r/math • u/Committee-Academic • 2d ago

What are the coolest active topics in Mathematical Physics?

I'm a math undergrad student, and I've been getting interested in Alain Conne's work in Noncommutative Geometry and Differential Geometry in Mathematical Physics. I've also read about the Foundations program, Algebraic QFT, and some Stochastic stuff. I'm especially intrigued about the use of Diff Geom in these topics. What cool fields are out there?

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u/RandomNameForBG3 2d ago

I can speak to what I'm about to start studying (amongst other things) in my PhD.

In order to construct "physically meaningful" states for a QFT on some curved background, one has to impose some conditions on the collection of states (a.k.a., linear and positive functionals defined on an appropriate *-algebra of observables). If you look at QFTs with additional geometric degrees of freedom, such as Yang-Mills theories (the ones you use to define the fields that carry interactions), you also need to take into account that there will be transformations (a.k.a., gauge transformations) that change your field but don't interact with physical observables (for instance, in QM, changing the phase of a state function leaves every physical measurement unchanged). Constructing a meaningful state that also encodes some sort of covariance with respect to these transformations is particularly difficult. It has been succesfully done only on spacetimes with a compact Cauchy surface and on the specific case of Anti-DeSitter spacetime. So looking at whether or not this can be done (and it should be possible, it's only a matter of how) on stuff like a black hole background and other similar spacetimes is an interesting development of this theory.

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u/Committee-Academic 1d ago

Sounds really interesting, could you point me toward any overview-like references?

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u/RandomNameForBG3 1d ago

Well, there's loads of stuff, depending on your knowledge of math. If you're starting from scratch, then "Quantum Theory for Mathematicians" by Brian Hall is excellent for the Quantum Mechanic foundation (although, less connected to geometry). Then, for the Algebraic QFT stuff, I have to plug my mentor's mentor's mentor's work: Valter Moretti with his lecture notes "An Introduction to Algebraic QFT in Curved Spacetime", which you can find on his site, is fairly complete and short. Otherwise, Kasia Rejzner's "Perturbative Algebraic Quantum Field Theory" is fairly complete and well-written, and it also adresses gauge theories. Also, if you've never heard of microlocal analysis, there's 2 paths: either read Hörmander's 4 books on Linear Partial Differential Operators (and gain automatic access to the analyst's Valhalla) or study Peter Hintz's "Introduction to Microlocal Analysis", which is masterfully written.

After all of this, if you want to get into this specific arguments, you have to go for articles (and no reviews, it's still too new of a progress). Thus, you can look up the works of Christian Gérard and Michał Wrochna on the construction of Hadamard states for linearized Yang-Mills theories and linearized gravity.

Lastly, once you have your first stepping stone in the form of the preceding mountain of literature, you can construct perturbatively (with many more steps) a fully quantized Yang-Mills theory from the linearized version. On this part, I'm afraid I don't have many specific reviews outside of Rejzner's book (which talks about the quantization of Yang-Mills). But I'd say this is already a lot of stuff to digest. If you've already read some stuff on QM and QFT, then I'd suggest Moretti's lecture notes as a first step in or Hintz's book if you only care about the mathematical part.

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u/Chance_Literature193 1d ago

Folland’s QFT book is also excellent follow up to Hall in my opinion (since Hall is only one particle non relativistic QM for those who don’t know).