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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.

Andrews
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Jonathan Gleason
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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 05 '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. – JT_NL May 05 '11 at 20:04
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    One of the major predictions of string theory, at least in mathematics, is the phenomenon of mirror symmetry. The Clay mathematics institute has put out a book on the subject, intended to be read by grad students in physics and maths (available as a free dl at http://www.claymath.org/publications/Mirror_Symmetry/). Have a casual look at it and see if you can get a feel for what kind of subjects they talk about. For the maths, at least, it shows you'll need a good grounding in differential and algebraic geometry (so, complex geometry, probably). – Gunnar Þór Magnússon May 05 '11 at 20:09
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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 05 '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: http://www.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 05 '11 at 21:16
  • pathway* (more text) – Max Muller May 05 '11 at 21:29
  • @user02138: but isn't there more interesting math related to string theory (e.g. algebraic geometry, mirror symmetry) than qft? I know of mathematicians working in math related to string theory but don't know of any working in math related directly to qft. – Eric O. Korman May 05 '11 at 22:00
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    Nearly every field you mentioned also shows up in QFT _and_ in a rigorous fashion. For example, the interplay between Chern-Simons Gauge (Field) Theory and the Jones Polynomial of Knots is a prime example of this nice interplay. Unfortunately, I know of no rigorous string theory per se, no axiomatic approach. However, mathematicians working on objects that appear in string theory is _very_ different. (They are working on formalizing the physics or determining its origin in well established mathematics.) – user02138 May 05 '11 at 22:11
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    You seem to be hurried by something. Why don't you relax and enjoy thinking about the topics you like best? – Bruno Stonek May 05 '11 at 23:20
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    @Bruno Stonek: Actually I do feel hurried. Even since high school, I have felt a huge pressure to do research, pressure from professors, peers, admissions committees, scholarships, graduate schools, etc. The truth is, I'm not interested in research at all unless it's something I have a real passion for, and because my passion is finding a theory of everything, I actually feel at a significant disadvantage to people who have already been doing research for a couple of years because their field requires much less prerequisites in comparison. – Jonathan Gleason May 06 '11 at 00:34
  • @ last comment by GleasSpty: yes exactly, that is why I answered how I did... I felt the same pressure and hurry as you and since I arrived at the conclusion that only very gifted students could be fast enough to pick up the prerequisites and contribute soon, I decided to switch to other fields more slowly paced. Besides, publishing in physics is really more rushed than in math. But as I commented below my answer, even mathematicians like A. Connes have made attempts to a Theory of Everything, pursuing very abstract pure maths like noncommutative geometries. – Javier Álvarez May 06 '11 at 01:40
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    Well, I just hope you don't forget to have fun learning... – Bruno Stonek May 06 '11 at 01:49
  • Maybe you also want to look at other fields. Although this stuff is called 'theory of everything' it is actually not (even if successful) what it sounds like. Its unlikely that any change in the perception of low energy physics (and even high energy physics) will emerge from that. You could argue that a 'real' theory of everything could be more related to 'non-equilibrium thermodynamics/dynamical systems/quantum chaos'. – lalala Feb 27 '17 at 21:21

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UPDATE+: Nature's article G. Ellis & J. Silk - Scientific Method: Defend the Integrity of Physics led to the conference in Münich "Why Trust a Theory?" about the dangers of non-empirical confirmation, defended by some popular string theorists basically to mend the scientific method so that their "theory" can be justified without direct evidence, with philosophers and physicists presenting arguments and raising the concern that there is A REAL PROBLEM in the community. The talks by David Gross, Massimo Pigliucci and Carlo Rovelli were specially insightful. I personally 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. A good skeptic, and now well-known, blog with up-to-date criticism about string 'theory' and some of its theorists' attitudes is P. Woit's Not Even Wrong. Other phenomenological quantum gravity skeptic blog is Sabine Hossenfelder's Backreaction.

String "theory" has produced an enormous mathematical framework of insights, hidden inter-connections and tools but it has failed so far to produce any empirical physical theory that actually solves the problems it was claimed to be able to solve. The question is not whether string theory is interesting mathematically, but whether it is a framework worth funding disproportionately with hype in the media by many important professionals selling it as a tested final theory (or even calling it a physics theory since there is still no known way to get a 4-dimensional limit without supersymmetry at low energy including the standard model of particle physics and non-perturbative general relativity with a positive cosmological constant). This slide by Rovelli in his Münich talk is a good summary:

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Original answer (years ago before the LHC results)

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.

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, Causal Dynamical Triangulations...). 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 SOME of the string theory community...... 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 SOME 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 article"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 Non-Commutative Geometry or 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 professor I had who was a string theorist 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".

Javier Álvarez
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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 06 '11 at 00:23
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    Also, great commment by the way. INCREDIBLY helpful. – Jonathan Gleason May 06 '11 at 00:27
  • Thank you, I have revised, edited, and added a couple of things, check it out. – Javier Álvarez May 06 '11 at 00:30
  • As I commented in my last revision of the answer.... your opinion and mine (and others) depend much on the philosophy you have regarding the ontology and value of mathematics and physics. PERSONALLY FOR ME THEY ARE THE SAME THING.... JUST DIFFERENT ASPECTS OF THE WORLD OUT THERE WHICH WE TRY TO UNDERSTAND...... Most of mathematics was developed to solve problems in physics. Even abstract maths pursued for their own sake happened to be of much use (Riemannian geometry, Hilbert spaces and others!!).... – Javier Álvarez May 06 '11 at 00:33
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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 06 '11 at 00:38
  • There are alternatives that see relativity/gravity as an emergent phenomenon not related to the other forces in the traditional way, and thus making an unification or quantum gravity impossible and senseless. The "Theory of Everything" is a very misleading term because it actually refers to a very particular kind of science: the structural unification of the different gauge field couplings with matter. For example the Noncommutative Geometry of Alain Connes is able to make a "Theory of Everything" of that kind unifying gauge/gravity via a noncommutative structure added to spacetime. – Javier Álvarez May 06 '11 at 00:51
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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 06 '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 06 '11 at 16:34
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    @ Max Muller: ... Axiomatic Quantum Field Theories are well defined in the free case, but up to the moment there seems to be NO rigorous formulation of an INTERACTING theory, that is, one with couplings between the fields, and that is precisely the interesting use of the Path Integral: the perturbative expansion of interacting fields. THERE MAY BE concrete examples and models where the Path Integral makes sense because as I said earlier it is a broader concept than it seems, depending on the kinematics of your model, for example when working with finite degrees of freedom like in Matrix models – Javier Álvarez May 06 '11 at 16:38
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    @ Max Muller: ... in the same manner, maybe within the realm of Topological Field Theories one can make sense of a precise and rigorous Path Integral taking into account that there are no local degrees of freedom on such models. A finite version of the Path Integral is well-defined in the sense of Lattice Field Theory (used to make numerical approximations and predictions) and discrete models of space-time like in Loop QGravity where the integral reduces to a finite summatory in the best cases. There are heuristic techniques turning the Path Int. into a Wiener or Gaussian integral. – Javier Álvarez May 06 '11 at 16:45
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    @ Max Muller:... about the work by Kontsevich, it is indeed wonderful, useful and very interesting. I do not know enough but it seems it cannot still be extended to ordinary Quantum Field Theory which is the model used to explain experimentally tested physics. In a sense, Topological (Quantum or no) Field Theories are better defined in a mathematically rigorous way than ordinary Quantum FT, precisely using the kind of work by Kontsevich and others regarding algebraic geometry and category theory. In conclusion, the Path Integral is a PHYSICAL CONCEPT to be defined precisely for each model. – Javier Álvarez May 06 '11 at 16:53
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    @ Max Muller: I recommend you read the book I mentioned by Zeidler, volume 1, chapters 7,8, 13 and beyond where he explains that finite quantum field theory is perfectly well-defined and ordinary QFT is just accepting the same formulas in a formal generalization to the functional continuous case (what he calls "magic formulas") Again, the well-defined functional integrals of Kontsevich are probably meaningful because of the topological nature of the models makes them isomorphic to well-defined mathematical constructions not having a continuous tower of infinite coupled local degrees of freedom – Javier Álvarez May 06 '11 at 16:59
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    @ Max Muller: if you get enough background in abstract algebraic geometry you can read a very interesting introduction to the Feynman Motivc integration and other constructions related to Kontsevich's work, in the book by **Marcolli** - *"Feynman Motives"*. – Javier Álvarez May 06 '11 at 17:19
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    @ Max Muller: if you want a quick Quantum Field Theory overview by a mathematician (treating the path integral as a formal asymptotic series) you should read http://arxiv.org/pdf/math-ph/0204014v1 – Javier Álvarez May 06 '11 at 17:32
  • @Javier Alvarez: Nice paper (the one in your last comment). First page made me chuckle. – Jonathan Gleason May 06 '11 at 17:57
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    @ GleasSpty Indeed, step 10 of a "lifecycle of a theoretical physicist"... @ Javier Thanks a lot for all the information you provided me with. Unfortunately, many of the things you said and refered to are way over my head at the moment. I will study the subject more carefully in the future. – Max Muller May 08 '11 at 16:25
  • @JavierÁlvarez: You mention some reference books for physicists needed to build the mathematical background, but what roadmap, and books would you suggest for a physicist looking to go to maths grad school, and later pursue physics? I am a physics undergrad in exactly the same situation, with very less background of mathematics, but I want to build a strong mathematics base which would be from self-study. Physics would be taken care of in my courses. So please could you tell me what books I should read from the bare minimum? Starting from topology, and group theory to algebraic topology.(cond) –  Nov 09 '12 at 17:34
  • @JavierÁlvarez: differntial geometry, representation theory, functional analysis etc...And also, what area should one pursue in grad school with the home of contributing to TOE in the future? –  Nov 09 '12 at 17:35
  • @ramanujan_dirac: go to my profile here (and in mathoverflow.com) and check out my answers concerning bibliography and references for studying different subjects (mainly geometry-related). You need a strong background in real analysis, try *Mathematical Analysis* by **Zorich** vol. 1 & 2, complex analysis, try **Freitag & Busam** *"Complex Analysis"* vol 1 & 2. Try the compendium by **Szekeres** *"A Course in Modern Mathematical Physics"* for a first approach to different advanced math. For geometry try by **Frankel** *"The Geometry of Physics"* and **Nakahara**'s "Geometry,Topology & Physics" – Javier Álvarez Nov 09 '12 at 19:14
  • @ramanujan_dirac: for a first intro in groups in physics try **Jones**' *"Groups, Representations & Physics"*. Once you have mastered all of those books (from the previous comment), you should jump into purely mathematical books, along with the TOE books you need. In particular **Nicolaescu**'s *"Geometry of Manifolds"*, **Huybrechts**' *"Complex Geometry"*, **Kreyszig** *"Functional Analysis"*, **Gilmore**'s *"Lie Groups, Lie Algebras with Applications"* and books on quantum field theory for mathematicians (**Ticciati**), string theory (**Becker, Schwarz**), loop gavity (**Rovelli**)... – Javier Álvarez Nov 09 '12 at 19:20
  • @ramanujan_dirac: if you go to math grad school you should pursue something related to geometry or topology, specially low dimensional topology (applications of QFT), symplectic geometry and complex (algebraic and differential) geometry, moduli problems in any of those, advanced complex analysis, mirror symmetry...... You can find a background to master searching in Google the book lists by Gerard 't Hooft and John Baez's advice. If you have any more detailed or particular doubt, do not hesitate to ask! – Javier Álvarez Nov 09 '12 at 19:23
  • @JavierÁlvarez: Thanks a lot for the answer, it always motivates me to see people who have changed to maths from physics, and muster the courage. A quick question though. Some of the books that you have mentioned such as Szekeres, Jones, or Nakahara are mathematical physics textbooks. Would I be able to acquire the rigour, and also the problem solving skills that is required in grad school? as most of these books concentrate, on having an intuitive feel of the maths as that is sufficient for physicists. I have already read parts of Nakahara, which I didn't find rigorous. (contd) –  Nov 10 '12 at 03:37
  • @JavierÁlvarez:Shoudn't I be reading books like Munkres, Hatcher, Dummit&Foote, Fulton and Harris, spivak etc. or would the books you mentioned help me to build the required mathematical skills and knowledge? This has been bugging me for a long time. –  Nov 10 '12 at 03:39
  • @ramanujan_dirac: Indeed, for passing qualifying exams in grad school you should study the standard references like **Folland** - *Real Analysis*, **Conway** - *Functions of One Complex Variable* **Dummit&Foote** - *Abstract Algebra*, **Munkres** - *Topology*, **Hatcher** - *Algebraic Topology*, **Tu** - *An Introduction to Manifolds*, **Evans** - *Partial Differential Equations*, **Fulton&Harris** - *Representation Theory, A First Course*, and for spezialization **Jost** - *Riemannian Geometry and Geometric Analysis*, **Arapura** - *Algebraic Geometry over the Complex Numbers* and the others. – Javier Álvarez Nov 10 '12 at 07:33
  • @ramanujan_dirac: also very good alternatives to some of those are **Bredon** - *Topology and Geometry* and **Rotman** - *Advanced Modern Algebra*, besides the ones I mentioned in my earlier comments. Mathematical Physics books like Szekeres, Nakahara and Jones are very useful coming from a background in physics in order to get a motivation for more abstract titles. Szekeres in particular is quite rigorous, just a compendium of adv. linear algebra, real and functional analysis and differential geometry. You should approach those subjects first from there and then jump into purely math books. – Javier Álvarez Nov 10 '12 at 07:36
  • @ramanujan_dirac: you should check out all the titles I mentioned, but do not forget that if you intend to work in TOE, you will end up being a mathematical physicist or "theoretical mathematician", that is, you will be doing research in the frontier where math tools are to be developed and much physical insight is to be needed. For example much of string theory needs nonrigorous maths like Feynman's path integral and many dualities like different types of mirror symmetry which are only partially proved. So you should develop both attitudes. – Javier Álvarez Nov 10 '12 at 07:42
  • These are the links you should consult for more advice on books: http://www.staff.science.uu.nl/~hooft101/theorist.html and http://math.ucr.edu/home/baez/advice.html and http://math.ucr.edu/home/baez/books.html – Javier Álvarez Nov 10 '12 at 07:50
  • Nice post. Given the ideas you develop, I was expecting to see a mention of Woit's blog. Even though the expression "Not even wrong" is in your post, you do not mention it. Is there a reason? – Did Jan 12 '14 at 16:41
  • @Did: thanks! I am aware of Woit but I believe he started as a computer scientist and was not that well-known in my country when I was interested in (and academically driven by) string theory, which was years ago. Nevertheless, I recognize his role as a critic quite useful to help keep our minds open. I mentioned "not even wrong" but as an old scientific quote which comes from Wolfgang Pauli (and was used in philosophy of physics before Woit). – Javier Álvarez Jan 12 '14 at 17:25
  • @Did: I preferred to emphasize the fact that there are other more modest alternatives to strings, some of whose researchers offer more technical arguments and scientific insight, and less propaganda. In particular, Rovelli changed my life with his highly conceptual book on quantum gravity: in no other book have I seen Einstein's hole argument explained, despite the fact that its implications on the meaning of general covariance is explicit in his 1915-16 articles. From that moment on, I saw that strings on flat Minkowski space-time miss the conceptual content of general relativity. – Javier Álvarez Jan 12 '14 at 17:34
  • Of course the phrase "Das ist nicht nur nicht richtig, es ist nicht einmal falsch!" was not coined by Woit... Still, regarding bibliographical issues: Woit's blog (2004) predates "The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next" (2006), which you (rightfully) mention. – Did Jan 12 '14 at 17:43
  • @Did: I was just not aware of him at that time and when I learnt about his writings I had already left these issues. But if you feel it is unfair for me not to have mentioned him, I would say that I once listened to Leonard Susskind underrate his knowledge of physics and despise his criticism, supporting this whole idea of the misguiding unscientific attitudes by some of the major advertisers of strings. After that I guess I subconsciously chose to quote and reference mostly from researchers in alternatives (Rovelli's dialogue is from 2003) to avoid fallacious ad hominem arguments. – Javier Álvarez Jan 12 '14 at 18:34
  • An alternative would be to judge by yourself Woit's expertise and (most importantly, if you ask me) the cogency of his views on the subject (or the lack thereof). These views being there for everyone to read and dissect and critique, and with your background, this should not take long. – Did Jan 12 '14 at 18:54
  • @Did: of course! please, excuse me if I expressed myself badly. What Susskind and others said made me more convinced of what Woit and others were arguing! We need more Woit's. I just meant that I had witnessed many discussions where the string supporters were undervaluing criticism from non-physicists and non-experts in gravity, that is why I always tried to look for the opinions of researchers in alternative quantum gravity theories (like loops, spinfoams, causal triangulations, asymptotic safeness, emergent gravity, topoi...) to avoid the ad hominem arguing of some string theorists. – Javier Álvarez Jan 12 '14 at 19:03
  • @Did the link to Woit's blog has been added to the first paragraph of my answer, as you suggested and we discussed. I had been uninterested in the "string theory" topic for long, but recently I have found Woit's blog as a very useful reference to keep updated. – Javier Álvarez Dec 18 '14 at 12:28
  • @JavierÁlvarez: I'm not sure whether you will notice my comment here, but I want to ask you a question. You didn't mention any classics on particle physics or standard model (e.g. Quarks and Leptons by F. Halzen, Heavy Quark Physics by A. Manohar, QCD and Collider Physics by R. Ellis, and Dynamics of the Standard Model by J. Donoghue). Do I not need to read them to be a string theorist? Assuming I will read most of the books you mentioned, should I still read the books I mentioned too? If so, which ones in particular? I'm a mathematician who want to also be a physicist..like Witten. – Math.StackExchange Mar 29 '15 at 15:19
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    @AranKomatsuzaki: the books you mention, and many others are less mathematical and more "phenomenological/experimental". Before or along with learning QFT, you may get the basics of real particle physics with Griffiths' "Introduction to Elementary Particles" and then something like Halzen-Martin "Quarks & Leptos". The more you know the better, but few "pure physics" books are easy to digest by some mathematicians as they tend to be too prosaic or not rigorous enough (e.g. more heuristics than formalism). Anyway, the other books you mention are only needed for a particle physicist I think. – Javier Álvarez Mar 30 '15 at 08:59
  • Thanks for your response. As a mathematician, I do not like phenomenological stuffs so much. They are tedious and mathematically not stimulating for me. I'm glad to know that these texts are not usually needed for a string theorist or those with interest in TOE. – Math.StackExchange Mar 30 '15 at 09:31
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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.

Ryan Budney
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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 06 '11 at 01:26
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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.)

Tim van Beek
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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 06 '11 at 19:23