Spin operator and spin states in Galilean covariant Fermi field theories

TL;DR

This paper investigates the properties of spin operators in Galilean covariant Fermi fields across different dimensions, revealing non-conservation of Dirac spin in higher dimensions and its conservation upon reduction, with implications for nonrelativistic quantum theories.

Contribution

It analyzes the behavior of spin operators in Galilean covariant Dirac fields, clarifies their conservation properties, and constructs spin states in nonrelativistic limits, providing a generalized framework.

Findings

Dirac spin is not conserved in (4+1) dimensions due to non-Galilean boosts.

Upon reduction to (3+1) dimensions, the Dirac spin becomes conserved.

Constructed one-particle spin states and discussed particle-antiparticle systems.

Abstract

Spin degrees of freedom of the Galilean covariant Dirac field in (4+1) dimensions and its nonrelativistic counterpart in (3+1) dimensions are examined. Two standard choices of spin operator, the Galilean covariant and Dirac spin operators, are considered. It is shown that the Dirac spin of the Galilean covariant Dirac field in (4+1) dimensions is not conserved, and the role of non-Galilean boosts in its nonconservation is stressed out. After reduction to (3+1) dimensions the Dirac field turns into a nonrelativistic Fermi field with a conserved Dirac spin. A generalized form of the Levy-Leblond equations for the Fermi field is given. One-particle spin states are constructed. A particle-antiparticle system is discussed.