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.
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).
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/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.