%0 Book Section %1 statphys23_0992 %A Wawrzycki, J.W. Jaroslaw %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K algebras geometry measurement noncommutative operator probability quantum statphys23 topic-11 %T Noncommutative geometry, probability and quantum mechanics %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=992 %X We present a geometric approach to quantum mechanics based on the noncommutative geometry. Thus, we have -- as in classical mechanics -- purely geometric data: a symplectic (noncommutative) manifold and Hamiltonian flow. State of the system is given by point of the manifold. The possible measurement results and their probability distributions are consequences of the geometric data. The only (crucial) diffrence in comparison to classical mechanics is that the algebra of smooth functions of the manifold is noncommutative. The occurence of probabilities is a consequence of this noncommutativnees. The first fundamental ingredient of the approach is the so called spectral triple subsuming all the geometric information of the (noncommutative) symplectic manifold. The second ingredient is the construction of point of the noncommutative manifold, which is motivated by a Grothendieck construction and may be considered a generalization of Gelfand-Naimark duality for commutative C*-algebras, the general duality principle. We discuss relation to the orthodox quantum theory.

@incollection{statphys23_0992, abstract = {We present a geometric approach to quantum mechanics based on the noncommutative geometry. Thus, we have -- as in classical mechanics -- purely geometric data: a symplectic (noncommutative) manifold and Hamiltonian flow. State of the system is given by point of the manifold. The possible measurement results and their probability distributions are consequences of the geometric data. The only (crucial) diffrence in comparison to classical mechanics is that the algebra of smooth functions of the manifold is noncommutative. The occurence of probabilities is a consequence of this noncommutativnees. The first fundamental ingredient of the approach is the so called spectral triple subsuming all the geometric information of the (noncommutative) symplectic manifold. The second ingredient is the construction of point of the noncommutative manifold, which is motivated by a Grothendieck construction and may be considered a generalization of Gelfand-Naimark duality for commutative C*-algebras, the general duality principle. We discuss relation to the orthodox quantum theory.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Wawrzycki, J.W. Jaroslaw}, biburl = {https://www.bibsonomy.org/bibtex/26f0d12a7e876a104a290a0890f52ff18/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {ee80f5c4987581727158204cb7d5fb6f}, intrahash = {6f0d12a7e876a104a290a0890f52ff18}, keywords = {algebras geometry measurement noncommutative operator probability quantum statphys23 topic-11}, month = {9-13 July}, timestamp = {2007-06-20T10:16:36.000+0200}, title = {Noncommutative geometry, probability and quantum mechanics}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=992}, year = 2007 }