"Principle of Indistinguishability" and equations of motion for particles with spin

TL;DR

This paper reviews and generalizes the principle of indistinguishability to derive equations of motion for particles with various spins, revealing the existence or absence of first-order equations for different Lorentz group representations.

Contribution

It extends the principle of indistinguishability to higher spins, derives new equations, and explores the existence of Dirac-like equations for various Lorentz group irreducible representations.

Findings

Derives O(p^{2j}) equations for representations containing spin j.

Shows no Dirac-like equations exist for most higher-spin representations except for specific cases.

Reproduces known equations and finds new multiplet descriptions for particles with spin 1/2, 1, 3/2.

Abstract

In this work we review the derivation of Dirac and Weinberg equations based on a ``principle of indistinguishability'' for the (j,0) and (0,j) irreducible representations (irreps) of the Homogeneous Lorentz Group (HLG). We generalize this principle and explore its consequences for other irreps containing spin >= 1. We rederive Ahluwalia-Kirchbach equation using this principle and conclude that it yields O(p^{2j}) equations of motion for any representation containing spin j and lower spins. We also use the obtained generators of the HLG for a given representation to explore the possibility of the existence of first order equations for that representation. We show that, except for j=1/2, there exists no Dirac-like equation for the (j,0)\oplus (0,j) representation nor for the (1/2,1/2) representation. We rederive Kemmer-Duffin-Petieau (KDP) equation for the (1,0)\oplus (1/2,1/2)\oplus(0,1) representation by this method and show that the (1,1/2)\oplus (1/2,1) representation satisfies a Dirac-like equation which describes a multiplet of j=3/2 and j=1/2 with masses m and m/2} respectively.