New wave equation for ultrarelativistic particles

TL;DR

This paper introduces a new wave equation for ultrarelativistic particles that incorporates spin, reproduces key results of Dirac theory, and allows testing Lorentz invariance violations at high energies.

Contribution

It presents a novel wave equation derived from first principles, including spin naturally, and capable of describing particles with potential Lorentz invariance violations.

Findings

Reproduces standard Dirac results in high-momentum limit

Contains plane wave solutions similar to neutrino oscillations

Allows testing Lorentz invariance violations at high energies

Abstract

Starting from first principles and general assumptions based on the energy-momentum relation of the Special Theory of Relativity we present a novel wave equation for ultrarelativistic matter. This wave equation arises when particles satisfy the condition, $p>>m$, i.e, when the energy-momentum relation can be approximated by, $E\simeq p+\frac{m^{2}}{2p}$. Interestingly enough, such as the Dirac equation, it is found that this wave equation includes spin in a natural way. Furthermore, the free solutions of this wave equation contain plane waves that are completely equivalent to those of the theory of neutrino oscillations. Therefore, the theory reproduces some standard results of the Dirac theory in the limit $p>>m$, but offers the possibility of an explicit Lorentz Invariance Violation of order, $\mathcal{O}((mc)^{4}/p^{2})$. As a result, the theory could be useful to test small departures from Dirac equation and Lorentz Invariance at very high energies. On the other hand, the wave equation can also describe particles of spin 1 by a simple substitution of the spin operators, $\boldsymbol{\sigma}\rightarrow\boldsymbol{\alpha}$. In addition, it naturally admits a Lagrangian formulation and a Hamiltonian formalism. We also discuss the associated conservation laws that arise through the symmetry transformations of the Lagrangian.