Asymmetric particle-antiparticle Dirac equation: first quantization

TL;DR

This paper introduces an asymmetric Dirac equation where particles and antiparticles with the same wave number have different energies and momenta, demonstrating Lorentz covariance and exploring its implications for particle properties and hydrogen atom solutions.

Contribution

It presents the derivation of a new asymmetric Dirac equation, analyzes its covariance properties, and connects it to the standard Dirac equation, expanding the theoretical framework of relativistic quantum mechanics.

Findings

The asymmetric Dirac equation is Lorentz covariant under proper transformations.

It reproduces standard Dirac predictions with parameter adjustments.

The hydrogen atom is solved within this new framework.

Abstract

We derive a Dirac-like equation, the asymmetric Dirac equation, where particles and antiparticles sharing the same wave number have different energies and momenta. We show that this equation is Lorentz covariant under proper Lorentz transformations (boosts and spatial rotations) and also determine the corresponding transformation law for its wave function. We obtain a formal connection between the asymmetric Dirac equation and the standard Dirac equation and we show that by properly adjusting the free parameters of the present wave equation we can make it reproduce the predictions of the usual Dirac equation. We show that the rest mass of a particle in the theoretical framework of the asymmetric Dirac equation is a function of a set of four parameters, which are relativistic invariants under proper Lorentz transformations. These four parameters are the analog to the mass that appears in the standard Dirac equation. We prove that in order to guarantee the covariance of the asymmetric Dirac equation under parity and time reversal operations (improper Lorentz transformations) as well as under the charge conjugation operation, these four parameters change sign in exactly the same way as the four components of a four-vector. The mass, though, being a function of the square of those parameters remains an invariant. We also extensively study the free particle plane wave solutions to the asymmetric Dirac equation and derive its energy, helicity, and spin projection operators as well as several Gordon's identities. The hydrogen atom is solved in the present context after applying the minimal coupling prescription to the asymmetric Dirac equation, which also allows us to appropriately obtain its non-relativistic limit.