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Differential geometry Definition

Geometric quantization

Builds on Symplectic manifoldHamiltonian vector fieldPoisson bracketHilbert spacede Rham cohomology

Definition

Let (M,ω)(M, \omega) be a symplectic manifold of dimension 2n2n and let ℏ>0\hbar > 0. Assume the integrality condition: the de Rham class of ω/(2πℏ)\omega / (2\pi\hbar) lies in the image of H2(M;ℤ)→H2(M;ℝ)H^2(M; \Z) \to H^2(M; \R). Geometric quantization consists of the following data and steps.

  1. (Prequantization) A prequantum line bundle is a Hermitian complex line bundle L→ML \to M, a vector bundle of rank one, with a connection ∇\nabla. The connection must preserve the Hermitian metric and have curvature −iℏω-\frac{i}{\hbar}\omega. The prequantum Hilbert space is the space of sections ss of LL with ∫M∣s∣2 ωn/n!<∞\int_M \abs{s}^2 \, \omega^n / n! < \infty. To a smooth function ff it assigns the operator f^=−iℏ∇Xf+f\hat{f} = -i\hbar \nabla_{X_f} + f on sections, where XfX_f is the Hamiltonian vector field of ff.
  2. (Polarization) A polarization is a subbundle P⊂TM⊗ℂP \subset TM \otimes \C of complex rank nn that is closed under the Lie bracket of vector fields. The complex-bilinear extension of ω\omega must vanish on PP.
  3. (Quantization) The quantum Hilbert space is the Hilbert space of square-integrable sections ss that are polarized: ∇Xs=0\nabla_X s = 0 for every section XX of PP.

Remarks

The integrality condition holds exactly when a prequantum line bundle exists. With consistent conventions, f↦f^f \mapsto \hat{f} sends the Poisson bracket to the commutator up to the factor ±iℏ\pm i\hbar, which is Dirac's quantization condition. Only functions whose flows preserve PP act directly on the polarized sections; in practice one also adds half-forms, the metaplectic correction, to obtain the correct spectra. Sign and factor conventions differ from author to author.

Example: take M=ℂnM = \C^n with the standard form and the Kähler polarization spanned by the vectors ∂/∂zˉj\partial / \partial \bar{z}_j. Then the quantum Hilbert space is the Segal–Bargmann space of holomorphic functions that are square-integrable for a Gaussian weight. For the sphere S2=ℂℙ1S^2 = \mathbb{CP}^1 with total area 2πℏk2\pi\hbar k, k≥1k \ge 1 an integer, the prequantum bundle is the line bundle of degree kk. The polarized sections are its holomorphic sections, a space of dimension k+1k + 1.

Non-example: for the sphere with total area πℏ\pi\hbar, the integrality condition fails, since ∫S2ω/(2πℏ)=12\int_{S^2} \omega / (2\pi\hbar) = \frac{1}{2} is not an integer, so no prequantum line bundle exists.

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  1. Hamiltonian vector field concept
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Further reading

  1. Web https://en.wikipedia.org/wiki/Geometric_quantization
About this page
TagCGEXQ8BAH5
URIm2://CGEXQ8BAH5
Kinddefinition
WritingW0 stub
MSC53D50
DefinesGeometric quantization
UsesSymplectic manifold, Hamiltonian vector field, Poisson bracket, Hilbert space, de Rham cohomology
Changed2 November 2026

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