At the moment I am studying operator algebras from a mathematical point of view. Up to now I have read and heard of many remarks and side notes that von Neumann algebras ($W^*$ algebras) are important in quantum physics. However I didn't see where they actually occur and why they are important. So my question is, where they occur and what's exactly the point why they are important.
Tell me more
×
Physics Stack Exchange is a question and answer site for
active researchers, academics and students of physics. It's 100% free, no registration required.
|
|
|||||
|
|
[Once again reading @Lubos' answer sparked these memories in my mind. Thanks for the inspiration @Lubos :)] @student - everything @Lubos says in this answer is valid. Given that von Neumann algebras are an exotic beast at present as far as their application in physics is concerned, I do know of three cases where they have had significant direct or indirect influence on theoretical physics.
Of course these are all rather specific and esoteric sounding applications so as @Lubos' noted, vNA's are far more being ubiquitous in theoretical physics. |
|||||||||
|
|
your impression that the von Neumann algebras are not being talked about by the physicists is entirely valid. The operators on the Hilbert space may perhaps satisfy the definition of a von Neumann algebra, but it doesn't make the specific results from this portion of maths useful in physics. The von Neumann algebras are not linked to any "specific piece" of interesting knowledge or mechanisms that physicists have to learn. An exception was algebraic or axiomatic quantum field theory which liked to talk about the von Neumann algebra but it has eventually become a fringe subdiscipline of theoretical physics. AQFT doesn't really work - it is not compatible with the most recent 40 years of fundamental physical insights about quantum field theory, such as the Renormalization Group. So the particular "focus" of the von Neumann algebra - in comparison with any algebra of operators on a linear space - are unlikely to be relevant for some important insights. Aside from quantum field theory, the notion of von Neumann algebras is also sometimes mentioned by physicists who study statistical physics and other fields but I think it is correct to say that only physicists who have gone through some math education in the past may be seen to "spontaneously" start to use the concept of von Neumann algebras. The algebras have surely not become a standard topic of undergraduate or graduate courses directed at theoretical physicists and I think that even most top theoretical physicists don't exactly know what the algebras are and aren't. Clearly, John von Neumann who introduced them would think that they would become much more important in physics in those 80 years than they have become. Von Neumann may be counted as one of the founding fathers of quantum mechanics; among these fathers, he was clearly the most mathematical (abstract) one and many of his hints simply didn't become standard. That's also true for some other concepts he introduced to quantum mechanics, including quantum logic. Still, he was a smart guy. Best wishes Lubos |
|||||||||||
|
|
as already mentioned, von Neumann algebras are at the heart of axiomatic approaches to quantum field theory and statistical mechanics, classical references to these topics are for the former (there are a lot more, of course)
and for the latter
The basic idea is that the observables of a physical theory should have some algebraic structure, for example it should be possible to scale them, that is measure c*A instead of A. Even more, one should be able to measure any (measurable, no pun intended) function of any observable A, which is possible if A is a memeber of a von Neumann algebra by Borel functional calculus. The philosophy of axiomatic quantum field theory in the sense of Haag-Kastler is therefore that a specific QFT is specified by a net of von Neumann algebras fulfilling a specific set of axioms, and that everything else can be deduced from this net of algebras (for an example see the page on the nLab here). As Lubos pointed out, this ansatz has been very succesful in proving a lot of model indepenent insights/theorems, like the PCT and spin/statistics theorem, but has not been successful in describing the standard model, as far as I know it is not possible to use this ansatz to calculate any number that could be compared to any experiment, which puts some criticism of string theory along these lines into perspective. On the other hand, it is possible to derive the Unruh effect and Hawking radiation using this framework in a much more rigorous fashion than it was done by the original authors, for more details see Robert M. Wald: "Quantum field theory in curved spacetime and black hole thermodynamics." (Although somewhat outdated, this is still a good place to start.) Two striking results where the deep connection between physical intuition and the (deep) mathematical theory of von Neumann algebras is visible involve the modular group of von Neumann algebras with a separating and cyclic vector:
The Bisognano-Wichmann theorem says that under specific conditions the modular group (of the von Neumann algebra associated with a wedge region in Minkowski space) coincides with the Lorentz boosts (that map the wegde onto itself), so here we have a very nontrivial connection of a mathematical object obtained from the structure theory of von Neumann algebras (modular theory) with an object coming from special relativity (a representation of the Lorentz group). |
|||
|
|
