Title: Dynamical chaos and the volume gap
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by Chris Coleman-Smith, Duke University
At the Planck scale, a quantum behavior of the geometry of space is expected. Loop quantum gravity provides a specific realization of this expectation. It predicts a granularity of space with each grain having a quantum behavior. In particular the volume of the grain is quantized and its allowed values (what is technically known as "the spectrum")have a rich structure. Areas are also naturally quantized and there is a robust gap in their spectrum. Just as Planck showed that there must be a smallest possible photon energy, there is a smallest possible spatial area. Is the same true for volumes?
These grains of space can be visualized as polyhedra with faces of fixed area. In the full quantum theory these polyhedra are fuzzed out and so just as we cannot think of a quantum particle as a little spinning ball we cannot think of these polyhedra as the definite Platonic solids that come to mind.
[The Platonic Solids, by Wenzel Jamnitzer]
It is interesting to examine these polyhedra at the classical level, where we can set aside this fuzziness, and see what features we can deduce about the quantum theory.
The tetrahedron is the simplest possible polyhedron. Bianchi and Haggard [1] explored the dynamics arising from fixing the volume of a tetrahedron and letting the edges evolve in time. This evolution is a very natural way of exploring the set of constant volume polyhedra that can be reached by smooth deformations of the orientation of the polyhedral faces. The resulting trajectories in the space of polyhedra can be quantized by Bohr and Einstein's original geometrical methods for quantization. The basic idea here is to map some parts of the smooth continuous properties of the classical dynamics into the quantum by selecting only those orbits whose total area is an integer multiple of Planck's constant. The resulting discrete volume spectrum gives excellent agreement to the fully quantum calculation. Further work by Bianchi, Donna and Speziale [2] extended this treatment to more complex polyhedra.
Much as a bead threaded on a wire can only move forward or backward along the wire, a tetrahedron of fixed volume and face areas only has one freedom: to change its shape. Classical systems like this are typically integrable which means that their dynamics is fairy regular and can be exactly solved. Two degree of freedom systems like the pentahedron are typically non integrable. Their dynamics can be simulated numerically but there is no closed form solution for their motion. This implies that the pentahedron has a much richer dynamics than the tetrahedron. Is this pentahedral dynamics so complex that it is actually chaotic? If so, what are the implications for the quantized volume spectrum in this case. This system has recently been partially explored by Coleman-Smith [3] and Haggard [4] and was indeed found to be chaotic.
Chaotic systems are very sensitive to their initial conditions, tiny deviations from some reference trajectory rapidly diverge apart. This makes the dynamics of chaotic systems very complex and endows them with some interesting properties. This rapid spreading of any bundle of initial trajectories means that chaotic systems are unlikely to spend much time 'stuck' in some particular motion but rather they will quickly explore all possible motions. Such systems 'forget' their initial conditions very quickly and soon become thermal. This rapid thermalization of grains of space is an intriguing result. Black holes are known to be thermal objects and their thermal properties are believed to be fundamentally quantum in origin. The complex classical dynamics we observe may provide clues into the microscopic origins of these thermal properties.
The fuzzy world of quantum mechanics is unable to support the delicate fractal structures arising from classical chaos. However its echoes can indeed be found in the quantum analogues of classically chaotic systems. A fundamental property of quantum systems is that they can only take on certain discrete energies. The set of these energy levels is usually referred to as the energy spectrum of the system. An important result from the study of how classical chaos passes into quantum systems is that we can generically expect certain statistical properties of the spectrum of such systems. In fact the spacing between adjacent energy levels of such systems can be predicted on very general grounds. For a non chaotic quantum system one would expect these spacings to be entirely uncorrelated and so be Poisson distributed (e.g the number of cars passing through a toll gate in an hour) resulting in most energy levels being very bunched up. In chaotic systems the spacings become correlated and actually repel each other so that on average one would expect these spacings to be quite large.
This is suggestive that there may indeed be a robust volume gap since we generically expect the discrete quantized volume levels to repel each other. However the density of the volume spectrum around the ground state needs to be better understood to make this argument more concrete. Is there really a smallest non zero volume?
The classical dynamics of the fundamental grains of space provide a fascinating window into the behavior of the very complicated full quantum dynamics of space described by loop quantum gravity. Extending this work to look at more complex polyhedra and at coupled netwworks of polyhedra will be very exciting and will certainly provide many useful new insights into the microscopic structure of space itself.
[1]: "Discreteness of the volume of space from Bohr-Sommerfeld quantization", E.Bianchi & H.Haggard. PRL 107, 011301 (2011), "Bohr-Sommerfeld Quantization of Space", E.Bianchi & H.Haggard. PRD 86, 123010 (2012)
[2]: "Polyhedra in loop quantum gravity", E.Bianchi, P.Dona & S.Speziale. PRD 83, 0440305 (2011)
[3]: "A “Helium Atom” of Space: Dynamical Instability of the Isochoric Pentahedron", C.Coleman-Smith & B.Muller, PRD 87 044047 (2013)
[4]: "Pentahedral volume, chaos, and quantum gravity", H.Haggard, PRD 87 044020 (2013)
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