- For the three basis elementsljofso(3), show that the momentum map
gives functionsμljthat are just the components of the angular momentum. - Show that the maps
lj∈so(3)→μlj
give a Lie algebra homomorphism fromso(3) to the Lie algebra of functions
on phase space (with Lie bracket on such functions the Poisson bracket).
Problem 3:
In the same context as problem 2, compute the Poisson brackets
{μlj,qk}
between the angular momentum functionsμljand the configuration space co-
ordinatesqj. Compare this calculation to the calculation of
π′(lj)ek
forπthe spin-1 representation ofSO(3) onR^3 (the vector representation).
Problem 4:
Consider the symplectic groupSp(2d,R) of linear transformations of phase
spaceR^2 dthat preserve Ω.
- Consider the group of linear transformations of phase spaceR^2 dthat act
in the same way on positions and momenta, preserving the standard inner
products on position and momentum space. Show that this group is a
subgroup ofSp(2d,R), isomorphic toO(d). - Using the identification betweenSp(2d,R) and matrices satisfying equa-
tion 16.10, which matrices give the subgroup above? - Again in terms of matrices, what is the Lie algebra of this subgroup?
- Identifying the Lie algebra ofSp(2d,R) with quadratic functions of the
coordinates and momenta, which such quadratic functions are in the Lie
algebra of theSO(d) subgroup? - Consider the function
1
2
∑d
j=1
(q^2 j+p^2 j)
What matrix does this correspond to as an element of the Lie algebra of
Sp(2d,R)? Show that one gets anSO(2) subgroup ofSp(2d,R) by taking
exponentials of this matrix. Is thisSO(2) a subgroup of theSO(d) above?