Quantum Mechanics for Mathematicians

time evolution), with the functionsμLhaving non-zero Poisson brackets with
the Hamiltonian function.

15.1 Vector fields and the exponential map

A vector field onM=R^2 can be thought of as a choice of a two dimensional
vector at each point inR^2 , so given by a vector-valued function

F(q,p) =

(

Fq(q,p)

Fp(q,p)

)

Such a vector field determines a system of differential equations

dq

dt

=Fq,

dp

dt

=Fp

Once initial conditions
q(0) =q 0 , p(0) =p 0

are specified, ifFqandFpare differentiable functions these differential equa-
tions have a unique solutionq(t),p(t), at least for some neighborhood oft= 0
(from the existence and uniqueness theorem that can be found for instance in
[48]). These solutionsq(t),p(t) describe trajectories inR^2 with velocity vector
F(q(t),p(t)) and such trajectories can be used to define the “flow” of the vector
field: for eachtthis is the map that takes the initial point (q(0),p(0))∈R^2 to
the point (q(t),p(t))∈R^2.
Another equivalent way to define vector fields onR^2 is to use instead the
directional derivative along the vector field, identifying

F(q,p)↔F(q,p)·∇=Fq(q,p)

∂

∂q

+Fp(q,p)

∂

∂p

The case ofFa constant vector is just our previous identification of the vector
spaceMwith linear combinations of∂q∂ and∂p∂.
An advantage of defining vector fields in this way as first-order linear differ-
ential operators is that it shows that vector fields form a Lie algebra, where one
takes as Lie bracket of vector fieldsX 1 ,X 2 the commutator

[X 1 ,X 2 ] =X 1 X 2 −X 2 X 1 (15.1)

of the differential operators. The commutator of two first-order differential op-
erators is another first-order differential operator since second-order derivatives
will cancel, using equality of mixed partial derivatives. In addition, such a
commutator will satisfy the Jacobi identity.
Given this Lie algebra of vector fields, one can ask what the corresponding
group might be. This is not a finite dimensional matrix Lie algebra, so expo-
nentiation of matrices will not give the group. The flow of the vector fieldX
can be used to define an analog of the exponential of a parameterttimesX: