# Projection factors and generalized real and complex Pythagorean theorems

## TL;DR

This paper introduces generalized Pythagorean theorems for real and complex inner product spaces using projection factors, with implications for quantum theory and probability interpretation.

## Contribution

It develops a unified framework connecting projection factors, Grassmann angles, and Pythagorean theorems in real and complex spaces, highlighting quantum applications.

## Key findings

- Complex Pythagorean theorems differ from real ones by not squaring measures.
- Projection factors relate to quantum probabilities in eigenspaces.
- The theorems may address the probability problem in Everettian quantum mechanics.

## Abstract

Projection factors describe the contraction of Lebesgue measures in orthogonal projections between subspaces of a real or complex inner product space. They are connected to Grassmann's exterior algebra and the Grassmann angle between subspaces, and lead to generalized Pythagorean theorems, relating measures of subsets of real or complex subspaces and their orthogonal projections on certain families of subspaces. The complex Pythagorean theorems differ from the real ones in that the measures are not squared, and this may have important implications for quantum theory. Projection factors of the complex line of a quantum state with the eigenspaces of an observable give the corresponding quantum probabilities. The complex Pythagorean theorem for lines corresponds to the condition of unit total probability, and may provide a way to solve the probability problem of Everettian quantum mechanics.