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Brst Formalism for Massive Vectorial Bosonic Fields

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Brst Formalism for Massive Vectorial Bosonic Fields Dedicated to Professor Oliviu Gherman’s 80th Anniversary LAGRANGEAN sp(3) BRST FORMALISM FOR MASSIVE VECTORIAL BOSONIC FIELDS R. CONSTANTINESCU, C. IONESCU Dept. of Theoretical Physics, University of Craiova, 13 A. I. Cuza Str., Craiova, RO-200585, Romania Received August 23, 2010 In the ’90s professor Gherman promoted many PhD theses in the field of the BRST quantization, a very modern topic at that time, and contributed to the creation of a strong research group in this field at his university. This paper summarizes one of the aspects developed by this group, the extended formalism for the Lagrangian BRST quantization. The procedure will be illustrated on a generalized version of the massive bosonic field with spin 1, model known as Proca field, named from another famous Romanian physicist. We shall start from the Hamiltonian formalism and we shall end with the Lagrangian quantum master action. This way, from Hamilton to Lagrange, has a double motivation: the Proca model is not covariant and the gauge fixing procedure in the BRST Lagrangian context is simpler following this way, as far as the ghost spectrum. Key words: extended BRST symmetry, Proca model, gauge fixing procedure. PACS: 11.10.Ef 1. INTRODUCTION The BRST quantization method is one of the most powerful tools for describing the quantization of constrained dynamical systems and, in particular, the gauge field theories. The BRST symmetry is expressed either as a differential operator s, or in a canonical form, by the antibracket ( ; ) in the Lagrangean (Batalin-Vilkovisky) case [1] and by the extended Poisson bracket [ ; ] in the Hamiltonian (Batalin-Fradkin- Vilkovisky) formulation [2]: s∗ = [∗;Ω] = (∗;S): (1) The BRST charge Ω and the BRST generator S are both defined in extended spaces generated by the real and by the ghost-type variables. In order to perform path- integral calculations in this frame, it is necessary to remove the redundant gauge variables, that is to say to gauge-fix the action by choosing a suitable gauge fermion Y . The elimination of gauge variables assures the BRST invariance of the action [3] but not of the measure. A BRST transformation of the coordinates could generate Rom. Journ. Phys., Vol. 55, Nos. 9–10, P. 961–970, Bucharest, 2010 962 R. Constantinescu, C. Ionescu 2 non-trivial terms in the action. These terms can be exponentiated (Fadeev-Popov trick), leading to what is called the ”quantum” action. The gauge fixing procedure in the standard BRST supposes the introduction of some supplementary variables from a non-minimal sector. As this extension is not al- ways very simple and can generate difficulties, an extended sp(2) BRST symmetry has been formulated, both in Hamiltonian [4] and in Lagrangian formalisms [5]. Later on, members of the Craiova research group proposed even more extended BRST sym- metries [6], [7]. Many interesting models of gauge theories have been successfully investigated using these extended formalisms. The simplest example of gauge theory is the abelian gauge field, or more pre- cisely the electromagnetic field. In the four dimensional space-time it is described by the quadripotential fAµ;µ = 0;1;2;3g and by the tensor field F µν = @µAν − @νAµ (2) Not all the components of the quadripotential Aµ(r;t) are independent. In the Hamil- tonian description, using the Dirac terminology, the electromagnetic field can be seen as a constrained dynamical system. In the early 1930’s Proca considered a model of massive bosonic field, assuming that the photon had some small but non-zero mass [8]. The Lagrangian action had the form: Z 1 1 SL = d4x(− F µνF + m2A Aµ − jµA ) (3) 0Pr 4 µν 2 µ µ It is known as Proca action and far from the sources (jµ = 0) it generates Euler- Lagrange equations of the form: ν ν µ 2 ν A − @ (@µA ) + m A = 0 (4) 2 µ We note that the mass term m AµA breaks the gauge invariance of the model and this is why a coupling with a scalar field φ is usually considered. A more general action, coupling the Proca field with a scalar field but also with its higher order derivatives, has been proposed in [9]. This last model is described by the action: Z Z 1 S [Aµ;A_ µ;φ] = d4xL = d4x (− F F µν− 0 4 µν (5) 1 − k @ F αλ@ F ρ + (@ φ − mA )(@µφ − mAµ)) λ ρ α 2 µ µ When the scalar field φ and the coupling constant k vanish, we recover the Proca equation (4). The sp(3) BRST formalism for the model (5) had been proposed in [10]. In this paper we shall obtain a gauge fixed Lagrangian in the sp(3) BRST 3 Lagrangean sp(3) BRST formalism for massive vectorial bosonic fields 963 formalism and its quantum version for the massive vectorial bosonic field, seen as the case k = 0 of the model described by (5). In order to avoid the lost of the gauge invariance, we shall keep the scalar field φ 6= 0 in (5) and at the end only the limit φ = 0 could be considered. The paper has the following structure: in the next sec- tion we shall make the canonical analysis of the model, then, in section 3, we shall demonstrate how the gauge fixed Lagrangian can be generated through the Hamilto- nian formalism. As conclusions of the paper, we shall see how the quantum master equations can be written for the massive abelian fields in our sp(3) BRST context. 2. THE CANONICAL ANALYSIS Let us consider the Lagrangian action (5) in the case k = 0. The canonical analysis of this action leads to the following irreducible first class constraints (1) G (x) ≡ F00(x) = p0(x) ≈ 0 (6) (2) i G (x) = −@ pi(x) + mpφ ≈ 0 (7) where we considered the conjugated momenta of the massive vectorial fields: @L pi ≡ (8) @A_ i The conjugated momenta attached to the scalar field φ has the form @L _ 0 pφ ≡ = φ − mA (9) @φ_ It do not generate a new constraint, allowing to express the ”velocity”: _ 0 φ = pφ + mA (10) The first class Hamiltonian will be: Z 1 1 H (A;φ,p;p ) = d3x F ijF − p pi + A (−@ip + mp )+ 0 φ 4 ij 2 i 0 i φ (11) 1 1 m2 + p p − (@ φ)(@iφ) − mAi(@ φ) + A Ai 2 φ φ 2 i i 2 i The gauge algebra of the constraints and of the Hamiltonian is expressed by the relations: (1) (1) 0 (1) (2) 0 [G (x);G (x )]x0=x00 = 0;[G (x);G (x )]x0=x00 = 0; (12) 964 R. Constantinescu, C. Ionescu 4 (2) (2) 0 [G (x);G (x )]x0=x00 = 0 (1) (2) (2) [H0;G (x)]x0=x00 = G (x); [H0;G (x)]x0=x00 = 0: (13) As we see, it is an abelian algebra and, moreover, the limit φ ! 0 do not affect this al- gebra. For simplicity reason, we shall introduce the condensed notation fG(∆);∆ = 1;2g where the superscript ∆ = 1 designates the primary constraints and ∆ = 2 the secondary ones. 3. THE sp(3) BRST SYMMETRY 3.1. THE sp(3) HAMILTONIAN FORMALISM To implement a sp(3) BRST symmetry supposes to find not one, but three anticommuting differentials so that: sT = s1 + s2 + s3;sasb + sbsa = 0;a;b = 1;2;3 (14) The relation (1) is generalized now as: sa∗ = [∗;Ωa] = (∗;S)a;a = 1;2;3 (15) As we mentioned, our aim is to obtain for our model a Lagrangian sp(3) symmetry and, in order to recover the Lagrangian gauge fixed action, we shall start from the Hamiltonian action and we shall use the equivalence between the two formalisms. There is a direct modality of obtaining the sp(3) BRST Lagrangean formalism, but it assumes the use of a very large spectrum of ghost generators. To avoid this unuseful extension, it is simpler at the classical level to construct the Lagrangean formalism following its equivalence with the Hamiltonian one [11]. The sp(3) Hamiltonian for- malism is easier to be built, without the need of introducing a nonminimal sector, and, on the other hand, the Lagrangian gauge fixed action is useful in the quantum des- cription of the model. So, we shall start by developing the sp(3) BRST Hamiltonian formalism [11] for our theory. The action (5) can be written in the following canonical form: Z i (0) Scan[A;p;u] = dt(A_ pi − H (A;p;u)); i = 1;··· ;n (16) where the Hamiltonian H(0)(q;p;u) has the form (0) (∆) (∆) H (A;p;u) = H0(A;p) + u G : (17) The Lagrange multipliers fu(∆);∆ = 1;2g play a key role in establishing an equiva- lence between the Hamiltonian and the Lagrangian formalisms. The condition (14) 5 Lagrangean sp(3) BRST formalism for massive vectorial bosonic fields 965 requires at the first step the introduction for each constraint G(∆) of the anticommut- (∆)a (∆) ing ghosts fQ ;a = 1;2;3g and of their conjugated momenta fPa ;a = 1;2;3g. As the original fields fA;pg are bosonic, the new variables must have the Grassmann (∆)a (∆) parities "(Q ) = "(Pa ) = 1. The next steps asks for the introduction of ghosts (∆)a (∆) of ghosts λ with the conjugated momenta πa and of ghosts of ghosts of ghosts η(∆) with their conjugated momenta π(∆). We shall use the following condensed notation for all these generators of the extended phase space: - for momenta (real momenta and ghost ones): (∆) (∆) (∆) (∆) (∆) (∆) (∆) PA ≡ fpi;PAg ≡ fpi;P1 ;P2 ;P3 ;π1 ;π2 ;π3 ;π g: (18) - for fields (real and ghost type): QA ≡ fAi;QAg ≡ fAi;Q(∆)1;Q(∆)2;Q(∆)3;λ(∆)1;λ(∆)2;λ(∆)3;η(∆)g (19) These generators have to obey to the relations (in the de Witt notation): j j (∆0) (∆)a ∆0∆ a pi;A = −δi ;[Pb ;Q ] = −δ δb ; (20) h (∆0) (∆)ai ∆0∆ a (∆0) (∆) ∆0∆ πb ;λ = −δ δb ;[π ;η ] = −δ : The BRST charges Ωa; a = 1;2;3 are defined as solutions of the master equa- tions h i Ωa;Ωb = 0; a;b = 1;2;3 (21) with boundary conditions @Ωa @Ωa @Ωa = δaG(∆); = "abcP(∆); = π(∆): (∆)b b (∆)b αb (∆) a (22) @Q Q=λ=0 @λ Q=λ=0 @η Q=λ=0 The solution of the problem (21)-(22) can be obtained using the homological per- turbations theory [3] and, by doing that in our case, we shall obtain the following concrete expressions for the BRST charges: Z 3 (1)b i (2)b (∆) (∆)b (∆) (∆) Ωa = d x(p0Q δba +(−@ pi +mpφ)Q δba +"abcPc λ +πa η ): (23) The second problem, consisting in finding the BRST Hamiltonian, has the solution given by [H;Ωa] = 0; a = 1;2;3 with the boundary condition HjP =Q=π=λ=π=η=0 = H0 966 R.
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