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 (Although somewhat outdated, this is still a good place to start.)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.
It depends on the viewpoints. von Neumann algebras are a technical tool of crucial mathematical relevance in quantum theory. Whether or not they have physical relevance is a different issue.
The reason for their technical relevance is the following: They are structures which embody both the algebraic and topological features of all manipulations of Quantum Theory in Hilbert space. The topological aspect is in particular relevant: von Neumann algebras are closed exactly with respect to the topologies used in spectral theory and the permitted operations are continuous with respect to them. For instance if an observable belongs, or is affilitated, to a von Neumann algebra, then its spectral measure does and all (Borel) functions of that observable do.
So, technically speaking, the structure of von Neumann algebra is the best for all mathematical descriptions or manipulations that Quantum Theory (Quantum Mechanics and Quantum Field Theory) needs in its Hilbert space formulation. They also permit to go behind and beyond some basic postulates of QM and to extend them. For instance, the fact that composite systems are formalized in terms of the tensor product can be proved here, using a more general definition of independent subsystems, under suitable hypotheses (in QFT, independence does not automatically imply the use of the tensor product as is well known dealing with type-III factors).
What is the physical weakness of a formalisation relying on von Neumann algebras? It is easy to understand: As soon as we assume that the physical observables of a quantum system are (exactly) the operators which belong or are affiliated to a certain von Neumann algebra (determined by the quantum system), we are assuming an incredibly strong postulate which does not have a justification and it cannot be controlled. In the practice, it is evidently false that every self-adjoint operator in a Hilbert space is an observable. Though, when dealing with systems without superselection rules or a non-abelian Gauge group, the relevant von Neumann algebra contains (as affiliated operators at list) every selfadjoint operator in the Hilberts space.
There is no general way in QT to associate an experimental measurement procedure to a arbitrarily given selfadjoint operator, so the postulate cannot be even checked. Think of $P^2+X^2$ when we are able to measure the position $X$ and the momentum $P$: If $P^2/2m+kX^2/2$ is the Hamiltonian of the system we know how to measure it, but if it is not, how we can measure it? (For instance the Hamiltonian may be $P^2/2m+ cX^4$.)
Another example is the Newton-Wigner position operator: it is clearly unphysical since, when assuming the standard ideal formulation of the collapse of the state, it does not satisfy basic requirements about causality. However it can be constructed with the basic observables of the considered elementary particle (in Wigner's sense) and it belongs to the von Neumann algebra generated by them. This example is important because we have here a self-adjoint operator which cannot define an observable though there is no superselction rule o Gauge group to justify this fact.
(A few, very personal, words about von Neumann are in my view necessary, since I read some negative comments quite inappropriate in my view. In my opinion he was one of the most outstanding applied and pure mathematicians of human kind. He can be compared with a very small number of other mathematicians, like Archimedes, for his productivity and relevance and influential ideas in an incredibly large and spread fields of pure and especially applied mathematics. Maybe he was not a pleasant person, with personal opinions I cannot say to share. However his book on mathematical foundations of QM contains exceptionally deep ideas, some of them completely understood only in this century: all modern view on quantum measurement relies in particular upon ideas by von Neumann, some of them already present in that book. )
Von Neumann algebras help understanding how physical properties of quantum systems evolve when more and more degrees of freedom are added.
To put it simply, if one builds a large quantum system by assembling $N$ elementary systems (e.g. qubits or spins 1/2), the resulting Hilbert state space $H_N$ is the tensor product of these $N$ 2D elementary Hilbert spaces. Its dimension is $2^N$. Now if one takes the $N\rightarrow \infty$ limit to model a macroscopic quantum system, the complete Hilbert space has an uncountable dimension which means it is non-separable.
There is however a theorem by von Neumann (let's call it the "sectorisation theorem") that tells that this huge Hilbert space is the direct sum of (uncountable many) separable subspaces. Each of these separable subspaces are generated by a reference vector in the complete Hilbert space (say all qubits in $|1\rangle$) and all the states obtained by changing a finite number of elementary states in the reference state.
Each reference state defines a classical sector, and one can show a number of properties on them by using von Neumann algebra concepts. In particular, there is a spontaneous decoherence mechanism that does not involve information loss to the system environment, that emerges from these properties.
This allows understanding what can be quantum and what can't. In particular it gives an explanation of the quantum/classical boundary from within the quantum formalism.
You can find more on this in arxiv/2209.01463