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June 22, 2008

Landsman on Quantization of Poisson Algebras Associated to Lie Algebroids

Posted by Urs Schreiber

In the last entry I mentioned the work in

N.P. Landsman, B. Ramazan
Quantization of Poisson algebras associated to Lie algebroids

summarizing it with the remarkable slogan

Quantization is Lie integration.

true at least in all the cases where the classical Poisson manifold to be quantized is the dual of a Lie algebroid or of a coadjoint orbit inside of that.

This article states and proves a theorem (two versions, actually) which asserts that the C*C *-algebraic deformation quantization of such a Poisson manifold is the groupoid C-star algebra of one of the Lie groupoids integrating the given Lie algebroid.

C *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 XX (maybe assuming for simplicity that it is actually symplectic) and find a non-commutative deformation of the commutative algebra of functions on XX 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 [0,1]\hbar \in [0,1] which interpolates between the Poisson algebra and the noncommutative C-star algebra at =1\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 *XT^* X, which is the dual to the tangent Lie algebroid TXT X.


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

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

I haven’t checked but would guess that Landsman’s discussion should generalize the coadjoint orbits of arbitrary Lie nn-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]

Posted at June 22, 2008 6:08 PM UTC

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Re: Landsman on Quantization of Poisson Algebras Associated to Lie algebroids

One thing about Landsman and Ramazan’s approach and also that of Eli Hawkins is that I don’t see where the idea of a polarization enters. Landsman and Ramazan’s introduction gives a wonderful summary of all things quantum; I love the lines

…writing down a path integral may be seen as an act of quantization.

(this makes it seem like a criminal offence!) and also

Initially, practically all of quantization theory was based on a single idea of Dirac, which he conceived of in 1926 during a Sunday walk near Cambridge.

Anyhow, in their review of geometric quantization, they look at prequantization, discover a flaw in the naive approach, and then conclude that one must adopt deformation quantization. Isn’t this a straw man argument?

Posted by: Bruce Bartlett on June 24, 2008 2:53 PM | Permalink | Reply to this

Re: Landsman on Quantization of Poisson Algebras Associated to Lie algebroids

Bruce said:

One thing about [the approach] of Eli Hawkins is that I don’t see where the idea of a polarization enters.

I’m not sure quite what you mean by that. In the abstract of his paper you find, amongst other things, the following.

Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra.

You might want to be a bit more explicit…

Posted by: Simon Willerton on June 24, 2008 5:17 PM | Permalink | Reply to this

Re: Landsman on Quantization of Poisson Algebras Associated to Lie algebroids

Hi Bruce,

it is true that in both cases the role of the Hilbert space is taken to be secondary, while the algebra (the total one or the image of the classical Poisson algebra under quantization) plays the primary role.

You write:

I don’t see where the idea of a polarization enters.

In the approach described in the Landsman article indeed no polarization is mentioned, nor needed to perform the deformation quantization.

I didn’t read that as saying that other approaches are less interesting. Just as emphasizing what certainly deserves to be emphasized: the deformation quantization of the Poisson Lie algebra on the dual of (a coadjoint orbit inside) a Lie algebroid is the groupoid algebra of the Lie groupoid integrating it.

That deserves to be said. It ought to be taught in school and promoted in TV commercials.

Eli Hawkins’ approach is, while closely related, a bit different. I am still trying to figure out how precisely Landsman’s and Hawkins’ point of view fit together.

Notice that in Eli Hawkins’s approach the notion of polatization plays a major, even the central role:

Have another look at Eli’s article. Its central contribution is the definition of a polarization of a symplectic groupoid. The concept is presented as definition 4.7 on p. 18. The following sections highlight different aspects of this notion of polarization.

Eli essentially says that, if you are intersted in the C *C^* algebra of quantum observables, there is a shortcut to geometric quantization which does not need to construct a Hilbert space as an intermediate step. The algebra is obtained from the groupoid algebra of a symplectic groupoid but is smaller: the choice of groupoid polarization cuts it down to the true algebra. This groupoid polarization plays the role analogous to the polarizartion on the prequantization Hilbert space.

And another point of all this is that it works in the case where we do not have a symplectic but just a Poisson manifold. Landsman makes some interesting remarks about the role of Poisson manifolds as the true context of classical mechanics, relating it to the presence of superselection sectors. He also points out that the phase space of the nonabelian charged particle is just Poisson, not symplectic.

Finally, a general comment to all those readers who, such as some of my fellow physics participants at the Hausdorff institute, are wondering what this polarization is all about in case it hasn’t become clear: in phys-speak, a choice of polarization is nothing but a choice of canonical coordinates and canonical momenta. But in a way that makes more general sense than the description given in essentially all the standard classical mechanics and/or QM textbooks.

Posted by: Urs Schreiber on June 24, 2008 5:25 PM | Permalink | Reply to this

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