wang
(Wang)
#1
8.2 The Lie algebra of rotations – angular momentum
The rotation groupSO(3) (as well as allSO(n) groups) is actually aLie group, which means
that its group elements can be parametrized in a continuous and differentiable manner by
real parameters, such asωandn, or also just by the entries of the matricesR.
The Norwegian mathematician Sophus Lie (1842 - 1899) proved two remarkable theorems
about Lie groups. Lie’s first theorem states that if the group multiplication is continuous
and once differentiable, then it is real analytic (i.e. infinitely differentiable and completely
given by its Taylor series expansion).
Lie’s second theorem is equally powerful. It states the equivalence (up to certain global
topological issues, which we will mention later) of Lie groups and Lie algebras.
A Lie algebra is obtained from a Lie group by Taylor series expanding the group around
its identity element. We are, of course, already familiar with doing thisfor rotations by a
small angleω,
R(n,ω) =I−
i
̄h
ωn·L+O(ω^2 ) (8.11)
The quantitiesLare the generators of the Lie algebra, and the multiplication law ofSO(3)
matrices yielding again orthogonal matrices then translates into the commutation relations
(8.9), while the associativity translates into the Jacobi identity,
[La,[Lb,Lc]] + [Lb,[Lc,La]] + [Lc,[La,Lb]] = 0 (8.12)
Lie proved that, conversely, if we have a Lie algebra defined by its commutation relations
(generalizing (8.9)) which satisfy the Jacobi identity, then there exists a unique globally well-
definedsimply connectedLie group, of which it is the Lie algebra. Practically, this means
that essentially all operations and constructions may be carried out at the level of the Lie
algebra, and we are then guaranteed that they will nicely carry over to the whole Lie group.
The advantage of the Lie algebra is that it is much much easier to handle.
8.3 General Groups and their Representations
A groupG,∗consists of a setGand a multiplication law∗satisfying the following axioms,
1. The multiplication closes: g 1 ∗g 2 ∈Gfor allg 1 ,g 2 ∈G;
2. Associativity, (g 1 ∗g 2 )∗g 3 =g 1 ∗(g 2 ∗g 3 ) =g 1 ∗g 2 ∗g 3 for allg 1 ,g 2 ,g 3 ∈G;
3. Gcontains an identity elementesuch thate∗g=g∗e=gfor allg∈G;
4. Everyg∈Ghas an inverseg−^1 so thatg∗g−^1 =g−^1 ∗g=e.
