Geometric Quantization

John Baez

August 31, 2018

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 requires that we choose a Hermitian line bundle \(L\) over \(X\), equipped with a \(\mathrm{U}(1)\) connection \(\nabla\) 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, the bundle \(L\) is determined up to isomorphism, but not canonically. After choosing \(L\), the curvature \(\omega\) determines the connection up to a gauge transformation locally, but not globally: we also need to choose its holonomies around noncontractible loops. So, it is best to consider \(L\) and \(\nabla\) as extra choices required for geometric quantization.

  3. The Hilbert space \(H_0\) of square-integrable sections of \(L\) is called the prequantum Hilbert space, \(H_0\). 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 classsical observables: there's a map sending any smooth function \(f \colon X \to \mathbb{R}\) to an operator \(Q(f) \) on \(H_0\), namely $$ Q(f) = i \hbar \nabla_{v_f} + M_f $$ where \(v_f\) is the Hamiltonian vector field associated to \(f\), \(\nabla_{v_f}\) is covariant differentation in the \(v_f\) direction using the connection \(\nabla\), and \(M_f\) is the operator of multiplication by \(f\). Prequantization takes Poisson brackets to commutators, just as one would hope: $$ [Q(f), Q(g)] = i \hbar Q[\{f,g\}]. $$

  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, we get by taking \(X = \mathbb{C}^n\) 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 \(\mathbb{C}^n\) 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, try Wikipedia, or some books on differential geometry and the math of classical mechanics.

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

© 2018 John Baez