A sheaf model of quantum mechanics

John Corbett – 24 February 1999

The standard Hilbert space formalism of quantum mechanics is reinterpreted via the topos of sheaves on the state space of the system. The logic is intuitionistic and the associated real numbers, the sheaf of Dedekind reals,contains quantum numbers that are labelled by the self-adjoint operators of the standard formalism. These numbers are taken as the values of physical quantities associated with the quantum system. A number of interesting consequences follow: Quantum particles have trajectories. If the dynamics, expressed in these numbers,is taken to be given by Newton's equations of motion then Heisenberg's operator equations of motion arise as local approximations. The von Neumann- Luders projection postulate for measurement, also known as collapse or reduction of the wave packet, is an approximation to the description of the passage of a particle through a slit.