SYMMETRY LIE-ALGEBRA OF THE 2 BODY SYSTEM WITH A DIRAC OSCILLATOR INTERACTION

A few years ago Moshinsky and Szczepaniak introduced a Dirac equation linear not only in the momentum but also in the coordinate, which they called the Dirac oscillator, as for the large component of the eigens tate with positive energy, it reduces to a normal oscillator with a st rong spin-orbit term. This problem has interesting degeneracies that w ere shown by Quesne and Moshinsky to be due to an o(4) circle plus o(3 , 1) symmetry Lie algebra. The equation was then generalized to a two particle system with a Dirac oscillator interaction, for which the deg eneracy disappears for states of parity -(-1)(j), with j being the tot al angular momentum, but remains for states of parity (-1)(j). We show that for the latter, the degeneracy is due tea u(3) symmetry Lie alge bra if we take states of spin 0 and 1 separately or to an o(4) symmetr y Lie algebra if we take them together. Furthermore we consider the no nrelativistic limit of our problem which reduces it to an operator (N) over cap - L . S where (N) over cap is the total number of quanta, L the orbital angular momentum and S the total spin, whose eigenvalues a re now s = 0 or 1. In this case the symmetry Lie algebra for the state s of parity (-1)(j) remains the one discussed above, but there is now degeneracy also for states of parity -(-1)(j), which is explained, by a reasoning similar to that for the single particle Dirac oscillator b y the symmetry Lie algebra o(4) circle plus o(3, 1) but now with a spi n s = 1 instead of s = 1/2.