On the 6j-symbols for SL(2,C) group

TL;DR

This paper investigates the 6j-symbols for SL(2,C), deriving their integral and sum representations using Feynman diagrams, focusing on infinite-dimensional unitary principal series representations.

Contribution

It rederives the 6j-symbols for SL(2,C) using Feynman diagrams, providing explicit integral and sum formulas for these symbols.

Findings

Derived 6j-symbols as complex plane integrals

Expressed 6j-symbols as Mellin-Barnes type sums

Connected results to earlier constructions by Ismagilov

Abstract

We study 6j-symbols, or Racah coefficients for tensor products of infinite-dimensional unitary principal series representations of the group SL(2,C). These symbols were constructed earlier by Ismagilov and we rederive his result (up to some slight difference associated with equivalent representations) using the Feynman diagrams technique. The resulting 6j-symbols are expressed either as a triple integral over complex plane with a symmetric kernel or as an infinite bilateral sum of integrals of the Mellin-Barnes type.