Некоторые аспекты лагранжева квантования модели фридмана-таундсенда Текст научной статьи по специальности «Математика»

UDC 539.12; 537.8; 530.1:51-72; 537.8

B. S. Merzlikin

SOME ASPECTS OF LAGRANGIAN QUANTIZATION OF FREEDMAN-TOWNSEND MODEL

We consider the simplest non-Abelian tensor gauge theory, which is called Freedman-Townsend model. Using the method of quantization of reducible theories we construct total action of the model under consideration. Also, following more convenient method for simple models we reproduce result which coincide with obtained above one.

Key words: tensor gauge theory, Batalin-Vilkovisky method, reducible theories, Freedman-Townsend model.

1. Introduction

It is common observation, that the model of field theory can be described by various sets of fields and actions, but the equations of motion must be the same. In this case one can say about the classical equivalence of both theories. For instance, the model of antisymmetric rank two tensor field (ATF) is classically dual to the unconstrained scalar one [1] at four dimensional space-time. It is interesting to note, that the model of rank three ATF doesn’t contain physical degrees of freedom [2]. Further study of two classically equivalent theories may be devoted to the problem of quantum equivalence of them. This question does nontrivial and interest for solving. In order to prove quantum equivalence of two models, one can show that quantum energy-momentum tensor of these model coincides or differs on topological invariant. Problems of quantum equivalence of various classically equivalent models have been discussed in the following papers [3-9].

The non-Abelian generalization of rank two ATF model were developed by Freedman and Townsend [10]. They have shown, that this model is classically equivalent to nonlinear sigma-model. The problem of quantum equivalence hasn’t discussed in this paper. Subsequently, this generalization was called Freedman-Townsend (FT) model. There are some reasons to study theories with ATF. First of all, generators of gauge transformation of models with ATF are dependent. It is significant, that in case of FT model generators are depend on equations of motion. Also, ATF is a component field of spinor chiral superfield and naturally arise in models of supergravity and superstring theories (e.g. [11]).

In the present paper some aspects of Lagrangian quantization of FT model is carried out. Using the method quantization of reducible theories [12] we construct the total action for the model under consideration. It is known, that for quantization of simple reducible model, like FT model, we can apply Faddeev-Popov procedure step by step canceling the reducibili-ty [8]. We employ this method to construction of total action of the model. It is interesting to note, that we obtain the same result as if we have used Batalin-Vilk-ovisky (BV) method [12]. We will use notations and conventions from original paper [12].

The paper can be outlined as follows. In Sec. 2 we consider in brief the FT model. Also, dependence of generators of gauge transformation is discussed in this section. In Sec. 3 we construct the total action of FT model using BV method and we obtain the same result in Sec. 4 using easier method. Results are summarized and briefly discussed in Sec. 5.

2. Non-Abelian tensor gauge theory

Let us consider non-abelian ATF B^v (x), taking on a value in Lie algebra g of some compact group G with generators T (^, v,... are space-time indexes,

i, j,. are representation one)

b,v (x) - b; v (x )T,

where Bv (x) = - B'v ^ (x) is a set of antisymmetric rank two tensor fields and

T‘ e g, i = 1,...,dimG,

[T, Tj ] = fkTk, tr(TTj) = 5. (1)

Coefficients f‘jk are totally antisymmetric structure constants of Lie algebra g of group G . Let us introduce auxiliary vector field

4- 4T‘.

One can define the field theory by the action

S[B, A] = jd4x ^-4b;vs^Fl + 2 4A^ J, (2)

where

Fi - dpi‘ - 6aa; + efkAaJApk. (3)

Here e is a coupling constant. Theory with action (2) is called Freedman-Townsend model [10].

The action (2) is invariant under following gauge transformation

(4)

§4 =0, (5)

where (x) = %a (x)T is G valued vector field and Dj = 8J8^ + ef'kiA^ is a covariant derivative. Equations of motion read

8S[ B, A] 8Bi,

= P^va|3 pi

(6)

85[В А] = ^ e;vaßDv¡Bj = о.

sa;

aß

(7)

It is obvious, that gauge transformations (4) are reducible on equation of motion (6)

SB =(Dij LJ - Dij LJ )|

uv V U ^V V T>U / I

= DDv-DvD’)=(FX)‘ L =0=0

'’V V ’/ V ’V" ■/ F’V =0 (8)

One can see that zero-valued eigenvectors of generators (4) are independent by-turn. Thus, the theory under consideration is the first-stage one.

3. Batalin-Vilkovisky quantization of Freed-man-Townsend model

General method of lagrangian quantization of gauge theories with dependent generators were developed by Batalin and Vilkovisky [l2]. Using this procedure we construct the total action for the model under consideration. Following terminology, FT model is first stage one. The model of non-abelian ATF in adjoint representation of gauge group is described by the action (2). It is invariant under transformation (4) and (5). As is mentioned above, gauge transformation (4) is reducible (8). Thus we have

Ral1 (x, y)Zj (y, Z) - 0,

Vr = ( D 5J-D|5! ) 54( x - y), ZY = DYS4(y - z).

(9) (10)

Here Zy jk S4 (y - z) is linearly independent zerovalued eigenvector of generators of gauge transformation. Let us start to construct of total action

Stt, = S0 + S + S h ,,

total 0 gauge ghost'

where S0 is an initial action (2), Sgauge and Sghost are gauge fixing action and action of ghost fields correspondingly. They have following form for arbitrary first stage theory

с = 8^ R‘ c“о

ghost 5ф^ a0 0

8У

8C7

"П0а, +^

8У

8C0a0

8У

8cia

ya0r< ai a1 1

nia +

8У

8CÏ

y=Соа-Х 0(Ф)+Сіа®*с;0 +с

а

с,

are

e (C0a ) = e (C0a ) = 1.

e (ci) = e (Ci) = e (c;) = 0,

6(n0a ) = 0, e(^)=l, e(ni) = e(n; ) = 1 and ghost numbers

gh(C0a ) = 1> gh(C0a ) = -l>

gh(C) = 2, gh(C) = -2, gh(C') = 1, gh(Y) = -1, gh(n ) = 1, gh(ni) = -1, gh(noa ) = °.

Using (9) and (10) one can find the following expressions for S

and S

gauge ghost

8У

8B.

Rij Y Cj +

Äaß Oy ^

= \a x

aß

8У

sei

8У

8C0a

+ '

8У

8СІ

7ijCj

^a° І

8У 8С "

л

(ll)

(12)

0a ^^1

Then, we choose gauge fixing function in the form

(13)

y‘ = Dijß B‘

a aß

and following notation for matrices ю 1 a and а

Taking into account the last notation and using (13) the gauge fermion reads

¥ =

\d4 x (C,

CJ). (14)

In order to allow for dependence on momentums n0 a, n1, n1, we have to add the term

1

Cl ia . s-u fi і /~in

0аП0 +T C1 П1 —niT C1 a b b

where a and b are arbitrary constant. Finally for the gauge fermion we have

¥ = Jd4 x(C0 a Dj Bia + CiD'lCf + C0a DijaC;j +

+—с; п; -П—с; ). 2b 11 1 2b 17

(15)

I 1 i „i a + 0 C0an0

2a

To construct Sgauge and SghOSt using (15), we have to find following variation derivatives

where ¥ is a gauge fermion, and © 1 and ct certain matrices. Vectors C0a, C0a are ghost fields of zero stage. Scalars C1, C1 are ghost fields of first stage and scalar C/ is an extraghost one. Vector field n0a is canonically conjugate momentum of zero stage. n1 and tcJ are momentums conjugated to first stage ghosts and to extraghost correspondingly. Fields listed above have the following Grassmann parity

S¥

sBT

S¥

SCT

sy

SCT

sy

SC^

sy

'aß w^Ga w^Ga UV1 0C1

Omitting calculus let us write an answer ([,] means antisymmetrization)

SY 8У

= - D’aC(,

SB'aß

= - n‘J CJ

^ [ ß'-'0a]’

SC'

__ = D\BJ aß + — n’oa + D'J a Cl

1

SC

2a

о

gauge

SY 1 8T - 1

^ = DaC0“+ —<, — = -DaC0ai- — ni. (16) SC; a 0 2b SC" 2b

Then, let us substitute (16) to (11) and (12) after this we have (we omit indices related to representation of algebra)

s =

ghost

Jd4* tr(tCMC¡- D,C,Z"C), (17)

\d'

s = \æx tr

gauge 1

Dr Ba\a +

2a

+DaC0anl -DaC0an[+ DaC{n0a;

. (18)

First of all we consider Sghost, with an allowance for (9) we get

sgh0SI =x tr(-D№c»a]((cl-Dq)-DacDC\ (19)

After opening of antisymmetrization on indices in first term of (19) for S,t we have

Sghost = \d‘

x tr

1

'P^0a DaC0P ) X

x(DaC0p - D pC0a ) - DaC1 DaC1

2( DbC0i

(20)

Let us start to consider Smi,ae in detail and rewrite

gauge

(18) in the form

^(Dp B“p + DaC)n0a + 1

л

2a

noanoa+DaCoani- DaC0a<

(21)

It is easy to see, that after some simplifications one can integrate (21) over momentum n0a and a functional integral will contain only

exp

ai

2

Jd4x tr(DpB“p + DaC;f

Now we can

study last two terms in (21). After integration over n and nj we obtain two functional delta-functions

5(DaC0a) and 5(DaC0). Then we use following trick. We put into delta-functions some scalar fields 5(DaC -p) and 8(DaC0“ - a). Because initial action don’t depend on p and o we integrate on them with the weight exp Jd4xpo . After this we obtain following result for S’ :

id4

S = |d4x tr

gauge 1

-- (De b“p + Dac;y -- 2( DC )( DaQ )

,

(22)

Finally, we sum up (2), (20), and (22) and get ex-pressi°n for Stotal :

Stotal [ B, A, Co, C0, Cl, C1, C[] =

S0[B,A] + jd4x trf-2(DpC0a-DaCop)(DaC0p -

-DeC0a) - DaClDaCl - -(DpBap + DaCD(DyBay +

+DaC’1) - -¿DC )(DaCoa ) I,

(23)

where a and b are arbitrary constants. The total action Stota¡ includes the initial action of fields B^v (x) and AK( x ), the action of all ghosts and the gauge broken action, which generate dynamic of field B^v ( x ).

4. Simplify Lagrangian quantization of Freedman-Townsend model

In this section by the example of FT model we show that simple theories can be quantized by consistent application of Faddeev-Popov procedure. Then we will follow the method of paper [8]. Also, we omit indexes related to representation of algebra.

Let us take a gauge fixing function in the covariant form

Xa [B] = DeBea. (24)

The gauge fixing function (24) transforms under (4) and (5) as follows

Xa [B ] = Xa + §Xa = Xa + »a^p >

Ю.

The gauge fixing action reads Sf [B, A] = -1 jd4x tr(xax“ ) =

= -2 jd4x tr(DYBTaDsBSa ).

(25)

It should be noted, that Dv“ \F =0 = 0 and as a

consequence the delta function 8(%a (B)) is ill-defined. Therefore the naive Faddeev-Popov procedure does not work directly. However one can define an analogue of delta function from the vector which obeys the constrain Da%a = 0. After that we can use the standard Faddeev-Popov prescription, of course if we define the group measure correctly. We will use delta function for covariantly transverse vector constructed in [8]

5(4)[Xa ] = jdK exp \_i \d4X tr(K“Xa ) ] X

x5(4)[ D a Ka ]det(^,), (26)

where Ka is also a covariantly transverse vector and

S(4)[Da] = Jd0 exp |". Jd4x tr(Ka, (27)

□0ap = dy D Vp.

Using (27) one can rewrite (26) in the form 5 (4)[Xa ] = fd 05(4)[Xa + DaO]det(^0). (28)

Following [8] we define the invariant group measure in the form

d ^ = 8[ D a^a ]det(q,)d ^ (29)

Taking into account (28) one can prove following identity

fdmx5(4) [xa(Bx)] = det-1(Q)det2po) - A-1, (30)

where we have used the notation

□1“p = DyDyn“p- [Dp, Da ]. (31)

Now we can consider the vacuum functional of the model under consideration. It should be reminded that our aim is construction of total action of the theory. We don’t study an effective action. Let us consider the integral

eir = \dBdA exp i(S0 [ B, A]). (32)

We input into the integral (32) the following unit

1 = A-fd mx5(4) [Xa (Bx )] =

= A - fddOS(4) [xa (Bx) + DaO - Va ] detq,. (33)

Here we used (28), (29) and introduced the arbitrary vector field Va, since (33) doesn’t depend on it. Of course we have to integrate over them at the end of calculation. After integration over gauge group we have

e‘r = fdB • dA • dO det^o - A • §(4) x

X [ (B) + DaO- Va] exp iSo [B, A]. (34)

Let us integrate (34) over the vector field Va with

the weight exp

—fd4x trV2

jdB • dA • dOdetH^- A x

<exp i

So[B, A] - 2 jd4x tr(xa (B) + DaO)2

JdO exp -2Jd4x tr(xa (B) + DaO)2

= det 2(^0)exp

(36)

\d4x tr(a(B)xa (B))

Taking account of (24), (30) and (36), we get

_ 3

jdB ■ dA ■ det 2 (□) det(Q) exp i[S0 [B, A] +

2\d4x tr(DYBayDpB“p)]. (37)

=

Let us rewrite determinants under an integral in (37) in the following form

det^) = JdC0dC0 exp iJd4x tr(C0a □^P C0p),

det-1 (□„) = JdC1dC1 exp ifd4x tr(Cj □„ Cj),

_ 1

det2(^0) = fdCj exp ifd4x tr(Cj^0 Cj'). (38)

Here, C0 a, C0 a are (anti-)ghosts of zero stage with the parities e(C0a ) = e(C0a ) = 1; C1, C1 are (anti-) ghosts of first stage and s(Cj) = s(Cj) = 0; Q' is an extraghost with s(Cj') = 0 . Finally using (38) we get e"B = jdB ■ dA ■ dC0 ■ dC0 ■ C1 ■ dC1 ■ dC; x X exp [iStotal [ B, A, C0, C0, C;, C;, C;']] ,

(39)

where the action Stotal has the form

Stoi [ B, A, C0, C0, Ci, Ci, C[] = 50[ B, A] + 1

+fd1

x tr

2 DBai Dp B rp + C0a □?> Cop +

+C □o Cl + c;q, c;

(40)

(35)

To obtain the final result, we should integrate over scalar field O. It is not difficult to see, that

The total action (40) constructed above coincide with the (23) one if we put b = -2 and a = -1 in (21). We used the method of the paper [8] to construct total action of FT model. Used method is more successful to study one loop correction to effective action and to analyze quantum equivalence between FT model and non-linear sigma model.

5. Summary

We briefly discussed Freedman-Townsend model [10] and noted that it is reducible theory of first stage (8). Using general procedure of quantization of theories with linearly dependent generators [12] we constructed the total action (23) of the model under consideration. We obtain the same result (40) using the method of paper [8]. Constructed action (40) is convenient to study loop corrections and quantum equivalence of classically dual FT model and non-linear sigma model.

Acknowledgements

I am indebted to I. L. Buchbinder for discussions and useful suggestion. I am appreciate financial support from Dynasty Foundation. The work is partially supported by FTP «Research and Pedagogical Cadre for Innovative Russia», contract P2596 and by the RFBR grant, project No. 09-02-00078-a.

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Tomsk State Pedagogical University.

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

E-mail: [email protected]

Received 14.03.2011.

Б. С. Мерзликин

НЕКОТОРЫЕ АСПЕКТЫ ЛАГРАНЖЕВА КВАНТОВАНИЯ МОДЕЛИ ФРИДМАНА-ТАУНДСЕНДА

Рассматривается простейшая неабелева тензорная калибровочная теория - модель Фридмана-Таунсенда.

С использованием метода квантования приводимых теорий построено полное действие для рассматриваемой модели. Также полученный результат воспроизведен более удобным методом для простых моделей.

Ключевые слова: тензорная калибровочная теория, метод Баталина-Вилковыского, приводимые теории, модель Фридмана-Таунсенда.

Мерзликин Б. С., аспирант.

Томский государственный педагогический университет.

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

E-mail: [email protected]