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 =
˙
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).
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