Quantum tetrahedra and simplicial spin networks

Details Quantum tetrahedra and simplicial spin networks Quantum tetrahedra and simplicial spin networks

1997-07-04

arxiv.org/abs/gr-qc/9707010v2

Nucl.Phys. B518 (1998) 714-728

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

11 pages, LaTeX, 1 figure.ps, typos corrected, references added and updated, a note added, E-mail and postal addresses changed

Scientific paper

10.1016/S0550-3213(98)00093-5

A new link between tetrahedra and the group SU(2) is pointed out: by associating to each face of a tetrahedron an irreducible unitary SU(2) representation and by imposing that the faces close, the concept of quantum tetrahedron is seen to emerge. The Hilbert space of the quantum tetrahedron is introduced and it is shown that, due to an uncertainty relation, the ``geometry of the tetrahedron'' exists only in the sense of ``mean geometry''. A kinematical model of quantum gauge theory is also proposed, which shares the advantages of the Loop Representation approach in handling in a simple way gauge- and diff-invariances at a quantum level, but is completely combinatorial. The concept of quantum tetrahedron finds a natural application in this model, giving a possible intepretation of SU(2) spin networks in terms of geometrical objects.

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