# Torsion axial vector and Yvon-Takabayashi angle: zitterbewegung, chirality and all that

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## Abstract

We consider propagating torsion as a completion of gravitation in order to describe the dynamics of curved-twisted space-times filled with Dirac spinorial fields; we discuss interesting relationships of the torsion axial vector and the curvature tensor with the Yvon-Takabayashi angle and the module of the spinor field, that is the two degrees of freedom of the spinor field itself: in particular, we shall discuss in what way the torsion axial vector could be seen as the potential of a specific interaction of the Yvon-Takabayashi angle, and therefore as a force between the two chiral projections of the spinor field itself. Chiral interactions of the components of a spinor may render effects of zitterbewegung, as well as effective mass terms and other related features: we shall briefly sketch some of the analogies and differences with the similar but not identical situation given by the Yukawa interaction occurring in the Higgs sector of the standard model. We will provide some overall considerations about general consequences for contemporary physics, consequences that have never been discussed before, so far as we are aware, in the present physics literature.

## 1 Introduction

The Dirac spinorial field theory has the spinor field as fundamental ingredient, defined in terms of its spinorial transformation law: the classical Dirac spinor is a column of 4 complex scalar fields, carrying spin-\(\frac{1}{2}\) representations of the Lorentz group; this means that the Dirac spinor has in total 8 real components, of which 6 removable by a Lorentz transformations, so that it has 2 real degrees of freedom alone [1, 2, 3]. Essentially, the idea is that the 6 removable components are the 3 components of the space velocity, removable by boosts, and the 3 components of the spin, removable by rotations: the two remaining components are the physical degrees of freedom and they are the module and what in literature is known as the Yvon-Takabayashi angle [4, 5, 6]. On the other hand, however, in any treatment of modern physics, like QFT for instance, after having boosted into the rest frame and after having rotated the spinor so to have it aligned along the third axis, the final form of the spinor field is a constant.

This apparent discrepancy comes from the fact that in modern treatments all we use are plane waves, for which the module is constant and the YT angle is zero [7]; still, we know that in the most general of cases there must be a varying module and a non-zero YT angle. So in general we know that a given quantity is present, although in the commonly accepted paradigms it is always set to zero.

A similar situation occurs when the Dirac theory is endowed with a dynamics, that is the Dirac spinor field is seen as solution of the Dirac equation and with conserved quantities as source of geometric field equations: a Dirac spinor field possesses three conserved quantities, that is the current and energy density as well as the spin density.

On the other hand, the geometric field equations come from the set-up of a geometrical background, and in general it is possible to see electrodynamics emerging as a gauge theory of the unitary phase transformation similarly to the fact that gravity emerges from the non-trivial structure of the space-time: when the background is constructed in its full generality, we witness the appearance of gauge strengths beside curvature as well as torsion [8].

Like in Maxwell theory one takes the charge as source of electrodynamic field equations, analogously in Einstein gravitation one takes the energy as source of gravitational field equations and in the same spirit one should take the spin as source of torsion field equations. But again in the usual treatment, as in the case of QFT, this is never done.

An apparent discrepancy emerges once again; and once again we may claim that in the most general case the spin exists and should be coupled, and it is natural to have it coupled to the torsion of the space-time. This is a second instance in which in general some dynamics is to be given although in common paradigms it is always absent.

As we will have the opportunity to discuss later on, the YT angle will indeed be connected to the spin of a spinor field, and henceforth there will be relationships between the YT angle and the torsion. This connection might well justify why in the normal paradigms neglecting both can be equivalent to neglecting only one of these two.

Still, even neglecting one of them is an artificial constraint, and it is interesting to study what happens when, in the most general circumstance, both torsion and YT angle are allowed to take place in the dynamics.

This is what we will do next in this paper.

## 2 Geometry of the spinor fields

^{1}implicitly. We use the metric to raise/lower Greek indices, tetrads to change Greek into Latin indices and the Minkowskian matrix to raise/lower Latin indices; procedure \(\overline{\psi }=\psi ^{\dagger }\varvec{\gamma }_{0}\) is used to pass from spinors to conjugate spinors in such a way that

^{2}given in terms of the velocity \(\vec {v}\) while \(\xi \) such that \(\xi ^{\dagger }\xi =1\) is some arbitrary semi-spinor field and \(\alpha \) is a generic unitary phase.

*q*and called spinorial connection. Remark that for now we have only defined the torsionless connection.

*R*Ricci scalar and \(F_{\mu \nu }\) Faraday tensor, and where

*X*is the strength of the interaction between torsion and the spin of spinor fields while

*M*and

*m*are the mass of torsion and the spinor field itself. Having defined the connection in the torsionless case it would seem we went in a case of restricted generality, but in reality we are still in the most general situation even if the connection has no torsion so long as torsion is eventually included in the form of a supplementary massive axial vector field [16].

This shows how module and YT angle link to curvature and torsion; at times this becomes a pure link between module and curvature and between YT angle and torsion.

In what follows we will proceed to deepen this link.

## 3 Torsion axial vector and Yvon-Takabayashi angle

*F*some function that vanishes when the YT angle vanishes, one can maintain the above effects of gravity while having

*F*in terms of the YT angle while (56) is the field equation describing how the torsion axial vector affects the YT angle dynamics.

Therefore the torsion axial vector is the potential of an interaction of the YT angle and, since this describes the relative motions of chiral components of the spinor field, then the torsion axial vector gives rise to a force between the chiral components of the spinorial field itself.

Because the YT angle appears as the argument of circular functions then we have to expect that some type of non-linearity will arise for the dynamics [21].

## 4 Zitterbewegung

For a given spinor the chiral components are given by the left-handed and right-handed projections designated by the *L* and *R* spinors,^{3} and when the spinor is taken in the spin-up eigenstate it means that both *L* and *R* have the same spin; however, they have opposite chirality: so, they have opposite projections of the velocity. In taking care of seeing the dynamics of *L* and *R* then, we observe that these two spinors are not a stable system unless some factor intervenes for which the velocity of the irreducible chiral parts is inverted; maintaining the spin unchanged, this can be done by transmuting *L* into *R* and viceversa and in no other way. What this implies is that the torsion must switch *L* and *R* with one another, and therefore it must have the character of a scalar or pseudo-scalar field.

Although torsion is an axial vector, the fact that it is a massive field, thus subject to no gauge transformation, implies that the torsion pseudo-scalar component cannot be transformed away, and such a torsional pseudo-scalar part is exactly the one isolated by the partially-conserved axial-vector current: the torsion pseudo-scalar is thus the torsion axial vector divergence. As a matter of fact any pseudo-scalar or scalar would give similar dynamics, but the torsion need not be assumed as it is already present.

With torsion, and specifically its pseudo-scalar degree of freedom, giving an interaction between *L* and *R* parts, we should expect that *L* and *R* display some oscillatory behaviour, which could be seen, in the momentum space, as an oscillation along the third axis of two point-like particles, and in the position space, as an oscillation around the third axis of two plane-fronted waves in general.

This stability condition \(M^{4}>8X^{2}\phi ^{2}m\) is certainly true for small values of the constant \(m\phi ^{2}\) and thus for any light enough particle or for small densities of the matter field.

According to the standard wisdom, quantum field theory is known to have a cut-off beyond which new physics should enter into consideration, and what in the present analysis we seem to witness is that such a cut-off may be given by the torsion mass with the new physics entailed by the dynamical effects due to the YT angle.

Stability condition \(M^{2}>4mXWs\) is certainly true for torsion with large mass or small coupling to spinors but also when \(XWs<0\) and therefore when the coupling of torsion to spinor fields is universally attractive.

In such case \(\mu \approx 2\sqrt{m}\sqrt{|XWs|}\) showing that the larger is the coupling the larger is the frequency of oscillation.

Notice that if the torsion-spin coupling is attractive no singular value is met, and therefore the expression for the effective Yvon-Takabayashi mass is always regular.

Of course, one should not limit oneself to this simplest solution, but discussing whether such a condition would hold in general would require solving the entire system of field equations, which is an extremely complicated procedure in general [22], and we leave it to future works.

Then, the small component is also related to the effects of zitterbewegung. As a consequence, the YT angle itself is related to the presence of zitterbewegung effects.

And zitterbewegung phenomena should be expected in situations where \(\beta \) has oscillatory behaviour [23].

## 5 Chirality

The picture that has emerged is one for which a spinor, albeit fundamental, is not irreducible but instead can be decomposed in terms of *L* and *R* spinors, themselves irreducible, and such that they describe two matter distributions having opposite velocities, and whenever stability conditions are valid they display oscillatory motion; these stability conditions involve some requirement on the torsion dynamics, which is seen as the force acting between the two chiral components, and which induces their relative oscillation. In this way, torsion can be interpreted as the force that prevents the chiral components to separate.

Nevertheless, the splitting of regular spinor fields can be useful for a further physical analysis of Eq. (76) and the equations that follow, an analysis that up to now has only resided in formal aspects in the literature.

*N*mass of the Higgs and

*Y*being the Yukawa coupling and having neglected higher-order Higgs terms.

It is very important now to mention that Eq. (84) for type-3 regular spinor fields is led to the Klein–Gordon equation with no potential; the effective approximation given by (85) yields \(N^{2}H\approx 0\) and (86) yields the standard Dirac equation for type-3 spinor fields. This shows that the type-3 regular spinor fields would simply not admit a coupling to the Higgs field, mining its place in the Standard Model. But on the other hand, type-1 and type-2 regular spinor fields do, as in these cases the non-linear term in \(\Phi Y^{2}/N^{2}\) does remain non-trivial. The Standard Model coupling to the Higgs field is an interaction that physically distinguishes between type-3 and other classes of regular spinor fields in Lounesto classification.

Because both torsion and the Higgs field have effective limit given by the NJL model, giving rise to an attraction between the chiral parts, then we may conclude that both torsion and the Higgs field can be seen as the mediators of the interaction in terms of which the chiral parts might form bound-states granting the stability of spinor fields.

This last form also shows that an even richer dynamics can be possible if we do not take into account effective approximations, considering torsion as propagating.

For some general treatment about the dynamics of the torsionally-induced axial-vector currents we refer to [27].

As a consequence of this analysis, we have all elements to build an analogy between the above mentioned prototypical version of the strong force and our model in the effective approximation. No effective approximation might be considered and in this case the analogy would have to be built with the present-day description of strong interactions. The picture that emerges is one for which we may see any spinor as a chiral bound state due to torsion attractive mediation in the same way in which we see any baryon as a bound state for gluon attractive mediation.

This picture is quite simple and rather intuitive, which is the reason why we think it is appealing. But aside from the theoretical value, it may be used to infer one possible effect that could be of phenomenological importance.

In the following we will proceed to discuss it.

## 6 Discussion

Phenomenologically, the effects of this supplementary torsionally-induced attraction depend on the specific values of the torsion coupling constant, and even more on the torsion mass; however, a question we might already ask is whether there are systems of fermions where some additional attraction might play a role. To this question, a first answer that comes in mind is the recently increased tension existing between the theoretically calculated and experimentally established radius of the proton [28].

Because theoretical computations over-estimate the radius of the proton, we think it is quite natural to ask the question about whether such an extra binding force could actually give rise to corrections that bring the theoretical result back to be compatible with the observed value and consequently releasing the tension with experiments.

This question may have an answer that is much closer than what we think new physics could be because, despite torsion would indeed be new physics compared to normal paradigms, nonetheless torsion is already part of the most general geometric background of field theories.

## 7 Conclusion

In this paper, we have studied how the torsion massive axial vector field relates to the YT angle, showing in what way the divergence of the torsion massive axial vector has the role of an attractive force between the two chiral parts binding them into pseudo-scalar states; thus we may see a spinor as a chiral bound state due to torsion like we see a baryon as a bound state due to gluons: this analogy is justified by the fact that in the effective limit our model reduces to the Nambu–Jona–Lasinio model. A difference with the NJL model is that the torsion is already massive, and thus it has partially-conserved axial-vector currents, those giving rise to the pseudo-scalar bound state, as just mentioned; this is important because in our model spinor fields can be massive even without the quantum anomaly introduced by radiative processes. Some phenomenological effect of this extra binding force can be sought into a correction diminishing the theoretical value of the radius of the proton releasing the tension with measurements.

With no electrodynamics, approximating gravity away allowed us to find for the torsion massive axial vector a ground state in which the YT angle had oscillations when certain stability conditions were valid: this oscillation can be interpreted as the oscillation of the chiral parts about their equilibrium configuration. From a rather figurative perspective such an oscillation is precisely what we would have to expect as a consequence of the zitterbewegung.

Opportunities for investigating more general dynamics might involve releasing the assumption of small YT angle, therefore allowing also a non-constant module. Also, one might proceed in studying non-trivial gravitational fields.

Of course the most general case would be to allow electrodynamics and see what happens in this situation.

We postpone this treatment to following works.

## Footnotes

## Notes

### Acknowledgements

Roldão da Rocha is grateful to CNPq (Grant No. 303293/2015-2) and to FAPESP (Grant No. 2017/18897-8) for partial financial support.

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