Group Theoretic Approach to Fermion Production

TL;DR

This paper introduces a universal group theoretic framework for describing fermion production via various interactions, simplifying calculations and providing a geometric interpretation, especially demonstrated for pseudo-scalar derivative coupling.

Contribution

It develops a novel $SU(2) imes U(1)$ group theoretic approach that unifies fermion production mechanisms and simplifies the expression for particle number density.

Findings

Universal equation of motion for fermion production derived

Simplified geometric interpretation of fermion dynamics provided

Application demonstrated for pseudo-scalar derivative coupling

Abstract

We propose a universal group theoretic description of the fermion production through any type of interaction to scalar or pseudo-scalar. Our group theoretic approach relies on the group $SU(2) \times U(1)$, corresponding to the freedom in choosing representations of the gamma matrices in Clifford algebra, under which a part of the Dirac spinor function transforms like a fundamental representation. In terms of a new $SO(3)$ ($\sim SU(2)$) vector constructed out of spinor functions, we show that fermion production mechanism can be analogous to the classical dynamics of a vector precessing with the angular velocity. In our group theoretic approach, the equation of motion takes a universal form for any system, and choosing a different type of interaction or a different basis amounts to selecting the corresponding angular velocity. The expression of the particle number density is greatly simplified, compared to the traditional approach, and it provides us with a simple geometric interpretation of the fermion production dynamics. For the purpose of the demonstration, we focus on the fermion production through the derivative coupling to the pseudo-scalar.