Quantum tetrahedra and simplicial spin networks

TL;DR

This paper introduces the concept of quantum tetrahedra linked to SU(2) representations, establishing a geometric interpretation of spin networks and proposing a combinatorial quantum gauge theory model.

Contribution

It presents a novel definition of quantum tetrahedra using SU(2) representations and develops a combinatorial model for quantum gauge theories with geometric insights.

Findings

Quantum tetrahedra are associated with SU(2) representations.

The geometry of a quantum tetrahedron exists only as a mean geometry due to uncertainty relations.

A combinatorial model of quantum gauge theory is proposed, linking spin networks to geometric objects.

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.