Constraints and Stability in Vector Theories with Spontaneous Lorentz

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II. BUMBLEBEE MODELS AND With this form, the Lagrange multiplier field λ decouples ELECTROMAGNETISM from the equations of motion for Bµ. The model given in (1) involving a vacuum-valued vec- Bumblebee models are field theories with spontaneous tor has a number of features considered previously in the Lorentz violation in which a vector field acquires a literature. For example, with the potential V and the nonzero vacuum value. For the case of a bumblebee cosmological constant Λ excluded, the resulting model field Bµ coupled to gravity and matter, with general- has the form of a vector-tensor theory of gravity consid- ized quadratic kinetic terms involving up to second-order ered by Will and Nordvedt [23, 24]. Models with poten- derivatives in Bµ, and with an Einstein-Hilbert term for tials (4) and (5) inducing spontaneous symmetry break- the pure-gravity sector, the Lagrangian density is given ing were investigated by Kosteleckýand Samuel (KS) [1], as while the potential (6) was recently examined in [7]. The special cases with a nonzero potential V , τ1 = 1, and 1 µ ν µ LB = (R − 2Λ) + σ1B B Rµν + σ2B BµR σ1 = σ2 = τ2 = τ3 = 0 are the original KS bumblebee 16πG models [1]. Models with a linear Lagrange-multiplier po- 1 µν 1 µ ν − τ B B + τ D B D B tential (4), σ1 = σ2 = 0, but arbitrary coefficients τ1, τ2, 4 1 µν 2 2 µ ν and τ3 are special cases (with a fourth-order term in Bµ 1 µ ν µ 2 + τ3DµB Dν B − V (BµB ∓ b )+ LM. (1) omitted) of the models described in Ref. [12]. 2 Since bumblebee models spontaneously break Lorentz In this expression, b2 > 0 is a constant, and in Riemann and diffeomorphism symmetry, it is expected that massless Nambu-Goldstone (NG) and massive Higgs modes spacetime Bµν = ∂µBν −∂ν Bµ. The quantities σ1, σ2, τ1, should appear in these theories. The fate of these modes τ2, and τ3 are fixed constants that determine the form of was recently investigated in [6, 7]. The example of a KS the kinetic terms for the bumblebee field. The term LM represents possible interaction terms with matter fields bumblebee was considered in detail. It was found that µ 2 for all three potentials (4), (5), and (6), massless NG or external currents. The potential V (BµB ∓ b ) has a minimum with respect to its argument or is constrained modes can propagate and behave essentially as photons. to zero when However, in addition, it was found that massive modes can appear that act as additional sources of energy and µ 2 BµB ∓ b =0. (2) charge density. In a linearized and static limit of the KS bumblebee, it was shown that both the Newtonian and This condition is satisfied when the vector field has a Coulomb potentials for a point particle are altered by the nonzero vacuum value presence of a massive mode. Nonetheless, with suitable choices of initial values, which limit the phase space of the Bµ = hBµi = bµ, (3) theory, solutions equivalent to those in Einstein-Maxwell µ 2 theory can be obtained for the KS bumblebee models. with bµb = ±b . It is this vacuum value that spontaneously breaks Lorentz invariance. Bumblebee models with other (non-Maxwell) values of There are many forms that can be considered for the the coefficients τ1, τ2, and τ3 are expected to contain µ 2 massless NG modes as well. However, in this case, since potential V (BµB ∓ b ). These include functionals involving Lagrange-multiplier fields, as well as both poly- the kinetic terms are different, a match with electrody- µ 2 namics is not expected. The non-Maxwell kinetic terms nomial and nonpolynomial functionals in (BµB ∓ b ) [1, 11]. In this work, three limiting-case examples are alter the constraint structure of the theory significantly, considered. They represent the dominant leading-order and a different number of physical degrees of freedom can terms that would arise in an expansion of a general scalar emerge. potential V , comprised of vector fields Bµ, which are To compare the constraint structures of different types not simply mass terms. They include examples that of bumblebee models with each other and with electro- are widely used in the literature. The first introduces dynamics, the flat-spacetime limit of (1) is considered. a Lagrange-multiplier field λ and has a linear form, The Lagragian density in this case reduces to

V = λ(B Bµ ∓ b2). (4) 1 1 µ L = − τ B Bµν + τ ∂ B ∂µBν 4 1 µν 2 2 µ ν which leads to the constraint (2) appearing as an equa- 1 µ ν µ 2 µ tion of motion. The second is a smooth quadratic poten- + τ3∂µB ∂ν B − V (BµB ± b ) − BµJ . (7) 2 tial For simplicity, interactions consisting of couplings with V = 1 κ(B Bµ ∓ b2)2, (5) 2 µ an externally prescribed current J µ are assumed, and where κ is a constant. The third again involves a a Minkowski metric ηµν in Cartesian coordinates with Lagrange-multiplier field λ, but has a quadratic form, signature (+, −, −, −) is used. Following a Lagrangian approach, second-order differ- 1 µ 2 2 µ V = 2 λ(BµB ∓ b ) . (6) ential equations of motion for B are obtained. They 3 are: In contrast to these early models, the KS bumblebee ν was proposed as a theory with physical Lorentz viola- (τ1 + τ3)[ Bµ − ∂µ∂ Bν ] ′ tion. Even if the NG modes are interpreted as photons −(τ2 + τ3) Bµ − 2V Bµ − Jµ =0. (8) in the KS model, and no massive modes are present, in- Here, V ′ denotes variation of the potential V (X) with teractions between the vacuum vector bµ and the matter respect to its argument X. Since the NG modes stay in current Jµ provide clear observable signals of physical the minimum of the potential, a nonzero value of V ′ indi- Lorentz violation. However, the presence of a potential cates the presence of a massive-mode excitation. Taking V also allows additional degrees of freedom to enter in the divergence of these equations gives the KS model. If arbitrary values of the coefficients τ1, τ2, and τ3 are permitted as well, the resulting theory can µ ′ ∂ [(τ2 + τ3) Bµ +2V Bµ + Jµ]=0. (9) differ substantially from electromagnetism. Since many of these models contain unphysical modes, Clearly, as expected, with V = V ′ = 0, τ = 1, and the 1 either as auxiliary or Lagrange-multiplier fields, con- remaining coefficients set to zero, the equations of motion straint equations are expected to hold. It is the nature of reduce to those of electrodynamics, and (9) reduces to the these constraints that determines ultimately how many statement of current conservation. However, if a nonzero physical degrees of freedom occur in a given model. With potential with V ′ 6= 0, or if arbitrary values of τ , τ , τ 1 2 3 Dirac’s Hamiltonian constraint analysis, a direct proce- are allowed, then a modified set of equations holds. dure exists for determining the constraint structure and In flat spacetime, the KS bumblebee has a nonzero the number of physical degrees of freedom in these mod- potential V and coefficients τ = 1, and τ = τ = 0. Its 1 2 3 els. equations of motion evidently have a close resemblance to those of electrodynamics. The main difference is that the KS bumblebee field itself acts nonlinearly as a source III. HAMILTONIAN CONSTRAINT ANALYSIS of current. Equation (9) shows that the matter current ′ Jµ combines with the term 2V Bµ to form a conserved Given a Lagrangian density L describing a vector field current. µ Bµ, the canonical Hamiltonian density is H = Π ∂0Bµ − Interestingly, if the matter current Jµ is set to zero, and a linear Lagrange-multiplier potential (4) is used, L, where the canonical momenta are defined as the KS model in flat spacetime reduces to a theory con- δL sidered by Dirac long before the notion of spontaneous Πµ = . (10) symmetry breaking had been introduced [25]. Dirac in- δ(∂0Bµ) vestigated a vector theory with a nonlinear constraint If additional fields, e.g., Lagrange multipliers λ, are con- identical to (2) with the idea of finding an alternative ex- tained in the theory, additional canonical momenta for planation of electric charge. In his model, gauge invari- these quantities are defined as well, e.g., Π(λ). (Note: ance is destroyed, and conserved charge currents appear here λ is not a spacetime index). In the Hamiltonian ap- only as a result of the nonlinear term involving V ′ for the proach, time derivatives of a quantity f are computed by Lagrange-multiplier potential. Dirac did not, however, taking the Poisson bracket with the Hamiltonian H, propose a theory of Lorentz violation. A vacuum value bµ was never introduced, and with Jµ = 0 no Lorentz- ∂f f˙ = {f,H} + . (11) violating interactions with matter enter in the theory. ∂t The idea that the photon could emerge as NG modes in a theory with spontaneous Lorentz violation came more The second term is needed with quantities that have ex- than ten years after the work of Dirac. First, Bjorken plicit time dependence, e.g., an external current J µ. proposed a model in which collective excitations of a In Dirac’s constraint analysis, primary and secondary fermion field could lead to composite photons emerging as constraints are determined, and these are identified as ei- NG modes [26]. The observable behavior of the photon in ther first-class or second-class. In the phase space away this original model was claimed to be equivalent to elec- from the constraint surface, the canonical Hamiltonian is trodynamics. Subsequently, Nambu recognized that the ambiguous up to additional multiples of the constraints. constraint (2) imposed on a vector field could also lead An extended Hamiltonian is formed that includes multi- to the appearance of NG modes that behave like photons ples of the constraints with coefficients that can be de- [27]. He introduced a vector model that did not involve a termined, or in the case of first-class constraints, remain symmetry-breaking potential V . Instead, the constraint arbitrary. It is the extended Hamiltonian that is then (2) was imposed as a nonlinear U(1) gauge-fixing condi- used in (11) to determine the equations of motion for the tion directly at the level of the Lagrangian. The resulting fields and conjugate momenta. gauge-fixed theory thus contained only three indepen- A system of constraints is said to be regular if the Ja- dent vector-field components in the Lagrangian. Nambu cobian matrix formed from variations of the constraints demonstrated that his model was equivalent to electro- with respect to the set of field variables and conjugate magnetism and stated that the vacuum vector can be momenta has maximal rank. If it does not, the system allowed to vanish to restore full Lorentz invariance. is said to be irregular, and some of the constraints are 4 typically redundant. Dirac argued that theories with pri- tained and only Lorentz boosts are spontaneously bro- mary first-class constraints have arbitrary or unphysical ken. For each type of model to be considered, all three degrees of freedom, such as gauge degrees of freedom. of the potentials in (4), (5), and (6) are considered. For These types of constraints therefore allow removal of two comparison (and use as benchmarks), electromagnetism field or momentum components. Dirac conjectured that and the theory of Nambu are considered as well. In each this is true as well for secondary first-class constraints. case, the explicit form of the Lagrangian is obtained from Based on this, a counting argument can be made. It (7) by inserting appropriate values for V , τ1, τ2, and τ3, states that in a theory with n field and n conjugate- and the conjugate momenta and Hamiltonian are then momentum components, if there are n1 first-class con- computed. For example, electrodynamics is obtained by straints and n2 second-class constraints, the number of setting V = 0, τ1 = 1, and τ2 = τ3 = 0. Conventional no- physical independent degrees of freedom is n−n1 −n2/2. tation sets Bµ = Aµ and Bµν = Fµν . The Hamiltonian is (Note: it can be shown that n2 is even). This counting given in terms of the four fields Aµ and their conjugate argument based on Dirac’s conjecture holds up well for momenta Πµ. The Lagrangian in Nambu’s model also theories with regular systems of constraints. However, starts with these same values (allowing U(1) invariance). counterexamples are known for irregular systems [22]. However, in this case, one component of Aµ is eliminated Once the unphysical modes have been eliminated, by in terms of the remaining three, using the nonlinear con- applying the constraints and/or imposing gauge condi- dition in (2). For the case of a timelike vector, the sub- 2 2 1/2 tions, the evolution of a physical system is determined stitution A0 = (b +Aj ) is made directly in L. The re- by the equations of motion for the physical fields and sulting Hamiltonian in Nambu’s model therefore depends momenta, subject to initial conditions for these quanti- only on three fields Aj and three conjugate momenta j ties. Any bumblebee theory that has additional degrees Π . In contrast, bumblebee models are defined with a of freedom in comparison to electrodynamics must there- nonzero potential V and have Hamiltonians that depend fore specify additional initial values. The subsequent evo- on all four fields Bµ and their corresponding conjugate µ lution of the extra degrees of freedom typically leads to momenta Π . Examples with a Lagrange-multiplier po- effects that do not occur in electrodynamics. However, tential involve a fifth field λ and its conjugate momentum (λ) in some cases, equivalence with electrodynamics can hold Π . However, in examples with a smooth quadratic po- in a subspace of the phase space of the modified theory. tential, there is no Lagrange multiplier, and the relevant µ For this to occur, initial values must exist that confine fields and momenta are Bµ and Π . the evolution of the theory to a region of phase space that matches electrodynamics in a particular choice of gauge. In general, the stability of a theory, e.g., whether the A. Electromagnetism Hamiltonian is positive, depends on the initial values and allowed evolution of the physical degrees of freedom. As The conjugate momenta in electrodynamics are discussed in the subsequent sections, most bumblebee models contain regions of phase space that do not have a j 0 Π = ∂0Aj − ∂j A0, Π =0. (12) positive definite Hamiltonian, though in some cases, restricted subspaces can be found that do maintain H > 0. 0 The latter constitutes a primary constraint, φ1 = Π ≈ 0. In a quantum theory, instability in any region of the clas- j 0 It leads to a secondary constraint, φ2 = ∂j Π − J ≈ 0, sical phase space might be expected to destabilize the which is Gauss’ law, since Πj can be identified as the elec- full theory. However, bumblebee models, with gravity tric field components Ej and J 0 is the charge density. In included, are intended as effective theories presumably these expressions and below, Dirac’s weak equality sym- emerging at or below the Planck scale from a more fun- bol “≈” is used to denote equality on the submanifold de- damental (and unknown) quantum theory of gravity. In fined by the constraints [22]. Both of the constraints φ1 this context, quantum-gravity effects might impose ad- and φ2 are first-class, indicating that there are gauge or ditional constraints leading to stability. However, in the unphysical degrees of freedom. Following Dirac’s count- absence of a fundamental theory, the question of the ulti- ing argument, there should be n−n1−n2/2=4−2−0=2 mate stability of bumblebee models cannot be addressed. independent physical degrees of freedom. These are the For this reason, in the subsequent sections, only the be- two massless transverse photon modes. havior of bumblebee models in classical phase space is The canonical Hamiltonian in electrodynamics is considered. The following sections apply Dirac’s constraint analy- 1 H = 1 (Πj )2 + Πj ∂ A + (F )2 + A J µ. (13) sis to a number of different bumblebee models, including 2 j 0 2 jk µ the KS bumblebee as well as more general cases with arbitrary values of the coefficients τ1, τ2, τ3. Since much of In the presence of a static charge distribution, with the literature has focused on the case of a timelike vec- J µ = (ρ, J~) = (ρ(~x), 0), no work is done by the exter- tor Bµ, this restriction is assumed throughout this work nal current, and the Hamiltonian is positive definite. To as well. With this assumption, there always exists an observe this, integrate by parts and use the constraint φ2 1 i 2 1 2 observer frame in which rotational invariance is main- (Gauss’ law) to show that H = 2 (Π ) + 2 (Fjk ) ≥ 0. 5

The equations of motion for the fields Aµ and momenta phase space in Nambu’s theory. To see that this follows, Πµ obtained from the extended Hamiltonian contain ar- consider the equations of motion in Nambu’s model, bitrary functions due to the existence of the first-class ˙ j 2 2 1/2 constraints. These can be eliminated by imposing gauge- Aj = Π + ∂j (b + Ak) , (17) fixing conditions. The evolution of the physical degrees j of freedom, subject to a given set of initial values, is then ˙ j k j j k l A Π = ∂ ∂kA − ∂ ∂kA − ∂lΠ 2 2 1/2 determined for all time. (b + Ak) j 0 A J j + 2 2 1/2 − J . (18) B. Nambu’s Model (b + Ak) Taking the spatial divergence of (18) and using current The starting point for Nambu’s model [27] is the con- conservation yields the nonlinear relation ventional Maxwell Lagrangian with U(1) gauge invari- Aj ance and a conserved current J µ. For the case of a time- j 0 l 0 ∂0(∂j Π − J )= −∂j (∂lΠ − J ) 2 . (19) 2 2 1/2 (b2 + A )1/2 like vector Aµ, the condition A0 = (b + Aj ) is sub- k stituted directly into the Lagrangian as a gauge-fixing j 0 This equation shows that if Gauss’ law, (∂j Π − J ) = 0, condition. The result is j 0 holds at t = 0, then ∂0(∂j Π − J )=0 as well at t = 0. Together these conditions and Eq. (19) are sufficient to L = 1 (∂ A )2 + 1 (∂ (b2 + A2 )1/2)2 − 1 (∂ A )2 2 0 j 2 j k 2 j k show that Gauss’ law then holds for all time. From this it 1 2 2 1/2 + 2 (∂j Ak)(∂kAj ) − (∂j (b + Ak) )(∂0Aj ) follows that H is positive over the restricted phase space, −(b2 + A2 )1/2J 0 − A J j . (14) which matches that of electrodynamics in a nonlinear k j gauge. Thus, by restricting the phase space to solutions Nambu claimed that this theory is equivalent to electro- with initial values obeying Gauss’ law, the equivalence of magnetism in a nonlinear gauge. He argued that a U(1) Nambu’s model with electromagnetism is restored. gauge transformation exists that transforms an electromagnetic field in a standard gauge into the field Aµ obey- µ 2 C. KS Bumblebee Model ing the nonlinear gauge condition AµA = b . The Hamiltonian in Nambu’s model is KS bumblebee models [1] in flat spacetime have a 1 1 j 2 2 j 2 2 1/2 Maxwell kinetic term and a nonzero potential V . The H = 2 (Π ) + (Fjk) + Π ∂j (b + Ak) 2 choice of a Maxwell form for the kinetic term is made to 2 2 1/2 0 j +(b + Ak) J + Aj J . (15) prevent propagation of the longitudinal mode of Bµ as a ghost mode. The KS Lagrangian is obtained from (7) by It depends on three field components Aj and their con- setting τ = 1 and τ = τ = 0. The constraint structures j 2 2 1/2 1 2 3 jugate momenta Π = ∂0Aj −∂j (b +Ak) . In this the- for models with each of the three potentials (4) - (6) are ory, there are no constraints, and therefore application of considered. For definiteness, the case of a timelike vector Dirac’s counting argument says that there are three phys- Bµ is assumed. ical degrees of freedom, which is one more than in electromagnetism. An extra degree of freedom arises because gauge fixing at the level of the Lagrangian causes Gauss’ 1. Linear Lagrange-Multiplier Potential j 0 law, ∂j Π −J = 0, to disappear as a constraint equation. A smilar disappearance of Gauss’ law is known to occur With a linear Lagrange-multiplier potential (4), an ad- in electrodynamics in temporal gauge (with A0 = 0 sub- ditional field component λ is introduced in addition to stituted in the Lagrangian) [28]. Indeed, the linearized the four fields B0 and Bj . The conjugate momenta are limit of Nambu’s model with a timelike vector field is 0 (λ) i electrodynamics in temporal gauge. Π = Π =0, Π = ∂0Bi − ∂iB0, (20) Observe that with with J~ = 0 and using integration and the canonical Hamiltonian is by parts, the Hamiltonian can be rewritten as 1 i 2 i 1 2 1 H = 2 (Π ) + Π ∂iB0 + 2 (∂iBj ) − 2 (∂j Bi)(∂iBj ) 1 j 2 1 2 j 0 2 2 H = (Π ) + (Fjk) − (∂j Π − J )(b + Ak). (16) 2 2 2 µ 2 2 +λ(B0 − Bi − b )+ BµJ . (21) In the absence of a constraint enforcing Gauss’ law, H Four constraints are identified as need not be positive definite. For example, if the ex- 0 φ1 = Π (22) tra degree of freedom in Aj causes large deviations from (λ) Gauss’ law, which are not forbidden by any constraint, φ2 = Π (23) then negative values of H can occur. i 0 φ3 = ∂iΠ − 2λB0 − J (24) However, equivalence between Nambu’s model and 2 2 2 electrodynamics can be established by restricting the φ4 = − B0 − Bj − b . (25) 6

The constraints φ1 and φ2 are primary, while φ3 and φ4 third component in Bj is an auxiliary field that is con- are secondary. All four are second-class. strained by the usual form of Gauss’ law when λ = 0. Applying Dirac’s algorithm to determine the number Note, however, that even with the phase space restricted of independent degrees of freedom gives n − n1 − n2/2= to regions with λ = 0, the matter sector of the theory 5−2−2/2 = 3. Hence, there is an extra degree of freedom will exhibit signatures of the spontaneous Lorentz viola- in the KS bumblebee model in comparison to electrody- tion through the interaction of the vacuum value bµ with namics. It arises due the presence of the extra field λ and the matter current J µ. the changes in the types of constraints. Unlike electro- It is clear from these results, that conservation of the magnetism, there are no first-class constraints in the KS matter current J µ is necessary for the stability of the bumblebee, which reflects the lack of gauge invariance. KS bumblebee model. Note, however, that the theories The constraint φ3 gives a modified form of Gauss’ law in lack local U(1) gauge invariance and that the current which the combination 2λB0 acts as a source of charge conservation could arise simply from matter couplings density. Since V ′ = λ in this example, any excitation that are invariant under a global U(1) symmetry. As a of the field λ is away from the potential minimum and result, photons in the KS bumblebee model appearing as therefore acts effectively as a massive Higgs mode [7]. NG modes are due to spontaneous Lorentz breaking, not In curved spacetime, such a mode can modify both the local U(1) gauge invariance. For further discussion of the gravitational and electromagnetic potentials of a point bumblebee currents, including in the presence of gravity, particle. However, here, in flat spacetime, the presence see Ref. [7]. In that work, there is also further discussion of λ leads only to modifications of the Coulomb potential. of the fact that the Lagrange-multiplier field can act as a The Hamiltonian with J~ = 0 reduces, after using φ3, source of charge density in the KS bumblebee model and φ4, and integration by parts, to that there can exist solutions (with nonzero values of λ) in which the field lines converge or become singular, even 1 j 2 1 2 2 in the absence of matter charge. This behavior has been H = (Π ) + (Bjk) − 2λB0 . (26) 2 2 referred to in the literature as the formation of caustics in The full phase space of the theory on the constraint sur- the KS model. However, as described in [7], it is simply a face includes regions in which H is negative due to the natural consequence of the fact that the bumblebee fields presence of the additional degree of freedom. For ex- themselves act as sources of current. Moreover, with the ample, consider the case with J 0 = 0 and initial val- phase space restricted to regions with λ = 0, the only j 2 2 1/2 ues [29] Bj = ∂j φ(~x) and Π = −∂j (b + (∂kφ) ) singularities appearing for the case of a timelike vector at t = 0, where φ(~x) is an arbitrary time-independent Bµ are those due to the presence of matter charge as in 2 2 1/2 scalar. These give Bjk = 0 and B0 = (b + (∂j φ) ) at ordinary electrodynamics with a 1/r potential. t = 0. Inserting these initial values in (26) reduces the 1 j 2 Hamiltonian to H = − 2 (Π ) at t = 0. The corresponding initial value for λ is 2. Quadratic Smooth Potential

1 2 2 −1/2 2 2 2 1/2 λ = − (b + (∂j φ) ) ∇~ (b + (∂kφ) ) . (27) A similar analysis can be performed for a KS bumble- 2 h i bee with the smooth quadratic potential defined in (5). Evidently, the Hamiltonian in the classical KS bumble- The parameter κ appearing in V is a constant. There- bee model can be negative when nonzero values of λ are fore, in this case, there are four fields B0, Bj , and their allowed. four conjugate momenta, However, if initial values are chosen that restrict the 0 j phase space to values with λ = 0, the resulting solutions Π =0, Π = ∂0Bj − ∂j B0. (31) for the vector field and conjugate momentum are equivalent to those in electromagnetism in a nonlinear gauge. There are two constraints, Examination of the equation of motion for λ, 0 φ1 = Π (32) 1 1 B j 2 2 2 0 λ˙ = ∂ (λB ) − ∂ J µ − λ j Πj + ∂ B , φ2 = ∂j Π − 2κB0 B0 − Bj − b − J , (33) B j j 2B µ (B )2 j 0 0 0 0 (28) where φ1 is primary, φ2 is secondary, and both are reveals that if the current J µ is conserved, and λ = 0 second-class. Dirac’s counting argument says there are at time zero, then λ will remain zero for all time. The n − n1 − n2/2=4 − 0 − 2/2 = 3 independent degrees Hamiltonian in this case is positive. The equations of of freedom, which again is one more than in electromag- j motion for Bj and Π are netism. The condition (2) does not occur as a constraint in this j B˙ j = Π + ∂j B0, (29) case. Instead, an extra degree of freedom appears as a ′ 2 2 2 j j massive Higgs excitation V =2κB0 B − B − b 6=0 Π˙ = ∂k∂kBj − ∂j ∂kBk − J +2λBj (30) 0 j away from the potential minimum. The constraint φ2 With λ = 0, these combine to give the usual Maxwell yields a modified version of Gauss’ law, showing that the equations describing massless transverse photons. The massive mode acts as a source of charge density. 7

The stability of the Hamiltonian with J~ = 0 can be a Lagrangian approach, the constraint (2) follows from examined. Using the constraints and integration by parts the equation of motion for λ. The on-shell equations of gives motion for Bµ are the same as in electromagnetism. In this case, the ﬁeld λ decouples and does not act as a 1 j 2 1 2 1 2 2 2 2 2 2 source of charge density. On shell, the potential obeys H = (Π ) + (Bjk) − κ(3B0 +Bj +b )(B0 −Bk −b ), ′ µ 2 2 2 V = 0, current conservation ∂µJ = 0 holds, and there (34) is no massive mode. This model provides an example of a which evidently is not positive over the full phase space. 2 2 2 theory with physical Lorentz violation due to the matter If a nonzero massive mode proportional to (B0 −Bj −b ) couplings with J µ. Nonetheless, in the electromagnetic is present, negative values of H can occur. sector, the theory is equivalent to electromagnetism in However, equivalence to electrodynamics does hold in the nonlinear gauge (2). a restricted region of phase space. To verify this, consider However, the Hamiltonian formulation of this model the equations of motion, involves an irregular system of constraints [22]. Thus, ˙ 2 2 2 −1 k depending on how the constraints are handled, Dirac’s 2κB0 = (3B0 − Bj − b ) 4κB0Bk(Π + ∂kB0) counting algorithm might not apply and equivalence with 2 2 2 µ +2κ∂k[Bk(B0 − Bl − b )] + ∂µJ , (35) the Lagrangian approach may not hold. The conjugate j momenta are B˙ j = Π + ∂j B0, (36) ˙ 0 j 0 2 2 2 0 Π = ∂j Π − J − 2κB0(B0 − Bj − b ), (37) Π =0, (40) j j Π˙ = ∂k∂kBj − ∂j ∂kBk Π = ∂0Bj − ∂j B0, (41) 2 2 2 j (λ) +2κBj(B0 − Bk − b ) − J . (38) Π =0. (42)

Combining these gives From these, four constraints can be identiﬁed,

− 0 2 2 2 2 2 2 1 φ1 = Π , (43) κ∂0(B0 − Bj − b )=(3B0 − Bj − b ) 2 2 2 µ φ = Π(λ), (44) × 2κB0∂k[Bk(B0 − Bl − b )] + B0∂µJ 2 j 2 2 2 0 2 2 2 l φ3 = ∂j Π − 2λB0 B − B − b − J , (45) −2κ(B0 − Bk − b )Bl(Π + ∂lB0) . (39) 0 j 2 φ = − 1 B2 − B2 − b2 . (46) This equation reveals that if the current J µ is conserved 4 2 0 j and (B2 − B2 − b2)=0at t = 0, then (B2 − B2 − b2)=0 0 j 0 j With φ4 ≈ 0, the constraint surface is limited to fields for all time. Therefore, with these conditions imposed, 2 2 2 obeying (B0 − Bj − b ) = 0, and φ3 reduces to Gauss’ the massive mode never appears, the Hamiltonian is pos- law. In this case, φ2 and φ4 can be identified as first- itive, and the phase space is restricted to solutions in class, while φ1 and φ3 are second-class. Dirac’s counting electromagnetism in the nonlinear gauge (2). algorithm then states that there are n − n1 − n2/2 = In theories with a nonzero massive mode, the size of the 5 − 2 − 2/2 = 2 independent degrees of freedom, which 2 mass scale κb becomes relevant. For very large values, matches electromagnetism, and the Hamiltonian is posi- perturbative excitations that go up the potential min- tive throughout the full physical phase space. However, imum would be expected to be suppressed. Since the if instead the squared constraint φ4 is replaced by the mass scale associated with spontaneous Lorentz violation ′ 2 2 2 equivalent constraint φ4 = (B0 − Bj − b ) that spans the is presumably the Planck scale, its appearance necessar- same constraint surface, then a different set of results ily brings gravity into the discussion. It is at the Planck holds. In this case, additional constraints appear from scale where quantum-gravity effects might impose addi- the Poisson-bracket relations that are not equivalent to tional constraints that could maintain the overall stabil- the set defined above, and Dirac’s counting algorithm ity of the theory. At sub-Planck energies, massive-mode fails to determine the correct number of degrees of free- excitations have been shown to exert effects on classical ′ dom. The resulting theory with φ4 replacing φ4 is not gravity. For example, as shown in Ref. [7], the gravita- equivalent to the Lagrangian approach. tional potential of a point particle is modified. However, Evidently, care must be used in working with a squared in the limit where the mass of the massive mode becomes ′ constraint equation. The constraints φ4 and φ4 are exceptionally large, it was found for the case of the KS redundant, and the Hamiltonian system is irregular. bumblebee model that both the usual Newtonian and Nonetheless, with these caveats, the KS model with a Coulomb potentials are recovered. squared Lagrange-multiplier potential provides a useful model of spontaneous Lorentz violation. It allows an im- plementation of the symmetry breaking that does not 3. Quadratic Lagrange-Multiplier Potential require enlarging the phase space to include a massive mode or nonlinear couplings with λ. The only physical The KS bumblebee model with a quadratic Lagrange- degrees of freedom in the theory are the NG modes that multiplier potential (6) involves five fields λ and Bµ. In behave as photons. 8

D. Bumblebee Models with (τ2 + τ3) =6 0 Four constraints are found for this model:

(λ) φ1 = Π , (51) In this section, the constraint analysis is applied to φ = −(B2 − B2 − b2), (52) bumblebee models in flat spacetime that have a La- 2 0 j grangian (7) with a generalized kinetic term obeying 1 j τ1 φ3 = −Bj Π + (∂j B0) (τ2 + τ3) 6= 0. Such models do not have a Maxwell form τ1 − τ2 τ1 − τ2 for the kinetic term. Throughout this section, arbitrary 1 0 τ3 values of τ1, τ2, and τ3 are used; however, it is assumed +B0 Π + (∂j Bj ) , (53) τ2 + τ3 τ2 + τ3 that discontinuities are avoided when these parameters appear in the denominators of equations. The three potentials in (4) - (6) are considered, and Bµ is assumed to 2 τ2 + τ3 2 φ4 = −λ(B0) − λ (Bj ) be timelike. Since the kinetic term is not of the Maxwell τ1 − τ2 form, it is not expected that the NG modes in these types 2 τ1τ3 τ1 (τ2 + τ3) 2 of models can be interpreted as photons. For this reason, − + (∂j B0) 2(τ − τ ) 2(τ − τ )2 the interaction term B J µ is omitted in this section. 1 2 1 2 µ 2 1 τ τ1τ3 The point of view here is that the generalized bum- + 3 + (∂ B )2 2 τ + τ τ − τ j j blebee models originate from a vector-tensor theory of 2 3 1 2 gravity with spontaneous Lorentz violation induced by 1 − (τ2 + τ3)Bj ∂k∂kBj the potential V . In this context, the vector fields Bµ 2 2 have no matter couplings and reduce to sterile fields in 1 τ3 − (τ1 + τ3)(τ2 + τ3)) − Bj ∂j ∂kBk a flat-spacetime limit. Nonetheless, NG modes and mas- 2 τ1 − τ2 sive modes can appear in this limit. Dirac’s Hamilto- 2 2 τ1 − (τ1 − τ2) nian analysis is used to examine the constraint structure + B0∂j ∂j B0 and the number of physical degrees of freedom associated 2(τ1 − τ2) τ3 0 τ1 j with these modes. Comparisons can then be made with − Bj (∂j Π )+ B0(∂j Π ) the results in electromagnetism and the KS bumblebee 2(τ1 − τ2) 2(τ1 − τ2) models. 1 1 τ1(τ2 + τ3) j − τ3 + Π ∂j B0 (τ1 − τ2) 2 τ1 − τ2

τ3 τ1 0 + + Π (∂j Bj ) τ1 + τ3 2(τ1 − τ2)

1. Linear Lagrange-Multiplier Potential 1 0 2 τ2 + τ3 j 2 + (Π ) − 2 (Π ) . (54) 2(τ2 + τ3) 2(τ1 − τ2)

Beginning with a model with the linear Lagrange- The constraint φ1 is primary, while φ2, φ3, and φ4 are sec- multiplier potential in Eq. (4), the Lagrangian is given ondary. All four are second-class. According to Dirac’s in terms of the five fields B0, Bj , and λ. From this the counting argument there are n−n1−n2/2=5−0−4/2= conjugate momenta are found to be 3 degrees of freedom in this model. The constraint φ2 shows that only three of the four 0 fields Bµ are independent. In the timelike case, it is Π = (τ2 + τ3)(∂0B0) − τ3(∂j Bj ), (47) natural to solve for B in terms of B . The first and j 0 j Π = (τ1 − τ2)(∂0Bj ) − τ1(∂j B0), (48) third constraints can be used, respectively, to fix Π(λ) to (λ) 0 j Π =0. (49) zero and to determine Π in terms of Bj and Π . The remaining constraint φ4 can be used to determine λ in j terms of Bj and Π . Interestingly, this leaves the same The canonical Hamiltonian is then given as number of independent degrees of freedom as in the KS bumblebee model with a similar potential. One might 2 2 have thought that switching from a Maxwell kinetic term, τ1 − (τ1 − τ2) 2 1 j 2 H = (∂j B0) + (Π ) which results in the removal of a primary constraint Π0 = 2(τ1 − τ2) 2(τ1 − τ2) 0, would have introduced an additional degree of freedom. τ1 j 1 2 + Π (∂j B0)+ (τ1 − τ2)(∂j Bk) However, instead, new secondary constraints appear that 0 τ1 − τ2 2 still constrain Π , though not to zero. As a result, B0 0 1 1 0 2 and Π remain unphysical degrees of freedom despite the − τ1(∂j Bk)(∂kBj )+ (Π ) 2 2(τ2 + τ3) change in the kinetic term. Since the generalized bumblebee model is not viewed τ3 0 τ2τ3 2 + Π ∂j Bj − (∂j Bj ) as a modified theory of electromagnetism (e.g., no cur- τ2 + τ3 2(τ2 + τ3) rent J µ is introduced), there is no analogue or modified 2 2 2 +λ(B0 − Bi − b ). (50) version of Gauss’ law as there is in the KS bumblebee 9 model. Nonetheless, in the constraint φ4, λ plays a simi- includes dependence on the momenta, while the second, lar role as a nonlinear source term for the other fields as it does in the KS bumblebee. Indeed, the constraint equa- τ1 − τ2 2 2τ1 − τ2 + τ3 2 HB = − (∂j B0) − (∂j Bj ) tion φ4 ≈ 0 reduces to the same modified form of Gauss’ 2 2 law as in (24) with J 0 = 0 in the limit where Π0 → 0 τ1 − τ2 2 − (∂iBj − ∂j Bi) , (60) and the coefficients τ1, τ2, τ3 take Maxwell values. Thus, 4 when considering initial values of the independent fields depends only on the fields B . B and Πj in the generalized bumblebee case, the con- µ j First consider H . From the condition for ghost-free straint φ can play a role similar to that of the modified B 4 propagation in (57), it follows that the first and third Gauss’s law in the KS bumblebee model. terms are nonpositive. The second term is nonpositive Restrictions on the coefficients τ , τ , τ can be found 1 2 3 as well if 2τ − τ + τ > 0, which implies α < 2. Thus by examining the freely propagating modes in the the- 1 2 3 H ≤ 0 if the conditions (57) hold and α< 2. ory. Investigations along these lines with gravity included B Next consider the momentum-dependent term H . have been carried out by a number of authors [12, 30]. Π Assuming the conditions (57) for ghost-free propagation, Since the theory with generalized kinetic terms has three the first term is nonnegative, while the second is nonpos- degrees of freedom, there can be up to three independent itive. Note that the two terms are not independent, since propagating modes. These include the NG modes asso- they are related by constraint φ . However, one choice of ciated with the spontaneous Lorentz breaking. To de- 3 initial values that makes both terms vanish (and there- termine their behavior, it suffices to work in a linearized fore satisfies φ ≈ 0) is limit and to look for solutions in the form of harmonic 3 waves. Carrying this out in the Hamiltonian formulation j 0 Π + τ1∂j B0 = Π + τ3∂j Bj =0. (61) requires combining the linearized equations of motion to form a wave equation for Bj . For physical propagation, The initial value of λ is then chosen to make φ4 vanish, i.e., to avoid signs in the kinetic term that give rise to 2 2 1/2 and B0 = (b + Bj ) is used to make φ2 ≈ 0. Conse- ghost modes, the condition (τ1 − τ2) > 0 must hold [30]. quently, with HΠ vanishing, if α < 2, and the condition In this case, two solutions are found that propagate as (57) holds, then there exist initial conditions with H < 0. transverse massless modes at the speed of light. However, To investigate the remaining cases, corresponding to a third longitudinal mode can be found as well. In an other possible values of α consistent with (57), use the observer frame with wave vector kµ = (k0, 0, 0, k3), it constraint φ3 to rewrite HΠ as obeys a zero-mass dispersion relation of the form 2 2 1 j 2 (τ1 − τ2)k0 + (τ2 + τ3)k3 =0. (55) HΠ = (Π + τ1∂j B0) 2(τ1 − τ2) For physical velocities, the ratio [B (Πj + τ ∂ B )]2 −α j 1 j 0 . (62) 2 2 τ2 + τ3 B2 α ≡ k0/k3 = − (56) 0 τ1 − τ2 must be positive, which together with the requirement of In any volume element, choose initial values for Bj of the ghost-free propagation gives form (B1,B2,B3)=(0, 0,B(~x)). It then follows that

(τ1 − τ2) > 0, (τ2 + τ3) < 0 (57) 1 1 2 2 2 HΠ = (Π + τ1∂1B0) + (Π + τ1∂2B0) Note in comparison that the KS bumblebee model has 2(τ1 − τ2) (τ + τ ) = 0, and therefore the third degree of freedom B2 2 3 + 1 − α (Π3 + τ ∂ B )2 . (63) does not propagate as a harmonic wave. Instead, it is b2 + B2 1 3 0 an auxiliary field that mainly affects the static potentials [7]. With this form, initial values of the components Π1 and The stability of the theory also depends on whether Π2 can be chosen that make the first two terms in this H is positive over the full phase space. Examining this expression vanish. The third term becomes negative for should include consideration of possible initial values at any α> 1, provided an initial value of B2 is chosen that t = 0 that satisfy the constraints. Using integration by obeys parts and φ2 ≈ 0, the Hamiltonian (50) can be written b2 as the sum of two parts, B2 > . (64) α − 1 H = HΠ + HB. (58) 3 The first, With HΠ < 0, and Π + τ1∂3B0 6= 0, the initial value of Π3 can then be made arbitrarily large so that the total 1 2 H = Πj + τ ∂ B initial Hamiltonian density H = HΠ + HB is negative, Π 2(τ − τ ) 1 j 0 1 2 even if HB > 0. 1 0 2 Thus, the Hamiltonian density H can take negative + Π + τ3∂j Bj , (59) 2(τ + τ ) initial values for any choice of the parameters τ1, τ2, τ3 2 3 10 satisfying the conditions (57) for ghost-free propagation. expressions for Π0 and Πj are the same as in Eqs. (47) The two examples with α< 2 and α> 1 are sufficient to and (48), respectively. There are no constraints in this cover all possible cases. model. Thus, according to Dirac’s counting algorithm Evidently a dilemma occurs in the generalized bum- there are n−n1 −n2 =4−0−0 = 4 independent degrees blebee model. If the coefficients τ1, τ2, τ3 are restricted of freedom. This is two more than in electromagnetism, to permit ghost-free propagation, then regions of the full and one more than in the KS bumblebee model. phase space allowed by the constraints can occur with These four degrees of freedom include three NG modes H < 0. This parallels the behavior in the KS bumble- and a massive mode. For arbitrary values of τ1, τ2, and bee model. With τ1, τ2, τ3 equal to Maxwell values, the τ3, all three NG modes can propagate, but with disper- allowed regions of phase space in the KS model include sion relations that depend on these coefficients. In con- solutions with H < 0. However, as demonstrated in a trast, in the KS model, with a Maxwell kinetic term, previous section, if initial values with λ = 0 are chosen, only two of the NG modes propagate as transverse pho- and current conservation holds, then λ = 0 and H > 0 tons. A massive mode occurs in either theory when ′ 2 2 2 for all time in the KS bumblebee model. V = 2κ(B0 − Bj − b ) 6= 0. In the generalized bum- Based on this, one could look for similar restrictions of blebee case, there is no analogue of Gauss’ law, and it the phase space in the case of the generalized bumblebee is possible for the massive mode to propagate. However, model. For example, the solutions with H < 0 described in the KS model with a timelike vector, the constraint above must typically have λ 6= 0 at t = 0 to satisfy the (33) provides a modified version of Gauss’ law, and the constraint φ4 ≈ 0. This suggests the idea of trying to massive mode is purely an auxiliary field that acts as a limit the choice of initial values to λ = 0 in an attempt nonlinear source of charge density in this relation. to exclude the possibility of solutions with H < 0 . The Hamiltonian for the generalized bumblebee has However, this idea seems unlikely to succeed in the the same form as in (50), but with the potential in the case of the generalized bumblebee model, since setting last term replaced by the expression in (5). With no λ =0at t = 0 is not sufficient to restrict the phase space constraints, the full phase space includes solutions with to solutions with λ = 0 for all time. This is because an unrestricted range of initial values. Thus, for any the equation of motion for λ has different dependence values of the coefficients τ1, τ2, τ3, there will either be on the other fields in the generalized bumblebee model propagating ghost modes or permissible initial choices for compared to the KS model. In particular, λ˙ is not pro- the fields and momenta with H < 0. portional to just λ itself. This is evident even in the linearized theory, with Bµ expanded as Bµ = bµ + Eµ. Applying the constraint analysis to the linearized theory j yields a first-order expression for λ in terms of Ej and Π 3. Quadratic Lagrange-Multiplier Potential equal to As a final example, the generalized bumblebee model 1 τ1 + τ3 j λ ≃ ∂j Π , (65) with a quadratic Lagrange-multiplier potential (6) can 2b τ1 − τ2 be considered as well. In this case there are ten fields µ (λ) while the equation of motion for λ in the linearized theory Bµ, Π , λ, and Π . The conjugate momenta are given is in (47) - (49). The Hamiltonian is the same as in (50), but with the potential replaced by (6). In this case, two 1 (τ2 + τ3)(τ1 + τ3) constraints are found, λ˙ ≃− (∂k∂k∂j Ej ). (66) 2b (τ1 − τ2) (λ) φ1 = Π , (67) The latter equation shows that (with non-Maxwell values 1 2 2 2 2 τ + τ 6= 0) λ˙ is independent of λ at linear order. There- φ2 = − (B0 − Bj − b ) . (68) 2 3 2 fore, even if λ =0at t = 0, nonzero values of λ can evolve over time. This makes it difficult to decouple regions of Constraint φ2 imposes the condition (2). However, it phase space with H > 0 in the generalized bumblebee involves a quadratic expression for this condition, and model purely by making a generic choice of initial val- therefore the system is irregular, and the same caveats ues. It would thus seem likely that the regions of phase must be applied as in the KS model. In particular, sub- ′ 2 2 2 space with H < 0 include solutions obeying λ = 0 at stitution of an equivalent constraint φ2 = (B0 − Bj − b ) t = 0. causes Dirac’s counting argument to fail. However, with φ1 and φ2 identified as first-class constraints, Dirac’s algorithm gives n − n1 − n2/2=5 − 2 − 0 = 3 degrees of 2. Quadratic Smooth Potential freedom. This is again one more than in the KS model. In this case, there is no massive mode, and λ decouples The generalized bumblebee model with a smooth completely. The three independent degrees of freedom quadratic potential (5) depends on four field components are the NG modes, which in the generalized bumblebee Bµ and their corresponding conjugate momenta. The can all propagate. However, even if values of τ1, τ2, and 11

TABLE I: Summary of constraints. Shown for each model are the number of primary (1o), secondary (2o), ﬁrst-class (FC), and second-class (SC) constraints, and the resulting number of independent degrees of freedom (DF). The last column indicates the µ regions of phase space that are ghost-free and have H > 0. Current conservation ∂µJ = 0 is assumed in the KS models.

Theory Kinetic Term Potential V Fields 1o 2o FC SC DF Ghost-Free, H > 0 1 F F µν A µ Electromagnetism − 4 µν – µ,Π 1 1 2 0 2 full phase space 1 F F µν A j ∂ j J 0 Nambu Model − 4 µν – j ,Π 0 0 0 0 3 subspace ( j Π = ) 1 B Bµν λ B Bµ b2 B µ λ (λ) λ KS Bumblebee − 4 µν ( µ ± ) µ,Π , ,Π 2 2 0 4 3 subspace ( = 0) τ τ τ 1 κ B Bµ b2 2 B µ B Bµ b2 ( 1 = 1, 2 = 3 = 0) 2 ( µ ± ) µ,Π 1 1 0 2 3 subspace ( µ = ) 1 λ B Bµ b2 2 B µ λ (λ) 2 ( µ ± ) µ,Π , ,Π 2 2 2 2 2 full phase space µ 2 µ (λ) General Bumblebee non-Maxwell λ(BµB ± b ) Bµ,Π , λ,Π 1 3 0 4 3 no subspace found τ τ τ 1 κ B Bµ b2 2 B µ (arbitrary 1, 2, 3) 2 ( µ ± ) µ,Π 0 0 0 0 4 no subspace found 1 λ B Bµ b2 2 B µ λ (λ) 2 ( µ ± ) µ,Π , ,Π 1 1 2 0 3 no subspace found

τ3 can be found that prevent these modes from propagat- spacelike vector Bµ. In this case, it is straightforward to ing as ghost modes, there are no other constraints in the show that the linearized KS model is equivalent to elec- theory that prevent initial-value choices that can yield trodynamics in an axial gauge [6]. However, additional solutions with H < 0. care is required in conducting a constraint analysis of the full nonlinear KS or generalized models, since B0 can vanish in the case of a spacelike vector, making addi- IV. SUMMARY & CONCLUSIONS tional singularities a possibility. Alternatively, an analysis in terms of the BRST formalism could be pursued, Table I summarizes the results of the constraint anal- which would be suitable as well for addressing questions ysis applied to electrodynamics, Nambu’s model, the KS of quantization. Lastly, an extension of the constraint bumblebee, and the generalized bumblebee. For each of analysis to a curved spacetime in the presence of gravity the bumblebee models, three types of potentials V are would be relevant, since ultimately bumblebee models are considered. The results show that no two models have of interest not only as effective field theories incorporat- identical constraint structures. In most cases, there are ing spontaneous Lorentz violation, but also as modified one or more additional degrees of freedom in comparison theories of gravity. For example, they are currently one to electromagnetism. These extra degrees of freedom are of the more widely used models for exploring implica- important both as possible additional propagating modes tions of Lorentz violation in gravity and cosmology and and in terms of how they alter the initial-value problem. in seeking alternative explanations of dark matter and In considering the stability of the bumblebee mod- dark energy. However, performing a constraint analy- els, it is not sufficient to look only at the propagating sis with gravity presents even greater challenges and is modes. The range of possible initial values must be ex- beyond the scope of this work. amined as well. In general, when the extra degrees of In summary, the constraint analysis presented here in freedom appearing in these models are allowed access to a flat-spacetime limit is useful in seeking insights into the full phase space, the Hamiltonians are not strictly the nature of theories with spontaneous Lorentz violation positive definite. However, in the KS models, it is pos- and what their appropriate interpretations might be. In sible to choose initial values for the fields and momenta particular, the KS bumblebee models offer the possibility that restrict the phase space to ghost-free regions with that Einstein-Maxwell theory might emerge as a result H > 0. In contrast, in models with generalized kinetic of spontaneous Lorentz breaking instead of through local terms obeying (τ2 + τ3) 6= 0, no such restrictions are U(1) gauge invariance. Indeed, in the flat-spacetime limit found. These theories either have propagating ghosts or of this model, with a timelike vacuum value, electromag- have extra degrees of freedom that evolve in such a way netism in a fixed nonlinear gauge is found to emerge in that makes it difficult to separate off restricted regions a well-defined region of phase space. of phase space with H > 0. In the end, it appears that only the KS models have a simple choice of initial values that can yield a physically viable theory in a restricted region of phase space. Acknowledgments The examples considered in this analysis all focused on the case of a timelike vector Bµ, which is the most widely We thank Alan Kosteleckýfor useful conversations. studied case in the literature, since it involves an observer This work was supported in part by NSF grant PHY- frame that maintains rotational invariance. A natural ex- 0554663. The work of R. P. is supported by the Por- tension of this work would be to consider models with a tuguese Funda¸cão para a Ciência e a Tecnologia. 12

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Erratum Appended to Published Version [Physical Review D 77, 125007 (2008)]

The Hamiltonian density term HB given in Eq. (60), in Section III D 1, is incorrect. A correct expression is 1 1 H = (τ − τ ) (∂ B )2 − (∂ B )2 − (τ + τ )(∂ B )2. (60) B 2 1 2 j k j 0 2 1 3 j j This term is used to examine the positivity of the Hamiltonian density H for the case of a model with a general kinetic term and a Lagrange-multiplier potential. The change in the term HB alters some of the conclusions that follow Eq. (60), which are stated in terms of a parameter α deﬁned in Eq. (56). A revised argument still assumes α > 0 and that Eq. (57) holds for ghost-free propagation. The argument and conclusions for α > 1 are unchanged, with the result that H can be negative. However, for 0 < α ≤ 1, a new examination has to be carried out. Using 2 2 Schwartz inequalities, it is found that HΠ ≥ 0 and (∂j Bk) − (∂j B0) ≥ 0. From the latter condition it follows that 1 2 HB ≥ − 2 (τ1 + τ3)(∂j Bj ) . For (τ1 + τ3) > 0, corresponding to α < 1, HB can be made arbitrarily negative, and solutions with H < 0 can therefore exist. However, for (τ1 + τ3) = 0, corresponding to α = 1, it follows that HB ≥ 0, 13 and that the Hamiltonian density H is nonnegative. The model with α = 1 and H ≥ 0 has a Lagrangian density 1 µ ν with a kinetic term proportional to L = − 2 (∂µBν )(∂ B ). 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 in Table I for the case with general values of τ1, τ2, τ3 and a linear Lagrange-multiplier potential. The model with the same kinetic term but with a quadratic Lagrange-multiplier (as discussed in Section III D 3) also has H > 0 when (τ1 + τ3) = 0. However, all of the other results in the paper, including the discussion of the KS model and the smooth quadratic potential, are unchanged by this correction.