Similarity-Projection structures: the logical geometry of Quantum Physics

TL;DR

This paper introduces similarity-projection structures as a generalized framework for quantum physics, capturing the logical and geometric aspects of quantum states and projections without relying on traditional linear algebra.

Contribution

It develops a formal framework that unifies classical and quantum logic through similarity-projection structures, highlighting the role of phase factors in quantum mechanics.

Findings

Provides a generalized formalism for quantum logic

Clarifies the role of phase factors in quantum physics

Introduces a linear algebra-free notion of self-adjoint operators

Abstract

Similarity-Projection structures abstract the numerical properties of real scalar product of rays and projections in Hilbert spaces to provide a more general framework for Quantum Physics. They are characterized by properties that possess direct physical meaning. They provide a formal framework that subsumes both classical boolean logic concerned with sets and subsets and quantum logic concerned with Hilbert space, closed subspaces and projections. They shed light on the role of the phase factors that are central to Quantum Physics. The generalization of the notion of a self-adjoint operator to SP-structures provides a novel notion that is free of linear algebra.