Quantum tetrahedra and simplicial spin networks

TL;DR

The paper links tetrahedral geometry to SU($2$) representations by defining a quantum tetrahedron via the closure of face spins, revealing quantum fluctuations that necessitate a mean-geometry interpretation. It then develops a kinematical, purely combinatorial simplicial gauge theory based on 4-valent dual graphs, where SU($2$) spin networks encode geometric data and diffeomorphism invariance is realized combinatorially. By tying quantum tetrahedra to 4-valent spin-network vertices, the work connects to Loop Quantum Gravity and Regge Calculus, and discusses a volume-like operator ${f U}$ with potential higher-dimensional interpretations. Finally, it sketches routes to dynamics—via a scalar constraint or Regge-like transition moves—toward a full quantum gravity framework and continuum limit.

Abstract

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.