This can be written as an exponential,R(θ) =eθL= cosθ 1 +Lsinθfor
L=
(
0 − 1
1 0
)
HereSO(2) is a commutative Lie group with Lie algebraso(2) =R. Note that
we have a representation onV =R^2 here, but it is a real representation, not
one of the complex ones we have when we have a representation on a quantum
mechanical state space.
In three dimensions the groupSO(3) is three dimensional and non-commu-
tative. Choosing a unit vectorwand angleθ, one gets an elementR(θ,w) of
SO(3), rotation by an angleθabout thewaxis. Using standard basis vectors
ej, rotations about the coordinate axes are given by
R(θ,e 1 ) =
1 0 0
0 cosθ −sinθ
0 sinθ cosθ
, R(θ,e 2 ) =
cosθ 0 sinθ
0 1 0
−sinθ 0 cosθ
R(θ,e 3 ) =
cosθ −sinθ 0
sinθ cosθ 0
0 0 1
A standard parametrization for elements ofSO(3) is in terms of 3 “Euler angles”
φ,θ,ψwith a general rotation given by
R(φ,θ,ψ) =R(ψ,e 3 )R(θ,e 1 )R(φ,e 3 ) (6.1)i.e., first a rotation about thez-axis by an angleφ, then a rotation by an
angleθabout the newx-axis, followed by a rotation byψabout the newz-
axis. Multiplying out the matrices gives a rather complicated expression for a
rotation in terms of the three angles, and one needs to figure out what range to
choose for the angles to avoid multiple counting.
The infinitesimal picture near the identity of the group, given by the Lie
algebra structure onso(3), is much easier to understand. Recall that for orthog-
onal groups the Lie algebra can be identified with the space of antisymmetric
matrices, so in this case there is a basis
l 1 =
0 0 0
0 0 − 1
0 1 0
l 2 =
0 0 1
0 0 0
−1 0 0
l 3 =
0 −1 0
1 0 0
0 0 0
which satisfy the commutation relations
[l 1 ,l 2 ] =l 3 ,[l 2 ,l 3 ] =l 1 ,[l 3 ,l 1 ] =l 2Note that these are exactly the same commutation relations (equation 3.5)
satisfied by the basis vectorsX 1 ,X 2 ,X 3 of the Lie algebrasu(2), soso(3) and
su(2) are isomorphic Lie algebras. They both are the vector spaceR^3 with the