Quantum mechanics in a metric sheaf: a model theoretic approach

Abstract

We study model-theoretical structures for prototypical physical systems. First, a summary of the model theory of sheaves, adapted to the metric case, is presented. In particular, we provide conditions for a generalization of the generic model theorem to metric sheaves. The essentials of the model theory of metric sheaves appeared in the form of Conference Proceedings. We provide a version of those results, for the sake of completeness, and then build metric sheaves for physical systems in the second part of the paper. Specifically, metric sheaves for quantum mechanical systems with pure point and continuous spectra are constructed. In the former case, every fiber is a finite projective Hilbert space determined by the family of invariant subspaces of a given operator with pure point spectrum, and we also consider unitary transformations in a finite-dimensional space. For an operator with continuous spectrum, every fiber is a two sorted structure of subsets of the Schwartz space of rapidly decreasing functions that includes imperfect representations of position and momentum states. The imperfection character is parametrically determined by the elements on the base space and refined in the generic model. Position and momentum operators find a simple representation in every fiber as well as their corresponding unitary operators. These results follow after recasting the algebraic properties of the integral transformations frequently invoked in the description of quantum mechanical systems with continuous spectra. Finally, we illustrate how this construction permits the calculation of the quantum mechanical propagator for a free particle.