Quantum Mechanics for Mathematicians

Using the formula
LX= (d+iX)^2 =diX+iXd

for the Lie derivative acting on differential forms (iXis interior product with
the vector fieldX), one has

(diX+iXd)ω= 0

and sincedω= 0we have
diXω= 0

WhenM is simply-connected, one-formsiXω whose differential is 0 (called
“closed”) will be the differentials of a function (and called “exact”). So there
will be a functionμsuch that

iXω(·) =ω(X,·) =dμ(·)

although such aμis only unique up to a constant.
Given an elementL∈g, aGaction onMgives a vector fieldXLby equation
15.10. When we can choose the constants appropriately and find functionsμL
satisfying
iXLω(·) =dμL(·)

such that the map
L→μL

taking Lie algebra elements to functions onM (with Lie bracket the Poisson
bracket) is a Lie algebra homomorphism, then this is called the moment map.
One can equivalently work with

μ:M→g∗

by defining
(μ(m))(L) =μL(m)

15.4 Examples of Hamiltonian group actions

Some examples of Hamiltonian group actions are the following:

• Ford= 3, an elementaof the translation groupG=R^3 acts on the
  phase spaceM=R^6 by translation

m∈M→a·m∈M

such that the coordinates satisfy

q(a·m) =q+a, p(a·m) =p(m)