Understanding Quantum Numbers and Symmetry Algebra for Keplerian Motion

[gerald V]

so(4) is the symmetry algebra of Keplerian motion. Its structure is well known. The principal quantum number n must be a positive integer. The associated Casimir operator has eingenvalues n^2 - 1 . The secondary quantum number j is integer and can take any value from zero to n-1. The eigenvalue of the "angular momentum squared" operator J^2 is j(j+1).

Here my question: How is this for so(2,2) over the field of real numbers? Which values can the principal and the secondary quantum numbers take and which are the eigenvalues of the associated operators (Casimir operator; operator corresponding to J^2)?

Many thanks in advance for any help!

 

[Paul Colby]

Hi, just an observation. $so(4)$ the orthogonal matrices in 4 dimensions is a compact group. This implies a finite discrete spectrum for the hermitian irreducible representations which may be found by the usual operator ladder method. The groups $so(2,2)$ is I believe not a compact groups so I would expect at least some of the Casimir operators to take on values in a continuum for a hermitian rep. The Lorentz group is an example of such. Finding the irreducible representations for the Lorentz group can be a real chore. If you're happy with non-hermitian representations then you may find a finite discrete spectra but not a hermitian one.

Hope this is helpful

 

[A. Neumaier]

The groups so(2,2) is I believe not a compact group

Yes, but the unitary representations are known, first through work of Harish-Chandra. Books on advanced representaiton theory of groups contain the details.

How is this for so(2,2) over the field of real numbers?

For which physics application do you need it?