Geometric Quantization

Abstract

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.