Symplectic Dirac Equation

TL;DR

This paper develops a phase space formulation of quantum mechanics using symplectic Dirac equations, deriving key relativistic equations and exploring electromagnetic interactions within this framework.

Contribution

It introduces a symplectic approach to Dirac equations in phase space, linking noncommutative star-product structures with relativistic quantum theory.

Findings

Derived phase space Klein-Gordon and Dirac equations.

Established a consistent physical theory using group theory and star-product formalism.

Analyzed Dirac equation with electromagnetic interaction in phase space.

Abstract

Symplectic unitary representations for the Poincar\'{e} group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Klein-Gordon and Dirac equations are derived in phase space. As an application, we study the Dirac equation with electromagnetic interaction in phase space.