Tempting look forward:
How do we understand a linear transformation with a given square matrix A?
Through its eigenvector analysis which decomposes the space into a direct
sum of eigenspaces, on each one of which matrix multiplication by A reduces
to scalar multiplication by the eigenvalue. When a matrix group acts on a
vector space as a group of linear transformations, then we can understand
its action by doing an eigenvector analysis to decompose the whole space
into a direct sum of subspaces on each of which the matrices act simply. For
a 1-dimensional Abelian matrix group, there is only one independent matrix
generator in its Lie algebra and its
eigenvectors in the representation are enough to decompose the whole space
on which it acts into eigenspaces. [See above Maple worksheet.]
Matrices that do not commute cannot be simultaneously diagonalized, i.e.,
have common eigenvectors, since on them, matrix multiplication reduces to
scalar multiplication which does commute. For a matrix group generated by
more than 1 basis matrix that don't commute, only one can be "diagonalized"
(find an eigenvector basis) but the others can be chosen to have a simple behavior on each such eigenspace. The three generating
"spin angular momentum" matrices S1 ,S2,
S3 of the rotation group do
not commute, so only one of them can be diagonalized by an eigenvector
analysis. But we can introduce the total spin angular momentum operator
S2 =S12 + S22
+ S32 which commutes with all the Si 's and decompose the space
in terms of simultaneous eigenvectors of the pair S2, S3.
For the representation of the rotation group on (0,2)-tensors over
R3
(i.e., matrices), this leads to the spin 0,1,2 subspaces of total spin angular
momentum, and we can further introduce bases of each eigenspace of total
spin angular momentum which are also eigenvectors of L3. These are called tensor
harmonics.
But wait. Suppose we want to understand how a function on the sphere
deviates from rotational symmetry? The infinite dimensional vector space of
functions on the sphere (add functions to get new functions, multiply by
numbers to get new functions, presto, its a vector space) is also a representation of the rotation group
under composition with the rotation, which rotates the function itself. It
is the corresponding partial derivative operators (now called orbital
angular momentum operators) L1
,L2, L3 and
L2 which represent S1
,S2, S3 and
S2 on that representation (the first 3 are the
corresponding vector
fields on
R3 of course tangent to the sphere,, the last is related to
the angular part of the Laplacian as we will later see), so if we decompose
the space of functions into eigenvectors (i.e., eigenfunctions) of the pair L2, L3
, then on each space the eigenvalue pairs are (-Ɩ
(Ɩ+1), i m) where Ɩ = 1/2, 1, 3/2, 2, ... and for each such
value, the second eigenvalue takes 2 Ɩ + 1 values m = -Ɩ, -Ɩ+1,
0, ...,Ɩ-1, Ɩ . The lowest eigenvalues
(0,0) correspond to invariance, and as one increases Ɩ, the deviation from
invariance gets more and more pronounced with more detailed structure on
smaller and smaller length scales compared to the radius of the sphere. The
eigenfunctions are the spherical harmonics, usually complex because we are
interested in complex wave functions in quantum mechanics, but can be taken
to be real for analyzing real functions on the sphere like Cole was doing in
the temperature distribution on the surface of a star.
Finally for vector or spinor or tensor fields on the sphere or in space, we
can consider the combined action of spin on the indices (use the symbol
Si) and orbital angular
momentum on the functional behavior under rotations (use the symbol
Li) and the vector sum of the
two yields "total angular momentum" Ji =
Li + Si which later we will see is the
Lie derivative with respect to the vector field generators of the rotations.
Because the rotations leave the dot product inner product invariant, all of
these decompositions of the representations are orthogonal with respect to the
natural inner products on the representation spaces.
So while there is not time to dwell on these exciting implications, they should
motivate why we take seriously vector fields interpreted as first order
derivative operators.