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I think that recently I've started to lean in my interest more towards operator algebras and away from differential geometry, the latter having many applications to physics. But while taking physics courses, it was also brought to my attention that operator theory is a very integral part of quantum mechanics. Are there applications of operator algebras in particular to quantum or relativity, or other fields of modern physics? What about applications of operator theory?

In a related vein, I've been trying to find some research problems at my level or only slightly higher so that I can get a flavor, either for these applications, or for the abstract subject itself, as it's studied today. I was wondering if anybody knows either of any such problems, or a source where I could find such problems. To be honest, being a first year graduate student going on second, I'm not even quite sure where I'd look to find research problems that I could definitely guarantee are open, let alone ones I could reach right now. In fact, I'm told that sometimes professors even mistake solved questions as being open. If you have good examples of solved problems that are recent and representative of what I might face in the future, those would be helpful too. Thanks.

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This has all been very helpful, but I am also interested in knowing about a good source of open questions, particularly those you would find suitable to give to a first or second year graduate student. Also, perhaps even more importantly, those you would give to a PhD student in his 3rd-5th year for his thesis. – Jeff Aug 10 '12 at 8:29
Quantum Information Theory and Statistical Mechanics. Check out the Vanderbilt math department's website (Center for Operator Algebras and Noncommutative Geometry) – r.g. Dec 31 '12 at 0:45
@Jeff Since you're at Caltech (I gather), go find Matilde Marcolli and speak with her. She's an expert on Noncommutative geometry, which basically amounts to applying operator algebras to...everything... – Alex Nelson Feb 12 '14 at 17:03
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The only applications to general relativity that I know of (my field!) is via Connes' noncommutative geometry, which is...complicated. Connes' work is freely (and legally!) available online. See also the math overflow thread Applications of Noncommutative Geometry, which may be interesting.

Operator algebras pop-up in Algebraic Quantum Field Theory too, which may be fun to look at. Actually, trying to discuss quantum fields on curved spacetime requires operator algebras.

The Physics.SX thread "Why are von Neumann Algebras important in quantum physics?" is also relevant.

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Ah thanks, this certainly gives me a flavor of the kind of applications there are. My physics education certainly doesn't let me talk with any authority about physics, but it does sort of confirm a fear I had coming out of my undergrad physics degree at Caltech. There, this course for mathematical physics is offered, and also one for stat mech/thermo. It seemed like discrete and differential geometry, PDEs, and other related fields appeared all over the field while more "abstract analysis" like Operator Algebras/Theory took a back seat. – Jeff Aug 8 '12 at 3:09
It sounds from the sources you linked that were I to pursue applications to physics intentionally, then there'd be many ways to do it though. Just the general, relatively non-abstract community in physics would not be very interested. I suppose in time as my interests reveal themselves during grad school, I will be able to better judge if this is the niche for me. – Jeff Aug 8 '12 at 3:12
@Jeff when you say "the general, relatively non-abstract community in physics would not be very interested" what do you care what other people think? The important thing is you enjoy doing what you do; if it's borderline pure mathematics with a physics aftertaste, wonderful! As long as you enjoy it, and you're not a crackpot ;) – Alex Nelson Aug 8 '12 at 4:44

You might look at Bratteli and Robinson, "Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States. Models in Quantum Statistical Mechanics

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Seconded. Both the volumes are my personal favourites. – RSG Aug 9 '12 at 4:38

Recently, operator space theory is used to understand/ create a mathematical foundation of Bell's Inequality, which is one of the fundamental concept in quantum mechanics (and quantum information). This is now an active field of research. You can check the paper (arxiv version) and the follow up papers for more information.

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"But while taking physics courses, it was also brought to my attention that operator theory is a very integral part of quantum mechanics."

Your are correct here, but this doesn't mean operator algebras "is a very integral part" of quantum mechanics. Much effort in modern mathematical physics has been devoted to the analysis of a single operator, and if you want to do research integral to quantum theory- you should learn partial differential equations and spectral theory of Schrodinger operators!

Operator Algebras were useful for studying the foundations of quantum mechanics and this approach was pioneered by Haag, Ruelle among others. But be aware that this is not the only "foundational approach" to quantum mechanics (and you need to be careful when devoting your early years to any single foundational approach- I am saying this not to sound rude but because I think it is a responsible thing to say). Furthermore, the algebraic quantum field theory approach of Haag and others has led to some problems with non-local effects that even recently people (mostly philosophers of physics) have tried to defend. But it is mostly "talking" than doing mathematics.

I don't want to sound like a bigot, but I wanted to stress that there is much much much more to quantum mechanics than operator algebras and it is a bit of a stretch to consider operator algebras as an integral part of quantum theory! You should first focus on the analysis of a single Schrodinger operator (first pioneered by Tosio Kato, Rellich, etc.) and there is considerable current progress in the field and is relativly more integral to quantum theory than an algebra of operators.

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