On the logical structure of physics

TL;DR

This paper proposes a novel logical framework for physics using Continuous Logic, modeling free theories with Gaussian potentials through pseudo-finite structures, unifying quantum and statistical mechanics.

Contribution

It introduces a model in Continuous Logic for free physical theories with Gaussian potentials, linking quantum and statistical mechanics within a unified logical framework.

Findings

Models quantum and statistical mechanics as domains in the same structure

Uses pseudo-finite fields to interpret physical theories logically

Explains Wick rotation as a scale transformation in the model

Abstract

One of the main claims of the paper is that Dirac's calculus and broader theories of physics can be treated as theories written in the language of Continuous Logic. Establishing its true interpretation (model) is a model theory problem. The paper introduces such a model for the fragment which covers ``free theories'', that is physical theories with Gaussian (quadratic) potential. The model is pseudo-finite (equivalently, a limit of finite models), based on a pseudo-finite field in place of the field of complex numbers. The advantage of this unusual setting is that it treats the quantum and the statistical mechanics as just domains in the same model and explains Wick rotation as a natural transformation of the model corresponding to a shift in scales of physical units.