Reading list recommendation for a hep-ph student to start studying QFT at a more mathematically rigorous level?

Question

Edition

On July 15 2021, the description of the question has been considerably modified to meet the requirement of making this question more OP-independent and thus more useful for general readers. But the central point is the same.

One sentence summarization

For a student initially working on a more phenomenological side of the high energy physics study, what is the recommendation of introductory reading materials for them to dive into a more mathematically rigorous study of the quantum field theory.

Elaboration

1. "phenomenology side of the high energy physics study" basically means that when this student uploads their article to arXiv, they will, in principle, choose hep-ph as the primary archive. This background also indicates that this student has, in general, already learned at least

   a. Undergraduate level physics classes;

   b. QFT from the first half of Peskin and Schroeder's classical textbook, or, maybe, Greiner's Field Quantization;

   c. Complex analysis like those covered in the first five chapters of Conway's Functions of One Complex Variable I;

   d. Special function;

   e. Elementary set theory;

   f. General topology like those covered in Amstrong's Basic Topology;

   g. Elementary vector space theory with no details about the operator algebra;

   h. Some elementary conclusions from group theory, with a focus on the Lie group and Lorentz group;

   i. Enough particle physics so that he/she is able to understand what the summary tables of the PDG are talking about, or he/she at least knows how to find the definition of some of those unknown symbols.

2. "more mathematically rigorous study of quantum field theory" may have different meanings for different people. Just to give an example, Prof. Tachikawa from IPMU had a talk titled "Mathematics of QFT, by QFT, for QFT" on his website (https://member.ipmu.jp/yuji.tachikawa/transp/qft-tsukuba.pdf), which the OP find really tasteful, of course, from a rather personal point of view. One may agree that all those studies covered there may be considered mathematically rigorous studies of quantum field theory. Or, one may agree on only part of them. Or, maybe the answerer thinks that some other researches also count. That is all OK. But, please specify your altitude before elaborating your answer. The answerer may also want to specify explicitly for which subfield they is recommending references. In the OP's understanding, "more mathematically rigorous study of quantum field theory" basically covers those topics

   a. Algebraic/Axiomatic QFT, Constructive QFT or the Yang-Mills millennium problem, which tries to provide a mathematically sounding foundation of QFT.

   b. Topological QFT or Cconformal QFT, which utilizes some fancy mathematical techniques to study QFT.

   c. Even some math fields stemming from QFT like those fields-awarding work done by Witten.

3. It is best that those "introductory reading materials" could focus on the mathematical prerequisites of that subfield. It will be really appreciated if the answerer could further explain why one has to know such knowledge before doing research in that subfield. Answers may choose from a really formal vibe or a physics-directing vibe when it comes to the general taste of their recommendation and may want to point out that difference explicitly. It is also a good recommendation if the answerer lists some review paper, newest progress, or commentary-like article in that subfield so that people wanting to dive into this subfield could get a general impression of that subfield.

Additional comments

1. In general, this is not a career advice post. But, if the answerer wants to say something about some real-life issues of both hep-ph or hep-th/math-ph research, or the transition between those two fields, please feel free and don't be shy.

2. I understand that there already exist some good similar recommendation list posts on this site, but, at least as I know, none of them focuses on the transition from hep-ph to hep-th/math-ph, which is the central point of this post. If there does exist such a post and someone has already offered a quite good answer, please commet.

3. Another way of formulating this question is the following. Please give a list of the introductory courses that a graduate student majoring in mathematical physics should learn, or maybe a list of those papers an advisor would recommend to a first-year graduate student majoring in mathematical physics. But, keep in mind that this student has already had accomplished a Master's level of hep-th study.

4. This question has also been crossposted on Math SE (https://math.stackexchange.com/questions/4194712/how-should-i-start-to-study-qft-at-a-somewhat-mathematically-rigorous-level-lik).

Answer 1

As Igor said, it's a bit late for that. If you just finished undergrad and you discovered a passion for QFT from a rigorous mathematical standpoint, what you should do, for example, is apply for the math PhD program at UVa and then do a thesis with me ;) Without expert help, trying to get into the subject by reading material in topology, differential geometry or algebraic geometry, as preparation for introductory courses that should hopefully equip you with the necessary mathematical background to finally be ready to start thinking about a research problem related to rigorous QFT, sounds like a recipe for not getting anywhere.

That being said, if you are serious about your goal, here is what you can do. Pick one of the QFT models that have been constructed rigorously and study that proof of existence until you understand it completely. I would recommend a result where the method is sufficiently general so by learning an example you actually get a feel for the general situation. This is in line with Hilbert's quote about the example that contains the germ of generality. In the present situation, this narrows the pick to a proof of construction of a QFT model using renormalization group methods (If you wonder why, see edit below).

As a rule, Fermionic models are considerably easier that Bosonic models, when it comes to rigorous nonperturbative constructions. I therefore think the best pick for you would be the article "Gentle introduction to rigorous Renormalization Group: a worked fermionic example" by Giuliani, Mastropietro and Rychkov. It is pretty much self-contained. If you know the Banach fixed point theorem, you're in business. In the article, they construct an RG fixed point, which in principle corresponds to a QFT in 3d which conjecturally is a conformal field theory. What they do not do is construct the correlations from the knowledge of that fixed point. As a consequence, they also do not prove conformal invariance of correlations. So here are two contemporary research problems for someone who did the "homework assignment" I just mentioned and would like to go further and prove something worthwhile.

If you prefer Bosonic models, then the other pick I would recommend is the article "Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions" by Chandra, Guadagni and myself. It is a toy model for the Bosonic analogue of the example considered by Giuliani, Mastropietro and Rychkov. The spacetime on which the fields are defined has a hierarchical structure which facilitates the multiscale analysis. There we rigorously constructed the RG fixed point and also the correlations for two primary fields, the elementary scalar field and its square. What we did not prove is conformal invariance, which is also conjectured to hold. The definition of conformal invariance in this setting is the same as in my previous answer:

What is a simplified intuitive explanation of conformal invariance?

Namely, just change Euclidean distance to the maximum of the $p$-adic absolute values of the components. Our article is also self-contained and also needs the Banach fixed point theorem together with some complex analysis and very minimal knowledge of $p$-adic analysis. The basics of $p$-adic analysis needed take a weekend to learn.

---

Edit addressing the OP's three new questions in the comments:

1. Why didn't I mention say TQFT or other approaches? Topological QFTs (the stress tensor vanishes completely) is a small subset of Conformal QFTs (the trace of the stress tensor vanishes) which themselves form a tiny subset of general QFTs. The study of these particular cases is certainly interesting but this relies on different tools that are specific to these particular cases and once you invest in learning these tools you will most likely be stuck with these particular cases for the rest of your research career. I proposed RG methods because I believe they cast a wider net and also should broaden your understanding of the subject. I think it would be easier to later specialize in say TQFT if that is where your taste leads you, rather than go the other way around: first develop expertise in say TQFT, and then learn some other method like the RG in order to escape from the narrow realm of TQFTs and study QFTs which are not topological. Next, a comment on "does this mean that all other method than rigorous renormalization group method have failed currently to construct a sensible QFT satisfying Wightman’s axioms (or its equivalent)?". The quick answer is no, this is not why I said you should choose an article which uses RG methods. I wrote my answer not just for you but also for other young people interested in rigorous QFT, from the hypothetical perspective of a PhD advisor talking to a beginning PhD student, starting the PhD thesis work now in 2021. As far as what method has been successful in proving the Wightman axioms for specific models, there are several. RG is one, as in the work of Glimm, Jaffe, Feldman, Osterwalder, Magnen and Sénéor on $\phi^4$ in 3d, and the work of Feldman, Magnen, Rivasseau and Sénéor for massive Gross-Neveu in 2d. The earlier work of Glimm, Jaffe, Spencer on $\phi^4$ in 2d used a single scale cluster expansion. Methods based on correlation inequalities and the more recent ones based on stochastic quantization typically allow you to prove most of the Osterwalder-Schrader axioms but not all, e.g., not Euclidean invariance.

2. Suppose we were having this discussion about geometric invariant theory instead of QFT, would you be asking me: I understand that Hilbert is the pioneering figure in the field of GIT, shouldn’t I start with some classical work at first? Sure, you could go read his 1893 Ueber die vollen Invariantensysteme but this might be more appropriate for a PhD in the history of math, rather than for doing research in this mathematical area now in 2021.

3. More or less yes, although I do not like your choice of words when you said that Axiomatic QFT "put QFT on the rigorous mathematical ground" in contrast to Constructive QFT which merely tries to "propose actual QFT models which satisfy those axioms". The ones who propose models are theoretical physicists. They come to you and say: here look at this Lagrangian it describes a model which is important for physics. Then you, say the constructive QFT mathematician, your job is 1) to prove that this model makes sense rigorously, by controlling the limit of removing cutoffs with epsilons and deltas but certainly no handwaving, and 2) to prove the limiting objects satisfy a number of properties like the Wightman axioms. Then the axiomatic QFT person can come and say: as a consequence of satisfying the axioms, here are these other wonderful properties that your model also satisfies, by virtue of this general theorem I proved the other day. Hope this clarifies the logical articulation of these different subareas of rigorous QFT.