Quantum Mechanics for Mathematicians

Definition(Flow of a vector field and the exponential map).The flow of the
vector fieldXonMis the map

ΦX: (t,m)∈R×M→ΦX(t,m)∈M

satisfying
d
dt

ΦX(t,m) =X(ΦX(t,m))

ΦX(0,m) =m

In words,ΦX(t,m)is the trajectory inMthat passes throughm∈Matt= 0,
with velocity vector given by the vector fieldXevaluated along the trajectory.
The flow can be written as a map

exp(tX) :m∈M→ΦX(t,m)∈M

called the exponential map.

If the vector fieldXis differentiable (with bounded derivative), exp(tX) will
be a well-defined map for some neighborhood oft= 0, and satisfy

exp(t 1 X) exp(t 2 X) = exp((t 1 +t 2 )X)

thus providing a one-parameter group of maps fromMto itself, with derivative
Xat the identity.

Digression.For any manifoldM, there is an infinite dimensional Lie group,
the group of invertible maps fromM to itself, such that the maps and their
inverses are both differentiable. This group is called the diffeomorphism group
ofMand writtenDiff(M). Its Lie algebra is the Lie algebra of vector fields.
The Lie algebra of vector fields acts on functions onMby differentiation.
This is the differential of the representation ofDiff(M)on functions induced in
the usual way (see equation 1.3) from the action ofDiff(M)on the spaceM.
This representation however is not one of relevance to quantum mechanics, since
it acts on functions on phase space, whereas the quantum state space is given
by functions on just half the phase space coordinates (positions or momenta).

15.2 Hamiltonian vector fields and canonical trans-

formations

Our interest is not in general vector fields, but in vector fields corresponding to
Hamilton’s equations for some Hamiltonian functionf, e.g., the case

Fq=

∂f

∂p

, Fp=−

∂f

∂q

We call such vector fields Hamiltonian vector fields, defining: