constraints and stability in vector theories with spontaneous lorentz violation

Constraints and stability in vector theories with spontaneous Lorentz violation

Robert Bluhm,1 Nolan L. Gagne,1 Robertus Potting,2 and Arturs Vrublevskis1,3

1Physics Department, Colby College, Waterville, Maine 04901, USA2CENTRA, Departamento de Fısica, Faculdade de Ciencias e Tecnologia, Universidade do Algarve, Faro, Portugal

3Physics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Received 2 March 2008; published 3 June 2008)

Vector theories with spontaneous Lorentz violation, known as bumblebee models, are examined in flat

spacetime using a Hamiltonian constraint analysis. In some of these models, Nambu-Goldstone modes

appear with properties similar to photons in electromagnetism. However, depending on the form of the

theory, additional modes and constraints can appear that have no counterparts in electromagnetism. An

examination of these constraints and additional degrees of freedom, including their nonlinear effects, is

made for a variety of models with different kinetic and potential terms, and the results are compared with

electromagnetism. The Hamiltonian constraint analysis also permits an investigation of the stability of

these models. For certain bumblebee theories with a timelike vector, suitable restrictions of the initial-

value solutions are identified that yield ghost-free models with a positive Hamiltonian. In each case, the

restricted phase space is found to match that of electromagnetism in a nonlinear gauge.

DOI: 10.1103/PhysRevD.77.125007 PACS numbers: 11.30.Cp, 04.40.Nr, 11.10.Ef

I. INTRODUCTION

Investigations of quantum-gravity theories have uncov-ered a variety of possible mechanisms that can lead toLorentz violation. Of these, the idea that Lorentz symmetrymight be spontaneously broken [1] is one of the moreelegant. Spontaneous Lorentz violation occurs when avector or tensor field acquires a nonzero vacuum expecta-tion value. The presence of these background values pro-vides signatures of Lorentz violation that can be probedexperimentally. The theoretical framework for their inves-tigation is given by the standard-model extension (SME)[2,3]. Experimental searches for low-energy signals ofLorentz violation have opened up a promising avenue ofresearch in investigations of quantum-gravity phenomenol-ogy [4,5].

Theories with spontaneous Lorentz violation can alsoexhibit a variety of physical effects due to the appearanceof both Nambu-Goldstone (NG) and massive Higgs modes[6–8]. In the context of a gravitational theory, these effectsinclude modifications of gravitational propagation, as wellas altered forms of the static Newtonian potential, both ofwhich may be of interest in theoretical investigations ofdark energy and dark matter. Many investigations to datehave concentrated on the case of a vector field acquiring anonzero vacuum value. These theories, called bumblebeemodels [1,9,10], are the simplest examples of field theorieswith spontaneous Lorentz breaking. Bumblebee modelscan be defined with different forms of the potential andkinetic terms for the vector field, and with different cou-plings to matter and gravity [11–18]. They can be consid-ered as well in different spacetime geometries, includingRiemann, Riemann-Cartan, or Minkowski spacetimes.

Much of the interest in bumblebee models stems fromthe fact that they are theories without local U(1) gauge

symmetry, but which nonetheless allow for the propagationof massless vector modes. Indeed, one idea is that bum-blebee models, with appropriate kinetic and potentialterms, might provide alternative descriptions of photonsbesides that given by local U(1) gauge theory. In thisscenario, massless photon modes arise as NG modeswhen Lorentz violation is spontaneously broken.However, in addition to lacking local U(1) gauge invari-ance, bumblebee models differ from electromagnetism (inflat or curved spacetime) in a number of other ways. Forexample, 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 typi-cally this may involve the introduction of ghost modes intothe theory. Further differences arise due to the presence ofa potential term V in the Lagrangian density for bumblebeemodels. It is this term that induces spontaneous Lorentzbreaking. It can take a variety of forms, which may involveadditional excitations due to the presence of massivemodes or Lagrange-multiplier fields that have no counter-parts in electromagnetism.The goal of this paper is to investigate further the extent

to which bumblebee models can be considered as equiva-lent to electromagnetism or as containing electromagne-tism as a subset theory. This question is examined here inflat spacetime. While gravitational effects are a feature ofprimary interest in bumblebee models, any equivalence ormatch to electrodynamics would presumably hold as wellin an appropriate flat-spacetime limit. In a Minkowskispacetime, the main differences between bumblebee mod-els and electromagnetism are due to the nature of theconstraints imposed on the field variables and in the num-ber of physical degrees of freedom permitted by the theory.To investigate these quantities, a Hamiltonian constraintanalysis [19–22] is used. This approach is particularly well

PHYSICAL REVIEW D 77, 125007 (2008)

1550-7998=2008=77(12)=125007(12) 125007-1 � 2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.77.125007

suited for identifying the physical degrees of freedom in atheory with constraints. It can be carried out exactly withall nonlinear terms included. It also permits examination ofthe question of whether the Hamiltonian is bounded frombelow over the constrained phase space.

II. BUMBLEBEE MODELS ANDELECTROMAGNETISM

Bumblebee models are field theories with spontaneousLorentz violation in which a vector field acquires a nonzerovacuum value. For the case of a bumblebee field B�

coupled to gravity and matter, with generalized quadratickinetic terms involving up to second-order derivatives inB�, and with an Einstein-Hilbert term for the pure-gravity

sector, the Lagrangian density is given as

LB ¼ 1

16�GðR� 2�Þ þ �1B

�B�R�� þ �2B�B�R

� 1

4�1B��B

�� þ 1

2�2D�B�D

�B�

þ 1

2�3D�B

�D�B� � VðB�B

� � b2Þ þLM: (1)

In this expression, b2 > 0 is a constant, and in Riemannspacetime B�� ¼ @�B� � @�B�. The quantities �1, �2,

�1, �2, and �3 are fixed constants that determine the form ofthe kinetic terms for the bumblebee field. The term LM

represents possible interaction terms with matter fields orexternal currents. The potential VðB�B

� � b2Þ has a mini-

mum with respect to its argument or is constrained to zerowhen

B�B� � b2 ¼ 0: (2)

This condition is satisfied when the vector field has anonzero vacuum value

B� ¼ hB�i ¼ b�; (3)

with b�b� ¼ �b2. It is this vacuum value that spontane-

ously breaks Lorentz invariance.There are many forms that can be considered for the

potential VðB�B� � b2Þ. These include functionals in-

volving Lagrange-multiplier fields, as well as both poly-nomial and nonpolynomial functionals in ðB�B

� � b2Þ[1,11]. In this work, three limiting-case examples are con-sidered. They represent the dominant leading-order termsthat would arise in an expansion of a general scalar poten-tial V, comprised of vector fields B�, which are not simply

mass terms. They include examples that are widely used inthe literature. The first introduces a Lagrange-multiplierfield � and has a linear form,

V ¼ �ðB�B� � b2Þ; (4)

which leads to the constraint (2) appearing as an equationof motion. The second is a smooth quadratic potential,

V ¼ 12�ðB�B

� � b2Þ2; (5)

where � is a constant. The third again involves a Lagrange-multiplier field �, but has a quadratic form,

V ¼ 12�ðB�B

� � b2Þ2: (6)

With this form, the Lagrange-multiplier field � decouplesfrom the equations of motion for B�.

The model given in (1) involving a vacuum-valuedvector has a number of features considered previously inthe literature. For example, with the potential V and thecosmological constant� excluded, the resulting model hasthe form of a vector-tensor theory of gravity considered byWill and Nordtvedt [23,24]. Models with potentials (4) and(5) inducing spontaneous symmetry breaking were inves-tigated by Kostelecky and Samuel (KS) [1], while thepotential (6) was recently examined in [7]. The specialcases with a nonzero potential V, �1 ¼ 1, and �1 ¼ �2 ¼�2 ¼ �3 ¼ 0 are the original KS bumblebee models [1].Models with a linear Lagrange-multiplier potential (4),�1 ¼ �2 ¼ 0, but arbitrary coefficients �1, �2, and �3 arespecial cases (with a fourth-order term in B� omitted) of

the models described in Ref. [12].Since bumblebee models spontaneously break Lorentz

and diffeomorphism symmetry, it is expected that masslessNG and massive Higgs modes should appear in thesetheories. The fate of these modes was recently investigatedin [6,7]. The example of a KS bumblebee was consideredin detail. It was found that for all three potentials (4)–(6),massless NG modes can propagate and behave essentiallyas photons. However, in addition, it was found that massivemodes can appear that act as additional sources of energyand charge 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 of

the coefficients �1, �2, and �3 are expected to containmassless NG modes as well. However, in this case, sincethe kinetic terms are different, a match with electrodynam-ics is not expected. The non-Maxwell kinetic terms alterthe constraint structure of the theory significantly, and adifferent number of physical degrees of freedom canemerge.To compare the constraint structures of different types of

bumblebee models with each other and with electrodynam-ics, the flat-spacetime limit of (1) is considered. TheLagrangian density in this case reduces to

L ¼ �14�1B��B

�� þ 12�2@�B�@

�B� þ 12�3@�B

�@�B�

� VðB�B� � b2Þ � B�J

�: (7)

For simplicity, interactions consisting of couplings with an

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externally prescribed current J� are assumed, and aMinkowski metric �� in Cartesian coordinates with sig-

nature ðþ;�;�;�Þ is used.Following a Lagrangian approach, second-order differ-

ential equations of motion for B� are obtained. They are

ð�1 þ �3Þ½hB� � @�@�B��

� ð�2 þ �3ÞhB� � 2V 0B� � J� ¼ 0: (8)

Here, V 0 denotes variation of the potential VðXÞ withrespect to its argument X. Since the NG modes stay inthe minimum of the potential, a nonzero value of V 0indicates the presence of a massive-mode excitation.Taking the divergence of these equations gives

@�½ð�2 þ �3ÞhB� þ 2V 0B� þ J�� ¼ 0: (9)

Clearly, as expected, with V ¼ V 0 ¼ 0, �1 ¼ 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 V0 � 0, or if arbitrary values of �1, �2, �3 areallowed, then a modified set of equations holds.

In flat spacetime, the KS bumblebee has a nonzeropotential V and coefficients �1 ¼ 1 and �2 ¼ �3 ¼ 0. Itsequations of motion evidently have a close resemblance tothose of electrodynamics. The main difference is that theKS bumblebee field itself acts nonlinearly as a source ofcurrent. Equation (9) shows that the matter current J�combines with the term 2V 0B� to form a conserved

current.Interestingly, if the matter current J� is set to zero, and a

linear Lagrange-multiplier potential (4) is used, the KSmodel in flat spacetime reduces to a theory considered byDirac long before the notion of spontaneous symmetrybreaking had been introduced [25]. Dirac investigated avector theory with a nonlinear constraint identical to (2)with the idea of finding an alternative explanation ofelectric charge. In his model, gauge invariance is de-stroyed, and conserved charge currents appear only as aresult of the nonlinear term involving V 0 for the Lagrange-multiplier potential. Dirac did not, however, propose atheory of Lorentz violation. A vacuum value b� was never

introduced, and with J� ¼ 0 no Lorentz-violating interac-

tions with matter enter in the theory.The idea that the photon could emerge as NG modes in a

theory with spontaneous Lorentz violation came more thanten years after the work of Dirac. First, Bjorken proposed amodel in which collective excitations of a fermion fieldcould lead to composite photons emerging as NG modes[26]. The observable behavior of the photon in this originalmodel was claimed to be equivalent to electrodynamics.Subsequently, Nambu recognized that the constraint (2)imposed on a vector field could also lead to the appearanceof NG modes that behave like photons [27]. He introduceda vector model that did not involve a symmetry-breaking

potential V. Instead, the constraint (2) was imposed as anonlinear U(1) gauge-fixing condition directly at the levelof the Lagrangian. The resulting gauge-fixed theory thuscontained only three independent vector-field componentsin the Lagrangian. Nambu demonstrated that his modelwas equivalent to electromagnetism and stated that thevacuum vector can be allowed to vanish to restore fullLorentz invariance.In contrast to these early models, the KS bumblebee was

proposed as a theory with physical Lorentz violation. Evenif the NG modes are interpreted as photons in the KSmodel, and no massive modes are present, interactionsbetween the vacuum vector b� and the matter current J�provide clear observable signals of physical Lorentz vio-lation. However, the presence of a potential V also allowsadditional degrees of freedom to enter in the KS model. Ifarbitrary values of the coefficients �1, �2, and �3 arepermitted as well, the resulting theory can differ substan-tially from electromagnetism.Since many of these models contain unphysical modes,

either as auxiliary or Lagrange-multiplier fields, constraintequations are expected to hold. It is the nature of theseconstraints that determines ultimately how many physicaldegrees of freedom occur in a given model. With Dirac’sHamiltonian constraint analysis, a direct procedure existsfor determining the constraint structure and the number ofphysical degrees of freedom in these models.

III. HAMILTONIAN CONSTRAINTANALYSIS

Given a Lagrangian density L describing a vector fieldB�, the canonical Hamiltonian density is H ¼��@0B� �L, where the canonical momenta are defined

as

�� ¼ Lð@0B�Þ : (10)

If additional fields, e.g., Lagrange multipliers �, are con-tained in the theory, additional canonical momenta for

these quantities are defined as well, e.g., �ð�Þ. (Note:here � is not a spacetime index.) In the Hamiltonianapproach, time derivatives of a quantity f are computedby taking the Poisson bracket with the Hamiltonian H,

_f ¼ ff;Hg þ @f

@t: (11)

The second term is needed with quantities that have ex-plicit time dependence, e.g., an external current J�.In Dirac’s constraint analysis, primary and secondary

constraints are determined, and these are identified aseither first class or second class. In the phase space awayfrom the constraint surface, the canonical Hamiltonian isambiguous up to additional multiples of the constraints. Anextended Hamiltonian is formed that includes multiples ofthe constraints with coefficients that can be determined, orin the case of first-class constraints, remain arbitrary. It is

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the extended Hamiltonian that is then used in (11) todetermine the equations of motion for the fields and con-jugate momenta.

A system of constraints is said to be regular if theJacobian matrix formed from variations of the constraintswith respect to the set of field variables and conjugatemomenta has maximal rank. If it does not, the system issaid to be irregular, and some of the constraints are typi-cally redundant. Dirac argued that theories with primaryfirst-class constraints have arbitrary or unphysical degreesof freedom, such as gauge degrees of freedom. These typesof constraints therefore allow removal of two field ormomentum components. Dirac conjectured that this istrue as well for secondary first-class constraints. Basedon this, a counting argument can be made. It states thatin a theory with n field and n conjugate-momentum com-ponents, if there are n1 first-class constraints and n2second-class constraints, the number of physical indepen-dent degrees of freedom is n� n1 � n2=2. (Note: it can beshown that n2 is even.) This counting argument based onDirac’s conjecture holds up well for theories with regularsystems of constraints. However, counterexamples areknown for irregular systems [22].

Once the unphysical modes have been eliminated, byapplying the constraints and/or imposing gauge conditions,the evolution of a physical system is determined by theequations of motion for the physical fields and momenta,subject to initial conditions for these quantities. Any bum-blebee theory that has additional degrees of freedom incomparison to electrodynamics must therefore specify ad-ditional initial values. The subsequent evolution of theextra degrees of freedom typically leads to effects that donot occur in electrodynamics. However, in some cases,equivalence with electrodynamics can hold in a subspaceof the phase space of the modified theory. For this to occur,initial values must exist that confine the evolution of thetheory to a region of phase space that matches electro-dynamics 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 > 0.In a quantum theory, instability in any region of the clas-sical phase space might be expected to destabilize the fulltheory. However, bumblebee models, with gravity in-cluded, are intended as effective theories presumablyemerging at or below the Planck scale from a more funda-mental (and unknown) quantum theory of gravity. In thiscontext, quantum-gravity effects might impose additionalconstraints leading to stability. However, in the absence ofa fundamental theory, the question of the ultimate stabilityof bumblebee models cannot be addressed. For this reason,

in the subsequent sections, only the behavior of bumblebeemodels in classical phase space is considered.The following sections apply Dirac’s constraint analysis

to a number of different bumblebee models, including theKS bumblebee as well as more general cases with arbitraryvalues of the coefficients �1, �2, �3. Since much of theliterature has focused on the case of a timelike vector B�,

this restriction is assumed throughout this work as well.With this assumption, there always exists an observerframe in which rotational invariance is maintained andonly Lorentz boosts are spontaneously broken. For eachtype of model to be considered, all three of the potentials in(4)–(6) are considered. For comparison (and use as bench-marks), electromagnetism and the theory of Nambu areconsidered as well. In each case, the explicit form of theLagrangian is obtained from (7) by inserting appropriatevalues for V, �1, �2, and �3, and the conjugate momentaand Hamiltonian are then computed. For example, electro-dynamics is obtained by setting V ¼ 0, �1 ¼ 1, and �2 ¼�3 ¼ 0. Conventional notation sets B� ¼ A� and B�� ¼F��. The Hamiltonian is given in terms of the four fields

A� and their conjugate momenta ��. The Lagrangian in

Nambu’s model also starts with these same values [allow-ing U(1) invariance]. However, in this case, one componentof A� is eliminated in terms of the remaining three, using

the nonlinear condition in (2). For the case of a timelike

vector, the substitution A0 ¼ ðb2 þ A2j Þ1=2 is made directly

in L. The resulting Hamiltonian in Nambu’s model there-fore depends only on three fields Aj and three conjugate

momenta �j. In contrast, bumblebee models are definedwith a nonzero potential V and have Hamiltonians thatdepend on all four fields B� and their corresponding con-

jugate momenta��. Examples with a Lagrange-multiplierpotential involve a fifth field � and its conjugate momen-

tum �ð�Þ. However, in examples with a smooth quadraticpotential, there is no Lagrange multiplier, and the relevantfields and momenta are B� and ��.

A. Electromagnetism

The conjugate momenta in electrodynamics are

�j ¼ @0Aj � @jA0; �0 ¼ 0: (12)

The latter constitutes a primary constraint, 1 ¼ �0 � 0.It leads to a secondary constraint, 2 ¼ @j�

j � J0 � 0,

which is Gauss’ law, since �j can be identified as theelectric field components Ej and J0 is the charge density.In these expressions and below, Dirac’s weak equalitysymbol ‘‘�’’ is used to denote equality on the submanifolddefined by the constraints [22]. Both of the constraints 1

and 2 are first class, indicating that there are gauge orunphysical degrees of freedom. Following Dirac’s count-ing argument, there should be n� n1 � n2=2 ¼ 4� 2�0 ¼ 2 independent physical degrees of freedom. These arethe two massless transverse photon modes.


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The canonical Hamiltonian in electrodynamics is

H ¼ 12ð�jÞ2 þ�j@jA0 þ 1

2ðFjkÞ2 þ A�J�: (13)

In the presence of a static charge distribution, with J� ¼ð�; ~JÞ ¼ ð�ð ~xÞ; 0Þ, no work is done by the external current,and the Hamiltonian is positive definite. To observe this,integrate by parts and use the constraint2 (Gauss’ law) toshow that H ¼ 1

2 ð�iÞ2 þ 12 ðFjkÞ2 � 0.

The equations of motion for the fields A� and momenta

�� obtained from the extended Hamiltonian contain arbi-trary functions due to the existence of the first-class con-straints. These can be eliminated by imposing gauge-fixingconditions. The evolution of the physical degrees of free-dom, subject to a given set of initial values, is then deter-mined 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 invarianceand a conserved current J�. For the case of a timelike

vector A�, the condition A0 ¼ ðb2 þ A2j Þ1=2 is substituted

directly into the Lagrangian as a gauge-fixing condition.The result is

L ¼ 12ð@0AjÞ2 þ 1

2ð@jðb2 þ A2kÞ1=2Þ2 � 1

2ð@jAkÞ2þ 1

2ð@jAkÞð@kAjÞ � ð@jðb2 þ A2kÞ1=2Þð@0AjÞ

� ðb2 þ A2kÞ1=2J0 � AjJ

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 electromag-netic field in a standard gauge into the field A� obeying the

nonlinear gauge condition A�A� ¼ b2.

The Hamiltonian in Nambu’s model is

H ¼ 12ð�jÞ2 þ 1

2ðFjkÞ2 þ�j@jðb2 þ A2kÞ1=2

þ ðb2 þ A2kÞ1=2J0 þ AjJ

j: (15)

It depends on three field components Aj and their conju-

gate momenta�j ¼ @0Aj � @jðb2 þ A2kÞ1=2. In this theory,

there are no constraints, and therefore application ofDirac’s counting argument says that there are three physi-cal degrees of freedom, which is one more than in electro-magnetism. An extra degree of freedom arises becausegauge fixing at the level of the Lagrangian causes Gauss’law, @j�

j � J0 ¼ 0, to disappear as a constraint equation.

A similar disappearance of Gauss’ law is known to occur inelectrodynamics in temporal gauge (with A0 ¼ 0 substi-tuted in the Lagrangian) [28]. Indeed, the linearized limitof Nambu’s model with a timelike vector field is electro-dynamics in temporal gauge.

Observe that with ~J ¼ 0 and using integration by parts,the Hamiltonian can be rewritten as

H ¼ 12ð�jÞ2 þ 1

2ðFjkÞ2 � ð@j�j � J0Þðb2 þ A2kÞ: (16)

In the absence of a constraint enforcing Gauss’ law, Hneed not be positive definite. For example, if the extradegree of freedom in Aj causes large deviations from

Gauss’ law, which are not forbidden by any constraint,then negative values of H can occur.However, equivalence between Nambu’s model and

electrodynamics can be established by restricting the phasespace in Nambu’s theory. To see that this follows, considerthe equations of motion in Nambu’s model,

_A j ¼ �j þ @jðb2 þ A2kÞ1=2; (17)

_� j ¼ @k@kAj � @j@kA

k � @l�l Aj

ðb2 þ A2kÞ1=2

þ AjJ0

ðb2 þ A2kÞ1=2

� Jj: (18)

Taking the spatial divergence of (18) and using currentconservation yields the nonlinear relation

@0ð@j�j � J0Þ ¼ �@j

�ð@l�l � J0Þ Aj

ðb2 þ A2kÞ1=2

�: (19)

This equation shows that if Gauss’ law, ð@j�j � J0Þ ¼ 0,

holds at t ¼ 0, then @0ð@j�j � J0Þ ¼ 0 at t ¼ 0 as well.

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� as a

ghost mode. The KS Lagrangian is obtained from (7) bysetting �1 ¼ 1 and �2 ¼ �3 ¼ 0. The constraint structuresfor models with each of the three potentials (4)–(6) areconsidered. For definiteness, the case of a timelike vectorB� is assumed.

1. Linear Lagrange-multiplier potential

With a linear Lagrange-multiplier potential (4), an addi-tional field component � is introduced in addition to thefour fields B0 and Bj. The conjugate momenta are

�0 ¼ �ð�Þ ¼ 0; �i ¼ @0Bi � @iB0; (20)

and the canonical Hamiltonian is


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H ¼ 12ð�iÞ2 þ�i@iB0 þ 1

2ð@iBjÞ2 � 12ð@jBiÞð@iBjÞ

þ �ðB20 � B2

i � b2Þ þ B�J�: (21)

Four constraints are identified as

1 ¼ �0; (22)

2 ¼ �ð�Þ; (23)

3 ¼ @i�i � 2�B0 � J0; (24)

4 ¼ �ðB20 � B2

j � b2Þ: (25)

The constraints 1 and 2 are primary, while 3 and 4

are secondary. All four are second class.Applying Dirac’s algorithm to determine the number of

independent degrees of freedom gives n� n1 � n2=2 ¼5� 2� 2=2 ¼ 3. Hence, there is an extra degree of free-dom in the KS bumblebee model in comparison to electro-dynamics. It arises due to the presence of the extra field �and the 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 3 gives a modified form of Gauss’ law inwhich the combination 2�B0 acts as a source of chargedensity. Since V 0 ¼ � in this example, any excitation of thefield � is away from the potential minimum and thereforeacts effectively as a massive Higgs mode [7]. In curvedspacetime, such a mode can modify both the gravitationaland electromagnetic potentials of a point particle.However, here, in flat spacetime, the presence of � leadsonly to modifications of the Coulomb potential.

The Hamiltonian with ~J ¼ 0 reduces, after using 3,4, and integration by parts, to

H ¼ 12ð�jÞ2 þ 1

2ðBjkÞ2 � 2�B20: (26)

The full phase space of the theory on the constraint surfaceincludes regions in which H is negative due to the pres-ence of the additional degree of freedom. For example,consider the case with J0 ¼ 0 and initial values [29] Bj ¼@jð ~xÞ and �j ¼ �@jðb2 þ ð@kÞ2Þ1=2 at t ¼ 0, where

ð ~xÞ is an arbitrary time-independent scalar. These give

Bjk ¼ 0 and B0 ¼ ðb2 þ ð@jÞ2Þ1=2 at t ¼ 0. Inserting

these initial values in (26) reduces the Hamiltonian toH ¼ � 1

2 ð�jÞ2 at t ¼ 0. The corresponding initial value

for � is

� ¼ �12ðb2 þ ð@jÞ2Þ�1=2½ ~r2ðb2 þ ð@kÞ2Þ1=2�: (27)

Evidently, the Hamiltonian in the classical KS bumblebeemodel can be negative when nonzero values of � areallowed.

However, if initial values are chosen that restrict thephase space to values with � ¼ 0, the resulting solutionsfor the vector field and conjugate momentum are equiva-lent to those in electromagnetism in a nonlinear gauge.

Examination of the equation of motion for �,

_� ¼ 1

B0

@jð�BjÞ � 1

2B0

@�J� � �

Bj

ðB0Þ2ð�j þ @jB0Þ;

(28)

reveals that if the current J� is conserved, and � ¼ 0 attime zero, then � will remain zero for all time. TheHamiltonian in this case is positive. The equations ofmotion for Bj and �j are

_B j ¼ �j þ @jB0; (29)

_� j ¼ @k@kBj � @j@kBk � Jj þ 2�Bj: (30)

With � ¼ 0, these combine to give the usual Maxwellequations describing massless transverse photons. Thethird component in Bj is an auxiliary field that is con-

strained by the usual form of Gauss’ law when � ¼ 0.Note, however, that even with the phase space restrictedto regions with � ¼ 0, the matter sector of the theory willexhibit signatures of the spontaneous Lorentz violationthrough the interaction of the vacuum value b� with the

matter current J�.It is clear from these results that conservation of the

matter current J� is necessary for the stability of the KSbumblebee model. Note, however, that the theories lacklocal U(1) gauge invariance and that the current conserva-tion could arise simply from matter couplings that areinvariant under a global U(1) symmetry. As a result, pho-tons in the KS bumblebee model appearing as NG modesare due to spontaneous Lorentz breaking, not local U(1)gauge invariance. For further discussion of the bumblebeecurrents, including in the presence of gravity, see Ref. [7].In that work, there is also further discussion of the fact thatthe Lagrange-multiplier field can act as a source of chargedensity in the KS bumblebee model and that there can existsolutions (with nonzero values of �) in which the field linesconverge or become singular, even in the absence of mattercharge. This behavior has been referred to in the literatureas the formation of caustics in the KS model. However, asdescribed in [7], it is simply a natural consequence of thefact that the bumblebee fields themselves act as sources ofcurrent. Moreover, with the phase space restricted to re-gions with � ¼ 0, the only singularities appearing for thecase of a timelike vector B� are those due to the presence

of matter charge as in ordinary electrodynamics with a 1=rpotential.

2. Quadratic smooth potential

A similar analysis can be performed for a KS bumblebeewith the smooth quadratic potential defined in (5). Theparameter � appearing in V is a constant. Therefore, in thiscase, there are four fields B0, Bj, and their four conjugate

momenta,

�0 ¼ 0; �j ¼ @0Bj � @jB0: (31)


125007-6

There are two constraints,

1 ¼ �0; (32)

2 ¼ @j�j � 2�B0ðB2

0 � B2j � b2Þ � J0; (33)

where 1 is primary, 2 is secondary, and both are secondclass. Dirac’s counting argument says there are n� n1 �n2=2 ¼ 4� 0� 2=2 ¼ 3 independent degrees of freedom,which again is one more than in electromagnetism.

The condition (2) does not occur as a constraint in thiscase. Instead, an extra degree of freedom appears as amassive Higgs excitation V0 ¼ 2�B0ðB2

0 � B2j � b2Þ � 0

away from the potential minimum. The constraint 2

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 ¼ 0 can beexamined. Using the constraints and integration by partsgives

H ¼ 12ð�jÞ2 þ 1

2ðBjkÞ2� 1

2�ð3B20 þ B2

j þ b2ÞðB20 � B2

k � b2Þ; (34)

which evidently is not positive over the full phase space. Ifa nonzero massive mode proportional to ðB2

0 � B2j � b2Þ is

present, negative values of H can occur.However, equivalence to electrodynamics does hold in a

restricted region of phase space. To verify this, consider theequations of motion,

2� _B0 ¼ ð3B20 � B2

j � b2Þ�1½4�B0Bkð�k þ @kB0Þþ 2�@k½BkðB2

0 � B2l � b2Þ� þ @�J

��; (35)

_B j ¼ �j þ @jB0; (36)

_� 0 ¼ @j�j � J0 � 2�B0ðB2

0 � B2j � b2Þ; (37)

_� j ¼ @k@kBj � @j@kBk þ 2�BjðB20 � B2

k � b2Þ � Jj:

(38)

Combining these gives

�@0ðB20 � B2

j � b2Þ ¼ ð3B20 � B2

j � b2Þ�1

� ½2�B0@k½BkðB20 � B2

l � b2Þ�þ B0@�J

� � 2�ðB20 � B2

k � b2Þ� Blð�l þ @lB0Þ�: (39)

This equation reveals that if the current J� is conservedand ðB2

0 � B2j � b2Þ ¼ 0 at t ¼ 0, then ðB2

0�B2j �b2Þ¼0

for all time. Therefore, with these conditions imposed, themassive mode never appears, the Hamiltonian is positive,and the phase space is restricted to solutions in electro-magnetism in the nonlinear gauge (2).

In theories with a nonzero massive mode, the size of themass scale �b2 becomes relevant. For very large values,

perturbative excitations that go up the potential minimumwould be expected to be suppressed. Since the mass scaleassociated with spontaneous Lorentz violation is presum-ably the Planck scale, its appearance necessarily bringsgravity into the discussion. It is at the Planck scale wherequantum-gravity effects might impose additional con-straints that could maintain the overall stability of thetheory. At sub-Planck energies, massive-mode excitationshave been shown to exert effects on classical gravity. Forexample, as shown in Ref. [7], the gravitational potential ofa point particle is modified. However, in the limit where themass of the massive mode becomes exceptionally large, itwas found for the case of the KS bumblebee model thatboth the usual Newtonian and Coulomb potentials arerecovered.

3. Quadratic Lagrange-multiplier potential

The KS bumblebee model with a quadratic Lagrange-multiplier potential (6) involves five fields, � and B�. In a

Lagrangian approach, the constraint (2) follows from theequation of motion for �. The on-shell equations of motionfor B� are the same as in electromagnetism. In this case,

the field � decouples and does not act as a source of chargedensity. On shell, the potential obeys V 0 ¼ 0, current con-servation @�J

� ¼ 0 holds, and there is no massive mode.

This model provides an example of a theory with physicalLorentz violation due to the matter couplings with J�.Nonetheless, in the electromagnetic sector, the theory isequivalent to electromagnetism in the nonlinear gauge (2).However, the Hamiltonian formulation of this model

involves 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

�0 ¼ 0; (40)

�j ¼ @0Bj � @jB0; (41)

�ð�Þ ¼ 0: (42)

From these, four constraints can be identified,

1 ¼ �0; (43)

2 ¼ �ð�Þ; (44)

3 ¼ @j�j � 2�B0ðB2

0 � B2j � b2Þ � J0; (45)

4 ¼ �12ðB2

0 � B2j � b2Þ2: (46)

With 4 � 0, the constraint surface is limited to fieldsobeying ðB2

0 � B2j � b2Þ ¼ 0, and 3 reduces to Gauss’

law. In this case, 2 and 4 can be identified as first class,while 1 and 3 are second class. Dirac’s counting algo-


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rithm then states that there are n� n1 � n2=2 ¼ 5� 2�2=2 ¼ 2 independent degrees of freedom, which matcheselectromagnetism, and the Hamiltonian is positivethroughout the full physical phase space. However, if in-stead the squared constraint 4 is replaced by the equiva-lent constraint 0

4 ¼ ðB20 � B2

j � b2Þ that spans the same

constraint surface, then a different set of results holds. Inthis case, additional constraints appear from the Poisson-bracket relations that are not equivalent to the set definedabove, and Dirac’s counting algorithm fails to determinethe correct number of degrees of freedom. The resultingtheory with 0

4 replacing 4 is not equivalent to theLagrangian approach.

Evidently, care must be used in working with a squaredconstraint equation. The constraints 0

4 and 4 are redun-dant, and the Hamiltonian system is irregular. Nonetheless,with these caveats, the KS model with a squared Lagrange-multiplier potential provides a useful model of spontane-ous Lorentz violation. It allows an implementation of thesymmetry breaking that does not require enlarging thephase space to include a massive mode or nonlinear cou-plings with �. The only physical degrees of freedom in thetheory are the NG modes that behave as photons.

D. Bumblebee models with ð�2 þ �3Þ � 0

In this section, the constraint analysis is applied tobumblebee models in flat spacetime that have aLagrangian (7) with a generalized kinetic term obeyingð�2 þ �3Þ � 0. Such models do not have a Maxwell formfor the kinetic term. Throughout this section, arbitraryvalues of �1, �2, and �3 are used; however, it is assumedthat discontinuities are avoided when these parametersappear in the denominators of equations. The three poten-tials in (4)–(6) are considered, and B� is assumed to be

timelike. Since the kinetic term is not of the Maxwell form,it is not expected that the NG modes in these types ofmodels can be interpreted as photons. For this reason, theinteraction term B�J

� is omitted in this section.

The point of view here is that the generalized bumblebeemodels originate from a vector-tensor theory of gravitywith spontaneous Lorentz violation induced by the poten-tial V. In this context, the vector fields B� have no matter

couplings and reduce to sterile fields in a flat-spacetime

limit. Nonetheless, NG modes and massive modes canappear in this limit. Dirac’s Hamiltonian analysis is usedto examine the constraint structure and the number ofphysical degrees of freedom associated with these modes.Comparisons can then be made with the results in electro-magnetism and the KS bumblebee models.

1. Linear Lagrange-multiplier potential

Beginning with a model with the linear Lagrange-multiplier potential in Eq. (4), the Lagrangian is given interms of the five fields B0, Bj, and �. From this the

conjugate momenta are found to be

�0 ¼ ð�2 þ �3Þð@0B0Þ � �3ð@jBjÞ; (47)

�j ¼ ð�1 � �2Þð@0BjÞ � �1ð@jB0Þ; (48)

�ð�Þ ¼ 0: (49)

The canonical Hamiltonian is then given as

H ¼��21 � ð�1 � �2Þ2

2ð�1 � �2Þ�ð@jB0Þ2 þ

�1

2ð�1 � �2Þ�ð�jÞ2

þ�

�1�1 � �2

��jð@jB0Þ þ 1

2ð�1 � �2Þð@jBkÞ2

� 1

2�1ð@jBkÞð@kBjÞ þ

�1

2ð�2 þ �3Þ�ð�0Þ2

þ�

�3�2 þ �3

��0@jBj �

��2�3

2ð�2 þ �3Þ�ð@jBjÞ2

þ �ðB20 � B2

i � b2Þ: (50)

Four constraints are found for this model:

1 ¼ �ð�Þ; (51)

2 ¼ �ðB20 � B2

j � b2Þ; (52)

3 ¼ �Bj

�1

�1 � �2�j þ �1

�1 � �2ð@jB0Þ

�

þ B0

�1

�2 þ �3�0 þ �3

�2 þ �3ð@jBjÞ

�; (53)

4 ¼ ��ðB0Þ2 � �

��2 þ �3�1 � �2

�ðBjÞ2 �

��1�3

2ð�1 � �2Þ þ�21ð�2 þ �3Þ2ð�1 � �2Þ2

�ð@jB0Þ2 þ 1

2

��23

�2 þ �3þ �1�3

�1 � �2

�ð@jBjÞ2

� 1

2ð�2 þ �3ÞBj@k@kBj � 1

2

��23 � ð�1 þ �3Þð�2 þ �3ÞÞ

�1 � �2

�Bj@j@kBk þ

��21 � ð�1 � �2Þ2

2ð�1 � �2Þ�B0@j@jB0

� �32ð�1 � �2ÞBjð@j�0Þ þ �1

2ð�1 � �2ÞB0ð@j�jÞ � 1

ð�1 � �2Þ�1

2�3 þ �1ð�2 þ �3Þ

�1 � �2

��j@jB0

þ�

�3�1 þ �3

þ �12ð�1 � �2Þ

��0ð@jBjÞ þ 1

2ð�2 þ �3Þ ð�0Þ2 � �2 þ �3

2ð�1 � �2Þ2ð�jÞ2: (54)


125007-8

The constraint 1 is primary, while 2, 3, and 4 aresecondary. All four are second class. According to Dirac’scounting argument there are n� n1 � n2=2 ¼ 5� 0�4=2 ¼ 3 degrees of freedom in this model.

The constraint2 shows that only three of the four fieldsB� are independent. In the timelike case, it is natural to

solve for B0 in terms of Bj. The first and third constraints

can be used, respectively, to fix �ð�Þ to zero and to deter-mine �0 in terms of Bj and �j. The remaining constraint

4 can be used to determine � in terms of Bj and �j.

Interestingly, this leaves the same number of independentdegrees of freedom as in the KS bumblebee model with asimilar potential. One might have thought that switchingfrom a Maxwell kinetic term, which results in the removalof a primary constraint�0 ¼ 0, would have introduced anadditional degree of freedom. However, instead, new sec-ondary constraints appear that still constrain �0, thoughnot to zero. As a result, B0 and �0 remain unphysicaldegrees of freedom despite the change in the kinetic term.

Since the generalized bumblebee model is not viewed asa modified theory of electromagnetism (e.g., no current J�

is introduced), there is no analogue or modified version ofGauss’ law as there is in the KS bumblebee model.Nonetheless, in the constraint 4, � plays a similar roleas a nonlinear source term for the other fields as it does inthe KS bumblebee. Indeed, the constraint equation4 � 0reduces to the same modified form of Gauss’ law as in (24)with J0 ¼ 0 in the limit where �0 ! 0, and the coeffi-cients �1, �2, �3 take Maxwell values. Thus, when consid-ering initial values of the independent fields Bj and �j in

the generalized bumblebee case, the constraint4 can playa role similar to that of the modified Gauss’s law in the KSbumblebee model.

Restrictions on the coefficients �1, �2, �3 can be foundby examining the freely propagating modes in the theory.Investigations along these lines with gravity included havebeen carried out by a number of authors [12,30]. Since thetheory with generalized kinetic terms has 3 degrees offreedom, there can be up to three independent propagatingmodes. These include the NG modes associated with thespontaneous Lorentz breaking. To determine their behav-ior, it suffices to work in a linearized limit and to look forsolutions in the form of harmonic waves. Carrying this outin the Hamiltonian formulation requires combining thelinearized equations of motion to form a wave equationfor Bj. For physical propagation, i.e., to avoid signs in the

kinetic term that give rise to ghost modes, the conditionð�1 � �2Þ> 0 must hold [30].

In this case, two solutions are found that propagate astransverse massless modes at the speed of light. However, athird longitudinal mode can be found as well. In an ob-server framewith wave vector k� ¼ ðk0; 0; 0; k3Þ, it obeys azero-mass dispersion relation of the form

ð�1 � �2Þk20 þ ð�2 þ �3Þk23 ¼ 0: (55)

For physical velocities, the ratio

� � k20=k23 ¼ � �2 þ �3

�1 � �2(56)

must be positive, which together with the requirement ofghost-free propagation gives

ð�1 � �2Þ> 0; ð�2 þ �3Þ< 0: (57)

Note in comparison that the KS bumblebee model hasð�2 þ �3Þ ¼ 0, and therefore the third degree of freedomdoes not propagate as a harmonic wave. Instead, it is anauxiliary 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 this shouldinclude consideration of possible initial values at t ¼ 0 thatsatisfy the constraints. Using integration by parts and2 �0, the Hamiltonian (50) can be written as the sum of twoparts,

H ¼ H� þH B: (58)

The first,

H� ¼ 1

2ð�1 � �2Þ ð�j þ �1@jB0Þ2

þ 1

2ð�2 þ �3Þ ð�0 þ �3@jBjÞ2; (59)

includes dependence on the momenta, while the second,

H B ¼ � �1 � �22

ð@jB0Þ2 � 2�1 � �2 þ �32

ð@jBjÞ2

� �1 � �24

ð@iBj � @jBiÞ2; (60)

depends only on the fields B�.

First consider H B. From the condition for ghost-freepropagation in (57), it follows that the first and third termsare nonpositive. The second term is nonpositive as well if2�1 � �2 þ �3 > 0, which implies�< 2. ThusH B 0 ifthe conditions (57) hold and �< 2.Next consider the momentum-dependent term H�.

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 the constraint 3. However, one choiceof initial values that makes both terms vanish (and there-fore satisfies 3 � 0) is

�j þ �1@jB0 ¼ �0 þ �3@jBj ¼ 0: (61)

The initial value of � is then chosen to make4 vanish, and

B0 ¼ ðb2 þ B2j Þ1=2 is used to make 2 � 0. Consequently,

with H� vanishing, if �< 2, and the condition (57)holds, then there exist initial conditions with H < 0.To investigate the remaining cases, corresponding to

other possible values of � consistent with (57), use theconstraint 3 to rewrite H� as


125007-9

H � ¼ 1

2ð�1 � �2Þ�ð�j þ �1@jB0Þ2

� �½Bjð�j þ �1@jB0Þ�2

B20

�: (62)

In any volume element, choose initial values for Bj of the

form ðB1; B2; B3Þ ¼ ð0; 0; Bð ~xÞÞ. It then follows that

H� ¼ 1

2ð�1 � �2Þ�ð�1 þ �1@1B0Þ2 þ ð�2 þ �1@2B0Þ2

þ�1� �

B2

b2 þ B2

�ð�3 þ �1@3B0Þ2

�: (63)

With this form, initial values of the components�1 and�2

can be chosen that make the first two terms in this expres-sion vanish. The third term becomes negative for any �>1, provided an initial value of B2 is chosen that obeys

B2 >b2

�� 1: (64)

With H� < 0, and �3 þ �1@3B0 � 0, the initial value of�3 can then be made arbitrarily large so that the totalinitial Hamiltonian densityH ¼ H� þH B is negative,even if H B > 0.

Thus, the Hamiltonian density H can take negativeinitial values for any choice of the parameters �1, �2, �3satisfying the conditions (57) for ghost-free propagation.The two examples with �< 2 and �> 1 are sufficient tocover all possible cases.

Evidently a dilemma occurs in the generalized bumble-bee model. If the coefficients �1, �2, �3 are restricted topermit ghost-free propagation, then regions of the fullphase space allowed by the constraints can occur withH < 0. This parallels the behavior in the KS bumblebeemodel. With �1, �2, �3 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 � ¼ 0 are chosen,and current conservation holds, then � ¼ 0 andH > 0 forall time in the KS bumblebee model.

Based on this, one could look for similar restrictions ofthe phase space in the case of the generalized bumblebeemodel. For example, the solutions with H < 0 describedabove must typically have � � 0 at t ¼ 0 to satisfy theconstraint 4 � 0. This suggests the idea of trying to limitthe choice of initial values to � ¼ 0 in an attempt toexclude the possibility of solutions with H < 0.

However, this idea seems unlikely to succeed in the caseof the generalized bumblebee model, since setting � ¼ 0 att ¼ 0 is not sufficient to restrict the phase space to solu-tions with � ¼ 0 for all time. This is because the equationof motion for � has different dependence on the other fieldsin the generalized bumblebee model compared to the KSmodel. In particular, _� is not proportional to just � itself.This is evident even in the linearized theory, with B�

expanded as B� ¼ b� þ E�. Applying the constraint

analysis to the linearized theory yields a first-order expres-sion for � in terms of Ej and �j equal to

� ’ 1

2b

��1 þ �3�1 � �2

�@j�

j; (65)

while the equation of motion for � in the linearized theoryis

_� ’ � 1

2b

ð�2 þ �3Þð�1 þ �3Þð�1 � �2Þ ð@k@k@jEjÞ: (66)

The latter equation shows that (with non-Maxwell values�2 þ �3 � 0) _� is independent of � at linear order.Therefore, even if � ¼ 0 at t ¼ 0, nonzero values of �can evolve over time. This makes it difficult to decoupleregions of phase space with H > 0 in the generalizedbumblebee model purely by making a generic choice ofinitial values. It would thus seem likely that the regions ofphase space with H < 0 include solutions obeying � ¼ 0at t ¼ 0.

2. Quadratic smooth potential

The generalized bumblebee model with a smooth qua-dratic potential (5) depends on four field components B�

and their corresponding conjugate momenta. The expres-sions for �0 and�j are the same as in Eqs. (47) and (48),respectively. There are no constraints in this model. Thus,according to Dirac’s counting algorithm there are n�n1 � n2 ¼ 4� 0� 0 ¼ 4 independent degrees of free-dom. This is two more than in electromagnetism, andone more than in the KS bumblebee model.These 4 degrees of freedom include three NGmodes and

a massive mode. For arbitrary values of �1, �2, and �3, allthree NG modes can propagate, but with dispersion rela-tions that depend on these coefficients. In contrast, in theKS model, with a Maxwell kinetic term, only two of theNG modes propagate as transverse photons. A massivemode occurs in either theory when V0 ¼ 2�ðB2

0 � B2j �

b2Þ � 0. In the generalized bumblebee case, there is noanalogue of Gauss’ law, and it is possible for the massivemode to propagate. However, in the KS model with atimelike vector, the constraint (33) provides a modifiedversion of Gauss’ law, and the massive mode is purely anauxiliary field that acts as a nonlinear source of chargedensity in this relation.The Hamiltonian for the generalized bumblebee has the

same form as in (50), but with the potential in the last termreplaced by the expression in (5). With no constraints, thefull phase space includes solutions with an unrestrictedrange of initial values. Thus, for any values of the coef-ficients �1, �2, �3, there will either be propagating ghostmodes or permissible initial choices for the fields andmomenta with H < 0.


125007-10

3. Quadratic Lagrange-multiplier potential

As a final example, the generalized bumblebee modelwith a quadratic Lagrange-multiplier potential (6) can beconsidered as well. In this case there are ten fields B�,�

�,

�, and �ð�Þ. The conjugate momenta are given in (47)–(49). The Hamiltonian is the same as in (50), but with thepotential replaced by (6). In this case, two constraints arefound,

1 ¼ �ð�Þ; (67)

2 ¼ �12ðB2

0 � B2j � b2Þ2: (68)

Constraint 2 imposes the condition (2). However, it in-volves a quadratic expression for this condition, and there-fore the system is irregular, and the same caveats must beapplied as in the KS model. In particular, substitution of anequivalent constraint 0

2 ¼ ðB20 � B2

j � b2Þ causes Dirac’scounting argument to fail. However, with 1 and 2

identified as first-class constraints, Dirac’s algorithm givesn� n1 � n2=2 ¼ 5� 2� 0 ¼ 3 degrees of freedom. Thisis again one more than in the KS model. In this case, thereis no massive mode, and � decouples completely. The threeindependent degrees of freedom are the NG modes, whichin the generalized bumblebee can all propagate. However,even if values of �1, �2, and �3 can be found that preventthese modes from propagating as ghost modes, there are noother constraints in the theory that prevent initial-valuechoices that can yield solutions with H < 0.

IV. SUMMARY & CONCLUSIONS

Table I summarizes the results of the constraint analysisapplied to electrodynamics, Nambu’s model, the KS bum-blebee, and the generalized bumblebee. For each of thebumblebee models, three types of potentials V are consid-ered. The results show that no two models have identicalconstraint structures. In most cases, there is one or moreadditional degrees of freedom in comparison to electro-

magnetism. These extra degrees of freedom are importantboth as possible additional propagating modes and in termsof how they alter the initial-value problem.In considering the stability of the bumblebee models, it

is not sufficient to look only at the propagating modes. Therange of possible initial values must be examined as well.In general, when the extra degrees of freedom appearing inthese models are allowed access to the full phase space, theHamiltonians are not strictly positive definite. However, inthe KS models, it is possible to choose initial values for thefields and momenta that restrict the phase space to ghost-free regions with H > 0. In contrast, in models withgeneralized kinetic terms obeying ð�2 þ �3Þ � 0, no suchrestrictions are found. These theories either have propagat-ing ghosts or have extra degrees of freedom that evolve insuch a way that makes it difficult to separate off restrictedregions of phase space with H > 0. In the end, it appearsthat only the KS models have a simple choice of initialvalues that can yield a physically viable theory in a re-stricted region of phase space.The examples considered in this analysis all focused on

the case of a timelike vector B�, which is the most widely

studied 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 aspacelike vector B�. In this case, it is straightforward to

show that the linearized KS model is equivalent to electro-dynamics in an axial gauge [6]. However, additional care isrequired in conducting a constraint analysis of the fullnonlinear KS or generalized models, since B0 can vanishin the case of a spacelike vector, making additional singu-larities a possibility. Alternatively, an analysis in terms ofthe BRST formalism could be pursued, which would besuitable as well for addressing questions of quantization.Lastly, an extension of the constraint analysis to a curvedspacetime in the presence of gravity would be relevant,since ultimately bumblebee models are of interest not onlyas effective field theories incorporating spontaneousLorentz violation, but also as modified theories of gravity.For example, they are currently one of the more widely

TABLE I. Summary of constraints. Shown for each model are the number of primary (1), secondary (2), first-class (FC), andsecond-class (SC) constraints, and the resulting number of independent degrees of freedom (DF). The last column indicates the regionsof phase space that are ghost-free and have H > 0. Current conservation @�J

� ¼ 0 is assumed in the KS models.

Theory Kinetic term Potential V Fields 1 2 FC SC DF Ghost-free, H > 0

Electromagnetism � 14F��F

�� A�, �� 1 1 2 0 2 Full phase space

Nambu model � 14F��F

�� Aj, �j 0 0 0 0 3 Subspace (@j�j ¼ J0)

KS bumblebee � 14B��B

�� ðB�B� � b2Þ B�, ��, �, �ð�Þ 2 2 0 4 3 Subspace (� ¼ 0)

ð�1 ¼ 1; �2 ¼ �3 ¼ 0Þ 12�ðB�B

� � b2Þ2 B�, �� 1 1 0 2 3 Subspace (B�B� ¼ b2)

12�ðB�B

� � b2Þ2 B�, ��, �, �ð�Þ 2 2 2 2 2 Full phase space

General bumblebee Non-Maxwell �ðB�B� � b2Þ B�, ��, �, �ð�Þ 1 3 0 4 3 No subspace found

(Arbitrary �1, �2, �3)12�ðB�B

� � b2Þ2 B�, �� 0 0 0 0 4 No subspace found12�ðB�B

� � b2Þ2 B�, ��, �, �ð�Þ 1 1 2 0 3 No subspace found


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used models for exploring implications of Lorentz viola-tion in gravity and cosmology and in seeking alternativeexplanations of dark matter and dark energy. However,performing a constraint analysis with gravity presentseven greater challenges and is beyond the scope of thiswork.

In summary, the constraint analysis presented here in aflat-spacetime limit is useful in seeking insights into thenature of theories with spontaneous Lorentz violation andwhat their appropriate interpretations might be. In particu-lar, the KS bumblebee models offer the possibility thatEinstein-Maxwell theory might emerge as a result of spon-

taneous Lorentz breaking instead of through local U(1)gauge invariance. Indeed, in the flat-spacetime limit ofthis model, with a timelike vacuum value, electromagne-tism in a fixed nonlinear gauge is found to emerge in awell-defined region of phase space.

ACKNOWLEDGMENTS

We thank Alan Kostelecky for useful conversations. Thiswork was supported in part by NSF Grant No. PHY-0554663. The work of R. P. is supported by thePortuguese Fundacao para a Ciencia e a Tecnologia.

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