Spinor superalgebra: Towards a theory for higher spin particles

TL;DR

This paper introduces a superalgebra framework S2(N/2) for higher spin particles, generalizing the Dirac algebra, with explicit matrix realizations and a new wave equation for spin 3/2 particles.

Contribution

It defines a novel superalgebra S2(N/2) for higher spins and provides matrix representations and a first-order wave equation for spin 3/2 particles.

Findings

S2(N/2) superalgebra generalizes spin representations

Matrix realization supports Lorentz symmetry for higher spins

Derivation of a first-order wave equation for spin 3/2

Abstract

We define a superalgebra S2(N/2) as a Z2 graded algebra of dimension 2N+3, where N is a positive, odd integer. The even component is a three-dimensional abelian subalgebra, while the odd component is made up of two N-dimensional, mutually conjugate algebras. For N = 1, two of the three even elements become identical, resulting in a four-dimensional superalgebra which is the graded extension of the SO(2,1) Lie algebra that has recently been introduced in the solution of the Dirac equation for spinn 1/2. Realization of the elements of S2(N/2) is given in terms of differential matrix operators acting on an N+1 dimensional space that could support a representation of the Lorentz space-time symmetry group for spin N/2. The N = 3 case results in a 4x4 matrix wave equation, which is linear and of first order in the space-time derivatives. We show that the "canonical" form of the Dirac Hamiltonian is an element of this superalgebra.