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 do they occur and what's exactly the point why they are important.
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 (Although somewhat outdated, this is still a good place to start.)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).
[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.
The entire program of knot theory and manifold invariants etc - as represented in Witten's work on TQFT's (topological quantum field theories) - owes in large part to Vaugh Jones' discovery of a knot invariant known as (obviously) the Jones Polynomial. I know only the vague outline of how he was led to this discovery but I do know that it happened in the course of his investigations on a particular class (type III?) of von Neumann algebras.
Connes' non-commutative geometry program also has its roots in the study of von Neumann's algebras if I'm not mistaken. Non-commutative geometry is coming of age with a large number of applications ranging from methods of unifying the Standard Model particles to understanding the quantum hall effect. NCG also arises naturally in string inspired models of cosmology and inflation, [Reference]
Finally, Connes and Rovelli put forward the intriguing "thermal time hypothesis" in order to try to resolve some of the dilemmas regarding the notion of "time" evolution and dynamics which arise in theories of quantum gravity where the Hamiltonian is a pure constraint - as is the case in the "Canonical Quantum Gravity" program. Their construction rests on a certain property of von Neumann algebras. To quote from their abstract:
... we propose ... that in a generally covariant quantum theory the physical time-flow is not a universal property of the mechanical theory, but rather it is determined by the thermodynamical state of the system ("thermal time hypothesis"). We implement this hypothesis by using a key structural property of von Neumann algebras: the Tomita-Takesaki theorem, which allows to derive a time-flow, namely a one-parameter group of automorphisms of the observable algebra, from a generic thermal physical state. We study this time-flow, its classical limit, and we relate it to various characteristic theoretical facts, as the Unruh temperature and the Hawking radiation.
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.
It seems to be a good time to bring this topic up to date, at least on an increasingly active development in the field of condensed matter physics, since this discussion is part of a very few on the topic of Von Neumann Algebra.
I insist that the topic mentioned is a current effort, and nothing has been brought to conclusion yet. However, for students/researchers interested in trying to answer question of mathematical physics using Von Neumann algebra, this is certainly a promising direction to follow, and hence important to mention already at this stage.
To begin with, let us recall briefly why the operator algebraic framework of quantum statistical mechanics is an important part of physics: the concept of phase and phase transition is a well defined mathematical phenomenon only in the thermodynamic limit. For instance, recall that a "spontaneous symmetry breaking" of an Ising model is always assumed to be "in the thermodynamic limit", because the two phases "all up" and "all down", have to be separated by an infinite energy gap, so that one is sure that no local operator (meaning no operator acting on finitely many spins) can switch from one state to the other. However in quantum mechanics, these broader terms of "symmetry breaking" and "phase" or "sector" are very often employed in a relaxed manner. The reason for this is that an infinite dimensional Hilbert space (for instance an infinite 2D lattice of spins) is an object requiring the tools of algebraic quantum statistical mechanics, in which states are no longer simply vector in a Hilbert space. Usually, researcher can avoid using this framework by explicitly finding a bound of some quantity like energy, depending on the size of the system, and making the size grow to "infinity". A lot of very rigorous and pioneering results have been found as such.
This being recalled, as it is commonly known in the entire quantum physics community now, classification of quantum phases of matter is a very active and challenging domain. Among the classes of phases are non invertible phases, commonly referred to as topological order, for which the mathematical framework is already known to be category theory. It has been the subject of massive attention in the past 20 years. However, like any quantum phase, to be mathematically well defined a topological ordered phase need to be studied in the thermodynamic limit and a classification of these phases is a mathematically complicated task: they are interacting long ranged entangled phases and finding an "index" (a robust mathematical quantity that distinguish phases) seem to require to work directly in the thermodynamic limit. This method has been very successful and straightforward (non ambiguous) in the last couple of years to classify symmetry protected phases (SPT), see for instance https://arxiv.org/abs/2111.07335 for a very comprehensible case of SPT. It has recently been brought forward (but in fact assumed from the beginning by a small community of researchers) that the classification of topological order should also be successful in the operator algebraic Framework, where the super-selection sectors, as defined here http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/DHR+superselection+theory , are expected to be the essential object, and to show rigorously the appearance of braided tensor categories. In this active effort of research, Von Neumann algebra plays an important role: https://arxiv.org/abs/1608.02618. In this same direction, the topological entanglement entropy is thought to be rigorously defined thanks to the Jones index: https://arxiv.org/abs/2106.15741. This is not so surprising to see V. Jones' work appear here, since topological order and topological entanglement entropy are, at this point, derived only from topological quantum field theory arguments, in which Jones' work on knots is central.