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Classical 6j-symbols and the tetrahedron
Justin Roberts
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Geometry & Topology 3 (1999)
21–66
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Abstract
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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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Keywords
6j–symbol, asymptotics,
tetrahedron, Ponzano–Regge formula, geometric
quantization, scissors congruence
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Mathematical Subject Classification
Primary: 22E99
Secondary: 51M20, 81R05
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Publication
Received: 9 January 1999
Accepted: 9 March 1999
Published: 22 March 1999
Proposed: Robion Kirby
Seconded: Vaughan Jones, Walter Neumann
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