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)

- Hellmut Baumgärtel: "Operatoralgebraic Methods in Quantum Field Theory".

and for the latter

- Ola Bratteli and Derek W. Robinson: "Operator algebras and quantum statistical mechanics." (two volumes).

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 (Although somewhat outdated, this is still a good place to start.)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 characterization of KMS states in statistical mechanics,

the Bisognano-Wichmann theorem connecting the automorphism of the modular group to the representation of the Lozentz group, for more ideas using modular theory see the paper "Modular theory for the von Neumann algebras of Local Quantum Physics" by Daniele Longo on the arXiv.

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).

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Tim van Beek