Extended Supersymmetric Quantum Mechanics of Fierz and Schur Type

TL;DR

This paper explores two methods for constructing N-extended Supersymmetric Quantum Mechanics using Fierz identities and Schur lemma, revealing their relations with Clifford algebra periodicity and Hamiltonian structure.

Contribution

It introduces two independent frameworks for N-extended supersymmetry in quantum mechanics, linking algebraic identities with Hamiltonian tensor products.

Findings

Fierz identities and Schur lemma provide consistent constructions for supersymmetric quantum mechanics.

The relations among N, D, and d are explicitly characterized.

Clifford algebra periodicity is encoded in the constructions.

Abstract

We discuss two independent constructions to introduce an N-extended Supersymmetric Quantum Mechanics. The first one makes use of the Fierz identities while the second one (divided into two subcases) makes use of the Schur lemma. The N supercharges Q_I are square roots of a free Hamiltonian H given by the tensor product of a D-dimensional Laplacian and a 2d-dimensional identity matrix operator. We present the mutual relations among N, D and d. The mod 8 Bott's periodicity of Clifford algebras is encoded, in the Fierz case, in the Radon-Hurwitz function and, in the Schur case, in an extra independent function.