The Integrability of Pauli System in Lorentz Violating Background

TL;DR

This paper investigates the integrability of a Pauli system influenced by Lorentz-violating backgrounds, analyzing how such violations affect constants of motion and the system's integrability in both two- and three-dimensional cases.

Contribution

It systematically examines the impact of Lorentz-violating terms on the integrability of the Pauli system at the non-relativistic level, identifying conditions for constants of motion.

Findings

Lorentz-violating terms can alter the system's integrability.

Spin-orbit interaction appears as a higher-order correction.

Different background couplings affect the existence of constants of motion.

Abstract

We systematically analyze the integrability of a Pauli system in Lorentz violating background at the non-relativistic level both in two- and three-dimensions. We consider the non-relativistic limit of the Dirac equation from the QED sector of the so-called Standard Model Extension by keeping only two types of background couplings, the vector a_mu and the axial vector b_mu. We show that the spin-orbit interaction comes as a higher order correction in the non-relativistic limit of the Dirac equation. Such an interaction allows the inclusion of spin degree non-trivially, and if Lorentz violating terms are allowed, they might be comparable under special circumstances. By including all possible first-order derivative terms and considering the cases a\ne 0, b\ne 0, and b_0\ne 0 one at a time, we determine the possible forms of constants of motion operator, and discuss the existence or continuity of integrability due to Lorentz violating background.