Derivation of equations for scalar and fermion fields using properties of dispersion-codispersion operators

TL;DR

This paper derives equations for scalar and fermion fields using a phase space quantum representation based on dispersion-codispersion operators, offering a novel operator-based approach to fundamental relativistic equations.

Contribution

It introduces a new method employing dispersion-codispersion operators to derive Klein-Gordon and Dirac equations from a phase space perspective.

Findings

Derived scalar field equation using second order operator relation.

Derived fermion field equation using first order operator relation.

Established a novel operator-based framework for relativistic quantum equations.

Abstract

We establish equations for scalar and fermion fields using results obtained from a study on a phase space representation of quantum theory that we have performed in a previous work. Our approaches are similar to the historical ones to obtain Klein-Gordon and Dirac equations but the main difference is that ours are based on the use of properties of operators called dispersion-codispersion operators. We begin with a brief recall about the dispersion-codispersion operators. Then, introducing a mass operator with its canonical conjugate coordinate and applying rules of quantization, based on the use of dispersion - codispersion operators, we deduce a second order differential operator relation from the relativistic expression relying energy, momentum and mass. Using Dirac matrices, we derive from this second order differential operator relation a first order one. The application of the second order differential operator relation on a scalar function gives the equation for the scalar field and the use of the first order differential operator relation leads to the equation for fermion field.