Quantum Mechanics for Mathematicians

and are given by

q(t) =q(0) cost+p(0) sint, p(t) =p(0) cost−q(0) sint

The exponential map is given by clockwise rotation through an anglet

q(exp(tXh)(m)) =q(m) cost+p(m) sint

p(exp(tXh)(m)) =−q(m) sint+p(m) cost

The vector fieldXhand the trajectories in theqpplane look like this

q

p

Fq=p

Fp=−q

Figure 15.1: Hamiltonian vector field for a simple harmonic oscillator.

and describe a periodic motion in phase space.
The relation of vector fields to the Poisson bracket is given by (see equation
15.2)
{f 1 ,f 2 }=Xf 2 (f 1 ) =−Xf 1 (f 2 )

and in particular

{q,f}=

∂f

∂p

, {p,f}=−

∂f
∂q
The definition we have given here ofXf(equation 15.2) carries with it a
choice of how to deal with a confusing sign issue. Recall that vector fields onM
form a Lie algebra with Lie bracket the commutator of differential operators. A
natural question is that of how this Lie algebra is related to the Lie algebra of
functions onM(with Lie bracket the Poisson bracket).