A semiclassical tetrahedron by Carlo Rovelli and Simone Speziale

In this paper the authors construct a macroscopic semiclassical state state for a quantum tetrahedron. They find that the expectation values of the geometrical operators representing the volume, areas and dihedral angles are peaked around assigned classical values with vanishing relative uncertainties.

The authors procedure to construct the semiclassical state is:

  • Choose a classical geometry A1 . . .A4, θ12, θ13, and compute the corresponding spins using Ai= ji,  j0(θ12) and k0(θ13) respectively using

semiclassical tetrahedron fig 1

and

semiclassical tetrahedron fig 2

  •  Pick up an auxiliary tetrahedron with ( 1/2 ) ji, j0, k0 as edge lengths, and calculate the two dihedral angles φ and χ using:

semiclassical tetrahedron fig 3

  • Choose a basis in Ij1…j4 , say |ji>ab, and take the linear combination with coefficients given by

semiclassical tetrahedron fig 4 – Gaussians with expectation value the required j0 or k0 and phase given by the corresponding dihedral angle φ or χ.

In the large spin limit, this state  encodes the classical values of the chosen geometry.

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