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Guide to learning mathematical physics

  1. Oct 10, 2014 #1
    Currently, I am a graduate student specializing in algebraic geometry. On the other hand, I have also become extremely interested in the mathematical physics. However, I am not sure what steps I should take to get to the modern frontiers of mathematical physics research. My mathematical knowledge covers basic graduate analysis, algebra and topology with an emphasis in algebraic geometry, and as for physics, I know up to basic quantum field theory at the level of P&S and basic general relativity at the level of Wald.

    However, unlike those fields that I have studied so far, I am not sure what to learn in order to learn the basics to get to the research level, i.e., able to fully comprehend and dissect research papers in the mathematical physics journals. Most mathematical physics books that I have seen so far are only mathematical methods used in physics. To re-emphasize, I don't simply want references of mathematics used in physics, I wish to know the fundamentals that mathematical physics have to master and concrete examples of modern topics in the field. However, if it is essential, I would like to know what the main relevant mathematical topics are.

    Thus, for my main question: What specific books/papers should I start reading to understand the fundamentals of mathematical physics at this point and in what order should I read/study them?

    As for side questions: I do not really understand the basic knowledge that a mathematical physicist should have. Do they specialize in a particular area of mathematics or is it mostly topology and geometry or must they know other applicable areas such as functional analysis as well and to what depth?
    Would me continuing to self-study algebraic geometry be compatible with learning mathematical physics at the same time? What main fields are there now and what advanced books/papers could I read regarding them after learning the fundamentals as addressed in my previous question? What are the most relevant mathematical topics? Off the top of my head, I can think of mostly functional analysis and topology and geometry.

    Last edited: Oct 10, 2014
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  3. Oct 10, 2014 #2


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    I don't think there's such thing as 'I want to learn mathematical physics', because mathematical physics is not really a particular topic. What you can learn, however, is physics with a mathematical taste. :)
  4. Oct 10, 2014 #3
    Oh alright. How would one approach learning the prerequisites to understand research papers, and the modern research areas in mathematical physics? On that note, what are the main modern research areas?
  5. Oct 10, 2014 #4


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    It's quite difficult to try to reach research level, if not quided by a researcher himself. I mean, to reach a profound level in a subject, you first need to learn the 'basics': QM, QFT, GR and then, according to the research group's interests, the niche where you'd get to work in a short time.

    Example: Solidly knowing QFT and GR at an advanced level (say Weinberg for standard QFT and Wald for GR) would get you on the proper path for quantum gravity which is most well known to be done under 'loop quantum gravity'.

    As for the mathematical background, there's never too much, but there's always too little: differential geometry however is a must (and I'd say algebraic topology and functional analysis).
    Last edited: Oct 10, 2014
  6. Oct 10, 2014 #5
    Thanks-you. Do you know specifically what kinds of fields there are? So what should I start off learning first and are there any specific good references/books that you would recommend? As for QFT, should I learn all the variants of it such as conformal field theory, topological qft, etc.
    It just seems so much more unclear as to what to prepare for in terms of doing mathematical physics. For example, to learn any other field thoroughly such as algebraic geometry, one would just search for relevant books and read them in order, occasionally picking up other books as references or to do more problems. But for this, I'm not even sure what to start with, besides just doing different kinds of math.
    I suppose one of the main things I'm confused about is, before one does any actual research, what exactly is the difference between the training/preparation for a mathematical physicist and a pure mathematician? It seems that mathematical physicists basically just learns mathematics, except it's not focused in a particular field and has some physical applications.Do mathematical physicists often get insights behind the way a physicist thinks about problems as well?
    Last edited: Oct 10, 2014
  7. Oct 10, 2014 #6


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    There are so many fields in theoretical physics, I don't know where to start.

    The theoretical physicist's carrer 'must' path is at a minimum level:
    Lagrange+Hamilton+H-J mechanics > electrodynamics + special relativity > quantum mechanics > quantum field theory > general relativity.

    These are already specialized topics, even active in the field of research. QFT is typically done as: standard QFT (at the level of Weinberg vol.1+2), then axiomatical QFT at the level of Bogolubov, Logunov, Todoriv (1975) supplmented by Haag (1992) for the algebraic approach.

    Only then you can tackle CFT or TQFT or SUSY-QFT.

    Prepare to apply pure mathematics to the study of nature.

    Physics is learnt in the minimal path I mentioned above in bolded characters. Physics is learnt from textbooks and lectures by teachers (usually professional physicists, already involved in research). For each theory of physics there are books on the subject which vary in difficulty from introductory to hardcore.

    First I'd argue that a mathematician's insights are always welcome in the field of physics. And their rigor in terms of not letting statements unproven. Ever since GW Leibniz, mathematicians are welcome to write (about) physics.
    When a mathematical physicist, you're already focused on a particular subtheory of physics and to that end you should master all mathematical tools which have a relevance for that topic. For example, a relativist should master differential geometry and its prereqs: linear algebra, real and complex calculus, set theory and point set topology. He's not truly expected to know very well the mathematical methods of quantum field theory, like functional analysis, probability theory.
  8. Oct 10, 2014 #7
    Thanks for the detailed response. So basically, a mathematical physicist chooses a particular subfield of theoretical physics and applies pure mathematics to it? It appears that quantum gravity appears to be the most active field for this right now. In such a way of characterizing it though, it seems like mathematical physics is more physics than mathematics since from your description, it doesn't seem like mathematical physicists would do as much research into the pure math in their field, however, I have heard many say that mathematical physics is basically just mathematics since there is a clear distinction between a theoretical physicist and a mathematical physicist.

    It seems that qft in general, and all those variants of qft are the most active right now, so for now, would it be a good idea to read over a book like Nakahara and learn the topics I haven't learned yet, and then focus on those parts of qft, as well as mastering the relevant mathematics? Regarding learning the mathematics, how in-depth should I go about it? For instance, I learned category theory from Awodey's book, but I'm not sure if I have to go even more in-depth and start reading research papers regarding the mathematical topics I learn about.

    And also, having taken the physics classes that you mentioned as an undergraduate (I was a physics major too, should I start reading the relevant mathematical physics papers after learning enough of the mathematics? As for preparation, I was referring to, for example, what to self-study to get ahead in terms of training and getting a head-start into the field.

    One more thing that confuses me about this in general, is really the depth in mathematics that one should go into (I mean, it shouldn't be as little as a regular theoretical physicist, but probably should be as in-depth as a mathematician specializing in that field), as well as the depth in the physics that one should go into for the same reasoning as above. Are there dedicated mathematical physics topics/books to learn from that is a balance of the two extremities?

    As for my previous question regarding how one should learn mathematical physics, I suppose I was mistaken. You gave a response regarding learning physics, but I was addressing mathematical physics in particular, which I can't really find dedicated books aside from mathematical methods. Would books like qft from a mathematical perspective be of this type?

    And finally (sorry for the long list of questions but I've been wondering about this for some time), is it commonplace for a mathematical physicist to also specialize in either a field in mathematics or physics besides mathematical physics? What I really wanted to do was work at the boundaries of mathematics and physics and it seemed like mathematical physics was the only place where I could do that, and moreover I see that the only places where the two fields coincide most obviously are in dedicated departments of topology and theoretical physics.

    Last edited: Oct 10, 2014
  9. Oct 10, 2014 #8


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    So many new questions, but too little to add from myself. It is an ambitous task/goal you've set up for yourself. Just remember that research *as teaching* one doesn't do by himself. Guidance is the most important factor in reaching knowledge goals.
    I hope someone else (working in the domain, really) can bring new. fresh insights into your desired answers.
  10. Oct 11, 2014 #9
    This will tell you everything you need to know, at least if you want to study the things Baez used to:


    All the stuff you need is under This Week's Finds, fun stuff, and seminar.

    Here's a general overview:

    Personally, I found out that it was way too hard to get really deeply into both math and physics. This is not just me. A string theorist told Atiyah he wanted to learn algebraic geometry and Atiyah said, "You can't."

    And he didn't mean him personally. It's just that algebraic geometry is a full time job to keep up with, as is string theory.

    Things like this are part of what lead me to quit math and physics for good, as far as coming up with any new results is concerned. I suffer from wanting to understand things very thoroughly, though, which makes me too slow. People who are less particular about that are capable of going further than I did. Look at Baez and how far he got, for example. My goal now is only to try to clear the way for more people like me, so that they can understand things more thoroughly and still learn fast enough to succeed where I failed.
  11. Oct 11, 2014 #10
    Thanks! There are a lot of useful things on the site. However, is there a particular reason he didn't mention any algebraic geometry texts in his books page? I heard that mathematical string theorists need to know a lot of k-theory and noncommutative geometry.
    So I guess if people do mathematical physics, then that's basically the only thing they specialize in?
    What exactly differentiates string theorists from people like Baez or mathematical physicists in general? Is it just the fact that mathematical physicists have to know a lot more pure math, but not necessarily specialize in those areas of pure mathematics?
  12. Oct 11, 2014 #11
    Not really what he knows.

    Apparently, yes.

    No, mathematical physics could mean a lot of things.

    Baez was in the loop quantum gravity camp, so slightly different math there. Also, he was a bit more of a mathematician, but still one of the few mathematicians who is that deep into physics.

    Maybe. Mathematicians who nominally work on string theory can be so far from the physics that they don't really even know what the actual string theorists are talking about. There are only a few people out there who really know both physics and math, Ed Witten being the canonical example.
  13. Oct 11, 2014 #12


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    Congratulations on entering grad school,
    In my opinion, a mathematical physicist should have all the basic knowledge in math and physics. I recommend learning them if you do not have one and do not underestimate the importance of them.
    For university math knowledge, group thoery and complex manifolds may be the most important in the field of mathematical physicist.
    Also, I think that special and general relativity can come before quantum mechanics.
  14. Oct 11, 2014 #13
    Oh alright, so basically, I need to know both the general pure mathematics and theoretical physics to contribute to research in mathematical physics. So what exactly, do mathematical physicists specialize in if they don't really specialize any subfield of either pure mathematics or theoretical physics?
    I find it quite ironic for Atiyah to have said that of string theorists when he himself, was a renown mathematical physics himself, which meant he must have known a lot of both pure mathematics and theoretical physics. Or I suppose that merely means that one cannot hope to master both sides, pure mathematics and theoretical physics, and must settle for a balance in order to contribute to mathematical physics.
    By the way, thanks a lot for recommending Baez's site, it was very helpful in terms of giving references for relevant books, do you know of any other similar source?
    So just to summarize, to make sure I actually understand, mathematical physicists work at the boundary of mathematics and physics and are distinct from pure mathematicians and theoretical physicists in that they do not specialize in a particular subfield of either math or physics, but needs to know a sufficient amount of both areas, and more importantly, they do not have the luxury to be able to specialize in both as pointed out by Atiyah?
    Also, I just had one more question. Regarding the modern topics of mathematical physics, are the variants of qft and string theory the most active right now?

    Thanks for all the help!
  15. Oct 11, 2014 #14
    I know nothing in these respects, but I do know that the head of the mathematics department at UCSB is a leading reasercher in both theoretical physics (in his case string theory) and in algebraic geometry.

    Here is a link to his web page where you can find links to class notes he has and published papers, I hope this helps.

  16. Oct 11, 2014 #15
    Different things. The problem is that mathematical physics isn't a very well-defined term. Could be conceivably any branch of physics or physics-inspired math.

    He knows some physics, but I don't think he knows physics all that well, on the same level as a specialist in string theory. I thought my adviser was almost like a physicist, but it turns out, yes, he may know some QFT and maybe even a little bit more than that, but he's not really a physicist. Any reasonably good grad student in physics would probably know more than he does about their area of specialization. I suspect Atiyah might be similar. He also said something to the effect that there's a huge gap between a mathematician and physicist, even someone like Penrose (his mathematical brother who became a physicist, after a PhD in algebraic geometry).

    You can always hope, but reality could strike and show you how hard it can really be. Most likely, you'd be stuck trying to find a balance.

    If you read his This Week's Finds, it has lots of papers that you can try to read and more books. I'm not sure if there's anyone else who has quite so good a presence on the web.

    Well, no, they usually do specialize in some branch of physics. Very specialized, just like most mathematicians. Most just have to find their little niche. Baez worked in loop quantum gravity. He had a few string theory ideas, but I don't think he knew string theory that well.

    I'm not sure. I think fluid mechanics might be a big thing, and that's in a pretty different direction.
  17. Oct 11, 2014 #16
    I suppose it's the fact that mathematical physics isn't a well-defined term that led me to these questions in the first place!
    I was actually looking into David Morrison, and it seemed like he's both a master mathematician and theoretical physicist, yet I don't think he really deals with mathematical physics itself per se.
    So basically mathematical physicists do specialize, but usually in branches of physics with direct applications from pure mathematics and vice versa?
    It would be the ideal field to go into for someone that loves both pure math and theoretical physics but dislikes the much more abstract topics in math, and also some more rather applied areas in physics like condensed matter?
    And finally, it seems like most mathematical physicists work in either functional analysis, symplectic topology, or algebraic geometry, along with their corresponding applications in string theory or fluid mechanics. So I guess I should concentrate in these areas? I was actually quite confused when I was introduced to the notions of the many variants of qft (axiomatic, algebraic, conformal field theory, etc.) do you have any insight into how this relates to math-phys and the topics as described above?

    Thanks again!
  18. Oct 12, 2014 #17
    Yes. Some people are able to have more breadth, but it can be hard these days.

    Maybe, but there are some pretty abstract topics in math that can come in. Condensed matter might be more interesting than you think. If I had stayed in math and could have my pick of anywhere to work, I'd probably be working at Microsoft station q on topological quantum computing, which is condensed matter theory but uses some of the same math as string theory and loop quantum gravity.

    I don't know. Sounds a little bit too broad for a grad student. Typically, you're pretty limited as a grad student if you want to graduate in a reasonable amount of time. Usually, you have to choose one or two main areas. It can be good to branch out a little bit, but in a few years, you can only do so much. You'll figure it out.

    I worked on topological quantum field theory, which is sort of like conformal field theory, with the metric thrown away, but I don't know too much outside of that. You probably know more QFT than I do already, since I more or less failed at learning it, at least to a level that I was satisfied with. I'm so particular about it, that I basically would have had to almost re-invent the whole subject to put it in a form that I find palatable--see Baez's seminar notes on quantization and categorification for a start on what I have in mind there, along with maybe Feynman and Hibbs. The problem with me is that I'm such a skeptic that I can't really move on until I understand it so well I feel like I could have invented it myself. I'm always asking, "why would you do that?" Maybe taking the road less traveled is good sometimes, but I think if you're not a super-genius, it can be untenable as far as academic careers are concerned.

    It turns out that you don't really have to know any physics to speak of to work on at least some aspects of TQFT. It was mainly out of curiosity that I tried to pursue physics, but topology took up most of my time.
  19. Oct 13, 2014 #18
    You're asking many great questions that I've asked myself and researched over the past couple of years regarding what "mathematical physics" is.

    Broadly speaking I would divide "mathematical physics" into two categories.

    Firstly there's the type of mathematical physics, where you are trying to put already existing, and well-understood physics on a rigorous foundation. Axiomatic QFT is an example of this and there are many people working on this type of stuff, usually housed in mathematics departments. The main tool which seems to be used is functional analysis. I personally don't find this too exciting so I don't know much about it.

    Then there's the other kind, which is much less well-defined, sometimes given it's own name: "Physical Mathematics". This is where people like the David Morrison you mention fit in. This is a very broad field and there are a lot less strict rules as to how to proceed. In this what usually happens is you try to study mathematical topics that show up a lot in physics, and use the tools of physics to come up with new mathematical results. The Feynman path integral is an extremely important tool for doing this. As an example, the Atiyah-Singer index theorem has a proof entirely based on path integrals and supersymmetric quantum mechanics. Understanding the low-level manifestation of this in the form of Gauss-Bonnet theorem is not too hard actually (see the Mirror Symmetry book I mention later). Other general topics which are very important for this field include topological QFTs, supersymmetric QFTs, CFT and string theory on the physics side, and differential geometry, topology and complex and algebraic geometry on the mathematics side (with many other subtopics depending on your specialty). Some early papers of Witten are a great starting point to see what kind of stuff is done here. ("Quantum Field Theory and the Jones Polynomial" is a classic for example). A couple of other very broad topics of investigation in this field which I'm somewhat familiar with include things like topological string theory and mirror symmetry.

    Overall, I think an excellent starting point, and a very comprehensive and pedagogical guide to a large subset of physical mathematics is the text "Mirror Symmetry" by K. Hori, C. Vafa, et al, which I find to strike a very good balance between mathematics and physics.

    Absolutely! Which is why I'm trying to get into this field for graduate school. I find that the best way to describe people who work in this field is that they are both mathematicians and physicists, but also neither! Both since their toolbox includes a wide variety of mathematics and physics, and neither because on the math side the standards of proof are much more relaxed, and on the physics side, since you aren't really predicting anything for experimentalists!

    I do have to warn you though: very seemingly abstract topics like category theory DO show up in mathematical physics (look up homological mirror symmetry), and similarly stuff that sounds very applied, like "condensed matter theory" is actually pretty interesting mathematically speaking as well (like homeomorphic mentions). The difference comes more from the general rules and conventions of the field, rather than including or excluding specific topics.

    Regarding research: The best possible way is to find an advisor who works in this field and ask him to guide you! Most likely he or she will start by mentioning some foundational book or paper to you which should get you started.
    Last edited: Oct 13, 2014
  20. Oct 13, 2014 #19
    That book does look interesting, I will look further into it!
    So what kind of prerequisites do you recommend to learn before tackling things like Witten's early papers? Do you also have in mind other good, relevant books or papers by other authors?
    In addition to the topics you mentioned above, are there any other topics such as fluid mechanics or CMT that u mentioned, that are as cutting edge at the frontier of modern research, or are things like string theory and qft variants the most popular right now?
    And finally, just to clarify, do you think most people that do mathematical physics also specialize concurrently in either another physics or mathematics field? Or do those that aspire to work at the boundary solely specialize in topics in mathematical physics itself?
    And on that note, should one just gain enough knowledge from both the physics and mathematics side, and then spend most of their time with cutting edge topics in mathematical physics such as mathematical string theory, cft, etc., which means that if I really get into mathematical physics and become extremely interested in it, it might do me better to stop focusing as much on algebraic geometry? As I mentioned earlier above, with the way I studied category theory from Awodey, and the way an aspiring category theorist would study (reading more advanced texts, going into research papers), exactly how much should I go into each subject relevant to the topics in mathematical physics? Surely, it should not be as much as one that plans to specialize in that subject, but I'm not sure exactly how much.

    Thanks for your insights!
  21. Oct 14, 2014 #20
    I would suggest that your first course of action should be to get familiar with path integrals. That's the most widely used tool at the starting stage. It's what Alvaraz Gaume used for the supersymmetric proof of Atiyah-Singer. It's what Witten used to derive the Jones polynomial from Chern-Simons theory. It's what's used to show that the topological string generates Gromov-Witten invariants. Once again, the MS book is excellent for this, and also for the general prerequisites, as it develops things very systematically (including the prerequisite physics and mathematics).

    Well I haven't heard much about the fluid mechanics and it's relation to mathematical physics. I heard Shiraz Minwalla did some interesting work relating string theory to fluid dynamics, but I don't know much beyond that.

    Regarding CMT, the key words to look out for include "topological phases of matter", "topological insulators", "topological quantum computation"

    I think mathematical physics is a sufficient specialization in itself. You may have the occasional person who has very broad research interests, but that's not the case for the majority.

    I'm still figuring this part out myself. I think you would be right in saying that you surely don't need to go in the same level of depth as a traditional mathematician would when you first start learning the subject. "Working knowledge" is the important word I think. One thing I noticed about my current mentor is that even though he may not know all the gritty details about the proofs in complex geometry, algebraic geometry etc, he does have an excellent intuitive understanding of what's going on and as a result, is able to use the stuff very fruitfully in his work. So while knowing every little detail may not be important, having an intuitive understanding is indispensable. I would also think that how deeply one does the math also probably depends on the specific department you work in, since this flavor of mathematical physics is pretty evenly divided with regards to whether people are working in mathematics or physics departments.
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