Tachyonic Dirac Equation Revisited

TL;DR

This paper compares two theoretical formulations of the tachyonic Dirac equation, highlighting their differences in symmetry properties and physical implications, and concludes that current theories cannot fully describe tachyonic neutrinos covariantly.

Contribution

It provides a detailed analysis of two distinct approaches to formulating the tachyonic Dirac equation, clarifying their mathematical and physical differences.

Findings

First approach violates parity and charge conjugation invariance.

Second approach is invariant under parity and charge conjugation.

Neither approach fully describes tachyonic neutrinos covariantly.

Abstract

In this paper, we revisit the two theoretical approaches for the formulation of the tachyonic Dirac equation. The first approach works within the theory of restricted relativity, starting from a Lorentz invariant Lagrangian consistent with a spacelike four-momentum. The second approach uses the theory of relativity extended to superluminal motions and works directly on the ordinary Dirac equation through superluminal Lorentz transformations. The equations resulting from the two approaches show mostly different, if not opposite, properties. In particular, the first equation violates the invariance under the action of the parity and charge conjugation operations. Although it is a good mathematical tool to describe the dynamics of a space-like particle, it also shows that the mean particle velocity is subluminal. In contrast, the second equation is invariant under the action of parity and charge conjugation symmetries, but the particle it describes is consistent with the classical dynamics of a tachyon. This study shows that it is not possible with the currently available theories to formulate a covariant equation that coherently describes the neutrino in the framework of the physics of tachyons, and depending on the experiment, one equation rather than the other should be used.