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

Abstract

We consider propagating torsion as a completion of gravitation in order to describe the dynamics of curvedtwisted 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.

Constraints on mapping the Lounesto’s classes

“…Remarkably enough, as one can see in Ref [6], class 6 spinors necessarily have or φ R = 0 or φ L = 0, evincing, thus, the impossibility (independent of the choice of the phases) to map class 6 into any other class. We emphasize, however, that a single-helicity spinor can never be regarded as a singular spinor, note that, for the algebraic spinor displayed in (21), the computation of the bilinears σ and ω provides σ = αβ * + α * β, (27) and…”

Covariant inertial forces for spinors

“…with u a the velocity vector and s a the spin axial-vector and where φ is a scalar and β is a pseudo-scalar known as module and Yvon-Takabayashi angle (we stress that the name Takabayashi can sometimes be spelled Takabayasi). One can easily prove that the directions are such that u a u a = −s a s a = 1 (7) u a s a = 0 (8) and we notice that the velocity vector has only the temporal component while the spin axial-vector has only the third component when the polar form (1) is taken in the case in which S is the identity: therefore the module and the Yvon-Takabayashi angle are the only two real degrees of freedom as is most manifest when the polar form with choice S = I is taken into account for the spinorial field.…”

“…specifying second-order derivatives of the gauge potentials, torsion and tetrad fields: these are the field equations that couple electrodynamics, torsion and gravity to the currents, spin and energy densities, respectively [8].…”

“…In [7] we have for the first time pointed out that, when the polar reduction is performed, some background quantities arise, which can be used, with the spin connection, to construct objects of a very specific type: these objects are built only in terms of entities arising from local frame and general coordinate transformations, and as such they should behave like connections, but nonetheless they have been proven to be tensors. In [8] we have discussed these tensors in one specific case involving electrodynamics and torsion-gravity. In this paper we deepen the discussion.…”

On the Mathematics of Coframe Formalism and Einstein–Cartan Theory—A Brief Review

“…The proof comes from the observation that Equations (19) and (30) coincide in π −1 (U β ) for both a horizontal (for which they are zero) and a vertical vector field.…”