Why the Quantum Tetrahedron?

Well in loop quantum gravity (LQG), the geometry of the physical space turns out to be quantized. When studying the spectral problem associated with the operators representing geometrical quantities such as area and volume, one finds two families of quantum numbers, which have a direct geometrical interpretation: SU(2) spins, labelling the links of a spin network, and SU(2) intertwiners, labelling its nodes. The spins are associated with the area of surfaces intersected by the link, while the intertwiners are associated with the volume of spatial regions that include the node, and to the angles formed by surfaces intersected by the links . A four-valent link, for instance, can be interpreted as a quantum tetrahedron: an elementary ‘atom of space’ whose face areas, volume and dihedral angles are determined by the spin and intertwiner quantum numbers.

The very same geometrical interpretation for spins and intertwiners can be obtained from a formal quantization of the degrees of freedom of the geometry of a tetrahedron, without any reference to the full quantization of general relativity which is at the base of LQG. In this case, one can directly obtain the Hilbert space H describing a single quantum tetrahedron. The states in H can be interpreted as ‘quantum states of a tetrahedron’, and the
resulting quantum geometry is the same as that defined by LQG.

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