|Vanishing dimensions. |
Source: arXiv:1406.2696 [gr-qc]
Some years ago, we discussed the “Evolving Dimensions”, a new concept in the area of physics beyond the standard model. The idea, put forward by Anchordoqui et al in 2010, is to make the dimensionality of space-time scale-dependent so that at high energies (small distances) there is only one spatial dimension and at small energies (large distances) the dimension is four, or possibly even higher. In between – in the energy regime that we deal with in everyday life and most of our experiments too – one finds the normal three spatial dimensions.The hope is that these evolving dimensions address the problem of quantizing gravity, since gravity in lower dimensions is easier to handle, and possibly the cosmological constant problem, since it is a long-distance modification that becomes relevant at low energies.
One of the motivations for the evolving dimensions is the finding that the spectral dimension decreases at high energies in various approaches to quantum gravity. Note however that the evolving dimensions deal with the actual space-time dimension, not the spectral dimension. This immediately brings up a problem that I talked about to Dejan Stojkovic, one of the authors of the original proposal, several times, the issue of Lorentz-invariance. The transition between different numbers of dimensions is conjectured to happen at certain energies: how is that statement made Lorentz-invariant?
The first time I heard about the evolving dimensions was in a talk by Greg Landsberg at our 2010 conference on Experimental Search for Quantum Gravity. I was impressed by this talk, impressed because he was discussing predictions of a model that didn’t exist. Instead of a model for the spacetime of the evolving dimensions, he had an image of yarn. The yarn, you see, is one-dimensional , but you can knit it to two-dimensional sheets, which you can then form to a three-dimensional ball, so in some sense the dimension of the yarn can evolve depending on how closely you look. It’s a nice image. It is also obviously not Lorentz-invariant. I was impressed by this talk because I’d never have the courage to give a talk based on a yarn image.
It was the early days of this model, a nice idea indeed, and I was curious to see how they would construct their space-time and how it would fare with Lorentz-invariance.
Well, they never constructed a space-time model. Greg seems not to have continued working on this, but Dejan is still on the topic. A recent paper with Niayesh Afshordi from Perimeter Institute still has the yarn in it. The evolving dimensions are now called vanishing dimensions, not sure why. Dejan also wrote a review on the topic, which appeared on the arxiv last week. More yarn in that.
In one of my conversations with Dejan I mentioned that the Causal Set approach makes use of a discrete yet Lorentz-invariant sprinkling, and I was wondering out aloud if one could employ this sprinkling to obtain Lorentz-invariant yarn. I thought about this for a bit but came to the conclusion that it can’t be done.
The Causal Set sprinkling is a random distribution of points in Minkowski space. It can be explicitly constructed and shown to be Lorentz-invariant on the average. It looks like this:
|Causal Set Sprinkling, Lorentz-invariant on the average. Top left: original sprinkling. Top right: zoom. Bottom left: Boost (note change in scale). Bottom right: zoom to same scale as top right. The points in the top right and bottom right images are randomly distributed in the same way. Image credits: David Rideout. [Source]|
The reason this discreteness is compatible with Lorentz-invariance is that the sprinkling makes use only of four-volumes and of points, both of which are Lorentz-invariant, as opposed to Lorentz-covariant. The former doesn’t change under boosts, the latter changes in a well-defined way. Causal Sets, as the name says, are sets. They are collections of points. They are not, I emphasize, graphs – the points are not connected. The set has an order relation (the causal order), but a priori there are no links between the points. You can construct paths on the sets, they are called “chains”, but these paths make use of an additional initial condition (eg an initial momentum) to find a nearest neighbor.
The reason that looking for the nearest neighbor doesn’t make much physical sense is that the distance to all points on the lightcone is zero. The nearest neighbor to any point is almost certainly (in the mathematical sense) infinitely far away and on the lightcone. You can use these neighbors to make the sprinkling into a graph. But now you have infinitely many links that are infinitely long and the whole thing becomes space-filling. That is Lorentz-invariant of course. It is also in no sensible meaning still one-dimensional on small scales. [Aside: I suspect that the space you get in this way is not locally identical to R^4, though I can’t quite put my finger on it, it doesn’t seem dense enough if that makes any sense? Physically this doesn’t make any difference though.]
So it pains me somewhat that the recent paper of Dejan and Niayesh tries to use the Causal Set sprinkling to save Lorentz-invariance:
“One may also interpret these instantaneous string intersections as a causal set sprinkling of space-time [...] suggesting a potential connection between causal set and string theory approaches to quantum gravity.”
This interpretation is almost certainly wrong. In fact, in the argument that their string-based picture is Lorentz-invariant they write:
“Therefore, on scales much bigger than the inverse density of the string network, but much smaller than the size of the system, we expect the Lorentz-invariant (3+1)-dimensional action to emerge.”Just that Lorentz-invariance which emerges at a certain system size is not Lorentz-invariant.
I must appear quite grumpy going about and picking on what is admittedly an interesting and very creative idea. I am annoyed because in my recent papers on space-time defects, I spent a considerable amount of time trying to figure out how to use the Causal Set sprinkling for something (the defects) that is not a point. The only way to make this work is to use additional information for a covariant (but not invariant) reference frame, as one does with the chains.
Needless to say, in none of the papers on the topic of evolving, vanishing, dimensions one finds an actual construction of the conjectured Lorentz-invariant random lattice. In the review, the explanation reads as follows: “One of the ways to evade strong Lorentz invariance violations is to have a random lattice (as in Fig 5), where Lorentz-invariance violations would be stochastic and would average to zero...” Here is Fig 5:
|Fig 5 from arXiv:1406.2696 [gr-qc]|
Unfortunately, the lattice in this proof by sketch is obviously not Lorentz-invariant – the spaces are all about the same size, which is a preferred size.
The recent paper of Dejan Stojkovic and Niayesh Afshordi attempts to construct a model for the space-time by giving the dimensions a temperature-dependend mass, so that, as temperatures drop, additional dimensions open up. This begs the question though, temperature of what? Such an approach might make sense maybe in the early universe, or when there is some plasma around, but a mean field approximation clearly does not make sense for the scattering of two asymptotically free states, which is one of the cases that the authors quote as a prediction. A highly energetic collision is supposed to take place in only two spatial dimensions, leading to a planar alignment.
Now, don’t get me wrong, I think that it is possible to make this scenario Lorentz-invariant, but not by appealing to a non-existent Lorentz-invariant random lattice. Instead, it should be possible to embed this idea into an effective field theory approach, some extension of asymptotically safe gravity, in which the relevant scale that is being tested then depends on the type of interaction. I do not know though in which sense these dimensions then still could be interpreted as space-time dimensions.
In any case, my summary of the recent papers is that, unsurprisingly, the issue with Lorentz-invariance has not been solved. I think the literature would really benefit from a proper no-go theorem proving what I have argued above, that there exist no random lattices that are Lorentz-invariant on the average. Or otherwise, show me a concrete example.
Bottomline: A set is not a graph. I claim that random graphs that are Lorentz-invariant on the average, and are not space-filling, don’t exist in (infinitely extended) Minkowski space. I challenge you to prove me wrong.