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  "body": "\n## Definition\n\nLet $(M, \\omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$ and let $\\hbar \u003e 0$.\nAssume the integrality condition: the [[de Rham class|GX74C07T0G]] of $\\omega / (2\\pi\\hbar)$ lies in the image of $H^2(M; \\Z) \\to H^2(M; \\R)$.\nGeometric quantization consists of the following data and steps.\n\n1. (Prequantization) A *prequantum line bundle* is a Hermitian complex line bundle $L \\to M$, a vector bundle of rank one, with a connection $\\nabla$. The connection must preserve the Hermitian metric and have curvature $-\\frac{i}{\\hbar}\\omega$.\nThe *prequantum Hilbert space* is the space of sections $s$ of $L$ with $\\int_M \\abs{s}^2 \\, \\omega^n / n! \u003c \\infty$.\nTo a smooth function $f$ it assigns the operator $\\hat{f} = -i\\hbar \\nabla_{X_f} + f$ on sections, where $X_f$ is the [[Hamiltonian vector field|C5EH3B4BEP]] of $f$.\n2. (Polarization) A *polarization* is a subbundle $P \\subset TM \\otimes \\C$ of complex rank $n$ that is closed under the Lie bracket of vector fields. The complex-bilinear extension of $\\omega$ must vanish on $P$.\n3. (Quantization) The *quantum Hilbert space* is the [[Hilbert space|HM2HHE3J51]] of square-integrable sections $s$ that are polarized: $\\nabla_X s = 0$ for every section $X$ of $P$.\n\n## Remarks\n\nThe integrality condition holds exactly when a prequantum line bundle exists.\nWith consistent conventions, $f \\mapsto \\hat{f}$ sends the [[Poisson bracket|2S0MXDT4YH]] to the commutator up to the factor $\\pm i\\hbar$, which is Dirac's quantization condition.\nOnly functions whose flows preserve $P$ act directly on the polarized sections; in practice one also adds half-forms, the metaplectic correction, to obtain the correct spectra.\nSign and factor conventions differ from author to author.\n\nExample: take $M = \\C^n$ with the standard form and the Kähler polarization spanned by the vectors $\\partial / \\partial \\bar{z}_j$.\nThen the quantum Hilbert space is the Segal–Bargmann space of holomorphic functions that are square-integrable for a Gaussian weight.\nFor the sphere $S^2 = \\mathbb{CP}^1$ with total area $2\\pi\\hbar k$, $k \\ge 1$ an integer, the prequantum bundle is the line bundle of degree $k$.\nThe polarized sections are its holomorphic sections, a space of dimension $k + 1$.\n\nNon-example: for the sphere with total area $\\pi\\hbar$, the integrality condition fails, since $\\int_{S^2} \\omega / (2\\pi\\hbar) = \\frac{1}{2}$ is not an integer, so no prequantum line bundle exists.\n",
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