A Geometrical Interpretation of Grassmannian Coordinates

TL;DR

This paper offers a geometric interpretation of Grassmannian anticommuting coordinates as representing the inherent indeterminacy at spacetime points caused by quantum foam, linking them to quantum wormholes and supersymmetry.

Contribution

It introduces a novel geometric perspective on Grassmannian coordinates as measures of spacetime indeterminacy related to quantum wormholes in quantum gravity.

Findings

Grassmannian numbers represent spacetime indefiniteness.

Displacement of wormhole mouths correlates with changes in Grassmannian coordinates.

Supersymmetric fields must be invariant on spacetime foam background.

Abstract

A geometrical interpretation of Grassmannian anticommuting coordinates is given. They are taken to represent an indefiniteness inherent in every spacetime point on the level of the spacetime foam. This indeterminacy is connected with the fact that in quantum gravity in some approximation we do not know the following information : are two points connected by a quantum wormhole or not ? It is shown that: (a) such indefiniteness can be represented by Grassmanian numbers, (b) a displacement of the wormhole mouth is connected with a change of the Grassmanian numbers (coordinates). In such an interpretation of supersymmetry the corresponding supersymmetrical fields must be described in an invariant manner on the background of the spacetime foam.