A topos for algebraic quantum theory Export

@misc{citeulike:2064724, abstract = {We show how a C*-algebra naturally induces a topos in which the family of its commutative subalgebras becomes a commutative C*-algebra. Its internal spectrum is a compact regular locale, and the Kochen-Specker theorem is equivalent to this spectrum having no points. (Quasi-)states become integrals, and self-adjoint elements become functions to the pertinent generalised real numbers (the interval domain). This provides a probabilistic interpretation of propositions in quantum theory. The topos-theoretic truth value of such a proposition is the collection of pure states of commutative subalgebras that make it true; in a physical interpretation these are the pure states for a classical observer making the proposition true. These results were motivated by a topos-theoretic approach of the Kochen-Specker theorem by Isham and co-workers. Our main tool is the use of the internal mathematics of a topos, such as the constructive Gelfand duality of Banaschewski and Mulvey, which simplifies the computations and provides very natural connections between internal and external reasoning.}, archivePrefix = {arXiv}, author = {Heunen, Chris and Spitters, Bas}, citeulike-article-id = {2064724}, citeulike-linkout-0 = {http://arxiv.org/abs/0709.4364}, citeulike-linkout-1 = {http://arxiv.org/pdf/0709.4364}, day = {27}, eprint = {0709.4364}, keywords = {algebraic-quantum-theory, quant-ph, topos-theory}, month = {Sep}, posted-at = {2009-04-20 03:36:39}, priority = {2}, title = {A topos for algebraic quantum theory}, url = {http://arxiv.org/abs/0709.4364}, year = {2007} }

TY - ELEC ID - citeulike:2064724 L3 - citeulike-article-id:2064724 N2 - We show how a C*-algebra naturally induces a topos in which the family of its commutative subalgebras becomes a commutative C*-algebra. Its internal spectrum is a compact regular locale, and the Kochen-Specker theorem is equivalent to this spectrum having no points. (Quasi-)states become integrals, and self-adjoint elements become functions to the pertinent generalised real numbers (the interval domain). This provides a probabilistic interpretation of propositions in quantum theory. The topos-theoretic truth value of such a proposition is the collection of pure states of commutative subalgebras that make it true; in a physical interpretation these are the pure states for a classical observer making the proposition true. These results were motivated by a topos-theoretic approach of the Kochen-Specker theorem by Isham and co-workers. Our main tool is the use of the internal mathematics of a topos, such as the constructive Gelfand duality of Banaschewski and Mulvey, which simplifies the computations and provides very natural connections between internal and external reasoning. TI - A topos for algebraic quantum theory KW - algebraic-quantum-theory KW - quant-ph KW - topos-theory AU - Heunen, Chris AU - Spitters, Bas PY - 2007/Sep/27/ UR - http://arxiv.org/abs/0709.4364 ER -