Unitary irreducible representations of \documentclass[12pt]{minimal}$\mathrm{SL(2,\mathbb {C})}$ SL (2,C) in discrete and continuous \documentclass[12pt]{minimal}$\mathrm{SU(1,1)}$ SU (1,1) bases

Abstract

We derive the matrix elements of generators of unitary irreducible representations of \documentclass[12pt]{minimal}\begin{document}$\mathrm{SL(2,\mathbb {C})}$\end{document} 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 \documentclass[12pt]{minimal}\begin{document}$J^3$\end{document}J3 and a continuous basis diagonalized by \documentclass[12pt]{minimal}\begin{document}$K^1$\end{document}K1, 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.