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This is going to be a relatively broad/open-ended question, so I apologize before hand if it is the wrong place to ask this.

Anyways, I'm currently a 3rd year undergraduate starting to more seriously research possible grad schools. I find myself in somewhat of a weird spot as my primary interests lie in physics, but I usually can't stand the imprecision with which most physicists do physics. I eventually would like to do work relevant to the quest of finding a theory of everything, but because I do not like the lack of rigor in physics, I have decided not to go to graduate school in physics. However, when looking at the research interests of faculty members, I've found that most institutions have zero, one, or occasionally two (mathematics) faculty members working in this area. Am I just not looking in the right places? Where I am to go if I am looking to get into a field like String Theory from a mathematician's perspective?

As a separate but related question, I've found the prerequites for string theory to be quite daunting. At this point, I feel as if it will be at least another year or two before I can even start learning the fundamentals of the theory (I won't even be taking a course in QFT until next year). To be honest, I am starting to feel a little scared that I won't have enough time to do my thesis work in a field related to string theory. Compared to algebraic topology or something, which I took last year, this year and next I could be learning more advanced aspects of the field so that by the time I got to grad school I could immediately jump in and start tackling a problem, whereas with string theory I feel as if I won't be able to really do this until my third year of grad school or so. Is this something I should actually be worried about, or am I worrying about nothing?

Also, if any of you have studied string theory, I would be interested in knowing what subjects I should study to prepare myself and textboks that you recommend for studying from. I would prefer textbooks about physics written by a mathematician or at least a great deal of mathematical rigor, although I am willling to compromise.

Thanks before hand for all the help/suggestions. I am interested to hear mathematicians' take on this.

EDIT: A comment made me think that I should point out that I am ultimately interested in a theory of everything, not string theory per se. At this point, because I have so much to learn, I think that if I head in the direction of string theory (learn things like QFT, GR, Conformal Field Theory, Supersymmetry, etc.) I can't go wrong. It won't be for awhile until I have to really make a choice between candidates for a TOE.

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I would suggest mathematical physics for you. String theory isn't as mathematically well defined as is, say, Constructive Quantum Field Theory, which has numerous axiomatic foundations (Wightman Axioms, Osterwalder-Schrader Axioms, etc.). –  user02138 May 5 '11 at 19:59
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There is this series "quantum field theory for mathematicians", Folland has also written a book about QFT. –  Jonas Teuwen May 5 '11 at 20:04
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Honestly, if you can't jump into research right when you get to grad school, this is not a problem. Some people don't even pick an area to go into until well into their second or maybe even third year. Using the first couple of years of grad school to get a foundation in your area is the norm rather than the exception. –  Matt May 5 '11 at 20:38
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Hi GleasSpty, as for the subjects you should study, there is a wonderful website created by Nobel laureate Gerard 't Hooft which contains a lot of helpful suggestions on the matter: staff.science.uu.nl/~hooft101/theorist.html . It suggests a rough patway towards learning the deepest parts of theoretical physics, including Quantum Field Theory and Superstring Theory. –  Max Muller May 5 '11 at 21:16
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Well, I just hope you don't forget to have fun learning... –  Bruno Stonek May 6 '11 at 1:49
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4 Answers 4

UPDATE: I fully endorse Carlo Rovelli's philosophy of science emphasizing the need of conceptual understanding to advance as the great thinkers of the past: A critical look at strings. It is remarkable that the critic I make below to certain attitudes of part of the theoretical physics community are highly illustrated by this other article of Rovelli about demystifying the problem of dark energy, which is the type of unscientific propaganda I talk about below: Why all these prejudices against a constant?.

I am going to try to answer your question by talking about my particular experience since I had interests and intentions similar to yours. At the end I give some book recommendations.

My interest in physics and mathematics began as a teenager with the desire to understand general relativity which led me to the dream of quantum gravity and unification in theoretical physics. Then I started to learn as much math as possible to have the necessary background to understand it all while starting my undergraduate degree (5 year Licenciatura in Spain) in theoretical physics. Through all those years I actually began to realize that my vocation and mind setting was more mathematical than physical: I arrived at this conclusion after realizing that I was interested in many more courses from the math curriculum than from the physics degree (which here is filled with compulsory laboratory courses and very applied subjects like electronics or solid state physics). My passion was actually more on the pure mathematics side with a strong flavor of theoretical-mathematical physics (understanding the structure of Nature). Eventually I understood general relativity and quantum field theory among the elective courses of my last year.

All along my undergraduate studies in theoretical physics I was aiming at getting into a Ph.D. program on String Theory.... but at the same time I was getting my own personal opinions about string theory because of the knowledge about the status of physics research nowadays and my own deep understanding of theoretical physics. Some years ago I came to the conclusion that string theory is not something worth pursuing right now for me.

(THIS IS ONLY MY OPINION BASED ON MY EXPERIENCE AND KNOWLEDGE OF PHYSICS)

Why is that so? First of all, the particle physics community now is expecting to get new physics from CERN's LHC and other projects related to dark matter, dark energy and cosmology; the theoretical physics advances for the next decade will be around fitting phenomenological models to the new data, none of them is realistically derived from string theory. Secondly and personally I have found string theory to be a very naive attempt of "final theory": it will not be a final theory nor a theory of everything because it is not even a theory yet. The string models are a TREMENDOUS framework of structures connected to each other and to almost every aspect of mathematics but are not rigorous enough because even Interacting Quantum Field Theory is not rigorous mathematically. At the present time there is no solid mathematical foundation for the Feynman Path Integral and other continuum problems of field theory which are the cornerstone of relativistic quantum theories nowadays (they are likely to be a continuum formal asymptotic approximation to some fundamental discrete physics like in Loop Quantum Gravity). Thirdly, the amount of background needed to study and tackle any problem of string theory is too much for a person to devote his/her career to: you pretty much have to be a multidisciplinary mathematician able to handle with easy branches from differential and algebraic topology, differential and algebraic geometry, complex analysis and even number theory; this besides the amount of physics needed, since ordinary quantum field theory per se is a whole world hard to master for a single person. Much of the progress of the last 30 years has been done more on the mathematical side (topological quantum field theories, mirror symmetry and lots of small advancements in pure math thanks to physical insights...). At this moment an amount of theorists are shifting their string-theoretic methods and knowledge to condensed matter physics (graphene...) and such... because the original objectives are kind of stalled (landscape multiverses, lack of contact with experiment, no clues on how to get a nonperturbative formulation of the theory and deduce the standard model of particle physics...).

Lastly, and most importantly, I personally do not like the attitude of much of the string theory community (in general)...... Just as Richard Feynman thought of string theory in the 80s, they are behaving VERY UNSCIENTIFICALLY..... The "theory" might be right in the end but they are not been modest or cautious at all........ There is too much propaganda regarding string theory focused on attracting young students and more funding.... This is unprecedented in physics..... and many great scientists of the past would agree I think. Too many books and documentaries talking about string theory like it were already the final theory, the final answer even though it has not made a single real prediction... A true scientist must be wary of this kind of declarations since despite how much promising and rich a theory is, string theory is NOT EVEN WRONG, as for the time being cannot be tested or falsified, and therefore is not a "theory" in the same respect as the theory of relativity or quantum theory. I find that kind of naming vey misleading.

Other very disturbing attitude related to this is the narrow-minded behavior of many researchers and departments, to the point of not allowing deviations from string theory. At the present time there is no compelling empirical reason for string theory to be better than loop quantum gravity, emergent gravity or other models of quantum gravity which are pursued at the same time. Nevertheless very few people work on them and are not encouraged for graduate students at most universities.... so more and more people end up working in a very tiny small corner of string theory, just trying to go on with their careers and get settled somewhere...... without caring too much that their work may be completely wrong or that other alternatives are as much valid as theirs. I recommend you read the books "Three Roads to Quantum Gravity" and above all "The Trouble with Physics" by Lee Smolin (many people think he exaggerates to defend his case, but read them and think for yourself). Other interesting articles regarding the academic situation in physics and string theory are "Why no new Einstein?" by Lee Smolin and "A dialog on quantum gravity" by Carlo Rovelli.

I must say that even though I am being very critic to string theory, I was very interested in it from the very beginning and talked to many professors working on it. Besides, I acknowledge its tremendous usefulness and impact on mathematics.

But as I learned more and more physics I have found Loop Quantum Gravity and particularly its Spin Foam version, much more attractive than string theory in spite of the fact of being still in a very early stage (remember, the greatest theorists for the past 30 years going to string theory...). Their attitude towards science is more conservative and they do not attempt a theory of everything but working out first a quantum gravity theory compatible with known relativity and quantum mechanics. Besides it is a better background independent, relational model. For example I get the feeling that the discrete Spin Foam formalism is in the good direction due to its finite sums of a discrete geometry dynamics as the final substrate of the Feynman Path Integral, therefore automatically regularizing the mathematical nonrigorous difficulties of ordinary quantum field theory (and string theory). Maybe it is just another theory or attempt, maybe both models are different sides of the story...

...

So I decided eventually to shift my career towards Pure Mathematics. Pursuing graduate studies in math is a more secure path right now. Some professors I had who are string theorists advised me that as well. Now, I can relax and study any field of mathematics I know will still be there after a decade or two, letting me fulfill a career as a researcher and/or teacher without any regrets. Thus, I see too risky a plan to study string theory right now. Nevertheless you can pay attention to advances and news about theoretical physics at the same time. I have been told by many Math professors that a mathematician can later become a string theorist (and publish in physics journals) but it is harder for a theoretical physicist to become a pure mathematician (and publish in math journals).

If you are interested in a Theory of Everything... that is far from being discovered in the near future... In my view the only Theory of Everything is MATHEMATICS..... as the final structure of Nature and everything we will be ever able to say about it will be mathematical. The problem of the "Physical Theory of Everything" is therefore to find out which is the particular structure of our Universe/Nature, at least as far as how much can be said from within. If you carry on with physics you will be looking for the last kind of theory of everything: the final mathematical model which embodies all the laws of Nature as we know it. If you choose pure mathematics you will be looking for the first kind of theory of everything: all that can be said about any logical structure conceivable in Nature.

Maybe you will find a similar view interesting in the kind of speculative and philosophical (but fun and enlightening!) papers on the structure of a final theory and the mathematical universe by Max Tegmark:

EDIT: in response to your comments and edit addition to your question, be sure that learning General Relativity, Quantum Field Theory, Conformal Field Theory, SuperSymmetry, Topological Field Theory and the like WILL BE VERY USEFUL AND INTERESTING WHATEVER PATH YOU TAKE AFTERWARDS. I must emphasize here that I learnt many of those subjects already and before deciding to switch to Pure Mathematics, since they are part of an undergraduate education in my country. I decided to forget about String Theory ONLY AFTER I knew enough physics to check out whether I liked/understood the theory or not. So my last advice would be to learn as much as possible and when the moment comes make up your mind depending on your personal tastes and philosophy. Therefore focusing right now on string theory is irrelevant since you will have to learn in the meantime as much physics and mathematics as needed for any other specialization in theoretical physics or pure mathematics.


If you wish to understand string theory there are many books available but you have to master a lot of mathematics as well.... as it is mentioned here. My personal advice would be to become a graduate student in mathematics with enough physics background (see below) and then read books like Deligne et al. - "Quantum Fields and Strings: A Course for Mathematicians", maybe after a softer (physical) introduction with Becker/Becker/Schwartz - "String Theory and M-Theory: A Modern Introduction", Johnson - "D-Branes" and Polchinski - "String Theory vol I, II".

A quick overview for physicists of the mathematics needed (for example to understand the book by Becker) is developed in the books by Nakahara - "Geometry, Topology and Physics" and Eschrig - "Topology and Geometry for Physics" but they are not at all enough for the others. Applications such as mirror symmetry are very well explained in Hori et al. - "Mirror Symmetry", Bridgeland et al. - "Dirichlet Branes and Mirror Symmetry" and Cox - "Mirror Symmetry and Algebraic Geometry".

You will need a strong background in general relativity which may be studied with Grøn/Hervik - "Einstein's General Theory of Relativity" and Ortin - "Gravity and Strings"; and also a deep knowledge of quantum field theory which can be studied in a mathematical style with Ticciati - "Quantum Field Theory for Mathematicians" and de Faria/de Melo - "Mathematical Aspects of Quantum Field Theory". The best references for mathematical Gauge theory are Naber - "Topology, Geometry And Gauge Fields: Foundations" and Naber - "Topology, Geometry And Gauge Fields: Interactions". Also, the huge tomes by Zeidler - "Quantum field theory I. Basics in Mathematics and Physics" and Zeidler - "Quantum Field Theory II. Quantum Electrodynamics" and their future volumes seem to me the BEST mathematical approach to modern theoretical physics. Some supersymmetry will be needed like in Wess/Bagger - "Supersymmetry and Supergravity" and Terning - "Modern Supersymmetry, dynamics and duality".

For other approaches to quantum gravity the book by Rovelli - "Quantum Gravity" is the best introduction to loop quantum gravity, and can be followed by the very mathematical but understandable book by Thiemann - "Modern Canonical Quantum General Relativity" which includes hundreds of pages of mathematical appendices with the needed formalisms. A good review of most approaches is Kiefer - "Quantum Gravity" with a conceptual overview by Oriti et al. - "Approaches to Quantum Gravity".

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Of all the criticism of String Theory, one of them that I don't quite understand is that it is unscientific. Personally, I have some problems with pure mathematics in that it seems a lot of people investigate mathematical questions just for the sake of it, just because it interests them, with no care for outside applications. If String Theory turns out not to be physics per se, would it not be of a pure mathematical interest as well? String Theory would be just like other mathematical fields, except that it also has a possiblity of being relevant to the real world. –  Jonathan Gleason May 6 '11 at 0:23
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Also, great commment by the way. INCREDIBLY helpful. –  Jonathan Gleason May 6 '11 at 0:27
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String theory is STILL unscientific because the definition of a scientific model requires it to explain well-known phenomena and make new testable/falseable predictions. Since the strings are not a THEORY yet, they make almost all possible universes and laws of physics possible, making almost of no use to everyday physical phenomena. IT IS MATHEMATICALLY RELEVANT and maybe the final theory of a multiverse reality, but it is NOT EXPERIMENTAL SCIENCE YET. It has remarkable insights into mathematics and that makes it valuable but being strict it is not science yet, so they should be more careful. –  Javier Álvarez May 6 '11 at 0:38
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@ Javier Alvarez: Impressive answer! I have a question regarding the Feynman Path Integral you mention. I don't know anything about the subject, but wikipedia states that Maxim Kontsevich "introduced moduli spaces of stable maps, which may be considered a mathematically rigorous formulation of the Feynman Integral for topological string theory." You say that there's no solid mathematical foundation for the Feynman Path Integral. Could you please explain why the work of Kontsevich is not enough to define the Feynman Path Integral rigorously? –  Max Muller May 6 '11 at 15:45
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@ Max Muller: all right, I was talking in general. As far as I know the typical Feynman Path Integral of Quantum Field Theory is not well defined because there exists no complex measure on the noncompact configuration space where it is heuristically defined. The Path Integral is a misguiding name for a very delicate formal technique which in the end is actually defined by what perturbative expansion you mean by it. It is actually a formal asymptotic series expansion where tricky renormalization techniques must be used to make sense. It is a VERY WELL developed subject anyway. –  Javier Álvarez May 6 '11 at 16:34
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Javier Alvarez gives a very nice response. I'd like to say something simply to offer some contrast.

My own bias is that if you're looking for a theory of everything, mathematics is not where you'll find it. Mathematics offers supportive framework but a convincing TOE is going to require an idea. Discovering ideas is a process of cross-pollenation: looking at experiments, various theories including even the long-dead and discarded ideas, not just the current fads of the day. Sometimes it's not enough to have in-principle understanding of theories -- it can be a huge help to your understanding to craft and complete experiments on your own. The set-backs you have in realizing an experiment give you feedback on your (lack) of understanding of what you're doing. It functions similarly to proofs in mathematics -- they're humbling and informative at the same time.

Look carefully at how some theories developed -- in particular the Lorentz-Einstein-Minkowski story that led to the development of the General Theory of Relativity. Look to established physics, chemistry even. Once you've got an understanding of the basics, then feel free to move on. The story of GR is IMO a wonderfully instructive one.

The question you're interested in likely has no good answer. And it begs the question, can there be a theory of everything? It's possible that understanding reality is like peeling an onion (with no eye protection). Every layer you peel back makes you cry, and makes peeling the next layer that much harder. And perhaps there isn't a finite number of layers to the onion.

I doubt there's much more to say other than learn as much as you can and contribute how you can. Try to make your contributions as relevant to your goals as possible. And try not to set unrealistic goals.

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I agree completely. Above all, the fact that the story of GR and also Quantum Mechanics are very instructive. The last paragraph advice is the best thing every enthusiastic student can do, I try to do it and encourage everyone to follow it. –  Javier Álvarez May 6 '11 at 1:26
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There are two ways to proceed as a pure mathematician: You can specialize in differential geometry and become an expert in general relativity: This is a theory that is mathematically rigorous, and a challenge.

You could also specialize in operator algebras, which is a huge and very active topic. There is an axiomatic, mathematically rigorous approach to quantum field theory, called axiomatic, local or algebraic quantum field theory, that uses and needs a lot of material from operator algebras. The cooperation of physicists and mathematicians has been rich and fruitful, much like in general relativity.

Have a look at AQFT in the nLab.

A good start is the classic book

  • Bogolyubov, Logunov, Oksak, Todorov: General principles of quantum field theory (Mathematical Physics and Applied Mathematics, 10. Dordrecht etc.: Kluwer Academic Publishers, 1990)

In both cases you'll have the opportunity to pursue an academic career in a math department, while continuing your quest to better understand the fundamental problem of a TOE.

Edit and Addendum: As you can read in the comments to other answers, there currently is no rigorous construction of an interacting field theory in 4D in local/algebraic/axiomatic QFT, which is the reason why most theoretical physicists don't actively pursue this approach anymore. But some do, it is certainly worth knowing the many nontrivial deep model independent results, and the leading string theorists know a lot about this approach, too. (And yes, Witten has also (co-)written papers about it.)

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Indeed, I forgot to mention in my comments the fact that the problem is mainly for 4D spacetime.... It would be speculative but intriguing to think about recent ideas regarding a scale dependent dimensionality of spacetime (decreasing down to the Planck lenght) which could be of much help to ease some of the problems of the mathematical foundation of the effective quantum field theory we know and use at available scales. –  Javier Álvarez May 6 '11 at 19:23
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It would be culpably stupid to plan a career in String Theory at this late stage. Don't you read the blogs? String Theory has been dying for ten years or more, and is only kept alive by the intransigence of its supporters. Try Loop Quantum Gravity if you want to make your mark.

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Wow, biased much? –  BBischof May 5 '11 at 22:02
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Well, at least we have a vocabulary lesson! –  Tyler May 5 '11 at 23:42
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Even if string theory completely dies as physics, it would still be of interest to mathematicians. –  Matt May 5 '11 at 23:47
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@Matt: but that goes against the interest of GleaSpty -- he says he's primarily interested in a "theory of everything" rather than String Theory proper. –  Ryan Budney May 6 '11 at 0:13
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While this post is strongly exaggerated, it still doesn't deserve -4! There is indeed a growing skepticism towards string theory. On the other hand, I want to point out that loop quantum gravity is in no better predicament. First, LQG is not a TOE but just a TOQG. Second, it seems it is an open problem even just to get classical GR out of LQG. Whereas this is the principal claim to fame of String theory that GR comes out "gratis"! –  Raskolnikov May 6 '11 at 9:44
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