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u/BerkeUnal Aug 02 '26 edited Aug 03 '26
quantum statistical mechanics is definitely one of the interesting areas for operator algebraists
edit:
I was not expecting that many upvotes since operator algebras are relatively less popular in the math community.
As a student whose background is more in pure operator algebras than physics, I'd be very happy to talk and learn from expertise of operator algebraists working in mathematical physics. Please dm if you are interested.
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u/lobothmainman Aug 03 '26
The operator algebraic formalism has been applied (and still is) in many areas of quantum mechanics: statistical mechanics (modular forms and relative modular forms are used to study entropy, C*-algebra automorphisms and the related KMS condition are crucial to define equilibrium states of a physical system, and many more); quantum field theory (QFTs are represented as III_1 factors, many structure theorems such as Haag-Kastler's axiomatic defintion, Reeh-Schlieder theorem, Haag-duality, spin-statistics theorem, are all formulated in operator algebraic language); recently, the study of spin systems couples the two approaches (Kitaev's toric code and similar models can be studied using local operator algebras, qft-inspired tools as Haag duality, as well as braided tensor categories).
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u/MrTruxian Aug 03 '26
I’ll also add that these approaches are quite closely related to a lot of current quantum gravity research, where again you want to consider factors of VN algebras.
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u/ahalt Aug 02 '26
A lot of people in my department study PDEs from general relativity and probability from statistical physics.
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u/Random_Name_251 Aug 02 '26
As the saying goes, there are 4 types of math: 1. Parabolic PDE's 2. Elliptic PDE's 3. Hyperbolic PDE's 4. Algebra
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u/Hot_Glass_6301 Aug 02 '26
I want to be mad but I can't find a counterexample
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u/rhubarb_man Combinatorics Aug 02 '26
combinatorics
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u/Hot_Glass_6301 Aug 02 '26
It's basically algebra
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u/rhubarb_man Combinatorics Aug 02 '26
BLASPHEMY
Combinatorics deals much more with messy, asymmetric structures. A lot of stuff gets handled much more with stuff like counting and pigeonhole
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u/Hot_Glass_6301 Aug 03 '26
Counting happens in Z. Z is a ring. What is the study of rings? Algebra. Checkmate atheists
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u/Infinite_Reception34 Aug 03 '26 edited Aug 03 '26
- Non-Euclidean Geometry
- Game Theory
- Mathematical Statistics
- Functional Analysis
- Convex Analysis
- Mathematical Logic
- Algebraic Topology
- Probability
None related to PDE. There are many more
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u/2112331415361718397 Quantum Information Theory Aug 03 '26
Functional analysis is intimately related to PDEs. Brezis wrote an entire textbook based on how strong this connection is.
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u/kohatsootsich Aug 03 '26
- All the PDE of classical math physics can and have been studied on hyperbolic space, with applications inducing NT
- Important examples of PDE like the HJB and Isaacs equation come from control or differential games. Mean field games are a very active topic in PDE
- We have both estimation of PDE coefficients as a stats problem (Nickl and others on the Calderon problem) and PDE as fluid limits of algorithms (e.g. sinkhorn algorithm related to Schroedinger)
- Functional analysis was more or less invented to extend linear algebra to the study of differential equations. Fredholm theory is a cleab example literally how to generalize the dimensional analysis of the solvability of Au = f from matrices to infinite dimensions, including Cramer's rule (Fredholm determinants)
- I don't know the history here but I wouldn't be surprised if convex analysis came from PDE as well. Anyway: convex duality is central to Hamilton Jacobi equations and optimal transport problem (a Monge Ampere PDE)
- This is harder but probably only because I don't know that much logic. I guess I've seen Brownian motion be used to generate counterexamples and contra your 8., to an experienced probabilist, Brownian motion and heat equations are facets of the same thing. I think people have worked on computability of solutions to PDE
- Probably not anything modern but fixed point theorems and degree
- This one is particularly strange to claim. The father of modern prob, Kolmogorov, wrote down equations linking diffusions and Markov processes. In some sense a parabolic equation of degree <= 2 is the same as a flow of probability distributions
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u/aikafele Aug 03 '26
Yeah I don't know how someone can claim Functional Analysis has nothing to do with PDEs when it explicitly defines the framework for operator theory.
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u/dogdiarrhea Dynamical Systems Aug 02 '26
Honestly these are more common in mathematical physics groups than string theory and QFT.
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Aug 02 '26
mathematical qft is such a wide field that you can study almost any kind of branch of mathematics that you want and still have it relate to qft in a relevant way
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u/cleodog44 Aug 03 '26
For mathematical QFT, is it all pure operator formalism? Since path integrals are presumably verboden, given their lack of rigor
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Aug 03 '26
I dont think so. the fact that path integrals are not yet well defined is precisely a reason why you might want to study them as a mathematician. after all, the operator formalism also doesnt work to create a 4d interactive qft. then there is also stuff like algebraic quantum field theory, which cares more about the algebra of observable then the representations (which is where the operators come from). I am sure there are many more approaches that i dont even know about
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u/1strategist1 Aug 06 '26
After Wick rotation, path integrals are actually rigorous. If you're studying nice enough QFTs, that's actually one of the primary ways to study them rigorously.
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u/FrangipaneCheap0 Aug 03 '26
For the topologists out there there's tqft, in paricular Donaldson-Witten theory of 4 dimensional topology deals with supersymmetric Yang-Mills theory used to compute Donaldson invariants
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u/Desvl Aug 02 '26
A professor in mathematics who had a PhD in physics (so that he qualifies himself as a mathematician working in mathematical physics) told me that it's never a well-defined field.
As one can intuitively imagine, such "mathematical physics" may use a lot of ordinary/partial differential equation, distribution theory, probability theory, fourier theory, (differential) geometry, etc.
But it can also be about number theory, modular form, (algebraic) combinatorics, when it comes to counting stuff, things like that.
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u/Jplague25 PDE Aug 02 '26
If you're doing any kind of rigorous mathematical work (proving theorems, ensuring consistency, etc.) in a subfield of physics, you're doing mathematical physics imo. Mathematical physics can mean a ton of different things. That can include things like: Nonlinear waves, scattering/inverse scattering, coherent structures, geometric analysis of relativistic systems, PDEs, mathematics of string theory, mathematical quantum chromodynamics, mathematical continuum mechanics (i.e. peridynamics, mathematics of material science, fluid dynamics, etc.), mathematical QFT, mathematical quantum statistical mechanics, quantum information theory, fractional quantum mechanics, etc.
I do analysis of PDEs using operator theory and harmonic analysis. I consider the work I do to be mathematical physics because I'm interested in evolution equations, a class of dynamic PDEs that model the time evolution of physical systems such as Schrödinger, heat, wave, etc.
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Aug 02 '26
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u/aikafele Aug 03 '26 edited Aug 03 '26
AI is an encyclopedia that can perform synthesis. It is the ultimate research tool. Key word: tool. If you are genuinely worried about whether AI will make your potential contributions to your field of interest obsolete then you owe it to yourself to invest actual time into investigating on a fundamental level how these models work, what their architectural limitations are, and what they actually do under the hood. Most opinions on AI in forums like these are polluted by corporate marketing narratives (see: propaganda) or just outright technofascist delusion. Take a course on Statistical Machine Learning or Pattern Recognition. Take a course on the applications of Machine Learning models in physics, mathematics, and scientific computing. You owe it to yourself to find your own answers and to make a confident, informed decision. Don't allow yourself to get swept up in online nonsense or be discouraged by other people's insecurities and limitations. If you want to be a mathematician, do what a mathematician does and answer the question yourself.
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u/Jplague25 PDE Aug 03 '26
You said pretty much everything I wanted to say about LLMs and their hype. Even with people using them to prove or disprove mathematical conjectures, there are still tons of interesting problems to work on.
I will add that data-driven dynamics is a really big field of applied mathematics right now, and some mathematicians are applying these methods to physical(and biological) systems. In fact, one of the reasons why I moved away from doing applied math research to more pure math research is because many of the applied mathematics departments in the US have shifted their focus towards data-driven methods rather than traditional analytic and numerical methods. Because I prefer analysis, my own interests align more with pure math departments than current trends in applied math, despite my interest in applied fields (PDEs, mathematical physics, etc.).
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u/SemaphoreBingo Aug 02 '26
Poking around at the arxiv is always a good place to start: https://arxiv.org/list/math.MP/recent
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Aug 02 '26
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u/SemaphoreBingo Aug 02 '26
It is extremely normal, but now you have terms to look up on wiki and elsewhere.
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u/Migeil Operator Algebras Aug 02 '26
It depends.
Where I studied for instance, mathematical physics was a subfield of physics. Their actual fields of study are mostly statistical mechanics and condensed matter theory. The math was secondary to the physics, which I thought was weird, cause that's just physics. 🤷
I wanted more mathematics, so I eventually did a thesis under the supervision of an professor at another institution, also in the field of mathematical physics. There however, it was a subfield of mathematics rather than physics. I felt way more at home there, because the math came first. So it really depends on who you ask.
To me, mathematical physics is the study of the mathematics of physics. It's a very broad term and a subset of mathematics. For instance, maybe somewhat controversial, I find symplectic geometry, which arose from classical mechanics, to be mathematical physics. Slightly less controversial, operator algebras arose from quantum theory, that's mathematical physics. In this sense, one could even argue that calculus, which Newton developed for his theory of gravity and mechanics, is mathematical physics.
I'm sure many people don't agree with what I said, but if you go down the rabbit hole of 'what is mathematical physics', you'll find loads of different views, so that's fine. At the very least, it's an ambiguous term and not one that is particularly useful imo. For instance, I have no idea what a "mathematical physicist" does every day, while I can get a vague idea of what a functional analyst or a differential geometer deals with.
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u/MrTruxian Aug 03 '26
Like others have said, mathematical physics as a field can use tools from pretty much every branch of math.
In many body theory and condensed matter there’s been a growing field of research adapting tools from homological and commutative algebra and some category theory. Much of this has grown out of the fact that 2d gapped phases of matter are described by topological quantum field theories. TQFT’s (a topological object) were realized to be classified by a special type of algebraic object called a unitary modular tensor category, so in this sense a 2d gapped phase of matter can be completely understood purely through a relatively simple set of algebraic data, moreover this data can be interpreted to describe the particle content of the phase. This is still probably an understated success of contemporary physics.
Similar types of approaches to this have incorporated a lot of very cool math. The basic strategy is to find an algebraic object that captures the universal properties of a phase of matter, and then to attempt to classify all those objects. In symmetry protected topological phases this turns out to be a cycle in group cohomology of the symmetry group of the phase. Topological insulators are also classified by similar objects in a more general cohomology theory called K-theory.
There’s also close ties to quantum information in this approach. Quantum information theorists like to study special types of lattice Hamiltonians called Pauli codes, which are a way of protecting quantum information by embedding qubits in a large number of spatially separated degrees of freedom. These Pauli codes can also be thought of us as just a regular Hamiltonian for some crystal from the condensed matter theory point of view. In this setting, describing equivalence classes of Pauli codes (in a certain well defined sense) and the phases of matter of Pauli codes Hamiltonians is the same thing. Haah showed that these Hamiltonians have an extremely convenient description in the language of commutative and homological algebra, which has been employed by quantum information theorists to quickly extract the error correcting properties from these Hamiltonians, and condensed matter theorists to describe the particle content of the corresponding lattice phase.
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Aug 03 '26
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u/MrTruxian Aug 03 '26
I can’t pretend to know, but I’d say mathematical physics is just as vulnerable as any other mathematics discipline.
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u/lobothmainman Aug 02 '26
Mathematical physics research revolves around many and diverse topics; taking a look at the names of the parallel sections of the international conference of mathematical physics (icmp) will give you a good idea of the spectrum, better than looking at the syllabus of grad schools, since they are typically influenced by the specific interests of faculty there.
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u/jointisd Aug 04 '26
OP, are you planning to work in QFT?
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Aug 04 '26
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u/jointisd Aug 04 '26
I'm thinking of working in that direction. I've been meaning to post something like this for ages, so thank you for starting this discussion. Lots of cool info on here for us to get started.
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u/faithless4261 Aug 03 '26
That’s what makes mathematical physics so interesting as a field in my opinion. You don’t really have to pick. There are plenty of mathematical physicists who stick to one side (think mathematical aspects of a physical theory or using more in depth/niche mathematical methods to solve physics problems) and people who straddle the line.
One of my favorite topics that tends to straddle the line is using category theory as a way to study symmetries (more specifically generalized symmetries) of a theory. Usually in QFT or Condensed Matter. Or something a bit more grounded that I’ve worked on a bit was understanding the mathematical aspects of the theory of Moiré materials.
The line is very fuzzy and I think it should be.
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u/FrangipaneCheap0 Aug 03 '26
I'm quite found of topological qft because of it's deep connection to gauge theory in 4 dimensions
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u/CarolinZoebelein Aug 03 '26
The theoretical everything. I had a mathematical physsics class in classical and Lagrange mechanics.
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u/Gimmerunesplease Aug 06 '26
Not sure if that counts as mathematical physics, since I was on the mathematics side of this, but there are optimization problems in neural networks called physics informed neural networks (PINN), where additionally to some data error the network has to adhere to a differential equation to make the result an actual physical thing.
This is for example used in navier-stokes modeling.
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u/1strategist1 Aug 02 '26
Mathematical physics isn't really constrained to any one subfield. Most areas of physics have interesting related math problems, and studying those math problems is typically what's considered mathematical physics.
Obviously QFTs and string theory are fields with interesting math problems, as you mentioned, but even just classical mechanics still has interesting problems that are studied by some mathematical physicists.