BS in math to PhD in physics?

Undecided Guy

Hi, I'm new. I just graduated with a BS in mathematics (pure bent) from a state university in the US. I ended up taking 7 graduate courses - two semesters of measure theory, a semester of functional analysis, two semesters of complex analysis, a semester of general topology and a semester of algebraic topology - receiving 6 As and one A-. In addition, I took a variety of undergraduate courses (hyperbolic geometry, probability, two semesters of applied PDEs, numerical analysis, number theory, undergraduate algebra, etc), all with high marks (gpa in the 3.8ish range).

However, what I have had precious little of is physics. Outside of the introductory course I was required to take, I've taken none. I've started studying Griffiths QM and EM books this summer. Getting to the end of my undergraduate degree and looking towards graduate school, I realize that I'd much rather be in physics than mathematics, for a variety of reasons. I'm going to obviously talk to professors in the physics department I just graduated from, but I was wondering if anyone could give me some advice on this:

I'd like to go into graduate school for physics. I do not want to do another undergraduate degree. Is it possible to make the transition from mathematics to theoretical physics? Will my pure mathematics training be entirely wasted?

Edit: Deep apologies if this would have fit better in the Career Guidance section.

Jorriss

1) Of course your pure mathematics degree wont be entirely wasted. Math on its own is very useful in physics and the level of math you've taken has assuredly pushed your ability to think.

2) It is possible to make the transition but you will need to study a lot and take the physics gre. Classical Mechanics, E&M and quantum mechanics at minimum. Thermodynamics and stat mech too to be competitive.

Given your background, you may appreciate a textbook like Cohen-Tannoudji more to learn quantum from.

Undecided Guy

Is it more theoretical? I'll admit, the mathematical sloppiness in Griffith's book has caused me quite a bit of stress (much to the humor of my physics friends).

Also, am I correct then in thinking that the sort of math I've taken is not directly applicable to physics, possibly outside of general problem solving techniques?

Edit: And thank you very much for your input. It's greatly appreciated.

Vanadium 50

Yes, most of what you learned in math is not directly applicable.

The question you have to ask yourself is "why should a graduate school take me?" and you need to make sure your application reflects that. You should also expect that if they do take you, they are going to make you spend some time - perhaps a year, maybe more - catching up.

Jorriss

It is more theoretical than Griffith but it still would not satisfy most pure mathematicians. You shouldn't be stressed out by the sloppy math though - it's physics, the aim is different.

For undergraduate physics and even most intro-grad courses, it's not useful. But there are niche theory areas that use pure math so, sure, it's possible your math could be useful but probably not most of it.

Undecided Guy

That does make things seem a little difficult to impossible. I can't say why any graduate school should take me. My experience is entirely in pure mathematics. I have complete confidence in my ability to do the work, but I think a lot of people can boast that. And a graduate program wouldn't have to waste time bringing them up to par. I was considering taking a job and trying to take a graduate physics course or two in the fall to prove myself, so to speak. I know I still need to speak with my department to see if there's anything they can do for me.

Out of curiosity, is there anything in particular you could suggest to someone serious about wanting to make such a transition?

Thank you very much for your help!

I have confidence that there is a firm mathematical foundation for the physics used; but shouldn't these details still be checked? After all, couldn't we end up in a lot of trouble if we're careless with, say, building a theory based on an inappropriate interchange of limiting operations?

Jorriss

They are checked, over and over again and by many mathematical physicists. But the point of physics textbooks is to explain physics, not math. It's largely unnecessary to explain mathematical details.

Mépris

There is one who goes by the handle of CarlBrannen on physicsgre.com - he went into a PhD program in physics from a math major. He may have lots of interesting insight to offer. Why not create a thread there and see what he has to say? Actually, there was a similar thread there a few weeks ago.

yenchin

I have made the same transition. I have a B.Sc. in pure math, followed by a M.Sc. in math [but my thesis was largely theoretical physics]. I am now pursuing a PhD in physics. I think you really have to convince the admission committee why you now want to switch to physics, for my case it was easier since my M.Sc. is already geared towards physics [having also taken a few physics courses like cosmology and quantum field theory during that period].

It is perfectly natural though to be uncomfortable with how physicists do things, with the lack of rigor and all, but as mentioned before by other posters, physics is physics and math is math, they are not the same thing. There are then two choices: if you are really bothered about rigor, stay in math, and you can still do lots of mathematical physics. If you want to switch to physics though, you have to make the transition at some point, and be comfortable with the sloppiness. Again, you should seriously ask yourself *why* would you want to switch, and whether you would be happier doing mathematical physics in math department.

If you do decide to switch, I assure you it would be a fun ride to pick up physics along the way. Eventually you will, hopefully, get used to the lack of rigor [yet you should still be able to switch to mathematician mode as and when necessary]. Remember that some parts of physics like quantum field theory does not even yet have a satisfactory rigorous foundation in math. But that is ok. Even in the early stage when calculus was first invented, it was not rigorous; epsilon and delta came much later.

deRham

I am no Vanadium, but my thought was along the lines of asking you what you'd like to do within physics, and how you arrived at it. You have done some serious study in mathematics, and you no doubt realize how different doing mathematics is from hearing a little bit about it. Thus, what convinced you that the kind of work you'll be doing in theoretical physics is what you would really like to do?

I think underlying the answer to that question is a huge part of the story of how you'll make the transition. It will expose the link between your current path and your future path. The mathematics you've learned is most certainly of interest to mathematical physicists, but your actual physics background will definitely need a lot of beefing up (something I can say even with no knowledge of theoretical physics admissions).

One option is of course to go into mathematical physics, and see if you can do something that is very directly related to what physicists do. Some mathematical physics is just interested in the structures that arise in physics, studied as mathematical objects, but there are undoubtedly those who interact very meaningfully with the physics community. Perhaps some study in physics (taking courses as a non-degree-seeking student) would help aid your plans.

Undecided Guy

There are two answers here. I'll give the more pertinent one to academics first. The thing that bothers me in math is the lack of practicality. When I took calculus, I was incredibly bothered by the lack of justifications. My professors would assure me that things were true, but I had no understanding for why things were true, or how anyone could be so confident that these arcane looking manipulations were even valid. I could follow the heuristic justifications, but they hardly seemed adequate (appeals to things like infinitesimal dx's just didn't sit right with me).

So I decided to pick up a copy of Rudin and teach myself analysis, on the promise that this clarified things. And it did. I felt like I had a firm grip on the precise mathematics of approximation and estimation. Differentiation, integration, continuity, approximation...all of these concepts had precise meanings in my mind. Moreover, I now felt like I could use them in the sort of calculusy, wave your hands way, but knowing all the while that I could insert the appropriate epsilon and delta as necessary. And this made going back to calculus problems fun. I was working with tools I understood very well, and could work with them in complete confidence of how the operated, without having to resort to faith or pedantic formalism.

And then I kept going in math. Results became so qualitative and non-constructive in nature that it was difficult for me to imagine how these things could possibly be useful. I appreciate solid proofs, and knowing a result holds almost everywhere sure does say a lot, but I was hoping that my mathematical education would run somewhat in parallel to the physical problems motivating its development. Instead, the desire for abstraction and generality seemed to move math out of any relation to reality.

I want to study physics because it seems to have the aspects I like about math, only firmly founded in something relevant. When I'm asked why I care about measures on compact topological groups, I don't have an answer for myself, much less anyone else. Physics doesn't seem to have the same problem. Its questions are motivated for obvious reasons.

I could keep going, but I feel like I've more or less beaten this horse to death. Math cares not at all about its relevance. Physics is grounded in it.

Of course, there is another reason, which isn't very good.

espen180

From what I've seen, current research in theoretical physics is very mathematical. Check out John Baez's column "This week's finds in mathematical physics" to see what I mean. I am currently wavering in the opposite direction. In order to be able to do theoretical physics, I am going deeper into abstract mathematics.

I think that with a solid math background, learning physics is greatly accelerated. Instead of grinding through elementary mechanics and E&M, one can begin at the level of Lagrangian mechanics and the covariant formulation of E&M. Quantum mechanics and be immediately understood at the general Hilbert space level without taking introductory wave mechanics etc.

denjay

One possibility is looking for an Applied Mathematics program that works very closely with a Physics group.

If you're completely set on applying to Physics programs, take some practice Physics GREs and see how you place. This should give you a very good idea of how much you know and how much work you need to do. Keep in mind that people typically spend 3 years going over the subject matter covered in the test and still some do not do well. If you think you can place well with much less time going over the subject matter then by all means pursue it.

ZapperZ

Start by reading this thread

http://www.physicsforums.com/showthread.php?t=64966

Zz.

deRham

I find it hard to say that myself, though. If you said engineering has reasons motivating themselves, that makes a lot of sense to me - building more earthquake-safe things, etc, is pretty self-explanatory. But it's not so easy for me to understand why you are interested in studying how light interacts with matter, geometric theories of gravitation, etc, etc, when you could study measures on locally compact groups. Lie groups are interesting objects, and the symmetries provided by Lie groups might find their way in some potentially less abstract form into a gauge theory text.

I would be careful about transitioning to physics before you try doing a lot of it. Doing physics is bound not to be the same as learning something (even nontrivial) about it. Doing a discipline ends up being about fiddling with some small aspect of it, not altogether unraveling the mysteries of the universe in one fell swoop.

I think mathematics does aim to study objects that are somehow significant. It just strives for greater generality than is often immediately needed in a physics context.

If you truly enjoy working with the mathematics to describe the objects physicists care about, in the fashion they do (and if the lack of rigor does not bother you now that you are able to fill in a lot of the rigor yourself), then perhaps physics is indeed a choice for you.

What sort of physics, though? Physics can be mathematical to varying degrees. Some physics enthusiasts would probably enjoy a lot of the mathematics you seemed not to. Others won't really care about that. You'll have to find that niche, and then start building your coursework accordingly.

It will be easier to move in a mathematical physics direction, but if that's not what you want (that is, probing more mathematical structures that have connections to physics but may not be directly applicable), you are best off just studying physics for its own sake. Gaining coursework as a non-degree-seeker is fine. I think physics graduate schools will be happy to accept someone with a math major and a fair amount of physics background without a physics degree, because it would mean you can obviously handle the math and the physics.

One reason I'm taking care to address all this so lengthily is that I really think you can ask about the "relevance" of any pure discipline, and I count physics as a pretty pure discipline. Doing a lot of stuff with the field will be what convinces you whether you're mistaken as to how interested you are.

I mean, let me ask - why not chemistry or engineering?

deRham

Studying pure physics isn't exactly the most practical path - trying to play with divergent integrals arising from calculations about particle interactions to gain fundamental knowledge about the fundamental particles constituting what we encounter in the universe is practical? I find that not much more practical than studying an abstract structure that has potential inspiration from things we have encountered in every day life. Perhaps you'd be more interested in using principles of quantum physics to solve some kind of subtle, complex engineering problem.

Your pure mathematics training is "wasted" only in the sense that you seem to want to go away from that sort of thing.

By the way, I'm only asking all these questions for potential benefit. It's quite possible you know what you're doing and will go on to a fine physics career, but as someone who has questioned his path many times, I like to offer cautionary remarks that have helped me when thinking about these sorts of things.

deRham

Lastly - what is your "not very good" reason? If you would like to message it privately, or just not discuss it, that's understandable.