Fractional helicity, Lorentz symmetry breaking, compactification and anyons

TL;DR

This paper develops covariant wave equations for massless fields using R-deformed Heisenberg algebra, revealing fractional helicity states that break Lorentz symmetry and lead to a theory of anyons in 2+1 dimensions.

Contribution

It introduces a novel framework for describing fractional helicity states and connects them to anyons via compactification, highlighting the absence of certain massless representations in 3+1D.

Findings

Infinite-dimensional representations correspond to fractional helicity states.

Fractional helicity states break 3+1D Lorentz invariance to 2+1D.

A consistent theory for anyons in 2+1D is constructed.

Abstract

We construct the covariant, spinor sets of relativistic wave equations for a massless field on the basis of the two copies of the R-deformed Heisenberg algebra. For the finite-dimensional representations of the algebra they give a universal description of the states with integer and half-integer helicity. The infinite-dimensional representations correspond formally to the massless states with fractional (real) helicity. The solutions of the latter type, however, break down the (3+1)$D$ Poincar\'e invariance to the (2+1)$D$ Poincar\'e invariance, and via a compactification on a circle a consistent theory for massive anyons in $D$=2+1 is produced. A general analysis of the ``helicity equation'' shows that the (3+1)$D$ Poincar\'e group has no massless irreducible representations with the trivial non-compact part of the little group constructed on the basis of the infinite-dimensional representations of $sl(2,\CC)$. This result is in contrast with the massive case where integer and half-integer spin states can be described on the basis of such representations, and means, in particular, that the (3+1)$D$ Dirac positive energy covariant equations have no massless limit.