Spontaneous breakdown of Lorentz symmetry in scalar QED with higher order derivatives
aa r X i v : . [ h e p - t h ] S e p Spontaneous breakdown of Lorentz symmetry in scalar QED with higher orderderivatives
Janos Polonyi a , Alicja Siwek ab a University of Strasbourg, High Energy Physics Theory Group, CNRS-IPHC,23 rue du Loess, BP28 67037 Strasbourg Cedex 2, France and b Wroclaw University of Technology, Institute of Physics,Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland (Dated: November 13, 2018)Scalar QED is studied with higher order derivatives for the scalar field kinetic energy. A localpotential is generated for the gauge field due to the covariant derivatives and the vacuum withnon-vanishing expectation value for the scalar field and the vector potential is constructed in theleading order saddle point expansion. This vacuum breaks the global gauge and Lorentz symmetryspontaneously. The unitarity of time evolution is assured in the physical, positive norm subspace andthe linearized equations of motion are calculated. Goldstone theorem always keeps the radiation fieldmassless. A particular model is constructed where the the full set of standard Maxwell equations isrecovered on the tree level thereby relegating the effects of broken Lorentz symmetry to the level ofradiative corrections.
Keywords: Effective theories, symmetry breaking, Higgs phenomenon, higher order derivatives
I. INTRODUCTION
We know no exact equation of motion in physics, all laws are inferred by ignoring some loosely attached part of thesystem considered. As a result, the equations of motion should be tested for the stability of solutions against addingsmall correction terms to the equations. Such an analysis is usually performed in the framework of the renormalizationgroup [1] and perturbation expansion can be used to establish that perturbing terms with higher mass dimension (weuse units c = ~ = 1) are less important at short distances.Nevertheless, it is an important difference whether the higher mass dimension arises from field amplitude or space-time derivative because the latter may modify the tree-level, normal mode structure and generate new degrees offreedom. In fact, a field theory with a real, single component scalar field characterized by a Lagrangian containing n d space-time derivatives of the field contains n d degrees of freedom. Once the new propagating degrees of freedomare present their interactions might well be non-negligible due to IR or UV divergences even if the coupling constantsin the bare Lagrangian are weak. There is another more fundamental change generated by these terms, spontaneoussymmetry breaking of space-time symmetries due to an inhomogeneous condensate, the subject of this work. The pointis that the renormalization group equations usually include quantum fluctuations only. The higher order derivativesterms may generate new relevant operators in the IR on the tree-level which lead to a vacuum with inhomogeneouscondensate. We do not embark on a general renormalization group study here, rather present a simple-minded analysisof the symmetry and the quasi-particle content of an Abelian gauge model in the leading order saddle point expansion.If the condensate consists of bosons with non-vanishing momentum, filling up the whole quantization volume, thenthe “wavy vacuum” breaks the space-time symmetries, in a manner similar to solids where the infinite inertia ofthe solid prevents the zero modes to restore the broken external symmetries. The result, expected from solid statephysics, is the appearance of several branches of the dispersion relations, different elementary excitations in the theory.Note that if translation invariance is broken at sufficiently short length scale to remain undetectable for the class ofobservables one uses then the vacuum appears homogeneous. We shall see that in models with gauge symmetry wherethe covariant derivative is supposed to acquire non-vanishing value in the condensate the inhomogeneity of the vacuummay be gauged away and we find a homogeneous condensate which simplifies the model enormously. The result issome kind of extension of Higgs-mechanism where the non-vanishing expectation value for the gauge field breaksLorentz symmetry. The resulting Goldstone modes remain in the gauge field sector and protect some components ofthe gauge field against mass generation.The model studied in this work is scalar QED where higher order (covariant) derivative terms are introduced for thecharged scalar field. The higher order terms of this model can be imagined either as smooth cutoff in defining an UVfinite theory or as originating from the elimination of some heavy particle and approximating the self energy of a scalarcharged particle by a polynomial of finite order in the momentum. Goldstone theorem protects the electromagneticfield against becoming massive, Maxwell equations are recovered in the linearized equation of motion, rendering theLorentz symmetry breaking effects to radiative corrections. The rather technical problem of proving unitarity of themodel within the physical, positive norm subspace is solved within perturbation expansion by assuring real energyspectrum for normal modes and preserving the physical subspace, consisting of states of positive norm during thetime evolution.The dynamical breakdown of space-time symmetries by higher order derivatives has already been studied in two [2],three [3] and four [4, 5] dimensional Euclidean models where periodically modulated condensate has been observedand several particle modes have been found corresponding to a single quantum field [5]. The present work can beconsidered as continuation of such inquiries for models defined in Minkowski space-time and equipped with gaugesymmetry. The spontaneous breakdown of relativistic symmetries has been considered within the scheme of emergingphotons [6] and the bumblebee models [7] where an external Mexican hat potential is assumed for the vector bosons.Our plan is less ambitious and starts with photons as elementary particles.Our results can be best summarized by comparing them with the conventional Higgs-mechanism where Goldstonemode arising from the spontaneous breakdown of global gauge invariance appears in the gauge field which becomesmassive. In our case the relativistic space-time symmetry is broken spontaneously as well, leaving behind three moreGoldstone modes. Two of them are non-vanishing helicity components of the gauge field and restore the conventional,massless radiation field of electrodynamics. The third soft mode resides in a certain combination of the vanishinghelicity component of the photon and the scalar field and is responsible for the preservation of the usual, long rangeCoulomb propagator for the temporal component of the gauge field. Therefore, despite the spontaneous breakdown ofinternal and external symmetries the free propagator and the normal modes of the electromagnetic field are equivalentwith those of conventional electrodynamics. The symmetry breaking influences radiative corrections and the dynamicsof the charged scalar field only.We start in this paper in section II by listing few salient features of scalar models with higher order derivatives.The issue of unitarity and the way it can be recovered by proving reflecting positivity in Euclidean space-time isdiscussed in Section III. Our model, scalar electrodynamics with higher order derivative for the charged scalar fieldis introduced in Section IV. The dynamics is discussed in static temporal gauge where the exceptional features ofthe time component of the gauge field can be dealt with in the easiest manner. Section V contains the constructionof the vacuum in the leading, tree-level order of the saddle point expansion. The stability of the vacuum and theunitarity within the physical subspace is shown in Section V C. The particle content of the theory is defined by thequadratic part of the Lagrangian which is explored in Section VI. Finally, Section VII is reserved to our summary.An Appendix contains the details of calculating the quadratic part of the action. II. UNITARITY AND HIGHER ORDER DERIVATIVES
Effective theories may or may not be unitary. In fact, the unitarity is lost when a particle, retained in an effectivetheory can lower its energy by the emission of other particles which have been eliminated in deriving the effective theory.Nevertheless, non-unitary effective theory remains a powerful approximation scheme when these decay processes arekinematically suppressed and make the life-time sufficiently long. But one would still prefer to recover the simplicityfollowing from unitarity in effective theories which tend to be rather complicated. For instance processes whose energyremains below the mass M of the particle eliminated should reflect unitary dynamics when considered for sufficientlylong time. Nevertheless the UV divergences and quantum anomalies of the underlying theory mix the high energyeffects into low energy sector. The most natural way of recovering a unitary effective theory is to place the UV cutoffbelow the eliminated particle mass, Λ < M . But this solution is not as simple as it seems. On one hand, smoothcutoff allows decay processes with small but non-vanishing probability, and on the other hand, sharp cutoff leads toartificial non-local, acausal dynamics at the length scale Λ − which is observable in this case.The hallmark of effective theories is the appearance of higher order derivatives in the Lagrangian, reflectingmomentum-dependent self energies of quasi-particles or form factors. The latter appear in vertices and have mainlyperturbative effects. But the former are in the quadratic part of the action in the fields and modify the structure ofquasi-particles. They are sometimes used as a kind of Pauli-Villars regulator which renders the effective theory UVfinite [8–10]. Even though the scale of this smooth cutoff is M , the non-unitary processes are not fully suppressed.Once the effective theory is rendered UV finite we may consider it as an extension of the class of potentially inter-esting consistent, microscopic models because its UV dynamics is well defined. Motivated by the search of possiblefundamental theories one naturally expects the complete suppression of non-physical, non-unitary processes.We consider in this section a model for a neutral scalar particle described by the field φ ( x ). The interaction verticeswill be kept in a momentum independent fashion only and the Lagrangian L = 12 φ ( x ) L ( − (cid:3) ) φ ( x ) − V ( φ ( x )) (1)where the real function L ( p ) represents the sum of the kinetic term and a momentum dependent self energy and issupposed to be a polynomial of order ( p ) n d which assumes the form L ( p ) − V ′ ( ¯ φ ) = Z − n d Y n =1 ( p − m n ) , (2)where Z is real, the potential has a minimum at φ = ¯ φ and the poles might appear in complex pairs. The role of thepoles p = m n can be seen more clearly by means of partial fraction decomposition [11], ZL ( p ) − V ′ ( ¯ φ ) = X n z n p − m n (3)where z n = Z/∂L ( m j ) /∂p . We assumed single roots in this equation. In case a root p = m n is of ℓ -th order thenthe right hand side may contain terms z kn / ( p − m n ) k with 1 ≤ k ≤ ℓ . Complex roots produce complex contributionsto the loop-integrals and lead to exponentially damped or increasing amplitudes in time and unitarity can be savedby a graph-by-graph modification of the theory only [12]. Another problem is seen for real roots when the kineticterm L ( p ) is a real function and it displays slope with alternating sign at its roots. Thus approximately half of thecontributions to the kinetic energy has the wrong sign, indicating that the Hamiltonian is unbounded from below.This instability can be cured by introducing negative norm states [10] but the unitarity within the physical, positivenorm subspace could not be established in a nonperturbative manner [13]. Therefore a careless truncation of the selfenergy may spoil unitarity and stability of the effective theory. III. UNITARITY AND REFLECTION POSITIVITY
A proposition to preserve the desired properties was put forward by starting with an effective theory in Euclideanspace-time [14], where it is usually derived perturbatively. The effective theory (1) should have a well defined Euclideanpath integral representation, a condition assured by imposing the constraint L ( − p E − p ) >
0. The safest is to uselattice regularization in Euclidean space-time where higher order derivatives can be represented as higher order finitedifferences. It is easy to see in lattice regularization that we need new variables to regain the usual description fortheories with higher order derivatives [15]. The Kolmogorov-Chapman equation expresses the group structure of thetime evolution in the Fock-space and can be written as e − S t − t [ φ (3) ,φ (1) ] = Z D [ φ (2) ] e − S t − t [ φ (3) ,φ (2) ] − S t − t [ φ (2) ,φ (1) ] (4)where the configurations φ ( j ) specify states in the field diagonal representation at time t j and the exp( − S t − t ′ [ φ, φ ′ ])denotes the matrix element of the Euclidean time evolution operator during the time interval t − t ′ . This equation canobviously be derived for any theory with nearest neighbor coupling in time. New variables φ ( x ) → φ a ( x ), a = 1 , . . . , n d which allow us to rewrite the action with higher derivative in a form with nearest neighbor coupling in time can beintroduced in the following manner. Start with a hyper-cubic lattice with lattice spacing a = 1 in each direction andconstruct an anisotropic lattice where the lattice spacing in the time direction is increased to n d by regrouping n d time slices of the original lattice. A natural choice is φ a ( x ) = ∂ a φ ( x ), the a -th order finite difference operator in timeacting on the original field where the finite difference is calculated from the center of the blocked time slice in a timereversal covariant manner assuming odd n d . The map φ ( x ) → φ a ( x ) of the Euclidean field variables is an invertiblelinear transformation which preserves the lattice regulated action, S E [ φ ] = S E [ φ a ] and the generator functional, Z E [ j ] = Z D [ φ ] e − S E [ φ ]+ R dxjφ = Y a Z D [ φ a ] e − S E [ φ a ]+ P x jφ (5)as long as the source is placed at the center of the block time slices. The transformation preserves its form inMinkowski space-time and provides the mapping whose inverse can be used after the Wick rotation of the blockedtime slice theory to real time.The signature of the norm of states created by the operator φ a ( x ) turns out to be σ [ φ a ] = ( − a . In order to preservethe orthogonality of field eigenvectors, h φ | φ ′ i = 0 for φ ( x ) = φ ′ ( x ) we have to use skew-adjoint field operators, whichpossess imaginary eigenvalues, in the negative norm sector and σ [ φ ] = ± T φ ( t ) = τ [ φ ] φ ( − t ), giving τ [ ∂ a φ ] = ( − a τ [ φ ]. Thisrelation suggests the equivalence of the internal Euclidean time reversal parity and the signature of the state createdby acting on time reversal invariant vacuum by any time reversal invariant combination ψ of elementary fields φ a , σ [ ψ ] = τ [ ψ ] . (6)One has to make sure that unitarity holds within the physical, positive norm subspace, too. This can be achievedby the reconstruction theorem of axiomatic quantum field theory, in particular by showing that the main nontrivialcondition of the theorem, reflection positivity holds in the linear space generated by the action of local operators withpositive time parity on the vacuum as long as both dynamics and vacuum respect time reversal invariance and theboundary conditions φ a ( t f , x ) = ( − a φ a ( t i , x ) are imposed where t i and t f denote the initial and final time. Animportant result of the argument [14] is the direct verification of Eq. (6). This relation indicates, as well, that thetrajectory of φ a in the path integral is real or imaginary for a even or odd, respectively. The vacuum may containcondensate as long as it is invariant under time reversal.This construction gives in the first sight more than expected, it eliminates non-unitarity altogether for theories(1)-(2). But the tacit assumptions the argument relies upon are the convergence of the Euclidean path integraland the possibility of its analytic extension, Wick rotation, back to real time. The former condition (i) imposes ℜ m n >
0. The latter assumption requires that the rotation of the frequency contour in the loop integrals is carriedout without passing singularities in the integrals. This conditions excludes poles from the quadrant ℑ m n · ℜ m n > L are real.Note that the exclusion of complex poles from the kinetic term restricts the space-time dependence of the pertur-bative Green functions to the sum of oscillatory terms e iωt excluding monotonic terms like e ωt . The functional spacein which the expectation values are constructed is tailored in this manner and the runaway solutions, characteristic ofunstable theories are excluded. This is in contrast to classical physics where the integration of the equations of motionis performed in an unlimited functional space of trajectories. Therefore the classical and the quantum, loop-expansionbased stability analysis disagree as far as the time-dependent instabilities are concerned. This eliminates the notoriousinstability problem of theories with higher order kinetic term [17]. IV. SCALAR ELECTRODYNAMICS
An important step towards more realistic models is the extension of previous discussion for gauge models. We nowturn to scalar electrodynamics, defined by the Lagrangian L = − Z dxF µν F µν + Z dx [ φ ∗ L ( − D ) φ − V ( φ ∗ φ )] , (7)with F µν = ∂ µ B ν − ∂ ν B µ and D µ = ∂ µ − ieB µ , L ( z ) being a polynomial of finite order and is supposed to possessseparate time and space inversion invariance. In relativistically covariant canonical quantization procedure one addsa gauge fixing term, L → L − ξ ( ∂A ) /
2, and imposes the canonical commutation relations[ A µ ( t, x ) , Π ν ( t, y )] = − ig µν δ ( x − y ) (8)where Π µ = ∂ L /∂∂ A µ and g µν = (1 , − , − , − µ = ν = 0 indicatesthat temporal photon states have negative norm. The Gupta-Bleuler quantization procedure or BRST symmetry canbe used to prove that usual QED, without higher order derivative is unitary in the physical subspace, spanned bystates with positive norm.With an A field represented by a self-adjoint field operator the field eigenstates are not orthogonal. Orthogonalityis assured if the operator A is skew-adjoint only [14]. The complication, induced by the use of the traditional self-adjoint representation is that non-orthogonality renders the path integral expression for the transition amplitudesrather complicated. How to recover then the standard path integral representation for gauge theories in Minkowskispace-time? The usual path integral over real field configurations A µ ( x ) can easily be found by treating A as anauxiliary, non-dynamical field either in static temporal or Coulomb gauge. The former will be imposed to establishunitarity in the physical subspace because the impact of non-vanishing vacuum expectation value for A on thedynamics and the similarity with spontaneous symmetry breaking can better be seen in static temporal gauge. Thelatter gauge will be used in clarifying the physical content of the theory since the dynamical degrees of freedom canbe traced easier in Coulomb gauge.We start with fields defined without initial or final conditions in time for −∞ < t < ∞ and carry out the gaugetransformation A µ → A µ + ∂ µ α , φ → e ieα φ and φ ∗ → e − ieα φ ∗ with α ( t, x ) = − Z dt ′ A ( t ′ , x ) (9)to arrive at temporal gauge A = 0 where the functional Schr¨odinger representation is constructed, using A ( x ) ascoordinates. The canonical momentum Π = ∂ A = − E satisfies the canonical commutation relations [ A j ( x ) , Π k ( y )] = iδ jk δ ( x − y ). Gauss’ law, ∇ E = ρ , where ρ is the electric charge density, the equation of motion for A is lost in thisgauge but can be regained as a constraint. In fact, it can easily be shown by the help of the canonical commutationrelations that G [ α ] = Z d x [ ∇ α ( x ) E ( x ) + α ( x ) ρ ( x )] (10)generates static gauge transformations hence it commutes with the gauge invariant Hamiltonian H , [ G ( x ) , H ] = 0.The average over static gauge transformations, P = Z D [ α ] e i R d x [ ∇ α ( x ) E + α ( x ) ρ ( x )] (11)projects into the subspace satisfying Gauss’ law for a given static charge distribution ρ ( x ).One is usually interested in transition amplitude between gauge invariant states, the latter constructed from agauge-noninvariant representative like a field eigenstate, | A , φ, φ ∗ i sym = P| A , φ, φ ∗ i . (12)It is enough to insert the projection operator P once only in the matrix element, h A f , φ f , φ ∗ f | e − itH | A i , φ i , φ ∗ i i sym = h A f , φ f , φ ∗ f |P e − itH | A i , φ i , φ ∗ i i = Z D [ α ] h A f , φ f , φ ∗ f | e i R d x [ ∇ α ( x ) E + α ( x ) ρ ( x )] e − itH | A i , φ i , φ ∗ i i (13)and one finds the path integral representation h A f , φ f , φ ∗ f | e − itH | A i , φ i , φ ∗ i i sym = Z D [ A ] D [ φ ] D [ φ ∗ ] e iS st [ A,φ,φ ∗ ] (14)where S st [ A, φ, φ ∗ ] is the usual action in static temporal gauge, ∂ A ( x ) = 0 , (15)and tA ( x ) = α ( x ) denotes the time-independent integral parameter of the projector. The integration is over config-urations A ( t i , x ) = A i ( x ), φ ( t i , x ) = φ i ( x ), φ ∗ ( t i , x ) = φ ∗ i ( x ), A ( t f , x ) = A f ( x ), φ ( t f , x ) = φ f ( x ), φ ∗ ( t f , x ) = φ ∗ f ( x ).If the projector is inserted at each time slice of the path integral expression for transition amplitude then the gaugeinvariant action is recovered, h A f , φ f , φ ∗ f | e − i ∆ tH P · · · P e − i ∆ tH | A i , φ i , φ ∗ i i sym = Z D [ A ] D [ φ ] D [ φ ∗ ] e iS [ A,φ,φ ∗ ] (16)∆ tA ( t, x ) playing the role of parameter α ( x ) in the projector inserted at time t .Temporal gauge, used in the Hamiltonian formalism after Eq. (9) is usually not accessible when boundary conditionsare imposed in time, as done in path integral expressions. Actually the field component A ( x ) represents a truephysical variable. We can see this by noting that A ( x ) cannot be transformed away from the path integral by gaugetransformation. In fact, setting A = 0 instead of integrating over A ( x ) on the right hand side of Eq. (16) removesthe projector P on the left hand side and the matrix element is changed, h· · ·i sym → h· · ·i .A generally applicable gauge choice is static temporal gauge, given by Eq. (15). Whatever gauge we use, thePolyakov line Ω( x ) = e − ie R tfti dtA ( t, x ) (17)denotes a physical, gauge invariant quantity which prevents us from reaching temporal gauge as soon as some boundaryconditions are imposed at the initial and final time. But the integrand of the path integral (14) remains unchangedunder global gauge transformation of the initial or final state by the phase factor 1 = exp 2 πi , represented by the shift A ( x ) → A ( x ) + 2 πe ( t f − t i ) . (18)Due to this discrete symmetry the integrand in Eq. (16) does not depend on the space-time independent component, A ( x ) = A and the variable A decouples in the limit t f − t i → ∞ . Nevertheless, the homogeneous component A remains a physical parameter when matrix elements among the vacuum are considered because the vacuum statedepends on A . In fact, eA acts as a chemical potential and one arrives at grand canonical ensemble where expectationvalues of observables are saturated by the total charge sector of the Fock space which minimizes H − eA R d xρ . V. SEMI-CLASSICAL VACUUM
Let us suppose that the model given by Eq. (7) is weakly coupled and saddle point expansion can be used toexplore its phase structure. The case of global symmetry, e = 0 in the absence of higher order derivative terms L ( p ) = p , is well known, the model supports homogeneous condensate for appropriately chosen local potential.Higher order derivative terms in the action may induce a condensation of particles with non-vanishing momentum,an inhomogeneous coherent state and a relativistic “band structure”, reminiscent of solid state physics is observed.When the interaction with the gauge field is turned on with L ( p ) = p then the usual Higgs phase can be found. Aninteresting variant of Higgs mechanism can be generated by the higher order derivatives terms. The point is that thepartial derivatives are turned into covariant derivatives in the minimal coupling scheme and contain the connectionterm which can induce a nontrivial local potential for the gauge field. The effective interaction, represented by thispotential may induce a non-vanishing expectation value for the gauge field. We call such a vacuum condensate thoughone should keep in mind that it is actually a coherent state only because our gauge particle, the photon is neutraland the Bose-Einstein condensation is not possible. A. Condensate
We follow the strategy of the saddle point approximation and for this end we separate the fields into the sum ofsaddle point and quantum fluctuations by writing φ = ¯ φ + χ and B µ = ¯ A µ + A µ , the first term in each expressionrepresenting the saddle point. When a non-vanishing value of the covariant derivative − D ¯ φ ( x ) = k ¯ φ ( x ) (19)is selected for the semi-classical vacuum by the kinetic energy of the charges then a gauge transformation can alwaysexchange contributions of the partial derivative and the connection term. One possibility is when the eigenvalue k inthis equation is provided by the partial derivative alone, ¯ φ ( x ) = ¯ φe − ipx , ¯ A µ = 0. By a suitable gauge transformationwe may rearrange the semi-classical vacuum into ¯ φ ( x ) = ¯ φ , e ¯ A µ = k µ . This is a remarkable simplification offeredby gauge invariance, the vacuum consisting of the condensate of particles of non-vanishing momentum can be madehomogeneous. We exploit this possibility and assume the homogeneity of the saddle point and the orthogonality ofthe fluctuations to the saddle point, Z dxχ ( x ) = Z dxA µ ( x ) = 0 . (20)Note that apart from broken global gauge invariance the gauge field condensate leads to the spontaneous breakdownof the Lorentz symmetry. When the function L ( p ) generates spacelike gauge field condensate, k < O (1 ,
2) and the excitation spectrum loses rotational invariance. We seek vacuum with non-relativistic Galilean O (3) invariance hence we restrict our attention to models with timelike gauge field condensate, e ¯ A µ = g µ k >
0. Hence there will be four combinations of fields playing the role of Goldstone bosons when ¯ φ, ¯ A µ = 0,corresponding to gauge rotations and Lorentz boosts. The number of massless particle modes is not necessarily thesame. On one hand, it may be smaller because either non-relativistic fields have half as many particle modes astheir relativistic counterparts [18] or some field combinations may control not particle-like excitations, with vanishingresiduum in the propagator at the “mass-shell”. On the other hand it may be more because higher order derivativeterms may generate several “bands”. Global gauge rotation is applied if necessary to make the scalar condensate, ¯ φ ,real. B. Fluctuations
According to section III classical stability analysis is sufficient for the homogeneous components of the fields and thestability of the fluctuations around the vacuum will be verified by checking the spectrum of the elementary excitationsin quantum theory. The energy-momentum tensor of a theory with polynomial, higher order derivative terms caneasily be obtained, it is the sum of the usual expression for the energy-momentum tensor plus terms containing higherorder derivatives of the fields. Therefore the energy density of the semi-classical homogeneous vacuum characterizedby ¯ A µ and ¯ φ is given by the Lagrangian up to a sign, U ( e ¯ A , ¯ φ ) = − ¯ φ L ( e ¯ A ) + V ( ¯ φ ) . (21)We assume at this point that L ( p ) is bounded from above and it assumes a maximal value at p = k thus theminimization with respect to ¯ A , 0 = ∂U ( e ¯ A , ¯ φ ) ∂e ¯ A = − ¯ φ L ′ ( e ¯ A ) (22)sets e ¯ A = k and e ¯ A µ = g µ, k as mentioned above. The separation of the kinetic and the potential energyterm in the Lagrangian (7) for the scalar field is not unique, the invariance of the action under the transformation L ( p ) → L ( p ) + ∆ L , V ( φ ) → V ( φ ) − ∆ Lφ can be used to set L ( k ) = 0. We assume the form L ( p ) = − k ( p − k ) , (23)the simplest polynomial satisfying our requirements. The scalar condensate ¯ φ is found by minimizing U ( k , φ ), i.e.solving the equation 0 = V ′ ( ¯ φ ) − L ( k ) (24)with the auxiliary condition that the first non-vanishing derivative of the potential at the vacuum is positive.Once the homogeneous field components are found we turn to the free theory by considering the quadratic part ofthe action. We use the decomposition χ = χ + iχ , and A = n A L + A T , n = p / | p | , followed by the separation ofthe static components ˜ χ a , ˜ A L and ˜ A T by writing χ a → χ a + ˜ χ a , A L → A L + ˜ A L and A T → A T + ˜ A T . The quadraticaction is written as a sum S (2) = S (2) + ˜ S (2) with S (2) = 12 Z d x ( χ , χ , A L , A T ) K K K L K K K L K L K L K LL
00 0 0 K T T χ χ A L A T ˜ S (2) = t f − t i Z d x ( ˜ χ , ˜ χ , ˜ A , ˜ A L , ˜ A T ) ˜ K K ˜ K L
00 ˜ K K L K K K L ˜ K L K LL
00 0 0 0 ˜ K T T ˜ χ ˜ χ ˜ A ˜ A L ˜ A T . (25)The momentum space representation of the quadratic form, K ( p ) = Z dxe ip ( x − y ) K ( x, y ) (26)is calculated in Appendix A with the result K = L + d ( p ) − V ′′ ¯ φ = K K = iL − d ( p ) = − K K L = −| p | [ z ( p ) L d ( p )] − = K L K L = i | p | [ z ( p ) L d ( p )] + = − K L K LL = p [ z ( p ) L d ( p )] + + ω ,K T T = ω − p , (27)where the notation f ± ( p ) = f ( p ) ± f ( − p ) has been introduced with L d ( p ) = L (( p + e ¯ A ) ) − L ( k ) , (28) z ( p ) = e ¯ φ/ ( p + 2 ωk ) and p = ( ω, p ) for the four dimensional fields. The three dimensional, static sector has quadraticforms ˜ K ( p ) = K ( p ) | ω =0 , obtained from Eqs. (27) and˜ K = − e ¯ φk p L d ( p ) | ω =0 = K ˜ K = 8 e ¯ φ k ( p ) L d ( p ) | ω =0 + p (29) C. Unitarity
We turn now to the question of unitarity of the time evolution within the positive norm subspace of the Fock-space.There are two circumstances requiring to go beyond the argument based on the reconstruction theorem for Euclideantheories [19]. One is that the manifest O (4)/Lorentz invariance of the Euclidean/Minkowski Green functions, oneof the numerous conditions of the theorem is lost in our case. Another point is that states belonging to excitationsgenerated by the time component of the gauge field have negative norm in Minkowski space-time and are thus non-physical. Rather than attempting to generalize the reconstruction theorem we choose a simpler argument, valid inany finite order of the perturbation expansion.The partial fraction decomposition of the propagator is now made in terms of ω rather than p and the realnessof the one-particle energies guarantees the unitarity of the perturbative model within the Fock-space with indefinitenorm. Perturbation expansion, based on the vacuum with homogeneous fields ¯ φ and ¯ A µ leads to a stable and unitarytheory if all solutions of the equation det K ( p ) = 0 of the quadratic form K of Eq. (25),det K ( p ) = 4 k ( ω − p ) n ω − ω ( p + 2 k ) + ω h k + 16 k p + 6 (cid:0) p (cid:1) + 4 V ′′ ¯ φ k i − ω h V ′′ ¯ φ (cid:0) e p ¯ φ + 2 k p − k (cid:1) + 4 (cid:0) p (cid:1) + 8 k (cid:0) p (cid:1) i + ω h V ′′ ¯ φ + 4 V ′′ ¯ φ k (cid:0) p (cid:1) + 2 e ¯ φ (cid:0) p (cid:1) − e ¯ φ k p i − e k p V ′′ ¯ φ − e (cid:0) p (cid:1) V ′′ ¯ φ o , (30)obtained for the kinetic term (23) have real frequency components, ω >
0. It is easy to see that this expression hasnegative or complex ω as roots, there are instable modes in the scalar particle, longitudinal gauge field sector. Theseinstabilities can be excluded by imposing the condition V ′′ ( ¯ φ ) = 0 (31)on the local potential which is not a natural relation, it requires a fine tuning to cancel the scalar particle scatteringamplitude at vanishing momentum.According to Goldstone theorem the minimization of the vacuum energy with respect to the strength of condensatecancels the gap for certain modes. Goldstone mode arising from the breakdown of global gauge invariance is made byEq. (24). As far as the three soft field combinations are concerned, which correspond to the breakdown of Lorentzsymmetry, let us introduce a mass term for the gauge field by the extension L → L + m B / U ( e ¯ A , ¯ φ ) → U ( e ¯ A , ¯ φ ) − m ¯ A / m = 0 and a third is a combinationof ∂ µ A µ , ¯ A µ A µ , χ and χ . To simplify matters we return in our discussion to scalar electrodynamics, m = 0 wherethe determinant of Eq. (30), whose vanishing identifies the normal mode dispersion relation, readsdet K ( p ) = 4 k ( ω − p ) ω ( ω − kω − p ) ( ω + 2 kω − p ) . (32)The energy spectrum is real, transverse gauge fields make up two Goldstone modes with ω = ±| p | . The scalar field,together with the longitudinal components of the gauge field produce the dispersion relations ω = σ k + σ p k + p , (33)where σ , σ = ±
1. The choice σ σ = − ω = 0 in Eq. (32) is never vanishing.Once the unitarity has been established in the whole Fock-space let us turn to the physical subspace. The argumentin Ref. [14] was presented for Yang-Mills-Higgs model, given by the Lagrangian (7) though some additional care isrequired in this case to draw conclusions for Minkowski space-time theories. The Wick rotation is more involvedfor gauge than for scalar fields because the norm of state created by A changes sign during Wick rotation betweenEuclidean and Minkowski space-time. This leads to the following two problems. One has already been mentioned inSection IV, the usual path integration formulas require the orthogonality of the field eigenstates and we should useskew-adjoint representation for A in Minkowski space-time. This amounts to integration over imaginary A fieldwhich is in an obvious conflict with the usual interpretation of A as the temporal component of a Hermitian quantumfield. The solution of this apparent contradiction is well known, the treatment of A as a non-dynamical, auxiliaryvariable. This is what happens in static temporal gauge where A is the (real) integral variable of the projectionoperator to restrict the dynamics into the subspace with Gauss’ law. Once the real, static A configurations areaccepted in the path integral of Eq. (14) then we may return to the gauge-free case, Eq. (16) in the calculactionof gauge invariant quantities. In other words, in the usual path integral formalism for real time, available for gaugetheories with higher order derivatives, as well, the temporal component of the gauge field is better to interpret asan auxilary variable to handle Gauss’ law rather than a quantum field handling physical excitations. The situationis reminiscent of conventional QED where elementary excitations, stability, renormalizability, etc. are trivial inrelativistic gauges but one has to go into another, physical gauge, usually chosen to be the Coulomb gauge to recoverunitarity in the physical subspace in an obvious manner.Another problem, caused by an exceptional feature of A ( x ) during Wick rotation is that Eq. (6) used to identifythe signature of the norm is not valid anymore for this component of the gauge field in Minkowski space-time.A generalization valid for gauge field is σ [ ψ ] = τ [ ψ ] π [ ψ ] , (34)where ψ is any local combination of the elementary bosonic fields ∂ a φ , ∂ a φ ∗ and ∂ a A and space inversion acts as P ψ ( t, x ) = ψ ( t, − x ) with π [ φ ] = − π [ A ] = 1. PT invariance yields the conservation of σ and assures unitarity withinthe positive norm, physical subspace [21]. VI. QUASI-PHOTONS
It has been established so far that our model has unitary time evolution within the positive norm subspace and istherefore a physically interpretable. The next question is its physical content which will be assessed by comparing itwith standard electrodynamics. The usual Higgs-mechanism renders photons massive. The Goldstone modes arisingfrom the spontaneous breakdown of the Lorentz invariance make three combinations of the fields soft. Two of themare the transverse, non-vanishing helicity components of the gauge field and they keep the radiation field massless,just as in standard electrodynamics. Two further soft field combinations are made up from the longitudinal gaugeand the scalar field components.The double pole of (23) may render the normal modes of the scalar field non particle-like because scatteringamplitude wave packets, constructed by this kind of excitations may be vanishing according to the reduction formulas.Thus we take the point of view that the scalar field corresponds to so far non-observed excitations and seek thedynamics of the gauge field only. To simplify matters further, we ignore radiative corrections due to the chargedscalar field and restrict ourselves to the O (cid:0) A (cid:1) part of the action where the normal modes are quasi-photons. Weconsider below two aspects of the model, the number of propagating, dynamical degrees of freedom and their dispersionrelation. It is worthwhile separating two different kinds of dynamics for the gauge field, first arising through the fieldstrength tensor in the Maxwell-action, the first term in the Lagrangian (7) and second, coming directly from theconnection term of the covariant derivative in the minimal coupling. The former, field strength tensor dynamicsrepresents conventional electrodynamics and the latter, connection term dynamics is the source of genuine quantumand topological effects.Let us first have a look into the Proca-theory, the simplest model with massive vector field and use the standardthree-dimensional notation A µ = ( ϕ, A ), j µ = ( ρ, j ), E = −∇ ϕ − ∂ A , H = ∇ × A . We separate the transverseand longitudinal components, A = A T + ∇ Φ, j = j T + ∇ κ where current conservation implies ∂ ρ + ∆ κ = 0. TheLagrangian L = 12 E − B − ρϕ + m ϕ − A ) + jA (35)0can be written as L = L T + L L where the first and the second term contain the transverse and longitudinal andtemporal components, L T = 12 ( ∂ A T ) − B − m A T + j T A T L L = 12 ( ∇ ϕ + ∂ ∇ Φ) + m ϕ − ( ∇ Φ) ] − ρϕ + ∇ Φ ∇ κ. (36)As is well known, the temporal component ϕ is not a dynamical degree of freedom and can be eliminated by solvingits algebraic equation of motion in time, ϕ = 1 m − ∆ ( ρ + ∂ ∆Φ) , (37)without generating non-local effects in time and the resulting effective Lagrangian for Φ is L L = − ρ m − ∆ ρ + m (cid:3) + m ) m − ∆ Φ + Φ (cid:20) ∆ m − ∆ ∂ ρ − ∆ κ (cid:21) . (38)For massless photon, m = 0, the equation of motion for Φ is the current conservation and longitudinal photons dropout from the field strength dynamics. But the mass term, arising from the connection term dynamics may bring thelongitudinal component back as a genuine dynamical variable. Gauge transformations may make the separation ofauxiliary and truly dynamical variables difficult. For instance, there are gauges, such as the static temporal gauge,where the longitudinal component appears to be dynamical but it drops out from gauge invariant observables. Whenhigher order derivative terms appear in the connection term dynamics then either the temporal or the transversecomponent of the gauge field may acquire non-trivial dynamics. The formal gauge invariance always makes the theoryredundant therefore one expects three dynamical, propagating degrees of freedom for the theory (7) from the photonfield, just as in the usual Higgs-mechanism. But their dispersion relations differ from those of the Higgs-mechanism,betraying the different underlying symmetry breaking patterns.Let us look into the dispersion relation of the model (7) in Coulomb gauge which offers a particularly clear view inour model with spontaneously broken Lorentz symmetry. The Lagrangian L = L T + L m is written as the sum of thetransverse part, given by the first equation in Eqs. (36), and the rest whose quadratic part is L (2)0 m = 12 ( χ , χ , ϕ ) K C χ χ ϕ (39)where χ = χ + iχ and K C = L + d ( p ) iL − d ( p ) [( p + 2 k ) z ( p ) L d ( p )] + − iL − d ( p ) L + d ( p ) − i [( p + 2 k ) z ( p ) L d ( p )] − [( p + 2 k ) z ( p ) L d ( p )] + i [( p + 2 k ) z ( p ) L d ( p )] − [(2 k + p ) z ( p ) L d ( p )] + + p . (40)The dispersion relation is defined by the roots of the determinant of the quadratic form,det K C ( p ) = 4 k p ( ω − kω − p ) ( ω + 2 kω − p ) . (41)Comparing this expression with Eq. (32), the determinant of the small fluctuations in static temporal gauge apartfrom the obvious absence of two massless modes, corresponding to non-vanishing helicity transverse modes of the gaugefield one notices the appearance of a new root, p , suggesting the emergence of a conventional Coulomb propagator.One can obtain a more detailed view of the normal modes by the inspection of the propagators. The inverse of K C is a full matrix with rather involved matrix elements. Matrix elements of K − C between the matter field contain thefactor ( ω − kω − p ) ( ω + 2 kω − p ) , indicating the non-particle like behavior. The matrix elements between thematter field and ϕ have the factors ( ω − kω − p )( ω + 2 kω − p ) and p in the denominators. Finally, the simplestinverse matrix element is the diagonal one for ϕ , ( K − C ) = 1 p , (42)confirming that the factor p in Eq. (41) corresponds to the unchanged Coulomb law.1The expectation of three dynamical, propagating components for the gauge field, mentioned after Eq. (38) aboveturned out to be wrong and the nontrivial dynamics for the longitudinal component, expected by analogy with theProca case, Eq. (38) was too naive. The higher order derivative terms render the nontrivial dispersion relation for thelongitudinal component a gauge artifact and the usual dispersion relation is recovered for the electromagnetic field.The surprising simplicity of (42) is the result of nontrivial cancellations. This can easiest be seen by calculatingthe A propagator directly. For this end we eliminate the charged field by its equation of motion what is simplest tocarry out in the complex χ basis, where L (2) = 12 ( χ ∗ , χ, ϕ ) K − K K K χχ ∗ ϕ (43)with K − = − (cid:3) − ik∂ ) /k , K = 2 e ¯ φ (2 k + i∂ )( (cid:3) − ik∂ ) /k and L = 2 e ¯ φ ( ∂ − k ) /k − ∆. The equationsof motion for χ ∗ and χ , 0 = K − χ + K A , and 0 = χ † K − + A K , used to eliminate the scalar field yield L (2) = 12 ϕD − ϕ (44)where D − = K −
12 ( K K − − K + K tr K − tr − K tr ) (45)gives D − = p after some cancellations. Therefore the deviation from usual electrodynamics and the impact of thehigher order derivative terms are seen by the charged scalar field in our approximation. VII. CONCLUSION
A novel spontaneous symmetry breaking is discussed in the framework of scalar QED which involves higher ordercovariant derivatives. One finds non-vanishing expectation value for the gauge field and unitary, physically acceptableinteractions in properly fine tuned models.The unitarity is proven in the physical, positive norm subspace in two steps. First it is assured in the whole Fockspace by fine tuning the self interactions for the charged scalar field. Second, it is shown that PT invariance makesthe physical subspace closed under time evolution.The particle content of our model is radically different than the one found in the conventional Higgs-mechanism.Goldstone theorem renders the radiation field massless. Furthermore, a particular model is proposed where allcomponents of the gauge field are massless and Maxwell equations are recovered in the linearized equations of motion.We sought in this work a vacuum which supports Galilean invariance, therefore the temporal component of thegauge field was allowed to develop vacuum expectation value. It acts as some dynamically generated chemical potentialfor the charged scalar particle. The scalar particle condensate remains electrically neutral due to the equal numberof particles and anti-particles it contains as a result of the higher order derivative terms in their dispersion relation.The status of Lorentz symmetry, broken by the vacuum expectation values to the Galilean group, is rather peculiarin the Abelian model. Despite the breakdown of Lorentz invariance Goldstone modes display relativistic dispersionrelations. Furthermore, three components of the gauge field become Goldstone modes corresponding to the spontenousbreakdown of relativistic symmetries and hence remain massless even if one starts with massive Proca action forphotons. The quadratic part of the Lagrangian in the fluctuations of the gauge field is identical to that of QED,leaving the Lorentz non-invariant part of the photon dynamics to be generated by radiative corrections. The deviationof this model from standard electrodynamics is due to radiative corrections only.There are numerous extensions one may consider. Similar models with non-Abelian gauge symmetry should leadto some massive gauge field components because Goldstone theorem cannot protect all components of the gauge fieldanymore against mass generation. Using a basis in internal space where the massless gauge bosons are diagonal theother, non-commuting components of the gauge field are charged and allow us to construct models with an unbroken U (1) subgroup, as in the Standard Model. It remains to be seen if natural models, requiring no fine-tuning can beconstructed by the eventual inclusion of charged fermions. Another issue, the scale-dependence of the breakdown ofLorentz invariance is interesting, too. Being a spontaneous symmetry breaking, it should be strong at low energy.But some interesting results about non-Lorentz invariant quadratic terms in gauge theories [20] suggests that certainLorentz symmetry breaking parameters of the dynamics tend to be suppressed in the low energy limit. A systematicrenormalization group study of the model would be needed to reveal the true scale dependence of this symmetrybreaking. Finally, extension for gravity opens new questions since the spontaneous breakdown of Lorentz symmetrymay generate massive gravitons by a gravitational Higgs-effect.2 Appendix A: Quadratic action in momentum space
To find the momentum dependence of the quadratic form K ( p ) we evaluate the quadratic action (25) for the testfunctions χ ( x ) = χ ′ e − ipx ,A µ ( x ) = A ′ µ e − ipx (A1)before gauge fixing for the sake of simplicity. The quadratic form of the O ( χ ∗ χ ) part can easily be written as K χ ∗ χ = 2 L d ( p ) − V ′′ ¯ φ (A2)by means of Eq. (24) with L d ( p ) introduced in Eq. (28).To find the other terms it is advantageous to represent the higher derivative kinetic term of the scalar field as apolynomial, L ( p ) = n d X n =0 c n p n . (A3)The block that mixes the scalar and the gauge field originates from the O ( B ) piece in L ( − D ) ¯ φ = n d X n =0 c n [ − ( ∂ − ie ¯ A − ieA ) ] n ¯ φ (A4)and we find for the O ( Aχ ) contributions12 Z dxdyχ ( x ) K χA ( x, y ) A ( y ) = ie Z dxχ ( x ) n d X n =0 c n n X ℓ =1 ( − ¯ (cid:3) ) · · · (2 A ( x ) ¯ ∂ + ∂A ( x )) · · · ( − ¯ (cid:3) ) ¯ φ (A5)where ¯ ∂ µ = ∂ − ie ¯ A µ , ¯ (cid:3) = ¯ ∂ µ ¯ ∂ µ , and the ℓ -th factor of the term O (cid:0) ( − D ) n (cid:1) is replaced by the O ( A ) part of − D in the right hand side. The choice (A1) leads to K χA ( p ) A ′ = 2 ie n d X n =0 c n n X ℓ =1 ( p + e ¯ A ) · · · ( − iA ′ k − ipA ′ ) k · · · ¯ φ, (A6)written as K χA ( p ) A ′ = 2 e (2 A ′ k + pA ′ ) k − n d X n =0 c n k n n − X ℓ =0 (cid:18) p + 2 p kk (cid:19) ℓ ¯ φ. (A7)The geometric series can be summed, K χA ( p ) A ′ = 2 e A ′ k + pA ′ p + 2 p k n d X n =0 c n k n (cid:20) (1 + p + 2 p kk ) n − (cid:21) ¯ φ, (A8)and we have K χA µ ( p ) = 2 e ¯ φ (cid:20) L d ( p ) 2 g µ k + p µ p + 2 p k (cid:21) (A9)The O (cid:0) A (cid:1) quadratic form for real field requires more care. Since it acts on real field it must be symmetrical. Weshall consider a complex plane wave component of the gauge field in the actual calculation of this term and carry outthe symmetrization at the end only. This term is the sum of two contributions. One of them is the standard Maxwellpiece K (1) A µ A ν ( p ) = − T µν p (A10)3where T µν = g µν − p µ p ν /p is the projection into the transverse polarization subspace. The other part is the O (cid:0) A (cid:1) contribution in ¯ φL ( − D ) ¯ φ = X n c n ¯ φ [ − ( ∂ − ie ¯ A − ieA ) ] n ¯ φ (A11)which will be written as the sum K (2) AA ( p ) + K (3) AA ( p ). The first term stands for the O (cid:0) A (cid:1) contributions of the − D factor, A ′ K (2) AA A ′ = 2 e n d X n =0 c n n X ℓ =1 ¯ φ ( − (cid:3) ′ ) · · · A ( x ) · · · ( − (cid:3) ′ ) ¯ φ (A12)which is vanishing, K (2) A µ A ν ( p ) = 2 g µν ¯ φ e L ′ ( k ) = 0 . (A13)The other contributions is for the product of two O ( A ) terms, A ′ K (3) AA ( p ) A ′ = − e n d X n =0 c n n − X ℓ =1 n X ℓ ′ = ℓ +1 ¯ φ ( − (cid:3) ′ ) · · · (2 A∂ ′ + ∂A ) · · · (2 A∂ ′ + ∂A ) · · · ( − (cid:3) ′ ) ¯ φ = 2 e n d X n =0 c n n − X ℓ =1 n X ℓ ′ = ℓ +1 ¯ φk · · · (2 A k + pA ) · · · ( k + p + 2 p k ) · · · (2 A k + pA ) · · · k ¯ φ (A14)what is written as A ′ K (3) AA ( p ) A ′ = 2 e ¯ φ (2 A k + pA ) k − n d X n =0 c n k n n − X ℓ =1 n X ℓ ′ = ℓ +1 (cid:18) p + 2 p kk (cid:19) ℓ ′ − ℓ − . (A15)The summation of this geometric series gives A ′ K (3) AA ( p ) A ′ = 2 e ¯ φ (2 A k + pA ) p + 2 p k k − n d X n =0 c n k n " n − X ℓ =1 (cid:18) p + 2 p kk (cid:19) n − ℓ − n + 1 . (A16)The resulting geometrical series in the square bracket can again be summed with the result A ′ K (3) AA ( p ) A ′ = 2 e ¯ φ (2 A k + pA ) ( p + 2 p k ) n d X n =0 c n k n (cid:20)(cid:18) p + 2 p kk (cid:19) n − − n p + 2 p kk (cid:21) , (A17)yielding finally A ′ K (3) AA ( p ) A ′ = 2 e ¯ φ L d ( p ) (2 A k + pA ) ( p + 2 p k ) (A18)and K A µ A ν ( p ) = − T µν p + e ¯ φ L d ( p ) (2 g µ k + p µ )(2 g ν k + p ν )( p + 2 p k ) + e ¯ φ L d ( − p ) (2 g µ k − p µ )(2 g ν k − p ν )( p − p k ) . (A19)after symmetrization. [1] K. G. Wilson, J. Kogut, Phys. rep. , 75 (1974).[2] P. Azaria, B. Delamotte, T. Jolicoeur, Phys. Rev. Lett. , 3175 (1990); M. Dufour Fournier, J. Polonyi, Phys. Rev. D61 ,065008 (2000). [3] G. Kohring, R. E. Schrock, Nucl. Phys. B295 , 36 (1988); S. Caracciolo et al, Nucl. Phys. B Proc. Suppl. , 815 (1993);J. L. Alonso et al, Phys. Rev. B53 , 2537 (1996); M. L. Plumer, A. Caille, J. Appl. Phys. , 5961 (1991); H. Kawamura,J. Phys. Soc. Jpn. , 1299 (1992); H. G. Ballesteros et al, Phys. Lett. B378 , 207 (1996); Nucl. Phys.
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