Free Relativistic Anyons with Canonical Spin Algebra

TL;DR

This paper explores a relativistic free particle with fractional spin in 2+1 dimensions, revealing that maintaining both free motion and translational invariance is incompatible due to the canonical spin algebra.

Contribution

It demonstrates that a free relativistic anyon with canonical spin algebra necessarily breaks translational invariance, highlighting a fundamental limitation in modeling such particles.

Findings

Poincaré invariance is broken to Lorentz invariance for these anyons.

Canonical spin algebra leads to the loss of translational symmetry.

Free relativistic anyons cannot preserve both free motion and translational invariance.

Abstract

We discuss a relativistic free particle with fractional spin in 2+1 dimensions, where the dual spin components satisfy the canonical angular momentum algebra $\left\{ S_\mu , S_\nu \right\}\,=\,\epsilon_{\mu \nu \gamma}S^\gamma $. It is shown that it is a general consequence of these features that the Poincar\`e invariance is broken down to the Lorentz one, so indicating that it is not possible to keep simultaneously the free nature of the anyon and the translational invariance.