What are strong graduate programs for researching foundations of quantum mechanics?
What do you mean? The foundations of quantum mechanics are pretty well established. Do you mean quantum field theory? String theory?
Atomic, molecular, and optical physics
The foundations of mathematics or of GR are very well established too. But there is still research on those things.
The stuff people argue about incessantly in our Quantum Physics forum? (Interpretations; Bell's Theorem and related things)
Those silly discussions are about interpretations, which is foundations too of course. But there is other stuff. For example, you can do projective geometry and its relation to quantum mechanics, or you can do the C*-algebraic approach to quantum theory. Those are all very interesting foundational stuff that don't go into interpretations.
I would love to read about such notions. Although google might help me out, but what references do you recommend?
First I need to tell you that before reading such books, you should be acquainted with QM already, and also with the mathematics. Anyway, as an introduction, there are two very good books with surprisingly little overlap, so they're both important:
https://www.amazon.com/Introduction-Mathematical-Structure-Quantum-Mechanics/dp/9812835229 This deals with C*-algebras and a bit of quantum logic.
https://www.amazon.com/Quantum-Theory-Mathematicians-Graduate-Mathematics/dp/1489993622 Very good book from an operator theory perspective.
Then there's also Varadarajan for the link with projective geometry. https://www.amazon.com/Geometry-Quantum-Veeravalli-Seshadri-Varadarajan/dp/0387493859/
If you tell me the math and physics you're comfortable with, I might be able to give you a quick introductin to thee books.
(Sorry for hijacking but I really find this interesting). I am a graduate student at UT Austin and my field of research is String Theory. In mathematics, I am quite familiar with rigged Hilbert Space formalism, differential geometry, topology and algebra. In physics I am comfortable with quantum mechanics, quantum field theory and general relativity (coordinate dependent/independent formalisms). I have taken graduate courses in physics topics I mentioned.
I have research experience in quantum information and algorithms (if that helps). Currently, I am also learning tensor networks and emerging spacetime geometry.
Cool! You should have no problems with the books then. The books don't do the rigged Hilbert space formalism though, although it is the best formalism for QM.
Anyway, Strocchi starts off immediately with the C*-algebraic approach. The idea is to make operators/observables the primary object of QM, and not the states. This results in a very natural approach to quantum mechanics. You basically see that QM is the exact same thing as classical mechanics, but only "made noncommutative". The usual Hilbert space formalism (and the rigged Hilbert space formalism actually) can be derived from the more natural C*-algebraic formalism. This is known as the Gelfand-Naimark-Segal construction.
As for quantum logic. The idea is that observables in QM are measurable functions to the Borel sigma-algebra of the reals. Quantum logic then approaches the subject by replacing this measurable function/sigma algebra by a more general structure. This structure essentially is projective geometry, which is quite surprising.
In Hall, quantum mechanics is derived from scratch but rigorously. Functional analysis is applied rigorously to the theory.
Interesting. I wonder how it pans out in CFTs where we have state-operator correspondence.
Again, seems interesting. I will certainly try to read these texts. Thank you very much.
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