Remark on charge conjugation in the non relativistic limit

TL;DR

This paper investigates the non-relativistic limit of charge conjugation in the Dirac equation, showing it remains well-defined and interpretable within Galilean relativity and complexified Lorentz groups, with implications for symmetry structures.

Contribution

It provides a detailed analysis of charge conjugation's non-relativistic limit, demonstrating its existence and form within Galilean relativity and complexified Lorentz groups, extending symmetry understanding.

Findings

Charge conjugation limit exists in non-relativistic regime.

The symmetry group is isomorphic to a semidirect sum of dihedral group and Z2.

Charge conjugation acts as complex conjugation of spatial coordinates.

Abstract

We study the non relativistic limit of the charge conjugation operation $\cal C$ in the context of the Dirac equation coupled to an electromagnetic field. The limit is well defined and, as in the relativistic case, $\cal C$, $\cal P$ (parity) and $\cal T$ (time reversal) are the generators of a matrix group isomorphic to a semidirect sum of the dihedral group of eight elements and $\Z_2$. The existence of the limit is supported by an argument based in quantum field theory. Also, and most important, the limit exists in the context of galilean relativity. Finally, if one complexifies the Lorentz group and therefore the galilean spacetime $x_\mu$, then the explicit form of the matrix for $\cal C$ allows to interpret it, in this context, as the complex conjugation of the spatial coordinates: $\vec{x} \to \vec{x}^*$. This result is natural in a fiber bundle description.