Level of Rigor in Mathematical Physics

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.

Solution 1:

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.

Solution 2:

The way I see it is that physicists use "hand-wavy" or "proof by intuition" because it is precisely that intuition that originally led them to use that mathematical model to describe the physical phenomenon in question. Physicists don't often concern themselves with which technical assumptions are needed because they are only interested in cases that arise in application, rather than pathological scenarios. As a crude example, Newton used calculus to describe basic mechanics without thinking about whether the relevant functions were differentiable everywhere, or whether the integrals converged absolutely, etc.

That we can, after the fact, rigorously prove many of the claims that physicists merely have intuition for is in many ways an affirmation of the incredible power of mathematics.

If you are interested in learning mechanics from a mathematical point of view, I suggest Mathematical Methods of Classical Mechanics by Vladimir Arnold.

Solution 3:

Unfortunately, the space between words in the construct "mathematical physics" is just half as stretched as in "mathematical art". Both things exist but what a professional mathematician would create as "mathematical art" and what a professional painter would do differ enormously and there are all shades in between. Both the level of rigor (the thing you noticed) and the relevance of the model to the material world (the thing you didn't mention) in the math. physics books can be anything from $0$ to $1$ completely independently of each other and many authors never tell you the exact point in $[0,1]^2$ they stand upon. "Feynman's lectures", say, are mostly at $(1,1)$ despite it is a pure physics book. I'll abstain from bringing up a $(0,0)$ example but you should be aware that it also exists.

You do not need to accept anything your mind revolts against. You can always change the textbook (or even the field of study). A book (or a lecturer) pursuing rigor will not always present a rigorous derivation of something, but will always either mention a reference to such derivation, or make a clear statement that at the current stage it is impossible to make perfect sense of this mathematically either at all, or without some particular leaps of faith, with full understanding that those leaps of faith seem to be correct as far as all our experience is concerned but if a counterexample arises in some range, it will invalidate the model and may invalidate the conclusions in that range as well. That's as much rigor as you can possibly expect there.

At last, it is worth mentioning that pure math. at high level is not as "rigor oriented" as one might think. It is true that you cannot claim a result until you have a watertight proof, but when mathematicians discuss how one could possibly approach a problem, the imagination, associations, and intuition play huge role. If you allow me a simple metaphor, the frontier of mathematics is not a set of theorems cut in stone but a cloud of ideas (many of which are half-baked). It is the ability to ride this cloud, which makes you a top mathematician, not just the skill of carving and masonry. Unfortunately, nobody knows how to take anyone else on this cloud ride without having him fall down or getting him lost in the mist within a minute, but it is known how to cut the cloud shapes out of stone and teach people to climb these shapes and to build them themselves. That's why we make students learn proofs and solve problems paying attention to every detail. The hope is that they'll eventually figure out how to fly. But until they are able at least to climb to the top of a simple brick pyramid of medium height, there is no chance they'll be able to stay in the lofts for long on any type of more advanced craft. Needless to say, no educationist will ever mention that...