In article <COLUMBUS.92Nov23103624@strident.think.com> columbus@strident.think.com (Michael Weiss) writes:
>SU(2) is the double cover of SO(3), and thereby hangs a tale, ably told in
>many places (e.g., Kaufmann's "Knots and Physics".)
>
>SU(2) x SU(2) is the double cover of SO(4). Am I correct in suspecting
>that this is related to the Dirac equation?
Probably, but there are plenty of things it's good for that I'm more
familiar. For example, Schrodinger's equation for the hydrogen atom is
exactly solvable but most central potentials are not. Why? The 1/r potential
has a hidden so(4) symmetry - not just the obvious SO(3) symmetry. At
the level of Lie algebras, so(4) = so(3) + so(3) by the facts you cite
above. The "extra" so(3) symmetry of hydrogen atom Hamiltonian
is most easily seen from the presence of 3 extra conserved quantities in addition to angular momenta -- these are the components of the "Runge-Lenz vector".
So as it turns out, there is a deep relationship between the hydrogen atom
Hamiltonian and the Laplace equation on S^3. There is a nice 1-1 correspondence
between the eigenfunctions of the two (if one counts only the bound
states of hydrogen, not the ionized states) since they are isomorphic as
representations of so(4).
For more variations on this theme take a peek at Guillemin and Sternberg's
nice tour of mathematics, Variations on a Theme by Kepler.
>SU(4) is the double cover of SO(6). Has nature made use of this delightful
>fact?
Probably, but I don't know where. I hadn't remembered this one! There are
some other nice Lie group coincidences and they all lead to beautiful
