Need some intuition behind spontaneous dimensional reduction

Question

In quantum gravity there is the notion of spontaneous dimensional reduction. Namely, at small scales, the dimension of spacetime is less than the ostensible four.

In the causal dynamical triangulation program, this dimensional reduction can be seen by using a diffusion process on the spacetime manifolds. It can be shown for small diffusion times, spacetime has a lower (spectral) dimension than four.

I can't really gain any intuition what it means for a spacetime to behave like a lesser dimensional spacetime at small scales. Are there any simple analogues that could give some insight into this?

Answer 1

There are a few papers investigating a classical setting for a kind of spontaneous dimensional reduction, using constrained Hamiltonian mechanics (aka Dirac's formulation) for "degenerate" systems.

These are systems whose matrix of Poisson brackets of constraints does not satisfy the usual constant-rank condition everywhere. Instead, there are "degeneracy surfaces" where that rank decreases. This turns some erstwhile 2nd class constraints into 1st class ones, and thus decreases the number of local degrees of freedom per space point.

Therefore if time evolution leads you inside such a degeneracy surface, you've observed a dynamical reduction of the number of degrees of freedom.

This is discussed in detail in https://arxiv.org/abs/hep-th/0011231. It's unclear whether a scenario of spontaneous compactification has been realised for a gravity theory in this setting, but some followups do discuss higher-dimensional Chern-Simons-type gravity theories which generically do admit degeneracy surfaces in this sense.

Answer 2

One argument is due to entropy. For a renormalizable field theory the entropy as the function of energy should scale with the energy on the power of $(d-1)/d$, where $d$ is the dimension of spacetime. On the other hand, according to the Beckenstein - Hawking formula, in the high energy regime, around black holes or on the Planck scale, it scales as $(d-2)/(d-3)$. These two functions match only, if the dimension of spacetime is $d= 3/2$.

In asymptotic safety, in loop quantum gravity, in causal sets, in CDT, the dimension reduction was calculated / observed. In many theories it is thought/calculated to be 2 exactly, which is different than 1.5.

Actually in CDT, both values can be observed, close to the UV fixpont candidates. One is more consistent with 2 and one is with 1.5. Time will tell which is true.

It is important to mention that it may be that Lorentz invariance is violated on the shortest scales and time scales differently as space. If the spacetime exhibits a fractal behavior, then measuring the distances on smaller and smaller scales, you will get longer distances, similarly as how I can prove to you that the length of the shore around the British Island (circumference) is infinitely large. This would mean that quantum gravity is like in the Hořava-Lifshitz model.

An intuitive but mathematically not precise description:

Let's say that you have a test particle which ought to travel distance $X$ in timestep $T$. By increasing the resolution you get distance $X'$. Your particle has now $T'$ steps to move. Let's say the magnifying factor is 2.

The problem is here, that with twice as big resolution $X' > X*2$, thus making $T' = T*2$ and performing $T'$ steps in the new units, you won't reach $X'$. It is because you have to travel a longer distance, more steps.

As a test particle travels a smaller distance, it will measure a different value for the Hausdorff and spectral dimensions, an effectively smaller value.

Upon request I can provide some links to articles for further reading.