Recursion and Symmetry Relations for the Clebsch-Gordan Coefficients of the Homogeneous Lorentz Group

Abstract

Recursion and symmetry relations are obtained for the Clebsch-Gordan coefficients associated with the coupling of two SL(2, C) principal-series representations. An explicit procedure based on the recursion relations which are not restricted to the case of principal-series representations is given for generating a set of coefficients from a single "initial" coefficient. A program for the study of a possible connection, through analytic continuation, between the coupling coefficients for finite-dimensional representations and those for the infinite-dimensional case is presented.

Emergent Cosmology from Quantum Gravity in the Lorentzian Barrett-Crane Tensorial Group Field Theory Model

“…In the following, we construct the necessary notions of SL(2, C) recoupling theory, mainly based on [96], which itself is a refinement of the work done in [127][128][129][130][131].…”

A Wick rotation for EPRL spin foam models

“…4 If we take the solution of (36) for large quantum number to be p = γ(j + 1) and k = j instead as in [3], we have to add an extra γ 2 to both j L and j R . (26). The EPRL Euclidean vertex amplitude can be expressed as a superposition of SU (2) {15j} symbols weighted by the product of four Euclidean booster functions…”

A primer of group theory for Loop Quantum Gravity and spin-foams

“…An intermediate choice of phase is the one of [DN67,Ras03], which has the advantage of simplifying the recursion relations satisfied by the Clebsch-Gordan coefficients [ARRW70b,ARRW70a]. The latter are now either real or purely imaginary.…”

“…Contrary to the usual SU (2)-Clebsh-Gordan coefficients, there is no consensual convention for this phase. The choice of Kerimov differs from that of Anderson [ARRW70b] or Speziale [Spe17].…”

“…When willing to define a graphical calculus for SL 2 (C) one encounters the difficulty of finding a good SL 2 (C)-equivalent to the 3jm-symbol of SU (2) recoupling theory, so that it would satisfy the good symmetry relations to be well-represented by a 3-valent vertex. This issue is investigated in [ARRW70b], but the symmetry relations are intricate and depend on the convention chosen for the phase κ. As a result there is no consensus about the definition of the rules of graphical calculus for SL 2 (C).…”