infinitesimal length e such that (for large N) Ne = T

(13.7)

and introducing a complete set of eigenstates of a^ at every intermediate point, we obtain (sfin\e-TH9\sin)

=

£ (*M\e-€H*M Me- cH «|**_i) • • • Me-€fr«h>(13.8)

si=±i

where the intermediate sums are for the values i = 2,3, • • •, N. Furthermore, we have also identified Sin = Si

Sfin = SM

(13.9)

Note that if e is small, then we can expand the individual exponents in Eq. (13.8) and write (si+1\e-eHi\Si)

Using the relations,

~ (*i+i|(l + eaaj - e7<73)|si)

(13.10)

354

CHAPTER

(s i+1 |CT 3 |Si)

=

13. ISING

- ( S i + Sj+i)

MODEL

(13.11)

which can be explicitly checked, we obtain (si+1\e-eH<;\Si)

{-(at + si+1))2 + ea(-{Si - sl+1))2 - e 7 - ( * + s i+1 )(13.12)

From the fact that for any i, s; = ± 1 , we also have the following identities. (I(si

+

s i + 1 )) 2 " = ( l ( s t

+ S l + 1 ))

(-(si + s i + i)) 2 n + 1 = -(Si + si+1) (±(8i - si+1))2n (^(Si + si+1))n(^(Si

= (±(8i - sl+1)f - si+1))m

= 0

2

forn>l forn>0 forn>l

(13.13)

for n, m > 1

Using the relations in Eq. (13.13), then, we can obtain (for constant parameters 6 and A)

355

13.1. ONE DIMENSIONAL ISING MODEL

e

(A(-(Si-St+1))2

+ 6-{Si +

l

Si+1))

£

oo

1

1

1

n =i

nl

2

2

1+E n=l

^ ( | ( « i " ' ^ l ) ) 2 " + ^ ( | ( » * + **+!))"

A",!, „=i n! 2 oo

/«2n

+ E^-T n=l (

2 n

)

2n+l

-i

T

( ^ + *; + I)) 2 "

!

2

1

4

~. 62n+1 , 1 . „ = 0 (2n + 1)! 2

2

,

oo

X2n+1

1

= i + (« -i)(^-^.)) + E( SrFI5i5 (^ + ^.) oo

^2n

-^

n=i (2nj! 2 = 1 + (e A - l ) ( - ( s ; - si+1))2 + sinhS (-(«< + s m ) ) + ( c o S h 6 - l ) ( ^ ( S i + S i + 1 )) 2

(13.14)

Let us now use the algebraic relation ( ^ + Si+i))2 + {\(si - si+1))2 = 1 Then, we can write Eq. (13.14) also as

(13.15)

356

CHAPTER 13. ISING MODEL

2

e(A(^(Si-si+1))

+ 8-{Si +

Si+1))

= cosh<5(-(si + s i + 1 )) 2 + e (-(si + sinh 6-(si + si+1)

si+1))2 (13.16)

We note that this has precisely the same form as the transition amplitude between two neighboring sites in Eq. (13.12) provided we make the identification cosh 8 -

eS + e-* =1 2

e A == ecu sinh 8 =

eS -

eS

•ej

(13.17)

Equivalently, with the identification e e6

=

ea

= l-€7

(13.18)

we can write

{si+1\e

exi

i\si)

= e

2

2

(13.19)

13.1. ONE DIMENSIONAL ISING MODEL

Consequently, using this identification, we obtain

Tre-THq

Si=±l

( A & i - « w ) ) 2 + « f ; ^ + «w))

=

^

=

e 2

i=i *

e

»=i

( - ^ M W + iE'i)

— ^

l

i=i

e

N

N

i=1

i=1

AT/K

= e 2

£

= e~2~ £

e

i=i

e-/3^

(13

provided we identify A = -2/3J

8 = -/3£

With these identifications, then we see that we can write

(13

358

CHAPTER 13. ISING MODEL

TH

Tre- i

NA = e~2~ Z(/3)

(13.22)

where Z(/3) represents the partition function for the one dimensional ising model. Once again, this shows that the quantum

fluctuations

in a quantum theory can be related to the thermal fluctuations of a classical statistical system.

13.2

The Partition Function To evaluate the partition function for the one dimensional

Ising model explicitly, let us rewrite the exponent in Eq. (13.2) in the partition function in a way that is easy to use. Note that

e

(/3JsiSi+1 - /3B-(si + si+1)) L = e =

[-2/3J x - ( - 1 + 1 - 8isi+1) -(3Bx *

ef3J

[-WJ(\{si

-(Si + si+1)} *

- si+i))2 ~ PBl-{Si + si+1)}

= ePJ [cosh/35 (\{Si +

S i + 1 ))

2

+ e-WJ^Si

- s i n h / 3 5 - ( S i + s i+1 )]

-

si+1))2 (13.23)

where we have used the identities in Eq. (13.13). It is now clear that

359

13.2. THE PARTITION FUNCTION

if we define a matrix operator K = ePJ fcosh pB 1 + e~2^Ja1

- sinh f3B a3

(13.24)

then, the matrix element of this operator between the eigenstates |s;) and |*i+i) of the a3 operator will be obtained using Eq. (13.11) to be / i»ri \ (PJ*i8i+i -pB-(si (si+1\K\si) = e 2

+ si+1))

(13.25)

Note that K is a 2 x 2 matrix and has the explicit form 1

K

=

ePJ (cosh pB - s i n h ^ E )

e~PJ

e-PJ

ePJ (cosh f3B + sinh f3B)

eP(J

- B)

e-/3J

e-0J ef3(J

\ (13.26)

+ B)

Prom this analysis, it is clear that we can derive the partition function for the one dimensional Ising model explicitly as follows.

Z(P)

=

E

N l (PJ Y, siSi+i - PB-(si 2 e -i

+ si+1))

S;=±l

=

E

(si\K\sN)(sN\K\sN_1)---(s2\K\s1)

= TrKN = \? + \$

(13.27)

360

CHAPTER 13. ISING MODEL

where Ai and A2 are the two eigenvalues of the matrix K in Eq. (13.26). The eigenvalues of the matrix K can be easily obtained from det(K -XI) e(3(J

or, d e t ,

-B)_x ^ j

=0 e-(3J

ej3{J +

) B)_x

, A 2 -2Ae/ 3 J C osh/3 J B + e W _ e - 2 ^ J

=

Q

2 5jr or., A -2Ae- cosh/35 + 2sinh2/3/ = 0

(13.28)

This is a quadratic equation whose solutions are easily obtained to be

A =

J

eP

coshpB±(e2PJcosh2/3B-2smh2pJ)-*

= e@J cosh/35 ± {e2PJ{l

+ sinh 2 (3B) - e2PJ + e~2PJf>

= ePJcosh(3B±(e2[3Jsmh2(3B

If we identify the two eigenvalues as

+ e-2PJ)'

(13.29)

13.2. THE PARTITION FUNCTION

361

Aj = ePJ cosh(3B + (e2PJ sinh2(3B + A2 = ePJ cosh(3B-(e2PJ

e-2PJ)*

sinh2(3B + e-2PJ)i

(13.30)

then, we note that since Ai > A2, for large N, we can approximately write Z(J3) =

TrK N

= Af + A ^ A f ei31 c o s h ^ S + (e2P J sinh 2 j3B + e " 2 ^ J ) ^ (13.31)

This method of evaluating the partitioon function is known as the matrix method and we recognize K as the transfer matrix for the system (see also section 3.3). We can now derive various quantities of thermodynamic interest. Let us note from Eq. (13.31) that we can write In Z(p)

= N In \ePJ cosh/35 + (e2PJ sinh 2 f3B + e"2/3"7)^] =

N f3J + ln{cosh f3B + ( s i n h 2 ^ + e" 4 / 3 , 7 ^}] (13.32)

Therefore, the average magnetization per site defined in Eq. (12.22)

362

CHAPTER 13. ISING MODEL

can now be derived from Eq. (13.32) to be 1

M

dF

1

= NdB =

d

/

1 i

N d B ^

~,a^

l n Z

^

1 dlnZ(/3) Np dB »r/o • ,

1 nn

2/3 sinh/3£ cosh/3£ .

cosh/35 + (sinh2 (3B + e" W ) s

N[3

sinh (3B '(sinh /3£ + e - W ) §

(13.33)

2

It is interesting to note that when the external magnetic field is switched off, the magnetization vanishes. In this one dimensional system, therefore, there is no spontaneous magnetization and consequently, it cannot describe the properties of a magnet. The magnetic susceptibility for such a system can also be easily calculated (see Eq. (12.23)) and takes the form

x= -

dM 8B

(3e2f3J

(13.34)

B=0

This shows that for |2/3J| < 1, the susceptibility obeys Curie's law. Namely, in this case,

x-P

=^

( 13 - 35 )

The absence of spontaneous magnetization in the present

363

13.2. THE PARTITION FUNCTION

system may appear puzzling because naively, we would have expected the configurations where all the spins are "up" or "down" to correspond to minimum energy states. These, being ordered, we would have expected spontaneous magnetization for the system. The lack of magnetization can actually be understood through the instanton calculation which we discussed earlier. Let us recall that for the doublewell potential (see section 7.4 as well as chapter 8)

the naive ground states would give (x) = ± a

(13.36)

The true ground state, as we have seen earlier in Eq. (7.53), is a mixture of these two states (the symmetric state) such that \-E/true

~~ "

(13.37)

In this case, we showed explicitly that the tunneling or the presence

364

CHAPTER 13. ISING MODEL

of instanton states contributes significantly leading to the mixing of the states and restoring the symmetry. In the one dimensional Ising spin system, we can correspondingly think of the following two configurations

as denoting the two ground states for which the magnetization is nonzero or (M)(3 + 0

(13-38)

However, in the present case, there are other spin configurations such as

TTTTIIII1 - o n e kink or one instanton

TTTlilTTT —two kinks or one instanton-anti-instanton

and so on which contribute significantly. It is worth recalling that in a thermodynamic ensemble, it is the free energy which plays the

13.2. THE PARTITION FUNCTION

365

dominant role. Even t h o u g h these configurations have higher energy, t h e y also are more disordered. Consequently, they will have a higher entropy and as a result can have a lower free energy. T h e consequence resulting from t h e contributions of these spin configurations is t h a t the t r u e ensemble average of magnetization vanishes. Namely, (M)%ue

= 0

(13.39)

This qualitative discussion can actually be m a d e more precise t h r o u g h the use of t h e p a t h integrals. As we have seen, p a t h integrals are defined by discretizing space-time variables.

In fact, space-time lattices are often used to

define a regularized q u a n t u m field theory. T h e continuum theory is, of course, obtained in t h e limit when t h e lattice spacing goes to zero. Viewed in this way, let us note t h a t

H = -JJ2sisi+i i

^

continuum

i

+ BJ2si i

i

/ dx (—(9s(x)) 2 + js(x))

+ constant

(13.40)

where a a n d 7 are two constants. Namely, we can think of t h e one dimensional Ising model as corresponding to a one dimensional free

366

CHAPTER 13. ISING MODEL

scalar field theory interacting with a constant external source in the continuum limit.

13.3

Two Dimensional Ising Model Let us next consider a two dimensional array of spins on

a square lattice interacting through nearest neighbors. Once again, let us use periodic boundary conditions along both the axes so that s

i

=

s

ilti2

=

S

h+N,i2

=

S

ii,i2+N

(1.0.41}

where we are using the notation that i = (11,12) denotes a point on the two dimensional lattice and we are assuming that N denotes the total number of lattice sites along any axis.

( * 1 > * 1)

The total number of points on the lattice is then obtained to be

n = N2

(13.42)

13.3. TWO DIMENSIONAL ISING MODEL

367

T h e spins are assumed t o take only the values ± 1 . T h a t is, f ° r a l l H,t2

Si = Sii,i2 = ± 1

(13.43)

T h e Hamiltonian describing t h e interaction of t h e spins is given by (see also Eq. (12.17)) N

H = -JY,sisj

= -J

{ij)

X (*»i,i2s*i+i,i2 + sh,i2sh,i2+i)

(13.44)

«i i*2=l

T h e symbol (ij) is introduced as a short h a n d for sites which a r e nearest neighbors. We can also think of t h e s u m in Eq. (13.44) as being t a k e n over all t h e links of t h e lattice. ( R e m e m b e r t h a t a Hnk connects two nearest neighbors on a lattice.) T h e partition function for t h e system described by t h e Hamiltonian in Eq. (13.44) can now b e defined t o b e

z(p)= X)

e_/3jy

(/3 J Y, SiSj)

= E

«i=±l

e

{ij)

Si=±l

(K Z

s s

i j)

Si=±l

E II eKSiSj

=

(13.45)

«*=±i (ij)

where we have defined K = f3J

(13.46)

368

CHAPTER 13. ISING MODEL

Note that we are discussing the simpler case when the spin system is not interacting with an external magnetic field.

This partition

function, as it stands, appears to be only slightly more complicated than that for the one dimensional case in Eq. (13.2). However, as we will see, this partition function is much more difficult to evaluate in closed form. Before going into the actual evaluation of this partition function, let us discuss some of the symmetries associated with this system.

13.4

Duality

Let us note that since

Si

= ±1

(13.47)

we can expand the exponent in the partition function in Eq. (13.45) to obtain eKSiSj

_

c o g n K

_j_

SiS,

g

j

n n

K

= cosh«;(l + SiS3;tanh/t)

Therefore, we can also write

(13.48)

369

13.4. DUALITY

Z{P) = =

II eKSi$i

£

«=±1 («>

£ JI cosh«;(l + SiSj tanh/c) «=±1 („)

= (cosh«)2n £

II( 1 + s i s j t a n n ' c )

(cosh/c) 2n £

nE(si*jtanhK)'

=

*=±1 (ij) 1=0

= (cosh/c) 2n £

nE(tanhK)'(siSj);

(13.49)

We see that we can simplify this expression by assigning a number lk = l{j — l^ = (0,1) to each link between the sites i and j and rewriting Z(/3) = (coshK) 2 n E(tanhK) / l + ' 2 + " £ h

U(^sj)lij

(13-50)

*i=±i (ij)

Let us next note that the product on the right hand side in Eq. (13.50) can simply be understood as the product of the spins at each lattice site with an exponent corresponding to the sum of the link numbers for links meeting at that site. Namely, for nearest neighbors, j ,

£ n ( W ' = £ n(s;)E^= £ n « r «i=±l (ij)

«i=±l i

«i=±l i

where we have defined, for nearest neighbors j ,

(13.51)

370

CHAPTER

13. ISING

n; = E hj

MODEL

(13.52)

i

The sum over the nearest neighbors can now be done to give

E m^i)1'1 = n E (*r = na + (-ID Si=±l (ij}

i Sj=±l

(13.53)

i

It is clear that the expression in Eq. (13.53) vanishes when ra; is odd. For evenTC;,on the other hand, it has the value

E I K W " = 2n

(13.54)

«i=±i

Putting everything back in Eq. (13.50), we obtain Z(p) = (2 cosh2 /c)n £ ( t a n h K ) ' 1 + / 2 + " = Z(/c) k

(13.55)

The constraint here is that the Zj.'s in Eq. (13.55) must satisfy £ /^ = 0

mod 2

(13.56)

3

for any four Mnks joining at a site. In other words, if li,h,h

and Z4

denote the link numbers for four links meeting at a common site, then

^1 + ^2 + ^3 + ^4 = 0

mod 2

(13.57)

13.4.

DUALITY

371

Let us next consider the dual lattice associated with our original lattice. It is constructed by placing a lattice site at the center of each plaquette of the original lattice.

-

2

1

/ - *3

Thus, each plaquette of the dual lattice encloses a given site of the original lattice and intersects the four links originating from that site. Let us also define a dual variable cr; at each site of the dual lattice and assume that it can take values ± 1 . Denoting by (1,2,3,4) the sites of the dual lattice which enclose the point k of the original lattice, we note that for every Hnk that is intersected by a dual link, we can define

h

=

- ( 1 - <7i
h

=

- ( 1 - 0-20-3)

h

=

o ( l - °"30"4)

' h

- ( 1 - 0-40-1)

(13.58)

We see that each of the Zjt's have the value 0 or 1 as required. Furthermore, we also have

372

CHAPTER 13. ISING MODEL

h+h

+h +h

= T ( 4 —
2 ^ 4 _ (°"i + a3)(o-2 + ^4))

=

0

mod 2

(13.59)

In other words, t h e constraint equation in Eq. (13.57) can b e naturally solved t h r o u g h t h e dual lattice variables. Going back t o t h e expression for t h e partition function in Eq. (13.55), we note using E q . (13.58) t h a t (tanh

K)'1

= (tanh

K)^

1

-™)

= e^t

1

-

a

^ )

(13.60)

where we have denned tanh>c = e - 2 K *

(13.61)

Substituting this back into E q . (13.55), we obtain (-2TIK* 2

+

n

K) = ( 2 c o s h K ) £ e

Z(K)

'

K*^2aiaj) (ij)

e-n-2K*^e

(2 cosh 2 K,)n Z(K*)

(e2**)"

(13.62)

373

13.4. DUALITY

This relation in Eq. (13.62) is quite interesting in t h a t t h e relation t a n h K = e~2K' which can also be written as sinh 2K sinh 2K* = 1

(13.63)

defines a transformation between strong a n d weak coupUngs (or high a n d low t e m p e r a t u r e s (see Eq. (13.46))). And we find t h a t the corresponding partition functions are related as well. Consequently, if there exists a single phase transition in this model (which was known from general arguments due t o Peirels), it m u s t occur at a unique point where K — Kc

— <

or, sinh 2/cc = 1 or, sinh 2KC = 1 p~^KC

p^^C

or, or,

KC

= 2

e2Kc =

A/2 + 1

= J(3C = ^ ( V 2 + l)

or, /3C =

^M^+1)

(13.64)

374

13.5

CHAPTER 13. ISING MODEL

High and Low T e m p e r a t u r e Expansions Quite often in statistical mechanics, t h e partition function

cannot be evaluated exactly. In such a case, we would like to study t h e system at very high t e m p e r a t u r e s as well as at very low t e m p e r a tures to see if any meaningful conclusion regarding the system can be obtained. In the language of field theory, we have seen in Eq. (12.51) t h a t t h e t e m p e r a t u r e can be related to t h e Planck's constant which in some sense measures t h e q u a n t u m coupling. Therefore, high a n d low t e m p e r a t u r e expansions are also known as strong coupling a n d weak coupling expansions (or approximations). Let us go back to the partition function for t h e 2-d Ising model. We have

(« E Z(K)

=

E

e

s s

i i)

{ij)

(13M)

where, as in Eq. (13.46), we have defined

If t h e t e m p e r a t u r e is high enough, t h e n K is small. We have seen in Eq. (13.55) t h a t we can write Z { K i ^ = E (tanhK)'^+(2 cosh 2 n)n fc=o,i

(13.66)

13.5. HIGH AND LOW TEMPERATURE EXPANSIONS

375

where the link numbers are assumed to satisfy Zi + Z2 + Z3 + Z4 = 0

mod 2

for any four links meeting at a lattice site. Since K is small for high temperatures, so is tanh K and the right hand side can be expanded in a power series in tanh K. TO do that, let us note that the link numbers, /it's, can only take values 0 or 1. Accordingly, let us postulate the rule that if Ik = 0, then we will not draw a bond connecting the two lattice sites whereas if I). — 1, then a bond will connect the sites. With this rule then, the constraint on the link numbers simply says that there must be an even number of bonds originating from a given lattice site. Consequently, we note that the first term on the right hand side of the expansion will correspond to the case where there are no bonds on the lattice.

1

The next term in the series will be of the form

: n : : — ^*?

376

CHAPTER

13. ISING

MODEL

In other words, the first nontrivial term in the series will correspond to the product of the weight factor tanh K over a single plaquette. The plaquette can be drawn in n-different ways on the lattice (recall the periodic boundary condition) and hence this term will come with a multiplicity of n. The next term in the series will represent the product of the weight factor tanh K over a plaquette involving two lattice lengths.

# 9

_>

(tanhK)

6

It is not hard to see that such a diagram can be drawn in 2ra different ways and hence this term will come with a multiplicity of In. At the next order the diagrams that will contribute are

13.5. HIGH AND LOW TEMPERATURE EXPANSIONS

EH

311

->• (tanh/t) 8

\B\

(tanh«)8

The combinatorics can be worked out in a straightforward manner for these graphs so that the high temperature expansionof the partition function will have the form Z(K)

— \ ' n (2 cosh n)

1 = 1 + n(tanh K) 4 + 2n(tanh K) 6 + -n(n + 9)(tanh K) 8 2 + •••

(13.67)

To obtain the low temperature expansion, let us note that when T is small, J

378

CHAPTER 13. ISING MODEL

is large. If T = 0, then we will expect all the spins to be frozen along one axis, say up. Therefore, the low temperature expansion would merely measure the deviation from such an ordered configuration. Namely, the low temperature expansion will be a measure of how many spins flip as T becomes nonzero but small. Thus, dividing the partition function by e2nK (which is the value of the partition function when all the spins are pointing along one direction), we have from Eq. (13.45) 7(K\

JS-E'

( K £ ( - 1 + S;S,))

w

< 13 - 68 >

To develop the right hand side diagrammatically, let us draw a cross on the lattice to represent a flipped spin. Thus, the first term on the right hand side will correspond to a diagram of the form

1

The next term in the series will correspond to the case where one of the spins on the lattice has flipped and will represent a diagram of the form

13.5. HIGH AND LOW TEMPERATURE EXPANSIONS

• x • •

379

•

In other words, a single flipped spin will interact with a nearest neighbor with weight e~2K and since there are four nearest neighbors for any site, the term would have a weight e - 8 ". Furthermore, the flipped spin can occur at any lattice site and hence this term will come with a multiplicity of n. The next term in the series will correspond to two flipped spins. Interestingly enough, this leads to two possibilities. Namely, the flipped spins can be nearest neighbors or they need not be. Diagrammatically, the two possibilities can be represented as

• x • x •

y

e~16fc

380

CHAPTER 13. ISING MODEL

In other words, in t h e first configuration, t h e interaction between the two nearest neighbor spins which are flipped does not contribute to t h e partition function.

F u r t h e r m o r e , the number of ways a pair of

flipped spins can occur as nearest neighbors is 2n. T h e multiplicity of t h e second diagram, obviously, will be \n(n

— 5). However, t h a t is

not t h e only kind of diagram which contributes an a m o u n t e~ 16K . In fact, there is another class of diagrams, namely, ones where there are three or four flipped spins which are nearest neighbors also contribute t h e same a m o u n t .

x x x •

•

X

•

e -16«

• e -16«

• • • •

•

X

X

•

•

•

X

X

•

•

e-16«

13.5. HIGH AND LOW TEMPERATURE EXPANSIONS

381

Thus, we can consistently derive a low temperature expansion of the partition function which has the form 4 g

=

i +

ne

-8*

+

2ne-l2K

+ ±n(n + 9 ) e " 1 6 K + • • •

(13.69)

It is clear now that if we denote, for low temperatures, K = K*

(13.70)

then, we can write the low temperature expansion also as * £ ) = 1+

ne-8«-

+ 2 n e - 1 2 - + + \n{n + 9)e-™«* + • • •

(13.71)

Thus, comparing Eqs. (13.67) and (13.71) we see that under the mapping tanhK = e- 2 K *

(13.72)

we have Z(K) 2

(2 cosh K)

71

_ Z(K*) ~ e2nK*

(13.73)

This is, of course, the duality relation that we have derived earher in Eq. (13.62). In the present case, we see explicitly that the duality mapping (transformation) really takes us from the high temperature expansion to the low temperature expansion and vice versa.

382

13.6

CHAPTER 13. ISING MODEL

Q u a n t u m Mechanical Model Before finding the correspondence of the two dimensional

Ising model with a quantum mechanical model, let us derive the transfer matrix for the system. Let us begin by writing the Hamiltonian for the system in a way that is better suited for our manipulations. Let us label the sites on a given row by 1 < i < N and the rows by 1 < m < N.

t; : : i : : m i — Then, we can write the interaction energy between the spins on a row, m, as H (m) = - J £ Si(m)si+1 (m)

(13.74)

Similarly, the interaction energy between two adjacent rows, say m and (m + 1), can be written as N

H(m, m + 1) = - J E Si(m)si(m + 1)

(13.75)

2=1

Given this, we can write (see Eq. (13.44)) the total energy of the system as

383

13.6. QUANTUM MECHANICAL MODEL

H =

£

{H(m) + H{m,m N

+ 1))

N

= - / E E ( s i( m )*«+i( m ) + Si(m)si(m + 1))

(13.76)

m=l i=l

If we desire, we can also add an external magnetic field at this point. However, let us ignore it for simplicity. The partition function will involve the exponent

e~PH = e

(-(3Jt(H(m) + H(m,m + l))) m=i

(13.77)

Let us, for simplicity, concentrate only on a single factor of this exponent. Namely, let us look at e-P{H(m)

+ H{m,m

+ 1))

N

PJ(Y,(si(m)si+i(m)

+ Si(m)si(m + 1)))

<=i

=

e

=

T[ePJSi(m)Si(m

+

l)e/3Js;(m)s;+i(m)

(13.78)

Let us next note that on every row there are N sites and if we introduce a two component eigenvector of a% at every site, then we can define a 2N dimensional vector space on every row through a direct product as

384

CHAPTER 13. ISING MODEL \s(m))

= \si) \s2)
\si,s2,---,sN)

(13.79)

We can define an inner product on such states as (s(m + l)\s(m))

=

(s'1,s'2,---,s'N\s1,s2,---,sN)

= t>Sls'16s2s'2---6SNs'N

(13.80)

Similarly, the completeness relation will take the form £

\s(m))(s(m)\ = I

(13.81)

S; = ± l

where I denotes the 2N X 2N identity matrix. With these preliminaries, let us now introduce the followN

ing 2

x 2

N

matrices. (There will be N of each.) <7i(i) = a3(i)

J(8)/(8i---(8)
= 7<8)/®---
(13.82)

Namely, all the entries in the above expression correspond to the trivial 2 x 2 identity matrix except at the i th entry. Let us also record here the product formula for matrices defined through direct products, namely,

13.6. QUANTUM MECHANICAL MODEL

385

{A ® B)(C ®D) =

AC®BD

(13.83)

We also note that the 0\ in the i th place acts on the vectors |s;) and, therefore, as a 2 x 2 matrix, we can write (s^ae

<Jl

|sj) = (s'^a cosh.9 + acri sinh0|s;) a cosh 9 a sinh 8

(13.84)

a sinh 9 a cosh 8 where a is a constant. On the other hand, we note that a term such as 0 /3Jsj(m)sj(Tn

+ 1)

can also be written as a 2 x 2 matrix of the form

JJ*i»'i-\

e

f3J -0J

e-/3J\ PPJ

(13.85)

Therefore, comparing the two relations in Eqs. (13.84) and (13.85), we note that we can identify (s'Jae^lsi)

= ePJ*i*'i

provided the following relations are true, namely,

(13.86)

386

CHAPTER 13. ISING MODEL a cosh 6 = e@J asinhO

= e^3

(13.87)

Equivalently, we can make the above identification provided tanhfl =

e"2/3*7

a = (2sinh2/3J)5

(13.88)

Consequently, it is clear that if we define a 2N x 2N matrix as

„ (*£>i(0) = (2sinh2/3J)^e

<=i

(13.89)

with tanh6» = e - 2 ^ J

(13.90)

then, we can write

(s(m + l)\Ki\s{m))=e

/ 3 / Q > ( m ) S i ( m + 1)) i

(13.91)

13.6. QUANTUM MECHANICAL MODEL

387

Let us also note that

•(*(m + l)|e(^»(* + 1M«))|«(m)) =

(s(m + 1)| coshfl + a3(i + l)cr3(z) sinh0|s(m))

=

(s(m + l)|s(m)) (cosh0 + s; + is;sinh0)

=

(s(m + l ) | s ( m ) ) e ( ^ + l S i )

=

(s(m + l ) | s ( m ) ) e ^ J s i s i + i

(13.92)

provided we make the identification § = /3J

(13.93)

Thus, defining

K2 = e

/3J(5>3(i + 1 M 0 ) <=i

(13.94)

we note that we can write N « E ^ i ( « ) /3JX>3(* + l)
which defines the transfer matrix for the system. Namely,

388

CHAPTER 13. ISING MODEL

(s(m + l)\K\s(m))

= (s(m +

l ^ l ^ m ) )

= (s(m + l)\Ki\s(m))

(3 J J2 SiSi+i e *=i

/3«7E s;(m)sj(m + 1) =

=

»=i

e

(3J^2si(m)si+1(m) e

-/3(ff(m) + £T(m,m + l))

i=i

(13.96)

The partition function can now be written as

Z(P)= E e

-f3H

Si=±l

=

E (s(l)\K\s(N))(s(N)\K\s(N

= TrKN

- 1)) • • • <«(2)|Jir|,(l)) (13.97)

This, therefore, is the starting point for the Onsager solution of the two dimensional Ising model. In field theory language, we are looking for a quantum Hamiltonian whose Euclidean transition amplitude will yield the partition function. Furthermore, in field theory, even if we are dealing with a theory on the lattice, we would prefer the time variable to be continuous. Thus, let us identify one of the axes, say the vertical one,

389

13.6. QUANTUM MECHANICAL MODEL

to correspond to time and we choose the separation between the rows to be e, a very small quantity.

1

•

•

•

The continuum time limit, of course, will be obtained by choosing e —> 0. Let us note that we are only changing the spacing among the rows. The spacing along a row, of course, is unchanged. At first sight, this may appear bothersome. But, let us recall that if there is a critical point in the theory, then the correlation lengths become quite large in this limit and in such a limit the lattice structure becomes quite irrelevant. Let us also note here that by making the lattice asymmetric, we have actually destroyed the isotropy of the system and, consequently, the couplings along different axes can, in principle, be different. Thus, allowing for different couplings along the two axes, we can write JV

H =

N

X) ^2(-J'si(m)si(m

+ 1) -

Jsi(m)si+1(m))

m=l i=l N

=

^(H{m) m=l

+ H{m,m

+ 1))

(13.98)

390

CHAPTER 13. ISING MODEL

where N

H(m)

= - J £ *t*i+i i=l JV

H(m,m

+ 1) = -J'J2sis'i

(13.99)

The partition function, in this case, will have the form -/3tf

=

£

=

£

e

-/3 mE= 1( % ) + H(m,m + 1))

(/3J£Sisi+1 + / 3 J ' £ s ^ ) N n e • •

(13.100)

s;=±l m=l

In the quantum field theory language, we can write the Euclidean time interval as TEUCI.

= Ne

(13.101)

and assume that there exists a quantum Hamiltonian Hq such that we can write Z(/3) = Tre-TEudHi =

£

=

Tre-NeHi

(-(l)|e-cfl«|.(JV)>...(«(2)|e-c^|.(l)>

S;=±l

=

E

f[{s{m

s;=±l m=l

+ l)\e-eHi\s{m))

(13.102)

391

13.6. QUANTUM MECHANICAL MODEL

Thus, comparing Eqs. (13.100) and (13.102), we recognize that the two can be identified if

eU

PY,(JsiSi+l =e i

u

{s(m + l)\e-
+

JS S

' i 'i) (13.103)

From our discussion of the transfer matrix in Eqs. (13.95) and (13.96), we immediately conclude that

Hq = ~ I > i ( 0 + W * + 1 M 0 )

(13.104)

i

where A is a constant parameter and as before, we can identify

eA =

pj

tanhe ~ e = e'2^J'

(13.105)

This relation is quite interesting in the sense that it brings out a relationship between the coupling strength as a function of the lattice spacing. In particular, we note that as we make the spacing between the rows smaller, the corresponding coupling between the rows becomes stronger. This is renormaUzation group behavior of the couplings in its crudest form.

CHAPTER 13. ISING MODEL

392

13.7

Duality in the Quantum System We have been able to relate the 2-d Ising model to a one

dimensional quantum mechanical system with a Hamiltonian Hq = ~ X > i ( * ) + Aa3(i + 1 ) M 0 )

•

•

(13.106)

• • • • • original lattice

1 2 3 - - JV •

• • • • • • dual lattice

Let us next consider the dual lattice corresponding to this one dimensional lattice and define the dual operators on the dual lattice as

M O =
(13-!07)

3= 1

It is easy to see that

MO = (M* + i)M*'))2 = ' MO = dl*3)2 = I 3=1

(13-108)

13.7. DUALITY IN THE QUANTUM SYSTEM

393

These results can be shown to follow from the basic commutation relations of the Pauli matrices, namely,

[°i(i)>
° =[
# * 7^ J

o\{i) = I = *Hi) Mz),<x 3 (i)] + = 0

(13.109)

Using these, we can also derive that for i ^ j , [A*I(0,A*I0")] = M * + iW0^3(j' + i W j ) ] = o

[»3(i),Mj)} = [n^(fc),n^i(0] = o

( l3 - n °)

On the other hand,

[A*I(*)»A*S(*)]+ =

M * + 1 ) ^ 3 ( 0 . n ^i(j)]+ i-1

=
= 0

(13.111)

Thus, //i(i) and /x3(0 also have the same algebraic properties as the Pauli matrices on the original lattice. Furthermore, let us note that by definition,

394

CHAPTER 13. ISING MODEL

fj,3(i + l)n3(i) =CTi(i+ l)

(13.112)

Using these, then, we note that we can write

Hq(x)

=

- ! > ! ( ; ) + A<73(; + I ) < 7 3 ( 0 ) JV

=

- A E ( / i i ( « ) + A-V3(» + l)A*3(»))

= Afl,(A _1 )

(13.113)

This is the self duality relation for this system. Namely, it maps the strong coupling properties of the system to its weak coupling properties. This shows, in particular, that the energy eigenvalues of this system must also satisfy the relation E(X) = \E(\~1)

(13.114)

For some finite value of A, if there is a phase transition such that the correlation lengths become infinite or that some energy eigenvalue becomes zero (zero mode), then the above duality relation implies that this must happen at A = 1. This is precisely how we had determined the critical temperature for the 2-d Ising model in Eq. (13.64). Let us also note here that since we have a quantum mechanical description

13.8. REFERENCES

395

of the 2-d Ising model, we can also develop a perturbation theory in the standard manner.

13.8

References

Huang, K., "Statistical Mechanics", John Wiley. Kogut, J., Rev. Mod. Phys., 5 1 , 659 (1979). Savit, R., Rev. Mod. Phys., 52, 453 (1980).

Index Anharmonic oscillator, 97

Connected diagram, 244

Anti-commutation relation,

Connected Greens function,

101, 103

244

Anti-instanton, 192

Continuous transformation, 285

Anti-periodic boundary

Continuum limit, 56

condition, 343

Coordinate basis, 11-12

Asymptotic equation, 211

Correlation function, 79, 82, 92

Baker-Campbell-Hausdorff

Critical exponent, 331

formula, 21

Double well, 187

Basis states, 15

Double well potential, 363

Bethe-Salpeter equation, 256

Double-well potential, 174

Born diagram, 258

Dual lattice, 371

Classical field, 249, 251

Duality, 368

Classical path, 30

in quantum systems, 392

Classical phase, 158-159

Effective action, 93, 249, 258

Classical statistical system, 344

Effective potential, 266

Classical trajectory, 39, 45, 63,

Entropy, 327

68, 169 Complex scalar field, 295

Euclidean action, 188 Euclidean equation, 189

397

INDEX

Euclidean field theory, 342

Functional integral, 36

Euclidean generating

Gaussian integral, 112

functional, 352 Euclidean rotation, 71

Generating functional, 71, 87, 236

Euclidean space, 78, 188

Global transformation, 286

Euler-Lagrange equation, 8, 44,

Goldstone mode, 317

63

Goldstone particle, 318

Fermi-Dirac statistics, 103

Goldstone theorem, 318

Fermionic oscillator, 101, 345

Grassmann variable, 106

Feynman diagram, 243

Greens function, 63, 234

Feynman Greens function, 65,

Harmonic oscillator, 43, 94, 337

97, 126, 228

Heisenberg picture, 11

Feynman path integral, 25, 27

Heisenberg states, 17

Feynman propagator, 118, 123,

Imaginary time, 74, 77, 188

232, 238

Instanton, 192

Feynman rules, 241, 268

Instantons, 187

Field, 1

Interaction picture, 12

Finite temperature, 323

Invariance, 281

Fourier series, 48

Ising model, 351

Fourier transform, 16, 64

one dimensional, 351

Free energy, 328, 347

two dimensional, 366

Free particle, 36, 50

Jost function, 208

Functional, 3

Left derivative, 107, 118

Functional derivative, 5

Local transformation, 286

398

INDEX

Loop expansion, 268 Many degrees of freedom, 221 Metric, 2 Mid-point prescription, 22

Quantum mechanical model, 382 Quantum statistical system, 342

Multi-instanton, 194, 215

Relativistic field theory, 226

Nicolai map, 145

Right derivative, 107, 118

Noether's theorem, 286

Saddle point method, 164, 168,

Normal ordering, 19-20 One loop effective potential, 272

170, 194, 200 Scalar field theory, 226 Schrodinger equation, 9, 156

One particle irreducible, 247

Schrodinger picture, 11

Operator ordering, 18-19

Schrodinger states, 17

Partition function, 326, 346,

Schwinger-Dyson equation, 256

358 Pauli principle, 104 Periodic boundary condition, 342

Semi-classical approximation, 162 Semi-classical methods, 155, 168, 188

Perturbation theory, 100

Shape invariance, 145

Perturbative expansion, 241

Slavnov-Taylor identities, 303

Phase transformation, 296

Space-time translation, 291

Phase transition, 330

Spontaneous magnetization,

Proper self energy, 262 Quantum correction, 238 Quantum fluctuation, 46

334, 362 Spontaneous symmetry breaking, 306, 315

399

INDEX

Steepest descent, 166, 168, 174

Ward identities, 300

Stirling's approximation, 168

Weyl ordering, 20

Stress tensor, 293

WKB approximation, 155, 161,

Supersymmetric Oscillator, 133 Supersymmetric quantum mechanics, 141

174 WKB wave function, 161-162, 174, 179

Supersymmetry, 133

Zero eigenvalue, 195

Symmetries, 281

Zero mode, 196, 208, 394

Temperature expansion, 374 high, 377 low, 377 Time evolution operator, 10 Time ordered product, 234 Time ordering, 81, 124 Time translation invariance, 194, 202 Transfer matrix, 361 Transition amplitude, 12, 22, 45, 50, 58, 70, 170 Tree diagram, 258 Vacuum functional, 87, 92, 116, 120, 224 Vacuum generating functional, 226