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Confused between mathematical physics and theoretical physics

  1. Mar 14, 2014 #1
    Hello!! This is my second thread in here. I'm just finished my high school and waiting for the results to get admission in an undergrad physics program.
    I was confused about career guidance and academic guidance feel free to move my thread if I'm in the wrong place

    I was wondering about major options between mathematical physics and Theoretical physics.
    To reduce the confusion I can simply define mathematical physics to be a branch of theoretical physics but that doesn't work and my question is deeper

    I had a look in these pages

    One of the reasons I fell in love with physics was because of it's mathematical accuracy and elegance and how equations can tell so much what happens in nature. I understand perfection in mathematics is most important skill needed to learn/know physics.

    Theoretical physics seemed a good way to go. All I know about it is it has many branches,,, eg. QM, GR, Particle physics etc.
    What I din't understand was about quantum mechanics
    GR is a geometric theory of gravity,, gravitational red shift and all tensor analysis and getting solutions to equations etc (I'm well versed with Special relativity)
    Particle physics is all about muon, meson, electron, etc and their interactions
    Then there is also QED!!!!
    (I don't know anything about codensed mater physics or statistical physics but i know it's an extension of thermodynamics but that's again very incomplete knowledge.)
    QM from what I 'Understand' is about solving Schrodinger equation for different systems
    But there sure seems something bigger which I don't understand/know

    I also love pure maths - Number Theory (Enough to fall in love with proof based maths :smile: ), Higher algebra, Combinatorics (very elementary), probability (also very elementary) etc..
    I've started Set theory from Schaum , I hope to do a lot more in future

    Mathematical physics uses rigorous maths techniques like those from pure maths to solve problems in physics
    Wiki article says that "mathematical physicists solves some problems which are considered already solved by theoreticians" This was disturbing. But again I don't know how much to trust wiki.
    but again my knowledge is limited

    If I decide to do mathematical physics there are some topics like quaternions which I could start learning earlier than what they expect in the college. Whereas I could look into some other topics which are prerequisites for theoretical physics earlier if i decide to do that
    In the past,, knowing the things before they teach it to you benefited more than i expected

    Also I like the fantasy world of any subject which has less or no applications in real world. I know it's bad and discouraging but still..

    I feel something is wrong with what I think and sensing that I have some misconceptions which I have tried to express but fail to see myself

    To do what I love I should know what I love and to know what I love I should study/learn/know more and more things

    so here are the questions
    1.) What am I missing? (something big?)
    2.) What should I do?

    I hope the questions are not too vague!

    3.) Can I become an undergrad maths teacher with a major in mathematical physics?
    4.) Am I over thinking or worring too much?
    (((I understand it just keeps getting more and more tough from this point in my life and job market is bad for a theoretician. I want to be prepared very well under any conditions)))

    If anything else pops in I'll add in the process as the thread develops

    My apologies for any mistakes or the length..
    And thank you to everybody in advance for taking out time to read all this and answering it
    And lastly your'e all awesome!!!
    Last edited: Mar 14, 2014
  2. jcsd
  3. Mar 14, 2014 #2
    I'm too busy to respond to everything but I'll toss some tidbits out there.

    The intense rigor invented by mathematicians is an artifact of their cultural view towards mathematics as a subject of study independent of the applications of math. If you want to start asking questions about, say, differential equations irrespective of using them, suddenly all you can rely upon is the precision of your arguments, and nothing else.

    A good example of the distinction arises in the path integral formulation of quantum mechanics. Looking in great detail at each piece of the path integral, such at its measure, raises alarming questions of rigor (part of the measure, for instance, clearly diverges in the limit). In other words, it's not clear how, looking at just the mathematics, one can arrive at the conclusions the physicists do. However, guided by physical intuition and observation, physicists can work with these mathematical devices to great effect (they underly one of the most accurate theories in all of physics) without answering these questions.

    Pragmatically speaking, this interest in rigor is usually useless, but it is aesthetically useful I suppose.

    EDIT: Most mathematicians seem to do mathematics because they share the cultural view that mathematics is really independent of its applications. Although I don't personally feel this way, if you happen to, it would be a mistake to disregard mathematical physicists as "solving problems that have already been solved.", and so I think it might be worth pursuing.
  4. Mar 14, 2014 #3


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    I recommend stopping that, and to take up Hrbacek & Jech instead. It's one of the best books on set theory: https://www.amazon.com/Introduction-Edition-Revised-Expanded-Mathematics/dp/0824779150

    One of the nice parts about undergrad is that you don't need to make the decision now. It seems like you love math and proofs, and it seems like you're into physics. So it would make sense to go for a double major math-physics. Take math classes like analysis and abstract algebra. And take physics classes like QM, relativity, CM, E&M, etc. Do some undergrad research as well! This will give you an excellent preparation for grad school, where you have to make the actual choice of what kind of physics you're going into.

    Also, I don't like to destroy your enthusiasm, but mathematical physics is a really tough field to get into. I don't mean the material, but the number of jobs available is very few. And if you somehow drop out of academia, then finding a good job as a mathematical physicist is going to be very tough on you. So you shouldn't only look at what you want to do, but you should also prepare a back-up plan for when things don't go your way!
    Last edited by a moderator: May 6, 2017
  5. Mar 15, 2014 #4
    Wow. this was helpful

    I found this later so I partially understand a better scenario
    Fortunately I have a lot of time under my belt and I feel better to realize my interests in an early age

    I'll have a look and try to understand what you are telling, perhaps it would change my views,, Thanks.

    I agree on this. And I have good backup plan(s). And It's a lonng story, and I don't want to drag the thread off-topic.In short my uncle wants me to join family business after undergrad but I'm adamant enough in wanting to seek a phd and an academic job, he's very supportive if things should go wrong but I'm trying to become independent by not relying on him too much(I have very bad social skills (=17 percentile according to a reputed local aptitude test center) so business isn't a very good option for me)

    Thank you!! I'll have a look..

    Yes I know about undergrad research. I'll search a thread about it. All I know that it adds a huge value for my application to grad school + Experience is also a valuable. And ultimately I want a research/academic based career so It's a must for me
    Scientists in India are taking a lot of efforts to promote pure science so initiating an Undergrad research wont be a very tough,, rest depends on my input..
    Also I have to take courses which the program offers or I have to search a program with my needs or self study it. Self studying wont add onto the certificate (does it matter?), but considering the long run its useful and People in academia are very good at heart when it comes to helping a student so guidance isn't a problem and there is also PF!!
    Double major seems a tough job even for very hardworking people and it adds to your years in education field

    Double major(maths-physics) can seek me employment in maths sector? I don't see a reason why not,, but again I'm confused
    There are tons of threads I found on double major maths-physics!
    So I'll have a look there, and decide accordingly

    I do not know anything about the academic job systems, I'll search more threads to get an idea

    I agree on micromass' answer more about option between theoretical and mathematical
    Though I'll give it a deeper thought

    Thank you to both of you
    Last edited by a moderator: May 6, 2017
  6. Mar 15, 2014 #5


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    OK, so working with you uncle is a good backup plan (although it's not one you particularly like).

    Note that for many jobs you're going to need social skills, especially for job interviews. So you might want to consider working on that (I know very well this is easier said than done).

    It shouldn't be a tough job. Math and physics are so closely related that a major in physics will basically be enough for a minor in math already. I think you could do a math and physics major in a very reasonable amount of time. It's not like math and art history which are so disjoint that you'll need tons of courses to pull it off.

    And you don't need to go for a double major right away. Start college with the basic math courses you didn't take like calculus, linear algebra, diff eq. And then take some proof based courses and see whether you like them enough to do a double major.

    A BS in math or physics is a very weak degree for the job market. You'll see many people here saying they got a BS, but who can't find a job. If you can't at least get a Masters or a PhD, then things will be tough.
  7. Mar 15, 2014 #6
    Its BSc in India and it had no scope in here except maybe lab-assistant or clerk to manage scientific documents only because he knows what science means! As you say masters or Phd is a minimum for a decent job.

    I've finished calculus where I know how to differentiate and integrate vectors I don't know which level is that but I know gradient, divergence, curl and laplacian and all that i liked it very much. diff eq(ODE), I know till Bernoulli equation which talks about higher powers of 'y'.. PDE,, I know the analogies that are there with algebra, and variable separable etc (without proofs)
    So I've to do less hard work as compared to others in undergrad which maybe compensated with other topics(I have particular liking towards quaternions ) As you say I've to take some proof based courses, which is why I started Set theory which is needed in many proof based fields in maths as I know it.

    Thanks for this info. I'll ask in the university as early as possible

    Yes I know it's a harsh truth :frown:
  8. Mar 15, 2014 #7
    I think the first thing to realize is that "mathematical physics" and "theoretical physics" are just words, and actually pretty vague ones. Don't get so hung up on the words. Think more about what you'd like to do.

    A mathematical physicist would be someone a little like me (although I quit), who has a PhD in math, but is interested in the connections with physics.

    Mathematicians use a lot of intuition, so that's not exactly true. It's just that they like to back the intuition up by proof. And they are not actually always as rigorous as people like to think. There are a lot of reasons to be rigorous. If you've done math research, you learn how easy it is to be wrong when making complicated arguments. So, you become afraid of making mistakes, and therefore you see the need for rigor. Many mathematicians also recognize the importance of having an intuitive picture in spotting errors, as well. Of course, I say this as someone is kind of put off by divorcing math from applications. So, I do think the obsession with rigor and de-emphasis of intuition has done a lot of harm. Yes, Weierstrass had a point, but we do not all need to emulate him. He was an important mathematician, but also a bit of a boring one.

    Part of the problem, I think, is that mathematicians don't do a very good job explaining the point of the rigor. People take an analysis class and wonder what the point was. Well, if you were paying attention, you would have realized that you now know why it's okay to differentiate power series term-by-term within the radius of convergence and stuff like that. After a little experience with weird examples of series, you might realize how questionable that really seems without proof. It wasn't just a matter of setting the foundations for calculus. Not just being able to prove the stuff that's already obvious in calculus. But people don't emphasize that, besides pointing out that there are pathological cases where our intuition needs to be refined more, such as continuous, nowhere-differentiable functions. They do better when pointing out the logical necessity of rigor. The usefulness of rigor, less so. I mean, what often happens is that they have to point out lots of "niceness" conditions that guarantee that everything works as we expect and nothing weird happens. But most functions you run into in practice are nice enough. So, for example, the work Dirichlet did on Fourier series was probably good enough for most physical purposes, even though people like Riemann and Lebesgue got greedy and wanted to be able to work with Fourier series for uglier and uglier functions and that's how huge amounts of modern analysis got started. But there's no telling where that stuff might be important later on. Riemann's motivation seems to have been partly to study number theory, which back then was useless, but today has become important in cryptography.

    It's not clear to me that rigor wouldn't help to develop new theories. Maybe if you knew how to make it precise, you'd understand it better and see how to go beyond it. I have no idea whether that would be the case, but if you are trying to solve a problem, you have to throw everything you've got at it until something works. Leave no stone unturned. And of course, there's a big curiosity factor here. It works so well, even though the foundations are shaky. Why is that? Makes you wonder. Well, maybe some of the better answers are more physical and less rigorous, but still. Also, to suggest that mathematicians "just look at the mathematics" is not quite fair, even when they are trying to be rigorous. It's not that rigor is their be-all and end-all. A mathematical physicist probably is interested in physics in the first place because he has at least some liking for physical intuition.

    I definitely don't share that view, and that's part of why I quit math.
  9. Mar 15, 2014 #8
    I understand very less of what homeomorphic says I read it 3-4 times..:confused:
    I've yet to get admission to undergrad so forgive my illiteracy

    But I get a fair idea of what I'm supposed to do!
    Thank you for all your inputs!
  10. Mar 17, 2014 #9


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    A few days ago I was talking with a condensed matter theorist at a grad school visit whose research has been described to me as very mathematical. Even so, he is not a mathematical physicist. Although his work uses many advanced math concepts in topology, geometry, and algebra, it is meant to provide physical intuition about systems rather than mathematical rigor. I think this is the real underlying difference.
  11. Mar 18, 2014 #10
    I always thought a mathematical physicist was someone who expresses their research with very advanced math, regardless of rigor. Then, a mathematical physicist is a physicist who works with the same level of rigor as a mathematician, when doing their derivations? A physicist using advanced or abstract math, but not caring for rigor or proving their mathematical statements is not a mathematical physicist?
  12. Mar 18, 2014 #11
    That's why I said they are just words. People mean different things by them in different contexts. Personally, I don't care about the words. Theoretical physics usually means the part of physics in which you don't do experiments, so that's clear. But the difference between theoretical and mathematical physics is just a pointless thing to get stuck on to me. There are just a bunch of physicists, with various degrees of rigor and level of math. The ones that use more advanced math and/or more rigor are more likely to be called mathematical physicists. Either rigor or higher level math. Rigor tends to mean more advanced math because it comes with the territory. To be rigorous in physics, you might have to know a lot of functional analysis and so on, which you can just get by with a minimal amount of if you are an ordinary physicist.

    It's like saying someone is "tall". One person might say you have to be one height in order to be considered tall, and another person might have a different minimum height to be considered tall.
  13. Mar 24, 2014 #12
    So the value of rigor comes when one is making complex arguments? What is rigor really? It seems to be at once a slavish obedience to strictly defined terms, and at the same time a refusal to allow any mathematical statement to escape careful scrutiny. If you rely only upon the argument, this is very useful. But if you are going to do an experiment at the end of the day, or if you have already done an experiment, I would argue that you need not be so concerned with either of those two aspects of rigor. When you begin to reason about black holes and have no experiments to look at, suddenly rigor becomes much more valuable if you intend to convince anybody of your arguments, but nature's predilection for outsmarting us reminds us that faulty premises invalidate even the most flawless deductions. And it is these premises which rely purely upon intuition.

    A power series is two things at once: a mathematical tool, and a symbolic representation of a physical quantity. Taken out of the latter context, certain possibilities are open to consideration and necessitate some level of rigor. But in the context of physics at least, one runs into nice functions, and when one doesn't, the theory is probably breaking down. Functional integrals in the context of the path integral formalism are a good example; they are conceptually constructed from the notion that a path is a function, and a function has a taylor series, which looks like a vector projected upon a set of coordinates (the coordinates being polynomials of increasing power). The second statement no doubt raises a mathematician's eyebrows; there are not-nice functions which are not differentiable everywhere, and they haven't got a convergent taylor series. But the physicist happily makes brazen assumptions and carries on calculating, and in this case, the physicist was Richard Feynman and he won a Nobel Prize.

    The mathematics is a symbolic representation of the physics; it does not exactly encapsulate it. The mathematics springs from a very large bundle of atoms called a human too; as an imperfect computer, it cannot fully appreciate every fastidious detail of the outside world. Its function appears to be both to organize the reasoning in a very convenient way (it is much easier to think about algebraic problems with short hand symbols and equations than with conventional language) and to allow for extremely precise statements. The main utility of mathematicians to physicists seems to be not rigor, but the fantastic foresight into organizationally useful tools; historically, this has been mainly in the form of algebraic advances.

    Rigor from an analytic point of view has been of somewhat less utility, since this mainly pertains to the precision of the calculations (how do I know my path integral truly converges?); seeing how this precision arises from more fundamental assumptions is unnecessary given that one need only develop a consistent set of rules for mapping a calculation technique to numerical results. If these results agree with all experiments, no further comment seems to be necessary.
  14. Mar 24, 2014 #13
    You don't have to be as concerned, but it might be cheaper to be rigorous first and rely less on expensive experiments. Also, I think you shouldn't just be concerned about experimental correctness of results, but also the understanding of results. Too much empiricism means you are just recording what happens instead of analyzing it and understanding it. Of course, maybe physical understanding is more important there than rigor, but I think rigor can also have some role to play.

    Intuition + experiment + interpretation of experimental data. But once you have your premises, don't you want to make sure your reasoning from that point on is solid? Also, being rigorous doesn't mean that you can't also check your answer against physical intuition or experiment as well.

    This is perfectly okay if you are talking about merely verifying your theory. Not many mathematicians are going to say the standard model is wrong and will predict the wrong experimental results in particle accelerators. But if you want to move beyond that to a new theory, it's not so clear that explaining and understanding the results from every possible point of view, including rigor, is going to be irrelevant. Perhaps it would give some clues about further things to investigate. Maybe not, but how do you know unless you try it?

    Also, you might want to know WHY it works, rather than saying "it works because we did the experiment and that's how it came out". Having a theory that just gives the correct experimental results is not the only goal. A theory that you can understand more thoroughly would be better.

    I'm not saying rigor is necessarily the answer. I'm just saying I don't know that it can be ruled out as having nothing to say.
  15. Mar 24, 2014 #14
    I missed an important point here, which is that mathematicians actually have no rules against that, stereotypes not withstanding. What they do have a rule for is that when you are done with your non-rigorous calculation, you should go back and try to prove it. Of course, they may be more prone to falling into "rigor-mortis", but they have no rule against playing around with things non-rigorously until they are finalizing their results.
  16. Mar 24, 2014 #15
    I suppose this raises an interesting philosophical point; understanding "what's going on" has taken much less importance in physics since the early 20's onward. It is very difficult to develop a physical picture of what is going on.

    Interpreting the mathematics to explain "what is going on" if it has no utility in the development of theory usually results in enormous confusion, since converting the "nonsense" of say, quantum mechanics, into sense is seemingly beyond the human capacity; case in point, the apparent motion of particles through imaginary time when they tunnel through barriers. That peculiar discovery may have had something to do with attention to rigor, as important results in path integral quantum mechanics are derived by converting Fresnel integrals into Gaussians by surreptitiously dropping the complex unit, which can be justified mathematically if one things of time as being complex*

    The point, however, is that physicists seem to have generally given up attempting "understand" what such results might mean. Constructing a rigorous framework in which to understand the results independent of the physics seems to provide little guidance in developing novel theories. A simpler example has to do with probability amplitudes: so the physicist says that a Euclidean norm should be used to compute probabilities from wave-vectors. This result is completely baseless saving only that it actually has serious predictive power over reality, and many other such results ultimately rest upon this kind of physical thinking.

    Of course, I will eat my words when some clever individual develops a powerful theory which identifies, say, complex time as something other than just a mathematical curiosity.

    *this knowledge is pretty fresh in my head, so somebody with more experience might identify errors or caveats in my statements, unfortunately.
  17. Mar 24, 2014 #16
    I'm not sure about the historical route to that rule, but it is physically plausible once you accept that electrons are really wave-like. Intensity is usually proportional to the square amplitude of a wave. It's natural to guess that if you replace intensity with probability, it will give you things like the interference pattern in the double-slit experiment, by analogy to the classical physics of waves. So, I wouldn't say it's quite baseless, apart from its predictive power. There's some prior reason to expect it to work ("prior" might get tricky to define here, but let's ignore that). Then, you still have to check it experimentally, of course, because what if it isn't quite like your analogy suggests?

    Here's a possible way in which rigor might help. It could be that in order to make it rigorous, you'd have to think about it in a completely new way. A way that could yield more than just rigor. That's completely hypothetical, of course, but it illustrates how innovation can come from a direction that no one expects, and therefore, it's hard to rule anything out.
  18. Mar 24, 2014 #17
    Just to put some examples about QFT:

    "Local Quantum Physics: Fields, Particles, Algebras" (Haag and Kastler)
    "Quantum Physics: A Functional Integral Point of View" (Glimm and Jaffe)
    "Introduction to Algebraic and Constructive Quantum Field Theory" (Baez, Segal, Zhou)

    Those three books I call "Mathematical Physics" because (almost) everything is stated and proved with mathematical rigour.

    Also you can look at DarMM posts about "Rigorous Quantum Field Theory".

    On the other hand, books like:

    "The Quantum Theory of Fields" (Weinberg, 3 volumes)
    "Mecanica Cuantica" (Galindo y Pascual)
    "Quantum Mechanics: A Modern Development" (Ballentine)

    I would call "Theoretical Physics".

    As homeomorphic wrote earlier, not everybody makes this clear cut separation. I like to do it because they really are two completely different things, and I prefer to have two different names.

    I just put some examples about QFT, but the same is true in any other branch of physics.
  19. Apr 2, 2014 #18
    This turned out better than I hoped. Although I understood very less about the debate that took place.

    matt thanks for the books. I'll search them up

    Edit: I'll re-read when I understand more about maths and physics, possibly I'll follow micromass's advice about a double major.
    Thank you to everyone
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