Pentahedral volume, chaos, and quantum gravity

We show that chaotic classical dynamics associated with the volume of discrete grains of space leads to quantal spectra that are gapped between zero and nonzero volume. This strengthens the connection between spectral discreteness in the quantum geometry of gravity and tame ultraviolet behavior. We complete a detailed analysis of the geometry of a pentahedron, providing new insights into the volume operator and evidence of classical chaos in the dynamics it generates. These results reveal an unexplored realm for the application of chaos in quantum gravity.

Figure 1

Closing a pentahedron into a tetrahedron. (a) A fiducial face labeling of the pentahedron. (b) The closure of the pentahedron upon continuing its sides. (c) The resulting tetrahedron with face labels and associated scalings.

Figure 2

Illustrative Pachner move for a pentahedron (the back face, 3, suppressed). (a) The Schlegel diagram of a 54-pentahedron. (b) Schlegel diagram of a subdominant quadrilateral pyramid. (c) Resulting Schlegel diagram of a 12-pentahedron upon completion of the move.

Figure 3

A partial phase diagram for the adjacency classes of a pentahedron with fixed $\gamma =1.7$. The central, shaded region is the parameter space in which the 54-pentahedron is constructible. Neighboring regions are labeled by the pentahedral classes that are constructible within them.

Figure 4

The unfolded spectrum for $\sim 1000$ consecutive eigenvalues of Sinai’s billiard (a prototypical chaotic billiard). The solid (dashed) curve is the prediction for a classically chaotic (integrable) system: Wigner’s surmise or Poisson’s distribution, respectively. This figure was adapted from Ref. 18.

Figure 5

The first face of a 54-pentahedron and its scaling into a triangle through the closure process described in the main text.