CC BY
53
EecmHUK ТГПУ (TSPUBulletin). 2011. 8 (110)
UDC 531; 531.12/.13:530.12; 537.8
L. L. Ryskina
BRST-BFV APPROACH TO CONSTRUCTION OF LAGRANGIAN FOR MASSIVE BOSONIC ANTISYMMETRIC FIELDS IN CURVED SPACE
We apply the BRST-BFV approach to Lagrangian formulation of bosonic totally antisymmetric tensor fields in arbitrary d-dimensional curved space. The obtained theories are reducible gauge models and the order of reducibility grows with the value of the rank of the antisymmetric field.
Key words: BRST-approach, massive bosonic fields, curved space, BRST-operator, Lagrangian, gauge transformation.
1. Introduction
As is well known an antisymmetric bosonic field ^...^ of rank-p will realize irreducible representation
of the Poincare group (in Minkowski spacetime) if the following equations are satisfied
(d2 - m2)i,.,, = 0 i,.*, = 0 (1)
When we turn to an arbitrary curved spacetime we suppose that conditions on ^... , which must be satisfied, tend to (1) in flat space limit. It tells us that the equations on ^ ^ in curved spacetime must be of the form
(V2 - m2)§ „ + terms with curvature = 0,
(2)
K , a+b } = nab , nab = diag(- +, +, ■• • •,+).
(3)
D»=E (v>
p=o
U ft...Up
) aUl+...aUp + | 0).
(5)
Now we want to realize equations (2) (with m = 0) as operator constraints in the Fock space. For this purpose let us define operators
l0 = D2 + X,
/, = -ia^ D
i ^.t
(6)
We will see that the “terms with curvature” are to be defined uniquely in process of Lagrangian construction.
The paper is organized as follows. In Section 1 we develop the BRST approach for massive antisymmetric bosonic fields. Section 2 is devoted to discussion of the results.
2. Lagrangian construction for massive fields
To avoid explicit manipulations with a big number of indices it is convenient to introduce the Fock space generated by fermionic creation and annihilation operators with tangent space indices
where D2 = g^v (D^Dv -T^vDCT ) and the operator X is responsible for the “terms with curvature” in the first equation of (2). Then one can show that the relations l0| ^} = 0, l1 | ^) = 0 reproduce the conditions on the fields determining the irreducible representation of the Poincare group (2). We treat the operators l0, l1 as the constraints in some unknown Lagrangian theory.
Let us define operator l0m) in the following form l0m) = l0 - m2 and in order to have a set of operators which is invariant under Hermitian conjugation and which form an algebra we add operator gm = m2.
In the set of operators we have one operator gm which is not a constraint neither in the bra nor in the ket-vector space. In this case in order to construct Lagrangian within the BRST approach (see e.g. [1-3]) we need to introduce additional (new) creation and annihilation operators and then construct extended operators o. ^ = o. + o , o = (l0m) , ^ , l^ gm ) , which
must satisfy two conditions:
1) they must form an algebra \Oi, Oj ] * Ok;
2) the operators which are not constraints must be zero (that is in the case under consideration we must have G = g + g = 0).
m o m o m ' ) +
The set of operators L0 , L , L form an algebra. After this one should construct BRST-operator
Qm =n„L:'+qL + qL+ qq,P„, Qm = o, (6)
As usual the tangent space indices and the curved indices are converted one into another with the help of the vielbein e^a, which is assumed to satisfy the relation V eVa = 0. Then one introduces a derivative operator
Dn=dn+m;‘Ka, DJ0)=aj0)=0 (4)
which acts on an arbitrary state vector in this Fock space | as the covariant derivative operator
a.
= a+a^ + f+f + qPi-pq, [Qmm] = 0, (7)
In the massive case the general state in the Hilbert space looks as follows
^> = Sn5(«+ )*■(Pi*)*’(f't *
xa*1"...a'""$*'*■**(x) | 0).
(8)
The sum in (8) ki is taken over k1, k4 running from 0 to 1 and over k2, k3, k0 running from 0 to infinity.
To construct Lagrangian for a field with a given spin p we restrict the fields | and the gauge parameters | A(l)\in the extended Fock space (8) as follows
l^) = P |A('°) = P |A('°), (8)
with operator um given in (7). If we omit these conditions then the Lagrangian (and the gauge transformations) will contain fields with all spins. One can show (see e.g. [1-4]) that Lagrangian can be written as
L = fd%(®\Qm |Ф),
(21)
which is invariant under the reducible gauge transformations
5|®) = QJA(0)), ... 5|A«) = Qm | A(i+1)^,
... 8|A(p-2)) = Qm I A(p-11). (22)
We note here that in the massive case the gauge symmetry is Stuckelberg one.
3. Summary
We have shown that the BRST approach, which was developed earlier for higher spin field models in flat and AdS spaces (see e.g. [5]), perfectly works for massless and massive bosonic antisymmetric fields in arbitrary curved space-time. The obtained theories are reducible gauge models, the corresponding Lagrangi-ans and gauge transformations are given by (21), (22) respectively for massive theories. In both the theories the order of reducibility grows with the value of the rank of the antisymmetric field. We automatically get a formulation with appropriate Stuckelberg fields. Like all the Lagrangians constructed on the base of the BRST approach, the obtained Lagrangians possess more rich gauge symmetry and contain more fields in comparison with those which are commonly used for description of the antisymmetric fields.
References
Buchbinder I. L., Krykhtin V. A. Gauge invariant Lagrangian construction for massive bosonic higher spin fields in D dimensions // Nucl.Phys. B (2005). P. 537-563.
Buchbinder I. L., Krykhtin V. A., Lavrov P. M. Gauge invariant Lagrangian formulation of higher spin massive bosonic field theory in AdS space // Nucl. Phys. B (2007). P. 344-376.
Buchbinder I. L., Krykhtin V. A. BRST approach to higher spin field theories // arXiv:hep-th/0511276.
Buchbinder I. L., Krykhtin V. A., Ryskina L. L. BRST approach to Lagrangian formulation of bosonic totally antisymmeric tensor fields in curved space // Mod. Phys. Lett. A 24. 2009. P. 401-414.
Noguchi A., Sugamoto A. Dynamical origin of duality between gauge theory and gravity // Tomsk State Pedagogical University Bulletin. 2004. Issue 7 (44). P. 59-61.
Tomsk State Pedagogical University.
Ul. Kievskaya, 60, Tomsk, Russia, 634061. E-mail: [email protected]
Received 14.03.2011.
Л. Л. Рыскина
БРСТ-БФВ ПОДхОД ДЛЯ ПОСТРОЕНИЯ ЛАГРАНЖИАНОВ МАССИВНЫх
бозонных антисимметричных полей в искривленном пространстве
Рассматривается БРСТ-БФВ подход к лагранжевой формулировке полностью антисимметричных бозонных полей в произвольном d-мерном искривленном пространстве. Полученные теории являются приводимыми калибровочными моделями, и степень приводимости растет с ростом ранга антисимметричного поля.
Ключевые слова: БРСТ-подход, массивные бозонные поля, искривленное пространство, БРСТ-оператор, лагранжиан, калибровочное преобразование.
Рыскина Л. Л., кандидат физико-математических наук.
Томский государственный педагогический университет.
Ул. Киевская, 60, Томск, Россия, 634061.
E-mail: [email protected]
