Algebraic approach to a $d$-dimensional matrix Hamiltonian with so($d+1)$ symmetry

TL;DR

This paper introduces a new algebraic framework using a spin-extended so(d+1,1) algebra to analyze a d-dimensional matrix Hamiltonian with spin 1/2 and so(d+1) symmetry, simplifying the derivation of integrals of motion.

Contribution

It presents a novel algebraic approach with additional operators to derive integrals of motion and the Laplace-Runge-Lenz vector for spin-1/2 systems in d dimensions.

Findings

Derived integrals of motion using the new algebraic framework

Connected Sturm and Schrödinger representations for the Hamiltonian

Extended the algebraic understanding of the Laplace-Runge-Lenz vector with spin

Abstract

A novel spin-extended so($d+1$,1) algebra is introduced and shown to provide an interesting framework for discussing the properties of a $d$-dimensional matrix Hamiltonian with spin 1/2 and so($d+1$) symmetry. With some $d+2$ additional operators, spanning a basis of an so($d+1$,1) irreducible representation, the so($d+1$,1) generators provide a very easy way for deriving the integrals of motion of the matrix Hamiltonian in Sturm representation. Such integrals of motion are then transformed into those of the matrix Hamiltonian in Schr\"odinger representation, including a Laplace-Runge-Lenz vector with spin. This leads to a derivation of the latter, as well as its properties in a more extended algebraic framework.