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.
Quantum Mechanics – Importance of von Neumann Algebras in Quantum Physics
mathematical physicsquantum mechanicsquantum-field-theoryquantum-information
Related Solutions
All of physics has two aspects: a local or even infinitesimal aspect, and a global aspect. Much of the standard lore deals just with the local and infinitesimal aspects -- the perturbative aspects_ and fiber bundles play little role there. But they are the all-important structure that govern the global -- the non-perturbative -- aspect. Bundles are the global structure of physical fields and they are irrelevant only for the crude local and perturbative description of reality.
For instance the gauge fields in Yang-Mills theory, hence in EM, in QED and in QCD, hence in the standard model of the known universe, are not really just the local 1-forms $A_\mu^a$ known from so many textbooks, but are globally really connections on principal bundles (or their associated bundles) and this is all-important once one passes to non-perturbative Yang-Mills theory, hence to the full story, instead of its infinitesimal or local approximation.
Notably what is called a Yang-Mills instanton in general and the QCD instanton in particular is nothing but the underlying nontrivial class of the principal bundle underlying the Yang-Mills gauge field. Specifically, what physicists call the instanton number for $SU(2)$-gauge theory in 4-dimensions is precisely what mathematically is called the second Chern-class, a "characteristic class" of these gauge bundles_
- YM Instanton = class of principal bundle underlying the non-perturbative gauge field
To appreciate the utmost relevance of this, observe that the non-perturbative vacuum of the observable world is a "sea of instantons" with about one YM instanton per femto-meter to the 4th. See for instance the first sections of
- T. Schaefer, E. Shuryak, Instantons in QCD, Rev.Mod.Phys.70:323-426, 1998 (arXiv:hep-ph/9610451)
for a review of this fact. So the very substance of the physical world, the very vacuum that we inhabit, is all controled by non-trivial fiber bundles and is inexplicable without these.
Similarly fiber bundles control all other topologically non-trivial aspects of physics. For instance most quantum anomalies are the statement that what looks like an action function to feed into the path integral, is globally really the section of a non-trivial bundle -- notably a Pfaffian line bundle resulting from the fermionic path integrals. Moreover all classical anomalies are statements of nontrivializability of certain fiber bundles.
Indeed, as the discussion there shows, quantization as such, if done non-perturbatively, is all about lifting differential form data to line bundle data, this is called the prequantum line bundle which exists over any globally quantizable phase space and controls all of its quantum theory. It reflects itself in many central extensions that govern quantum physics, such as the Heisenberg group central extension of the Hamiltonian translation and generally and crucially the quantomorphism group central extension of the Hamiltonian diffeomorphisms of phase space. All these central extensions are non-trivial fiber bundles, and the "quantum" in "quantization" to a large extent a reference to the discrete (quantized) characteristic classes of these bundles. One can indeed understand quantization as such as the lift of infinitesimal classical differential form data to global bundle data. This is described in detail at quantization -- Motivation from classical mechanics and Lie theory.
But actually the role of fiber bundles reaches a good bit deeper still. Quantization is just a certain extension step in the general story, but already classical field theory cannot be understood globally without a notion of bundle. Notably the very formalization of what a classical field really is says: a section of a field bundle. The global nature of spinors, hence spin structures and their subtle effect on fermion physics are all enoced by the corresponding spinor bundles.
In fact two aspects of bundles in physics come together in the theory of gauge fields and combine to produce higher fiber bundles: namely we saw above that a gauge field is itself already a bundle (with a connection), and hence the bundle of which a gauge field is a section has to be a "second-order bundle". This is called gerbe or 2-bundle: the only way to realize the Yang-Mills field both locally and globally accurately is to consider it as a section of a bundle whose typical fiber is $\mathbf{B}G$, the moduli stack of $G$-principal bundles. For more on this see on the nLab at The traditional idea of field bundles and its problems.
All of this becomes even more pronounced as one digs deeper into local quantum field theory, with locality formalized as in the cobordism theorem that classifies local topological field theories. Then already the Lagrangians and local action functionals themselves are higher connections on higher bundles over the higher moduli stack of fields. For instance the fully local formulation of Chern-Simons theory exhibits the Chern-Simons action functional --- with all its global gauge invariance correctly realized -- as a universal Chern-Simons circle 3-bundle. This is such that by transgression to lower codimension it reproduces all the global gauge structure of this field theory, such as in codimension 2 the WZW gerbe (itself a fiber 2-bundle: the background gauge field of the WZW model!), in codimension 1 the prequantum line bundle on the moduli space of connections whose sections in turn yield the Hitchin bundle of conformal blocks on the moduli space of conformal curves.
And so on and so forth. In short: all global structure in field theory is controled by fiber bundles, and all the more the more the field theory is quantum and gauge. The only reason why this can be ignored to some extent is because field theory is a compex subject and maybe the majority of discussion about it concerns really only a small little perturbative local aspect of it. But this is not the reality. The QCD vacuum that we inhabit is filled with a sea of non-trivial bundles and the whole quantum structure of the laws of nature are bundle-theoretic at its very heart. See also at geometric quantization.
For an expanded version of this text with more pointers see on the nLab at fiber bundles in physics.
I had the same question, when studying the subject. Let me tell you, what I was told - it relates to the functional calculus:
Recall that in quantum mechanics, as we usually learn it, a measurement is a projective measurement, i.e. the outcomes of a measured observables are eigenvalues of the observable and we "update" the state according to the knowledge obtained (i.e. we project it into the Hilbert space). We can of course use the whole formalism of POVMs instead if you know about this, but still, projective measurements remain important special cases. For this reason, we don't actually need our Hermitian observables, but we need the spectral calculus and the spectral theorem. You want all spectral projections of an observable to belong to your space, since if you measure the observable, your results will be updated according to the eigenprojections. And here is the problem: C*-algebras in generally do not contain all their projections, von Neumann algebras do. So the "physical consequence" is that you actually have all your measureable quantities inside the algebra of operators you call "observables". I believe that is as physical as it gets. Since von Neumann algebras can always be seen as closures of C*-algebras in some topology, I would not expect there to be a much deeper reasons, although I'd love to know them myself, if there are.
Other reasons mentioned to me refer to the structure of von Neumann algebras (and its lattice of projections) and how this enters different scenarios in physics, but in this case, I would say that the reason to study von Neumann algebras is rather technical than physical.
Finally, let me point out that it is not a priori clear why we should study C*-algebras at all - I mean, the only physical quantities are the Hermitian operators, but generically, our algebras will contain many nonhermitian elements. In my view, this means there is no reason to study either C*-algebras or von Neumann algebras, but one would actually have to study Jordan algebras (the set of Hermitian elements of the bounded operators on some Hilbert space forms such a Jordan algebra, or more precisely, a Jordan operator algebra). Since these algebras are however nonassociative (which is inconvenient) and can nearly always be embedded into some associative algebra, we study the associative algebras. So, in a sense, studying C*-algebras is already "a technical thing".
Best Answer
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 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).