International Journal of Theoretical Physics

, Volume 46, Issue 2, pp 319–382

Quantum Observables Algebras and Abstract Differential Geometry: The Topos-Theoretic Dynamics of Diagrams of Commutative Algebraic Localizations

Article

DOI: 10.1007/s10773-006-9223-z

Cite this article as:
Zafiris, E. Int J Theor Phys (2007) 46: 319. doi:10.1007/s10773-006-9223-z

Abstract

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), à la Mallios, in a topos-theoretic environment, and hence, the extension of the “mechanism of differentials” in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection.

Key Words

quantum observablescommutative algebrassheavestopos theorygrothendieck topologyadjunctionadjoint functorscohomology of algebrasconnectioncurvaturequantum dynamicsdifferential geometry

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of Athens, PanepistimioupolisAthensGreece