r^. 2 6. j A ! < i - -

%?«/ , — ! • • • • ; •

IC/70/1

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

S-MATRDC FOR GAUGE-INVARIANT FIELDS

E.S. FRADKIN

and

I.V. TYUTIN

INTERNATIONAL

ATOMIC ENERGYAGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1970 MIRAMARE-TRIESTE

IC/70/1

I n t e r n a t i o n a l Atomic Energy Agency

and

United Nations Educational Scientif ic) and Cul tura l Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

S-MATRIX FOR GAUGE-INVARIANT FIELDS*

E.S. FRADKIN

and

I.V. TYUTIN

Lebedev Institute of Physics, Academy of Sciences, Moscow, USSR.

ABSTRACT

A method has been suggested (and appl ied to the Yang-Mills f i e l d

and to the gravi ta t ional f i e ld ) for constructing the generating

functional (S-matrix) for f ie lds possessing an invariance

group* The uni ta ry and gauge-independence of the S-matrix on the

mass shel l i s seen e x p l i c i t l y .

MIRAMARE - TRIESTE

January 1970

This paper was the subject of a lecture given by Professor E.S. Fradkin at the International Centre forTheoretical Physics, Trieste, on 8 January 1970.

- 1 -

1, Introduction

There has lately "been considerable intensification

in studying the theories partially or completely invariant

under non-Abelian groups of transformations. This is in

connection with the discovery of vector mesons and the

classification of them into multiplets, with the use of

vector mesons for description of the form factors of

partioles and in the current algebra approach,. Introduction

of intermediate vector "bosons is used in many schemes of

weak interaction* An important example of the theory

with a non-Abelian group of invarianoe is the gravitational

field.

In the present paper a procedure for constructing the

Feynman rules is proposed for the theories possessing a

gauge group, to which theories the massless Yang-Mills and

gravitation fields belong. It is known that some additional

(gauge) condition must be imposed on the dynamical variables

in order that a consistent quantum field theory may be

formulated on the basis of the Lagrangian density invariant

under a local transformations group. In covariant gauges

this can conveniently be done by use of the Lagrange

multiplier method. The basic idea of the method proposed

- 2 -

is to choose the Lagrange multiplier in suoh a way that

one is led to free equations of the motion for the

additional field. This fact provides the unitarity of

the S-matrix in the physical subspace. The Feynraan rules

obtained coincide with those proposed in refs,[2,3»4,5j•

The difference of the method under consideration from

that of ref a. [2,3,4,5.3 i s "that w e sucoeed in obtain ing

a set of consistent dynamical equations whiqh completely

describe the theory. These equations, on the one hand,

make i t possible to eluoidate the reason for the appearance of

additional diagrams, and, on the other hand, guarantee

the unitarity of the physical p-matrix, § 2 is devotedthe

to the construction of/ p -matrix for the masslessan ^

Tang-Mills field in)arbitrary gauge and to the proof of* the

the gauge invarianoe of the S-raatrix. In §3/const ruction

of Feynman rules in the Coulomb and axial gauges is con-

sidered on the basis of a oanonioal quantisation procedure.

The S-matrix obtained coincides with that found in § 2,

which fact reaffirms i ts unitarity. It is shown, further-

more, that taking the additional conditions consistently

into account makes i t possible to obtain self-consistent

equations for the massless Tang-Mills field in the

presence of external source.

m»

- 3 -

In § 4 the Peynman rules for the gravitational field

are constructed in covariant gauges. These rules coincide

with those suggested in refs.[2,3,5j » In the framework

of our approach we also ohtain the S-matrix for a non-

covariant (Dirac) gauge for which the Feynman rules have

"been obtained "by Popov and Faddeev KJ hy a method closely

connected with the canonical formulation of gravitation

field, and "by our method the equivalence of the S-rnatrix

in covariant and non-covariant gauges is proved.

We use the following notations. The Greek /*,, v » A »*••

and the Latin i , j , K indices takes the values 0,1,2,3

and 1,2,3» respectively. In §'s 2-3 Quy means the

Minkowski tensor (+, - - -) and S'lLr- means the unit tensor.

By the summation over the repeating indices is every-.

where meant: au HA= % *> - <X-S- \ dy 5 °/%X » D s *>I»JU •

n = J ^ , In § 4 <\ means the metric tensor, and

the Minkowski tensor is designated as £ ^ , The usual

summation over the repeating indices means: a. o = / , ^ u. **

¥Q use the system of units "K = C => 1 in §'s 2-3 and

• " i in §4 (where K is gravitational cons-

tan t ) . Every paragraph is supplied with a separate

formula numeration. Reference to a formula of the other/"

paragraph is supplied with the number of the paragraph.

2# The general theory of construction of Feynaan

rules for massless Yang-Mills field. The gauge invariance

of S-matrix . |_

In this paragraph the general theory for constructing

• unitary p -matrix for/massless Yang-Mills fp.eld.

is considered.the :

The classical Lagrangian for/ Yang-Mills field has

the formi

is field strengh tensor

r

•f are the structure constants of,/arbitrary finite-dimeu-

Bional compact Simple hie groupyt The are imaginary and totally

antisymmetric and satisfy the Jaco"by identity

Lagrangian (1) is invariant under arbitrary g8ugeA q

transformations depending on coordinates which for H* have• » • • • . : ' . '••' • ••

the formi '

and for u^ are

, • - 5 - >'• •

The matrices p/"J= $ f«J fomthe adjoint repreaeo-

tatian of ,and U are the set of the group parameters. In

(5) and (6) these parameters are arbitrary functions of coor-

dinates.

For infinitesimal transformational

' $%)*?*•*&*(*) ' . (7)

the gauge transformations have the form

u6)i.B the covariant differentiation

I D the case of Abelian groups formulas (4) and (5) are the gauge

transformations of electromagnetic field*

H^H^rli j fj,^^ -. (9)

With the help of Lagrangian (1) the classical field equations

are obtaineds

According to invariance of the Lagrangian under 6auSe trans-

formations it follows that gauge variation of the- action

1 must be equal to zero. 'the ., . • . (12)the ., .

^functions U (*) are arbitrary, from (12)an important identity follows

- 6 -

It Is necessary to note that identity (13) is valid for

arbitrary functions /L^.(12) may "be obtained with tie

help of covariant differentiation (10).

t ia known that the theory with Lagrangian (1)

permits no direct transition to canonical formalism (both

in the classical and in the quantum theories). This ia

connected with the following. Let us find the canonical

.^momenta: • , • ','••

In the canonical theory the relations must fulfil

denotes ' awhere [_ * J /a commutator in the quantum theory and^Poisson

traoket in the classical theory./In canonical variables the equation kc^o i s a s

fOllOWS: A € A-0 • (X6)

and it is in contradiction withthe-/

A Similar situation - is found when constructing/$ -matrix

in the theory with Lagrangian (1). Usually one writes the

-matrix as an expansion in normal products of the free

fields. The corresponding Lagrangian is equal to the total

Lagrangian when the coupling constants vanish. In the present

case we obtain , v , „ •

M (V U (91)

- 7 - .. . .

[8] • :

However, one can prove J, that in the framework of the axioms

of modern quantum field theory the Lorentz-covariant operator

»i/W is equal to zero when it satisfies (21), (91)., 3 ,

For the correct construction of the theory of/Yang-Mills

field in the framework of quantum theory of canonical forma-

lism of the classical theory it is necessary impose

an addional (gauge) condtion. For example, in , electrody-

namics one uses the Lorentz condition . —

p

or the Coulomb gauge conditionwNow the gauge invariance of the theory consists in

dindependence of physical observables, iD particular of p -

matrix elements, on the choice of gauge conditions.

It is convenient to introduce the gauge conditions

in the theory with the help of the Lagrange multiplier tK 7 .

In quantum theory the Lagrange multiplier Q must be consi-

dered as a new dynamical variable and it is necessary thattheft

the physical observables should not depend on\m -field.

For example, the correct formulation of quant.um electro-

dynamics in Lorentz gauge is obtained with^help of the

Lagrangian:

It turns out thatyfTi -field is free .When calculating

the $-matrix we must use the transversal photon propagator.

The g-matrix is unitary injphysical subspace which is determined

- 8 -

with help of the conditionj

Furthermore i t is known that in electrodynamics v V -mat-

rix i s gauge invariant, i . e . i t i s independent of the choice

of gauge condition.

Ws shall use a similar procedure . in th« theories with an

arbitrary gauge group.t h e . • • ; . '

The basic idea ofJ method proposed for constructing the

Q -matrix in theories with a gauge group consists in

choosing the" Lagrange multiplier such that the additional

(fictitious) 5 -field should be free.This means that j[ Q -fieldthWso.thjeW

•. ia not involved in the scattering and/ n -matrix is unitary

in the physical subspace.

Let us -first.. take- the Lagrange multiplier for lang-

Millfl theory in the form'. (1.7)

Then / Ay,-field satisfies the Loreotz condition

and we muswse the transverse., propagator for the

in perturbative calculations.

However, as was first noted by .fteynman j_l|, if

one takes / - [^ as an interaction Lagrangian,the£[ -matrix

is non-unitary in the physical subspace. This originates

from the fact that in the case of Lagrangian (19)thel-field

satisfies the equation

.. (20)

«• Q m

The fictitious 0 -field is not free and does take part in

the scattering.

Thus we conclude that the Lagrange multiplier must de-

pend On the A^-field.

Now we are ib a position to proceed to the concrete. • • • ^ f i . o f

- construction of/£> -matrix. Consider a class/gauges which

i s described by -an arbitrary function—

I

For example, > q can be equal to <L/L or

Later we shall impose a condition on the function "Y'<t . Let

us choose the action in the form(22)

y ' the/)QThe part V ^ is added to specify the gauge otlhu -field.

We shall consider the case o ^ O only for the gauge function

chopse the function © ^ f r o m the condition that the

Pi -field obeys the free field equation.

The variation of (22) over /L and tf results in the

following field equations;

- 10 -

In (36) L is tie

With the help of the identity (13) one obtains from (26)

( 2 8 )

We impose the restriction 00 y> that the operator

should be a nonsingular differential operator.

If we choose a function $-y as ;

. • ' (30)

or in a syntoolical notioni

then £| -field satisfies the free field equation

Note that the ^-function satisfying (35) does exist and thaiof r

the determinant/JiLis not zero,at least in the framework of

.perturbation theory;.

Since the fictitious ^-field satisfies .the free field

equation (33)» iD the physical subspace

- l i -

-matrix is unitary and the classical field equation

L -field is satisfied* . " •

!' (35)the.

I D order to obtain en expression for] S - matrix wethe < r •

use the connection "between! p -matrix and the generating• ~f \ r I

functional of Green functions £

i.u the

(A are the set of free field operators describing^ phy-

sical system for ~t~^ ~~ oO . .

For the generating functional we use the representation

in the form of^functional integral over/complete set of

fields 1>1O1 ' ' ,Thus in the model under consideration we obtain for

the generating functional the following expression*

(37)

When calculating (37) we pass from the integration over—— Q which

to the integration over gw and for the Jacobian^arisee we

use the expression

- 12 -

ID the gauges o(= Q ' (which include the

W/JM^C' andy Coulomb gauge ^/) =

the expression for the generating functional has the form:

'(39)

The expressions (37) and (59) for generating.functionalthat

indicate^the correct Feynman rules for the perturbative

calculation Q-y- the Green functions are governed by -thefollowing considerations:

a) thBre exist the two usual vertexes for the interaction

of vector mesonsJ

which describeb) there exist additional vertexes < 1 the interaction

the . Aof vector mesons with]fictitious Q -field; its form is deter-

mined bythe ' Q

c) /fictitious rj-particles always occur in closed loops,

every loop possessing an/additional 'factor /-) and the propa-the / v '

gator of/B -field being

a thenfLet us pass to]proof of the gauge invariance of / p -matrix,

a * ttew A r±he\ ,i , e . to/proof of the independence of/p-matrix on^ (X 3 U/i,

and on the type of gauge condition j in (39)•he rf

J jj -

when d-0 * i ' 6 * o f the c l l o i c e o f t h e

thillrst we prove that J jj -matrix i s independent of

• . . . . ; . . . :,± .^-.. , .

^ we use the method of ref. [2], Define the func-tion Ay(7l) by the relation:

theIn (43) p is an element of|gauge group, /L is defined

by (5) end q Mlp) is the measure of group integratiofip^r

compact simple groups (which we consider) ciiAlo) is defined

uniquely and has the property

Property (44) makes it possible to prove the gauge invarian-

It is sufficient to know the function i-iyl'yonly for

field satisfying the condition (4-2). In this case the group

integral is concentrated in the neighbourhood of the unit

element of the group. By use of (7) and (5') we obtain:

x yor morQ details of the gauge group resulting from a gi

Fl2l 'ven simple Lie group eeeL j , _,

- 14 -

ID (45) we omit the inessential infinite determinant of Athe . the

One can see from (45) thaty expression (39) for/generat

ing functional can "be rewritten: in the form

(46)

Consider now another gauge:

l(*)^0 „ (42t h e J ' \>7 • , .

Multiply^expression (46)^the function 9^w/ and perform

the gauge t ransformation! I'-.1 ;

Taking into account.tliegauge invar iance of Zl-u * £*% t

and / the proper ty (44) we o b t a i n i x

/ l w

Consider the term ft* Ju in more d e t a i l s , A f t e r pass ing onto

the masafshell ( i . e . a f t e r passing from the generat ing funct ionalthe Lt o ! p -matr ix according to (36)) t h i s term has the form

In terms of diagramsQoperating on /ju means operating on an

external line of the diagram. The exti/ -»

of the form fry which cancels out Q .

^ J/J

external line of the diagram. The external line has a pole

Q

x The Jacooian ^ J/Jffi) ±B

D operating op the product of .:. fields at- the same

point means that U operates directly on the vertex. .ID the

framework of the perturbation theory the vertex has no pole

of the form /^.Therefore we can perform?integration by

parts in ,the expressions of the type (4S.)>which then vani&hes.the ' .•

Thus,expression (48) can be rewritten on/mass/shell

(ID.a.) as

Now according to (43) the group integral is equal to

one and we obtain ••

(50)

Relation (50) means that p -matrix is independent of

' the type of gauge condition (42).

Below we shall consider three particular gau6es a n d

the. ^^ei

give the proof of the independence of/p -matrix on/oT5 .

1. Axial gauge. This gauge is defined by the con-

With the help of (27), (2$), (52) we obtain

A-i-i^ . (52)

As expression (39) for the generating functional contains

"?) is effectively equal to one.

- 16 -

Thus in axial gauge the generating functional is

The Feynman rules have no additional diagrams. The treetheAfl

propagator of { AL -field is

pk 4 ^ 1 C 5 4 ). 2. Coulomb gaupe. This gauge i s defined by ;the condi-

t i o n (S&L &ei° OsJ)

With the help of (27), (28), (32) we obtain'

By use of (59) the expression for the generating functional

may "be calculated to give ' A

The Feynman rules in Coulomb gauge, are:

a) the free propagator of /L is equal to the well-knownthe ' the .

propagator of/electromagnetic field ir Coulomb gauge:

V--0(58)

b) besides/usualvertexes (40) the vertex of the/I

interaction of H H with/fictitiousft. 0* field exists:

(59)

, • * •

- 17 -thet

c) the linea of / 5 -field occur only in closed loops,

every loop possessing an additional factor ("*J , and the pro

pagatorofyo -field is *_•-,4 •

t

= 0 (60)

In the following paragraph we shall construct generatingthe "the

functionala (53) a d (57) in/axial and Coulom"b gauges "by use

of the canonical quantization procedure.

5) General ffeynman gauges. The general Feynman gauge

is defined by the gauge function [s.£* cJlto M )

added

We obtain the Lorentz gauge by means of the/Lagrangian- term

(2>) Vith (i~/l(^J<"^v1)aXi^ "wither. L for the-Eaynman gauge.

With the help of (27), (28), (?2), and (39) we obtain:

(63)

theInJtransversal gauge we have

.Now let us prove that the 0 -matrix (6?) is independent• the " . i

-Beplace the variables in/functional integrals' <63X -

•

: - 18 -

being infinitesimal. Retaining the terms of first

order in W we have '

(69)

5*p fa t)R e c a l l i n g t h a t t ) P 7 ( X l d y m e a n s

'let US integrate "by parts in (63) and (70) so that no ;

derivative should operate on the expression \& njltfj*

Taking into account the antisymmetry of j and the rela-

tions

- 19 -

(72)

We obtain

Using the Jacoby indentity (4) we find that the change of

the expression pP*VV in (69) compensates the Jacobian /u*\.

Thus we have

' L ^ (75)

We can prove again that the following replacement is

true on the mass/shell(76)

Finally we obtain p., ,•

» rules for oalculati.S the geDeratiDg fu.o

(63) H.

,as- followas

- 20 -

a) the free propagator of the vector mesons has the

form:

ID) "besides the usual vertexes (40) there sxista the

( 7 9 )

/) the Qinteraction of L with/fictitious D -field

c) the lines of D -field occur only in closed loopsthe.

and the propagator of/ D -field is

•- S — 2 (80)

theThe Peynman rules for^massless Yang-Mills field were

obtained also by Popov and Faddeev Fzl Dewitt /"jiand

Mandelstam/4 by otherir\Ctl^o4^e

3- The construction of the j -matrix in canonical

formalism.the -JJ a

In this paragraph we construct / p>-matrix for/massless Ya-

ng-Mills field in axial and Coulomb gauges in the framework

of the canonical quantization procedure and the interaction

representation. The corresponding Feynman rules coincide

with those found in 2.

Consider the Lagrangian:

is the external current on which no restriction of

- 21 -

t h e oonservat- ion 1taw--?type i s imposed.-— '•'-••

Besides the difficulty with the canonical formalism

there is also another problem for Lagrangian (1).

Let us write the field equations obtained from (1)

varying the J .

identity (2.1?) leads to

Obviously we can not satisfy O ) since the external source

Ju does not depend on flu ,

Below we shall show that the gauge condition gives a

possibility to avoid this difficulty as well,

1) Coulomb gauge. Yang-Mills field in 'the Coulomb

gauge is defined both by the Lagrangian and by the gauge

condition

The gauge condition (4) can be introduced into the theory with

the help of •< a •• Lagrange multiplier just as this was

done in S 2. It can be proved that the corresponding field

.equations are consistent in the presence of the external sour-

ce as well. However the method of the Lagrange multiplier

does not permit canonical quantization of the theory.

For the purpose of The canonical quantization we

- 22 -

consider (4) as a constraint excluding one of the dynamical

variables. This exclusion must be made beforepinding/field

equations.

Thus let us choose

as independent variables ,

With the help of (4) we obtain the expression for ^

(6)

Expression (6) should be substituted in (1) and the

Lagrangian should be varied only over /} , fl.» and j\ .

The corresponding field equations are:

tf£ (7)

quantizing, just as when quantizing any essentially

nonlinear theory, the question about.the ordering of the

Don-commutative variables arises.

Later we shall consider this question and show that

with the heip of the corresponding symmetrization procedure

of multipliers, quantum expressions can always be represented

in the form coinciding with the corresponding classical

expressions.

Therefore for the time being we assume that the multipliers

- 25 -

in quantum theory expressions are commutative as they are

in classical theory.

Let us find the canonical momenta

=o,

• It is shown that (7) is the constraint equation and it

should be used for excluding ft0 when constructing the Hamil-

toDian.

Let us decompose ^W into transversal and longitudinal

componentsi __

C + ' I " (ID

Then equation (7) has the formu J •

tfrffi^® ; (13)This can "be solved with the help of V -function

>0

The *^) -function was considered first "by Schwinger

Rewriting (7) in the form

and using (14) and (15) we obtain

- 2 4 - '

The expression for |L will be also necessary for us

Now let us find the expression for . Vyi}^ .Using the field

equations we obtain:

• T*A Q I

UWhen deducing (21) we expressed U ^ with the help of equa

tions (8) and (12) .Expressing^ in. terms of L and GL ,

using (19) for /|K and (8) for C»K we obtain after hard|K and (8) for C»K

algebraic work

( 2 2 )

It is not difficult to prove that equations (20), (22)

a r e c o a a i a t e B t : ^ f y fy^^Thus we can see that consistent use of the gauge condi-

tion enablesAto obtain consistent field equations in the pre-

sence of external source as well. The correct form of thea*]

field equations was obtained by SchwingerYwith the help of

a • different method,

Note that in the case J. 5 Q or in the case when J^

is a current of matter (i.e» when the equation C7? H — A

is true) the field equations for Yang-Mills field written

ia h -dimensional £orm have the usual form.

- 25 -

The Hamiltonian in the Coulomb gauge is equal to

(to within a total space derivativ

In (24) it is assumed that /)_ is expressed with theT

help of (6) and Q0K is expressed with the help of (12).Let us pass to interaction representation. For this

purpose |-j should be split in%« ft free Hamiltonian n0 and

an interaction Hamiltonian H^^ We choose the total H with

for H o * d

, C 2 6 )

(27)

• , (28)

In (25) - (28) we use the designation U K instead of M K

for free . Yang-Mills field in interaction representation.

keeping the designation /;« for the Heisenberg fields." • //G

With the help of (25)-(28) the field equations 14^ andthe propagator can be obtained!

(29)

(I

- 26.-

(30)

I D (30) IJJmeans the Dyson I -ciironological product:

The Interaction Hamiltonian is obtained:

Beiations between £/; and ft* are given by usual formulas

connecting interaction gnd Heisenberg representations. Let us

define ,

'^ " + ' . (35)

where p means p-matrix in interaction representation

Then we have

/ ) V U > • (37)

When passing from the classical expressions of (15),

(18)-type or from field equations to corresponding quantum

expressions we assume that all the operators should be

- 27 -

expressed in terms of the canonical variables /) and Qo

t and the productsof operators should be understood- - as

Ijj -products, the exact meaning of which is defined by

For example

the theOne can prove using/commutation relations (27) and/field

equations (29) that the Heisenberg operators defined by

(35) -(59) satisfy field equations (7), (8) and (22).).

One can also show (on which we do not dwell) that the

symmetrization procedure of Heisenberg operators n^ and MCt<

can be suggested in such a way that the expressions of type

(15), (18) in the quantum case should formally coincide with

the corresponding classical expressions (see also, J_15j)«

Let us pass to constructing the Feynman rules. Calcula-

te for this,, purpose the generating functionalJ (se* <Uso [t£J)

(41)

In (40) the exponent operates on ? 0 according to the rules:

- 28 -

is the generating functional of the free Green func-

tions: .

\y rr- —p-

Let us passjin (45) from ij)-product to Wickfs /-product,

i.e. instead of the Dyson propagator we shall use the Wick

propagator.

(45)

'• J \7 Jd (46)

C 4 7 )

Perform in (40) the functional differentiation

only in the first factor in (47). Then one can substitute

and put ^

For the functional expression of A arisen we use the

relation ^ n

Then (40) transforms

In (50) the Uo means

- 29 -

W in (51) coincides with freepropagator in the COUIOEQD gauge (2.58).

Combining (50) and (5l) we obtain the following expres-

sion of tlie generating functional:

Expression (55) coincides with expression (2.57) for

the generating functional in the Coulomb gauge obtained in

2.• • The • the;;

2) Axial gauge. / axial g auge ie defined by/condition

VChoosing

as independent variables

we obtain the field equations

and the canonical momenta

- 3 0 - •

W . (57)

(55) is the constraint equation from which^we find the

expression for

After the calculation analogous to that in the Coulomb gauge

we obtain the field equations in the *-j -dimensional form '

. Vy ffvr + 5r " i ^ 3 V/- - 0 ;; (60)Equations (60) are non-contradictory.

Hamiltonian in axial gauge ia equal to (to within a total

space derivative)* : .

Let us pass to the interaction representation

(65)

With the help of (62) we find

(64)

where we use the*definition

(55)

- 31 -

Expression ^66') coincides with free propagator (2»54) and we

will call it the Wick propagator.

* - (67)

(67) coincides with the Wick propagator.

(68) differs from the Wick propagator (the first term) by

the contact term.

The interaction Hamiltonian is*

i<

that

The generating functional is equal to

C70)

Let us pass from ~C -product t o ^ / -product.

^ .KjtA

(73)>w

- 32 -

(75) the exponent operates on Z~Oj according to rules

(42)» (51). Making the substitution

"* ^ " >

> and using the relation

we obtain finally the expression for the generating func-

tional

which coincides with (2.55). '

The results of this paragraph can "be regarded as

an additional proof of the fact that the method for con.-

struct ing the Feyuman rules developed in £ 2 results injunita-

p-matrix*

- 33, -

4. The construction of the Feynman rules for

gravitational field.

In this paragraph we shall obtain the general rulesthe ^ the

for construction of^ p -matrix foij^ gravitational field, provetheuf

gauge invariance o f ^ -matrix and in more detail consider

two covariant gauges (harmonic condition and its linearized

form) and the Dirac [18] non-covariant gauge .The Feyn-

man rules found "by us coincide with those of [2,5,5j.

The classical gravitational field is described by the

action [19] *

Here N i s the me^^ tensor, §"ZcUc §f\} , ^ %h~° *V

and ^ y is the curvature tensor of second rank (Ricci ten

sor)

is Christoffel symbol

As we shall show later, the most convenient choice

of the concrete form of the dynamical variables depends

on the gauge condition.The variables we shall use belong to the following

class:

(5)3(p-3 3

- 34- -

In general case we shall designate the variables

a n d dlf) a s ^7**/ * Th-Q Einstein equations for gravitational

field can tie obtained from (1) by two.methods:

In the formalism of first order exp. (1) should be

varied by ^ y and I considered as independent variables.

From the equations obtained by varying of (1) by / ^

expression (4-) for [y^ can be deduced.

In the second order formalism the variation should be

made only by 2^i/ . It is assumed in this case that(l)

is expressed only in terms of 3£W with the help of (4).

Both methods lead of course to the same field equations

As / is known,(1) and (2) are invariant under the

gauge transformations of *&M and / y\ the infinitesimal

form of which is ;

1 VA -1 v> s V y> ' v > v ' v * W I j i c V V > ? (7)

^jare arbitrary.infinitesimal functions of X •

In the formalism of the first order the gauge transforma-

tions of ^«v and fy^ should be madeiin the formalism of

the second order it is only the gauge transformations o

that should be made.

The gauge variation of (1) has the form

35 -

In the case ^ d v - * ) / ^ tlie first term in (8) should be

assumed as

The second term in (8) is absent in the second-order forma-

lism«\ve used the following designations!

T ;" fn/*M - (12)

(12) exists in the second order formalism only. The diffe-

rential operators R we find with the help of (6)-(7):

for the case

for the case

As the 3 (*) are arbitrary we obtain an important identity

from (8);

(16)

- 36. -

Note that (16) is satisfied "by arbitrary %u)/ and I .

According to (16) four identities exist between the

Einstein equations. As noted in m 2;this means that

four additional (gauge) conditions should "be imposed on ^^

y\ for consistent construction of the quantum theory.

As well as in £ 2 we shall use the method of Lagrange-

multipliers. Consider the class of gauges determined by

the function:

Three concrete forms of the gauge functions will be con-

sidered later* ;

Lagrangian is ^

| J " { V 6 C18>where tfuy is Minkowski tensor. The case of^O will be con-

sidered only for those gauge functions (17) which depend

linearly on "*W and are independent of I y^ •

f I d 13Varying (18) wit;h respect to Sf y f I ^ and 13 we

obtain the field equations:

5V

in the case of the second order formalism. In the first

order formalism the following equation should be added:

r

<-:=**,r m

- 37 . -

With the help of identity (16) we obtain the consequence

of field equations:

We impose the restriction on fy that in the limitM

the operator C??^ should be a nonsingular differential

operator Q ^ . Choose #p-function in the form

Then 5 -field satisfies the free equation

Therefore ^ -matrix is unitary and the Einstein

equations are valid in the ^physical subspace.

The generating functional is equal to

(26)

In the gauge

i.e. f o r ^ - p ,we have

- "38 -

In (26) and (27) olfyjP) is equal to

jV ....in the second order formalism and to •*"

r ' ' ( 2 9 )

in the firat order formalism.

Let us prove that p -matrix corresponding to generating

functional (26) and (27) is independent of the choice of

the variables of the functional integration belonging to

class (5). Suppose we integrate over ^ ^ in (36) • The corres-

ponding Jacobian has in general the form,

where V % "$A are nupibgrs and

T h e n we h a v e * / » , / , i , « ( v n V „ iil4}f\j.d 9 D / / S W " ^ ?

m^\v)^W^'^^ + w M& W j a

We can see from (25') that 2?J corresponds to the

field equations for 3f V; ty^and 6 which are obtained

from (20)-(22) by the substitution &£-> 6^ .

- 39 - ' • '

/ * <ac>

vx

.* v •

It ia seen from (19«) and (22•) that fi satisfies the

free equation . , • .

Let us write f-Xfr)) in the form

f e W j ) ^ 1 ^ ) (52)

She expression (19*) acquires then the form

It is clear that in the second term of (53) integrat-

ing by parts can change the direction of the Q operation

(at least in the perturbation theory) and one can use (251)•

Finally we obtain that the field equations (2O')-(22')

coincide with (20)-(22),

Thus generating functional (26*) leads to the same

field equations of 3W)Pv)O 6 aDd consequently to the

sane $ -matrix as generating functional (26) does.

Note also that all the Heieenberg operators belonging

to class (5) must lead to the same p -matrix according to

- -'4P -

the Borchers theorem. **

Now we pass to the proof of the gauge invariance.

Prove at first the p -matrix being independent upon the

type of gauge condition

i.e. the independence of p -matrix corresponding to generating

functional (27) upon the form of the function V M •

Define the function /Lfe;P) by relationj

(35) £ is a° element of the coordinate gauge trans-

formation group, dulvj is the measure of group integra-

tion with the property (2.44)3SC. •

As well as in ^ 2 the property (2.44) enablesAto prove

the gauge invariance of/W?)' 7/,

We must "know the function ZLfoP/ only for ^^ and

satisfying condition (34).In this case the group integral is

concentrated in the neighbourhood of the unit element. The

gauge transformations have th.e form (6)-(7) and

' * We ignore the question of the meaning of Sfyy ifl 'the opera-

tor function of 2fu^ .See also the analogous statement for the

case of nonlinear chiral Lagrangians in L20J*

For more details of the coordinate gauge transformation

group see

, \

- 41 -

As well as in ? 2 we obtain

Keeping in mind (58) tiie expression (27) can be rewritten

in the form:

In (39) we use d(%\ P) instead of <?/ fay P) since we

have proved the independence of p -matrix on choice of

integration variables in class (5)» The form of ^uy/ will

be specified later.

Consider another gauge condition

Multiply (39) by ^fc-f^, VJ and perform the gauge trans-.

The quantities Lct Uy ,Aj - ' are invariant. Furthermore

the following substitution can be made on massfehellt

: as s discussed in b 2. * .

Then we obtain - «. \rx/r/> ^fir/li 'Sr P)v

^ v 3in (42) is the Jacobian

S ) / (43)

- 42. -

If we choose 2 ^ so that

the group integral is equal to one according to definition

Condition (44) is satisfied for example try

in the second order formalism (see (28)) and by

in the first order formalism (see (39)).

Thus we prove that p -matrix of the gravitation field

is independent of the type of gauge condition (34)•

The proof of independence of ^ -matrix O D « ( ^ for the

case when the gauge function is fixed can "be made similarly to

£ 2. For this purpose substitution (6) or (61) should be

made in expression (26) with

and the variation of the term Sp-w/t in (26) is compensated

by the arising Jacobian. We omit the corresponding cumbersome

calculations.

Let us pass to consideration of some particular gauges.

1, Harmonic condition. Consider the class of the gauges

determined by the function

- 43; -

*>, I", f-fi?"The harmonic condition corresponds to

(50)

We shall uso V as independent variables. By means of

(with (3~ y^ ), (23);and (24) we find

The generating functional is equal to

I D transversal ea^Se ( " >) we have

The Feynman rules for calculation of generating func-

tional (53) ia "the powers of

(55)

is taken as interaction Lagrangian. • •

b) U(o) is Lagrangian of the linearized theory

V

- 44. -

The lowering and rising of indices in (57) are made

by the Minkowslp. tensorstLand d •

c) The free propagator of (\( is calculated from (57)

The Feynamn rules for gravitation field in the gauge

H ~Q was also obtained by Popov and Faddeev |_2j#

2) Linearized form of the harmonic condition. Consider-

the class of gauges described by the function •'.

( 5 9 )

We choose ^u^ . ells independent variables. One obtains

with the help of (13) (with g = Q ) ,,(32), and (24)

ctiThe generating functional is (s**

The Peynman rules for the perturbation calculation of (61)

in powers of

consist

LL ( C63)

-45' -

should be taken as the interaction Lagrangian .

b) Ufa) is Lagrangian of the linearized theory

The rising of indices in (64) is made "by the Minkowski

tensor o .

c)the free propagator of ylu\/ is calculated from

The Feynman rules for J, = - 1 was also given by Mandels-

Dira.'cr gauge FlS 1 Consider the following set of gauge

conditions: A

(67)

Here: , V

£f 'W • (68)

In gauge (66), (67) it is natural to use the first-order

formalism. We choose J' and j y^ as independent variables.

With the help of (14) (with ^~Q ), (15), and (23) one finds:

(69)6 , > * ' • > • . , •

, M

- 46, -

( 7 2 )

From (68) -(73) we obtain

The generating functional is

According to the general proof given in this paragraph

generating functional (75) on masashell is equivalent to

(53), (5^)),and (61).

Now we transform exp.(75).Using (25) and (74) the field

equations for 5 can t>e obtained:

The only physical solution of (76) is S>U^LQ .Thus

<\\ and p// satisfy (20)-(21) with 6ut=C, i*6* relations

(4) for py\ are valid and V satisfies the usual Einstein

equations. '

Substituting (4) into (69) one obtains:

- 47 -

Here (ffvis three-dimensional Ohristoffel symbol and

t* /1? , Let us find the expression for / ,v with the

help of Einstein equations for j?' and substitute it into

the relation:

P

which must be true according to (66). Erom (79) one obtains

(BO)

Here K —*- ^LK. * I\IK $•& three-dimensional curvature tensor

of second rank.

Finally the expression for 6 ^ takes the form

(82)

Thus we can see that the expression

i ftp:,. « .. .-. &.. i ^

for the generating functional with gauge conditions (66),

(67) can be used instead of (75). Generating functional (85)

leads to the same field equations for 1 , V§\ a

consequently to the same S -matrix as (75) does.

Now let us integrate in (83) otfer all the j L^

except I K, . One can show £.6 J that hQ takes the form

(84)

Here nfiw°)) is the Hamiltonian of gravitation field

the explicit form of which we do not need, 7T are the

canonical momenta for

p<> (85)

Let us pass from the integration over j and ~j'

in (83) to the integration over 7f , gu^ • The Jacobian

., arising can be omitted. The proof of this fact is analogous

to that of the possibility of arbitrary choice of the

functional integration variables belonging £o class (5)«

The final expression for the generating functional

in gauge (66 or 86) and (67) is

C87)

Exp. (87) has been obtained by Popov and Faddeevi-°J

with the help of the other method closely connected with

•che canonical quantization procedure.

Note once more that \ -matrix corresponding to (87)

is equal to that in covariant gauges.

of (&) p ^ ^Aobi'm:(cc) COM4in iU- •W"l« ^^ fly Pine1- J;

- 49 -

5« Conclusion

The present paper has been devoted to constructing

the S -matrix in the theories invariant under gauge

groups.Though the only cases considered were those of

Yang-Mills field and gravitation, the method developed

can be in principle applied to arbitrary theories * \

particularly in the cases when no connection with the

canonical scheme can be traced. Furthermore, the method

suggested proves to be convenient for constructing the

perturbation expansion of the p -matrix in theories par-

tially invariant under gauge group, the power of divergence

in p -matrix being considerably reduced.

In the paper the fields were considered paying.no

attention to possible interactions with other particles.

The latter would not affect our consideration however.

We would like to discuss briefly the problems which

have not been yet solved.

1) Owing to divergences there is an important problem

to arrange^ the regularization-which would not affect the

group properties of the theory. Recall thfct the gauge

non-invariant regularization in electrodynamics creates

the photon mass.

From the modern view the photon mass arising is due

to the Schwinger terms or in the end due to a singular

ae) The theories of Yang-Mills and gravitation fields are

apparently the only gauge theories of physical interest |12J

- 50 -

character of the field operators product in coinciding

points. In the nonlinear theories this problem becomes even

more complicated. The Schwinger terms affect even

the renormalization constant^for instance in the Case of

Yang-Mills field.

2. There is an interesting question of whether

the gravitation field is renormalizable in the framework

of perturbation theory,;3C7 It is convenient to treat this '

problem using the variables (\ and Pj! in the first

order formalism where there are two vertices: a vertex

and the vertex responsible for the interaction of K-

with the fictitious ft -field. The formal estimate of

degrees of growth leads to the conclusion that thejtheory

ie of unrenormalizable type. ' -;

2» Since we have two different-results concerning

-matrix of massive Yang-Mills field, a - detailed check

and comparison of them is necessary .This work is now in

progress.

ACKNOWLEDGMENTS

The authors are grateful to the participants of the theoretical seminar in P.N. LebedevInstitute for useful discussion. One of the authors (EF) thanks Drs, Popov and Faddeev forbeing so kind as to have given an opportunity of acquaintance with the article [6] prior topublication, and to Professor E.R. Caianiello for his kind hospitality. He is also gratefulto Professor Abdus Salam for hospitality at the International Centre for Theoretical Physics, Trieste.

x) We mean here the usual perturbation expansion with

respect to a coupling constant and not the method of

Fradkin-Efimov

• •• - 5 1 - . .

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