Invariant vector fields and the prolongation method for supersymmetric quantum systems

TL;DR

This paper reviews the symmetries of supersymmetric quantum systems using the vector field prolongation method, introduces supersymmetry generators forming Lie superalgebras, and explores related models like the Jaynes-Cummings system.

Contribution

It presents a novel approach to supersymmetry analysis without Grassmann variables, utilizing matrix realizations and dynamical symmetry concepts.

Findings

Lie superalgebras of symmetries and supersymmetries are constructed.

The method applies to models like the supersymmetric harmonic oscillator and spin systems.

Connections between supersymmetric models and the Jaynes-Cummings model are established.

Abstract

The kinematical and dynamical symmetries of equations describing the time evolution of quantum systems like the supersymmetric harmonic oscillator in one space dimension and the interaction of a non-relativistic spin one-half particle in a constant magnetic field are reviewed from the point of view of the vector field prolongation method. Generators of supersymmetries are then introduced so that we get Lie superalgebras of symmetries and supersymmetries. This approach does not require the introduction of Grassmann valued differential equations but a specific matrix realization and the concept of dynamical symmetry. The Jaynes-Cummings model and supersymmetric generalizations are then studied. We show how it is closely related to the preceding models. Lie algebras of symmetries and supersymmetries are also obtained.