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

TL;DR

This paper derives matrix elements of SL(2,C) representations using bases from SU(1,1) and SU(2), providing explicit state definitions and a unified differential framework for discrete and continuous series.

Contribution

It introduces a unified differential approach to derive matrix elements of SL(2,C) representations across multiple bases and series, including new explicit state definitions.

Findings

Explicit matrix elements for SL(2,C) in various bases

Unified differential framework applicable to all cases

Explicit state functions related to SU(1,1) and SU(2) matrix elements

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.