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.

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Correspondence to John Harding.

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Harding, J., Heunen, C. Topos Quantum Theory with Short Posets. Order (2020). https://doi.org/10.1007/s11083-020-09531-6

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Keywords

  • Topos
  • Von Neumann algebra
  • Internal logic
  • Presheaf
  • Daseinisation
  • Automorphism