Dimensional Reduction

Dimensionally reduced scientist.

“Science is the only news,” Steward Brand wrote. My reading of this sentence is that science, the exploration of nature and natural law, is the ultimate source of inspiration. Developing a model and studying its properties can be like discovering a new world, and the discoveries that are the most fascinating are the ones that are surprising and unintuitive.

Probability amplitudes and wavefunctions are examples of such surprising and unintuitive properties, examples that are now a century old and that have changed the way we think about the world. Holography is a more recent example. And, gathering momentum in the quantum gravity community right now, is dimensional reduction.

Dimensional reduction means that on short distances the dimension of space-time decreases. To quantify what this means one has to be very careful with defining “dimension.”

The way we normally think about the dimension of space is to picture how lines spread out from a point. How quickly the lines dilute into their environment tells us something about the spheres we can draw around the point. The dimension of these spheres can be used to define the “Hausdorff dimension” of a space. The faster the lines dilute with distance, the larger the Hausdorff dimension.

The notion of dimension that is relevant for the effect of dimensional reduction is not the Hausdorff dimension, but instead the “spectral dimension.” The spectral dimension can be found by first getting rid of the Lorentzian signature and going to Euclidean space. And then to watch a random walker who starts at one point, and measure the probability for him to return to that point. The smaller the average return probability, the higher the probability he’ll get lost, and the higher the number of dimensions. One can define the spectral dimension from the average return probability.

Normally, for a flat, classical space, both notions of dimension are identical. However, there have been several approaches toward quantum geometry that found that the spectral dimension at short distances goes down from four to two. The return probability for short walks is larger than expected. One says that the spectral dimension “runs”, meaning it depends on the distance at which space-time is probed.

Surprising. Unintuitive.

This strange behavior was first found in Causal Dynamical Triangulations (hep-th/0505113), where one does a numerical simulation of an actual random walk in Euclidean space. But in other approaches one does not need a numerical simulation; it is possible to study the spectral dimension analytically as follows.

The behavior of the random walk is governed by a differential equation, the diffusion equation, in which there enters the metric of the background space-time. In approaches to quantum gravity in which the metric is quantized, it is then the expectation value of the operator that the metric has become which enters the diffusion equation. From the diffusion equation one calculates the return probability for the random walk.

This way, one can then infer the spectral dimension also in Asymptotically Safe Gravity (hep-th/0508202). Interestingly, one finds the same drop from four to two spectral dimensions. Yet another indication comes from Loop Quantum Gravity, where the scaling of the area operator with length changes at short distances. It is somewhat questionable whether the notion of a metric makes sense at all in this regime, but if one nevertheless constructs the diffusion equation from this scaling, one again finds that the spectral dimension drops from four to two (0812.2214). And Horava-Lifshitz gravity is maybe the best studied case where one finds dimensional reduction (0902.3657).

Surprising. Unintuitive. It is difficult to interpret this behavior. Maybe a good way to picture it, as Calcagni, Eichhorn and Saueressig suggested, is to think of the quantum fluctuations of space-time hindering a particle’s random walk and slowing it down. It wouldn’t have to be that way. Quantum fluctuations could also be kicking the particle around wildly, thus increasing the spectral dimension rather than decreasing it. But that’s not what the theory tells us. One shouldn’t take this picture too seriously though, because we’re talking about a random walk in Euclidean space, so it’s not an actual physical process.

It seems strange that such entirely different approaches to quantum gravity would share a behavior like this. Maybe our theories are trying to teach us a lesson about a very general property of quantum space-time. But then again, the spectral dimension does not say all that much about the theory. There are many different types of random walks that give rise to the same spectral dimension. And while these different approaches to quantum gravity share the same scaling behavior for the spectral dimension, they differ in the type of random walk that produces this scaling (1304.7247).

So far, this is an entirely theoretical observation. It is interesting to speculate whether one can find experimental evidence for this scaling behavior. In fact, this recent paper by Amelino-Camelia et al aims to “explore the cosmological implications” of running spectral dimensions. At least that is what the first sentence of the abstract says. If you read the second sentence though you’ll notice that what they actually explore are modified dispersion relations. And while modified dispersion relations lead to a running spectral dimension, the opposite is not necessarily the case. But is there any better indication for a topic being hot than that people use it in the first sentence of an abstract to draw the readers interest?