nLab
geometric quantization

Idea

With a symplectic manifold regarded as a classical mechanical system, geometric quantization is one formalization of the notion of quantization of this to a quantum mechanical system.

The idea is to

1. realize the symplectic form as the curvature of a $U(1)$-principal bundle with connection (which requires the form to have integral periods);

2. choose a polarization – a splitting of the abstract phase space into “coordinates” and “momenta”;

and then form

1. a Hilbert space of states as the space of sections of the associated line bundle which depend only on the “coordinates” (not on the “momenta”);

2. associate with every function on the symplectic manifold – every Hamiltonian – an operator on this Hilbert space.

The approach is due to Kirillov (“orbit method”), Kostant and Souriau. It is closely related to Berezin quantization? and the subject of coherent states.

In a long term project Alan Weinstein and many of his students have followed the idea that the true story behind this prescription crucially involves symplectic Lie groupoids: higher symplectic geometry. See geometric quantization of symplectic groupoids for more on this.

Overview

Geometric quantization is a marvelous tool for understanding the relation between classical physics and quantum physics. However, it’s a bit like a power tool – you have to be an expert to operate it without running the risk of seriously injuring your brain. Here’s a brief sketch of how it goes. This is pretty terse; for the details you’ll have to read the series of articles on geometric quantization on the sci.physics.research archive.

1. We start with a classical phase space: mathematically, this is a manifold $X$ with a symplectic structure $\omega$.

2. Then we do prequantization: this gives us a Hermitian line bundle $L$ over $X$, equipped with a $U(1)$ connection $D$ whose curvature equals $i\omega$. $L$ is called the prequantum line bundle.

   Warning: we can only do this step if $\omega$ satisfies the Bohr–Sommerfeld condition, which says that $\omega /2\pi$ defines an integral cohomology class. If this condition holds, $L$ and $D$ are determined up to isomorphism, but not canonically.

3. The Hilbert space $H_0$ of square-integrable sections of $L$ is called the prequantum Hilbert space. This is not yet the Hilbert space of our quantized theory – it’s too big. But it’s a good step in the right direction. In particular, we can prequantize classical observables: there’s a map sending any smooth function on $X$ to an operator on $H_0$. This map takes Poisson brackets to commutators, just as one would hope. The formula for this map involves the connection $D$.

4. To cut down the prequantum Hilbert space, we need to choose a polarization, say $P$. What’s this? Well, for each point $x\in X$, a polarization picks out a certain subspace $P_x$ of the complexified tangent space at $x$. We define the quantum Hilbert space, $H$, to be the space of all square-integrable sections of $L$ that give zero when we take their covariant derivative at any point $x$ in the direction of any vector in $P_x$. The quantum Hilbert space is a subspace of the prequantum Hilbert space.

   Warning: for $P$ to be a polarization, there are some crucial technical conditions we impose on the subspaces $P_x$. First, they must be isotropic: the complexified symplectic form $\omega$ must vanish on them. Second, they must be Lagrangian: they must be maximal isotropic subspaces. Third, they must vary smoothly with $x$. And fourth, they must be integrable.

5. The easiest sort of polarization to understand is a real polarization. This is where the subspaces $P_x$ come from subspaces of the tangent space by complexification. It boils down to this: a real polarization is an integrable distribution $P$ on the classical phase space where each space $P_x$ is Lagrangian subspace of the tangent space $T_{x}X$.

6. To understand this rigamarole, one must study examples! First, it’s good to understand how good old Schrödinger quantization fits into this framework. Remember, in Schrödinger quantization? we take our classical phase space $X$ to be the cotangent bundle $T^{*}M$ of a manifold $M$ called the classical configuration space. We then let our quantum Hilbert space be the space of all square-integrable functions on $M$.

   Modulo some technical trickery, we get this example when we run the above machinery and use a certain god-given real polarization on $X=T*M$, namely the one given by the vertical vectors.

7. It’s also good to study the Bargmann–Segal representation, which we get by taking $X={ℂ}^n$ with its god-given symplectic structure (the imaginary part of the inner product) and using the god-given Kähler polarization. When we do this, our quantum Hilbert space consists of analytic functions on ${ℂ}^n$ which are square-integrable with respect to a Gaussian measure centered at the origin.

8. The next step is to quantize classical observables, turning them into linear operators on the quantum Hilbert space $H$. Unfortunately, we can’t quantize all such observables while still sending Poisson brackets to commutators, as we did at the prequantum level. So at this point things get trickier and my brief outline will stop. Ultimately, the reason for this problem is that quantization is not a functor from the category of symplectic manifolds to the category of Hilbert spaces – but for that one needs to learn a bit about category theory.

Basic Jargon

Here are some definitions of important terms. Unfortunately they are defined using other terms that you might not understand. If you are really mystified, you need to read some books on differential geometry and the math of classical mechanics before proceeding.

• complexification: We can tensor a real vector space with the complex numbers and get a complex vector space; this process is called complexification. For example, we can complexify the tangent space at some point of a manifold, which amounts to forming the space of complex linear combinations of tangent vectors at that point.

• distribution: The word “distribution” means many different things in mathematics, but here’s one: a “distribution” $V$ on a manifold $X$ is a choice of a subspace $V_x$ of each tangent space $T_{p}X$, where the choice depends smoothly on $x$.

• Hamiltonian vector field: Given a manifold $X$ with a symplectic structure $\omega$, any smooth function $f:X\to ℝ$ can be thought of as a “Hamiltonian”, meaning physically that we think of it as the energy function and let it give rise to a flow on $X$ describing the time evolution of states. Mathematically speaking, this flow is generated by a vector field $v(f)$ called the “Hamiltonian vector field” associated to $f$. It is the unique vector field such that

  $$\omega (-,v(f))=df$$\omega(-, v(f)) = d f

  In other words, for any vector field $u$ on $X$ we have

  $$\omega (u,v(f))=df(u)=uf$$\omega(u,v(f)) = d f(u) = u f

  The vector field $v(f)$ is guaranteed to exist by the fact that $\omega$ is nondegenerate.

• integrable distribution: A distribution on a manifold $X$ is “integrable” if at least locally, there is a foliation of $X$ by submanifolds such that $V_x$ is the tangent space of the submanifold containing the point $x$.

• integral cohomology class: Any closed $p$-form on a manifold $M$ defines an element of the $p$th de Rham cohomology of $M$. This is a finite-dimensional vector space, and it contains a lattice called the $p$th integral cohomology group of $M$. We say a cohomology class is integral if it lies in this lattice. Most notably, if you take any $U(1)$ connection on any Hermitian line bundle over $M$, its curvature $2$-form will define an integral cohomology class once you divide it by $2\pi i$. This cohomology class is called the first Chern class, and it serves to determine the line bundle up to isomorphism.

• Poisson brackets: Given a symplectic structure on a manifold $M$ and given two smooth functions on that manifold, say $f$ and $g$, there’s a trick for getting a new smooth function $\{f,g\}$ on the manifold, called the Poisson bracket of $f$ and $g$.

  This trick works as follows: given any smooth function $f$ we can take its differential $df$, which is a $1$-form. Then there is a unique vector field $v(f)$, the Hamiltonian vector field associated to $f$, such that

  $$\omega (-,v(f))=df$$\omega(-,v(f)) = d f

  Using this we define

  $$\{f,g\}=\omega (v(f),v(g))$$\{f,g\} = \omega(v(f),v(g))

  It’s easy to check that we also have $\{f,g\}=dg(v(f))=v(f)g$. So $\{f,g\}$ says how much $g$ changes as we differentiate it in the direction of the Hamiltonian vector field generated by $f$.

  In the familiar case where $M$ is ${ℝ}^{2n}$ with momentum and position coordinates $p_i$, $q_i$, the Poisson brackets of $f$ and $g$ work out to be

  $$\{f,g\}=\sum _i\frac{df}{d{p}_i}\frac{dg}{d{q}_i}-\frac{df}{d{q}_i}\frac{dg}{d{p}_i}$$\{f,g\} = \sum_i \frac{d f}{d p_i} \frac{d g}{d q_i} - \frac{d f}{d q_i}\frac{d g}{d p_i}

• square-integrable sections: We can define an inner product on the sections of a Hermitian line bundle over a manifold $X$ with a symplectic structure. The symplectic structure defines a volume form which lets us do the necessary integral. A section whose inner product with itself is finite is said to be square-integrable. Such sections form a Hilbert space $H_0$ called the “prequantum Hilbert space”. It is a kind of preliminary version of the Hilbert space we get when we quantize the classical system whose phase space is $X$.

• symplectic structure: A symplectic structure on a manifold $M$ is a closed $2$-form $\omega$ which is nondegenerate in the sense that for any nonzero tangent vector $u$ at any point of $M$, there is a tangent vector $u$ at that point for which $w(u,v)$ is nonzero.

• $U(1)$ connection: The group $U(1)$ is the group of unit complex numbers. Given a complex line bundle $L$ with an inner product on each fiber $L_x$, a $U(1)$ connection on $L$ is a connection such that parallel translation preserves the inner product.

• vertical vectors: Given a bundle $E$ over a manifold $M$, we say a tangent vector to some point of $E$ is vertical if it projects to zero down on $M$.

The only way to learn the rules of this Game of games is to take the usual prescribed course, which requires many years, and none of the initiates could ever possibly have any interest in making these rules easier to learn. — Hermann Hesse, The Glass Bead Game

Definition

Prequantum line bundle

(…)

Quantum states

(…)

Quantum operators

Let $\nabla :X\to BU(1{)}_{\mathrm{conn}}$ be a prequantum line bundle $E\to X$ with connection for $\omega$. Write ${\Gamma }_X(E)$ for its space of smooth sections, the prequantum space of states.

Definition

For $f\in C^{\mathrm{\infty }}(X,ℂ)$ a function on phase space, the corresponding quantum operator is the linear map

$$\hat{f}:{\Gamma }_X(E)\to {\Gamma }_X(E)$$\hat f : \Gamma_X(E) \to \Gamma_X(E)

given by

$$\psi \mapsto -i{\nabla }_{v_f}\psi +f\cdot \psi ,$$\psi \mapsto -i \nabla_{v_f} \psi + f \cdot \psi \,,

where

• $v_f$ is the Hamiltonian vector field corresponding to $f$;

• ${\nabla }_{v_f}:{\Gamma }_X(E)\to {\Gamma }_X(E)$ is the covariant derivative of sections along $v_f$ for the given choice of prequantum connection;

• $f\cdot (-):{\Gamma }_X(E)\to {\Gamma }_X(E)$ is the operation of degreewise multiplication pf sections.

Polarizations

For $(X,\omega )$ a symplectic manifold, a polarization is a foliation of $X$ by Lagrangian submanifolds with respect to $\omega$.

After a choice of prequantum line bundle $\nabla$ lifting $\omega$, a Bohr-Sommerfeld leaf of a polarization is a leaf on which the prequantum line bundle is not just flat, but also trivializable as a circle bundle.

Quantum state space

(space of sections of the prequantum line bundle whose covariant derivative along the polarizaiton leaves vanishes…)

Examples

Cotangent bundles – Schrödinger representation

For the particle propagating on the line with respect to some standard action functional, the phase space is the cotangent bundle $T^{*}ℝ\simeq {ℝ}^2$, where the isomorphism is given by choosing standard coordinates, $\{q,p\}$.

The symplectic form is the canonical volume form, which in these coordinates reads

$$\omega =dp\wedge dq.$$\omega = d p \wedge d q \,.

A prequantum line bundle for this is given by the trivial line bundle equipped with the connection that is given by the globally defined 1-form

$$\theta :=q\wedge dp.$$\theta := q \wedge d p \,.

A section of this complex line bundle is canonically identified simply with a $ℂ$-valued smooth function on ${ℝ}^2$.

A choice of foliation of phase space is given by constant-$q$-slices

$${\Lambda }_q=\subset {ℝ}^2.$$\Lambda_q = \subset \mathbb{R}^2 \,.

The polarization condition is that the covariant derivative along the leafs? vanishes

$${\nabla }_{\partial /\partial p}\psi =0,$$\nabla_{\partial/\partial p} \psi = 0 \,,

which with the above choice of connection is equivalently

$$\frac{\partial }{\partial p}\psi +iq\psi =0.$$\frac{\partial}{\partial p} \psi + i q \psi = 0 \,.

The solutions to this differential equation are of the form

$$\psi (q,p)=\psi (q,0)\exp (ipq).$$\psi(q,p) = \psi(q,0) \exp(i p q) \,.

The space of these quantum states is (noncanonically) identified with the space of complex functions on the line by evaluating at $p=0$.

Since we have the hamiltonian vector fields

$$v_q=\frac{\partial }{\partial p}$$v_q = \frac{\partial}{\partial p}

$$v_p=-\frac{\partial }{\partial q}$$v_p = -\frac{\partial}{\partial q}

the action of the quantum operators $\hat{q}$ and $\hat{p}$ on these states is

$$\hat{q}\psi =-i{\nabla }_{\partial /\partial p}\psi +q\psi =q\psi$$\hat q \psi = - i \nabla_{\partial/\partial p}\psi + q \psi = q \psi

and

$$\hat{p}\psi =i{\nabla }_{\partial /\partial q}\psi +p\psi =i\frac{\partial }{\partial q}\psi .$$\hat p \psi = i \nabla_{\partial/ \partial q} \psi + p \psi = i \frac{\partial}{\partial q} \psi \,.

Related concepts

• reductions deformations resolutions in physics

• quantization

  • deformation quantization

  • geometric quantization, geometric quantization of symplectic groupoids

  • path integral

  • semiclassical approximation

References

General

A comprehensive review is

• A. Echeverria-Enriquez, M.C. Munoz-Lecanda, N. Roman-Roy, C. Victoria-Monge, Mathematical Foundations of Geometric Quantization Extracta Math. 13 (1998) 135-238 (arXiv:math-ph/9904008)

A brief review is

• William Gordon Ritter, Geometric Quantization (arXiv:math-ph/0208008)

The above “Overview” and “Basic Jargon” sections are taken from

• John Baez, Geometric Quantization (web)

Some useful talk notes include

• Eva Miranda, From action-angle coordinates to geometric quantization and back (2011) (pdf)

Refinements

References on geometric quantization of symplectic groupoids are

• Eli Hawkins, A Groupoid Approach to Quantization (arXiv)