Four based SO+(1,2) holonomies reconstruct a unique strictly convex tetrahedron in dS^3 or AdS^3, with det G selecting the model and recovering Euclidean cases via SU(2).
Classical 6j-symbols and the tetrahedron
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abstract
A classical 6j-symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally used simply to express the symmetry of the 6j-symbol, which is a purely algebraic object; however, it has a deeper geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a striking (but unproved) asymptotic formula relating the value of the 6j-symbol, when the dimensions of the representations are large, to the volume of an honest Euclidean tetrahedron whose edge lengths are these dimensions. The goal of this paper is to prove and explain this formula by using geometric quantization. A surprising spin-off is that a generic Euclidean tetrahedron gives rise to a family of twelve scissors-congruent but non-congruent tetrahedra.
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UNVERDICTED 2representative citing papers
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Generalized Minkowski Theorem for Tetrahedra in ${\rm dS}^3$ and ${\rm AdS}^3$
Four based SO+(1,2) holonomies reconstruct a unique strictly convex tetrahedron in dS^3 or AdS^3, with det G selecting the model and recovering Euclidean cases via SU(2).
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Causal structure in spin-foams
Proposes a causal EPRL spin-foam model where the two-complex orientation encodes causality and aids semiclassical geometry reconstruction.