What's algebraic approach to QM good for?

Question

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 haven't really seen a working application of this approach, i.e. an example of a (preferably physical) 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 physical substance (even if it's sometimes masqueraded as having connections with physics).

edit: by "standard formalism" I mean "observables, i.e. operators, acting on a Hilbert space of physical states", either in Schroedinger picture (time dependence of states) or Heisenberg (time dependence of operators). C*-algebraic approach starts from a C*-algebra and defines states as positive functionals on the elements of algebra, time evolution as a *-automorphism etc. (for an introduction to this formalism, see http://hal.archives-ouvertes.fr/docs/00/12/88/67/PDF/qds.pdf)

This might be of less interest to non-physicists than a typical MO post, but I hope it's still relevant.

Answer 1

If you want to consider quantum statistical physics properly this approach is necessary. The KMS condition gives a generalization of the quantum Gibbs postulate that allows for the treatment of phase transitions, coexistence of multiple phases, etc.

The functional-analytic approach is also important for other reasons (besides the rigged Hilbert space that is necessary even to understand a free particle): Fock space depends on it, as does axiomatic quantum field theory, and pages 12-13 of Bratteli and Robinson v.1 give some quick background to the KMS approach (both volumes are worth looking at). A book by Sewell called Quantum mechanics and its emergent macrophysics also gives quite a bit of relevant physical background.

BTW, the KMS condition is not as obscure as it might at first seem (the Wikipedia article is one of the few places I recall seeing that demystifies it). In the Heisenberg picture observables evolve under the time evolution map

$\tau_t : A \mapsto e^{iHt/ \hbar}Ae^{-iHt/ \hbar}.$

The appropriate generalization of the classical Gibbs rule is

$\langle A \rangle = Z^{-1}\mbox{Tr}(e^{-\beta H} A), \quad Z := \mbox{Tr}(e^{-\beta H}).$

To see this, consider the projection observable $\Pi_k := \lvert k \rangle \langle k \rvert$. We have that

$\langle \Pi_k \rangle = Z^{-1}\mbox{Tr}(e^{-\beta E_k} \lvert k \rangle \langle k \rvert) = Z^{-1}e^{-\beta E_k}$

in accordance with classical statistical physics. Now for generic observables $A$ and $C$, we have that

$\left \langle \tau_t(A)C \right \rangle = Z^{-1}\mbox{Tr}(e^{-\beta H} e^{iHt/\hbar}Ae^{-iHt/\hbar}C)$

$= Z^{-1}\mbox{Tr}(Ce^{iH(t+i\hbar\beta)/\hbar}Ae^{-iHt/\hbar})$

$= Z^{-1}\mbox{Tr}(Ce^{iH(t+i\hbar\beta)/\hbar}Ae^{-iH(t+i\hbar\beta)/\hbar}e^{-\beta H})$

$= \left \langle C\tau_{t+i\hbar\beta}(A) \right \rangle$

which gives the KMS condition:

$\left \langle \tau_t(A)C \right \rangle = \left \langle C\tau_{t+i\hbar\beta}(A) \right \rangle.$

Answer 2

The algebraic approach subsumes the standard formulation of QM (everything that can be done in the latter can be done in the former).

One important feature of the algebraic framework is that it allows you to handle inequivalent representations of the algebra of observables (which invariably come up in field theory, or any theory with an infinite number of degrees of freedom).

An example of a problem tackled only in the algebraic framework is the perturbative construction of interacting quantum field theory on arbitrary globally hyperbolic spacetimes. The construction is perturbative because every quantity of interest is taken to be a formal power series in the interaction strength. AFAIK, this construction has only been done using algebraic methods.

Answer 3

I think one answer ist to gain conceptual insight. Since Dirac it was usual to quantize via
Classical theory in a Hamiltonian framework => Quantum theory
Since there are a lot of problems with this approach (to QFT), it would be nice not to have to take the route over the classical theory. Hence the algebraic theory was born.

Well, I attended a workshop on algebraic QFT last year - it is not my research area, though. They said that they are still looking very hard for a working model in 4d (they have good models for dimension 2 and 3). That's also the reason why you did not encounter any concrete computable examples.
That's a critical issue - if they don't find a good model in the future, the field might be as good as dead (as a physics discipline).