Realistic interpretation of Grassmann variables

TL;DR

This paper redefines Grassmann integrals as limits of sums over specific geometric contours, providing a more intuitive geometric interpretation and extending the class of functions that can be integrated.

Contribution

It introduces a geometric framework for Grassmann integrals using contours in infinite-dimensional space, combining wedge and Clifford products for a more rigorous interpretation.

Findings

Grassmann integrals can be represented as limits over specific contours.

The geometric interpretation clarifies the rescaling properties of Grassmann integrals.

Non-analytic functions can be integrated with this new approach, though their exact values depend on hidden parameters.

Abstract

The goal of this paper is to define the Grassmann integral in terms of a limit of a sum around a well-defined contour so that Grassmann numbers gain geometric meaning rather than symbols. The unusual rescaling properties of the integration of an exponential is due to the fact that the integral attains the known values only over a specific set of contours and not over their rescaled versions. Such contours live in infinite dimensional space and their sides are infinitesimal, and they make infinitely many turns. Finally, two different products are used: anticommutting wedge product and a Clifford dot product (the wedge product is used in the finite part of the integral and the Clifford dot product is used between the finite and infinitesimal parts). The integrals of non-analytic functions will become well-defined, although their specific value is unknown due to the various hidden parameters.