Polar decomposition of a Dirac spinor

TL;DR

This paper explores local decompositions of Dirac spinors into Majorana components, analyzing their properties, symmetries, and implications for gauge theories and high-temperature superconductivity models.

Contribution

It introduces a novel local decomposition framework for Dirac spinors, including specific 2+1 dimensional examples and connections to supersymmetric and condensed matter models.

Findings

Decomposition into Majorana and Dirac-algebra components is feasible.

Identifies local reparametrisation and electromagnetic invariances.

Proposes a dynamical nonabelian gauge theory framework.

Abstract

Local decompositions of a Dirac spinor into `charged' and `real' pieces psi(x) = M(x) chi(x) are considered. chi(x) is a Majorana spinor, and M(x) a suitable Dirac-algebra valued field. Specific examples of the decomposition in 2+1 dimensions are developed, along with kinematical implications, and constraints on the component fields within M(x) sufficient to encompass the correct degree of freedom count. Overall local reparametrisation and electromagnetic phase invariances are identified, and a dynamical framework of nonabelian gauge theories of noncompact groups is proposed. Connections with supersymmetric composite models are noted (including, for 2+1 dimensions, infrared effective theories of spin-charge separation in models of high-Tc superconductivity).