Axiomatization of physics

Question

I'm studying mathematical physics and I often suffer from the fact that "big theories" in physics never have explicit axioms (quantum mechanics, QFT, GR, ...), or at least fundamental rigorous principles.

I am not talking about the fact that physics aims to describe the real world, but about the fact that the mathematical models used are often not well defined (at least to my actual knowledge).

Is this really the case? Is there some area of math that aims to axiomatise some physical models? Could you recommend some literature?

Answer 1

In Mathematics, you start by axioms. then you choose good definitions and then you prove theorems. in general the result you get don't change with time.

In Physics, you start by Principles. Then you check the result you can predict by experience or test. In some cases, you have to change the principle if some experience give unexpected result.

In Philosophy, you start by Postulates. But every one has own postulates and own conclusions.

Answer 2

This is far from a complete answer, but it illustrates the basic fact that such axiomatization is very much a work in progress.

One Millennium Prize problem asks us to prove each compact simple gauge group is associated with a certain theory that would be of great interest to physicists. Reading that article, you'll find out about the Wightman axioms, which are one of several choices for axioms that such a theory ought to satisfy. We're not entirely sure which axioms are a sensible choice. If the problem is ever solved, it will likely involve saying, "Let's go with this list of axioms instead".

Why is physics in this situation? I'll only mention one contributing factor.

When a physical theory is shown with either empirical evidence, a thought experiment or a calculation to have a serious problem, physicists patch it to address this problem. Mathematicians aren't generally familiar with a need to do this. They arguably should be, but only in a historical sense; nowadays, "use these axioms and you'll be fine, and never mind the false starts before we got there" is a standard approach in undergraduate or higher mathematics education. When physicists have to "compromise" their axioms (if they even think of it in those terms; to be fair, mathematics itself largely didn't before about 1900, despite Euclid's influence), it's kind of like when Russell's paradox forced mathematicians to patch set theory. How to patch it has been the subject of some controversy ever since!

Mathematical theories often have equivalent choices for their axioms, such that which you'll use depends on what's convenient in your context, especially from the standpoint of pedagogy. In physics, the way problems historically evolve the subject compound this issue. Let's pretend it's 1800 for a second, so electromagnetism, special relativity, quantum mechanics etc. aren't an issue. Do you use Newtonian, Lagrangian, Hamiltonian or Jacobian mechanics? Horses for courses, but they're equivalent. As new physics is added, you often have to jump around between different choices of standpoint for further insights. This is an awkward analogy, but if a choice of axioms were like a basis of a vector space, it would be as if you had to rotate to another basis to get a now useful perspective.

Having said all that, I think you'll be interested to read about specific examples of axiomatisation, though I don't recommend specific texts. You'll want to read about the examples I've mentioned so far, Haag-Kastler and Mach.