The Platonic solids of quantum gravity

Hal Haggard, CPT Marseille
Title: Dynamical chaos and the volume gap 
PDF of the talk (8Mb) Audio [.wav 37MB] Audio [.aif 4MB]

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
PDF of the talk (14Mb) Audio [.wav 41MB] Audio [.aif 4MB]

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.

Bianchi space-times in loop quantum cosmology

Brajesh Gupt, LSU 

Title: Bianchi I LQC, Kasner transitions and inflation
PDF of the talk (800k) Audio [.wav 30MB] Audio [.aif 3MB]

by Edward Wilson-Ewing

The Bianchi space-times are a generalization of the simplest Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological models.  While the FLRW space-times are assumed to be homogeneous (there are no preferred points in the space-time) and isotropic (there is no preferred direction), in the Bianchi models the isotropy requirement is removed.  One of the main consequences of this generalization is that in a Bianchi cosmology, the space-time is allowed to expand along the three spatial axes at different rates.  In other words, while there is only one Hubble rate in FLRW space-times, there are three Hubble rates in Bianchi cosmologies, one for each of the three spatial dimensions.

For example, the simplest Bianchi model is the Bianchi I space-time whose metric is given by

ds² = - dt² + a₁(t)² (dx₁)² +  a₂(t)² (dx₂)² +  a₃(t)² (dx₃)²,

where the a_i(t) are the three scale factors.  This is in contrast to the flat FLRW model where there is only one scale factor.

It is possible to determine the exact form of the scale factors by solving the Einstein equations.  In a vacuum, or with a massless scalar field, it turns out that the i^th scale factor is simply given by the time elevated to the power k_i: a_i(t) = t^ki, where these k_i are constant numbers and are called the Kasner exponents.  There are some relations between the Kasner exponents that must be satisfied, so that once the matter content has been chosen, one Kasner exponent between -1 and 1 may be chosen freely, and then the values of the other two Kasner exponents are determined by this initial choice.

In addition to allowing more degrees of freedom than the simpler FLRW models, the Bianchi space-times are important due to the central role they play in the Belinsky-Khalatnikov-Lifshitz (BKL) conjecture.  According to the BKL conjecture, as a generic space-like singularity is approached in general relativity time derivatives dominate over spatial derivatives (with the exception of some small number of “spikes” which we shall ignore here) and so spatial points decouple from each other.  In essence, as the spatial derivatives become negligible, the complicated partial differential equations of general relativity reduce to simpler ordinary differential equations close to a space-like singularity.  Although this conjecture has not been proven, there is a wealth of numerical studies that supports it.

If the BKL conjecture is correct, and the ordinary differential equations can be trusted, then the solution at each point is that of a homogeneous space-time.  Since the most general homogeneous space-times are given by the Bianchi space-times, it follows that as a space-like singularity is approached, the geometry at each point is well approximated by a Bianchi model.

This conjecture is extremely important from the point of view of quantum gravity, as quantum gravity effects are expected to become important precisely when the space-time curvature nears the Planck scale.  Therefore, we expect quantum gravity effects to become important near singularities.  What the BKL conjecture is telling us is that understanding quantum gravity effects in the Bianchi models, which are relatively simple space-times, can shed significant insight into the problem of singularities in gravitation.

What is more, studies of the BKL dynamics show that for long periods of time, the geometry at any point is given by the Bianchi I space-time and during this time the geometry is completely determined by the three Kasner exponents introduced above in the third paragraph.  Now, the Bianchi I solution does not hold at each point eternally, rather there are occasional transitions between different Bianchi I solutions called Kasner or Taub transitions.  During a Kasner transition, the three Kasner exponents rapidly change values before becoming constant for another long period of time.  Now, since the Bianchi I model provides an excellent approximation at each point for long periods of time, understanding the dynamics of the Bianchi I space-time, especially at high curvatures when quantum gravity effects cannot be neglected, may help us understand the behaviour of generic singularities when quantum gravity effects are included.

In loop quantum cosmology (LQC), for all of the space-times studied so far including Bianchi I, the big-bang singularity in cosmological space-times is resolved by quantum geometry effects.  The fact that the initial singularity in the Bianchi I model is resolved in loop quantum cosmology, in conjunction with the BKL conjecture, gives some hope that all space-like singularities may be resolved in loop quantum gravity.  While this result is encouraging, there remain open questions regarding the specifics of the evolution of the Bianchi I space-time in LQC when quantum geometry effects are important.

One of the main goals of Brajesh Gupt's talk is to address this precise question.  Using the effective equations, which provide an excellent approximation to the full quantum dynamics for the FLRW space-times in LQC and are expected to do the same for the Bianchi models, it is possible to study how the quantum gravity effects that arise in loop quantum cosmology modify the classical dynamics when the space-time curvature becomes large and replace the big-bang singularity by a bounce.  In particular, Brajesh Gupt describes the precise manner of how the Kasner exponents ---which are constant classically--- evolve deterministically as they go through the quantum bounce.  It turns out that there is some sort of a Kasner transition that occurs around the bounce, the details of which are given in the talk.

The second part of the talk considers inflation in Bianchi I loop cosmologies.  Inflation is a period of exponential expansion of the early universe which was initially introduced in order to resolve the so-called horizon and flatness problems.  One of the major results of inflation is that it generates small fluctuations that are of exactly the form that are observed in the small temperature variations in the cosmic microwave background today.  For more information about inflation in loop quantum cosmology, see the previous ILQGS talks by William Nelson, Ivan Agullo, Gianluca Calcagni and David Sloan, as well as the blog posts that accompany these presentations.

Although inflation is often considered in the context of isotropic space-times, it is important to remember that in the presence of matter fields such as radiation and cold dark matter, anisotropic space-times will become isotropic at late times.  Therefore, it is not because our universe appears to be isotropic today that it necessarily was some 13.8 billion years ago.  Because of this, it is necessary to understand how the dynamics of inflation change when anisotropies are present.  As mentioned at the beginning of this blog post, there is considerably more freedom in Bianchi models than in FLRW space-times, and so the expectations coming from studying inflation in isotropic cosmologies may be misleading for the more general situation.

There are several interesting issues that are worth considering in this context, and in this talk the focus is on two questions in particular.  First, is it easier or harder to obtain the initial conditions necessary for inflation? In other words, is more or less fine-tuning required in the initial conditions?  As it turns out, the presence of anisotropies actually makes it easier for a sufficient amount of inflation to occur.  The second problem is to determine how the quantum geometry effects from loop quantum cosmology change the results one would expect based on classical general relativity.  The main modification found here has to do with the relation between the amount of anisotropy present in the space-time (which can be quantified in a precise manner) and the amount of inflation that occurs.  While there was a monotonic relation between these two quantities in classical general relativity, this is no longer the case when loop quantum cosmology effects are taken into account.  Instead, there is now a specific amount of anisotropy which extremizes the amount of inflation that will occur, and there is a turn around after this point.  The details of these two results are given in the talk.

Reduced loop quantum gravity

Tuesday, Mar 12th.
Emanuele Alesci, Francesco Cianfrani
Title: Quantum reduced loop gravity
PDF of the talk (4Mb) Audio [.wav 29MB] Audio [.aif 3MB]

By Emanuele Alesci, Warszaw University and  Francesco Cianfrani,Wrocław University

We propose a new framework for the loop quantization of  symmetry-reduced sectors of General Relativity, called Quantum Reduced Loop Gravity, and we apply this scheme to the inhomogeneous extension of the Bianchi I cosmological model (a cosmology that is homogeneous but anisotropic). To explain the meaning of this sentence we need several ingredients that will be presented in the next sections. But let us first focus on the meaning of “symmetry reduction”: this process simply means that a if a physical system has some kind of symmetry we can use it to reduce the number of independent variables needed to describe it. Symmetry then in general allows to restrict the variables of the theory to the true independent degrees of freedom of it. For instance, let us consider a point-like spinless particle moving on a plane under a central potential. The system is invariant under 2-dimensional rotations on the plane around the center of the potential and as a consequence the angular momentum is conserved. The angular velocity around the origin is a constant of motion and the only “true” dynamical variable is the radial coordinate of the particle. Going to the phase space (the space of positions and momenta of the theory), it can be parameterized by the radial and angular coordinates together with the corresponding momenta, but the symmetry forces the momentum associated with the angular coordinate to be conserved. The reduced phase-space associated with such a system is parameterized by the radial coordinate and momentum, from which, given the initial conditions, the whole trajectory of the particle in the plane can be reconstructed. The quantization in the reduced phase space is usually easier to handle than in the full phase space and this is the main reason why it is a technique frequently used in order to test the approaches towards Quantum Gravity, whose final theory is still elusive. In this respect, the canonical analysis of homogeneous models (Loop Quantum Cosmology) and of spherically-symmetric systems (Quantum Black Holes) in Loop Quantum Gravity (LQG) has been mostly performed by first restricting to the reduced phase space and then quantizing the resulting system (what is technically known as reduced quantization). The basic idea of our approach is to invert the order of “reduction” and “quantization”. The motivation will come directly from our analysis and, in particular, from the failure of reduced quantization to provide a sensible dynamics for the inhomogeneous extensions of the homogeneous anisotropic Bianchi I model. Hence, we will follow a different path by defining a “quantum” reduction of the Hilbert space of quantum states of the full theory down to a subspace which captures the relevant degrees of freedom. This procedure will allow us to treat the inhomogeneous Bianchi I system directly at the quantum level in a computable theory with all the ingredients of LQG (just simplified due to the quantum-reduction).

To proceed, let us first review the main features of LQG.

 Loop Quantum Gravity
 LQG is one of the most promising approaches for the quantization of the gravitational field. Its formulation is canonical and thus it is based on making a 3+1 splitting of the space-time manifold. The phase space is parameterized by the Ashtekar-Barbero connections, and the associated momenta, from which one can compute the metric of spatial sections. A key point of this reformulation is the existence of a  gauge invariance (technically known as SU(2) gauge invariance), which together with background independence, lead to the so-called kinematical constraints of the theory (every time there is a symmetry in a theory an associated constraint emerges implying that the variables are not independent and one has to isolate the true degrees of freedom). The quantization procedure is inspired by the approaches developed in the 70s to describe gauge theories on the lattice in the strong-coupling limit. In particular, the quantum states are given in terms of spin networks, which are graphs with "colors" in the links between intersections. An essential ingredient of LQG is background independence. The way this symmetry is implemented is a completely new achievement in Quantum Gravity and it allows to define a regularized expression (free from infinities) for the operator associated with the Hamiltonian constraint asssociated with the dynamics of the theory.  Thanks to a procedure introduced by Thiemann, the Hamiltonian constraint can be approximated over a certain triangulation of the spatial manifold. The limit in which the triangulation gets finer and finer gives us back the classical expression and it is well defined on a quantum level over s-knots (classes of spin networks related by smooth deformations). The reason is that s-knots are diffeomorphisms invariant and, thus, insensitive to the characteristic length of the triangulation. This means that the Hamiltonian constraint can be consistently regularized and, by the way, the associated algebra is anomaly-free. Unfortunately, the resulting expression cannot be analytically computed, because of the presence of the volume operator, which is complicated. This drawback appears to be a technical difficulty, rather than a theoretical obstruction, and for this reason our aim is to try to overcome it in a simplified model, like a cosmological one.

 Loop Quantum Cosmology 
 Loop Quantum Cosmology (LQC) is the best theory at our disposal to threat homogeneous cosmologies. LQC is based on a quantization in the reduced phase space, which means that the reduction according with the symmetry is entirely made on a classical level. Once that the classical reduction is made, one then proceeds with a quantization of the degrees of freedom left with LQG techniques. We know that our Universe experiences a highly isotropic and homogeneous phase at scales bigger than 100Mpc. The easiest cosmological description is the one of Friedmann-Robertson-Walker (FRW), in which one deals with an isotropic and homogeneous line element, described by only one variable, the scale factor. A generalization can be given by considering anisotropic extensions, the so-called Bianchi models, in which there are three scale factors defined along some fiducial directions. In LQC one fixes the metric to be of the FRW or Bianchi type and quantizes the dynamical variables. However a direct derivation from LQG is still missing and it is difficult to accommodate in this setting inhomogenities because the theory is defined in the homogeneous reduced phase space.

 Inhomogeneous extension of the Bianchi  models:
 We want to define a new model for cosmology able to retain all the nice features of LQG, in particular a sort of background independence by which the regularization of the Hamiltonian constraint can be carried on as in the full theory. In this respect, we consider the simplest Bianchi model, the type I (a homogeneous but anisotropic space-time), and we define an inhomogeneous extension characterized by scale factors that depend on space. This inhonomogeneous extension contains as a limiting case the homogeneous phase in an arbitrary parameterization. The virtue of these models is that they are invariant under what we called a reduced-diffeomorphism invariance, which is the invariance under a restricted class of diffeomorphisms preserving the fiducial directions of the anisotropies of the Bianchi I model. This is precisely the kind of symmetry we were looking for! In fact, once quantum states are based on reduced graphs, whose edges are along the fiducial directions, we can define some reduced s-knots, which will be insensitive to the length of any cubulation of the spatial manifold (we speak of a cubulation because reduced graphs admit only cubulations and not triangulations). Therefore, all we have to do is to repeat Thiemann's construction for a cubulation rather than for a triangulation. But does it give a good expression for the Hamiltonian constraint?? The answer is no and the reason is that there is an additional symmetry in the reduced phase space that prevents us from repeating the construction used by Thiemann for the Hamiltonian constraint. Henceforth, the dynamical issue cannot be addressed by standard LQG techniques in reduced quantization.

 Quantum-Reduced Loop Gravity

 What are we missing in reduced quantization? The idea is that we have reduced the gauge symmetry too much and that is what prevents us from constructing the Hamiltonian. We therefore go back and do not reduce the symmetry and proceed to quantize first. We then impose the reduction of the symmetry at a quantum level.  Hence, the classical expression of the Hamiltonian  constraint for the Bianchi I model can be quantized according with the Thiemann procedure. Moreover, the associated matrix elements can be analytically computed because the volume operator takes a simplified form in the new Hilbert space. Therefore, we have a quantum description for the inhomogeneous Bianchi I model in which all the techniques of LQG can be applied and all the computations can be carried on analytically. This means that for the first time we have a model in which we can explicitly test numerous aspects of loop quantization: Thiemann's original graph changing Hamiltonian, the master constraint program, Algebraic Quantization or the new deparameterized approach with matter fields can all be tested. Such a model is a cuboidal lattice, whose edges are endowed with  quantum numbers and with some reduced relations between those numbers at vertices. In two words we have a sort of hybrid “LQC” along the edges with LQG relationships at the nodes, but with a graph structure and diagonal volume! This means that we have an analytically tractable model closer to LQG than LQC and potentially able to threat inhomogeneities and anisotropies at once. Is this model meaningful? What we have to do now is “only” physics: as a first test try to work out the semiclassical limit. If this model will yield General Relativity in the classical regime, then we can proceed to compare its predictions with Loop Quantum Cosmology in the quantum regime, inserting matter fields and analyzing their role, discussing the behavior of inhomogeneities and so on.. We will see..