Bohr-Sommerfeld quantization of space

We introduce semiclassical methods into the study of the volume spectrum in loop gravity. The classical system behind a 4-valent spin network node is a Euclidean tetrahedron. We investigate the tetrahedral volume dynamics on phase space and apply Bohr-Sommerfeld quantization to find the volume spectrum. The analysis shows a remarkable quantitative agreement with the volume spectrum computed in loop gravity. Moreover, it provides new geometrical insights into the degeneracy of this spectrum and the maximum and minimum eigenvalues of the volume on intertwiner space.

Figure 1

The space of shapes for a tetrahedron, $\mathcal{P}(\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2})$. The darkened contours are quantized level sets of the classical volume squared, $Q$. One quantized level set is hidden from view. The inset depicts a tetrahedron corresponding to the dot on the $q_3$ level set. The point corresponding to the tetrahedron with the largest possible volume, given the face area constraints, is also marked with a dot. The shape space coordinates are indicated at the top of the sphere.

Figure 2

The action integral $I(Q)$ for the same parameters as Fig. 1. The quantized levels $q_n$ shown satisfy the Bohr-Sommerfeld quantization condition. The corresponding orbits are shown in Fig. 1.

Figure 3

A comparison of the Bohr-Sommerfeld and loop gravity volume spectra. On the left: configuration with spins $\{j,j,j,j+1\}$. On the right: configuration with spins $\{4,4,4,j\}$ and $j$ varying in its allowed range. The Bohr-Sommerfeld values of the volume of a tetrahedron are represented as dots and the eigenvalues of the loop-gravity volume operator as circles. Recall that the spins and areas are related by $A_l=j_l+1/2$.

Figure 4

Comparison of the semiclassical scaling of the largest volume eigenvalue (dark line) to the Bohr-Sommerfeld (dots) and loop gravity (circles) spectra. The first three spins are given by $j_1=j_2=j_3=j$, and the largest spin $j_4$ is given by the closest integer or half integer to $\sqrt{3}j$ that respects the Clebsch-Gordan conditions.

Figure 5

Our characterization of the shape space using the quartics $P_0(x)$ and $P(x)$, defined by Eqs. (75, 50). The straight lines are given by $y=324Q^{2}x$ and the quartic curves by $y=P(x,0)\equiv P_0(x)$. The upper left panel shows the generic case in which the fixed volume determines four distinct roots. The upper right panel shows the coalescence of the middle roots at maximum volume and $m=0$. The lower two panels show the coalescence of other pairs of roots at zero volume and $m=1$ (these are two distinct cases).

Figure 6

The real values of the Lambert $W$ function.

Figure 7

Comparison of the semiclassical scaling of the largest and smallest volume eigenvalues (dark lines) to the Bohr-Sommerfeld (dots) and loop gravity (circles) spectra. The four spins are equal, $j_1=j_2=j_3=j_4=j_0$, and the corresponding phase space contains flat configurations. This explains the poorer agreement of the Bohr-Sommerfeld spectra at small eigenvalues.