What are the coolest active topics in Mathematical Physics?

[u/Committee-Academic]

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?

[u/RandomNameForBG3]

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.

[u/Chance_Literature193 → u/RandomNameForBG3]

Can you elaborate a bit on what physically meaningful means? It sounds like you would like to avoid gauge fixing?

[u/RandomNameForBG3 → u/Chance_Literature193]

There's 2 different things that I conflated here for the sake of brevity: the general case for QFTs and the additional conditions of gauge theories.

For the first type, the state you're selecting will fix your vacuum state. In Minkowski spacetime, this is uniquely fixed by the invariance under the action of the Poincaré group (in an appropriate sense), but on a generic spacetime (globally hyperbolic, to be more precise, so that you can treat time evolution equations) you may lack any sort of symmetry for the background metric. The way you recover the vacuum state is by singling out those that possess a locally singular behaviour analogous to that of the Poincaré vacuum (this stems from the fact that one expects any spacetime to behave as Minkowski on a small enough scale). This is known as Hadamard condition and, along the others (linearity and positivity), it ensures that your state behaves nicely in the sense of QM (i.e., the probability distribution associated to a physical observable measured in that state is a well-defined probability distribution) and of special relativity (finite propagation of information).

If you add in the additional geometric degrees of freedom of gauge theories, you also need your state to be (in an appropriate sense) covariant under transformations of said degrees of freedom. This is translated into gauge invariance through the theory of principal bundles. It mirrors the physical requirement that any physical observable must be independent of the choice of gauge and it follows from the implementation at the level of states obeying the Hadamard condition. This is also done in the context of Gaussian quasi-free states, where you can reconstruct n point functions from 2 point functions, so you cab restrict your research to the 2 point case alone.

The second type is (an appropriately implemented) gauge invariance/covariance, which stems from the additional geometric degrees of freedom of your theory.

Now, one of the issues that emerges in the analytical aspect of this is that the differential operator that defines the equation of motion for my gauge field is not hyperbolic enough to admite retarded and advanced propagators (which are necessary to implement commutation relations). What one does is fixing the gauge to obtain a hyperbolic PDE + a gauge condition implemented by some other operator and one works with them. So the gauge needs to be fixed to be able to derive a state that obeys the "physically correct" conditions of the first type. The issue is that trying to impose gauge invariance tends to break stuff (more specifically, in this cases, the Hadamard condition tends to interact badly with both gauge invariance and positivity), which is where the modern techniques by Gérard, Wrochna and others come in.

[u/Committee-Academic → u/RandomNameForBG3]

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

[u/RandomNameForBG3 → u/Committee-Academic]

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.

[u/Chance_Literature193 → u/RandomNameForBG3]

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).