12 - Introduction to the Weyl calculus

Summary

Motivation, definitions, basic properties

Quantization

The basic concept from which we would like to start is that of quantization. This name originates in the observation, at the turn of the twentieth century, that energy is exchanged in discrete units. Mathematically, quantization refers to a procedure by which one passes from functions on the phase space of classical mechanics in the Hamiltonian formulation (the cotangent bundle of a manifold) to operators on a Hilbert space, the phase space of quantum mechanics. As this is not a physics textbook, we will not motivate – let alone explore – this quantization problem in any generality or depth. Moreover, we assure the reader that this chapter is self-contained (up to knowing the Fourier transform and calculus) and that no knowledge of physics will be required to follow it. From a mathematical perspective the calculus of pseudodifferential operators (ΨDOs), which is a result of such a quantization procedure, is an essential tool in elliptic PDEs, through microlocal techniques, whereas Fourier integral operators (FIOs) arise naturally in hyperbolic PDEs.

The only information we start from is the following basic list of correspondences on the phase space ℝ^2d for the variables (x, ξ) (we will not discuss here the motivation for these correspondences from physics):

x_j ↦ X_j,

ξ_j ↦ D_j,

1 ↦ Id.

Here X_j is the (unbounded) operator on L²(ℝ^d) given by X_j(f)(x) = x_jf(x) and

alternatively, D_jf = (ξ_jf)^∨ at least for Schwartz functions.