Two-Dimensional Supersymmetry: From SUSY Quantum Mechanics to Integrable Classical Models

TL;DR

This paper combines two-dimensional supersymmetric quantum mechanics frameworks to construct and analyze SUSY-extended classical models with integrable properties, emphasizing symmetry operators and classical integrals of motion.

Contribution

It introduces a unified approach to 2D SUSY quantum models, including those not separable, and constructs their classical limits with explicit symmetry and integrability features.

Findings

Explicit quantum symmetry operators for SUSY-extended models

Classical integrals of motion derived for SUSY-extended systems

Connection established between SUSY models and Hamilton-Jacobi method

Abstract

Two known 2-dim SUSY quantum mechanical constructions - the direct generalization of SUSY with first-order supercharges and Higher order SUSY with second order supercharges - are combined for a class of 2-dim quantum models, which {\it are not amenable} to separation of variables. The appropriate classical limit of quantum systems allows us to construct SUSY-extensions of original classical scalar Hamiltonians. Special emphasis is placed on the symmetry properties of the models thus obtained - the explicit expressions of quantum symmetry operators and of classical integrals of motion are given for all (scalar and matrix) components of SUSY-extensions. Using Grassmanian variables, the symmetry operators and classical integrals of motion are written in a unique form for the whole Superhamiltonian. The links of the approach to the classical Hamilton-Jacobi method for related "flipped" potentials are established.