Constraints and degrees of freedom in Lorentz-violating field theories

Many current models which"violate Lorentz symmetry"do so via a vector or tensor field which takes on a vacuum expectation value, thereby spontaneously breaking the underlying Lorentz symmetry of the Lagrangian. To obtain a tensor field with this behavior, one can posit a smooth potential for this field, in which case it would be expected to lie near the minimum of its potential. Alternately, one can enforce a non-zero tensor value via a Lagrange multiplier. The present work explores the relationship between these two types of theories in the case of vector models. In particular, the na\"ive expectation that a Lagrange multiplier"kills off"one degree of freedom via its constraint does not necessarily hold for vector models that already contain primary constraints. It is shown that a Lagrange multiplier can only reduce the degrees of freedom of a model if the field-space function defining the vacuum manifold commutes with the primary constraints.

Many current models which "violate Lorentz symmetry" do so via a vector or tensor field which takes on a vacuum expectation value, thereby spontaneously breaking the underlying Lorentz symmetry of the Lagrangian. To obtain a tensor field with this behavior, one can posit a smooth potential for this field, in which case it would be expected to lie near the minimum of its potential. Alternately, one can enforce a non-zero tensor value via a Lagrange multiplier. The present work explores the relationship between these two types of theories in the case of vector models. In particular, the naïve expectation that a Lagrange multiplier "kills off" one degree of freedom via its constraint does not necessarily hold for vector models that already contain primary constraints. It is shown that a Lagrange multiplier can only reduce the degrees of freedom of a model if the field-space function defining the vacuum manifold commutes with the primary constraints.

I. INTRODUCTION
Many classical field theories are constructed in such a way that the "most natural" solutions to the equations of motion involve a non-zero field value. This paradigm, where an underlying symmetry of the Lagrangian or Hamiltonian is spontaneously broken by the solutions of the equations of motion, has proven to be both compelling and fruitful over the years. In the context of particle physics, the best-known example is the Higgs field [1][2][3][4]; in the context of condensed-matter physics, this paradigm underlies the modern theory of phase transitions, most notably the Ginzburg-Landau theory of superconductivity [5]. Many non-linear sigma models can also be thought of in this way, if one view the model's target manifold as being embedded in some higher-dimensional space in which the fields are forced to a non-zero value. Such models have been used in the study of both particle physics [6,7] and in ferromagnetism [8].
In more recent years, this paradigm has also been used to study possible observational signatures of Lorentz symmetry violation in the context of modified gravity theories. Such models include a new vector or tensor field, and have equations of motion that are satisfied when the metric is flat and the new tensor field is constant but non-zero. In this sense, Lorentz symmetry is spontaneously broken in these theories, as the tensor field takes on a vacuum expectation value that has non-trivial transformation properties under the Lorentz group. The vacuum state of such a model is often said to be "Lorentzviolating", though it would be more accurate to say that Lorentz symmetry is spontaneously broken in the model. Examples of such models can be found in [9][10][11][12][13][14], as well as a particularly early example in [15].
In general, all of these models (Lorentz-violating or otherwise) have the property that the fields, collectively denoted by ψ α , will satisfy an equation of the form * mseifer1@conncoll.edu f (ψ α ) = 0 in the "vacuum", appropriately defined. Here, f is a real-valued function of the fields; thus, if the fields ψ α are specified by N real numbers, the equation f (ψ α ) = 0 will generically specify an (N −1)-dimensional hypersurface in field space. I will therefore refer to this function as the vacuum manifold function.
Two broad classes of models in which some fields have non-trivial background values can readily be conceived of. In one class, the constraint f (ψ α ) = 0 is enforced exactly via the introduction of a Lagrange multiplier λ in the Lagrange density: where we have denoted the kinetic terms for the fields ψ α schematically. I will call such models Lagrange-multiplier (LM) models.
In the other class, the fields ψ α are assigned a potential energy that is minimized when the field is non-zero. In particular, if we define a potential V (ψ α ) ∝ f 2 (ψ α ), and write down a Lagrange density then the lowest-energy state of the theory would be expected to occur when f (ψ α ) = 0 and ∇ψ α = 0. I will call such models potential models. A natural question then arises: for a given collection of fields ψ α and a given kinetic term (∇ψ α )(∇ψ α ), what is the relationship between the models (1) and (2)? In particular, one might expect the following statement to be true: Conjecture. In a potential model such as (2), the fields can take on any value in the N -dimensional field space. By contrast, in a Lagrange-multiplier model, the fields are constrained to an (N − 1)-dimensional subspace of field space. Thus, the number of degrees of freedom of a Lagrange-multiplier model (1) should be one fewer than the degrees of freedom of the corresponding potential model (2).
The main purpose of this article is to show that this naïve conjecture not is true in general; if ψ α includes a spacetime vector or tensor field, it may be false. In such models, the fields may need to satisfy certain constraints due to the structure of the kinetic terms; adding a new "constraint" to such theories, in the form of a Lagrange multiplier, does not automatically reduce the number of degrees of freedom of the theory.
To demonstrate this, I will analyse the degrees of freedom of two types of symmetry-breaking models. After a brief description of Dirac-Bergmann analysis in Section II, I will first analyse a multiplet of Lorentz scalar fields with an internal symmetry, followed by a vector field (Sections III and IV respectively.) For the vector fields, the analysis will depend on the structure of the kinetic term chosen, and so three distinct sub-cases will need to be treated. In each case, I will examine a model where the fields are assigned a potential energy, and one where the fields are directly constrained via a Lagrange multiplier.
The bulk of the explicit analysis in this work will be done in the context of a fixed, flat background spacetime; however, I will briefly discuss these models in the context of a dynamical curved spacetime in Section V. In that section, I will also discuss the implications of these results for the broader relationship between these two classes of models. More general tensor fields will be examined in a forthcoming work [16].
Throughout this work, I will use units in whichh = c = 1; the metric signature will be ( − + + + ). Roman indices a, b, c, . . . will be used to denote spacetime tensor indices; i, j, k, . . . will be used to denote spatial indices, where necessary. Greek indices α, β, γ, ... will be used exclusively to denote indices in field space. All expressions involving repeated indices (either tensor indices or field space indices) can be assumed to obey the Einstein summation convention.

II. DIRAC-BERGMANN ANALYSIS
Our primary tool for finding the number of degrees of each model will be Dirac-Bergmann constraint analysis [17]; my methods and nomenclature below will draw heavily from the later work of Isenberg & Nester [18]. I will briefly summarize the method here, and then illustrate it in more detail via the example theories described in Section III.
The method of Dirac-Bergmann analysis involves the construction of a Hamiltonian which generates the timeevolution of the system. In the process of this construction, one may need to introduce constraints among various variables, thereby reducing the number of degrees of freedom of the system. One may also discover that the evolution of certain field combinations is undetermined by the equations of motion (for example, gauge degrees of freedom). These combinations of fields, which we will collectively call "gauges", must be interpreted as unphysical, again reducing the number of physical degrees of freedom of the model. If, as is usual, we count a field degree of freedom as a pair of real-valued functions (e.g., a field value and its conjugate momentum) that can be freely specified on an initial data surface, then the number of degrees of freedom N dof can be inferred quite simply once the above analysis is complete: In general, of course, the number of fields and the number of conjugate momenta will be the same. Moreover, in the particular field theories I will be considering in this work, I will not find any "gauges", so the last term in (3) will vanish. Thus, for my purposes, the above equation reduces to

III. SCALAR MULTIPLET FIELDS
The first case we will consider is a multiplet of N Lorentz scalars: ψ α = φ α , with α = 1, 2, 3, . . . , N . We wish to construct a model where these scalars "naturally" take on values in some (N − 1)-dimensional hypersurface, defined by f (φ α ) = 0 (with f a real-valued function.) The "potential model" for this field will be derived from the Lagrange density where κ is a proportionality constant; the LM model for this field will be where λ is a Lagrange multiplier field. In both cases, we have chosen a time coordinate t and performed a 3+1 decomposition of the tensors; a dot over a quantity (e.g., φ α ) will denote its derivative with respect to this time coordinate, while spatial derivatives will be denoted with either ∇ or ∂ i depending on the expression.

A. Potential model
The degrees of freedom for the model (5) are particularly easy to count. The momenta conjugate to the fields are all well-defined: The Hamiltonian density is therefore (8) Nothing further is required here; we have no primary constraints on the initial data, and the Hamiltonian obtained by integrating H 0 over space will generate the field dynamics. We thus have the N degrees of freedom one would expect.

B. Lagrange-multiplier model
Counting the degrees of freedom for the model (6) requires a bit more effort. As the kinetic term of (6) is the same as that of (5), the momenta conjugate to the scalars φ α are again defined by (7). The difficulty arises due to the Lagrange multiplier λ. From the perspective of the model, it is just another field; but its associated momentum vanishes automatically: We thus have a primary constraint, Φ = 0, on this theory. The "base Hamiltonian density" H 0 = π αφα − L must then be modified to obtain the "augmented Hamiltonian density" H A by adding this primary constraint multiplied by an auxiliary Lagrange multiplier u λ : 1 We now need to ensure that this constraint is preserved by the time-evolution of the system; in other words, we must have̟ = {̟, If this Poisson bracket does not vanish identically, this demand will yield a secondary constraint Ψ 1 = 0. The demand that this constraint be preserved may lead to new secondary constraints Ψ 2 = 0, Ψ 3 = 0, and so forth, which must themselves be conserved. We will refer to the stage at which a secondary constraint arises as its "order". In other words, if Ψ 1 ensures the preservation of a primary constraint, it is a "first-order secondary constraint"; if Ψ 2 ensures the preservation of Ψ 1 , it is a "second-order secondary constraint"; and so on. In this process, it may occur that the preservation of these constraints allows us to determine the auxiliary Lagrange multiplier u λ introduced above. The process is continued until all constraints are known to be automatically conserved or a contradiction is reached.
With this in mind, we derive the secondary constraints for this model. We first have Thus, Ψ 1 = f (φ α ) = 0 is a secondary constraint. We repeat this procedure, obtaining another secondary constraint: where we have defined δ β f = δf /δφ β . (Higher derivatives will be defined similarly.) Ψ 2 must also be conserved, which leads to a third secondary constraint: Finally, demanding that Ψ 3 be conserved allows us to determine the auxiliary Lagrange multiplier u λ ; this is because Ψ 3 is itself dependent on λ: When the dust settles, we obtaiṅ So long as δ α f = 0 when f = 0, this allows us to determine the previously-unknown auxiliary Lagrange multiplier u λ . Thus, the process terminates here.
Having determined the Hamiltonian and its constraints, we can count the degrees of freedom. We have N + 1 fields, namely the multiplet φ α (α = 1, . . . , N ) and the Lagrange multiplier λ; one primary constraint Φ = ̟ = 0; and three secondary constraints, {Ψ 1 , Ψ 2 , Ψ 3 } = 0. The single auxiliary Lagrange multiplier is determined, which means that there are no unphysical gauge degrees of freedom. With N + 1 fields and four constraints, the number of degrees of freedom of the model (6) is therefore Thus, the Lagrange-multiplier theory (6) has one less degree of freedom than the corresponding potential theory (5); in this case, the naïve conjecture outlined in the introduction holds true.

IV. VECTOR FIELDS
We now consider the case of a vector field A a which spontaneously breaks Lorentz symmetry. We will again restrict our attention to the case of flat spacetime. As the motivation behind these models is usually a spontaneous breaking of Lorentz symmetry, we want our Lagrange density to be a Lorentz scalar, without any prior geometry specified.
In any such Lagrange density, we can identify a set of "kinetic" terms L K that depend on the derivatives of A a . The most general kinetic term that we can write down which is quadratic in the field A a is 3 we can eliminate one of c 2 or c 3 via an integration by parts. We will therefore set c 2 = 0 in what follows. The familiar "Maxwell" kinetic term We can now perform the usual 3 + 1 decomposition of the Lagrange density, writing A 0 for the t-component of A a and A (or A i ) for its spatial components. The kinetic term (16) then becomes where c 13 ≡ c 1 + c 3 . The momenta conjugate to A 0 and A are then and These equations can be inverted, to find the velocitieṡ A 0 and˙ A, so long as c 1 + c 3 = 0 and c 1 = 0. If either of these expressions vanishes, (19) and (20) will instead yield constraint equations; we will have to handle these cases separately. For the potential term, meanwhile, our desire for the Lagrangian to be a Lorentz scalar implies that the only possible form for the vacuum manifold function is one which sets the norm of A a to some constant b. For simplicity's sake, we will therefore choose f (A a ) to be of the following form: where b is a constant. Depending on the sign of b, the "vacuum" manifold will consist of timelike vectors (b < 0), spacelike vectors (b > 0), or null vectors (b = 0). 4 Our potential model will then be where κ is again a proportionality constant; the LM model will be For compactness, I will denote the four-norm of A a as A 2 = −A 2 0 + A 2 ; the norm of the spatial part on its own will always be denoted by A 2 .

Potential model
Performing a Legendre transform on L P (22) to obtain the Hamiltonian density, we obtain a base Hamiltonian density of The resulting theory has four fields (A 0 and A) and no constraints; so the process terminates here, and the base Hamiltonian is the complete Hamiltonian for the model. Counting the degrees of freedom, we find that

Lagrange multiplier model
For the Lagrange multiplier model (23), we have a primary constraint associated with λ: We must therefore augment the Hamiltonian with an auxiliary Lagrange multiplier u λ to enforce this constraint: We now find the secondary constraints required for the primary constraint to be conserved under time-evolution. Taking the Poisson brackets of each constraint with the Hamiltonian in turn, we obtain three secondary constraints: where All three of the quantities Ψ 1 , Ψ 2 , and Ψ 3 must vanish for the model to be consistent. When we take the Poisson bracket of Ψ 3 with H A , we will obtaiṅ This means that for generic initial data, for which we can solve (32) for u λ . 5 Thus, the auxiliary Lagrange multiplier u λ is determined via the self-consistency of 5 The full expression for {Ξ, H A } is complicated and not terribly the theory. We have five fields (A 0 , A, and λ), and self-consistency generates four constraints (one primary, three secondary); and so the total number of degrees of freedom of this model is Again, as expected from the conjecture, we have lost one degree of freedom to the Lagrange multiplier. A related analysis was performed by Garfinkle, Isenberg, and Martin-Garcia [19] in the case of Einsteinaether theory. In such models, the vector field is constrained to satisfy A a A a = b = −1, i.e., the vector field is unit and timelike. In [19], the time component A 0 of the vector field was explicitly eliminated from the Lagrangian after performing a 3 + 1 decomposition, leaving the components of A as the three dynamical fields. It was found in that case that the model did not contain any extra constraints on these three dynamical fields, if (assuming c 2 = c 4 = 0, as we have done here) Such models were called "safe" by the authors of [19]; such a model would be expected to contain the three degrees of freedom present in A. Models with c 1 = 0 were called "endangered", in that they contained additional constraints on the initial data; such models would contain fewer than three degrees of freedom. Finally, models with c 1 = 0 and c 3 /c 1 > 0 were called "conditionally endangered", since the constraint structure of the equations differed at various points in configuration space.
To connect this to the present work, we note that (33) is equivalent to or, since c 1 = 0 and −A 2 In the case where c 3 /c 1 ≤ 0 and b < 0, this is guaranteed to hold, and we therefore have no additional constraints and three degrees of freedom. However, if c 3 /c 1 > 0, there can be non-generic points in configuration space where the number of constraints changes.

Potential model
We now consider a vector field A a with a "Maxwell" kinetic term, for which c 1 = −c 3 in (16). We again have illuminating, so we will not present it here. However, from the form of Ξ and H A , we can see that it will depend on A 0 , A, Π 0 , Π, and λ-but it will be independent of both ̟ and (more importantly) u λ . four independent fields, namely the four components of A a . In this case, the canonical momentum Π 0 defined in (19) vanishes automatically, giving us a constraint: The other three canonical momenta Π defined in (20) have an invertible relationship with the corresponding field velocities: Thus, the base Hamiltonian density H B , given by must be augmented by an auxiliary Lagrange multiplier term enforcing the constraint Π 0 = 0. After simplification, this yields Once again, we must ensure that the primary constraint Φ = Π 0 = 0 is conserved by the equations of motion; this again produces a series of secondary constraints: So long as −3A 2 0 + A 2 − b = 0, the demand that the secondary constraint Ψ 1 be preserved by the evolution determines the auxiliary Lagrange multiplier u 0 uniquely. We therefore have four fields, two constraints (one primary, one secondary), and no undetermined Lagrange multipliers; counting the degree of freedom therefore yields

Lagrange multiplier model
We now apply the same process to the vector model with a Lagrange multiplier, (23). With the addition of the Lagrange multiplier λ, we must also introduce a conjugate momentum ̟. As in the scalar LM model, this vanishes identically, yielding a second primary constraint: Including this constraint with an auxiliary Lagrange multiplier u λ in the Hamiltonian density gives us the augmented Hamiltonian density for the model: We now derive the secondary constraints and see if their time-evolution fixes the auxiliary Lagrange multipliers u 0 and u λ : Assuming A 0 = 0, the requirement that both (48) and (49) vanish determines the auxiliary Lagrange multipliers u λ and u 0 . The degree of freedom counting is therefore five fields (four components of A a , plus λ); four constraints (two primary, two secondary); and no undetermined auxiliary Lagrange multipliers, for a result of This is a surprising result: the number of degrees of freedom of the theory when the vector field is "constrained" to a vacuum manifold determined by f (A a ) = 0 is exactly the same as when it is "allowed" to leave this vacuum manifold. In other words, the Lagrangemultiplier "constraint" does not actually reduce the degrees of freedom of the model.
We can again connect this model to the terminology of [19], as we did for the "general" LM case in Section IV A 2. In that work, for a model of a timelike vector field with c 1 = 0 and c 13 = 0 (i.e., c 3 /c 1 = −1), the number of constraints was found to be zero for all points in configuration space; such a model was therefore "safe", with three degrees of freedom at all points in field space. This is in agreement with our work here: so long as the vector A a is constrained to be timelike (b < 0), we will always have A 0 = 0, and the above analysis holds.

Potential model
In this case, we have a Lagrange density with c 1 = 0 and c 3 = 0; such a field is called a "V -field" by Isenberg & Nester [18]. When we calculate the conjugate momenta in this case, (19) allows us to solve for the velocityȦ 0 = Π 0 /c 3 , but (20) becomes a set of three constraints: The augmented Hamiltonian density for the potential model (22) is then where u is a vector of auxiliary Lagrange multipliers enforcing the primary constraints (50). Enforcing these primary constraints under time evolution then yields a set of three secondary constraints Ψ: (52) The time-evolution of these secondary constraints, written out in terms of spatial components, is theṅ We require thatΨ i = 0. This can be guaranteed in equation (53) via an appropriate choice of u j so long as the matrix M ij is invertible. For general field values, this inverse can be calculated to be and so we can solve the equation˙ Ψ = 0 for u so long as A 2 − b = 0. 6 The generic theory therefore has four fields, six constraints (three primary, three secondary) and degree of freedom.

Lagrange multiplier model
As in Section IV B 2, the switch from a potential Vfield model to a Lagrange-multiplier V -field model does 6 It is notable that this set of field values is precisely the vacuum manifold. This property becomes more important in the context of tensor models involving potentials and Lagrange multipliers, and will be discussed more extensively in an upcoming work [16]. not actually "kill off" any degrees of freedom. The augmented Hamiltonian density now contains one more auxiliary Lagrange multiplier u λ , which (as before) enforces the constraint ̟ = 0: We thus have four primary constraints, Φ = 0 (as defined in (50)) and ̟ = 0. Requiring that˙ Φ = 0 anḋ ̟ = 0 then yields four secondary constraints, which I will denote by Ψ and Ψ: The time-evolution of these secondary constraints is theṅ These equations determine all four auxiliary Lagrange multipliers u and u λ so long as A = 0 and λ = 0; in this case, we have Thus, for generic initial data, we are done. We have five fields, four primary constraints, and four secondary constraints; and so the number of degrees of freedom is As for the Maxwellian vector theory in Section IV B 2, the addition of a Lagrange multiplier to a V -field model does not reduce its degrees of freedom. This analysis is again in agreement with the work of Garfinkle, Isenberg, & Martin-Garcia [19]. For a model with c 1 = 0, they find that Einstein-aether theory contains additional initial data constraints on the three dynamical fields A, and is therefore "endangered". In the present work, we have confirmed this result: this model does indeed contain fewer than three degrees of freedom. 7

A. Generalization
We have found that a field theory model in flat spacetime may or may not "lose" a degree of freedom when a constraint is added to the system via a Lagrange multiplier. Specifically, scalar models (Section III) and general vector models (Section IV A) lose a degree of freedom when we replace a potential with a Lagrange multiplier; but Maxwell-type and V -type vector models (Sections IV B & IV C, respectively) retain the same number of degrees of freedom regardless of whether the field values are governed by a potential or by a Lagrange multiplier.
There is an obvious difference between these cases. In those models where there are no primary constraints in the potential model, a Lagrange multiplier eliminates a degree of freedom. In contrast, in the models where the potential model does contain primary constraints, the field theory retains the same number of degrees of freedom when a constraint is imposed via a Lagrange multiplier.
The reason for this difference can be traced to a particular feature of the models we have examined. In those models containing primary constraints, the conservation of the first-order secondary constraints leads to an equation that determines the auxiliary Lagrange multiplier u λ (Eqns. (48) and (60) for the Maxwell-like and Vfield models, respectively). In those models without primary constraints, u λ is only determined once we require that higher-order secondary constraints (specifically, the third-order secondary constraints in Eqs. (13) and (30)) be conserved.
To extend this to a general statement, we first note that the primary constraints for a potential model and its corresponding Lagrange multiplier model are simply related. If the primary constraints for the potential model are a set of M functions {Φ 1 , · · · , Φ M }, then the primary constraints for the corresponding Lagrange multiplier model will simply be {Φ 1 , · · · , Φ M , ̟}, where ̟ is the conjugate momentum to the Lagrange multiplier λ. Moreover, ̟ will commute with all of the primary constraints that derive from the potential model, since none of these constraints depend on λ.
The augmented Hamiltonian density will then be the base Hamiltonian density with terms added to impose the constraints: (Here and in what follows, repeated capitalized Roman indices are summed from 1 to M .) The first-order secondary constraint required in order to maintain ̟ = 0 under time evolution will then be where ψ α here stands for the collection of fields in the model. In addition, there will be a set of first-order sec-ondary constraints Ψ I (I = 1, . . . , M ), each derived from the requirement thatΦ I = 0; these are given by Now consider the time-evolution of the first-order secondary constraints. The time derivative ofΨ λ will be independent of u λ , though it will generally depend on the other auxiliary Lagrange multipliers u I : Note that {f (ψ α ), ̟} = 0 since f (ψ α ) is independent of λ. Meanwhile, the expressionΨ I (for arbitrary I) will contain terms of the forṁ Equations (68) and (69) together imply that if then the equations for conservation of the constraints (Ψ I = 0 andΨ λ = 0) do not contain u λ , leaving this auxiliary Lagrange multiplier undetermined at this stage. If this occurs, then we must proceed to find additional second-and higher-order secondary constraints. Since we have more than two additional constraints, but only one additional degree of freedom from λ itself, we conclude that in such cases, the Lagrange-multiplier model will have fewer degrees of freedom than the potential model. 8 This condition (70) can be greatly elucidated via use of the Jacobi identity. Specifically, we have (71) for any primary constraint Φ I . Since ̟ commutes with the rest of these primary constraints, the last term automatically vanishes; and applying (66) and (67) yields the equation Thus, the equation (69) will leave u λ undetermined, and the Lagrange multiplier will reduce the degrees of freedom of the model, so long as i.e., the vacuum manifold function f (ψ α ) commutes with all the primary constraints.

B. Lagrange-multiplier models in dynamical spacetimes
The number of degrees of freedom of a field theory in flat spacetime is not always simply related to the number of degrees of freedom it possesses in a curved, dynamical spacetime. It is well-known that diffeomorphisminvariant field theories have primary constraints corresponding to the non-dynamical nature of the lapse and shift functions; when we pass to a dynamical spacetime, we both introduce new fields (the ten metric components) as well as new constraints. 9 Perhaps less well-known, but equally important, is that degrees of freedom which are unphysical (gauge or constraint) in flat spacetime can become "activated" in a minimally coupled curvedspacetime theory [18]. This occurs due to the fact that the covariant derivative of a tensor field (unlike that of a scalar) depends on the derivatives of the metric. The "minimally coupled" kinetic term for a tensor field therefore contains couplings between the metric derivatives and the tensor field derivatives, which can turn equations that were constraints or gauge degrees of freedom in flat spacetime into dynamical equations in curved spacetime, and vice versa.
In light of these facts, we might then ask how much of the above analysis would carry over to dynamical spacetimes. Given the critical role played by the constraints in this analysis, it is natural to ask whether a Lagrangemultiplier model in a dynamical curved spacetime would lose any degrees of freedom relative to the corresponding potential model in a dynamical curved spacetime.
The condition (73) sheds some light on this question. We know that if u λ remains undetermined when we require conservation of the first-order secondary constraints, then we will in general have to find higher-order secondary constraints, leading to a reduction of the degrees of freedom of the theory relative to the corresponding potential model. This will occur when the vacuum manifold function f (ψ α ) commutes with the primary constraints of the theory.
Any diffeomorphism-invariant theory, when decomposed into 3+1 form, will contain terms involving the lapse N and shift N a ; these are related to the spacetime metric g ab and the induced spatial metric h ab by where t a is the vector field we have chosen to correspond to "time flow" in our decomposition. We can then write down the Einstein-Hilbert action in terms of this induced metric, the lapse, and the shift. As the lapse and shift can be arbitrarily specified, the are effectively "gauge quantities" corresponding to diffeomorphism invariance; thus, their time derivatives do not appear in the Lagrange density of the theory when it is decomposed. In the Dirac-Bergmann formalism, there are therefore primary constraints on the momenta conjugate to these quantities: The question is then whether the vacuum manifold function f (ψ α ) commutes with these primary constraints. But this is easy enough to see, since Thus, the question of whether the Lagrange multiplier reduces the number of constraints is reduced to the question of whether the vacuum manifold function depends on the lapse and shift. In particular, for a collection of scalar fields in curved spacetime (the dynamical-spacetime analogue of Section III), the vacuum manifold function will be independent of the metric, and so there is no way for the lapse or shift functions to enter into it. We would therefore expect that a Lagrange-multiplier model containing N scalars would have fewer than N degrees of freedom attributable to the scalars. 10 However, for a function of a vector field A a , the norm of the vector field A a will depend on the lapse and shift functions: where A t = t a A a and A ⊥ a = h a b A b . Any function of the spacetime norm of A a will therefore depend on the lapse and shift, and so the vacuum manifold function will not commute with the primary constraints of the theory. Given the results stated above, it seems unlikely that the Lagrange multiplier would reduce the number of degrees of freedom of such a theory.
It is interesting to note that this coupling occurs even if the flat-spacetime theory does not contain any primary constraints, as in the general vector models described in Section IV A. Since the conservation of the first-order secondary constraints determines the auxiliary multiplier u λ in this case, rather than giving rise to further constraints, one would conclude that the number of degrees of freedom of a general vector theory in curved spacetime would not be reduced by the presence of a Lagrange multiplier, in contrast to the situation in flat spacetime. In fact, this is confirmed by known results. A model consisting of a vector field in a curved spacetime with a "generic" kinetic term (as in Section IV A) will contain two "metric" degrees of freedom and three "vector" degrees of freedom, regardless of whether the vector is forced to a non-zero expectation value by a Lagrange multiplier [22] or by a potential [18,23].
C. Potential models in the low-energy limit In classical particle mechanics, it is common to think of a constrained system in relation to an unconstrained system with a potential energy. In the limit where the potential energy becomes infinitely strong, it can be shown that the dynamics of the unconstrained system reduce to those of a system constrained to lie only in the minimum of the potential [24]. It is therefore common, in the analysis of constrained systems, to simply include one or more Lagrange multipliers that enforce the constraints. In general, each Lagrange multiplier reduces the number of degrees of freedom of the system by one.
One might think that this general picture could be carried over to field theory. In particular, a set of fields in a potential could be thought of as possessing a certain number of massive modes (corresponding to oscillations in field-space directions in which the potential increases) and a certain number of massless modes (oscillations in field-space directions in which the potential is flat.) One could then construct a low-energy effective field theory in which the massive modes have "frozen out", reducing the number of degrees of freedom of the model. In this low-energy limit, one would expect the fields to always lie in their vacuum manifold, effectively being constrained there. Hence, one would think that the Lagrange-multiplier version of a potential theory would nicely correspond to the low-energy behavior of the corresponding potential theory.
The results of this work, however, show that the picture is not so simple. While this simple picture holds for scalar fields in flat spacetime, it seems quite unlikely that the low-energy limit of a Maxwell-type or V -type vector field in a potential would correspond to a model with the same kinetic term but containing a Lagrange multiplier. One would expect the low-energy limit to have fewer degrees of freedom than the full potential model; but in these cases, the Lagrange-multiplier models and the corresponding potential models have the same number of degrees of freedom. While the low-energy limit of some such models has been investigated [23,25,26], the Lagrange-multiplier models would necessarily have a different behavior.
In fact, this feature was noted in [23] in the context of a vector field with a "Maxwell" kinetic term. In Section IV.C of that work, it was noted that the Lagrangemultiplier model only corresponded to the low-energy ("infinite-mass") limit of the potential model if the Lagrange multiplier λ was set to zero by fiat. However, for a generic solution λ will not vanish; the vanishing of the secondary constraint in Eq. (46) requires that λ = − ∇ · Π/2A 0 . In other words, one must restrict the class of solutions under considerations-i.e., further reduce the number of degrees of freedom-to obtain the low-energy limit of a potential model from the corresponding Lagrange-multiplier model. This work shows that this lack of direct correspondence is a common feature of models in which tensor fields take on a vacuum expectation value.

Appendix: Poisson brackets and functionals
In calculating the time-evolution of a field quantity in Hamiltonian field theory, one would like to take the Poisson bracket of a field ψ α (x) with the Hamiltonian H to find the time-evolution of the field at x: ψ α (x) = {ψ α (x), H}. (A.1) However, one does have to be careful with this notation, as the Poisson bracket is only rigorously defined on realvalued field functionals, not on functions of space like ψ α . Specifically, we have (A.2) where π α is the conjugate field momentum to ψ α (and a summation over α is implied), and the functional derivatives are implicitly defined via the relation and similarly for π α , where the ellipses stand for higherorder derivatives of ψ α (or π α ), and the partial derivatives of f (and their gradients) are evaluated at the point z. Here and throughout, I will use partial derivatives ∂ to denote the variation of a locally constructed field quantity with respect to one of its arguments, while the δ notation will be reserved for functional derivatives. Under this extension, the Poisson bracket of a local field quantity f (ψ α (x), ∇ψ α (x), . . . ) with the Hamiltonian H = Hd 3 x is "really" the Poisson bracket of the functional F x with H. Restricting attention to quantities that only depend on the fields ψ α and π α and their first derivatives, this Poisson bracket is d dt [f (ψ α , ∇ψ α , π α , ∇π α )] = {F x , H} where all the field quantities are now evaluated at x. Note that this definition implies that that timeevolution "commutes" with spatial derivatives when we take the Poisson bracket, as one would expect. For example, suppose that f = ψ α and H = Hd 3 x, where H is locally constructed from the fields. Then we have This definition can be extended straightforwardly to quantities depending on higher derivatives of ψ α and π α , and the above-mentioned commutativity extends to such cases as well. It can also be extended to the Poisson brackets of two local field quantities f (x) and g(y) by defining functionals F x and G y and following the same procedure. In such cases, the resulting Poisson bracket will contain a factor of δ 3 (x−y). However, in the interests of clarity, we will elide these factors when we take the Poisson bracket of two such quantities; in other words, we will take it as understood that the first argument of such a Poisson bracket is evaluated at x, the second at y, and that the result is multiplied by δ 3 (x − y) or its derivatives.

ACKNOWLEDGMENTS
I would like to thank D. Garfinkle, T. Jacobson, and J. Tasson for helpful discussion and correspondence on this subject. I would also like to thank Perimeter Institute for their support and hospitality during the period over which the majority of this research was conducted.