BS in math to PhD in physics?

  • #1
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.
 

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  • #2
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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?
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.
 
  • #3
Given your background, you may appreciate a textbook like Cohen-Tannoudji more to learn quantum from.
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.
 
  • #4
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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.
 
  • #5
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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).
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.

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?
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.
 
  • #6
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.
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!

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.
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?
 
  • #7
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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?
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.
 
  • #8
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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.
 
  • #9
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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.
 
  • #10
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Out of curiosity, is there anything in particular you could suggest to someone serious about wanting to make such a transition?
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.
 
  • #11
. Thus, what convinced you that the kind of work you'll be doing in theoretical physics is what you would really like to do?
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.
 
  • #12
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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.
 
  • #13
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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.
 
  • #14
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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.
Start by reading this thread

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

Zz.
 
  • #15
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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 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?
 
  • #16
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The thing that bothers me in math is the lack of practicality.
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.
 
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  • #17
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Lastly - what is your "not very good" reason? If you would like to message it privately, or just not discuss it, that's understandable.
 
  • #18
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Physics has a very practical side.

My impression is that the people driving progress in the computer industry, for example, are usually trained as physicists, not electrical engineers. That's a reason to get excited about physics.

Nuclear fusion is another practical goal.

Even the less practical goals in physics seem to have more philosophical significance than pure math does. The fact that it is dealing with reality makes it seem inherently more relevant.

I don't think I'd feel any better in string theory or loop quantum gravity than I do in pure math, though.

Here's the general argument for the importance of mathematics:

1) Math research is cheap, so it doesn't have to produce that much in order to be worthwhile.
2) It's impossible to predict applications. Who would have suspected number theory would be useful in internet cryptography? People didn't have any idea the internet would ever exists, throughout most of its history.
3) The applied stuff has deep connections to the pure stuff. Because everything in math is interconnected, it can't really be divided into pure and applied. So, some of the pure stuff has some influence on the practical stuff.

Be that as it may, some people, including myself would feel better if they knew that what they were doing was more directly relevant. Also, I'm not convinced that just any old approach to math is as good as any other. More specifically, I don't think the mathematical community has the right balance of pure vs. applied stuff going on right now. I think a lot more thought needs to be put into making the pure stuff more relevant. Much more contact with physics, biology, and chemistry is needed than what I have seen. If only 200 mathematicians in the world can understand something, and only after spending years and years learning it, I see a danger that no scientist will ever be able to learn it and apply it outside math.

Math is interesting, but it's really, really hard work. So, it seems fairly depressing to me if that hard work ends up having no value to society.
 
  • #19
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I see no problem with someone being a little disgruntled with where his/her mathematics work is going - there's a reason not everyone pursues mathematics, even if talented at mathematics, as a career. Picking physics for the very practical side is, to me, a pretty good goal, but one that should be done without trying to pick a field where one's mathematics training will be useful (the reference to physics having "aspects of math" that the poster likes got me a little concerned ... because so do lots of other fields).

I think discussing what "reality" is happens to be more of a matter of philosophy itself than either physics or mathematics. Physics studies physics - a description of "reality" is to me the point of studying almost any discipline, not just physics. Physics might isolate some salient features of reality to study that are fundamental in some sense, no doubt. Nevertheless, a lot of physics probably studies subtle phenomena that probably won't impact me very much in my lifetime, unless I'm vastly mistaken. Depends very much what kind of physics.
 
  • #20
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Nevertheless, a lot of physics probably studies subtle phenomena that probably won't impact me very much in my lifetime, unless I'm vastly mistaken. Depends very much what kind of physics.
Er... have you seen a smartphone lately?

You ARE aware that the largest percentage of practicing physicists are in condensed matter/material science, aren't you? This is the very same field that not only is involved in many of the modern electronics that you are using, but also has produced some of the most fundamental aspects of physics that permeated all through the rest of physics (look up spontaneous broken symmetry, and where the Higgs mechanism came from, for example).

Zz.
 
  • #21
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I already acknowledged I am aware there are very practical sides to physics, but that there are sides that aren't primarily concerned with practicality, and I'm commenting on someone's plan who claimed to be interested in physics after barely some introductory work, having done lots of mathematics clearly, and believing physics has the "aspects of math" he is interested in.

That is what concerned me.

If someone were to claim interest in physics primarily for its contributions to society, and with significant exposure to the kind of work that would entail, I'd not raise this doubt.

I do have friends who went pretty strongly into physics because their electrical engineering direction led them to it. However, their original training, motivations, and attitudes were quite different from someone with primarily mathematical training. Further, what each side can tolerate doing for long periods of time seems to differ greatly, no matter what they value as an end result.

Hopefully that makes it clear my point is not in conflict with yours; that said, thanks for the info - I wasn't aware of what the largest percentage of practicing physicists does; it's always nice to learn something new.
 
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  • #22
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Career choices are always hard because you never know what your chosen career actually entails until it's too late. It's hard to get an idea, even if you talk to people who already are doing it, although that is your best bet.

Personally, my motivation is always to understand things conceptually. I think that has a wide range of application and can often be turned in a practical direction. My primary interest in science is its beauty, but practical applications are still a secondary goal.

I do find that a lot of the things that appeal to me about math are there in physics. I've studied physics, computer science, electrical engineering, and math. Math happens to be the one that I went the furthest in, by far, but I never saw any of these disciplines as being all that different from each other, at least as far as the theory part goes. It's all pretty much math to me. The part of all these things that interests me is conceptualizing, and it doesn't seem like it's any different in the different fields. It's just different contents. The thought process is the same. Less conceptual thinkers might not realize the similarity as much as I do.
 
  • #23
I advise you not to leave mathematics. I had a similar crisis, mine being brought on by taking real analysis, complex analysis and Abstract algebra in my first year. However when I actually sat down to do physics I realised that Theoretical Physics seems to be more focused on calculations than understanding. I don't know how to convey what I mean by understanding, but let it suffice to say that a physicist is happy when his findings agree with experiment whereas a mathematician cares more for the reasoning and logic of the process. Physics, even Theoretical is a poor substitute for Mathematics in that respect.

What is more Mathematics is infinite there will always be problems to solve, whereas Physics may only be finite. Yet even if we did have a fundemental "Theory of Everything", we will not be able to determine why this theory is the case and not something else. Theoretical Physics is a branch of mathematics which aims to determine a set of axioms for the physical world. I think most of the understanding in theoretical physics comes from the mathematics.
 
  • #24
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While I'm in the mathematics camp myself, I think "understanding" is a term of potentially many meanings, and to some, the idea of furnishing results where calculations match with experiment could be more rewarding in terms of saying they "understand" the phenomenon than theorizing about different structures that model the phenomenon, as terrific a goal as this may be. I personally love the idea of theorizing until the mathematics elegantly reflects fundamental physical phenomena in a "natural way," perhaps more "natural" than the original form these things were discovered in, but that's probably why I studied mathematics in the first place.

Homeomorphic raises a good point, which is a better articulated version of something I wanted to get at, which is that the various "mathematical" disciplines share a lot more than one may realize at first. Of course, the issue is how they are practiced by the actual professionals can differ hugely - even the difference in lingo is a huge gap, when describing the very same things. I wanted to bring up that if physics shares some of the things one may like about mathematics, well so can things engineers, chemists, and many others work on - it depends how mathematical one wishes to get. Love for the discipline should ultimately be decided based on the particulars of the content rather than just the general flavor (as opposed to the specific flavor of a particular sub-region of the field...aka how it feels working in that area in practice...).
 
  • #25
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Zz, I don't want to drag this thread off-topic but are you implying that the job prospects of physicists with training in theoretical/experimental condensed matter or materials' science actually have *good* job prospects? I'm guess all those high technology firms, like nVidia for example, employ some of them?

I advise you not to leave mathematics. I had a similar crisis, mine being brought on by taking real analysis, complex analysis and Abstract algebra in my first year. However when I actually sat down to do physics I realised that Theoretical Physics seems to be more focused on calculations than understanding. I don't know how to convey what I mean by understanding, but let it suffice to say that a physicist is happy when his findings agree with experiment whereas a mathematician cares more for the reasoning and logic of the process. Physics, even Theoretical is a poor substitute for Mathematics in that respect.
That's not just physics! The same could be said for any field that uses mathematics heavily.
What is more Mathematics is infinite there will always be problems to solve, whereas Physics may only be finite. Yet even if we did have a fundemental "Theory of Everything", we will not be able to determine why this theory is the case and not something else. Theoretical Physics is a branch of mathematics which aims to determine a set of axioms for the physical world. I think most of the understanding in theoretical physics comes from the mathematics.
Is the universe finite? If yes, and if we go by your definition of what theoretical physics is about, then you're correct.

I googled to check if my initial hunch was correct and it turns out that it is "uncertain whether the universe if finite or not."

In any case, let's say mathematics consisted of an infinite number of things and physics did not. In either (math or physics) case, one scientist will be focusing on one part of that huge set. So, what does it matter anyway?
 

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