Considerations concerning the generalization of the Dirac equations to unstable fermions

TL;DR

This paper extends the Dirac equations and spinors to describe unstable fermions, ensuring Lorentz covariance, and explores their properties, including probability density and time reversal behavior.

Contribution

It introduces a generalized formulation of Dirac equations and spinors for unstable fermions, maintaining Lorentz covariance and relating to the stable case.

Findings

Derived generalized adjoint Dirac equations and spinors.

Established the relation between unstable and stable fermion cases.

Evaluated probability density for unstable fermions.

Abstract

We discuss the generalization of the Dirac equations and spinors in momentum space to free unstable spin-$1/2$ fermions taking into account the fundamental requirement of Lorentz covariance. We derive the generalized adjoint Dirac equations and spinors, and explain the very simple relation that exists, in our formulation, between the unstable and stable cases. As an application of the generalized spinors, we evaluate the probability density. We also discuss the behavior of the generalized Dirac equations under time reversal.