I am a physics/math undergrad and I have recently become familiar with some more rigorous formalisms of mechanics, such as Lagrangian mechanics and Noether's Theorem. However, I've noticed that the writing on mathematical physics (at a level that I can understand) that I've been able to find is not nearly as rigorous as math writing. It often relies on heuristic reasoning or assumptions. I understand that this is sort of how physics is often done (at least this is how it is taught at my uni), but I was wondering if more advanced physics/mathematical physics is as rigorous as pure math? Or is this lack of rigor something I will just have to accept as I move on in physics?
Making assumptions in physics doesn't bother me, but I feel that sometimes I see arguments in mechanics that are very hand-wavy. I feel I'd get a better understanding if the author would explicitly state whichever assumptions are needed: then the argument could take the form of a proof instead of heuristic reasoning.
I appreciate any insight into the subject. Thanks for any help and sorry if this question is too vague.
Best Answer
It varies. A lot. The vast majority of physics you are likely to encounter at undergrad will be in the category of "things which can be formulated into theorems and rigorously proved". There are notable exceptions. For example, the foundational assumptions of statistical physics (around mixing and ergodic theory) are used fairly unjustifiably.
Things can be much more shaky closer to the forefront of physics research. In particular, quantum field theories, string theories and their ilk are treated rather more confidently than their foundations allow. Yet certain results about them are proved rigorously, with and without assumptions that everything is suitably well behaved. If you are a rigorous, meticulous person there are many many areas of research which are very thorough.
There is a difference between the above and things like "assuming that the solution to this equation is continuously differentiable" in mechanics, or (in some cases) "assume that this PDE has a differentiable solution" perhaps (with standard but tedious proofs, or difficult and off-topic proofs) where it's simply that it would be a completely different course to discuss the foundations. (Though this would also be the case for the above anyway.) I agree that providing references or quoting theorems would be nice here. It's not done often because it's considered unnecessary or boring or off-topic...
The simple point is that the answers to the above issues are not known but at the same time are almost certainly expected not to be pressing issues because making these handwaving assumptions gives good physics. There are plenty of corner cases and exceptions of course, which is why one major type of physics research amounts to finding exceptions to rules. Even when (to some extent at least) rigorous results are proved, it may not immediately be obvious what the loopholes are (supersymmetric theories come to mind, as related to Coleman-Mandula).
Ultimately, I feel strongly that teachers should make clear the distinction between known but off topic and unknown but physically plausible; but do be prepared to find a community which has to make assumptions because it is grounded in experiment rather than axioms. It would be insane to refuse to listen to anyone in particle physics in the last century simply because axiomatic QFT is hard.