Asymmetric particle-antiparticle Dirac equation: second quantization

TL;DR

This paper develops a relativistic quantum field theory based on asymmetric Dirac fields, where particles and antiparticles have different energies, yet the theory remains consistent and empirically equivalent to standard QED, offering new insights into matter-antimatter asymmetry.

Contribution

It introduces an asymmetric Dirac equation and constructs a consistent, renormalizable QED framework where particles and antiparticles differ in energy and momentum, expanding the standard model.

Findings

Particles and antiparticles with the same wave number have different energies and momenta.

The asymmetric Dirac QED is empirically equivalent to standard QED.

The theory respects CPT symmetry and offers potential explanations for matter-antimatter asymmetry.

Abstract

We build the fully relativistic quantum field theory related to the asymmetric Dirac fields. These fields are solutions of the asymmetric Dirac equation, a Lorentz covariant Dirac-like equation whose positive and "negative" frequency plane wave solutions' dispersion relations are no longer degenerate. At the second quantization level, we show that this implies that particles and antiparticles sharing the same wave number have different energies and momenta. In spite of that, we prove that by properly fixing the values of the relativistic invariants that define the asymmetric Dirac free field Lagrangian density, we can build a consistent, fully relativistic, and renormalizable quantum electrodynamics (QED) that is empirically equivalent to the standard QED. We discuss the reasons and implications of this non-trivial equivalence, exploring qualitatively other scenarios in which the asymmetric Dirac fields may lead to beyond the standard model predictions. We conjecture that this non-degeneracy in the energies for particles and antiparticles may lead to a fully relativistic understanding of the asymmetry between matter and antimatter in the present day universe as well as to an alternative way of modeling the gravitational interaction between a particle and an antiparticle. We give a complete account of how the asymmetric Dirac fields and the corresponding annihilation and creation operators transform under improper Lorentz transformations (parity and time reversal operations) and under the charge conjugation operation. We also prove that the present theory respects the CPT theorem.