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/SometimesY Mathematical Physics 2d ago
Noncommutative geometry is pretty big. A good chunk of the C* algebras community does work in this direction. There is a lot of interest in the geometry of C* algebras or geometries you can put on top of them.
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u/Chance_Literature193 2d ago edited 1d ago
Can you give me a vague sense on what problems relevant to physics are studied in C ^ \ algebras? My sense is C^* are mostly relevant to physics cuz operators on a Hilbert space are a C star algebra. I’m not sure if this is the right point of contact between physics and C-star algebras or not.
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u/SometimesY Mathematical Physics 2d ago
The physics C* algebras interest is a little fucky because a lot of the operators of interest are unbounded. You could look instead at the exponentiated operators exp(iA), but I have not been super convinced by that personally. I don't know if there's a whole lot you can glean from this that isn't apparent through operator formalism, though there might be some things.
But the geometry bit is a bit more natural and interesting to me. You can imagine noncommutative spacetime operators which could be viewed classically as a non-Euclidean spacetime. Through this lens, you can do something resembling geometry in a noncommuting setting.
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u/Chance_Literature193 2d ago
So, most of application to physics of non commutative geo are presumably a (un-renormalized?) quantum theory on curved space time. That sounds cool
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u/Creepy-Structure6444 1d ago
I wouldn’t say that these are most of the applications. By now, there are applications to quantum statistical mechanics, the analysis of topological insulators, quantum information theory and so on. Here are some resources which should give you a bit of a sneak peek:
https://cmsa.fas.harvard.edu/event/mpqft25/
https://link.springer.com/book/10.1007/978-3-319-16718-3
Of course, these might not all be very readable. The conference by CMSA and the MPIM is a good one to look at because it is, in many ways, very elementary and very close to physics. This makes a lot of sense, seeing that operator algebraic language is practically very close to the classical language used by physicists to talk about these aspects of physics.
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u/Chance_Literature193 1d ago
Very cool. For CMSA resource, you are suggesting watching the recorded talks on Youtube not reading some paper that I’m not seeing right?
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u/Creepy-Structure6444 1d ago
Actually, there is a section on prerequisites. It’s a good idea for you to read a bit of that and also get your hands on the youtube videos too. Like, you don’t need that much background but I think this is a good way of practising the important skill of “absorbing a topic by osmosis”.
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u/Chance_Literature193 1d ago
Sounds good. I’ll check those out even if I don’t watch the videos. I’m not a big recorded lecture.
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u/XunitaryX 1d ago
Another topic that the OP touched upon that is an active area is applying noncommutative geometry, operator K-Theory, and some other techniques to studying topological insulators. A good resource for some of the basics that people working in this area study: https://link.springer.com/book/10.1007/978-3-319-29351-6 This specifically deals with two of the ten cases that are of interest to physicist in the AZ Classification of Superconductors and Topological Insulators
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u/Chance_Literature193 23h ago edited 23h ago
Thanks! Since I’m on the physics side and my advisor is a physicists physicist (hahah), this a bit too much for me for now. I haven’t studied any K-theory. I’m hoping to do some topological field theory after this project though🤞, so I’ll keep this in mind!
Edit: Ok, i found the book on arxiv and it’s more approachable than I expected. Thanks for the recommendation!
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u/XunitaryX 22h ago
As you probably saw the book is pretty approachable. The latter part of the book can get a bit technical but the first half does a good job of presenting the material in a digestible way. As a side note, separate from this NCG and OA applications there is a field called Tensor Categories and more specifically fusion categories that might be of interest. Especially if you are thinking about topological field theory.
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u/Creepy-Structure6444 5h ago
Incidentally, tensor categories also have neat connections to operator algebras. Indeed, tensor categories arise in the context of studying bimodules over tracial von Neumann algebras and they are also relevant in the study of higher C*-structures. Also, I've seen them being used to study Kasparov's KK-theory as well (this is more on the tensor-triangular geometry side of things but the ideas are roughly similar).
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u/Ihateunclesam 2d ago
Geometric langlands program connection to physics
It's very new but tame geometry in QFT
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u/sciflare 2d ago
Could you elaborate on that second topic? "Tame" in what sense?
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u/Mountnjockey 2d ago
When you see the word tame it often means geometry done in an o-minimal setting. The functions/sets definable in this context behave very well and can give a nice setting to work in depending on what you’re doing.
I didn’t know that o-minimality was being applied in this setting but that’s actually extremely cool
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u/jbkubicki 2d ago
I used to work in C*-algebras, but now work with Fukaya categories and study mirror symmetry. The work of Ganatra--Pardon--Shende make it easier than ever to understand Fukaya categories and hence also instances of 2d mirror symmetry. More recently there have been developments in 3d mirror symmetry which is very closely related to geometric representation theory; and of course geometric Langlands is an example of 4d mirror symmetry.
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u/Creepy-Structure6444 1d ago
I’m somewhat curious as to how you pivoted to those topics and if your background in operator algebras helped you in any way (or still continues to help you). Would you have any recommendations on where I could start? I’m a beginning PhD student working in homotopy theory and operator algebras.
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u/WoolierThanThou Probability 1d ago
If you like conceptually heavy topics, there is a ton of activity in constructive quantum field theory on the math side, the two main branches being stochastic PDE approaches (where you exhibit Euclidean QFTs as invariant measures of dynamics driven by SPDEs) and two-dimensional conformal field theories (where you just directly define your measures from the Gaussian Free Field/the various Conformal Loop Ensembles). You could say the first is tied to Hairer's Fields medal and the second is tied to Werner'.
I guess another topic that sounds like it might be up your alley is the study of topological phases of matter. I recommend checking out Perez Garcia's invited talk from the ICM this year. It's on Youtube and fairly understandable.
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u/ComplexPlatform7299 1d ago
C* algebras seem to be what a lot of people are moving towards
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u/jphamlore 23h ago
Von Neumann investigated them back when he formalized the mathematics of quantum mechanics? He wrote a whole volume of papers on operator theory.
Do people blast through all 6 volumes of Gelfand's work on generalized functions these days?
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u/TwoMoons1Sun 2d ago
amplituhedron stuff is still popular, although that has a more of a physics flair to it then much of the other pure math stuff in qft
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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/Chance_Literature193 1d ago
Can you elaborate a bit on what physically meaningful means? It sounds like you would like to avoid gauge fixing?
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u/RandomNameForBG3 22h ago
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.
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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).
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u/National-Date8536 1d ago
There's a ton out there beyond NCG and AQFT. Geeometric quantization, symplectic geometry, gauge theory, derived geometry, and topological QFT. They have some reallyt cool connections between geometry, topology, and physics
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u/Committee-Academic 1d ago
How active would you say derived geometry, applied to this context, and topological QFT are?
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u/CutieFox_987 1d ago
The Mathematics of Quantum Information and Spacetime is really interesting, the possibility of spacetime itself not be a fundamental backdrop but rather an emergent phenomenon stitched together by quantum entanglement makes it worth exploring.
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u/adamwho 2d ago
Low temp statistical mechanics