Advice on doing physics under the umbrella of mathematics and the converse

Question

In the current scenario of research in QFT and string theory (and related mathematical topics), which of the following would an undergraduate student, like me, be advised to do and why if s/he is interested in both the foundations and pushing the frontiers of these subjects and doesn't differentiate between them or thinks that different set of skills and attitudes are required (plus the student is not so strict about mathematical rigor and even precise, logical, hand waving arguments suffice to convince him/her)?

1. Get into a maths department for graduate studies and work under people having similar interests and probably also take advice from people in physics department.

2. Get into a physics department for graduate studies and work under people having similar interests and probably pick up the mathematics one needs along the way.

Moreover, what would such a student majoring both in mathematics and physics advised to concentrate on during his/her undergraduate education?

It seems to me after reading this post and general experiences of mine that often mathematicians are more willing to accept physical ideas than physicists willing to accept mathematical ideas. Also, the number of people willing also seem to be more in mathematics, at least to me. Also, it seems that some physicists often develop a kind of hatred for mathematics and always remain skeptical that any mathematics can ever do good to physics. On the basis of this, I am inclined towards option 1. Please feel free to correct me.

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Comments

I think concerning the topics you are interested in, mathematicians and physicists help each other to make large progress in both fields. Mathematicians like it to see when their work has implications for fundamental physics questions, and the physicists appreciate it when the mathematicians can help with rigorously proofing their findings. Both fields gain a lot from these "positive" interactions. Shing-Tung Yau beautifully describes this in his "Shape of Inner Space" book, spiced with interesting and funny anecdotes and personal experiance.

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So it is probably not that important if you approach these topics from the mathematicians or physicists side (and it is probably not irreversible?) and you should just do what feels best for you personally and interests you most.

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Just a comment about the last thoughts, my experience is exactly the opposite; I know many physicist (all theoreticians) that love to learn and work with advance mathematics.

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I think you should be aware that the set of topics, and the skills acquired, in math or physics context will be very different. Even if the subjects are logically related, the choice of problems to work on the approach to these problems will be very different. There is some overlap between what math or physics people are doing under the umbrella of ST or QFT, but that overlap is not all that large.

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Answer 1

Frankly I think Yuji's deleted answer hits the nail on the head, in that concerning yourself with the sociology of various departments is rather a bad idea. "Mathematician"'s answer is essentially the opposite of this advice, making very generic claims about the nature of researchers in various departments, and I believe this is poor advice.

The exact purview of various university departments are not constant across different universities. I know physicists working in at least 5 different departments (physics, mathematics, materials, chemistry, engineering), and mathematicians working in a similar number (mathematics, physics, computer science, biology, engineering). Even talking in generalities, theoretical physics can be found in either its own department, or as part of physics or mathematics departments depending on the university. Making claims about what research papers will look like based on the departmental affiliation of the author is crazy. I myself wrote Theorem...Proof type papers while in a Materials department, and wrote papers without such structure while in a maths department.

When choosing where to do a PhD, you should choose it almost entirely based on the supervisor, not other factors. You'll be shaped far more by your supervisor and their group than by any other factor in grad school. Forget about what department they are in. Many universities don't even include the department on your degree. When asked what their PhD is in, pretty much everyone gives the area not the department. The factors which should really concern you are things like whether a potential supervisor wants to work with you and whether you think they would be good to work with, whether their research is the kind of thing you are interested in, whether they give their students much of their time or whether they are absentee supervisors, the funding situation, etc. None of this is department specific.

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Comments

+1: One thing you might want to worry about, which I've seen cause trouble for some interdisciplinary people, is being able to teach enough physics (or mathematics) courses to get hired by a physics (or mathematics) department. So make sure you take a reasonable range of courses in graduate school. This shouldn't be that onerous, since I think this will also very likely help you with your research.

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Answer 2

If you're not bothered by the need for rigour, I'd recommend option 1, mathematics. You are going to want to get a job afterwards, and in theoretical areas, physics departments seem to get caught up in fads. There was a great fad for hiring in string theory for quite a while, and people pursuing other theoretical directions were out of luck. Now, there seems to be a fad for phenomenology, and people pursuing other theoretical directions are out of luck.

Mathematics departments are much less susceptible to short-term fads like this. If you're doing substantial mathematics (i.e., some of your papers read Theorem ... Proof ...) that's related to theoretical physics, you should be able to get a job.

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Comments

Most string theorists I know are in Mathematics departments.

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We must know a different section of the string community. I see only a tiny minority of string theorists being employed in math dpt., mostly because anyone with physics-only education would have hard time teaching math courses, but also because the fraction of the community with interest in mostly mathematical research (as opposed to using math for physics purposes) is pretty small.

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@Moshe: whence my advice to take enough courses to be able to teach in both math and physics departments. You may not need all that many ... if you can teach some of the basic courses and a few advanced undergrad courses, I think you should be considered hireable.

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That's a good advice for interdisciplinary people. I just don't see all that much overlap between string theory and math, both in methodology and goals. There is some overlap, for some people very large overlap, but I think they are a minority. It is not enough to classify ST as a truly interdisciplinary field, say like biophysics, which seems to me implied in this discussion.

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@Moshe: I think there are a number of string theorists in math departments, and a lot of people in math study quantum field theory. I agree that the goals and methodology of math ST people and physics ST people can be quite different.

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@Moshe: Maybe. I've no idea what the true distribution is like. My sample size is relatively small (10 or 12) and is not all that geographically diverse.

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