aa r X i v : . [ phy s i c s . g e n - ph ] M a y Chern-Simons Extension of ESK Theory
Luca Fabbri
DIME, Sez. Metodi e Modelli Matematici, Università di Genova,Via all’Opera Pia 15, 16145 Genova, ITALY (Dated: May 25, 2020)The commonly-known Chern-Simons extension of Einstein gravitational theory is written in termsof a square-curvature term added to the linear-curvature Hilbert Lagrangian. In a recent paper, weconstructed two Chern-Simons extensions according to whether they consisted of a square-curvatureterm added to the square-curvature Stelle Lagrangian or of one linear-curvature term added to thelinear-curvature Hilbert Lagrangian. The former extension gives rise to the topological extension ofthe re-normalizable gravity, the latter extension gives rise to the topological extension of the least-order gravity. This last theory will be written here in its torsional completion. Some consequencesfor cosmology and particle physics will be addressed.
I. INTRODUCTION
When investigating the issue of including topologicalterms in Einstein gravity one is immediately faced with aproblem of homogeneity. The Chern-Simons enlargmentof Einsteinian gravitation in its original form is that of[1], and it consists in adding one specific square-curvaturecontribution to the linear-curvature Hilbert Lagrangian.Homogeneity can be restored in two ways: either withthe inclusion of a square-curvature term into the square-curvature Stelle Lagrangian [2, 3] or with the inclusion ofa linear-curvature term into the linear-curvature HilbertLagrangian. In [4] we have discussed both cases finding,in the former case, the topological re-normalizable gravi-tation, and, in the latter case, the topological least-orderderivative gravitation. In the first instance, the modifica-tion proposed by Jackiw and Pi was such that the diver-gence of the gravitational field equations did not lead tothe conservation of the energy, but to one constraint thatwas not verified in general, while we found that when theenergy is that of the Dirac spinors, such a constraint willbe verified indeed. Not so in the second instance, wherethe modification we proposed is such that the divergenceof the gravitational field equations does not yield energyconservation but a constraint that is not verified, and inthis situation it cannot be verified on general grounds.Of this last, least-order derivative gravitational theorywith topological term we will here consider the torsionalcompletion. The torsional completion of gravitation, thatis the Sciama-Kibble completion of the Einstein theory,is what we obtain when we allow torsion to be sourced bythe spin in the same way in which curvature is sourced bythe energy density of the matter field distributions [5–8].Of all possible torsion completions of gravity, here wewill take into account the one where torsion is completelyantisymmetric so to fit its source given by the completelyantisymmetric spin that pertains to the Dirac field [9–15].This type of spin-torsion coupling has important effectsranging from cosmology up to particle physics [16–19].In this paper, the torsion completion of gravity will beperformed for the theory that is dynamically defined withthe least-order of the derivative, namely the one linear inthe curvature or equivalently the one linear in derivatives and squares of the connection, for both the leading termgiven by the Hilbert Lagrangian and the topological termgiven by the Chern-Simons type of extension.
II. SCIAMA-KIBBLE TORSION COMPLETIONOF THE EINSTEIN GRAVITY
We begin by specifying that in the background, metricproperties will be defined in terms of the metric g αν = g να and g αν = g να such that g αν g νσ = δ σα and which can alsobe used as a tool for raising/lowering indices in tensorialquantities. Tetrads are defined so that e rα e sν g να = η rs and e αr e νs g να = η rs where η is the Minkowskian matrix, as wellas e αr e sα = δ sr and e αr e rν = δ αν in such a way that these pairof dual tetrads allow the passage from coordinate (Greek)indices to Lorentz (Latin) indices. Clifford matrices γ a are defined by { γ a , γ b } = 2 η ab I so that the [ γ a , γ b ] = 4 σ ab defines the generators of the complex Lorentz algebra andrelationship i σ ab = ε abcd πσ cd implicitly defines π (thismatrix is usually denoted as gamma matrix with an indexfive, but in space-time this index has no meaning, and sowe employ a notation with no index), then we have γ i γ j γ k = γ i η jk − γ j η ik + γ k η ij + iε ijkq πγ q (1)with { γ a , σ bc } = iε abcd πγ d (2) [ γ a , σ bc ] = η ab γ c − η ac γ b (3)and { σ ab , σ cd } = [( η ad η bc − η ac η bd ) I + iε abcd π ] (4) [ σ ab , σ cd ] = η ad σ bc − η ac σ bd + η bc σ ad − η bd σ ac (5)and the γ will be used for the adjoint spinor operation.This triplication of formalisms may appear to be futilebut in fact it is very important, because coordinate tensorfields are defined to be what transforms according to themost general coordinate transformation. Then the tetradbases convert coordinate tensor fields into Lorentz tensorfields, transforming according to local Lorentz transfor-mations. Finally, the generators of the complex Lorentzlgebra can be exponentiated to yield the local complexLorentz group in terms of which spinor fields transform.With the concept of scalar product between spinors itis possible to form the bi-linear spinor quantities Σ ab = 2 ψ σ ab π ψ (6) M ab = 2 iψ σ ab ψ (7) S a = ψ γ a π ψ (8) U a = ψ γ a ψ (9) Θ = iψ π ψ (10) Φ = ψψ (11)and they are all real tensors, and they verify ψψ ≡ Φ I + U a γ a + i M ab σ ab −− Σ ab σ ab π − S a γ a π − i Θ π (12)and then σ µν U µ S ν π ψ + U ψ = 0 (13) i Θ S µ γ µ ψ +Φ S µ γ µ π ψ + U ψ = 0 (14)and Σ ab = − ε abij M ij (15) M ab = ε abij Σ ij (16)with M ab Φ − Σ ab Θ = U j S k ε jkab (17) M ab Θ+Σ ab Φ = U [ a S b ] (18)alongside to M ik U i = Θ S k (19) Σ ik U i = Φ S k (20) M ik S i = Θ U k (21) Σ ik S i = Φ U k (22)and with the orthogonality relations M ab M ab = − Σ ab Σ ab = Φ − Θ (23) M ab Σ ab = − (24)and U a U a = − S a S a = Θ +Φ (25) U a S a = 0 (26)as geometric identities called Fierz identities. Notice thatwhile defining six bi-linear spinor quantities maintains acertain symmetry in the expression of the Fierz identities,(15, 16) show that of the two antisymmeric tensors onlyone is actually needed. In the general case where at least one between Θ and Φ is not identically zero (17, 18) canbe combined to give the expressions Σ ab = 2 φ (cos βu [ a s b ] − sin βu j s k ε jkab ) (27) M ab = 2 φ (cos βu j s k ε jkab +sin βu [ a s b ] ) (28)where we have introduced S a = 2 φ s a (29) U a = 2 φ u a (30)and Θ = 2 φ sin β (31) Φ = 2 φ cos β (32)for simplicity, and showing that in this case no antisym-metric tensor at all is needed. With these expressions allFierz identities (19-22) and (23, 24) reduce to be trivialwhile (25, 26) reduce to u a u a = − s a s a = 1 (33) u a s a = 0 (34)so that u a is time-like and s a is space-like in such a case.This is important because this means that one can al-ways find some combination of up to three Lorentz boostsbringing u a to have the temporal component only. Thistells us that u a is the velocity vector and that a rest frameis possible. However, these Fierz identities also tell that s a is the Pauli-Lubanski axial-vector, and in general it isalways possible to find some combinations of up to twoLorentz rotations bringing it to have only the third of itscomponents. A consequence of this is that s a is the spinaxial-vector and one can always find a frame where it isaligned with the third of the axes. In this situation, thelast rotation is the one around the third axis. As such itcan be used to act onto the global phase of the spinorialfield. Putting all together allows one to see that generalspinors can always be written according to ψ = φe − i β π S (35)in chiral representation, for some local complex Lorentztransformation S , with the β and φ being a pseudo-scalarand scalar fields called Yvon-Takabayashi angle and mod-ule, and where the spinor is said in polar form. In such aform, the initial real components are re-arranged intothe configuration in which the true degrees of freedomgiven by the YT angle and module remain isolated fromthe components that can always be transferred into theframe given by the velocity and spin, or equivalently bythe rapidities and angles that are encoded in the Lorentztransformation. Notice that the YT angle has zero mass-dimension and so the module inehrits the full / mass-dimension that characterizes spinors. The reader who is2nterested in the details of the construction of this polardecomposition can have a look at references [20–29].The most general connection that can be defined shallbe indicated by Γ σαν and the corresponding most generalcovariant derivative will be assumed to satisfy D ν g ασ = 0 known as metric-compatibility condition. The antisym-metric part in the two lower indices Γ σαν − Γ σνα = Q σαν iscalled torsion tensor and as anticipated we will assume itcompletely antisymmetric, so that it will be with no lossof generality that we can write Q σαν = W π ε πσαν as theHodge dual of an axial-vector. The metric-compatibilityand torsion’s complete antisymmetry together imply that Γ σαν = g σρ [( ∂ α g νρ + ∂ ν g αρ − ∂ ρ g αν )+ W π ε πραν ] (36)where the part that is entirey written in terms of the par-tial derivatives of the metric is the symmetric connectionwhose associated covariant derivative satisfies conditions ∇ ν g ασ ≡ identically. The passage to tetradic formalismis possible in terms of the spin connection Ω acπ for whichthe corresponding covariant derivative is assumed to sat-isfy D ν e cα = 0 similarly to what was done before. Becauseof the different type of indices torsion can not be definedfor the spin connection. Nonetheless, the vanishing of thecovariant derivative of the tetrads can be written as Ω abπ = ξ νb ξ aσ ( ∂ π ξ σi ξ iν +Γ σνπ ) (37)linking the connection to the spin connection, so that itis still possible to decompose the spin connection into thesum of a torsion term plus the torsionless spin connection,that is the one whose associated covariant derivative stillsatisfies ∇ ν e cα ≡ identically as above. Finally, spinorialconnections can also be defined as Ω µ with correspondingcovariant derivative assumed to satisfy D µ γ a = 0 also inanalogy to what done before. This condition can then beworked out to give the following expression Ω µ = iqA µ I + Ω abµ σ ab (38)as the decomposition in terms of the spin connection withan additional vector field, and again we could decomposethe spinorial connection into the sum of a torsional termplus the torsionless spinorial connection, with associatedcovariant derivative ∇ µ γ a ≡ identically as above.When the polar form is taken into the spinorial deriva-tive and considering that we can formally write S ∂ µ S − = i∂ µ α I + ∂ µ θ ij σ ij (39)we can define ∂ µ α − qA µ ≡ P µ (40) ∂ µ θ ij − Ω ijµ ≡ R ijµ (41)needed to write D µ ψ = ( − i D µ β π + D µ ln φ I − iP µ I − R ijµ σ ij ) ψ (42)as spinorial covariant derivative, from which D µ s i = R jiµ s j (43) D µ u i = R jiµ u j (44) valid as general geometric identities. Notice that despite(40, 41) contain the same information of connection andgauge potential nonetheless they are proven to be tensorsand invariant under the gauge transformation. As we saidabove, writing spinor fields in polar form allows to makea re-configuration in which the degrees of freedom remainisolated from the components transferable into the gaugeand the frames, and now we see that the components thatare transferred into gauge and frames combine with gaugepotential and spin connection, with no alteration to theirinformation content while rendering the resulting objectstrue tensors invariant under gauge transformations. Thusthey have been called gauge-invariant vector momentumand tensorial connection. To have more details see [30].Then the commutator of spinorial covariant derivativesjustifies the definitions R ijµν = ∂ µ Ω ijν − ∂ ν Ω ijµ +Ω ikµ Ω kjν − Ω ikν Ω kjµ (45) F µν = ∂ µ A ν − ∂ ν A µ (46)that is the Riemann curvature and the Maxwell strength.Taking the commutator with polar variables gives qF µν = − ( D µ P ν − D ν P µ − P α Q αµν ) (47) R ijµν = − ( D µ R ijν − D ν R ijµ ++ R ikµ R kjν − R ikν R kjµ − R ijα Q αµν ) (48)in terms of the Maxwell strength and Riemann curvature,so that they encode electrodynamic and gravitational in-formation filtering out gauge and frame information. Butthe other hand, the gauge-invariant vectorial momentumand tensorial connection (40, 41) contain the informationabout electrodynamics and gravity and also about gaugeand frames. Because there exist non-zero gauge-invariantvectorial momentum and tensorial connection which stillgive zero Maxwell strength and Riemann curvature, thenthere exists a type of information that is related to gaugeand frames but which is encoded in covariant objects.The Dirac spinor dynamics is given by the Dirac spinorfield equations i γ µ D µ ψ − mψ = 0 (49)and by multiplying it by γ a and γ a π and by ψ splittingreal and imaginary parts gives D α Φ − ψ σ µα D µ ψ − D µ ψ σ µα ψ ) = 0 (50) D ν Θ − i ( ψ σ µν πD µ ψ − D µ ψ σ µν π ψ )+2 mS ν = 0 (51)which are called Gordon decompositions.In them it is possible to plug the polar form getting B µ − P ι u [ ι s µ ] + D µ β +2 s µ m cos β = 0 (52) R µ − P ρ u ν s α ε µρνα +2 s µ m sin β + D µ ln φ = 0 (53)with R aµa = R µ and ε µανι R ανι = B µ and which couldbe proven to be equivalent to the polar form of the Dirac3pinorial field equations. Notice that the Dirac spinorialfield equations are real equations and that is as many asthe vectorial equations given by the (52, 53) above, thusspecifying all space-time derivatives of the two degrees offreedom given by YT angle and module. The reader whois interested in details may have a look at reference [31].An example of gauge-invariant vector momentum andtensorial connection that are non-zero but which have aMaxwell strength and a Riemann curvature null is in [32].For general summaries, readers might look at [33, 34]. III. CHERN-SIMONS TOPOLOGICALEXTENSION
The Chern-Simons type of topological extension of theEinstein gravitation is done by adding to the Hilbert La-grangian L = R ijµν e µi e νk η jk (54)a term of the form L = kbD µ K µ (55)with b pseudo-scalar and K µ axial-vector. Homogeneityrequires this term to have mass-dimension, and the waywe have to do this is to take b as a pseudo-scalar of zeromass dimension and K µ an axial-vector of unitary mass-dimension. The polar form of spinor fields shows that wedo have such objects, the Yvon-Takabayashi angle β andthe dual of the completely antisymmetric component ofthe tensorial connection B µ alone. So we may write L = kβD µ B µ (56)as the option for the topological term. Altogether wehave L = R + kβD µ B µ (57)which is indeed homogeneous, and it will be taken as theLagrangian for torsion-gravity. Then, we end up having L = R + kβD µ B µ − iψ γ µ D µ ψ + mψψ (58)as the Lagrangian of torsion-gravity for a space-time filledwith spinorial matter. For more details, see [4].In polar form it becomes L = R + kβD µ B µ − φ [ s µ ( D µ β + B µ ) ++2 u µ P µ − m cos β ] (59)as it can be checked straightforwardly.Variation of this Lagrangian with respect to the tetradsgives the gravitational field equations R νσ − g νσ R − k [ D µ ( βB µ ) g νσ − ε µναη D µ βR σαη ] == i ( ψ γ ν D σ ψ − D σ ψ γ ν ψ ) (60) while the variation with respect to torsion gives Q ανµ + k ε ανµσ D σ β = − i ψ { σ αν , γ µ } ψ (61)and with respect to the spinor gives i γ µ D µ ψ − k D · Bφ − e i π β i π ψ − mψ = 0 (62)as the full set of gravity and matter field equations.In polar form we have that R νσ − g νσ R − k [ D µ ( βB µ ) g νσ − ε µναη D µ βR σαη ] == φ ( u ν P σ + s ν D σ β − R σij ε νijk s k ) (63)with W σ − kD σ β = φ s σ (64)and − P ι u [ ι s µ ] + B µ + D µ β +2 s µ m cos β = 0 (65) − P ρ u ν s α ε µρνα + R µ − s µ kD · Bφ − ++ D µ ln φ +2 s µ m sin β = 0 (66)again as it is quite direct to verify.Taking the divergence of (60) and (61) and employingthe Jacobi-Bianchi cyclic identities gives D α [ ε σαρν R µρν D σ β − g αµ D ν ( βB ν )] −− Q ηρµ ε σρπν D σ βR πνη ++ R ρπηµ ε ρπηα D α β = − D µ βD ν B ν (67)and D σ β ( ε σαρν R ηρν − ε σηρν R αρν ) = ε αηµσ Q µσρ D ρ β (68)where also (62) have been thoroughly used.These are a pair of restrictions over the derivatives ofthe Yvon-Takabayashi angle and tensorial connection.Notice that with no tensorial connection we would ob-tain the theory that has been presented in [35–37]. IV. TORSIONLESS DISCONTINUITY
We have presented the general ESK theory for a space-time filled with Dirac spinor matter fields with topologi-cal extension of Chern-Simons type. And that is, we havegiven the topological extension of a spinor interaction inpresence of gravity with torsion. Or then again, we haveconsidered the torsional extension of [4]. Therefore, it isnatural to assume that in the torsionless limit we shouldobtain the results that were obtained in reference [4].Nonetheless, this is generally not possible because herethe presence of torsion as an independent degree of free-dom makes one more way of varying the Lagrangian, andultimately one more field equation, namely (64), and onemore constraint, namely (68). Additional field equationsand constraints are not necessarily going to be identically4atisfied in the torsionless limit, creating a condition thatcan be called torsionless discontinuity [38]. In this case,the limit of torsionlessness does not at all coincide withthe theory one would have had in total absence of torsion.Consequently, an interesting question is now, what arethe effects of the torsionless limit in this case? Will thistheory be continuous? To answer this let us simply takethe limit W σ → in all field equations and constraints.To do that, we replace all the covariant derivatives withtheir torsionless covariant derivatives and specify that alltensorial connections will be intended to be torsionless aswell. We immediately have that (64) gives k ∇ σ β = − φ s σ (69)locking the YT angle to the spin density and R νσ − g νσ R + k [ −∇ µ ( βB µ ) g νσ ++ ∇ µ β ( ε µσαη R ναη + ε µναη R σαη )] == φ ( u ν P σ + u σ P ν + s ν ∇ σ β + s σ ∇ ν β −− R σij ε νijk s k − R νij ε σijk s k ) (70)which have also been symmetrized using the constraints.In fact the constraints read ∇ α [( ε σαρν R µρν ∇ σ β + ε σµρν R αρν ∇ σ β ) −− g µα ∇ ν ( βB ν )] = −∇ µ β ∇ ν B ν (71)and ∇ σ β ( ε σαρν R ηρν − ε σηρν R αρν ) = 0 (72)as it can be easily checked.Notice that these are exactly all the field equations andconstraints of [38] plus the additional field equation (69)and constraint (72). So the limit of torsionlessness doesnot give the theory in absence of torsion, and torsionlessdiscontinuity arises in the topological torsion-gravity.Notice however that if we were to take also the spinlesslimit s σ → then we will see further simplifications.In fact in this case (69) would give ∇ ν β = 0 so that (72)would be identically verified, hence continuity recovered.However, we would also have that the other constraintwould have the only solution ∇ µ ∇· B = 0 so that R νσ − g νσ R − Λ g νσ = φ ( u ν P σ + u σ P ν ) (73)where
2Λ = kβ ∇· B was defined. Because β and ∇· B areconstant, then these equations are just the usual gravita-tional field equations but with an effective cosmologicalconstant that has been generated by the topology. V. NONLOCALITY
In the last two sections we have witnessed that in tak-ing the divergence of the field equations to get conserva-tion laws we instead obtained constraints, as is expected from the fact that the tensorial connection does not verifyany system of field equations. And that its presence doesprovide the conditions to generate an effective cosmolog-ical constant, as expected for fields that do not vanish atinfinity. So, we must also expect a nonlocal behaviour.Consider for example the situation of a spinor that dis-plays a flipping of the spin s a in general. The process canbe described by a rotation of angle θ = ωt around the firstaxis. The standard situation with no flip of the spin isobtained when s a points toward a fixed direction. In thisinstance, we can still use the above solutions but taking ω = 0 in general. We may call R ′ jiµ the tensorial connec-tion in the former case and R jiµ the tensorial connectionin the latter case. Because of (43) it is clear that in bothcases we will have R ′ jiµ = R ijµ +∆ R ijµ where ∆ R ijµ has ∆ R t = ω with all the other components being equal tozero identically. Suppose now that a measurement pro-cess intervenes so that the former solution reduces to thelatter solution. Such a process clearly would require that ω → as the reduction of the solutions. So we would alsohave R ′ jiµ → R jiµ as the reduction for the correspondingtensorial connections. This would hold for one particle.If then the above solution is meant to describe a stateof two particles with opposite spin, then the reduction ofone spin would imply the reduction of the other spin forspin conservation. As discussed above, the transmissionof the reduction from one spin to the other will be donein terms of the tensorial connection. Such a transmissiondoes not need to be causal because tensorial connectionsdo not have any genuine form of dynamical propagation.It is however important to specify that any mechanismtaking place via the tensorial connection need not respectcausality because the tensorial connection is not governedby field equations, and not because causality is violated inthe dynamics. In fact, causality is still ensured for fieldsthat propagate, and more in general the causal structureis granted for fields that are solutions of field equations.The possibility of having nonlocal effects may be givenwithout violating the causal propagation because we aredescribing them with objects that have no dynamics. VI. CONCLUSION
In this paper, we have constructed the Sciama-Kibbletorsional completion of the Einsteinian gravitational the-ory that is obtained by extending with the Chern-Simonstopological terms the Hilbert Lagrangian. The result wasa theory in which the topological term was formed by apurely spinorial degree of freedom, the Yvon-Takabayashiangle, and the divergence of a purely geometric degree offreedom, the dual of the completely antisymmetric partof the tensorial connection that also contained the torsionaxial-vector. This term therefore is the product of a pairof dynamically coupled variables and as such it accountsfor an interaction between the two. As a consequence, acheck on the conservation laws failed to give the conser-vation of the energy, and it yielded some constraints that5ere not identically verified in general circumstances.We found that in the limit of torsionlessness the theoryhad discontinuities, since the constraints were still not ingeneral satisfied. It was only by taking the spinless limitthat continuity would be ensured by the total lack of anyconstraint. In this case, we found that a term giving aneffective cosmological constant was eventually generated.We concluded with generic remarks about nonlocality.The lack of dynamical field equations for the tensorialconnection, together with its non-trivial structure at in-finity, and its nonlocal properties, all points toward the fact that the tensorial connection is rather unique in thepanorama of physical theories developed in recent times.And it all fits with the fact that the tensorial connec-tion is an object that is not coupled to sources at all.It all appears to indicate that the tensorial connectionhas the character of an object that cannot describe phys-ical degrees of freedom, and yet it is found to be generallydifferent from zero and manifestly fully covariant.Such a seemingly contraddictory object might turn outto be intriguing enough to require further studies.A follow up is already being worked on. [1] R.Jackiw, S.Y.Pi, “Chern-Simons modification of generalrelativity”,
Phys.Rev.D , 104012 (2003).[2] K.S.Stelle, “Classical Gravity with Higher Derivatives”, Gen.Rel.Grav. , 353 (1978).[3] K.S.Stelle, “Renormalization of Higher Derivative Quan-tum Gravity”, Phys.Rev.D , 953 (1977).[4] L.Fabbri, “Homogeneous Chern-Simons Einstein Grav-ity”, arXiv:2004.04051[5] F.W.Hehl, P.Von Der Heyde, G.D.Kerlick, J.M.Nester,“General Relativity with Spin and Torsion: Foundationsand Prospects”, Rev.Mod.Phys. , 393 (1976).[6] I.L.Shapiro, “Physical aspects of the space-time torsion”, Phys.Rept. , 113 (2002).[7] R.T.Hammond, “Torsion gravity”,
Rept.Prog.Phys. , 599 (2002).[8] H.I.Arcos, J.G.Pereira, “Torsion gravity:A Reappraisal”, Int.J.Mod.Phys.D , 2193 (2004).[9] C.Laemmerzahl, A.Macias, “On the dimensionality ofspace-time”, J. Math. Phys. , 4540 (1993).[10] J.Audretsch, C.Lammerzahl, “ConstructiveAxiomatic Approach To Space-time Torsion”, Class. Quant. Grav. , 1285 (1988).[11] L.Fabbri, “On a completely antisymmetric Cartan torsiontensor”, In Annales de la Fondation de Broglie,Special Issue on Torsion (2007) .[12] L.Fabbri, “On the Principle of Equivalence”,
In Contemporary Fundamental Physics,Einstein and Hilbert: Dark Matter (2012) [13] L.Fabbri, “On the problem of Unicity in Einstein-Sciama-Kibble Theory”,
Annales Fond. Broglie , 365 (2008).[14] L.Fabbri, “On the consistency of Constraints in MatterField Theories”, Int.J.Theor.Phys. , 954 (2012).[15] L.Fabbri, “Least-order torsion-gravity for fermion fields,and the nonlinear potentials in the standard models”, Int.J.Geom.Meth.Mod.Phys. , 1450073 (2014).[16] L.Fabbri, “Singularity-free spinors in gravity with propa-gating torsion”, Mod.Phys.Lett.A , 1750221 (2017).[17] L.Fabbri, “A geometrical assessment of spinorial energyconditions”, Eur.Phys.J.Plus , 156 (2017).[18] L.Fabbri, “On geometric relativistic foundations ofmatter field equations and plane wave solutions”,
Mod.Phys.Lett.A , 1250028 (2012).[19] L.Fabbri, “On a purely geometric approach to theDirac matter field and its quantum properties”, Int.J.Theor.Phys. , 1896 (2014).[20] P.Lounesto, Clifford Algebras andSpinors (Cambridge University Press, 2001).[21] R.T.Cavalcanti, “Classification of Singular Spinor Fields and Other Mass Dimension One Fermions”,
Int.J.Mod.Phys.D , 1444002 (2014).[22] J.M.Hoff da Silva, R.T.Cavalcanti, “Revealing howdifferent spinors can be: the Lounesto spinorclassification”, Mod.Phys.Lett.A , 1730032 (2017).[23] J.M.Hoff da Silva, R.da Rocha, “Unfolding Physicsfrom the Algebraic Classification of SpinorFields”, Phys. Lett. B , 1519 (2013).[24] R.Abłamowicz, I.Gonçalves, R.da Rocha, “BilinearCovariants and Spinor Fields Duality in QuantumClifford Algebras”,
J. Math. Phys. , 103501 (2014).[25] W.A.Rodrigues, R.da Rocha, J.Vaz, “Hiddenconsequence of active local Lorentz invariance”, Int.J.Geom.Meth.Mod.Phys. , 305 (2005).[26] J.M.Hoff da Silva, R.da Rocha, “From Dirac Action toELKO Action”, Int.J.Mod.Phys.A , 3227 (2009).[27] R.da Rocha, J.M.Hoff da Silva, “ELKO, flagpole andflag-dipole spinor fields, and the instanton Hopffibration”, Adv. Appl. Clifford Algebras , 847 (2010).[28] R.da Rocha,L.Fabbri,J.M.Hoff da Silva,R.T.Cavalcanti,J.A.Silva-Neto, “Flag-Dipole Spinor Fields in ESK Grav-ities”, J.Math.Phys. ,102505(2013).[29] L.Fabbri, “A generally-relativistic gaugeclassification of the Dirac fields”, Int.J.Geom.Meth.Mod.Phys. ,1650078(2016).[30] L.Fabbri, “Covariant inertial forces for spinors”, Eur.Phys.J.C , 783 (2018).[31] L.Fabbri, “Torsion Gravity for Dirac Fields”, Int.J.Geom.Meth.Mod.Phys. ,1750037(2017).[32] L.Fabbri, “Polar solutions with tensorial connection ofthe spinor equation”, Eur.Phys.J.C , 188 (2019).[33] L.Fabbri, “General Dynamics of Spinors”, Adv. Appl. Clifford Algebras , 2901 (2017).[34] L.Fabbri, “Spinors in Polar Form”, arXiv:2003.10825[35] M.Lattanzi, S.Mercuri, “A solution of the strong CPproblem via the Peccei-Quinn mechanism through theNieh-Yan modified gravity and cosmologicalimplications”, Phys.Rev.D ,125015 (2010).[36] O.Castillo-Felisola, C.Corral, S.Kovalenko, I.Schmidt,V.E.Lyubovitskij, “Axions in gravity with torsion”, Phys.Rev.D , 085017 (2015).[37] L.Fabbri, “Re-normalizable Chern-Simons Extension ofPropagating Torsion Theory”, arXiv:2004.14776[38] Luca Fabbri, “A discussion on the most general torsion-gravity with electrodynamics for Dirac spinor matterfields”, Int.J.Geom.Meth.Mod.Phys. , 1550099 (2015)., 1550099 (2015).