Reducible Stueckelberg symmetry and dualities

We propose a general procedure for iterative inclusion of Stueckelberg fields to convert the theory into gauge system being equivalent to the original one. In so doing, we admit reducibility of the Stueckelberg gauge symmetry. In this case, no pairing exists between Stueckelberg fields and gauge parameters, unlike the irreducible Stueckelberg symmetry. The general procedure is exemplified by the case of Proca model, with the third order involutive closure chosen as the starting point. In this case, the set of Stueckelberg fields includes, besides the scalar, also the second rank antisymmetric tensor. The reducible Stueckelberg gauge symmetry is shown to admit different gauge fixing conditions. One of the gauges reproduces the original Proca theory, while another one excludes the original vector and the Stueckelberg scalar. In this gauge, the irreducible massive spin one is represented by antisymmetric second rank tensor obeying the third order field equations. Similar dual formulations are expected to exist for the fields of various spins.


Introduction
Since the original Stueckelberg's work [1], the idea remains attractive for decades concerning inclusion of auxiliary fields into the action in such a way that modified theory becomes gauge invariant while it is still equivalent to the original one. The reviews and further references can be found in [2], [3].
In the constrained Hamiltonian formalism, the Stueckelberg idea has transformed into a method of converting the second class constraints into the first class ones [4], [5]. The conversion is achieved by extending the phase space by extra dimensions, that can be understood as introduction of Stueckelberg fields. The local existence theorem for the conversion procedure has been proven in the article [6], the global proof of the conversion existence can be found in [7]. The starting point of the Hamiltonian conversion is a complete system of the constraints including primary and secondary ones of all the generations. The conversion variable is assigned to every second class constraint. Given the complete system of constraints, the Hamiltonian conversion works as a systematic iterative procedure which is proven unobstructed. Unlike the Hamiltonian counterpart, the common practice of including Stueckelberg fields in Lagrangian formalism seems more art than science. Most often this works as a "Stueckelberg trick", which implies that the action is split into gauge and non-gauge parts. The Stueckelberg gauge symmetry of the original fields is assumed to remain the same as for the gauge invariant part of the action, while the transformations of Stueckelberg fields are chosen to compensate the non-invariance of the rest part. The choice of this split into gauge and non-gauge parts is an art, and it can be ambiguous. It is even unclear, why such a split is always possible. From the Hamiltonian perspective, this would mean to assume each second class constraint to be decomposed into the first class part and the "symmetry breaking part". The Hamiltonian conversion method proceeds from any complete set of constraints, not assuming the possibility of any decomposition of the constraints.
Recently, a systematic procedure has been proposed for covariant inclusion of the Stueckelberg fields [8] in Lagrangian formalism. The starting point for the method is the involutive closure of the original Lagrangian system of field equations. The original equations can be non-involutive, i.e. they can admit the lower order consequences. Completion of the system of the field equations by their consequences is understood as an involutive closure, if the completed system does not admit any further lower order consequences. In principle, the involutive closure can include also the higher order consequences. Completion of the Hamiltonian constrained system by the secondary constraints is an example of the involutive closure. The involutively closed form of the field equations allows one to count the degree of freedom number in an explicitly covariant manner [9]. The procedure of the article [8] allows one to iteratively include Stueckelberg fields for any field theory proceeding from the involutive closure of the original Lagrangian equations, and it is proven to be unobstructed. This procedure implies inclusion of independent consequences into the involutive closure of Lagrangian equations. Given this starting point, one arrives at the irreducible Stueckelberg gauge symmetry.
In this article, we consider inclusion of Stueckelberg fields proceeding from the involutive closure which involves a reducible set of consequences of Lagrangian equations. This leads to two main distinctions from the case of independent consequences. First, there is no pairing anymore between the Stueckelberg fields and gauge parameters. Second, the Stueckelberg gauge symmetry turns out reducible. There are no obstructions to inclusion of the Stueckelberg fields in the reducible case, much like to the irreducible one. To exemplify the general procedure, we consider the third order involutive closure of the Proca equations when the original equations are also complemented, besides the first order consequence, by the antisymmetric combinations of the derivatives of the Lagrangian equations. This leads to inclusion, besides the usual Stueckelberg scalar, of the Stueckelberg field, being the second rank antisymmetric tensor. Full Stueckelberg symmetry mixes the original vector with all the Stueckelberg fields. This reducible gauge symmetry admits different gauge fixing conditions. The simplest gauge kills all the Stueckelberg fields reducing the dynamics to the original Proca equations. The alternative gauge fixing condition is also admissible such that kills the Stueckelberg scalar and the original vector field, while all the dynamics is described by the antisymmetric tensor B µν obeying the third order equation, In Minkowski space, any transverse vector is a divergence of the antisymmetric tensor, In a sense, B µν is a "potential" for the transverse vector A µ . The non-Lagrangian equations (1) can be viewed as a reformulation of the Proca model in terms of the potential, such that automatically accounts for the transversality condition. Under the proposed procedure of inclusion of the Stueckelberg fields, both dual formulations, (1) and (2), are included into a uniform Lagrangian theory even though one of them is non-Lagrangian by itself. Imposing appropriate gauge fixing conditions, one can switch from the vector formulation to the dual one, and vice versa. As explained in the conclusion, it seems to be a general phenomenon which extends to other representations and goes beyond the free level.
The article is organized as follows. In the next section, the general scheme of inclusion of the Stueckelberg fields is outlined for the case of reducible Stueckelberg gauge symmetry. In Section 3, the general procedure is exemplified by unconventional inclusion of Stueckelberg fields in the Proca model such that leads to reducible gauge symmetry. The results and further perspectives are discussed in the Conclusion.

Inclusion of Stueckelberg fields with reducible gauge symmetry
As a preliminary, let us explain the strategy of including Stueckelberg fields implemented in this Section. First, the Lagrangian equations are complemented by the consequences such that the entire system is involutive. Once the completed system is non-Lagrangian, the second Noether theorem does not apply, and the gauge identities arise, being unrelated to the gauge symmetry.
The general structure of gauge algebra is known for not necessarily Lagrangian field equations [10], [11]. For the case when the non-Lagrangian system is a completion of the Lagrangian one, the gauge algebra has some specifics which are detailed as the second step. As the third step, we introduce the Stueckelberg fields with two goals. First, the involutive system should be zero order of the expansion of Lagrangian Stueckelberg equations. Second, the gauge identities of the involutive closure of the original system should be reproduced as zero order (in Stueckelberg fields) of Noether identities for Stueckelberg action. This defines zero order of gauge symmetry generators and the first order of the action. The existence of all the higher orders can be proven along the similar lines to the irreducible case [8].
In this Section, we use the condensed notation. All the condensed indices are supposed to include numerical labels and the space-time points. Summation over the condensed index implies integration over x. The partial derivatives are understood as variational.
Consider a theory of fields φ i with the action S(φ). Lagrangian equations read In this article, we consider a theory where the original action does not have gauge symmetry. This means that any identity between the field equations (4) has a trivial generator which vanishes on Inclusion of the Stueckelberg fields in the gauge invariant actions will be considered elsewhere.
Let us complement the field equations (4) by their differential consequences, where Γ α i (φ) are supposed to be local differential operators. The generators Γ of the consequences are considered equivalent if they lead to the same τ . Hence, the equivalence relation reads The completed system is assumed involutively closed, i.e. all the lower order consequences are already contained among equations (8). Obviously, the involutive closure (8) is equivalent to the original system, because all their solutions coincide. By construction, the involutively closed system enjoys gauge identities while there are no gauge symmetry. Let us assume the set of the generators Γ α i of consequences i.e. certain combinations of Γ's reduce to the trivial gauge generators (5). This results in the identities between the consequences (6): The generators of identities are considered equivalent if they differ by the trivial generator vanishing on shell, The operators Z A α are assumed to constitute the generating set for the null-vectors of the conse- (11) can admit further reducibility, i.e. certain combinations of the identity generators Z A α reduce to the trivial null-vectors (12).
In principle, the generating set of the second null-vectors Z 1A 1 A can be over-complete in its own turn. In this article, we do not consider this option assuming no further reducibility.
The set of identities (9), (11) between the equations of involutive closure (8) is assumed complete. This means, any set of identities, labeled by some condensed index I, reduces to the linear combination of identities (9), (11), Hence, the generators Λ I i , Λ I α of any identity between the equations of the system (8) reduce to the linear combinations of the generators Γ and Z modulo trivial generators: Relation (10) leads to the identities between the identities (9), (11), because certain combination of the identity generators is trivial.
Also notice that the set of the identity generators Z A α is over-complete (13). This leads to further identities between the identities (11). These second level identities are irreducible, as their generators Z 1A 1 A are assumed independent. Any set of identities, being labeled by the condensed index I 1 , between the identities of identities is supposed generated by Z 1A 1 A : Even though original action has no gauge symmetry, the involutive closure (8)  seek for the Stueckelberg action, and its gauge symmetry, as the power series in ξ: where the first order is defined by the completion functions (6) Once the gauge identities (9), (11) are to be converted into the Noether identities of the action (17), corresponding gauge parameters ǫ α and ǫ A are introduced The gauge symmetry of the Stueckelberg action is equivalent to the Noether identities between the equations, Let us expand the action (17)  with the identities (9), (11), we find zero order of the Stueckelberg gauge transformations, where Γ i α are the generators of consequences of Lagrangian equations (6) included into the involutive closure of original system, and Z α A are the generators of the identities (11) between τ α . The dots stand for the ξ-depending terms. The generators Γ i α of the consequences (6) where dots stand for the ξ-depending terms, Z α A are the null-vectors for the generators of consequences (10), and Z 1 A A 1 are the generators of reducibility for Z α A , see (13). The gauge parameters of symmetry for symmetry are denoted ω A and ω A 1 . The gauge identities of identities in the original system (8) are reducible again. At the level of Stueckelberg theory this leads to the gauge symmetry of the parameters ω from the transformation above. This symmetry of symmetry in zero order in ξ is generated by the same operators as in the corresponding identities of identities of the original system. Hence, the next level gauge symmetry reads The second order in ξ of the Stueckelberg action (17), and the first order of the gauge transformations (19), can be found from Noether identities (20) at the first order, given the previous order (18), (21). Once the previous order is found, it is substituted into the expansion of the Noether identity up to the next order. This allows one to find the next order, etc. In this way, all the orders of the action and gauge generators are iteratively found. Up to the second order in ξ, the Stueckelberg action reads The similar procedure applies to iteratively solving order by order the identities for identities proceeding from zero order (22), (23).
Given the regularity of the gauge algebra of the involutive closure (8) described at the beginning of this section, no obstructions can arise to the iterative inclusion of the Stueckelberg fields at any order. This can be proven by the tools of homological perturbation theory as described for the irreducible case in the article [8]. The main distinction of this proof from the usual homological perturbation theory procedures of gauge theories [12] is the unusual grading, where positive resolution degree is assigned to the Stueckelberg fields and their anti-fields, unlike the other fields. The aspect of reducibility can be accounted for in the homological perturbation theory with this grading in a natural way. This issue will be addressed elsewhere. From the point of view of the application in specific models, only the fact is important that the described procedure for including the Stueckelberg fields is unobstructed at all iteration steps.
In this Section, we exemplify the general method of inclusion of Stueckelberg fields with reducible gauge symmetry by the case of Proca model. The usual Stueckelberg scalar corresponds to the completion of the Proca system by the first order consequence -transversaliity condition. This is sufficient to make the Proca system involutive. However, the system can be completed also by the third order consequences, and it remains involutive. This option of the third order involutive closure, being treated by the procedure of previous section, leads to inclusion of the antisymmetric second rank tensor as the Stueckelberg field. The third order consequences turn out obeying the gauge identities of their own (cf. (11)), so we arrive at reducible Stueckelberg symmetry. This is no surprise once the antisymmetric tensor is introduced. The Stueckelberg action includes four derivatives, while the theory remains equivalent to the original Proca system. Besides exemplifying the general method, this case may have some interest of its own, as it demonstrates the scheme for constructing dual formulations for the fields of the same spin.
Consider the Proca Lagrangian for massive vector field A µ in d = 4 Minkowski space, The Proca equations δS are not involutive as such, as they admit the first order differential consequence The system (26) These equations mean that the strength tensor of original field A µ obeys Klein-Gordon equation. For the involutive closure of Proca system (26), (27), (28), these identities read The consequences τ µν (28) are reducible in the sense of identity (11). This identity reads where ε µνρλ is Levi-Chivita symbol. These identities are reducible in their own turn, as the divergence of the l.h.s. identically vanishes for any τ µν . It is the second level identity (cf. (13)): The identities (29) With all the ingredients at hands, following the general procedure of Section 2, we iteratively construct the Stueckelberg action, generators of gauge symmetries, and symmetries of symmetries.
Once the original action is quadratic and the identity generators (33) are field-independent, the procedure terminates at the first iteration. The Stueckelberg action and reducible gauge symmetry transformations read where (36) are the gauge symmetry transformations of the original gauge parameters ǫ, while (37) is the gauge symmetry of the second level gauge parameters ω.
Consider the Lagrangian equations for Stueckelberg action (25), These equations involve the fourth order derivatives, so equivalence with the original Proca theory may seem doubtful. However, these equations enjoy the reducible gauge symmetry (35). This symmetry admits gauge fixing conditions This gauge eliminates all the Stueckelberg fields and reduces the system to Proca equations (26).
It is interesting to notice another admissible gauge fixing for the symmetry (35): As this gauge fixing kills scalar ϕ and vector field A µ , equations (38)-(39) reduce to third-order equation (1), while (40) becomes its differential consequence. Let us detail fixing of the gauge parameters by conditions (42). Taking variation of (42) we arrive at the conditions So, the gauge conditions (42) restrict the gauge parameters by the relations Once ǫ = 0, the second of these equations means ǫ µν = ε µνλρ ∂ λ ω ρ , where ω λ is arbitrary. Substituting that into the last relation we see that the difference between the gauge parameter ǫ µ and ω µ obeys free Maxwell equations. Maxwell equations have unique solution modulo the gradient of arbitrary scalar ∂ µ ω, given the Cauchy data. So, the general solution of equations (44) reads where ω µ , ω are arbitrary functions. This means, the gauge conditions (42) fix parameters ǫ, ǫ µ , ǫ µν modulo symmetry of symmetry (36). The ambiguity of this type always remains unfixed at the level of field equations for original fields in the case of reducible gauge symmetry. In the BRST formalism, this ambiguity is fixed by imposing gauge conditions on the ghosts and introducing ghosts for ghosts [12].
Admissibility of the gauge fixing condition such that kills the original vector field means that A µ can be considered as a pure gauge from the viewpoint of the action (25)

Conclusion
Let us summarize and discuss the results. First, we propose a systematic way for inclusion of Stueckelberg fields such that guarantees equivalence of the resulting gauge theory to the original system. The starting point for inclusion of Stueckelberg fields is the involutive closure of original Lagrangian equations (8). If the closure includes an over-complete set of consequences (see (11)), the Stueckelberg symmetry turns out reducible. In any case, the Stueckelberg theory is iteratively constructed for any involutive closure of Lagrangian equations without obstructions at any stage, be the consequences (6) reducible or not. In this sense, the covariant method is a complete analogue to the Hamiltonian method of conversion of the second class constraints into the first class ones.
The interesting option for inclusion of Stueckelberg fields is to start with the involutive closure of the higher order than it is minimally sufficient. This option is exemplified in Section 3 by the third order involutive closure of Proca model, where the added consequences are reducible.
Following the general procedure of inclusion of the Stueckelberg fields, we arrive to the higher derivative Stueckelberg action (34) which is equivalent to the first derivative Proca action. This Stueckelberg model for massive spin 1 turns out comprising two dual field theoretical realizations for the same irreducible representation. The first one is the original Proca model, and the second one is the third order formulation (1) in terms of the antisymmetric tensor field. The field B µν can be considered as a potential for the original transverse vector (cf. (3)). Notice that various dual formulations are studied once and again for the same spin representation. For the most recent results on this topic and further references we refer to the article [13]. Important motivation for studying dual formulations is that they are inequivalent, in general, w.r.t. inclusion of interactions.
Among the examples of this sort, we can mention the representation of the massless spin 2 by the third rank tensor field with Young diagram of the hook type [14]. Unlike the representation of the same spin by the symmetric second rank tensor, the hook does not admit inclusion of consistent interactions [15]. Similar phenomena are observed among the higher spin gravities. In particular, the long known light-cone analysis of the higher spin vertices in Minkowski space [16] demonstrates admissibility of the interactions such that are missing among the deformations of Fronsdal's Lagrangians for symmetric tensors. There is a growing evidence that Lagrangians for dual formulations of higher spins can admit these vertices. For the recent results, discussion of the area, and further references we refer to [17], [18]. Notice that the considered dual formulations are typically connected to each other algebraically, hence all the actions are of the same order.
Proposed scheme of inclusion Stueckelberg fields proceeds from the involutive closure of the original Lagrangian equations. If the starting point is the higher order closure of the original system, corresponding Stueckelberg field, being candidate for the dual to the original field, would be involved in the Lagrangian with higher derivatives. This dual would be connected to the original field by a differential relation, like a potential (cf. (3)). So, this scenario of inclusion Stueckelberg fields can serve as a tool for constructing a different type of dual formulations. For example, if the original fields are symmetric, the second order Lagrangian equations can admit the third order involutive closure with differential consequences, being the tensors of hook type. Corresponding higher derivative Stueckelberg Lagrangian has to be equivalent to the original one by construction, while the hook tensors would serve as dual to the original fields, following the pattern of Section 3. These dual models can have their chances for consistent interactions as the potentials can be less obstructive to deformations than corresponding strength tensors.