Singular Hamiltonians in Models with Spontaneous Lorentz Symmetry Breaking

PHYSICAL REVIEW D 100, 065017 (2019)

Singular Hamiltonians in models with spontaneous Lorentz symmetry breaking

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

(Received 15 May 2019; published 23 September 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. One common way to construct such a model is to posit a smooth potential for this field; the natural low-energy solution of such a model would then be excepted to have the tensor field near the minimum of its potential. It is shown in this work that some such models, while appearing well posed at the level of the Lagrangian, have a Hamiltonian which is singular on the vacuum manifold and are therefore ill posed. I illustrate this pathology for an antisymmetric rank-2 tensor field and find sufficient conditions under which this pathology occurs for more general field theories.

DOI: 10.1103/PhysRevD.100.065017

I. INTRODUCTION Many (though not all) of the above-cited models share The prospect of finding new physics via Lorentz sym- two features. First, they accomplish the spontaneous break- metry violation has been of significant interest over the past ing of Lorentz symmetry by assigning a potential energy couple of decades. In many such models, Lorentz sym- VðΨÞ to some Lorentz tensor Ψ. This potential is metry is broken spontaneously; one postulates the existence constructed in a “Higgs-like” way, with a set of minima of a new fundamental field that is not a Lorentz scalar and forming a vacuum manifold in field space. Since the field assigns dynamics to this field that obey Lorentz symmetry Ψ is a Lorentz tensor but we want the underlying but lead it to take on a nonzero “vacuum expectation value.” equations of motion to obey Lorentz symmetry, we have The existence of this nonzero Lorentz vector or tensor field to construct the potential out of one or more Lorentz scalars then provides a preferred geometric structure in spacetime. that are dependent on Ψ. The vacuum manifold will be In effect, such a field would provide a “hook” upon determined by a set of conditions on these Lorentz scalars. which one can “hang” frame-dependent effects. In the For example, if we wish to construct a model in which a presence of a Yukawa-like couplings between this new field Lorentz vector field Aa spontaneously breaks Lorentz and conventional matter fields, the results of experiments symmetry, it is not hard to see that the potential VðAaÞ would depend on the relative orientation in spacetime of the must be some function of the 4-vector norm A Aa, and the ’ a observer s 4-velocity and the new field, leading to frame- vacuum manifold will be the set of all 4-vectors of a dependent effects. Such a field is frequently called a particular norm. “ ” Lorentz-violating field, though this is something of a Second, the kinetic terms in the Lagrangians for these misnomer; the postulated field would still transform theories must be constructed with care. Assuming that the between frames via the standard Lorentz transformation underlying Lagrangian is second order and gives rise to laws. The dynamics of such fields in flat spacetime have second-order equations of motion, the kinetic terms will been studied in their own right [1–4] and similar models typically involve a contraction of the spacetime derivative have also been developed in the context of curved space- ∇ Ψ with itself. However, one cannot usually simply – a b1bn time as possible modifications to general relativity [5 7]. ∇ Ψ ∇aΨb1bn write down a kinetic term of the form a b1bn and call it a day. Since the Minkowski metric is indefinite, such a kinetic term will lead to terms with the “wrong sign” *[email protected] of the kinetic energy in the Hamiltonian, which will generically lead to instabilities in the classical solutions. Published by the American Physical Society under the terms of Instead, the kinetic term is usually constructed out of more the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to complicated combinations of the field derivatives, such that the author(s) and the published article’s title, journal citation, the problematic terms in the kinetic energy do not appear and DOI. Funded by SCOAP3. (via cancellations between various contractions). Again,

2470-0010=2019=100(6)=065017(12) 065017-1 Published by the American Physical Society MICHAEL D. SEIFERT PHYS. REV. D 100, 065017 (2019) returning to the case of a 4-vector field for illustration, it is II. HAMILTONIAN FIELD THEORY 1 a b not hard to see that a kinetic term of the form 2 ∇aAb∇ A 2 In classical mechanics, the construction of a Hamiltonian 1 _ 2 _⃗ _ will include terms of the form 2 ½A0 − A , which is from a Lagrangian Lðqα; qαÞ is a relatively straightforward indefinite. However, the time derivatives that arise from process. One performs a Legendre transform on the “Maxwell” kinetic term − 1 F Fab (with F ¼ 2∇ A ) Lagrangian, by defining the conjugate momenta pα ¼ 4 ab ab ½a b ∂ ∂ _ are all of the same sign, evading this problem. L= qα; inverting these relationships to write the velocities _ The price one pays for this good behavior, however, is qα in terms of the coordinates qα and the momenta pα; that the time derivatives of some fields do not appear at all and finally writing the Hamiltonian as Hðqα;pαÞ¼ _ − _ _ in the Lagrangian. This leads to the model being con- pαqα Lðqα; qαÞ, now viewing qα as a function of qα strained; certain canonical field momenta must vanish and pα. One can then find the evolution of the coordinates ’ identically, and so one cannot freely specify both the and momenta via Hamilton s equations, or via the Poisson values of the fields and their time derivatives at some bracket, initial moment t0. df The purpose of this work is to illustrate and explore ¼ff; Hgð1Þ a potential incompatibility between these two features. dt Specifically, a model which contains a constrained field where for any quantities f and g we have with a vacuum manifold, while appearing well posed at the level of the Lagrangian, may in fact have a Hamiltonian that X ∂f ∂g ∂g ∂f becomes singular on the vacuum manifold. This raises ff; gg ≡ − : ð2Þ ∂qα ∂pα ∂qα ∂pα serious doubts about the viability of such a theory; it α predicts that the field, if perturbed slightly from its vacuum In field theory for some set of fields ψ α, one would like manifold, could not smoothly evolve back to the vacuum to follow the same procedure, starting from a Lagrange manifold. This conflict does not occur in all Lorentz- density Lðψ α; ψ_ α; ∇ ψ αÞ. However, an important difference violating field models but instead seems to arise when the i “ ” can arise when one attempts to perform the Legendre number of constraints exceeds the codimension of the transform. It can happen that, due to the structure of the vacuum manifold in field space. kinetic terms, the field velocities cannot all be written in The work is structured as follows. A brief summary of terms of the field momenta. In general, this implies that one the techniques of Hamiltonian field theory pertinent to or more of the equations of motion for the field are actually this result is presented in Sec. II. Section III contains an constraint equations, not evolution equations. illustration of how a singular Hamiltonian can arise in a One can, however, still attempt to construct a Lorentz-violating field theory, by examining the dynamics Hamiltonian that generates the time evolution of the system of a rank-2 antisymmetric tensor field Bab in a Lagrangian [in the sense of (1)] via a construction due to Dirac and that is designed to spontaneously break Lorentz symmetry. Bergmann [8]. A brief description of the method can also Finally, Sec. IV discusses how this pathology could arise in be found in a paper by Isenberg and Nester [9], and the a general field theory. notation used here will largely follow the notation in that ℏ Throughout this work, I will use units in which ¼ work. The construction proceeds as follows: 1 − c ¼ ; the metric signature will be ð þþþÞ. Roman (i) Define the field momenta via the natural generali- … indices a; b; c; will be used to denote spacetime tensor zation: indices; i; j; k; … will be used to denote spatial indices, where necessary. Greek indices α; β; γ; … will generally ∂L πα ¼ : ð3Þ only be used to denote indices in field space. All expres- ∂ψ_ α sions involving repeated indices (either tensor indices or field-space indices) can be assumed to obey the Einstein One can then attempt to invert these relationships to summation convention, unless explicitly stated otherwise; find ψ_ α as a function of πα, ψ α, and their derivatives. in particular, for a significant fraction of Sec. IV B and in However, it may occur that certain equations (or the Appendix, the field-space summations will be written combinations of equations) in (3) do not contain any out explicitly. While I will be working exclusively in the velocities. These equations must be thought of as realm of flat spacetime, I will still use the symbol ∇a constraining the initial data and can be written in (or ∇i) to denote spacetime (or spatial) derivatives; the the form symbol ∂ will be reserved for partial derivatives of ⃗ functions of fields (such as the field potential energy or ΦIðψ α; ∇ψ α; παÞ¼0 ð4Þ the Lagrangian density) with respect to their arguments. δ The symbol will generally denote either variations of for I ¼ 1; 2; …;M. The functions ΦI are known as functionals or functional derivatives. the primary constraints.

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(ii) Construct the base Hamiltonian density via a determined, and (since the number of fields and momenta Legendre transform on the Lagrange density: will be the same) we will have α H0 ≡ παψ_ − L: ð5Þ no: of 1 no: of N ¼ − : ð8Þ dof fields 2 constraints It can be shown that all of the velocities ψ_ α will vanish in this process. However, the evolutionR 3 H generated by this base Hamiltonian H0 ¼ d x 0 III. ANTISYMMETRIC RANK-2 TENSOR will not, in general, preserve the constraints (4).To obtain a Hamiltonian which preserves the con- A. Action straints, one must “augment” the base Hamiltonian To illustrate the problems which can arise in a model density by adding the constraints to it, each containing both constraints and a vacuum manifold, we multiplied by an as-yet-undetermined Lagrange consider the case of a rank-2 antisymmetric tensor field multiplier u : I Bab ¼ B½ab. In four-dimensional spacetime, there are two possible invariants that can be constructed from this field, H ≡ H Φ A 0 þ uI I: ð6Þ which we will denote as X and Y,

ab ab This latter quantity is the augmented Hamiltonian X ¼ B Bab Y ¼ B Bab; ð9Þ density. (iii) If the constraints are to beR preserved by the aug- where 3 mented Hamiltonian HA ¼ d xHA, it must be the case that fΦ ;H g¼0 for each constraint. If this 1 I A Bab ≡ ϵabcd Poisson bracket does not vanish identically, this will 2 Bcd: ð10Þ yield a secondary constraint ΨI ¼ 0, which the ab initial data must also obey. Similarly, this secondary (Note that B Bab ¼ −X.) We consider an action of the constraint must also be conserved, so we demand form Ψ 0 f I;HAg¼ as well; this will then generate further Z secondary constraints, which must in turn be pre- 1 4 − abc − served, and so forth. The requirement that all of the S ¼ d x 12 FabcF VðX; YÞ ; ð11Þ constraints so generated be preserved may determine some or all of the hitherto undetermined Lagrange where multipliers, in which case we can replace them in the augmented Hamiltonian with an expression F ≡ 3∂ B : ð12Þ written solely in terms of the fields and the mo- abc ½a bc menta. It is also conceivable that we may find The Euler-Lagrange equations derived from this action will an inconsistent model (i.e., one of the constraint then be equations cannot be preserved under the time evolution generated by HA). 1 When the dust settles, one is left with a Hamiltonian H ∂ cab − ab − Bab 0 A 2 cF VXB VY ¼ ; ð13Þ which generates the time evolution of the fields. We can use this Hamiltonian to count (in a simplified way) the number where VX ≡∂V=∂X and VY ≡∂V=∂Y. of degrees of freedom (d.o.f.) of the system; specifically, Since we will be attempting to construct a Hamiltonian for this model, we will need to perform a 3 þ 1 decom- 1 0 number of number of position. Given a choice of time coordinate t ¼ x on our Ndof ¼ 2 þ fields momenta spacetime, we can decompose the field Bab into spatial ⃗ number of number of undetermined vectors P⃗ and Q, corresponding to its “electric” and − − : “ ” constraints Lagrange multipliers magnetic parts respectively, ð7Þ 1 i 0i i ϵijk jk P ¼ B Q ¼ 2 B ; ð14Þ The undetermined Lagrange multipliers that remain after this process are associated with gauge d.o.f. and so are where ϵijk is the volume element on a constant-t hyper- unphysical. In the present work, however, the models under surface in spacetime. In terms of these, the kinetic term in consideration will have all of their Lagrange multipliers the Lagrangians above can be rewritten as

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1 1 _ 2 − abc ⃗− ∇⃗ ⃗ − ∇⃗ ⃗ 2 Thus, we have three primary constraints, corresponding to 12 FabcF ¼ 2 ½ðQ × PÞ ð · QÞ ; ð15Þ ⃗ ⃗ the three components of Φ ≡ ΠP ¼ 0. The augmented Hamiltonian will therefore require three Lagrange multi- while the invariants X and Y become pliers, which we will assemble into a vector u⃗; this allows 2 2 us to write the augmented Hamiltonian compactly as X ¼ −2P⃗ þ 2Q⃗ ;Y¼ −4P⃗· Q:⃗ ð16Þ _ H Π⃗ ⃗− L ⃗ Π⃗ For the sake of concreteness in what follows, we will A ¼ Q · Q þ u · P 1 1 want to have an explicit form for the potential V. To support 2 ⃗ ⃗ 2 ¼ Π⃗ þ Π⃗ · ð∇ × P⃗Þþ ð∇ · Q⃗Þ Lorentz violation, the tensor Bab must have a nonzero 2 Q Q 2 expectation value which can couple to other matter fields. ⃗ Π⃗ We therefore want to construct a potential such that there þ VðX; YÞþu · P: ð20Þ exist solutions to (13) where B is nonzero but constant. ab ⃗ A potential V which is linear in the invariants X and Y will We must now see whether the primary constraints Φ ¼ 0 lead only to solutions where Bab ¼ 0, and we must there- are closed under the time evolution of the system, thereby fore construct a potential which is quadratic in the obtaining secondary constraints and/or values for the invariants (and hence quartic in the fields), Lagrange multipliers u⃗. The preservation of the primary constraints leads to a set of secondary constraints Ψ⃗: 1 2 1 2 VðBabÞ¼ κ1X þ κ2XY þ κ3Y þ λ1X þ λ2Y; ð17Þ 2 2 ⃗_ ⃗ 0 ¼ Φ ¼fΠP;HAg κ λ “ ” where the i and i coefficients determine the shape of ⃗ ∂V ⃗ ¼ ∇ × Π⃗ − ≡ Ψ: ð21Þ the potential. Q ∂ ⃗ Equation (13) will then be satisfied for a constant tensor P 0 1 field if and only if VX ¼ VY ¼ ,or However, preservation of the secondary constraints leads to an equation involving the unknown Lagrange multipliers u⃗: κ1 κ2 X λ1 þ ¼ 0: ð18Þ κ2 κ3 Y λ2 ∂ 0 Ψ_ ∇⃗ Π⃗ − V ¼ i ¼ ð × QÞi ;HA 2 ∂Pi κ1κ3 − κ ≠ 0 Assuming that 2 , the solutions to (18) will be 2 ⃗ ∂V ∂ V those where X and Y both have a particular value − ∇ × − u κ λ ¼ ⃗ ∂ ∂ j determined by the i and i coefficients. This solution ∂Q i Pi Pj space is the vacuum manifold of our model; it will be a 2 ∂ V ⃗ four-dimensional manifold in the six-dimensional field − Π⃗ ∇ × P⃗ : 22 ∂ ∂ ½ Q þ j ð Þ space.2 More generally, the dimension of the vacuum Pi Qj manifold will be 4 plus the dimension of the solution ⃗ space of (18). For example, for the antisymmetric tensor If we define a vector v as models discussed in Refs. [10,11], the invariant Y is 2 ⃗ ∂V ∂ V ⃗ undetermined, and thus the vacuum manifold is five v ≡ ∇ × Π⃗ ∇ × P⃗ 23 i ⃗ þ ∂ ∂ ½ Q þ j ð Þ dimensional. ∂Q i Pi Qj

B. Constructing the Hamiltonian and a matrix Mij as From the kinetic term in (15), we can find the conjugate ∂2V momenta for the fields P⃗and Q⃗, the former of which can be M ij ¼ ∂ ∂ ; ð24Þ seen to vanish: Pi Pj

δL δL _ then Eq. (22) reduces to the equation Π⃗ 0 Π⃗ ⃗− ∇⃗ ⃗ P ¼ _ ¼ ; Q ¼ _ ¼ Q × P: ð19Þ δP⃗ δQ⃗ Mijuj þ vi ¼ 0: ð25Þ

1The “if” part of this statement is obvious. To see the “only if” This equation will determine some or all of the components ≠ 0 α βB 0 α β ∈ R part, suppose that Bab and Bab þ ab ¼ for some ; . of u⃗; the number of components so determined is equal to ϵabcd αB − β 0 Contracting this equation with yields ab Bab ¼ , the rank of the matrix M. and these two equations together imply that α ¼ β ¼ 0. 2 2 M It can be shown that this manifold is homeomorphic to TS , What remains is to find an expression for ij. Using the the tangent bundle on the sphere. chain rule, it is not hard to show that

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M −4 δ 16κ 32κ 16κ M ij ¼ VX ij þ 1PiPj þ 2PðiQjÞ þ 3QiQj; ij would not be of full rank. This would leave one or more components of u⃗ undetermined in (25); it would ð26Þ also require that certain components of v⃗(those not in the range of Mij) vanish automatically, leading to additional where VX ¼ ∂V=∂X ¼ κ1X þ κ2Y þ λ1. To solve (25),we need to invert M . By taking an ansatz of the form constraints. The iterative constraint-generation procedure ij described in Sec. II would therefore have to continue; −1 assuming that it did not lead to an inconsistency, the ðM Þ ¼ Aδ þ BP P þ 2CP Q þ DQ Q ð27aÞ jk jk j k ðj kÞ j k resulting theory would necessarily have fewer d.o.f. than the theory constructed for a generic point in field space. and requiring that M ðM−1Þ ¼ δ , we find that the ij jk ik This “loss” of a d.o.f. at certain points in field space inverse exists for a generic point in field space, with was noted in Ref. [12] in the context of a vector field 1 model with an unorthodox kinetic term. It has also been A − noted in certain vector field models in curved spacetime ¼ 4 ð27bÞ VX [9,13]. While these features of those other models are 1 troubling, one could perhaps argue that the singularities B −κ 4 ⃗2 κ κ − κ2 ¼ Q ½ 1VX þ Q ð 1 3 2Þ ð27cÞ of those models occur at nongeneric points in field VX space that in some sense are well separated from 1 “typical” field configurations. In that sense, those ⃗ ⃗ 2 C ¼ ½−κ2V − 4ðP · QÞðκ1κ3 − κ Þ ð27dÞ models might still be viable. V Q X 2 X What makes the singularity in the present case especially 1 vexing, however, is that we cannot make such an argument. D −κ 4 ⃗2 κ κ − κ2 ¼ ½ 3VX þ P ð 1 3 2Þ; ð27eÞ The Hamiltonian is singular when VX ¼ 0, and by defi- VXQ nition VX ¼ 0 holds for all points in the vacuum manifold. where The above arguments imply that the evolution between field configurations “on” the vacuum manifold and field “ ” Q ≡ 2 − 4 κ ⃗2 2κ ⃗ ⃗ κ ⃗2 configurations off the vacuum manifold is rather ill VX VXð 1P þ 2P · Q þ 3Q Þ posed, since Hamilton’s equation for P⃗is 2 ⃗2 ⃗2 ⃗ ⃗ 2 þ 16ðκ1κ3 − κ2Þ½P Q − ðP · QÞ : ð27fÞ δ dPi HA − M−1 At a generic point in field space, this is well defined, and ¼ ¼ ð Þijvj: ð30Þ dt δΠPi so we can invert (25) to determine the three Lagrange multipliers u⃗ in terms of the other fields. The overall This casts serious doubt on the viability of such a field Hamiltonian density for the system would then be theory as a candidate for dynamical Lorentz symmetry violation. The field configurations with v⃗≠ 0 are generic in 1 1 2 ⃗ ⃗ 2 H ¼ Π⃗ þ Π⃗ · ð∇ × P⃗Þþ ð∇ · Q⃗Þ þ VðX;YÞ field space, both on and off the vacuum manifold. Most A 2 Q Q 2 small perturbations away from the vacuum manifold would − Π M−1 ⃗≠ 0 “ ” ð PÞið Þijvj: ð28Þ therefore have v as they evolve back towards the vacuum manifold. But since M−1 becomes singular as Further, if this inverse is well defined, we can count the the fields approach the vacuum manifold, we are forced to ⃗ number of d.o.f. of the theory. We have six fields (P⃗and Q⃗), conclude that dP=dt will diverge as the fields evolve back ⃗ towards the vacuum manifold. three primary constraints ΠP ¼ 0, and three secondary constraints given in (21). Thus, the number of d.o.f. for a This statement may seem to be at odds with the work of general point in field space is Altschul et al. [4]. In that work, the authors linearized the equations of motion (13) about a constant background 1 tensor for which V ¼ V ¼ 0. The authors allowed for the 6 − 3 3 0 3 X Y Ndof ¼ 2 ð þ þ Þ¼ : ð29Þ presence of “massive modes,” field configurations for which V ≠ 0. However, their results showed that at linear It is evident, however, that the inverse matrix (27) is not order the field invariants X and Y (and thus the value of V well defined when either Q or VX vanishes. This presents itself) are constant in time. In their terminology, the massive a dilemma. If we start with an initial-data configuration modes are “nonpropagating.” Altschul et al. also raised the satisfying the constraints and for which Q and VX are question of whether these modes might become “propa- nonvanishing, then the Hamiltonian (28) becomes singular gating modes” at higher orders in perturbation theory. My if the fields ever evolve to a point where VX or Q vanishes. present work answers this question in the affirmative; in Alternately, one could construct a Hamiltonian under the fact, it is the evolution of the field from V ≠ 0 to V ¼ 0 that assumption that VX and/or Q vanish. In this case, the matrix leads to the pathology I have found.

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_ In constructing linearized versions of a nonlinear field A⃗do not appear. However, the 3 × 3 matrix M from that theory, it is often implicitly assumed that the solutions to ij work [defined analogously to (24) here] is the linearized equations have a one-to-one correspondence with “small solutions” of the full nonlinear equa- 2 ∂ V tions of motion. A theory for which this correspondence M 4κ δ a − 2 ij ¼ ∂ ∂ ¼ ½ ijðAaA bÞþ AiAj: ð33Þ can be drawn is said to be linearization stable [14]. Ai Aj

However, not all models have this property; in many a models, one can find solutions to the linearized equations This can be seen to have a rank of 1 if AaA ¼ b and 3 M that do not correspond to small solutions of the full otherwise. Since has full rank off the vacuum manifold but has a nontrivial null space on the vacuum manifold, the nonlinear equations. Showing whether a given set of M nonlinear differential equations is linearization stable is inverse for becomes singular on the vacuum manifold, a complex question, and a full discussion would be leading to the same pathology we found in the antisym- beyond the scope of this paper. However, the fact that metric tensor case. solutions with V ≠ 0 remain small in the linearized theory In the other two models discussed in Ref. [12], however, [4], but can diverge in the nonlinear theory, would suggest the number of constraints is smaller. For a general kinetic that the antisymmetric tensor evolution equations (13) are term of the form not in fact linearization stable. L ∇ ∇a b ∇ ∇b a More recently, Hernaski has also examined the conse- K ¼ c1ð aAbÞð A Þþc3ð aAbÞð A Þð34Þ quences of spontaneous Lorentz symmetry violation in the context of an antisymmetric rank-2 tensor [15]. That work and the same potential VðAaÞ, there is one primary constraint if c1 ¼ −c3 (this is the familiar Maxwell kinetic used general symmetry considerations to find the most ≠− ≠ 0 general form of an effective Lagrangian for the Nambu- term) and no primary constraints if c1 c3 and c1 . Goldstone modes arising from this sort of spontaneous The pathology therefore seems to depend on the number Lorentz symmetry breaking. While this construction does of primary constraints in the model. Specifically, both the not involve the problematic massive modes of the model, V-field model and the antisymmetric tensor model have the the fact that the “low-energy limit” of an action (11) is so ill property that on the vacuum manifold the rank of the matrix ’ M is less than the number of primary constraints. In both posed means that it is unclear if Hernaski s effective M Lagrangian could actually correspond to the low-energy cases, the matrix is constructed by taking the second derivatives of the potential V with respect to the “con- limit of a model involving a fundamental tensor field ⃗ ⃗ breaking Lorentz symmetry. However, in the construction, strained fields”: P for the antisymmetric tensor and A for Hernaski remained agnostic about the mechanism by which the V-field. It is the failure of this matrix to be full rank on this vacuum expectation value arose, and other mechanisms the vacuum manifold that leads to a singular Hamiltonian. could possibly still give rise to such an effective Lagrangian It is not hard to show that the rank of any matrix (see Sec. V). constructed in such a way is bounded above by the number of invariants used to construct the potential V. Suppose we have a potential VðX1;X2; …;XNÞ, where the quantities XA IV. GENERAL FIELD THEORIES are in turn functions of some subset of the field variables ψ ψ ψ … ψ A. Invariants and constraints α ¼f 1; 2; ; ng, with n>N. The analogous n × n matrix will then be Confronted with this problem, two natural questions arise: why does this pathology occur, and does it affect 2 ∂ V other tensor field models? A similar pathology was noted in M αβ ¼ ∂ψ ∂ψ : ð35Þ Ref. [12] for the “V-field,” a model consisting of a vector α β field A governed by the action a If we imagine diagonalizing this matrix, we can see that the Z null space of this matrix corresponds to the “directions” in 4 ∇ b∇ a − S ¼ d x½ aA bA VðAaÞ; ð31Þ field space in which the potential is flat. More explicitly, we can use the chain rule to rewrite Mαβ as where VðAaÞ¼κðAaA − bÞ2. This Lagrangian can be a ∂2 ∂ ∂ ∂ ∂2 integrated by parts to cast it in the alternate form M V XA XB V XA αβ ¼ ∂ ∂ ∂ψ ∂ψ þ ∂ ∂ψ ∂ψ ; ð36Þ Z XA XB α β XA α β S d4x ∇ Aa 2 − V A : ¼ ½ð a Þ ð aÞ ð32Þ where a summation over A and B is understood. In the vacuum manifold, the second term in (36) will vanish In this form, it is evident that the model has three (since the vacuum manifold, by definition, extremizes V constraints, since the velocities of the spatial components with respect to of all its arguments.) The first term,

065017-6 SINGULAR HAMILTONIANS IN MODELS WITH SPONTANEOUS … PHYS. REV. D 100, 065017 (2019) X meanwhile, will have a rank of at most N, the number of 1 L ¼ ½Pαβψ_ αψ_ β þ2Qα βψ_ α∇ ψ β þR α β∇ ψ α∇ ψ β: invariants used to construct V. This implies that in the K 2 i i i j i j α;β≤n vacuum manifold the rank of Mαβ will be less than n, the dimension of the space it is defined on.3 ð37Þ Effectively, this means that if the potential has “too many flat directions” we risk the rank of this matrix being The quantities Pαβ, Qαiβ,andRiαjβ are assumed to be too small on the vacuum manifold.4 But for a given tensor numerical coefficients, independent of the fields and of field, there are only a limited number of independent spacetime coordinates. Here and in what follows, we will Lorentz invariants that can be constructed from it, and this need to explicitly write out the summations over field indices; number of invariants places an upper bound on the number repeated indices should not be assumed to be summed if the of “nonflat” directions that V can have. It is therefore summation is not stated explicitly. However, the summations possible that a given tensor field may not have enough over the spatial indices i and j will remain implicit. invariants to reduce the nullity (and increase the rank) of Via various field and coefficient redefinitions, it can be Mαβ sufficiently. shown (see Appendix) that a set of kinetic terms of this This illustrates why the matrix M is not of full rank in form can always be rewritten in the form either the antisymmetric tensor model or the V-field model. In both cases, we are constructing the matrix M by taking 1 X X X L P ψ_ S ∇ ψ ψ_ S ∇ ψ the derivatives of V with respect to three constrained fields: K ¼ 2 αβ α þ αiγ i γ β þ βiδ i δ α β≤ γ≤ δ≤ the “electric vector” P⃗for the antisymmetric tensor or the ; m n n X 1 X spatial components of Aa for the V-field. But there are only Q ψ_ ∇ ψ R ∇ ψ ∇ ψ þ αiβ α i β þ2 iαjβ i α j β; ð38Þ two invariants that can be constructed out of an antisym- α;β>m α;β≤n metric tensor Bab and only one that can be constructed out of a vector field A , and so the rank of M decreases when a where Pαβ is a nondegenerate diagonal matrix and we are on the vacuum manifold. m ≤ n. The advantage of this form is that it is particularly simple to identify the primary constraints. For α ≤ m, B. Constraint structure we have Given the above features of the antisymmetric tensor and ∂L X V-field models, one might conjecture that any model which π ≡ K P ψ_ S ∇ ψ has “more constraints than invariants” would exhibit a α ∂ψ_ ¼ αα α þ αiδ i δ ; ð39Þ α δ≤ similar pathology. However, the picture is not so simple. n In particular, the structure of the constraints was critical to which can be easily inverted to find the velocities in terms the argument; the problematic conditions followed from the α preservation of the secondary constraints, and the preserva- of the momenta and derivatives. For >m, meanwhile, tion of the primary constraints did not determine any of the we have Lagrange multipliers uα in any way. In this section, I will X therefore proceed through the constraint algebra for a more πα ¼ Qαiβ∇iψ β; ð40Þ general field theory to see which features of these models led β>m to this pathology and whether this simple picture of the pathology arising from more constraints than invariants which can easily be seen to be a constraint equation: might hold under more general circumstances. X Suppose we consider a field theory in terms of some set Φα ≡ πα − Qαiβ∇iψ β ¼ 0: ð41Þ of fields ψ α (α 1; …;n) for which the dynamics are given ¼ β>m by a Lagrangian that is quadratic in these fields’ derivatives, both spatial and temporal. Suppose, further, that the The number of primary constraints in this model is there- “kinetic terms” L of the Lagrangian depend only on K fore n − m. these derivatives, so we have Let us now suppose that the full Lagrangian density of the model is of the form

3In both of the explicit models under consideration, the second L L − ψ term in (36) is of rank n when VX ≠ 0; in fact, it works out to be ¼ K Vð αÞð42Þ proportional to δαβ. This may not occur in a more general case. 4 Note that if we consider all of the fields ψ α, rather than just a with L of the form given in (38) and Vðψ αÞ a potential that subset, the nullity of Mαβ is precisely the dimension of the K vacuum manifold in field space, and the rank of Mαβ is its does not depend on any field derivatives. Then, the base codimension in field space. Hamiltonian density of this model will be

065017-7 MICHAEL D. SEIFERT PHYS. REV. D 100, 065017 (2019) X X X 1 −1 where vα represents all terms that do not depend on the H0 ¼ Pαβ παπβ − παSαiβ∇iψ β 2 Lagrange multipliers uα. α;β≤m α≤m β≤n The first term in the parentheses in (48) is the one that 1 X − R ∇ ψ ∇ ψ ψ caused the pathology in the cases of the antisymmetric 2 iαjβ i α j β þ Vð αÞ: ð43Þ α;β≤n tensor and the V-field; the rank of the matrix

2 The augmented Hamiltonian density can then be obtained ∂ V ¼ Mαβ ð49Þ by adding a set of Lagrange multiplier terms, ∂ψ α∂ψ β X X was different on the vacuum manifold than on a general H uα πα − Qα β∇ ψ β ; LM ¼ i i ð44Þ point in field space. However, we can see from the above α>m β >m that in a more general model the derivative terms in (48) can also help to determine the Lagrange multipliers; the number to (43). of Lagrange multipliers that are determined by requiring The first requirement we will need to impose to M reproduce the pathology found in the previous section is that (48) vanishes is not necessarily just the rank of αβ. For a model to have the vacuum manifold pathology to require that the coefficients Qαiβ in (38) vanish for α; β >m. To see this, note that the time evolution of a described in the previous sections, it is sufficient for these derivative terms to vanish; in other words, R α β ¼ 0 for primary constraint Φα will be given by i j α; β >m.5 _ In summary, any Lagrangian of which the kinetic terms Φα ¼fΦα;H0gþfΦα;H g; ð45Þ LM are of the form (38) (or can be put into this form) will suffer R R 3 3 from the vacuum manifold pathology described above if where H0 ≡ d xH0 and HLM ≡ d xHLM. The latter the coefficients Qα β ¼ 0 and R α β ¼ 0 when both Poisson bracket can be evaluated to be i i j α and β are greater than m and when the potential V X is constructed from fewer than n − m-field quantities fΦα;H g¼− Qα β∇ uβ: ð46Þ LM i i depending on the fields ψ α (with α >m). Both the β >m antisymmetric tensor model and the V-field model satisfy these criteria. The kinetic term (15) is of the form (38);a As noted above, the pathology in the previous section arises term corresponding to the R α β term does exist in the from the preservation of the secondary constraints, not the i j ⃗ ⃗ 2 primary constraints. If the Lagrange multipliers enter at this kinetic terms [specifically, the term ð∇ · QÞ ], but it only stage, then the preservation of the primary constraints will involves the “unconstrained” fields Q⃗ and not the con- at least partially determine them, and the chain of logic will strained fields P⃗. Similarly, the kinetic term for the V-field diverge at this stage. Thus, for a model to follow the same Lagrangian (32), when decomposed into its time and space logical chain, we must have these terms vanishing for all α, components, is and so we must have Qαiβ ¼ 0. In the language of Dirac, this means that the primary constraints are all first class, _ ⃗ ⃗ 2 LK ¼ðA0 − ∇ · AÞ ; ð50Þ since they all mutually commute with each other. Both the antisymmetric tensor model and the V-field model have which does not contain any terms corresponding to the this property. second or third summations in (38) at all. Making this assumption, the secondary constraints The above-listed conditions appear to be sufficient for Ψ ≡ Φ α f α;H0g can be calculated to be this pathology, but they may not be necessary. It is entirely possible that a model for which some of the primary ∂ X Ψ − V − S ∇ π R ∇ ∇ ψ constraints were second class, or for which the constrained α ¼ ∂ψ ð αiβ i β þ iαjβ i j βÞ: ð47Þ α β≤n fields appeared with spatial derivatives, could still have a Hamiltonian which became singular on the vacuum mani- These secondary constraints will in turn need to be fold. However, in either case, the Lagrange multipliers _ could not be solved for algebraically, and so the analysis preserved, i.e., Ψα ¼fΨα;HAg¼0. In the pathological cases described above, the Lagrange multipliers entered of the Hamiltonian would not be nearly as straightforward. into the analogous equation. To see where they enter here, It is also unclear whether any physically well-motivated models with these features exist. we can calculate the Poisson bracket fΦα;HAg; we obtain X ∂2 5 R Ψ_ − V R ∇ ∇ Note that we can take iαjβ to be symmetric under the α ¼ vα uβ þ iαjβ i juβ ; ð48Þ exchange of i and j, since R α β∇ ψ α∇ ψ β ¼ R α β∇ ψ α∇ ψ β up ∂ψ α∂ψ β i j i j i j j i β>m to total derivatives.

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V. DISCUSSION Moreover, there are no cross-couplings between the derivatives (either spatial or temporal) of C00 and C0 ; this means We have shown that a model which spontaneously i that the appropriate Qα β and R α β coefficients vanish in breaks Lorentz symmetry may, unless constructed with i i j care, have pathological evolution that is not immediately order for the general result of Sec. IV B to hold. evident at the level of the Lagrangian. It is important to note Given these features of linear cardinal gravity, one might that this technique is qualitatively different from much of be concerned that it runs the risk of suffering from the same the previous literature. Many investigations into the viabil- vacuum-manifold pathology as the antisymmetric rank-2 ity of such models have focused on their phenomenology, tensor field. However, there is one important distinction: investigating the stability of the new fields (e.g., Ref. [16]) while there are only two independent invariants one can or investigating their interactions with matter (e.g., construct from an antisymmetric tensor field (X and Y), Ref. [17]). A framework applicable to some such models there are four independent invariants that can be con- in a gravitational context was also put forward in Ref. [18]. structed from a symmetric rank-2 tensor field: Such phenomenological studies usually lead to a set of a constraints on the parameters of the model. However, in X1 ¼ Ca ð52aÞ general, these phenomenological constraints, no matter b a how stringent, will leave behind a small region of viable X2 ¼ Ca Cb ð52bÞ parameter space and cannot rule out a “fine-tuned” version of a model. The methods of the present work, in contrast, b c a X3 ¼ Ca Cb Cc ð52cÞ have the potential to completely rule out a model; the pathological behavior of the antisymmetric tensor model in b c d a X4 C C C C : 52d Sec. III exists regardless of the values of the parameters ¼ a b c d ð Þ κ1; κ2; κ3; λ1; λ2 of the model. f g “ ” This result has serious implications for the construction Thus, there appear to be just enough invariants for a of such models. If we wish to build a model with a Lorentz- cardinal gravity model to avoid the vacuum-manifold violating field, how can we reliably evade these patholo- pathology discussed in this work, so long as the potential gies? Assuming that the model under consideration can be has nontrivial dependencies on all four of these invariants. It must be emphasized, however, that having no more written in the form (38), with Qαiβ ¼ 0 and Riαjβ ¼ 0 when α; β >m, then the only way to avoid the pathology is to constraints than invariants is a necessary, not sufficient, condition to obtain a nonsingular Hamiltonian that can be ensure that the rank of the matrix Mαβ is sufficiently high. extended throughout all of field space; it is possible that This may not always be possible. For example, for the case other, more subtle pathologies occur in such a model. of an antisymmetric 2-tensor B , there are only two ab In the face of these difficulties, it is important to note independent Lorentz invariants that can be constructed that there are other methods by which models can include from it, namely X and Y as defined in (9). Since the kinetic “naturally nonzero” fields. One could, for example, postu- term used in (11) gives rise to three constraints, it is simply late that B is not a fundamental field but is instead a not possible to write down a potential for B that does not ab ab function of some other fundamental field which gains a give rise to pathological evolution. One could possibly vacuum expectation value. For example, a recent work by change the kinetic term so that the model had no more than Assunção et al. [19] introduced a model in which B is a two primary constraints; however, this could very well give ab spinor condensate. It is not immediately clear whether the rise to instabilities, as discussed in the Introduction. present results could be generalized to such models. For tensor fields of higher rank or different symmetry Another method to avoid these difficulties would be to type, a similar set of considerations would have to come simply constrain the field to be nonzero via the use of a into play. As an example, Kostelecký and Potting’s linear Lagrange multiplier λ, cardinal gravity model [2] involves a symmetric rank-2 Z tensor field Cab in a flat background with a potential 1 VðC Þ and the standard kinetic terms for a massless spin-2 4 − abc − λ − ab S ¼ d x 12 FabcF ðX bÞ ; ð53Þ field:

1 where X is defined as in (9) and b ≠ 0 is a constant. The L − ∇ ∇c ab − ∇ ∇a λ K ¼ 4 ½ cCab C aC C equation of motion for is then simply X ¼ b, and thus the “ ”6 ab ab c tensor field Bab would be nonzero in vacuum. þ 2∇aC∇bC − 2∇bC ∇cCa : ð51Þ

6 3 1 It is worth noting here that Hernaski’s effective low-energy Performing a þ decomposition, we find that the time Lagrangian [15] could equally well arise from a “fundamental” derivatives of C00 and C0i do not appear in this Lagrangian, Lagrangian of this type, rather than from a Lagrangian involving and thus this model will have four primary constraints. a potential for the fundamental field Bab.

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This method introduces a new field to the model as well It must be said that some authors (myself included) find as a new primary constraint; more importantly, it also the use of Lagrange multipliers to be somewhat inelegant. changes the algebra of the primary constraints in important This prejudice arises from classical particle mechanics, ways [12]. If we perform Dirac-Bergmann analysis on this where a constrained model can often be viewed as a limit: Lagrangian, the augmented Hamiltonian density can be one imagines a model where a potential energy is mini- shown to be mized on the constraint surface and then takes the limit of 1 1 this model as the potential becomes infinitely strong. From ⃗2 ⃗ ⃗ ⃗ ⃗ ⃗ 2 HA ¼ ΠQ þ ΠQ · ð∇ × PÞþ ð∇ · QÞ þ λðX − bÞ this perspective, Lagrange multipliers are just an ad hoc 2 2 approximation to a more fundamental theory. However, the ⃗ þ u⃗· ΠP þ uλϖ: ð54Þ situation is a lot more nuanced than that in classical field theory, particularly in the presence of models with primary Here, ϖ is the conjugate momentum to λ; it is identically constraints [12]. The present work shows that the use of zero, and so the equation ϖ ¼ 0 is enforced by a fourth Lagrange multipliers and/or composite fields may be Lagrange multiplier uλ (in addition to the three Lagrange unavoidable if one wants to model Lorentz symmetry ⃗ multipliers u⃗ enforcing the constraint ΠP ¼ 0.) If the violation with a tensor field, particularly one of higher primary constraints are to be preserved, their Poisson rank, and that Lagrange-multiplier models may not be brackets with the augmented Hamiltonian HA must vanish. relatable to a “more fundamental” potential model at all. The secondary constraints are found to be Finally, one must remember that all of this analysis has been performed at the classical level. Ideally, one would ⃗ ⃗ ⃗ ⃗ Ψ ≡ 4λP − ∇ × ΠQ ð55Þ want to take quantum effects into account as well. In the case of a set of scalar fields with a spontaneously broken and internal symmetry, it is possible to take a fundamental quantum Lagrangian and calculate an effective potential for Ψ ≡ −ðX − bÞ: ð56Þ the expectation value of the field. The expectation value of These secondary constraints must in turn be preserved the field will then obey a version of the classical equation of under time evolution; after some algebra, it can be shown motion, with V replaced by the effective potential, and so that there is a well-defined parallel between classical models and quantum models of these fields. However, it is still an ⃗_ ⃗ ⃗ ⃗ open question whether an effective potential formulation Ψ ¼ 4½uλP þ λu⃗þ ∇ × ðλQÞ ð57Þ exists in the case of spontaneously broken Lorentz sym- and metry, and so it is unclear whether or not a similar correspondence between the classical and quantum pictures _ ⃗ ⃗ ⃗ ⃗ ⃗ 8 Ψ ¼ 4½P · u⃗− 4Q · ðΠQ þ ∇ × PÞ: ð58Þ can be drawn. It may be that the pathologies found in the present work are an indication that such a correspondence These equations uniquely determine the Lagrange multi- does not actually exist for some fields undergoing sponta- ⃗ pliers u⃗and uλ, so long as P ≠ 0 and λ ≠ 0: neously Lorentz symmetry breaking; more research on this subject is needed. 1 − ⃗ ∇⃗ λ ⃗ λ ⃗ Π⃗ ∇⃗ ⃗ uλ ¼ 2 ½P · × ð QÞþ Q · ð Q þ × PÞ ð59Þ P ACKNOWLEDGMENTS 1 ⃗ − ⃗ ∇⃗ λ ⃗ The author would like to thank J. Tasson, B. Altschul, u ¼ λ ½uλP þ × ð QÞ: ð60Þ V. A. Kostelecký, Q. Bailey, and D. Garfisnkle for dis- For generic points in the vacuum manifold, the Hamiltonian cussions during the preparation of this work. This research is nonsingular, and the vacuum manifold pathology does was supported in part by Perimeter Institute for Theoretical not arise. In some sense, this is not a surprise; the pathology Physics. Research at Perimeter Institute is supported by the in the potential model arises when the field evolves onto or Government of Canada through Industry Canada and by the off of the vacuum manifold, but the field is “stuck” on the Province of Ontario through the Ministry of Economic vacuum manifold in the Lagrange-multiplier model.7 Development & Innovation.

7The above derivation also gives us the number of d.o.f. of the APPENDIX: CANONICAL FORM OF FIELD model (53). We have seven fields in the original Hamiltonian (the THEORY LAGRANGIAN components of Bab and the Lagrange multiplier λ), four primary ⃗ Consider a Lagrangian for which the kinetic term is of constraints (ΠP ¼ 0 and ϖ ¼ 0), and four secondary constraints ⃗ 1 the form (37) (reproduced here): (Ψ ¼ 0 and Ψ ¼ 0.) There are therefore 7 − 2 ð8Þ¼3 d.o.f. for this model. Note that, as discussed at length in Ref. [12], adding a Lagrange multiplier does not eliminate a d.o.f. from this model. 8I thank B. Altschul for this observation.

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1 X where the arrays Qα β and R α β have been redefined L P ψ_ ψ_ 2Q ψ_ ∇ ψ R ∇ ψ ∇ ψ i i j K ¼ 2 ½ αβ α β þ αiβ α i β þ iαjβ i α j β: after the field transformation. α;β≤n (ii) By splitting the second sum in (A2) into three parts, integrating by parts, and applying the symmetry of In Sec. IV B, it was stated that a set of kinetic terms of this Qαiβ, we can show that form can always be put into the form (38). This Appendix describes how this may be accomplished. X From the form of (37), it is fairly evident that we can take Qαiβψ_ α∇iψ β P α β α β≤ αβ to be symmetric under the exchange of and . Less ; n X X X X evident, but equally important, is that Qα β can also be i ¼ Qαiβψ_ α∇iψ β þ Qαiβψ_ α∇iψ β taken to be symmetric under this exchange. The antisym- α≤ β≤ α≤ β m n Xm >m metric part of this array (when contracted with ψ_ α∇iψ β) þ Qαiβψ_ α∇iψ β can be expressed in terms of total derivatives: α β≤ X X mm can redefine the fields ψ α (via an invertible linear i transformation) so that the matrix Pαβ becomes

diagonal. Moreover, we can reorder these fields so (iii) Since Pαβ is a diagonal, nondegenerate matrix for P ≠ 0 α ≤ P 0 α that αα for all m, and αα ¼ for >m. α; β ≤ m, it has an inverse. By then defining The kinetic terms of the Lagrangian then become X X X −1 ˜ 1 Sα β ≡ Pαγ Qγ β; A5 L P ψ_ ψ_ Q ψ_ ∇ ψ i i ð Þ K ¼ 2 αβ α β þ αiβ α i β γ≤m α;β≤m α;β≤n 1 X R ∇ ψ ∇ ψ þ 2 iαjβ i α j β; ðA2Þ we can then show that the first two summations in α;β≤n (A2) are equal to

1 X X 1 X X X P ψ_ ψ_ Q ψ_ ∇ ψ P ψ_ S ∇ ψ ψ_ S ∇ ψ 2 αβ α β þ αiβ α i β ¼ 2 αβ α þ αiγ i γ β þ βiδ i δ α;β≤m α;β≤n α;β≤m γ≤n δ≤n X 1 X X Q ψ_ ∇ ψ − P S S ∇ ψ ∇ ψ þ αiβ α i β 2 αβ αiγ βiδ i γ j δ: ðA6Þ α;β>m α;β≤m δ;γ≤n

Effectively, what we have done here is simply “completed the square” of the first two summations in (A2). (iv) The last sum in (A6) can then be absorbed into the last term in (A2), yielding a set of kinetic terms of the desired form: 1 X X X X 1 X L P ψ_ S ∇ ψ ψ_ S ∇ ψ Q ψ_ ∇ ψ R ∇ ψ ∇ ψ K ¼ 2 αβ α þ αiγ i γ β þ βiδ i δ þ αiβ α i β þ 2 iαjβ i α j β: ðA7Þ α;β≤m γ≤n δ≤n α;β>m α;β≤n

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