## June 20, 2008

### Eli Hawkins on Geometric Quantization, I

#### Posted by Urs Schreiber

Recently I had mentioned Eli Hawkins’ Groupoid approach to quantization. Today at HIM he gave the first of a two-part lecture on this. This first one was on basics of geometric quantization. Next Friday we’ll here the corresponding groupoid version.

Here are some of the interesting aspects of today’s talk, including a remarkable slogan on the relation of quantization and Lie integration.

Before coming to a couple of interesting facts, a

Quick reminder of some concepts

Given a manifold $X$ and a bivector $\pi \in \Lambda^2 T X$ we obtain an antisymmetric product on functions on $X$ given by

$\pi : C^\infty(X) \otimes C^\infty(X) \to C^\infty(X)$ $(f,g) \mapsto \{f,g\} := (\pi(f)g) \,.$

So if in a local chart we have $\pi = \pi^{ij} \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^j}$ then $\{f,g\} = \pi^{ij} (\frac{\partial}{\partial x^i} f) ( \frac{\partial}{\partial x^j} g) \,.$

The tensor $\pi$ is called a Poisson tensor if this bracket is a Lie bracket, i.e. if it satisfies the Jacobi identity. A manifold equipped with a Poisson tensor is a Poisson manifold.

In the game called “geometric quantization” one wants to stat with a Poisson manifold and constrcuct from it in a natural way a Hilbert space and/or the algebra of bounded operators on that Hilbert space.

Moreover, one wants to send the commutative algebra of functions on $X$ to some usually non-commutative sub-algebra of that big algebra in such a way that the commutator in this non-commutative algebra is “approximated” by the Poisson bracket.

The following example seems to be very fundamental and important, but I wasn’t really aware of it before. Now I am.

Geometric quantization of the dual of a Lie algebra.

For $g$ a Lie algebra, the dual vector space $g^*$ is naturally equipped with a Poisson structure:

first consider linear functions $f_1, f_2 : g^* \to \mathbb{R}$ on $g^*$. These are nothing but elements of $g$. So a natural bracket operation on them is the Lie bracket. This extends, apparently, uniquely to a Poisson bracket on all functions on $g^*$.

Let $(g^*, \pi_{[\cdot,\cdot]})$ be $g^*$ regarded as a Poisson manifold with this particular Poisson structure. Then we can try to apply geometric quantization, or some generalization of it which also applies to Poisson manifolds. Let $C_0(g^*)$ be a space of functions on $g^*$ that go to 0 at infinity in a sufficiently nice way. The quantization process should send this commutative algebra to some possibly non-commutative algebra.

$C_0(g^*) \stackrel{quantization}{\mapsto} A \,.$

What is that algebra? Apparently, turning some crank or other and making some non-canonical choices which one has to make in this quantization business, that algebra turns out to be the convolution algebra of one of the Lie groups $G$ integrating $g$ (the choices corresponding to which Lie group one gets)

$C_0(g^*) \stackrel{quantization}{\mapsto} C^*(G) \,.$

This convolution $C^*$-algebra is the Lie analog of the group algebra of a finite group.

Eli did not explain many of the details going into this quantization of algebras. He says that this statement can be found in the book

N. P. (Klaas) Landsman
Mathematical topics between classical and quantum mechanics

which I haven’t seen yet. But details are in

N.P. Landsman, B. Ramazan
Quantization of Poisson algebras associated to Lie algebroids
arXiv:math-ph/0001005 .

In there the statement is actually generalized to Lie algebroids:

Geometric quantization of the dual of a Lie algebroid.

The fiberwise dual $A^*$ of the vector bundle $\array{ A &\to& T X \\ & \searrow \swarrow \\ & X }$ underlying a Lie algebroid $A$ also inherits a Poisson structure $\pi_{[\cdot,\cdot]}$ from the bracket on sections of $A$ given by the Lie algebroid structure of $A$.

So we can ask what the quantization of the Poisson manifold $(A^*, \pi_{[\cdot, \cdot]})$ is. As you can now guess if you meditate over the above statement for a second is that the quantized algebra of functions $C_0(A^*)$ on $A^*$

$C_0(A^*) \stackrel{quantization}{\mapsto} C^*(Gr) \,,$

is the $C^*$-convolution algebra of $Gr$, where $Gr$ is one of the Lie groupoids integrating the Lie algebroid $A$.

Crucial example

In particular, take $A = T X$ to be the tangent Lie algebroid of some manifold $X$. Then the dual space $A ^* = T^* X$ is the cotangent bundle.

The Poisson structure induced on that cotangent bundle by the above general procedure is just what you’d expect: it’s the actually symplectic structure of the symplectic 2-form $d \alpha$, where $\alpha$ is the canonical 1-form on $T^* X$.

We already know what the quantization of this Poisson manifold $T^* X$ should be. After all, this is the example that got the business of quantization started in the 1930s! The algebra we want to see come out is that of operators on square integrable functions on $X$.

And indeed, by the above statement we find that the algebra of functions on $T^* X$ is sent to the convolution algebra of the groupoid integrating the tangent Lie algebra – which is the fundamental groupoid of $X$ or one of its quotients, such as the pair groupoid of $X$:

$C_0(T^* X) \stackrel{quantization}{\mapsto} C^*(PairGroupoid(X))$

but that indeed is $K(L^2(X))$ – compact operators on the Hilbert space of square integrable functions on $X$, in a direct generalization of how the groupoid algebra of the pair groupoid of any finite set of cardinality $n$ is that of $n \times n$ matrices.

This deserves a slogan.

This is too good to remain unnoticed, so it deserves a catchy slogan. Maybe this one:

Quantization is nothing but Lie integration.

Hm…

Quantization is nothing but Lie $n$-tegration.

Well, in the above context this is true for Poisson manifolds which are total spaces of Lie algebroids. So this cannot be true generally without further modification. But it’s still suggestive.

Bisections and inner automorphisms

Eli Hawkins mentions that from a Lie groupoid $Gr$ over $X$ one obtains the Lie algebroid differentiating it (or rather, directly, it’s Lie-Rinehart pair, I would say) by taking the Lie algebra of the group of bisections.

A bisection here is define to be a section of the source map of the Lie groupoid which is also a section of the target map. Bisections have an obvious notion of composition which makes them into a group.

In more categorical terms this means, in other words, that a bisection is an arrow field on the groupoid which is the component map of a natural transformation $\eta$ which related the identity automorphism of the groupoid to some other automorphism $\alpha$.

$\array{ & \nearrow \searrow^{Id} \\ Gr &\Downarrow^\eta& Gr \\ & \searrow \nearrow_{\alpha} } \,.$

Drawn this way, it is clear which group these bisections sit in: the inner automorphism 2-group of $Gr$.

In fact, we have talked about this way of getting Lie algebroids from differentiating the inner automorphism groups of their Lie groupoids a lot here, in the general context of “Arrow-theoretic differential theory”.

For instance the discussion of the fact that the Lie algebroid of $Gr$ comes from the Lie algebra $Lie(INN(Gr))$ is discussed in section 4.1 of Tangent categories, and also on Supercategories.

Posted at June 20, 2008 3:56 PM UTC

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Read the post Landsman on Quantization of Poisson Algebras Associated to Lie Algebroids
Weblog: The n-Category Café
Excerpt: A quick review of Landsman's result on strict deformation quantization of Poisson manifolds dual to Lie algebroids: the quantum algebra is nothing but the groupoid algebra of the Lie groupoid integrating the Lie algebroid.
Tracked: June 22, 2008 6:47 PM

### Re: Eli Hawkins on Geometric Quantization, I

The Poisson structure on the dual of a Lie algebra is a wonderful thing. It’s often called the ‘Kirillov–Kostant’ Poisson structure.

Here are a couple of nice facts about it.

First, if $g$ is any Lie algebra, we can think of the symmetric algebra $S g$ as the algebra of polynomial functions on $g^*$. This becomes a Poisson algebra in the way you described, where the bracket of linear functions (i.e. elements of $g$) is just the obvious thing (their Lie bracket).

Deformation quantization commands us to ask this question: can we find a nice deformation of $S g$ into a 1-parameter family of noncommutative algebras, depending on Planck’s constant $\hbar$?

The answer is yes! Note that there is a 1-parameter family of Lie algebras $g_\hbar$. These all have $g$ as their underlying vector space, with the bracket being a rescaled version of the bracket in $g$:

$[x,y]_\hbar = \hbar [x,y]$

The universal enveloping algebra $U g_\hbar$ is the deformation quantization we’re looking for! It’s typically a noncommutative algebra except at $\hbar = 0$, when it reduces to our polynomial algebra $S g$. And, if we take the commutator in $U g_\hbar$, differentiate it with respect to $\hbar$, and set $\hbar = 0$, we get back the Poisson bracket on $S g$. So, it’s a deformation quantization in the technical sense.

Second, if our Lie algebra comes from a Lie group $G$, the Poisson manifold $g^*$ is foliated by ‘coadjoint orbits’: orbits of the action of $G$. Some of these coadjoint orbits will be ‘integral’, meaning the symplectic structure defines an integral element of deRham cohomology. We get a line bundle over any integral coadjoint orbit, which is almost enough to do geometric quantization and get a representation of $G$.

It works very nicely when $G$ is a compact simple Lie group. Then the coadjoint orbits are actually Kähler manifolds! So, we can go ahead and do geometric quantization of the integral coadjoint orbits, and we get precisely all the irreducible finite-dimensional representations of $G$.

There’s also a very nice way to go back from finite-dimensional irreducible representations of $G$ to the coadjoint orbit it came from: just take the orbit of any highest-weight vector. In fact, if you don’t know much about this ‘highest-weight’ baloney, geometric quantization is a nice way to get a feel for it.

So, both deformation quantization and geometric quantization are very beautiful when $g^*$ comes from a compact simple Lie group. It’s even more fun to see how the two approaches to quantization fit together, in this case… but that’s another story.

Posted by: John Baez on June 26, 2008 12:28 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, I

The universal enveloping algebra $U_{g_\hbar}$ is the deformation quantization we’re looking for!

Well, yes - but, no! :-)

This is the formal deformation quantization, where we deform in the world of formal power series in one variable, canonically called $\hbar$.

One thing that is nice about formal deformation quantization is that it has been solved: Kontsevich showed that every Poisson manifold has a formal deformation quantization.

One thing that is bad about formal deformation quantization is that it is just formal. For many applications, notably for doing really physics, we want to pick a concrete value for $\hbar$. Such as $\hbar = 1$. But in formal deformation quantization this is not possible.

What Eli Hawkins talked about, and what that work by Landsman is concerned with is something even better: $C^*$-algebraic deformation quantization. Here we deform, of course, in the world of $C^*$-algebras and get an honest answer (if at all) for $\hbar = 1$.

While $C^*$-algebraic deformation quantization apparently currently does not have a general solution for all Poisson manifolds, it does have a general solution for all those that are (coadjoint orbits in) duals of Lie algebroids, in particular for the case of duals of Lie algebras that you mention.

And there the result is “better” than in formal deformation quantization: as Landsman and Ramazan prove:

The $C^*$-algebraic deformation quantization of the Poisson algebra of functions on the dual of a Lie algebroid is the groupoid $C^*$-algebra of a Lie groupoid integrating that Lie algebroid.

As Eli tells me, given that $C^*$-algebraic deformation quantization we can recover the formal deformation quantization from it by “Taylor expanding” the $\hbar$-dependent quantization map from $[0,1]$ to $C^*$-algebras around $\hbar = 0$.

So the $C^*$-algebraic deformation quantization knows more than formal deformation quantization. The universal enveloping algebra of the Lie algebra $g$ is “just an approximation” to the group algebra of the group integrating $G$.

Or, as Eli put it in his talk: Landsman’s theorem shows that the universal enveloping algebra really wants to be the group(oid) $C^*$-algebra.

I find this $C^*$-algebraic deformation quantization quite remarkable, that’s why I was talking about it here and here. It shows that if we go beyond formal deformation quantization of (coadjoint orbits in) Lie algebroids, then quantization is Lie integration. That’s remarkable.

Posted by: Urs Schreiber on June 26, 2008 2:25 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, I

Urs wrote:

This is the formal deformation quantization…

The universal enveloping algebra I described is not a C*-algebra, but it’s not merely ‘formal’: it involves polynomials in $\hbar$, which can be evaluated at any value of $\hbar$.

Posted by: John Baez on July 2, 2008 1:54 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, I

Yes, in that case, true. But in general the output of formal deformation quantization need not have this property.

I’ll try to check again with Eli what precisely the relation between the universal enveloping algebra of a Lie algebra and the $C^*$-convolution algebra of a group integrating it is. My understanding was that the former is a kind of linearization of the latter in a precise sense. I’ll check again.

Posted by: Urs Schreiber on July 2, 2008 2:01 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, I

Okay, I have just checked. the statement is apparently this one (hope I get the details right):

Let $X$ be a Poisson manifold, $\star_\hbar$ a formal deformation quantization and $Q_\hbar$ a $C^*$-algebraic deformation map, sending functions on $X$ to elements of some $C^*$-algebra. Then the formal deformation quatization is \emph{asymptotic} to the $C^*$-algebraic onein the sense that for all intergers $n$ and for all functions $f$ and $g$ we have that

$\hbar^{-n} ( Q_\hbar(f)Q_\hbar(g) - Q_\hbar(f \star_\hbar^{(n)} g) )$

converges in the $C^*$-algebra norm to 0 as $\hbar \to 0$, where $\star_\hbar^{(n)}$ is the truncation of $\star_\hbar$ at order $n$.

Posted by: Urs Schreiber on July 2, 2008 3:35 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, I

Urs wrote:

Yes, in that case, true. But in general the output of formal deformation quantization need not have this property.

Right, sometimes it´s ‘purely formal’ and we can´t evaluate the results at nonzero values of $\hbar$. I was only talking about the quantization of the dual of a Lie algebra, which is a very beautiful special case. I like this beautiful special case because I like Lie algebras and their representations.

I wrote:

So, both deformation quantization and geometric quantization are very beautiful when $g^*$ comes from a compact simple Lie group. It’s even more fun to see how the two approaches to quantization fit together, in this case… but that’s another story.

Let me say a bit more about this story — the even more beautiful, more special case when $g$ is the Lie algebra of a compact simple Lie group.

I explained how deformation quantization gives us an algebra $U_\hbar g$ for each value of $\hbar$, and how geometric quantization gives us an irreducible representation for each integral coadjoint orbit in $g^*$. But how do these ideas fit together?

First of all, $U_\hbar g$ becomes is a $*$-algebra if we decree all the elements of $g$ to be skew-adjoint:

$x^* = -x$

for all $x\in g$.

It is not a $C^*$-algebra, but there is a very easy trick for completing it to get a $C^*$-algebra. This trick is sometimes called the ‘universal enveloping $C^*$-algebra’ trick.

Beware: the term ‘universal enveloping $C^*$-algebra’ has nothing to do with the universal enveloping algebra of a Lie algebra! You can take the universal enveloping $C^*$-algebra of any $*$-algebra (as long as some condition holds). But today, just to confuse you, I’m going to apply this very general trick to the universal enveloping algebra of a Lie algebra!

How does it work? Easy: we take a $*$-algebra $A$ and consider all possible $*$-representations $R$ of $A$ as bounded operators on Hilbert spaces. Then, put a norm on it where the norm of any element $a$ is defined to be the supremum of the norms of the operators $R(a)$. If this supremum is finite and nonzero for all nonzero $a$, we can complete $A$ using this norm and get a $C^*$-algebra.

(The ‘condition’ I mentioned earlier is that the supremum be finite and nonzero for all nonzero $a$. Often it’s not!)

In the case of the universal enveloping algebra of the Lie algebra of a compact Lie group, this trick works for all $\hbar \ne 0$. So, we can complete $U_\hbar g$ and get a $C^*$-algebra.

Even better, the resulting $C^*$-algebra is just a direct sum of matrix algebras, one for each irreducible representation of $g$.

Even better, there´s one irreducible representation of $g$ for each integral coadjoint orbit in $g^*$!

So, we are getting a very nice quantum analogue of the process of foliating the Poisson manifold $g^*$ by symplectic leaves — that is, coadjoint orbits. We´re getting a big $C^*$-algebra of quantum observables that´s a direct sum of little $C^*$-algebras, one for each integral coadjoint orbit.

Posted by: John Baez on July 3, 2008 4:09 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, I

I should add — just to connect with what Eli and Urs are talking about — that the C*-algebra I described above can also be thought of as a convolution algebra of generalized functions on the simply-connected version of the compact simple Lie group whose Lie algebra is $g$.

But, I forget what kind of generalized functions!

If use $L^2$ functions on $G$ we get a very nice convolution algebra, but it´s not a C*-algebra: it´s a blend of a $*$-algebra and a Hilbert space called an H*-algebra! In fact, this is why H*-algebras were invented in the first place.

There are also convolution algebras consisting of $L^1$ functions and continuous functions, but neither of those can be quite right, because our convolution C*-algebra must have an identity element (unlike the H*-algebra I just mentioned). And, the identity of our convolution C*-algebra corresponds to the delta function at the identity of $G$.

So, our convolution C*-algebra has to contain at least a few generalized functions.

I used to know this stuff better…

Posted by: John Baez on July 3, 2008 5:18 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, I

I should add — just to connect with what Eli and Urs are talking about — that the C*-algebra I described above can also be thought of as a convolution algebra of generalized functions on the simply-connected version of the compact simple Lie group whose Lie algebra is g.

But, I forget what kind of generalized functions!

Does it contain all the “translation operators” (i.e. convolution with a dirac delta supported on some group element)?
And do these operators generate the whole of your C*-algebra (in norm topology)?

Posted by: Yemon Choi on July 3, 2008 11:46 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, I

Yemon Choi:

Does it contain all the “translation operators” (i.e. convolution with a dirac delta supported on some group element)?

Actually I think my construction of the ‘universal enveloping C*-algebra’ was incorrect… it’s a nice general construction, but I think it fails here, because if $x$ is an element of $U_\hbar g$, the supremum of $\|R(x)\|$ over all $*$-representations $R$ of $U_\hbar g$ will rarely be finite!

There’s certainly a nice C*-algebra that contains all your ‘left translation operators’: for example, the algebra of all bounded operators on $L^2(G)$ that commute with right translations. Linear combinations of left translation operators aren’t norm dense in this C*-algebra, but they could be dense in the weak operator topology.

Posted by: John Baez on July 9, 2008 5:35 PM | Permalink | Reply to this
Read the post Eli Hawkins on Geometric Quantization, II
Weblog: The n-Category Café
Excerpt: Eli Hawkins explains his method of getting a quantum algebra from the convolution algebra of sections on a symplectic groupoid.
Tracked: June 27, 2008 5:49 PM

### Re: Eli Hawkins on Geometric Quantization, I

I started collecting (links to) this material at the $n$Lab entry

geometric quantization

Much more hopefully to be said there, eventually. But I did at least import John’s old notes on geometric quantization. Hope that’s okay. This way they become hyperlinked within the $n$Lab.

Posted by: Urs Schreiber on August 18, 2009 9:17 PM | Permalink | Reply to this

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