Topos Quantum Theory with Short Posets

Abstract

Topos quantum mechanics, developed by Isham et. al. [2,6,7,14,15,23,24,25,26], creates a topos of presheaves over the poset V(N ) of Abelian von Neumann subalgebras of the von Neumann algebra N of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the nonexistence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of N and measures on the spectral presheaf; and (d) a model of dynamics in terms of V(N ). We consider a modification to this approach using not the whole of the poset V(N ), but only its elements V(N ) * of height at most two. This produces a different topos with different internal logic. However, the core results (a)-(d) established using the full poset V(N ) are also established for the topos over the smaller poset, and some aspects simplify considerably. Additionally, this smaller poset has appealing aspects reminiscent of projective geometry.