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topos approach to quantum mechanics

There are several approaches using topos theory to formulate and study foundations of quantum mechanics. Some people say “quantum topos theory” what is abit unfortunate as it is not about some quantum analogue of a topos (while such an object may be desired, say in noncommutative geometry), but about using the usual topoi to model quantum mechanics.

The main proposal is from Chris Isham and his school, see higher category theory and physics.

• Andread Doering, C.J. Isham, A Topos Foundation for Theories of Physics I-IV quant-ph/0703060

• Kunji Nakayama, Sheaves in quantum topos induced by quantization, arxiv/1109.1192

See also Kochen-Specker theorem for the topos theoretic interpretation summarizing

• Jeremy Butterfield, John Hamilton, Chris Isham, A topos perspective on the Kochen-Specker theorem, I. quantum states as generalized valuations, Internat. J. Theoret. Phys. 37(11):2669–2733, 1998, MR2000c:81027, doi; II. conceptual aspects and classical analogues Int. J. of Theor. Phys. 38(3):827–859, 1999, MR2000f:81012, doi; III. Von Neumann algebras as the base category, Int. J. of Theor. Phys. 39(6):1413–1436, 2000, arXiv:quant-ph/9911020, MR2001k:81016,doi; IV. Interval valuations, Internat. J. Theoret. Phys. 41 (2002), no. 4, 613–639, MR2003g:81009, doi

Another, more recent variant of using topos theory in quantum foundations is due Nijmengen school, see Bohr topos for detail.

• Chris Heunen, Nicolaas Landsman, Bas Spitters, A topos for algebraic quantum theory arXiv:0709.4364