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

TL;DR

The work develops a spin-extended so($d+1$,1) algebra to analyze a $d$-dimensional matrix Hamiltonian with spin 1/2 and so($d+1$) symmetry, integrating Sturm- and Schrödinger-representation perspectives. By introducing $d+2$ additional operators forming an so($d+1$,1) irrep, it derives integrals of motion in Sturm representation and maps them to the Schrödinger picture to obtain a Laplace-Runge-Lenz vector with spin, together with its algebraic properties. The analysis yields a spinful generalization of the LRL vector, encapsulated in relations such as $ ilde{oldsymbol{A}}^2 = 2Higl(oldsymbol{J}^2 + rac{1}{8}d(d-1)igr) + rac{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}}}{ ext{where}}oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}}^2$, and a $d=3$ special-case coupling $ oldsymbol{J}cdot ilde{oldsymbol{A}} = rac{1}{2}oldsymbol{ abla}$-like constant. These results extend dynamical symmetry analyses of Coulomb-like systems to a spinful matrix setting and offer a path to higher-spin generalizations.

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 Schrödinger 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.