A use of geometric calculus to reduce Berezin integral to the limit of a Riemann sum

TL;DR

This paper demonstrates how Berezin integrals over Grassmann variables can be rigorously realized as limits of geometric calculus integrals using Clifford algebra, linking superspace and spinors within this framework.

Contribution

It introduces a novel approach using geometric algebra to interpret Berezin integrals as limits of geometric calculus integrals, providing a new perspective on superspace and spinors.

Findings

Berezin integrals can be realized as limits of geometric calculus integrals.

The formalism connects Grassmann variables with Clifford algebra structures.

It offers a rigorous geometric interpretation of superspace concepts.

Abstract

Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann numbers are endowed with a second associative product coming from a Clifford algebra structure, we show how Berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus. We then show how the concepts of spinors and superspace transform into this framework.