# Mathematical formalization of physics

Let me first give a vague definition of "theory"/"physical theory", see also. A (physical)theory is a collection of rules and notions that were successful in predicting a behaviour of an idealised physical system. This question is about the theories that have stood the test of time.

I would like to know if it is possible to express physical theories, as defined above, in the language of (higher-)categories and then to consider the "category of all the theories". Since I am not an expert this question will be quite vague and speculative. It looks that the book Differential cohomology in a cohesive infinity-topos, gives a partial answer to my question, but it is quite long and sometimes difficult to read. Would you kindly provide me with some shorter references?

Very naively, when you "take the limit $$c \to +\infty$$" in Special Relativity(SR) you recover Classical Mechanics(CM) and when you "take the limit $$\hbar \to 0$$" in Quantum Mechanics(QM) you also recover Classical Mechanics. In the same fashion by considering flat metric in General Relativity(GR) you recover Special Relativity. It appears that, in an appropriate setting (if such a setting would exist), that SR is a unique solution to a deformation problem and there is a canonical arrow SR $$\to$$ CM, and similarly for QM $$\to$$ CM and GR $$\to$$ SR.

Question Is it possible to interpret SR, CM, QM, and GR as objects of some well defined category? This question is only about the physics that are contained in the first 3 volumes of Landau and Lifshitz. However electromagnetism is not considered in this question.

EDIT : I would like to add two links to physics.stackexchange. I really liked Schreiber's answer to this question as well as to this question.

• What abstract existence results would you want to show? "For any two theories, there is a weakest theory that implies both, and a strongest theory that is implied by both of them"? Sep 29 '20 at 14:20
• You may want to read n-cat lab for this : ncatlab.org/nlab/show/HomePage Sep 29 '20 at 14:25
• A prerequisite would be getting a rigorous mathematical understanding of the known theories of physics. As of now that pretty much isn't the case for any realistic quantum field theory. Sep 29 '20 at 14:30
• Let alone quantum gravity!!! Sep 29 '20 at 14:30
• There is a rigorous mathematical understanding of general relativity, and in what sense classical mechanics can be seen as its limit. Section 3.2 of this paper by David Malament has an introduction to how that works: arxiv.org/abs/gr-qc/0506065 Sep 29 '20 at 14:52

John Baez (2004) discusses quantum theory and general relativity from a unified perspective provided by category theory.

Faced with the great challenge of reconciling general relativity and quantum theory, it is difficult to know just how deeply we need to rethink basic concepts. Here we propose that quantum theory will make more sense when regarded as part of a theory of spacetime. This calls for a category-theoretic perspective — in particular, one that de-emphasizes the primary role of the category of sets and functions, relevant for classical physics.

The category Hilb, whose objects are Hilbert spaces and whose morphisms are linear operators between these, plays an important role in quantum theory. The category $$n$$Cob, whose objects are ($$n$$− 1)-dimensional manifolds and whose morphisms are $$n$$-dimensional manifolds, plays an important in general relativity.

While an individual manifold is not very much like a Hilbert space, the category $$n$$Cob turns out to have many structural similarities to the category Hilb. The most puzzling features of quantum theory all arise from ways in which Hilb resembles $$n$$Cob more than the "classical physics" category Set, whose objects are sets and whose morphisms are functions.

• I'm not convinced by this paper's claims that these categories "play an important role" in the physical theories, but it is a bold and impressive attempt by Baez in the direction of the question here. Sep 29 '20 at 17:26
• @Matt F.: isn't quantum mechanics 'all' about Hilb and quantum field theory (at least in an attempt to a formalization) 'all' about Cob? Oct 1 '20 at 14:47
• I must unfortunately point out that this answer, nor the work of Baez pointed to, actually answers the OP's question. In none of the categories discussed in Baez's work could SR, CM, QM, and GR be objects. Oct 1 '20 at 14:50

It's not clear what it would really mean to formalize a broad physical theory such as quantum mechanics. Such a theory has some general scaffolding and then a lot of more specific stuff. Formalizing the scaffolding is typically easy but vacuous. Physical theories are not the same kind of beast as mathematical theories, and it's wrong to imagine them as sort of defective mathematical theories that could be perfected and formalized if only the physicists were a little more rigorous.

Some references for relativity:

Andreka et al., "On logical analysis of relativity theories," Hungarian Philosophical Review, 2010/4, pp.204-222, http://arxiv.org/abs/1105.0885

Hawking and Ellis, pp. 58-61: Spacetime is a Hausdorff manifold with a nondegenerate metric that has signature +--- and satisfies certain conditions on integrability and differentiability. They also introduce the Einstein field equation and stress-energy tensor, and two postulates (causality and conservation of energy-momentum), but these are sort of vacuous unless you have some separate theory of the matter fields that you intend to plug in to GR.

An effort has been made to formalize Newton's Principia and put it into a computer proof system:

Fleuriot and Paulson, "A combination of nonstandard analysis and geometry theorem proving, with application to Newton’s Principia," Lecture Notes in Computer Science 1421 (2006) 3, https://www.cl.cam.ac.uk/~lp15/papers/Isabelle/fleuriot-princip-CADE.pdf

People have come up with various axiomatizations of quantum mechanics. Review: Kapustin, https://arxiv.org/abs/1303.6917

Very naively, when you "take the limit c→+∞" in Special Relativity(SR) you recover Classical Mechanics(CM)

The Galilean limit of SR is a lot more subtle than this, especially when you consider electricity and magnetism. See Marc De Montigny, Germain Rousseaux, "On the electrodynamics of moving bodies at low velocities," http://arxiv.org/abs/physics/0512200 and work by Baccetti, Tate, Visser, Le Bellac, Levy-Leblond, Montigny, Rousseaux.

I don't think it is possible using current knowledge to perform the overall unification of physical theories that you suggest.

However, on a much smaller scope similar things are possible. For example, one can understand the matching between an effective low-energy quantum field theory and its underlying high-energy quantum field theory in terms of a category whose objects are the (formal) Lagrangian densities built out of the relevant fields, and whose morphisms are Legendre transformations and the formal integrating out of the high-energy degrees of freedom; then the matching is given by a morphism between the high- and low-energy theories such that a certain diagram commutes.

• Could you please give me a few references?
– A413
Oct 1 '20 at 16:16
• I'm afraid there aren't any published references which make the categorical angle to matching explicit. It's hiding in plain sight e.g. in section 3 (Fig. 2) of arXiv:1001.4783 or in section 2 of arxiv:1105.5309, but I'm not aware of an explicit statement outside of some unpublished notes.
– gmvh
Oct 2 '20 at 6:24
• Field theories are very interesting, however my question is about the physics that are contained in the first 3 volumes of Landau and Lifshitz, with exception of electromagnetism.
– A413
Oct 2 '20 at 8:45
• At the time of my reply it wasn't; there was even a mention of quantum gravity in the question.
– gmvh
Oct 8 '20 at 7:14
• After checking the edit history, you will see that the question(the text after the word "Question" in the bold font) always was: " Is it possible to interpret SR, CM, QM, and GR as objects of some well defined category?". It is true, however, that in the paragraph above this question there was a mention of quantum gravity. However it was intended as a motivation for this question, not as an end goal. I decided to remove it to avoid confusion.
– A413
Oct 8 '20 at 7:54