Level of Rigor in Mathematical Physics

Question

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.

Answer 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.

Answer 2

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...

Answer 3

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.

Answer 4

I was also raised by physicists. Many mathematical discussions in my major (physics) courses left me dissatisfied. In particular, the treatment of vector calculus and variational calculus seemed mystical. Since then, I've seen what was hidden then and I understand why my teachers and texts did not show me the deeper aspects of that math. Another topic that comes to mind is that of point charges in electromagnetism, the plethora of dirac delta functions and formulas such as $\frac{d}{dx}\theta(x) = \delta (x)$ stood in stark contrast to my real analysis course. On the one hand, in electromagnetics, we were teased for not knowing the derivative of a discontinuous function was a distribution called a function. On the other hand, I lost points in my math course for failing to say $c \in \mathbb{R}$ at the point in the proof where it matters. The dissimilarity of the disciplines was manifest in those semesters of my undergraduate. What I see now is this: if the physics courses really did the math then they would never get that far into physics. They have a way of doing math with notation which is unreasonably successful in the majority of instances. So, why should they teach math in their courses which distracts from the physical content of the course? I'm guessing the underlying reasoning is something like that, but I will admit it is a popular opinion that the professors just don't know the math so they can't teach it rigorously. Arrogance of youth or truth? I suppose it depends on your school.

Ok, so to summarize, the standard core topics in physics can be made rigorous with study of advanced calculus, variational calculus, perhaps a course covering the theory of distributions. And, a good PDE course. Much of what is hand-waved could be fixed with known math. You'll find these things as you go on because you care.

The other case is the edge of things. I think Sharkos and Fedja's answers are fantastic, really think about what they say. There are aspects of physics which will remain hand-wavy because at the base of it all the equations which frame physics do not entirely describe physics. It is an art, there is some common intuition which goes beyond the mathematical description (in my opinion, this is obviously not the sort of thing we can prove). This is especially apparent in classical mechanics where you meet some of these creatures with a deeper intuition than your own. But, it's not just mechanics. There is a difference between physics students and a physicist. For the real physicist it's hard-wired in some sense. I've met a few, I'm not one. The reason in part we don't say all those parameters which clarify the problem is that our thinking in physics is a bit fuzzy. Physical intuition keeps us on point so a lack of mathematical clarity is not as dangerous as it might seem.

To answer your question, I don't think it's usually the case that graduate physics is more mathematically rigorous. On some points, perhaps. On other points, the mathematical heresy cuts even deeper.

Answer 5

The classical mechanics initiated by Newton was very much non-rigorous at the time of its birth. After three hundred years it looks like mathematics, due to the services rendered by hundreds of mathematicians in the mean time. Electrodynamics, as developed by Maxwell and others were filled with ugly mathematics (like Dirac-delta function) at the early stage, but after one hundred years with the advent of "calculus of differential forms on smooth manifolds" the situation is getting better. After another hundred years, one can hope for a far better Electrodynamics than today. This is the way physics and mathematics go together.

Answer 6

I completed my PhD in pure math (differential/Riemannian geometry) a few years ago, and have recently turned my attention to quantum field theory and string theory, purely out of interest and curiosity.

I've learnt that physicists seem to use a different style of logical reasoning and derivations in their work, and a different process in the development of their body of knowledge. They largely use the physics of a problem to guide their derivations and reasoning. They use the scientific process of hypothesis and experiment to guide the development of physics - what passes as acceptable knowledge in physics is ultimately that which is validated by experiment to an acceptable degree of accuracy. Physicists are interested in the final result of their reasoning of explaining the universe around us to an acceptable degree of precision, not necessarily in the level of mathematical rigour they displayed in getting to this result. They do physics not mathematics. These things take some getting used to and I understand that it couldn't really be any other way.

However, what frustrates me is I've noticed that physicists often use their own mathematical notations, definitions, terminology and even mathematical reasoning, even when clearer, simpler and in general better ways of doing things have been developed and highly refined within the mathematics community over a significant period of time. Physicists also often seem to assume large amounts of universal symbols, notation and terminology - they often proceed with doing a significant amount of work, derivations, reasoning, etc, with undefined terms and mathematical notation, assuming that the reader already knows the notation and terms as though they are unambiguously and universally accepted in the wider physics community and set in stone. I've read some theoretical physics books where the authors have used, and significantly drawn upon throughout the whole book, certain mathematical symbols and notations which they never actually defined in the first place.

You asked the question: "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?"

My experience is that mathematical rigour is less taken into consideration as you get into more advanced mathematical/theoretical physics. Things get very confusing and if you don't understand or are unable to verify something you're reading, it becomes very difficult to discern if it's due to just not following along or that what you're trying to understand or verify actually isn't well founded mathematically.

Answer 7

I think that is the difference between mathematical physics and theoretical physics. I have a course in mathematical physics and it is mostly concerned with mathematical methods for solving partial differential equations. I think that many problems in mathematical physics are problems that are inspired by physics, for example the $n$-dimensional wave equation. Mathematical physics tries to formalize many of the ideas that come up in theoretical physics, for example renormalisation. Mathematical physics applies the same rigour as other disciplines in mathematics.

Theoretical physics on the other hand is more concerned with the physics than with the mathematics. They are trying to find models for physial processes from which predictions can be made. They can then be tested in experiments.

You can read more about the difference here: Wikipedia: Mathematical vs. theoretical physics