Quantum Theory for Mathematicians pp 467-482 | Cite as

# Geometric Quantization on Euclidean Space

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## 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 for

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

*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 2*n*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*).## Keywords

Hilbert Space Symplectic Manifold Momentum Operator Quantization Scheme Geometric Quantization
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Copyright information

© Springer Science+Business Media New York 2013