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.1 The quantum real number interpretation of quantum mechanics.The quantum real number (qr-number) interpretation of quantum mechanics is based on the claim that the change from classical physics to quantum physics is achieved by changing in the type of numerical values that the physical variables can possess. It assumes that the properties of microscopic entities differ from those of classical because their numerical values are not standard real numbers but are Dedekind real numbers in a spatial topos built upon the quantum state space of the microscopic entities. The Dedekind reals differ from standard real is that each holds true to a non-trivial extent given by an open subsets of the quantum state space. These are its conditions, they replace the individual states of standard quantum theory. The internal logic is intuitionistic. The "direct connection between observation properties and properties possessed by the independently existing object" [10] is cut, an indirect connection is made through the experimental measurement processes. The interpretation builds on the fact that any experimental measurement has a limited level of accuracy. Within this level of accuracy an attribute's qr-number value is approximated by a standard real number.Its mathematical structure is built from elements of standard quantum mechanics: we start from von Neumann's assumption [37] that to any quantum system we can associate a Hilbert space H . H is the carrier space of a unitary representation of a Lie group G, the symmetry group of the quantum system. Then, as in the standard interpretation, the physical attributes (measurable qualities) of the quantum system are represented by essentially self-adjoint operators that act on a dense subset D ⊂ H . The set of operators form a * -algebra A and the state space E S (A ) of the system is the space of normalized linear functionals on A . The state space has the weak topology generated by the real -valued functions a Q : E S (A ) → R given by a Q (ρ) = Tr(ρ.Â) : ∀ρ ∈ E S (A ) and labeled by the operators ∈ A . A typical open set is a finite intersection of open sets N (ρ 0 ;...