Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Tachyonic Dirac Equation Revisited

Version 1 : Received: 13 May 2020 / Approved: 14 May 2020 / Online: 14 May 2020 (11:38:12 CEST)

How to cite: Nanni, L. Tachyonic Dirac Equation Revisited. Preprints 2020, 2020050236 (doi: 10.20944/preprints202005.0236.v1). Nanni, L. Tachyonic Dirac Equation Revisited. Preprints 2020, 2020050236 (doi: 10.20944/preprints202005.0236.v1).

## 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.

## Keywords

Dirac equation; tachyon; non-Hermitian operator; superluminal Lorentz transformations; CPT symmetry

## Subject

PHYSICAL SCIENCES, General & Theoretical Physics

Comment 1
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The commenter has declared there is no conflict of interests.
Comment: I have read your paper with interest. I like the way to have put up the Dirac wavefunction in Eq. (9) and (10) as a 4x4 wavefunction. Except in the work I have done myself, in all my life, I have never seen the Dirac wavefunction written in the manner you have written. I want direct you to my latest work this @ https://doi.org/10.13140/RG.2.2.35874.56008. You must realize that, all those four solutions can be lumped up as a solution representing a single particle. I am sure that if you infuse this idea into your theory, you will have something interesting. I must say that --- you have had a glimpsed at the future of the Dirac wavefunction!
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