A Topos theory for quantum mechanics

Any quantum system that has an algebra of physical attributes represented by a *-algebra, A, on a Hilbert space that carries a unitary representation of its symmetry group has its own real number system for the values of its attributes. They are called quantum real numbers (qr-numbers). When its state space εS(A) has the weak topology generated by the real - valued functions aQ : εS(A) → ℝ given by a(ρ) = Tr(ρ:Â) : ∀ρ ∈ εS(A) and  ∈ A, the qr-numbers are sections of ℝD(εS(A)), the sheaf of Dedekind real numbers in the spatial topos Shv(εS(A)). The open subsets of εS(A) are the conditions of the system, the internal logic is intuitionistic. The standard real number value of a physical attribute obtained in a measurement is a constant qr-number approximations to the attribute’s actual qr-number value. Each quantum particle with positive mass moves in a spatial continuum that is isomorphic to ℝD(εS(A))³. This continuum is sufficiently non-classical that 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 so that Einstein locality is retained in quantum space.