Constraints and Stability in Vector Theories with Spontaneous Lorentz Violation
Robert Bluhm, Nolan L. Gagne, Robertus Potting, Arturs Vrublevskis
aa r X i v : . [ h e p - t h ] D ec Constraints and Stability in Vector Theories with Spontaneous Lorentz Violation
Robert Bluhm , Nolan L. Gagne , Robertus Potting , and Arturs Vrublevskis , Physics Department, Colby College, Waterville, ME 04901 CENTRA, Departamento de F´ısica, Faculdade de Ciˆencias e Tecnologia, Universidade do Algarve, Faro, Portugal Physics Department, Massachusetts Institute of Technology, Cambridge, MA 02139
Vector theories with spontaneous Lorentz violation, known as bumblebee models, are examined inflat spacetime using a Hamiltonian constraint analysis. In some of these models, Nambu-Goldstonemodes appear with properties similar to photons in electromagnetism. However, depending onthe form of the theory, additional modes and constraints can appear that have no counterparts inelectromagnetism. An examination of these constraints and additional degrees of freedom, includingtheir nonlinear effects, is made for a variety of models with different kinetic and potential terms, andthe results are compared with electromagnetism. The Hamiltonian constraint analysis also permitsan investigation of the stability of these models. For certain bumblebee theories with a timelikevector, suitable restrictions of the initial-value solutions are identified that yield ghost-free modelswith a positive Hamiltonian. In each case, the restricted phase space is found to match that ofelectromagnetism in a nonlinear gauge.
I. INTRODUCTION
Investigations of quantum-gravity theories have un-covered a variety of possible mechanisms that can leadto Lorentz violation. Of these, the idea that Lorentzsymmetry might be spontaneously broken [1] is one ofthe more elegant. Spontaneous Lorentz violation occurswhen a vector or tensor field acquires a nonzero vacuumexpectation value. The presence of these background val-ues provides signatures of Lorentz violation that can beprobed experimentally. The theoretical framework fortheir investigation is given by the Standard-Model Exten-sion (SME) [2, 3]. Experimental searches for low-energysignals of Lorentz violation have opened up a promisingavenue of research in investigations of quantum-gravityphenomenology [4, 5].Theories with spontaneous Lorentz violation can alsoexhibit a variety of physical effects due to the appear-ance of both Nambu-Goldstone (NG) and massive Higgsmodes [6, 7, 8]. In the context of a gravitational theory,these effects include modifications of gravitational prop-agation, as well as altered forms of the static Newtonianpotential, both of which may be of interest in theoreti-cal investigations of dark energy and dark matter. Manyinvestigations to date have concentrated on the case ofa vector field acquiring a nonzero vacuum value. Thesetheories, called bumblebee models [1, 9, 10], are the sim-plest examples of field theories with spontaneous Lorentzbreaking. Bumblebee models can be defined with differ-ent forms of the potential and kinetic terms for the vectorfield, and with different couplings to matter and gravity[11, 12, 13, 14, 15, 16, 17, 18]. They can be consideredas well in different spacetime geometries, including Rie-mann, Riemann-Cartan, or Minkowski spacetimes.Much of the interest in bumblebee models stems fromthe fact that they are theories without local U (1) gaugesymmetry, but which nonetheless allow for the propaga-tion of massless vector modes. Indeed, one idea is thatbumblebee models, with appropriate kinetic and poten-tial terms, might provide alternative descriptions of pho- tons besides that given by local U (1) gauge theory. Inthis scenario, massless photon modes arise as NG modeswhen Lorentz violation is spontaneously broken. How-ever, in addition to lacking local U(1) gauge invariance,bumblebee models differ from electromagnetism (in flator curved spacetime) in a number of other ways. For ex-ample, the kinetic terms need not have a Maxwell form.Instead, a generalized form as considered, for example, invector-tensor theories of gravity can be used, though typ-ically this may involve the introduction of ghost modesinto the theory. Further differences arise due to the pres-ence of a potential term V in the Lagrangian density forbumblebee models. It is this term that induces sponta-neous Lorentz breaking. It can take a variety of forms,which may involve additional excitations due to the pres-ence of massive modes or Lagrange-multiplier fields thathave no counterparts in electromagnetism.The goal of this paper is to investigate further the ex-tent to which bumblebee models can be considered asequivalent to electromagnetism or as containing electro-magnetism as a subset theory. This question is examinedhere in flat spacetime. While gravitational effects area feature of primary interest in bumblebee models, anyequivalence or match to electrodynamics would presum-ably hold as well in an appropriate flat-spacetime limit.In a Minkowski spacetime, the main differences betweenbumblebee models and electromagnetism are due to thenature of the constraints imposed on the field variablesand in the number of physical degrees of freedom per-mitted by the theory. To investigate these quantities, aHamiltonian constraint analysis [19, 20, 21, 22] is used.This approach is particularly well suited for identifyingthe physical degrees of freedom in a theory with con-straints. It can be carried out exactly with all nonlinearterms included. It also permits examination of the ques-tion of whether the Hamiltonian is bounded from belowover the constrained phase space. II. BUMBLEBEE MODELS ANDELECTROMAGNETISM
Bumblebee models are field theories with spontaneousLorentz violation in which a vector field acquires anonzero vacuum value. For the case of a bumblebeefield B µ coupled to gravity and matter, with general-ized quadratic kinetic terms involving up to second-orderderivatives in B µ , and with an Einstein-Hilbert term forthe pure-gravity sector, the Lagrangian density is givenas L B = 116 πG ( R − σ B µ B ν R µν + σ B µ B µ R − τ B µν B µν + 12 τ D µ B ν D µ B ν + 12 τ D µ B µ D ν B ν − V ( B µ B µ ∓ b ) + L M . (1)In this expression, b > B µν = ∂ µ B ν − ∂ ν B µ . The quantities σ , σ , τ , τ , and τ are fixed constants that determine the form ofthe kinetic terms for the bumblebee field. The term L M represents possible interaction terms with matter fieldsor external currents. The potential V ( B µ B µ ∓ b ) has aminimum with respect to its argument or is constrainedto zero when B µ B µ ∓ b = 0 . (2)This condition is satisfied when the vector field has anonzero vacuum value B µ = h B µ i = b µ , (3)with b µ b µ = ± b . It is this vacuum value that sponta-neously breaks Lorentz invariance.There are many forms that can be considered for thepotential V ( B µ B µ ∓ b ). These include functionals in-volving Lagrange-multiplier fields, as well as both poly-nomial and nonpolynomial functionals in ( B µ B µ ∓ b )[1, 11]. In this work, three limiting-case examples areconsidered. They represent the dominant leading-orderterms that would arise in an expansion of a general scalarpotential V , comprised of vector fields B µ , which arenot simply mass terms. They include examples thatare widely used in the literature. The first introducesa Lagrange-multiplier field λ and has a linear form, V = λ ( B µ B µ ∓ b ) . (4)which leads to the constraint (2) appearing as an equa-tion of motion. The second is a smooth quadratic poten-tial V = κ ( B µ B µ ∓ b ) , (5)where κ is a constant. The third again involves aLagrange-multiplier field λ , but has a quadratic form, V = λ ( B µ B µ ∓ b ) . (6) With this form, the Lagrange multiplier field λ decouplesfrom the equations of motion for B µ .The model given in (1) involving a vacuum-valued vec-tor has a number of features considered previously in theliterature. For example, with the potential V and thecosmological constant Λ excluded, the resulting modelhas the form of a vector-tensor theory of gravity consid-ered by Will and Nordvedt [23, 24]. Models with poten-tials (4) and (5) inducing spontaneous symmetry break-ing were investigated by Kosteleck´y and Samuel (KS) [1],while the potential (6) was recently examined in [7]. Thespecial cases with a nonzero potential V , τ = 1, and σ = σ = τ = τ = 0 are the original KS bumblebeemodels [1]. Models with a linear Lagrange-multiplier po-tential (4), σ = σ = 0, but arbitrary coefficients τ , τ ,and τ are special cases (with a fourth-order term in B µ omitted) of the models described in Ref. [12].Since bumblebee models spontaneously break Lorentzand diffeomorphism symmetry, it is expected that mass-less Nambu-Goldstone (NG) and massive Higgs modesshould appear in these theories. The fate of these modeswas recently investigated in [6, 7]. The example of a KSbumblebee was considered in detail. It was found thatfor all three potentials (4), (5), and (6), massless NGmodes can propagate and behave essentially as photons.However, in addition, it was found that massive modescan appear that act as additional sources of energy andcharge density. In a linearized and static limit of the KSbumblebee, it was shown that both the Newtonian andCoulomb potentials for a point particle are altered by thepresence of a massive mode. Nonetheless, with suitablechoices of initial values, which limit the phase space of thetheory, solutions equivalent to those in Einstein-Maxwelltheory can be obtained for the KS bumblebee models.Bumblebee models with other (non-Maxwell) values ofthe coefficients τ , τ , and τ are expected to containmassless NG modes as well. However, in this case, sincethe kinetic terms are different, a match with electrody-namics is not expected. The non-Maxwell kinetic termsalter the constraint structure of the theory significantly,and a different number of physical degrees of freedom canemerge.To compare the constraint structures of different typesof bumblebee models with each other and with electro-dynamics, the flat-spacetime limit of (1) is considered.The Lagragian density in this case reduces to L = − τ B µν B µν + 12 τ ∂ µ B ν ∂ µ B ν + 12 τ ∂ µ B µ ∂ ν B ν − V ( B µ B µ ± b ) − B µ J µ . (7)For simplicity, interactions consisting of couplings withan externally prescribed current J µ are assumed, anda Minkowski metric η µν in Cartesian coordinates withsignature (+ , − , − , − ) is used.Following a Lagrangian approach, second-order differ-ential equations of motion for B µ are obtained. Theyare: ( τ + τ ) [ B µ − ∂ µ ∂ ν B ν ] − ( τ + τ ) B µ − V ′ B µ − J µ = 0 . (8)Here, V ′ denotes variation of the potential V ( X ) withrespect to its argument X . Since the NG modes stay inthe minimum of the potential, a nonzero value of V ′ indi-cates the presence of a massive-mode excitation. Takingthe divergence of these equations gives ∂ µ [( τ + τ ) B µ + 2 V ′ B µ + J µ ] = 0 . (9)Clearly, as expected, with V = V ′ = 0, τ = 1, and theremaining coefficients set to zero, the equations of motionreduce to those of electrodynamics, and (9) reduces to thestatement of current conservation. However, if a nonzeropotential with V ′ = 0, or if arbitrary values of τ , τ , τ are allowed, then a modified set of equations holds.In flat spacetime, the KS bumblebee has a nonzeropotential V and coefficients τ = 1, and τ = τ = 0. Itsequations of motion evidently have a close resemblanceto those of electrodynamics. The main difference is thatthe KS bumblebee field itself acts nonlinearly as a sourceof current. Equation (9) shows that the matter current J µ combines with the term 2 V ′ B µ to form a conservedcurrent.Interestingly, if the matter current J µ is set to zero,and a linear Lagrange-multiplier potential (4) is used,the KS model in flat spacetime reduces to a theory con-sidered by Dirac long before the notion of spontaneoussymmetry breaking had been introduced [25]. Dirac in-vestigated a vector theory with a nonlinear constraintidentical to (2) with the idea of finding an alternative ex-planation of electric charge. In his model, gauge invari-ance is destroyed, and conserved charge currents appearonly as a result of the nonlinear term involving V ′ for theLagrange-multiplier potential. Dirac did not, however,propose a theory of Lorentz violation. A vacuum value b µ was never introduced, and with J µ = 0 no Lorentz-violating interactions with matter enter in the theory.The idea that the photon could emerge as NG modes ina theory with spontaneous Lorentz violation came morethan ten years after the work of Dirac. First, Bjorkenproposed a model in which collective excitations of afermion field could lead to composite photons emerging asNG modes [26]. The observable behavior of the photon inthis original model was claimed to be equivalent to elec-trodynamics. Subsequently, Nambu recognized that theconstraint (2) imposed on a vector field could also leadto the appearance of NG modes that behave like photons[27]. He introduced a vector model that did not involve asymmetry-breaking potential V . Instead, the constraint(2) was imposed as a nonlinear U(1) gauge-fixing condi-tion directly at the level of the Lagrangian. The resultinggauge-fixed theory thus contained only three indepen-dent vector-field components in the Lagrangian. Nambudemonstrated that his model was equivalent to electro-magnetism and stated that the vacuum vector can beallowed to vanish to restore full Lorentz invariance. In contrast to these early models, the KS bumblebeewas proposed as a theory with physical Lorentz viola-tion. Even if the NG modes are interpreted as photonsin the KS model, and no massive modes are present, in-teractions between the vacuum vector b µ and the mattercurrent J µ provide clear observable signals of physicalLorentz violation. However, the presence of a potential V also allows additional degrees of freedom to enter inthe KS model. If arbitrary values of the coefficients τ , τ , and τ are permitted as well, the resulting theory candiffer substantially from electromagnetism.Since many of these models contain unphysical modes,either as auxiliary or Lagrange-multiplier fields, con-straint equations are expected to hold. It is the nature ofthese constraints that determines ultimately how manyphysical degrees of freedom occur in a given model. WithDirac’s Hamiltonian constraint analysis, a direct proce-dure exists for determining the constraint structure andthe number of physical degrees of freedom in these mod-els. III. HAMILTONIAN CONSTRAINT ANALYSIS
Given a Lagrangian density L describing a vector field B µ , the canonical Hamiltonian density is H = Π µ ∂ B µ −L , where the canonical momenta are defined asΠ µ = δ L δ ( ∂ B µ ) . (10)If additional fields, e.g., Lagrange multipliers λ , are con-tained in the theory, additional canonical momenta forthese quantities are defined as well, e.g., Π ( λ ) . (Note:here λ is not a spacetime index). In the Hamiltonian ap-proach, time derivatives of a quantity f are computed bytaking the Poisson bracket with the Hamiltonian H ,˙ f = { f, H } + ∂f∂t . (11)The second term is needed with quantities that have ex-plicit time dependence, e.g., an external current J µ .In Dirac’s constraint analysis, primary and secondaryconstraints are determined, and these are identified as ei-ther first-class or second-class. In the phase space awayfrom the constraint surface, the canonical Hamiltonian isambiguous up to additional multiples of the constraints.An extended Hamiltonian is formed that includes multi-ples of the constraints with coefficients that can be de-termined, or in the case of first-class constraints, remainarbitrary. It is the extended Hamiltonian that is thenused in (11) to determine the equations of motion for thefields and conjugate momenta.A system of constraints is said to be regular if the Ja-cobian matrix formed from variations of the constraintswith respect to the set of field variables and conjugatemomenta has maximal rank. If it does not, the systemis said to be irregular, and some of the constraints aretypically redundant. Dirac argued that theories with pri-mary first-class constraints have arbitrary or unphysicaldegrees of freedom, such as gauge degrees of freedom.These types of constraints therefore allow removal of twofield or momentum components. Dirac conjectured thatthis is true as well for secondary first-class constraints.Based on this, a counting argument can be made. Itstates that in a theory with n field and n conjugate-momentum components, if there are n first-class con-straints and n second-class constraints, the number ofphysical independent degrees of freedom is n − n − n / n is even). This countingargument based on Dirac’s conjecture holds up well fortheories with regular systems of constraints. However,counterexamples are known for irregular systems [22].Once the unphysical modes have been eliminated, byapplying the constraints and/or imposing gauge condi-tions, the evolution of a physical system is determinedby the equations of motion for the physical fields andmomenta, subject to initial conditions for these quanti-ties. Any bumblebee theory that has additional degreesof freedom in comparison to electrodynamics must there-fore specify additional initial values. The subsequent evo-lution of the extra degrees of freedom typically leads toeffects that do not occur in electrodynamics. However,in some cases, equivalence with electrodynamics can holdin a subspace of the phase space of the modified theory.For this to occur, initial values must exist that confinethe evolution of the theory to a region of phase space thatmatches electrodynamics in a particular choice of gauge.In general, the stability of a theory, e.g., whether theHamiltonian is positive, depends on the initial values andallowed evolution of the physical degrees of freedom. Asdiscussed in the subsequent sections, most bumblebeemodels contain regions of phase space that do not have apositive definite Hamiltonian, though in some cases, re-stricted subspaces can be found that do maintain H > τ , τ , τ . Since much ofthe literature has focused on the case of a timelike vec-tor B µ , this restriction is assumed throughout this workas well. With this assumption, there always exists anobserver frame in which rotational invariance is main- tained and only Lorentz boosts are spontaneously bro-ken. For each type of model to be considered, all threeof the potentials in (4), (5), and (6) are considered. Forcomparison (and use as benchmarks), electromagnetismand the theory of Nambu are considered as well. In eachcase, the explicit form of the Lagrangian is obtained from(7) by inserting appropriate values for V , τ , τ , and τ ,and the conjugate momenta and Hamiltonian are thencomputed. For example, electrodynamics is obtained bysetting V = 0, τ = 1, and τ = τ = 0. Conventional no-tation sets B µ = A µ and B µν = F µν . The Hamiltonian isgiven in terms of the four fields A µ and their conjugatemomenta Π µ . The Lagrangian in Nambu’s model alsostarts with these same values (allowing U(1) invariance).However, in this case, one component of A µ is eliminatedin terms of the remaining three, using the nonlinear con-dition in (2). For the case of a timelike vector, the sub-stitution A = ( b + A j ) / is made directly in L . The re-sulting Hamiltonian in Nambu’s model therefore dependsonly on three fields A j and three conjugate momentaΠ j . In contrast, bumblebee models are defined with anonzero potential V and have Hamiltonians that dependon all four fields B µ and their corresponding conjugatemomenta Π µ . Examples with a Lagrange-multiplier po-tential involve a fifth field λ and its conjugate momentumΠ ( λ ) . However, in examples with a smooth quadratic po-tential, there is no Lagrange multiplier, and the relevantfields and momenta are B µ and Π µ . A. Electromagnetism
The conjugate momenta in electrodynamics areΠ j = ∂ A j − ∂ j A , Π = 0 . (12)The latter constitutes a primary constraint, φ = Π ≈ φ = ∂ j Π j − J ≈ j can be identified as the elec-tric field components E j and J is the charge density. Inthese expressions and below, Dirac’s weak equality sym-bol “ ≈ ” is used to denote equality on the submanifold de-fined by the constraints [22]. Both of the constraints φ and φ are first-class, indicating that there are gauge orunphysical degrees of freedom. Following Dirac’s count-ing argument, there should be n − n − n / − − H = (Π j ) + Π j ∂ j A + 12 ( F jk ) + A µ J µ . (13)In the presence of a static charge distribution, with J µ = ( ρ, ~J ) = ( ρ ( ~x ) , φ (Gauss’ law) to show that H = (Π i ) + ( F jk ) ≥ A µ and momentaΠ µ obtained from the extended Hamiltonian contain ar-bitrary functions due to the existence of the first-classconstraints. These can be eliminated by imposing gauge-fixing conditions. The evolution of the physical degreesof freedom, subject to a given set of initial values, is thendetermined for all time. B. Nambu’s Model
The starting point for Nambu’s model [27] is the con-ventional Maxwell Lagrangian with U(1) gauge invari-ance and a conserved current J µ . For the case of a time-like vector A µ , the condition A = ( b + A j ) / is sub-stituted directly into the Lagrangian as a gauge-fixingcondition. The result is L = ( ∂ A j ) + ( ∂ j ( b + A k ) / ) − ( ∂ j A k ) + ( ∂ j A k )( ∂ k A j ) − ( ∂ j ( b + A k ) / )( ∂ A j ) − ( b + A k ) / J − A j J j . (14)Nambu claimed that this theory is equivalent to electro-magnetism in a nonlinear gauge. He argued that a U(1)gauge transformation exists that transforms an electro-magnetic field in a standard gauge into the field A µ obey-ing the nonlinear gauge condition A µ A µ = b .The Hamiltonian in Nambu’s model is H = (Π j ) + 12 ( F jk ) + Π j ∂ j ( b + A k ) / +( b + A k ) / J + A j J j . (15)It depends on three field components A j and their con-jugate momenta Π j = ∂ A j − ∂ j ( b + A k ) / . In this the-ory, there are no constraints, and therefore application ofDirac’s counting argument says that there are three phys-ical degrees of freedom, which is one more than in elec-tromagnetism. An extra degree of freedom arises becausegauge fixing at the level of the Lagrangian causes Gauss’law, ∂ j Π j − J = 0, to disappear as a constraint equation.A smilar disappearance of Gauss’ law is known to occurin electrodynamics in temporal gauge (with A = 0 sub-stituted in the Lagrangian) [28]. Indeed, the linearizedlimit of Nambu’s model with a timelike vector field iselectrodynamics in temporal gauge.Observe that with with ~J = 0 and using integrationby parts, the Hamiltonian can be rewritten as H = (Π j ) + 12 ( F jk ) − ( ∂ j Π j − J )( b + A k ) . (16)In the absence of a constraint enforcing Gauss’ law, H need not be positive definite. For example, if the ex-tra degree of freedom in A j causes large deviations fromGauss’ law, which are not forbidden by any constraint,then negative values of H can occur.However, equivalence between Nambu’s model andelectrodynamics can be established by restricting the phase space in Nambu’s theory. To see that this follows,consider the equations of motion in Nambu’s model,˙ A j = Π j + ∂ j ( b + A k ) / , (17)˙Π j = ∂ k ∂ k A j − ∂ j ∂ k A k − ∂ l Π l A j ( b + A k ) / + A j J ( b + A k ) / − J j . (18)Taking the spatial divergence of (18) and using currentconservation yields the nonlinear relation ∂ ( ∂ j Π j − J ) = − ∂ j (cid:20) ( ∂ l Π l − J ) A j ( b + A k ) / (cid:21) . (19)This equation shows that if Gauss’ law, ( ∂ j Π j − J ) = 0,holds at t = 0, then ∂ ( ∂ j Π j − J ) = 0 as well at t = 0.Together these conditions and Eq. (19) are sufficient toshow that Gauss’ law then holds for all time. From this itfollows that H is positive over the restricted phase space,which matches that of electrodynamics in a nonlineargauge. Thus, by restricting the phase space to solutionswith initial values obeying Gauss’ law, the equivalence ofNambu’s model with electromagnetism is restored. C. KS Bumblebee Model
KS bumblebee models [1] in flat spacetime have aMaxwell kinetic term and a nonzero potential V . Thechoice of a Maxwell form for the kinetic term is made toprevent propagation of the longitudinal mode of B µ as aghost mode. The KS Lagrangian is obtained from (7) bysetting τ = 1 and τ = τ = 0. The constraint structuresfor models with each of the three potentials (4) - (6) areconsidered. For definiteness, the case of a timelike vector B µ is assumed.
1. Linear Lagrange-Multiplier Potential
With a linear Lagrange-multiplier potential (4), an ad-ditional field component λ is introduced in addition tothe four fields B and B j . The conjugate momenta areΠ = Π ( λ ) = 0 , Π i = ∂ B i − ∂ i B , (20)and the canonical Hamiltonian is H = (Π i ) + Π i ∂ i B + ( ∂ i B j ) − ( ∂ j B i )( ∂ i B j )+ λ ( B − B i − b ) + B µ J µ . (21)Four constraints are identified as φ = Π (22) φ = Π ( λ ) (23) φ = ∂ i Π i − λB − J (24) φ = − (cid:0) B − B j − b (cid:1) . (25)The constraints φ and φ are primary, while φ and φ are secondary. All four are second-class.Applying Dirac’s algorithm to determine the numberof independent degrees of freedom gives n − n − n / − − / λ andthe changes in the types of constraints. Unlike electro-magnetism, there are no first-class constraints in the KSbumblebee, which reflects the lack of gauge invariance.The constraint φ gives a modified form of Gauss’ law inwhich the combination 2 λB acts as a source of chargedensity. Since V ′ = λ in this example, any excitationof the field λ is away from the potential minimum andtherefore acts effectively as a massive Higgs mode [7].In curved spacetime, such a mode can modify both thegravitational and electromagnetic potentials of a pointparticle. However, here, in flat spacetime, the presenceof λ leads only to modifications of the Coulomb potential.The Hamiltonian with ~J = 0 reduces, after using φ , φ , and integration by parts, to H = 12 (Π j ) + 12 ( B jk ) − λB . (26)The full phase space of the theory on the constraint sur-face includes regions in which H is negative due to thepresence of the additional degree of freedom. For ex-ample, consider the case with J = 0 and initial val-ues [29] B j = ∂ j φ ( ~x ) and Π j = − ∂ j ( b + ( ∂ k φ ) ) / at t = 0, where φ ( ~x ) is an arbitrary time-independentscalar. These give B jk = 0 and B = ( b + ( ∂ j φ ) ) / at t = 0. Inserting these initial values in (26) reduces theHamiltonian to H = − (Π j ) at t = 0. The correspond-ing initial value for λ is λ = −
12 ( b + ( ∂ j φ ) ) − / h ~ ∇ ( b + ( ∂ k φ ) ) / i . (27)Evidently, the Hamiltonian in the classical KS bumble-bee model can be negative when nonzero values of λ areallowed.However, if initial values are chosen that restrict thephase space to values with λ = 0, the resulting solutionsfor the vector field and conjugate momentum are equiv-alent to those in electromagnetism in a nonlinear gauge.Examination of the equation of motion for λ ,˙ λ = 1 B ∂ j ( λB j ) − B ∂ µ J µ − λ B j ( B ) (cid:0) Π j + ∂ j B (cid:1) , (28)reveals that if the current J µ is conserved, and λ = 0at time zero, then λ will remain zero for all time. TheHamiltonian in this case is positive. The equations ofmotion for B j and Π j are˙ B j = Π j + ∂ j B , (29)˙Π j = ∂ k ∂ k B j − ∂ j ∂ k B k − J j + 2 λB j (30)With λ = 0, these combine to give the usual Maxwellequations describing massless transverse photons. The third component in B j is an auxiliary field that is con-strained by the usual form of Gauss’ law when λ = 0.Note, however, that even with the phase space restrictedto regions with λ = 0, the matter sector of the theorywill exhibit signatures of the spontaneous Lorentz viola-tion through the interaction of the vacuum value b µ withthe matter current J µ .It is clear from these results, that conservation of thematter current J µ is necessary for the stability of theKS bumblebee model. Note, however, that the theorieslack local U(1) gauge invariance and that the currentconservation could arise simply from matter couplingsthat are invariant under a global U(1) symmetry. As aresult, photons in the KS bumblebee model appearing asNG modes are due to spontaneous Lorentz breaking, notlocal U(1) gauge invariance. For further discussion of thebumblebee currents, including in the presence of gravity,see Ref. [7]. In that work, there is also further discussionof the fact that the Lagrange-multiplier field can act as asource of charge density in the KS bumblebee model andthat there can exist solutions (with nonzero values of λ )in which the field lines converge or become singular, evenin the absence of matter charge. This behavior has beenreferred to in the literature as the formation of caustics inthe KS model. However, as described in [7], it is simply anatural consequence of the fact that the bumblebee fieldsthemselves act as sources of current. Moreover, with thephase space restricted to regions with λ = 0, the onlysingularities appearing for the case of a timelike vector B µ are those due to the presence of matter charge as inordinary electrodynamics with a 1 /r potential.
2. Quadratic Smooth Potential
A similar analysis can be performed for a KS bumble-bee with the smooth quadratic potential defined in (5).The parameter κ appearing in V is a constant. There-fore, in this case, there are four fields B , B j , and theirfour conjugate momenta,Π = 0 , Π j = ∂ B j − ∂ j B . (31)There are two constraints, φ = Π (32) φ = ∂ j Π j − κB (cid:0) B − B j − b (cid:1) − J , (33)where φ is primary, φ is secondary, and both aresecond-class. Dirac’s counting argument says there are n − n − n / − − / V ′ = 2 κB (cid:0) B − B j − b (cid:1) = 0away from the potential minimum. The constraint φ yields a modified version of Gauss’ law, showing that themassive mode acts as a source of charge density.The stability of the Hamiltonian with ~J = 0 can beexamined. Using the constraints and integration by partsgives H = 12 (Π j ) + 12 ( B jk ) − κ (3 B + B j + b )( B − B k − b ) , (34)which evidently is not positive over the full phase space.If a nonzero massive mode proportional to ( B − B j − b )is present, negative values of H can occur.However, equivalence to electrodynamics does hold ina restricted region of phase space. To verify this, considerthe equations of motion,2 κ ˙ B = (3 B − B j − b ) − (cid:2) κB B k (Π k + ∂ k B )+2 κ∂ k [ B k ( B − B l − b )] + ∂ µ J µ (cid:3) , (35)˙ B j = Π j + ∂ j B , (36)˙Π = ∂ j Π j − J − κB ( B − B j − b ) , (37)˙Π j = ∂ k ∂ k B j − ∂ j ∂ k B k +2 κB j ( B − B k − b ) − J j . (38)Combining these gives κ∂ ( B − B j − b ) = (3 B − B j − b ) − × (cid:2) κB ∂ k [ B k ( B − B l − b )] + B ∂ µ J µ − κ ( B − B k − b ) B l (Π l + ∂ l B ) (cid:3) . (39)This equation reveals that if the current J µ is conservedand ( B − B j − b ) = 0 at t = 0, then ( B − B j − b ) = 0for all time. Therefore, with these conditions imposed,the massive mode never appears, the Hamiltonian is pos-itive, and the phase space is restricted to solutions inelectromagnetism in the nonlinear gauge (2).In theories with a nonzero massive mode, the size of themass scale κb becomes relevant. For very large values,perturbative excitations that go up the potential min-imum would be expected to be suppressed. Since themass scale associated with spontaneous Lorentz violationis presumably the Planck scale, its appearance necessar-ily brings gravity into the discussion. It is at the Planckscale where quantum-gravity effects might impose addi-tional constraints that could maintain the overall stabil-ity of the theory. At sub-Planck energies, massive-modeexcitations have been shown to exert effects on classicalgravity. For example, as shown in Ref. [7], the gravita-tional potential of a point particle is modified. However,in the limit where the mass of the massive mode becomesexceptionally large, it was found for the case of the KSbumblebee model that both the usual Newtonian andCoulomb potentials are recovered.
3. Quadratic Lagrange-Multiplier Potential
The KS bumblebee model with a quadratic Lagrange-multiplier potential (6) involves five fields λ and B µ . In a Lagrangian approach, the constraint (2) follows fromthe equation of motion for λ . The on-shell equations ofmotion for B µ are the same as in electromagnetism. Inthis case, the field λ decouples and does not act as asource of charge density. On shell, the potential obeys V ′ = 0, current conservation ∂ µ J µ = 0 holds, and thereis no massive mode. This model provides an example of atheory with physical Lorentz violation due to the mattercouplings with J µ . Nonetheless, in the electromagneticsector, the theory is equivalent to electromagnetism inthe nonlinear gauge (2).However, the Hamiltonian formulation of this modelinvolves an irregular system of constraints [22]. Thus,depending on how the constraints are handled, Dirac’scounting algorithm might not apply and equivalence withthe Lagrangian approach may not hold. The conjugatemomenta are Π = 0 , (40)Π j = ∂ B j − ∂ j B , (41)Π ( λ ) = 0 . (42)From these, four constraints can be identified, φ = Π , (43) φ = Π ( λ ) , (44) φ = ∂ j Π j − λB (cid:0) B − B j − b (cid:1) − J , (45) φ = − (cid:0) B − B j − b (cid:1) . (46)With φ ≈
0, the constraint surface is limited to fieldsobeying ( B − B j − b ) = 0, and φ reduces to Gauss’law. In this case, φ and φ can be identified as first-class, while φ and φ are second-class. Dirac’s countingalgorithm then states that there are n − n − n / − − / φ is replaced by theequivalent constraint φ ′ = ( B − B j − b ) that spans thesame constraint surface, then a different set of resultsholds. In this case, additional constraints appear fromthe Poisson-bracket relations that are not equivalent tothe set defined above, and Dirac’s counting algorithmfails to determine the correct number of degrees of free-dom. The resulting theory with φ ′ replacing φ is notequivalent to the Lagrangian approach.Evidently, care must be used in working with a squaredconstraint equation. The constraints φ ′ and φ areredundant, and the Hamiltonian system is irregular.Nonetheless, with these caveats, the KS model with asquared Lagrange-multiplier potential provides a usefulmodel of spontaneous Lorentz violation. It allows an im-plementation of the symmetry breaking that does notrequire enlarging the phase space to include a massivemode or nonlinear couplings with λ . The only physicaldegrees of freedom in the theory are the NG modes thatbehave as photons. D. Bumblebee Models with ( τ + τ ) = 0 In this section, the constraint analysis is applied tobumblebee models in flat spacetime that have a La-grangian (7) with a generalized kinetic term obeying( τ + τ ) = 0. Such models do not have a Maxwell formfor the kinetic term. Throughout this section, arbitraryvalues of τ , τ , and τ are used; however, it is assumedthat discontinuities are avoided when these parametersappear in the denominators of equations. The three po-tentials in (4) - (6) are considered, and B µ is assumed tobe timelike. Since the kinetic term is not of the Maxwellform, it is not expected that the NG modes in these typesof models can be interpreted as photons. For this reason,the interaction term B µ J µ is omitted in this section.The point of view here is that the generalized bum-blebee models originate from a vector-tensor theory ofgravity with spontaneous Lorentz violation induced bythe potential V . In this context, the vector fields B µ have no matter couplings and reduce to sterile fields ina flat-spacetime limit. Nonetheless, NG modes and mas-sive modes can appear in this limit. Dirac’s Hamilto-nian analysis is used to examine the constraint structureand the number of physical degrees of freedom associatedwith these modes. Comparisons can then be made withthe results in electromagnetism and the KS bumblebeemodels.
1. Linear Lagrange-Multiplier Potential
Beginning with a model with the linear Lagrange-multiplier potential in Eq. (4), the Lagrangian is givenin terms of the five fields B , B j , and λ . From this theconjugate momenta are found to beΠ = ( τ + τ )( ∂ B ) − τ ( ∂ j B j ) , (47)Π j = ( τ − τ )( ∂ B j ) − τ ( ∂ j B ) , (48)Π ( λ ) = 0 . (49)The canonical Hamiltonian is then given as H = (cid:18) τ − ( τ − τ ) τ − τ ) (cid:19) ( ∂ j B ) + (cid:18) τ − τ ) (cid:19) (Π j ) + (cid:18) τ τ − τ (cid:19) Π j ( ∂ j B ) + 12 ( τ − τ )( ∂ j B k ) − τ ( ∂ j B k )( ∂ k B j ) + (cid:18) τ + τ ) (cid:19) (Π ) + (cid:18) τ τ + τ (cid:19) Π ∂ j B j − (cid:18) τ τ τ + τ ) (cid:19) ( ∂ j B j ) + λ ( B − B i − b ) . (50) Four constraints are found for this model: φ = Π ( λ ) , (51) φ = − ( B − B j − b ) , (52) φ = − B j (cid:20) τ − τ Π j + τ τ − τ ( ∂ j B ) (cid:21) + B (cid:20) τ + τ Π + τ τ + τ ( ∂ j B j ) (cid:21) , (53) φ = − λ ( B ) − λ (cid:18) τ + τ τ − τ (cid:19) ( B j ) − (cid:18) τ τ τ − τ ) + τ ( τ + τ )2( τ − τ ) (cid:19) ( ∂ j B ) + 12 (cid:18) τ τ + τ + τ τ τ − τ (cid:19) ( ∂ j B j ) −
12 ( τ + τ ) B j ∂ k ∂ k B j − (cid:18) τ − ( τ + τ )( τ + τ )) τ − τ (cid:19) B j ∂ j ∂ k B k + (cid:18) τ − ( τ − τ ) τ − τ ) (cid:19) B ∂ j ∂ j B − τ τ − τ ) B j ( ∂ j Π ) + τ τ − τ ) B ( ∂ j Π j ) − τ − τ ) (cid:18) τ + τ ( τ + τ ) τ − τ (cid:19) Π j ∂ j B + (cid:18) τ τ + τ + τ τ − τ ) (cid:19) Π ( ∂ j B j )+ 12( τ + τ ) (Π ) − τ + τ τ − τ ) (Π j ) . (54)The constraint φ is primary, while φ , φ , and φ are sec-ondary. All four are second-class. According to Dirac’scounting argument there are n − n − n / − − / φ shows that only three of the fourfields B µ are independent. In the timelike case, it isnatural to solve for B in terms of B j . The first andthird constraints can be used, respectively, to fix Π ( λ ) tozero and to determine Π in terms of B j and Π j . Theremaining constraint φ can be used to determine λ interms of B j and Π j . Interestingly, this leaves the samenumber of independent degrees of freedom as in the KSbumblebee model with a similar potential. One mighthave thought that switching from a Maxwell kinetic term,which results in the removal of a primary constraint Π =0, would have introduced an additional degree of freedom.However, instead, new secondary constraints appear thatstill constrain Π , though not to zero. As a result, B and Π remain unphysical degrees of freedom despite thechange in the kinetic term.Since the generalized bumblebee model is not viewedas a modified theory of electromagnetism (e.g., no cur-rent J µ is introduced), there is no analogue or modifiedversion of Gauss’ law as there is in the KS bumblebeemodel. Nonetheless, in the constraint φ , λ plays a simi-lar role as a nonlinear source term for the other fields as itdoes in the KS bumblebee. Indeed, the constraint equa-tion φ ≈ J = 0 in the limit where Π → τ , τ , τ take Maxwell values. Thus,when considering initial values of the independent fields B j and Π j in the generalized bumblebee case, the con-straint φ can play a role similar to that of the modifiedGauss’s law in the KS bumblebee model.Restrictions on the coefficients τ , τ , τ can be foundby examining the freely propagating modes in the the-ory. Investigations along these lines with gravity includedhave been carried out by a number of authors [12, 30].Since the theory with generalized kinetic terms has threedegrees of freedom, there can be up to three independentpropagating modes. These include the NG modes asso-ciated with the spontaneous Lorentz breaking. To de-termine their behavior, it suffices to work in a linearizedlimit and to look for solutions in the form of harmonicwaves. Carrying this out in the Hamiltonian formulationrequires combining the linearized equations of motion toform a wave equation for B j . For physical propagation,i.e., to avoid signs in the kinetic term that give rise toghost modes, the condition ( τ − τ ) > k µ = ( k , , , k ), itobeys a zero-mass dispersion relation of the form( τ − τ ) k + ( τ + τ ) k = 0 . (55)For physical velocities, the ratio α ≡ k /k = − τ + τ τ − τ (56)must be positive, which together with the requirement ofghost-free propagation gives( τ − τ ) > , ( τ + τ ) < τ + τ ) = 0, and therefore the third degree of freedomdoes not propagate as a harmonic wave. Instead, it isan auxiliary field that mainly affects the static potentials[7].The stability of the theory also depends on whether H is positive over the full phase space. Examining thisshould include consideration of possible initial values at t = 0 that satisfy the constraints. Using integration byparts and φ ≈
0, the Hamiltonian (50) can be writtenas the sum of two parts, H = H Π + H B . (58)The first, H Π = 12( τ − τ ) (cid:0) Π j + τ ∂ j B (cid:1) + 12( τ + τ ) (cid:0) Π + τ ∂ j B j (cid:1) , (59) includes dependence on the momenta, while the second, H B = − τ − τ ∂ j B ) − τ − τ + τ ∂ j B j ) − τ − τ ∂ i B j − ∂ j B i ) , (60)depends only on the fields B µ .First consider H B . From the condition for ghost-freepropagation in (57), it follows that the first and thirdterms are nonpositive. The second term is nonpositiveas well if 2 τ − τ + τ >
0, which implies α <
2. Thus H B ≤ α < H Π .Assuming the conditions (57) for ghost-free propagation,the first term is nonnegative, while the second is nonpos-itive. Note that the two terms are not independent, sincethey are related by constraint φ . However, one choice ofinitial values that makes both terms vanish (and there-fore satisfies φ ≈
0) isΠ j + τ ∂ j B = Π + τ ∂ j B j = 0 . (61)The initial value of λ is then chosen to make φ vanish,and B = ( b + B j ) / is used to make φ ≈
0. Conse-quently, with H Π vanishing, if α <
2, and the condition(57) holds, then there exist initial conditions with H < α consistent with (57), use theconstraint φ to rewrite H Π as H Π = 12( τ − τ ) (cid:26) (Π j + τ ∂ j B ) − α [ B j (Π j + τ ∂ j B )] B (cid:27) . (62)In any volume element, choose initial values for B j of theform ( B , B , B ) = (0 , , B ( ~x )). It then follows that H Π = 12( τ − τ ) (cid:26) (Π + τ ∂ B ) + (Π + τ ∂ B ) + (cid:18) − α B b + B (cid:19) (Π + τ ∂ B ) (cid:27) . (63)With this form, initial values of the components Π andΠ can be chosen that make the first two terms in thisexpression vanish. The third term becomes negative forany α >
1, provided an initial value of B is chosen thatobeys B > b α − . (64)With H Π <
0, and Π + τ ∂ B = 0, the initial value ofΠ can then be made arbitrarily large so that the totalinitial Hamiltonian density H = H Π + H B is negative,even if H B > H can take negativeinitial values for any choice of the parameters τ , τ , τ α < α > τ , τ , τ are restrictedto permit ghost-free propagation, then regions of the fullphase space allowed by the constraints can occur with H <
0. This parallels the behavior in the KS bumble-bee model. With τ , τ , τ equal to Maxwell values, theallowed regions of phase space in the KS model includesolutions with H <
0. However, as demonstrated in aprevious section, if initial values with λ = 0 are chosen,and current conservation holds, then λ = 0 and H > H < λ = 0 at t = 0 to satisfy theconstraint φ ≈
0. This suggests the idea of trying tolimit the choice of initial values to λ = 0 in an attemptto exclude the possibility of solutions with H < λ = 0 at t = 0 is not sufficient to restrict the phase spaceto solutions with λ = 0 for all time. This is becausethe equation of motion for λ has different dependenceon the other fields in the generalized bumblebee modelcompared to the KS model. In particular, ˙ λ is not pro-portional to just λ itself. This is evident even in thelinearized theory, with B µ expanded as B µ = b µ + E µ .Applying the constraint analysis to the linearized theoryyields a first-order expression for λ in terms of E j and Π j equal to λ ≃ b (cid:18) τ + τ τ − τ (cid:19) ∂ j Π j , (65)while the equation of motion for λ in the linearized theoryis ˙ λ ≃ − b ( τ + τ )( τ + τ )( τ − τ ) ( ∂ k ∂ k ∂ j E j ) . (66)The latter equation shows that (with non-Maxwell values τ + τ = 0) ˙ λ is independent of λ at linear order. There-fore, even if λ = 0 at t = 0, nonzero values of λ can evolveover time. This makes it difficult to decouple regions ofphase space with H > H < λ = 0 at t = 0.
2. Quadratic Smooth Potential
The generalized bumblebee model with a smoothquadratic potential (5) depends on four field components B µ and their corresponding conjugate momenta. The expressions for Π and Π j are the same as in Eqs. (47)and (48), respectively. There are no constraints in thismodel. Thus, according to Dirac’s counting algorithmthere are n − n − n = 4 − − τ , τ , and τ , all three NG modes can propagate, but with disper-sion relations that depend on these coefficients. In con-trast, in the KS model, with a Maxwell kinetic term,only two of the NG modes propagate as transverse pho-tons. A massive mode occurs in either theory when V ′ = 2 κ ( B − B j − b ) = 0. In the generalized bum-blebee case, there is no analogue of Gauss’ law, and itis possible for the massive mode to propagate. However,in the KS model with a timelike vector, the constraint(33) provides a modified version of Gauss’ law, and themassive mode is purely an auxiliary field that acts as anonlinear source of charge density in this relation.The Hamiltonian for the generalized bumblebee hasthe same form as in (50), but with the potential in thelast term replaced by the expression in (5). With noconstraints, the full phase space includes solutions withan unrestricted range of initial values. Thus, for anyvalues of the coefficients τ , τ , τ , there will either bepropagating ghost modes or permissible initial choices forthe fields and momenta with H <
3. Quadratic Lagrange-Multiplier Potential
As a final example, the generalized bumblebee modelwith a quadratic Lagrange-multiplier potential (6) canbe considered as well. In this case there are ten fields B µ , Π µ , λ , and Π ( λ ) . The conjugate momenta are givenin (47) - (49). The Hamiltonian is the same as in (50),but with the potential replaced by (6). In this case, twoconstraints are found, φ = Π ( λ ) , (67) φ = −
12 ( B − B j − b ) . (68)Constraint φ imposes the condition (2). However, itinvolves a quadratic expression for this condition, andtherefore the system is irregular, and the same caveatsmust be applied as in the KS model. In particular, sub-stitution of an equivalent constraint φ ′ = ( B − B j − b )causes Dirac’s counting argument to fail. However, with φ and φ identified as first-class constraints, Dirac’s al-gorithm gives n − n − n / − − λ decouplescompletely. The three independent degrees of freedomare the NG modes, which in the generalized bumblebeecan all propagate. However, even if values of τ , τ , and1 TABLE I: Summary of constraints. Shown for each model are the number of primary (1 o ), secondary (2 o ), first-class (FC), andsecond-class (SC) constraints, and the resulting number of independent degrees of freedom (DF). The last column indicates theregions of phase space that are ghost-free and have H >
0. Current conservation ∂ µ J µ = 0 is assumed in the KS models.Theory Kinetic Term Potential V Fields 1 o o FC SC DF Ghost-Free, H > − F µν F µν – A µ , Π µ − F µν F µν – A j , Π j ∂ j Π j = J )KS Bumblebee − B µν B µν λ ( B µ B µ ± b ) B µ , Π µ , λ , Π ( λ ) λ = 0)( τ = 1, τ = τ = 0) κ ( B µ B µ ± b ) B µ , Π µ B µ B µ = b ) λ ( B µ B µ ± b ) B µ , Π µ , λ , Π ( λ ) λ ( B µ B µ ± b ) B µ , Π µ , λ , Π ( λ ) τ , τ , τ ) κ ( B µ B µ ± b ) B µ , Π µ λ ( B µ B µ ± b ) B µ , Π µ , λ , Π ( λ ) τ can be found that prevent these modes from propagat-ing as ghost modes, there are no other constraints in thetheory that prevent initial-value choices that can yieldsolutions with H < IV. SUMMARY & CONCLUSIONS
Table I summarizes the results of the constraint anal-ysis applied to electrodynamics, Nambu’s model, the KSbumblebee, and the generalized bumblebee. For each ofthe bumblebee models, three types of potentials V areconsidered. The results show that no two models haveidentical constraint structures. In most cases, there areone or more additional degrees of freedom in comparisonto electromagnetism. These extra degrees of freedom areimportant both as possible additional propagating modesand in terms of how they alter the initial-value problem.In considering the stability of the bumblebee mod-els, it is not sufficient to look only at the propagatingmodes. The range of possible initial values must be ex-amined as well. In general, when the extra degrees offreedom appearing in these models are allowed access tothe full phase space, the Hamiltonians are not strictlypositive definite. However, in the KS models, it is pos-sible to choose initial values for the fields and momentathat restrict the phase space to ghost-free regions with H >
0. In contrast, in models with generalized kineticterms obeying ( τ + τ ) = 0, no such restrictions arefound. These theories either have propagating ghosts orhave extra degrees of freedom that evolve in such a waythat makes it difficult to separate off restricted regionsof phase space with H >
0. In the end, it appears thatonly the KS models have a simple choice of initial valuesthat can yield a physically viable theory in a restrictedregion of phase space.The examples considered in this analysis all focused onthe case of a timelike vector B µ , which is the most widelystudied case in the literature, since it involves an observerframe that maintains rotational invariance. A natural ex-tension of this work would be to consider models with a spacelike vector B µ . In this case, it is straightforward toshow that the linearized KS model is equivalent to elec-trodynamics in an axial gauge [6]. However, additionalcare is required in conducting a constraint analysis ofthe full nonlinear KS or generalized models, since B canvanish in the case of a spacelike vector, making addi-tional singularities a possibility. Alternatively, an anal-ysis in terms of the BRST formalism could be pursued,which would be suitable as well for addressing questionsof quantization. Lastly, an extension of the constraintanalysis to a curved spacetime in the presence of gravitywould be relevant, since ultimately bumblebee models areof interest not only as effective field theories incorporat-ing spontaneous Lorentz violation, but also as modifiedtheories of gravity. For example, they are currently oneof the more widely used models for exploring implica-tions of Lorentz violation in gravity and cosmology andin seeking alternative explanations of dark matter anddark energy. However, performing a constraint analy-sis with gravity presents even greater challenges and isbeyond the scope of this work.In summary, the constraint analysis presented here ina flat-spacetime limit is useful in seeking insights intothe nature of theories with spontaneous Lorentz violationand what their appropriate interpretations might be. Inparticular, the KS bumblebee models offer the possibilitythat Einstein-Maxwell theory might emerge as a resultof spontaneous Lorentz breaking instead of through localU(1) gauge invariance. Indeed, in the flat-spacetime limitof this model, with a timelike vacuum value, electromag-netism in a fixed nonlinear gauge is found to emerge ina well-defined region of phase space. Acknowledgments
We thank Alan Kosteleck´y for useful conversations.This work was supported in part by NSF grant PHY-0554663. The work of R. P. is supported by the Por-tuguese Funda¸c˜ao para a Ciˆencia e a Tecnologia.2 [1] V.A. Kosteleck´y and S. Samuel, Phys. Rev. D , 1886(1989); Phys. Rev. D , 683 (1989); Phys. Rev. Lett. , 224 (1989).[2] V.A. Kosteleck´y and R. Potting, Phys. Rev. D , 3923(1995); D. Colladay and V.A. Kosteleck´y, Phys. Rev. D , 6760 (1997); Phys. Rev. D , 116002 (1998).[3] For a review of the Standard Model Extension, see R.Bluhm, in Special Relativity: Will It Survive the Next 101Years? eds., J. Ehlers and C. L¨ammerzahl, (Springer,Berlin, 2006), hep-ph/0506054.[4] Tables summarizing experimental data in tests of Lorentzand CPT symmetry are given in V.A. Kosteleck´y and N.Russell, arXiv:0801.0287.[5] For recent reviews of experimental and theoretical ap-proaches to Lorentz and CPT violation, see V.A. Kost-eleck´y, ed.,
CPT and Lorentz Symmetry IV (World Sci-entific, Singapore, 2008) and the earlier volumes in thisseries,
CPT and Lorentz Symmetry III (World Scientific,Singapore, 2005);
CPT and Lorentz Symmetry II , WorldScientific, Singapore, 2002;
CPT and Lorentz Symmetry ,World Scientific, Singapore, 1999.[6] R. Bluhm and V.A. Kosteleck´y, Phys. Rev. D , 065008(2005).[7] R. Bluhm, S.-H. Fung. and V.A. Kosteleck´y, Phys. Rev.D, in press, arXiv:0712.4119.[8] V.A. Kosteleck´y and R. Potting, Gen. Rel. Grav. ,1675 (2005).[9] V.A. Kosteleck´y and R. Lehnert, Phys. Rev. D ,065008 (2001); V.A. Kosteleck´y, Phys. Rev. D , 105009(2004); Q.G. Bailey and V.A. Kosteleck´y, Phys. Rev. D , 045001 (2006).[10] The inspiration for this name (given by Kosteleck´y)comes from the real-life bumblebees. For a long timeit was thought that with their large bodies and littlewings it was theoretically impossible for them to fly, andyet they do. Similarly, in bumblebee models the usualtheoretical explanation for massless photons, local U(1)gauge invariance, does not apply. Nonetheless, masslessNG modes do propagate (as photons in the KS models)due to the spontaneous Lorentz violation in these models.[11] B. Altschul and V.A. Kosteleck´y, Phys. Lett. B , 106(2005). [12] T. Jacobson, arXiv:0801.1547.[13] P. Kraus and E.T. Tomboulis, Phys. Rev. D , 045015(2002).[14] J.W. Moffat, Intl. J. Mod. Phys. D , 1279 (2003).[15] B.M. Gripaios, JHEP , 069 (2004).[16] S.M. Carroll and E.A. Lim, Phys. Rev. D , 123525(2004).[17] O. Bertolami and J. Paramos, Phys. Rev. D , 044001(2005).[18] J.L. Chkareuli, C.D. Froggatt, and H.B. Nielsen,hep-th/0610186; J.L. Chkareuli, C.D. Froggatt, J.G. Je-jelava, H.B. Nielsen, arXiv:0710.3479.[19] P.A.M. Dirac, Lectures on Quantum Mechanics , YeshivaUniversity, New York, 1964.[20] A. Hanson, T. Regge, and C. Teitelboim,
ConstrainedHamiltonian Systems , Accademia Nazionale Dei Lincei,Roma, 1976.[21] D.M. Gitman, I.V. Tyutin,
Quantization of Fields withConstraints , Springer-Verlag, New York, 1991.[22] M. Henneaux and C. Teitelboim,
Quantization of GaugeSystems , Princeton University Press, Princeton, NJ,1992.[23] C.M. Will and K. Nordtvedt, Astrophys. J. , 757(1972); R.W. Hellings and K. Nordtvedt, Phys. Rev. D , 3593 (1973).[24] C.M. Will, Theory and Experiment in GravitationalPhysics , Cambridge University Press, Cambridge, Eng-land, 1993.[25] P.A.M. Dirac, Proc. R. Soc. Lon.
A209 , 291, (1951).[26] J.D. Bjorken, Ann. Phys. , 174 (1963).[27] Y. Nambu, Prog. Theor. Phys. Suppl. Extra 190 (1968).[28] See, for example, R. Jackiw, Rev. Mod. Phys. , 661(1980).[29] M.A. Clayton, gr-qc/0104103; C. Eling, T. Jacobson, andD. Mattingly, gr-qc/0410001.[30] J.W. Elliott, G.D. Moore, and H. Stoica, JHEP ,066 (2005); E.A. Lim, Phys. Rev. D , 063504 (2005);M.L. Graesser, A. Jenkins, M.B. Wise, Phys. Lett. B , 5 (2005); C. Eling, Phys. Rev. D , 084026 (2006);M.D. Seifert, Phys. Rev. D , 064002 (2007); T.R. Du-laney, M.I. Gresham, and M.B. Wise, arXiv:0801.2950. Erratum Appended to Published Version [Physical Review D , 125007 (2008)]The Hamiltonian density term H B given in Eq. (60), in Section III D 1, is incorrect. A correct expression is H B = 12 ( τ − τ ) (cid:2) ( ∂ j B k ) − ( ∂ j B ) (cid:3) −
12 ( τ + τ )( ∂ j B j ) . (60)This term is used to examine the positivity of the Hamiltonian density H for the case of a model with a generalkinetic term and a Lagrange-multiplier potential. The change in the term H B alters some of the conclusions thatfollow Eq. (60), which are stated in terms of a parameter α defined in Eq. (56). A revised argument still assumes α > α > H can be negative. However, for 0 < α ≤
1, a new examination has to be carried out. UsingSchwartz inequalities, it is found that H Π ≥ ∂ j B k ) − ( ∂ j B ) ≥
0. From the latter condition it follows that H B ≥ − ( τ + τ )( ∂ j B j ) . For ( τ + τ ) >
0, corresponding to α < H B can be made arbitrarily negative, andsolutions with H < τ + τ ) = 0, corresponding to α = 1, it follows that H B ≥ H is nonnegative. The model with α = 1 and H ≥ L = − ( ∂ µ B ν )( ∂ µ B ν ). It has recently been examined in arXiv:0812.1049 by S.M.Carroll, T.R. Dulaney, M.I. Gresham, and H. Tam. It provides a counterexample to the conclusions summarized inTable I for the case with general values of τ , τ , τ and a linear Lagrange-multiplier potential. The model with thesame kinetic term but with a quadratic Lagrange-multiplier (as discussed in Section III D 3) also has H > τ + τ3