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If we compare the geometric prequantization of the symplectic manifold (R2, dp∧ dq) (see Example 6.3.4) with the classical Schrödinger quantization then it is clear that the corresponding operators δ q do not agree, and the Hilbert representation spaces are different. More precisely, the Hilbert space of the first consists of functions of q and p simultaneously, in the second case the Hilbert space consists of functions depending on the q only. The obvious way to derive the Schrödinger quantization from geometric prequantization is to restrict the attention to functions on R2 which are independent of coordinate p, but then they no longer belong to the Hilbert space of geometric prequantization (except if they are identically zero) because the integral over p diverges. However, if we restrict our attention to functions independent of p and integrate over the q(R) instead of over the q and p(R2) then we get the Schrödinger quantization.
KeywordsLine Bundle Symplectic Manifold Quantum Operator Cotangent Bundle Geometric Quantization
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