Constraints and Degrees of Freedom in Lorentz-Violating Field Theories

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L LM aavco rtno edwhich field tensor or vector a ia naptnilmdlsc as such model potential a In vo,oecnpstasmooth a posit can one avior, P aummnfl function manifold vacuum oeta models potential ( = ( = ∇ ∇ ψ ψ f ( N α ollege ( α ψ )( )( N − α ∇ ∇ and 0 = ) ψ 1) elnmes h equation the numbers, real ∗ ψ ψ N α dtheories ld α dmninlsbpc of subspace -dimensional α dmninlfil space. field -dimensional ) (1) + ) . r sindapotential a assigned are f − V ( ψ V λf hudb n fewer one be should ( Lagrange-multiplier α ψ N ( ∇ ψ senforced is 0 = ) ( α ∇ ψ ψ − α ) ψ α α ) 1)-dimensional ∝ . , ψ ) )( α (2) , α f ∇ .Iwill I 0. = 2 h fields the , ψ nldsa includes ( ψ α α ,what ), ,and ), (1) (2) ψ λ α 2 spacetime vector or tensor field, it may be false. In such If, as is usual, we count a field degree of freedom as models, the fields may need to satisfy certain constraints a pair of real-valued functions (e.g., a field value and due to the structure of the kinetic terms; adding a new its conjugate momentum) that can be freely specified on “constraint” to such theories, in the form of a Lagrange an initial data surface, then the number of degrees of multiplier, does not automatically reduce the number of freedom Ndof can be inferred quite simply once the above degrees of freedom of the theory. analysis is complete: To demonstrate this, I will analyse the degrees of free- 1 no. of no. of dom of two types of symmetry-breaking models. After N = + dof 2 fields momenta a brief description of Dirac-Bergmann analysis in Sec- tion II, I will first analyse a multiplet of Lorentz scalar no. of no. of − − . (3) fields with an internal symmetry, followed by a vector constraints gauges field (Sections III and IV respectively.) For the vector fields, the analysis will depend on the structure of the In general, of course, the number of fields and the number kinetic term chosen, and so three distinct sub-cases will of conjugate momenta will be the same. Moreover, in need to be treated. In each case, I will examine a model the particular field theories I will be considering in this where the fields are assigned a potential energy, and one work, I will not find any “gauges”, so the last term in (3) where the fields are directly constrained via a Lagrange will vanish. Thus, for my purposes, the above equation multiplier. reduces to The bulk of the explicit analysis in this work will be no. of 1 no. of N = − . (4) done in the context of a fixed, flat background spacetime; dof fields 2 constraints 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 III. SCALAR MULTIPLET FIELDS for the broader relationship between these two classes of models. More general tensor fields will be examined in a The first case we will consider is a multiplet of N forthcoming work [16]. Lorentz scalars: ψα = φα, with α = 1, 2, 3,...,N. We Throughout this work, I will use units in which ¯h = wish to construct a model where these scalars “naturally” c = 1; the metric signature will be ( − + + + ). Roman take on values in some (N −1)-dimensional hypersurface, indices a, b, c, . . . will be used to denote spacetime tensor defined by f(φα) = 0 (with f a real-valued function.) indices; i, j, k, . . . will be used to denote spatial indices, The “potential model” for this field will be derived from where necessary. Greek indices α, β, γ, ... will be used the Lagrange density exclusively to denote indices in field space. All expres- 1 α a α α 2 sions involving repeated indices (either tensor indices or L = − ∂aφ ∂ φ − κf(φ ) (5) field space indices) can be assumed to obey the Einstein 2 1 summation convention. = (φ˙α)(φ˙α) − (∇~ φα) · (∇~ φα) − κf(φα)2, 2 h i where κ is a proportionality constant; the LM model for II. DIRAC-BERGMANN ANALYSIS this field will be 1 L = − ∂ φα∂aφα − λf(φα) (6) Our primary tool for finding the number of degrees 2 a of each model will be Dirac-Bergmann constraint analy- 1 = (φ˙α)(φ˙α) − (∇~ φα) · (∇~ φα) − λf(φα), sis [17]; my methods and nomenclature below will draw 2 h i heavily from the later work of Isenberg & Nester [18]. I where λ is a Lagrange multiplier field. In both cases, we will briefly summarize the method here, and then illus- have chosen a time coordinate t and performed a 3+1 trate it in more detail via the example theories described decomposition of the tensors; a dot over a quantity (e.g., in Section III. φ˙α) will denote its derivative with respect to this time The method of Dirac-Bergmann analysis involves the coordinate, while spatial derivatives will be denoted with construction of a Hamiltonian which generates the time- either ∇~ or ∂ depending on the expression. evolution of the system. In the process of this construc- i tion, one may need to introduce constraints among various variables, thereby reducing the number of degrees of A. Potential model 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 The degrees of freedom for the model (5) are particu- of freedom). These combinations of fields, which we will larly easy to count. The momenta conjugate to the fields collectively call “gauges”, must be interpreted as unphys- are all well-defined: ical, again reducing the number of physical degrees of ∂L πα = = φ˙α (7) freedom of the model. ∂φ˙α 3

The Hamiltonian density is therefore process, it may occur that the preservation of these constraints allows us to determine the auxiliary Lagrange α α 1 α α α α α 2 H0 = π φ˙ − L = π π + (∇~ φ ) · (∇~ φ ) + κf(φ ) . multiplier uλ introduced above. The process is contin- 2 ued until all constraints are known to be automatically h i (8) conserved or a contradiction is reached. Nothing further is required here; we have no primary With this in mind, we derive the secondary constraints constraints on the initial data, and the Hamiltonian ob- for this model. We first have tained by integrating H0 over space will generate the field dynamics. We thus have the N degrees of freedom one δH 0 =̟ ˙ = {̟,H } = − A = f(φα). (11) would expect. A δλ α Thus, Ψ1 = f(φ ) = 0 is a secondary constraint. We B. Lagrange-multiplier model repeat this procedure, obtaining another secondary constraint: Counting the degrees of freedom for the model (6) re- ∂f(φα) δH Ψ˙ = {Ψ ,H } = A = (δ f)πβ ≡ Ψ , (12) quires a bit more effort. As the kinetic term of (6) is 1 1 A ∂φβ δπβ β 2 the same as that of (5), the momenta conjugate to the α β scalars φ are again defined by (7). The difficulty arises where we have defined δβf = δf/δφ . (Higher deriva- due to the Lagrange multiplier λ. From the perspective tives will be defined similarly.) Ψ2 must also be con- of the model, it is just another field; but its associated served, which leads to a third secondary constraint: momentum vanishes automatically: Ψ˙ 2 = {Ψ2,HA} ∂L ̟ = =0 ≡ Φ. (9) = −δ f −∇2φα + λδ f + παπβ(δ f) ≡ Ψ . ∂λ˙ α α αβ 3 (13) We thus have a primary constraint, Φ = 0, on this theory. α α The “base Hamiltonian density” H0 = π φ˙ − L must Finally, demanding that Ψ3 be conserved allows us to then be modified to obtain the “augmented Hamiltonian determine the auxiliary Lagrange multiplier uλ; this is density” HA by adding this primary constraint multiplied because Ψ3 is itself dependent on λ: 1 by an auxiliary Lagrange multiplier uλ: Ψ˙ 3 = {Ψ3,HA} 1 α 2 α 2 α HA = H0 + uλ̟ = (π ) + (∇~ φ ) + λf(φ )+ uλ̟ ∂Ψ3 δHA ∂Ψ3 δHA ∂Ψ3 δHA 2 = γ γ − γ γ + . (14) h i (10) ∂φ δπ ∂π δφ ∂λ δ̟ We now need to ensure that this constraint is preserved When the dust settles, we obtain by the time-evolution of the system; in other words, we 3 2 must have̟ ˙ = {̟,HA} = 0, where HA ≡ HAd x. Ψ˙ = πγ (δ f) 3∇2φα − 4λδ f + ∇2 (δ f) If this Poisson bracket does not vanish identically, this 3 αγ α α R α β demand will yield a secondary constraint Ψ1 = 0. The + π π (δαβγ f) − (δαf)(δαf)uλ. (15) demand that this constraint be preserved may lead to So long as δαf 6= 0 when f = 0, this allows us to deter- new secondary constraints Ψ2 =0, Ψ3 = 0, and so forth, which must themselves be conserved. We will refer to the mine the previously-unknown auxiliary Lagrange multi- stage at which a secondary constraint arises as its “or- plier uλ. Thus, the process terminates here. Having determined the Hamiltonian and its con- der”. In other words, if Ψ1 ensures the preservation of a primary constraint, it is a “first-order secondary constraints, we can count the degrees of freedom. We have N +1 fields, namely the multiplet φα (α =1,...,N) and straint”; if Ψ2 ensures the preservation of Ψ1, it is a “second-order secondary constraint”; and so on. In this the Lagrange multiplier λ; one primary constraint Φ = ̟ = 0; and three secondary constraints, {Ψ1, Ψ2, Ψ3} = 0. The single auxiliary Lagrange multiplier is determined, which means that there are no unphysical gauge 1 Here and throughout, we will need to distinguish between the degrees of freedom. With N + 1 fields and four con- “real” Lagrange multiplier that appears in the original La- straints, the number of degrees of freedom of the model grangian and the “auxiliary” Lagrange multipliers that are used (6) is therefore to construct a Hamiltonian for the model. In general, we will only have one real Lagrange multiplier at a time, which we will 1 denote with λ; auxiliary Lagrange multipliers will be denoted Ndof = (N + 1) − (4) = N − 1. with the symbol u, possibly with subscripts or diacritical marks. 2 2 We are playing a bit fast and loose with notation here; in Hamil- tonian field theory, the Poisson bracket is only rigorously defined Thus, the Lagrange-multiplier theory (6) has one less de- for a functional with a single real value, not for a field which is gree of freedom than the corresponding potential theory a function of space. A more rigorous definition of what we mean (5); in this case, the na¨ıve conjecture outlined in the in- by an expression like {̟,HA} is given in the Appendix. troduction holds true. 4

IV. VECTOR FIELDS possible form for the vacuum manifold function is one which sets the norm of Aa to some constant b. For sim-

We now consider the case of a vector field Aa which plicity’s sake, we will therefore choose f(Aa) to be of the spontaneously breaks Lorentz symmetry. We will again following form: restrict our attention to the case of flat spacetime. As f(A )= AaA − b (21) the motivation behind these models is usually a sponta- a a neous breaking of Lorentz symmetry, we want our La- where b is a constant. Depending on the sign of b, the grange density to be a Lorentz scalar, without any prior “vacuum” manifold will consist of timelike vectors (b < geometry specified. 0), spacelike vectors (b > 0), or null vectors (b = 0).4 In any such Lagrange density, we can identify a set of Our potential model will then be “kinetic” terms LK that depend on the derivatives of Aa. 2 2 ~2 2 The most general kinetic term that we can write down LP = LK − κf(Aa) = LK − κ(−A0 + A − b) , (22) 3 which is quadratic in the field Aa is where κ is again a proportionality constant; the LM a b a 2 b a Lk = c1∂aAb∂ A + c2(∂aA ) + c3∂aAb∂ A . (16) model will be 2 ~2 However, since LLM = LK − λf(Aa)= LK − λ(−A0 + A − b). (23)

a 2 b a a b b a (∂aA ) = ∂aAb∂ A + ∂a A ∂bA − A ∂bA , (17) For compactness, I will denote the four-norm of Aa as A2 = −A2 + A~2; the norm of the spatial part on its own 0 we can eliminate one of c2 or c3 via an integration by will always be denoted by A~2. parts. We will therefore set c2 = 0 in what follows. The familiar “Maxwell” kinetic term 1 A. General case: c13 =6 0, c1 =6 0 L = − F F ab, K 4 ab 1. Potential model 1 with Fab =2∂[aAb] corresponds to c1 = −c3 = − 2 . We can now perform the usual 3+1 decomposition of Performing a Legendre transform on LP (22) to obtain the Lagrange density, writing A0 for the t-component of the Hamiltonian density, we obtain a base Hamiltonian ~ Aa and A (or Ai) for its spatial components. The kinetic density of term (16) then becomes 2 2 2 1 1 c − c c3 2 2 2 ~ 2 1 3 ~ ~ ~ 1 c1 c1 ˙ 2 ˙ HB = Π0 − Π + ∇A0 − Π · ∇A0 LK = c13 A˙ 0 − ∇~ A0 − A~ − c3A~ · ∇~ A0 2c13 2c1 2c1 c1 2 2 2 c1 i j c3 j i 2 2 c1 c3 − (∂iAj )(∂ A ) − (∂iAj )(∂ A )+ κ A − b . + (∂ A )(∂iAj )+ (∂ A )(∂j Ai), (18) 2 2 2 i j 2 i j (24) where c ≡ c + c . The momenta conjugate to A and 13 1 3 0 ~ A~ are then The resulting theory has four fields (A0 and A) and no constraints; so the process terminates here, and the base 0 ∂LK Hamiltonian is the complete Hamiltonian for the model. Π = = c13A˙ 0 (19) ∂A˙ 0 Counting the degrees of freedom, we find that 1 and N =4 − (0) = 4. (25) dof 2 ∂LK ˙ Π=~ = −c A~ − c ∇~ A . (20) ˙ 1 3 0 ∂A~ 2. Lagrange multiplier model These equations can be inverted, to find the velocities ˙ A˙ 0 and A~, so long as c1 + c3 6= 0 and c1 6= 0. If either For the Lagrange multiplier model (23), we have a pri- of these expressions vanishes, (19) and (20) will instead mary constraint associated with λ: yield constraint equations; we will have to handle these ∂L cases separately. ̟ = =0. (26) For the potential term, meanwhile, our desire for the ∂λ˙ Lagrangian to be a Lorentz scalar implies that the only

4 As spontaneous symmetry breaking implies a non-zero ﬁeld value, one would normally exclude the case b = 0 to ensure that 3 This follows the notation of [10], with the coeﬃcient c4 from that Aa = 0 is not in the vacuum manifold. However, this is not reference set equal to zero. necessary for the analysis which follows. 5

We must therefore augment the Hamiltonian with an the theory. We have five fields (A0, A~, and λ), and auxiliary Lagrange multiplier uλ to enforce this con- self-consistency generates four constraints (one primary, straint: three secondary); and so the total number of degrees of freedom of this model is H = Π0A + Π~ · A~ − L + u ̟ A 0 LM λ 1 2 2 2 Ndof =5 − (4) = 3. (34) 1 2 1 ~ 2 c1 − c3 ~ c3 ~ ~ 2 = Π0 − Π + ∇A0 − Π · ∇A0 2c13 2c1 2c1 c1 Again, as expected from the conjecture, we have lost one c c − 1 (∂ A )(∂iAj ) − 3 (∂ A )(∂j Ai) degree of freedom to the Lagrange multiplier. 2 i j 2 i j 2 A related analysis was performed by Garfinkle, Isen- + λ(A − b)+ uλ̟ (27) berg, and Martin-Garcia [19] in the case of Einstein- aether theory. In such models, the vector field is con- We now find the secondary constraints required for the a primary constraint to be conserved under time-evolution. strained to satisfy AaA = b = −1, i.e., the vector field Taking the Poisson brackets of each constraint with the is unit and timelike. In [19], the time component A0 of Hamiltonian in turn, we obtain three secondary con- the vector field was explicitly eliminated from the La- straints: grangian after performing a 3 + 1 decomposition, leaving the components of A~ as the three dynamical fields. It 2 ~2 ̟˙ = {̟,HA} = A0 − A + b ≡ Ψ1; (28) was found in that case that the model did not contain any extra constraints on these three dynamical fields, if Ψ˙ 1 = {Ψ1,HA} (assuming c2 = c4 = 0, as we have done here) 1 0 1 =2 A0Π + A~ · Π+~ c3∇~ A0 ≡ Ψ2; (29) c3 c13 c1 c1 6=0, ≤ 0. (35) c1 and Such models were called “safe” by the authors of [19]; such a model would be expected to contain the three de- 2 ~2 A0 A ~ Ψ˙ 2 = {Ψ2,HA} =2λ − + Ξ ≡ Ψ3, (30) grees of freedom present in A. Models with c1 = 0 were c13 c1 ! called “endangered”, in that they contained additional constraints on the initial data; such models would con- where tain fewer than three degrees of freedom. Finally, models 2 with c 6= 0 and c /c > 0 were called “conditionally en- 1 0 2 1 ~ ~ 1 3 1 Ξ ≡ 2 Π − 2 Π+ c3∇A0 dangered”, since the constraint structure of the equations c13 c1 differed at various points in configuration space. c3 0 c1 − c3 2 + A~ · ∇~ Π − A0∇~ · Π~ + A0∇ A0 To connect this to the present work, we note that (33) c1c13 c1 is equivalent to c3 ~ ~ ~ ~ ~ 2 ~ − A · ∇ ∇ · A + A · ∇ A. (31) 2 ~2 ~2 c1 c1A0 − c1A − c3A 6=0, (36) All three of the quantities Ψ ,Ψ , and Ψ must vanish 2 ~2 1 2 3 or, since c1 6= 0 and −A0 + A = b under the constraint, for the model to be consistent. c3 2 When we take the Poisson bracket of Ψ3 with HA, we b 6= − A~ . (37) will obtain c1 In the case where c /c ≤ 0 and b< 0, this is guaranteed 2 ~2 3 1 A0 A to hold, and we therefore have no additional constraints Ψ˙ 3 = {Ξ,HA} + 2λ − ,HA ( c13 c1 ! ) and three degrees of freedom. However, if c3/c1 > 0, there can be non-generic points in configuration space 2 ~2 A0 A = {Ξ,HA} +2uλ − where the number of constraints changes. c13 c1 !

1 0 1 ~ ~ ~ B. Maxwell case: c13 = 0, c1 =6 0 +4λ 2 A0Π + 2 A · Π+ c3∇A0 (32) c13 c1 This means that for generic initial data, for which 1. Potential model 2 ~2 A0 A 6= , (33) We now consider a vector field Aa with a “Maxwell” c c 13 1 kinetic term, for which c1 = −c3 in (16). We again have 5 we can solve (32) for uλ. Thus, the auxiliary Lagrange multiplier uλ is determined via the self-consistency of illuminating, so we will not present it here. However, from the form of Ξ and HA, we can see that it will depend on A0, A~, Π0, Π,~ and λ—but it will be independent of both ̟ and (more 5 The full expression for {Ξ,HA} is complicated and not terribly importantly) uλ. 6 four independent fields, namely the four components of Including this constraint with an auxiliary Lagrange mul- 0 Aa. In this case, the canonical momentum Π defined in tiplier uλ in the Hamiltonian density gives us the aug- (19) vanishes automatically, giving us a constraint: mented Hamiltonian density for the model:

Π0 =0 ≡ Φ1. (38) 1 2 HA = − Π~ + Π~ · ∇~ A0 2c1 The other three canonical momenta Π~ defined in (20) c − 1 (∂ A )(∂iAj ) − (∂ A )(∂j Ai) have an invertible relationship with the corresponding 2 i j i j field velocities: 2 0 + λ(A − b)+ u0Π + uλ̟. (45) ~ ~˙ ~ Π= c1 −A + ∇A0 . (39) We now derive the secondary constraints and see if their time-evolution fixes the auxiliary Lagrange multi- Thus, the base Hamiltonian density HB, given by pliers u0 and uλ: ˙ HB = Π0A˙ 0 + Π~ · A~ − L (40) Φ˙ 1 = {Π0,HA} = ∇~ · Π+2~ λA0 ≡ Ψ1 (46) ˙ 2 must be augmented by an auxiliary Lagrange multiplier Φ2 = {̟,HA} = −A + b ≡ Ψ2 (47) 0 term enforcing the constraint Π = 0. After simplifica- Ψ˙ 1 = {Ψ1,HA} = −∇~ · λA~ +2A0uλ +2λu0 (48) tion, this yields Ψ˙ 2 = {Ψ2,HA} =2A0u0 − 2A~ · Π+~ ∇~ A0 (49) 1 2 HA = − Π~ + Π~ · ∇~ A0 2c1 Assuming A0 6= 0, the requirement that both (48) and c1 i j j i − (∂iAj )(∂ A ) − (∂iAj )(∂ A ) (49) vanish determines the auxiliary Lagrange multipliers 2 uλ and u0. The degree of freedom counting is therefore 2 2 0 + κ(A − b) + u0Π . (41) five fields (four components of Aa, plus λ); four constraints (two primary, two secondary); and no undeter- Once again, we must ensure that the primary constraint mined auxiliary Lagrange multipliers, for a result of Φ = Π0 = 0 is conserved by the equations of motion; this again produces a series of secondary constraints: 1 N =5 − (4) = 3. dof 2 0 2 Π˙ = {Π0,HA} = ∇~ · Π+4~ κ(A − b)A0 ≡ Ψ1 (42) This is a surprising result: the number of degrees of ˙ Ψ1 = {Ψ1,HA} freedom of the theory when the vector field is “con- 2 2 2 strained” to a vacuum manifold determined by f(Aa)=0 = −8κ ∇~ · ((A − b)A~) − (−3A + A~ − b)u0 0 is exactly the same as when it is “allowed” to leave h 1 this vacuum manifold. In other words, the Lagrange- + A0A~ · Π~ − ∇~ A0 . (43) c1 multiplier “constraint” does not actually reduce the degrees of freedom of the model. 2 ~2 So long as −3A0 + A − b 6= 0, the demand that the sec- We can again connect this model to the terminology ondary constraint Ψ1 be preserved by the evolution deter- of [19], as we did for the “general” LM case in Section mines the auxiliary Lagrange multiplier u0 uniquely. We IVA2. In that work, for a model of a timelike vector therefore have four fields, two constraints (one primary, field with c1 6= 0 and c13 = 0 (i.e., c3/c1 = −1), the one secondary), and no undetermined Lagrange multipli- number of constraints was found to be zero for all points ers; counting the degree of freedom therefore yields in configuration space; such a model was therefore “safe”, with three degrees of freedom at all points in field space. 1 This is in agreement with our work here: so long as the Ndof =4 − (2) = 3. 2 vector Aa is constrained to be timelike (b < 0), we will always have A0 6= 0, and the above analysis holds. 2. Lagrange multiplier model

C. V -field case: c13 =6 0, c1 = 0 We now apply the same process to the vector model with a Lagrange multiplier, (23). With the addition of 1. Potential model 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: In this case, we have a Lagrange density with c1 = 0 and c3 6= 0; such a field is called a “V -field” by Isenberg & ∂L Nester [18]. When we calculate the conjugate momenta ̟ = =0 ≡ Φ2 (44) ˙ ∂λ˙ in this case, (19) allows us to solve for the velocity A0 = 7

0 Π /c3, but (20) becomes a set of three constraints: not actually “kill oﬀ” any degrees of freedom. The augmented Hamiltonian density now contains one more aux- ~ ~ ~ Φ ≡ Π+ c3∇A0 =0. (50) iliary Lagrange multiplier uλ, which (as before) enforces the constraint ̟ = 0: The augmented Hamiltonian density for the potential model (22) is then 1 0 2 c3 j 2 HA = Π − (∂iAj )(∂ Ai)+ λ(A − b) 2c3 2 0 ˙ HA = Π A0 + Π~ · A~ − LP + ~u · Π+~ c3∇~ A0 + ~u · Π+~ c3∇~ A0 + uλ̟. (57) 1 0 2 c3 j = Π − (∂iAj )(∂ Ai) ~ 2c3 2 We thus have four primary constraints, Φ = 0 (as de- ˙ 2 2 ﬁned in (50)) and ̟ = 0. Requiring that Φ~ = 0 and + κ(A − b) + ~u · Π+~ c3∇~ A0 , (51) ̟˙ = 0 then yields four secondary constraints, which I where ~u is a vector of auxiliary Lagrange multipliers en- will denote by Ψ~ and Ψ: forcing the primary constraints (50). Enforcing these pri- ~˙ ~ ~ 0 ~ ~ ~ ~ ~ mary constraints under time evolution then yields a set Φ= {Φ,HA} = ∇Π − c3∇ ∇ · A − 2λA ≡ Ψ, (58) of three secondary constraints Ψ:~ 2 ̟˙ = {̟,HA} = A − b ≡ Ψ. (59)

˙ 0 2 Φ=~ {Φ~ ,HA} = ∇~ Π − c3∇~ ∇~ · A~ − 4κ(A − b)A~ ≡ Ψ~ . The time-evolution of these secondary constraints is then (52) ˙ Ψ=~ {Ψ~ ,H } =2 ∇~ (λA ) − u A~ + λ~u (60) The time-evolution of these secondary constraints, writ- A 0 λ ten out in terms of spatial components, is then h 1 0 i Φ=˙ {Ψ,HA} = A~ · ~u − A0Π . (61) c3 Ψ˙ = {Ψ ,H } = M u − 4κ∂ (A2 − b)A i i A ij j i 0 These equations determine all four auxiliary Lagrange 8κ 0 ~ + A0Ai Π + ∇~ · A~ , (53) multipliers ~u and uλ so long as A 6= 0 and λ 6= 0; in this c3 case, we have where 1 1 0 uλ = A~ · ∇~ (λA0)+ λA0Π (62) 2 A~2 c3 Mij ≡ 4κ δij A − b +2AiAj . (54) and We require that Ψ˙ = 0. This can be guaranteed in i 1 equation (53) via an appropriate choice of uj so long ~u = u A~ − ∇~ (λA ) . (63) λ λ 0 as the matrix Mij is invertible. For general field values, h i this inverse can be calculated to be Thus, for generic initial data, we are done. We have five fields, four primary constraints, and four secondary −1 1 2AiAj M = δij − (55) constraints; and so the number of degrees of freedom is ij 4κ(A2 − b) ~2 1+2A 1 ˙ Ndof =5 − (8) = 1. (64) and so we can solve the equation Ψ=0~ for ~u so long as 2 A2 − b 6= 0.6 The generic theory therefore has four fields, As for the Maxwellian vector theory in Section IV B 2, six constraints (three primary, three secondary) and the addition of a Lagrange multiplier to a V -field model does not reduce its degrees of freedom. 1 Ndof =4 − (6) = 1 (56) This analysis is again in agreement with the work of 2 Garfinkle, Isenberg, & Martin-Garcia [19]. For a model degree of freedom. with c1 = 0, they find that Einstein-aether theory contains additional initial data constraints on the three dynamical fields A~, and is therefore “endangered”. In the 2. Lagrange multiplier model present work, we have confirmed this result: this model does indeed contain fewer than three degrees of freedom.7 As in Section IV B 2, the switch from a potential V - field model to a Lagrange-multiplier V -field model does 7 The cases A~ = 0 and λ = 0 were excluded from the above analysis. In this case, one would have to look at the time evolution of the quantities in (60) and (61), generate one or more second- 6 It is notable that this set of field values is precisely the vacuum order secondary constraints, and attempt to solve these for the manifold. This property becomes more important in the context auxiliary Lagrange multipliers. In any event, this would generate of tensor models involving potentials and Lagrange multipliers, a model with no more than one degree of freedom (if the resulting and will be discussed more extensively in an upcoming work [16]. model was even consistent at such points in configuration space.) 8

V. DISCUSSION ondary constraints ΨI (I =1,...,M), each derived from the requirement that Φ˙ I = 0; these are given by A. Generalization ΨI = {ΦI ,HA}. (67) We have found that a field theory model in flat spacetime may or may not “lose” a degree of freedom when a Now consider the time-evolution of the first-order sec- ˙ constraint is added to the system via a Lagrange multi- ondary constraints. The time derivative of Ψλ will be plier. Specifically, scalar models (Section III) and general independent of uλ, though it will generally depend on vector models (Section IVA) lose a degree of freedom the other auxiliary Lagrange multipliers uI : when we replace a potential with a Lagrange multiplier; ˙ α but Maxwell-type and V -type vector models (Sections Ψλ ⊃ uJ {Ψλ, ΦJ } + uλ{Ψλ,̟} = uJ {f(ψ ), ΦJ }. (68) IVB & IVC, respectively) retain the same number of degrees of freedom regardless of whether the field values Note that {f(ψα),̟} = 0 since f(ψα) is independent of are governed by a potential or by a Lagrange multiplier. λ. Meanwhile, the expression Ψ˙ I (for arbitrary I) will There is an obvious difference between these cases. In contain terms of the form those models where there are no primary constraints in the potential model, a Lagrange multiplier eliminates a Ψ˙ I = {ΨI ,HA}⊃ uJ {ΨI , ΦJ } + uλ{ΨI ,̟}, (69) degree of freedom. In contrast, in the models where the potential model does contain primary constraints, the Equations (68) and (69) together imply that if field theory retains the same number of degrees of freedom when a constraint is imposed via a Lagrange multi- {ΨI ,̟} =0, (70) plier. The reason for this difference can be traced to a par- then the equations for conservation of the constraints ticular feature of the models we have examined. In those (Ψ˙ I = 0 and Ψ˙ λ = 0) do not contain uλ, leaving this models containing primary constraints, the conservation auxiliary Lagrange multiplier undetermined at this stage. of the first-order secondary constraints leads to an equa- If this occurs, then we must proceed to find additional tion that determines the auxiliary Lagrange multiplier second- and higher-order secondary constraints. Since we uλ (Eqns. (48) and (60) for the Maxwell-like and V - have more than two additional constraints, but only one field models, respectively). In those models without pri- additional degree of freedom from λ itself, we conclude mary constraints, uλ is only determined once we require that in such cases, the Lagrange-multiplier model will that higher-order secondary constraints (specifically, the have fewer degrees of freedom than the potential model.8 third-order secondary constraints in Eqs. (13) and (30)) This condition (70) can be greatly elucidated via use be conserved. of the Jacobi identity. Specifically, we have To extend this to a general statement, we first note that the primary constraints for a potential model and its {{ΦI ,HA},̟} + {{HA,̟}, ΦI } + {{̟, ΦI },HA} =0 corresponding Lagrange multiplier model are simply re- (71) lated. If the primary constraints for the potential model for any primary constraint ΦI . Since ̟ commutes with are a set of M functions {Φ1, ··· , ΦM }, then the pri- the rest of these primary constraints, the last term auto- mary constraints for the corresponding Lagrange multi- matically vanishes; and applying (66) and (67) yields the plier model will simply be {Φ1, ··· , ΦM ,̟}, where ̟ is equation the conjugate momentum to the Lagrange multiplier λ. α Moreover, ̟ will commute with all of the primary con- {ΨI ,̟} = −{f(ψ ), ΦI }. (72) straints that derive from the potential model, since none of these constraints depend on λ. Thus, the equation (69) will leave uλ undetermined, and The augmented Hamiltonian density will then be the the Lagrange multiplier will reduce the degrees of free- base Hamiltonian density with terms added to impose dom of the model, so long as the constraints: α {f(ψ ), ΦI } =0, (73) HA = H0 + uI ΦI + uλ̟. (65) (Here and in what follows, repeated capitalized Roman i.e., the vacuum manifold function f(ψα) commutes with indices are summed from 1 to M.) The first-order sec- all the primary constraints. ondary constraint required in order to maintain ̟ = 0 under time evolution will then be

δHA Ψ = {̟,H } = − = f(ψα), (66) 8 λ A δλ It is also conceivable that uλ could remain undetermined even after the process of finding the constraints is completed. This where ψα here stands for the collection of fields in the would also reduce the number of degrees of freedom in the final model. In addition, there will be a set of first-order sec- counting. 9

B. Lagrange-multiplier models in dynamical their time derivatives do not appear in the Lagrange den- spacetimes sity of the theory when it is decomposed. In the Dirac- Bergmann formalism, there are therefore primary con- The number of degrees of freedom of a field theory in straints on the momenta conjugate to these quantities: flat spacetime is not always simply related to the number ∂L ∂L of degrees of freedom it possesses in a curved, dynam- Π ≡ =0, Πa ≡ =0. (75) ical spacetime. It is well-known that diffeomorphism- ∂N˙ ∂N˙ a invariant field theories have primary constraints corre- The question is then whether the vacuum manifold func- sponding to the non-dynamical nature of the lapse and tion f(ψα) commutes with these primary constraints. shift functions; when we pass to a dynamical spacetime, But this is easy enough to see, since we both introduce new fields (the ten metric components) as well as new constraints.9 Perhaps less well-known, δf δf {f(ψα), Π} = , {f(ψα), Π } = (76) but equally important, is that degrees of freedom which δN a δN a are unphysical (gauge or constraint) in flat spacetime can become “activated” in a minimally coupled curved- Thus, the question of whether the Lagrange multiplier spacetime theory [18]. This occurs due to the fact that reduces the number of constraints is reduced to the ques- the covariant derivative of a tensor field (unlike that of tion of whether the vacuum manifold function depends on a scalar) depends on the derivatives of the metric. The the lapse and shift. In particular, for a collection of scalar “minimally coupled” kinetic term for a tensor field there- fields in curved spacetime (the dynamical-spacetime ana- fore contains couplings between the metric derivatives logue of Section III), the vacuum manifold function will and the tensor field derivatives, which can turn equations be independent of the metric, and so there is no way for that were constraints or gauge degrees of freedom in flat the lapse or shift functions to enter into it. We would spacetime into dynamical equations in curved spacetime, therefore expect that a Lagrange-multiplier model containing N scalars would have fewer than N degrees of and vice versa. 10 In light of these facts, we might then ask how much of freedom attributable to the scalars. the above analysis would carry over to dynamical space- However, for a function of a vector field Aa, the norm times. Given the critical role played by the constraints of the vector field Aa will depend on the lapse and shift in this analysis, it is natural to ask whether a Lagrange- functions: multiplier model in a dynamical curved spacetime would a a 2 ((t − N )Aa) lose any degrees of freedom relative to the corresponding A A gab = A habA − a b a b N 2 potential model in a dynamical curved spacetime. a ⊥ 2 The condition (73) sheds some light on this ques- ⊥ ⊥ At − N Aa = A habA − , (77) tion. We know that if u remains undetermined when a b 2 λ N we require conservation of the first-order secondary con- a ⊥ b straints, then we will in general have to find higher-order where At = t Aa and Aa = ha Ab. Any function of secondary constraints, leading to a reduction of the de- the spacetime norm of Aa will therefore depend on the grees of freedom of the theory relative to the correspond- lapse and shift, and so the vacuum manifold function will ing potential model. This will occur when the vacuum not commute with the primary constraints of the theory. manifold function f(ψα) commutes with the primary con- Given the results stated above, it seems unlikely that the straints of the theory. Lagrange multiplier would reduce the number of degrees Any diffeomorphism-invariant theory, when decom- of freedom of such a theory. posed into 3+1 form, will contain terms involving the It is interesting to note that this coupling occurs even lapse N and shift N a; these are related to the spacetime if the flat-spacetime theory does not contain any primary metric gab and the induced spatial metric hab by constraints, as in the general vector models described in Section IV A. Since the conservation of the first-order 1 secondary constraints determines the auxiliary multiplier gab = hab − (ta − N a)(tb − N b), (74) N 2 uλ in this case, rather than giving rise to further con- a straints, one would conclude that the number of degrees where t is the vector field we have chosen to correspond of freedom of a general vector theory in curved space- to “time flow” in our decomposition. We can then write time would not be reduced by the presence of a Lagrange down the Einstein-Hilbert action in terms of this induced multiplier, in contrast to the situation in flat spacetime. metric, the lapse, and the shift. As the lapse and shift can In fact, this is confirmed by known results. A model be arbitrarily specified, the are effectively “gauge quan- consisting of a vector field in a curved spacetime with tities” corresponding to diffeomorphism invariance; thus,

10 As there is no coupling between the kinetic terms of the scalar 9 See [20, 21] for a detailed description of the Hamiltonian formu- and the metric, it seems likely that there would also still be two lation of general relativity. degrees of freedom attributable to the metric itself. 10 a “generic” kinetic term (as in Section IV A) will con- class of solutions under considerations—i.e., further retain two “metric” degrees of freedom and three “vector” duce the number of degrees of freedom—to obtain the degrees of freedom, regardless of whether the vector is low-energy limit of a potential model from the corre- forced to a non-zero expectation value by a Lagrange sponding Lagrange-multiplier model. This work shows multiplier [22] or by a potential [18, 23]. that this lack of direct correspondence is a common feature of models in which tensor ﬁelds take on a vacuum expectation value. C. Potential models in the low-energy limit

In classical particle mechanics, it is common to think Appendix: Poisson brackets and functionals of a constrained system in relation to an unconstrained system with a potential energy. In the limit where the po- In calculating the time-evolution of a field quantity tential energy becomes infinitely strong, it can be shown in Hamiltonian field theory, one would like to take the that the dynamics of the unconstrained system reduce Poisson bracket of a field ψα(x) with the Hamiltonian H to those of a system constrained to lie only in the mini- to find the time-evolution of the field at x: mum of the potential [24]. It is therefore common, in the α α analysis of constrained systems, to simply include one or ψ˙ (x)= {ψ (x),H}. (A.1) more Lagrange multipliers that enforce the constraints. In general, each Lagrange multiplier reduces the number However, one does have to be careful with this notation, as the Poisson bracket is only rigorously defined on real- of degrees of freedom of the system by one. α One might think that this general picture could be car- valued field functionals, not on functions of space like ψ . ried over to field theory. In particular, a set of fields Specifically, we have in a potential could be thought of as possessing a cer- 3 δG1 δG2 δG1 δG2 tain number of massive modes (corresponding to oscil- {G1, G2}≡ d z − , δψα(z) δπα(z) δπα(z) δψα(z) lations in field-space directions in which the potential Z (A.2) increases) and a certain number of massless modes (os- where πα is the conjugate field momentum to ψα (and a cillations in field-space directions in which the potential summation over α is implied), and the functional deriva- is flat.) One could then construct a low-energy effective tives are implicitly defined via the relation field theory in which the massive modes have “frozen out”, reducing the number of degrees of freedom of the δG δG = d3z δψα(z). (A.3) model. In this low-energy limit, one would expect the δψα(z) fields to always lie in their vacuum manifold, effectively Z being constrained there. Hence, one would think that the To extend the definition (A.2) of a Poisson bracket to Lagrange-multiplier version of a potential theory would a local field quantity f(ψα(x), ∇ψα(x),... ) constructed nicely correspond to the low-energy behavior of the cor- from field quantities at a fixed point x, one introduces responding potential theory. the functional The results of this work, however, show that the pic- 3 α α 3 ture is not so simple. While this simple picture holds for Fx ≡ d y f(ψ (y), ∇ψ (y),... )δ (x − y) . (A.4) scalar fields in flat spacetime, it seems quite unlikely that Z the low-energy limit of a Maxwell-type or V -type vector The functional derivatives in (A.2) then become field in a potential would correspond to a model with the same kinetic term but containing a Lagrange mul- δFx ∂f 3 ∂f 3 tiplier. One would expect the low-energy limit to have = δ (x − z) − ∇a δ (x − z) δψα(z) ∂ψα ∂(∇ ψα) fewer degrees of freedom than the full potential model; a but in these cases, the Lagrange-multiplier models and + ... (A.5) the corresponding potential models have the same num- and similarly for πα, where the ellipses stand for higher- ber of degrees of freedom. While the low-energy limit order derivatives of ψα (or πα), and the partial deriva- of some such models has been investigated [23, 25, 26], tives of f (and their gradients) are evaluated at the point the Lagrange-multiplier models would necessarily have a z. Here and throughout, I will use partial derivatives ∂ to different behavior. denote the variation of a locally constructed field quan- In fact, this feature was noted in [23] in the context tity with respect to one of its arguments, while the δ of a vector field with a “Maxwell” kinetic term. In Sec- notation will be reserved for functional derivatives. tion IV.C of that work, it was noted that the Lagrange- Under this extension, the Poisson bracket of a local multiplier model only corresponded to the low-energy field quantity f(ψα(x), ∇ψα(x),... ) with the Hamilto- (“infinite-mass”) limit of the potential model if the La- nian H = Hd3x is “really” the Poisson bracket of the grange multiplier λ was set to zero by fiat. However, functional F with H. Restricting attention to quantities for a generic solution λ will not vanish; the vanishing x that only dependR on the fields ψα and πα and their first of the secondary constraint in Eq. (46) requires that derivatives, this Poisson bracket is λ = −∇~ · Π~ /2A0. In other words, one must restrict the 11

d [f(ψα, ∇ψα, πα, ∇πα)] = {F ,H} dt x ∂f ∂f δH = d3z δ3(x − z) − ∇ δ3(x − z) ∂ψα a ∂(∇ ψα) δπα(z) Z a ∂f ∂f δH − δ3(x − z) − ∇ δ3(x − z) (A.6) ∂πα a ∂(∇ πα) δψα(z) a ∂f δH ∂f δH ∂f δH ∂f δH = + ∇ − − ∇ , ∂ψα δπα(x) ∂(∇ ψα) a δπα(x) ∂πα δψα(x) ∂(∇ πα) a δψα(x) a a (A.7)

where all the field quantities are now evaluated at x. cases as well. It can also be extended to the Poisson Note that this definition implies that that time- brackets of two local field quantities f(x) and g(y) by evolution “commutes” with spatial derivatives when we defining functionals Fx and Gy and following the same take the Poisson bracket, as one would expect. For ex- procedure. In such cases, the resulting Poisson bracket ample, suppose that f = ψα and H = Hd3x, where H will contain a factor of δ3(x−y). However, in the interests is locally constructed from the fields. Then we have of clarity, we will elide these factors when we take the R Poisson bracket of two such quantities; in other words, δH ∂H {ψα,H} = = , (A.8) we will take it as understood that the first argument of δπα(x) ∂πα such a Poisson bracket is evaluated at x, the second at 3 α y, and that the result is multiplied by δ (x − y) or its evaluated at x. Meanwhile, if f = ∇aψ , we have derivatives.

δH ∂H {∇ ψα,H} = ∇ = ∇ a a δπα(x) a ∂πα α ACKNOWLEDGMENTS = ∇a ({ψ ,H}) . (A.9)

This fact simplifies the calculation of the Poisson brackets I would like to thank D. Garfinkle, T. Jacobson, and J. considerably. Tasson for helpful discussion and correspondence on this This definition can be extended straightforwardly to subject. I would also like to thank Perimeter Institute quantities depending on higher derivatives of ψα and πα, for their support and hospitality during the period over and the above-mentioned commutativity extends to such which the majority of this research was conducted.

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