The Platonic solids of quantum gravity

Hal Haggard, CPT Marseille
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)

Coarse graining theories

Tuesday, Nov 27th. 2012
Bianca Dittrich, Perimeter Institute 
Title: Coarse graining: towards a cylindrically consistent dynamics
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by Frank Hellmann

Coarse graining is a procedure from statistical physics. In most situations we do not know how all the constituents of a system behave. Instead we only get a very coarse picture. Rather than knowing how all the atoms in the air around us move, we are typically only aware of a few very rough properties, like pressure, temperature and the like. Indeed it is hard to imagine a situation where one would care about the location of this or that atom in a gas made of 10^23 atoms. Thus when we speak of trying to find a coarse grained description of a model, we mean that we want to discard irrelevant detail and find out how a particular model would appear to us.
 
The technical way in which this is done was developed by Kadanoff and Wilson. Given a system made up of simple constituents Kadanoff's idea was to take a set of nearby constituents and combine them back into a single such constituent, only now larger. In a second step we could then scale down the entire system and find out how the behavior of this new, coarse grained constituent compares to the original ones. If certain behaviors grow stronger with such a step we call them relevant, if they grow weaker we call them irrelevant. Indeed, as we build ever coarser descriptions out of our system eventually only the relevant behaviors will survive.

In spin foam gravity we are facing this very problem. We want to build a theory of quantum gravity, that is, a theory that describes how space and time behave at the most fundamental level. We know very precisely how gravity occurs to us, every observation of it we have made is described by Einsteins theory of general relativity. Thus in order to be a viable candidate for a theory of quantum gravity, it is crucial that the coarse grained theory looks, at least in the cases that we have tested, like general relativity.

The problem we face is that usually we are looking at small and large blocks in space, but in spin foam models it is space-time itself that is built up of blocks, and these do not have a predefined size. They can be large or small in their own right. Further, we can not handle the complexity of calculating with so many blocks of space-time. The usual tools, approximations and concepts of coarse graining do not apply directly to spin-foams.

To me this constitutes the most important question facing the spin foam approach to quantum gravity. We have to make sure, or, as it often is in this game, at least give evidence, that we get the known physics right, before we can speak of having a plausible candidate for quantum gravity. So far most of our evidence comes from looking at individual blocks of space time, and we see that their behaviour really makes sense, geometrically. But as we have not yet seen any such blocks of space time floating around in the universe, we need to investigate the coarse graining to understand how a large number of them would look collectively. The hope is that the smooth space time we see arises like the smooth surface of water out of blocks composed of atoms, as an approximation to a large number of discrete blocks.

Dittrich's work tries to address this question. This requires bringing over, or reinventing in the new context, a lot of tools from statistical physics. The first question is, how does one actually combine different blocks of spin foam into one larger block? Given a way to do that, can we understand how it effectively behaves?

The particular tool of choice that Dittrich is using is called Tensor Network Renormalization. In this scheme, the coarse graining is done by looking at what aspects of the original set of blocks is the most relevant to the dynamics directly and then keeping only those. Thus it combines the two steps, of first coarse graining and then looking for relevant operators into a single step.

To get more technical, the idea is to consider maps from the boundary of a coarser lattice into that of a finer one. The mapping of the dynamics for the fine variables then provides the effective dynamics of the coarser ones. If the maps satisfy so called cylindrical consistency conditions, that is, if we can iterate them, this map can be used to define a continuum limit as well.

In the classical case, the behaviour of the theory as a function of the boundary values is coded in what is known as Hamilton's principal function. The use of studying the flow of the theory under such maps is then mostly that of improving the discretizations of continuum systems that can be used for numerical simulations.

In the quantum case, the principal function is replaced by the usual amplitude map. The pull back of the amplitude under this embedding then gives a renormalization prescription for the dynamics. Now Dittrich proposes to adapt an idea from condensed matter theory called tensor network renormalization.

In order to select which degrees of freedom to map from the coarse boundary to the fine one, the idea is to evaluate the amplitude, diagonalize and only keep the eigenstates corresponding to the n largest eigenvalues.

At each step one then obtains a refined dynamics that does not grow in complexity, and one can iterate the procedure to obtain effective dynamics for very coarse variables that have been picked by the theory, rather than by an initial choice of scale, and a split into high and low energy modes.

It is too early to say whether these methods will allow us to understand whether spin foams reproduce what we know about gravity, but they have already produced a whole host of new approximations and insights into how these type of models work, and how they can behave for large number of building blocks.