Ultrarelativistic Decoupling Transformation for Generalized Dirac Equations

TL;DR

This paper develops an ultrarelativistic decoupling transformation for generalized Dirac equations, revealing effective gravitational interactions for high-energy particles, including tachyons, and providing insights into their relativistic limits.

Contribution

It introduces a new ultrarelativistic decoupling transformation for Dirac equations, applicable to tachyons and high-energy particles, expanding understanding of their gravitational interactions.

Findings

Effective gravitational attraction for tachyons at high energy

Ultrarelativistic limit characterized by mass and potential perturbations

Transformation applicable to free Dirac particles and tachyons

Abstract

The Foldy--Wouthuysen transformation is known to uncover the nonrelativistic limit of a generalized Dirac Hamiltonian, lending an intuitive physical interpretation to the effective operators within Schr\"{o}dinger--Pauli theory. We here discuss the opposite, ultrarelativistic limit which requires the use of a fundamentally different expansion where the leading kinetic term in the Dirac equation is perturbed by the mass of the particle and other interaction (potential) terms, rather than vice versa. The ultrarelativistic decoupling transformation is applied to free Dirac particles (in the Weyl basis) and to high-energy tachyons, which are faster-than-light particles described by a fully Lorentz-covariant equation. The effective gravitational interactions are found. For tachyons, the dominant gravitational interaction term in the high-energy limit is shown to be attractive, and equal to the leading term for subluminal Dirac particles (tardyons) in the high-energy limit.