Grassmann angle formulas and identities

TL;DR

This paper introduces Grassmann angles that generalize traditional angles between subspaces, providing formulas and identities that relate to volume, projections, and quantum probabilities, applicable to real and complex spaces.

Contribution

It develops new formulas and identities for Grassmann angles, extending classical trigonometric identities to high-dimensional and complex subspaces, with applications in geometry and quantum theory.

Findings

Derived formulas for Grassmann angles in arbitrary bases

Generalized Pythagorean identities for high-dimensional subspaces

Connections to volume, quantum probabilities, and Clifford products

Abstract

Grassmann angles improve upon similar concepts of angle between subspaces that measure volume contraction in orthogonal projections, working for real or complex subspaces, and being more efficient when dimensions are different. Their relations with contractions, inner and exterior products of multivectors are used to obtain formulas for computing these or similar angles in terms of arbitrary bases, and various identities for the angles with certain families of subspaces. These include generalizations of the Pythagorean trigonometric identity $\cos^2\theta+\sin^2\theta=1$ for high dimensional and complex subspaces, which are connected to generalized Pythagorean theorems for volumes, quantum probabilities and Clifford geometric product.