Axiomatic field theory and quantum electrodynamics : the massive case

A NNALES DE L ’I. H. P., SECTION A

O. S TEINMANN

Axiomatic field theory and quantum electrodynamics : the massive case

Annales de l’I. H. P., section A, tome 23, n

^o

1 (1975), p. 61-97

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61

Axiomatic field theory ^and quantum electrodynamics :

the massive

case

O. STEINMANN

Universitat Bielefeld, Fakultat fur Physik, Bielefeld, Germany

Vol. XXIII, ^n° 1, 1975, Physique théorique.

ABSTRACT. - Massive quantum

electrodynamics

^{of the electron is formu-}

lated as an LSZ

theory

^{of the}

electromagnetic

^field and the electron-

positron

^fields

1/1, ~.

The interaction is introduced with the

help

^{of mathe-}

matically

^{well defined}

subsidiary

conditions. These are:

1 )

gauge invariance of the first

kind,

^{assumed to} be

generated by

^{a conserved}

current j ; ²⁾

^the

homogeneous

^Maxwell

equations

^{and a} massive version of the inhomo- geneous Maxwell

equations; 3)

^a

minimality

^condition

concerning

^the

high

momentum behaviour of the

theory.

^{The «}

inhomogeneous

^Maxwell

equation »

^{is a} linear differential

equation connecting F /lV

^{with the}

current j .

No

Lagrangian,

^no non-linear field

equations,

^{and no}

explicit expression of ju

^{in terms are} needed. It is shown in

perturbation theory

^{that the}

proposed

conditions fix the

physically

^relevant

(i.

^e.

observable) quantities

of the

theory uniquely.

1. INTRODUCTION

Any

attempt ^to fit quantum

electrodynamics (henceforth

^called

QED)

into the framework of axiomatic field

theory

encounters two types ^of

problems, namely

1 )

^the

general problem

^of

characterizing particular

^{models without}

using

^such

mathematically

dubious notions as

Lagrangians,

^non-linear

equations

^of

motion,

^or

equal

^time commutators of

interacting ^fields, 2) problems specific

^to

QED,

^{which are} connected with the

vanishing photon

^mass. The best known of these are the infrared

divergences

^{of the}

S-matrix,

which prevent the

application

^{of the LSZ formalism in its esta-}

Annales de l’Institut Henri Poincaré - Section A - Vol.

blished form. It is

known, furthermore,

^that

QED

^{does not even fit into}

the more

general Wightman

^framework

[7]:

ⁱⁿ

formulating QED

^{as a}

field

theory

^{one must violate one or several of}

Wightman’s

^axioms.

In the present work ^we shall deal

only

with the first type ^of

problems.

The difficulties mentioned under

point 2)

will be avoided

by giving

^the

photon

^{a small}

non-vanishing

^{mass A. We}

hope

^to make the limit A -~ 0 the

subject

^{of a}

subsequent

paper.

We consider the

QED

^{of a}

charged spin 1 2 ^particle,

^{called «}

electron »,

with mass m &#x3E; 0. This

theory

^{we want to} formulate as an LSZ

theory

of the

electromagnetic

^field

Fuv

^{and the}

electron-positron

^fields

~, ifj.

The interaction shall be

specified

^{with the}

help

^of

mathematically

^well-

defined

subsidiary

conditions.

In Section 2 we consider the LSZ formulation of the

theory

ⁱⁿ

question.

In

particular

^we discuss the GLZ

theorem,

^which

permits

^the

complete

characterization of the

theory by

its retarded functions.

In Section 3 we propose ^{a set} of conditions

singling

^out

QED

^{from all}

possible §-theories.

^{We do not use}

Lagrangians

^or non-linear field

equations.

The conditions are formulated

directly

for the field The introduction of a vector

potential

^{is not} necessary,

though

^{later on it will}

turn out to be useful as a means of

algebraic simplification.

^{Hence no}

mention will be made of _gauge transformations of the second kind.

They

cannot be fitted

easily

into the axiomatic

frame,

because in

general they

do not respect translation invariance. Since

they

^{leave the} observables

completely untouched,

^their

physical significance

^is anyway ^not

clear,

^so that their absence should not be considered a flaw of our formalism

( 1 ).

Our conditions are of the

following

^{kind. shall}

satisfy

^{the homo-}

geneous and ^a massive version of the

inhomogeneous

^Maxwell

equations.

The latter are linear

equations

^{between and the}

electromagnetic

current so that no distributionistic difficulties arise. An

explicit

expres- sion

of ju

^{in terms of}

~, ~

^{is not} needed. The

coupling

^to

~, ~

is achieved cia the commutation relations of these fields with the _space

integral over jo,

the

charge Q.

^In

addition,

^we

postulate

^a

minimality

condition for the behaviour of certain

physically important quantities

^at

large

^4-momenta.

We have no

general proof

^that these conditions do indeed

specify

^a

theory.

^{In Section 4 we}

show, however,

^that

they

^can be satisfied in all finite orders of

perturbation theory

^and determine there the

physically

relevant

quantities uniquely (2).

^{For this we follow} the methods

developed

(1) The author is aware that this attitude towards the gauge group is not in accordance with the fashion of the day.

(~) Perturbative QED ^has already been discussed in a similar vein, but on a lower level of rigour, ^{some time} ago by Nishijima [3]. Nishijima considered directly ^{the case} A ⁼ 0, disregarding the infrared problems.

Annales de l’lnstitut Henri Poincaré - Section A

for the

simpler

^{case of a}

single

scalar field in Ref.

[2],

henceforth

quoted

as B. The results of B will be used

freely.

^Their

generalizations

^{to the} present

case are

mostly straightforward

^{and will not be}

explicitly proved. Equa-

tion

(n. m.)

and Theorem n. m. of B will be

quoted

^as

Eq. (B.

^n.

m.)

^and.

Theorem B. n. m.

respectively.

Section 5 is devoted ^to a brief discussion of observable fields other than and

ju.

As was to be

expected

^{we do not}

gain

any fundamental new

insights,

but

reproduce merely

^{in a}

mathematically

clean way ^some well-known results of the canonical formalism.

Also,

the expert reader will

easily perceive

^{that our} methods of

proof

have often been

inspired by

^{the corres-}

ponding

canonical considerations.

2. THE LSZ FORMALISM

The

generalization

^{to massive}

QED

^of the formalism discussed in B for a scalar field does not present any fundamental difficulties. Therefore

we

only

collect here without

proofs

the facts and notions which we shall need later on.

We consider a

theory

^{of two}

four-component spinor

^fields

~, ~

^{and a}

real

antisymmetric

^{tensor field which}

satisfy

^{all the}

Wightman

axioms

[4] [5].

^In

particular

^{we assume} invariance under the orthochronous Lorentz group

Lt, including

^the

parity

component. Our

theory

^{will also}

be

C-invariant, hence,

^{due to the CTP}

theorem,

^also

T-invariant,

^{so that}

the latter invariance need not be

postulated explicitly.

We use the

following

notations: arguments of

~-fields appearing

ⁱⁿ

retarded

products

^{and their vacuum}

expectation

^{values are denoted}

by

^{small latin letters :} x, y, ^..., p, q, ^..., arguments ^of

~-fields by

^barred

small latin letters: x, y, ^..., p, q, ^..., arguments of

by

^small

script

^{letters: ...,}

_~,

~, ^.... The bar over a variable is not part ^of the

variable,

^but

signifies only

^the occurrence of this variable as a ment. The same variable x may ^occur in a

given

mathematical

expression

once with a

bar,

^{once without a}

bar,

^e. g. in different

x-dependent

^factors

of a

product.

^Small

greek

^{letters :}

~,

~, ..., p, (D, ^... stand for variables which _may be arguments of any type ^of field. As in B we use

capitals

^to

denote sets of small letters of the same

character,

^e. _g. X

= {

^...,

~c" }.

I X ~

stands for the number of elements in the set X

(note

^{that this conven-}

tion differs from B,

where ! X ) I

had another

meaning).

We use Dirac matrices yo, ... , y~ with

gllv the Minkowski tensor defined with

signature ( + - - - ).

yo is hermi-

tian,

yt ₂₃ ^are anti-hermitian. Vector indices are raised and lowered with

Vol. XXIII, n° ^{1 - 1975.}

the metric tensor Indices

occurring

^{once as}

subscript,

^{once as} super-

script

^{are summed over. For V~ an}

arbitrary

^{4-vector we define}

The fields are connected

by

The field

Fuv

^shall

^satisfy

^the

homogeneous

^Maxwell

equation (HM equa-

tion)

^{_ _ .. _ _ _ u}

with ~03B1

a

= 20142014 .

^.

For

large positive

^or

negative

^times

~,

^{F are assumed to} converge in the LSZ sense

[6]

^to free fields

Fex,

^{where ex} stands for in in the

case t -~ 2014 ~c, for out in the case t ^~ + oo. The

asymptotic

fields shall

satisfy

^{the Dirac}

equations

and the Klein-Gordon

equation

respectively, ^with

(2.7)

The Fourier transform _{of any} field is defined as

Since we shall

mostly

^{work in} p-space ^we shall henceforth omit the tilde in

~p.

^{It will}

usually

^{be clear from the} argument of cp and from the context

whether we are in _x-space or p-space. In ^case of doubt this will be

explicitly specified.

The free fields are in p-space of the

form, dropping

^the super-

script

^{ex for the moment:}

=

0(± Po)~( P2 - m2)

^and

They satisfy

the anticommutation relations

from which the anticommutators of the creation

operators § - , $ -

^{and the}

. Annales de l’lnstitut Henri Poincaré - Section A

destruction

operators ~+, ~+

^are

easily

^obtained. For the free electro-

magnetic

field Fex we have

We assume

asymptotic completeness:

where ~

^{is the total} Hilbert _space of the

theory, ~ex

^{the Fock} space of the fields The

identity operator

ⁱⁿ

~’n

^{can be written}

with

Eap

^the

orthogonal projection

^onto the a-electron

03B2-positron subspace

of the Fock _space,

E~

^the

projection

^{onto the}

y-photon

space in the F"~ Fock _space. These

projections

^{have the} representations

Here

cv( p) _ ( p2

⁺

m2)1/2,

⁼

(.2 + A2)1 i2. ~ 10) 0 I

^{is the}

projec-

tion onto the

vacuum

⁰

).

^The arrows over the

products

ⁱⁿ

(2.17)

^mean that in

171

the factors stand in order of

ascending

indices i, ⁱⁿ

fi

in order of

descending

^i.

The GLZ theorem

[7]

^{tells us that we can} characterize the

theory by specifying

its retarded functions

[8].

^{Due to the}

anticommutativity

^of

spinor

fields there are some trivial

changes

^of

sign,

^{relative to the} scalar case, in the definition of retarded

products.

In Ruelle’s formal definition

[9]

of a retarded

product (or

^a

generalized

^retarded

product)

as a sum over

permuted products

of fields

multiplied

^with

appropriate

step

functions,

(3) Note that the mass in the definition of ð:t ^{is In or A} according ^to the character of its argument. We hope that this illegitimate notation will not lead to confusion.

we have an additional minus

sign

^{for the terms} in which the

ordering

^of

the

spinor

variables differs from a standard

ordering (defined

^{as the order-}

ing

exhibited in the argument ^of

R( ... )) by

^{an odd}

permutation.

^{In the}

axiomatic definition of B we

replace Eq. (B. 2.22) by

...,

Çn }

^{is a set} of variables of

arbitrary

type. ^{The tensor} indices of F and the

spinor

indices of

~, ~

^{are to} be considered part ^{of the}

corresponding

^variables

~,.

This convention will be used

throughout

^this

paper. In ^cases where

explicit

exhibition of these indices is

desirable, they

will be shown as

subscripts standing

in front of the variable. For

instance,

the

expression

/l"X in the argument ^{of a retarded}

product

^{or function}

means that x is the argument ^{of a field while} x denotes the _p-com- ponent of the 4-vector ^x.

The sum in the

right-hand

^{side of}

(2.19)

^{extends over all}

partitions

of E into two

complementary

^subsets

3L

^and An anticommutator

occurs if both factors

R(~, 3L)

^and

R(’1,

^{contain an} odd number of

spinors,

^a commutator

otherwise. ~L

^{= t 1 is the}

parity

^{of the}

spinor

variables in the

ordering E:L, 1], Ep

^as

compared

^{to their}

ordering in ç,

1], E. Within

E~,

the variables stand in the same order as in E. The alter- native

sign

in the left-hand side is

positive

^if

both ç and 1]

^are

spinor variables, negative

^otherwise.

The retarded

function

^{is the vacuum}

expectation

^{value of}

R(E:).

Amputation

^{of a} retarded function with respect ^{to a}

photon

variable ~ in p-space ^means

multiplication

^with

(~ 2014 A2}.

For the electron momenta p, q

we use two different

amputation prescriptions:

The

amputated

^{variables are}

separated

^{from the}

non-amputated

^ones

by

a semi-colon. We shall never have occasion to use the

completely

unampu- tated r-functions. Therefore we can

again drop

the index n in r", ^{with the}

understanding

that henceforth

r(...)

^{will stand for}

n( ... ).

The reduction formulae for matrix elements of fields and retarded pro- ducts between in-states look

exactly

^{like in the scalar case}

(see

(B.

2 . 32)),

with the r"-functions used.

Annales de l’lnstitut Henri Poincaré - Section A

The GLZ

equations

^are obtained from the relations

(2.19) by forming

the vacuum

expectation value, inserting

^the

identity representation (2.16)- (2.18)

^{on the}

right-hand

^{side and}

expressing

^the

resulting

^{in matrix ele-}

ments with the

help

^of the reduction formulae. We obtain the

following equations,

^written in p-space, for the

totally n-amputated

r-functions :

8L and the

sign

^on the left-hand side are as in

(2.19).

pi,

p2, S2,

^{are called}

external

variables, S, T, 2,

internal variables. Let

NL

be the number of external

spinor

variables in the left-hand r-factor of the

integrand.

^Then

(4) 8~

⁼

( -

^{1 The tensor} indices of the internal F-variables _are,

according

to

(2 .18),

upper indices in the left _r, lower indices in the

right

^{r. With} respect

to the

spinor

indices matrix

multiplication

^is

implied.

^For the variable _sl,

e. g., this

looks,

^{written out}

explicitly :

with ⁼

By

^a

simple generalization

of Theorems B . 2 . 1 and B. 2. 2 a solution

{ r(Q)}

^{of the GLZ}

equations (2.21)

^{defines a field}

theory

of the desired type,

provided

that the distributions

r(Q) satisfy

^the

following

conditions :

a) Reality.

- The relation

(2.3)

^{leads to}

Here

yq~

^{acts on the}

^spinor

^index

^belonging

to q ^and

analogously

^for

yp~.

The

ordering

of the variables in the argument of rp need ^not be as shown here. It can be

arbitrary,

^{but it must be the same on} both sides

(5).

^In

(2.22)

(4) Because of the well-known fermion superselection ^rule only ^{the r with an even number} of spinors ^{can be non-zero. Hence} G~ ⁼ G~ except in the trivial case of an odd number of external spinor variables, ^{in which case} both sides of (2 . 21 ) ^vanish identically.

(5) ^{Unless noted} otherwise, this remark will also apply ^{to similar} expressions ^{in the} future.

Vol.

we have

assumed P ~ = Q ~

^{as will be the case in}

QED,

^{due to}

charge

conservation.

b)

Covariance. ^- Under orthochronous Poincaré transformations ; transforms like the

corresponding product

of classical fields

F, ~.

c) Symmetries.

^{- We have}

where the minus

sign applies

^{if the}

exchanged

^{variables are both}

spinor variables,

^the

plus sign

in all other cases.

(2.23)

^{does not hold if one of the}

exchanged

variables is the foremost

standing

^{one. This}

exceptional

^first

variable will henceforth be called the ^«

distinguished

variable »

... ).

We shall

occasionally

^{take the}

liberty

^{of not}

putting

^{it at} the front of the argument.

Note

^{that the two r in}

(2 . 23)

represent different functions

if pt

^{and 1}

are not of the same type, ^e. g.

r(x, y) ~ r( x, y).

^We

apologize

^{for this}

possibly confusing

notation which has been introduced to avoid an even more

confusing proliferation

of indices.

d) Support.

^- In x-space ^...,

~n)

vanishes outside the ^set

(ç - ç)

^E

V +,

^di.

e)

^{Mass shell} restriction. The restriction of r to the mass shell in several or all of its variables exists and satisfies the smoothness _pro- perty

(B . 2 . 43),

^{which we will not} repeat ^{here. It} guarantees ^{the local} existence of the

integrals

ⁱⁿ

(2 . 21 ).

Since this condition is

automatically

satisfied

by

^the

perturbative

construction of B we will not consider it _any further.

f)

Normalization

ql’

^the

2-point functions.

^{- For}

p2 (m

⁺

A)2

^and

jt2 ^4A2 respectively

^{we have}

with

Fi analytic.

^{The other}

2-point

functions vanish

identically.

As a further condition we have the HM

equation (2.4)

^{which has no}

equivalent

ⁱⁿ the scalar case of B. Before

formulating

^{it as a condition}

for r we must make a preparatory remark. The fields defined

by

^{a solution}

of

(2 . 21 )

^{with all the necessary}

properties

^are

explicitly given by

^their

Annales de l’lnstitut Henri Poincaré - Section A

Haag expansion.

^{Let be} any local field of our

theory,

^{i. e.}

Fuy,

^{or any}

of the fields ^to be introduced later on. Then we have in n-snace

Summation over

corresponding spinor

^{and tensor indices in r and the in}

fields is understood. Because of the relations

following

from the Dirac

equations (2.5)

^{we can}

replace

^r

by

^{r" in}

(2.26).

The combinatorial coefficient in front of the

integral acquires

^{then the}

additional factor

(2m)°‘( -

Since the fields

~, ~, ^F,

determine the

theory completely,

the r-functions

are

physically

^relevant

only

^{in so far as}

they

contribute to the

Haag

expan- sions of these fields. More

exactly:

^{two sets} of retarded

functions {r1}

and { s~2 ~

^{for which}

~’ i ~P ~ Jf, P, Q) = ~~ P, Q)

^for

A2, p~

⁼

^{qh -}

^rn2

are

physically equivalent.

After this side remark we return to the HM

equations.

g) Homogeneous

^Maxwell

equations.

^{- We must have}

Condition

(2.28)

reflects that the ra

occurring

^{in it}

is,

^for

k0

&#x3E;

0,

apart

from a numerical factor the matrix and

for

’0

0 a similar matrix element with the

one-photon

^{state on the}

right. (2.29) stipulates

^that every ^term in the

Haag expansion

^{of F satisfies}

the HM

equation. (2.28)

^{is not a} consequence of

(2.29)

^{and the}

asymptotic condition,

^{because the}

Haag expansion

presupposes

validity

^of

(2.28).

We end this section with a remark

concerning

the normalization condi- tions

(2.24).

^In

anticipation

^{of the limit A -~ 0 to be}

performed

^{at a later}

occasion it v’ould be desirable to

generalize

these conditions to

with Z2 &#x3E; 0 an

arbitrary

^{function of A. An} additional factor Z- 2)’ must

then be inserted in the GLZ

equations (2. 21)

^{and the}

Haag expansion (2.26).

We can,

however,

^{at once find a}

solution {

^{of the}

generalized

^{case from}

a

solution r}

^{with Z = 1, to wit:}

Hence we can put ^{Z = 1 without}

restricting generality.

3. FIXING THE

INTERACTION

From the collection of theories covered

by

^the

general

^{formalism of}

Section 2 we want to

single

^out

QED by appropriate subsidiary

conditions.

Firstly

^{we demand} gauge invariance of the first kind. _{A gauge} transfor- mation of the first kind is a substitution

with a a real number. These transformations form an Abelian _group.

Invariance of the

theory

under this _group means existence of a continuous

unitary representation U(oc)

^with

By

Stone’s theorem there exists a

self adjoint

^operator

Q’

^{such that}

Gauge

symmetry ^{shall be}

generated by

^a conserved current, the ^electro-

magnetic

^{current. This means that}

in §

there exists a vector field

satisfying Wightman’s ^axioms,

^{which is local relative to}

^F, gl,

^{and is}

conserved: _ - ,- ",

such that

with e a real number which will serve as

coupling

^{constant. For the exact}

mathematical sense in which the

integral (3.9)

^must be understood we Annales de l’lnstitut Henri Poincaré - Section A

refer to the review article

by

^Orzalesi

[10]

^{and the}

original

papers

quoted

there.

Q

is assumed to commute with

jP-.

The

current

^{shall be}

coupled

^{to the field}

F p-v ^through

^{the «}

^inhomoge-

neous Maxwell

equation » (I M equation)

Note that this

equation

is linear in the distributions F

and j,

hence mathe-

matically meaningful.

^{We need no}

explicit expression for j

^{in terms}

of t/1

and

~.

A final condition

concerning

^the

high

^{momentum behaviour of the}

theory

^will yet ^{have to be} introduced. But first we want to transcribe the conditions formulated _up to now into

properties

^of the retarded functions.

We consider now also retarded

products containing j/l-fields.

^Their argu- ments will be denoted

by

^barred

script

^{letters :}

~

^...,

~,

^....

j-variables

are never

amputated.

It is well known that _gauge invariance

implies vanishing

^{of the}

Wightman functions,

^and hence the retarded

functions,

^with

unequal

^numbers

of 03C8

and §

variables: ^.

The conservation

equation (3 . 8)

^{and the I M}

equation (3.10)

^{can be}

translated

by analogy

^to

(2.29)

into conditions on the

Haag

coefficients

of j

^{and F.}

For

fixing

the numerical value of e we need a normalization condition:

This we obtain

by inserting

the definition

(3.9)

^of

Q

^into

and

expressing

^the

resulting

matrix element

of jo

with the reduction formula.

The result is

.. ~ ...~ . ~ a~

.rU wlrl ^{. i}

which

generalizes by

covariance to

Conversely,

^{let us assume}

(3 .11 ), (3.14),

^{and the}

validity

^{of the} IM equa- tion and the conservation

equation

^{for the}

Haag

coefficients of F

and j respectively.

^{From the latter two}

assumptions

^{we find at once that the}

fields F

and j

themselves

satisfy

^{the IM}

equation (3.10)

^{and the}

divergence

condition

(3.8) respectively.

^But

(3. 8) implies

^{that the} operator

Q

^defined

Vol.

by (3.9)

annihilates the vacuum

[10]. Making

^{use of results due to Kraus}

and Landau

[11]

^we find furthermore

with e, f;

^{h as} yet undetermined

complex

numbers and the

charge conjugate

^{of In}

deriving

^these

equations

^we have used that

[Q,

must

satisfy

^{the Dirac}

equation

^{and that}

Q

^{is a} Lorentz scalar. As a conse-

quence of

(3.11)

the functions

0 ~ I 0 &#x3E; and 0 I 0 &#x3E; vanish,

hence _

This

implies / = 0, because ( 0 ! 0 )

^{does not vanish}

identically.

Insertion of

(3.15)

^into

0 I [Q, I 0 &#x3E; gives (3.12),

^{and the}

’

normalization condition

(3.13)

shows that e has the desired real value.

We have then also

Commuting Q

^{once to the}

right,

^{once to the left}

in ( 0 0 )

we obtain

hence h ⁼ 0. From their

respective Haag expansions

^we

easily

^{find the}

commutators

of Q

^{with the}

interacting

^{fields :}

i. e. we recover the relations

(3.6). According

^to Kraus and Landau

[11] ]

the operator

Q

^is

self-adjoint.

This is then also true for

Q’ =

^and

equation (3.5)

defines the desired

unitary representation

^{of the} gauge group.

As a result of these considerations we find that our conditions on the r-functions are

equivalent

^{to the} operator conditions formulated at the

beginning

of this section.

We come now to the

minimality requirement

^at

high

^momenta

already

alluded to. The conditions discussed until now do not determine the

theory uniquely.

We need yet ^{a condition}

corresponding

^{to the small distance condi-}

tion of B. A direct

generalization

^of this condition to our case, translated into a p-space

form,

^{reads as follows. We} define the

asymptotic degree

^of

the _p-space distribution

r(Q),

abbreviated

AD(r),

^{as the real}

number 03B2,

for which

Annales de l’lnstitut Henri Poincaré - Section A

for all G &#x3E; 0. Here 0 ~ _x~, is a distribution in Q and the limits

~ -+ JJ must be taken in ~’. The

asymptotic degree

is connected with the _x-space

scaling degree

^{of B}

by

where

r(Q)

^and

r(E)

^are Fourier transforms of each other. The smoothness condition of B becomes then : the distiibutions

r(Q)

shall have the mini- mal AD

that is compatible

with the conditions

already

enumerated. We call this

condition _{the first} minimality

^condition.

Unfortunately

^{it will turn out} in Section 4 that at least in

perturbation theory

^first

minimality

still does not determine the

theory uniquely :

^the

theory

is unrenormalizable in the

terminology

^of

B, Chapter

^{VIII. In order}

to escape this

predicament

^{we examine more}

closely

^which

objects

^{of the}

theory

^are

physically relevant, meaning

^that

they

^enter into measurable

quantities. Obviously

^{these are} the matrix elements of observables between

physical

^{states. We do not wish to} discuss here what are the most

general

observables of the

theory.

^{For the moment we note}

only

that the fields

Fuy and ju

^are observables

(after integration

^{over real test}

functions),

^{but not}

the

fields ~

^and

~.

^More

general

observable

fields,

^e. g. local

polynomials

of

~, ~,

will be discussed in Section 5.

constitutes a

complete

^{set of states.} Hence it suffices to consider the matrix elements of

F, j,

^{between in-states}

(6),

and these matrix elements are,

according

^to the reduction

formulae,

determined

by

the restrictions

to the mass shell

Zf

⁼

A2,

^,

p2 = q2 - m2

^of

ra(~ ; Jf, P, Q)

^and

5i, P, Q) respectively.

^More

generally

^{we include}

^retarded products

of F

and j

^fields among the

prospective

observables and define

We demand now

minimality of

within the class

of r-functions

admitted

by

the earlier conditions. This condition we call the second mini-

mality

condition. Excluded from the second

minimality

^{condition are the}

2-point

^functions

r( p, q)

^and

r( p, q),

for which

rMS

does not exist. The functions

r(P, Q) with I P - ~ I Q &#x3E;

¹ must,

however,

^be

included,

^because

they

^will

play

^an essential role in our later discussion of the consequences of second

minimality,

^{and because}

they

^are

closely

related:..to certain S-matrix

elements,

^{which are} also observable

quantities.

In the

following

^{section we} shall show in

perturbation theory

^that

second

minimality

determines the r~

uniquely,

whilst the off-mass-shell continuation of ra remains

strongly ambiguous. These ambiguities

do not,

however,

influence the

physical

^{content of the}

theory.

(6) ^Note that these matrix elements also determine the matrix elements of polynomials,

and even more complicated functions, of the smeared fields.

Vol.

Let us

briefly recapitulate

^our conditions: the

theory

is determined

by

^{a set}

{ ^{J?, P,} Q)}

^of distributions which

i) satisfy

^{the GLZ}

equations (2 . 21 )

and conditions

a)-g)

of Section

2, ii) satisfy

^the gauge condition

(3 .11 ),

iii)

^{taken as}

Haag

coefficients of F

or j satisfy,

ⁱⁿ

analogy

^to

(2.29),

the IM

equation (3.10)

and the conservation

equation (3.8), iv) satisfy

the normalization condition

(3.14),

v) satisfy,

the second

minimality

^condition.

In order to

keep

^the

unphysical

parts ^{of r as smooth as}

possible

^{we can}

also demand first

minimality.

^{For the sake of}

simplicity

^we shall do this in the present paper, ^even

though

^{it is not} necessary.

For the _purposes of the

following

section it turns out to be convenient

to reformulate the

theory

^somewhat

by

the introduction of a vector

potential. We define ~

From the conservation

of j

and the Maxwell

equations (2.4)

^and

(2.10)

we derive ^_ .~

Conversely,

^if

Au

^{is a conserved local}

^field,

^then

^(3.23)

^and

^{(3. 24)}

^define

local fields

Fuv and ju satisfying

the Maxwell

equations

^{and current conser-}

vation. Hence

knowledge

^{of A is}

equivalent

^to

knowledge

^{of F}

and j.

For

large

^times

A~

^{converges in the LSZ sense} towards free vector

fields with ^.

Ain can take over the part ^of F~~ in

asymptotic completeness:

^{the Fock}

space of

A in,

^{is the}

§’"

^{of our}

theory.

We can therefore reformulate massive

QED

^{as an LSZ}

theory

^{of the}

fields

A, ~, ~

^{instead of}

~.

^This

theory

^is

again

characterized

by

its

n-amputated

^functions

^{;(3i, P,} Q).

Unless noted

otherwise, script

letters: x, ...,

~,

^...,

^~,

^{... in the argument} of r-functions will hence- forth denote A-variables.

Amputation

^with respect ^{to an}

A-momentum k

means

multiplication

^with

(fz2 - ^A~).

The GLZ

equations

^{of the new} formulation look almost

exactly

^like

the former GLZ

equations (2.21).

^The

only difference,

^{aside from the}

reinterpretation

^{of the}

photon

^{variables as}

A-variables,

^{is the}

replacement

of the factor

( - 2A2)-’’ by ( - 1 )- y

^{in the} combinatorial coefficient ^on the

right-hand

^side.

Similarly

^we obtain the new

Haag expansions

^from

(2.26)

Annales de l’Institut Henri Poincaré - Section A

by dropping

the factor

(2A~

^and

replacing by

^Condi-

tions

a)-e)

and the normalizations

(2.24)

^{are taken over}

unchanged,

^while

(2.25)

^is

replaced by

with N the second-rank tensor

The new version of condition

g)

expresses conservation of A :

The normalization condition

(3.14)

becomes in view of

(3.24)

The _gauge condition

(3 .11 )

^{and the two}

minimality

conditions are taken

over

unchanged.

In connection with condition

(3.29)

it is convenient to write the GLZ

equations

^{in a}

slightly

^more

complicated

way.

By (3.29)

^{we have}

Therefore we can introduce factors

N(f;)

^{into the}

integrals

^{of the GLZ}

equations

^without

changing anything.

^The

equations

^{read then}

(see (2 . 21 )

for definitions of the

symbols)

The reverse of

(3.32)

^{holds as follows : let =}

Nu

^then

= 0 for

jt2

^{= A2.}

Using

^{this we can} prove

THEOREM 3. L - Let

{r(Q)}

be a solution

of

^{the GLZ}

equations (3 . 33)

Vol.

satisfying

^all

subsidiary

conditions stated

before

^{with the}

possible exception

of (3. 29). ^{Then .}

solves

(3.33)

^and

satisfies

^all

subsidiary

conditions

including (3.29).

The

proof

is trivial and will not be

given

^here.

In

considering only

solutions of the form

(3.34)

^{we are not}

restricting generality:

any admissible solution is

physically equivalent

to, i. ^e.

yields

the same

Haag

coefficients for

A, ~,

^{as, a} solution of the form

(3.34),

where we can set r ⁼ r’. This is so because these

Haag

coefficients are not

changed by multiplication

^with any

A-variable,

^{due to}

(3.29)

and

(3.30). Hence,

^if

using

^{the GLZ}

equations

in the form

(3.33),

^{we can}

ignore

^condition

(3.29),

^{since it can}

eventually

be satisfied with the

help

of Theorem 3.1. In Section 4 we shall therefore not take

(3 . 29)

^{into account.}

4. PERTURBATION THEORY

In this section we solve the model defined in Section 3 in

perturbation theory.

^We

expand

^the

quantities

^{of the}

theory,

ⁱⁿ

particular

the retarded

functions,

into formal _power series in the

coupling

constant e, ^e. g.

We will not discuss convergence of this

series,

^but

only

derive the coeffi- cients _r6

in

_every finite order.

In zeroth order the fields are

free,

^{i. e.}

All other _ro vanish.

In orders a ~ 1 ^we follow

exactly

^the

procedure

established in B for the scalar _case,

ignoring

^{for the moment the}

divergence

^condition

(3.8)

and the second

minimality condition,

which have no

equivalents

^{in the}

scalar case. The

generalization

^{of the}

procedure

^{to our model}

being straight-

forward we

give

^here

only

^{the merest} outline. The

only point

^whose genera-

Annales de l’Institut Henri Poincaré - Section A

lization is not obvious : the

proof

that covariance of r can be

satisfied,

^will

be dealt with in an

appendix.

The

expansions (4 .1 )

^are substituted into the GLZ

equations (3.33)

and the terms of a

given

order 7 collected on both sides. This

yields equations

where

Ia is,

ⁱⁿ very abbreviated

notation,

of the form

For the

meaning

^of the dotted parts ^see

(3.33).

^The

important

^{feature of}

this

expression

is that it does not contain _ra:

Ia

^{can be}

computed

^from

the _{~, r (1,}

by

summation. This allows a recursive determination of _r~.

Let rr, z

^7, be known and

satisfy

^the

subsidiary conditions, possibly

with the two

exceptions

noted above. Then

Ia

^{can be}

calculated, and fy

^is

obtained as solution of the linear functional

equation (4.3),

where in the

course of the solution full account must be taken of the

subsidiary

^condi-

tions. The basic idea behind the solution is

that,

^{due to} the

postulated

x-space support ^of r, we must have

if

(~ 2014 r~) ~ 9_,

V - the closed backward _cone, and

if

(~ 2014 r~) ~ V +.

This fixes _r~ outside the

manifold ~

⁼ ~. The result must be continued onto this manifold in such a way that our conditions are

satisfied. That such a continuation exists has been shown in B. There we

have also discussed to what extent first

minimality

restricts the

possible

solutions. The solution is

unique

^{if -}

4,

otherwise it is ambi- guous, the number of

ambiguities increasing

^with

increasing

^The

ambiguous

parts ^of ra ^are solutions of the

homogeneous equation

and are of the form

with ~

^a

polynomial

^with

appropriate

symmetry and covariance _pro-

perties.

^{For this}

h~

^{we have}

provided that B ~

^0.

deg B

^{is the}

degree

^{of the}

polynomial B.

In first order we find ⁼ 0 for all

Q,

^{and first}

minimality implies

the

vanishing

^{of all} r 1 except ^the

3-point

^function

q)-and

^the

functions connected to it

by permutations

^{of the}

arguments for

^which

the normalization condition

(3.31)

^must be satisfied. This is achieved

by

choosing

_"T

Namely,

^for po ⁼ the relation

holds because

(p -

⁺

m)

⁼

p2 - m2

^{vanishes on the mass shell.}

1"1 ^as

given by (4.10)

^{has the}

asymptotic degree - 4,

which is the minimal

possible

^{value for an}

expression

of the form

(4. 8). (4.10)

satisfies also the

as yet

ignored divergence

^condition

(3.30),

^since p - q,

p2

⁼

q2 ^{= m2}

^{we find}

(4.10)

^{is the}

only

^ansatz of the form

(4.8),

^with

asymptotic degree - 4, satisfying

both these conditions.

Note

that r

I as

given by (4.10)

^{saturates condition}

(3.31).

^{Hence we}

have for a &#x3E; 1 the

requirement

We must now turn to the

remaining

conditions in orders a &#x3E;

1,

^{i. e. conser-} vation of A and second

minimality,

^{which are not covered}

by

the results of B. In the remainder of this section we prove

THEOREM 4.1. - In all

finite

^{orders 6}

of perturbation theory

^{there exist}

distributions

~(Jf, P, Q) satisfying

^all the conditions enumerated in Sec- tion

3, including

^condition

(3.30)

and second

minimality.

^The

corresponding

mass shell restrictions

~s

^are

uniquely

determined.

The

proof

of this theorem is attained in several steps.

First we

introduce,

^{as an}

auxiliary

mathematical construct, ^{a one-} parameter

family

of LSZ theories of vector fields and

spinor

^fields

03B403C8, 03B403C8,

5 is a parameter

characterizing

^a

particular theory

^{of the}

family.

^{It lies in}

the interval 1. The

couples

^{are related}

by (2.3).

^{The GLZ}

equations

^{of the}

ð-theory

^look

exactly

^like

(3.33),

except that the kernels

are

replaced by

^{A A}

The

distributions 03B4r satisfy

^{the same}

reality,

symmetry, covariance, ^and x-space support conditions as the

physical

^{r. The}

2-point

^normaliza-

tions

(2.24)

^{hold for}

general ~,

while the 2-A-function satisfies

F3 regular

ⁱⁿ

~2

^4A2.

Annales de l’lnstitut Henri Poincaré - Section A