Prequantization - Knowlpedia

Definition

Let $(M,\omega)$ be a symplectic manifold and $\hbar>0$ . A prequantization consists of a Hermitian line bundle $L\to M$ with a compatible connection whose curvature is the prescribed scalar multiple of $\omega$ , together with the induced quantization map on smooth sections of $L$ . With the conventions

\iota_{X_f}\omega=-df, \qquad F_\nabla=\frac{i}{\hbar}\omega,

the Kostant–Souriau operator is

Q_{\mathrm{pre}}(f)=i\hbar\nabla_{X_f}+f,

and satisfies $[Q_{\mathrm{pre}}(f),Q_{\mathrm{pre}}(g)]=i\hbar Q_{\mathrm{pre}}(\{f,g\})$ on a common invariant domain.

Integrality condition

Such a line bundle with connection exists precisely when the de Rham class $[\omega/(2\pi\hbar)]$ is integral, meaning that it lies in the image of $H^2(M;\mathbb Z)\to H^2_{\mathrm{dR}}(M;\mathbb R)$ . This is the prequantization condition on the symplectic form. Different prequantum bundles can remain when this condition holds; their ambiguity is governed by flat Hermitian line bundles.

Why prequantization is not yet quantization

The prequantum Hilbert space is generally too large and the representation of observables is typically reducible. Geometric quantization therefore adds a polarization and, in many treatments, a half-form correction to select the physical state space. “Prequantization” names the line-bundle and operator stage before that reduction.

Equivariant lifts

A Hamiltonian group action need not lift to the prequantum line bundle as an action preserving its Hermitian structure and connection. When a lift exists, it differentiates to the Kostant–Souriau operators associated with the comoment map. Failure to lift can leave only a projective unitary representation of the original group or require passage to a central extension.

References

B. Kostant, “Quantization and Unitary Representations,” in C. T. Taam, ed., Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics 170, Springer, 1970, 87–208. DOI record. Relevant: prequantum line bundles and infinitesimal symmetry operators.
N. M. J. Woodhouse, Geometric Quantization, 2nd ed., Oxford University Press, 1992. Publisher record. Relevant: Chapters 5–6, prequantization and polarized sections.