Proper relativistic position operators in 1+1 and 2+1 dimensions

TL;DR

This paper investigates relativistic position operators in 1+1 and 2+1 dimensions, revealing their roles in conserved quantities and Lorentz generators within Dirac theory using covariant representations.

Contribution

It introduces and compares canonical and covariant position operators, showing their distinct roles in conservation laws and Lorentz symmetry in lower-dimensional Dirac theories.

Findings

The particle position operator yields conserved Lorentz generators in 1+1 dimensions.

The canonical position operator's mass moment requires an unphysical spin-like operator for conservation.

In 2+1 dimensions, orbital and spin angular momenta are conserved separately when using the particle position operator.

Abstract

We have revisited the Dirac theory in 1+1 and 2+1 dimensions by using the covariant representation of the parity-extended Poincar\'e group in their native dimensions. The parity operator plays a crucial role in deriving wave equations in both theories. We studied two position operators, a canonical one and a covariant one that becomes the particle position operator projected onto the particle subspace. In 1+1 dimensions the particle position operator, not the canonical position operator, provides the conserved Lorentz generator. The mass moment defined by the canonical position operator needs an additional unphysical spin-like operator to become the conserved Lorentz generator in 1+1 dimensions. In 2+1 dimensions, the sum of the orbital angular momentum given by the canonical position operator and the spin angular momentum becomes a constant of motion. However, orbital and spin angular momentum do not conserve separately. On the other hand the orbital angular momentum given by the particle position operator and its corresponding spin angular momentum become a constant of motion separately.