aa r X i v : . [ phy s i c s . g e n - ph ] A p r Geometry, Zitterbewegung, Quantization
Luca Fabbri
DIME, Sez. Metodi e Modelli Matematici, Università di Genova, Via all’Opera Pia 15, 16145 Genova, ITALY (Dated: April 28, 2020)In the most general geometric background, we study Dirac spinor fields with particular emphasisgiven to the explicit form of their gauge momentum and the way in which this can be inverted so togive the expression of the corresponding velocity; we study how zitterbewegung affects the motionof particles, focusing on the internal dynamics involving the chiral parts; we discuss the connectionsto field quantization, sketching in what way anomalous terms may be gotten eventually.
I. INTRODUCTION
After nearly seventy years since the first experimentalconfirmation, quantum field theory (QFT) has yet to faila phenomenological test. Whether it is the correction tothe magnetic moment of fermions or the energy splittingin hydrogen-like systems, QFT has always provided veryprecise predictions matching to less than a part in severaltrillions the most accurate of measurements.Nonetheless, despite its success when compared to ourobservations, QFT is still lacking a proper mathematicaldefinition: conceptual problems start from the fact thatall calculations are done by expanding fields in terms ofplane waves, which are not square integrable (and in factthey do not really exist); they continue with the fact thatin such an expansion the coefficients are reinterpreted asa pair of creation/annihilation operators, which still lacka definition, and for that matter their set of commutationrelations might make no sense [1]; and they end with thefact that for all these calculations we employ the so-calledinteraction picture, which has been demonstrated not toexist in general in the context of a Lorentz-covariant fieldtheory at all [2]. In the face of these issues, the fact thatperturbative expansions do not converge, or that each oftheir terms is finite only up to a certain regularization orrenormalization, looks like a minor problem indeed.Of all these conceptual issues, the lack of some propermathematical definition of the creation/annihilation op-erators was felt particularly by Schwinger, who took thisunsatisfactory situation to prompt himself into finding adifferent formulation for field quantization: his efforts ledhim to the construction of the so-called source theory [3].Nevertheless, again, this is not a solution: Schwinger’ssource theory is in fact a prototypical version of the pathintegral, whose measure has never been defined too.More in general, if we were to go back to the roots ofquantization, we would see that the first problem wouldbe involving the use of plane waves, which are not squareintegrable and as such they can not represent a particle.This point, however, may constitute a possible avenuefor our way out. As it is, QFT might be too dramatic inits first assumption that only plane waves should be used,thus leaving out too much information, so that some ofthe lost information must be reintroduced, and this couldbe done through quantization. In other words, quantiza-tion fills the gaps left by too strong approximations en- forced by having the particle described with plane waves.Were this the case, any theory of fields employing onlythe particular solution given by the plane wave but laterquantized through some subsidiary conditions should bereplaceable by a theory of fields employing general solu-tions with no subsidiary condition to be implemented.We do not know whether this is the case. On the otherhand, some literature does exist which follows this path:a first example is the one given by Koba and Welton, whofaced the treatment of the anomalous magnetic momentof the electron and the Lamb shift, respectively, in termsof semi-classical considerations [4, 5]; more systematic isthe work of Barut and co-workers, Dowling above all, whostudy the electron in electrodynamic self-interaction, andwith no references to quantization, recovering the abovementioned results, on anomalous magnetic moment andLamb shift, in general [6, 7]; the deepening on Lamb shiftand new results on spontaneous emission/absorption arealso presented in [8–11]; a result on vacuum polarizationis also given in [12] and discussed with the same spirit.In fact, the concept of vacuum polarization might comehelpful in visualization. Because a plane wave describesa freely propagating point particle and quantization ac-counts for radiative processes involving virtual loops thenmuch in the same way in which quantization fills the gapsleft by working with plane waves all virtual loops fill thegaps left by working with point particles; in addition, thesurrounding cloud of virtual loops gives an effective sizeto what would otherwise be a mere point particle.In the perspective outlined above, any theory of pointparticles whose dynamics is corrected in terms of virtualloops should be replaceable by a theory of extended fieldswhose dynamics is comprehensive enough to contain thecorrections attributed to the virtual loops.Although the lack of any exact solution makes it diffi-cult to know what are all the dynamical effects that couldreplace the corrections due to virtual processes, one of thepossibilities that has been considered is zitterbewegung,as addressed by Hestenes [13], and with more details byRecami and Salesi [14, 15]. Zitterbewegung is a dynamiceffect due to the relative motion between left-handed andright-handed semi-spinor projections of spinors.As such it cannot be present for a point particle, whoselack of size means lack of internal structures. Still, it canbe, and in fact it is, present in general for extended fields,and therefore it does make sense to consider it as whatight give rise to dynamical effects, among which somecan be mimicking the corrections of virtual loops.Consequently, having in mind the idea of reproducingquantum corrections, and following the hint that such anendeavour might be done in terms of zitterbewegung, itis wise to start from the most comprehensive dynamics.In this paper we will do this, defining the most generaldynamics for matter fields, deriving some of the possibleconsequences of zitterbewegung, and seeing what connec-tion there can be with known quantum corrections.
II. FUNDAMENTAL THEORETICALGENERALITIESA. Dirac Spinorial Field
To begin, we recall that γ a are Clifford matrices, fromwhich [ γ a , γ b ] = 4 σ ab and i σ ab = ε abcd πσ cd are the defini-tions of the σ ab and the π matrix (this matrix is usuallyindicated as gamma with an index five, but since in thespace-time this index has no meaning we use a notationwith no index so to avoid confusion): given ψ as a Diracspinor field, we define the bi-linear quantities given by M ab = 2 iψ σ ab ψ (1)with S a = ψ γ a π ψ (2) U a = ψ γ a ψ (3)as well as Θ = iψ π ψ (4) Φ = ψψ (5)and which, despite being written only with spinor fields,are all real tensors. From the metric we define the sym-metric connection as usual with Λ σαν and with it we definethe spin connection Ω abπ = ξ νb ξ aσ (Λ σνπ − ξ σi ∂ π ξ iν ) in such away that with the gauge potential qA µ we can define Ω µ = Ω abµ σ ab + iqA µ I (6)needed to write the spinorial covariant derivative ∇ µ ψ = ∂ µ ψ + Ω µ ψ (7)in which for the moment no torsion is defined: the com-mutator of spinorial covariant derivatives can be used tojustify the definitions of space-time and gauge curvature R ijµν = ∂ µ Ω ijν − ∂ ν Ω ijµ +Ω ikµ Ω kjν − Ω ikν Ω kjµ (8) F µν = ∂ µ A ν − ∂ ν A µ (9)which are again in the torsionless case. The Lagrangianwe will consider is given according to the standard L = ( ∂W ) − M W + R + F −− iψ γ µ ∇ µ ψ + XS µ W µ + m Φ (10) with R trace of the space-time curvature and F squareof the gauge curvature and where the generality we tem-porarily lost when we defined torsionless connections cannow be restored by including torsion as an axial vector W µ with curl given by ( ∂W ) µν for the sake of simplicity.As it has been discussed first of all by Wigner and morerecently by Lounesto and Cavalcanti [16, 17], spinor fieldscan be classified in two large classes, in terms of which aspinor such that Θ = Φ = 0 is called singular and it is thesubject of many studies [18–24], while a spinor such thateither Θ = 0 or Φ = 0 is called regular and it is the centerof attention of the present work: we can always write themost general regular spinor in terms of a generic complexLorentz transformation S according to ψ = φe − i β π S (11)called polar form [25], and such that with it we have M ab = 2 iψ σ ab ψ = 2 φ (cos βu j s k ε jkab +sin βu [ a s b ] ) (12)in terms of S a = ψ γ a π ψ = 2 φ s a (13) U a = ψ γ a ψ = 2 φ u a (14)such that u a u a = − s a s a = 1 and u a s a = 0 and representingthe velocity vector and the spin axial-vector as well as Θ = iψ π ψ = 2 φ sin β (15) Φ = ψψ = 2 φ cos β (16)being a scalar and a pseudo-scalar known as module andYvon-Takabayashi angle and which are the only two realdegrees of freedom of the spinor field. Because generally S ∂ µ S − = i∂ µ θ I + ∂ µ θ ij σ ij (17)where θ is a generic complex phase and θ ij = − θ ji are thesix parameters of the Lorentz group, then we can define ∂ µ θ ij − Ω ijµ ≡ R ijµ (18) ∂ µ θ − qA µ ≡ P µ (19)being real tensors called tensorial connection and gaugevector momentum respectively, with which we have ∇ µ ψ = ( ∇ µ ln φ I − i ∇ µ β π − iP µ I − R ijµ σ ij ) ψ (20)and also ∇ µ s i = R jiµ s j (21) ∇ µ u i = R jiµ u j (22)identically: from the commutator of the covariant deriva-tives we deduce that the curvatures are such that R ijµν = − ( ∇ µ R ijν −∇ ν R ijµ + R ikµ R kjν − R ikν R kjµ ) (23) qF µν = − ( ∇ µ P ν −∇ ν P µ ) (24)2elling that the tensors defined in (18, 19) do not generateany curvature tensor that is not already generated by thespin connection and the gauge potential. The Lagrangianabove gives rise to field equations that in the polar formare transcribed into the geometric field equations ∇ k R kaa g ρσ −∇ i R iσρ −∇ ρ R σii + R iki R kσρ ++ R ρik R kσi − R iki R kaa g ρσ −− R ika R kai g ρσ = [ M ( W ρ W σ − W α W α g ρσ ) ++ ( ∂W ) g ρσ − ( ∂W ) σα ( ∂W ) ρα ++ F g ρσ − F ρα F σα −− φ [( XW −∇ β ) σ s ρ +( XW −∇ β ) ρ s σ −− P σ u ρ − P ρ u σ ++ R σij ε ρijk s k + R ρij ε σijk s k ]] (25)with ∇ P µ −∇ σ ∇ µ P σ = − q φ u µ (26)and ∇ ( XW ) µ −∇ α ∇ µ ( XW ) α + M XW µ = 2 X φ s µ (27)alongside to the matter field equations ε µανι R ανι − P ι u [ ι s µ ] ++2( ∇ β/ − XW ) µ +2 s µ m cos β = 0 (28) R aµa − P ρ u ν s α ε µρνα ++2 s µ m sin β + ∇ µ ln φ = 0 (29)specifying all the first-order derivatives of the module andthe YT angle [26], and which can be proven to be equiv-alent to the original Dirac spinor field equations [27].To see that, we start by considering the Lagrangian wehave written in (10) and then we vary it with respect tothe spinor field, getting the field equations i γ µ ∇ µ ψ − XW µ γ µ π ψ − mψ = 0 (30)which we then multiply by γ a π and γ a and by the con-jugate spinor, splitting real and imaginary parts, to getthe four real vectorial field equations given according to i ( ψ ∇ α ψ − ∇ α ψψ ) −∇ µ M µα −− XW σ M µν ε µνσα − mU α = 0 (31) ∇ α Φ − ψ σ µα ∇ µ ψ − ∇ µ ψ σ µα ψ ) ++2 X Θ W α = 0 (32) ∇ ν Θ − i ( ψ σ µν π ∇ µ ψ − ∇ µ ψ σ µν π ψ ) −− X Φ W ν +2 mS ν = 0 (33) ( ∇ α ψ π ψ − ψ π ∇ α ψ ) − ∇ µ M ρσ ε ρσµα ++2 XW µ M µα = 0 (34) and in which we now plug the polar form (11) obtaining −∇ µ ln φM µσ + ( ∇ µ β − XW µ ) M πν ε πνµσ ++ P σ Φ+ R ανρ M πκ ε ανρµ ε πκσµ −− R aµa M µσ − mU σ = 0 (35) −∇ σ ln φ Φ+( ∇ σ β − XW σ )Θ − P µ M µσ −− R ανρ ε ανρσ Θ − R aσa Φ = 0 (36) ∇ σ ln φ Θ+( ∇ σ β − XW σ )Φ ++ P µ M πκ ε πκµσ − R ανρ ε ανρσ Φ ++ R aσa Θ+ mS σ = 0 (37) ∇ µ ln φM πκ ε πκµσ +( ∇ µ β − XW µ ) M µσ −− P σ Θ − R ανρ M µσ ε ανρµ + R aµa M πκ ε πκµσ = 0 (38)as a straightforward substitution shows: the second andthird, after inserting the bi-linear quantities, become ∇ α ln φ cos β − ( ∇ α β − XW α ) sin β ++ P µ ( u ρ s σ ε ρσµα cos β + u [ µ s α ] sin β ) ++ R µαµ cos β + R ρσµ ε ρσµα sin β = 0 (39) ∇ ν ln φ sin β +( ∇ ν β − XW ν ) cos β ++ P µ ( u ρ s σ ε ρσµν sin β − u [ µ s ν ] cos β ) −− R ρσµ ε ρσµν cos β + R µνµ sin β + ms ν = 0 (40)and after diagonalization ε µανι R ανι − P ι u [ ι s µ ] −− XW µ + ∇ µ β +2 s µ m cos β = 0 (41) R aµa − P ρ u ν s α ε µρνα ++2 s µ m sin β + ∇ µ ln φ = 0 (42)in general. Conversely, from these and then consideringthe general identities given by the expressions σ µν u µ s ν π ψ + ψ = 0 (43) is µ γ µ ψ sin β + s µ γ µ π ψ cos β + ψ = 0 (44)it is possible to see that i γ µ ∇ µ ψ − XW σ γ σ π ψ − mψ == [ i γ µ P ρ u ν s α ε µρνα ++ P ι u [ ι s µ ] γ µ π + P µ γ µ −− is µ γ µ m sin β − s µ γ µ π m cos β − m I ] ψ = 0 (45)showing that when the spinor is in polar form these fieldequations are valid: as any spinor can always be writtenin polar form then also these field equations are valid ingeneral. So (30) is equivalent to (28, 29) in general [28]. B. Gauge Momentum
As we already said, the objects (18, 19) are real tensorsand, because of the information content that can be de-duced from their definition, they contain all information3ormally contained within the connection and the gaugepotential, and thus we called them tensorial connectionand gauge vector momentum, respectively: in particularwe have that R ijk has a trace defined as R a = R cac (46)and its completely antisymmetric part has dual as B a = ε aijk R ijk (47)so that the non-completely antisymmetric traceless part Π ijk = R ijk − ( R i η jk − R j η ik ) − ε ijka B a (48)is such that Π aia = 0 and Π ijk ε ijka = 0 hold; instead, for P a we have irreducibility. However, it is possible to have P a written in terms of R ijk according to the expression P µ = m cos βu µ − ( ∇ k β − XW k + B k ) s [ k u µ ] −− ( ∇ k ln φ + R k ) s j u i ε kjiµ (49)although it is only a link between P a and the two vectorialparts of R ijk and not a link to the full tensor; this is clear,because the only occurrence of the full R ijk is within thefield equations (25) but even there it is always either inderivatives or in products. Only (28, 29) contain the pureforms of R a and B a needed for (49) to be expressed.To see that, consider (28, 29) in terms of R a and B a B µ − P ι u [ ι s µ ] +2( ∇ β/ − XW ) µ +2 s µ m cos β = 0 (50) R µ − P ρ u ν s α ε µρνα +2 s µ m sin β + ∇ µ ln φ = 0 (51)and then contract the first by u µ and s µ and the secondby u ν s α ε ναµρ eventually getting P s + ( ∇ β − XW + B ) u = 0 (52) P u + ( ∇ β − XW + B ) s − m cos β = 0 (53) P ρ + P ss ρ − P uu ρ + ( ∇ ln φ + R ) µ s α u ν ε µανρ = 0 (54)which are now easier to manipulate since in the last one P ρ appears isolated and the other occurrences P s and
P u can be substituted in terms of the other two expressionsgiven above: if the replacement is made then we obtain P ρ = ( ∇ β − XW + B ) us ρ − ( ∇ β − XW + B ) su ρ ++ m cos βu ρ − ( ∇ ln φ + R ) µ s α u ν ε µανρ (55)in general. Therefore (28, 29) imply (49) in general [29]. C. Velocity
Expression (49) gives the gauge momentum in terms ofthe module and the Yvon-Takabayashi angle, but also interms of the velocity and the spin: because both momen-tum and spin are supposed to be constants of motion, itis interesting to invert it for the velocity. For this, define m cos β − ( ∇ β − XW + B ) k s k = X (56) ( ∇ β − XW + B ) k = Y k (57) − ( ∇ ln φ + R ) k = Z k (58) in terms of which the momentum is written as P a = ( Xη ak + Y k s a + Z i s j ε ijka ) u k (59)in the form of a matrix containing only the spin appliedto the velocity: when inverted it will give the velocity asa product of a specific spin-dependent matrix applied tothe momentum. To get such inversion, have (59) dottedinto Z i s j ε ijka and Z a so to obtain P a Z i s j ε ijka = XZ i s j u a ε ijka ++( Z + | Z · s | ) u k − Z · u ( Z k + Z · ss k ) (60) P · Z + P · sZ · s = XZ · u (61)and after having (61) substituted into (60) and the resultsubstituted back into (59) we finally end up with u k = (1+ Z /X + | Z · s | /X ) − [ η ka ++ s a s k (1+ | Z · s | /X )+ Z a Z k /X ++( s a Z k + Z a s k ) Z · s/X + Z i s j ε ijka /X ] P a /X (62)giving the velocity as product of a specific spin-dependentmatrix further applied onto the momentum in general.More specifically we can introduce also ζ k = Z k /X (63)and write u k = (1+ ζ + | ζ · s | ) − [ η ka ++ s a s k (1+ | ζ · s | )+ ζ a ζ k ++( s a ζ k + ζ a s k ) ζ · s + ζ i s j ε ijka ] P a /X (64)as the most compact form we can have again in general. III. PHYSICAL EFFECTSA. Internal Dynamics
In the previous section, we have introduced the polarform of spinors in dimensions. Of course, one may alsoconsider what the polar form would look like when timeis a parameter, and so in dimensions: in such a case, aspinor would have two complex components, hence realcomponents, and because the spinor transformation lawwould contain only rotations, up to of these compo-nents can be removed. This spinor in polar form wouldhave a single degree of freedom, the module. Therefore,if the non-relativistic spinor ought be obtained as a limitof the general spinor, this limit must account for the factthat the Yvon-Takabayashi angle has to vanish beside thefact that the velocity spatial part has to vanish [25, 27].Because when the spatial part of the velocity is equalto zero but the Yvon-Takabayashi angle is different fromzero we are still unable to obtain the full non-relativisticlimit, then we must conclude that the Yvon-Takabayashiangle describes the motion of what remains even in rest4rame, which has to be the intrinsic motion. That is, it isthe motion describing the internal dynamics of a spinor.We notice that in the definition of the non-relativisticlimit, given by the β → and ~u → above, there appearsno gauge momentum (49) at all: as a matter of fact, themomentum in non-relativistic limit is given by E = m − ( X ~W − ~B ) · ~s (65) ~P = − ( XW − B ) ~s − ( ~ ∇ ln φ − ~R ) × ~s (66)showing that the energy does not reduce to the mass andthe spatial momentum does not vanish. The usual limitgiven by P a → ( m,~ can only be obtained by neglectingthe spin content of the spinor, that is if the macroscopicapproximation is also implemented for field distributions.Again, this makes sense, because non-relativistic limitmeans small spatial momentum only if the momentum isfree from spin contributions, and that is from the internaldynamics. Therefore, it is reasonable that such limit canbe obtained in this way only by requiring that the internaldynamics be concealed inside the field distribution, as itwould normally happen for macroscopic approximations.Having obtained some insight from the non-relativisticlimit, let us see what happens for velocities that are largein general. In this case we can use the expression of thevelocity given by (64) in terms of gauge momentum andspin while depending on module and Yvon-Takabayashiangle in general. Considering that momentum and spinare constants of motion, module and Yvon-Takabayashiangle are the only variables. This tells us that spinorialfields can be seen as very peculiar types of fluid for whichthe velocity depends on density and internal dynamics.Nonetheless, (64) is too complicated to get meaningfulinformation. To simplify, we study limiting cases, and asspecial case we take ζ a to be small so that we have u k ≈ ( η ka + s a s k + ζ i s j ε ijka ) P a /X (67)to the first-order perturbative. Notice that the ζ a poten-tial contains the gradient of the logarithm of the density,so it can be regarded as the de Broglie-Bohm quantumpotential, in the first-order derivative form, and contain-ing also the Yvon-Takabayashi angle contributions.Differently from the non-relativistic case, based on theSchrödinger equation, for which the quantum potential issecond-order derivative, in this most general case, basedon Dirac equations, the quantum potential is first-orderderivative. And differently from the non-relativistic case,where only the module is present, in this most general ofcases, both the module and the Yvon-Takabayashi anglegive some contribution to the quantum potential.As a consequence, we regard the ζ a vector as the quan-tum potential in the most general form possible.If we were to take the approximation of small Y a then m cos βu k ≈ (cid:16) Y b s b m cos β (cid:17) ( η ka + s a s k + ζ i s j ε ijka ) P a (68)and for small Yvon-Takabayashi angle mu k ≈ (1+ Y b s b /m )( P k + P a s a s k − P a ζ i s j ε aijk ) (69) with a contribution scaling the momentum in terms of theYvon-Takabayashi angle plus a contribution changing thedirection of the momentum in terms of both module andYvon-Takabayashi angle. That is, as quantum potential.Considering only the spatial part, cases of small spatialmomentum allow us to better see the effects of the spincontributions. In such cases we have ~u ≈ (1 − ~Y · ~s/m ) ~Z × ~s/m (70)with the spin contributions that are in fact explicit.If torsion were negligible and B a = R a = 0 [28] then ~u ≈ (1+ ~ς · ~ ∇ β/m ) ~ ∇ ln φ × ~ς/m (71)having introduced ~ς = ~s/ as the usual expression of spin.It is interesting to notice that in this case we can com-pute the magnetic moment obtaining the expression ~µ = 12 Z ~r × q ~U dV == q m Z (1+ ~ς · ~ ∇ β/m )[ ~r × ( ~ ∇ φ × ~ς )] dV == q m Z (1+ ~ς · ~ ∇ β/m )2 φ ~ςdV == q m ~ς (cid:16) h ~ς · ~ ∇ β/m i (cid:17) (72)and so that we can finally write ( g − / ≈ h ~ς · ~ ∇ β/m i (73)in terms of the common form of the gyromagnetic factor.To first order, this would agree with the α/ π term ifon average the gradient of β along the spin were equal tothe fine-structure constant α over half Compton length,and this is remarkable since β is generally expected to beof the order of the fine-structure constant α [29].On the other hand, going beyond order-of-magnitudeevaluations requires the Yvon-Takabayashi angle β to beknown in terms of exact solutions and for the time beingthis task appears to be out of the possibilities.More in general, the spin contributions are as in ~u ≈ (1 − ~Y · ~s/m ) ~Z × ~s/m (74)showing that the correction to the magnetic moment willdepend not only on the Yvon-Takabayashi angle but alsoon torsion and on the B a axial vector, altogether collectedinto the Y a axial vector. While the very presence of thisterm depends on the ζ a vector, it is necessary for furthercorrections that the terms X or Y a be present as well.While the ζ a vector is the quantum potential providingquantum mechanical effects, the X or Y a terms are whatprovides quantum field theoretical corrections. B. Chirality
When we consider again the non-relativistic limit, thefirst condition is that of boosting into the rest frame, and5n this frame the assumption of rotating the spinor so toalign its spin along the third axis is equivalent to requirethat in the polar form S be the identity: in this instance ψ = φ e i β e − i β (75)in chiral representation or ψ = φ √ cos β − i sin β (76)in standard representation. The remaining condition de-mands that the Yvon-Takabayashi angle vanishes ψ = φ √ (77)showing that the lower component is zero. This allows asingle condition to represent the non-relativistic limit.When the spinor is written in standard representation,its lower component is for this reason also called smallcomponent. As for the same spinor in chiral representa-tion, the lower component is the right-handed componentand the upper component is the left-handed component.So the Yvon-Takabayashi angle, which is related to thesmall component, gives the phase opposition between theright-handed and left-handed components of spinors.Because zitterbewegung effects are known to arise fromthe existence of the small component, or more in generalfrom the interplay of the chiral components, then we caninfer that zitterbewegung must be linked to the presenceof the Yvon-Takabayashi angle quite generally.It is also interesting to see that the bi-linear quantities Θ and Φ when written in terms of the left-handed and theright-handed components L and R have expressions givenby Θ = i ( L † R − R † L ) and Φ = ( L † R + R † L ) so that, for sucha reason, they assume a remarkably clear interpretation.Considering that the Yvon-Takabayashi angle is relatedto the scalar Θ / Φ while the module is Θ +Φ we interpretthe module and the Yvon-Takabayashi angle as the meanof the chiral components and the standard deviation fromthe mean of the chiral components, respectively.This recovers the interpretation of the module and theYvon-Takabayashi angle as describing an averaged mate-rial distribution with its internal structure. IV. APPLICATIONSA. Anomaly
Having the expression of the velocity (64), an applica-tion of some importance can be found for the electrody- namic potential. This potential has the form V = qU k A k (78)and consequently we can write V = 2 qφ (1+ ζ + | ζ · s | ) − [ P a A a ++ P a s a A k s k (1+ | ζ · s | )+ P a ζ a A k ζ k ++( P a s a A k ζ k + P a ζ a A k s k ) ζ · s ++ P a A k ζ i s j ε ijka ] /X (79)quite straightforwardly. As (64) is the expression of themost general form of the velocity, then it is clear that theabove is the most comprehensive electrodynamic poten-tial density. Its integral over the volume must contain thecomplete information about all electrodynamic effects.In the previous section we have seen that in some ap-proximation, it is possible to find the magnetic momentcorrection (73), which should furnish the correct value inspecific cases of Yvon-Takabayashi angle. Just the same,we also commented that this can be done only when exactsolutions are found, and such task is difficult in general.Nonetheless, since (79) is the electrodynamic potentialdeveloped in its most general form then it should containfull information about all electrodynamic effects.In the simplest from for plane waves it reduces to V pw = 2 qφ P a A a /m (80)which means that ∆ V = V − V pw contains the informationabout all effects arising from general solutions and whichwould not be obtained by analyses employing plane wavesolutions. So ∆ V has the information found in non-zeroYvon-Takabayashi angles or in non-constant modules.That ∆ V may encode electrodynamic effects not ob-tainable with the use of plane waves means that in it wemight find all of the anomalous terms due to the radiativecorrections normally arising from field quantization.This idea is not a new suggestion, as anomalies to thegyromagnetic factor have already been studied by usingaffine structures or spinorial interactions [30, 31]. V. CONCLUSION
In this paper, we have studied the Dirac spinor field inpolar form giving the field equations, combining them asto get the explicit form of the momentum, and invertingit as to obtain the explicit form of the velocity: with suchtools, we established that the Yvon-Takabayashi angle iswhat describes the internal dynamics, defined as relativemotion between chiral parts, that this is connected to theeffects of zitterbewegung, arising from spin contributions,and that both Yvon-Takabayashi angle and module cometogether to form the ζ a vector, which is the most generalform of quantum potential providing quantum mechani-cal effects; we also established that the Yvon-Takabayashiangle takes part in the Y a axial vector, which appears toencode the information about the anomalous terms that6rise as quantum field theory corrections. Consequently,we have conjectured that the anomalous behaviour fieldsdisplay might be obtained by considering solutions thatare more general than the simplest mere plane waves.We have shown that this might happen for the anomalyof the magnetic moment, and we have recalled that in theliterature it has already been discussed that this may alsohappen in the case of the running of coupling constantsfor the renormalization group; other cases were exhibitedin the literature mentioned in the introduction. The idea,expressed also in the introduction, that effects describedby quantization in terms of plane wave solutions shouldalso be described solely in terms of more general solutionshas therefore some evidence, although the actual proof of this conjecture requires exact solutions, which we lack.Our point in the present paper, however, was much lessambitious. What we wanted to do was merely to presenta problem and conjecture a possible way out, that is thatthe problems of quantization may be altogether circum-vented by an approach that does not involve quantizationat all but which recovers its effects in terms of more gen-eral approaches involving fields displaying more generalinternal dynamics, and to this purpose, we provided themost general form for the Dirac spinor field theory.This paper is intended to lay the grounds as a startingpoint for any future work that aims to face the problemof finding general solutions with internal dynamics whichmay recover the effects of quantization. [1] R.F.Streater, A.S.Wightman, PCT, Spin andStatistics, and All That (Princeton, 2000).[2] R.Haag, “On quantum field theories”,
Mat. Fys. Med. , 12 (1955).[3] J.Schwinger, Particles, Sources and Fields (Addison-Wesley, 1989).[4] Z.Koba, “Semi-classical treatment of the reactive correc-tions”,
Prog. Theor. Phys , 319 (1949).[5] T.A.Welton, “Some observable effects of the quantummechanical fluctuations of the electromagnetic field”, Phys. Rev. , 1157 (1948).[6] A.O.Barut, J.P.Dowling, “QED based on self-fields:a relativistic calculation of g-2”, Z. Naturforsch. A , 1051 (1989).[7] A.O.Barut, J.P.Dowling, “Quantum electrodynamicsbased on self-energy, without second quantization:the Lamb shift and long-range Casimir-Poldervan der Waals forces near boundaries”, Phys. Rev. A , 2550 (1987).[8] A.O.Barut, J.Kraus, Y.Salamin, N.Unal, “Relativistictheory of the Lamb shift in self-field quantum electrody-namics”, Phys. Rev. A , 7740 (1992).[9] A.O.Barut, J.Kraus, “Non-perturbative quantum electro-dynamics: the Lamb shift”, Found. Phys. , 189 (1983).[10] A.O.Barut, J.F.Van Huele, “Quantum electrodynamicsbased on self-energy: Lamb shift and spontaneousemission without field quantization”, Phys. Rev. A , 3187 (1985).[11] A.O.Barut, Y.I.Salamin, “Relativistic Theory of Sponta-neous Emission”, Phys. Rev. A , 2284 (1988).[12] I.Acikgoz, A.O.Barut, J.Kraus, N.Unal, “Self-field quan-tum electrodynamics without infinities. A new calcula-tion of vacuum polarization”, Phys.Rev.A ,126(1995).[13] D.Hestenes, “The Zitterbewegung interpretation of quan-tum mechanics”,
Found. Phys. , 1213 (1990).[14] G.Salesi, E.Recami, “About the kinematics of spinningparticles”, Adv. Appl. Clifford Algebras , S253 (1997).[15] E.Recami, G.Salesi, “Kinematics and hydrodynamics ofspinning particles”, Phys. Rev. A , 98 (1998).[16] P.Lounesto, Clifford Algebras and Spinors (Cambridge University Press, 2001).[17] R.T.Cavalcanti, “Classification of Singular SpinorFields and Other Mass Dimension One Fermions”,
Int.J.Mod.Phys.D , 1444002 (2014).[18] J.M.Hoff da Silva, R.da Rocha, “Unfolding Physics fromthe Algebraic Classification of Spinor Fields”, Phys. Lett. B , 1519 (2013).[19] R.Abłamowicz, I.Gonçalves, R.da Rocha, “Bilinear Co-variants and Spinor Fields Duality in Quantum CliffordAlgebras”,
J. Math. Phys. , 103501 (2014).[20] R.da Rocha, R.T.Cavalcanti, “Flag-dipole and flagpolespinor fluid flows in Kerr spacetimes”, Phys. Atom. Nucl. , 329 (2017).[21] R.da Rocha, J.M.Hoff da Silva, “ELKO, flagpole and flag-dipole spinor fields, and the instanton Hopf fibration”, Adv. Appl. Clifford Algebras , 847 (2010).[22] C.H.Coronado Villalobos, J.M.Hoff da Silva, R.da Rocha,“Questing mass dimension 1 spinor fields”, Eur.Phys.J.C , 266 (2015).[23] R.T.Cavalcanti,J.M.Hoff da Silva,R.da Rocha,“VSR sym-metries in the DKP algebra: the interplay between Diracand Elko spinor fields”, Eur.Phys.J.Plus , 246 (2014).[24] 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).[25] L.Fabbri, “A generally-relativistic gaugeclassification of the Dirac fields”, Int.J.Geom.Meth.Mod.Phys. ,1650078(2016).[26] D.Hestenes, “Real Spinor Fields”, J.Math.Phys. , 798 (1967).[27] L.Fabbri, “Torsion Gravity for Dirac Fields”, Int.J.Geom.Meth.Mod.Phys. ,1750037(2017).[28] L.Fabbri, “General Dynamics of Spinors”, Adv. Appl. Clifford Algebras , 2901 (2017).[29] L.Fabbri, “Covariant inertial forces for spinors”, Eur.Phys.J.C , 783 (2018).[30] S.Capozziello, D.J.Cirilo-Lombardo, M.De Laurentis,“The Affine Structure of Gravitational Theories:Symplectic Groups and Geometry”, Int.J.Geom.Meth.Mod.Phys. , 1450081 (2014).[31] S.Capozziello, D.J.Cirilo-Lombardo, A.Dorokhov,“Fermion interactions, cosmological constant and space-time dimensionality in an unified approach based onaffine geometry”, Int.J.Theor.Phys. , 3882 (2014)., 3882 (2014).