Categories of Physical Processes

TL;DR

This paper reconstructs quantum theory from a categorical perspective, deriving core principles like the Born rule and Schrödinger picture, and discusses the integration of classical limits and quantum processes.

Contribution

It introduces a novel categorical framework for quantum physics based on symmetric monoidal functors and the GNS construction, providing foundational insights.

Findings

Derives the Born rule and quantum pictures from categorical principles

Establishes a connection between symmetries and group representations in quantum theory

Proposes a framework for quantum Markov processes, including wave function collapse

Abstract

We study the mathematical foundations of physics. We reconstruct textbook quantum theory from a single symmetric monoidal functor $GNS : \mathbf{Phys} \longrightarrow \ast\mathbf{Mod}$, based on the Gelfand-Naimark-Segal construction and the notion of representability. We derive the probabilistic interpretation of quantum mechanics, including the Born rule, the Schr\"odinger and Heisenberg pictures, the relation between symmetries and group representations, and a theory of quantum Markov processes, including wave function collapse. Inclusion of the classical limit and deformation quantization is briefly sketched. Gauge symmetry and extended locality cannot currently be accommodated, due to conceptual difficulties discussed in an appendix.