A Geometrical Interpretation of Grassmannian Coordinates

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.