Spectral dimension of quantum geometries

Johannes Thürigen, Albert Einstein Institute 
Title: Spectral dimension of quantum geometries 
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By Francesco Caravelli, University College London

One of the fundamental goals of quantum gravity is understanding the structure of space-time at very short distances, together with predicting physical and observable effects of having a quantum geometry. This is not easy. Since the introduction of fractal dimension in Quantum Gravity, and its importance emphasized in the work done in Causal Dynamical Triangulations (Loll et al. 2005) and Asymptotic Safety (Lauscher et al. 2005), it has become more and more clear that space-time, at the quantum level, might have a radical transformation: the number of effective dimensions might change with the energy of the process involved. Various approaches to Quantum Gravity have collected evidences of a dimensional flow at high energies, which was popularized by Carlip as Spontaneous Dimensional Reduction (Carlip 2009, 2013). (The use of the term reduction is indeed a hint that a dimensional reduction is observed, but the evidences are far from conclusive. We find dimensional flow more appropriate.)

Before commenting on the results obtained by the authors of the paper discussed in the seminar
(Calcagni, Oriti, Thuerigen 2013), let us first step back for a second and spend some time introducing the concept of fractal dimension, which is relevant to this discussion.

The concept of non integer dimension was introduced by the mathematician Benoit Mandelbrot half a century ago. What is this fuss about fractals and complexity? What is the relation with spacetimes, quantum space-times?

Everything start from an apparently simple question asked by Mandelbrot: What is the length of the coast of England (or more precisely, Cornwall)? As it turned out, the length of the cost of England, depended on the lens used to magnify the map of coast, and depending on the magnifying power, the length changed with a well defined rule, known as scaling, which we will explain shortly.

There are several definitions of fractal dimension, but let us try to keep things as easy as possible, and see why a granular space-time might indeed imply a different dimensions at different scales (i.e., our magnifying power). The easy case is the one of a regular square lattice, which for the sake of clarity we consider infinite in any direction.

                                                          Source: Manny Lorenzo

 The dimension of the lattice might look two dimensional, as the lattice is planar: it can be embedded into a two dimensional surface (this is what is called embedding dimension). However, if we pick any point of this lattice, and count how many points are at a distance “d” from it, we will see that the number of points increases with a scaling law, given by*:

N ~ d^gamma .

If d is not too big, the value of gamma changes if the underlying structure is not a continuum, or is granular, and gamma can take non-integer values. This can be interpreted in various ways. For the case of fractals, this implies that the real dimension of fractals is not integer. Analogously to the case of the number of points within a certain distance d, it is possible to define a diffusion operation which will do the work of counting for us. However, the counting process depends on the operator which defines the diffusion process: how a swarm of particles move on the underlying discrete space. This is a crucial point of the procedure.

In the continuum, the technology is developed to the point that it can  to show that such an operator can be defined precisely**. The problem then is that the scaling not precise: for too long times, the scaling relation is not exact (as curvature effect might contribute). Thus, the time given to the particle to diffuse has to be appropriately tuned. This is what the authors define in Section 2 of the paper discussed in the talk and is a standard procedure in the context of the spectral dimension. Of course, what discussed insofar is valid for classical diffusion, but the operator can be defined for quantum diffusion as well, which is, put in simple terms, described by a Schroedinger unitary evolution like in ordinary quantum mechanics.

It is important to understand that the combinatorial description of a manifold (how these are represented in the discrete setting), rather than the actual geometry, plays a very relevant role. If you calculate the fractal dimension of these lattices, although at large scale they give the right fractal dimension, on small scale they do not. This shows that in fact discreteness does have an effect on the spectral dimension, and that results do indeed depend on the number of dimensions. But more importantly the authors observe that the spectral dimension, even in the classical case, depends on the precise structure of the underlying pseudo-manifold, i.e. how the manifold is discretized. If you combine this with the fact that insofar the fractal dimension is the global observable saying in how many dimensions you live in (concept very important for other high energy approaches), the interest might be quite well justified.

The case of a quantum geometry, considered using Loop Quantum Gravity (LQG), is then put forward at the end. The definition is different from the one given previously (Modesto 2009, assuming that the scaling is given by the area operator of LQG), and it leads to different results.

Without going into the details (described anyway quite clearly in the paper), probably it is noteworthy to anticipate the results and explain the difficulties involved in the calculation. The first complication comes from the calculation itself: it is in fact very hard to calculate the fractal dimension in the full quantum case. However, in the semiclassical approximation (when the geometry is in part quantum and in part classical), the main "quantum" part can be neglected. The next issue is that, in order to claim the emergence of a clear topological dimension, the fractal dimension has to be constant for a wide range of distances of several orders of magnitude. It is important to say that, if you use the fractal dimension as your definition of dimension, it is not possible to assign a given dimensionality unless the number of discrete points under consideration is large enough. This is a feature of the fractal dimension which is very important for Loop Quantum Gravity in many respects, as there as been for long time a discussion on what is the right description of classical and quantum spacetime. Still, this approach gives the possibility of a bottom-up definition of dimension (in the top-down, there would not be any dimensional flow).

As a closing remark, it is fair to say that this paper goes one step further into defining a notion of fractal dimension in Loop Quantum Gravity. The previous attempt was made by Modesto and collaborators using a rough approximation to the Laplacian. That approximation exhibited a dimensional flow towards an ultraviolet 2-dimensional space, which seems to be not present using a more elaborated Laplacian.

 *For a square lattice, if d is big enough, \gamma is equal to two: this is the Haussdorf dimension of the lattice, and indeed this dimension can be defined through the following equation: gamma=\partial log(N)/ \partial d

** Using the technical terminology, this is the Seeley-De Witt expansion of the heat kernel on curved manifolds. This is usually called spectral dimension. The first term of the expansion depends explicitly on the spectral dimension, while in the terms at higher orders there are also contributions from the curvature.