2D on the planck scale ?

Carlip, a cosmologist at University of California, in his just submitted preprint argued that, the universe at the imaginably smallest scale is perhaps 2D, one being space while the other being time. He called this dimensional reduction. In his scenario, this reduction is random and non-systematic, and thus indicating Lorentz invariance violation: going in different direction encounters distinct physics. He called for further investigations. Here I give a roster of some key concepts:

(1)Definition of dimension by random walk: "In particular, the return probability K(x, x, s) is
K(x, x; s) ∼ (4πs)−dS/2." Here ds is just the dimension, and K(x,x,s) gives the return probability of a random walker in space s.

(2)Some claimed evidences:

• Causal Dynamical Triangulations;
• Renormalization Group Analysis;
• Loop quantum gravity;
• High temperature strings;
• Anisotropic scaling models

(3)Wheeler-DeWitt equation, which is the Schodinger equation for metric tensor.
(4)Strong coupling limit.

At much smaller scales, on the other hand, the proper description is far less obvious.
While clever experimentalists have managed to probe some features down to distances close to the Planck scale [2], for the most part we have neither direct observations nor a generally accepted theoretical framework for describing the very small-scale structure of spacetime. Indeed, it is not completely clear that “space” and “time” are even the appropriate categories for such a description. But while a complete quantum theory of gravity remains elusive, we do have fragments:
approximations, simple models, and pieces of what may eventually prove to be the correct theory. None of these fragments is reliable by itself, but when they agree with each other about some fundamental property of spacetime, we should consider the possibility that they are showing us something real. The thermodynamic properties of black holes, for example, appear so consistently that it is reasonable to suppose that they reflect an underlying statistical mechanics of quantum states. Over the past several years, evidence for another basic feature of small-scale spacetime has been accumulating: it is becoming increasingly plausible that spacetime near the Planck
scale is effectively two-dimensional. No single piece of evidence for this behavior is in itself very convincing, and most of the results are fairly new and tentative. But we now have hints from a number of independent calculations, based on different approaches to quantum gravity, that all point in the same direction. Here, I will summarize these clues, provide a further piece of evidence in the form of a strong-coupling approximation to the Wheeler- DeWitt equation, and discuss some possible implications.