Quantum Mechanics for Mathematicians

Takingato be the corresponding element in the Lie algebra ofG=R^3 ,

the vector field onMcorresponding to this action (by 15.10) is

Xa=a 1

∂

∂q 1

+a 2

∂

∂q 2

+a 3

∂

∂q 3

and the moment map is given by

μa(m) =a·p(m) (15.12)

This can be interpreted as a functiona·ponMfor each elementaof the

Lie algebra, or as an elementp(m) of the dual of the Lie algebraR^3 for

each pointm∈M.

• For another example, consider the action of the groupG=SO(3) of
  rotations on phase spaceM=R^6 given by performing the sameSO(3)
  rotation on position and momentum vectors. This gives a map fromso(3)
  to vector fields onR^6 , taking for example

l 1 ∈so(3)→Xl 1 =−q 3

∂

∂q 2

+q 2

∂

∂q 3

−p 3

∂

∂p 2

+p 2

∂

∂p 3

(this is the vector field for an infinitesimal counter-clockwise rotation in

theq 2 −q 3 andp 2 −p 3 planes, in the opposite direction to the case of

the vector fieldX (^12) (q (^2) +p (^2) )in theqpplane of section 15.2). The moment
map here gives the usual expression for the 1-component of the angular
momentum
μl 1 =q 2 p 3 −q 3 p 2
since one can check from equation 15.2 thatXl 1 =Xμl 1. On basis elements
ofso(3) one has
μlj(m) = (q(m)×p(m))j
Formulated as a map fromMtoso(3)∗, the moment map is
μ(m)(l) = (q(m)×p(m))·l
wherel∈so(3).

• While most of the material of this chapter also applies to the case of a
  general symplectic manifoldM, the case ofMa vector space has the
  feature thatGcan be taken to be a group of linear transformations of
  M, and the moment map will give quadratic polynomials. The previous
  example is a special case of this and more general linear transformations
  will be studied in great detail in later chapters. In this linear case it turns
  out that it is generally best to work not withMbut with its dual space
  M.