Effective Lagrangians in Quantum Electrodynamics (Lecture Notes in Physics)

Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, M~Jnchen,K. Hepp, Z~Jrich R. Kippenhahn, MLinchen, H. A. ...

Author: W. Dittrich | M. Reuter

60 downloads 964 Views 6MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, M~Jnchen,K. Hepp, Z~Jrich R. Kippenhahn, MLinchen, H. A. Weidenm~ller, Heidelberg and J. Zittartz, K61n

220 Walter Dittrich Martin Reuter

Effective Lagrangians in Quantum Electrodynamics

Springer-Verlag Berlin Heidelberg New York Tokyo

Authors Walter Dittrich Martin Reuter Institut fQr Theoretische Physik der Universit~t Tebingen D-7400 Tf3bingen, ER.G.

ISBN 3-540-15182-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-38?-15182-6 Springer-Verlag New York Heidelberg Berlin Tokyo Library of Congress Cataloging in Publication Data. Dittrich, Walter. Effective Lagrangians in quantum electrodynamics. (Lecture notes in physics; 220) Bibliography: p. 1. Quantum electrodynamics. 2. Lagrangian functions. I. Reuter, Martin, 1958-. II. Title. II1. Series. QC68O.D53 1985 537.6 85-2527 ISBN 0-387-15182-6 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Veflag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210

PREFACE

With

these notes we would

subject

of effective

Although

this topic

vides

Lagrangians

of QED which

Moreover,

studying vacuum problems

considerations

To make

In contrast

our computations

general

concepts

these sections

in typing

T~bingen,

for similar

and other more

compli-

electrodynamics

as possible,

In particular,

has the

many of them

this is true

The reader who is mainly

could omit the rather

technical

for the

interested

derivations

in a first reading.

We wish to thank Christel skill

are

can still be done analytically.

as transparent

3rd and 4th sections.

tech-

texts.

in QED, where matters

to the latter,

in great detail.

calculational

preparation

(QED).

it also pro-

found in standard

can be a helpful

that many calculations

are presented

important

in quantum chromodynamics

cated theories. advantage

of several

to the

electrodynamics

in its own right,

are usually not

fairly well understood,

an introduction

for quantum

is interesting

us with an example

niques

2nd,

like to provide

Kienle

the various

September

1984

for her endless

versions

patience

and

of the manuscript.

W. Dittrich M. Reuter

of

in

TABLE OF CONTENTS

(I)

Introduction

I

(2)

The Electron Propagator in a Constant External Magnetic Field

28

(3)

The Mass Operator in a Constant External Magnetic Field

37

(4)

The Polarization Tensor in a Constant External Magnetic Field

56

(5)

One-Loop Effective Lagrangian

73

(6)

The Zeta Function

98

(7)

Two-Loop Effective Lagrangian

121

(8)

Renormalization Group Equations

147

(9)

Applications

167

and Discussion

APPENDIX (A)

Units, Metric, Gamma Matrices

183

(B)

One-Loop Effective Lagrangian of Scalar QED

186

(C)

The Casimir Effect

197

(D)

Derivatives

206

(E)

Power Series and Laurent Series of K(z,v)

210

(F)

Contact Term Determination in Source Theory

228

~G)

One-Loop Effective Lagrangian as Perturbation Series

232

(H)

Summary of the Most Important Formulae

235

REFERENCES

of W[A]

242

(I) INTRODUCTION The problem of the existence of a stable electron dates back to the very beginning of electrodynamics: if it is assumed to be an extended charge distribution, it is unstable due to the repulsive electrostatic forces, and if one assumes a point charge, one finds a divergent self energy. Already at the beginning of this century, attempts were made to solve this problem by generalizing Maxwell's equations 1912, Born and Infeld 1934)

(Mie

Because these equations can be de-

rived via a variational principle from a Lagrangian density L, it is natural to generalize the expression for L. In doing so, the following points must be taken into account: (i) In order to generate Lorentz covariant equations,

L must

be a Lorentz scalar, i.e., it must be a function of invariant combinations of the field quantities. (ii) L must be gauge invariant. (iii) In the limit of small field strengths, L(O)

=

L has to approach

~I (~2_~2), which is the Lagrangian leading to

Maxwell's equations. The electromagnetic field has only two gauge invariant Lorentz scalars, viz. =

=

*The original papers of this section are cited in the list of references under ref. [I].

2

where

*F ~" = ~I ~ , p o

Fp ° is the dual

(note that ~ • ~ is a pseudoscalar,

field strength which

under a parity

transformation).

F and G 2 only.

Born and Infeld used the following,

bitrary

function of the invariants

Thereby,

E ° has the meaning

fields much weaker

of second order. relativistic reduces

v < < c.

This

levant

the square

mechanics,

for classical

with a finite

self-acceleration Whereas

these

to the

L = mc211-(1-v2/c2)1/2], I 2 L = ~ mv for

v = c is of no importance

field strength E ° is irre-

theory could account self energy;

classical

first attempts

This

fields

however,

other noto-

during

such as the

could not be resolved. a non-linear

electro-

the development

in the thirties

it became

to understand

of quantum electrodynamics

(QED),

of the interaction

electrons,

electro-

in the context

the relativistic positrons

of re-

apparent

can give rise to non-linear

is easiest

between

for a stable electron

electrodynamics,

to construct

speculative,

quantum mechanics

effects.

in analogy

formula

of charged particles,

that quantized matter magnetic

only terms

electrodynamics.

within

dynamics were highly lativistic

the maximum

For

the ordinary Maxwell

of a free particle,

size and finite

difficulties

field strength.

root and keeping

to the non-relativistic

Indeed this non-linear

rious

of a maximum

Just as the limiting velocity

in classical

quite ar-

as their Lagrangian:

function was chosen

Lagrangian

its sign

L can be a function of

than Eo, one recovers

Lagrangian by expanding

which

Thus,

changes

tensor.

quantum

theory

and the electro-

magnetic field. Electromagnetic fields can be both macroscopic (external) fields and radiation fields

(light quanta). It is

necessary to formulate this theory as a quantum field theory. This means: (i) All particles are described as excitation states of fields; thus, there is an electromagnetic field as well as a fermi field for the electrons and positrons in the theory. (ii) The fields are not ordinary functions of space and time, but are non-commuting operators. This gives rise to uncertainty relations:

If the field is precisely known in

a point, its conjugate momentum is completely arbitrary. In general, the product of the uncertainties is given by Planck's constant. Whether it is possible to use the classical concept of a particle or a field depends on the physical situation under consideration. For the moment, let us imagine a radiation field. This can be visualized as a system of an infinite number of coupled oscillators (one oscillator at each point of space)

[21]. By intro-

ducing normal-coordinates, the system decouples and one gets an infinite set of free oscillators with the Hamiltonian H--

1

C~9.z :

7 where Pi' qi and m i denotes momentum, amplitude and frequency, respectively, of the i-th oscillator. Quantizing this system, one obtains for the possible energy eigenvalues

E: Z

where N i denotes

the number of quanta

in the mode

us look a little closer at the vacuum state. theory,

there would not be much

is the state with vanishing In quantum

theory,

matters

N i = O for all modes fluctuations

zero-point

appropriately observable (photon-)

energy

vacuum:

each other because

magnetic

field.

Appendix

C.

Other observable photon

However,

for example,

ideal

consequences of the vacuum in atomic

radiation generally,

and field free space.

for every matter

plates

in more detail

fluctuations levels

in

of the

or the anoma-

of the vacuum being

This is not only true for the

local

field.

This means

fluctuations

of the electromagnetic

but also charge

fluctuations

due to the creation

of electron-positron

or, more

that the vacuum

(which can be interpreted as virtual

annihilation

[20].

of the electro-

field strengths

sequent

effect

conducting

field, but also for the electron-positron,

does not only contain

of the

of the electron.

Thus we are forced to give up the concept a particle

there are

the Casimir

This point will be discussed

moment

is infinite,

structure

of the fluctuations

field are the Lamb shift

lous magnetic

which contribute

E ° is eliminated by

scale.

two uncharged,

even if

the zero-point

of modes

of this non-trivial

consider,

As was shown by Casimir, attract

Usually

the energy

consequences

left with

the number

diverges.

choosing

because

oscillators,

Because

simply

all over the space.

are not so simple,

i, we are still

½ ~i"

In a classical

the ground-state

field strength

of all the harmonic

an energy E ° = [ this

to say:

i. Now let

pairs.

photons),

and the subIt is these

charge fluctuations of the quantized electron-positron (Dirac) field with an external

(i.e. unquantized) electromagnetic field

which we now want to study in some detail. In the source-free regions of space this external field obeys in absence of matter fields the classical Maxwell equations ~

F p~ = O, which, as

stated above, can be derived from the variational principle

(1.1)

Our aim is to find an effective action Weff[A] = W(°)[A] + W(1)[A], where W (I) describes the non-linear effects induced by the quantized fermion fields. The new equations of motion are then given by

vq#

=

0

To be precise, for the fermions we assume Dirac particles

(1. z)

[37,39]

described by the Lagrangian (for our conventions, see appendix A) o

W

(1.3) o

The current jP of the electrons and positrons can be obtained from the action W~ = /d4x L~ via

~ ] ~ f~' ~'¢ ~] It

is

this

current

however,

we a r e

plicitly

contains

stop

after

an action

to which

not

electrons

W(1)

so that

at it

a purely generates

V~](" generalized

[~]

Because

we h a v e

non-linear)

a

current

must be conserved

that,

e_~x-

we w o u l d

looking

for

classical

level.

o n e now d e f i n e s

to A , i.e. je(g)I

~,

(1.2)

F vp = 0 (F pv i s for

of j~

(1.5)

read

are

the A

(in field

antisymmetric!),

(1.5)

of

by

(i .6)

to>

these for

value

0¢

=
If

vacuum expectation

of motion

of

we a r e

which

the presence

equations

Because

I f we w a n t e d

Instead,

respect

c-,

in a theory

couples;

simulates

Maxwell

averaged

~.

A~(x)

[A] w h i c h

~-- < 0 1

equations ~

field

Ap

9 , - F "#

field

interested

the

with

(1.4)

external

(1.3).

W(1)

upon differentiating

the

Dirac

derived

functional

the

the

actually

the

having

= ~ (~,

general

highly

only. the

vacuum

to be consistent:

This is indeed the case if W (I) is a gauge invariant functional of Ap, i.e., if we have

~VrH) E ~,4 ] = ~y$141[ ~] where A A is a gauge transform of A:

(1.8)

A -- n r Using the chain rule,

0

=

ga,~

(1.9)

-

(1.8) can be exploited as follows

!"

]

a=o

I~ cyJ (1.1o)

=- =

~?

9r < o ,

>' & l - x , F~,/o>

In the third line,

(1.5) was used. Now we have shown that a

gauge invariant effective action functional leads to physically acceptable equations of motion and our remaining task is to solve eq. W (I)

(1.5) together with the boundary condition

[F v = O] = 0 for W (I)

To this end we first redefine the current appearing on the right-hand side of (1.6). As is well kown

[37], when quantizing

the electron field by imposing the anti-commutation relations

(1.11) one obtains an infinite total charge for the vacuum state. Subtracting this infinite quantity from the charge operator corresponds

to replacing @yP@ by the current

e.

i~ = ~ ~ ~ _

~

- -z [~-,

- ~ ~ ~, ~''","l ~

~t]

(1.12)

which is gauge invariant and fulfills

< o, i,', ~.,~, ~ 0 7" ''-°

:

0

(1.13)

The vacuum expectation value now reads

XL--, ~ (1.14) Hereby T denotes the time ordering operator m

I0

and the coincidence limit has to be performed symmetrically with respect to the time coordinate:

-

$

Now i t

X°t~>X °

is convenient

(propagator,

to i n t r o d u c e

two-point function)

X °~ < X °

the f e r m i o n Green's

function

defined by

(1.17) which i s the r e s o l v e n t

o f the D i r a c o p e r a t o r

[37]:

(1.18) with

"J~',F¢ --- ~

%

-- ~___%

(1.19)

(Recall that the Dirac equation resulting from (1.3) reads [Y~

+ m]~ = O~. . Hence the current expectation value is given

by

~.~,~0>'~ = ~

.6~ ~t~X ",,~ G~.~cx ~,, (1.20)

where the symmetric limit is understood in the sequel. The defining equations for W (I) now are

(1.21)

~/,lj[~,, =07 = O As is demonstrated in appendix D, this problem is solved by

we%~t ]

=

i ~ £,~ ( ~ - e

~'n &,+)-I (1.22)

with G÷[A] the propagator in an external field

1

(1 .z3)

w h i c h i s c o n n e c t e d w i t h G+ = G+[F v = 0] by

G+[#]

= ~+(1-

e ~

G+)

-1 (1.24)

10

The symbol Tr denotes fd4x tr, i.e. the trace in both spinor and configuration space. In the language of Feynman diagrams, (1.21) and (1.22), respectively,

are represented by a single

electron loop in an external field (a "short-cut" propagator G[A]):

Q

X

(1.2S)

(The double line denotes the presence of an external field). The evaluation of (1.22)

for a given potential A (x) will, in

general, be an extremely complicated task. Simple solutions are known only for a very limited class of fields

(constant fields,

laser fields, weak fields, slowly varying fields, etc.). The first people who discussed effective actions like (1.22) were Heisenberg and Euler [I], as well as Weisskopf 1936. Then, in 1951, Schwinger in which he evaluated W(1)[A]

[I], in

[3] published a classical paper for several types of fields. He

used the so-called proper-time method which reduces the calculation of (1.22) to a one-dimensional problem of ordinary particle quantum mechanics. method to calculate Lagrangian)

In chapter (5), we will use a similar

L (I) (frequently called Heise~berg-Euler

defined by

--.

d~ x ~p_ ( 4 )

(1.26)

for a constant magnetic field. Now, looking back to the early works of Mie and Born and Infeld, we see that today the motivations for studying non-linear genera-

11 lizations of Maxwell's equations for the vacuum are quite different. The main objective when studying effective actions is to learn something about the structure of the vacuum which, in this approach, is probed by an external electromagnetic field. However, from the modern point of view, the problem of a stable electron is not an issue which can be discussed in terms of pure electrodynamics; instead, it should be solved within an up to now unknown fundamental theory of matter and its interactions. The problem of the diverging self-energy, for example, is not really solved within the present theory but is hidden behind sophisticated renormalization schemes. At this point it might be interesting to look at a system closely related to the quantized fermions in an external, i.e., classical, electromagnetic field, viz. a quantized matter field in presence of a classical background gravitational field. For simplicity's sake, we consider a free scalar field ~(x) with the classical action [48]

where the metric tensor field is a prescribed function of x. (The most general Lagrangian for ~ would also contain a term ~R~ 2 with R being the scalar curvature). theory where g ~ ( x )

In a semiclassical

is treated classically whereas ~{x) is

treated quantum mechanically, the vacuum expectation value of the matter field

12 acts as a source on the right-hand side of Einstein's

equations:

=

Obviously,

this

is

the analogue o f

right-hand

s i d e s e t equal t o zero i s

o f the u s u a l F i n s t e i n - H i l b e r t

WC° Eg'] =

(1.6).

Eq.

(1.29)

with

the

o b t a i n e d as the v a r i a t i o n

action

16,rG

as

0

(1.31)

If we now define the effective

action W (I) by (1.32)

our g e n e r l i z e d equations

(1.29)

are g i v e n by (1.33)

which i s

clearly

to Maxwell's linear is

the analogue o f

equations,

(1.2).

Einstein's

a l r e a d y at the p u r e l y

However,

equations

classical

in contrast

are h i g h l y

level.

non-

But the s t r a t e g y

the same i n both cases: because one does n o t want to t r e a t

the m i c r o s c o p i c ~(x))

explicitly,

simulating

Ap(x)

degrees o f freedom

their

one d e r i v e s presence

for

(the quantum f i e l d s

effective

equations

the m a c r o s c o p i c ,

~(x)

or

o f motion

classical

field

or g~v(x).

For the equations covariantly

(1.29) to be consistent,

conserved,

IO> must be

because the left-hand side of (1.29) is.

18 By a reasoning analogous to (1.10) one can show [49], that is indeed covariantly conserved if W (1) is invariant under general coordinate transformations, sary for W (I) of electrodynamics

just as it was neces-

to be gauge invariant for the

induced vacuum current to be conserved. This is one example of the correspondence between gauge invariance in electrodynamics and general covariance in gravitation theory. Finally we mention that also in this case, W (I) can be expressed via the matter field propagator:

Now G+[g] denotes the propagator in presence of a gravitational field described by g~v(x), and G+ is the corresponding flat space-time propagator.

(The factor of -I/2 which is not present

in (1.22) is due to the fact that ¢ is an uncharged scalar field, whereas @ was a Dirac field). For a comprehensive introduction to these questions, After

~his

see [48].

digression,

let us return to our original problem

of quantized fermions in a classical background electromagnetic field, which we now reconsider from the path integral point of view (for an introduction,

see [24,25]). For the moment, let us

consider an arbitrary field theory with fields {0} and Lagrangian L({¢}). Transition amplitudes then can be expressed as functional integrals of the general form

with

the

action

S[{¢}]

= ld4x

L({¢}).

Now a s s u m e

that

the

set

14

{¢} can be d i v i d e d i n two s u b s e t s are "light"

field

c o m p o n e n t s whose dynamics we d i r e c t l y

(the photon field,

or the classical

{~H} a r e " h e a v y " f i e l d s influence

[50].

A (x),

(the electron

the dynamics o f the l i g h t

observable

{~L} and {¢H}, where {~L}

in our case),

field fields

(1.35)

while

in our case) which but are not directly

S i n c e t h e {¢H} a r e h i d d e n from v i e w , i t

c o n v e n i e n t to w r i t e

observe

is

i n t h e form

(1.36)

where t h e e f f e c t i v e

a c t i o n Wef f f o r t h e l i g h t

fields

i s de-

f i n e d by

Clearly, plete

the effective

description

action,

if

exactly

known, g i v e s a com-

o f t h e d y n a m i c s o f {~L} w i t h o u t

any r e f e r e n c e

to the heavy fields. Now l e t

us c o n s i d e r s e v e r a l

over the heavy field

e x a m p l e s o f such i n t e g r a t i o n s

components [ 5 0 ] .

f e r m i o n s i n an e x t e r n a l

field

w,°, -2a* f - ;

First

(1.37))

for the

we have

(1.3s)

]

and W(1) i s g i v e n by ( n o t e t h a t exp(iW ( ° ) ) of

of all,

cancels

on b o t h s i d e s

15

.~p (i W'~'C,~])

=

(1.39) (Recall (1.3)). According to the general rules for the path integral quantization of Fermi fields [12], @ and @ are anticommuting classical fields forming a Grassmann algebra; hence a Gauss-type integral like (1.39) can be evaluated to be [12]

(1.40)

This gives

w"'coa = -i =

+

ga cld; (GC~ -4) i £.,.,. de4 GC~]

(1.41)

where we used the (formal) identity det(exp G) = exp (Tr G). Because action functionals are defined only up to a constant, we may exploit this freedom to replace (1.41) by

(i .42)

which is identical to the previously derived result (1.22) and thus vanishes for F ~ = O. Obviously, the notion of integrating out unobserved degrees of freedom together with the rules for

16

the integration over Grassmann fields leads us back to the results already derived in a more pedestrian manner. As another example of an effective Lagrangian, we mention the four-fermion interaction of the type L - ~GF J~ J theory of weak interactions. The current J tonic part £ £

in the Fermi

consists of a lep-

and a hadronic part h~. A typical contribution to

is, for instance

describing the destruction of a neutrino and creation of an electron. The terms appearing in L have all the graphical representation

% where the ~i's are arbitrary fermions (e,Ve,~,~ ,... , quarks). Due to the fact that G F has dimension (mass)

-2

, this field theory

is non-renormalizable; nevertheless, it describes to a very good approximation weak interaction phenomena at low energies. As is generally believed, the "fundamental" theory of electro-weak interactions is the renormalizable Glashow-Weir~erg-Salam gauge theory [51] in which, in addition to the fermions, the fundamental Lagrangian also contains gauge and Higgs bosons. The 4-fermion vertex is now replaced by the exchange of a heavy gauge ÷

boson W- or Z:

17

'/'I

~]",

Z

"-/-,,.

Because of the large mass of the gauge bosons, the forces mediated by them are very short-ranged; thus in the low energy limit (roughly E < 80 GeV), Fermi's point interaction is recovered. It is in this sense that the non-renormalizable J

J~-

interaction can be regarded as an effective long wavelength or low energy effective Lagrangian of the renormalizable GlashowWeinberg-Salam model. reads

In a symbolic path-integral notation, this

[SO]:

(1.43)

Another possible application of the effective action concept is Adler's induced gravity approach to quantum gravity [50]. As has been long known, a quantum field theory of gravitation based upon the Einstein-Hilbert Lagrangian (1.30) is non-renormalizable because Newton's constant G has dimension (mass) -2. Now it is tempting to assume that there is some fundamental, renormalizable theory of gravitation which, upon integrating out unobserved matter fields, yields as an effective low energy (or long-wavelength) theory the Einstein-Hilbert action. At present, however,

18

this approach is far from having been completely worked out; for a further discussion~ the reader is invited to read the review article of Adler [50]. In the preceeding discussion we developed the intuitive notion of effective Lagrangians as describing the dynamics of "light" fields in interaction with "heavy" fields hidden from direct observation. But there is still another way to look at functionals like W(1)[A]. As already explained, the fermion vacuum is characterized by a continuous creation and subsequent annihilation of (virtual) electron-positron pairs. Owing to the energy-time uncertainty principle AE • At ~ ~, the maximum life-time of such a pair is about ~/2mc2~ where m is the electron's mass. If we apply a sufficiently strong external electric field to the vacuum, it is possible for this field to separate the electron from the positron so that no recombination takes place.

In energetical

terms this means that each of the particles must aquire an energy of at least mc 2 during its life-time ~/2mc 2. Then the virtual electron (positron) is converted into a real electron (positron). Of course, this is not a "creatio ex nihilo" because the energy corresponding to the rest-mass of the created particles is extracted from the external field. As we shall see in the later chapters, for the pair production rate to be significant, electric field strengths of about 1016 V/m are necessary; this tremendous number explains why one usually assumes the vacuum to be an insulator. In fact, at field strengths large enough, the vacuum becomes a conducting medium!

19 In other words, in presence of an external field the vacuum state

IO> which contains no real particles can become unstable,

i.e., it is energetically preferable that containing real particles. state

If we prepare our system to be in the

Io> in the remote pa~t (t ÷-~),

plitude A remain in the ground-state

IO> decays into states

then the probability am-

~ A for the system to

Io> must not equal unity. Now, of

course, the question arises: which is the functional dependence of the vacuum persistence amplitude A on the external field described by the vector potential A (x)? One way to answer this question is to refer to standard texts on path-integral methods in field theory [25,26,12,51] where it is shown that this amplitude is given by exactly the path-integral

(1.39),

i.e., it is expressed by the effective action as

Thus, knowing W (I), we can calculate the probability of pairs being created as I-I
(1.44)

using standard perturbation theory [37,39]. In the interaction picture the relevant S-matrix element is given by

with the interaction Hamiltonian

(1.46)

20

Expanding the exponential leads us to the perturbation series

'~-

]I

[X"~>

= ~C×',x")['a*'k £~k~-~L, ') 1 e~p[-isE'OL(°'ko k"+ J (~.)~ cos C%~)

(2.45)

Furthermore, we need matrixelements of the following form

<x'l

e~e~-~'s [cx'~'rr.

7T'.~'"~rr~cx.~ rG.z }

IT I~

IX"'>

C%_~-) From t h i s equationp t o g e t h e r w i t h

qFr r

K~

~

(~_~) =

i

and

=

36 follow the important relations

<~'le~p {-~ I:~'°'TroTr ° + o,'"rr, ~~+ o~ rr~ ]}(1, Y'~. r'rr', a

c.~.,~"~

c o s Ce ~ s c~_)

F o r the propagator

G+(x',x")

=

<%'l

=

{

w,-~-,-

I×">

e -~'sc:kt

i~) <x~l e

ITS'+ ~.~--~'e

~'ds 0

-z'sTT

C~- ~',r)(x">

we thus get

(~÷c~',x"}

(2.47a)

whe re

0

; 6 "s 2 cos ~

-- ~'~a~

(2.47b)

c o ~ ~:

We shall use g(k) in this form in the following sections, in order to calculate the mass operator and the polarization tensor; furthermore, formula (2.46) allows us to easily transform the thus calculated mass operator from the momentum into the position space representation.

37 (3) The Mass Operator in a Constant External Magnetic Field In deriving the electron propagator in the last section, we did not take into account any radiative corrections,

i.e., G+,

according to (2.47) can be considered as the first term of an expansion of the complete two-point function,

i.e., of

the dressed propagator

~+cx~x") = I

i < o i T ?(~'~ ~t×") IO>

(3.1)

in powers of the coupling constant ~. For now, if we consider the case of a v a n i s h i n g

external field, then G~(x',x")

I = G+(x'

is translation invariant and we can define its Fourier transform by

I

=

~-t- (~)

The first terms of the perturbation series for G~(p) are presented by the following Feynman graphs:

J

G÷ cp) --

P >

+

+

I-

.

.

-

.

"

-X")

38

The latter diagram is thereby one-particle-reducible,

since

it can be divided into two diagrams by cutting on___eeinner propagator,

i.e., such a reducible diagram can be obtained

through iteration of simpler graphs. It is therefore useful to introduce the "proper" self-energy part of the electron or mass operator Z(p), by the sum of all contributing irreducible graphs. Iteration of Z then gives

t

I

G+

=

w h e r e m° r e p r e s e n t s electron,

If

the unrenormalized

we l i m i t

ourselves

(bare)

now t o

the

mass o f

first

order

the i n ~,

then only the diagram

has to be calculated, which, according to the usual Feynman rules, leads to

w h e r e we chose t h e Feynman gauge f o r D+pv = g#v D+. The s u p e r s c r i p t second-order entire

approximation

mass o p e r a t o r

the consideration structure

of

of

G~(2)(p).

~.

in

(2) the

in

For this

Z (2)

coupling

We now w i s h

radiative

the photon propagator:

to

constant

determine

corrections purpose,

indicates

the e of

what

the

influence

has on t h e p o l e we d e f i n e

a mass m b y

39

i.e., as the pole of the propagator G+ (2) (p) in the presence of interactions with the photon field. Now we expand the mass

operator

according _

to +

with constants

and~)

= quadratic

and h i g h e r

order

terms

in

(~+m).

In terms of these q u a n t i t i e s , G~CZ) (p) yields G IC~)

where we have used the customary notation A ~ ~m. Near the mass shell,

i.e., for ~ ~ - m

or p2+m2 ~ O, this

reduces to

from which we can conclude, because of the defining equation for m as pole of G~(2)(p) on the mass shell, mass m o is associated with the physical

that the bare

(renormalized) mass m by

The fact that we are confronted with a non-vanishing mass operator Z(2)(p)l~=_m when considering interactions between the electromagnetic

field and the Dirac field forces us to redefine

40

the mass parameter

of the theory according

to the last equation,

so that the pole of the electron propagator physical

(observed)

mass m. It should be noted that the neces-

sity of such a renormalization the quantity

is given by the

6m is divergent.

is independent

of whether

or not

Thus we obtain

-1

p

Near the mass shell,

+ ",.. +

C approaches

zero quadratically

so that

we obtain

In order for the normalization to be retained be one, which

[39],

functions

the fields

~o' ~o' which

yield renormalized

a renormalization in QED,

divergent

of the pole

appearing

according

with residue

is independent

though,

quantities.

function

at ~ = -m must

the case for B + O. This

leads

in (3.1) as bare wave

to

fields ~ , ~ which

electron propagator

or not;

the residue

is not, however,

us to interpret

of the electron wave

give us a pole

one. The necessity of whether

in the of such

Z 2 is divergent

both ~m and Z 2 are established

to be

41

Finally,

for the renormalized dressed electron propagator,

we obtain perturbatively,

G I(~..)

c p) =

+

=

~

to order e 2"

ItZl

c p)

i

__

I

+

OCe~-)

where we took advantage of the fact that both C(~) and B are of order e 2, i.e., up to this order, neglected.

the term B.C(~) can be

If we compare the last expression with the original

equation for the unrenormalized electron propagator G~(p) = (~ + m o + Z(~)) -I, we conclude that in the process of renormalization,

the bare mass m o has been replaced by the physical

mass m and the function C(@) has taken the place of the unrenormalized mass operator Z (2) where C agrees with I (2) from the quadratic term in (@+m) on, but does not contain constant linear terms. Therefore, operator Z R(2)(~] = C(@) e n ~ J

to calculate the renormalized mass (which we shall again refer to by Z(p)

in the following), we can also proceed by starting with the ansatz

and c h o o s i n g

leading physical

the

counter

terms

t o A = B = O. ~ ( p - k ) mass m.

and

is

c.t.

in

thereby

such a manner t h a t

parametrized

by the

42 Following Tsai [7], we now want to calculate the diagram

(3.4)

explicitly, whereby we shall use the electron propagator in the external field indicated by the double line in (3.4). In space representation,

7-

c,~',,~"~

i e "~~

-

(3.4) leads to the expression

'

""

with the free photon propagator

o(~'k

z: kx

4' (3.6)

and

G+ ('~"X") = q~)(x', .~.,) O '~?d'~ (2~r} e i~'ael-ae'l 9 ("P)

where ,9(P)

(3.7)

is given by (2.47b).

Substituion gives

Z cx' .',; = ~c*', ,,") Jc:znff'+ [~-~ e ~Pc'~'-x'~ 7" cio)

(3.8)

43

with

+C.~.

J~i.-j- kL;e

Using

(2.~7b)

and t h e p r o p e r

(3.9)

time integral

I

(3.1o) o

we o b t a i n oo

• {~ 3L,., C ~=' (''':'+ cF'-~): ÷ -~-#JcP-~}~') o

•

~

'~(p-~

k]}) +

°. e .

~ ol'k e - ; s z ( k L ~Z) (zrn ~-

.

_ ~s, (~'*(p-k), ~+ ~ ,

(p-~)i)

e

. 9-h ~CDs ~ [~_ ~p-~),,

~c~;E -~

(3.11)

44 Now we perform a transformation of variables for the Sl, s 2 integration:

oo 0

o

0

,t

(3.12) 0

The exponential functions in (3.11) can then be rearranged as follows :

)

-(Pl-

(3.13)

with

and

•. =

'?.x.t~-,tA.)

-F

y,

C0f--''~~)c o ~'~:" ~ y y .+

~.~nyly

(3.15)

P .L.

With these rearrangements it follows from ( 3 . 1 1 ) 4 0

i s ~'(:~. k)

o

C~-~EP-k),

co~ >,

+c.~. (3.16)

45

Further simplification

yields

_~-~y

The remaining k-integration first need the formulae

~_~

(3.17)

is simple to do; for this, we

[36]

,

4>0 (3.18)

J',~,<

~',:,.c,qr-~ :

(.~)

S.

,

~>o

Then (3.19)

(3.zo)

With we g e t ,

by s h i f t i n g

the i n t e g r a t i o n

variables

I~ [' I<.- ,-,- $=,,, v /v

-

c-,:) C4~)z

So f a r we o b t a i n

Cp)~

C-~) z

(e'w)z

for

't S= e

7's ( c ~ + , u . ~ ' )

z

]

(3.21)

1

the mass o p e r a t o r

- £s C'u..,,,,.%~,)

o1~ j~l~ °-{

o

e ("-~'~'~Y

+ '~;"~Y/Y

(3.22)

46 using the abbreviations

~"~-~Y g,

(3.23)

(~-~)~y

-~ ~_

s'~y/y

With the aid of the formulae given in Appendix A, the Dirac algebra of the A's can be further simplified.

El÷ e- ~ ' ~ ¥ ~

1 ~ . = - - 2 e ;~-~'/ The next term yields,

(~.2~)

accordingly

~-'~,./

~

(3.2s)

e-2 ~~>' I

The last term A 3 can be written in the form

I~% = --~ e ~''~Y ~" If we put the A's into

~-'~ (3.22), it follows that

e _~¢ay j~

+ (~-~)

e •

with the fine structure constant ~ = e2/(4~). The remaining integrations cannot be carried out in a closed form; it is, however, amplitudes

easy with the help of the transformation

(2.46) to convert

(3.27) into a space representation.

For this we use the two auxiliary equations

47

e ~pc~L~D

= CoS~

< ~'1 e-"~-c~-~ P'~ e- ; ~ ~" Ix"> (3.28)

and "

_~:~

~Lx"l

~c~',~',> I ~(;.~7 , e'r

~

c~ ~,, ~- 6 re-)

--

(3.29)

where we set

•Jn:~.,~ ~

:=

( , t - ~ ) ,~',,. y (3.30)

From this, we get for cos 13

(3.31)

= tc'-~'~y in abbreviated

* ~ "~vl×t

Zk- k

form

A.-- o - ~ ) % ~ ( ~ - ~ ) ~ y ~,'.×/y ÷ ~ - ~ v / ) '~ With

(3.28) and (3.29) the mass operator takes the space re-

presentation

48

7-- cxr,x") = <x'~ ~ c with

o~

(3.32)

~> Ix">

4 z S "m- ,u.

(3.33) C I denotes the contribution of the first two terms in the parenthesis of (3.27)) i.e., of

to ~.(E). Now we can use

(3.28) with

(3.31) and get

C I = {o--)~,y + ~ "~-v/z ] ~-~ e - ~"~c~'~t~ •

• e - ~ ~ rri. e "~'y O+ ~-~"~-'Y> (-t-'o.) cos y ...(-

(3.34)

¢x.r~.,,,y/y

e

(~+e

with

since from this definition and

--

the

first

~

line

of (3.34)

,

again

~

--

follows,

Similarly we get for the contribution of the third term in the parenthesis of (3.27), because of (3.29)

49

~su~,~~- ~su}~ ~

_z;~.~y (3.36)

Finally, we must evaluate C3:

~Trj.

~S- z

~

e

First, we need

from which we obtain

= ~ - { {(t-.,.)o,,y÷~,.:'~yly~ .e-~E"~ E~C,-c.c.)s'~y -- a - ~ c~-w e

+

A-

We can now substitute this into the above expression for C3: ~4~ y -~

~-$Z

e

--t~

~

z'5c ~ t -

-4-

a

With these representations

2"~crz~z [

(~-~ (4- ~.,~) a,

~

(3.37)

y of Ci, we get from (3.32) as end

result for the unrenormalized mass operator

50

Z Cn-) =

a~-

~_,-~-'~y

_ •4 -

[I.-

,

"

+ (_-,(-~1 e .

~_.

-

a"~'~Y

~TF,-

,.(.-

(.t-,u.)

(

-~ --~ +

.38a)

with

and

=

C.t-~) •

The c o u n t e r after

terms

turning

conditions

-,- :~,,.,.c.,-.,.,.~.,.,.:.,,.~,×~y c.t.

are n o w to be so d e t e r m i n e d

off the field,

are fulfilled,

Za ~'Tr--,'- ~

+ .,.,_,. (- ~,,,~¥ z.,)a

the mass

i.e.,

Z

operator

(3.38c)

that,

renormalization

that

=o

(3.39)

, B "-" 0

and

(3.40) "qrir..-t.- 1~.

is valid.

First

~

~

o

in the l i m i t i n g

case y = e B s u + O:

51

just as

©(g=o~

= " ~ , , ~ - ~ E ','--L ~ ' ]

+ ~

[

-c,,-~q

~ ,,,: --~0

--"'~ 0

Furthermore,

it follows that

A CB-o) =C4-~.) + ~ c ~ - - - ) - - 7 - c~,~y + I

=

1'

t,

-...., -f

From

(3.38) we thus get the mass operator ~(B=O)

: = ~o for

the field-free case

~o

-

a,r

o

So the value of ~o

e")"

(3.41)

(without counterterms) "~

ooo

" ;Z-rr

on the mass shell is

z'S'~'~

~

C-~

-o ~

..,.. (3.42)

,

"-s- o

But according to (3.39), this should vanish,

i.e., we must

choose the first part of the counter terms in the form C . ' ~ . C'f~ ~

__ (..,,f÷ "IZ_)

(3.43a)

so that ~o(p = -m) = O. To fulfill (3.40), we need the derivative

"a.~

•e

.z.,-r

--~ 0

c- ,~) ~c-~- ~: c-.~o') •

L~ + ~4- ~ : * ~

52 which

gives

the expression

o.~

°

on the mass shell

(without

c.t.)

o

I

In order to eliminate part of the counter

this as well, we choose the second

terms as

- - ~-4,~,~~

whereby we make sure that using the factor Our end result, a constant

(~r)

still remains

(3.43b)

valid,

by

(m+yH). then,

external

~

-

(3.39)

(~- ~ ) S1

~ s

for the renormalized

magnetic

mass operator

in

field reads:

o

(-9

-

-e-

One can check that in the free field case the conventional spectral

representation

[29,39] :

of the mass operator O0

is obtained

,yl~ ?-

(3.45)

f --~7--(H-~'t- ~-t-CM÷~)~" We observe appears,

that for M ÷ m the well-known

which has its origin

infrared

in the masslessness

divergence of the photon.

53 As an important

application

to the renormalization

of the 2-1oop effective

are now going to explicitly

where

~nr denotes

without

of the mass operator with regard

evaluate

the unrenormalized

counter terms.

From equation

Lagrangian,

we

the mass displacement

mass operator,

i.e.,

(3.42) we read off

immediately: 4

f 4 ~'t4.) e. ~..'rr"

which

apperently

o

S

o

diverges

used in this form.

for s + o and therefore

Furthermore,

integration would be divergent, To obtain a regularization

duce finite

Here,

s1

lower limits

i.e.,

for u = o, the s-

at the upper limit as well.

scheme

the variable

cannot be

for u = o, the exponential

damping of the integrand vanishes,

we first reverse

-z's ~'ZC'~Z"-- CZ')

consistent with

substitution

section

(3.12),

and intro-

variables

from the

for both integrals:

and s 2 are again the integration

proper

time representations

gator.

Now we can put s' = 0 (but s > O) without making o o

integral become

proper

of the electron

and photon propathe

divergent:

~1::~, ~ In this form,

(7)

i.e.,

~-+S&

(3.47)

as a function of the lower limit o f the

time integration

of the electron propagator

we shall

54

use ~m later on. To calculate the (convergent)

integral

(3.47), we introduce

a new substitution for the s2-integration

~' which, with

leads to

The u-integration is now elementary,

×

~Ot×

olx

x~e_.

---- ~ c x

and, with the aid of

e_.

)C

~

~c

t

Po~

K

we get

-4-

Now, in the first and second term, we do one, respectively two, integrations by parts,

and calculate the integrals in

the third and fourth term. The result

is:

55 oL~

F

CSo~ = ~

e -~-~=°

L- I t ; . . ' = . ) "

t--

Since

we

are

this

1

e -~° £.,:,~Z;o)

q-

i ~

1 + . ~ d $~ e :L(z"~zs°)z So

interested

exponentials

v ~

in the

and d i s r e g a r d

all

limit

so + o

terms

which

~ So

]

we e x p a n d vanish

the

for s o

-~ O "

result

for

gives

So Performing

the e x p o n e n t i a l

integral

[36]

leads

to

--

% =

which,

for s

÷ o, 0

oo

Here,

-~C~

reduces

or, f o r

the

[36]

- ~'~)~,

displacement

~ w (~o) -

to

Z

C = In y is E u l e r ' s

the mass

-- £ ~ L - ; t ~ L , ' ~ ) S o )

constant.

our

final

reads:

1

3oc.'m

change

So,

in m

2

:

+

E

(3.48a)

56

~-rr

-i ~-,~ z.S'o

(3.48b)

This equation corresponds exactly to Schwinger's [3] and Ritus'

[4] results, but was achieved by completely different

methods.

(4) The Polarization Tensor in a Constant External Magnetic Field A further important building block for higher-order processes in QED is the polarization tensor, which we shall calculate to the order ~ in this section. First, however, we consider the completely dressed photon propagator without external field, defined by I

:b_~,,,~ c~-x') = i < o l T R ~

g~c~')Io>

(4.1)

and whose Fourier transform can be written as

The perturbation series of D~ with respect to ~ thus contains the graphs I

56

~-rr

-i ~-,~ z.S'o

(3.48b)

This equation corresponds exactly to Schwinger's [3] and Ritus'

[4] results, but was achieved by completely different

methods.

(4) The Polarization Tensor in a Constant External Magnetic Field A further important building block for higher-order processes in QED is the polarization tensor, which we shall calculate to the order ~ in this section. First, however, we consider the completely dressed photon propagator without external field, defined by I

:b_~,,,~ c~-x') = i < o l T R ~

g~c~')Io>

(4.1)

and whose Fourier transform can be written as

The perturbation series of D~ with respect to ~ thus contains the graphs I

57

where the last diagram is one-particle-reducible, i.e., one only has to cut one inner line in order to get two simpler diagrams. So, analogous to the definition of the mass operator in the last section, it is convenient here to introduce a proper self-energy part of the photon or polarization tensor ~ ~(k) as the sum of all contributing one-particle-irreducible graphs without external propagators. By explicitly calculating the diagram of second order in e given by

v (k)= we

shall

ie" ~' a(~.j

see t h a t

n~v _(2)(k)

. (k) ={9e, with

a scalar

~'v G.+cl°-

can be w r i t t e n

k - k~k,)

polarization

t o assume a p o w e r s e r i e s

function expansion

as

I T ( ~ ' C k :) R(2)(k2) of

the

for

form

TTr~'CK~) = TG ÷ 7& k a + ~ (k=) z + By i t e r a t i o n of R(2)

order

.

.

.

.

.

we get for the photon propagator up to

e2 :

r ,

=.

w h i c h we w i s h

~,,

k ~ ( i + ~ ÷ ~ k~+---)

+

Lo. s.

58

which,

near the mass

shell k 2 = 0 (note that the photon pole

has not been shifted by the interaction),

,~o)

(~ k

+

=

~
we again must require

at k 2 = 0 be one; field A

in (4.1) as a bare

calculated with

/(z)

=

÷7"~, ~ .

function

that the residue

to be

of the pole

this can be achieved by considering field A

is related to the renormalized pagator

of the wave

-I

~'+p

A

Do

which,

according

the to

so that the photon pro-

it ~'

.

:Z 3

-,I

+ Io . ~/Acp

has the residue then

+ 0 (e ~} + Io~3.

one. The associated polarization

function

is

59

Following our line of reasoning in the last section, we can obtain the renormalized polarization tensor, which we now again shall denote be 9 p~) (2) , by replacing the original definition of ~[2)(k)" by pv " (4.1 ') and,

after

the c o u n t e r

evaiuating terms

c.t.

the

integral

~ pv (Z)

these in

ference

observations,

non-trivial

[6].

choosing

-- 0

the p r e s e n c e to

trace,

i n such a way t h a t

U After

and t h e

(4.2)

we want now t o e x p l i c i t l y

of a constant

T h e r e b y we s h a l l

magnetic

limit

graph of the perturbation

field,

ourselves series,

calculate with to the

i.e.,

refirst

to

but shall again take into account the external field to all orders of the coupling constant. With the practical notation

=

,0 6.?..rr) ~

60

we

then h a v e

to e v a l u a t e

Ti-c"(l<) =--<~ ~r <~,,, 9cp) r,, /,,,, whereby

g is a g a i n

C~ ( p - Ix: ) ~> + C. ~. (4.4)

g i v e n by

o

CoS E

with

z = eBs.

Substitution

results

in

--<<' %..,, (t..) = -~ <>'~" !~>. <e>~p~-,'~.,,,'..¢,-~.l~.lr'~]+--,- { ~ O.,,.-~l~,,> e _

with

'~ cp- t'cL

z I = eBs I a n d

N o w we

-c~,.

introduce

(4.s) z 2 = eBs 2.

new

variables ~-v

and get

of i n t e g r a t i o n

61

as well

V:

x~-x"

:~=

~ ~B.r~

~z:

e~g'z

as ao

2..,

~

.... To simplify function

~:

"=:

--

(4.7)

oo

="

o

o

=:

(4.5), we

appearing

(4.8)

--o

first note

-I that

in it can be put

the exponential

in the

form

with

= z,

~-v~ k~ +

COS ~v--co.t~

2.

k~.

(4.1o)

and

~,

Now we are allowed oo

to put

(4.9)

into

(4.5)

kj " ~ 2 ( 4 . 1 1

and obtain

4

(4.12)

•

Frr~"-~P"l~

,r(p-kL o,.~?

]~>

- ~r

+c.~.

~r~ 12~-~p-.~,,) e ~'t-

)

62 We can now perform the p-integration very easily, by s h i f t i n g the integration variable and using equation

(3.19). The

simplest integral needed in (4.12) is

<e -''~" > = .~J,,~d'~~,,e[-,,,~(e,,te'-~.)~ ~+~ro-~,_, ~. ~*~'k'~ K)'} ] C-~l

I

"~

co~' ~"

(4.1 3)

c..o,.~?

Fur the rmo re

<~-~.* p,,.~= P-~-~.~[_ ~m.-~,.>'+ ~'-~'~'c~. -

,..,.,_~~.~-}~ p, (4.14)

_- ~+v ~.'k, <e-~'~

>

since the integral vanishes over an odd function. Analogously we get

t

<"

K. <e-;'~" >

~'r P"~- <~-;"~'>[C~"v

< e

[P,v-IP-~,~

t~'~'- ~ %"'-] ~ .k~ ~ . ~ , ~

(4.1s) ; K-r k,,~ 1

.L

97.. ' -J whereby we introduced the definitions

(o Oo )

_

(4.16)

"

_

°.tO)

With the help of (4.13) we can now eliminate the factor (cos ~ cosq)

-I

which leads to

63 ~0

4

+C.~. 0

(4.17)

--1

with 4

With t h i s ,

we must evaluate the f o l l o w i n g t r a c e

{r<,e %t[.~e

--~p,,e

qre*.~'yi(~e -~p-k),,e

co~?

e.~f,

#.

coa,.~

+ -&r < e-e'?'~ qP. % ~Cp-kk3

4 ~ .

co~

Before we begin to solve t h i s problem, we d e r i v e two u s e f u l trace relations which will prove helpful. The first reads:

(4.19) By way of proof, we take advantage of the fact that only the components F12 = -F21 = B of the field strength tensor do not vanish, i.e., t h a t

//

64

T )r" : 9"r 9, -9,., 9F, is valid. With the formulae

given in Appendix A we then obtain

~-)

:-~o~ In addition,

~,.~, + ( ~ r. ~,~ ~

we need the relation

(4.20)

- ~'--~,, ~ [T~ ~ ~ * -

~ ~ ~~-~ ~ - ~='; ~,"~" ~" ~ ~,~ ~ ~ ~rf~

Supplied with

and

(4.19)

the individual tr Si,

sums of (4.18), which we shall denote

i -- I...5.

~ z'~

(4.20) we can now begin calculating as

First we get

- ~<~-~'Yr e

I

~

~<e-~Y .>

I ----;

Here, we used the cyclicity of ¥I Y2 with respect sum we now need

of the trace and the antisymmetry

to the indices

I and 2. In the second

65

(Li..zo;

T

We again used the special form of the field strength tensor in the second line. If this is put into the above equation and the addition theorem of trigometric functions is used, it follows that

Itshould be noted that the second term on the right-hand side is an odd function in v, which after s u b s t i t u t i o n into (4.17) makes no contribution.

If we still use (4.21)

then, again thanks to the addition theorem ~r ~ ,

------~

To further evaluate

S %V ~

~ - C / ~ p , + odd function

(4.18), we must go back to the integrals

(4.14) and (4.15); for tr $2, we get, for example

='.

with

C

(4.22)

66

C' ,4(s =

< ~-"'~" >-' [ <,~-"'~" ~,,"~,; > - 4 <~-"'r' ~;,> t

and

~ ~'~'

,~, ~'~,~

Due to the cyclicity of the trace it then fellows

It can easily be shown that

'7"ku e

=

~k~f

~i ~

and "0 is valid,

With

and

and, substituted

into the above equation,

leads to

67 this becomes

-tr S'~ _ i

=:

~cr{e

~t~sf

E ~"

s)

~.,'+.,23z +~B~

whe r e, with the help of (4.22) we can put the B's into the form

% %--.

.+ o~oz % . j , , ' , , .

P.valuation

of

the remaining

terms

of

(4.18)

is

accomplished

analogously and following a longer but elementary calculation, leads to the following expression for the function I~v

(4.2s) J"*

__• - ~:=~ ~v - v ~

+

CoS' ~ - V - - c ~ "

4_ -e

~. ~

~-

l

t[

~,)[k,~<,-

%

K = ~ - K,, t K,,~]

68

Further simplification of ( 4. 17)

can be achieved by inte-

g r a t i o n by p a r t s oO 0

=:ST+

{

0

(BT = boundary terms). Differentiation gives

(~,~ ~

'(

--

+ co.~ ~,,

I~

so t h a t ,

after

"-

-PCo~~V J- ~

I~,~

-

~" "[ I - - @:,C_.c~ ]

a short intermediary calculation,

we f i n a l l y

get oo

0

s

c~a

4

T

~.

(4. 26)

H

69

Accordingly,

-i'

4

(4.27)

With the help of these two integrations by parts, the integral representation

(4.17) of the polarization tensor can be put

into a very simple form after some rearrangement: ~

(4.28a)

-1 2 with

~

~o_ ~), k,,L~,,r ~ ) ~

:,.~

and

(4.28c)

N.r : _ ~ Q ¢ . • ( ~ _ v~.+ v ~-T~/='~ ~-v

N :2..=

~'~ ~

s~

70 Here, we have incorporated the boundary terms BT in the c.t. Now we must set the counter terms so that the renormalization condition for the polarization tensor

=

kZ~o

:~-'~ 0

is fulfilled.

k~-~o

0

(4.29)

~---~0

To that end, we first investigate

in the case of a vanishing external field; by a power series expansion z = eBs we find

=

~O-vU

so that kI can again be joined with k~ to make k 2, This was to be expected,

since after tu~ningoff the magnetic field, a

preferred space direction no longer exists. So (4.30)

and

It follows that

No

-- 4 - v

q- -"* o

Z~

NI= 0

71

The vanishing direction terms

of N 1 and N 2 again garantees

exists

in (4.28a)

for B = O. So if one chooses

~S~ ~

fulfilled

Our end result

where

)

the renormalization

7th section

of the polarization

in order to derive

the 2-1oop effective

Lagrangian.

tation for the polarization

a simple

tensor

expression

But first, we should

return to the case B = O and construct

(4.32),

(4.29).

(4.28c).

We shall use this representation

From

conditional

then reads

the N i are given by

in the

the counter

in the form

= then one has

that no preferred

a spectral

for

like to

represen-

tensor.

one gets for a vanishing

magnetic

field

with

o

where we took advantage is even.

Next,

o

of the fact that the integrand

we perform a partial

integration

in v

with respect

to v

- ~

oT

v=o

72 4 O

~-I< a i~v vZ(~ - v ~

2 m

For the s-integration, 4

~-c~)_

~ K~ ~

I~s

(,,-~÷

-it again replaces

m

2

~ c , - ~ , ~ ) [ ~ ~-'w_,,~] -~

~T

0

I

4

~oIT

4--V~

4_V~

Z

0

(4.33) Now we use

as the new variable the desired spectral

of integration representation

and thus get from

(4.33)

of the polarization

function:

Trc~DBecause

¥ z'~ c~o(-xo) it

6-~0 +

X-Xo

± iE

X~Xo

follows

f o r the imaginary p a r t

~

)

(4.35)

73 In this section we have assembled all the building blocks needed to calculate the one and two - loop effective Lagrangians of spinor QED following in the next chapters.

(5) One-Loop Effective Lagrangian Our first goal in this section is the derivation of an integral expression for the effective Lagrangian in the one-loop approximation.

The central subject we shall be interested in

is the vacuum amplitude in the presence of an external field which, in the framework of this approximation can be written as (see section (I))

with iWc~'[~]

=

--

t-7 £~ ( 4 - e ~ G . , . ) - t

C

/ G+col)

(s.z)

Here G+ = G+[O]is the electron propagator in the field-free case, connected with G+[A] by

G+C~-] = G+C~- e ~ & + ) -1

(s.3)

furthermore, Tr indicates the trace both in spinor and configuration space. The 1-1oop effective action W (I), i.e., the effective Lagrangian L (I), introduced in (5.I) is the formal expression for the effect which an arbitrary number of 'external photon lines' can have on a single Fermion loop, i.e., W (I) and i (I) represent

73 In this section we have assembled all the building blocks needed to calculate the one and two - loop effective Lagrangians of spinor QED following in the next chapters.

(5) One-Loop Effective Lagrangian Our first goal in this section is the derivation of an integral expression for the effective Lagrangian in the one-loop approximation.

The central subject we shall be interested in

is the vacuum amplitude in the presence of an external field which, in the framework of this approximation can be written as (see section (I))

with iWc~'[~]

=

--

t-7 £~ ( 4 - e ~ G . , . ) - t

C

/ G+col)

(s.z)

Here G+ = G+[O]is the electron propagator in the field-free case, connected with G+[A] by

G+C~-] = G+C~- e ~ & + ) -1

(s.3)

furthermore, Tr indicates the trace both in spinor and configuration space. The 1-1oop effective action W (I), i.e., the effective Lagrangian L (I), introduced in (5.I) is the formal expression for the effect which an arbitrary number of 'external photon lines' can have on a single Fermion loop, i.e., W (I) and i (I) represent

74 the sum of all graphs of the type (compare with Appendix G)

(5.4)

which one abbreviates as

0

(s.s)

The wavy lines in (5.4) symbolize interactions with the external field which are considered to all orders of the coupling constants and not real photons of the radiation field. Note that (5.2) vanishes for F

='

0 so that the vacuum

amplitude then becomes

~ ° =

='~

e

On the other hand, if F v ~ O, then

as it must be.

the possiblity exists for

IA 12 ~ 1 to produce electron-positron pairs by means of the external field. As already discussed in the introduction, the probability for this is

~)= ,/-I
=

I-

e

~

75

For small values of W, one can expand the exponential function and only consider the linear term:

So, for the pair production probability w per unit space and time, it follows:

Here, we have always written W(L) rather than W(1)(L(1)),

to

signify that, in order to determine L exactly diagrams with an arbitrary number 1 of loops had to be summed up, i.e., it is + ....

+--with

the

classical

In section in

the

But

8, we s h a l l

limiting

let

(5.7)!

culated

case

for

We n o t e

explicitly

of strong

u s now r e t u r n

representation to

Maxwe11-Lagrangian

to

this first

in Appendix

D of

i(1) most of all

(8.7)

L(°).

perform

such

a summation

fields. again simple, the

t h e W( I )

and look

for

non-trivial functional

with

respect

an explicit contribution

derivative to

the

calpotential

A :

In addition

we n e e d

the

the electron propagator:

proper

time

representation

[5]

of

76

G+ E~I] -.,, ~F~-~ cO

(s.9)

-Ls E"~z- ~r " ]

o

We now show that the Ansatz

~/C~,=_ ~~ ~_C~)= - ~ 4 ~

e --~S~'zZ 77 [ e -~'~=~ ]

(5.I0)

o fulfills eq.

We s h a l l

later

(5.8) and t h e r e b y

determine this

g i v e s W(1) constant

(to within

so t h a t

a constant).

the action

vanishes for vanishing external field as well. To calculate the derivative from (5.10), we use the chain rule in the form

=-e ~ where ~

= p8 - eA B was used , and we get:

i

~-~-

With

6" o%r~%

77

follows

_,~('.~La;')

oo O

(5.11) Here,

in the second line we took advantage

of the fact that

the trace of an odd number of gamma matrices With this,

the validity

therefore write

of (5.10)

vanishes.

is demonstrated,

for the unrenormalized oo

and one can

Lagrangian

O whereby

the trace

further evaluating

tr only refers

to the spinor

this expression,

index.

we calculate

Before

the derivative

4 oo

= ~r <×I c ~ - ~

¢

]~s e

e

Ix>

o

Here, we again used the fact, of gamma matrices to the simple

vanishes.

that the trace of an odd number

A comparison with

result

Now we return to eq.

we can also write

as

(5.12) which,

because

of

(5.9)

leads now

78

(5.14) 0

In section (2),

<x(i. ~-

we

found

~'1

so that, due to ~(x,x)=1, we get the gauge invariant diagonal element

=

I~ ~ s---~ -I~,_~

c ~

(S.lS)

Here, the k-integration was performed with the help of ( 3 . 1 9 ) . Substitution into (5.14) gives

or, with

follows

~__C4) 4 T

eJS

-- z* ~z

This expression a p p a r e n t l y renormalization.

diverges

for

Before we b e g i n w i t h

to g i v e a n o t h e r method f o r

s ÷ o and thus r e q u i r e s

that,

the d e r i v a t i o n

of

however, we want (5.16).

We b e g i n at

79

with g(k) according to (2.47b) and we calculate the diagonal element (which, again, is gauge invariant) 4 C~) ~

¢ Here, in the second line, the second and third term make a contribution,

do

not

since the integrand in these terms is

odd in k. From (5.13) it follows

oo

0

Integration then gives

~OC~ ~0~,~',~I~'~z~(~'~'£)

0

O i.e., exactly

(5.16). We have chosen the integration bounda-

ries so that L (I) vanishes for m ÷ ~. This is in accord with the physical requirement that for infinite fermion mass all non-linear effects must vanish, since in this case, the creation of virtual electron-positron pairs becomes impossible.

80 Now we return to

(5.16)

inspite of the apparent according to

(5.16)

and attempt to get a finite result divergence.

First we note that

for B ÷ o does not,

as required,

L (I)

vanish,

instead the value oo

o

arises.

Since

the addition of a constant of L has no physical

meaning, we can subtract this term from

e

(5.16)

LcegslC ;ce

to get

)_ I

(5.17)

which now does fulfill

Ic but still diverges expands

CB=o)

=

O

for s + o. This becomes

clear when one

the above expression:

S~

....

~

$

for

S÷O

The divergence which stems then from the quadratic term in B can now be gotton r i d of by renormalization of charge and f i e l d strength. We should keep in mind t h a t in our e n t i r e c a l c u l a t i o n s up u n t i l now, we should have a c t u a l l y w r i t t e n mo, eo, B° instead of m, e, B, in order to indicate that the bare parameters which f i r s t parametrize the theory do not have to coincide with the observable q u a n t i t i e s . Furthermore, we r e c a l l t h a t the e n t i r e Lagrangian in one-loop approximation contains, besides L( I ) , the c l a s s i c a l part

81

(s.18) as well.

So we can write

+m~

°

e

with

~=

o

Here we subtracted order to combine

the divergent

it with

L (°).

T

e

(s.2o)

term and added it again in

If we now introduce

with

(5.21)

the renormalized because

field strength

of eoB o = eB from

(5.19)

B, and charge

e, it follows

82

+

= o'--R.

(5.22)

+ ocp.

with the gauge invariant renormalized Lagrangian

=-~ and

~- ~

~

o

T h a t L (1) r e a l l y

is

finite

becomes c l e a r

if

one e x p a n d s t h e

integrands

4 ~'CeIS~)co~-Cegs)+~Cel~s)-1~~',

~"I

(e,R,)~S + - ' -

(5.24)

So it is decisive for the renormalization of L that the divergent part of L (I) has exactly the structure of L (°), i.e., is quadratic in B. The fact that the mass need not be renormalized is a special characteristic of the one-loop approximation which, as we shall see, does not apply to L (2) We can get another equivalent form of L~1)c from (5.23) by rotating the integration path by s + -is according to the i~ rule (m 2 ÷ m 2 -i~), which because of cot(ix) = -i coth(x) leads

~_~ It tic

to

C~) -

becomes c l e a r

~r • in

-~-e this

~ ~_.~(5.25)

representation

fields of arbitrary strength,

that

for

purely

magne-

Im L(1)(B) and thus the

83 pair production probability disappears. This is a result of the fact that a purely magnetic field cannot transfer energy to a charged particle

(in our case to the virtual electron

and positrons). In all our previous calculations we have always presumed a pure magnet field; however, the contrary limiting case of a pure electric field can easily be derived from this by considering that L as a gau~invariant Lorentz scalar, must be written as a functionof the only twogauge invariant Lorentz scalars of the Maxwell field

The conversion from pure electric to pure magnetic field takes place by the substitution B + TI E

since G 2 vanishes

in both cases and the necessary change in the si'~n of F is caused by exactly this substitution. So, from (5.25) we get o
84

~__L 2_eB

(5.27)

Here ~ represents the Riemann Zeta function in two arguments. (An equivalent form is given in [40]). We shall derive this result in the next section using a completely different method. It remains to be observed that according to [9], eq.

(5.27)

can also be written in the form

,,f~_,~ ('IS) -

B21r~

4 + z,e_~

I

(s.28)

whereby a n~aerical evaluation of L(~) became possible [9,10,11]. ~e want to use (5.28) to calculate LR(1) in the limiting case of strong fields, i.e., for (eB)/m 2 >> I. First we show that for those values of the field strength,

the integral over the logarithm

of the gamma function only yields a constant 4

[9]

¢

'/

1

4 ~+~-~

'H"O ~)' c~,< (~r--O 4

-

4•

"-/-(#

= - ± C

(b..=(e~)/~) •

85 Here, C is Euler's number.

By only considering

the dominant

terms for large field strength from (5.28), we get the asymptotic form of the one-loop effective Cg) ~ or,

with

Lagrangian

.....

a = e2/4~:

B ~

-~--z + ~{ y b 4 ) - ~ ÷ . ~ £J

We shall come back to this expression when we examine the Lagrangian Next,

(5.29)

in the next section

for massless

spinor QED.

let us look how LR(I) behaves as a function of B. Equation

(5.27) has been evaluated numerically

by several authors

It turns out that L~ I) is a monotonically of B for all B > O

increasing

Bcr ~ m2/e ~ 4.4

1,1,1

0

J

0

[11]).

.

10

20 30 H(HcR)

*All diagrams are taken from ref.

[11].

40

50

If

field strength"

1013 Gauss, we get the following diagram*

2

~

function

(this can also be shown analytically

we plot L~ I) against B in units of the "critical

[10,11].

86 Hereby

LR(1) is given in units of B 2cr. Note that in order to

obtain the complete effective

Lagrangian

one has to add the dominating

classical

to the above diagram.

Lef f = L (°) + L (I) + .., 2 contribution -I/2 B

The sum is a monotonically

decreasing

function B. Making the substitution function for O < E

for L~I)(E).

B÷-i

E in (5.27), one gets a complex

Its real part has the following behaviour

< 50 Ecr with Ecr ~ m2/e = 1.7

electrical

critical

1016 V/cm being the

field strength:

I.U

-2

0

10

20

30

40

50

E(EcR)

It is interesting

that Re L~I)(E) possesses

a maximum at about

E ~ 3 Ecr , which is not resolved in the above plot;

investigating

the range O < E < 5 Ecr reveals the following structure negative

real part of L~I)(E):

for the

87

6 A

4

"' ,'7

O

A

~2

v

I,LI

~0

-2 0

i

i

i

1

2

3

i

4

5

E(EcR) Note

that

obtain

L~1)r now is measured

the total effective

wellian

contribution

in the magnetic function

case,

of E because

than compensated

E a n d B) and i t s that usual

L e f f = L o) only

the

Legendre

one has

the complete

Lagrangian

the quantum corrections

is

transform

just as

is a monotonic L~ I) are more

the considered

monotonic

i n E a n d B,

a t E = O a n d B = O. T h i s

potential

Then,

2

(at least within ÷ L 1)

to

to add the Max-

I/2 E 2 to the above values.

extremum is

effective

Lagrangian,

for by I/2 E

Thus it is shown that

in units of 10 -4 E 2 Again, cr"

Vef f calculated

range

for

respectively,

in turn

implies

from Lef f v i a

the

[37]

(s.3o)

has a unique minimum

for E = O and B = O, respectively.

findings

importance,

are of some

because

These

they show that the

88

phenomenon of spontaneous in a pure-electric-

symmetry breaking

or pure-magnetic-field

[51,53] case

does not occur

in quantum elec-

trodynamics. This means vacuum,

the following:

The true ground-state,

of a field theory

minimizes fulfils

the expectation

(we assume

uniquely

to

(5.31),

given by

value of the Hamiltonian,

(5.31)

quantum

is not degenerate.

Determining

Either

or there are several criterion. Until

field theories

in which

[O> satis-

those

the vacuum state

one assumed

this not to be

since the work of Nambu and Goldstone symmetry

can occur also in relativistic

this point by a simple example. scalar

states

The latter case is referred

we know that as a resultat of spontaneous

of a complex

the vacuum is

1960 one only discussed

(To be precise,

the case.°) However,

illustrate

it

t h e vacuum s t a t e

two Eases can occur.

to as vacuum degeneracy.

a degeneracy

i.e.,

(5.31)

= 1.

fying this minimalization

relativistic

I0> which

<

1~> with <~1~#>

according

as that state

true

= I).

for all

is defined

i.e.,

breakdown,

theories. Consider

[55]~ such

Let us

the theory

field defined by the Lagrangian

with

(s.ss) and ~ and ~ being

two real constants.

ground state of this model

Now let us try to find the

in the lowest

("tree")

To this end we study the vacuum expectation

approximation.

value of the field

8g

operator ¢(x):

©c~c~

=

Obviously, ~cl is a classical

(s.34)

(c-number) field. If the ground-

state is assumed to be translational invariant (which is usually done), we must have

C~cl C~' = CO.S"~ ~ (~eL ~ ~ i.e., ~cl(X) must be independent of the space-time point x. The magnitude of the constant ~cl remains to be determined. To lowest order, i.e., when radiative correction are neglected, it is given by the value minimizing the classical potential

~V '~4~

¢ = ¢'or..

(5.33):

=0

This leads to

I c~,. I~-

I~,:t c~'~l~=.,Z~"

or

CS.3S) with an arbitrary phase angle

~

[O,2~). We see that one does

not get a unique ground-state, but an infinite number of vacua parametrized by the a n g l e ~ .

This still can be considered from

a different point of view. It is easy to see that (5.32) with (5.33) is invariant under the phase transformation

¢c~

~

~

c~c~

c5.3~)

with a constant a ~ ~. Due to Noether's theorem, this symmetry of L gives rise to the conservation of the quantum number carried

90

by the complex

~-field.

However,

that a given vacuum state

returning

to (5.35),

is characterized

it is clear

by one value Of ~,

by multiplying

(5.35) with exp(i~),

another;

given the theory based on a given vacuum,

thus,

vacuum is not invariant

under the transformations

This is a simple example Lagrangian

we come from one vacuum

of a theory where

to

this

(5.36)!

a symmetry

is not shared by the ground-state

i.e.,

of the

due to a vacuum de-

generacy. At first sight,

it appears

in electrodynamics

because

tors and, by acquiring they would

single

that such a phenomenon the fields ~ and ~

a non-vanishing

out a direction

the vacuum would be broken. kind of domain structure invariance

randomly

to the domain structure indeed proposed Now,

a unique E=B=O, as

solution so

a quantum

where,

fields.

of a Lorentz

microscopic

This w o u l d be similar

of ferromagnetica.

(Such a vacuum was

chromodynamics;

see section indicate

(9)).

that such a

for

a vacuum

level,

the

of course,

minimalization

degeneracy,

if

electrodynamics of

any,

H=f would

provides

r~,E2÷B2,, have

to

viz.

arise

effect.

The situation expectation

but exhibits

of

should not occur in QED up to the one loop

At the classical

and

level,

value,

invariance

one could conceive

of the above calculations

symmetry breaking level.

vacuum expectation

and thus Lorentz

However,

oriented

for quantum

our results

(or A ) are vec-

for the vacuum which restores

at a macroscopic

"bubbles" with

can not occur

in QED, where

value

the vacuum seems

for ~ and ~, has

as we shall discuss

to have a vanishing

to be contrasted with QCD,

in section

(9), one finds

91

+ After

now h a v i n g

(1) , l e t

LR is

O.

us

discussed

turn

responsible

to

for

consider

its

the

the

the

properties

imaginary pair

case

part,

creation

ofapure

of

the

which,

of

real via

an external

magnetic

field.

part

of

w = 2Im field.

Looking

L, First

let

us

at

eq.

(5.25), we see that the integrand in the integral for LR(1)(B)

is real and has no singularities on the path of integration. This

means t h a t

L R(1 ) ( B )

is

purely

real

and so w = 2 Im LR(1)

vanishes. The fact that a magnetic field does not produce pairs can be understood in simple physical terms. Because the Lorentz force is always perpendicular to the velocity, it does not transfer energy from the field to a particle; hence the field can not supply the energy 2mc 2 necessary for a virtual electron pair to become a real one. To deal with the electric field, we return to (5.23) and substitue E ÷ - i B ; ~(~

the result is y 0

(5.37)

-

17

The imaginary part of this integral is easily calculated using the method of the residues. First we take advantage of the fact that the imaginary part of

e

= cos("~S)

-

~ ~'>~'l'~zs)

is an odd function of s, whereas the real part is even. Because s -3 times the square bracket in (5.37) is also odd, we have

92

-('f)

#O

4' ~rz

II

~(e Es/co'//., tees) -

e

- ~s ( e E s J z -

1]

(5.38)

with the integration contour now being the whole real axis of the complex s-plane. Note that the integrand has poles on the imaginary axis due to the coth-function:

/

~e s

(The crosses denote the positions of the poles). Without altering the integral we may close the contour by a semi-circle in the lower half-plane; hence Im LR(1) is given by (-2~i) times the sum of the residues of the poles on the negative imaginary axis: c4)

~:

=

[

-- £ ~ z s

~

s3 e

,

(s.39)

(The same result would aiso be obtained by starting from (5.26) and choosing an integration contour which passes the poles due

g3 to the cot-function in the upper half-plane). coth(ax)

Recalling that

= I/ax + ..., we immediately obtain

,,i,t~~ ,'~.. f r "

=, . - z ~- ~ , . -

-

-

2.7r £

-

e ~

(5.40)

~=i

This sum can not be done in closed form; nevertheless, to get simple expressions

for the limiting cases of very strong

or very weak fields. For strong fields

(eE >>m2),

in (5.40) is close to unity and exploiting ,~-~"

=

it is easy

C.Z)

--

the exponential

[36]

,

one obtains

(5.41)

This result could also have been derived from (5.29) by substituting E ÷ - i B . For very weak fields

(eE<< m2), the contributions of the n = 2,3...

terms in (5.40) can be negelected and one ends up with

~E) e ~

e~

(5.42)

For intermediate values of E, numerical methods are necessary; then gets the following plot

[11]

one

g4

6

"-~4 ¢~ I,,J iii v

,.,e v

0

0

20

40

60 E(Ec~)

80

100

The values are again expressed in units of Ecr ~ m2/e = 1.7.1016V/cm. Now that we have explicit formulae for the pair creation probability 2 Im L~ I), it is natural to ask for which values of E pair creation becomes significant.

The region where such effects

become observable is reached for fields with about 10 -2 Ecr = 1014V/cm. However, this is by many orders of magnitude away from the field strengths which can be produced by experimentalists.

These have

a magnitude of, say, 108 V/cm, giving a rate for the pair production of about IO-(IO8)(!) which is far from being experimentally observable. Nevertheless,

in nature there are field strengths with magnitudes

near Ecr , namely at the surface of a heavy nucleus. however,

In this case,

the strong interactions dominate and the QED effects

are at best small corrections.

(Furthermore,

the above results

95 are, strictly speaking, valid only for constant fields. But for non-constant fields, the order of magnitude of Im L is presumably not very different from that of constant fields). Finally let us look a little closer at the functional form of (5.42). Due to the appearance of the factor I/e in the argument of the exponential,

Im LR(1)(eF < < m 2) is a non-analytic function

in e. (The same is true for every individual term in the sum (5.40)). This means that Im L (I) vanishes in every finite order of perturbation theory around e = O. In computing Im L (I) we did a geniunely non-perturbative calculation, because the functional W (I) represents the sum of an infinite number of Feynman diagrams.

(Cf. appendix G).

Another interesting point is revealed by reinstating ~ (and c) in (5.42); the exponential then becomes

e x p ~- '~" C~5 '~ ~ e~

E

(5.43)

Obviously, Im L is also non-analytical in N. This means that pair creation cannot be computed by starting from a classical configuration

and then calculating small quantum fluctuations

(of order ~, say) around this configuration. So pair production is an intrinsically qunatum mechanical process, which is not seen in any finite order of ~. This situation is similar to that of a particle with energy E which tunnels through a potential barrier V(x); the amplitude for a tunneling process to occur is given by the Gamow factor

96 featuring the same non-analytic ~-dependence

as (5.43). Hence

tunneling does not occur in every finite order of an expansion in ~. The notion of pair creation being a tunneling process can be made more precise by using Dirac's hole theory.

For E = O,

the vacuum is characterized by a completely filled lower Dirac sea, whereas the upper is completely empty: 4

£

0

Y//////// Now we switch on a constant electric field ~ = I~I ex directed along the positive x-axis; the corresponding potential

is

A ° = -I~Ix and the electrostatic energy for an electron is W = -eA ° = el~Ix the energy levels

(e >0).

Hence we get the following shift of

(as a function of the spatial coordinate x):

E

O

)X

97 Consider an electron of the lower Dirac sea sitting at x I with energy E; if it succeeds in tunneling through the forbidden region between x I and x 2 it may occupy a level of the upper Dirac sea at x 2 and would then be accelerated by ~ in the negative x-direction.

When leaving Xl, however,

it leaves

behind a hole, which will be accelerated in the opposite direction. It is in this sense that e+/e - creation can be considered as a tunneling process. Note that if IE[ increases, Ix2-xll

the distance

decreases and hence the tunneling probability gets larger.

Up t o now we have always worked either with a pure electric or a pure magnetic field. For the sake of Completeness we finally also write down L~ I) for the case when electric and magnetic fields are present simultaneously:

(e,

~)

--

.....

t~Tr~" o

e

S "~

(5.44)

-

1

+ { e 's"

with F and 0

]

given above.

~or a derivation using the elegant proper-time method, is referred to Schwinger's paper

(I) , possible to write LR

[3]. We see that it is indeed

and hence W(1)j entirely in terms of

the gauge invariant Lorentz scalars discussion in section

F and G 2. According to our

(I), this gauge invariance of W (I) assures

the consistence of the generalized Maxwell equations Eq.

the reader

(1.6).

(5.44) could serve as a starting point for the actual

98

computation of the non-linear effects mentioned in section (I). This task is complicated by the fact that the integral

C5.44)

cannot be solved in closed form. For the application to the photon splitting process,

for instance, see Adler [2].

With these remarks we terminate our discussion of L~ I) and turn in the next section to the promised derivation of the explicit representation

(5.27).

Remark : Our result (5.29) is identical with Ritus'

As p r o o f ,

one u s e s

differentiates substituting andr'(2)

the

with the

one t h e n

functional

respect

easily gets

to

z and then

obtainable the

equation

desired

values relation

[4]

[56],

sets

z = -1.

[36]

for

between

After

~(2),

~(-1)

~' ( - 1 )

a n d ~' (2) .

(6) The Zeta Function In this section the concept of the ~-function regularization [12,15] will be applied to the one-loop effective Lagrangian of spinor QED; thereby we shall show that these methods produce

98

computation of the non-linear effects mentioned in section (I). This task is complicated by the fact that the integral

C5.44)

cannot be solved in closed form. For the application to the photon splitting process,

for instance, see Adler [2].

With these remarks we terminate our discussion of L~ I) and turn in the next section to the promised derivation of the explicit representation

(5.27).

Remark : Our result (5.29) is identical with Ritus'

As p r o o f ,

one u s e s

differentiates substituting andr'(2)

the

with the

one t h e n

functional

respect

easily gets

to

z and then

obtainable the

equation

desired

values relation

[4]

[56],

sets

z = -1.

[36]

for

between

After

~(2),

~(-1)

~' ( - 1 )

a n d ~' (2) .

(6) The Zeta Function In this section the concept of the ~-function regularization [12,15] will be applied to the one-loop effective Lagrangian of spinor QED; thereby we shall show that these methods produce

99

exactly eq.

(5.27), but

with much less calculatory effort

than by dimensional regularization. We begin directly at

(6.1) where, as will be seen, we can do without the additional normalization factor 6+[0]. This can also be written as

(6.2) so t h a t one now must c a l c u l a t e the determinant o f the propagator. The usual d e f i n i t i o n o f ' d e t A' as product o f the eigen-

value of an operator A is unfitting here, however,

since this

product diverges in our case. We will see that one can get the correct generalization of the determinant definition by considering the following: One considers an operator A with positive,

real, discrete eigen-

values {an} , i.e. Afn(X ) = anf(X ) is valid and one defines its associated Zeta function by

~'fl

:=

z

o-C.s

=

z

where n runs over all eigenvalues.

e-- ~

~

If one chooses for A the

Hamilton operator of the harmonic oscillator for example, one gets

(6.3)

then

(apart from the zero point energy) exactly the Riemann

Zeta function. By formal differentiation,

now follows:

100

~I(o) = -- ~ 2 ~ e ~

e -s2~

This suggests the definition

(6.4) which we s h a l l e x c l u s i v e l y be u s i n g in the f o l l o w i n g . The advantage o f t h i s method i s t h a t ~ ( 0 ) many o p e r a t o r s o f p h y s i c a l i n t e r e s t

is n o t s i n g u l a r f o r

.

Now we bring W[A] into a form which leads only to differential operators of the second order, allowing a simple calculation of the associated Zeta function later on; first, it follows from (6.1):

(6.5)

Now we use

_ ~ ~= 7ra_ ~e ¢r ~ T ; ~

= rr ~_ e ~ 6--3

and g e t I--~ l ~ C ~ , + ~ )

+ TY ~

(~--~) (6.6)

If we denote the trace in spinor (coordinate) space as try(tr x) and the i-th eigenvalue of the matrix A as EWi(A) , then

101

Z', .~.,,. [c,~-'-,-~T') Z,+

-

eB

~-'~]

=- ~",( ~*r ,C,,,- [(,,--~+~hz;,,- ~ ' a ~ ] 4-

~'-.- ,,-" eB 0 - ,.~.T+ ;~=,f

Substitution into

O

rr.z.+ e

1

,'="- z ' ~ ¢ r "¢ + ¢

(6.6) gives then

I~ 2.1,,.C'v-+~) + ~ .&,~(~,,,,.,-~) = .2.. ~rx

["£-,,,.(",'~'+nLeB) +-e",-('~"-z+~Tz+eB)']

or, with detyx -= det:

£,.. o~a. x C',-+~ + ~",,, ~e. f,,--,~,~ (6.7)

= :7 L~e,~ C,'~-,-,T~--e B) + .2 £.,,,de~ C',,,-z-,-~'÷ eB ) But the determinant of (m + ~) is a Lorentz scalar which means in particular that it does not depend on the sign of the ~, i.e. it is

From (6.7) it follows

(6.8)

-F

102 which, put into (6.5), gives:

~/c~ [~] =

-i [Z~ ~e~ (~+~'+ e ~)+ t ~ e ~

~*~-e~)]

(6.93

This equation is, however, not correct regarding the dimensions-, since in (6.1) we left out the G+[O] in the denominator, the right side of (6.8) now has the dimension while the left side in our units

(mass) 2,

(I~ = c = I) is dimensionless.

Thus, we introduce an arbitrary parameter p with the dimension of a mass

and replace

(6.9) by

(6.10)

With the determinant definition

(6.4) becomes

co)] (

with f~_

~

where we again substituted m 2 -is for m 2 Wick rotation by the substitution t ÷ T

(S) (6.12) Now we perform a = it and get for

(6.123

f~. (S)

"J~')~--Z('iw- + / r . -t- ¢ 1~)

"~

e ~ ) £s)

(6.13)

with the Euclidean momentum ~E" In order to calculate the zeta

103

function

(6.12) or (6.13) according

to(6.3) we need the spectrum

of the operator

=

*~

+(T ~ 9-

e

~)~

,.

_,

--

or i t s

--

e~)

(6

j_

14)

Euclidean analogue

9 ~-

H:,_=

-

_

+ 2 (p-e~)~ i s known from o n e - p a r t i c l e

The s p e c t r u m o f t h e o p e r a t o r quantum m e c h a n i c s ; t h e r e in a constant z-direction,

(6.15)

J,..

it

i s shown [44]

magnetic field

B

=

that

a particle

moving

B~ and h a v i n g no momentum i n

i s d e s c r i b e d by a H a m i l t o n o p e r a t o r

H-~c

,,

having the eigenvalues

Thus, t h e l a s t

t e r m o f t h e sum o f ( 6 . 1 5 )

(e~) I f we i m a g i n e f u r t h e r

C3~-~ ~ )

,

has t h e e i g e n v a l u e s

9"L E IN

t h a t we e n c l o s e

the field

(6.16) in a very large

normalization

volume ~ = L4 o f E u c l i d e a n s p a c e - t i m e , t h e n we 2 2 can a p p r o x i m a t e (-~ -~3) by p l a n e waves w i t h e i g e n v a l u e s (k o2+k3)2 , k o , k 3 ~ ~ ,

t h e n has

and d e n s i t y

(L/2~) 2. A l l t o g e t h e r

one

104

(6.17)

In continuum approximation, replaced by integrals, oo

the sum in (6.3) can be partially

leading to oo

]-s

z

(6.18)

Now we shall first evaluate the k-integrations;

to do so,

we use the formula [36]

~--"

C.f+ x:~.)

4

0

(6.19)

and evaluate

(6.18) for those values of s for which the inte-

grals exist.

(B designates the Beta function or the Euler

integral of the first kind) Thereafter,

~2 can be analytically

continued to a meromorphic function of the whole complex plane. First we consider two special cases of (6.19): co

=

furthermore,

C~) G-s

3

105

(6.21)

Applying (6.20) to (6.18) gives

f~(~)=]~ ~ ~

with

(6.21)

it follows

Z2 c ~ ' = / ' - ~

~o_ ~e'B

~¢~,s-t}

- ~~OIKs ~C ~ +

that

- ~ ( ~ ' ~ - ~ ) BC~,~-~) ~ (6.22)

If one uses the two functional

equations of t h e

Beta function

[361 4--.Z~

.F,c ,,. # ) -

.2_

:B ( ~

,~)

t h e n one g e t s ,

_

~4~-S"

-- ~-4

~- ~ ~ - 4 ) - ~ C~-~)~-~

(6.z3)

106

Substitution

=

into ~ (6.22) gives

~

e

~

~

('a.n-)3 S--I

Ca_.e~) ~-s

(6.24)

If one keeps in mind that the Riemann Zeta function in two arguments has the representation

[36]

(6.25) then it is clear that for the second sum one gets

and for the first

=

z

~

~)~-~

I=4

Our end result for ~2 thus reads

where we set ~Z

C~;= 2.e~ The derivative

of this equation is

(6.27)

107

_ (s_~) -~ 4~,~ ('2e B) CSeB) + C~- "0 - ' ~ ar ( ~ - - 0

¢~- ~ s-~'<-' Ez _f"a--,<< tj- t "-' ] .z,,,
or, for s -- O: I

To calculate Zeta function

¢(-I,q) we need the following property

of the

[36] I

C%* ~) £,~ ÷.L) Here,

Bn d e n o t e s

the

Bernoulli

~IN

polynomials.

(6.29)

In particular

so that one gets

(6.30)

108

From (6.28) together with (6.27) it thus follows

t(.

ff~o)

=

--s"c

-

(eB) ~ 4,.~r~ { ~-" *''~ "~"

~.e~

-

)-

~z -

~z

-

(6.31)

From this, according to (6.11) one can calculate via Wt~EWI

=

~f~Co)

the effective action and with

-

Z

the effective Lagrangian

(6.32) =,,

~.s

+

~

"-

~

/A.~

~ j-tC_ ~

)

This expression for L (1) still depends on the arbitrary parameter ~, which can be fixed by additional conditions for L. In the case of massive QED studied here, ~=m can be chosen as reference mass. With this choice (6.32) can be put except for a constant, in the form

109

".5,2tr *

(6.33)

-%,~-

4g&8) z f~(-4, "**~eB) t

which can be proven by simply multiplying out eq. Apparently

(6.33).

(6.33) agrees exactly with the form obtained by

dimensional regularization

(5.27). In particular this means

that the result obtained by Zeta function regularization is finite without additional subtraction of divergent counterterms (as in (5.17) and (5.19)

for example) and that it vanishes

for B = O. Thus (6.4) appears as a useful generalization of the concept of determinants for operators like G+. A further advantage of this method is that one can very easily specialize the above calculation to the case of vanishing Fermion mass

(m=O), while this is impossible in the integral represen-

tation (5.25) for L (I) without making the integral divergent on the upper limit. For this special case, one reads from (6.24)

C ~ ) % ~-~

~o

~=o

(6.34)

4 C ~ ) ~ ~-4 Here,

in the

last

line,

the representation

of the normal Riemann Zeta function was used. Differentiation of (6.34) gives

110

(

e~,

C~.)~_

[-

¢:~e%)~-s f C;-.~)/~z s

C~ - ~ ) - ~

c-~-',)-~ c _ ~ " - ~ f~c~-,)/~

+

~

+ ¢~_~,-'¢~_~_~f-~ ~c~-~) ~,/,,.z~ ~ ] Now we set s -- 0 again:

Yi~°'=

Since

~

~--% E- ~ )

for the Riemann

is valid

~-~)+~:~°~) ~ - ~ , ~ , c ~ s )

Zeta function

(z e ~ arbitrary),

in one or two arguments

then

and thus

J"~ ¢o) = -.m.ST__ Since

= --__r~ the one-loop

effective

is then given by

f~ (o)

Lagrangian

for QED with massless

Fermions

111

',,,=o _

If we would replace minator

1 + "1,2...,,,~I([_ 4)]

3.~.rrz ~,'~

~

(6.3s)

3_~,

the undetermined

parameter

p in the deno-

of the logarithm by the electron mass m, then this

would be precisely of the massive

the expression

(5.29)

theory in the limiting

But since there exists no natural with m = 0 with respect strengths,

for the Lagrangian

case of strong

fields!

mass scale in the theory

to which one can measure

the field

it is not possible with the help of the renormali-

zation conditions Nevertheless,

applied to i [28] to eliminate

the physical

content

of

(6.35)

~ from i~l~.

can not depend

on p, i.e.

d must be valid;

Bcr)) =

here we have indicated with the arguments

i also has an implicit

dependence

the choice of a particular a special

(6.36)

o

renormalized

value

on p via e and B, since with for ~, one also decides

coupling constant

with

the physical

coupling

on

or field strength

which as we have seen in the case of the massive coincides

that

constants

theory,

exactly

or field strengths

for p=m [31,32].

With

(6.36) we are led to the so-called

equation

renormalization

group

112

This partial differential equation tells us that an infinitesimal change in ~ can be compensated by a corresponding change in a and a rescaling of B. The above equation can be further simplified by utilizing the relationship between the renormalized quantities e, B and the bare eo, Bo:

(6.s8)

Here we have used the same notation as in section

(5), without,

however, having had to explicitly perform the given infinite scale transformation for charge and field strength in the framework of the Zeta function regularization,

Furthermore,

a = Z 3 ~o

and eB = e ° Bo, so that for the derivatives required in (6.37) we get

and

-

@

Here we used the fact that the bare quantities e o and B ° are naturally independent of the parameter 1.1 which is only introduced during the regularization process. Now we define

113

(6.39) and g e t

which, together with (6.37) gives

(6.40)

0 Note that this equation contains only renormalized quantities. By now substituting the known one-loop approximation for L, we can directly calculate the function 8¢. From

mgE~ with

U

= - i + 4a~rc-~)

it follows

~'rr +

~j: Co,c_)

114

I

--0 The validity

(S We shall

of

(6.36)

.~ ca,)

-

implies

z

oc

m

shown a g a i n t h a t with relatively m eth o d s

be t a k e n i n t o

temperature

the Zeta function little

[19].

difficulty

The e f f e c t s

account

result i n

section

the one-loop effective

the case of finite

other

(6.41)

return to this important

Now we want t o c o n s i d e r for

that

[15-17]

[18]

Lagrangian

as w e l l ;

regularization the

results

of finite

(8).

it

will

reproduces,

a c h i e v e d by

temperature

by p e r f o r m i n g

be

can

the substitution

in (6.18)

__~C~K° Here,

8 = I/kT

a periodicity thermal

(6.42) (k = Boltzmann's interval

Zeta function

~=~

constant,

in the Euclidean

T = temperature)

time T = it. Then the

is

e=o

+~+

~o

C~>

~I~3

is

~ ~ ÷ ~ B 7 -~ I ~

115

with a three-dimensional normalization volume ~3" From this one could caluclate with

~("~ (:B,T)

-~

--

._sC" .F't l

(o)

the Lagrangian of the theory with massive fermions at finite temperature and B ~ O. Since the integral and the sum in the above formula cannot be represented in closed form, we limit ourselves in the following to a massless QED (m=O) and vanishing external field (B=O). To get the relevant Zeta function for this, we go back to the equation (6.15), which now reduces to

2 2 ME = P E '

where

~ has

M

the

spectrum

Because of (6.42), then

--S

Introduction of polar coordinates gives

#~

= 2 p _ _ ~ ~: ~,~ •e - : O

~,~

~ ~ [~,.,,i~+ 0

[ ~c~+÷JI

k~?-~

~ ~q-~

The integration can be performed for suitable values of s with (6.19)

With

(6.25) and the identity

116

P~ Fky)

Cx, y) =

(6.43)

P(x+y)

if follows

Ca-Tr)~ Since

F has a pole

As can be seen

Furthermore,

~ ~

F' c ~

for s = 0,

from the e x p a n s i o n

one needs

and 4

Substitution

into

(6.44)

gives

(6.4s) Now we again use

(6.29): 4

!

117 By means of the relation [36]

this can be transformed into a Bernoulli polynomial of the 4th order

fc-a,"

I

"

"--

I

q-

4

:t-

i'

~'- -'iS-

4

9r=- ~

It follows for (6.15)

So we get the expression for the Lagrangian

(6.46)

3~0 b

360

7r~ k~ T ~

which, however, does not yet represent the correct final result. Since from the thermodynamic viewpoint, we are dealing with both an electron and positron gas, each with two spin projection possibilities, an additional factor 2 must be introduced into

(6.46):

~

Tr~k~T-~

This is precisely the result given in [19] for the Fermi-Dirac case (i.e., spinor QED). It is interesting that this expression

118

can be written as

-,,,,,=

B~'

o

o

with the right Fermi distribution. the integral

(By way of proof, we use

[36]

y X~-'d~ = C~-.2_+-~)

o ~F"<+,f

C--~TF]

for n = 2 and B 4 = -I/30.)

In order to have statistics,

also a system that satisfies B o s e - E i n s t e i n

we calculate

in Appendix B the one-loop effective

Lagrangian of scalar QED; without

taking temperature

then we get for the Zeta function effects

into account

From this we get the c o r r e s p o n d i n g equation for T > 0 by the substitution oo

[15-17]

(6. so)

C-Z,r) Therefore

the corresponding Zeta function is

=7" ~T_.Z" which again cannot be calculated in closed form for arbitrary values of the mass and field strength. selves

As before, we limit our-

therefore to m = 0 and B = 0

Now we again introduce polar coordinates

119 oo ----

3

e'---~o

Since the integrand is an even function of ~, we obtain ot~ oo

~ c~ - ,ff~ ~c~,,)~ ~7-,~..~~

02~ ~' [ ~.~.~+ K'] -~ (6.53)

o

The integral of the sum for I = O seems now, positive values of s, to diverge the volume ~3' which includes

for large enough,

for k = O. If we assume

all fields

is very large but

finite, then the k - i n t e g r a t i o n has an infrared very small value 3-2s

e. The integral

or in t h e a n ~ y t i c a l

vanishes

cut-off at a

is then p r o p o r t i o n a l

to

3 continuation s ÷ O, to ~ . But this

in the limiting case e +

volume becoming arbi~arilylarge

O, i.e.,

in a n o r m a l i z a t i o n

[15]. Thus we only take the

first terms into account in (6.53),

and calculate

it just as

we do the corresponding one for ~

4~)~ ~ ~ Taking the derivative

that

gives

which with the above calculated values and ~(-3)

1/120

120 (from ~(l-2m) = -B2m/2m , m=2, B 4 = -I/30)

jc~(~

leads to

~-% --~

From

.

#I

:

(compare Appendix),

co,

for massless,

scalar QED with vanishing

external field and finite temperature one then gets:

~,~), ,(~) - : o C~

: o , 7-) -

~6

I#

7T~k ~ T ~

(6.54)

which again coincides with the result in [19], achieved in another manner.

In order to re-write

(6.54), we use the integral

[36]

0

for ~ = 1 and ~ = 4; with r(4) = 6 and ~(4) -- 74/90

(for example

from [36 ] )

jc£r~) =

I~...I

.2

C P--~) Thus the Lagrangian,

{

i.e. the negative

free energy per unit

volume is

(6.ss) with the correct Bose-Einstein distribution function. We have now shown that, also when taking temperature effects into account,

the Zeta function regularization represents an

efficient method for evaluating determinants

as they occur in

121

calculating one-loop vacuum effects.

Its particular advantage

lies in the fact that, aside from calculatory simplification in comparison

to known Green's functions methods,

infinite counter

terms must never be resorted to, as was the case for example, in our treatment in the last section. To conclude, we should like to mention another formal aspect related to eq.

(6.$2)

(or its analogue in spinor QED).

In

switching to a finite temperature, we have replaced the variable k ° which can take on arbitrary real values by the expression 2__~ 8 £ ' which takes on only discrete values for fixed 8

So we

have made the time component of the momentum four vector over which

we

integrate,

discrete and thus introduced a new para-

meter into the theory in the form of 8. One can now ask whether an analogous substitution in a space component of the momentum vector also takes on a physical meaning.

In Appendix

C, we shall show that this is, in fact, so if one considers the Casimir effect

[20-23] in the light of path integral quan-

tization followed by Zeta function regularization.

Thereby,

the distanc~ a between the two plates introduced into the vacuum would play an analogous role to the parameter

8.

(7) Two-Loop Effective Lagrangian Thanks to the preparations

in sections

(3) and (4), we can now

quite simply derive a compact expression for the two-loop effective Lagrangian L (2) of spinor QED. The complete generating functional for the Green's functions of this theory [41] serves as starting point here, which,

in the presence of an external

121

calculating one-loop vacuum effects.

Its particular advantage

lies in the fact that, aside from calculatory simplification in comparison

to known Green's functions methods,

infinite counter

terms must never be resorted to, as was the case for example, in our treatment in the last section. To conclude, we should like to mention another formal aspect related to eq.

(6.$2)

(or its analogue in spinor QED).

In

switching to a finite temperature, we have replaced the variable k ° which can take on arbitrary real values by the expression 2__~ 8 £ ' which takes on only discrete values for fixed 8

So we

have made the time component of the momentum four vector over which

we

integrate,

discrete and thus introduced a new para-

meter into the theory in the form of 8. One can now ask whether an analogous substitution in a space component of the momentum vector also takes on a physical meaning.

In Appendix

C, we shall show that this is, in fact, so if one considers the Casimir effect

[20-23] in the light of path integral quan-

tization followed by Zeta function regularization.

Thereby,

the distanc~ a between the two plates introduced into the vacuum would play an analogous role to the parameter

8.

(7) Two-Loop Effective Lagrangian Thanks to the preparations

in sections

(3) and (4), we can now

quite simply derive a compact expression for the two-loop effective Lagrangian L (2) of spinor QED. The complete generating functional for the Green's functions of this theory [41] serves as starting point here, which,

in the presence of an external

122 field described by the potential A ~ can be written as

7_ [j 7, 7]

=

. e;

One o b t a i n s tiation

arbitrary

with

n-point

respect

mann)

c-number

equal

to

zero

to

sources

the

functions

from this

by differen-

commuting or anti-commuting

j and n(n)

thereafter.

iw[]+n]

and putting

The n o r m a l i z a t i o n

the

(Grag-

sources

constant

Nv , w h i c h

can be identified with the vacuum amplitude <0+

lo_>j'n'5=°-

is to be chosen in such a manner that Z[O,O,O]

= I. Thus it

follows field

from

(7.1) for the vacuum amplitude

(see also our discussion

N v --- < o + l oR - ~ This

=

~

formula contains

order

(I)):

(7. 2)

C

radiative

~=0_

corrections

to Nv o f a r b i t r a r y

which one can calculate perturbatively, i . e , by expanding the l e f t ex-

ponential series

in section

in the external

function.

into

consider

If

account,

one o n l y takes

one gets p r e c i s e l y

the quantum f l u c t u a t i o n s

perturbations

the O. term o f the

the quadratic term

back!

Since we

r e p r e s e n t e d by J as s m a l l

compared t o the c l a s s i c a l

now expand W[A+J]

(5.1)

around A and t e r m i n a t e

background f i e l d

A, we

the T a y l o r s e r i e s

after

123

According to Appendix D, the required functional derivatives are

In the framework of this approximation, we get for the vacuum amplitude:

With the aid o f

the identity[~1]

o~r[-- c ~P[

~ ~ ~ g3 + ~ J' ~ c ~ - ~ - ~ ¢ + ~ ~.~.,, c~-g~; -~ ~" ~- .I~ C~.~- I ~ ) --'t

this simplifies to

i~/c~lEel

We only want to study the simplest correction beyond W (I) and therefore take only into account the first terms of the logarithmic

124

series of the second exponential function, which leads to a quadratic term in e; then the exponent <j>D+HD+<j> proportional to e 4 does not contribute anything to this approximation. The re fore

,

Z

] (7.5)

e z

The first term added to Feynman

L 11)""

can be characterized by the

diagram

(7.6)

which suggests that the interaction of the electron with the external field is taken into account to all orders, while the interaction with the photon field is only considered up to the order e 2. Analogously,

the graph

(7.7)

represents the third part of (7.5) which, however, as we now wish to show, contributes

nothing for constant external fields.

First, it is clear that for B (x) = const also the diagonal matrix element of the propagator

(cf. (2.47)) and with it <j~> is

independent of the space-time point x. On the other hand, <jA> is a four-vector field which, under

an

125

arbitrary

Lorentz

tranformation

C£~) t r a n s f o r m s

according

to

(7.8) If the x-dependence

vanishes,

then

which, however, implies <.A J > = 0 according to (7.8) From ( 7 . 5 ) , tion

then

follows

the e x p r e s s i o n

for

the

qed.

first

correc-

of the Lagrangian going beyond L ( I )

which is in accord with the 2-1oop diagram

(7.6). In particu-

lar, in the Feynman gauge with

this means c~)

e z

Transformation

to the momentum representation by

gives, due to ~(x,x')

~(x',x)

= I, the gauge invariant expression

-

.

~'~' ._~+ ( x - ~')

cJcP-% )

126 N o w we

introduce

integral

and use

~a~' ~ + c~-~') ~ -

the

substitution

D+(x'-x)

~ = x' - x into

~'CP-~ - la~f ~+(f) e- ~fcr-~

~.CZ)

results

e-"~c~-~

e

in

et

• 3>+

So L ( z )

no

(p-~-) e-'~×cP-9 )

longer

is d e p e n d e n t

invariant~

as was

to be e x p e o t e d

background

field.

If we n o w

integration

which

x'-

= D+(x.x')

= ~+(p-~) Substitution

the

variables,

then

is in c o m p l i a n c e

with

on x and

is thus

because

set k

: = p-q

the m o m e n t u m

of the

translational assumed

and choose

space

p and k as

Feynman

p-k

P If we

recall

then we

can

the

definition

also w r i t e

of the p o l a r i z a t i o n

(7.10)

in the

form

constant

tensor

diagram

127

~'ol"~ ..~..,.(k){_,:e, ~'c.E~+ {rE~tgcp)~?~cp-k)]} = L ~-__Ek We found in section

~

I-F ('J > CK)

(7.11)

(4) for the unrenormalized polarization

tensor

(7.12)

] with

•= " ~

+ T--C"i-

kt

+

2. ~ ~-..,.:.,.,_i~

K~

The counter terms c.t. were left out because we have to renormalize L (2) again anyway. = g~ll~t

4~

It follows from (7.12), because gp = 4 and

= 2 for the required trace: 4

t~ ?Ck) =

~ ~ 0

~ --I

=e4.

Furthermore,

~o

~.

it is useful to set Z

= ~+o~,

~, ÷ ~ v,~_

(7.13)

with

c t - = ~ (4-v') (7.14)

128 If one chooses propagator

the proper

time representation

for the photon

in (7.11)

4 K z- i~

--

then it follows

i

l'~ste o

z's'Cl< L ;~:)

for the Lagrangian )

(7.15) • , ~ , -~'sC c~~,2+ ~ ) ~sCk:-,r) 0

Now the k-integration

~d~K

can be easily performed with

-,~'k~ -,'~ C ~

C.O~.)~ @

~

=Z

a%

d

(3.19):

+b~) k~ll

If we set A I: = s' + as and ~;: = s' + bs,

then

o~

_

~

d

-t

~) otK °

~ fl, C~oj

.( -.,,~a ~ e -

-i

z" gz CK~) 2 )

C/eKe) z

z

/

(7.16) 4

4

C¢~r)~ C~'+e,~)z C~% b ~ ) Analogously,

we c a l c u l a t e

(7.17) ~(__~_~)~

" t Z

C Substitution

,

7.

7..

I<.~gives us for I

q

"1

129 oo

5N0 + Nz l (s~÷s~) (~,+ sb)z

4-

The integration over s' is still elementary and gives

1

[36]

d

[C% No- N.,) '~

=--

e~

Cb -~)

+

4

"~

Z,',,t b

From (7.15) we then get for L (2) (7.18)

with 4

bt

"

b(~,-b) -4- Cb._.~) ~

= where

.-/

F-.. = C%tqo-N..,.) cxCb-~) + ( 3 No-+

~
(7.19)

b K.,+ k~ .Z-'~ ~

4

(b-oO ~

Ckt+Nz)

From (4.~8c), we first get

~4~) b Co.-b)

130 Further, the definitions (7.14) give

Cb-o.)Z

F"&( cos"~v-co~ e ) - ( ' ~ - v ' ) ~ ~.z~ ~qz

o,. C b _c~)

4

-- ~ ~

b CoL - - b )

= ce.~ ~ v - ~ o s

Together w i t h

K~

-

:~

÷

~1

~'~ ~

~C~v-c~)-~.-~

the above e q u a t i o n s ,

it

(~4-v ~) C c o ~ v - c o ~

~)

~ ~z,,,,.¢]

follows

from

(7.20):

4- .2 C~s' ~ v - e o . r . ) - C 4 - v ~') :~ ~'.'~ ~ (7.21)

Cco~ ~ v--cos ~) r% We can then write for the unrenormalized Lagrangian 4 -- ~ , ~ S

z where the s-integration for s + O apparently diverges. order to regularize so > O

(7.22), we introduce a finite lower limit

for the proper time integrals of the electron propagators

contained in the polarization tensor (4. 8)

In

(4.5).

must be replaced by

2

As a result,

131 which,

for the Lagrangian,

leads

to

~.

(~')----"~ ~.so s

=

kce, v)

or -~-~o 4-- S

oK)

-- C ~ ) ~

I

~o

_ £.~,L~S

olv e

o

Here we used the fact that K is a symmetrical v (compare

(7.21)).

we first subtract K(z,v);

thereby,

To now get a convergent

function

integral

the term constant with respect

L(21B=O) = O. In the next

step

for s + O ,

and now

, we also take

term in z of K out of the integral.

also even in z, the series expansion

in

to z = eBs from

L (2) is changed only by a constant

has the property the quadratic

(7.24)

~.c~,v)

Since

of the remaining

K is

integrands

thus begins with a term proportional vanishes series

for s + O, i.e. leads

expansion

to ~3 z4 = (eB)4s which s to a convergent s-integral. The

of K is performed

explicitly

in Appendix

E and

gives

Ix, c~, vJ =

Koz C~ v) + O ( { ")

NozC~,v)

1_vZ

with

=

Now we can write

(7.24)

(7.25) as

4-a~ls

-C Tr

a

4 (~.~.~

ol v

~ .aso

o

e-

(7.26)

132 Analogously to L (I), we shall later get rid of the divergent integral over the separated quadratic term by means of a charge and field strength renormalization;

furthermore,

a mass renorma-

lization will prove necessary. This means that we have until now not calculated with the physical parameters e, B, m, but with bare quantities eo, Bo, m o. We therefore write

(7.26)

from now on in the form 4-~solS

*Q.So

0

with

~-~ls

and z = % B o S .

If

one c o n s i d e r s

_~ (7.21),

it

becomes e v i d e n t

that

the v-integration in (7.27) diverges for s o = 0 at the upper limit

(i.e., for v ÷ 1 ) .

If we calculate the Laurent series of

K = K(v2), we find a non-vanishing singular part

(compare

Appendix E) :

So

¢ ~/--Vz with

--

+ ~cof e--9_~

(7.29)

Now we regularize the v-integration by also taking the term f(z)/(1-v 2) out of the integral

133

4- a.~/s

-- ":VCSo)

4- C4,rr)~

~,s

o

-,

4- v

2

(7.30)

4

In the third term s o could be set equal to zero, since now both integrations converge

(i)

For s ÷ o, (K-Ko2) as well as f vanish proportionally to s

(ii)

4

For v ÷ I, (K-Ko2) is exactly compensated for by f(z) /

(I-v2).

Now we turn our attention to the double integral over the singular part of the Laurent series, where the v-integration can easily be performed

4 - - V ~0

4--~,~/~

c~s. jr (eo~oS) e

,~ 0

Now

•,t- e . % / s

o

4

V-- 4

2

v+,, I[o

4

---Z ~-~o

Z

"9---~o

_ ,f--.3_~o/S

c::4vV z-

4--

134 so

that it follows

a'r'+'r~~

~-~

lEe.15°s ) Z

•4 " &E~,r.), ~ (7.51)

=In the the

Z-., ÷

second

integrand

integral

12,

vanishes

like

z~O) To r e a r r a n g e

and thus

s o can be s e t slns

for

equal

s÷o.

to

(Note

zero because f(z)~z 4 for

1 t we w r i t e

get oo

~ ~-O~o

- - ~. , v ~ z ~ 125o

This

integral

integration

converges

for

s o = 0 and y i e l d s

in which the boundary

terms

vanish,

e~

~.Se

oo

q.~

-t ~"~e~ ,c,°

"

after

a partial

135

According t o

(5.23) oo

£ ( ~ o , eo ~o) and a c c o r d i n g

to

~

(3.48a)

~ it

holds

for

t h e mass d i s p l a c e m e n t

that

where s o has the same significance as in (7.23), i.e. represents the lower limit of the electron propagator proper-time tion. Hence we obtain for 11

C,'~o~ eo~o)

~o~o'v,.,. 9

e,.)

For (7.31) one can then write

G

r-+trF

'~ ,~40

integra-

136

with

z = eoBo s.

Because =

it._~'~

of

(7.30), we have

"~'~o~+

For the whole

OLo ~rs.~ +F~-~"

two-loop

order in the coupling ation field)

effective constant

Lagrangian,

~owe

o4s

"J

or,,

(i.e. calculated

~ between

one gets,

+

--"~:~,-LKca,v)_ K,,c~, ,,~_.,-,,, ~,~_,3-

+

the fermion and radi-

keeping

(5.19)

. . . . .

S~

~o~

av

as~ $

e

e

in mind

0 ( < ~)

_c~ (-ll

-~ ce°~,o s)

to the first

+

~

C~o, eo ~o)

2(eo~o~) ~+ o~(~°.~.,~o)[ <~ c~.,e.~.)

o

oo

- ~'

~_~_.~ - i ~ , %

~z 4

0

o

If we use c_~

it follows

that

~,~

~

,~- ~ , 1 ~

137

_

_

e

oo -

-

4

S----a-

-

+

v

Kc~:,v)-- Ko~(:z-,v)--- 4_vZ

e

+0(e.'J

°

with

z = eoBoS.

(i)

Now we p e r f o r m

The r e n o r m a l i z e d

the f o l l o w i n g

(physical)

mass

renormalization:

is defined

by

9~L=...W.Zo-F(~,t~ (ii)

The r e n o r m a l i z e d duced

by

e

charge :

eo

(7.33)

and field

~J

allows

is intro-

"~" _

wich

strength

(7.34)

~

for eB = eoB o.

He re

=

7G-~, & &-- e

cgw)~

FOr

s0

+ o ) the e x p o n e n t i a l

.,('&S,,ls s e__

~

integrals

reduce

O

to

dv

+ 0(~.~>

a logarithm

[36] :

-~

Ko

4

~.( , ~ z So

4 W -z

2

(7.3s)

138

With

(7.33) and (7.34) it follows from (7.32) cn

Co~

C../P(-0

with

.C~)

(7.36)

+ O c t _ ~)

4

"1

-

,)-~

-l-v, ]

o~

~-,~

c- i ~

Pots - ~ ' , ~ ~ e ~ c o ~ ce~s) ÷ ½ ce~s) ~( 7 . 3 7)

- 1]

and z = eBs. Here, the derivative of L~ I) was explicitly inserted. Furthermore,

it should be noted that because of 6m~s

for each function g~ o, sg(mo) = sg(m) + O(s 2) is valid and that trivially s ° = s + O(~2); this means that one may replace the factors s ° before the integrals for L~ 2) by s and the mass m ° by m. With

(7.37) then, we have been successful in deriving a re-

latively lucid and simple expression for the effective Lagrangian to the first order in the coupling constant s between electron and photon field (i.e., according to (7.6) in twoloop approximation).This

expression is equivalent

to Ritus'

[4], but was arrived at in a completely different manner. In order to further evaluate

(7.37), we limit ourselves

limiting case of strong fields, i.e. eB/m2>>1.

to the

Since in all

the above formulae, m 2 is to be understood as an abbreviation for m 2 -i~ , we now rotate the integration paths for the electron

proper-times

s I and s 2 by the substitution s1,2÷-is1,2 ,

13g

to (4. 6)

which according

leads to s + - i s

and v ÷ v ; a c c o r d i n g l y

we must replace

> -~ z~J~ ~" c o . ~ ~.

Co~ and get

F -2- ~'.,.:-,d,~ ~- [o~1~ ~ v - w , , ~ ÷. ~ z ~

--'f~~

KC (-£~' V)

( c~I~ ~ . ~ . , , , . ~ ~ -

-F

~- )

2.(.co~h ~v-co~k~.) +C~-v~.)e ~-.,,:,.,d,,

-,~6~ ~ [:z(:~.,,~,~v-c~'k,-)+(~-v~)c~l~

[,<,a_(-~"~, v.)

(.~)c-~,v) =

~.

:]

~.~,,,,.kz~]

(o,,:h ~ v - co~h~-) ( ~ - v ~) ~- s ~

-2

Z~ It

÷v)

( 4 - v ~) [c,,=~, i ~ v - c.~=l, -~ )

(7.38)

2~ z

follows, for the Lagrangian

£ 0

0

¢x:)

"b

(7.39)

+ ~ c o ~ ~-- ~__~

140 We

can also use z = eBs instead of s as integration

O

~

~,

dve

c-~,v)

variable

~/_v~ -I-- 2

R

j-~

e~m L:~C_~6~_%~:z_ d ' "-, 4 "~

0

'5~..

z

c4~: --&-E ''~

,,,,.z

~z

I f we are i n t e r e s t e d i n an e v a l u a t i o n o f t h i s s t i l l

exact

equation only to the order B21n(B) then only the two underl i n e d terms A and C make a c o n t r i b u t i o n since the e x p o n e n t i a l with eB <<m2 o f the integrand i s small,

and thus i t s behaviour f o r

large values o f z becomes important. integration limit

Zo>O and consider at the end the limes

Zo÷O then we get, f o r the f i r s t oo C~-,r)"~

-

2~

I f we introduce a lower

o

Ce ,)

contribution:

~o ~

EzC-

~

.~o) -(-

O C,~ z)

(7.41)

- - .~ 2 .n.-s

The second contribution

is

0

Since both integrals

converge

even without

the exponential

factor,

141

we c a n

ignore

these

in

the

case

of

strong

fields

and

then

have

+ 0 c ~ z) whe r e -*" -2- co-~-h ~ ~

q

+

I 1

co-L-~ ~-

7_

}

0

Partial

integration in the second term gives

a n d thus eo

~o

~-o : ~

4 + 0(~o) 3 which finally leads to

-

~

Ce~) 2 £,n

oc

eS_

( e &)~ .£,.~ e.~

= 2_ %,2.rrz

+ 0 (~z)

(7.42)

,~.=

This all results in the following asymptotic form of the twoloop effective Lagrangian

~. c & )

~

,q+6"

d.. z,,~ z ~'.rr z

e...%

(7.43)

142

This result is identical to the one Ritus [4] gets from his calculation. It is interesting that (7.43) has the same dependence on B as the asymptotic form (5.29) of L~ I) for large values of B; retaining only terms of order B 2 in(B), the ratio B-independent

is

for eB >>m2:

%

~c~

L~2)/L~I)

~

O~

(7.44)

As was to be expected, the two-loop effective Lagrangian, which contains an additional factor of ~ due to the photon coupling, is about two orders of magnitude smaller than the Euler-Heisenberg (one-loop) Lagrangian L~ I). So, bearing in mind the smallness of the effects associated with L~ I)- , it appears extremely unlikely that effects due to radiative corrections of the oneloop calculations can be detected experimentally in the near future; whether there are astrophysical objects with magnetic fields large enough for radiative corrections to be measurable, further development must show. Thus, the refinement of one-loop calculations being of minor importance,

the significance of

L~ 2) is of a more conceptional nature. Another point should be mentioned in this context.

If one wants

to do a realistic calculation of A including corrections due to the fluctuations of the photon field, one should also take into account particles other than electrons and positrons;

admit-

tedly, they are the lightest charged particles known and hence can be created from the vacuum with the least effort, but for

143

muons, say, the creation rate calculated from

LR(1) for

a given

value of E is of about the same order of magnitude as the radiatively corrected creation rate for electrons.

(It is only for

very strong fields that the mass of the fermion becomes unimportant, cf. eq.

(5.41)). It thus appears that in principle

one should include all the other charged particles occuring in nature into one's calculation. Neglecting a particle of mass M is a valid approximation only as long as eE >>M 2. The inclusion of other leptons would be quite straightforward; however, owing to the strong interactions, the analogous computations for hadrons would be a formidable task. Now,

instead

Of

dwelling further upon the explicit form (7.37),

let us discuss some of the salient features of our two-loop calculation. First of all, because QED is a renormalizable theory, it is clear from the very beginning that by renormalizing B, e and the fermion mass, it must be possible to get a finite effective action W (2) or Lagrangian L (2). What is less clear is what a mass renormalization looks like at the level of an effective Lagrangian. Diverging terms arising in the calculation of L (2) with a field dependence proportional to B 2 simply can be absorbed in L (O) to produce a renormalized classical Lagrangian L~O); this was already done when calculating L (I). However, already in eq.

(7.30) it turns out that there are diverging

terms with a complicated field dependence

(through the function

f(z) given by (7.29)) which can not be incorporated in the classical Maxwell Lagrangian. Because the only further parameter available for a redefinition is m, it should be possible to separate off a factor of 6m(So) from these terms. As we saw,

144

this is indeed the case and now the power of renormalizability forces the coefficient of 6m(So) to be exactly SL(1)/Sm (see the equations prior to (7.33)). Because this is the only funtional form which allows us to absorb the divergent terms into a structure already present in Leff' namely i(1)(mo ). We think that this is an impressive example of how in a non-trivial case the fact that a theory is renormalizable constrains the appearance of divergencies. Another interesting point is that it was also possible to construct LR(2) in a gauge invariant way. The explicit dependence on the vector potential A

vanishes in our calculation at the point

where the Fourier transform of G+(x)x' I A), which contains the gauge dependent factor ~(x,x')

(cf. eq.

(2.47)), is inserted in

(7.9). Because of ~(x,x') ~(x',x) = I, which immediately follows from (2.16), the only remaining field dependence of L (2) resides in the trivially gauge invariant factor z ~ eBs. Strictly speaking, this only means that the unrenormalized

L (2) is gauge in-

variant; our calculation shows, however, that renormalization can be done in a way which conserves this property. According to section (I), this implies that the renormalized expectation value of the induced vacuum current is still conserved and that the generalized Maxwell equations

(1.6) are consistent.

(This remark

might appear trivial for the case of QED; nevertheless, spinor field theories

there are

(containing Y5-couplings) where there is

already at the one loop level no regularizing scheme which preserves the conservation property of <j~>, i.e., the gauge invariance of Wef f. These phenomena are known as chiral anomalies. For an introduction within the above context, see Jackiw [56].)

145

It is necessary to emphasize that the relatively compact representation (7.37) for LR(2) (the corresponding expressions in [4] look much more complicated) could only be obtained because of the decomposition (1.57) of the two-loop diagram together with Tsai's representation for the polarization tensor. One also could think of decomposing the two-loop diagram into an electron mass operator calculated to first order in ~, as computed in section (3), and an electron propagator, both to all orders in the coupling to the external field, of course. Symbolically:

It turns out, however, that this choice would be much less favourable than our decomposition (1.57); L (2) expressed in this way would contain a two-fold parameter integral coming from and one further coming from the electron propagator. None of these can be done in an easy analytical way, whereas in our approach, one of the three (the s'-integration prior to eq. (7.18)) could be calculated in terms of elementary functions. Asa by-product of the renormalization of i (2), we got an explicit expression for the renormalization constant Z 3 as a function of the proper time cut-off So(See (7.35)). In the next section,

146

this will be used to calculate the QED 8-function up to order 2

. This is usually done using the photon polarization tensor;

however, as will be seen, there is a close analogy between electrodynamics at short distances and electrodynamics with strong fields [4]. Thus it is not surprising at all that ief f contains the same information as the polarization function of the corresponding order.

In concluding let us recall that the calculations in this section can also be interpreted in the framework of Schwinger's Source Theory; here, the determination of suitable contact terms takes the place of the renormalization. We shall go into the details of these methods in Appendix F.

147

(8) Renormalization Group Equations In this section we shall further improve our preceding oneand two-loop calculations for the effective Lagrangian of QED with the aid of a perturbation series for the renormalization group equation in the asymptotic region, i.e., for eB > > m 2 [4,31]. Here it will be shown, for example, that in this field strength region, pair production propability undergoes no radiative corrections in a pure electric field in one-loop approximation. Just as this also applies for the npoint functions of the theory, so there are also various renormalization group equations for L for various renormalization schemes [32]: a Callan-Symanzik equation for 'On-Shell' renormalization, a 't Hooft-Weinberg equation for dimensional regularization, etc. For the actual perturbation calculation, we shall use here the Callan-Symanzik equation, since we have performed all preceding renormalizations according to the mass shell scheme. First, however, we shall describe the relation between both types of equations and demonstrate in particular that 2 the corresponding g-functions coincide to the order m .

It was shown by 't Hooft with the help of dimensional regularization [12,13] that for every renormalizable field theory, a renormalization scheme with the following properties exists: (i)

The bare quantities

(besides the mass) can be chosen in-

dependent of the renormalized mass.

148

(ii)

The bare mass

is proportional

('multiplicative

This method (MS).

mensional

n in space,

it is important

in order to assure

(ii),

that,

arbitrary

parameters

to the following

and m ° as a function

subtraction of di-

mass parameter

for an arbitrary

dimension

of the theory have

as the bare ones for n=4. leads

by minimal

that in the framework

an initially

the renormalized

same dimension (i) and

as renormalization

regularization,

is introduced

mass.

mass renormalization').

is described

Furthermore,

to the renormalized

This

Laurent

of the renormalized

the

together with series

quantities

for

o

~R and mR:

co=4

~o

bzc~g @

"~,o = '~Iz [ I ÷ 7e:~ e~-÷) L ]

Here,

~R is dimensionless

and the coefficients If we had used the

(8.2)

for every n (s o has dimension

a t and b I are independent 'On-Shell'

scheme,

o R and m R would have been identical

quantities

~ and m, which

satisfy

of m R •

to the physical

is now not the case because

of ~ i.a..For

4-n!)

then the renormalized

quantities

arbitrariness

(mass)

a particular

of the

value of ~ which must

an equation of the form ~ ~ K(~)m for dimensional

reasons,

149 the renormalized mass or charge will be equal to the physical. Accordingly,

there is a function K with m R = ~(a)m,

so that

it follows that

t~_,.)~ ]

o(.. [ ' I + ~

(8,3)

E

¢--,t

C'~L- ~)'~

The observable quantities are defined here by

4+

-ff-C

o)

eq. on page 39; unlike m R and eR, they are independent of p. Kaminski

[31] used this dimensional scheme to calculate the

effective Lagrangian in QED; the result is an expression which is

dependent on

which

coincides

shell

renormalization

The i n d e x

'R'

with

in

'LR' it

the

corresponding

only

has

for

a special

a different

earlier

sections;

suggests

as well

a s mR a n d e R d e p e n d s

result

that on t h e

value

meaning the

of

the

~ = K(~)m

here

renormalized

choice

of

mass

~.

than

in

the

Lagrangian

The r e n o r m a -

lization condition for i now reads

The

'bare Lagrangian'

on ~. Therefore, of

[o and its arguments are not dependent

the right side of (8.4) must also be independent

150

/ ~d ~

c~

z~,

~,

-~,)~) =

0

Because eRB R = eoB o = eB, this can also be written as

d

k The implicit dependencies of ~ according to the chain rule lead to

or to t-~ with

(8.6)

~

the t'Hooft Weinberg B-function

~ ~ , ~ )

~ 0

and

(8.8) Now we separate the Maxwell-Lagrangian

from [R

(8.9) (~ B) z

er,:~ and introduce new, dimensionless variables

(8.1o) So

151

'9

"a~' with

The renormalization

group equation for L(t,~R,p):= LR(eB,~R,mR,~)

thus reads

[-~ ~

~

~R

~ 7 ce~

or

This is the most general form of the 't Hooft-Weinberg equation for the effective Lagrangian valid for arbitrary values of B. If we limit ourselves to the limiting case eB>>m 2, then one can assume that L is independent of m R and so, of p as well. This assumption is plausible since, as we shall see, a certain analogy exists between the electrodynamics and the electrodynamics of short distances,

of strong fields i.e. of large mo-

menta. At arbitrarily high momenta, however, it is to be expected that each fixed, finite mass parameter becomes meaningless. (Analogous arguments are often used to establish the independence o f the coefficients a£ and b£ from mR, compare then becomes

[12]). Eq.

(8.11)

152

[..-L ~

'9

e ~ -t ,, d,..~ ) --- 0

We now substitute ~ = K(~)m into L

, ~ ) --0 which with

d._

e.~

= C_2),.~

'9

'9

leads to the following renormalization group equation for strong fields

(8.12) Here, the function [32]

~tW which is independent of ~ is de-

fined by (8.7) or because ~R = Z3~o (for n=4) by

(8.13)

~'~= z~c~J Eq.

~/~

(8.12) is very similar to Ritus'

[4] (homogenous)

Symanzik equation for strong fields

E ~ " ~-~--'~+ ~Or

~--~-] ~ £ ~ o

=o

Callan-

153

"3

(8.~4)

He re,

where and

the

'R'

7th sections.

(~c~(v') This the

index

help

However,

- ~i

function

again

~

-

2

was c a l c u l a t e d

of

the polarization

has

the

the

same m e a n i n g

B-function

~(~

in

the

now d e f i n e d

~ z

b y de R a f a e l tensor

is

as

5th by

(8.1s) and Rosner

to the

order

a3;

[42] the

with result

is

It is now interesting to note that 6cs coincides with gtw to the order 2 ;

to prove this we calculate

With ~ = <(a=~R(~=<m))m it now follows that

~ cs

C~)

4

=

~oz

~ + ~

~0~ ~c /

~ rO~

154

=-I+

=

With

4

+--~

<

~w c#~)

this we get a relation between

Btw and Bcs

[31]

O,~.('CaC)

The second term w i t h i n the paranthesis

leads to a term of

the 3rd order in ~, so that

(8.16)

thus gives us

.f A comparison with

(6.41)

(8.18)

oC

shows, moreover,

that

We shall now use the results of our two-loop calculation to calculate -I

2 8cs to the order e . We found in

(7.35)that

Z

This is now expressed by the physical mass m

155

whereby,

only the terms constant,

linear and quadratic in

0

need be taken into account: ~°

- ~

Therefore

( C-Y=

~cs C ~ o )

"

S"

~

~

"+ O c ~ )

"~"~.-2 )

~ S C,,t,,, ,~)

"8'm

(8.19) -n- + 1" C'~--J

Now we have to replace the bare coupling constant

physical one

~=

~o

with a = aoZ3(a o)

~sC~o~ =

~o E

:~'c~.)] -~

With o¢ 2 /~

it

follows f r o m

(8.19)

.

for 6cs dependent on a

by the

156

in complete accord with the result obtained from the study of the polarization

function

The above calculation between polarization

(8.16).

is an example of the close relationship function and effective

Lagrangian;

in

general I = L/L (°) in the limiting case of strong magnetic fields,

contains the same information as ~ (q2) does for

large space-like

arguments

(q2 >>m2).

But £ can be perturbatively

calculated diagrams

from a small number of topologically different 2 3 [4]: in the order ~,~ ,~ ,... one needs 1,2,10...

graphs for ~; for £, however, In order to illustrate

only 1,1,3, ....

these facts, we show that H(q 2) coin-

cides with £ for large space-like momenta with logarithmic exactness

in the order ~ and 2 .

By means of dimensional

regularization we get [32] for q2>>mR2:

W e substitute now q 2 ÷ e B

and multiply with L (°)

z

c- ,jE-

:..

_

z 2 +

' :..

~-OC~)

157

If

we s e t ~ = Km and c o n s i d e r

strong

fields,

only

the

terms

dominant

for

then

eB

(8.2o) oCz~ z

On the other hand, it follows from (5.29) and (7.43) that

Herewith, we have shown the equality of £ and H in the given approximation. Following Ritus [4], we shall now use the Callan-Symanzik equation (8.14)

F~ ~ with

(8.21)

158

to derive

an improved

form of [ for strong

fields.

So we

write

- f_ i = @ furthermore,

=

(8.22)

it is known from the perturbation

in higher orders

[42],

that one can make

calculation

the Ansatz

for £

(8.2s) with

¢= ,~ ~ Because

~ =e ~"

of ~z 2

it follows

@~

-_

~

that

Substitution

of (8.22)

and

(8.23)

gives:

~--Z

After performing

--~, "

- z -

the derivatives,

1<~--~

we get

@.)"Z%=K~ "

=-J~-, ~

d(~)

159

which can be simplified as

~o~ --

-

/7"

%

.t.1

C,~-)

n=z

{, =q

-

c,,,--',) 7__ o . ~ , 7

".=z

Z

,~

~" t='f

k-O

)Z ~

~ -

27_. a..,, K ~

k:

=

K-o

(8.24)

0

A comparison of coefficients with e shows

while

the vanishing of the terms constant with respect to z

of the second and fourth term leads to OID

-Z or to

~-_.v_

(8.25)

~=2. D=4

If we now compare the coefficients we get

(~k,f ~

of the various powers

K-,f

R-4

"~-" (~'-.~)q fo ~'k_ i 2--4.

=

~"k

(-~"- 4)O.io

~

k = z, %,-r, from

(8.25)

one.

If we define aoo

_;

;

(8. 269

J=4

Here the sum over i terminates

of e,

.

.

.

.

after i -- k-l,

since

that k = l+i and the smallest value

it follows

for £ is exactly

= I, then we can generalize

(8 26) to

K-4

OK4-----

)

Ot-'l) C l i o ~ k-;"

'

k--4, l,'3, ....

(8,27)

i=O this

reproduces

namely all

into account the underlined

= -aoo61 = - B l a n d s i m u l t a n e o u s l y term (without

t h e t e r m s f o r g = 1)

in (8.24). Here we have derived a first recursion the coefficients

takes

formula for

in (8.23); a second can be obtained by a

160

comparison of coefficients

in (8.24) with respect to zk,

k~2:

"i't.'=.Z

K=O

oo __

~

4.

"'-

~-'

+ Z (~,. (~Jl 7- (-~) e,r- .,>Z Ct.n,

~K

= 0

implies

so t h a t of

a comparison

magnituds

of

coefficients

with

respect

to

the

orders

of

z'=K

results. As a consequence

of the renormalization

expansion coefficients

group equation,

the

ank must satisfy the following re-

cursion relations

'1~1.--4

f=o

(8.28)

Ko,,,,~

= 7-

~=~.

(i-'~)°%,k_~ (~n-i

With the aid of this equation,

,

t<>~ Z

we can show that in (8.23),

to all orders in a, the coefficient

of the highest rower of

in(eB/m 2) (this is a11 = -61 and an,n_1, n-- 2,3,...) termined by 61and 62 alone. For this we use

_--

Cl-4~ O-i,k_ ~ ~_~

is de-

161

and calculate

(~- 4) ~ , ~ - ~

" W - - I ' i "~1. - - ~ -

N ,= ",,'1,

'I C ~ - ~ )

Q...-I,

~,_-z

e,~..--2.

N=~--I

_-

/%.,

C~-~)

~,.,.-'~, -,.-4-

/ V = " n . - ,7.

~-_,,~-?~

So

O.

,"v'c- .f

which, as claimed, depends only on ~I and ~2"

[8.29)

162

Analogously,

ofie c o u l d c a l c u l a t e

a n , n _ 2 and w o u l d h a v e t h e n

in each order of a the coefficients power of ln(eB/m2),

to the second-highest

etc.

We now define

(~)* L . -

~--- C

e_.,..,._. ~

~

I,'~, .... ( 8 . 3 0 )

~--t+~

The s i m p l e s t

examples are

+(~) q,,, ~ + (~)~%-+ ~ -, .... j.2

,,~.~

g.~

z

3

~.g"

J. 4

(~)~L.~ =

(~) L,~-_ So L I is the sum of the highest powers of z in every order of a, L2, the sum of the second-highest orders, etc. With (8.30), we can also write the series for £ R ~

( ÷ ) " ( L . c,a +c~,o)

as (8.31)

with

X . = d. ~% ~%

_

- * ~

--

rr

~ "~r~

z

The expansion in ~ no longer signifies an expansion according to the number of photons;

rather

(~)nL

n

is the sum of those

logarithms whose power is smaller by n than the largest possible in the corresponding order of ~.

163

With

(8.29), we can sum up the leading logarithms to any order

in closed form

=

l

% C-~)"'c.-,)-~ (L " - ~

oo

r

K

2:::: -~x

~'n

k='t

"-"

I(

~-~ . E ~ ( - t - × )

~

x c(-4,

+-~) (8.32)

Here, we have used the series

.~.,,,.(4-~:) = - -

~- ~I xk k=1

, x'e(-.r,+ 4).

One can perform an analogous calculation for L 2 too, but the result would contain the coefficients a20 and ~3 not calculated here; the same applies for the higher L n. If we consider only LI, then (8.31) can be simplified to

With the help of this representation, we can now evaluate the domain of validity of our improved,

asymptotic form.

If we leave out L I, then, with logarithmic accuracy,

~

= ~-)<

164

This formula is only applicable if the deviation from the 'unperturbed'

Lagrangian i (0) (i.e. form £R --I) is small,

i.e. for X" ~

'"IT"

ql'tr~ ~-

If L I is also taken into consideration, must

then its contribution

be small with respect to (l-x), i.e.

I must be valid. Because of x < 1, this means _

~ - .~.,,~ ( - ~ - × ) "IT

_

M.- ~ . ~ .... // "7 4-x

~/-.

(4-

~')

or

4 (8.33)

(~--~)

This means that the improved asymptotic form for L is valid in an enlarged region of x; x<< I must not necessarily ~pply and larger x's are possible, buth they must still satisfy (8.33).

So, neglecting L 2 and higher terms, we get

-~

d_

~_ £ ~ --I-×+-~ ~-~"~(~-×)+ ~,o "CTT.~.tz

165

with ~I = ~ and 82 =

according to (8.21). The coefficient

a10 results from (5.29)

and (8.23) to

0~ o It thus follows,

-

77"z

for the improved asymptotic form

~ir~z

~-z

or

f

-

~_Ez S

~%

~

(8.34)

The third term apparently contains a singularity at B = B D with ,n~

~t ~

in the proximity of which

(8.34) apparently is not valid.

Field strengths of the size of B D do not, however, have any physical meaning,

since they are still extremely large, even

compared to the 'critical field strength'

m2/e.

To conclude, we want to return for a moment to the case of the

166

pure electric field. For eE >> m 2 we get from (5.29), or for the second term in (8.34), by means of the substitution B÷!E 1

E~

o~.

(,~. --" ~)

=

w.. E ~

(8.3Sb)

We now establish that the radiative corrections to an arbitrary high order in ~

contained in L I do not correct the ima-

ginary part of the one-loop effective Lagrangian; the last t e r m in (8.34) leads, in fact, to

...,

,gl#

~Tl-~m,e L

An imaginary part could only appear in this expression for

W_

eE

which, on the one hand, would contradict the prerequisite x < I in (8.32) and on the other, would require unrealistically large field strengths E >>> m2/e. So (8.53b) undergoes no radiative corrections in the framework of the approximation of

(8.34).

167

This example

clearly

group equations.

shows

Knowing

the usefulness

of the renormalization

only the coefficients

B I and 62

(note

that the latter could also be obtained by calculating H (q2) at the two-loop ficient,

level, where

of course),

the case of B = O is completely

we were

able to sum up the leading

thms of every order of perturbation our former one-loop dity with

respect

the typical

calculation

its range of vali-

of the field.

This

leads

to

and Discussion

In this final section we want other authors

and,

to make contact with the work of

as a further example

vacuum of QED,

law due the presence

discuss

of the quantized restricted

case of constant

or magnetic

electric

be shown that the leading B 2 in B or

are not constant.

from the Maxwell

terms

the non-

of the Coulomb

fermions. our calculations fields.

for strong

to the

However,

fields,

it can

i.e.,

those

E 2 in E, are the same even if the fields

In a heuristic

(for a more rigorous

illustrating

the corrections

Up to now, we have always

of order

and to thus improve

by extending

to the strength

logari-

in (in B) term in (8.34).

(9) Applications

trivial

theory

suf-

discussion,

way this can be shown as follows see

[57] and

[58]):

One starts

Lagrangian

(9.1) and scales

the electromagnetic

coupling

e out of the fields:

167

This example

clearly

group equations.

shows

Knowing

the usefulness

of the renormalization

only the coefficients

B I and 62

(note

that the latter could also be obtained by calculating H (q2) at the two-loop ficient,

level, where

of course),

the case of B = O is completely

we were

able to sum up the leading

thms of every order of perturbation our former one-loop dity with

respect

the typical

calculation

its range of vali-

of the field.

This

leads

to

and Discussion

In this final section we want other authors

and,

to make contact with the work of

as a further example

vacuum of QED,

law due the presence

discuss

of the quantized restricted

case of constant

or magnetic

electric

be shown that the leading B 2 in B or

are not constant.

from the Maxwell

terms

the non-

of the Coulomb

fermions. our calculations fields.

for strong

to the

However,

fields,

it can

i.e.,

those

E 2 in E, are the same even if the fields

In a heuristic

(for a more rigorous

illustrating

the corrections

Up to now, we have always

of order

and to thus improve

by extending

to the strength

logari-

in (in B) term in (8.34).

(9) Applications

trivial

theory

suf-

discussion,

way this can be shown as follows see

[57] and

[58]):

One starts

Lagrangian

(9.1) and scales

the electromagnetic

coupling

e out of the fields:

168 giving now

_

Y

T/''''

~,

(9.5)

Note that in the complete interacting QED Lagrangian this is the only term containing e, because the vertex now simply reads ~A~ instead of e ~A~. The next step is to "renormalization-groupimprove"

(9.3) by replacing e with the running coupling constant

e(u) to first order in ~. This function is determined as the solution of (8.7) when including only the O(~)-term in the Bfunction. For e2(~) E 4~ ~(U), one gets [12] the scaling equation

e z(/"°)

=

e~/.)

" I - et('/'°), ~'fr" describing hereby

e2(lJo ) i s

where the masses

by F

fields

"~

coupling

are

c h a n g e s when t h e constant.

sufficiently the

(9.4)

' '

an i n t e g r a t i o n

are negligible,

magnitude 2

how t h e

~"

strong

length

(recall

e~. (~ ~)

dim F = ( m a s s )

!)

p varies;

In our applications, so t h a t

o r mass s c a l e

o f F 2 - F1jv F ljv. T h e r e f o r e 2

scale

we r e p l a c e

fermionic is set by the 4 !a i n ( 9 . 4 )

to obtain

e =(Y",) 7-e cn

=

(9.s)

with an arbitrary reference mass ~o" The last step is to replace e 2 in (9.3) by the field dependent running coupling constant e 2 ( F 2 ) :

-t

~,, T.,~,~

(9.6)

=

"T~. 7 .)`" F 1 - e'U,.)

"~

The o n e - l o o p

part

is

(we s c a l e

back

T z

e2(po ) - e 2 into

the

fields)

e z

eZT z

(9.7)

yielding for a pure magnetic field, for instance,

~c~

(eB) z

After

making the

order

B2 l n ( B )

derived the

in

special

the

(5)

above "derivation"

F2 in F for mation,

thus

arbitraryly

because,

choice

go = m t h i s

same a s o u r o l d

section

to be constant;

B

for of

constant (9.7)

as already

up t o

(5.29)

fields only.

d i d we h a v e

we may a s s u m e varying

equation

is

that

fields.

mentioned,

(9.7) This exact

terms

of

w h i c h was

A t no p o i n t

to

demand the is

is

correct important

formulae

for

of fields

to

order

inforL(1)

are known only for a very limited class of fields. The above manipulations can be justified by noting that the ansatz

(9.6) leads to the correct trace anomaly of the energy

momentum tensor; for a thorough discussion of this point, see Pagels and Tomboulis [57]. Now that we have established the Lagrangian (9.7) for strong but otherwise arbitrary fields, we can set up the generalized

170

Maxwell

equations

distribution.

and try to solve them for a given source

In general,

5.

[ z'°t

they are of the form

'°-

= o

with L (I) given by (9.7); hereby J current.

Of particular

interest

are the problems

statics where we have J (X) = Jo(~) ~(~) = -~ A°(~).

is a classical

source charge

of electro-

6 o , ~ = O and ~(x) =

This leads us to evaluate

(9.1o)

Note that changing Po to Po' in (9.7) gives rise only to a subdominant O(B2)-term; The physical

therefore we may set Po = m from now on.

content of the variational

easily visualized

in terms of the

problem

(9.10) is

dielectric "constant"

of

the vacuum

C C-"'E) ---- ~

- -

~IZ.~ . ~ .

and the displacement ~

~(~)

vector

~

(9.12)

In terms of these quantities,

~iV

~

=

This is a well-known media.

(9.11)

(9.10) simply reads

7o equation

(9.13) from electrostatics

Looking back at the microscopic

of polarizable

origin of E(~), we see that

171

the effect of the vacuum fluctuations such that the vacuum responds

of the electron

to an external

electric

if it were some sort of crystal which possesses dent dielectric

"constant".

come non-linear

due to the logarithm

To summarize,

Obviously,

caused by the electrons, vation but which

(9.13) with

(9.11),

the case where

and

the non-linear

ge-

the dynamics

from direct

being

obser-

of the A -field - for

arbitrary,

electrical

field.

into the effects produced by the second term

let us look at a specific

example.

Jo contains

isolated charge

only a single

= O (together with a compensating -Q at infinity)

be-

(9.11)

- the non-linearities

are hidden

but otherwise

To get some insight in eq.

which

influence

strong and static,

equations

equations

in (9.11).

(9.12), we have solved the problem of finding of Maxwell's

field as

a field depen-

Maxwell's

we can say that in deriving

neralizations

field is

spherical

We consider Q at

shell of charge

[56,57]:

(9.14) Making the s p h e r i c a l l y

,f.~.

the equation is a solution

Q-

I

symmetric

/-.t,m ?.z

(9.13)

ansatz

t

t

is solved provided

of the transcendental

Qc ) £

,/

(9.15)

that the function Q(r)

equation

(9.16)

172

The physical interpretation of Q(r) is that it is the charge lying within a sphere of radius r centered at x = O. The value Q(r) is always ~

than Q because the vacuum polarization

effects Screen the charge. If we let r ÷ ~ ,

Q(r) approaches the

(macroscopically) observed charge Q [37]. We thus got an implicit equation for

the modification of Coulomb's law by the

electron fluctuations:

(~(~)

(9,17)

We stress that this equation is derived for strong fields, and hence short distances r, only.

f"Strong" and "short" refer

to the scale set by m 2 and m -I, respectively). Of course, for extremely high field strengths, the one-loop approximation becomes invalid because the inequality x << I (cf. section (899 does not hold any longer. In this region it becomes advantageous to use the renormalization group improved Lagrangian (8.34) because of its greater domain of validity. We have now investigated the modification of Coulomb's law at very short distances; the contrary limiting case of larger r, however, can also be treated in a relatively simple way. Equivalently, we will ask for the effective Lagrangian for weak, but otherwise arbitrary, fields. In the weak field limit, e2F

FUV/m 4

becomes small due to the smallness of e. This implies that the contributions of

173 and those of higher order diagrams contained in W (I) (see appendix G) can be neglected relative to the diagram with only two vertices:

(9.18)

(The wavy lines are interactions with the external field, no photons])

Following appendix G, this leads us to the expression

(wf = weak field)

(9.19)

for the weak field limit of the one-loop effective action where H

is nothing but the order-e 2 polarization tensor (without

external field) derived in section (4). In momentum space it is given by (4.34):

(k) = (

I T (k ~1 -

k'-- k r

TT(k')

(9.20a)

oc k~. ~ c~ 1 3.T f(~:) k % ~ - iE 4#

(9. ZOb)

(9.ZOc)

As a c o n s e q u e n c e

of

the

particular

W( w fI ) can be written in terms of F

tensor

structure

of

(9.20a),

only and therefore is gauge

invariant, as it must be. Substituting the Fourier transform of (9.20) into (9.19) and adding the classical Maxwellian term, we get for the weak field effective Lagrangian

174

where~as o f eq.

usual,

(9.6).

~ ~ -a t2 + ~2.

This

There are two m a j o r

is

t h e weak f i e l d

differences

between

counterpart these

two

Lagrangians: (i)

The equations of motion derived from (9.6), i.e., eq. (9.13) in the static case, are non-linear due to the logarithm appearing in e(E). The equations of motion resulting from (9.21) for the weak field case are linear, because Leff is quadratic in the fields wf

(ii) The strong field Lagrangian of F ( x ) ;

(9.6) is a local function

the weak field Lagrangian is non-local due to

the ~ -operator in the square bracket in (9.21). Next, let us apply (9.21) to the Coulomb problem; specializing to the static case yields the field equation

For J o ( ~ )

we assume two p o i n t

°7oJ = 0 charges with separation

(9.22) r:

(9.23)

The variation in (9.22) then gives the equation of motion

175 (9.24) where

-:

"/ + ~

.L_~= zF~z

&,,,

(9.2s)

V

Making use of (9.24) and the position space representation of the resolvent

(t-~2) -I, one easily calculates the potential

eff o energy V = - f d3x Lwf ( A )

associated with the interaction of

the two point charges. One finds

Vc~)= - ~--~[~÷ ~~ ~,,T-~c~g'c,u e ~

j

(9.26)

The second term in the above bracket is the well-known Uehling correction of the Coulomb potential. Eq.

(9.26) was derived in

the weak field limit and thus should be valid at large distances. Obviously, the quantum corrections vanish for r ÷ ~

and the clas-

sical I/r-behaviour is recovered. Because the equations of motion (9.24) are linear,

(9.26) takes the form of a superposition of

Yukawa potentials. To summarize the two limiting cases discussed above [59], we can say that the QED vacuum behaves as a linear but spatially nonlocal medium at large distances

(i.e., for weak fields), whereas

it behaves as a local but non-linear medium at short distances (i.e., for strong fields). The calculation of an effective Lagrangian for arbitrary, but weak, fields is a relatively simple

176

matter, because bnly the diagram (9.18) has to be considered. Larger calculational effort is needed for strong fields; in this case, all orders in the coupling of the fermion loop of the external field must be taken into account. Even in the one- or two-loop approximation this summation over an infinite set of diagrams can be done in closed form only for very special types of fields, constant fields, as considered here, or laser fields (i.e., plane waves)

[3], for instance. Our computations in sec-

tions(5) to (8) are to be understood as a modest attempt to understand the QED vacuum at high field strengths or, equivalently, at short distances. As we have seen, the simplest non-linear, local quantum correction to the Maxwell Lagrangian arises from the diagram (5.4) which correctly describes the physics for strong fields as long as L (I) (~,~) is still well below L (°) (~,~), i.e., as long as the quantum corrections are small relative to the classical contribution.

The point where they would become

equal is the well-known Landau singularity when the denominator of eq.

[12]. It appears

(9.5) for the running coupling

(to

one-loop order) vanishes. Fields of this intensity, however,

are

of no physical significance, because they are many orders of magnitude stronger than the "critical field strength" m2/e, which is already well beyond the magnitude of all laboratory physics. (See the discussion following eq.

(8.34)). If one wants to pro-

ceed a little further towards the Landau singularity, one may use the renormalization group improved Lagrangian derived in section (8). Equation

(8.54) is still valid for values of B

too large for the simple L(1)(B) to be applicable

[4]. Also

this improved equation has the typical non-linear but local

177

form, of course.

(We assume, inspite of their derivation for

constant fields, these equations to also be valid for slowly varying fiels). Another type of correction to the Heisenberg-Euler Lagrangian which must be considered is the modification due to the quantized radiation field, the lowest order one, L (2), being represented by the two-loop diagram (7.6). As was shown in section (7), for constant ~- or ~-fields its contribution is about two orders of magnitude smaller than that of L(1); this should remain qualitatively true also for, at least slowly (on the scale of m-l), varying fields. Again L (2) is a non-linear, but local, function

of the fields. In the weak field domain these

corrections are taken into account (for otherwise arbitrary fields) by using a radiatively corrected polarization tensor in (9.19). The result is again a non-local Lagrangian quadratic in the fields, i.e., the equations of motion remain linear. Having now discussed in some detail the use of effective Lagrangians to characterize the vacuum of QED, let us finally briefly consider related problems in quantum chromodynamcis

(QCD), which

is believed to be the correct theory of strong interactions [60,12,51,531.

One of the salient features of this theory is

asymptotic freedom, i.e., the fact that the coupling constant decreases when the mass scale is increased or the length scale is decreased. This behaviour is caused by the fact that the sign of the leading term in the QCD B-function is changed compared to the QED 8-function. To lowest order, the coupling g scales as

178

9zfl ")

(9.2y)

with (Nf = number of quark flavours)

4

(9.28)

This is the QCD analogue to eq.

(9.4). However,

for Nf being

sufficiently small, b o is a positive constant so there is a crucial difference between

(9.27) and (9.4): the sign of the

quantum corrections in the denominators

is such that the coup-

ling increases in QED for increasing mass scale ~, but decreases in QCD for increasing ~. Or, stated in terms of a characteristic length scale - I :

in QED the coupling decreases when the dis-

tance becomes larger, but it increases in QCD. Just as in QED, we can derive effective Lagrangians

for this

theory; they are the formal expression for the interactions of an external color field with the quark/gluon vacuum fluctuations. A new feature appearing here is the possibility of gluongluon interactions;

this means that in one loop calculations

not only fermion, but also gluon loops be considered

(and ghost loops) must

[61]. To get a first understanding of the effects

associated with these Lagrangians, we can use the classical Yang-Mills Lagrangian

-

[12,25]

zeta

(9.29)

179 with the coupling scaled out and replace g2 by the function g2(F2) obtained by substituting F 2 for p corresponding manipulations

in (9.27).

(cf. the

in QED). The result reads [62]

t

,~, #

4

7

= JLa..s,( E L 2~,--) E I + .~ bo C~ .~

EZ"-~ z "1 ,l.,c~

J

(9.30)

f

E L ~z

with

"P4.

(9.31)

e

and E 2 -= ~a . ~a and similar for B 2. It is easy to show that the quantity

~ defined

in

(9.3])

is

renormalization

group invariant,

i.e., it does not change when varying Po" Hence ~ is a physical, i.e.,

observable,

quantity.

As a f u n c t i o n

o f E 2 - B 2,

Lef f has

qualitatively the following behaviour

.~

...

EL

to)

~Ba

:'1

Now let us ask where we can trust the above curve. We know that in QCD, perturbation theory is valid at short distances,

i.e.,

180

for strong fields (on the scale of ~). In this region the deviation of Lef f from the classical Young-Mills Lagrangian is small. Following the arguments of Adler [62,63], there is another region where

(9.30) should be correct; this is the domain near the origin,

because for E 2 - B 2 very small, i.e, ~ in (9.27) very small, g2(~) is again small (but negative!!) and perturbation theory (in this case a one loop calculation) should be possible.

If we

accept this argument, we know that Lef f has negative slope near the origin but positive slope for strong fields. Despite the fact that (9.31) becomes untrustworthy in the intermediate region, we can conclude that Lef f must possess a minimum for a non-vanishing value of E 2 - B 2 (its continuity assumed). One may therefore assume that (9.30) interpolates qualitatively correctly in the strong coupling regime E 2 - B 2 ~ K 2

It is therefore sensible

to use (9.30) as a model of the QCD vacuum for arbitrary fields. Having an effective Lagrangian, it is natural to set up the modified equations of motion for the color fields (the analogue of (1.6)) and to try to solve them in presence of external sources, just as we did in QED when investigating the modification of Coulomb's law. What one would like to explore using this method is the question of color confinement; referring to the quarks, this means that the static potential between two quarks should rise at least linearly with the distance because then the energy to separate two quarks would be infinite. Surely, the discussion of confinement purely in terms of the potential of static color sources

(i.e., infinitely heavy quarks) is not the whole story,

but it is a first step in understanding the non-perturbative effects in QCD.

181

The general program of the so-called leading-log model based upon

[62,63]

(9.30) is quite similar to what we did in QED star-

ting from (9.7). One sets up the static equations of motion for the fields coupled to two static point color charges corresponding to a quark/anti-quark pair, solves them and calculates the interaction energy of the quarks as a function of their distance. However,

due to the non-linear character of the equa-

tions, this is a highly non-trivial task which requires sophisticated numerical methods.

The details of this procedure are too

complicated to be reported here in a few words, the reader to the literature

[62,63]. Nevertheless,

of these computations are very encouraging: quark potential nearly,

the results

they show an inter-

increasing with the distance stronger than li-

i.e., at this level we can conclude

quark pair must be permanently confined! appear,

so we must refer

that a quark/anti-

For this result to

it is decisive that Lef f possess a minimum away from the

origin. Due to the argument of Adler concerning the second domain where perturbative calculations

are trustworthy,

this minimum is

already a consequence of (9.27). This is in contrast to the situation in QED, where, as we saw in section

(5), ief f is a mo-

notonic function of the fields with a unique minimum at E = B = O. The fact that in QCD the energy density is minimalized for a non-vanishing field (this is formally expressed by a non-vanishing "gluon-condensate"

F ~a

JO> ~ O) gave rise to the so-called

Copenhagen vacuum which describes a sort of domain structure domains being uniformly magnetized) medium.

For a detailed discussion,

(the

similar to a ferromagnetic see [64].

182

Another

consequence

of the altered sign in the ~-function

that the QCD vacuum behaves

as a linear,

medium at short distances,

while

as a local)

medium

but non-linear

but spatially

at large distances [59]. This

is

non-local

it behaves

is just opposite

to

the vacuum of electrodynamics. With these remarks we close our outlook approach

to quantum chromodynamics,

could well be an alternative simulations.

However,

that it is preferable

which,

the complexities to first study

Lagrangian

in certain cases,

to the commonly

used Monte

Carlo

of QCD once more suggest

a relatively well-understood

theory,

such as electrodynamics, within

turning

to problems

confinement.

on the effective

this context before

like the vacuum structure

of QCD and quark

Appendix A: U n i t s , M e t r i c , Gamma M a t r i c e s We use e x c l u s i v e l y t h e Heaviside-Lorentz System w i t h n a t u r a l units, i.e.,

a

=

we s e t h = c

=

1 . Then t h e f i n e s t r u c t u r e r e a d s

e2/(4n).

We have, a s m e t r i c

s o t h a t t h e f i e l d s t r e n g t h t e n s o r c a n be w r i t t e n a s

The y - m a t r i c e s , which s a t i s f y by d e f i n i t i o n t h e anti-commutation relations

we use i n t h e s t a n d a r d r e p r e s e n t a t i o n [ 3 7 1

with the Pauli matrices

I f we now d e f i n e

184

then

~--Co ~ ~° )

(i,j,k = 1 , 2 , 3

(i

Of special importance is az

which is

( ~'~ often

0

again

cyclic),

= 1,2,3).

(;- o) I

designated

by o

3

We write the frequently appearing scalar product of the y matrices with a four-vector a as

For arbitrary four-vectors ai, the identities apply

If one chooses in particular vectors with only one non-vanishing component, then it follows from the above equations that

furthermore,

that

185

'~? ~r/. = - - L F

~r ~ J ~

= ~ ~.b

186

Appendix B: One-Loop Effective Lagrangian of Scalar QED In this section, with the help of a path integral [24-28], we shall derive the equation analogous to (6.1) of scalar electrodynamics

and then evaluate it by means of a Zeta-function.

The one-loop approximation thereby appears as the lowest correction of the classical Lagrangian in an expansion with respect to powers of ~, which is equivalent to an expansion with respect to the number of loops.

(A simple proof of this

is given, for example, in [28].) Scalar QED describes charged, spinless particles with mass m associated with the complex scalar field ~ which interact with the electromagnetic field A ~. So the classical Lagrangian is

in order to quantize this system, first we couple the fields ~,~*, A to external sources n n*, J:

(B.2)

In addition, a gauge-fixing term was added here to L. The corresponding action is defined by

The classical solutions ~o' ~o' Ao are those fields for which S becomes stationary-

187

One sees t h a t sources

If

~o'

n, ~

~o and A o a r e

and J ;

functionally

d e p e n d e n t on t h e

symbolically

we now s e t

CaCa¢,an] e t h e n we can show [ 2 5 ]

that

the

of

Green's

functional

functions of

the n-point

the

equal

to

converge,

of

zero. m

2

the

the

generating

theory,

'connected'

functions

differentiation

Z is

while

Green's

in question

Z or W followed

W is

the

functions;

are obtained by s e t t i n g

(In order for the path integral

must be r e p l a c e d

functional

this

of

generating means t h a t

by f u n c t i o n a l of

in

the

sources

(B.3)

to

2 by m - i t ) .

From W the generating functional F (effective action) of the one-particle irreducible, amputated (IPI) Green's function can be obtained by means of a Legendre transformation. To this end, we define the classical fields

188

(B.4)

which are apparently

functionally

dependent on q, q

If one imagines this relation between inverted,

(¢c,¢c,Ac)

9r

and J.

and (rl,rl ,J)

then it holds for F [25,27]

(B.8)

By differentiation

With analogous

of F we get the sources back

formulae

for the derivation with respect to

~c and Ac, we then get all together

(B.6)

The meaning of the classical

fields is clarified by

189

10 _ > "z, 7", ~

(B. 7)

< o . to_> 7' "~'] where we put

(B.3) into (B.4)

Path integrals

(compare

[25]).

like those in (B.3) cannot be precisely cal-

culated for interacting to approximation

theories; we must, therefore,

methods.

One possibility

resort

is offered by the

saddle point method in which one first expands S around a stationary point

(~o,~,Ao),

and terminates

it after the qua-

dratic term

+

Here,

[0]

~

[~o,~o,Ao]

w e re i n t r o d u c e d denotes write

for

the quadratic

out later.

is required vanishes

,o.

and

the sake of shortness;

Since the classical

the saddle point

it

furthermore,

term of the Taylor series

t o be s t a t i o n a r y ,

so t h a t

+ ....

follows

method

action

the linear from (B.3)

SZ

w h i c h we s h a l l

S for

(@o,~o,Ao)

term of this

expansion

in the approximation

of

190

This particular notation signifies that @o' @: and A ° are dependent

on

the

sources

rl, q

and J.

So,

with

(B.9)

Before further evaluating (B.9), l e t us consider the so-called ~ree approximation of (B.8) in which we neglect the term of the order IT and simply set

From (B4), it follows then that

+

~-----~o

+

~

.. = o ~

6"~G-:I .~ ~oC.y) ~--0

C~c C,c ~

,

l

OCt)

191

The fields (B.4) in this approximation are, then, identical with the solutions of the classical equations of motion. For the effective action r it follows that

* +ot~ r ( ~,~:,eo)] +oct)

So the

IPI-Green's

r

"

functions

g e n e r a t e d by the c l a s s i c a l

i n the

" d~t; tree

~} * or,;

approximation

are

action!

We now turn back to (B.8) and calculate the term proportional to ~ (one-loop approximation); for this, the following definitions are useful A

~ =:

~t + ~ +

oc~ ~) ,

Since (~o,¢o,Ao) satisfies the classical equations of motion, we have

192

Furthermore,

So, for the effective action F it follows that 4~

The one-loop correction

£1

to the effective action is then

(B. 10)

&cc,,

,¢; , nJ ]

We shall evaluate the path integral only for the special case

=

=

0

(~)=-Z

t

X_,

if we define the one-loop effective potential V I by

then

Here, the functional S 2 is

q-

193

+ ,2+ ~c-o~

~.~ a~'co~

(B,12)

wi th

After

a shift of the variables

(B.11)

and

(B.12)

show that the path integral

the effective

potential

ignore.

it follows

t÷~

Thus,

of integration

V I by only from

(B.11)

a

(A÷A

+ Ac) ,

over A changes

constant which we can after a Wick

rotation

= it and with ~ = I that

V~c~> =

-.__~

.~-~

with

and

~

--_ ( ~ ' ~

___~ 2 + 7r~

~+*~+~e--~¢*~¢

c~.i~

194 The Gaussian path integral in (B.13) can be easily evaluated

[12] and gives --I

In order to make this equation right, also with respect to the dimensions, we again introduce an at first arbitrary factor

2

With (6.4) it follows that

(B. 14)

= - --~-~ --:

--

-n- -~

~'f ~H

,,.~It

CO) Co)

According to (6.17), the spectrum of ~E/~ 2 is

so that, in analogy to (6.18) it follows for the Zeta-function that

Cs-)--/u}s--~-

~..: ~

ca.-)"

'~,-=0

From this, with the integral

=

(6.20) and (6.21), we get

L: ~--~-~-

195

. a ~ _.9_

= 2~

~

(~

~)a-s

( s - s F "~ ~C~-'~,~r)

)

~'=

ae~

The derivative results in

= 4~,r---'-"

C.~-~,F ~ .~c~-~, ~.)

C~-~)

+ So, for

s

=

O,

Because of (6.30), we get

- ~ rc-~,~.~ = ~ L

~+ ~

C,,~'l z = ka---j~-)

4 4~

and thus

In accordance with (B. 14), t h e

one-loop e f f e c t i v e Lagrangian

is then given by

~ °'c ~) = - V~C~) = -.s'z-~CCo)

4

(B. 15)

196

Here,

we h a v e

exception written

of

again

chosen

an unimportant

U = m as mass constant,

is

(B.15)

With the

can also

be

as

~1~c~c~, - ~4~z~f ~ ' ~ - ~ce~,~ (4. ~ ~ this

scale.

exactly

a dimensional

the

result

regularization!

obtained

)-~by Dittrich

~-,,~"

(~

I ~ ~B [8]

using

)

197

Appendix C: The Casimir Effect In this appendix, the analogy mentioned in section 6 between L(1) at finite temperature and the Casimir effect will be more closely investigated. The

Casimir effect is a non-classical electromagnetic,

attractive or repulsive force which occurs between electrically neutral conductors in a vacuum. The size of this force was first calculated by Casimir [20] for the case of ideal conducting, infinitely extended, parallel plates; his result was a force

+

=----

(c.I)

where a is the distance between the plates and the negative sign indicates that the plates attract each other. This force apparently depends only on the fundamental constants ~ and c apart from the distance between the plates; not, however, of the coupling constant ~ between the~Maxwell and the matter field. Its quantum mechanical character is revealed by the fact that F vanishes in the classical limit ~ ÷ 0 . Casimir% derivation of (C.I) was based on the concept of a quantum electrodynamic (particle) vacuum representing the zero-point oscillations of an infinite number of harmonic oscillators. As a result, one gets the total vacuum energy by summation over the zero-point energies I ~

of all allowed

modes with wave number vector ~ and polarization o

198

If we evaluate

(C.2) for the case of two plane parallel plates

at distance a from each other, one does get a divergent total energy E(a), but the energy difference E(a)-~(a+~a) is finite (~a = infinitesimal change in the plate distance), leading also to a finite force per unit area

_

To c a l c u l a t e is usually

(c. 3)

9e this

energy difference

introduced,

i.e.,

(C.2)

or force,

a UV-cut-off

i s r e p l a c e d by

and, in the end result, the limit b + 0

is considered.

This derivation of (C.I), however, can give the impression that the appearance of the Casimir force is linked to the existence of the zero-point fluctuations of the quantized electromagnetic field. Since it has been speculated that the real Hamiltonian of the field [21] could not contain the term (C.2), this would mean that the Casimir force would also not appear. That these assumptions do not apply was shown by Schwinger [22] in the framework of Source Theory [29], which does totally without the concept of zero-point fluctuations of the field, i.e., a structured vacuum. Besides, the Casimir effect was proven experimentally [23].

199 In the following, we shall consider the problem according to Hawking

[15] from the viewpoint of path integral quantization

and Zeta-function regularization.

Here, it is again unnecessary

to refer to the vacuum oscillation.

For reasons of simplicity,

we wish to consider the Casimir effect only for a real, scalar field theory which is defined by (~ = c = I!)

~Z~ ('¢) = _ '~'f /~ ¢ ~ ¢ _

.4.~,,w~+2_ VL¢.¢)

(C.4)

with the arbitrary potential V. In order to carry over the result obtained into QED, we shall only have to multiply F by a factor 2 (corresponding to the two polarization states of the photon).

First, we couple the

field ¢ to an external source J

According

to

[25],

we c a n t h e n w r i t e

J or the action W[J]

the

vacuum amplitude

in the form

(c.6) where we guarantee the convergence of the path integral by the substitution m that

IO_> or

2

+m

2

-i~, ~>O. Until now, we have assumed

IO+> describes a vacuum which is not'disturbed'

the presence of certain geometries,

by

i.e., the path integral

(C.6) is, without restriction by boundary conditions,

to be

taken over all fields ~. This changes as soon as we introduce two plates into the vacuum, z-axis

for example, perpendicular

(points of intersection:

to the

z = O and z = a) and require

that only those fields should contribute to the path integral

200 which vanish on the plate

surfaces,

i.e.,

for which

it holds

that

[C.7) for arbitrary

(Xo,X I,x2).

In QED such boundary

be fulfilled by the use of perfect of

conducting

conditions surfaces.

can

Instead

(C.6), we now get

4.o.+1o->~

= e

.,; w [~, c~1 )

(c.8]

/Fa suggests that the path integral is only to be taken

where

over the restricted With

this, we have

action

space of functions represented

for the most general

tric parameter In order

we now choose

the vacuum amplitude

case as a function

a and as a functional

to approach

the conditions

a partial

(C.7). or the

of the geome-

of the external

source J.

of the QED Casimir

J = 0 as well as a free

field ~. Following

Fa defined by

(V = ~), masslees

effect, (m = O)

integration:

(c.9) The Gauss i n t e g r a l gives [12]

W (e,-) (C. 10)

Here,

N is a (divergent)

since

it only contributes

constant which we shall a

non-physical

set = I,

additive

constant

201 to W(a). By writing

~E/Fa,

with eigenfunctions

in Fa can be used to evaluate the de-

terminant.

Furthermore

we mean that only eigenvalues

(in keeping with the ie requirement),

~I£ =

a Wick rotation t ÷ i T was made, i.e.,

a2

T

+

&.

With (6.4), it follows that

[-Y'

=

-~.1~

(o)}]

(c.11)

~ The operator

- []E/Fa has the spectrum

and thus, the Zeta function

f _ QEi ~ ( ~ ) =

~

~

~=4

(c.12)

Here, the factor 2 makes allowance for the two polarization possiblities of the photon, which, in our simple model, have no analogue.

Furthermore, AT E is a normalization volume in

three dimensional

(O,1,2) space, where the Euclidean time T E

is linked to a (Minkowski) normalization time interval T by T E = iT. Dropping the term independent of a (n = O) in (C.12) simply leads to the subtraction of an (infinite) constant of

W(a). A comparison with (6.52) now shows, that ~_ []E/F a differs from ~(~)

only in that the discretization of the k-integration

was not performed in the time component, but in the space component;

furthermore, one can see that the parameter a

202

corresponds

to

6 = 1/kT.

Further evaluation of (C.12) now takes on exactly the same form as in section 6: o
a~.=4

Pcs)

The derivative

is

#T

I

From

(C.11) we get

----e

with

- i vc~) 7 -

__Z

7ao~ 3 The a p p e a r a n c e

of the

plitude

us

allows

placement

to

phase 'factor identify

and to write,

for

c(a) the

e -i¢(a)T

in the

v a c u u m am-

as

the

vacuum energy

force

per

surface

unit

dis-

203

which leads to

Fz 2

,+o

"

4 O(~l

or, after putting in ~ and c

Ez. 2.4-0

~e ~q"

(C.13)

This is precisely Casimir's result which we have now completely derived without the concept of a structured vacuum! We should now like to briefly demonstrate the relation between the above-described method and Schwinger's approach [22] which, by means of a causal analysis in the framework of source theory, beginning with the vacuum amplitude in presence of an external source J

] = e i

E3]

arrives at the following differential expression for the action:

=

Here,

t~t

refers

to v a r i a t i o n

of t h e p l a t e

distance,

and Da

is the Green's function of the probelm, defined by

Integration of (C.14) gives, disregarding ditive constant,

an

unimportant ad-

204

(c.ls)

In [22] the evaluation of (C.14)

or (C.15)

follows in such a

manner that the Green's function Dais expIicitly calculated and inserted into (C.14).

If one is only interested in W(a), how-~

ever, then knowledge of D a is not necessary,

since the deter-

minant in (C.15) can be calculated by means of a Zeta-function whose construction requires only the spectrum D a agrees with that of - ~ E / F a

. But this

analogous to our calculation.

means much less calculatory expense in comparison with

This

[22].

In concluding, we wish to mention again that one can naturally also get (C.13) from the formula for the vacuum energy often used in relation to the confinement problem example

(compare,

[45,46]):

E =-

~..o ~ , & , & :', , ,

(S E = Euclidean action).

~c~{] e - s E ~ ]

In our case,

E=--.L~- ",'~'~ 4

=-

Z ~

~ .... £ ~

F~et (_- tn~ l ~ )] -'~

for

205

W

.4

"tt- z

Tr z

7 2 _ o o,_s

which again gives

4

,q

'gq

2_4-o

O ""

206

Appendix D: Derivatives

of W[A]

We now calculate the first two functional

derivatives

of

[47]

~Wf~-~

=-

(~-

7; ~

e ~'/~

~+)(D.1)

with respect to the potential A. To do so, we first write the logarithm in (D.I)

2~

C,-~)

= - x-

~ ×~-

~ ×s .....

o 4

= -

,~a~, × ( 4 +

:t~ + (ax.) ~ ' - - - )

o ,,4

= in t h e

_

~,~

form

x

"/

6

1

4 0

where,

at

iWE~l

the end,

-

e'

was s e t e ~ u a l t o ~e. So (D.1)

t-~ ~.~ ( 4 -

-

~'

~ t.

gives

~+)

~ E ~ ~+ c~- ~' -,~ ~+1-~ ?

o

(D.2) (St. 3)

o

e o

207

NOW we show that, for the derivative of the propagator in an external field

~G+Cx,YI~) is valid.

For t h i s ,

we b e g i n w i t h

the d e f i n i n g

equation for

G+[A]

to which the integral equation

is equivalent. Taking the derivative results in

~?G+c~,y(n)

+ ~a~ G+cz-~)efvl~¢t~ 8G+(~'Yiri~ It

then

follows that

~ L&~-~)-G+c~-~)e~(~] ~+c~'Yln)-~c~ - G÷c~-~le~ ~÷¢~,~lRJ w r i t t e n as a m a t r i x , t h i s equation reads

or simply

208

which immediately leads to

6"G+[ ~,] =

(

~-

O+ e~,~)-

If we now return to space representation,

then (D.3) follows

exactly, qed. So, for the derivative of (D.2) we get e

o

~ I:Pc~)

e o e

c~)

e,

O

9e'

In the fourth line, the cyclicity of the trace was utilized, and in (*) we made use of

e

209

= ~+(4-e'

N ~ + ) -4 '71q ~ + ~ 4 - e ' ~ . )

-~

i.e., of

From (D.4), it follows for the second derivative:

"-I

,:S,Iqt'.', c~'.o

So, in summary, we have found

• ~w~~] Z - -

CD.S) 4

210

Appendix E: Power Series and Laurent Series of K(z,v) In this appendix we calculate the power series of K = K(z) to the quadratic term, necessary for the renormalization of L (2), and the singular part of the Laurent series of K = K(v 2) about v 2 = I. Here, K(z,v)

is given by (7.19) with

(7.21).

We first turn our attention to the power series of

I(~I

--

~'~z

t.

I

%

(.n-v.) (co= ~v-co~-~)

/

(E.1)

-Z-a" ( c o ~ . .

~.,.:,~ i~ - ~ )

-I-

--

- .~,

Then

Z.'I- v ~

2

,.

~

.?. ( c.os e v - - c o r e)

Expansion of D I and D 2 gives

o,._!

~ - 2

6"

--- ( . ~ - v ' - ) - ~- ~'v~C,t-v"~ * ~~ o

=

G

~

-

+"

V~ + ~ " t

-

-'"

: ~ + ( ~ v ~- ~ v ~ _ ~ v 6) + - - -

211

~_ =

a[~"

~0~ ~ v - o ~ ) ] -~ .~ ,~ ~+V ~. -~ ~ ~v ~ ~- - ~ -

~~ -

:

42.

4-v

z

4 CV ~ -~7/ ~

~0

-b

--"

_1_

.....

4-V

~

So

(,,- v ~-)

and thus 46

For the

expansion

of

CB, we n e e d

(E.3) ,%'..,>~:~

.2. {co~. ~=v- co~ ~:)- C~-v~) :z ~.,:,,. :a 4-- ~"> "~'+C~-v ~ ) + . . . . . ~

C4-v a) 4- . . . . I

-~ C,~-v'-) + ~4 ~ '(-~- v ~)

212

.,4

"~--.---~.~.~°,~.=,;,,,~_~)= 4~ C - { ) ~* -[4- %6 ~"----~ -'l

~-V-

.I

which, together with

.12

~ ~7.,,_v~)

(E.2), finally yields

K.~= C ~ C ~ (E.4)

+ 0 C~:") Now we s e t

.Z_.z .2 ('c~s" zv-cos- ~-), - ( . 4 - v ~) c , = ~. . ..r,>~ z

__: __4~ ~

~ ~- :B

..T',,2,~~

with

then yields

-2

=. (~.F

~ (~_v~)~ and from

~d+ ~-J ~-~+v~ ~

(~-v ~)

__..

I

213

2

=

~ [~- -~- ~

~o

it follows, together with (E.3), that

I~- -~-- P-. ( c°x ~ v - c ° x ~ ) -

(4-v~) c°~'~"

Z¢~zz'~

~ o C ~ ° - ~'~ v % V

+-..

}

So the expansion for K 2 reads 'I- . . . .

K~- ~~~ ~C,~_v~)@ (~+f~+ )14+ ~ ~ -C -~t - +v ~)~~

This can be simplified to

t.~

~ ~,.,~

~-~

C._v~Z

~ ~~- v 2

-

50 ~~

+ O C ~ ~)

}

C~.s)

Last of all, we still need the expansion of In(b/a), which gives

~"~o ~cs_v~)£~_v ~)

~C,_v~.)[.1+ therefore

...,.zo-C¥~v"l]

~ O c~ c )

+0c@6")]

214

Ko. -~'~ ~ q

=

4_v t

which, together with

-~-v"

4zo

4-v ~

~o

+ 0(~)

leads to the very simple end

(E.4),

result

t<=_ ~,, .+ k.z -&~. I~ "-- '- K o z

--t-- 0 C-?-~")

with

~-'P ['~o ~

~

4_V

t- . 2 ~ z

(E.6)

z

Now the divergent part of the Laurent series of K = K(v 2) about v

2

= I must be determined. We begin with

46 8 ~

r-

C o ~ ' ~ V - - V C o ~ .E • ~*:"~- ~-V

.Jr ( c.c,:~. ..r;.~ e - ~-) ~(c~

"f ev-co.~

~)-(~-v

4--V z

..~Z(y )

~) ~~.;-',~"

and set

4-v z

Ce¢ , c v ~

cog~-

.,'I--V ~" !

We now need the following values of fi and fi

1

215

.I{ ( v )

----

Trz ( v )

=

r~_(~)

=

Apparently,

--

~

~

v

--

cos ~ - v -

co6~

~v--v~-

~

co~-,co~,~-V

c o z ~-

0

both fl and f2 have a simple

then, with the de L'Hospital

zeroes

for v = I;

rule

..~I(41 I.

4

2

-I-

co£- ~

This means

~-- V z

E DC,~ a

+

(for v ÷ I regular terms).

To evaluate

~ [ ~ (-c'°" "~¥ - C°S"~') -- C4-v~') ~- ~'~"e" % ] -'~ we need

ely)

--

~(<) =

~ Ccc~v-c°~)-~

0

4-vt) ~ ~

216

~ t (.V)

=

~t' (4.)

=

~rrr CV)

0

-

=

Because of f"(1) + O, T 2 has a second order pole for v + 1 ,

i.e.,

an expansion exists in the form

I

"2_

C,l-v~:) z

I-

+ Ho + . . . .

,t-v z

the

following

only

/

w h e r e b y we s h a l l

in

concern

ourselves

the coefficients

A'_2 and A" I. The de L'Hospital

rule now gives

f

Iq_ z =

=

ZV'-~ ;~4

c,,- , h "

-6.;,,~

c3 "C 4) with gCv)

cj

Cv)

: = (1-vZ)2;

taking the derivative,

_=

( 4 - v ~) ~

c~ C4)

-

O

ej / cv)

=

-~vC4-v

5 ' ~"f)

=

0

~)

with

we get

217

Oj ~ ( v)

-tt(,~-v~)+

=

?v z

oj, c~l

in (4J

~-

21+v

__

:~ff-

Then

~ ~_~ = 9 "~) We calculate

--

_

4-

the next

coefficient

V.=_~ .,I

~

4--~

F

from

z

.~_,,~

....

cs-~j ~ - ~.fz [ac co~ ~ - ~ v---..t

~,.'~ e~,

( ' , - v ~] ~cv)

N u m e r a t o r and d e n o m i n a t o r have a t h r e e - f o l d

Eu-~zl~s_~ "

]

~)-cs-,,,~ ~-.,.,..,,.~].

{',-v ~-) E ~ ( c c ~ ~v-c_o~)-c-~- v'~

V "~ ~

The denominator

,~!z

of this

tv= ~) :

zero

so that

is

[c~_.~l'~

+ ~ c~-~,j"~ r

(v~ -h

218

_

Then

~-~ - -

42 ~=(r-2~--e-eos~)

We now write

(.,~--V ~) ~"

( .~-- V ~-)

and get

(E.7)

Now we examine the expression

c~ which,

( 4 _ v ~) ~- ~ for v = I, exhibits

a first order zero;

in the numerator and denominator

this means that it can be expanded in

a power series of the general

form

219 E

AS w i l l

~

be shown, we must c a l c u l a t e

and 12 o f t h i s

series,

E (4-v z) ~ ~ --,2 ~- gg',t ~well

the c o e f f i c i e n t s

Again~ wmth the de L ~ H o s p i t a l

lo~ 11 rule,

( ~ - v z) :k , ~ oz

V - + .l

as

....

~]~¢v=4)

(e.8)

4 {

as

4--V z V --~ "t

.2 (co# ~,¢- co.r :~) - ( 4 - v

z) ~- r,2~ ~.

4" "(-t) (E.9)

and

V--~4

O- v'J ~ =

~.;~ v-~

("-v')

S(~zv-co~)-(~-v') ?::~-~ (4-v~) ~° ?; ~

(~-v'J 2 ~

220

h m (1)

~sz~

Here,

It

[V~

:=

h

~-,I

=

['1 I~

(. V)

=

h"

(~)

---

0

h " Cv~

=

S. 21-1- v ( . ~ - v " ) -

[q m C'q

"=-

- 4-

h ""

(v)

=

%. : ~ C . 4 - v " )

h "'

(4)

=

-

follows

( 4 - V") "~

0

¢ C 4 - v " ) z . + 2~-V ~ ( 4 - v " )

-

k.8 v "~

- - C '-,,-~v z

C" q-

that 4

(F.. I o)

With

~

: =

I - v

2

, the

power

series

for

in(b/a)

£~b_o~ = ~'~ ( ~÷ 4" t ÷~%c ~+---)

reads

221

Here,

(E.11) and

L.z.

~-=- %9_..

"¢

(E.12) set

Now we

Kz--46 &s

~'~ ~ and

-Tr ' e c v >

get

<2~ ( v l

:=

3.. (co~ ~:V- cos ~:) -- (4-vZJ cot ~. e ~ z ~

0 (~1

[V)

=

)l (41

=

2_ ~ 4 ~

2 ~

(c~.

~.~-~ :~v

<~m [11 = (~jm Cv)

:

dp"' (I)

=

~_ ~-°r c ~

~v

~ - ~ )

222 Apparently

the numerator of T 3 has a single

zero, while the

denominator has a four-fold zero at v = I. Therefore

the

Laurent series of T 3 takes the form

C_~

I3 =

C-~ +

£~

+

C_.

6a

+Co~

E

Thus

'

L~C_~ + Ea

_

L~

C_~

-t-L,, C _ z

+

0

(~-o)

6

and

b K~

4,~: "~ ~ L.C-~

g~E

- r~

gz

@-v'-) ~

LzC-~ +L.C_~ +

e

4- ---

-4-

+ --_I

(E.13)

4-v-~

with

•? 4 . . 2

-- ?.~.~

O4_~

=--

This means

L.,

~-,>.- %-

EL.C-.-+L,C_. ]

(E.14)

that we must explicitly calculate the coefficients

C_2 and C_3. By means of our usual method,

C_~ = -~:'~- O,-v'P V--~4

v.--, 4

~Lv) (~>(V) .~z (.v)

(aQ~)"cv=4)

223 The n e c e s s a r y

derivatives

are

C~+),%_~,: [~:,,~+,4,',¢' + c~%"-,- ,,-~,'+" -,- ~,+" ]c v-~

+~#'¢"+ --

6

C¢"(.I)z

=2~- ~

(

s;~a -

~:c~,~.~) ~

so t h a t

--~-~£ C,_~

~i~

Cco~i~.~£~÷-~)

-"

Accordingly

--

V'--*,l

4_V~.

--

~*"~"~ L V--~,¢

~ ~cv~

"l- v ~

= &... (.,-p)'~ ¢c,,~- c_~ ~,\, v~'l

("Z-v ~)-rr =<-~'~ ~¢.v) + c v ) - C_%--~' z (. v)

v-*~

C~-v') 7e Z~v)

Co-v~) ¢ z].., ~v--~)

~ ÷"'] ~v:.,)

224

The derivatives

are

= [-P""'-~ + ~-F'"'¢+ -,o -F'V"+ -,o-~V " ÷ s--F'-~" + Y-F"-]u > = -2o -f"(.o -F "(.,, =

~'0 :~~'~,~

C~'~,,.,.~--~-co;~)

With

o~lll (-F

-~ ~'~ = E-~"-~ + ~-P".p ÷ ~-F'-F" +-~-;"] C-,) = 0

and

( " - v ~) Ic-) : -- 2. C-, - v " - )

"(.,) :

U' - v ~) c+,~(~) leading

- .2

=

0

; 4/"

~> 3

to

S.-vS~ ~]'' (<~ = [c+-,.)"~z +s-c<_ ~.~"~c~, ~_~o c.-v'~'-fz" +~o C.-~')"~c~' + s-C~-v')'f ~""~- C + - v ~ z''l]C~)

= 5" C~-vg'C~> -/~" ( ~ ) = ~ 21+0

~

(~;~-~Co=~)

z

If we put all of this into C_2 , we get

C_~= E- ~,~o ~ , , + - ~ , ~ + ) ~ - i -~ ~-~-~E+'~',~,~-cc-~-.~'+,,-+)+ + .2_ co; + (~,~',, ~ + - ~z ) ]

_

225

=/-t"

3 ~,:~ ~

"~

(~

~. ;e,, e - e ) + c~; e- (. $ ~ . . / % - e ~ )

C~S -F:. ~.:.~ ~-

-

We can now calculate the K. with 1

(E.14); it follows for K

-2

S'.,:,~ ~ - ( c e . ~ ~ . . ~ , , : . ~ . ~ - ~-.)

a-

which,

~z

5"~:',~e ( 4 - ~-¢_~-) z

together with (E.7) leads to

+

O )

i.e., K(z,v) has only a simple pole at v = I[ (This fact i s decisive for the renormalizability of L(2)). For K=I we need ~7 ( ~ ~. ~'~.t ~ - ~-)

s~ and

C~-~c,~)~

~7. ~ . ~ C ~ - , e o , ~ )

~-~.,~

226

s~e

the

(~-:~-~)

+ c o : e C z ~ % - ~ ~)

/4-

Cog ~. ~'¢~ £: - £

s u m is

LaC_+ L ~ C - a

=-

g

+

+ ~ " ~ ~ t"~- ~-co'E-z-) ~

cot ~. ~'.~, ~- - ~

+

cot ~ (~'~,~ ~ -

~)

so that it follows that

?d'-'1

.~.,t ~ ( L.z C_.~ + L.~ C-z ) +

LCO.~ Z . ~ t ~ -

~ ¢ ~ (4-

-~

+

~=co~:~-) ~

C o S ~.- g ~

~-

- -

co~.~

C r ¢ ~ - ~ ~)

+

,2. ~ ~.~-,.~e C ~ - ~ - c . ~ - )

From this, we get with (E.7) ..p

I

~- ('co~ ;~. ~,:.~ i - ÷ )

;

This expression can be considerably simplified:

--:~

227

?+e* R = ' ~ ' ~

- -

~.

C~-~°'~)]-~[2~'~ ~ " ' ~

- ~

~

~'~,..~ ~

All together, then,

~co~ This is the desired result which we can also write in the form

~cv~= .'/-V "~ z [

,.~¢'t~z~ :~

+ ~CO~ ~

+ (regular terms).

228

Appendix F: Contact Term Determination in Source Theory In the following, the calculation of the two-loop effective Lagrangian in the 7th section is interpreted in the framework of Schwinger's source theory [29,30,47]. In this phenomenological theory, there is no difference made between bare and renormalized quantities; all masses, charges, field strengths etc. which appear are the physical, i.e.,observable parameters. Applied to QED, this means, for example, that the fe~dons should always propagate with their observable mass m, thereby guaranteeing that one substitutes g(p) + ct for the propagator g(p) and defines the contact terms ct at the end of all calculations in such a manner that the pole of g is given by m. On the level of the mass operator, this 'on-shell'-condition is fulfilled by the substitution

The c t

are

t o be d e t e r m i n e d

thereby

from

0 Analogously, f o r

the polarization

tensor,

we s e t

and require that ~ pv (k 2 = O) = O. Our considerations in the third and fourth section show that this is equivalent to the procedure in conventional operator field theory.

229

In Source Theory, we now replace

(7.10) by

(F.I)

and define ct~ in such a way that the fermions always propagate with their physical mass; the ct's result from the re-

quirements

(i)

~.. c~3

CB= o) = 0 (F.2)

(ii) By means of the substitution p = : p' + ~, we can write

(F.I)

somewhat more symmetrically:

L

e •

~'~ d C.lTrJ ~-

At the end, a divergent

constant was incorporated

into ct'

The symmetry of L (2) in both electron propagators

leads to

CtlG = ct2G, which because of (F. 3) implies CtlL = ct2L. Thus it follows that

Z

eL) =

~¢~)

+

~-ct~

!

230

i.e., we only have to calculate the additional term of one to L (2) and can multiply it with 2 to get ct~÷ct~.

electron

We now use the fact that we can also write

L (2) as a folding

of an electron propagator with the mass operator calculated to the order

--

~ C.l.~.) ~

We again introduce contact terms

-"

The c ~

~

JC~-i~

are only responsible for Z (~ = -m) = O, since this

conforms to a propagation with the physical mass, the notation from section 3 or 7, c ~ above formula is not yet correct,

i.e., with

is equal to -6m. The

since c ~

only takes the

mass displacement of one electron of the loop (namely, of the electron in E)

into account; to be correct,

it would have to

read --

Z

,) t'~-,,; '~

= %~c.z)

c~:~

,~,~

-

(s':.4~)

The calculation of L (2)

OF.4) is now accomplished exactly like in

the 7th cahpter; but now, e and B are not renormalized. stead,

in order to fulfill

In-

(F.2), ct' is appropriately chosen.

231

(m, e, B are now the observable quantities

in the entire cal-

culation!) This leads precisely to the subraction of Ko2 , after the singular part of the Laurent series has been removed from the integral. for (F.4):

From our earlier results, we get

(note that c ~

= -6m) , p - V ~.-

o

G

:~.

This,

~TT ~

~

however,

ventional Source

~T

o ==L

--~'~"

is

~,

identical

renormalization

Theory

and Operator

- .

to

2 ,r¢e~s) ~

the

result

(7.37),

i.e.,

Field

Theory

÷ce~.~k~e~)-~l

achieved in this are

by con-

example,

equivalent!

too,

232

Appendix G: One-Loop Effective Lagrangian as Perturbation Series In this appendix,

the expression used in section 5

W c'' F~] = -

I-7 ~

C~-

e

~G+)

-~

for the one-loop effective action will be derived without the use of functional methods

[41], i.e., by calculating all rele-

vant graphs in this approximation with the well-known Feynman rules and then summing them up properly.

For that purpose, let

us first consider the exact effective action F[A], which represents the generating functional of the one-particle-irreducible

(IPl), amputated N-point functions F(N)(xl,...,XN) ;

i.e., if we take the derivatives of this functio~,al N times with respect to A and then set A = O, then we get the corresponding Green's function.

Conversely,

F[A] must then be able

to be written as a sum over the F (N)

N

This is still an exact equation~

in which the F (N) contain

graphs of an arbitrary high order in e 2. If we wish to evaluate the above sum only in one-loop approximation,

then the

only term contributing to F (N) is represented by the graph

FI

233 E x p r e s s e d by f o r m u l a e , t h i s

means

p CNI

/.,...p,,, C~,,-,XN) = (-'4) C~-")! i v [Ce$#~) G+cx,,~,.) (eg7.~) ~" , x ~ . ) .

The individual factors have the following origin: (I) The factor (-I) appears since a closed Fermion loop is present. (2) The factor (N-I)! takes into account that, after fixing one photon line, a total of (N-I)! topologically inequivalent graphs can be created by permutation of the remaining photon lines. (3) One factor (eye) is assigned to each vertex, (4) A free propagator G+ is assigned to each fermion line. (S) The fermion loop requires a Dirac-trace. Furthermore,

it should be noted that r (N) does not, by de-

finition, contain the external propagators. loop approximation,

For F[A] in one-

i.e., W(1)[A], we thus get

.......

e

.

234

I (The factor ~ in fromt of the sum follows

Z = exp(iW).)

Furthermore,

from the definition

it follows in matrix n o t a t i o n

W- 4

-

{

,~-.t

i

lq

N=~

4

='+ T ¢-" I~t%' <x,l£,,,t C~-e,ff~.~)lx,'> 4

--- -.

~Z,~

Uf-e#~.)

Here, we have used the series

--

,

X~

(-4,4)

-vL=~

We can also write our result as

d ~vc"EPc2 = -

t~ .~.~ Ca- e ~ F ' ~ + ) - ~

i.e., we obtain the same expression as the one Fried derived with functional methods,

qed.

[41]

235

Appendix H: Summary of the Most Important Formulae Here we summarize the most important dual sections

in synoptical

formulae of the indivi-

form; explanations

tions are to be found in the respective

of these equa-

chapters.

2nd Section (2.1)

Definition of the fermion propagator

./ ,

(2.6)

, Tr= p-e#

('~'T/-) z=__ 7/"z + ~ ~ v T ~ especially

for ~ = B~:

(,..~//_) 2

yfz+ e ~ _

(2.12) Differential

s

equation for G+ in space representation

I !

(2.47) Electron propagator

in an external constant magnetic

field

(

(straight path of integration)

X"

o

e c~s~

236

3rd Section (3.5)

Space representation of the mass operator (Feynman Gauge)

~c~',,~", -- ~ ' "d' (~+~,e,x',~+c,~'-~", ~ + O(e '~) (3.6)

free photon propagator

(3.8)

transition to momentum representation

c×', ×", =

~C×',x'9oc~.,*~a-~e~pC, - ~L~ ,,, ~

(-P)

4

(3.~)7_(p> = ~ ~r Om~"~~ 'KK~-;t C~Ce- k ~ + Oct", (3.44)

Tsai's stant

representation external

o f the mass o p e r a t o r

magnetic

in a con-

field

~cn'~-~-£'~"~ y°(s.~'~'~c-~s~,.'- ~_i~e_,.~ E~÷e--'~':~Y +c~-~,~-~'×~+(~-,~,~-~~

,,

A " = C'-'u-) ~ ~- 2~.(~-,~-~ ~,:',y ~ y / y (3.45)

Spectral

representation

of

the

y

~

_

+ ,.~(~'~..,yly)" mass o p e r a t o r

for

B = 0

237

4th Section (4.1') Polarization tensor in momentum representation

Definition of the polarization function

~.., Ck)= cOr.,k~- K,. K.,)TFC~.~; (4.32) Tsai's representation of the polarization tensor in a constant external magnetic field 4

_

ts~ z

kz

cos ~v N2:----

' +

~v ~

~

~v

~

(~v-~@)

+

(4.34) Spectral representation of the polarization function for B = 0

238

4

Kz÷ N L.z'~ 5th Section

(s.1)

One-loop vacuum amplitude

(5.13)

"&r G + C×, × I ~

=

(5.23) Integral representation Lagrangian electric

~)=

of the one-loop effective

for a constant magnetic and vanishing

field 4

~.~ ~-~e oS

(s.2s) -1

Ms

o (5.26) Integral representation

of the one-loop effective

Lagrangian for a constant electric and vanishing magnetic

field

239

.-,C")

E)

--

~

-4

--~%

V

(5.27) Representation of LR(1)(B) by the Riemann Zeta-function

-i.,,i. z

(5.29) asymptotic form

./

CT ['P-~" .~---Y

6th Section

(6.3)

Definition of the zeta-function

(6.4)

Definition

of

the

determinant

7th Section (7.2)

Vacuum amplitude

(D.5)

= - IT ..¢_~(~-e~

G.,.) -'4

Functional derivatives of W[A]

240

~ ~

(7.9)

Two-loop e f £ e c t L v e Lagrangian f o r c o n s t a n t f i e l d s

(7.~o)

(7.37) Integral representation ~or LR{Z)

L oo

-~'zs

_ _

46" ?r 3

_

~

. .

z ~F ceB~) ~

+(~Bs)c~ce~s)-2_]

o

--

+

~co'~ ~--~

~--Vz

= s-~.--'-~ L ( 4 - v ~ )

(cos, ~ v - - c o s - e )

-!

.

.

.

.

.

.

241

F.,~(qiV )

(7.43)

: ~

=

-

-

asymptotic

~.

(..~,

form

,-

~,~;~

~~:

8th Section

(8.11) 't Hooft-Weinberg equation for L R = - ~I B 2 L

'9

=0

Z, ~p (8.14)

C~

Callan-Symanzik

~

~es C#-) = Z~ Z

/"a-.,, eL

equation

for

References [I]

G. Mie, Ann. Phys. 37, 511 (1912); 39, I (1913), M. Born, L. Infeld, Proc. Roy. Soc. A 144, 425 (1934), W. Heisenberg, H. Euler, Z. Phys. 98, 714 (1936), V. Weisskopf, K. Danske Vidensk. Selsk.Mat.-fys.Medd. 14, No. 6 (1936), M. Greenman and F. Rohrlich, Phys.Rev D8, 1103 (1973) [2] S.L. Adler, Ann. Phys. (N.Y.) 67, 599 (1971) [3] J. Schwinger, Phys.Rev. 82, 664 (1951) [4] V.I. Ritus, JHTP 42, 774 (1976) [5] W. Dittrich, J.Phys. A, Vol. 10, No. 5, 833 (1977) [6] W.-Y. Tsai, Phys.Rev. D10, 2699 (1974) [7] W.-Y. Tsai, Phys.Rev. DIO, 1342 (1974) [8] W. Dittrich, J.Phys. A, Vol. 9, No. 7, 1171 (1976) [9] K.-H. Zimmermann, Diplomarbeit, Univ. TNbingen (1978) [10] W. Dittrich, W.-Y. Tsai, K.-H. Zimmermann, Phys.Rev. D19, 2929 (1979) [11] S.R. Valluri, D. Lamm, W.J. Mielniczuk,

Phys.Rev

D25,

2729 (I 982) [12] P. Ramond, "Field Theory", Benjamin Cummings, Reading/ Mass., (1981) [13] G. Leibbrandt, Rev.Mod. Phys. 47, 849 (1975) [14] J.C. Collins, Phys.Rev. DIO, 1213 (1973) [15] S.W. Hawking, Com.Math. Phys. 55, 133 (1977) [16] G. Ghika, H. Visinescu, Nuovo Cimente 46A, 25 (1978) [17] G.W. Gibbons, Phys. Letters 6°A, 385 (1977) [18] s. Weinberg, Phys.Rev. D9, 3320 (1974) [19] W. Dittrich, Phys.Rev. D19, 2385 (1978) [20] H.B.G. Casimir, Proc.kon.Ned.Akad.Wetenschap. 5_[, 793 (1948) [21] E.G. Harris, "A Pedestrian Approach to Quantum Field Theory", J. Wiley, New York [22] J. Schwinger, Letters in Math. Phys. ~, 43 (1975) [23] B.V. Deryagin, I.I. Abrikosava, JETP ~, 73 (1956) [24] R.P. Feynman, A.R. Hibbs, "Quantum Mechanics and Path Integrals", McGraw-Hill, New York [25] E.S. Abers, B.W. Lee, Phys.Reports 9C, I (1973 [26] J.C. Taylor, "Gauge Theories of Weak Interactions", Cambridge University Press (1976) [27] R. Jackiw, Phys. Rev. D9, 1686 (1974)

243

[28] S. Coleman, E. Weinberg, Phys.Rev. D7, 1888 (1973) [29] J. Schwinger, "Particles, Sources and Fields", AddisonWesley Co., Reading/Mass. (1970/1973) [30] W. Dittrich, Fortschritte der Physik 26, 289 (1978) [31] J.Z. Kaminski, Act. Phys.Austr. 53, 287 (1981) [32] R. Coquereaux, Ann. Phys. 125 (1980) [33] C. Nash, "Relativistic Quantum Fields", Academic Press

(1978) [34] F. Ravndal, "Scaling and Renormalization Groups", Nordita (1976) [35] J.C. Collins, A.J. Macfarlane, Phys.Rev. DIO, 1201 (1974 [36] I.S. Gradstein, I.M. Ryshik, "Tables of Series, Products and Integrals", Harri Deutsch, Thun, Frankfurt/M. (1981) [37] J.D. Bjorken, S.D. Drell) "Relativistische Quantenmechanik", "Relativistische Quantenfeldtheorie", Bibliograph. Inst., Mannheim [38] V. Schanbacher, Diplomarbeit, Univ. TNbingen (1979) [39] K. Nishijima, "Fields and Particles", W.A. Benjamin (1969) [40] W.J. Mielniczuk, J.Phys. AIS, 2905 (1982) [41] H.M. Fried, "Functional Methods and Models in Quantum Field Theory", MIT Press, Cambridge (1972) [42] E. de Rafael, J.L. Rosner, Ann. Phys. 82 369 (1974) [43] M.R. Brown, M.J. Duff, Phys.Rev. D11, 2124 (1975) [44] A. Messiah, "Quantenmechanik", de Gruyter (1976) [45] M.J. Teper, "Instantons @-Vacua, Confinement,...", Rutherford Laboratory (1980) [46] R.J. Hughes, Nucl. Phys. B186, 376 (1981) [47] W. Dittrich, "Lectures on Schwinger's Source Methods in Quantum Field Theory", Univ. TNbingen, WS 1975/76 [48] N.D. Birrell, P.C.W. Davies, "Quantum Fields in Curved Space", Cambridge Univ. Press (1982) [49] S. Weinberg, "Gravitation and Cosmology", John Wiley (1972) [50] S.L. Adler, Rev.Mod.Phys. 54, 729 (1982) [51] P. Becher, M. B6hm, H. Joos, "Eichtheorien der starken und elektroschwachen Wechselwirkung"~ Teubner, Stuttgart

(1981) [52] H. Mitter, Physik. Bl~tter 24, 159 (1968); 2_J, 348 (1968)

244

[53] C. Itzykson, J.B. Zuber, "Quantum Field Theory", McGrawHill, New York (1980) [54] J.S. Dowker, R. Critchley, Phys.Rev. D13, 3224 (1976) [55] Y. Nambu, Phys.Rev. Lett. 4, 380 (1960), J. Goldstone, Nuovo Cimento 19, 154 (1961) [56] R. Jackiw, Lectures given at Les Houches, 1983, MIT print CTP 1108 [57] H. Pagels, E. Tomboulis, Nucl. Phys. B143, 485 (1978) [58] A.B. Migdal, Nucl.Phys. B52, 483 (1973) [59] S.L. Adler, "Dynamical Applications of the gauge-invariant effective action formalism", to appear in the Bryce DeWitt 6Oth Birthday Quantum Gravity Volume [60] F.J. Yndur~in, "Quantum Chromodynamics", Springer (1983) [61] W. Dittrich, M. Reuter, Phys. Lett. 128B, 321 (1983) [62] S.L. Adler, "Non-Abelian Statics", Talk presented at the Maxwell Symposium, Amherst (1981) [63] S.L. Adler, Phys.Rev. D23, 2905 (1981); Phys.Lett. 11OB, 302 (1982); ibid. 113B, 405 (1982); ibid. 117B, 91 (1982); Nucl.Phys. B217, 381 (1983), S.L. Adler in workshop "NonPerturbative QCD", Stillwater, Oklahoma 1983; K.A. Milton and M.A. Samuel, Eds. Progress in Physics Vol. 8, Birkh~user Boston-Basel-Stuttgart, 1983 [64] H.B. Nielsen in Proceedings of the 3rd Adriatic Summer Meeting on Particle Physics, I. Andri~ et al., eds. NorthHolland Publishing Co., 1981

Lecture Notes in Physics Vol. 195: Trends and Applications of Pure Mathematics Mechanics. Proceedings, 1983. Edited by PG. Ciarlet M. Roseau. V, 422 pages. 1984.

to and

Vol. 196: WOPPLOT 83. Parallel Processing: Organization and Technology. Proceedings, 1983. by J. Becker and I. Eisele. V, 189 pages, 1984.

Logic, Edited

Vol. 197: Quarks and Nuclear Structure. Proceedings, Edited by K. Bleuler. VIII, 414 pages. 1984.

1983.

Vol. 198: Recent Progress in Many-Body Theories. Proceedings, 1983. Edited by H. Kiimmel and M. L. Ristig. IX, 422 pages. 1984. Vol. 199: Recent Developments dynamics. Proceedings, 1983. D. Jou and G. Lebon. XIII, 485 Vol. 200: H.D. Seiten. 1984.

Zeh,

Die

der

Zeitrichtung.

V, 86

Vol. 201: Group Theoretical Methods in Physics. Proceedings, 1983. Edited by G. Denardo, G. Ghirardi and T. Weber. XXXVII, 518 pages. 1984. Vol. 202: Asymptotic Geometry. Proceedings, 213 pages. 1984.

Vol. 204: Y. Waseda, Novel Application (Resonance) X-Ray Scattering for Structural tion of Disordered Materials. VI, 183 pages.

in

of Anomalous Characteriza1984.

Vol. 205: Solutions of Einstein’s Equations: Techniques Results. Proceedings, 1983. Edited by C. Hoenselaers W. Dietz. VI, 439 pages. 1984.

and and

Vol. 206: Static Critical Phenomena in Inhomogeneous Systems. Edited by A. Pekalski and J. Sznajd. Proceedings, 1984. VIII, 358 pages. 1984. Vol. 207: S. W. Koch, sitions in Equilibrium 148 pages. 1984.

Dynamics of First-Order and Nonequilibrium

Vol. 208: Supersymmetryand QCD. Proceedings, 1984. V, 389 pages. 1984.

Phase TranSystems. Ill,

Supergravity/Nonpetturbative Edited by P Roy and V. Singh.

Vol. 209: Mathematical and Computational Methods in Nuclear Physics. Proceedings, 1983. Edited by J. S. Dehesa, J.M.G. Gomez and A. Polls. V, 276 pages. 1984. Vol. 210: Cellular Structures 1983. Edited by J. E. Wesfreid 1984.

in Instabilities. and S. Zaleski.

Discrete

Spin

Models,

VII,

Vol. 215: Computing in Accelerator Design and Operation. Proceedings, 1983. Edited by W. Busse and R. Zelazny. XII, 574 pages. 1984. Vol. 216: Applications chanics. Proceedings, pages. 1985.

of Field Theory to Statistical Me1984. Edited by L. Garrido. VIII, 352

Vol. 217: Charge Density Waves in Solids. Proceedings, 1984. Edited by Gy. Hutiray and J. Solyom. XIV, 541 pages. 1985.

Vol. 219: Fusion Reactions Proceedings, 1984. Edited pages. 1985. Vol. 220: Quantum

Behavior of Mass and Spacetime 1983. Edited by F J. Flaherty. VI,

Vol. 203: C. Marchioro, M. Pulvirenti, Vortex Methods Two-Dimensional Fluid Dynamics, Ill, 137 pages. 1984.

Classical,

Vol. 218: Ninth International Conference on Numerical Methods in Fluid Dynamics. Edited by Soubbaramayer and J. l? Boujot. X, 612 pages. 1985.

in Nonequilibrium ThermoEdited by J. Casas-Vazquez, pages. 1984.

Physik

Vol. 214: H. Moraal, 251 pages. 1984.

Proceedings, VI, 389 pages.

Vol. 211: Resonances - Models and Phenomena. Proceedings, 1984. Edited by S. Albeverio, L. S. Ferreira and L. Streit. VI, 369 pages. 1984. Vol. 212: Gravitation, Geometry and Relativistic Physics. Proceedings, 1984. Edited by Laboratoire “Gravitation et Cosmologie Relativistes”, Universite Pierre et Marie Curie et C.N.R.S., lnstitut Henri Poincare, Paris. VI, 336 pages. 1984. Vol. 213: Forward Electron Ejection in Ion Collisions. Proceedings, 1984. Edited by K.O. Groeneveld, W. Meckbach and I. A. Sellin. VII, 165 pages. 1984.

Below the Coulomb by S.G. Steadman.

W. Dittrich, M. Reuter, Effective Electrodynamics. V, 244 pages.

Barrier. VII, 351

Lagrangians 1985.

in