Gaugeon formalism for the second-rank antisymmetric tensor gauge fields

... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . We present a BRST symmetric gaugeon formalism for the second-rank antisymmetric tensor gauge fields.A set of vector gaugeon fields is introduced as a quantum gauge freedom. One of the gaugeon fields satisfies a higher-derivative field equation; this property is necessary to change the gauge-fixing parameter of the antisymmetric tensor gauge field. A naive Lagrangian for the vector gaugeon fields is itself invariant under a gauge transformation for the vector gaugeon field. The Lagrangian of our theory includes the gauge-fixing terms for the gaugeon fields and corresponding Faddeev–Popov ghost terms. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index B05


Introduction
The standard formalism of canonically quantized gauge theories [1][2][3][4][5] does not consider the gauge transformations which transform the field operators to those of different gauges in a Lorentz-covariant manner [1].There is no such gauge freedom in the quantum theory, since the theory is defined only after the gauge-fixing.The whole state-vector space defined in a particular gauge is not broad enough to realize the quantum gauge freedom.
Recently, gaugeon formalisms for the Abelian second-rank antisymmetric tensor gauge fields were considered by Upadhyay and Panigrahi (Ref.[31]; in the framework of the "very special relativity" of Ref. [32]), and by Dwivedi [33].They introduced a vector gaugeon field which would play the role of the quantum gauge freedom of the antisymmetric tensor gauge field.The vector gaugeon field itself has a property of gauge fields.It has a gauge invariance.In fact, the Lagrangians given in Refs.[31,33] are invariant under the gauge transformation of the vector gaugeon field.So, we should fix the gauge before quantizing the vector gaugeon field.However, the authors of Refs.[31,33] did not fix the gauge.Thus, their vector gaugeon field was not quantized.Namely, their theories are incomplete as a gaugeon formalism for the antisymmetric tensor gauge fields; they do not permit the quantum-level gauge transformation, which is an essential ingredient of the gaugeon formalism.
The aim of this paper is quantizing the vector gaugeon field and obtaining a correct gaugeon theory for the second-rank antisymmetric tensor gauge field.
The paper is organized as follows.In Sect.2, we first review the standard formalism for the covariantly quantized antisymmetric tensor gauge field.Then, we show that the vector gaugeon field must be a massless dipole field, that is, its propagator has a term proportional to 1/(p2 ) 2 .In Sect.3, we covariantly fix the gauge of the massless dipole vector field and quantize the system.In Sect.4, incorporating the massless dipole vector field as the gaugeon field, we present a correct gaugeon theory of the second-rank antisymmetric tensor gauge field.Section 5 is devoted to summary and comments.

Standard formalism
A Faddeev-Popov quantization of the antisymmetric tensor gauge field [34,35] was first performed by Townsend [36].He revealed that the Faddeev-Popov (FP) ghosts themselves have gauge invariance and thus ghosts for the ghosts are necessary.His theory, however, violates unitarity because of inappropriate ghost contents.To ensure unitarity, counting of ghosts should have been improved.The correct ghost-counting was given by Kimura [37] and Siegel [38].In the BRST quantization scheme [2][3][4], Kimura [37] has introduced the correct number of FP ghosts and Nakanishi-Lautrup's fields (B-fields) [1] which form an off-shell nilpotent BRST symmetry [4,5].The unitarity of the theory is assured by Kugo-Ojima's mechanism of BRST quartets [4,5].Kimura also gave canonically quantized theories of the antisymmetric tensor gauge fields of third rank [39] and of arbitrary rank [40]. 1n the path integral formalism, Siegel [38] gave the precise ghost-counting by a careful application of 't Hooft averaging to the arbitrary rank antisymmetric tensor gauge fields. 2 In this section, we review Kimura's theory as a standard formalism.
The classical (gauge-unfixed) Lagrangian of a second-rank antisymmetric tensor gauge field B μν is given by where the third-rank antisymmetric tensor F λμν is the field strength of B μν defined by The tensor F λμν and thus the Lagrangian in Eq. ( 1) are invariant under the gauge transformation where μ is an arbitrary vector field.Thus, to obtain a quantized theory, we need gauge-fixing and appropriate ghosts and auxiliary fields.Note that the second term on the right-hand side of Eq. ( 3) is invariant under a "gauge transformation" μ → μ + ∂ μ with an arbitrary scalar function .This is the origin of why we need ghosts for ghosts in the quantized theory of the antisymmetric tensor gauge theories.The quantum Lagrangian given by Kimura [37] is where α and β are real parameters, B μ is (partly) a B-field imposing a gauge condition ∂ μ B μν = αB ν + • • • on B μν as a field equation, c μ and c * μ are FP ghosts, and scalar fields φ, φ * , d, d * , and η play the roles of ghosts for ghosts or B-fields.One may expect these roles by observing the following BRST transformations under which Kimura's Lagrangian in Eq. ( 4) is invariant: These BRST transformations satisfy the off-shell nilpotency δ 2 B = 0.The corresponding BRST charge Q B(K) can be written as in D-dimensional space-time, and With these field contents, Kugo-Ojima's quartet mechanism [4,5] works and all negative-norm states are removed from the physical subspace by Kugo-Ojima's subsidiary condition, In particular, the fields η and d are necessary in correct mode-counting; without these fields the longitudinal modes of B μ and c * μ could not form a BRST quartet.The field equations for the zero-ghost-number fields derived from Eq. ( 4) are from which we also have We regard Eq. ( 9) as the Lorenz-like gauge condition 4 for the gauge field B μν and α as a gaugefixing parameter.Now we consider possibility of changing the gauge-fixing parameter α by an appropriate q-number gauge transformation, which would be given by where the vector field Y μ is a would-be gaugeon field and τ is a real parameter.One possibility is that Y μ satisfies so that the gauge condition in Eq. ( 9) transforms under Eq.( 12) as Thus the gauge-fixing parameter changes from α to α + τ .From Eq. ( 13) together with Eq. ( 10), we presume the field equation for the gaugeon field Y μ to be which suggests that the gaugeon for the antisymmetric tensor B μν would be a massless dipole field.

Quantum theory of a massless dipole vector field 3.1. Classical theory
Here we consider the quantization of the massless dipole vector field Y μ , whose classical equation is given by Eq. (15).To avoid a higher-derivative Lagrangian we imitate the Froissart model [48] describing a dipole scalar field.Simply generalizing the Froissart model to our case, we adopt as a starting Lagrangian, where ε is a sign factor ε = ±1, and Y * μ is an auxiliary vector field.We call this model a massless vector-Froissart model. 5The field equations derived from Eq. ( 16) are from which we also have From Eqs. ( 17) and ( 20) we obtain the desired equation for Y μ :

Gauge-fixing
To quantize the Lagrangian in Eq. ( 16) we need appropriate gauge-fixing terms since the Lagrangian is invariant under the gauge transformation where is an arbitrary scalar function.Our gauge-fixed Lagrangian is where Y * and Y are scalar B-fields and β is a gauge-fixing parameter.The field equations derived from Eq. ( 23) are which lead to higher-derivative field equations for Y μ , The higher derivative of the field equations suggests multiple pole propagators.In fact, we have 5 A brief report of the quantization of this model was given by one of the authors (M.A.: Ref. [49]).

BRST symmetry
Because of the higher-derivative field equations, the state-vector space of the quantum theory derived from Eq. ( 23) is not positive definite.We must remove all negative-norm states from the physical subspace.This was done for the scalar Froissart model by introducing BRST symmetry [22,50].We again imitate here the Froissart model with BRST symmetry.We introduce vector FP ghosts K μ and K * μ , together with scalar FP ghosts K and K * , and define our Lagrangian by Note that the first term on the second line is invariant under the "gauge transformations" , where θ and θ * are arbitrary Grassmann odd functions.The remaining terms on the second line are the gauge-fixing terms for the gauge freedom; K * and K play the role of the B-fields.These gauge-fixing terms are necessary for the vector FP ghosts to have propagators.
The Lagrangian in Eq. ( 30) is invariant under the BRST transformations, which clearly satisfy the off-shell nilpotency δ 2 B = 0.The BRST invariance of Eq. ( 30) is easily confirmed when we rewrite Eq. ( 30) as The corresponding BRST charge can be expressed by Figure 2 shows the field contents and their BRST transformations of the quantized vector-Froissart model.All of the negative-norm states are removed by Kugo-Ojima's quartet mechanism; any physical states satisfying Q B(vF) |phys = 0 are zero-norm states.

Gaugeon formalism 4.1. Lagrangian and field equations
Combining the Lagrangians of Kimura's theory from Eq. ( 4) and the massless vector-Froissart model of Eq. ( 30), we present the Lagrangian of the gaugeon formalism for the second-rank antisymmetric tensor gauge field B μν : where a is a real parameter.The third and fourth terms of the right-hand side of Eq. (34) show that the term As seen later, the gauge-fixing parameter α of Kimura's theory in Eq. ( 4) can be identified through the parameter a as The field equations derived from Eq. ( 35) are for bosonic fields, and for fermionic fields.We emphasize here that we have chosen the gauge-fixing parameter The fields in the parentheses represent corresponding fields of Refs.[31,33]; there are no counterparts of our fields Y , Y * , K, and K * .Instead, two BRST-singlet fields Z and Z were introduced as ghosts for ghosts in Refs.[31,33].

BRST symmetry
The Lagrangian in Eq. ( 35) is invariant under the BRST transformations which are defined by for the fields of the standard formalism sector, and for the fields of gaugeon sector.Field contents and their BRST transformations are shown in Figure 3. (Fields introduced in Refs.[31,33] are also shown in the figure for comparison.)Because of the off-shell nilpotency δ 2 B = 0, the BRST invariance of the Lagrangian is easily understood when we rewrite Eq. ( 35) as The corresponding BRST charge Q B can be written as With the help of the charge, we can define the physical subspace of the whole state-vector space by

q-number gauge transformations
The Lagrangian in Eq. ( 35) permits the q-number gauge transformation, where we vary the gaugefixing parameter a.Under the field redefinitions with τ being a real parameter, the Lagrangian in Eq. ( 35) becomes where Φ A collectively represents all fields and â is defined by The form invariance in Eq. ( 45) concludes that the field equations transform gauge covariantly under the q-number gauge transformations in Eq. ( 44): ΦA satisfies the same field equation as Φ A if the parameter a is replaced by â.
It should be noted that the q-number gauge transformations in Eq. ( 44) commute with the BRST transformations in Eqs.(39) and (40).As a result, the BRST charge is invariant under the q-number gauge transformations: The physical subspace V phys is, therefore, also invariant under the q-number gauge transformation:

Gauge structure of the state-vector space
In addition to the BRST charge in Eq. ( 42), the Lagrangian in Eq. ( 35) has several conserved BRST-like charges.We focus here on the following three charges: 9/13

Fig. 1 .
Fig. 1.Field contents and BRST transformations of Kimura's theory.The arrows represent the directions of the BRST transformations.The fields of odd ghost numbers are fermionic, while those of even ghost numbers bosonic.
Figure1shows the field contents and their BRST transformations.

Fig. 2 .
Fig. 2. Field contents and BRST transformations of the BRST symmetric vector-Froissart model.The arrows represent the directions of the BRST transformations.

/ 13 Fig. 3 .
Fig.3.Field contents and BRST transformations of the gaugeon formalism.The arrows show the BRST transformations.The fields in the parentheses represent corresponding fields of Refs.[31,33]; there are no counterparts of our fields Y , Y * , K, and K * .Instead, two BRST-singlet fields Z and Z were introduced as ghosts for ghosts in Refs.[31,33].