A Topos Theory Foundation for Quantum Mechanics

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.

In Proceedings QPL 2011, arXiv:1210.0298. Dedicated to the memory of Professor Angas Hurst

References in corpus (5)

• A Topos Foundation for Theories of Physics: IV. Categories of Systems
• A topos for algebraic quantum theory
• A Topos Foundation for Theories of Physics: III. The Representation of Physical Quantities With Arrows
• Quantum Mechanics interpreted in Quantum Real Numbers
• The Born rule as structure of spectral bundles (extended abstract)