Quantization Schemes for Euclidean Space

Abstract

One of the axioms of quantum mechanics states, “To each real-valued function f on the classical phase space there is associated a self-adjoint operator $\hat{f}$ on the quantum Hilbert space.” The attentive reader will note that we have not, up to this point, given a general procedure for constructing $\hat{f}$ from f.If we call $\hat{f}$ the quantization of f,then we have only discussed the quantizations of a few very special classical observables, such as position, momentum, and energy.