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

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.

Solution 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.

------Post Edit------

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.

Solution 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.

Solution 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.