I am studying the theory of $C^*$-algebras, von Neumann algebras and unbounded operators in courses on Functional Analysis and Opertor Algebras. Now I want to apply this knowledge to (algebraic) quantum mechanics. I heard that one can "affiliate" unbounded operators to $C^*$-algebras and von Neumann algebras of special types. In particular, many operators in quantum mechanics are unbounded and many authors use the $C^*$-algebraic description of quantum mechanics. How can i connect now unbounded operators to $C^*$-algebras in the context of quantum mechanics? Are there good books where I can study this? Is is possible to describe quantum mechanics with affiliated $C^*$-algebras? Another approach to describe quantum mechanics is via the Weyl-$C^*$-algebra, the $CCR$ and $CAR$ relations and Fock spaces. I know the book from Bratteli about this, but are there also other good book about this topic? Can one connect this two approches of quantum mechanics with each other?

Thank you very much about hints, links and answers on this question.

  • $\begingroup$ A similar question was asked on MathSciNet a few years back: mathoverflow.net/questions/55988/… $\endgroup$ – Strottos Dec 13 '14 at 1:30
  • $\begingroup$ Maybe this paper could be of interest as well: arxiv.org/pdf/0903.5446v1.pdf $\endgroup$ – Strottos Dec 13 '14 at 1:39
  • $\begingroup$ @Strottos : Already thank you for this comments. I also read the link you posted, but I don't see the answer I want to have, but almost ;) (but it is wel quite interesting). $\endgroup$ – ArgPraTos35 Dec 13 '14 at 10:28
  • $\begingroup$ You're welcome. It's not something I've kept up to date with these days but I became very interested in the AQFT stuff around 3 years ago for a bit. It seems that the C* approach is used more often there than in Quantum Mechanics. Be interested to hear of any other references where this is useful if anyone else knows though. $\endgroup$ – Strottos Dec 14 '14 at 11:47
  • $\begingroup$ Thanks a lot. Do you other good book other than Bratelli, Strocci, etc. which refer to C* algebras and connections with unbounded operators? $\endgroup$ – ArgPraTos35 Dec 15 '14 at 16:12

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