Pentahedral volume, chaos, and quantum gravity by Haggard

This week I have been reviewing a couple of papers by Hal Haggard:

  • Pentahedral volume, chaos, and quantum gravity
  • Dynamical Chaos and the Volume Gap

These are related to the posts:

In these papers the author shows that chaotic classical dynamics associated to 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.  A detailed analysis of the geometry of a pentahedron, provides new insights into the volume operator and evidence of classical chaos in the dynamics it generates.

An outgrowth of quantum gravity has been the discovery that convex polyhedra can be endowed with a dynamical phase space structure. This structure was utilized to perform a Bohr-Sommerfeld quantization of the volume of a tetrahedron, yielding a novel
route to spatial discreteness and new insights into the spectral properties of discrete grains of space. Many approaches to quantum gravity rely on discretization of space or spacetime. This allows one to control, and limit, the number of degrees of freedom of the gravitational field being studied. Attention is often restricted to simplices. These papers study of grains of space more complex than simplices.

The Bohr-Sommerfeld quantization relied on the
integrability of the underlying classical volume dynamics, that is, the dynamics generated by taking as Hamiltonian the volume, H = Vtet. In general, integrability is a special property of a dynamical system exhibiting a high degree of symmetry. Instead, Hamiltonians with
two or more degrees of freedom are generically chaotic. Polyhedra with more than four faces are associated to systems with two or more degrees of freedom and so ‘Are their volume dynamics chaotic?’

The answer to this question has important physical consequences for quantum gravity. Prominent among these is that chaotic volume dynamics implies that there is generically a gap in the volume spectrum separating the zero volume eigenvalue from its
nearest neighbors. In loop gravity, it is convenient to
work with a polyhedral discretization of space because it
allows concrete study of a few degrees of freedom of the
gravitational fi eld;


however, what is key is the spectral discreteness of the geometrical operators of the theory. This is because the partition functions and transition amplitudes that defi ne the theory are expressed as sums over these area and volume eigenvalues. The generic presence
of gaps in the spectra of these operators ensures that these sums will not diverge as smaller and smaller quanta are considered; such a theory should be well behaved in the ultraviolet regime.

The author looks at the classical volume associated to a single pentahedral grain of space. He provides  evidences that the pentahedral volume dynamics is chaotic.  A new formula for the volume of a pentahedron in terms of its face areas and normals is found and it is shown that the volume dynamics is adjacency changing, General results from random matrix theory are used to argue that a chaotic volume dynamics implies the generic presence of a volume gap.

Consider a single pentahedral grain of space. An examination
of the classical volume dynamics of pentahedra relies on turning the space of convex polyhedra living in Euclidean three-space into a phase space. This is accomplished with the aid of two results:

  • Minkowski’s theorem states that the shape of a polyhedron is completely characterized by the face areas A and face normals n. More precisely, a convex polyhedron is uniquely determined, up to rotations, by its area vectors. The space of shapes of polyhedra with N faces of given areas A is:



The space PN naturally carries the structure of a phase space, with Poisson brackets,


This is the usual Lie-Poisson bracket if the A are interpreted physically as angular momenta, i.e. as generators of rotations.

To study the pentahedral volume dynamics on P5, with
H = Vpent, it is  necessary to find this volume as a function of the area vectors. This can be done as shown in the diagram below:


The main findings of the paper are summarized in a pentahedral phase diagram:



The presence of a volume gap in the integrable case of a tetrahedron has already been established. Furthermore, a chaotic pentahedral volume dynamics strongly suggest that there will be chaos for polyhedra with more faces, which have an even richer structure in their phase space. Consequently, the argument presented in this paper provides a very general mechanism that would ensure a volume gap for all discrete grains of space.


In this paper the authors have completely solved the geometry of
a pentahedron speci ed by its area vectors and defined its
volume as a function of these variables. By performing a numerical integration of the corresponding volume dynamics they have given early indicators that it generates a chaotic flow in phase space. These results uncover a new mechanism for the presence of a volume gap in the spectrum of quantum gravity: the level repulsion of quantum
systems corresponding to classically chaotic dynamics. The generic presence of a volume gap further strengthens the expected ultraviolet finiteness of quantum gravity theories built on spectral discreteness.


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