I'm trying to construct a curriculum for studying.
I’d like to know a little bit about how the approaches of various book. And what are the pros and cons of different approaches. Here I have a few physics books: https://drive.google.com/drive/folders/1YNeG78ZDeVsEQ4ISguWdXg8pfy_Z1S0-?usp=sharing
You can also find a google document in the Drive folder.
Costello's book is one of the few I have direct experience with. In the corner of mathematical physics near mine, people are looking very hard at Costello's approach to renormalization.
I've been wanting to read Costello's book for a long time. If you'd like to work through it, I might be able to follow along for the first chapter or so, although I won't be able to devote too much time to it.
Yes. My vague impression is that Costello took the machinery of cutoffs, effective actions, and renormalization group flow that physicists use to work with QFTs, and made those his definition of a QFT.
Also, one thing that also is unclear to me is.. if I study quantum field theory, does that help me understand the various attempts people are making to come up with a theory of everything. such as m theory, E8, noncommutative geometry...
As I understand (the little that I do) that there is some problem with gravity and Standard Model with is a QFT..
So, although I haven't studied it, I have faith that Costello's framework should at least start off pretty close to what physicists do, although the translation between Costello's language and typical physics textbook language might not be obvious without some mathematical sophistication. And I'm sure Costello throws in some extra technical machinery to make things mathematically precise.
Costello really starts to diverge from the usual physics language is in the later parts of the book, where he starts talking about cochain complexes and homotopies between QFTs. I think this machinery enters in the section on the Batalin-Vilkovisky formalism.
I have the vague impression that all this homotopy theory machinery does still correspond to what physicists are doing when they study gauge theories! See for example the dictionary (in Section 4.2 of my copy) that labels fields of different "homological degree" as what physicists would call "ghosts," "ordinary fields," "anti-fields," and "anti-ghosts."
However, I would expect the translation between Costello's language and standard physics language to become especially difficult at this point and beyond.
"The thing is, the introduction in these books say absolutely nothing to me"—you mean the introduction to Costello's book?
Oh—that's just one of those linguistic differences between math and physics. In my copy, "torsor" only appears in Theorem 1.5.1, as a comment on the term "principal bundle"; if you already know what a principal bundle is, you can ignore the "torsor" thing.
A "torsor" is, more or less, a group where you've forgotten which element is the identity element. You can think of a torsor for a group G as the smallest possible set that has a G symmetry. For example, a circle is a torsor for the 2d rotation group, a plane is a torsor for the 2d translation group, and the set {orange, lemon, lime} is a torsor for the 3-element permutation group.
When you encounter unfamiliar words like that in the mathematical QFT literature, I recommend just looking them up—or asking a nearby mathematician, if you can. They're often just unfamiliar words for familiar things.
It's the physics stuff I think I'm missing a bit too much in some sense, like "Let T (n) (E , S^cl) be the space of quantizations of the classical theory that are defined modulo hbar+1"
Which book is that "space of quantizations" line from?
How much QFT background do you already have? For example, my copy of Costello's book starts, "According to Wilson, a quantum field theory at energy up to Λ is described by the energy Λ effective action S [Λ] . This effective action is obtained by averaging over processes occurring at energies greater than Λ. The effective action S [Λ'] at some lower energy Λ' is obtained by averaging S [Λ] over fields with energy between Λ' and Λ."
When you read that, do you nod and say "yeah, that sounds right," or do you feel confused?
A big question of mine is though.. How different are these different approaches, like Connes' noncommutative geometry vs Costello's thing vs the finite QED thing vs all the other things that are out there..
"I have a vague idea of what an action is." Great! In that case, I would recommend two things.
First, if you haven't already, you should get comfortable with classical mechanics from the Hamiltonian and Lagrangian points of view.
Although it doesn't look like it at first glance, quantum field theory draws very heavily from classical mechanics.
Second, to get a quick feel for the basics of the functional integral approach to QFT, I highly recommend Alex Barnard's excellent notes on Richard Borcherds's "Lectures on Quantum Field Theory."
"How different are these different approaches"? That's a great question. There are many different approaches to QFT, and it's often difficult to see the connections between them until you understand several of them pretty well. I think that's one of the things that makes QFT so hard to learn.
I'd say that most approaches are trying to do the same thing: build a "quantum version" of a classical field theory. That's called "quantization." As you may know, quantization isn't a well-defined problem with a correct answer. It's like going from the macroscopic description of a material like air or water to a microscopic description in terms of molecules.
There could be many different microscopic descriptions consistent with the macroscopic behavior, and there's no purely mathematical way to choose between them.
The difference between some approaches to QFT—for example, the Fock space approach, the functional integral approach, and the algebraic QFT approach—is the difference between different recipes for quantization.
When you apply various quantization recipes to a classical field theory, you get quantum theories which look very different at first—but they turn out to be the same! You can see that as evidence that our mathematical approaches to QFT are on the right track.
My advice would be to look at several different approaches, with that perspective in mind. Eventually, when you learn enough, you'll be able to see the analogies between the approaches and start knitting them together.
By the way, I think Scharf's framework in Finite Quantum Electrodynamics is very closely related to the algebraic QFT (AQFT) approach. This approach is well-established for investigating the mathematical foundations of QFT, but it has limitations that make it unusable, in its present state, for the kinds of computations physicists need to do to make contact with experiment.
Scharf also talks a lot about the S-matrix approach. This approach developed in the 1970s or so, at a time when other approaches seemed to have hit an impasse. It tries to understand a QFT entirely in terms of its "S-matrix," which describes the things you'd actually measure in a particle accelerator experiment.
The S-matrix approach fell out of favor when other approaches got moving again, but it's making a comeback in the work of Nima Arkani-Hamed and others.
I'm not on Math.SE very much, so the most reliable way to contact me is by e-mail. You can find my address in the University of Toronto math department directory.
As you said earlier, "I'd like to get it right on the first try! :D I mean learn a point of view that ends up lining up with my goals."
It would be very helpful if you could send me a description of your goals. It's okay if you don't have too many details; but the more details, the better!
I'll try to formulate some understandable words. I'll send you email and share the Google doc with your email. We can chat there. I actually find the constantly updating sharing mechanism to be an excellent way to chat.
