Is learning (theoretical) physics useful/important for a mathematician?

Question

I'm starting to read The Princeton Companion to Mathematics, at the beginning it says:

A proper appreciation of pure mathematics requires some knowledge of applied mathematics and theoretical physics.

Some of my professors have told me that modern Mathematics require some knowledge about Quantum Mechanics and theoretical Physics.

I attended the second and third day of the José Adem Memorial Lecture Series by Matilde Marcolli at the beginning of this year, they were about Number theory, Quantum statistical Mechanics and Quantum Field theory. I did not understand a single word but somehow the few things I understood had a big impression on me.

Please give me some examples of pure Mathematics that require/use Physics.

Answer 1

As a graduate student of pure mathematics, I question how valid the statement that a proper appreciation of pure math requires some knowledge of applied mathematics and theoretical physics. Perhaps part of this comes from: where is the line that separates applied mathematics and pure mathematics? If we include the whole of calculus within applied mathematics, than this is without a doubt true. But I do a lot of work in analytic number theory, and I have a hard time coming up with theoretical physics that I employ. I also have a hard time separating theoretical physics from pure mathematics - both are largely mystical, loosely formatted, and open. One might argue that theoretical physicists concern themselves more with descriptions of the natural world while mathematicians only concern themselves with what is consistent rather than what is possible... but I don't know how I feel about that either.

I will say, however, that I think both applied math and theoretical physics rely heavily on 'pure math.' I think of fields such as Lie Theory, which I consider a pure math. Lie Groups are one of the fundamental tools used in theoretical quantum physics these days - the existence of many particles and symmetries is often suspected because of mathematics and thoughts birthed within Lie Algebras.

But perhaps it is nice to know the awesome power of pure mathematics sometimes. When learning about Lie Algebra, or any sort of Abstract Algebra, I think it might be very enriching to learn about the solutions to the quantum harmonic oscillator and/or particle in a box situations. While these can be done analytically (or through divine inspiration, as seems to be what my old professor expected of us), Dirac employed a very cool use of algebra to solve these things. This included group-like behavior and the creation of 'ladder operators,' and more can be found at wikipedia. In this sense, I do appreciate pure mathematics more because of this knowledge of applied mathematics.

A large amount of group theory can also be applied to quantum mechanics. The astounding properties of the Pauli spin matrices might seem flukelike, but they can be predicted and analyzed from a group-theoretic context as well. I suspect many find is vastly satisfying to be able to predict experimental results from entirely theoretical pursuits.

As a very pure mathematician, one of the questions I am often asked is 'Why do people care about what you do?' While I might come up with something good to say, I think it is reasonable to say that the majority of the work that pure mathematicians do will never find an application that people would declare 'useful.' Thus whenever a bit of pure mathematics is 'applied,' this re-convinces people that funding the study of pure mathematics is a worthwhile endeavor. I rely on that belief for my own funding, so in that I appreciate applied mathematics a whole lot :P.

I will end with one more example. I again refer to Dirac, because I happen to know a lot about his life and how he went about his research. Dirac championed the use of projective geometry (and what I think we would now classify as differential geometry) to discover physics. Here is a transcript of a talk he gave about this subject at one time. Geometry is an interesting thing, because it's often visually based or well-grounded on intuition. Although I am not a geometer, I nonetheless am very pleased whenever I can use a physical situation (even quantum physical, slightly less intuitive) to better interpret some sort of geometric situation. Similarly, it is nice to be able to apply a geometric intuition to an apparently non-geometric problem from the applied sciences. I really encourage a quick glance through the transcript.

This is what I have for now.

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I happened across a few links that I think talk about this subject nicely. One is a site that addresses how physics does not come from math alone - I take this as an example of how math is too general (in general) for physics. The second is a paper that talks about three very high-level math things that arose out of physical interpretations. A nice addendum, I hope.

Answer 2

It is certainly possible to study all kinds of topics in pure mathematics without any knowledge of physics, because you will always find literature/researchers who are used to explain the key concepts to fellow mathematicians without any knowledge in physics.

But here are some examples of useful interconnections, in arbitrary order:

1. The most prominent example of recent history is probably the work of Edward Witten on low dimensional topology that earned him the Fields medal. Wittens work is inspired by thinking about quantum field theory and extensions, using heuristic tools like path integrals. He was very successful, so that other mathematicians have tried to learn tools from theoretical physics, too, which led to seminars which led, e.g., to the two volume book "Quantum Fields and Strings: A Course for Mathematicians". His methods resulted in reducing the length of some proofs of Donaldson from 100+ pages to < 10 pages.

2. Classical mechanics is the study of symplectic manifolds (this is a simplification, of course). Much of the work of Poincaré on ordinary differential equations was inspired by the study of classical mechanics.

3. Principle bundles is about the study of both classical and quantum gauge theories, which encompasses electromagnetics, for example. The close ties have led Dale Husemöller to (co-) author "Basic Bundle Theory and K-Cohomology Invariants (Lecture Notes in Physics)", which is an extension of his classic textbook "Fibre bundles".

4. Quantum mechanics is about linear operators on Hilbert spaces. Most of linear functional analysis has been developed in order to understand quantum mechanics, espacially everything that von Neumann did.

5. Differential geometry on Lorentzian manifolds is the study of the theory of general relativity = gravitation.

6. Operator algebras (C* and von Neumann algebras) is about axiomatic quantum statistical mechanics, see Robinson, Bratteli: Operator Algebras and Quantum Statistical Mechanics 1 and two. And it is about axiomatic quantum field theory, see Haag: "Local Quantum Physics". The structure theory of von Neumann algebras finds deep applications here, for example the classification of factors is applied to local algebras, modular theory is connected to representations of the Poincaré group etc. You already mentioned the work of Connes et. alt. about noncommutative geometry and their application to quantum field theory and the standard model. Noncommutative geometry is also applied in order to construct quantum versions of spacetimes (i.e. quantum geometry).

7. Fluid dynamics is about solutions to the Navier-Stokes equations. The flow of ideal fluids forms infinite dimensional manifolds, see "V.I. Arnold, ; B.A. Khesin: Topological methods in hydrodynamics."

8. Someone else than me should explain the applications of complex and algebraic geometry to string theory. (And n-categories.)

All of these topics can be studied without mentioning theoretical physics, although I think it would be a shame to do that.

Answer 3

I suggest to read Missed Opportunities by Freeman Dyson an essay that describes missed opportunities in mathematics due to the fact that mathematicians were not interested in new developments in physics (e.g. investigating the symmetries of the new Maxwell equations).

It is certainly wrong that a knowledge of physics is required for mathematics, but a widespread ignorance of developments in areas that use mathematics leads to missed opportunities and missed intuition.

Edited to add: It might be worthwhile to note that the quote does not say that mathematics or doing mathematics requires knowledge of physics, but the proper appreciation of mathematics requires it. The book sets out to give an overview of mathematics and its development and requires some physics knowledge to do so. The fact that it is perfectly possible to work happily and productively in many branches of pure mathematics without getting your neurons dirty with physical intuition is not in contradiction with this.

Answer 4

Just a couple of isolated -- and manifestly non-expert -- remarks.

1) I am right there with you wondering about the usefulness of physics knowledge in mathematics. Here is my own story: I had the good fortune of attending one of the finest public high schools in the United States: Central High School. They had pockets of strength all around (in particular my senior year English teacher was one of the most learned people I have ever met; once I compared Greg Kuperberg to him on MO, and Greg was suitably flattered!) but in retrospect it is fair to say that their programs and math science were head and shoulders above the rest. I took every Advanced Placement science course they had to offer -- so Biology, Chemistry and two years of Physics -- and got college credit for all of them. But somehow I wasn't thrilled with these courses: there was a lot of memorization and solving of problems calibrated to be so-and-so difficult. As an undergraduate I went to the University of Chicago, which has one of the most extensive "core programs" among US universities. Moreover as a math major one had to take four non-math courses in the Physical Sciences Collegiate Division beyond core requirements. But because of all this AP, I was able to place out of all my core requirements and still have something left over to apply to the major requirements: the only "science" classes I took as an undergraduate were two quarters of computer science (in Pascal, if you can believe it). Instead I took, as you might expect, a bunch of math courses but also -- as you might or might not expect, depending upon how well you know me -- a bunch of humanities courses of various types: Shakespeare, classical film theory, Paradise Lost...(Each of these courses was a real treat and a welcome change of pace. I felt much more comfortable solving two problem sets a week and writing a paper every two weeks than solving four problem sets a week. Now that I think about it, maybe things haven't changed that much: I still find writing rewarding and even somewhat relaxing and make it a substantial part of my "work time".)

As a result of this, although I got some kind of training in the very basics of physics as 17 year-old, I have taken zero serious, university-level physics courses. For a while I was somewhat amused and somewhat self-conscious of this and asked around to see how common this was. I met a few people who had taken as little physics as I had; I can't remember anyone who admitted to taking less.

How do I feel about this state of affairs as an adult mathematician? I can definitely say that I wish I knew more physics than I do, or at least had the kind of robust, thoughtful understanding of the stuff I learned in high school that I do for basic mathematical concepts. I really didn't understand electricity and magnetism very well (let's be honest: deep understanding is neither required nor tested in AP courses): e.g. things like capacitors are very mysterious to me. This lack of understanding does come up in my work sometimes: once I refereed an arithmetic geometry paper in which there was some harmonic analysis on graphs that the authors wanted to phrase in the language of electric circuits. That was a complete no-go for me, and I insisted that they write in the language of mathematics, not physics. (I also borrowed a book on circuits from a colleague of mine who is a real expert in...ahem...capacity theory on algebraic curves. But I never read it.)

So, yes, there are aspects of mathematics related to my own research interests that I don't understand very well because of my lack of physics background. Of course there are plenty more aspects of mathematics related to my own research interests that I don't understand very well because of my lack of mathematics background in various topics: sometimes I fantasize about how dangerous I would be if I really understood algebraic stacks or derived categories, for instance. The point being that no one (almost no one...a few people active here and/or on MO come close enough to earn my admiration and envy) has a perfect background in everything that you would need to understand all the math you want to understand. My decision to read and write about Madame Bovary instead of, say, learn about quantum mechanics was the right one for me at the time. I am happy to be the guy who has spent his adult life knowing something about Flaubert -- it's part of who I am. Probably some day I will really need to learn some physics...and I don't think learning undergraduate physics is going to be any harder than learning about algebraic stacks, even if I wait until I'm in my 40's to do it.

2) [much shorter] Some parts of contemporary "theoretical physics" would be better thought of as being branches of mathematics: any differences that may exist there are more sociological than intellectual. This is already fairly true about string theory from what I have seen: string theorists are mathematicians who are bundled up together and working furiously on a tightly bunched collection of mathematical problems. It's like they are attending a working conference that has lasted for $20$ years: they're too busy talking to each other and making day-by-day progress to write things up according to the much more sedate standards of mainstream mathematical publishing, so it's a big flurry of arxiv preprints with arguments intended to convince each other rather than be formally complete...but does anyone really think these people aren't mathematicians? (The non-string theory physicists sure view them that way, from all I've heard.)

The OP mentioned Matilde Marcolli. I would say she's not even a physicist in the nominal sense of the string theorists: she's a mathematician through and through (I wish I were a mathematician like she is a mathematician). I just consulted her wikipedia article and was mildly surprised to learn that her undergraduate degree was in physics (but her thesis was on self-equivalences of fiber bundles!). It would be very interesting to hear more about her own background in physics and her take on the subject: from my outside perspective, it seems like it is mostly a source of interesting problems and intuition.

Answer 5

No, one does not need to know anything about physics to learn mathematics. But learning theoretical physics has at least two advantages. Theoretical physics could be seen as a set of good examples of mathematics, so this has some didactic value. And, if you are a math researcher, a lot of new math were born out of physical problems! So it's probably a good idea to keep an eye out for what's going on in the world of physics.

Answer 6

One example where I think some kind of physical or real-world intuition might be indispensable is in the study of differential equations. (OK, most of such study is already semi-applied in nature and arguably not "pure math", but pure mathematicians do occasionally draw on the theory, or need to understand specific equations.)

A central theme in understanding the solutions of ODEs and PDEs, even without solving the equations directly, is identifying quantities related to solutions that either don't change, or change in some regular fashion (e.g. they decrease over time, or whatever your name is for the variable you are interested in). You can think of these as just formal expressions that you want to find--- but in DEs of interest to physics, they often are physically meaningful things: quantities that are conserved or dissipated in some physically understandable way as time goes on. So in the literature, even in the pure math literature, you will see integral expressions for things that are arbitrarily given names like "energy" or "entropy", and then it's shown that for solutions of whatever DE you are interested in, they behave in some regular fashion. And you can understand this on a formal level, but I think it helps to have a background in physics and applications to understand why you might write these things down.

(As a concrete example, the Lyapunov method of showing that a dynamical system is stable involves finding an auxiliary function $V$ with specified properties related to the system you started with. You can think of it just as a formal thing that helps you do what you want--- prove stability--- but it is often called a "potential," and in applications you often find that functions that capture the application's sense of "potential energy", will satisfy the technical conditions you need in order to apply the method.)

Answer 7

A more simple example than the very good examples given above is: find the geometric shape with the minimal face area that is based upon the vertices of a cube. A mathematician would have to do a great deal of calculations, but a physicist would just drawn the cube in a bucket full of soap water.

P.S. In his book "The Music of the Primes" the author, Marcus du Sautoy, tells us about how the Quantum Mechanics helped to predict and give new information about pure mathematic research. It is a book easy to read and I enjoyed it a lot.

Answer 8

My own knowledge of pure mathematics is very limited, but I have observed that ideas from physics at least motivate "pure math" results. For instance, I have heard that Riemann's original proof the Riemann Mapping Theorem was based on intuition about electrostatic forces, as is Dirichlet's principle (http://en.wikipedia.org/wiki/Dirichlet's_principle). Similarly, some properties of harmonic functions stems from the intuition that an initial charge distribution on the boundary must stabilize.

So, in my own experience I have not encountered mathematics where knowledge of theoretical physics is "required" (but I have not dabbled in mathematical physics, where I'm sure this is true), but even at my elementary level I've seen a lot of mathematical ideas which are motivated by physics. Even more vaguely, I've found that some physical principles prove useful in grasping the content of mathematical theorems -- for example, thinking about degrees of freedom involved, special cases, etc.

Answer 9

I came across this site after reading the question posted here and some of the answers. I really enjoyed it. There are additional links on the page pursuing the discussion, providing examples to consider.

Enjoy! Chicken-or-egg? Math-Physics?

Answer 10

There's a big difference between mathematical physics and theoretical physics. The former is essentially a branch of mathematics. A physicist would motivate, for example, generalized functions, by pointing out that ever real measurement involves an integral (average) of an "observable field" with respect to a "test function." While you can certainly master generalized functions without ever speaking of measurements, I can't help thinking you will have a limited perspective the subject.

When I visited Cambridge, the pure mathematicians would send offprint request to the applied mathematicians by post. But then, to the former, DAMPT was the department of awful mathematicians trying to be physicists. Vive la différence.