Classical Supersymmetric Mechanics

TL;DR

This paper explores a supersymmetric mechanical model derived from (1+1)-dimensional field theory, analyzing solutions, symmetries, and oscillatory behaviors of bosonic and fermionic variables within Grassmann algebra frameworks.

Contribution

It introduces a method to solve equations of motion for supersymmetric mechanics with Grassmann algebra variables, including a layer-by-layer approach for finitely generated algebras.

Findings

Complete solutions for certain potentials using symmetry analysis

Decomposition of variables into real components for arbitrary potentials

Insights into oscillatory motion via Floquet theory

Abstract

We analyse a supersymmetric mechanical model derived from (1+1)-dimensional field theory with Yukawa interaction, assuming that all physical variables take their values in a Grassmann algebra B. Utilizing the symmetries of the model we demonstrate how for a certain class of potentials the equations of motion can be solved completely for any B. In a second approach we suppose that the Grassmann algebra is finitely generated, decompose the dynamical variables into real components and devise a layer-by-layer strategy to solve the equations of motion for arbitrary potential. We examine the possible types of motion for both bosonic and fermionic quantities and show how symmetries relate the former to the latter in a geometrical way. In particular, we investigate oscillatory motion, applying results of Floquet theory, in order to elucidate the role that energy variations of the lower order quantities play in determining the quantities of higher order in B.