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.

• 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 \(T_x X\) where the choice depends smoothly on \(x \in X\).

• Hamiltonian vector field — Given a manifold X with a symplectic structure \(\omega\), any smooth function \(f\colon X \to \mathbb{R}\) 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

  $$ \omega(\cdot,v_f) = df $$

  In other words, for any vector field \(u\) on \(X\) we have

  $$ \omega(u,v_f) = df(u) = u(f) $$

  The vector field \(v_f\) is guaranteed to exist and be unique by the fact that \(\omega\) is nondegenerate.

• Hermitian line bundle — a "Hermitian line bundle" over a manifold \(X\) is a complex line bundle \(p \colon L \to X\) such that each fiber \(L_x\) is equipped with a complex inner product, which varies smoothly as a function of \(x \in X\).

• integrable distribution — A distribution on a manifold \(X\) is "integrable" if at least locally, there is a foliation of \(X\) by submanifolds such that each subspace \(V_x\) is the tangent space of the submanifold containing the point \(x \in X\)

• integral cohomology class — Any closed \(p\)-form on a manifold \(X\) defines an element of the \(p\)th deRham cohomology of \(X\). This is a 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 \(\mathrm{U}(1)\) connection on any Hermitian line bundle over \(X\), 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 \(f,g \colon M \to \mathbb{R}\), there's a trick for getting a new smooth function \( \{f,g\} \) on that 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

  $$ \omega(\cdot,v_f) = df $$

  Using this we define

  $$ \{f,g\} = \omega(v_f, v_g) $$

  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 \(\mathbb{R}^{2n}\) with momentum and position coordinates \(p_i, q_i \), the Poisson brackets of \(f\) and \(g\) work out to be

  $$ \{f,g\} = \sum_i \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} - \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} $$

• 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 \(X\) is a closed 2-form \(\omega\) which is nondegenerate in the sense that for any nonzero tangent vector \(v\) at any point of \(M\), there is a tangent vector \(u\) at that point for which \(\omega(u,v)\) is nonzero.

• \(\mathrm{U}(1)\) connection — The group \(\mathrm{U}(1)\) is the group of unit complex numbers. Given a Hermitian line bundle \(L\), a \(\mathrm{U}(1)\) connection on \(L\) is a connection on \(L\) such that parallel translation preserves the inner product on the fibers.

• vertical vectors — Given a bundle \(E\) over a manifold \(M\), we say a tangent vector \(v\) to some point of \(E\) is vertical if it projects to zero down on \(M\): that is, if the projection \(p \colon E \to M\) has \(dp(v) = 0\).

© 2018 John Baez
baez@math.removethis.ucr.andthis.edu

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