Covariant basis induced by parity for the $(j,0)\oplus (0,j)$ representation

TL;DR

This paper constructs a covariant basis for operators on the $(j,0)igoplus(0,j)$ Lorentz representations, analyzing parity properties and providing tools for interaction terms and multipole moment calculations in gauge theories.

Contribution

It introduces an explicit covariant basis for these representations, including an algorithm for arbitrary $j$, and explores implications for multipole moments and causality.

Findings

Explicit covariant basis for $(j,0)igoplus(0,j)$ representations.

Predicts all multipole moments are determined by Lorentz algebra and gyromagnetic factor.

Demonstrates causality of wave propagation in electromagnetic backgrounds.

Abstract

In this work, we build a covariant basis for operators acting on the $(j,0)\oplus(0,j)$ Lorentz group representations. The construction is based on an analysis of the covariant properties of the parity operator, which for these representations transforms as the completely temporal component of a symmetrical tensor of rank $2j$. The covariant properties of parity involve the Jordan algebra of anti commutators of the Lorentz group generators which unlike the Lie algebra is not universal. We make the construction explicit for $j=1/2,1$ and $3/2$, reproducing well-known results for the $j=1/2$ case. We provide an algorithm for the corresponding calculations for arbitrary $j$. This covariant basis provides an inventory of all the possible interaction terms for gauge and non-gauge theories of fields for these representations. In particular, it supplies a single second rank antisymmetric structure, which in the Poincar\'e projector formalism implies a single Pauli term arising from gauge interactions and a single (free) parameter $g$, the gyromagnetic factor. This simple structure predicts that for an elementary particle in this formalism all multipole moments, $Q^{l}_{E}$ and $Q^{l}_{M}$, are dictated by the complete algebraic structure of the Lorentz generators and the value of $g$. We explicitly calculate the multipole moments, for arbitrary $j$ up to $l=8$. Comparing with results in the literature we find that only the electric charge and magnetic moment of a spin $j$ particle are independent of the Lorentz representation under which it transforms, all higher multipoles being representation dependent. Finally we show that the propagation of the corresponding spin $j$ waves in a electromagnetic background is causal.