Constraints, Symmetry Transformations and Conserved Charges for Massless Abelian 3-Form Theory

We demonstrate the existence of the first-class constraints on the massless Abelian 3-form theory which generate the classical gauge symmetry transformations for this theory in any arbitrary D-dimension of spacetime. We write down the explicit expression for the generator in terms of these first-class constraints. Using the celebrated Noether theorem, corresponding to the gauge symmetry transformations, we derive the Noether conserved current and conserved charge. The latter is connected with the first-class constraints of the theory in a subtle manner as we demonstrate clearly in our present investigation. We comment on the first-class constraints within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the conserved (anti-)BRST charges are the generalizations of the above generator for the classical gauge symmetry transformation. The standard Noether conserved (anti-)BRST charges are found to be non-nilpotent. We derive the nilpotent versions of the (anti-)BRST charges. One of the interesting observations of our present endeavor is the result that only the nilpotent versions of the conserved (anti-)BRST charges lead to the annihilation of the physical states by the operator form of the first-class constraints at the quantum level which is consistent with the Dirac quantization condition for the systems that are endowed with any kind of constraints. We comment on the existence of the Curci-Ferrari (CF) type restrictions from different theoretical angles, too.


Introduction
One of the most successful theories in the realm of theoretical high energy physics (THEP) is the standard model of particle physics (SMPP) where the agreements between theory and experiments are outlandish (see, e.g.[1][2][3][4][5] and references therein).This successful theory (describing three * out of four fundamental interactions of nature) is based on the quantum field theory of an interacting non-Abelian 1-form (p = 1) gauge theory where the weakly interacting neutrinos are taken to be massless.However, the modern experiments have established conclusively that the neutrinos are massive.Hence, it is very clear that the theory of SMPP is not complete and one has to go beyond the realm of the validity of SMPP (as far as the theoretical aspects of the SMPP are concerned).At present, the superstring theories are the most promising candidates which are expected to provide (i) the quantum theory of gravity, (ii) the complete unification of all the four fundamental interactions of nature, and (iii) the precise theoretical framework in THEP whose lower energy limit is the theoretical physics of SMPP (see, e.g.[6][7][8][9][10] and references therein).One of the key theoretical observations in the context of (super)string theories is the appearance of a tower of higher p-form (p = 2, 3, ...) basic gauge fields in their quantum excitations.
Against the backdrop of the above paragraph, it is but natural to think about the possibility of developing a quantum field theory that is based on the higher p-form (p = 2, 3, ...) gauge fields which provide a theoretical set-up beyond the theoretical regime of the SMPP in THEP.Hence, the quantum field theory, associated with the higher p-form (p = 2, 3...) gauge fields, is important and essential to be studied thoroughly because of its connections with the (super)string theories.The central objective of our present investigation is to study the constraint structures of the massless Abelian 3-form gauge theory and establish that they belong to the first-class variety in the terminology of Dirac's prescription for the classification scheme of constraints (see, e.g.[11][12][13][14][15] for details).The first-class constraints generate the classical local, continuous and infinitesimal gauge symmetry transformation of the above theory as they are found to be present in the standard expression for the generator of this specific gauge transformation (see, e.g.[16,17]).We have derived the Noether conserved current and charge (corresponding to the classical local, continuous and infinitesimal gauge symmetry transformation) and established that the latter is nothing but the standard definition of the generator in terms of the first-class constraints.
We have elevated the classical local, continuous and infinitesimal gauge symmetry transformation to the quantum level within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism [18][19][20][21] and shown that traces of the first-class constraints of the original massless Abelian 3-form theory are present in the physicality criteria (i.e.[Q the total quantum Hilbert space of states within the framework of BRST formalism which is consistent with the Dirac quantization condition (see, e.g.[11][12][13][14][15]) for theories endowed with any kind of constraints.At this stage, it is pertinent to point out that, in the context of the massless D-dimensional Abelian 3-form theory, the standard Noether theorem leads to the derivations of the conserved (anti-)BRST charges [Q (a)b ] but they are found to be non-nilpotent.In our earlier work [22], we have developed a systematic method to derive the nilpotent versions of the conserved (anti-)BRST charges [Q (1) (a)b ] from the standard conserved but non-nilpotent (i.e.Q 2 (a)b = 0) Noether conserved (anti-)BRST charges.In our present endeavor, we exploit this proposal [22] to obtain the off-shell nilpotent versions of the conserved (anti-)BRST charges and demonstrate that the physically criteria w.r.t.these latter charges lead to the appearance of the first-class constraints (of the original starting theory) in their operator forms which annihilate the physical states (| phys >) at the quantum level.This observation is consistent with the Dirac quantization condition for the theories that are endowed with any kind of constraints [11][12][13][14][15].We lay emphasis on the fact that the standard Noether conserved (anti-)BRST charges (Q (a)b ) do not lead to the annihilation of the physical states by the operator form of the first-class constraints of the classical D-dimensional massless Abelian 3-form theory.In addition, the non-nilpotent behavior of the (anti-)BRST charges creates problem in the precise discussion on the topic of BRST cohomology.We claim here that this observation is general and it is true for any BRST-quantized theory which is endowed with the non-trivial CF-type restriction(s).All the higher (non-)Abelian p-form (p = 2, 3, ...) gauge theories, discussed within the framework of BRST formalism, belong to this category.
Our present investigation is essential, interesting and important on the following counts.First of all, in all our previous works related with the theory under consideration (see, e.g.[23][24][25][26] for details), we have not demonstrated the existence of the classical local, continuous and infinitesimal gauge symmetry transformation in the terminology of the firstclass constraints in a systematic manner.Second, our present BRST-quantized massless Abelian 3-form theory is endowed with three CF-type redirections.We wish to demonstrate that the standard Noether conserved (anti-)BRST charges are non-nilpotent in our present case, too, as we have shown in the context of the modified amssive and massless Abelian 2-form theories [22,27].We obtain the nilpotent versions of the (anti-)BRST charges for our present case following the proposal of our earlier work [22].Third, our present endeavor is a modest step towards our main goal of proving the 6D Stückelberg-modified massive Abelian 3-form gauge theory to be a massive field-theoretic model for the Hodge theory within the framework of BRST formalism.In fact, such a proof is expected to provide a systematic theoretical basis for the existence of the quantum fields with negative kinetic terms (but with well-defined mass).These exotic fields are a set of possible candidates for the dark matter (whose massless limit corresponds to dark energy).Finally, we wish to study the higher p-form (p = 2, 3...) gauge theories up to p = 5 forms because these tower of quantum fields are relevant in the context of the quantum self-dual superstring theories that are consistently defined in the D = 10 dimensions of spacetime.
The theoretical contents of our present endeavor are organized as follows.In Sec. 2, we show the existence of the first-class constraints for the massless Abelian 3-form theory in any arbitrary D-dimension of spacetime.We write down the generator for the classical gauge symmetry transformation in terms of these first-class constraints and the non-trivial canonical commutators are mathematically expressed.Our Sec. 3 deals with the derivations of the Noether conserved current and charge where we establish a connection between the first-class constraints and the Noether conserved charge.Our Sec. 4 is devoted to a very brief discussion on the BRST approach to the D-dimensional massless Abelian 3-form theory where we recapitulate the bare essentials of our earlier work [28] and show that the standard Noether (anti-) BRST charges are non-nilpotent.In Sec. 5, we derive the off-shell nilpotent versions of the (anti-)BRST charges from the standard non-nilpotent Noether (anti-)BRST charges.Our Sec. 6, contains the theoretical material on the physicality criteria w.r.t. the off-shell nilpotent versions of the (anti-)BRST charges where we show the appearance of the operator form of the first-class constraints.Finally, we make some concluding remarks in our Sec.7 and point out a few future directions for further investigation(s).
In our Appendix A, we show the existence of the CF-type restrictions from the requirement of the absolute anticommutativity of the nilpotent (anti-)BRST charges.

Convention and Notations:
The background flat D-dimensional Minkowskian spacetime manifold is endowed with the metric tensor η µν = diag (+1, −1, −1, ...) so that the dot product between two non-null vectors X µ and Y µ in this space is: where the Greek indices µ, ν, λ, ... = 1, 2, ..., D − 1 correspond to the time and space directions and the Latin indices i, j, k, ... = 1, 2, 3, ..., D − 1 stand for the space directions only.Einstein's summation convention is adopted throughout our present endeavor.The convention of the left-derivatives w.r.t.all the fermionic fields has been taken into consideration in the entire text of our present paper.We always denote the (anti-)BRST symmetry transformations by the symbols s (a)b and the corresponding (anti-)BRST charges are represented by Q (a)b .The nilpotent [s 2 (a)b = 0] (anti-)BRST symmetry transformations are fermionic in nature and they anticommute with all the fermionic fields and commute with the bosonic fields of our theory.We have also adopted the convention of the field derivatives as: , for the D-dimensional Abelian higher p-form (p = 2, 3, ...) fields.

First-Class Constraints: Massless Abelian 3-Form Theory in Any Arbitrary Dimension of Spacetime
We begin with the free Lagrangian density (L 0 ) for the free Abelian 3-form [ gauge theory in any arbitrary D-dimension of the flat Minkowskian spacetime (which is described by only the kinetic term), namely; where !] H µνλζ (with the explicit form: In the above, the quantity Π 0ij ≈ 0 is the primary constraint on the theory where the Dirac notation of the symbol for the idea of weakly zero has been taken into account.It is straightforward to note that the Euler-Lagrange (EL) equation of motion (EoM) is true where the choices: ν = 0, λ = j and ζ = k have been taken into consideration.It is clear from equation (3) that we have the time derivative (in the natural units: = c = 1, ∂ 0 = ∂ t ) on the primary constraint Π 0ij ≈ 0 [cf.Eq. ( 2)] as follows [34] where Π ijk = 1 3 H 0ijk is the expression for the purely space components of the conjugate momenta.There are no further constraints on the theory after the derivation of the secondary constraint (i.e.∂ i Π ijk ≈ 0) in Eq. ( 4).It is interesting to point out that both the primary as well as the secondary constraints are expressed in terms of the components of the canonical conjugate momenta [defined in Eq. (2) w.r.t. the gauge field A µνλ ].Hence, they commute with each-other leading to their characterization as the first-class constraints in the terminology of Dirac's prescription for the classification scheme of constraints (see, e.g.[11][12][13] for details).Thus, we note that we have: Π 0ij ≈ 0 and ∂ i Π ijk ≈ 0 as the primary and secondary constraints, respectively, on our theory.
One of the key signatures of a classical gauge theory is the existence of the first-class constraints on it (see, e.g.[11][12][13][14][15] for details).The expression for the generator of the local, continuous, infinitesimal classical gauge symmetry transformations, in terms of the first-class constraints, can be written as (see, e.g.[16, 17.] for details): In the above expression, the totally antisymmetric nature of Π 0ij and Π ijk has been taken into account along with that of Λ 0i and Λ ij where a set of local (i.e.spacetime dependent) † The origin of the totally antisymmetric field-strength tensor H µνλζ from H (4) = dA (3) automatically demonstrates that: (2) is the underlying gauge symmetry of H (4) because of the nilpotency (i.e.d 2 = 0) of the exterior derivative.Here the 2-form Λ (2) = [(dx µ ∧ dx ν )/2!] Λ µν defines the antisymmetric (Λ µν = − Λ νµ ) tensor gauge transformation parameter.In explicit tensorial language, we have the transformation: functions (Λ µν (x) ≡ Λ 0i (x), Λ ij (x)) are the antisymmetric (Λ µν = − Λ νµ ) tensor gauge symmetry transformation parameters.Using the Gauss divergence theorem, we can reexpress the above generator, in its more transparent and useful form, as The above generator should be able to generate the infinitesimal and local gauge symmetry transformation: under which the starting Lagrangian density (L 0 ) remains invariant (i.e.δ g L 0 = 0).Using the following non-trivial equal-time canonical commutators it can be easily checked that we have the following where (A 0ij , A ijk ) are the independent components of the totally antisymmetric gauge field A µνλ .We would like to point out that all the rest of the other brackets [except (7)] of the theory are trivially zero.The above observation (8) establishes that the first-class constraints generate the local, infinitesimal and continuous gauge symmetry transformations.
We end this section with the following concluding remarks.First, we note that the Lagrangian density/Lagrangian of a gauge theory is always singular which implies that there are constraints on any arbitrary gauge theory.Second, for a gauge theory, the constrains are of the first-class variety in the terminology of Dirac's prescription for the classification scheme of constraints [11][12][13][14][15]. Finally, the local, infinitesimal and continuous gauge symmetry transformations owe their origin to the existence of the first-class constraints on the theory.Hence, the existence of the first-class constraints is a decisive signature of a theory to be an example of gauge theory.Thus, the massless Abelian 3-form theory, described by the Lagrangian density (1), is a simple example of a gauge theory.

Noether Conserved Charge and First-Class Constraints of the Theory Under Consideration
In this section, we establish a connection between the Noether conserved charge (corresponding to the infinitesimal gauge symmetry transformation) and the first-class constraints of the theory under consideration.In this context, first of all, we note that the Noether conserved current, for the massless Abelian 3-form gauge theory, is because of the observation that ∂ µ J µ = 0 due to the EoM (∂ µ H µνλζ = 0) and the totally antisymmetric property of H µνλζ and the symmetric property of the two successive ordinary derivatives on the gauge symmetry transformation parameter (Λ µν ).The above Noether conserved current leads to the definition of the conserved charge: Since our theory is endowed with the constraints (e.g.
, we have to be careful in expanding the r.h.s. of the above charge.In other words, we can not set the constraints strongly equal to zero.Thus, we have which is nothing but the final form of the generator [cf.Eq. ( 6)] for the gauge symmetry transformation ( in the case of the massless Abelian 3-form gauge theory (with the identifications: It is clear that, in a subtle manner, the first-class constraints are hidden in (6) as well as in the expression for the Noether conserved charge Q corresponding to the Noether conserved current.
We conclude this short section with the following useful remarks.First, there exists a connection between the standard generator of the infinitesimal, local and continuous gauge symmetry transformation that is expressed in terms of the first-class constraints and the Noether conserved charge (that is computed from the Noether conserved current) by using the celebrated Noether theorem.Second, in the computation of the conserved charge, the constraints should not be set equal to zero strongly if they appear in the zeroth component of the Noether conserved current.Finally, a close and careful look at expressions for the standard generator [16,17] for the gauge symmetry transformation and the conserved charge demonstrates that both are one and the same.Thus, the Noether conserved charge is nothing but the generator for the infinitesimal, local and continuous gauge symmetry transformation for our D-dimensional massless Abelian 3-form theory.

BRST Approach: Some Key Points
For our paper to be self-contained, we recapitulate the bare essentials of our earlier work [28] and highlight some of the issues that have not been pointed out strongly in [28].This section is divided into two subsections.In Subsec.4.1, we write down the coupled (but equivalent) Lagrangian densities and show their (anti-)BRST invariance with a bit of emphasis on the CF-type restrictions.The subject matter of Subsec.4.2 is concerned with the derivations of the standard Noether conserved currents and charges.We lay emphasis on the non-nilpotency property of the standard Noether (anti-) BRST charges that are the generators for the (anti-)BRST symmetries.

Coupled Lagrangian Densities: (Anti-)BRST Invariance
We begin with the description of the coupled ‡ (but equivalent) (anti-) BRST invariant Lagrangian densities that incorporate into themselves the gauge-fixing and FP-ghost terms.These Lagrangian densities for the massless Abelian 3-form theory have been derived in our earlier work [28] which are nothing but the generalizations of the original Lagrangian density [L 0 = (1/24) H µνλξ H µνλξ ].The explicit form of the coupled (but equivalent) (anti-) BRST invariant Lagrangian densities are as follows (see, e.g.[28]) where, as pointed out earlier, the field-strength tensor H µνλξ is derived from: H (4) = d A (3) .The gauge-fixing term owes its origin to the co-exterior derivatives δ = ∓ * d * because we note that the following 2-form defines the gauge-fixing term (i.e.∂ λ A λµν ) for the 3-form gauge field A λµν where we have taken a minus sign on the r.h.s. for the sake of brevity which is true for the even dimensional flat Minkowskian spacetime manifold.Here, there is a room for its generalization because we can add/subtract a 2-form Φ where the 1-form Φ (1) = dx µ φ µ defines a vector field φ µ .This is why, we have the gauge-fixing terms in L B and L B as: ), respectively.The fermionic (anti-)ghost fields ( Cµν ) C µν are the generalizations of the classical level gauge symmetry transformation parameter Λ µν and they carry the ghost numbers (−1) + 1, respectively.The bosonic (anti-)ghost fields ( βµ )β µ are endowed with the ghost numbers (−2) + 2, respectively.On the other hand, we have the fermionic (anti-)ghost fields ( C2 ) C 2 ‡ The (anti-)BRST invariant Lagrangian densities are coupled because of the existence of the non-trivial CF-type restrictions [cf.Eq. ( 20) below] which connect some of the auxiliary as well as the basic fields of both the Lagrangian densities in a very specific fashion.Hence, some of these connected fields are not independent.These Lagrangian densities are also equivalent from the point of view of the symmetry transformations as both of them respect the (anti-)BRST symmetry transformations provided the validity of the (anti-)BRST invariant CF-type restrictions are taken into account (see, e.g.[28] for details).
that carry the ghost numbers (-3)+3, respectively.Furthermore, there are bosonic auxiliary fields (B, B 1 , B 2 ) and the fermionic auxiliary fields (f µ , fµ , F µ , Fµ ) along with a bosonic vector field (φ µ ) in our theory.It can be checked that the auxiliary fields B and B 2 carry the ghost number (+2) and (−2), respectively, because we observe that: B = (∂ • β) and B 2 = − (∂ • β).However, the auxiliary field B 1 = (∂ • φ) carries the ghost number equal to zero.The fermionic auxiliary fields in pairs (f µ , F µ ) and ( fµ , Fµ ) are endowed with the ghost numbers (+1) and (− 1), respectively.Similarly, the (anti-)ghost fermionic fields ( C1 )C 1 carry the ghost numbers (−1) + 1, respectively.The Nakanishi-Lautrup type auxiliary fields (B µν , Bµν ) are invoked to linearize the gauge-fixing terms in L B and L B , respectively, and they are connected to each-other by the (anti-)BRST invariant CF-type restriction (i.e. The generalizations of the classical gauge symmetry transformation to their quantum counterparts (anti-)BRST symmetry transformations [s (a)b ] are as follows (see, e.g.[28]) which are found to be off-shell nilpotent [s 2 (a)b = 0] of order two.Hence, these quantum transformations are fermionic in nature and they transform the bosonic fields into the fermionic fields and vice-versa.It is straightforward to check that the Lagrangian densities L B and L B transform, under the (anti-) BRST symmetry transformations, to the total spacetime derivatives as: Hence, the action integrals S 1 = d D x L B and S 2 = d D x L B remain invariant (i.e s ab S 1 = 0, s b S 2 = 0) under the (anti-)BRST symmetrry transformations, respectively.We dwell a bit on the anticommutativity property of the (anti-)BRST symmetry transformations and lay emphasis on the CF-type restrictions.It turns out that the absolute anticommutativity property (i.e.{s b , s ab } = 0) is satisfied for all the fields of our theory except fields A µνλ , C µν and Cµν because we observe the following: However, for the above fields, we have the validity of the absolute anticommutativity property (i.e.{s b , s ab } = 0) of the (anti-)BRST transformations provided we use the following CF-type restrictions which have been derived by using the superfield approach to BRST formalism [23,24]: Using the (anti-)BRST symmetry transformations [cf.Eqs. ( 15), ( 16)], it can be checked that the CF-type restrictions are (anti-)BRST invariant quantities, namely; Hence, these restrictions on our theory are physical and they are connected with the geometrical objects called gerbes [28,35].The coupled Lagrangian densities lead to the following EL-EoMs w.r.t. the fields: B µν , Bµν , C µ , Cµ respectively: It is clear, from the above, that we obtain the following: Thus, we have already derived the bosonic CF-type restriction: and we argue that the other two equations in (23) lead to the derivations of the fermionic CF-type restrictions: It is clear, from the fermionic relationships in (23), that the ghost numbers can be conserved iff we take into account the following non-trivial § combinations of the pairs ( fµ , Fµ ) and (f µ , F µ ) of the fermionic § There is a trivial solution where we can take, in a straightforward fashion, the combinations: f µ +F µ = 0 and fµ + Fµ = 0.However, these combinations are not (anti-)BRST invariant.Hence, they can not be a set of physical restrictions on our theory.
auxiliary fields and the (anti-)ghost fields ( C1 )C 1 , namely; to make the r.h.s. of ( 23) equal to zero absolutely.However, the requirement of the (anti-)BRST invariance of the CF-type restrictions ensure that only the positive signs on the r.h.s. of ( 24) are permitted.We would like to lay emphasis that the r.h.s. of (24) can not be anything other than the derivatives on the (anti-)ghost fields ( C1 )C 1 .In other words, no other independent (anti-)ghost fields of our theory are permitted on the r.h.s. of (24).We end this subsection with the concluding remarks that the coupled Lagrangian densities can yield the CF-type restrictions from the EL-EoM.In other words, the latter are responsible for the existence of the coupled Lagrangian densities L B and L B .These Lagrangian densities are equivalent from the point of view of the symmetry considerations as both of these respect the (anti-)BRST symmetry transformations on the submanifold of the quantum Hilbert space of fields where all the three (anti-)BRST invariant CF-type restrictions (i.e. ) are satisfied (see, e.g.[28] for details).

Conserved Currents and Charges: Key Issues
According to the celebrated Noether theorem, the symmetry invariance of the action integral leads to the derivations of the conserved Noether currents and corresponding charges depending on the number of continuous symmetry transformations that are respected by the theory.The continuous, infinitesimal and off-shell nilpotent (anti-)BRST symmetry transformations [cf.Eqs.(15), (16)] lead to the following expressions for the Noether conserved (anti-)BRST currents [28]: The conserved (anti-)BRST charges (i.e.Q ab = d D−1 x J 0 ab and Q b = d D−1 x J 0 b ) from the above currents are as follows: These conserved (anti-)BRST charges are the generators for the infinitesimal, continuous and off-shell nilpotent (anti-)BRST symmetry transformations ( 15) and ( 16) because the nilpotent (anti-)BRST symmetries for any generic field (Φ) of the coupled (anti-)BRST invariant Lagrangian densities L B and L B can be written as follows: where the superscripts (±) signs on the square bracket correspond to the (anti)commutator for the generic field Φ being fermionic/bosonic in nature.
The relationship ( 28) is very general and it is applicable to any continuous symmetry transformation and its generator as the conserved Noether charge.For instance, it is obvious that: In view of these relationships, it can be checked that we have the following which demonstrate that the standard Noether conserved (anti-)BRST charges are not offshell nilpotent (i.e.Q 2 (a)b = 0) of order two.This happens due to the existence of the non-trivial CF-type restrictions (20) on our theory.In the above equations ( 29) and ( 30), the l.h.s. of each equation has been computed by the direct applications of the (anti-)BRST transformations ( 15) and ( 16) on the conserved (anti-)BRST charges (26) and (27).
We wrap-up this subsection with the following remarks.First of all, we have noted that the standard Noether (anti-)BRST charges are computed from the infinitesimal, continuous and off-shell nilpotent (anti-)BRST symmetry transformations.However, these charges are found to be conserved but not off-shell nilpotent of order two.Second, one has the freedom to use the appropriate EL-EoMs and the Gauss divergence theorem to convert the non-nilpotent standard versions of the (anti-)BRST charges.Third, it turns out that the standard Noether theorem always leads to the derivations of the non-nilpotent (anti-) BRST charges for the systems that are endowed with the non-trivial CF-type restrictions [22].Finally, the BRST-quantized systems, with trivial CF-type of restriction(s), lead to the existence of the conserved and off-shell nilpotent versions of the (anti-)BRST charges due to the standard Noether theorem.In this context, mention can be made of the BRSTquantized theory of the D-dimensional Abelian 1-form gauge theory.

Nilpotent (Anti-)BRST Charges: Explicit Derivation from the Non-Nilpotent Noether Charges
We follow our proposal [22] to obtain, in a systematic manner, the off-shell nilpotent versions of the (anti-)BRST charges [Q (a)b ] from the non-nilpotent versions of the (anti-)BRST charges ( 26) and ( 27) which have been derived by exploiting the standard theoretical techniques of Noether's theorem.In this context, first of all, we note that, in the expression for the BRST charge (27), we can do the following where we have dropped a total space derivative term due to Gauss's divergence theorem because all the physical fields vanish off as x −→ ± ∞.At this stage, we use the following EL-EoM that is derived from the Lagrangian density L B , namely; Taking ν = 0, λ = j and ξ = k, we obtain the following where we have used the totally antisymmetric property of the field-strength tensor H µνλξ .The substitution of (33) into the r.h.s. of (31) yields to the following: This term will be present in the off-shell nilpotent version of the BRST charge Q b .An application of the BRST symmetry transformation ( 16) on (34) produces the following: This term should cancel out from an appropriate term of Q b [cf.Eq. ( 27)] when the BRST transformation is applied on it.For this purpose, we modify the following appropriate term of Q b : It is clear that the application of s b on the second term of the r.h.s. of (36) cancels (35).Thus, this term will be present in the expression for the off-shell nilpotent version of the BRST charge Q b .In other words, so far, we have obtained two terms of Q b as: It is straightforward to note that if we apply the BRST symmetry transformations ( 16) on (37), it will yield zero.In what follows, to obtain the explicit expression for Q (1) b , we shall exploit the theoretical potential and power of (i) the Gauss divergence theorem, (ii) the BRST symmetry transformations, and (iii) the appropriate equations of motion from L B .
At this juncture, we focus on the first term on the r.h.s. of (36).Because of the antisymmetry property in i and j, we can write this term, taking into account the integral, as where we have applied the Gauss divergence theorem and dropped a total space derivative term.The following EL-EoM emerging out from L B , namely; with the choices: ν = 0, λ = j lead us to obtain the following: This term will be present in the nilpotent version of the BRST charge Q b .Hence, this will be the third term in addition to the two terms of (37).The BRST symmetry transformations on (40) leads to: − 2 (∂ 0 F i − ∂ i F 0 ) ∂ i C 2 which should cancel out from some appropriate term of Q b when s b is applied on it.For this purpose, we modify the following: It is clear that when s b will act on the second term on the r.h.s. of (41), we shall get the cancellation with: Thus, the second term on the r.h.s. of (41) will be present as the fourth term in Q b .We concentrate now on the first term of (41) which is present inside the integral and can be equivalently written as: We use the following EL-EoM ¶ from L B , namely; We would like point out that when a BRST-quantized theory is endowed with a large number of fields, it becomes cumbersome to choose the EL-EoMs that ought to be used to obtain the off-shell nilpotent versions of the (anti-)BRST charges from their counterparts non-nilptent versions.However, we have pointed out in our earlier work [22] that one has to start from the El-EoM for the gauge field in a given BRSTquantized gauge theory.After that, it becomes obvious, in a sequence, which are the El-EoMs ought to be used to obtain the off-shell nilpotent versions of the (anti-)BRST charges from the non-nilpotent Noether (anti-)BRST charges.For instance, in our case, we have used only the El-EoMs ( 32), ( 39) and (43).
With the choice: ν = 0, we obtain the following equality: Thus, finally, we obtain a BRST invariant quantity in the following explicit form: This will be the fifth term of b in addition to all the BRST invariant terms of our original Noether conserved BRST charge Q b .Ultimately, we obtain the following expression for the conserved and off-shell nilpotent version of the BRST charge [Q b ] 2 = 0) where the l.h.s.can be explicitly checked to be zero.
We dwell a bit, at this juncture, on the derivation of the off-shell nilpotent version of the anti-BRST charge [Q (1) ab ] from the non-nilpotent Noether conserved anti-BRST charge Q ab [cf.Eq. ( 26)].Exactly as we have performed the systematic exercise to obtain the off-shell nilpotent version of the BRST charge [Q (1) b ] from Q b [cf.Eq. ( 27)], we exploit the interplay of (i) the appropriate EL-EoMs from the Lagrangian density L B , (ii) the Gauss divergence theorem, and (iii) the anti-BRST symmetry transformations (15) to obtain Q (1) ab from Q ab .We do not think it is proper to repeat the similar kinds of computations as we have performed in the case of the BRST charge because it will only be an academic exercise.Thus, we straight away write down the expression for the off-shell nilpotent version of the anti-BRST charge as: It is straightforward to check that s ab Q (1) ab } = 0 which implies that the anti-BRST charge Q (1) ab ) 2 = 0] of order two.The off-shell nilpotent versions of the (anti-)BRST charges are very essential and important as we shall see in the next section.Besides it, as far as the BRST cohomology is concerned, the nilpotency property is very crucial and important (see, e.g.[34] for details).
We wrap-up this section with the final remark that the non-nilpotent Noether conserved (anti-)BRST charges can be converted into the conserved and nilpotent (anti-)BRST charges by exploiting the interplay of (i) the appropriate EL-EoMs, (ii) the Gauss divergence theorem, and (iii) the applications of the (anti-)BRST symmetry transformations at appropriate places.It is interesting to point out that, primarily, it is the EL-EoMs that are used to obtain the off-shell nilpotent versions of the (anti-)BRST charges from the nonnilpotent but conserved standard Noether (anti-)BRST charges.As a consequence, the resulting off-shell nilpotent versions of the (anti-)BRST charges are found to be conserved because the use of EL-EoMs and the Gauss divergence theorem would not violate the conservation law according to the basic principles behind the Noether theorem and ensuing conservation law (corresponding to the continuous symmetry transformations).
6 Physicality Criteria: Constraints at Quantum Level Unlike the classical starting Lagrangian density (L 0 ) which is endowed with the first-class constraints: Π 0ij ≈ 0, ∂ i Π ijk ≈ 0, the (anti-)BRST invariant coupled (but equivalent) Lagrangian densities [cf.Eqs. ( 12), ( 13)] do not possess, in their explicit form, these first-class constraints.For instance, it can be readily checked that the following explicit expressions for the canonical conjugate momenta, w.r.t. the gauge field A µνλ , namely; imply that the following relationships are true: Thus, we note that the original primary constraint (i.e.Π 0ij ≈ 0) has been tracked with the space components (B ij , Bij ) of the Nakanishi-Lautrup auxiliary fields B µν and Bµν , respectively.In exactly similar fashion, we observe that the following EL-EoMs from the Lagrangian densities L B and L B [cf. Eqs. ( 12), ( 13)], respectively, are as follows: Making the choices: ν = 0, λ = j and ξ = k, we obtain the following where we have taken H 00jk = 0 (i.e.strongly equal to zero) at the quantum level of our discussion because the coupled (but equivalent) Lagrangian densities L B and L B , do not possess any component of momenta (w.r.t. the gauge field A µνλ ) equal to zero.Taking into account the expression for the secondary constraint (∂ i Π ijk ≈ 0) of the original classical gauge theory, we find that (51) leads to the following explicit relationships: The above equations [cf.Eq. ( 52)] demonstrate that the secondary constraint (∂ i Π ijk ≈ 0) of the original classical gauge theory has been traded with the specific combinations of the derivatives on the Nakanishi-Lautrup type auxiliary fields that are present on the l.h.s. of equation ( 52).According to Dirac's quantization method of theories that are endowed with constraints, it is an essential requirement that the operator form of these constraints must annihilate the physical states (i.e.|phys >) at the quantum level.For our massless D-dimensional free Abelian 3-form theory, this essential requirement, according to Dirac's quantization method, implies that the following conditions must be satisfied for the well-defined quantum theory.At this stage, it will be noted that the physical states (i.e.| phys >) of our Abelian 3-form gauge theory correspond to only the gauge fields (and their canonical conjugate momenta).The stage is now set to capture the Dirac-quantization conditions (53) within the framework of BRST formalism where the off-shell nilpotent versions of the (anti-)BRST charges (i.e.Q (a)b ) play a very important role.The nilpotency property is very crucial when we discuss the physical states (i.e.|phys >) because the gauge transformed states can be shown to be the exact states w.r.t. the nilpotent versions of the (anti-)BRST charges.The nilpotency property ensures that such states are trivial as far as the (anti-)BRST charges are concerned (see, e.g.[34]).The physicality criteria, w.r.t. the nilpotent (anti-)BRST charges [Q (a)b ], are as follows (see, e.g.[14,15,31,34,35] for details): ab | phys >= 0.
In our BRST-quantized free massless Abelian 3-form theory, the physical gauge fields (as well as the associated Nakanishi-Lautrup auxiliary fields) and the (unphysical) (anti-)ghost fields (as well as the associated auxiliary (anti-)ghost fields) are decoupled.In other words, there are no interaction terms between them in our entire BRST-quantized free massless Abelian 3-form theory that is described by the coupled (anti-)BRST invariant Lagrangian densities [cf.Eqs. ( 12),( 13)].Right from the beginning, the (anti-)ghost fields are redundant in some sense.Thus, the total inner product space of the states of our BRST-quantized theory is precisely the tensor product of the inner product spaces of the states that are generated by the action of all the physical fields as well as the (anti-)ghost fields on their respective vacua.In other words, when the polynomials of the physical as well as the (anti-)ghost fields operate on their respective vacua, we obtain, ultimately, the inner product of the spaces of the physical states i.e. |phys >) and the ghost states.In physical terms, it boils down to the statement that the total quantum Hilbert space of states is the inner product of the physical states and the ghost states where the physical states (i.e.| phys >), as is obvious, carry the ghost number equal to zero.It is self-evident that the (anti-)ghost field operators of the conserved and nilpotent versions of the (anti-)BRST charges [cf.Eqs.(46),(47)] will act on the ghost states in the total quantum Hilbert space of states which will be, obviously, not equal to zero.Thus, the physicality criteria [cf.Eq. ( 54)], w.r.t. the conserved and nilpotent versions of the (anti-)BRST charges Q (a)b , will require that the field operators with the ghost number zero (which are the coefficients of the independent basic (anti-)ghost fields and/or the derivative(s) on them) would operate on the physical states to yield zero.In other words, first of all, if we concentrate on the the above physicality criterion w.r.t. the nilpotent BRST charge Q (1) b [cf.Eq. ( 46)], we obtain the following quantization conditions (in terms of the first-class constraints) on the physical states (i.e.| phys >) of the total quantum Hilbert space of states of our theory, namely; A close look at the nilpotent version of the BRST charge [cf.Eq. ( 46)] shows that the Nakanishi-Lautrup auxiliary field B ij is the coefficient of the totally antisymmetric combination (i.e.
) of the derivatives on the independent basic ghost fields and the totally antisymmetric combination (i.e. ) of the derivatives on the Nakanishi-Lautrup auxilairy fields is the coefficient of the independent basic ghost field C ij .Hence, the physicality criterion w.r.t. the nilpotent BRST charge Q b ) because the auxiliary field B 2 carries a ghost number equal to (-2).On the other hand, the physicality criterion w.r.t. the off-shell nilpotent version of the anti-BRST charge Q (1) ab [cf.Eq. ( 47)] implies that we have the following quantization conditions on the physical states of our theory: In other words, the coefficients of the combination (i.e.∂ 0 Cij + ∂ i Cj0 + ∂ j C0i ) and the basic anti-ghost field Cij , carrying the ghost numbers equal to zero, have annihilated the physical states (existing in the total Hilbert space of states).It should be noted that the physical states (i.e.| phys >) do not acquire any non-zero ghost number after the operation of the Nakanishi-Lautrup type auxiliary fields Bij and the specific combination of the derivatives on these auxiliary fields (i.e.∂ 0 Bjk + ∂ j Bk0 + ∂ k B0j ), respectively.It is an interesting observation that the terms like: − B ∂ 0 C2 , 3 C2 ∂ 0 B, etc., do not contribute to the quantization conditions on the physical states [in the physicality criterion w.r.t.(47)] because the auxiliary field B carries the ghost number (+2).Thus, ultimately, we note that the Dirac-quantization conditions are satisfied with the nilpotent (anti-)BRST charges in the case of our theory where the constraints are: Π 0ij ≈ 0, ∂ i Π ijk ≈ 0 at the classical level.We lay emphasis on the fact that the standard Noether conserved charges Q (a)b [cf.Eqs.(26), (27)] can not lead to the annihilation of the physical states by the operator forms of both the first-class constraints.To be precise, we obtain the quantization condition: B ij |phys >= 0 and/or Bij |phys >= 0 which corresponds to the annihilation of the physical states only by the operator form of the primary constraint of our classical D-dimensional massless Abelian 3-form gauge theory.
We lay stress on the observations (55) and (56) which imply that the nilpotent BRST and anti-BRST charges lead to the same quantization conditions on the physical states as far as the first-class constraints of the original classical gauge theory are concerned.We would like to mention here, in passing, that all the rest of the terms of the nilpotent versions of the (anti-)BRST charges Q  46)] and/or with the auxiliary (anti-)ghost fields [e.g.− B F 0 in (46)] that carry individually some non-zero ghost numbers.Hence, these terms would not lead to any quantization conditions on the physical states.The exceptions are the pairs of terms (B 0i f i , B 1 f 0 ) and ( B0i fi , B 1 f 0 ) that are present in the expressions for the nilpotent BRST charge [cf.Eq. ( 46)] and the anti-BRST charge [cf.Eq. ( 47)], respectively.A noteworthy point, at this stage, is one of the key observations that these pairs ( B0i fi , B 1 f 0 ) and (B 0i f i , B 1 f 0 ) are basically associated with the auxiliary (anti-)ghost fields ( fµ )f µ which are not independent (in the true sense of the word) as they participate in the specific kinds of the (anti-)BRST invariant CF-type restrictions on our theory.In the next paragraph, we comment on these specific terms and explain the reasons behind their nil contributions to the quantization conditions on the physical states.
We end this section with the following concluding remark.We point out that we have not taken (B 0i |phys >= 0, B 1 |phys >= 0) and/or ( B0i |phys >= 0, B 1 |phys >= 0) on the physical states because (i) the pairs of the auxiliary fields (B 0i , B 1 ) and/or ( B0i , B 1 ) are not associated with the independent basic (anti-)ghost fields of our theory (even though they carry the ghost numbers equal to zero), and (ii) we further observe that the canonical conjugate momenta [i.e.Π µ (φ) ] w.r.t. the Lorentz-vector φ µ field, from the Lagrangian densities L B and L B , are connected with the pairs of auxiliary fields (B 0i , B 1 ) and/or ( B0i , B 1 ) because we have the following: The above equation shows that the pairs (B 0i , B 1 ) and/or ( B0i , B 1 ) are not a set of constraints on our massless Abelian 3-form theory (at the classical level).Rather, they are equal to Π i (φ) = B 0i or B0i and Π 0 (φ) = B 1 .However, the field φ µ has appeared only at the quantum level and there is no trace of it at the classical level.To be precise, there are no constraints at the quantum level for the coupled (anti-)BRST invariant Lagrangian densities (when our classical gauge theory is properly BRST-quantized).Hence, the pairs (B 0i , B 1 ) and/or ( B0i , B 1 ), even though endowed with the ghost numbers equal to zero, can not impose any condition(s) on the physical states (i.e.|phys >) of our theory because they are not associated with the independent basic (anti-)ghost fields in the expressions for the nilpotent versions of the (anti-)BRST charges.Furthermore, we would like to point out that the pair of terms ( B0i fi , B 1 f 0 ) and (B 0i f i , B 1 f 0 ), in the expressions for the nilpotent versions of the (anti-)BRST charges, are associated with the specific components of the auxiliary (anti-)ghost fields ( fµ )f µ which are not independent as they are constrained to obey the (anti-)BRST invariant CF-type restrictions: fµ + Fµ = ∂ µ C1 , f µ + F µ = ∂ µ C 1 .This is the key reason behind our claim that the physicality criteria w.r.t.Q

Conclusions
In our present endeavor, the emphasis is laid on the constraint structures of the Ddimensional free massless Abelian 3-form theory at the classical as well as at the quantum level.For the latter case, we have exploited the theoretical beauty and strength of the BRST formalism.At the classical level, the starting Lagrangian density [cf.Eq. ( 1)] is endowed with the primary and secondary constraints that are found to belong to the first-class variety within the ambit of Dirac's classification scheme of constraints [11][12][13].For the BRST-quantized theory, these constraints appear in their operator form when we demand that the physical states of the quantum theory are those that are annihilated by the conserved and off-shell nilpotent versions of the (anti-)BRST charges.This observation is consistent with the Dirac-quantization scheme for theories that are endowed with any kind of constraints [11][12][13][14][15] because the operator form of these constraints must annihilate the physical state.This statement is mathematically captured in the requirement Q (a)b |phys >= 0 within the framework of BRST formalism where the conserved and nilpotent (anti-)BRST charges are denoted by Q (a)b for this purpose.
One of the interesting observations of our present endeavor is the fact that the standard Noether theorem does not lead to the derivations of the off-shell nilpotent (anti-)BRST charges even though they are derived from the continuous, infinitesimal and off-shell nilpotent (anti-)BRST symmetry transformations.In our earlier work [22], we have developed a systematic theoretical method to obtain the off-shell nilpotent versions of the (anti-)BRST charges from the non-nilpotent Noether (anti-)BRST charges.In a recent work [27], we have shown that our method is also applicable in the context of the (anti-)co-BRST charges.In our present investigation, we have exploited our earlier proposal [22] to obtain the off-shell nilpotent versions of the (anti-)BRST charges which are very useful in the context of the physicality criteria (cf.Sec. 6) where we have demonstrated the appearance of the operator form of the first-class constraints of the original classical theory.In fact, the operator form of the first-class constraints are found to annihilate the physical states at the quantum level which is consistent with the Dirac quantization scheme applied to theories (endowed with any kind of constraints).Thus, we have been able to discuss the first-class constraints of our theory at the classical level as well as at the quantum level (within the framework of the BRST formalism through the physicality criteria w.r.t. the off-shell nilpotent versions of the conserved (anti-)BRST charges).
One of the key signatures of a BRST-quantized theory is the existence of the CF-type restriction(s).In our present case, there are three such restrictions [cf.Eq. ( 20)].In our present endeavor, we have laid a bit of emphasis on the derivation of the CF-type restrictions from the EL-EoMs [cf.Eq. (32)] that have been derived from the coupled (but equivalent) (anti-)BRST invariant Lagrangian densities L B and L B , respectively.We have also pointed out the existence of the CF-type restrictions in the context of (i) the absolute anticommutativity of the (anti-)BRST symmetry transformations [cf.Eqs.(19) (20)], and (ii) the proof of the equivalence [28] of the coupled Lagrangian densities L B and L B from the point of view of (anti-)BRST symmetry transformations (cf.footnote on Page 7).For the BRST-quantized theories, the esistence of the CF-type restrictions is fundamental as the existence of the first-class constraints in the context of classical gauge theories.The CF-type restrictions have been shown to be connected with the geometrical objects called gerbes [28,36] which, primarily, prove the independent identity of the BRST and anti-BRST symmetry transformations and their corresponding charges because of their absolute anticommutativity properties.The absolute anticommutativity property of the nilpotent (anti-)BRST charges has been proven in our Appendix A and the existence of the CF-type restrictions has been demonstrated.
We would like to make a few brief comments on the physicality criteria (cf.Sec. 6) w.r.t. the conserved and nilpotent versions of the (anti-)BRST charges where it has been shown explicitly that the quantization conditions [cf.Eqs.(55),(56)] on the physical states (i.e.|phys >) of the BRST-quantized theory are obtained from (i) the field operators (of the above nilpotent (anti-)BRST charges) that are endowed with the ghost numbers equal to zero, and (ii) these ghost number zero operators are either the coefficients of the independent basic (anti-)ghost fields or the derivatives on the latter fields which do not participate in any kind of the (anti-)BRST invariant CF-type restrictions.It has also been demonstrated that a few zero ghost number operators do not contribute [cf.Eq. (57) and ensuing discussions] to the quantization conditions on the physical states (i.e.|phys >) if they happen to be the coefficients of the auxiliary (anti-)ghost fields (or the derivatives on them).To be more precise, the auxiliary (anti-)ghost fields, whose coefficients are the field operators with the ghost number equal to zero, would not contribute to the quantization conditions on the physical states, even if, the auxiliary (anti-)ghost fields participate or do not participate in the (anti-)BRST invariant CF-type restrictions of our theory.This statement is sacrosanct in the context of the physicality criteria w.r.t. the conserved and nilpotent versions of the (anti-)BRST charges of a properly D-dimensional BRST-quantized Abelian p-form massless and/or Stückelberg-modified massive gauge theory [26].
We plan to pursue our study of the higher p-form gauge theories up to, at least, p = 5 form theories as they are relevant to the self-dual superstring theories that are consistently defined in the D = 10 dimensions of spaetime.In other words, we know that up to p = 5 form gauge fields can exist in the D = 10 dimensions of spacetime.The BV formalism of the Abelian 3-form theory has been done in superspace [37] where the extended (anti-) BRST symmetries (including the shift symmetry) have been obtained for specific Lagrangian densities.It will be nice to apply our present insights and understandings in the context of these Lagrangian densities, too.Furthermore, the knowledge gained in our present endeavor will be useful in the completion of earlier work on such analysis and discussion for the Stückelberg-modified massive Abelian 3-form theory [26] which is yet to be completed.It is gratifying to state that this work has been completed now with the help of our experience and understanding of our present endeavor.The Abelian 3-form theory has natural connection with the M-theory where the BRST analysis of the ABJM model has been carried out (see, e.g.[38]).We can also study ABJM theory in the light of our present endeavor where the emphasis has been laid on the constraint structure.
b ]| phys >= 0) w.r.t. the conserved and nilpotent (anti-)BRST charges [Q (1) (a)b ] when we demand that the physical states (| phys >), in the total quantum Hilbert space of states, are those that are annihilated by the conserved and off-shell nilpotent (anti-)BRST charges [Q (1) (a)b ].It has been explicitly demonstrated, in our present endeavor, that the operator form of the first-class constraints of the original theory annihilate the physical states (i.e.| phys >) of observations in the above equation (55) because the auxiliary field B ij and the specific combination of the cyclic derivatives on the Nakanishi-Lautrup auxiliary fields (i.e.∂ 0 B ij + ∂ i B j0 + ∂ j B 0i ), carrying the ghost number equal to zero, are connected with the constraints that are associated with the gauge field (which is the physical field in our theory).It is interesting to point out that the terms like:B 2 ∂ 0 C 2 , −3 C 2 ∂ 0 B 2 ,etc., which are also associated with the independent basic ghost field C 2 in equation (46), do not contribute to the quantization conditions on the physical states (in the physicality criterion w.r.t.Q b are either connected with the basic (anti-)ghost fields [e.g. 2 (∂ 0 b would not lead to the quantization conditions: B0i |phys >= 0, B 0i |phys >= 0 and B 1 |phys >= 0. [29][30][31][32][33]rength tensor H µνλζ for the totally antisymmetric gauge field A µνλ .Here the symbol: d = dx µ ∂ µ (with d 2 = 0, µ = 0, 1, 2, ...D − 1) stands for the exterior derivative of the differential geometry[29][30][31][32][33].The totally antisymmetric nature † of H µνλζ forces the spacetime to be D ≥ 4. It can be explicitly checked that the starting Lagrangian density (L 0 ) is singular.Hence, there are constraints on the theory.The conjugate momenta, w.r.t. the gauge field A µνλ , are: