Covariant Quantization of CPT-violating Photons
CCovariant Quantization of CPT-violating Photons
D. Colladay, P. McDonald, J. P. Noordmans, and R. Potting New College of Florida, Sarasota, FL, 34243 CENTRA, Departamento de F´ısica,Universidade do Algarve, 8005-139 Faro, Portugal (Dated: June 28, 2018)
Abstract
We perform the covariant canonical quantization of the CPT- and Lorentz-symmetry-violating pho-ton sector of the minimal Standard-Model Extension, which contains a general (timelike, lightlike,or spacelike) fixed background tensor k µAF . Well-known stability issues, arising from complex-valued energy states, are solved by introducing a small photon mass, orders of magnitude belowcurrent experimental bounds. We explicitly construct a covariant basis of polarization vectors, inwhich the photon field can be expanded. We proceed to derive the Feynman propagator and showthat the theory is microcausal. Despite the occurrence of negative energies and vacuum-Cherenkovradiation, we do not find any runaway stability issues, because the energy remains bounded frombelow. An important observation is that the ordering of the roots of the dispersion relations isthe same in any observer frame, which allows for a frame-independent condition that selects thecorrect branch of the dispersion relation. This turns out to be critical for the consistency of thequantization. To our knowledge, this is the first system for which quantization has consistentlybeen performed, in spite of the fact that the theory contains negative energies in some observerframes. PACS numbers: 11.30.Cp, 11.30.Er, 12.60.-i, 41.20.Jb a r X i v : . [ h e p - t h ] O c t . INTRODUCTION The covariance of physical laws under boosts and rotations is at the basis of the stan-dard model (SM) of particle physics and general relativity [1]. This Lorentz symmetry isclosely related to the invariance under the combined action of charge conjugation, parityinversion, and time reversal, i.e. CPT symmetry [2, 3]. Over the last few decades the inter-est in the possibility of breaking Lorentz and CPT symmetry has been growing. This riseis motivated by theories that attempt to unify general relativity with quantum mechanicsand that exhibit mechanisms of Lorentz and CPT breaking [4–6]. The detection of a corre-sponding experimental signal would provide profound new physical insights and could pointus to the correct theory of quantum gravity.In this context, the Standard-Model Extension (SME) has proven to be a tool of greatvalue. It is a framework that incorporates Lorentz- and CPT-violating effects into the SM [7],gravity [8], and for matter-gravity couplings [9], by extending the Lagrangian to include allpossible Lorentz- and CPT-violating operators consisting of the conventional fields. Becauseof its generality it allows for broad experimental searches [10] as well as general theoreticalconsiderations of Lorentz- and CPT-violating effects.Of particular interest is the pure-gauge matter sector of the minimal SME, whichincludes only superficially renormalizable operators. Here, Lorentz violation (LV) can beintroduced, either while preserving CPT, or while violating it. In this paper we consider theChern-Simons-like operator of mass dimension three that causes both CPT- and Lorentz-symmetry breaking, parametrized by a fixed background vector k µAF [11]. Although boundedobservationally to minute levels [10, 11], this term has received intensive attention in theliterature, as it arises as a radiative correction from the fermion sector in the presence ofa LV axial-vector term [12]. It is thus important to establish both in the fermion and thephoton sector, whether such effects impede a rigorous quantization, and if not, in what waythe standard procedures have to be modified. While the quantization of the fermion sectorwas implemented successfully in the past [13], the situation is more ambiguous in the photonsector. Furthermore, while CPT-violating effects are strongly bounded in the photon sector,this is not at all the case in the gluon and weak gauge-boson sector [10]. Although weonly consider the abelian case here, our analysis may lead to important implications for LVnon-abelian theories as well. 2n this paper, we thus perform the covariant quantization of Maxwell-Chern-Simonstheory. Covariant quantization is extremely useful in performing quantum-field-theoreticcalculations, as the formulas retain explicit covariance throughout the computational pro-cedure. In a previous work, it was discussed how this can be implemented for the CPT-preserving case [14, 15]. In Ref. [16] the quantization for purely timelike k AF was discussedand applied to calculate vacuum-Cherenkov-radiation rates, whereas in Ref. [17, 18], atten-tion was restricted to the (massless) case of purely spacelike k µAF in an axial gauge. Althoughsome of the present results were already presented in Ref. [16], the approach we take here ismore general and rigorous, while we consider spacelike, lightlike, as well as timelike valuesof k µAF in a general class of covariant gauges.As in Ref. [14], the introduction of a mass regulator turns out to be necessary for aconsistent quantization. This is phenomenologically feasible, due to the fact that ultra-tightobservational bounds on k µAF [10] allow the choice of a photon mass sufficiently large tofix quantization problems, while simultaneously agreeing with current experimental bounds.Furthermore, the introduction of a photon mass is often used to regulate infrared divergencesthat turn up in loop diagrams in both conventional calculations, and in the context of LVeffects [19]. We note that the introduction of a photon mass in the context of the SME hasbeen studied in the presence of both CPT-preserving and CPT-violating terms at the levelof the equations of motion and the propagator in Ref. [20].The outline of this paper is as follows. In section II we introduce the Lorentz- and CPT-violating model, including a nonzero photon mass that is introduced through the St¨uckelbergmechanism. Subsequently, we find a covariant basis of normalized and orthogonal eigenvec-tors of the equation-of-motion operator in section III. These polarization vectors satisfymodified orthogonality relations when fixed to be on shell. In section IV we analyze theequations of motions in momentum space, and show that for the case of timelike k µAF theintroduction of a nonzero mass parameter avoids a region in three-momentum space thathas no corresonding real energy solutions. We also find a condition on k µAF that guaranteesenergy positivity. Energy positivity and its connection to stability is further discussed insection V, where we find a way to distinguish different branches of the dispersion relationin any observer frame. We derive a relation between the momentum-space propagator andthe polarization vectors in section VI. The field operator is then quantized in terms of cre-ation and annihilation operators in section VII. Subsequently, the commutator of fields at3pacelike separation is worked out in section VIII, and it is shown that the theory satis-fies microcausality. In section IX the Feynman propagator is derived and in section X weanalyze the space of states in the context of BRST quantization. Finally we present ourconclusions in section XI. Some of the more detailed analyses of the dispersion relation, thephoton group velocity, and the energy lower bound are relegated to the appendices. II. CPT-VIOLATING PHOTON SECTOR OF THE SME
CPT violation in the photon sector of the power-counting renormalizable part of theSME is given by the Lagrangian L A,k AF = − F µν F µν + 12 k κAF (cid:15) κλµν A λ F µν , (1)where k µAF is an arbitrary real-valued and fixed background vector with the dimensions ofmass. The CPT-violating term in Eq. (1) is gauge invariant up to total derivative terms,which, in the absence of topological obstructions, do not influence the physics.The theory in Eq. (1) breaks so-called particle Lorentz symmetry, while it is invari-ant under observer Lorentz transformations [7]. Observer Lorentz transformations are justtransformations of the coordinates of the reference frame of the observer and thus transformboth k µAF and the fields. Particle Lorentz transformations, on the other hand, affect onlythe particle fields, but leave the tensor k µAF unchanged. This corresponds to changing theorientation and/or velocity of the experimental system in absolute space.In this paper, we will consider the cases of spacelike, lightlike, and timelike k µAF . As iswell known [11], for timelike k µAF the dispersion relation following from (1) has a tachyoniccharacter: there are (small) momenta for which there are no corresponding real solutionsfor the energy, signaling an unstable theory that does not permit a consistent quantization(although proposals have been made for fermionic theories [21]).As was noted first in [22], a way around this problem is to introduce a small mass termfor the photon through the St¨uckelberg mechanism [23]. The gauge-fixed photon Lagrangianbecomes L A = − F µν F µν + 12 k κAF (cid:15) κλµν A λ F µν + 12 m γ A µ A µ − ξ ( ∂ µ A µ ) , (2)where ξ > m γ is the small photon mass. For spacelike,lightlike, as well as timelike values of k µAF , a nonzero photon mass is useful. In addition to4ts use as a regulator for infrared divergences [19], it allows for the introduction of so-calledconcordant frames [7].Concordant frames are observer frames in which the LV effects can be treated as smallperturbations to the Lorentz-symmetric physics. For nonzero values of k µAF , this cannotbe the case in all frames, because k µAF changes under the action observer Lorentz transfor-mations. Therefore, the size of its components is in principle unbounded, if one allows forarbitrary observer frames. To be compatible with experimental constraints, Earth’s rest-frame is then presumed to be in a concordant frame. However, to say anything meaningfulabout the size of the components of k µAF , we need a nonzero photon mass, since it is the onlyother dimensionful parameter in Eq. (2). As mentioned in the introduction, the requiredsize of photon mass lies many orders of magnitude below its experimental bounds. We willdiscuss this in more detail in Section IV. III. POLARIZATION VECTORS
In momentum space, the classical equation of motion, corresponding to the Lagrangianin Eq. (2), reads (cid:2) ( p − m γ ) η µν − (1 − ξ − ) p µ p ν − i(cid:15) αβµν ( k AF ) α p β (cid:3) e ( λ ) ν ( (cid:126)p ) ≡ S µν e ( λ ) ν ( (cid:126)p ) = 0 , (3)where e ( λ ) ν ( (cid:126)p ) are the eigenvectors of the equation-of-motion operator S µν . The index λ runsover 0 , , + , − , labeling the gauge mode, and three physical modes, respectively. Contractionof Eq. (3) with p µ yields ( ξ − p − m γ )( p · e ( λ ) ) = 0 , (4)demonstrating that there is a gauge mode satisfying p − ξm γ = 0. This expression alsoestablishes that the physical polarization vectors, corresponding to the remaining modes,satisfy p · e ( λ ) = 0. The contraction of Eq. (3) with ( k AF ) µ gives a similar expression thatdemonstrates the fact that the physical polarization modes either obey the conventionaldispersion relation p = m γ , or the corresponding polarization vectors satisfy k AF · e ( λ ) = 0.These facts are confirmed by the explicit expressions for the polarization vectors in Eq. (9)and by the functions defining the dispersion relations in Eq. (10).When the eigenvectors e ( λ ) ν ( (cid:126)p ) satisfy Eq. (3), they are functions of the three-momentum (cid:126)p , since p = p ( (cid:126)p ) is fixed by the dispersion relation. As shown in Refs. [14, 16],5uantization can be carried out referring only to these on-shell polarization vectors. How-ever, it turns out to be useful to consider e ( λ ) ν ( p ) as functions of both p and (cid:126)p that satisfy S µν e ( λ ) ν ( p ) = Λ λ ( p ) e ( λ ) µ ( p ) , (5)where Λ λ ( p ) is the eigenvalue of S µν belonging to the polarization mode λ . The relationΛ λ ( p ) = 0 , (6)can then be imposed to enforce the equation of motion. Each of the resulting dispersionrelations has two solutions, corresponding to the conventional positive and negative ener-gies. Usually, one then uses the positive root of Λ λ ( p ) to define the energy of the on-shellpolarization vectors. However, in Section IV we show that in the present LV case, the signof the roots of Λ + ( p ) is invariant only in concordant frames [13], i.e. frames where thecomponents of k µAF are small compared to the photon mass. In non-concordant frames thissign can depend on the size and direction of (cid:126)p . We will discuss this issue in more detail inSection IV. For now, we let E λ ( (cid:126)p ) denote the root of Λ λ ( p ) that is positive in a concordantframe. Substituting the solution p = E λ ( (cid:126)p ) in the expression for e ( λ ) ν ( p ) then gives therelevant on-shell polarization vector that satisfies the equation of motion in Eq. (3).We determine the explicit solutions for the polarization vectors e ( λ ) µ ( p ) of Eq. (5) byexpanding in the four basis vectors u µ = p µ N , u µ = (cid:15) µνρσ p ν n ρ ( k AF ) σ N , u µ = (cid:15) µνρσ p ν ( u ) ρ ( k AF ) σ N , u µ = p k µAF − ( p · k AF ) p µ N . (7)Here, the four-vector n µ is an arbitrary, observer-covariant, four-vector with at least onecomponent perpendicular to the subspace formed by p µ and k µAF . Note that it is generallynot possible to use only a single n µ vector to cover all of momentum space due ultimatelyto the theorem that “the hair on a sphere cannot be combed”, i.e. it is not possible tofind a single, smooth, non-vanishing vector field on a sphere. This problem exists even inthe conventional case in which one tries to construct a set of polarization vectors for thetransverse, massless photons. There is always at least one direction in momentum spacefor which the polarization vectors must be non-smooth. This geometrical impediment toconstructing a single, global frame field forces one to choose another external vector m µ ina different direction than n µ to define the polarization vectors in a small cone.6he N i in Eq. (7) are normalization factors, that we choose to be real. The basisvectors are orthogonal in the sense that u i · u j = 0 if i (cid:54) = j . However, u and u becomelightlike if p = 0 while u , and u become lightlike if ( p · k AF ) = p k AF . Note that thisconstruction completely fails when p µ ∝ k µAF . This is related to the existence of singularpoints on the on-shell energy surfaces there. We will discuss these singular points in moredetail below Eq. (11).Using the basis in Eq. (7) and noticing that (cid:15) αβµν ( k AF ) α p β ( u ) ν = N u µ , (8a) (cid:15) αβµν ( k AF ) α p β ( u ) ν = N − ( p k AF − ( p · k AF ) ) u µ , (8b)it becomes straightforward to determine the polarization vectors. They are given by e (0) µ ( p ) = u µ , (9a) e (3) µ ( p ) = u µ , (9b) e ( ± ) µ ( p ) = 1 √ (cid:18) u µ ± iN − (cid:113) ( p · k AF ) − p k AF u µ (cid:19) , (9c)where the square roots in the expressions for e ( ± ) µ ( p ) are defined by the conventional prin-cipal value. The eigenvalues corresponding to the eigenvectors, as defined in Eq. (5), areΛ ( p ) = 1 ξ ( p − ξm γ ) , (10a)Λ ( p ) = p − m γ , (10b)Λ ± ( p ) = p − m γ ± (cid:113) ( p · k AF ) − p k AF . (10c)These observer-scalar functions of p µ and k µAF define the dispersion relations for each of thepolarization modes by fixing Λ λ ( p ) = 0. Substituting the resulting solutions p = E λ ( (cid:126)p )into the expressions in Eq. (9) thus gives the polarization vectors that solve Eq. (3). Notethat the basis set in Eq. (9) is valid in a larger part of momentum space, in the sense thatthe four-vectors solve the off-shell condition in Eq. (5). This fact will be convenient for theanalyses of microcausality and the Feynman propagator.Excluding the hypersurfaces in momentum space where p = 0 or ( p · k AF ) − p k AF = 0,it is always possible to choose the normalization factors in Eq. (7) such that the polarizationvectors in Eq. (9) are normalized to +1 or −
1. The resulting values for N i , which we choose7o be real, are determined by | N | = | p | , | N | = | p (( n · k AF ) − n k AF ) + n ( p · k AF ) + k AF ( n · p ) − p · k AF )( n · p )( n · k AF ) | , | N | = | ( p · k AF ) − p k AF | , | N | = | p ( p k AF − ( p · k AF ) ) | . (11)As mentioned previously, there are hypersurfaces in momentum space where the def-initions in Eqs. (7), (9), and (11) are invalid. First, momenta that satisfy p = 0 yieldpolarization vectors e (0) µ ( p ) ∝ e (3) µ ( p ) that are lightlike. This is not a serious problem since p = 0 does not intersect the mass-shell of modes λ = 0 ,
3, because of the nonzero pho-ton mass. Ultimately we only need to be able to define the on-shell polarization vectors,while the off-shell polarization vectors are very convenient, but not necessary. The moreinteresting, perturbed physical λ = ± states are not problematic at p = 0.More serious singular points occur when ( p · k AF ) = p k AF , which happens, for exam-ple, when p µ ∝ k µAF . This only becomes an issue if the mass shell of a physical polarizationmode intersects the momentum-space hypersurface on which the singular points lie. When k AF ≤
0, the singular hypersurface never intersects the mass shell. In the case k AF >
0, anintersection occurs for all transverse polarization modes if p µ = ς K µ ≡ ς m γ k µAF (cid:112) k AF (12)with ς either 1 or −
1. At these two momenta, the dispersion relations of the modes λ = 3 , + , − are solved simultaneously, while p µ = ς K µ also satisfies ( p · k AF ) = p k AF .Furthermore, at p µ = ς K µ , the physical polarization vectors in Eq. (9) all vanish. In fact,the LV term in Eq. (3) also vanishes in these cases, so any polarization vector orthogonal to p µ will satisfy the equation of motion there. We will choose to define the polarization vectorsat p µ = ς K µ by taking some limit p → ς K of the off-shell polarization vectors evaluatedat (cid:126)p = ς (cid:126) K . At this value of the spatial momentum, the polarization vectors are given by e (3) µ ( p , ς (cid:126) K ) = ε ( p − ς K )˜ N (cid:32) ςη µ | (cid:126)k AF | + η µi ˆ k iAF (cid:112) k AF p m γ (cid:33) , (13a) e ( ± ) µ ( p , ς (cid:126) K ) = 1 √ | (cid:126)k AF | ˜ N (cid:16) (cid:15) µ ρσ (˜ u ) ρ ( k AF ) σ ± iε ( p − ς K ) | (cid:126)k AF | ˜ u µ (cid:17) , (13b)8here ε ( x ) = x/ √ x , ˜ u µ = (cid:15) µ ρσ n ρ ( k AF ) σ , and ˜ N and ˜ N are normalization factors thatsatisfy | ˜ N | = ( (cid:126)k AF × (cid:126)n ) , (14a) | ˜ N | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)k AF − ( p ) k AF m γ (cid:12)(cid:12)(cid:12)(cid:12) . (14b)Note that these definitions fail for (cid:126)k AF = (cid:126)
0. If this is the case, it is always possible to makea small observer Lorentz transformation to a frame in which (cid:126)k AF (cid:54) = (cid:126) p = ς K , since the factor ε ( p − ς K ) becomes undefined, however, we are free to pick thepositive sign that results from approaching the singular point from the direction p > ς K and use it to make a choice at the singular point. The price we pay for doing this, is that welose manifest observer Lorentz covariance along a singular line (on shell this corresponds tothe two singular points). This does not cause any issues in the current paper as a completebasis of polarization vectors at each momentum is all that is required for a covariant fieldexpansion, they need not be continuous through the singular point. In fact, it is not possibleeven in the conventional massless photon case to find a smooth set of transverse polarizationvectors that globally covers momentum space due to the topological obstruction involvedin “combing the hair on a sphere”. This means that manifest observer invariance is neverpossible as certain choices have to be made as to how the necessary discontinuities are placedin momentum space. An example of the physical effect of this obstruction can be observedin Berry’s Phase [24] in which a helicity state, adiabatically transported through a closedloop in momentum space, picks up a non-trivial phase proportional to the solid angle of theloop. This would not happen if a globally defined frame field of helicity states was possible.We can now summarize the orthogonality of the polarization vectors (evaluated at thesame four-momentum) by e ( λ ) ∗ ( p ) · e ( λ (cid:48) ) ( p ) = g λλ (cid:48) , (15)9ith g = diag(1 , − , − , −
1) for p > − , , − , −
1) for p < p · k AF ) − p k AF > − sgn( u ) σ for p < p · k AF ) − p k AF < . (16)In Eq. (16), is the 2 × σ is the usual Pauli matrix σ = . (17)The indices λ and λ (cid:48) in Eq. (15) label the rows and columns of g and run over 0 , , + , − in that order. At the on-shell singular points in Eq. (12), where ( p · k AF ) − p k AF = 0,the lower-right 2 × p = E λ ( (cid:126)p ), Eq. (15) does not represent an orthogonalityrelation for on-shell eigenvectors. In Ref. [16], such a relation was derived. With a slightchange in normalization relative to this reference, it is given by e ∗ ( λ (cid:48) ) µ ( (cid:126)p ) (cid:104) ( E λ ( (cid:126)p ) + E λ (cid:48) ( (cid:126)p )) (cid:0) η µν − (1 − ξ − ) δ µ δ ν (cid:1) − (1 − ξ − ) p i ( δ µi δ ν + δ µ δ νi ) − ik κAF (cid:15) κ µν (cid:105) e ( λ ) ν ( (cid:126)p ) = g λλ (cid:48) Λ (cid:48) λ ( p ) | p = E λ . (18)where Λ (cid:48) λ ( p ) is the derivative of Λ λ ( p ) with respect to p . The reason for choosing analternate normalization will become clear when we perform the quantization. Note that theonly relevant g λλ (cid:48) for the on-shell states is the diagonal one, provided m γ (cid:54) = 0.The fact that Λ (cid:48) λ ( p ) | p = E λ indeed corresponds to the normalization in Eq. (15) caneasily be seen by considering the p -derivative of Eq. (5), which reads S (cid:48) µν e ( λ ) ν ( p ) + S µν e (cid:48) ( λ ) ν ( p ) = Λ (cid:48) λ ( p ) e ( λ ) µ ( p ) + Λ λ ( p ) e (cid:48) ( λ ) µ ( p ) , (19)where the primes denote derivatives with respect to p . After contracting this equation with (cid:15) ∗ ( λ ) µ ( p ) and substituting p = E λ ( (cid:126)p ) everywhere, the second term on both the left-hand sideand the right-hand side vanishes. Inspection of the explicit expression for S (cid:48) µν then revealsthat we have obtained Eq. (18) for the case λ = λ (cid:48) , confirming the factor Λ (cid:48) λ ( p ) | p = E λ in that10quation. The derivation of Eq. (18) in Ref. [16] was done for E λ ( (cid:126)p ) (cid:54) = E λ (cid:48) ( (cid:126)p ). However,using the fact that Eq. (15) holds for on-shell eigenvectors with degenerate energies, we canuse Eq. (19) to show that Eq. (18) also holds if E λ ( (cid:126)p ) = E λ (cid:48) ( (cid:126)p ).In a similar way an orthogonality relation for polarization vectors with opposite three-momenta can derived [16]. As long as E λ ( (cid:126)p ) (cid:54) = − E λ (cid:48) ( − (cid:126)p ) it holds that e ( λ (cid:48) ) µ ( − (cid:126)p ) (cid:104) ( E λ ( (cid:126)p ) − E λ (cid:48) ( − (cid:126)p )) (cid:0) η µν − (1 − ξ − ) δ µ δ ν (cid:1) − (1 − ξ − ) p i ( δ µi δ ν + δ µ δ νi ) − ik κAF (cid:15) κ µν (cid:105) e ( λ ) ν ( (cid:126)p ) = 0 . (20)Note that there is no complex conjugate on the left-side polarization vector in this relation. IV. ANALYSIS OF THE DISPERSION RELATION
In the previous section we found explicit expressions for the eigenvectors of S µν inEq. (9). These become the on-shell photon polarization vectors if we substitute for p theconcordant-frame positive root E λ ( (cid:126)p ) of Λ λ ( p ), with Λ λ ( p ) defined in Eq. (10). In this sectionwe investigate the dispersion relations, given by Λ λ ( p ) = 0, and in particular the reality,degeneracy, and positivity of their roots [25].The full dispersion relation of the CPT-odd photons is given by det( S ) = (cid:81) λ Λ λ ( p ) = 0,and thus by 1 ξ ( p − ξm γ )( p − m γ )(( p − m γ ) − p · k AF ) + 4 p k AF ) = 0 . (21)The left-hand side is an eighth order polynomial in p and as such has eight (possiblycomplex and/or degenerate) roots, which we label by ω , . . . , ω . Because there is no termproportional to the seventh power of p in Eq. (21), Vieta’s formulas tell us that the sum ofall roots vanishes, i.e. (cid:88) i =1 ω i = 0 . (22)The polynomial in Eq. (21) can be factorized in three separate polynomials, two of whichare Λ ( p ) and Λ ( p ), while the third one is given byΛ T ( p ) = Λ + ( p )Λ − ( p ) = ( p − m γ ) − p · k AF ) + 4 p k AF . (23)Since Eq. (21) is invariant under p → − p , all roots come in pairs such as ω ( (cid:126)p ) = − ω ( − (cid:126)p ). In concordant frames, one root of each pair is positive, while the other is negative,11.g. if ω ( (cid:126)p ) >
0, then ω ( (cid:126)p ) <
0. We apply the usual redefinition to the concordant-framenegative-energy solutions, i.e. E ( (cid:126)p ) = ω ( (cid:126)p ) = − ω ( − (cid:126)p ) = (cid:113) (cid:126)p + ξm γ , (24a) E ( (cid:126)p ) = ω ( (cid:126)p ) = − ω ( − (cid:126)p ) = (cid:113) (cid:126)p + m γ , (24b) E + ( (cid:126)p ) = ω ( (cid:126)p ) = − ω ( − (cid:126)p ) , (24c) E − ( (cid:126)p ) = ω ( (cid:126)p ) = − ω ( − (cid:126)p ) . (24d)This also defines our labeling of the roots of the different Λ λ ( p ) functions. Although it isnot evident at this point, two of the roots ( ω and ω ) correspond to the polarization mode λ = +, while the other two ( ω and ω ) belong to λ = − . Together they are the roots of thefourth-order polynomial Λ T ( p ) in Eq. (23) and obey (cid:88) i =5 ω i = 0 . (25)Since the roots of Λ ( p ) and Λ ( p ) are trivial and need no further discussion, we will committhe rest of this section to Λ + ( p ) and Λ − ( p ).When not restricting to an observer frame where k µAF has a convenient form, theexplicit expressions for the roots of Λ ± ( p ) are unwieldy and provide little insight about theissues we want to discuss (except for lightlike k µAF ). However, even without such explicitexpressions, it is possible to show that if k AF < m γ , then Λ + ( p ) and Λ − ( p ) have two real andnon-degenerate roots each. Moreover, these four roots are all different, except at two pointsin momentum space for timelike k µAF . Similarly, we can show that the energy is boundedfrom below and that no negative energies occur if ( k AF ) < m γ .To prove the statements in the previous paragraph, we define the following functionsof p : f ( p ) = 12 (cid:0) Λ + ( p ) + Λ − ( p ) (cid:1) , (26a) f δ ( p ) = 12 (cid:0) Λ + ( p ) − Λ − ( p ) (cid:1) , (26b)where we now view Λ ± ( p ) as functions of p by considering them at fixed (cid:126)p . It follows thatΛ ± ( p ) = f ( p ) ± f δ ( p ) and that for a root, ω , of Λ ± ( p ), f ( ω ) = ∓ f δ ( ω ), i.e. in a plot theintersections of f ( p ) with ∓ f δ ( p ) correspond to the roots of Λ ± ( p ), while intersections of( f ( p )) and ( f δ ( p )) correspond to the roots of the polynomial Λ T ( p ). Examples of suchplots are given in Figs. 1 and 2. 12he derivatives with respect to p of the functions in Eq. (26) are given by f (cid:48) ( p ) = 2 p p →∞ −→ ∞ , (27a) f (cid:48) δ ( p ) = 4 (cid:126)k AF p − k AF ( (cid:126)p · (cid:126)k AF ) f δ ( p ) p →∞ −→ | (cid:126)k AF | , (27b)where the limiting values are for p to positive infinity. Taking p to negative infinity willgive the same result with opposite sign. The derivative f (cid:48) δ ( p ) shows that if f δ ( p ) ∈ R , then f δ ( p ) is an increasing (decreasing) function for p larger (smaller) than k AF ( (cid:126)p · (cid:126)k AF ) /(cid:126)k AF .To analyze the functions further, we will make a distinction between k AF ≤ k AF > Timelike case – If k AF >
0, a typical plot of the functions f ( p ) and ± f δ ( p ) looks likethe plot in Fig 1(a). The corresponding plots of ( f ( p )) and ( f δ ( p )) are shown in figureFig 1(b). From the limiting values of the derivatives in Eqs. (27), together with the factthat f δ ( p ) is real and non-negative if k AF >
0, it is easily seen that f δ ( p ) always intersects f ( p ) at two different points. These points correspond to the two roots of Λ − ( p ): ω and ω . This thus establishes that Λ − ( p ) always has two non-degenerate roots if k AF >
0. Oneof these roots is positive, while the other one is negative and these signs are the same in anyobserver frame. Moreover, E − ( (cid:126)p ) is bounded from below, as shown in Eq. (C5a).In Fig 1(a), − f δ ( p ) also intersects f ( p ) twice, once for positive p and once fornegative p . However, there are two other possible scenarios. The intersections can be onthe same side of the vertical p = 0 axis, or the curve of − f δ ( p ) might lie entirely below theone of f ( p ). In the former case, the roots of Λ + ( p ) have the same sign, while in the lattercase they both have a non-vanishing imaginary part. These three scenarios are summarizedby ( i ) − f δ (0) > f (0) → ω , ω ∈ R , sgn( ω ) = − sgn( ω ) , ( ii ) − f δ (0) < f (0) and ∃ p : − f δ ( p ) > f ( p ) → ω , ω ∈ R , sgn( ω ) = sgn( ω ) , ( iii ) ∀ p : − f δ ( p ) < f ( p ) → ω , ω ∈ C . It turns out that a sufficient observer non-invariant condition for scenario ( i ) is( k AF ) < m γ . (28)The fact that − f δ (0) > f (0) if this condition holds is shown in Appendix A. We also findthere that, if ( k AF ) > m γ , then there exist a range of (generally small) three-momenta forwhich − f δ (0) < f (0), so either scenario ( ii ) or ( iii ) applies.13n Appendix A we show that if the observer Lorentz invariant condition k AF < m γ (29)holds, we can always find a p for which − f δ ( p ) > f ( p ). This shows that both roots ofΛ + ( p ) are real and non degenerate if Eq. (29) is satisfied, while their sign is guaranteed todiffer if Eq. (28) applies. Eq. (29) can be enforced on the theory in any observer frame.However, the size of k AF changes when performing an observer Lorentz boost. Therefore itcan only be satisfied in a subset of frames, which we can call concordant frames. In otherwords, Eq. (28) provides a quantitative definition of a concordant frame in the case where k µAF is the only Lorentz-violating coefficient. The fact that such a definition is possiblehinges on the introduction of a nonzero photon mass.In non-concordant frames, the signs of the two roots of Λ + ( p ) can thus be equal. Ifthey are both negative, the energy (given in Eq. (24)) is also negative. However, since thisonly happens for a limited range of | (cid:126)p | values (see Eq. (A4)), E + ( (cid:126)p ) must be bounded frombelow. In fact, in Eq. (C5b), we determine that E + ( (cid:126)p ) ≥ (cid:113) m γ − k AF − | (cid:126)k AF | .If Eq. (29) is satisfied, the only degeneracy in the dispersion relation for the λ = ± modes can come from a root of Λ − ( p ) being equal to a root of Λ + ( p ). This requires f δ ( p ) = 0while p simultaneously has to solve p = m γ . It follows that the roots of Λ − ( p ) and Λ + ( p )become equal if p µ = ς m γ k µAF √ k ≡ ς K µ , (30)which are points in momentum space where the LV term disappears from the equation ofmotion, as already discussed in Section III. Spacelike/lightlike case – If k AF ≤
0, a typical plot of the functions in Eqs. (26) lookslike the one in Fig. 2(a). One clearly sees that the square root in f δ ( p ) becomes imaginaryfor values of p between x − and x +1 , with x α = k AF ( (cid:126)p · (cid:126)k AF ) (cid:126)k AF + α (cid:114) k AF (cid:16) ( (cid:126)p · (cid:126)k AF ) − (cid:126)p (cid:126)k AF (cid:17) (cid:126)k AF . (31)However, we show in Appendix A that | x α | < (cid:112) (cid:126)p + m γ for all values of (cid:126)p and k µAF . Thismeans that Λ + ( p ) and Λ − ( p ) always have two real roots each, if k µAF is spacelike or light-like. As in the case of timelike k µAF , we find, by investigating when − f δ (0) < f (0) (seeAppendix A), that the condition in Eq. (28) is sufficient to make sure that the roots of14 AF ( k AF ) sgn( k AF ( (cid:126)p · (cid:126)k AF )) domain roots sgn( ω , ω , ω , ω ) k AF < < m γ + or − R (+ , − , + , − ) k AF < > m γ + R (+ , − , + , − ) or (+ , + , + , − ) k AF < > m γ − R (+ , − , + , − ) or ( − , − , + , − )0 < k AF < m γ < m γ + or − R (+ , − , + , − )0 < k AF < m γ > m γ + R (+ , − , + , − ) or (+ , + , + , − )0 < k AF < m γ > m γ − R (+ , − , + , − ) or ( − , − , + , − ) k AF > m γ > m γ + or − C n.a.TABLE I. The different conditions on k µAF in the three columns on the left give different possibilitiesfor the sign and domain of the roots of the λ = ± dispersion relations. The latter are summarizedin the two right-most columns. Which of the options in the right-most column is realized, isdetermined by | (cid:126)p | and the angle between (cid:126)p and (cid:126)k AF . If | (cid:126)p | is in the interval in Eq. (A4) and theangle satisfies Eq. (A5), then three of the four roots will have the same sign (provided ( k AF ) > m γ ). Λ + ( p ) have opposite signs. On the other hand, if ( k AF ) > m γ then there exist observerframes in which both roots have the same sign. As in the timelike case, E + ( (cid:126)p ) can thusbecome negative, however in appendix B we show that E ± ( (cid:126)p ) ≥ (cid:113) m γ − k AF ∓ | (cid:126)k AF | forspacelike/lightlike k µAF .We summarize our findings regarding the signs and the domain of the roots in Table I.It is clear that k AF < m γ is a necessary condition for a consistent physical theory in allobserver frames. This shows that introducing a nonzero photon mass is unavoidable if k µAF is timelike. This was already found in Ref. [22]. In addition, both for spacelike, lightlike,and timelike k µAF , a nonzero photon mass allows for a quantitative definition of concordantframes, in the sense that in frames where Eq. (28) is satisfied, energy positivity is guaranteed(some additional issues related to energy positivity are discussed in Section V).It is interesting, therefore, to compare the current experimental bounds on k µAF and m γ . For the photon mass, the particle data group (PDG) quotes as the best possible bound[26] m γ < × − GeV . (32)This limit is inferred from the absence of a perturbed structure of large-scale magneticfields that would result from a significant nonzero photon mass (see Refs. [27, 28]). The15erification of certain properties of galactic magnetic fields might allow for an improvementof Eq. (32) by nine orders of magnitude [28, 29]. Nevertheless, this result is still manyorders of magnitude from the best bounds on k µAF , which follow from cosmological searchesfor birefringence [10]: k AF < − GeV , (33)where k AF is defined in Sun-centered inertial reference frame [10]. It follows that the as-sumption of a nonzero photon mass, that permits the construction of a phenomenologicallyviable model for photons with CPT-violation (with k AF < ( k AF ) < m γ ), is entirely consis-tent with experimental observations. Moreover, the St¨uckelberg mechanism can be used tointroduce the mass in a gauge-invarant manner, at least at the level of pure QED.This does not mean, however, that we can always ignore the negative energies in thetheory, even for practical purposes. This is illustrated for example by assuming that thesizes of the photon mass and k µAF are comparable with the best achievable bounds, such that m γ ∼ − GeV, as mentioned below Eq. (32). Frames moving with respect to Earth witha relativistic γ -factor up to γ ∼ m γ /k AF ∼ can then be considered to be concordantframes (i.e. there are no negative energies in these frames). Inversely, this means that therest frame of ultra-high-energy cosmic-ray protons could easily be a non-concordant frame,since these protons have energies up to 10 TeV, corresponding to γ = 10 . Note that thebound m γ < − GeV is quite speculative [28], since it depends on several assumptionsabout galactic magnetic fields. Taking the PDG value in Eq. (32) for the photon mass avoidsany potential problems with non-concordant frames. These values are discussed further inthe context of Cherenkov radiation in Ref. [16].
V. ENERGY POSITIVITY AND STABILITY
In the previous section we found that there exist (strongly boosted, but possibly phys-ically relevant) observer frames in which the λ = + polarization mode of the LV photonhas negative energies for a certain range of three momenta (see Eq. (A4) in Appendix A).Nevertheless, the energy remains bounded from below.It lies outside the scope of this paper to rigorously address if such a theory can be fullyconsistent, however, the point of view one often takes in this respect is to regard the theoryas an effective theory. The effective theory is only valid up to a certain energy scale, or,16quivalently, describes particles restricted to have concordant rest frames. Above this energyscale, unknown higher-dimensional nonrenormalizable operators become relevant. These areconjectured to prevent negative energies in all observer frames. This is discussed in detailin Ref. [13].It was also noticed in Ref. [13] that negative-energy issues are closely related to thestability of the theory. Photons with the λ = + polarization mode can have spacelikemomenta and can thus be emitted by an electron or positron traveling fast enough in vacuum.Since an appropriate Lorentz boost of a spacelike momentum can change the sign of its zerothcomponent, this corresponds to the existence of negative energies in some frame. Althoughalready discussed in Ref. [13], we emphasize once again that allowing for vacuum-Cherenkovradiation in an observer Lorentz invariant theory is thus equivalent to accepting that thetheory has negative energies in some frame. The alternative of assuming the existenceof nonrenormalizable, higher-order operators that prevent negative energies in all observerframes, also prevents vacuum-Cherenkov radiation in nature.A common problem in generic theories with negative energies seems to be that there isno simple separation between the positive- and negative-energy branches of the dispersionrelation. This impedes the canonical quantization of the theory. In a Lorentz-symmetrictheory, one can easily select one of the branches by using the sign of the roots, which isinvariant under Lorentz transformations if all on-shell momenta are timelike. Since themodel with k µAF that we consider here, allows for spacelike momenta and negative energies,the sign of the root can no longer be used to select a particular branch of the dispersionrelation. It turns out, however, that a function still exists whose sign, when evaluated at aroot of Λ λ ( p ), is the same in any observer frame. It is given byΛ (cid:48) λ ( p ) = ∂ Λ λ ( p ) ∂p . (34)This function can therefore be used to separate the branches in an observer-covariant wayeven when a single branch dips into the negative energy region. The key fact that makes thiswork is that the ordering of the roots of the dispersion relation is observer Lorentz invariant.To prove that this function indeed has the correct properties, we first label the rootsof Λ T ( p ) = Λ + ( p )Λ − ( p ) as before, such that ω and ω are the roots on the right-hand sidein Figs. 1 and 2 (these are positive in concordant frames), while ω and ω are the roots onthe left-hand side in the same figures (these are negative in concordant frames). We observe17hat we can write Λ (cid:48) T ( ω j , (cid:126)p ) = (cid:34) ∂∂p (cid:89) i =5 ( p − ω i ) (cid:35) p = ω j = (cid:89) i (cid:54) = j ( ω j − ω i ) . (35)By inspection of Figs. 1 and 2, together with the considerations in Appendix A, it is nothard to establish that Λ (cid:48) T ( ω , (cid:126)p ) , Λ (cid:48) T ( ω , (cid:126)p ) > , (36a)Λ (cid:48) T ( ω , (cid:126)p ) , Λ (cid:48) T ( ω , (cid:126)p ) < . (36b)For example, Λ (cid:48) T ( ω , (cid:126)p ) = ( ω − ω )( ω − ω )( ω − ω ) . (37)From Figs. 1 and 2 it is easy to see that | ω | > | ω | , | ω | > | ω | , ω >
0, and ω <
0, exceptat p µ = ς K µ for timelike k µAF (see Eq. (12)), where ω = ω = − ω . If the roots are notdegenerate, the product of the last two factors in Eq. (37) is smaller than zero. Moreover, ω − ω is also always larger than zero, because the ordering of these roots is the same inany observer frame. This follows directly from considerations in Appendix A. We concludethat Λ (cid:48) T ( ω , (cid:126)p ) <
0. The sign of Λ (cid:48) T ( ω j , (cid:126)p ) evaluated at the other three roots is determinedsimilarly and the result corresponds to Eq. (36).Subsequently, we note thatΛ (cid:48) T ( ω , , (cid:126)p ) = Λ − ( ω , , (cid:126)p )Λ (cid:48) + ( ω , , (cid:126)p ) , (38a)Λ (cid:48) T ( ω , , (cid:126)p ) = Λ + ( ω , , (cid:126)p )Λ (cid:48)− ( ω , , (cid:126)p ) . (38b)Examination of Figs. 1 and 2 reveals that Λ − ( ω , , (cid:126)p ) < + ( ω , , (cid:126)p ) >
0. Combiningthis with Eq. (36), we conclude thatΛ (cid:48) + ( ω , (cid:126)p ) , Λ (cid:48)− ( ω , (cid:126)p ) > , (39a)Λ (cid:48) + ( ω , (cid:126)p ) , Λ (cid:48)− ( ω , (cid:126)p ) < . (39b)At p µ = ς K µ , the expression in Eq. (37) vanishes and the derivation of Eqs. (39) fails.As discussed before, the LV term disappears from the equation of motion in that case andat (cid:126)p = ς (cid:126) K we have Λ (cid:48)± ( p , ς (cid:126) K ) = 2 p ± ε ( p − ς K ) | (cid:126)k AF | , (40)18ith ε ( x ) = x/ √ x . At p = ς K , ε ( p − ς K ) is undefined. However, it can be definedusing the same limiting procedure that was used for the polarization vectors in Eq. (13)with ε ( p − ς K ) = 1 and thus sgn(Λ (cid:48)± ( p , ς (cid:126) K )) (cid:12)(cid:12)(cid:12) p → ς K = sgn ( ς K ), because |K | > | (cid:126)k AF | for timelike k µAF . Therefore, also in the degenerate case, the sign of Λ (cid:48)± ( p ) is an observerLorentz invariant quantity. In fact, it corresponds to the sign of p .From this and from Eqs. (39) we thus conclude that the sign of Λ (cid:48)± ( p ), evaluated atone of its roots, corresponds to the sign of that root in a concordant frame and is an observerLorentz invariant quantity. This obviously holds for all functions Λ λ ( p ), since for the otherpolarization modes Λ (cid:48) , ( p ) ∝ p . The fact that it also holds for the polarization modes λ = ± is directly related to the fact that the ordering of the roots stays the same in any observerframe (provided k AF < m γ ), as becomes clear from the considerations below Eq. (37).More insight as to why the sign of Λ (cid:48) λ ( p ) is an observer Lorentz invariant quantity canbe gained from considering the group velocity, defined by (cid:126)v ( λ ) g = ∂E λ ( (cid:126)p ) ∂(cid:126)p . (41)The size of (cid:126)v ( ± ) g is related to the sign of Λ (cid:48)± ( p ). To show this, we perform an observer-Lorentz-transformation on Λ (cid:48) λ ( p ) and obtain ∂ Λ λ ( p ) ∂p (cid:12)(cid:12)(cid:12)(cid:12) p = E λ ( (cid:126)p ) −→ γ (cid:20) ∂ Λ λ ( p ) ∂p − (cid:126)β · ∂ Λ λ ( p ) ∂(cid:126)p (cid:21) p = E λ ( (cid:126)p ) = γ (cid:16) (cid:126)β · (cid:126)v ( λ ) g (cid:17) (cid:20) ∂ Λ λ ( p ) ∂p (cid:21) p = E λ ( (cid:126)p ) , (42)where γ = 1 / (cid:113) − (cid:126)β is the relativistic boost factor. We used the fact, clarified in Ap-pendix B, that (cid:126)v ( λ ) g = − (cid:104) ∂ Λ λ ( p ) ∂(cid:126)p (cid:14) ∂ Λ λ ( p ) ∂p (cid:105) p = E λ ( (cid:126)p ) . It is clear that if | (cid:126)v ( λ ) g | <
1, then Λ (cid:48) λ ( p ) hasthe same sign in any observer frame. In Appendix B we show explicitly that | (cid:126)v ( λ ) g | < (cid:48)± ( p ) as an observer-Lorentz-invariant way of selecting a particular branch of the dispersion relation. For exam-ple: (cid:90) d p (cid:90) dp h ( p )sgn (cid:0) Λ (cid:48) + ( p ) (cid:1) θ (cid:0) Λ (cid:48) + ( p ) (cid:1) δ (Λ + ( p )) = (cid:90) d p Λ (cid:48) + ( ω ) h ( ω ) , (43)where h ( p ) is an arbitrary function of p . Incidentally, this shows that d p Λ (cid:48) λ ( p ) is an observerLorentz invariant phase-space factor, that can be used to replace the usual d p p , which isused in the Lorentz-symmetric case. 19 I. PROPAGATOR AND POLARIZATION VECTORS
In this section we derive a relation between a sum over bilinears of polarization vec-tors and the propagator in momentum space. Since a propagator in coordinate space isa Green’s function of the equation-of-motion operator, the momentum-space propagator P µν ( p ) satisfies S µν ( p ) P νρ ( p ) = − iδ ρµ , (44)with S µν ( p ) defined in Eq. (3). Therefore, P µν ( p ) = − i ( S − ) µν = − i (cid:18) Adj( S )det( S ) (cid:19) µν . (45)The determinant of S is given on the left-hand side of Eq. (21), while the adjugate matrixcan be determined explicitly in terms of traces of powers of S [20]. The result is iP µν ( p ) = 1Λ T ( p ) (cid:104) ( p − m γ ) η µν + 4 ( p µ p ν k AF + k µAF k νAF p − ( p µ k νAF + p ν k µAF )( p · k AF )) p − m γ +2 i(cid:15) µναβ ( k AF ) α p β (cid:105) − (1 − ξ ) p µ p ν ( p − ξm γ )( p − m γ ) , (46)with Λ T ( p ) given in Eq. (23).If there are four orthogonal polarization vectors, we can derive a relation between theexpression in Eq. (46) and the polarization vectors as defined in Eq. (9). To show this, wedefine a matrix U that has the polarization vectors as its columns, i.e. its entries are definedby U ab = e ( b ) a a, b ∈ , , , , (47)where we identify e ( b ) with e (0) , e (3) , e (+) , e ( − ) for b = 0 , , , SU ) ab = Λ b ( p ) e ( b ) a (the matrix S here corresponds to S µν , with its first index up and itssecond index down). Assuming that the polarization vectors are normalized according toEq. (11), we conclude that ( U † ηSU ) ab = Λ b ( p ) g ab , (48)with η the Minkowski metric and g given in Eq. (16). If all the polarization vectors areorthogonal then U has an inverse: ( g − U † ηU ) ab = δ ab , (49)and we can write ( S − ) ab = ( U D − U † η ) ab , (50)20here D ab = Λ b ( p ) g ab . Writing with Lorentz indices and once again labeling polarizationsby λ = 0 , , + , − , this becomes iP µν ( p ) = ( S − ) µν = (cid:88) λλ (cid:48) g λλ (cid:48) e ( λ ) µ e ( λ (cid:48) ) ∗ ν Λ λ ( p ) . (51)This sum, containing bilinears of the polarization vectors, is thus equal to the expression inEq. (46).Off shell, the form of two of the four terms in Eq. (51) depends on the sign of ( p · k AF ) − p k AF . This follows from the dependence of g λλ (cid:48) on this same sign (see Eq. (16)).We write the relevant terms as P µνT ( p ) = g ++ e (+) µ e (+) ν ∗ Λ + ( p ) + g −− e ( − ) µ e ( − ) ν ∗ Λ − ( p ) for ( p · k AF ) − p k AF > g + − e (+) µ e ( − ) ν ∗ Λ + ( p ) + g − + e ( − ) µ e (+) ν ∗ Λ − ( p ) for ( p · k AF ) − p k AF < . (52)Notice that each of the terms in Eq. (52) has a branch cut in the complex p plane, dueto the square root in the expression for Λ ± ( p ). This seems to hamper the definition ofan appropriate contour integral to implement the boundary conditions of for example theFeynman propagator. However, the expression in Eq. (46), and therefore the entire sumin Eq. (51), has no such branch cuts. In fact, if we put in the explicit expressions for thepolarization vectors, Λ ± ( p ), and components of g , we find that P µνT ( p ) = 1Λ T ( p ) (cid:2) ( p − m γ ) η µν + 2 i(cid:15) µναβ ( k AF ) α p β (cid:3) + ( p − m γ )( p µ p ν k AF + k µAF k νAF p − ( p µ k νAF + p ν k µAF )( p · k AF ))(( p · k AF ) − p k AF )Λ T ( p ) , (53)for both positive and negative ( p · k AF ) − p k AF . This expression has no branch cuts in thecomplex p plane. Note that the dependence on n µ introduced to define the polarizationvectors has dropped out of the above expression. VII. QUANTIZATION
Using the polarization vectors that follow from the equation of motion, given in Eq. (9),we can give the explicit mode expansion of the photon field: A µ ( x ) = (cid:90) d (cid:126)p (2 π ) (cid:88) λ (cid:48) λ ( p ) (cid:104) a λ(cid:126)p e ( λ ) µ ( (cid:126)p ) e − ip · x + a λ † (cid:126)p e ( λ ) ∗ µ ( (cid:126)p ) e ip · x (cid:105) p = E λ ( (cid:126)p ) , (54)21here Λ (cid:48) λ ( p ) is the derivative with respect to p of Λ λ ( p ), defined in Eq. (10). This normal-ization differs from the conventional one and corresponds to the one chosen in Eq. (18). Asmentioned below Eq. (43), in this way the phase space factor in this expression for A µ ( x )is observer Lorentz invariant. The complex weights a λ(cid:126)p and a λ † (cid:126)p become annihilation andcreation operators on a Fock space, when we quantize the theory.To perform the quantization, we compute the canonical conjugate of A µ ( x ) in theusual way by taking derivatives of the Lagrangian with respect to the time derivative of thephoton field. This results in a canonical momentum, given by π µ ( x ) = F µ ( x ) + (cid:15) µαβ ( k AF ) α A β ( x ) − η µ ξ ∂ ν A ν ( x ) . (55)We then impose the following equal-time commutation relations on the fields:[ A µ ( t, (cid:126)x ) , π ν ( t, (cid:126)y )] = iδ νµ δ ( (cid:126)x − (cid:126)y ) , (56a)[ A µ ( t, (cid:126)x ) , A ν ( t, (cid:126)y )] = 0 . (56b)This implements the standard canonical quantization in a covariant manner, as is done inthe conventional Gupta-Bleuler method. From the imposed commutation relations and theexpression for the canonical momentum, we find the following commutation relations [16]:[ ˙ A µ ( t, (cid:126)x ) , A ν ( t, (cid:126)y )] = − [ A µ ( t, (cid:126)x ) , ˙ A ν ( t, (cid:126)y )] = i ( η µν − δ µ δ ν (1 − ξ )) δ ( (cid:126)x − (cid:126)y ) , (57a)[ ˙ A µ ( t, (cid:126)x ) , ˙ A ν ( t, (cid:126)y )] = i (cid:2) (cid:15) µνλ ( k AF ) λ + (1 − ξ ) (cid:0) δ µ δ νj + δ µj δ ν (cid:1) ∂ jx (cid:3) δ ( (cid:126)x − (cid:126)y ) . (57b)In order to see what the commutation relations in Eqs. (57) imply for the oscillators a λ(cid:126)p and a λ † (cid:126)p in the mode expansion in Eq. (54), note that the latter can be inverted using theorthogonality relations (18) and (20) as [16] g λλ (cid:48) a λ (cid:48) (cid:126)q = i (cid:90) d xe iq · x (cid:104) ↔ ∂ (cid:0) η µν − (1 − ξ − ) δ µ δ ν (cid:1) − (1 − ξ − ) q j ( δ µj δ ν + δ µ δ νj ) + 2 k AF κ (cid:15) κ µν (cid:105) e ∗ ( λ ) ν ( (cid:126)q ) A µ ( x ) , (58a) g λλ (cid:48) a λ (cid:48) † (cid:126)q = − i (cid:90) d xe − iq · x (cid:104) ↔ ∂ (cid:0) η µν − (1 − ξ − ) δ µ δ ν (cid:1) − (1 − ξ − ) q j ( δ µj δ ν + δ µ δ νj ) + 2 k AF κ (cid:15) κ µν (cid:105) e ( λ ) ν ( (cid:126)q ) A µ ( x ) , (58b)where q = E λ ( (cid:126)q ) in both expressions. Using Eq. (58), together with the commutationrelations in Eqs. (57), it can be shown that the oscillators satisfy the commutation relations22 a λ(cid:126)p , a λ (cid:48) † (cid:126)q ] = − (2 π ) g λλ (cid:48) Λ (cid:48) λ ( p ) δ ( (cid:126)p − (cid:126)q ) (cid:12)(cid:12) p = E λ ( (cid:126)p ) , (59a)[ a λ(cid:126)p , a λ (cid:48) (cid:126)q ] = [ a λ † (cid:126)p , a λ (cid:48) † (cid:126)q ] = 0 . (59b)The normalization of the polarization vectors in Eqs. (11), together with the normalizationfactor 1 / Λ (cid:48) λ ( p ) in the definition of the photon field, makes sure that the right-hand side ofEq. (59a) is always positive if λ = λ (cid:48) = 3 , + , − , while it is negative if λ = λ (cid:48) = 0. This holdsin all observer frames, and follows from the fact, discussed in Section V, that Λ (cid:48) λ ( p ) | p = E λ ( (cid:126)p ) is always positive.We define the one-particle state by | (cid:126)p, λ (cid:105) = a λ † (cid:126)p | (cid:105) , (60)where | (cid:105) is the vacuum state that is annihilated by a λ(cid:126)p . As in the usual case, the one-particlestates with λ = 0 have negative norm, while the other polarizations have a positive norm.This holds in any observer frame, due to the normalization in Eq. (59) and the on-shellform of g λλ (cid:48) , given in Eq. (16). The consistency of the quantization in arbitrary observerframes thus crucially depends on the fact that the sign of Λ (cid:48) λ ( p ) is an observer Lorentzinvariant quantity. One might think that a different choice for the normalization of thepolarization vectors or the photon field could invalidate this statement. However, to keepcovariant transformation properties for the photon field these changes have to be relatedand a different choice leads to the same conclusion.As in the conventional Gupta-Bleuler method one can now go on and implement agauge-fixing condition on the Hilbert space of physical states, such that no negative-normstates appear in physical observables. In Section X we show, in the context of BRSTquantization, that this can be done consistently.Finally we note that, although the theory contains negative-energy states in someobserver frames, the vacuum is stable in the sense that it is not possible to create physicalparticles from nothing. This follows from the fact that frames exist in which the theorydoes not contain any negative-energy states (concordant frames). In such frames energyconservation prohibits the mentioned process. Observer Lorentz invariance then impliesthat it must be forbidden in any observer frame. A similar argument shows that a chargedparticle emitting Cherenkov radiation will stop doing so after a while. In concordant frames23his happens when it has lost energy to the point that no more photons (with spacelikemomenta) can be emitted [13]. This depends on the fact that the energy is bounded frombelow in all observer frames. VIII. CAUSALITY
The notion of causality is closely related to relativity and Lorentz symmetry. In quan-tum field theory one usually considers microcausality, i.e. the local (anti)commutativity ofobservables for spacelike separations. In the present case the theory is microcausal if D µν ( x − y ) = [ A µ ( x ) , A ν ( y )] = 0 for ( x − y ) < . (61)In this section we will confirm by explicit calculation that Eq. (61) holds.Using the commutation relation of the creation and annihilation operators in Eqs. (59),we find that we can write D µν ( z ) = − (cid:90) d p (2 π ) (cid:88) λ g λλ Λ (cid:48) λ ( p ) (cid:2) e ( λ ) µ ( (cid:126)p ) e ( λ ) ν ∗ ( (cid:126)p ) e − ip · z − e ( λ ) µ ∗ ( (cid:126)p ) e ( λ ) ν ( (cid:126)p ) e ip · z (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) p = E λ ( (cid:126)p ) , (62)where z = x − y . Using Eq. (43), the fact that e ( λ ) µ ∗ ( − p ) e ( λ ) ν ( − p ) = e ( λ ) µ ( p ) e ( λ ) ν ∗ ( p ), andΛ (cid:48) ( p ) p →− p −→ − Λ (cid:48) ( p ), we can write this as D µν ( z ) = − (cid:90) d p (2 π ) (cid:88) λλ g λλ e ( λ ) µ ( p ) e ( λ ) ν ∗ ( p ) sgn(Λ (cid:48) ( p )) δ (Λ λ ( p )) e − ip · z , (63)It is straightforward to check, by explicit calculation or by using the relation in Eq. (51),that this is equal to D µν ( z ) = − (cid:90) d p (2 π ) (cid:90) C dp (2 π ) P µν e − ip · z = i (cid:90) d p (2 π ) (cid:90) C dp (2 π ) Adj(S) µν det( S ) e − ip · z , (64)where the contour in the complex p plane encircles all poles in the clockwise direction, P µν is given in Eq. (46), and Adj( S ) µν = det( S )( S − ) µν is the adjugate matrix of S , whoserelation to P µν is given in Eq. (45). Note that P µν contains double and triple poles at p µ = ς K µ , with K µ defined in Eq. (12) and ς = ±
1. The result of calculating the residuesat these higher-order poles corresponds to the definitions of the polarization vectors at thementioned momenta, given in Eq. (13).The expression in Eq. (64) is manifestly observer Lorentz covariant. Therefore, wecan calculate its components in a particular frame. If z = ( x − y ) <
0, we can go to an24bserver frame, where z = 0. In this frame we perform the contour integration. Realizingthat det( S ) = ξ (cid:81) i =0 ( p − ω i ), we get that D µν ( z ) | z =0 = − ξ (cid:90) d p (2 π ) (cid:88) i =1 (cid:34) Adj( S ) µν (cid:81) j (cid:54) = i ( p − ω j ) e − i(cid:126)p · (cid:126)z (cid:35) p = ω i . (65)This result is not valid at the two points in momentum space p µ = ς K µ (see Eq. (12)) when k AF >
0. At these two momenta, the functions Λ λ ( p ) with λ = 3 , + , − have degenerateroots. Therefore, the expression in Eq. (64) has a triple pole. However, performing the p contour integration at the fixed three-momentum value (cid:126)p = ς (cid:126) K in the appropriate way givesidentically zero.Away from p µ = ς K µ , we use that every component of the numerator in Eq. (65) is apolynomial in p . Furthermore, it is easy to show that (cid:88) i =1 ( ω i ) n (cid:81) j (cid:54) = i ( ω i − ω j ) = n = 0 , . . . ,
61 if n = 7 (cid:80) i =1 ω i if n = 8 . (66)Using the explicit expression for Adj( S ), that follows from Eq. (45) and Eq. (46), it becomesclear that D µν ( z ) | z =0 = 0 , (67)i.e. every compenent of D µν ( z ) vanishes in an observer frame where z = 0. Therefore,since D µν ( z ) is observer Lorentz covariant, we conclude that it vanishes in any frame with( x − y ) <
0, i.e. the fields commute for spacelike separation, confirming microcausality.Notice that we also confirmed Eq. (56b). The other commutation relations in Eq. (57) canbe derived in a completely analogous way.
IX. FEYNMAN PROPAGATOR
We take the Feynman propagator in coordinate space to be equal to the vacuumexpectation value of the time-ordered product of fields at x and y , i.e. D µνF ( x − y ) = θ ( x − y ) D µν + ( x − y ) + θ ( y − x ) D µν − ( x − y ) , (68)with D µν + ( x − y ) = (cid:104) | A µ ( x ) A ν ( y ) | (cid:105) , (69a) D µν − ( x − y ) = (cid:104) | A ν ( y ) A µ ( x ) | (cid:105) . (69b)25otice that the ± signs in these definitions have nothing to do with the λ = ± polarizations,rather they correspond (in concordant frames) to the positive and negative energy modes.This is further clarified when we look at the explicit expressions for D µν ± ( x − y ) that followfrom inserting Eq. (54). They are given by D µν ± ( z ) = − (cid:90) d p (2 π ) (cid:88) λλ (cid:48) g λλ (cid:48) θ ( ± Λ (cid:48) λ ( p ))sgn( ± Λ (cid:48) λ ( p ))(2 π ) δ (Λ λ ( p )) (cid:15) ( λ ) µ (cid:15) ( λ (cid:48) ) ν ∗ e − ip · z , (70)with z = x − y . Due to the Heaviside stepfunction θ ( ± Λ (cid:48) λ ( p )), discussed at the end ofSection IV, D µν ± ( z ) is only non vanishing if ± Λ (cid:48) λ ( p ) >
0, which in concordant frames isequivalent to ± p > θ ( z ) = i π (cid:82) e − iτz dττ + iε , wecan write the Feynman propagator as D µνF ( z ) = (cid:90) C d p (2 π ) P µν e − ip · z = − i (cid:90) d p (2 π ) (cid:34) ( p − m γ ) η µν Λ T ( p ) − iε + 4 ( p µ p ν k AF + k µAF k νAF p − ( p µ k νAF + p ν k µAF )( p · k AF ))( p − m γ + iε )(Λ T ( p ) − iε )+ 2 i(cid:15) µναβ ( k AF ) α p β Λ T ( p ) − iε − (1 − ξ ) p µ p ν ( p − ξm γ + iε )( p − m γ + iε ) (cid:35) e − ip · z , (71)where, on the first line, the integration contour in the complex p plane goes above (below)the poles ( ω ) for which Λ (cid:48) λ ( p ) | p = ω is positive (negative). After the second equality sign, thisis represented by a Feynman ε prescription. X. BRST AND THE SPACE OF STATES
The structure of the space of states is most clearly established in the BRST formalism[31] by completing the photon Lagrangian (2) with the contributions for the St¨uckelbergscalar field φ as well as the (anticommuting) ghost and antighost fields c and ¯ c (with ghost-numbers 1 and −
1, respectively): L St¨uck = L A + L φ + L gh (72)where L A is given by (2), while L φ = ( ∂ µ φ ) − ξm γ φ (73)26nd L gh = − ¯ c ( ∂ + ξm γ ) c . (74)Note that the antighost field ¯ c is defined to be anti-hermitian (¯ c † = − ¯ c ), while all otherfields are hermitian. Lagrangian (72) can now be obtained from the Lagrangian L (cid:48) St¨uck = − F µν F µν + 12 k κAF (cid:15) κλµν A λ F µν + 12 m γ ( A µ − m γ ∂ µ φ ) + ξ B + B ( ∂ µ A µ + ξm γ φ ) − ¯ c ( ∂ + ξm γ ) c (75)upon integrating out the (auxiliary) Nakanishi-Lautrup field B [32].Lagrangian (75) changes by a total derivative under the BRST transformation s definedby sA µ = (cid:15)∂ µ c (76) sφ = (cid:15)m γ c (77) s ¯ c = (cid:15)B (78) sB = sc = 0 (79)where (cid:15) is some constant infinitesimal Grassmann-valued parameter. The BRST transforma-tions (76)–(79) are generated by the action of the nilpotent BRST charge Q B = (cid:82) d (cid:126)xj B = (cid:82) d x ( B ∂ c − ∂ B c ), where j Bµ is the conserved Noether current. The space of physicalstates is defined by the space of closed states (those that are annihilated by Q B ) of ghostnumber zero modulo the exact states (those that are in the image of Q B ). Restricting our-selves to ghost number zero (no ghost excitations), it follows from (77) that one-particlestates created by the field φ are unphysical. Moreover, we see from (76) and (77) that thelinear combination χ µ = A µ − m γ ∂ µ φ (80)(the Proca field) is BRST invariant, and thus any one-particle states created by χ µ areclosed. Finally, we see from (78) that any one-particle excitations of the Nakanishi-Lautrupfield B are exact. Using the equations of motion for B and φ , it follows that on-shell we canreplace B → ∂ µ χ µ . From this we see that the physical one-particle states can be taken tocorrespond to the three transverse polarizations of χ µ (which coincide with the transversepolarizations e ( i ) µ ( (cid:126)p ), i = + , − ,
3, of A µ ). The exact one-particle states correspond to theremaining longitudinal mode of χ µ . 27t is worthwhile to point out that the quantization is unaffected by the Lorentz-violating k AF term. XI. DISCUSSION
In this paper, we performed the covariant quantization of Lorentz- and CPT-violatingMaxwell-Chern-Simons theory for spacelike, lightlike, as well as timelike k µAF . To avoidimaginary energies and for regularization purposes, a non-zero photon mass was introducedthrough the St¨uckelberg mechanism. This can be done well below any observational con-straints.We found explicit expressions for a set of four orthogonal and normalized polarizationvectors, whose definition is valid in almost all of four-momentum space. These polarizationvectors are eigenvectors of the equation-of-motion operator and have the functions Λ λ ( p ),defined in Eq. (10), as their eigenvalues. The relations Λ λ ( p ) = 0 determine the dispersionrelations for the different polarization modes and their solutions fix the on-shell polarizationvectors. The hypersurface in momentum space where the definitions of the polarizationvectors are invalid only intersects the relevant mass shells at two singular points and onlyfor timelike k µAF . This corresponds to the vanishing of the LV term in the original Lagrangian.We highlighted the treatment of these singular points throughout the paper.We discussed several properties of the dispersion relation. In particular, we showedthat it has eight nondegenerate roots, except at the two singular points, where it has two setsof three degenerate roots (and two nondegenerate ones). We confirmed that the observer-Lorentz-invariant condition k AF < m γ guarantees the reality of all the roots. Moreover, wederived an observer Lorentz non-invariant condition, ( k AF ) < m γ , that makes sure that allenergies are positive. Since the latter condition cannot be maintained in arbitrary observerframes, the sign of the roots cannot be used to select a branch of the dispersion relation, asis done in the usual, Lorentz-symmetric, case. However, we found that the sign of Λ (cid:48) ( p ) canbe used instead. This fact is closely related to the observer invariance of the root orderingand the group velocity being smaller than unity.Being able to unambiguously identify the different branches of the dispersion relationin all observer frames, allowed us to construct an observer Lorentz covariant mode expansionof the photon field in terms of the polarization vectors for the different modes. Using the28esulting explicit expression for the photon field, we performed the quantization of the theory.We also derived the Feynman propagator and showed that the theory is microcausal. Finallywe showed, in the context of BRST quantization, that the three transverse modes are thephysical ones.One obvious direction into which one can extend the present work is investigating themassless limit. We expect that taking the limit m γ → k µAF (for timelike k µAF ,the imaginary energies will reappear). All the more because the λ = 3 polarization modeseems to decouple in a gauge-invariant theory, because e (3) µ ∝ p µ in that case, resulting intwo physical states. As in the Lorentz-symmetric case, in the massless limit it is not possibleto find a basis of four orthogonal covariant polarization vectors in the general class of gaugeswe consider in this paper. However, we expect that it is possible, using BRST quantization,to show that the non-covariant components of the field are unphysical and decouple.A second option for follow-up work is to include interactions and quantum effects. Thelatter might introduce other, possibly higher-dimensional, LV coefficients through radiativecorrections. To go beyond tree-level one has to consider the effect of such effective LVcoefficients.Finally, one could try to apply the methods of the present work to the CPT-even k F term of the minimal SME or even include higher-dimensional kinetic terms for the photon,which have been categorized in Ref. [33]. Note that, in the latter case, one would haveto find a way to consistently deal with spurious Ostrogradski modes [34] that arise due tohigher-order time derivatives in the Lagrangian.Presently, the fact that the covariant quantization of the present Lorentz- and CPT-violating theory is possible, at least with a non zero photon mass (well below observationalconstraints), despite the presence of negative-energy states in some observer frames, is animportant result of this paper. It is of relevance, in particular, to considerations of vacuumCherenkov radiation, for which such negative-energy states are unavoidable. Moreover, theexplicit expressions for the polarization vectors, their bilinears, and the Feynman propagator,in arbitrary observer frames, pave the way for calculations of LV observables involving k µAF .29 CKNOWLEDGMENTS
We thank V. A. Kosteleck´y for helpful suggestions. This work is supported inpart by the Funda¸c˜ao para a Ciˆencia e a Tecnologia of Portugal (FCT) through projectsUID/FIS/00099/2013 and SFRH/BPD/101403/2014 and program POPH/FSE.
Appendix A: Reality and sign of the roots of Λ ± ( p ) In this Appendix we give the details of some statements made in the main text, con-cerning the sign and the possible complex-valuedness of the roots of Λ ± ( p ). We do this byconsidering the functions defined in Eq. (26) and plotted in Figs. 1 and 2 for the case oftimelike and spacelike/lightlike k µAF , respectively.First of all, we discuss the condition | x α | < (cid:113) (cid:126)p + m γ (A1)for spacelike and lightlike k µAF , which is used below Eq. (31). The points p = x α are thepoints where the branches of ± f δ ( p ) start (see Fig. 2) and their expressions are given inEq. (31). Eq. (A1) reflects the condition that these points stay inside the curve of f ( p ). Alittle algebra shows that Eq. (A1) holds if (cid:126)p > − (cid:126)k AF m γ ( k AF √ sin θ + α (cid:112) − k AF cos θ ) , (A2)where θ is the angle between (cid:126)p and (cid:126)k AF . Since the expression on the right is always negative,we see that Eqs. (A1) and (A2) are always satisfied and thus that Λ ± ( p ) always have tworeal roots each for spacelike and lightlike k µAF .Next, we consider the inequality − f δ (0) > f (0) , (A3)which, for k AF >
0, is sufficient to make sure that the signs of ω and ω are different. For k AF < f δ (0) is not real. However, in that case it is obvious from Fig. 2and the argument following Eq. (A1) that the signs of ω and ω will differ. It is easy to seethat − f δ (0) = f (0) if | (cid:126)p | = (cid:113) ( k AF ) − (cid:126)k AF sin θ ± (cid:113) ( k AF ) − (cid:126)k AF sin θ − m γ . (A4)30f k µAF is spacelike, the first square root can become imaginary. This corresponds to f δ (0)being imaginary, for which case ω and ω differ in sign. If both square roots in Eq. (A4)are real, Eq. (A4) defines an interval for | (cid:126)p | , outside of which Eq. (A3) is satisfied and Λ + ( p )has two roots of opposite sign. Inside of the interval, however, the signs of ω and ω are thesame and after the redefinition of one of the roots, the theory can contain states of negativeenergy. It is clear from Eq. (A4) that there will be no negative energies if ( k AF ) < m γ ,because the second square root is imaginary in that case. This confirms that the conditionin Eq. (28) implies energy positivity. Furthermore, the second square root is real ifcos θ > m γ − k AF (cid:126)k AF . (A5)For angles satisfying this inequality and | (cid:126)p | in the interval in Eq. (A4), the two roots ofthe dispersion relation Λ + ( p ) = 0 have the same sign. Since we redefine the energies asin Eq. (24), the theory will contain negative-energy photons if ω ( (cid:126)p ) <
0. This happens ifthe extremum of f δ ( p ) lies to the left of p = 0, i.e. if k ( (cid:126)p · (cid:126)k AF ) <
0. The momentumof the photons with negative energy thus lies in a cone around the direction defined by − sgn( k ) (cid:126)k AF (and not in the opposite direction).Finally, we show that all roots of Λ + ( p ) are real if Eq. (29) holds, i.e. if k AF < m γ . (A6)For lightlike and spacelike k µAF , this was already evident from the considerations followingEq. (A1). In the following, we show it for timelike k µAF . To achieve this, we ascertain thatif k AF < m γ , we can always find a value of p for which − f δ ( p ) > f ( p ) , (A7)meaning that − f δ ( p ) must intersect f ( p ) at two different values of p , corresponding tothe two real roots of Λ + ( p ). To proof this, we start with an ansatz for p : p = a | (cid:126)p | , (A8)with a a dimensionless factor. At this value of p , we find f ( a | (cid:126)p | ) = (cid:126)p ( a − − m γ , (A9a) − f δ ( a | (cid:126)p | ) = − | (cid:126)p |√ X ≡ − | (cid:126)p | (cid:113) a (cid:126)k AF − ak AF | (cid:126)k AF | cos θ + (cid:126)k AF cos θ + k AF . (A9b)31e do not gain much insight by solving − f δ ( a | (cid:126)p | ) > f ( a | (cid:126)p | ) for a directly. However, weeasily find that − f δ ( a | (cid:126)p | ) = f ( a | (cid:126)p | ) for | (cid:126)p | = √ X ± (cid:113) X + ( a − m γ − a . (A10)Furthermore, − f δ ( a | (cid:126)p | ) − f ( a | (cid:126)p | ), as a function of | (cid:126)p | , is a parabola that opens upward if a <
1. So, if a < | (cid:126)p | is in the interval defined by Eq. (A10), then − f δ ( a | (cid:126)p | ) < f ( a | (cid:126)p | ).It follows that if we can find an | a | < | (cid:126)p | does not exist and therefore − f δ ( a | (cid:126)p | ) > f ( a | (cid:126)p | )for any | (cid:126)p | .We find that the argument of the second square root in Eq. (A10) vanishes if a = ( k AF ) m γ + (cid:126)k AF | (cid:126)k AF | k AF cos θ ± k AF (cid:118)(cid:117)(cid:117)(cid:116) ( m γ − k AF ) (cid:32) m γ + (cid:126)k AF ( k AF ) − cos θ (cid:126)k AF ( k AF ) (cid:33) . (A11)It is straightforward to check that the square root is real if k AF < m γ and that the absolutevalue of first term is smaller than one in that case. Eq. (A11) thus defines an a -interval forwhich the second square root in Eq. (A10) is imaginary. Therefore, near the center of thisinterval, there are values of | a | < − f δ ( a | (cid:126)p | ) > f ( a | (cid:126)p | ). This means that − f δ ( p )intersects f ( p ) at two different values of p . We conclude that Λ + ( p ) always has two realroots if k AF < m γ . Appendix B: Group velocity
In this appendix, we consider the group velocity of the different modes of the photon.It is defined in Eq. (41) as (cid:126)v ( λ ) g = ∂E λ ( (cid:126)p ) ∂(cid:126)p . (B1)We will show that (cid:126)v ( λ ) g = − (cid:20) ∂ Λ λ ( p ) ∂(cid:126)p (cid:30) ∂ Λ λ ( p ) ∂p (cid:21) p = E λ ( (cid:126)p ) (B2)and that | (cid:126)v ( λ ) g | <
1. For the modes with λ = 0 , λ = ± modes in the remainder of this appendix.The fact that Eq. (B2) holds, follows easily by realizing that Λ T ( p ) = Λ + ( p )Λ − ( p ) is apolynomial in p , which allows us to write it asΛ T ( p ) = ( p − E + ( (cid:126)p ))( p + E + ( − (cid:126)p ))( p − E − ( (cid:126)p ))( p + E − ( − (cid:126)p )) , (B3)32here we used the energy redefinitions, given in Eq. (24). From Eq. (B2), it follows that,for λ = + , − , ∂E ± ( (cid:126)p ) ∂(cid:126)p = − (cid:20) ∂ Λ T ( p ) ∂(cid:126)p (cid:30) ∂ Λ T ( p ) ∂p (cid:21) p = E ± ( (cid:126)p ) = − (cid:20) ∂ Λ ± ( p ) ∂(cid:126)p (cid:30) ∂ Λ ± ( p ) ∂p (cid:21) p = E ± ( (cid:126)p ) , (B4)confirming Eq. (B2). This equality does not hold if p µ = ς K µ . In that degenerate case, thegroup velocity, as given in Eq. (B1), becomes ill-defined, as can be seen from explicit calcu-lations for purely timelike k µAF , or from the analysis at the end of this section. However, wecan assign a value to the right-hand side of Eq. (B2) by employing some limiting procedure,as described below Eqs. (13). This is in fact the quantity we need in Eq. (42).It remains to be shown that | (cid:126)v ( λ ) g | <
1. To this affect we define w µ ± ≡ ∂ Λ ± ( p ) ∂p µ = 2 (cid:32) p µ ± ( p · k AF ) k µAF − k AF p µ (cid:112) ( p · k AF ) − p k AF (cid:33) , (B5)such that (cid:126)v ( λ ) g = − [ (cid:126)w λ /w λ ] p = E λ ( (cid:126)p ) . It follows that | (cid:126)v ( λ ) g | < w µλ , evaluated on shell,is timelike. We determine that on shell w λ is given by w λ (cid:12)(cid:12) p = E λ ( (cid:126)p ) = 4( m γ − k AF ) . (B6)Therefore, w λ > k AF < m γ . The latter is a necessary condition for the theory to beconsistent, as already mentioned in Section IV. We thus conclude that the absolute value ofphoton group velocity is smaller than unity for the cases we consider. For the degenerate caseof p µ = ς K µ , this statement is invalid, because Eq. (B2) does not hold (the group velocitybecomes ill-defined). However, Eq. (B6) shows that the quantity relevant for Eq. (42), whichis the right-hand side of Eq. (B2), is smaller than unity, even if p µ = ς K µ . We note that amethod to desingularize the classical group velocity at the singular points exists [35]. Appendix C: Energy lower bound
Using the expression for the group velocity implied by Eqs. (B2) and (B5), we willdetermine the lowest value the photon energy can reach in a particular observer frame. Forthe polarization modes with λ = 0 , λ = ± modes. Tofind the stationary points of the energy as a function of (cid:126)p , we will determine the (cid:126)p values for33hich (cid:126)w ± in Eq. (B5), vanishes. These points correspond to the lower bound for the energy,unless the energy at the singular point (given by m γ | k AF | / (cid:112) k AF ) is smaller.It is straightforward to establish that (cid:126)w ± vanishes if (cid:126)k AF = (cid:126)
0, if k AF = 0, or if (cid:126)p ∝ (cid:126)k AF ,i.e. when ( (cid:126)p · ˆ k AF ) is either | (cid:126)p | or −| (cid:126)p | . In all of these cases, the dispersion relation can besolved exactly. For purely timelike and purely spacelike k µAF , the concordant-frame-positiveenergy solutions are given by E ± ( (cid:126)p ) | (cid:126)k AF = (cid:126) = (cid:113) (cid:126)p + m γ ∓ | k AF || (cid:126)p | , (C1a) E ± ( (cid:126)p ) | k AF =0 = (cid:114) (cid:126)p + m γ + 2 (cid:126)k AF ∓ (cid:113) (cid:126)k AF + m γ (cid:126)k AF + ( (cid:126)p · (cid:126)k AF ) , (C1b)such that (cid:126)v ( ± ) g (cid:12)(cid:12) (cid:126)k AF = (cid:126) = (cid:126)p ∓ | k AF | ˆ pE ± ( (cid:126)p ) , (C2a) (cid:126)v ( ± ) g (cid:12)(cid:12) k AF =0 = (cid:126)p (cid:113) (cid:126)k AF + m γ (cid:126)k AF + ( (cid:126)p · (cid:126)k AF ) ∓ ( (cid:126)p · (cid:126)k AF ) (cid:126)k AF E ± ( (cid:126)p ) (cid:113) (cid:126)k AF + m γ (cid:126)k AF + ( (cid:126)p · (cid:126)k AF ) . (C2b)The group velocity for the purely timelike case in Eq. (C2a) can only vanish for the λ = +mode, in which case | (cid:126)p | = | k AF | , giving a energy lower bound of (cid:113) m γ − ( k AF ) . For the λ = − mode, the energy lower bound for the purely timelike case is the energy at thesingular point ( (cid:126)p = (cid:126) k AF = 0, thegroup velocity vanishes if (cid:126)p = 0 (we will deal with (cid:126)p ∝ (cid:126)k AF seperately). It follows that theminimal energy is then given by (cid:113) m γ + (cid:126)k AF ∓ | (cid:126)k AF | .Having dealt with the special cases of purely timelike and purely spacelike k µAF , weproceed to the general case where the LV four vector has both nonzero time and spacecomponents. As mentioned earlier, the group velocity then only vanishes if (cid:126)p is (anti)parallelto (cid:126)k AF . The corresponding expressions for E ± ( (cid:126)p ) are given by: E ± ( (cid:126)p ) | (cid:126)p ∝ (cid:126)k AF = (cid:113) (cid:126)p + m γ + (cid:126)k AF ± k AF ( (cid:126)p · ˆ k AF ) ∓ | (cid:126)k AF | if k AF ( (cid:126)p · ˆ k AF ) | (cid:126)k AF | ≤ (cid:112) (cid:126)p + m γ (cid:113) (cid:126)p + m γ + (cid:126)k AF ∓ k AF ( (cid:126)p · ˆ k AF ) ± | (cid:126)k AF | if k AF ( (cid:126)p · ˆ k AF ) | (cid:126)k AF | ≥ (cid:112) (cid:126)p + m γ . (C3)The lower expression only applies for timelike k µAF , because k AF ( (cid:126)p · ˆ k AF ) > | (cid:126)k AF | (cid:112) (cid:126)p + m γ requires k AF >
0. For k AF ( (cid:126)p · ˆ k AF ) = | (cid:126)k AF | (cid:112) (cid:126)p + m γ , which also requires timelike k µAF andcorresponds to the singular point of Eq. (12), Eq. (C3) gives E ± ( (cid:126)p ) = (cid:112) (cid:126)p + m γ , showing34hat the energy is continuous through the singular point. However, the group velocity thatfollows from Eq. (C3) is given by (cid:126)v ( ± ) g (cid:12)(cid:12) (cid:126)p ∝ (cid:126)k AF = (cid:126)p ± sgn(ˆ p · ˆ k AF ) k AF ˆ pE ± ( (cid:126)p ) ± | (cid:126)k AF | if k AF ( (cid:126)p · ˆ k AF ) | (cid:126)k AF | ≤ (cid:112) (cid:126)p + m γ (cid:126)p ∓ sgn(ˆ p · ˆ k AF ) k AF ˆ pE ± ( (cid:126)p ) ∓ | (cid:126)k AF | if k AF ( (cid:126)p · ˆ k AF ) | (cid:126)k AF | ≥ (cid:112) (cid:126)p + m γ , (C4)which is clearly not continuous through the singular point, because approaching from belowgives (cid:126)v ( ± ) g = (cid:18) | (cid:126)k AF || k AF | ± √ k AF m γ (cid:19) ˆ p , while approaching from above gives (cid:126)v ( ± ) g = (cid:18) | (cid:126)k AF || k AF | ∓ √ k AF m γ (cid:19) ˆ p .Using methods described in Ref. [35], one can nevertheless rigorously define the group ve-locity at the singular points. Here we instead choose to check explicitly if the energy at thesingular point is the smallest energy value.We can thus use Eq. (C4) to determine the lower bound for the energies, unless theminimal energy is reached exactly at the singular point, which we check explicitly. Investi-gating when the expression for the group velocity vanishes, while simultaneously satisfyingthe condition on the right, and comparing to the energy at the singular point, we come tothe conclusion that the lower bound for the photon energies in the λ = ± modes is given by E − ( (cid:126)p ) | min = m γ | k AF | √ k AF if | (cid:126)k AF | ≤ ( k AF ) √ ( k AF ) + m γ (cid:113) m γ − k AF + | (cid:126)k AF | if | (cid:126)k AF | ≥ ( k AF ) √ ( k AF ) + m γ , (C5a) E + ( (cid:126)p ) | min = (cid:113) m γ − k AF − | (cid:126)k AF | , (C5b)for timelike, lightlike, as well as spacelike k µAF . Eqs. (C5) also capture the results for thepurely timelike and purely spacelike case.These results show, first of all, that the energy has a finite, albeit observer dependent,lower bound. Furthermore, we confirm the results of appendix A, that the energy of the λ = − mode is always positive, while E + ( (cid:126)p ) can become negative if ( k AF ) > m γ .35 IG. 1. (a) Typical plot of f ( p ) and ± f δ ( p ) for timelike k AF . The plots are exaggerated inthe sense that for physically viable values of k µAF in concordant frames and for experimentallyattainable values of (cid:126)p , both f δ ( p ) and − f δ ( p ) are nearly horizontal and very close to the p -axis. Black arrows indicate p values, colored arrows indicate values of the corresponding function.(b) Corresponding plot for ( f ( p )) and ( f δ ( p )) . The latter stays above the p axis, whichcorresponds to the square root being always real.FIG. 2. (a) Typical plot of f ( p ) and ± f δ ( p ) for spacelike or lightlike k µAF . The plots areexaggerated in the sense that for physically viable values of k µAF in concordant frames and forexperimentally attainable values of (cid:126)p , the two branches of both f δ ( p ) and − f δ ( p ) are nearlyhorizontal and very close to the p axis, while their starting points are also very close together.Black arrows indicate p values, colored arrows indicate values of the corresponding function. (b)Corresponding plot for ( f ( p )) and ( f δ ( p )) . The latter goes below the p axis, which correspondsto the square root becoming imaginary.
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