On the logical structure of physics and continuous model theory

Journal article

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 core model-theoretic challenge. 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 quantum and 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.

Authors: Zilber, B

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Monatshefte für Mathematik
• Volume: 208
• Issue: 4
• Pages: 839-880
• Publication date: 2025-05-13
• Acceptance date: 2025-03-25
• DOI: 10.1007/s00605-025-02081-2
• EISSN: 1436-5081
• ISSN: 0026-9255
• Language: English
• Keywords: Hilbert space; Continuous model theory; 05C66; Wick rotation; 81P05