Linear Differential Equations for a Fractional Spin Field

TL;DR

This paper develops a system of linear differential equations for fractional spin fields using infinite-dimensional representations of the SL(2,R) group, establishing gauge invariance and deriving a corresponding field action.

Contribution

It introduces a novel vector differential equation framework for fractional spin fields based on SL(2,R) representations, including gauge symmetry and an invariant action.

Findings

Formulation of differential equations for fractional spin fields.

Identification of gauge symmetry in the vector system.

Construction of a gauge-invariant field action.

Abstract

The vector system of linear differential equations for a field with arbitrary fractional spin is proposed using infinite-dimensional half-bounded unitary representations of the $\overline{SL(2,R)}$ group. In the case of $(2j+1)$-dimensional nonunitary representations of that group, $0<2j\in Z$, they are transformed into equations for spin-$j$ fields. A local gauge symmetry associated to the vector system of equations is identified and the simplest gauge invariant field action, leading to these equations, is constructed.