INTERNATIONAL CENTRE FORstreaming.ictp.it/preprints/P/70/001.pdf · 2005-02-19 · IC/70/1...
Transcript of INTERNATIONAL CENTRE FORstreaming.ictp.it/preprints/P/70/001.pdf · 2005-02-19 · IC/70/1...
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 - . .
REFERENCES
1. Feynman R.P., Acia Phys. Polon.,1963, 24^ 697.
2. Faddew L.D., Popov V.N., Phys. Lett., 1967, 2£B, JO.
Popov V.K., Faddeev L.D., Preprint ITP, Kiev, 1967.
3. DeWitt B.S., Phys. Rev., 1967, 162, 1195;1239.
4. Mandelstam S., Phys. Rev., 1968, 175*..'153O«,
5. Mandelstam S., Phys. Rev., 1968, 175, 1604.
6. Popov V.N., Faddeev L.D., in press.
7. Yang C.N., Mills B.I,., Phys. Rev., 195^, $6^ 191.
8. Wightman A.S., Garding L., Arkiv for Fysik,1964,28. 129.
Strocchi P., Phys. Bev.,1967,162, 1429;1968,166,- 1302.
9- Fradkin E.S., DAN USSR, 1954,^8^ 47; 1955, 100, 897;
Trudy PhlAN, 1965, 29. 7, .Moscow.^•"^V"'" Ut*T C
10. Feynman R.P., Phys. Rev.,1951, Sfk. 108.
11. Naimark M.A., Normirovannie koltza, Nauka, Moscow,1968.
12. DeWitt B.S., in "Relativity, Groups .and Topology", Gordon
and Bre^h, 1964.
13. Schwinger J., Phys. Sev.,1962, 125, 1043; 127, 324.
14. Schwinger J., in Theoretical Physics, IAEA, Vienna,1963.
15. Fradkin, E.S. , Tyutin, I .V., Phys. Let t . , 1969, 3QB, N8.
16. Pradkin, E .S . , the ar t io le «* the "book "Problems of Theoretical
Physios", Nauka, Mosoow 1969.
17. Fradkin, E.S. , Tyutin, I .V., preprint , CNB Laboratoro di
Cibernetica Hapoli 1969.
18. Dirac P.A.M., Phys. Rev., 1959, 114, 924.
19. Landau L.D., .Lifshitz,- S . M . , The f i e l d theory,Moscow,1960.
2 0 . Coleman S . , Wess J.fZ'tunino B., ' Phys. Rev. ,1969, 177A' 2239.
2 1 . Fradkin E . g . , Nucl. Phys . , 1963, 4 ^ 624; 1966, £6*. 588
Efimov G.V., JETP,1963,__44,2107; ffuovo Cimento, 1964,12,1046.