The quantum relativistic two-body bound state. II. The induced representation of SL(2,C)

It was shown in I [J. Math. Phys. 30, 66 (1989)] that the eigenfunctions for the reduced motion of the quantum relativistic bound state with O(3,1) symmetric potential have support in an O(2,1) invariant subregion of the full spacelike region. They form irreducible representations of SU(1,1) [in the double covering of O(2,1)] parametrized by the unit spacelike vector n _μ taken in I as the direction of the z axis (the spectrum is independent of this choice), for which this O(2,1) is the stabilizer. Lorentz transformations move these representations on an orbit whose range is the single-sheeted hyperboloid covered by this spacelike vector, providing a set of induced representations of SL(2,C). From linear combinations of functions from the irreducible representations of SU(1,1), the representations of the SU(2) subgroup of SL(2,C) on the orbit are extracted and the differential equations that are the eigenvalue equations for the Casimir operators of SL(2,C) are solved. It is found that these SU(2) representations form a basis for the principal series in the canonical representations of Gel'fand. There is a natural scalar product, obtained from group integration on SL(2,C), for which the canonical basis forms an orthogonal set, and the representation is unitary. Since the scalar product [over the O(2,1) invariant measure space] of SU(1,1) irreducible representations is invariant under the action of the little group, the remaining group measure [on the coset space SL(2,C)/SU(1,1)] is the volume on the hyperboloidal spacelike hypersurface dμ_n=d ⁴nδ(n²-1). The family of Hilbert spaces (ℋ_n) that carries the representations of O(3,1) is therefore embedded in a larger Hilbert space ℋ with measure d⁴ydμ _n, where the {y} are the space-time coordinates of the restricted region associated with n_μ. The representations with nonrelativistic limit coinciding with the known Schrödinger solutions for corresponding spherically symmetric potential problems are in the double covering (half-integer values for the lowest L level) of O(3,1).