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 {Aµ,µ = 0,1,2,3} 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 section 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 algebra. For simplicity reason, we shall introduce the condensed notation {G(∆),∆ = 1,2} 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 formalism 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 description 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 {u(∆),∆ = 1,2} play a key role in establishing an equivalence 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 {Q ,a = 1,2,3} and of their conjugated momenta {Pa ,a = 1,2,3}. As the original fields {A,p} are bosonic, the new variables must have the Grassmann (∆)a (∆) parities ε(Q ) = ε(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 ≡ {pi,PA} ≡ {pi,P1 ,P2 ,P3 ,π1 ,π2 ,π3 ,π }. (18) - for ﬁelds (real and ghost type):

QA ≡ {Ai,QA} ≡ {Ai,Q(∆)1,Q(∆)2,Q(∆)3,λ(∆)1,λ(∆)2,λ(∆)3,η(∆)} (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 equations h i Ωa,Ωb = 0, a,b = 1,2,3 (21) with boundary conditions ∂Ωa ∂Ωa ∂Ωa = δaG(∆), = εabcP(∆), = 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 +εabcPc 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

H|P =Q=π=λ=π=η=0 = H0 966 R. Constantinescu, C. Ionescu 6

It leads to the sp(3) BRST invariant Hamiltonian Z 3 (1)a (2) (1)a (2) (1) (2) H = H0 + d x (Q Pa + λ πa + η π ). (24)

By choosing for the gauge-ﬁxing fermion function the form Z 3 i (1) Y = d x (∂ Ai)π (25) we obtain the following gauge ﬁxing term Z 1 abc 3 i (1) i (2)a K = ε [Ωa,[Ωb,[Ωc,Y ]]] = d x (−p0(∂ Ai) − (∂iPa )(∂ Q )+ 3! (26) (1) i (2)a (1) i (2) + (∂iπa )(∂ λ ) − (∂iπ )(∂ η )). ¯ 0 T ¯ 0 The functions Y and Ya which satisfy to the relations s Y = saYa have the concrete forms 3 1 Z X Y¯ = d3x( [P (1)(∂iA ) − (∂ π(1))(∂iλ(2)a)− 3 a i i a=1 (27) 3 3 X (1) i (2)b X (1) i (2)c − (∂iπa )(∂ Q )] − εbcd(∂iπd )(∂ Q ) b=1 b=1 respectively Z 0 3 (1) i (1) i (2)c (1) i (2)b Ya = d x(Pa (∂ Ai) − εabc(∂iπb )(∂ Q ) − (∂iπ )(∂ λ )δba. (28)

(2) (1)a (2) (1)a (2) (1) By eliminating the auxiliary ﬁelds Pa , Q , πa , λ , π and η on the basis of their equations of motion, we shall obtain the following covariant gauge ﬁxed action Z 4 1 µν 1 2 µ µ SK = d x(− FµνF + m AµA + p0(∂ Aµ)+ 4 2 (29) (1) µ (2)a (1) µ (2)a (1) µ (2) + (∂µPa )(∂ Q ) − (∂µπa )(∂ λ ) + (∂µπ )(∂ η )). The action (29) is invariant to the BRST transformations

(2)b (2)b saAµ = [Aµ,Ωa] = ∂µQ δba,saφ = [φ,Ωa] = mQ δba, (1)c (1)c (1)b (2)c (2)c (2)b saQ = [Q ,Ωa] = εabcλ ,saQ = [Q ,Ωa] = εabcλ , (30) (1)c (1)c c (1) (2)c (2)c c (2) saλ = [λ ,Ωa] = δaη ,saλ = [λ ,Ωa] = δaη , (1) (1) (2) (2) saη = [η ,Ωa] = 0,saη = [η ,Ωa] = 0. 7 Lagrangean sp(3) BRST formalism for massive vectorial bosonic ﬁelds 967

3.2. THE LAGRANGIAN FORMALISM VIA THE HAMILTONIAN ONE

If in (16) we shall look at the momenta {pi,i = 1,··· ,n} as auxiliary variables, we can eliminate them on the basis of their equation of motion. We obtain the action Z i (1) (2) S0[A,φ,u] = dt L0(A ,∂iAj,φ,u ,u ) (31) where the Lagrange multipliers {u(∆), ∆ = 1,2} are considered now as real ﬁelds. Starting from (31) we will develop the sp(3) BRST Lagrangean formalism [7]. The complete spectrum of the antiﬁelds in our case is given by

∗ (∆)∗ (∆)∗ ∗ ∗ (∆)∗ (∆)∗ (∆)∗ (∆)∗ QAa ≡ {Qa ,ua } = {Aµa,φa,ua ,Qab ,λab ,ηa ,a,b = 1,2,3}, (32) ¯ ¯(∆) (∆) ¯ ¯ (∆) ¯(∆) ¯(∆) (∆) QAa ≡ {Qa ,u¯a } = {Aµa,φa,u¯a , Qab ,λab ,η¯a a = 1,2,3}. (33) ¯ ¯(∆) (∆) ¯ ¯ (∆) ¯(∆) ¯(∆) (∆) QA ≡ {Q ,u¯ } = {Aµ,φ,u¯ , Qa ,λa ,η¯ , a = 1,2,3}. (34) It is well known that the Lagrangian dynamics is generated in a ”anticanonical” structure. The generator of the Lagrangian BRST symmetry is

µ S = S0[A ,φ] + ···

Because S0 is unique, we shall consider that S is unique too, and we will introduce three antibracket structures which have the same properties as in the standard theory: δrF δlG δrF δlG (F,G)a = A ∗ − ∗ A δQ δQAa δQAa δQ A ∗ The functionals F and G are dependent on Q and QAa. On the basis of graduation properties and of Grassmann parities [7] we deﬁne the pairs canonically conjugated in respect with these antibrackets

∗ B B (QAb,Q )a = −δAδba (35) ∗ A i (∆) where QAa are as in (32) and Q represents the ﬁelds of the theory (real A ,u and ghosts QA)

QA ≡ {Ai,φ,u(1),u(2) ≡ A0,Q(∆)a,λ(∆)a,η(∆), a = 1,2,3}. (36)

For the Lagrange multipliers we shall have ε(u(∆)) = 0,gh(u(∆)) = 0. We note that some antiﬁelds of the theory have a canonical conjugate in the antibracket structure (35) while other antiﬁelds do not have canonical pairs (33). So, the BRST differentials {sa, a = 1,2,3} will present the following decomposition

sa∗ = (sa)can ∗ +V a∗ = (∗,S)a ∗ +V a∗, a = 1,2,3 (37) 968 R. Constantinescu, C. Ionescu 8 where the non-canonical operators Va have the form [7]

r r a ε(QA) abc ∗ δ ε(QA)+1 ab ¯ δ V ∗ ≡ (−) ε QAc ∗ +(−) δ QAb ∗ . (38) δQ¯Ab δQ¯A

The nilpotency condition for sa (37) leads to the master equations

1 (S,S)a + V aS = 0, a = 1,2,3. (39) 2 For our irreducible theory, the proper solution of the master eqs. (39), till terms linear in the antiﬁelds, is Z (∆) (∆) ∗ i (2)a (1)∗ ˙ (1)a (2)∗ ˙ (2)a (1)a S = S0 + dt −u G + Aia∂ Q + ua Q + ua (Q − Q ) abc (∆)c (∆)∗ ab ∗(∆) (∆) ¯ i (2)a (1) ˙ (1)a (2) ˙ (2)a (1)a (40) + ε λ Qab + δ λab η + Aia∂ λ +u ¯a λ +u ¯a (λ − λ ) ¯ i (2) (1) (1) (2) (2) (1) ¯(2) (2)a ¯(2) (2) +Ai∂ η +u ¯ η˙ +u ¯ (η ˙ − η ) + Qa λ + Q η .

On the basis of graduation rules and Grassmann parities we can identify the following variables (∆) (∆)∗ (∆) (∆) (∆) (∆) Pa ≡ ua ,πa ≡ u¯a ,π ≡ u¯ . (41) These identifications will be very useful in the gauge fixing procedure. For example, on the basis of the identifications (41), we can consider the following form of the gauge fixing functional: µ (1) Y = (∂µA )¯u . (42) The following relations are valid [12]:

δr 1 δr(∂ Aα) A∗ = ( ε V V Y ) = − α u(1)∗, (43) µa δAµ 2 abc b c δAµ a

δr(V Y ) δr(∂ Aα) A¯ = a = − α u¯(1) (44) µa δAµ δAµ a δrY δr(∂ Aα) A¯ = = − α u¯(1) (45) µ δAµ δAµ δL 1 u(1) = − ( ε V V Y ) = ∂ Aµ (1)∗ abc b c µ (46) δua 2 the remaining antiﬁelds vanishing because of the choice (42) for Y . 9 Lagrangean sp(3) BRST formalism for massive vectorial bosonic ﬁelds 969

4. CONCLUSIONS

Following the path from Hamilton to Lagrange, we obtained a sp(3) Lagrangian gauge fixed action depending on the original fields and on some antifields. The iden- tification (41), as well as the choice (42) of the gauge fixing function, as it was sug- gested by the Hamiltonian formalism, drastically reduced the spectrum of the supplementary ghost-type fields required by a pure sp(3) Lagrangian approach. Practically, the gauge fixed action will take the form:

¯ ¯ µ A (1)∗ (1) (1) ∗ ¯ ¯ S1Y = S1[A ,φ,Q ,ua ,u¯a ,u¯ ,Aµa,Aµa,Aµ] (47)

(1) ∗ ¯ ¯ where u ,Aµa, Aµa,Aµ comply the relations (43)-(46). It leads to an effective action which is sa-invariant and that can be further used in the path integral. This path integral can be written as Z L µ (2)a (2)a (2) (1)∗ (1) (1) ¯ ZY = DA DφDQ Dλ Dη Dua Du¯a Du¯ exp(iS1Y ). (48)

If we introduce the condensed notations

A µ (2)a (2)a (2) (1)∗ (1) (1) φ ≡ {A ,φ,Q ,λ ,η ,ua ,u¯a ,u¯ } (49) we can consider the following BRST transformations

A A0 A A εA φ → φ = φ − (saφ )µa(−1) . (50) where µa are small fermionic constant parameters. The Jacobian of these transformations can be approximated through the supertrace (because it involves both fermionic and bosonic ﬁelds, the Jacobian is a superdeterminant) and the integrating measure will be ∂r ∂rS A εA a A0 Dφ → 1 − (−1) A ∗ + V S µa Dφ = ∂φ ∂φAa (51) a A0 = (1 − ∆ S)µa Dφ .

In the previous relation, the operators ∆a,a = 1,2,3 have the form

∂r ∂rS ∆a ≡ (−1)εA + V a = ∂φA ∂φ∗ Aa (52) ∂r ∂rS ∂r ∂r = (−1)εA + εabcφ∗ + (−1)εA δabφ¯ A ∗ Ac ¯ Ab ¯ ∂φ ∂φAa ∂φAb ∂φA 970 R. Constantinescu, C. Ionescu 10 and they are nilpotent

∆a∆b + ∆b∆a = 0, a,b = 1,2,3 (53)

Such operators determine the sp(3) quantum master equations in the sp(3) symmet- ric formulation of the gauge theories:

a i W 1 a a ∆ e ~ = 0 ⇔ (W,W ) = i ∆ W, a = 1,2,3 2 ~ where 2 W = S1Y + ~W1 + ~ W2 + ... represents the ”quantum action”, the ~−order corrections being generated by the integrating measure.

Acknowledgements. The authors are grateful to professor Oliviu Gherman for the support offered along the early period of their scientiﬁc activity.

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