A unified treatment of the groups SO(4) and SO(3,1)

The irreducible representations of the group SO(4) in which the SO(3) subgroup is reduced are studied by an explicit construction of the operators and the basis in the spinor representation. The basis function which is formally identical with that for the coupling of two angular momenta j1 and j2 is expressible in terms of a hypergeometric function and strongly resembles that for the irreducible representations of the groups SO(3,1). For the Lorentz group, the bases for the unitary representations which require unphysical values of j1 and j2 are found to be analytic continuation of those for SO(4). The realization of the unitary irreducible representations of the group SO(4) in the Hilbert space of these functions leads, for appropriate unphysical values of j1, j2, to the Gelfand-Naimark formula for the principal and complementary series of the representations of SO(3,1). The matrix elements for finite transformations of SO(4) and SO(3,1) can be evaluated in this approach in a unified manner by using standard properties of the hypergeometric function. These turn out to be a finite sum of 3F2-functions which, as expected, are polynomials for SO(4) and infinite series for SO(3,1). A number of special matrix elements is calculated from the general formula and this agrees with results obtained previously. (author)