Topos Quantum Theory with Short Posets

Abstract

Topos quantum mechanics, developed by Döring (2008); Döring and Harding Houston J. Math. 42(2), 559–568 (2016); Döring and Isham (2008); Flori 2013)); Flori (2018); Isham and Butterfield J. Theoret. Phys. 37, 2669–2733 (1998); Isham and Butterfield J. Theoret. Phys. 38, 827–859 (1999); Isham et al. J. Theoret. Phys. 39, 1413–1436 (2000); Isham and Butterfield J. Theoret. Phys. 41, 613–639 (2002), creates a topos of presheaves over the poset $\mathcal {V}(\mathcal {N})$ of Abelian von Neumann subalgebras of the von Neumann algebra $\mathcal {N}$ 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 $\mathcal {N}$ and measures on the spectral presheaf; and (d) a model of dynamics in terms of $\mathcal {V}(\mathcal {N})$. We consider a modification to this approach using not the whole of the poset $\mathcal {V}(\mathcal {N})$, but only its elements $\mathcal {V}(\mathcal {N})^{*}$ 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 $\mathcal {V}(\mathcal {N})$ 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.