A Topos Theory Foundation for Quantum Mechanics

TL;DR

This paper introduces a topos-theoretic framework for quantum mechanics using quantum real numbers, aiming to clarify paradoxes and provide a more consistent description of quantum attributes and phenomena.

Contribution

It develops a topos-based foundation for quantum mechanics with quantum real numbers, offering a new perspective on quantum attributes and resolving paradoxes.

Findings

Quantum real numbers form non-classical spatial continua.

Quantum trajectories can pass through multiple slits without contradiction.

Einstein locality is preserved in the quantum space model.

Abstract

The theory of quantum mechanics is examined using non-standard real numbers, called quantum real numbers (qr-numbers), that are constructed from standard Hilbert space entities. Our goal is to resolve some of the paradoxical features of the standard theory by providing the physical attributes of quantum systems with numerical values that are Dedekind real numbers in the topos of sheaves on the state space of the quantum system. The measured standard real number values of a physical attribute are then obtained as constant qr-number approximations to variable qr-numbers. Considered as attributes, the spatial locations of massive quantum particles form non-classical spatial continua in which a single particle can have a quantum trajectory which passes through two classically separated slits and the two particles in the Bohm-Bell experiment stay close to each other in quantum space so that Einstein locality is retained.