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. 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¢

= ' & 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,

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 =7"

~

" ' "

Cx,,I ....

~/"*N (z~J

N

(1.47)

(Recall that each j~(x) contains a factor of e). Applying Wick's theorem to the right-hand side of (1.47), we see that we have to sum up an infinite sequence of terms represented by diagrams like

where the wavy lines denote interactions with the external field (no photons). This summation can be done explicitly [53] and the result is again (1.44).

(Note that (1.47) also contains

disconnected pieces like the second diagram above; owing to the "connectedness lemma" [41,25], these are not present

in

W = -i in function equation:

we make t h e A n s a t z

_

.,,t x,,fit A

~t

with

of G+:

(,~x ~l

xf

(2.14)

and

(2.1 5)

~(/~Cx'):'- -- Km ~-J'~(xU~"Jv just as in (2.7). It is easy to show that the integral in (2.14) is independent

of the choice of the path of integration, the integrand vanishes.

since the curl of

If one chooses a straight line as inte-

gration path,

..~(-e~ = X ~ + - ~ ( . ~ x

")

(

~-e Co,~1

one finds that the second term of the integrand gives no contribution. For a straight integration path, then

In addition, we need the derivative upper limit of the path integral: X t

of (2.14) with respect to the

31 If we then substitute the Ansatz

(2.13) into (2.12) we obtain

a differential equation for &+ [A']:

[(tg'-

e¢t')>'+ ::x"]. A+Cx:x"l lo,') = <Scx'-x"j

(2.18)

With the definition of A', eq. (2.15), we then end up with the defining equation for &+ [A']:

[-- ¢"52+

3.t_z_~_e

z

x~ :F'Z2~x,]Zi.(xl~')= ~tx~ (2.19)

Obviously, for a constant field the operator in the square bracket in (2.18) and (2.19) is translational invariant since it contains A (x) only in a gauge invariant form as F

. Furthermore, use has

been made of rotational invariance with respect to the z-axis which implies F P V ( X ~ v - X v S p ) A + ( X ] A ') = O. (See [29] for details). Finally, F 2~v stands for F ~

F~ .

Equation (2.19) can be solved by a Fourier Ansatz

[43]

A+( k I gt')

(2.203

thereby converting (2.19) into

[" kZ

eZ ~f.*) F22,, ~C~') ,M.~ ~'t] A.+. -~ +

+ %-" 7"

( k

I£ ¢)

-- 1

(2.213

Now we shall try to solve this equation by an Ansatz of the form

~'~

~.+(kllBi.'):

i I~ts

-- MC ,'s) - i s C ~ Z - f z e

•)

e

with

(2.223 (2.23)

~c~s~ -- k ~ , ~

(,:s) k r~ + ¥ ~ s )

. X.~= X ~

Inserting (2.22) into (2.21) then yields

0

(2.24)

32 This equation for determining X and Y takes the form

0o

_

i ~s

9c~'s) e

1Cc z,s)

= 1

(2.25)

0

(2.26)

If we were now to set

9 (,'s)

vr rc

this would, give

(2.273 that is,

(2.25)

is solved by our Ansatz

if there are solutions

of (2.26) with

4'co

o

=

,

The relation

k F"l -i-

(2.26)

e

(2.28)

reads in our case

the integration path according to is ÷ s and

(2.23),

~/+

oo

X(~?S)"~,~V(~s)]k-- "~ "~r("~"z)(~.d.~O= /'~'1'(: z'$)(2.29)

e. z

If we rotate we use

=

it follows

e ~

that

Xc~) T ~ X c s )

=

2

(s~

(2.30)

2

'X

= ~'

As one can easily prove by differentiation,

these equations

are solved by

(2.31)

Ycs~ Evidently X(O) conditions

in

~ ~r /~ cos ( e T a )

= Y(O)

= 0 is valid so that the first of the

(2.28) is satisfied.

In order to further evaluate tage of the special

these solutions,

we take advan-

form of the field strength tensor:

33

(ooo)

(o o)

e -~ o o

0

4

o o

0 (2.32)

Then we find

It

i s now c o n v e n i e n t

to i n t r o d u c e

cz,,-- (a °, O~ 0 , ~ )

the notation

~

,

(o,..6),, = - - ct~b°+o..'~b ~ ,

:

= (0, cC~

c~ ~,

o)

( o . b & : = cCb "~+e,.Zb~"

for arbitrary Lorentz vectors

(2.34)

a, b.

Then

~ (Note t h a t

X = (X 6).)

(e Bs)

Accordingly

z

(2.3s.)

we g e t

Ycz~ = { £r & c~s c~eTs~ (2.36)

= L, c~sCeBs) Let us recall the definition

~c~'s) -- t~Ic~'s)+ 7 ' s ( ~ K , ' £ ) = Thus, i t

becomes c l e a r

the second c o n d i t i o n

that (2.28):

kXt,';;l< + Y c , ' s ~ - , , ' s ( 3 Z ~ ' a ) the solutions

(2.31)

also fulfill

34 From (2.20),

(2.23) and (2.31) it now follows for the

scalar Green's function &+: oo 0

~

C~..rr}~.

-~(~;'+kZ

~-~*~-'~

~

(z.38)

Cos

(2.39)

wi th But beginning directly with the propagator

&+CX',X"II@) = < X ' l

lx'~> (2,40)

- z's (3~ ~--

= i j'c4s

~'~)

#

a comparison with

(2.38) shows that

~ C

-;,,('¢+'"*.,.T "~""' J (2.41)

From this important formula, further useful relations can

be

derived; let us substitute s ÷ aas; then Q

cost% ~)

(2.42)

Ccx.c~)

Because < x l l e - ~'s%- Tr ~~1~"~

= o" ~ , - e~c~',)~ <x'te -'~'Tr'~'>

(2 43)

we get

=

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



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',~,
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

+

=

~. <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

'+ ~'-~'~'c~. -

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

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

>

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

t



~'r P"~- > 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


with

according

as that state

true

= I).

~'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

(eE0).

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-~'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 = 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>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~

[,>> 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

get

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