Sheaves in Quantum Topos Induced by Quantization

Abstract: This paper shows that quantization induces a Lawvere-Tierney topology on (hence, a sheaf topos in) the quantum topos. We show that a quantization map from classical observables to self-adjoint operators on a Hilbert space naturally induces geometric morphisms from presheaf topoi related to the classical system into a presheaf topos, called the quantum topos, on the context category consisting of commutative von Neumann algebras of bounded operators on the Hilbert space. By means of the geometric morphisms, we define Lawvere-Tierney topologies on the quantum topos (and their equivalent Grothendieck topologies on the context category). We show that, among them, we can uniquely select the most informative one, which we call the quantization topology. We furthermore construct sheaves induced by the quantization topology. This can be done in an elementary and self-contained way, because the quantization topology has a quite simple expression.