Geometrical Properties and Propagation for the Proca Field Theory

Properties and Propagation of Proca the Field 21and differential operations D µ are defined through the connections Γ ρ αβ ;the metric tensor are to be such that they can be locally reduced to theMinkowskian form, and the covariant derivatives applied upon the metrictensors are required to vanish according to what is called metricity conditionD µ g = 0, as it has been discussed in reference [3]. Then, requiringthis condition of metricity for any connection leads to the complete antisymmetryof Cartan torsion tensor Q αµρ , as explained in reference [4].Further, we will define Riemann tensor with G αβµν antisymmetric inboth the first and the second couple of indices, with one independentcontraction given by the Ricci tensor G λ αλβ = G αβ, whose contraction isgiven by the Ricci scalar G αβ g αβ = G and this will set our conventionup.Riemann curvature tensor, Ricci curvature tensor and scalar, togetherwith Cartan torsion tensor verify the identities(D ρ Q ρµν + G νµ − 1 ) (2 gνµ G − G µν − 1 )2 gµν G ≡ 0 (1)andD µ(G µρ − 1 )2 gµρ G(− G µβ − 1 )2 g µβG Q βµρ + 1 2 Gµκβρ Q βµκ ≡ 0(2)which are geometric identities, called Jacobi-Bianchi identities.We remark that from the metric tensor it is possible to define theLevi-Civita tensor ε βµρσ for which D µ ε βµρσ = 0 identically; in turn,since torsion is completely antisymmetric then we can writeQ βµρ = ε βµρσ W σ (3)in terms of what is known to be the axial torsion vector, useful in thefollowing.Within this background, to define matter fields that can be classifiedaccording to the value of their spin we have to consider that a givenmatter field of spin s possesses 2s + 1 degrees of freedom, which have tocorrespond to the 2s + 1 independent solutions of a system of equationsthat specify the highest-order time derivative for all components of thefield, called system of matter field equations; however, since it may happenthat field equations are not enough to determine the correct rankof the solution, restrictions need to be imposed in terms of equations

Properties and Propagation of Proca the Field 23to be such that the relationshipsandD ρ S ρµν + 1 2 (T µν − T νµ ) = 0 (4)D µ T µρ − T µβ Q βµρ − S βµκ G µκβρ = 0 (5)are verified, implying the whole set of field equationsandG σρ − 1 2 gσρ G = − 1 2 T σρ (6)Q νσρ = S νσρ (7)to be such that the conservation laws (1) and (2) are satisfied automatically.This determines the setting of the fundamental field equations inminimal coupling, that is taking the least-order derivative in the fieldequations.3 Propagation and Geometrical PropertiesHaving settled this background, and because the background has restrictionsthen matter fields will be restricted correspondingly, as alsoexplained in [6].To consider which matter vector fields could possibly be definedwithin this background, we see that in the case of a vector V µ it is possibleto define beside the standard covariant derivative given in terms ofthe connection another most special differential operation written in theform Z µν = ∂ µ V ν −∂ ν V µ in terms of no additional field and called exteriorderivative or curl; this curl can be generalized up to the differential operatorgiven by Z ρµ = D ρ V µ −D µ V ρ that is formally the curl but writtenwith respect to the most general connection with torsion: according tothis most general dynamical term, the most general Proca Lagrangianwill have kinetic terms given not only by the standard Z µα Z µα termbut also by the additional Z ηρ Z µα ε µαηρ term whose contributions arerelevant because in presence of torsion it cannot be written as a totaldivergence in the Lagrangian and therefore neglected. The most general

24 L. Fabbriaction will then be defined in terms of two parameters given by a parameterλ accounting for the additional terms of this generalization andthe mass m of the matter field.By varying this action with respect to the field involved we get theProca matter field equationsD µ Z µα + λ 2 D µZ ηρ ε µηρα + m 2 V α = 0 (8)which specify the second-order time derivative for only the spatial components,but which also develop the constraintm 2 D µ V µ − λ 4 Q ρµνD ρ Z αβ ε αβµν − 1 2 Qραβ D ρ Z αβ −− λ 2 D µQ ρ βν Z ραε αβµν − 1 2 D ρQ ραβ Z αβ = 0 (9)in which the terms with the second-order time derivative of spatial componentscan be removed by means of field equations showing that thisis a real constraint, which can then be plugged back into the field equationsallowing them to specify the second-order time derivative of allcomponents showing that these are true field equations; the conservedquantities are given by the energyand the spinS ραβ = 1 2T αµ = − 1 2 gαµ m 2 V 2 + ( 14 gαµ Z ρη Z ρη − Z µθ Z α θ)++D ρ V µ ( Z ρα + λ 2 Z σθε σθρα) (10)[V α (Z ρβ + λ 2 Z σθε σθρβ )− V β (Z ρα + λ 2 Z σθε σθρα )](11)so that, whereas the conditionV α (Z ρβ + λ 2 Z σθε σθρβ )+ V ρ (Z αβ + λ 2 Z σθε σθαβ )= 0 (12)ensures the complete antisymmetry of the spin: this form of the spin withthe energy is such that one the field equations are considered then theconservation laws (10) and (11) are satisfied and thus the Jacobi-Bianchiidentities verified.We notice that the field equations for the spin express the spin interms of the covariant derivatives of the field written in terms of the

Properties and Propagation of Proca the Field 25spin itself, and therefore the field equations for the spin actually expressthe spin as an implicit relationship that can eventually be made explicitgiving as result the inverted relationship( ) 1 + λS ραβ 2= (∂ σ V ν − ∂ ν V σ ) V σ ε νραβ (13)3λin which the complete antisymmetry of the spin is manifest, and hencetorsion is writeable in terms of the torsionless derivatives of the field;then it can be plugged back into the field equations to account for theback-reaction of torsion onto the field, or equivalently the autointeractionof the field with itself.Further we can consider the condition of completely antisymmetricspin in its only independent contraction, of which we can take the divergence,thus obtaining two expressions from which it is possible to derivethe conditionsV α V α = 0, Z ρβ Z ρβ + λ 2 Z ρβZ σθ ε σθρβ = 0 (14)and alsoor equivalentlyV ρ S ραβ = 0, Z ρβ S ρβα = 0 (15)V ρ W β − V β W ρ = 0, Z ρβ W α + Z βα W ρ + Z αρ W β = 0 (16)which are orthogonality conditions of the field and its exterior derivativeswith the torsion tensor; these orthogonality conditions will considerablysimplify the expression of the field equations we are going to work out.Now, to study the propagation, it is possible to see that we have thenumber of degrees of freedom still correct; then, the calculation of thecharacteristic equation gives3m 2 n 2 − ( 1 + λ 2) Z ρα Z ρβ n α n β = 0 (17)from which we have that causal propagation is always possible, but byintroducing Z 0i = E i and Z ij = −ε ijk B k then whenever(n 0 ) 2 | ⃗ E| 2 − 2n 0 (⃗n · ⃗E × ⃗ B) − (|⃗n · ⃗E| 2 − |⃗n × ⃗ B| 2 ) < 0 (18)

26 L. Fabbriwe have that acausal propagation may occur and whenever|⃗n · ⃗E| 2 | ⃗ E| 2 + 2(⃗n · ⃗E)(⃗n · ⃗B)( ⃗ E · ⃗B) −−|⃗n · ⃗E| 2 | ⃗ B| 2 − |⃗n| 2 | ⃗ E · ⃗B| 2 < 0 (19)there is no propagation whatsoever for the present Proca massive vectorfield.On the other hand, to study the geometrical properties, we noticethat torsion imposes through the condition of complete antisymmetryV α (Z ρβ + λ 2 Z σθε σθρβ )+ V ρ (Z αβ + λ 2 Z σθε σθαβ )= 0 (20)an additional constraint that does reduce to 2 the highest number ofdegrees of freedom, leaving this generalization of the Proca massive vectorfield with the same degrees of freedom of a massless vector field andtherefore overdetermined.Therefore, although not in terms of the propagator but rather interms of the torsional restrictions, this present generalization of theProca massive vector field is not defined correctly, and the most generalProca matter field is to be considered that in its original form.4 ConclusionIn this paper, we have considered the Proca vector field in which the dynamicalterm is written in terms of the exterior derivative or equivalentlythe curl, calculated with respect to the most general metric connectionwith completely antisymmetric Cartan torsion tensor; also, another dynamicalterms has been included in the field equations: this more generalfield equations have been discussed from the perspective of propagationand the geometrical properties. The model is defined in terms of theparameter λ beside its mass m as usual.It has been shown that, regarding the propagation, such a field mayalways have propagating failure or be acausal, while about the geometricproperties, it never has the correct number of degrees of freedom.As this discussion has underlined, any attempt to pursue a generalizationof the Proca massive vector field is inconsistent, and the mostgeneral Proca matter field is the one originally introduced.

Properties and Propagation of Proca the Field 27Références[1] L. V. de Broglie, C. R. Acad. Sci. 198, 135 (1934).[2] G. A. Proca, in “Scientific publications”, Paris, 1988.[3] F. W. Hehl, P. Von Der Heyde, G. D. Kerlick and J. M. Nester,Rev. Mod. Phys. 48, 393 (1976).[4] L. Fabbri, Annales Fond. Broglie 32, 215 (2007).[5] G. Velo and D. Zwanziger, Phys. Rev. 188, 2218 (1969).[6] L. Fabbri, Annales Fond. Broglie 33, 365 (2008).(Manuscrit reçu le 2 avril 2010)