Abstract
Topos quantum mechanics, developed by Isham et. al., 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 non-existence 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 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.
Highlights
Isham and Butterfield [23, 24, 25, 26] introduced a topos approach to quantum mechanics and showed that the Kochen-Specker theorem is equivalent to the non-existence of a global section of a certain presheaf
A von Neumann algebra N is associated to a quantum system as in standard quantum mechanics
We have shown that many of the core results in the topos approach over V(N ) are retained upon restriction to a topos over the simpler poset V(N )∗
Summary
Isham and Butterfield [23, 24, 25, 26] introduced a topos approach to quantum mechanics and showed that the Kochen-Specker theorem is equivalent to the non-existence of a global section of a certain presheaf. This is applied in the case when N is the von Neumann algebra B(H) of bounded operators on a Hilbert space H, and in this case V(N ) is denoted V(H) The spirit of this topos approach is that elements of V(N ) give classical “snapshots” of the quantum system. In [16, 18] it was shown that V(N )∗ can be treated graphically and has similarities reminiscent of projective geometry This extends to a treatment of morphisms between the projections of von Neumann algebras N and M that become certain order preserving maps between the posets V(N )∗ and V(M)∗, and certain geometric morphisms between their toposes.
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