Unfree gauge symmetry in the Hamiltonian formalism

The constrained Hamiltonian formalism is worked out for the theories where the gauge symmetry parameters are unfree, being restricted by differential equations. The Hamiltonian BFV-BRST embedding is elaborated for this class of gauge theories. The general formalism is exemplified by the linearized unimodular gravity.


Introduction
If the gauge variation of action identically vanishes under the condition that the gauge parameters obey differential equations, the gauge symmetry is said unfree. The general structure of unfree gauge symmetry algebra has been recently established in ref [1]. The extension of the BV (Batalin-Vilkovisky) formalism to unfree gauge symmetry is proposed in [2].
While the phenomenon of unfree gauge symmetry is well-known in terms of Lagrangian formalism, it has been so far unclear how the unfree symmetry reveals in the corresponding Hamiltonian formalism. Even in well-studied models, like unimodular gravity, the transversality condition on the diffeomorphism parameters is not evident from the viewpoint of the Poisson algebra of Hamiltonian constraints. This problem is noticed in the literature, see [7] and references therein.
In this article, we work out the general Hamiltonian description of unfree gauge symmetry. In section 2, we list the basic features of Lagrangian description of general unfree gauge algebra as this is essential for constructing the Hamiltonian analogue. In section 3, we establish the general structure of unfree gauge symmetry in the constrained Hamiltonian formalism; in section 4, we construct corresponding Hamiltonian BRST (Becchi-Rouet-Stora-Tyutin) complex; the section 5 exemplifies the general formalism by the model of linearized unimodular gravity.

Generalities of unfree gauge symmetry in Lagrangian formalism
In the reference [1], it is noticed that the unfree gauge algebra is generated by four key ingredients: the action functional S; the generators of unfree gauge symmetry Γ i α ; the mass-shell completion functions τ a ; the operators of gauge parameter constraints Γ a α . The first two generating structures -the action, and the gauge symmetry generators -are the key ingredients of any gauge symmetry algebra, be the gauge parameters constrained, or not. The other two generating elements, τ a and Γ a α , are special for the unfree gauge symmetry. Let us non-rigorously explain their role in the dynamics, for a more systematic exposition we refer to [1], [2].
The main distinctive feature of theories with the unfree gauge symmetry is that the local quantities exist such that vanish on-shell while they are not spanned by the l.h.s. of Lagrangian equations (EoM's). In other words, the generating set for the ideal of on-shell vanishing local functions is not exhausted by the l.h.s. of EoM's, it includes some other quantities, denoted as τ a : where ≈ means on-shell equality, and θ(φ) are local. Here we use the DeWitt condensed notation.
The local quantities τ a (φ) are supposed independent, and they do not reduce to a linear combination of Lagrangian equations, τ a = θ i a ∂ i S. The quantities τ a are called the mass-shell completion functions. Examples of the completion functions can be found in [1], [2]. In general, modified Noether identities involve both Lagrangian equations and completion functions: With appropriate regularity assumptions (see in [1], [2]), relations (1), (2) define the unfree gauge symmetry of the theory. In particular, the gauge variation of the fields, is a symmetry of the action provided that the gauge parameters ǫ obey the equations The operators of gauge parameter constraints Γ a α (φ) are supposed independent. This means that the kernel of Γ a α (φ) is, at maximum, finite dimensional, Here, M is understood as a moduli space of the field theory. Given (2), (5), on shell τ a ≈ Λ a , Λ a ∈ M. In principle, the modular parameters Λ a can be included into the definition of τ a , so the completion functions can be considered on-shell vanishing without loss of generality. Also notice that the relations τ a ≈ 0 hold true for any solution of the EoM's with corresponding modular parameter, though these relations are not differential consequences of EoM's.
The modified Noether identities (2) along with corresponding regularity assumptions lead to the compatibility conditions involving higher structures, which define the full unfree gauge symmetry algebra. This algebra is more general than the one with unconstrained gauge parameters.
Corresponding gauge formalism is worked out in references [1], [2], we do not address it here, listing only the basic relations.
Consider the theory of fields φ i (x), where i is a discrete index (it can be spinorial, tensorial, isotopic index), and x µ are coordinates of space-time. Suppose the differential consequences of EoM's can be linearly combined into the gradient of local quantity τ , being the scalar function of fields and their first derivatives with respect to x µ : whereΓ i µ is a matrix whose entries are linear differential operators with the field-depending coefficients. As the derivatives of τ vanish on-shell, it is an on-shell constant, τ (φ, ∂φ) ≈ Λ = const.
The specific value of the constant Λ is determined by the boundary conditions or asymptotics of fields, not by initial data. For example, if the fields are supposed vanishing at spacial infinity of the space-time, then Λ = 0 irrespectively to the initial data, as τ (0, 0) = 0. So Λ should be understood as a modular parameter rather than an integral of motion. This modular parameter can be included into definition of the completion function, so one can set τ ≈ 0 without restricting generality. Once the modified Noether identity (2) reads as (6) in this example, the gauge parameter should be the vector field ǫ µ (x). The gauge parameter constraint operator Γ a α of (2) is identified by (6) as ∂ µ , so the equation (4) reduces to the transversality condition ∂ µ ǫ µ = 0 for the gauge parameter ǫ µ . The unfree gauge transformations (3) are generated by the conjugate to the operatorΓ involved in the modified Noether identity (2): Given the modified Noether identity (6), the transformation (7) leaves the action invariant: The unimodular gravity is covered by this example. The role of the modular parameter is played by the cosmological constant, while the completion function is the scalar curvature. Unfree gauge symmetries of some higher spin field theories, see [12], [13], [14], [15], also follow the pattern of this example, though the completion functions are tensors in these models, not scalars.

Unfree gauge symmetry transformations in the Hamiltonian formalism
Consider the action of Hamiltonian theory with primary constrains, The role of fields here is played by canonical variables q i , p i , and Lagrange multipliers λ α . In the previous section, the structures are described that define the unfree gauge symmetry for the general action. In this section, we detail these structures for the specific action (9) and find thereby the Hamiltonian form of unfree gauge symmetry transformations.
For the action (9), the EoM's read The constraints T α (q, p) are supposed irreducible. At this point we accept an auxiliary assumption that the differential consequences of the equations do not fix λ α as functions of q, p. Assuming that the Dirac conjecture holds true for theory (9), this means there are no second-class constraints.

Now, our primary objective is to identify completion functions (1) and gauge identities (2) for
EoM's (10), (11). We begin with applying the Dirac-Bergmann algorithm to equations (10), (11): Once the Lagrange multipliers are not defined by conservation of the primary constraints, the r.h.s. of the above relation should be a linear combination of the primary constraints and the secondary ones. Let us assume that the irreducible generating set {τ a (q, p)} can be chosen for the secondary constraints. This means the local differential operators Γ, W exist such that The irreducibility assumption of the secondary constraints τ a is twofold. First, all the constraints should be independent: where A ab = −A ba , A αβ = −A βα . Second, the kernel of the operator Γ a α should be, at most, finite dimensional, in the sense of relation (5). The only difference with the Lagrangian formalism is that Γ in (13) involves derivatives only by space coordinates, while the Lagrangian counterpart can differentiate also by time.
If Γ a α admitted the dual differential operatorΓ α a such that then the secondary constraints τ a would be the differential consequences of the original equations (10), (11). In the opposite case, τ a reduce to the element of the kernel of the differential operator Γ. In this case τ are considered as completion functions, and hence the gauge symmetry should be unfree. Once the kernel is finite of Γ a α , completion functions τ a (q, p) can be redefined by adding modular parameters Λ a to make τ vanishing on shell: Then τ a still vanish on-shell and viewed as secondary constraints, though they are not differential consequences of the EoM's, and hence the gauge symmetry should be unfree.
The next simplifying assumption is that no tertiary constraints appear. This means that the time derivatives of secondary constraints reduce on-shell to the combinations of themselves and the primary ones:τ Off shell, the time derivatives of primary and secondary constraints identically reduce to the linear combination of constraints and EoM's (10): Since the secondary constraints τ a are not the differential consequences of the primary ones (11), the above relations are modified gauge identities (2) rather than usual Noether identities between the variational equations. The identities (2) The constraints on the gauge parameters (4) are defined by the coefficients at τ a in the modified gauge identities (2). Given specific identities (18), (19), the constraints on gauge parameters read: The unfree gauge transformations (20), (21) have been deduced above by using the gauge identities (18), (19) for the theory (9) with the involution relations (13), (17). By direct variation, one can verify that the action (9) Upon substitution {T α , H T }, {τ a , H T } from relations (13), (17), the variation reads Once the gauge parameters obey equations (21), the integrand reduces to the total derivative, so the action is indeed invariant under the unfree gauge variation (20), (21).
Let us discuss the constraints imposed on the gauge parameters ǫ α and ǫ a by equations (21).
Equations (21) defineǫ a in terms of ǫ α . As the kernel of Γ a α is at maximum finite in the sense of relation (5), the time evolution of ǫ a is completely controlled by ǫ α , while the latter parameters are unconstrained by the equations. As the equations (21) have the structureǫ a = f a (ǫ b , ǫ β ), they admit any initial data for ǫ a , so these parameters are arbitrary at initial moment.
Alternatively, equations (21) can be considered as constraints imposed on the parameters ǫ α , defining some of them in terms of the rest of ǫ α andǫ a , ǫ a . If all the constraints (21) are explicitly resolved by excluding some of the gauge parameters ǫ α , then the gauge transformations of canonical variables (20) will includeǫ a , while the variation of λ α will involveǫ a . In this way, the firstorder unfree gauge symmetry (20), (21) is replaced by the second-order gauge symmetry with unconstrained gauge parameters. If the spacial locality is not an issue, the constrained Hamiltonian equations (10), (11) always admit the unconstrained parametrization of gauge transformations with higher order time derivatives of gauge parameters [16]. Also notice that any linear system of local field equations admits unconstrained local parametrization of gauge symmetry, possibly with higher derivatives, though the transformations can be reducible [14]. So, all these facts lead to the conjecture that the unfree gauge symmetry can be always equivalently replaced by local higher order reducible gauge symmetry. This conjecture will be addressed elsewhere.
Also notice that the Lagrange multipliers cannot contribute to the on-shell invariants, as δ ǫ λ α begins withǫ α (20), while the parameters ǫ α are not constrained by equations (21).
Let us now detail involution relations (13), (17). As H T = H(q, p) + λ α T α (q, p), the structure functions W (q, p, λ), Γ(q, p, λ) in (13), (17) are at most linear in λ α : By introducing uniform notation for primary and secondary constraints T A = (T α , τ a ), A = (α, a), and accounting for (24), (25), the involution relations (13), (17) are brought to the following form: The above involution relations include both primary and secondary constraints on an equal footing and merely correspond to a general first-class system. These relations, per se, do not reveal any indication of the equations imposed on the gauge parameters (21). At the level of action (9), however, the differences exist as the primary constraints are included into the action with the Lagrange multipliers, while the secondary ones are not. It is the asymmetry which leads to equations on gauge parameters (21). With this regard, we mention the long-known idea that the secondary first-class constraints τ a can be included into the action with their own Lagrange multipliers λ a , where λ A = (λ α , λ a ). If we begin with this action, it will have the usual first-order gauge symmetry, with unconstrained gauge parameters ǫ A = (ǫ α , ǫ a ). The introduced multipliers λ a can be con- To begin with the Hamiltonian BRST embedding of the theory, we briefly describe the minimal ghost sector in the BFV formalism. Every first-class constraint T A be it primary, or secondary, is assigned with a pair of canonically conjugate ghosts 1 with usual ghost numbers The BRST charge in the minimal sector is defined as where . . . meanP -depending terms that are iteratively defined by the equation {Q min , Q min } = 0. fixing. Therefore, the non-minimal sector of the theory includes the Lagrange multipliers λ α to the primary constraints, and the Lagrange multipliers π α to the independent relativistic gauge conditionsλ α − χ α (q, p) = 0. The corresponding canonical ghost pair is introduced for every pair of the Lagrange multipliers, so the complete non-minimal sector reads Given the extended set of variables, the complete BRST charge reads With this charge, the gauge-fixed BRST invariant Hamiltonian is defined in the usual way, The partition function Z Ψ , being defined by the Hamiltonian H Ψ , does not depend on the choice of gauge conditions included in Ψ due to usual reasons of the Hamiltonian BRST formalism [18].
Consider Z Ψ for the simplest case when the Hamiltonian H Ψ is at most squared in ghost variables. The path integral for the partition function reads: where ϕ = q, p, λ α , π α , C a ,P a , C α ,P α , P α ,C α . The integral by P α results in δ(P α +Ċ α ), which removes the integral overP α . The result reads where ϕ ′ = q, p, λ α , π α , C a ,P a , C α ,C α . The integral over anti-ghostsP a would enforce con-straintsĊ a + Γ a b C b + Γ a α C α = 0. This is quite a natural phenomenon: once gauge variations (20) induced by primary and secondary constraints are unfree, being restricted by equations (21), the ghosts should obey the same conditions as the gauge parameters do. The constraint on ghosts is the cornerstone for the extension of the BV formalism for the theories with unfree gauge symmetry [1], [2]. Here, we see that they naturally arise from the Hamiltonian BFV-BRST quantization.
The higher spin analogues [12], [13] admit similar solutions. For higher spins, this may be even more essential because the cosmological constant plays the role of interaction parameter for s > 2.
Given the gauge identity (38), which involves the completion function τ , the action (37) should enjoy unfree gauge symmetry. It does, in full accordance with the general prescription (3), (4): cf. (8). So, the action is gauge invariant off-shell under the condition ∂ᾱǫᾱ = 0.
By Legendre transform of (37), we get the Hamiltonian action where α, β = 1, 2, 3, η αβ = −δ αβ , h = η αβ h αβ , Π = η αβ Π αβ , λ α = h 0α . Conservation of primary constraints T α leads to the secondary constrainṫ The secondary constraint conserves by virtue of the primary ones: This symmetry can be verified by direct computation. Variation (45) of action (41) reads It is a symmetry indeed once ǫ 0 obeys equation (46). As we see, the general procedure of Sec- Consider the BFV construction for the model following the general prescription of Section 4.
The ghosts of minimal sector are assigned to all the constraints (cf. (29)), while the non-minimal sector is assigned only to the primary constraints (see (32)). The BRST charge (30), (33) for the linearized unimodular gravity reads: Impose three independent gauge fixing conditions, Introduce gauge fermion Ψ =C α χ α + λ αP α , and define gauge-fixed Hamiltonian H Ψ (34), where H is the original Hamiltonian (42). For H Ψ (50), partition function (35) reads where ϕ = h αβ , Π αβ , λ α , π α , C 0 ,P 0 , C α ,P α , P α ,C α ; H and T α are the original Hamiltonian and primary constraints (42). Integrating over P α ,P α , Π αβ we get Lagrangian representation for Z, where φ ′ = hᾱβ, π α ,P 0 , Cᾱ,C α , and L is the original Lagrangian. This representation of partition function has been deduced by Hamiltonian BFV-BRST quantization of the model. It appears to be a reasonable adjustment of Faddeev-Popov (FP) recipe to the case. Among the ghost terms, the first one represents constraint imposed on ghosts, withP 0 being the Lagrange multiplier. The ghost constraint mirrors the transversality condition imposed on the diffeomorphisms (46). As the gauge parameters are unfree, it is natural to have the corresponding ghosts constrained. The FP term (52) is not Poincaré covariant because the gauge is fixed by independent condition (49), being 3d vector. If the gauge condition was a 4d vector, the vector components would have to be redundant, to avoid "over-rigid" gauge fixing. This would require some extra ghosts. In the covariant formalism, this issue is considered in [2], while the Hamiltonian analogue will be addressed elsewhere.

Concluding remarks
The field theories with unfree gauge symmetry represent a special class of models where the gauge parameters have to obey differential equations. Every known example of these theories (see [3]- [15] and references therein) admits an "almost equivalent" analogue without constraints on gauge parameters. The subtle difference is that the models with unfree gauge symmetry comprise dynamics with arbitrary modular parameters, which are involved as integration constants, while the analogues explicitly involve fixed modular parameters. The example of such a parameter is a cosmological constant in unimodular gravity. It is the distinction which is behind the constraints on gauge parameters (21). In terms of Hamiltonian formalism, these constraints on gauge parameters have been previously unknown even in the examples, not to mention the general theory. We have worked out the general Hamiltonian BFV-BRST formalism with a due account for the unfree gauge symmetry. As we see by examples, it corresponds well to the extension of BV method to the unfree gauge symmetry [2], though these two schemes do not mirror each other. In the BV scheme the equations on parameters are directly accounted for as constraints on the corresponding ghosts, while the Hamiltonian formalism accounts for the conditions (21) indirectly, by an adjusted structure of the non-minimal ghost sector.