A New Recursion Relation for the 6j-Symbol

Abstract

The 6j-symbol is a fundamental object from the re-coupling theory of SU(2) representations. In the limit of large angular momenta, its asymptotics is known to be described by the geometry of a tetrahedron with quantized lengths. This article presents a new recursion formula for the square of the 6j-symbol. In the asymptotic regime, the new recursion is shown to characterize the closure of the relevant tetrahedron. Since the 6j-symbol is the basic building block of the Ponzano-Regge model for pure three-dimensional quantum gravity, we also discuss how to generalize the method to derive more general recursion relations on the full amplitudes.Comment: 10 pages, v2: title and introduction changed, paper re-structured; Annales Henri Poincare (2011

Generating functions for coherent intertwiners

“…j P p (j i , j)P q (j i , j) . (B14) 14 In details, 12 {j i }, j β |{j i }, W 4 = (−1) J −2j β 4 i=1 (2j i )! (j β + j 3 − j 4 )!…”

“…While some basic properties have been known for several decades, a need for new results involving more and more spins have appeared and have led to some interesting progress. They come from various areas of physics, like quantum information [4,5], semi-classical approximations for quantum angular momenta [6,7], and quantum gravity [8][9][10][11][12][13][14][15].…”

Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions

“…From this approach the full expansion can be computed in principle, however in a very lengthy way. We derive a recursion relations of the Ward-Takesaki type, which is surprisingly similar to the one invented in [36,37] however in very different context, that, basically can be used in the asymptotic expansion to derive the NLO in a more concise way. Moreover, we can show explicitly that the consecutive terms in the expansion (1.1) are of the conjectured 'sin/cos' form.…”

“…We will now derive a recursion relation for the full 6j symbol using a similar idea as in [36,37] that, we hope, can serve to compute the NLO expansion in more concise way.…”

“…since detG is constrained to vanish. Similar to [36,37], detG can be expanded as a sum over perturbations: 50) with the convention that θ ij = θ ji and θ ii = 0. Using (5.50), equation (5.49) can be rewritten as:…”