Spectral presheaves as quantum state spaces

Abstract

For each quantum system described by an operator algebra of physical quantities, we provide a (generalized) state space, notwithstanding the Kochen–Specker theorem. This quantum state space is the spectral presheaf . We formulate the time evolution of quantum systems in terms of Hamiltonian flows on this generalized space and explain how the structure of the spectral presheaf geometrically mirrors the double role played by self-adjoint operators in quantum theory, as quantum random variables and as generators of time evolution.