Квантование массивных p-форм в искривленном пространстве-времени Текст научной статьи по специальности «Математика»

UDC 530.1; 539.12; 537.8

E. N. Kirillova

QUANTIZATION OF MASSIVE P-FORMS IN CURVED SPACE-TIME

We consider massive p-forms in arbitrary D-dimensional curved space-time. Quantization of these models has been performed. The massive theories of p-forms do not possess gauge invariance, in contrary of massless theories. We restore the gauge invariance in massive p-forms models with help of the multi-step Stuckelberg procedure and we evaluate the effective actions. The result is presented in terms of d'Alembertians acting on p-forms.

Key words: quantum fields in curved space-time, antisymmetric tensor fields, p-forms, gauge field theories, effective action.

1. Introduction

Totally antisymmetric tensor fields (ATFs), or p-forms, have various applications. For example, they appear in supersymmetry, supergravity and string models, in quantum chromodynamics, in models of quark confinement and in cosmology (see refs in [1]).

Different aspects of massive ATFs have been considered in [2-7]. Massive models have specific features: they do not possess the gauge invariance in contrast to massless theories. This fact leads to complications in quantization of massive p-forms models. To simplify the calculations of the effective actions in massive theories it is useful to reformulate them as gauge theories. After that, one can apply the quantization methods of gauge theories in massive models.

In paper [1], the effective actions in massive ATF models in four-dimensional curved space-time are constructed. The effective actions are presented in terms of d'Alembertians acting on a scalar, vector, second- and third-rank ATFs. The obtained results have been used in the paper [6] for the purpose of studying the problem of quantum equivalence of classically equivalent theories. In paper [7], the unified quantization scheme in terms of p-forms in massive second- and third-rank ATF models in arbitrary fourdimensional curved space-time have been presented.

In this paper, quantization of massive p-form models in arbitrary D-dimensional curved space-time has been performed. The effective actions in these models have been presented in terms of d'Alembertians acting on p-forms. These results can be used for an investigation of quantum equivalence of classically equivalent theories.

The paper is organized in the following way. In Section 2, some required information about p-forms is placed. Also the massive p-form models in D-dimensional curved space, p < D, are described there. Section 3 is devoted to quantization of these models. The effective actions in these models are evaluated. In the Summary the presented results are discussed briefly.

2. Models

( p)

Let us consider p-form B in arbitrary D-dimensional curved space-time, p < D:

(p) i

B =— V B (x) dxMl a ... a dX

„ » ...^p ' 7

(1)

with the exterior derivative d which obeys d 2 = 0

(p) 1

dB=-----------^ (dB)^p+1 (x) dxHi a••• a dx^p+l • (2)

( p+1)!

Hi-Hp+1

(3)

(p+i)

This expression defines the field strength F

(p+i) (p)

F = dB with the components

(p+1) (p) £+1

F № = (dB) =EHrXB a =

p v=1 Mi---Mv -Mp+1

= (P + 1)(-1) PBtMi.., p, p.]-

The adjoint exterior derivative operator 5 (52 = 0) possesses the components

(4)

(SB) = -B

/ LU ...U„ i VI

(5)

'ft--+p-1 Wi--Vp-1 •

Combining exterior derivative d with co-derivative

5, one can obtain de Rham Laplasian A:

A = d 5 + 5d, (6)

with the properties

(AA, B) = (A, AB) = (dA, dB) + (SA, SB) . (7)

It should be note that Laplasian s action on a

( p)

form B

AB =-VvV B +Ÿ Rv B

ft... v v±j Vj... V .'L-i ft I

Vl

p p

ft... ft-1v ft+1... V p

-y y Rv p B

vp

ft ft _ ft...ft-lvft+1...ft-lpft+1... Vp

(8)

r=1 s=1,s ^ p

differs from d'Alembertians action on a form, that is

□ = -A = —d8 - Sd .

(9)

( p ) ( p )

Define the inner product of two p-forms A, B by

( p ) ( p ) 1

B

ft-. p

(A ,B) = p * dBxy/ig(X)V. p

The inner product is symmetric,

(A, B) = (B, A).

(11)

Generalizing Maxwell’s action, we can write the

( p)

classical action for p-form B [5] in arbitrary D-dimensional curved space-time:

, (p) i (p+i) (p+i) i (p) (p)

Scl[B] = —( F , F ) = --{dB,dB) =

1

2( p +1)!

( p+i)

J f, i.., ,,iF "-1

(12)

where F is defined by (2)-(4). The action (12)

is invariant under the gauge transformations

(p) (p) (p) (p-i)

B ^ B' = B - d E , because of d = 0. Let us add to the action (12) a massive term, so we get a generalized Proca action:

, (p) i (p) (p) m2 (p) (p)

Scl [ B ] = — {dB, dB) +— ( B, B ) =

1

2( p +1)!

- j dDx.

2 V|g( x)lFv

^1-M- p+1

B*

(13)

+—\dD 2 p!

It is easy to see that the kinetic part of action (13)

is invariant under the gauge transformations

(p) (p) (p) (p-i)

B ^ B = B - d E , but the massive term violates this symmetry. Evaluation of the effective action is rather convenient if the gauge invariance in the classical action (13) is restored (see [5, 8]). It could be restored via insertion of the Stuckelberg extra fields in the classical action. This process has been considered in components in the papers [5, 1, 6, 7].

Let us modify the classical action (13), inserting in

(p-1)

it the derivative of extra arbitrary (p - 1)-form C . The kinetic term does not change under this operation, because of d 2 = 0:

(p) (p-i)

( p )

(14)

1 (p-i) 1 (p) (p)

Scl [ B, C ] = Scl [ B + — d C ] = — (dB, dB) + m 2

m (p) i (p-i) (p) i (p-i)

+—(B +—d C , B + — d C ).

2 m m

It should be noted that the new action (14) is invariant under the interrelated gauge transformations of p-

(p) (p-1)

form B and (p - 1)-form C :

(p) (p) (p) (p-i) (p-i) (p-i) (p-i) (p-i)

B ^ B' = B - d \ , C ^ C = C + m % , (15)

and under the gauge transformations of the Stuckel-

(p) (p) (p)

berg extra fields (provided B ^ B' = B ):

(p-1) (p-1) (p-1) (p-2)

C ^ C = C - d A ,

(p-2) (p-2) (p-2) (p-3)

A ^ A' = A + d ç ,

and so on, up to (p - k) = 0.

Now we proceed to quantization of model (14).

3. Quantization of massive p-forms in arbitrary D-dimensional curved space-time

(p)

We begin with the path integral over the forms B ,

( p -1)

C , corresponding to the classical action (14):

(p) (p-1) (p) (p-1) (p) (p-1)

I[B, C ] = jDB DC exp(iS[B, C ]). (17)

The naive path integral (17) contains infinite gauge volumes related to the invariance of the action (14) under the gauge transformations (15), (16). To extract the gauge group volumes from the integral (17) we use the Faddeev-Popov multistep procedure (see, e. g., [9, 8]) in a form adapted for Abelian theories with linearly dependent generators [8, 10]. For the fields B^v, B . (2- and 3-form) this procedure has been performed in paper [1, 6, 7]. Here we carry out the quantization for (p)

p-form B in arbitrary D-dimensional curved spacetime, p < D .

Let us denote

(p) (p) (p-i) (p-i)

B = a, C = p . (18)

Then the gauge transformations (15) take the form:

(p) (p) (p) (p-i) (p-i) (p-i) (p-i) (p-i) a — a' = a-d a , |3 —> P' = p + m a , (19)

and we can rewrite the path integral (17) as

(p) (p-l) r (p) (p-l) (p) (p-l)

/[ a, p ] = Ip =JD a D p exp(iS[ a, p ]). (20)

Next, we choose the gauge fixing function uniformly at every quantization step:

(p-1) (p) (p-1) (p-2)

K =5a + m( P+ d p ).

( p-2)

Here p is a gauge parameter of (p -1) -level

gauge transformations (lower than (19)). The last term

(p-1)

in K (21) is absent if (p - 2) < 0. Such situation arises at lowest quantization level, that is

(0) (1) (0)

K = 8a+ m( P + 0).

(21)

(22)

With help of (21) and (19) we construct the “senior” Faddeev-Popov determinant

(p-1) (p-2) _ (p-1)

A-- = J D a D ß 5 [ K

(23)

where 5 [... ] is delta-function (to distinguish from coderivative 5 (5)), and transformed gauge fixing func-

(p-1) (p-1)

tion K' is gauge fixing function K (21) depended

(p) (p-1)

on transformed variables a', p' (19):

(p-1) (p-i) (p) (p-i) (p-i)

( p-i)

(16)

K' = K (a', ß' ) = K + (-8d + m ) a .

(24)

Next, we insert the “unity” (see (23)) in the path integral (17), according to Faddeev-Popov method,

(p-1) (p-2) _ (p-1)

1 = Ap-i j D a D pg[ K ], (25)

to extract from the integral (17) the gauge group volumes related to the gauge transformations (19). So, (17) takes the form:

. (p) (p-i (p-i (p—) „ (p-i (p) (p-1)

Ip = Ap-1 j Da D p D a D p S[ K ]exp(/S[ a, p ]) .(26)

Now we carry out in (26) inverse to (19) transformations

( p) a'

( p)

■ a ;

( p-i) в'

( p-i) Then K' -

( p-i) в .

( p-i)

(p-2) (p-1) (p) (p-1)

+d P ) - X ]exp(¿S[ a, p ]).

( p-i)

When comparing the path integral (30) with the classical action (14), we can see that the exponential expression in (30) is nothing other than

(p-1) (p-2)

iScl [ P , m p ]. So let us define

(p-1) (p-1) (p-2) (p-2)

P = B , m p = C , (31)

therefore we can write, instead of (30), taking into account (17),

(p) (p-i)

(p-1) (p-2)

(27)

i ( p)

K and the classical action Sc [ B ]

(13) does not change under (19). After that the inte-

( p-i)

grand in (26) does not depend on the parameter a ,

, (p-i)

and ID a corresponds the multiplicative divergence

which could be eliminated when redefining Ip (26).

( p-i)

Adding an arbitrary (p -1) -form % with the weight

i (p-i) (p-i)

exp(-------( x , X )) to the argument of delta-function

2a

and taking into account (21), we have

r (p) (p-l) (p-2) (p-l)

Ip = Ap-! J D a D p D p D x x

i (p-1) (p-1) _ (p ) (p-1)

Xexp(-— ( X , X ))S[Sa+ m( p +

2a

(32)

(33) and so on

I [ B, C ] = A p-1 Det 2(n, +m2) I [ B , C ], or,

1

I =A , Det 2(d + m2)I ,.

p p-\ \ p / p-\

1

In turn I , =A ,Det 2(d , + m2)I 2,

p-1 p-1 v p-1 s p - 2 -■

up to

1

II =A0Det 2(n1 +m2)I0,

where zeroth-order generating functional is

(0) -i I0 = Z[B] - Z0 = Det 2(d0+m2) . (34)

Let us assume that we have extracted the divergences related to the gauge transformations (19) of all levels from Faddeev-Popov determinants of p-1-order up to 0. Then we have a recurrence formula for p-or-der generating functional:

( p)

(28)

The integration of (28) over x yields in exponential quantity (see the expression (14) also):

1 m2 (p) 1 (p-1) (p) 1 (p-1)

i{— (d a,d a) +-(a +— d p , a +— d p ) -

2 2 m m

1 (p) (P-) (P-2) (p) (p-1) (p-2)

-----[(8a + m( P +d P ),8a + m( P + d p ))]}. (29)

2a

Zp — Ap _i A p-2... A0 Det xDet _12 (□ _ + m2 )... Det~1/2 (□„ +m2 ).

Zp - Z [ B ] = Ap-1 Det (dp + m ) Zp-1,

or, substituting Zp-1, Zp-2, and taking account of (34),

<-1/2(Dp + m2) x

_i..v-o ■■■-,• (35)

In short form,

(p) p „ p-1

Z[B ] - Zp = n Det (Dp-, +m ) n Ak. (36)

F s=0 F k=0

Proceeding to calculation of Faddeev-Popov determinants let us return to A p-1 (23). Note that the trans-

(p-1)

( p)

tegral over D a in (28)-(29) can be factorized:

( p )

( p )

( p)

Taking account of the properties of Laplasian (6)- formed gauge fixing function K' (24) is ivariant un-

(8) and those of the inner product (10), (11) we can see der lower degree transformations of type (19) with pa-

that the parameter a should be equal to 1. Then an in- (p-2)

rameter a :

(P-1) (P-1) (P-1 (P-2) (P-2) (p-2) (p-2) (p-3)

a ^ a' = a - d a , P ^P' = P+ m a . (37)

Extracting from (23) infinite gauge volumes related to these gauge transformations we construct gauge

(p-2)

fixing function K of the same structure as (21) but

(p-2)

of lower order. Through the agency of K the next Faddeev-Popov determinant A 2 of form (23) is con-

(0)

íD a exp{— [a,(□ +m2) a)]} = Det 2(n +m2). j 2 p p

So, the path integral (28) takes the form:

-i „ r (P-1) (P-2)

Ip = Ap-1 Det 2(Dp +m2)\D в D p x

i (P-1 (P-) „ (P— (p-2) (P— (P-2)

xexp{-- [(d p ,d p ) + m2( в +d P , P + d p )]}. (30)

structed. And so on, up to the K (22). — 26 —

Let us begin with zeroth-order Faddeev-Popov determinant А0. The gauge transformations which are

(0)

caused K (22) have the form

(1) (1) (1) (0) (0) (0) (0) (0)

a — a' = a-d a, p —> p' = p + m a ,

(0) (0) (0) (0)

and K' (22), (24) is simply K' = K + (□ 0 + m ) a , be-

(0)

cause of d8 a = 0. Then, in according to (23),

(0)

Д 0 = Det(□ 0 + m ) а .

(39)

(0)_ (0)

(1) (0)_ (1) (0)_ (О)

A-1 =A0 J D a D ßö [K'] D a 8 [K'].

(0)_ (0)

(4О)

factorized and we get

А2 = A-1 Det(□ 2 + m2 ) .

Next, we

(2) (1)

generate

(2)

the

new

“unity”

Now we insert the “unity” 1 = Д 0J D а б [K'] in expression (23) to find Аі:

Further we carry out the inverse to (38) transforma-

(i)

tions in the integrand of (40). The function K' is in-

(0) (0)

variant under these transformations, whereas K' ^ K.

Now the integrand of (40) does not depend on the pa-

(0) , (0)

rameter a , and I D a is the multiplicative divergence

which could be eliminated when redefining Д-1 in

such a way (see (22) and (24)):

(1) (0) _ (2) (1) (0)

Д-1 = Д 0 j D a D в 8 [ 8a+ m( в + d в ) +

, (1) _ (1) (0)

+(-8d + m )a]8[8a+ m в].

By means of second delta-function we evaluate the (0) (1) integral over D b , and integrating over D a yields us

Д1 = Д Q1Det(□ j+ m2) = Det-1(n0 + m2) Det(□ j+ m2).

, (1) (0)_ (1) Generating the “unity” 1 = A1 J D a D в 5 [K'], we

insert it in the next Faddeev-Popov determinant Д-1 of type (23) to eliminate divergences related to the gauge transformations (19) for p = 2 :

(2) (1) (1) (0)_ (2) _ (1)

Д-1 =Д1 J D a D в D a D p 8[K '] 8[K ']. (41)

Let us perform inverse to these transformations in the integrand of (41):

(2) (2) (2) (1) (1) (1) (1) (1) a — a' = a- d a, p —> P' = p + m a,

(1) (1)

then K' — K , and the rest expression stays invariant.

(i)

Now the dependence on a vanishes in the integrand,

and we can get out of correspondent infinite volume.

(i)

(38) 1 = Д2 J D a D в 5[K'] and insert it into the Faddeev-

Popov determinant A-1 of type (23), operating by analogy with p = 2 case, and so on. Eventually we have p such procedures (2p gauge parameters, p gauge functions, p Faddeev-Popov determinants) until highest order free of infinite gauge volumes A p-1 (which is needed in (33)) is derived. Obviously, an arbitrary order determinant Ak has the form:

A k = A l-Pet (D k + m 2). (42)

Or, in detail,

Ak = Det(D k + m2)Det- (D k-1+ m2 )x

xDet (□ k-2 + m2)Det— (□ k-3+ m2)... x xDet(a 2+ m1')Det^x(u 1+ m1')Det(u 0 + m2), (43a)

if k is even, and

Д k = Det (□ k + m2) Det “*(□ k-1+ m2) x xDet (□ k-2 + m2)Det— (□ k-3+ m2)... x xDet(□ j+ m2 )Det-1 (□ 0 + m2), (43b)

if k is odd. When combining (43a) and (43b), we can write

Лк = nDet(-1)l (□ -i +m2) .

(44)

There is product of determinants Ak from A0 (39) to Ap-1 (44) in the expression for p-order generating

(p)

functional Z[B] - Zp (36):

A = flAk . (45)

k=0

Let us consider in detail the case of even (p - 1) (i. e. oddp in (45) as provided by (43):

A = Ap_[ Ap_2 Ap-3 Ap_4... A A^ A0 =

' i '' 2 ” (p-i)/2

= Det (□ p _j+ m1 )Det-1 (□ p _ 2+ m2 )Det (□ p_3+ m2) x xDet-1 (□ p_4 + m2)...Det (□ 0 + m2) Det (□ p_2 + m2) x xDer1 (□ p _3+ m 2)Det (□ p _4+ m1)...Det~1 (□ 0 + m2) x xDet (□ p-3+ m2) Det “*(□ p-4+ m2)Det (□ p-5+ m2)... x

Further we integrate over ^ ß with second delta-func- xDet(□ 0+ m2)Det(□ p-4 + m2)Det-1(^ p-5 + m2) x

p-5

xDet(□ 6 + m2)...Det *(□ 0+ m2)...Det(□ 0 + m2).

tion, the dependence on b vanishes also. The infinite

r (0)

gauge volume I D p can be removed when redefining the determinant. Then the integral over D a is So for odd p we have

A ~ Ap-1 Ap-2

...A 0 = Det (□ p-1+ m2) Det (□ p-3+ m2) x

xDet(□ p-5 + m2)...Det(□ 0 + m2). (46)

Now consider odd (p- 1) (i. e. evenp) in (45):

A = Ap-i Ap-2 Ap,_3 Ap-4... A^A =

' i " 2 ” p/2 = Det (□ p _j+ m2) Det(□ p _ 2+ m2 )Det (□ p_3+ m2) x

xDefl(u 4 + m2)...Det (□ 0 + m2) Det (□ 2 + m2) x

P/2

Zp=nDerm(n 2s +m2)X

P s=0 P

( p-2)/2

n De?+1,2(□

p-(2k+1)

+m2).

(51)

And for oddp we have:

Zp = Det-yi (□ p +m2) Det-yi (□ p-1 + m2)... x xDet"^(□j +m2)Det^'^(□o + m2)Det(□ p1+ m2) x xDet (□ 3+ m2) Det (□ -5 + m 2)...Det (□ 0 + m2) =

xDet *(□ p_3+ m2)Det(□ p_4+ m2)...Det *(□ 0 + m2)x

= Det 1/2 (□ p +m2)Det+1/2(□ . +m2) x

xDet <P p _3+ m2) Det p _ 4+ m2)Det (□ p_5+ m 2)...x xDet-1/2(□ p-2 +m2) Det+1/2(□ p-3 +m2)... x

xDet (□ 0 + m2)Det (□ _4 + m 2)Det “*(□ _5 + m2) x

xDet ~1/2 (□1 +m2) Det+1/2 (□„ +m2),

xDet (□ _6 + m2)...Det “*(□ 0 + m2)...Det (□ 0 + m2) x xDet(□ 0 + m2) Det (□ 0 + m2).

Therefore, for even p we obtain A = Ar-i Ap-i-A* = Det(n p-i+ m2)Det{n m2) x

xDet(a p-5+ m2')...Demu ^ m2).

Let us combine the expressions (46) h (47):

A = Ap_t Ap_2...A0 = Det(n p_.+ m2)Det(n p—+ m2)x xDet(a x_p + m2)...Det (□ p.+ m2), where p0 = (1 + (-1)p) / 2, or, briefly,

or,

( P-1)/2

Zp = n Det-^(□p-,s +m2) X

F s=0 F

( P-1)/2

x n Det+1/2(□

+1/^p-(2k+i) +m2).

(52)

p-i

(2 p-(-1)p-3)/4

A = n.At = n Det- (q,-(2k+i) +m).

(47)

(48)

(49)

(50)

When using the definition p0 (49), one can combine (51) and (52):

( P-1+Po)/2

Zp = n

F s =0

( P-1-Po)/2

x n Det

-1/2(□p-2s + m2) x

Det

+1/2 (r-y , 2

'(□

p-(2 k+1)

+m ).

(53)

(p > (p>

Next, the correlation Z [ B ] = Zp = exp(zT[ B ]) yields the effective action (p) (p)

r[ B ] = r p =-iTr ln Z[ B ].

Further we apply given formula for product of Fad-deev-Popov determinants in the expression forp-order p _ i[(P 2 jr i^^

generating functional (36): P 2

Taking into account the derived expression (53) for p-order generating functional Zp, we have

( P-1+Po)/2

P -2s

+m2') ■

(p) p

Z[ B ] = Zp =n Det-1/2(n _ + m2) X

(2 p-(-1)p -3)/4

x n

s_0 ( P-1-Po)/2

- X Tr ln(R

s_0

Det “1/2(p

‘p-(2k+1)

+m2).

P-(2i+i) +m )]> P < D. (54)

For p = 2, for example, the formula (54) yields

We begin with the case of even p:

Zp = Det-y2 (□ p +m2) Det-y2 (□ p _1 +m2)... x xDet ~1/2 (□j +m2) Det ~1/2 (□0 + m2) Det (□ p-1+ m2) x xDet (□ p-3+ m2) Det (□ p-5 + m 2)...Det (□ 1+ m2) =

= Det~1/2 (□ p +m2)Det+1/2(□ p-1 +m2) x xDet ~1/2 (□ p-2 +m2) Det+1/2 (□ p-3 +m2)... x xDet+1/2 (□1 +m2) Det ~y 2 (□„ +m2), or,

r2 = — [Tr ln(H2+m2) + Tr ln(□0 +m2) -Tr ln(□1 +m2)], and for p = 3

r3 = ^ [Tr ln(^3 +m2) + Tr ln^j +m2) -

-Tr ln (^2 + m2) - Tr ln (□0 + m2)].

The concretization of expression (54) is actualized by means of generalized zeta-functions which correlate to operators (-□ p + m2) [6], in Euclidian formulation. Zeta-functions regularization technique have been used, e. g., in [11].

k=0

k=0

4. Summary

We have brought the detailed derivation of effective actions in massive p-forms theories in arbitrary D-dimensional curved space-time in terms of d’Alembertians acting on p-forms. The gauge transformations, Faddeev-Popov determinants and gauge fixing functions in Faddeev-Popov multistep procedure have been represented uniformly in the each stage of calculations.

To restore gauge invariance it needs additional Stückelberg fields. Quantization procedure is multistep as well as in massless case (see, e. g., [12]). But quantization of massive p-forms is more complicate than that of massless p-forms because of there are two additional interrelated fields in massive theory (instead of single field in massless theory) at each stage. This fact complicates calculations considerably.

Unified approach to quantization of massive p-form models in D-dimensional curved space simplifies application of massive ATF models to various aspects of quantum field theory, for example, to study the problem of quantum equivalence of classically equivalent theories or using that in supersymmetry theories.

Particular role of (anti)symmetric tensor fields in special manifold bas been underlined, for example, in paper [13].

This work will be continued for the purpose of studying the problem of quantum equivalence of classically equivalent massive p-forms models in arbitrary D-dimensional curved space-time.

It is known that massive rank-2 and massive rank-3 ATF models in curved space-time is classically equivalent to massive vector field and to massive scalar field with minimal coupling to gravity, correspondingly. In the paper [6], corresponding quantum equivalence is proven via the exact equality of the corresponding effective actions in four-dimensional curved space. Similar procedure can be applied to the effective actions in massive p-forms theories in arbitrary D-dimensional curved space-time.

Acknowledgements. The author is grateful to professor I. L. Buchbinder for a statement of the problem. The research was partially supported by Grant for Russian Leading Scientific Schools, project No 3558.2010.2.

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Tomsk State Pedagogical University.

Ul. Kievskaya, 60, Tomsk, Russia, 634061.

E-mail: [email protected]

Received 14.03.2011.

Е. Н. Кириллова

КВАНТОВАНИЕ МАССИВНЫх Р-ФОРМ В ИСКРИВЛЕННОМ ПРОСТРАНСТВЕ-ВРЕМЕНИ

Рассматриваются массивные />-формы в произвольном D-мерном искривленном пространстве-времени. Производится квантование этих моделей. В отличие от безмассовых теорий массивные теории />-форм не обладают калибровочной инвариантностью. Мы восстанавливаем калибровочную инвариантность в массивных моделях />-форм с помощью многоступенчатой процедуры Штюкельберга и оцениваем эффективное действие. Результат представлен в терминах Даламбертианов, действующих на />-формы.

Ключевые слова: квантовые поля в искривленном пространстве-времени, антисимметричные

тензорные поля, калибровочные полевые теории, эффективное действие.

Кириллова Е. Н., кандидат физико-математических наук, доцент.

Томский государственный педагогический университет.

Ул. Киевская, 60, Томск, Россия, 634061.

E-mail: [email protected]