Evolving Dimensions

That the space-time of Einstein's Special and General Relativity might not be fundamental plays a central rôle in our quest for quantum gravity. There are many possibilities how the fundamental structure of space-time may be different from the four-dimensional continuum; discretization and additional space-like dimensions are among those that have received the bulk of attention. No matter what the modification though, one has to make sure that deviations from the experimentally extremely well confirmed Standard Model of particle physics and General Relativity become important only at scales that we have not yet tested, typically at high energies or short distances.

The idea that space-time might not be higher-dimensional on short distances but instead be lower-dimensional has been around for some while, inspired by results from causal dynamical triangulation. In a paper last year, Anchordoqui et al proposed to examine the possibility of lower dimensionality at small distances for its phenomenology in their paper

Vanishing Dimensions and Planar Events at the LHC
Luis Anchordoqui, De Chang Dai, Malcolm Fairbairn, Greg Landsberg, Dejan Stojkovic
arXiv:1003.5914v2 [hep-ph]

Greg Landsberg gave a talk about this work on our last year's workshop on Experimental Search for Quantum Gravity (recording of the talk here). The basic idea is that the dimensionality of space changes with distance in such a way that it is 3-dimensional on scales we have tested it, lower dimensional on distances shorter than we have probed yet (about 1/1000 of a femtometer) and possibly higher-dimensional on distances larger than we can observe. The picture suggested is that of a (one-dimensional) string being knitted, and the knitted sheet (2-dimensional) being crumpled to a ball (3-dimensional). The authors dubbed this "evolving dimensionality." The merit of having a smaller number of space-like dimensions at small distances or high energies is that it improves the renormalizability of quantum field theories and esp. that of quantum gravity. (In contrast to additional dimensions which actually make the problem worse.)

The above paper as well as two recent follow-up papers, arXiv:1012.1870 [hep-ph] and arXiv:1102.3434 [gr-qc], looked at the phenomenological consequences of the evolving dimensions. Most interesting, they predict that at high energies the outgoing particles in scattering events should have an increased probability of being aligned in a plane. And the latest paper investigates the modification of the gravitational wave background. This modification is due to the early universe having been lower-dimensional if the idea is true, which would prohibit the propagation of gravitational waves. Both predictions are for all I know unique to this particular model.

But the question that springs to mind immediately is: What about Lorentz invariance? If one has a lower number of dimensions at short distances, these dimensions need to be oriented somehow relative to the four-dimensional continuum that must be reproduced at large distances. This orientation necessarily breaks Lorentz invariance. The problem is then that violations of Lorenz invariance are extremely tightly constrained already. I was thus curious to see how the model of evolving dimensions avoids these constraints.

The way this is achieved is that there is no model. Instead, it's in the authors words "not a concrete model, but rather a conceptual new paradigm." The papers offer pictures and analogies instead of a mathematical description of the new fundamental structure of space-time and the dynamics of quantum fields in it. The most recent paper addresses the issue of Lorenz invariance as follows:

"For random orientation of lower-dimensional planes/lines (see e.g. Fig. 2 ), violations of Lorentz invariance induced by the lattice become non-systematic, and thus evade strong limits put on theories with systematic violation of Lorentz invariance."

Unfortunately, this claim is not backed up by any argument and the figure does not represent a Lorentz-invariant random orientation. (The average spacings are approximately of the same size which is not boost invariant). From Causal Sets we know there are Lorentz-invariant 'sprinklings,' but these are sets of points and not distributions of planes. I also don't see from the picture if and how these planes end when they meet and it remains unclear how the length scales on which the dimensionality changes, supposedly a property of the space-time structure, is defined Lorenz-invariantly. Most problematic however is that the previous paper (arXiv:1012.1870) talked about the loss of energy into the background. This necessitates an interaction and that interaction should be described by an operator coupling the fields to the, oriented, background. I would then suspect this interaction falls among the already highly constrained Lorenz-invariance violations. It doesn't matter if these orientations average out on large distances if the effect that one looks for necessitates one is in a regime where one is sensitive to the distance it is not averaged out. This is very difficult to say though without a model.

However, in the recent paper on gravitational waves, one doesn't actually need Lorenz-invariance since one is concerned with cosmology and has a preferred frame - the restframe of the CMB - at hand anyway. So I wrote to one of the authors of the paper, Dejan Stojkovic from the University of Buffalo, who explained that they consider the model to be breaking boost-invariance but not rotational invariance. With that, the length scales on which dimensionality changes can be well defined without much effort. The question of Lorentz invariance violating operators however remains open. Dejan also readily admits that their new paradigm still needs work and explains how the first paper came about:

"I had this idea since 2003 while intensively working on higher dimensional theories. It crossed my mind that instead of making things more complicated at high energy (and hoping that the problems will miraculously disappear) we could instead make things less complicated - thus evolving dimensions (at short distances we have less dimensions, while at large distances we have more). However, I could not come up with a Lagrangian and would not dare to make it public.

Then at the meeting in Heidelberg, after diner and several beers, I told our friends, who intensively worked on extra dimensions, that the LHC is much more likely to find less rather than more dimensions, and after first 10 minutes of disbelief, they liked the idea and convinced me that in order to make a prediction rather than post-diction, the paper must go out NOW."

In summary: The idea of evolving dimensions is very interesting and makes predictions that are, for all I know, unique to this particular setting. At present it however lacks a mathematical model for the new fundamental structure and the dynamics of quantum fields in it.