Constraints and Degrees of Freedom in Lorentz-Violating Field Theories

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2-7-2019

Constraints and Degrees of Freedom in Lorentz-violating Field Theories

Michael D. Seifert Connecticut College, [email protected]

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This Article is brought to you for free and open access by the Physics, Astronomy and Geophysics Department at Digital Commons @ Connecticut College. It has been accepted for inclusion in Physics, Astronomy and Geophysics Faculty Publications by an authorized administrator of Digital Commons @ Connecticut College. For more information, please contact [email protected]. The views expressed in this paper are solely those of the author. PHYSICAL REVIEW D 99, 045003 (2019)

Constraints and degrees of freedom in Lorentz-violating field theories

Michael D. Seifert* Department of Physics, Astronomy, and Geophysics, Connecticut College, 270 Mohegan Ave., New London, CT 06320, USA

(Received 10 December 2018; published 7 February 2019)

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 nonzero 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” 1 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.

DOI: 10.1103/PhysRevD.99.045003

I. INTRODUCTION transformation properties under the Lorentz group. The vacuum state of such a model is often said to be “Lorentz- Many classical field theories are constructed in such a violating”, though it would be more accurate to say that way that the “most natural” solutions to the equations of Lorentz symmetry is spontaneously broken in the model. motion involve a nonzero field value. This paradigm, where Examples of such models can be found in [9–14], as well as an underlying symmetry of the Lagrangian or Hamiltonian a particularly early example in [15]. is spontaneously broken by the solutions of the equations In general, all of these models (Lorentz-violating or of motion, has proven to be both compelling and fruitful otherwise) have the property that the fields, collectively over the years. In the context of particle physics, the best- denoted by ψ α, will satisfy at least one equation of the form known example is the Higgs field [1–4]; in the context fðψ αÞ¼0 in the “vacuum”, appropriately defined. Here, f of condensed-matter physics, this paradigm underlies the is a real-valued function of the fields; thus, if the fields ψ α modern theory of phase transitions, most notably the are specified by N real numbers, the equation fðψ αÞ¼0 Ginzburg-Landau theory of superconductivity [5].Many will generically specify an (N − 1)-dimensional hypersur- nonlinear sigma models can also be thought of in this way, if face in field space. I will therefore refer to this function as one view the model’s target manifold as being embedded in the vacuum manifold function. some higher-dimensional space in which the fields are forced Two broad classes of models in which some fields have to a nonzero value. Such models have been used in the study nontrivial background values can readily be conceived of. of both particle physics [6,7] and in ferromagnetism [8]. In one class, the constraint fðψ αÞ¼0 is enforced exactly In more recent years, this paradigm has also been used via the introduction of a Lagrange multiplier λ in the to study possible observational signatures of Lorentz Lagrange density, symmetry violation. Such models include a new vector or tensor field and have equations of motion that are L ∇ψ α ∇ψ α λ ψ α satisfied when the metric is flat and the new tensor field is LM ¼ð Þð Þþ fð Þ; ð1Þ constant but nonzero. In this sense, Lorentz symmetry is spontaneously broken in these theories, as the tensor field where we have denoted the kinetic terms for the fields ψ α takes on a vacuum expectation value that has nontrivial schematically. I will call such models Lagrange-multiplier (LM) models. ψ α * In the other class, the fields are assigned a potential [email protected] energy that is minimized when the field is nonzero. In ψ α ∝ 2 ψ α Published by the American Physical Society under the terms of particular, if we define a potential Vð Þ f ð Þ and the Creative Commons Attribution 4.0 International license. write down a Lagrange density, Further distribution of this work must maintain attribution to ’ the author(s) and the published article s title, journal citation, L ∇ψ α ∇ψ α − ψ α and DOI. Funded by SCOAP3. P ¼ð Þð Þ Vð Þ; ð2Þ

2470-0010=2019=99(4)=045003(12) 045003-1 Published by the American Physical Society MICHAEL D. SEIFERT PHYS. REV. D 99, 045003 (2019) then the lowest-energy state of the theory would be heavily from the later work of Isenberg and Nester [18]. expected to occur when fðψ αÞ¼0 and ∇ψ α ¼ 0. I will I will briefly summarize the method here, and then illustrate call such models potential models. it in more detail via the example theories described in A natural question then arises: for a given collection of Sec. III. fields ψ α and a given kinetic term ð∇ψ αÞð∇ψ αÞ, what is the The method of Dirac-Bergmann analysis involves the relationship between the models (1) and (2)? In particular, construction of a Hamiltonian which generates the time- one might expect the following statement to be true: evolution of the system. In the process of this construction, Conjecture: In a potential model such as (2), the fields one may need to introduce constraints among various can take on any value in the N-dimensional field space. By variables, thereby reducing the number of d.o.f. of the contrast, in a Lagrange-multiplier model, the fields are system. One may also discover that the evolution of certain constrained to an (N − 1)-dimensional subspace of field field combinations is undetermined by the equations of space. Thus, the number of degrees of freedom (d.o.f.) of a motion (e.g., gauge d.o.f.). These combinations of fields, Lagrange-multiplier model (1) should be one fewer than the which we will collectively call “gauges”, must be inter- d.o.f. of the corresponding potential model (2). preted as unphysical, again reducing the number of The main purpose of this article is to show that this naïve physical d.o.f. of the model. α conjecture is not true in general; in particular, if ψ includes If, as is usual, we count a field d.o.f. as a pair of real- a spacetime vector or tensor field, it may be false. In such valued functions (e.g., a field value and its conjugate models, the fields may need to satisfy certain constraints momentum) that can be freely specified on an initial data due to the structure of the kinetic terms; adding a new surface, then the number of d.o.f. Ndof can be inferred quite “constraint” to such theories, in the form of a Lagrange simply once the above analysis is complete, multiplier, does not automatically reduce the number of d.o.f. of the theory. 1 no:of no:of no:of To demonstrate this, I will analyze the d.o.f. of two types N ¼ þ − dof 2 fields momenta constraints of symmetry-breaking models. After a brief description of Dirac-Bergmann analysis in Sec. II, I will first analyze no:of − : ð3Þ a multiplet of Lorentz scalar fields with an internal gauges symmetry, followed by a vector field (Secs. III and IV respectively.) For the vector fields, the analysis will depend In general, of course, the number of fields and the number on the structure of the kinetic term chosen, and so three of conjugate momenta will be the same. Moreover, in distinct sub-cases will need to be treated. In each case, I will the particular field theories I will be considering in this examine a model where the fields are assigned a potential work, I will not find any “gauges”, so the last term in (3) energy, and one where the fields are directly constrained via will vanish. Thus, for my purposes, the above equation a Lagrange multiplier. reduces to The bulk of the explicit analysis in this work will be done in the context of a fixed, flat background spacetime; 1 however, I will briefly discuss these models in the context no:of − no:of Ndof ¼ : ð4Þ of a dynamical curved spacetime in Sec. V. In that section, I fields 2 constraints will also discuss the implications of these results for the broader relationship between these two classes of models. III. SCALAR MULTIPLET FIELDS More general tensor fields will be examined in a forth- coming work [16]. The first case we will consider is a multiplet of N Lorentz Throughout this work, I will use units in which scalars: ψ α ¼ ϕα, with α ¼ 1; 2; 3; …;N. We wish to ℏ ¼ c ¼ 1; the metric signature will be ð− þ þþÞ. construct a model where these scalars “naturally” take Roman indices a; b; c; … will be used to denote spacetime on values in some (N − 1)-dimensional hypersurface, tensor indices; i; j; k; … will be used to denote spatial defined by fðϕαÞ¼0 (with f a real-valued function). indices, where necessary. Greek indices α; β; γ; … will be The “potential model” for this field will be derived from used exclusively to denote indices in field space. All the Lagrange density expressions involving repeated indices (either tensor indices or field space indices) can be assumed to obey the 1 L − ∂ ϕα∂aϕα − κ ϕα 2 Einstein summation convention. ¼ 2 a fð Þ 1 ϕ_ α ϕ_ α − ∇⃗ϕα ∇⃗ϕα − κ ϕα 2 II. DIRAC-BERGMANN ANALYSIS ¼ 2 ½ð Þð Þ ð Þ · ð Þ fð Þ ; ð5Þ Our primary tool for finding the number of degrees of each model will be Dirac-Bergmann constraint analysis where κ is a proportionality constant; the LM model for this [17]; my methods and nomenclature below will draw field will be

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1 1 L − ∂ ϕα∂aϕα − λ ϕα H H ϖ πα 2 ∇⃗ϕα 2 λ ϕα ϖ ¼ 2 a fð Þ A ¼ 0 þ uλ ¼ 2 ½ð Þ þð Þ þ fð Þþuλ : 1 ϕ_ α ϕ_ α − ∇⃗ϕα ∇⃗ϕα − λ ϕα ð10Þ ¼ 2 ½ð Þð Þ ð Þ · ð Þ fð Þ; ð6Þ We now need to ensure that this constraint is preserved where λ is a Lagrange multiplier field. In both cases, we by the time-evolution of the system; in otherR words, we have chosen a time coordinate t and performed a 3 þ 1 3 2 must have ϖ_ ¼fϖ;HAg¼0, where HA ≡ HAd x. If decomposition of the tensors; a dot over a quantity (e.g., this Poisson bracket does not vanish identically, this ϕ_ α ) will denote its derivative with respect to this time demand will yield a secondary constraint Ψ1 ¼ 0. The coordinate, while spatial derivatives will be denoted with demand that this constraint be preserved may lead to new ⃗ Ψ 0 Ψ 0 either ∇ or ∂i depending on the expression. secondary constraints 2 ¼ , 3 ¼ , 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 A. Potential model words, if Ψ1 ensures the preservation of a primary con- The d.o.f. for the model (5) are particularly easy to count. straint, it is a “first-order secondary constraint”;ifΨ2 The momenta conjugate to the fields are all well-defined, ensures the preservation of Ψ1,itisa“second-order secondary constraint”; and so on. In this process, it may ∂L πα ϕ_ α occur that the preservation of these constraints allows us to ¼ _ α ¼ : ð7Þ ∂ϕ determine the auxiliary Lagrange multiplier uλ introduced above. The process is continued until all constraints are The Hamiltonian density is therefore known to be automatically conserved or a contradiction is 1 reached. α _ α α α ⃗ α ⃗ α α 2 H0 ¼ π ϕ − L ¼ ½π π þð∇ϕ Þ · ð∇ϕ Þ þ κfðϕ Þ : With this in mind, we derive the secondary constraints 2 for this model. We first have ð8Þ δH 0 ϖ_ ϖ;H − A f ϕα : 11 Nothing further is required here; we have no primary ¼ ¼f Ag¼ δλ ¼ ð Þ ð Þ constraints on the initial data, and the Hamiltonian obtained Ψ ϕα 0 by integrating H0 over space will generate the field Thus, 1 ¼ fð Þ¼ is a secondary constraint. We repeat dynamics. We thus have the N d.o.f. one would expect. this procedure, obtaining another secondary constraint, ∂ ϕα δ _ fð Þ HA β Ψ1 Ψ1;H δβf π ≡ Ψ2; B. Lagrange-multiplier model ¼f Ag¼ ∂ϕβ δπβ ¼ð Þ ð12Þ Counting the d.o.f. for the model (6) requires a bit more β effort. As the kinetic term of (6) is the same as that of (5), where we have defined δβf ¼ δf=δϕ . (Higher derivatives ϕα the momenta conjugate to the scalars are again defined will be defined similarly.) Ψ2 must also be conserved, by (7). The difficulty arises due to the Lagrange multiplier which leads to a third secondary constraint, λ. From the perspective of the model, it is just another field, but its associated momentum vanishes automatically, _ 2 α α β Ψ2 ¼fΨ2;HAg¼−δαf½−∇ ϕ þ λδαfþπ π ðδαβfÞ

∂L ≡ Ψ3: ð13Þ ϖ ¼ ¼ 0 ≡ Φ: ð9Þ ∂λ_ Finally, demanding that Ψ3 be conserved allows us to We thus have a primary constraint, Φ ¼ 0, on this theory. determine the auxiliary Lagrange multiplier uλ; this is α _ α Ψ λ The “base Hamiltonian density” H0 ¼ π ϕ − L must then because 3 is itself dependent on , be modified to obtain the “augmented Hamiltonian den- Ψ_ Ψ sity” HA by adding this primary constraint multiplied by an 3 ¼f 3;HAg 1 auxiliary Lagrange multiplier uλ, ∂Ψ δ ∂Ψ δ ∂Ψ δ 3 HA − 3 HA 3 HA ¼ ∂ϕγ δπγ ∂πγ δϕγ þ ∂λ δϖ : ð14Þ 1Here and throughout, we will need to distinguish between the “real” Lagrange multiplier that appears in the original Lagrangian and the “auxiliary” Lagrange multipliers that are used to construct 2We are playing a bit fast and loose with notation here; in a Hamiltonian for the model. In general, we will only have one Hamiltonian field theory, the Poisson bracket is only rigorously real Lagrange multiplier at a time, which we will denote with λ; defined for a functional with a single real value, not for a field auxiliary Lagrange multipliers will be denoted with the symbol u, which is a function of space. A more rigorous definition of what possibly with subscripts or diacritical marks. we mean by an expression like fϖ;HAg is given in the Appendix.

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When the dust settles, we obtain ⃗ and A (or Ai) for its spatial components. The kinetic term (16) then becomes _ γ 2 α 2 α β Ψ3 ¼ π ½ðδαγfÞð3∇ ϕ − 4λδαfÞþ∇ ðδαfÞþπ π ðδαβγfÞ 1 c1 2 c1 _ 2 _ − ðδαfÞðδαfÞuλ: ð15Þ L _ 2 − ∇⃗ − ⃗ − ⃗ ∇⃗ K ¼ 2 c13ðA0Þ 2 ð A0Þ 2 A c3A · A0 c1 c3 So long as δαf ≠ 0 when f ¼ 0, this allows us to determine i j j i þ ð∂iAjÞð∂ A Þþ ð∂iAjÞð∂ A Þ; ð18Þ the previously unknown auxiliary Lagrange multiplier uλ. 2 2 Thus, the process terminates here. ⃗ Having determined the Hamiltonian and its constraints, where c13 ≡ c1 þ c3. The momenta conjugate to A0 and A we can count the d.o.f. We have N þ 1 fields, namely the are then multiplet ϕα (α ¼ 1; …;N) and the Lagrange multiplier λ, Φ ϖ 0 ∂L one primary constraint ¼ ¼ , and three secondary Π0 K _ Ψ Ψ Ψ 0 ¼ _ ¼ c13A0 ð19Þ constraints, 1 ¼ 2 ¼ 3 ¼ . The single auxiliary ∂A0 Lagrange multiplier is determined, which means that there 1 are no unphysical gauge d.o.f. With N þ fields and four and constraints, the number of d.o.f. of the model (6) is ∂L therefore ⃗ K ⃗_ ⃗ Π ¼ ¼ −c1A − c3∇A0: ð20Þ ⃗_ 1 ∂A 1 − 4 − 1 Ndof ¼ðN þ Þ 2 ð Þ¼N : _ These equations can be inverted to find the velocities A0 ⃗_ Thus, the Lagrange-multiplier theory (6) has one less d.o.f. and A so long as c1 þ c3 ≠ 0 and c1 ≠ 0. If either of these than the corresponding potential theory (5). In this case, the expressions vanishes, (19) and (20) will instead yield conjecture outlined in the Introduction holds true. constraint equations; we will have to handle these cases separately. IV. VECTOR FIELDS For the potential term, meanwhile, our desire for the Lagrangian to be a Lorentz scalar implies that the only We now consider the case of a vector field A which a possible form for the vacuum manifold function is one spontaneously breaks Lorentz symmetry. As the motivation which sets the norm of A to some constant b.For behind these models is usually a spontaneous breaking of a simplicity’s sake, we will therefore choose fðA Þ to be Lorentz symmetry, we want our Lagrange density to be a a of the following form: Lorentz scalar, without any prior geometry specified. In any such Lagrange density, we can identify a set of a fðAaÞ¼A Aa − b; ð21Þ “kinetic” terms LK that depend on the derivatives of Aa. The most general kinetic term that we can write down 3 which is quadratic in the field Aa is where b is a constant. Depending on the sign of b, the “vacuum” manifold will consist of timelike vectors (b<0), a b a 2 b a 0 0 4 Lk ¼ c1∂aAb∂ A þ c2ð∂aA Þ þ c3∂aAb∂ A : ð16Þ spacelike vectors (b> ), or null vectors (b ¼ ). Our potential model will then be However, since 2 2 ⃗2 2 LP ¼ LK − κfðAaÞ ¼ LK − κð−A0 þ A − bÞ ; ð22Þ a 2 b a a b b a ð∂aA Þ ¼ ∂aAb∂ A þ ∂a½A ∂bA − A ∂bA ; ð17Þ where κ is again a proportionality constant; the LM model we can eliminate one of c2 or c3 via an integration by parts. will be We will therefore set c2 ¼ 0 in what follows. The familiar “ ” Maxwell kinetic term L L − λ L − λ − 2 ⃗2 − LM ¼ K fðAaÞ¼ K ð A0 þ A bÞ: ð23Þ 1 ab LK ¼ − FabF ; 4 For compactness, I will denote the four-norm of Aa as 2 A2 −A2 A⃗ 1 ¼ 0 þ ; the norm of the spatial part on its own will with F 2∂ A corresponds to c1 −c3 − . 2 ab ¼ ½a b ¼ ¼ 2 always be denoted by A⃗. We can now perform the usual 3 þ 1 decomposition of the Lagrange density, writing A0 for the t-component of Aa 4As spontaneous symmetry breaking implies a nonzero field value, one would normally exclude the case b ¼ 0 to ensure that 3 This follows the notation of [10], with the coefficient c4 from Aa ¼ 0 is not in the vacuum manifold. However, this is not that reference set equal to zero. necessary for the analysis which follows.

045003-4 CONSTRAINTS AND DEGREES OF FREEDOM IN … PHYS. REV. D 99, 045003 (2019) A. General case: c ≠ 0, c ≠ 0 2 ⃗2 13 1 _ A0 A Ψ2 ¼fΨ2;HAg¼2λ − þ Ξ ≡ Ψ3; ð30Þ c13 c1 1. Potential model L Performing a Legendre transform on P (22) to obtain where the Hamiltonian density, we obtain a base Hamiltonian 1 1 density of Ξ ≡ Π0 2 − Π⃗ ∇⃗ 2 2 ð Þ 2 ð þ c3 A0Þ c13 c1 2 2 1 1 2 c − c 2 − H Π2 − Π⃗ 1 3 ∇⃗ c3 ⃗ ∇⃗Π0 − ∇⃗ Π⃗ c1 c3 ∇2 B ¼ 2 0 2 þ 2 ð A0Þ þ ðA · A0 · Þþ A0 A0 c13 c1 c1 c1c13 c1 c3 c1 − Π⃗ ∇⃗ − ∂ ∂i j − c3 ⃗ ∇⃗ ∇⃗ ⃗ ⃗ ∇2 ⃗ · A0 2 ð iAjÞð A Þ A · ð · AÞþA · A: ð31Þ c1 c1 c3 j i 2 2 − ð∂iAjÞð∂ A ÞþκðA − bÞ : ð24Þ 2 All three of the quantities Ψ1, Ψ2, and Ψ3 must vanish for the model to be consistent. ⃗ Ψ The resulting theory has four fields (A0 and A) and no When we take the Poisson bracket of 3 with HA,we constraints, so the process terminates here, and the base will obtain Hamiltonian is the complete Hamiltonian for the model. 2 ⃗2 Counting the d.o.f., we find that _ A0 A Ψ3 ¼fΞ;HAgþ 2λ − ;HA c13 c1 1 2 4 − 0 4 2 ⃗ Ndof ¼ ð Þ¼ : ð25Þ A0 A 2 ¼fΞ;HAgþ2uλ − c13 c1 1 1 4λ Π0 ⃗ Π⃗ ∇⃗ 2. Lagrange multiplier model þ 2 A0 þ 2 A · ð þ c3 A0Þ : ð32Þ c13 c1 For the Lagrange multiplier model (23),wehavea primary constraint associated with λ, This means that for generic initial data, for which

2 ⃗2 ∂L A0 A ϖ ¼ ¼ 0: ð26Þ ≠ ; ð33Þ ∂λ_ c13 c1 5 We must therefore augment the Hamiltonian with an we can solve (32) for uλ. Thus, the auxiliary Lagrange multiplier uλ is determined via the self-consistency of the auxiliary Lagrange multiplier uλ to enforce this constraint, ⃗ theory. We have five fields (A0, A, and λ), and self- 0 ⃗ ⃗ consistency generates four constraints (one primary, H ¼ Π A0 þ Π · A − L þ uλϖ A LM three secondary), and so the total number of d.o.f. of this 1 1 2 − 2 2 ⃗2 c1 c3 ⃗ 2 c3 ⃗ ⃗ model is ¼ Π0 − Π þ ð∇A0Þ − Π · ∇A0 2c13 2c1 2c1 c1 1 c1 c3 − ∂ ∂i j − ∂ ∂j i N ¼ 5 − ð4Þ¼3: ð34Þ 2 ð iAjÞð A Þ 2 ð iAjÞð A Þ dof 2 λ 2 − ϖ þ ðA bÞþuλ : ð27Þ Again, as expected from the conjecture, we have lost 1 d.o.f. to the Lagrange multiplier. We now find the secondary constraints required for the A related analysis was performed by Garfinkle et al. [19] primary constraint to be conserved under time-evolution. in the case of Einstein-aether theory. In such models, the Taking the Poisson brackets of each constraint with the a vector field is constrained to satisfy AaA ¼ b ¼ −1; i.e., Hamiltonian in turn, we obtain three secondary constraints, the vector field is unit and timelike. In that work, the time component A0 of the vector field was explicitly eliminated 2 ⃗2 ϖ_ ¼fϖ;HAg¼A0 − A þ b ≡ Ψ1; ð28Þ from the Lagrangian after performing a 3 þ 1 decomposition, leaving the components of A⃗as the three dynamical _ Ψ1 ¼fΨ1;HAg 5 1 1 The full expression for fΞ;HAg is complicated and not 0 ⃗ ⃗ ⃗ ¼ 2 A0Π þ A · ðΠ þ c3∇A0Þ ≡ Ψ2; ð29Þ terribly illuminating, so we will not present it here. However, c13 c1 from the form of Ξ and HA, we can see that it will depend on A0, A⃗, Π0, Π⃗, and λ—but it will be independent of both ϖ and (more and importantly) uλ.

045003-5 MICHAEL D. SEIFERT PHYS. REV. D 99, 045003 (2019) fields. It was found in that case that the model did not 1 ⃗2 ⃗ ⃗ HA ¼ − Π þ Π · ∇A0 contain any extra constraints on these three dynamical 2c1 fields, if (assuming c2 ¼ c4 ¼ 0, as we have done here) c1 − ∂ ∂i j − ∂ ∂j i 2 ½ð iAjÞð A Þ ð iAjÞð A Þ c3 ≠ 0 ≤ 0 2 2 0 c1 ; : ð35Þ þ κðA − bÞ þ u0Π : ð41Þ c1

Such models were called “safe” by the authors of [19]; such Once again, we must ensure that the primary constraint a model would be expected to contain the 3 d.o.f. present Φ ¼ Π0 ¼ 0 is conserved by the equations of motion; this ⃗ in A. Models with c1 ¼ 0 were called “endangered”, in that again produces a series of secondary constraints, they contained additional constraints on the initial data; _ 0 ⃗ ⃗ 2 such models would contain fewer than 3 d.o.f. Finally, Π ¼fΠ0;HAg¼∇ · Π þ 4κðA − bÞA0 ≡ Ψ1 ð42Þ models with c1 ≠ 0 and c3=c1 > 0 were called “condition- ” ally endangered , since the constraint structure of the _ Ψ1 ¼fΨ1;H g equations differed at various points in configuration space. A To connect this to the present work, we note that (33) is −8κ ∇⃗ 2 − ⃗ − −3 2 ⃗2 − equivalent to ¼ · ððA bÞAÞ ð A0 þ A bÞu0 1 2 ⃗2 ⃗2 ⃗ ⃗ ⃗ c1A0 − c1A − c3A ≠ 0; ð36Þ þ A0A · Π − ∇A0 : ð43Þ c1 2 ⃗2 or, since c1 ≠ 0 and −A0 þ A ¼ b under the constraint, 2 ⃗2 So long as −3A0 þ A − b ≠ 0, the demand that the c3 2 b ≠− A⃗: ð37Þ secondary constraint Ψ1 be preserved by the evolution c1 determines the auxiliary Lagrange multiplier u0 uniquely. ≤ 0 0 We therefore have four fields, two constraints (one primary, In the case where c3=c1 and b< , this is guaranteed one secondary), and no undetermined Lagrange multi- to hold, and we therefore have no additional constraints pliers; counting the d.o.f. therefore yields and 3 d.o.f. However, if c3=c1 > 0, there can be nongeneric points in configuration space where the number of con- 1 straints changes. 4 − 2 3 Ndof ¼ 2 ð Þ¼ : B. Maxwell case: c =0, c ≠ 0 13 1 2. Lagrange multiplier model 1. Potential model We now apply the same process to the vector model We now consider a vector field A with a “Maxwell” with a Lagrange multiplier, (23). With the addition of the a λ kinetic term, for which c1 ¼ −c3 in (16). We again have Lagrange multiplier , we must also introduce a conjugate ϖ four independent fields, namely the four components of Aa. momentum . As in the scalar LM model, this vanishes In this case, the canonical momentum Π0 defined in (19) identically, yielding a second primary constraint, vanishes automatically, giving us a constraint, ∂L ϖ ¼ ¼ 0 ≡ Φ2: ð44Þ Π0 ¼ 0 ≡ Φ1: ð38Þ ∂λ_

The other three canonical momenta Π⃗defined in (20) have Including this constraint with an auxiliary Lagrange multi- an invertible relationship with the corresponding field plier uλ in the Hamiltonian density gives us the augmented velocities, Hamiltonian density for the model,

⃗ ⃗_ ⃗ 1 2 Π c1 −A ∇A0 : ⃗ ⃗ ⃗ ¼ ð þ Þ ð39Þ HA ¼ − Π þ Π · ∇A0 2c1 H c1 Thus, the base Hamiltonian density B, given by − ∂ ∂i j − ∂ ∂j i 2 ½ð iAjÞð A Þ ð iAjÞð A Þ _ ⃗ ⃗_ λ 2 − Π0 ϖ HB ¼ Π0A0 þ Π · A − L ð40Þ þ ðA bÞþu0 þ uλ : ð45Þ must be augmented by an auxiliary Lagrange multiplier We now derive the secondary constraints and see if their 0 term enforcing the constraint Π ¼ 0. After simplification, time-evolution fixes the auxiliary Lagrange multipliers u0 this yields and uλ,

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Φ_ Π ∇⃗ Π⃗ 2λ ≡ Ψ 0 ⃗ ⃗_ ⃗ ⃗ 1 ¼f 0;HAg¼ · þ A0 1 ð46Þ HA ¼ Π A0 þ Π · A − LP þ u⃗· ðΠ þ c3∇A0Þ 1 c3 Π0 2 − ∂ ∂j _ 2 ¼ ð Þ ð iAjÞð AiÞ Φ2 ¼fϖ;HAg¼−A þ b ≡ Ψ2 ð47Þ 2c3 2 2 2 ⃗ ⃗ þ κðA − bÞ þ u⃗· ðΠ þ c3∇A0Þ; ð51Þ _ ⃗ ⃗ Ψ1 ¼fΨ1;HAg¼−∇ · ðλAÞþ2A0uλ þ 2λu0 ð48Þ where u⃗ is a vector of auxiliary Lagrange multipliers enforcing the primary constraints (50). Enforcing these _ ⃗ ⃗ ⃗ Ψ2 ¼fΨ2;HAg¼2A0u0 − 2A · ðΠ þ ∇A0Þ: ð49Þ primary constraints under time evolution then yields a set of three secondary constraints Ψ⃗, Assuming A0 ≠ 0, the requirement that both (48) and (49) _⃗ ⃗ ⃗ 0 ⃗ ⃗ ⃗ 2 ⃗ ⃗ vanish determines the auxiliary Lagrange multipliers uλ Φ ¼fΦ;HAg¼∇Π −c3∇ð∇ · AÞ − 4κðA − bÞA ≡Ψ: and u0. The d.o.f. counting is therefore five fields (four ð52Þ components of Aa, plus λ), four constraints (two primary, two secondary), and no undetermined auxiliary Lagrange The time-evolution of these secondary constraints, written multipliers, for a result of out in terms of spatial components, is then

1 _ 2 5 − 4 3 Ψ ¼fΨ ;H g¼M u − 4κ∂ ððA − bÞA0Þ Ndof ¼ ð Þ¼ : i i A ij j i 2 8κ 0 ⃗ ⃗ þ A0AiðΠ þ ∇ · AÞ; ð53Þ This is a surprising result: the number of d.o.f. of the c3 theory when the vector field is “constrained” to a vacuum where manifold determined by fðAaÞ¼0 is exactly the same as when it is “allowed” to leave this vacuum manifold. In 2 Mij ≡ 4κ½δijðA − bÞþ2AiAj: ð54Þ other words, the Lagrange-multiplier “constraint” does not _ actually reduce the d.o.f. of the model. We require that Ψi ¼ 0. This can be guaranteed in Eq. (53) We can again connect this model to the terminology of via an appropriate choice of u so long as the matrix M is “ ” j ij [19], as we did for the general LM case in Sec. IVA 2.In invertible. For general field values, this inverse can be ≠ 0 that work, for a model of a timelike vector field with c1 calculated to be 0 −1 and c13 ¼ (i.e., c3=c1 ¼ ), the number of constraints was found to be zero for all points in configuration space; 1 2A A M−1 δ − i j such a model was therefore “safe”, with 3 d.o.f. at all points ð Þij ¼ 2 ij 2 ; ð55Þ 4κðA − bÞ 1 þ 2A⃗ in field space. This is in agreement with our work here: so long as the vector A is constrained to be timelike (b<0), _ a Ψ⃗ 0 ⃗ we will always have A0 ≠ 0, and the above analysis holds. and so we can solve the equation ¼ for u so long as 2 6 However, the work here also implies that a version of the A − b ≠ 0. The generic theory therefore has four fields, six Einstein-aether theory with a spacelike vector field would constraints (three primary, three secondary) and “ ” only be conditionally safe , since it is possible for A0 to 1 vanish. 4 − 6 1 Ndof ¼ 2 ð Þ¼ ð56Þ

C. V-field case: c13 ≠ 0, c1 =0 degree of freedom.

1. Potential model 2. Lagrange multiplier model In this case, we have a Lagrange density with c1 ¼ 0 and As in Sec. IV B 2, the switch from a potential V-field c3 ≠ 0; such a field is called a “V-field” by Isenberg model to a Lagrange-multiplier V-field model does not and Nester [18]. When we calculate the conjugate momenta actually “kill off” any d.o.f. The augmented Hamiltonian _ in this case, (19) allows us to solve for the velocity A0 ¼ density now contains one more auxiliary Lagrange multi- 0 Π =c3,but(20) becomes a set of three constraints, plier uλ, which (as before) enforces the constraint ϖ ¼ 0,

⃗ ⃗ ⃗ 6 Φ ≡ Π þ c3∇A0 ¼ 0: ð50Þ 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 The augmented Hamiltonian density for the potential multipliers and will be discussed more extensively in an upcom- model (22) is then ing work [16].

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1 c3 H Π0 2 − ∂ ∂j λ 2 − V. DISCUSSION A ¼ 2 ð Þ 2 ð iAjÞð AiÞþ ðA bÞ c3 A. Generalization ⃗ ⃗ þ u⃗· ðΠ þ c3∇A0Þþuλϖ: ð57Þ We have found that a field theory model in flat spacetime may or may not “lose” a d.o.f. when a constraint is added to We thus have four primary constraints, Φ⃗¼ 0 [as defined in the system via a Lagrange multiplier. Specifically, scalar _ models (Sec. III) and general vector models (Sec. IVA) lose (50)] and ϖ ¼ 0. Requiring that Φ⃗¼ 0 and ϖ_ ¼ 0 then a d.o.f. when we replace a potential with a Lagrange Ψ⃗ yields four secondary constraints, which I will denote by multiplier; but Maxwell-type and V-type vector models Ψ and , (Secs. IV B and IV C, respectively) retain the same number _ of d.o.f. regardless of whether the field values are governed Φ⃗ Φ⃗ ∇⃗Π0 − ∇⃗ ∇⃗ ⃗ − 2λ⃗≡ Ψ⃗ ¼f ;HAg¼ c3 ð · AÞ A ; ð58Þ by a potential or by a Lagrange multiplier. There is an obvious difference between these cases. In ϖ_ ϖ 2 − ≡ Ψ ¼f ;HAg¼A b : ð59Þ those models where there are no primary constraints in the potential model, a Lagrange multiplier eliminates a d.o.f. The time-evolution of these secondary constraints is then In contrast, in the models where the potential model does contain primary constraints, the field theory retains the ⃗_ ⃗ ⃗ ⃗ Ψ ¼fΨ;HAg¼2½∇ðλA0Þ − uλA þ λu⃗ð60Þ same number of d.o.f. when a constraint is imposed via a Lagrange multiplier. 1 _ ⃗ 0 The reason for this difference can be traced to a particular Ψ ¼fΨ;HAg¼A · u⃗− A0Π : ð61Þ c3 feature of the models we have examined. In those models containing primary constraints, the conservation of the These equations determine all four auxiliary Lagrange first-order secondary constraints leads to an equation that ⃗ multipliers u⃗and uλ so long as A ≠ 0 and λ ≠ 0; in this determines the auxiliary Lagrange multiplier uλ [Eqs. (48) case, we have and (60) for the Maxwell-like and V-field models, respectively]. In those models without primary constraints, uλ is 1 1 ⃗ ∇⃗ λ λ Π0 only determined once we require that higher-order secon- uλ ¼ ⃗2 A · ð A0Þþ A0 ð62Þ A c3 dary constraints [specifically, the third-order secondary constraints in Eqs. (13) and (30)] be conserved. and To extend this to a general statement, we first note that 1 the primary constraints for a potential model and its ⃗ ⃗− ∇⃗ λ u ¼ λ ½uλA ð A0Þ: ð63Þ corresponding Lagrange multiplier model are simply related. If the primary constraints for the potential model Thus, for generic initial data, we are done. We have five are a set of M functions fΦ1; …; ΦMg, then the primary fields, four primary constraints, and four secondary con- constraints for the corresponding Lagrange multiplier straints, and so the number of d.o.f. is model will simply be fΦ1; …; ΦM; ϖg, where ϖ is the λ 1 conjugate momentum to the Lagrange multiplier . 5 − 8 1 Moreover, ϖ will commute with all of the primary Ndof ¼ 2 ð Þ¼ : ð64Þ constraints that derive from the potential model, since As for the Maxwellian vector theory in Sec. IV B 2, the none of these constraints depend on λ. addition of a Lagrange multiplier to a V-field model does The augmented Hamiltonian density will then be the not reduce its d.o.f. base Hamiltonian density with terms added to impose the This analysis is again in agreement with the work of constraints, Garfinkle et al. [19]. For a model with c1 ¼ 0, they find that H H Φ ϖ the Einstein-aether theory contains additional initial data A ¼ 0 þ uI I þ uλ : ð65Þ constraints on the three dynamical fields A⃗and is therefore “endangered”. In the present work, we have confirmed this (Here and in what follows, repeated capitalized Roman result: this model does indeed contain fewer than 3 d.o.f.7 indices are summed from 1 to M.) The first-order secondary constraint required in order to maintain ϖ ¼ 0 under time evolution will then be 7The cases A⃗¼ 0 and λ ¼ 0 were excluded from the above analysis. In this case, one would have to look at the time evolution δHA α of the quantities in (60) and (61), generate one or more second- Ψλ ¼fϖ;HAg¼− ¼ fðψ Þ; ð66Þ order secondary constraints, and attempt to solve these for the δλ auxiliary Lagrange multipliers. In any event, this would generate α a model with no more than 1 d.o.f. (if the resulting model was where ψ here stands for the collection of fields in the even consistent at such points in configuration space). model. In addition, there will be a set of first-order

045003-8 CONSTRAINTS AND DEGREES OF FREEDOM IN … PHYS. REV. D 99, 045003 (2019) secondary constraints ΨI (I ¼ 1; …;M), each derived from Note that this result implies that a model that is “already _ ” the requirement that ΦI ¼ 0; these are given by constrained by primary constraints from its kinetic term could potentially be “further constrained” by the introduc-

ΨI ¼fΦI;HAg: ð67Þ tion of a Lagrange multiplier. This did not occur in any of the vector models from Sec. IV, as all of those models ψ α Φ ≠ 0 Now consider the time-evolution of the first-order containing primary constraints also had ffð Þ; Ig . “ ” secondary constraints. The time derivative of Ψλ will be Examples of already-constrained models in which a ’ independent of uλ, though it will generally depend on the Lagrange multiplier further reduces the model s d.o.f. other auxiliary Lagrange multipliers uI, can be found in the next section.

_ α Ψλ ⊃ uJfΨλ; ΦJgþuλfΨλ; ϖg¼uJffðψ Þ; ΦJg: ð68Þ B. Lagrange-multiplier models in dynamical spacetimes The number of d.o.f. of a field theory in flat spacetime is α α Note that ffðψ Þ; ϖg¼0 since fðψ Þ is independent of λ. not always simply related to the number of d.o.f. it _ Meanwhile, the expression ΨI (for arbitrary I) will contain possesses in a curved, dynamical spacetime. It is well- terms of the form known that diffeomorphism-invariant field theories have primary constraints corresponding to the nondynamical _ ΨI ¼fΨI;HAg ⊃ uJfΨI; ΦJgþuλfΨI; ϖg: ð69Þ nature of the lapse and shift functions; when we pass to a dynamical spacetime, we both introduce new fields (the ten Equations (68) and (69) together imply that if metric components) as well as new constraints.9 Perhaps less well-known, but equally important, is that d.o.f. which fΨI; ϖg¼0; ð70Þ are unphysical (gauge or constraint) in flat spacetime can become “activated” in a minimally coupled curved- then the equations for conservation of the constraints spacetime theory [18]. This occurs due to the fact that the _ _ (ΨI ¼ 0 and Ψλ ¼ 0) do not contain uλ, leaving this covariant derivative of a tensor field (unlike that of a scalar) auxiliary Lagrange multiplier undetermined at this order. depends on the derivatives of the metric. The “minimally If this occurs, then we must proceed to find additional coupled” kinetic term for a tensor field therefore contains second- and higher-order secondary constraints. Since we couplings between the metric derivatives and the tensor have more than two additional constraints, but only one field derivatives, which can turn equations that were additional d.o.f. from λ itself, we conclude that in such constraints or gauge d.o.f. in flat spacetime into dynamical cases, the Lagrange-multiplier model will have fewer d.o.f. equations in curved spacetime, and vice versa. than the potential model.8 In light of these facts, we might then ask how much of This condition (70) can be greatly elucidated via use of the above analysis would carry over to dynamical space- the Jacobi identity. Specifically, we have times. Given the critical role played by the constraints in this analysis, it is natural to ask whether a Lagrange- ffΦI;HAg; ϖgþffHA; ϖg; ΦIgþffϖ; ΦIg;HAg¼0 multiplier model in a dynamical curved spacetime would lose any d.o.f. relative to the corresponding potential model ð71Þ in a dynamical curved spacetime. The condition (73) sheds some light on this question. for any primary constraint Φ . Since ϖ commutes with the I We know that if uλ remains undetermined when we require rest of these primary constraints, the last term automatically conservation of the first-order secondary constraints, then vanishes; and applying (66) and (67) yields the equation we will in general have to find higher-order secondary Ψ ϖ − ψ α Φ constraints, leading to a reduction of the d.o.f. of the theory f I; g¼ ffð Þ; Ig: ð72Þ relative to the corresponding potential model. This will occur when the vacuum manifold function fðψ αÞ com- Thus, the Eq. (69) will leave uλ undetermined, and the mutes with the primary constraints of the theory. Lagrange multiplier will reduce the d.o.f. of the model, Any diffeomorphism-invariant theory, when decom- so long as posed into 3 þ 1 form, will contain terms involving the a α lapse N and shift N ; these are related to the spacetime ffðψ Þ; Φ g¼0; ð73Þ I metric gab and the induced spatial metric hab by f ψ α i.e., the vacuum manifold function ð Þ commutes with 1 all the primary constraints. gab ¼ hab − ðta − NaÞðtb − NbÞ; ð74Þ N2

8 It is also conceivable that uλ could remain undetermined even after the process of finding the constraints is completed. This 9See [20,21] for a detailed description of the Hamiltonian would also reduce the number of d.o.f. in the final counting. formulation of general relativity.

045003-9 MICHAEL D. SEIFERT PHYS. REV. D 99, 045003 (2019) where ta is the vector field we have chosen to correspond one would conclude that the number of d.o.f. of a general to “time flow” in our decomposition. We can then write vector theory in curved spacetime would not be reduced by down the Einstein-Hilbert action in terms of this induced the presence of a Lagrange multiplier, in contrast to the metric, the lapse, and the shift. As the lapse and shift can be situation in flat spacetime. In fact, this is confirmed by arbitrarily specified, the are effectively “gauge quantities” known results. A model consisting of a vector field in a corresponding to diffeomorphism invariance; thus, their curved spacetime with a “generic” kinetic term (as in time derivatives do not appear in the Lagrange density of Sec. IVA) will contain 2 “metric” d.o.f. and three “vector” the theory when it is decomposed. In the Dirac-Bergmann d.o.f., regardless of whether the vector is forced to a nonzero formalism, there are therefore primary constraints on the expectation value by a Lagrange multiplier [22] or by a momenta conjugate to these quantities, potential [18,23]. ∂L ∂L Π ≡ ¼ 0; Πa ≡ ¼ 0: ð75Þ C. Potential models in the low-energy limit ∂N_ ∂N_ a In classical particle mechanics, it is common to think of a The question is then whether the vacuum manifold function constrained system in relation to an unconstrained system α fðψ Þ commutes with these primary constraints. But this is with a potential energy. In the limit where the potential easy enough to see, since energy becomes infinitely strong, it can be shown that the dynamics of the unconstrained system reduce to those of a δf δf f ψ α ; Π ; f ψ α ; Π : 76 system constrained to lie only in the minimum of the f ð Þ g¼δ f ð Þ ag¼δ a ð Þ N N potential [24]. It is therefore common, in the analysis Thus, the question of whether the Lagrange multiplier of constrained systems, to simply include one or more reduces the number of constraints is reduced to the question Lagrange multipliers that enforce the constraints. In gen- of whether the vacuum manifold function depends on the eral, each Lagrange multiplier reduces the number of d.o.f. lapse and shift. In particular, for a collection of scalar fields of the system by one. in curved spacetime (the dynamical-spacetime analogue of One might think that this general picture could be carried Sec. III), the vacuum manifold function will be independent over to field theory. In particular, a set of fields in a of the metric, and so there is no way for the lapse or shift potential could be thought of as possessing a certain functions to enter into it. We would therefore expect that a number of massive modes (corresponding to oscillations Lagrange-multiplier model containing N scalars would in field-space directions in which the potential increases) have fewer than N d.o.f. attributable to the scalars.10 and a certain number of massless modes (oscillations in However, for a function of a vector field Aa, the norm of field-space directions in which the potential is flat). One the vector field Aa will depend on the lapse and shift could then construct a low-energy effective field theory in functions, which the massive modes have “frozen out”, reducing the number of d.o.f. of the model. In this low-energy limit, one 2 ððta − NaÞA Þ would expect the fields to always lie in their vacuum A A gab ¼ A habA − a a b a b N2 manifold, effectively being constrained there. Hence, one A − NaA⊥ 2 would think that the Lagrange-multiplier version of a ⊥ ab ⊥ − ð t a Þ ¼ Aa h A 2 ; ð77Þ potential theory would nicely correspond to the low-energy b N behavior of the corresponding potential theory. a ⊥ b where At ¼ t Aa and Aa ¼ ha Ab. Any function of the The results of this work, however, show that the picture spacetime norm of Aa will therefore depend on the lapse is not so simple. While this simple picture holds for scalar and shift, and so the vacuum manifold function will not fields in flat spacetime, it seems quite unlikely that the low- commute with the primary constraints of the theory. Given energy limit of a Maxwell-type or V-type vector field in a the results stated above, it seems unlikely that the Lagrange potential would correspond to a model with the same multiplier would reduce the number of d.o.f. of such a kinetic term but containing a Lagrange multiplier. One theory. would expect the low-energy limit to have fewer d.o.f. than It is interesting to note that this coupling occurs even the full potential model, but in these cases, the Lagrange- if the flat-spacetime theory does not contain any primary multiplier models and the corresponding potential models constraints, as in the general vector models described have the same number of d.o.f. While the low-energy limit in Sec. IVA. Since the conservation of the first-order of some such models has been investigated [23,25,26], the secondary constraints determines the auxiliary multiplier uλ Lagrange-multiplier models would necessarily have a in this case, rather than giving rise to further constraints, different behavior. In fact, this feature was noted in [23] in the context of a “ ” 10As there is no coupling between the kinetic terms of the vector field with a Maxwell kinetic term. In Sec. IV C of scalar and the metric, it seems likely that there would also still be that work, it was noted that the Lagrange-multiplier model 2 d.o.f. attributable to the metric itself. only corresponded to the low-energy (“infinite-mass”) limit

045003-10 CONSTRAINTS AND DEGREES OF FREEDOM IN … PHYS. REV. D 99, 045003 (2019) of the potential model if the Lagrange multiplier λ was set to where πα is the conjugate field momentum to ψ α (and a zero by fiat. However, for a generic solution λ will not summation over α is implied), and the functional deriva- vanish; the vanishing of the secondary constraint in Eq. (46) tives are implicitly defined via the relation ⃗ ⃗ requires that λ ¼ −∇ · Π=2A0. In other words, one must Z restrict the class of solutions under consideration—i.e., δ — δ 3 G δψα further reduce the number of d.o.f. to obtain the low- G ¼ d z δψ α ðzÞ: ðA3Þ energy limit of a potential model from the corresponding ðzÞ Lagrange-multiplier model. This work shows that this lack of direct correspondence is a common feature of models in To extend the definition (A2) of a Poisson bracket to a which tensor fields take on a vacuum expectation value. local field quantity fðψ αðxÞ; ∇ψ αðxÞ; …Þ constructed from field quantities at a fixed point x, one introduces the ACKNOWLEDGMENTS functional I would like to thank D. Garfinkle, T. Jacobson, and J. Tasson for helpful discussion and correspondence on this Z 3 α α 3 subject. This research was supported in part by Perimeter Fx ≡ d y½fðψ ðyÞ; ∇ψ ðyÞ; …Þδ ðx − yÞ: ðA4Þ Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the The functional derivatives in (A2) then become Ministry of Economic Development & Innovation. δ ∂ ∂ APPENDIX: POISSON BRACKETS Fx f 3 f 3 α ¼ α δ ðx − zÞ − ∇a α δ ðx − zÞ AND FUNCTIONALS δψ ðzÞ ∂ψ ∂ð∇aψ Þ In calculating the time-evolution of a field quantity in þ … ðA5Þ Hamiltonian field theory, one would like to take the Poisson bracket of a field ψ αðxÞ with the Hamiltonian H to find the and similarly for πα, where the ellipses stand for higher- time-evolution of the field at x, order derivatives of ψ α (or πα), and the partial derivatives of α α f (and their gradients) are evaluated at the point z. Here and ψ_ ðxÞ¼fψ ðxÞ;Hg: ðA1Þ throughout, I will use partial derivatives ∂ to denote the However, one does have to be careful with this notation, as variation of a locally constructed field quantity with respect δ the Poisson bracket is only rigorously defined on real- to one of its arguments, while the notation will be valued field functionals, not on functions of space like ψ α. reserved for functional derivatives. Specifically, we have Under this extension, the Poisson bracket of a local field f ψ α x ; ∇ψ α x ; … H Rquantity ð ð Þ ð Þ Þ with the Hamiltonian ¼ Z H 3 “ ” 3 δG1 δG2 δG1 δG2 d x is really the Poisson bracket of the functional Fx fG1;G2g ≡ d z − ; δψ αðzÞ δπαðzÞ δπαðzÞ δψ αðzÞ with H. Restricting attention to quantities that only depend on the fields ψ α and πα and their first derivatives, this ðA2Þ Poisson bracket is

d α α α α ½fðψ ;∇ψ ;π ;∇π Þ¼fFx;Hg dt Z ∂f ∂f δH ¼ d3z δ3ðx−zÞ−∇ δ3ðx−zÞ ∂ψ α a ∂ð∇ ψ αÞ δπαðzÞ a ∂f 3 ∂f 3 δH − α δ ðx−zÞ−∇a α δ ðx−zÞ α ðA6Þ ∂π ∂ð∇aπ Þ δψ ðzÞ ∂f δH ∂f δH ∂f δH ∂f δH ¼ α α þ α ∇a α − α α − α ∇a α ; ðA7Þ ∂ψ δπ ðxÞ ∂ð∇aψ Þ δπ ðxÞ ∂π δψ ðxÞ ∂ð∇aπ Þ δψ ðxÞ where all the field quantities are now evaluated at x.

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Note that this definition implies that time-evolution This fact simplifies the calculation of the Poisson brackets “commutes” with spatial derivatives when we take the considerably. Poisson bracket, as one would expect. For example, This definition can be extended straightforwardly to R α α suppose that f ¼ ψ α and H ¼ Hd3x, where H is locally quantities depending on higher derivatives of ψ and π , constructed from the fields. Then we have and the above-mentioned commutativity extends to such cases as well. It can also be extended to the Poisson δH ∂H brackets of two local field quantities fðxÞ and gðyÞ by fψ α;Hg¼ ¼ ; ðA8Þ δπαðxÞ ∂πα defining functionals Fx and Gy and following the same procedure. In such cases, the resulting Poisson bracket will α δ3 − evaluated at x. Meanwhile, if f ¼ ∇aψ ,wehave contain a factor of ðx yÞ. However, in the interests of clarity, we will elide these factors when we take the Poisson δH ∂H bracket of two such quantities; in other words, we will take f∇ ψ α;Hg¼∇ ¼ ∇ a a δπαðxÞ a ∂πα it as understood that the first argument of such a Poisson bracket is evaluated at x, the second at y, and that the result ∇ ψ α 3 ¼ aðf ;HgÞ: ðA9Þ is multiplied by δ ðx − yÞ or its derivatives.

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