The n-Category Café

$C^*$-algebraic deformation quantization

As nicely reviewed in the introduction, there are several ways to make precise what deformation quantization is supposed to be.

In some way or other, one wants to start with a Poisson manifold $X$ (maybe assuming for simplicity that it is actually symplectic) and find a non-commutative deformation of the commutative algebra of functions on $X$ such that the commutator in the deformed algfebra is “approximated to first order” by the original Poisson bracket.

One way to approach this is by considering algebras in formal power series of a single formal parameter $\hbar$. That “formal” approach was famously shown by Kontsevich to always admit a solution to the deformation problem, we discussed that in That shift in dimension.

But in a sense this solution is unsatisfactory precisely because it is “formal”: there is in general no guarantee that one can replace $\hbar$ by an actual number (in particular by the number 1!) and get converging expressions. But that is need to really have a “physical quantization” in the ordinary sense.

Landsman and Ramazan review a different approach to the deformation quantization, one which does not suffer from this problem: they define a deformation of the Poisson algebra as “continuous field of C-star algebras” indexed by $\hbar \in [0,1]$ which interpolates between the Poisson algebra and the noncommutative C-star algebra at $\hbar = 1$ quantizing it.

Poisson manifolds from duals of Lie algebroids

Lots of relevant Poisson manifolds turn out to be duals of Lie algebroids, with the Poisson structure being essentially nothing but the Lie bracket on the Lie algebroid. This is best seen from the archetypical example, the canonical Poisson manifold $T^* X$, which is the dual to the tangent Lie algebroid $T X$.

In

N. P. Landsman
Strict deformation quantization of a particle in external gravitational and Yang-Mills fields
Journal of Geometry and Physics, Volume 12, Issue 2, p. 93-132.

this example is generalized to the situation of the general charged particle, in which case the Lie algebroid in question is the Atiyah Lie algebroid of the principal bundle giving the background field – and the quantum algebra is the groupoid algebra of the transport groupoid aka gauge groupoid of this bundle. (One has to think of this algebra as acting on sections of this bundle regarded as equivariant functions on the total space of the bundle. More on that later.)

Coadjoint orbits

In the special case that the bundle in question lives over a point, so that its total space is nothing but the structure group itself, the Lie algebroid in question is just a Lie algebra and the corresponding quantum algebra is the convolution $C^*$-algebra of the group – the continuous analog of the group algebra of a finite group.

In this case it is well known that the Poisson manifold $g^*$ is foliated by other Poisson manifolds: the coadjoint orbits. Their quantization corresponds to irreducible parts of the quantization of the full thing. There is a big interesting theory about coadjoint orbits.

I notice that one nice way to think of a coadjoint orbit in a Lie algebra is as the orbit in its Chevalley-Eilenberg DGCA of a homogeneous element under the inner automorphism group along the lines described here:

the orbit is of the form

$$exp([d,\iota_v]) t$$ for t in $g^*$, $v$ any Lie algebra element, $\iota_v$ the derivation on $CE(g)$ obtained by contracting with it and $d$ the differential on $CE(g)$.

This immediately generalizes to a notion of coadjoint orbits on arbitrary Lie algebroids and in fact Lie $n$-algebroids.

I haven’t checked but would guess that Landsman’s discussion should generalize the coadjoint orbits of arbitrary Lie $n$-algebroids. That should give a huge source of examples. So I am led to wonder if the slogan at the beginning of this post actually holds in complete generality.

[more later]