Unitary irreducible representations of SL(2,C) in discrete and continuous SU(1,1) bases

Abstract: We derive the matrix elements of generators of unitary irreducible representations of SL(2,C) with respect to basis states arising from a decomposition into irreducible representations of SU(1,1). This is done with regard to a discrete basis diagonalized by J^3 and a continuous basis diagonalized by K^1, and for both the discrete and continuous series of SU(1,1). For completeness we also treat the more conventional SU(2) decomposition as a fifth case. The derivation proceeds in a functional / differential framework and exploits the fact that state functions and differential operators have a similar structure in all five cases. The states are defined explicitly and related to SU(1,1) and SU(2) matrix elements.