Estados de equilíbrio na teoria quântica em topoi

Abstract

The main purpose of this work was to study equilibrium states in the topos approach to quantum theory. This is a relatively new approach to quantum theory then we also tried to present it in a self-contained way -at least from a mathematical point of view. Concerning equilibrium, we focused on two features: time translation invariance (stationary states) and the KMS condition. We characterize invariant states (in particular stationary states) by means of the so called "measures on the spectral presheaf". We couldn't do the same for the KMS condition and, as argued in this work, we think that there are good reasons to believe that it is not possible.Then, we studied the KMS condition using pseudo-states. For this purpose we introduced the idea of a "topos associated to a state" and proved that in the topos of any state there are at least two propositions (clopen subobjects of the spectral presheaf) which are true (in the sense of the pseudo-state) if, and only if, the state is KMS. One of these propositions is not of the kind "daseinization of a projector" and it can be seen as a way of giving physical meaning to a proposition which is not in the interpretation of the PL(S) language.

Step-by-step derivation of the algebraic structure of quantum mechanics (or from nondisturbing to quantum correlations by connecting incompatible observables)

“…Specker's principle follows from the fact that any finite set of pairwise compatible observables has a finite product [52,58,59] in the category of observables. The definition of product goes as follows.…”