Remarks on superdifferential equations

TL;DR

This paper clarifies the use of 'superdifferential equations' in literature, argues that Grassmann-valued differential equations require different analysis methods, and illustrates this with relevant examples.

Contribution

It distinguishes between types of superdifferential equations and proposes analyzing Grassmann-valued ones via standard differential equations on Grassmann algebra bundles.

Findings

Grassmann-valued differential equations cannot be effectively described by supergeometric techniques.

Standard differential equations on Grassmann algebra bundles are suitable for Grassmann-valued equations.

Examples demonstrate the relevance of the proposed analysis methods.

Abstract

We show that the term `superdifferential equation' has been employed in the literature to refer to different types of differential equations with even and odd variables. It is justified on physical and mathematical grounds that a subclass of them, the hereafter called Grassmann-valued differential equations, cannot be effectively described through supergeometric techniques. Instead, we analyse them in terms of standard differential equations on Grassmann algebra bundles. Our considerations are illustrated through examples of physical and mathematical relevance.