Lecture 19.

Handouts:
12. Homework 4 Solutions
13. Homework 5

Having derived the rotation matrices and their generators last time,
we discussed their noncommutativity; showed the commutation relations
of the angular momentum generators; and discussed Lie algebras and Lie
groups, of which the angular momentum generators and rotations are an
example. 

We first demonstrated, visually, that finite rotations do not commute:
that rotating a book 90 degrees first about the x-, then the y-axis
gives a different result from doing the same two rotations in opposite
order. 

Our work with symmetries and conserved charge operators, as well as
the Baker-Hausdorff lemma, tells us that noncommutation with a unitary
transformation operator U is related to noncommutation with its
generating charge Q. We thus explicitly checked for noncommutation
among the angular momentum generators, finding the commutation
relations [ J_i,  J_j] = i\, \epsilon_ ijk\  J_k.

These kind of commutation relations, where we have a set of Hermitian
operators T_i whose commutators always give i times some linear
combination of the T_i's, is called a Lie algebra. It has a very
rich mathematical structure, with it's chief virtue being that it is a
vector space. This makes it simpler than the symmetry transformations:
for example, the number of possible rotations in three-space is
infinite, and rotations have complicated composition laws (that is,
it's hard to guess a priori what x-rotation by 90 degrees followed by
y-rotation by 90 degrees will give you). The number of angular
momentum generators \theta  J \dot \hat{n}  is also infinite,
but in a simpler way: it is just the vector space of all linear
combinations of J_x, J_y, and J_z. This is just a 3-dimensional
vector space; an arbitrary Lie algebra has dimension given by the
number of independent generators T_i.

The set of unitary transformations obtained by exponentiating i
times all linear combinations of the basis generators T_i is called
a Lie group.  We discussed the 4 properties of a group (closure,
associativity, identity, and inverse) as well as the ``missing''
property, that groups need not be commutative. 

Lie group elements and Lie algebra elements are in one-to-one
correspondence; in fact, we usually specify a rotation by stating its
angle and axis, which describes the Lie algebra element that generates
the rotation by exponentiation. All information about the Lie group is
contained in the Lie algebra; information about the group's
multiplication table (which specifies the result of successive
transformations) is equivalent to information about the algebra's
commutation relations. (This connection you have seen already in the
Baker-Hausdorff lemma.)

Physicists generally prefer to work with the Lie algebra, because 1)
it is a vector space; 2) Lie algebra elements are Hermitian operators,
and hence observables of the physical system; and 3) if the Lie group
is a symmetry, its generators in the Lie algebra commute with the
Hamiltonian (by the QM Noether's theorem). Thus eigenstates of the Lie
algebra elements are physical energy eigenstates, and the structure of
the Lie algebra tells us about degeneracies in the energy spectrum of
our physical system.

With this motivation, we will look for eigenstates of the angular
momentum operators (which generate rotations).

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KB