Geometric Quantization on Euclidean Space

Abstract

In this chapter, we consider the geometric quantization program in the setting of the symplectic manifold ${\mathbb{R}}^{2n},$ with the canonical 2-form ω = dp _j ∧ dx _j . We begin with the “prequantum” Hilbert space ${L}^{2}({\mathbb{R}}^{2n})$ and define “prequantum” operators Q _pre(f). These operators satisfy

$$\displaystyle{Q_{\mathrm{pre}}(\{f,g\}) = \frac{1} {i\hslash }[Q_{\mathrm{pre}}(f),Q_{\mathrm{pre}}(g)]}$$

for all f and g. Nevertheless, there are several undesirable aspects to the prequantization map that make it physically unreasonable to interpret it as “quantization.” To obtain the quantum Hilbert space, we reduce the number of variables from 2n to n. Depending on how we do this reduction, we will obtain either the position Hilbert space, the momentum Hilbert space, or the Segal–Bargmann space. Each of these subspaces is preserved by the prequantized position and momentum operators, and by certain other operators of the form Q _pre(f).