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Can a pure mathematician be a professional theoretical physicist?

  1. Sep 11, 2011 #1
    the title says everything. I'm studying pure-mathematics as a freshman but I also like to understand subjects like GR and QM. I have a good understanding of Newtonian mechanics and the classic theory of electrodynamics but I have found quantum mechanics and general relativity so confusing and unintuitive. SR is way easier though, but I can't fully understand it in a level that I can say I've mastered it like Newtonian mechanics. Can a pure mathematician study the TP courses on his/her own and be a professional theoretical physicist? If yes, what courses should he/she study on his/her own?
  2. jcsd
  3. Sep 11, 2011 #2
    I have a pure mathematics background as undergraduate, but is now pursuing a PhD in physics. Yes, I have seen people who did math PhD but later on works on theoretical physics while being employed by math department. GR and QM will become easier once you have the appropriate mathematics background [e.g. differential geometry for GR].

    I sort of agree with what Spivak says in his https://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322" that mathematicians find more advanced physics *easier* than elementary physics. GR was easier for me than say, Newtonian mechanics. :tongue:

    But if it is clear to you at this point that you would want to eventually work in theoretical physics [it wasn't clear for me back then], why not consider switching over to physics now as a freshman?
    Last edited by a moderator: May 5, 2017
  4. Sep 11, 2011 #3
    Thanks for the comment. well, It's clear for me that I do like to study GR and QM and I find theoretical physics the most astonishing and fascinating area that mathematics can be applied to but I doubt if I can say that I like physics more than math. Physics is amazing, it deals with real situations while mathematics deals with abstract and sometimes -one may argue that most of the times deals with- non-intuitive concepts and I also know that some theoretical physicists like Edward Witten even has been given prestigious awards like fields medal but still I think that an average pure mathematician has a lot better understanding of mathematics than a fairly professional theoretical physicist in general.
  5. Sep 11, 2011 #4
    Sounds like you might do fairly well at mathematical physics? I mean of course mathematicians understand mathematics better than theoretical physicists - because that's what they work in and spend all day on.

    I would answer a resounding NO. Why? Just look at the question - a pure mathematician is a pure mathematician, not a theoretical physicist - now sure, they could in addition also be physicists, but certainly not just by taking some additional courses.

    In my world, there is no such thing as a pure mathematician without some equivalent of a PhD in mathematics, and the same goes for theoretical physics. Usually, "equivalent of" could be dropped, but there are some geniuses who can conduct PhD level research in mathematics as youngsters.

    So what you are really asking, as per my interpretation of the words, is whether you can get a PhD + significant research depth and experience in pure mathematics and physics by taking some physics courses, and the answer is simply no.

    Can a pure mathematician learn some physics? Yes. There are many good books in physics which are geared towards mathematicians, although these usually assume very good command over mathematics.

    Another piece of advice - if you want to become a professional theoretical physicist, you'll need a great deal of depth of thought in certain areas of physics. Consider that some of your cohorts will have spent time both in class, outside, etc reading and working on physics topics. Reading a book helps, but you should spend hours and hours through a long period of time understanding these things, especially if you want to work in them as a professional. Being a professional in something takes like a thousand times the work it does to get a basic class out of the way.

    Now if your question is just - can someone doing a math major self-study enough physics to go on to physics grad school and then concentrate in physics to become a professional in physics, that's a different story. Sure. I believe you can learn stuff without classes if you really make sure you get the intuition and think hard about it. But you asked about a pure mathematician and theoretical physicist, which are different from math and physics majors respectively.
  6. Sep 12, 2011 #5
    so, maybe I should change my question to this one: Can a pure mathematics graduate go after a PhD in physics? How hard is that? Is it possible to self-study the bare minimum required to start studying physics in the PhD level if you hold a M.S degree in pure mathematics?
  7. Sep 12, 2011 #6
    The problem with doing that is you need to be able to convince the graduate school that you are fit to do a PhD in physics. By just self-studying you don't have anything to prove that you are ready for graduate study in physics. If you want to walk this path, it is best to take as many physics courses that you can during your undergrad or M.S., or you could do a physics related thesis for your M.S.
  8. Sep 12, 2011 #7

    Vanadium 50

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    If you're a freshman, and want to do graduate work in physics, why don't you want to do undergraduate work in physics?
  9. Sep 12, 2011 #8


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    I like this breed of students who clearly want to work in field X, are fully capable of doing their academic studies in field X without any problems intellectually or administratively, yet insist on doing A, B, and C and wonder if A, B, and C can lead them to still do X?
  10. Sep 12, 2011 #9
    I ask the same question as Vanadium 50. I'd say study physics, and then take math courses as they are helpful and add value first. This obsession with doing a major in order to learn something about the subject is understandable, but misguided. With maturity, you will learn to hand pick what is good for you, or at least ask people who know it well. You should aim to master basic undergraduate linear algebra first if you'd like to understand quantum mechanics, and also crash course yourself on Fourier analysis basics. At some point, you should take real analysis and complex analysis, and if possible, some complex geometry. Some group theory too. But just stuffing these math topics in early is not necessarily useful - learn what you care about, see the beginnings of it in a physics context, and then perhaps add to your toolkit with some rigorous math coursework. Else you'll just forget all the math, if your motivation is really physics!

    I do agree it's possible to enter a physics PhD after a math undergraduate degree, but just as developing mathematical intuition is definitely a transition to be made, so is developing intuition for the basics of physics.

    Your reason to pursue mathematics, as opposed to physics, was that mathematicians have a better understanding of mathematics. Sure, but physicists have a better understanding of physics - which would be what you'd want if you want a PhD in physics. Else, you can study some aspects of mathematical physics when you do a math PhD - it certainly won't be the same, but it may suit someone who cares more about mathematics.
  11. Sep 12, 2011 #10
    OK, this sort of post may be a little mysterious to someone just beginning undergraduate studies. The point is that if you want to do research in physics eventually, you should start slowly working up to thinking like a researcher. Which is why I think you should learn what you care about, and let yourself recognize the need for the tools along the way, rather than brute forcing. In research, there is an infinite amount of stuff you can be working on, and you have to let your interests guide what technology you pick up. It's fine to take some things on faith, but not everything.
  12. Sep 12, 2011 #11
    You will simply not survive research if you do this. You usually have to learn / think several times beyond the bare minimum required to figure a problem out before you'll figure it out. Not to mention that figuring out what problems are interesting requires immersing yourself in the field.

    Breaking news - there is no well-defined bare minimum. It is the attitude of "bare minimum," and not the lack of specific knowledge, which will kill you.

    Just consider this - would an expert who has spent 40 years in your field figure out your PhD thesis in a few weeks? If so, it's kind of trash. And experts can figure out a lot of things in a few weeks. You'd be surprised how hard it is to come up with an interesting problem that you can also make progress on.

    If you do not develop a serious attitude towards physics, you will not survive in the long term as a physics professional.
  13. Sep 12, 2011 #12

    Ben Niehoff

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    If you give yourself a strong physics background, you can do it. Which basically means you should major in physics, or double major in physics and math, or choose one as a major and the other as a minor, or do a major in mathematical physics if your school offers it. In any case, you had better take a healthy load of physics classes.

    I guarantee you are no expert in Newtonian mechanics. Some of the very deepest mathematical insights of physics lie in ordinary classical mechanics, and it will take much time and study before you will appreciate them.

    The mathematician's approach to mathematics and the physicist's approach are very different. Mathematicians are concerned with moving slowly and defining everything carefully so that everything makes sense at every step. Theoretical physicists plow roughshod through mathematical formalisms in search of something they can use, sometimes discovering new areas where the mathematicians must come years later to clean up and give precise definitions. To a physicist, the definitions are not as important as the results, because the ultimate test is experiment, and the sensibility of any mathematical formalism is measured by its predictions.

    If you study pure math, you will find what physicists do makes you very uncomfortable. In my experience, theoretical physicists tend to learn mathematics much more quickly than mathematicians, but we do not learn it very carefully, because we think of it as a tool rather than an end in itself. I once tried to audit a course in differential geometry offered by the math department...I found it excruciatingly slow, and decided instead to read some books on my own and learned three courses' worth of differential geometry in a few weeks. But I can't remember what "second-countable and Hausdorff" means, and most of the time I don't care.

    Also, theoretical physicists must be very strong in computation in a way that I think mathematicians typically need not be. Mathematicians tend to focus on proving facts about mathematics, and this requires synthesis of many disparate definitions into a logical whole, but rarely requires long and complex calculations. Physics requires much more computation of specific examples, actually solving complex equations rather than simply proving that they must hold. Your computational ability must be top-notch if you want to make any meaningful progress.
  14. Sep 12, 2011 #13
    On a related note, how difficult would it be to get a top-tier graduate school in Physics without having an official Physics degree? I for example, am a Math major right now but my main area of interest seems to be in Theoretical Physics, so I'm going to be taking all the upper-level courses such as Quantum, Classical E&M and others and might even be able to add a grad course or two on Quantum Theory. Why am I not doing an official Physics major then? Because of the course requirements such as lab, and math classes for which I already know the material to, and hence instead of those, I could be taking more advanced classes in Theory.

    I think this is another indication that I should go into Theoretical Physics, as I've always been told that my computation skills are rather strong, however from the little experience I have, I also enjoy doing proofs.
  15. Sep 12, 2011 #14
    For theoretical physics, I think there is quite a bit of value in taking a wide range of math courses. It's quite possible a math degree would prepare one for PhD research in theoretical physics better than a physics degree, depending on the research topic and undergrad math vs. physics program.

    This is almost certainly true in other fields as well given how important interdisciplinary research is nowadays. In short, I think these can be pretty darn good questions :smile:.
  16. Sep 12, 2011 #15

    Ben Niehoff

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    This is a quite silly attitude. You need those lab classes. They will give you valuable understanding of how your theoretical work relates to the rest of the field. A theoretical physicist without an understanding of experiment is just an armchair philosopher.
  17. Sep 12, 2011 #16
    Yes, but in order to get an understanding of experiment do you really need a solid 3 semesters worth of labs with heavy reports due every week? I understand the need to take a lab, and I will eventually take a lab or two, but right now I want to develop a more solid understanding and want to get ahead in theory. Also, I'm going to start research sometime this semester, which might help me develop an understanding of experiment in a more realistic way.

    Also, your stance is a bit debatable as well. Theoretical Physics programs in the UK, for example, usually don't have any experimental components to them, so would you call all Theoretical Physicists from the UK "armchair philosophers"?
  18. Sep 13, 2011 #17


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    For those of you in the know, what about people like Roger Penrose or John Baez? Didn't they have their formal training in mathematics and then switch to mathematical physics?
  19. Sep 13, 2011 #18
    I think Chiro brings up a good point, Penrose and Baez both have quite a strong grasp on pure mathematics and Theoretical Physics. Both have Phds in mathematics, (Penrose's was certainly pure mathematics) and work in mathematics departments but are very much respected in the field of physics. I think at the "highest" level mathematics and physics converge. Furthermore I believe that a Pure Mathematician would have a better shot at grasping high energy physics than a Physical theorist would have of understanding p adic number theory for instance.
  20. Sep 13, 2011 #19
    You cited the example of a course, and then claimed to have read several books and learned more quickly - this is simply stating that courses are methodical and slow. I don't think that's an indication of how mathematicians actually learn when they are forced to research.

    Usually, a course is conducted more slowly, because the point is to get intuition for something, not just use its results as quickly as possible. This holds true of physics courses too.

    For instance, if I flipped your logic in an analogous direction, what if a mathematical physicist needed some intuition as to a bunch of physics, and to use its language, but really was only proving mathematical results? They could probably learn physics more quickly than that material would be presented in a graduate course in physics, as relevant to their research, which after all, isn't in physics, much as the physics researcher is only using the mathematics as a tool.
  21. Sep 14, 2011 #20
    I was going to point out something like that.

    I do think there is some truth in his statement but it does not hold true for Physics graduates *only*. I'm certain the same thing could be said about an applied math grad.
  22. Sep 14, 2011 #21
    I feel like I can speak to this topic somewhat (at least the process), as I got my B.S. in mathematics, but am currently pursuing a Ph.D. in physics.

    One thing you have to realize is that you will have to pass qualifying examinations in physics before you can move on to Ph.D. candidacy. Meaning that, one way or another, you need to be prepared for the content those exams cover. You can be accepted into some physics Ph.D. programs without all the necessary coursework to pass those exams, but you will then be expected to either take remedial undergraduate courses or self-study to fill in those gaps.

    Seeing as how you are a freshman, I think that it makes sense to at least double major - that way you have more options when you graduate. (Or you may decide to focus on just one after you have a little more of both subjects under your belt.) I understand that you don't want to take certain courses, but, that's part of life. You will take courses in graduate school that you may not be interested in, either. And labs are a crucial part of a physics degree; depending on the lab course, you will sometimes talk about topics not covered in class, and be expected to read research papers describing the experiments, theory, and put the experimental methods together and interpret the results yourself. That is much more similar to real research than following a course given by an instructor.

    Also, take a breadth. You're just a freshman. I knew several of my fellow mathematics majors who thought they liked quantum mechanics or general relativity, because they liked some of the relevant mathematics topics. When they actually took courses on these topics from the physics department, they changed their tune.

    Best of luck!
  23. Sep 14, 2011 #22
    Just out of curiosity, why did they change their minds?
  24. Sep 15, 2011 #23
    ^ Because they don't care about using hardcore crazy physics equations to describe what's going on, and they care about pretty math pictures.
  25. Sep 15, 2011 #24
    For whatever reason, I think they thought just because they knew the math behind certain physical concepts that the physics would present little challenge. But being able to prove that, for a simple example, the set of square-integrable functions is a vector space has little bearing on whether or not you can actually solve quantum mechanical problems. So, when a professor would drop a line like "Hilbert space is important in quantum mechanics", meaning "Hilbert space is an important concept in constructing a mathematically sound theory of quantum mechanics", they would hear it as "If I know how to do proofs about properties of square-integrable functions, or the Cauchy-Swarz inequality, or whatever, I will be good at quantum physics." Then, they get to class, and realize that, while the "abstract" mathematics is very important, there is still a lot of solid physics going on. That's the best reason I can come up with. To make a long story long, I think it boiled down to the fact that the type of math they enjoyed doing, while present, didn't play a large enough role in the course.

    Haha, this, in some sense, was the case. It's kind of like taking an Analysis class; you talk about integrals, when they are defined, some important inequalities, and some methods for calculating them, but you very rarely actually calculate them. Following along with what I said above, it's the same idea: while properties of a square-integrable functions may have been interesting to them, actually finding a square integrable function that satisfies the Schrodinger Equation was not. And you have to be able to do both.
    Last edited: Sep 15, 2011
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