What's algebraic approach to QM good for?

[neworder1]

The algebraic formulation of quantum mechanics (and related stuff, like quantum thermodynamics & dynamical systems etc.) via C*-algebras provides a viewpoint based mostly on abstract functional analysis. However, I've yet to see a working application of this approach, i.e. an example of a problem which is difficult to solve or even to formulate in the standard formalism, while considerably easier to tackle with all this algebraic stuff. Any ideas? Without such concrete examples the whole field seems to be interesting mathematically, perhaps, but lacking any substance.

 

[crackjack]

They are studied not because they will simplify life for us, but because they allow us a deeper look at the theory. When intuition is fixed by rigor, we might uncover subtle points that were earlier overlooked.

 

[Fredrik]

I don't know enough about these things to answer myself, but this article looks really interesting. I have only had a quick look at it. I intend to return to it when I have studied some more math.

 

[Gerenuk]

I know undergrad QM pretty well, but have no idea about the algebraic approach.

Could you tell me an example what kind of insight the algebraic approach has yielded so far?

 

[crackjack]

Zhu and Klauder, "Classical symptoms of quantum illnesses", Am J Phys (1993) vol 61, pp. 605
(DrDu's suggestion)

 

[Physics Monkey]

One place where something like the algebraic approach mentioned above has been useful is in the study of two dimensional conformal field theories. For example, vertex algebras and related objects have permitted a mathematically more rigorous study of some interesting features of string theory, 2d conformal field theory, affine lie algebras, quantum groups, etc. These algebras are also more directly physically relevant via their connection to the fractional quantum hall effect.

One interesting characteristic of this approach is that it dispenses with some of the usual trappings we associate with a quantum field theory such as an explicit Lagrangian. One can work with sensible quantum field theories which do not appear to have a sensible Lagrangian description, at least not directly. In this sense, the algebraic approach has helped expand and clarify what we mean by the phrase "quantum field theory".

 

[0xDEADBEEF]

The main point is, that we still don't know what we are doing in quantum mechanics. The algebra of classic systems is actually more complicated than that of quantum systems. It is not clear yet how to convert any given classical system into a quantum one.

In many fields rigor is not helpful, in physics it almost always is. There is a reason why you need three semesters before you can integrate. Physics is about truth much more than about application.

 

[Hurkyl]

To add some generalities...

Some people have good algebraic intuition, and an algebraic formulation will help those people.

Algebra often relates to synthesis -- algebraic objects often organize a lot of low-level data into one, useful unit, or otherwise present the same data in a form that, once learned, is easier to work with than handling the low-level details directly.

Good notation is often very powerful. e.g. compare working with L²(R) as a Hilbert space in bra-ket notation to working with it as a collection of square-integrable complex-valued functions.