Geometric and algebraic approaches to quantum theory

TL;DR

This paper explores a geometric formulation of physical theories, including quantum mechanics, using the set of states, and demonstrates how this approach can lead to new models and generalizations beyond conventional frameworks.

Contribution

It introduces a geometric approach to quantum theory based on state sets, connecting it with algebraic methods and enabling the construction of models with arbitrary symmetries.

Findings

Provides a geometric formulation of quantum and classical theories.

Shows how to derive models with prescribed symmetry groups.

Extends quantum theory using Jordan algebras within the geometric framework.

Abstract

We show how to formulate physical theory taking as a starting point the set of states (geometric approach). We discuss the relation of this formulation to the conventional approach to classical and quantum mechanics and the theory of complex systems. The equations of motion and the formulas for probabilities of physical quantities are analyzed. A heuristic proof of decoherence in our setting is used to justify the formulas for probabilities. We show that any physical theory theory can be obtained from classical theory if we restrict the set of observables. This remark can be used to construct models with any prescribed group of symmetries; one can hope that this construction leads to new interesting models that cannot be build in the conventional framework. The geometric approach can be used to formulate quantum theory in terms of Jordan algebras, generalizing the algebraic approach to quantum theory. The scattering theory can be formulated in geometric approach.