[Paper Review] Remarks on the naturality of quantization

Tatyana Foth, Alejandro Uribe|arXiv (Cornell University)|Oct 1, 2004

Geometry and complex manifolds3 references4 citations

TL;DR

This paper investigates the naturality of geometric quantization by analyzing the curvature of a natural $L^2$ connection on a bundle of quantum Hilbert spaces parametrized by compatible almost complex structures on a compact integral symplectic manifold. It shows that in the semi-classical limit, the curvature is never central, implying no projectively flat connection exists, and links the connection to the Schrödinger equation via parallel transport.

ABSTRACT

Hamiltonian quantization of an integral compact symplectic manifold M depends on a choice of compatible almost complex structure J. For open sets U in the set of compatible almost complex structures and small enough values of Planck's constant, the Hilbert spaces of the quantization form a bundle over U with a natural connection. In this paper we examine the dependence of the Hilbert spaces on the choice of J, by computing the semi-classical limit of the curvature of this connection. We also show that parallel transport provides a link between the action of the group Symp(M) of symplectomorphisms of M and the Schrodinger equation.

Motivation & Objective

• To examine the dependence of quantum Hilbert spaces on the choice of compatible almost complex structure $J$ in geometric quantization.
• To determine whether the natural $L^2$ connection on the bundle $\mathcal{E}_N \to \mathcal{J}$ is projectively flat in the semi-classical limit.
• To explore the link between symplectic transformations and the Schrödinger equation through parallel transport in the quantum bundle.
• To compute the semi-classical asymptotics of the curvature of the natural connection on $\mathcal{E}_N \to \mathcal{J}$.

Proposed method

• Define a natural connection $D^N$ on the bundle $\mathcal{E}_N \to \mathcal{J}$ via orthogonal projection $\Pi_{N,J}$ in the fixed $L^2(M, L^{igotimes N})$ space.
• Compute the curvature $\Upsilon^{(N)}_J(A,B)$ of $D^N$ as an operator on the fiber $\mathcal{E}_{J,N}$ using the formula $\Upsilon^{(N)}_J(A,B) = \Pi_{N,J} \circ [\partial_A, \partial_B] \circ \Pi_{N,J}$.
• Analyze the semi-classical limit ($N \to \infty$) of the curvature using oscillatory integral techniques and asymptotic analysis of Hermite FIOs.
• Use the Euclidean model (Bargmann-Fock representation) to compute curvature explicitly in the flat case, leveraging quadratic Hamiltonians and creation/annihilation operators.
• Establish $G$-invariance of curvature and symplectic form $\Omega$ on $\mathcal{J}$ to extend local results globally.
• Relate the curvature to the symplectic structure on $\mathcal{J}$ via the Kähler form $\Omega$ and the symbol $\chi_{A,B}(x) = \operatorname{Tr}(A_x J_x B_x)$.

Experimental results

Research questions

• RQ1 Is the natural $L^2$ connection on the bundle $\mathcal{E}_N \to \mathcal{J}$ projectively flat in the semi-classical limit?
• RQ2 What is the semi-classical asymptotic behavior of the curvature of the natural connection on $\mathcal{E}_N \to \mathcal{J}$?
• RQ3 How does the curvature of the connection relate to the symplectic structure on the space of almost complex structures $\mathcal{J}$?
• RQ4 Can parallel transport in $\mathcal{E}_N \to \mathcal{J}$ be used to realize the action of $\mathrm{Symp}(M)$ on quantum states via the Schrödinger equation?
• RQ5 Is the curvature of the connection central in the semi-classical limit, or does it carry non-trivial operator-valued content?

Key findings

• The semi-classical limit of the curvature of the natural connection $D^N$ on $\mathcal{E}_N \to \mathcal{J}$ is never central, implying the connection is not projectively flat.
• In the Euclidean case (Bargmann-Fock model), the curvature on the space of quadratic Hamiltonians is explicitly computed: $\operatorname{Curv}_{H^+_{m,l}; H^-_{r,s}} = -8i(\delta_{mr}\delta_{ls} + \delta_{ms}\delta_{lr})$.
• The curvature vanishes on the subspaces $\mathfrak{p}^+$ and $\mathfrak{p}^-$ of quadratic Hamiltonians, corresponding to $H^+_{m,l}$ and $H^-_{m,l}$, respectively.
• The curvature is proportional to the identity operator on the fiber $H_J^{(k)}$ only if the variation of $J$ lies in the subspace $\mathfrak{p}$, and even then, the proportionality factor is not central in the full operator algebra.
• The curvature is $G$-invariant and the symplectic form $\Omega$ on $\mathcal{J}$ is preserved, allowing global extension of local curvature computations.
• Parallel transport along paths in $\mathcal{J}$ realizes the action of the symplectomorphism group $\mathrm{Symp}(M)$ on quantum states and provides a geometric realization of the Schrödinger equation in the semi-classical limit.