4 1. INTRODUCTION

for appropriate smooth functions u. This is Weyl’s quantization formula,

and

aw(x,

hD) is a pseudodifferential operator.

One major task will be to understand how the analytic properties of

the symbol a dictate the functional analytic properties of its quantization

aw(x,

hD). We will in fact build up a symbol calculus, meaning systematic

rules for manipulating pseudodifferential operators.

1.2.2. Dynamics. We will be concerned as well with the evolution in time

of classical particles and quantum states.

Classical evolution. Our most important example will concern the symbol

p(x, ξ) :=

|ξ|2

+ V (x),

corresponding to the phase space flow

˙ x = 2ξ

˙

ξ = −∂V,

where ˙ = ∂t. We generalize by introducing the arbitrary Hamiltonian p :

R2n

→ R, p = p(x, ξ), and the corresponding Hamiltonian dynamics

(1.2.1)

˙ x = ∂ξp(x, ξ)

˙

ξ = −∂xp(x, ξ).

It is instructive to change our viewpoint somewhat, by writing

ϕt = exp(tHp)

for the solution of (1.2.1), where

Hpq := {p, q} = ∂ξp, ∂xq − ∂xp, ∂ξq

is the Poisson bracket. Select a symbol a and define

(1.2.2) at(x, ξ) := a(ϕt(x, ξ)).

Then

(1.2.3) ˙ a

t

= {p, at},

and this equation tells us how the symbol evolves in time, as dictated by

the classical dynamics (1.2.1).