Random versus holographic fluctuations of the background metric. I. (Cosmological) running of space-time dimension

A profound quantum-gravitational effect of space-time dimension running with respect to the size of space-time region has been discovered a few years ago through the numerical simulations of lattice quantum gravity in the framework of causal dynamical triangulation [hep-th/0505113] as well as in renormalization group approach to quantum gravity [hep-th/0508202]. Unfortunately, along these approaches the interpretation and the physical meaning of the effective change of dimension at shorter scales is not clear. The aim of this paper is twofold. First, we find that box-counting dimension in face of finite resolution of space-time (generally implied by quantum gravity) shows a simple way how both the qualitative and the quantitative features of this effect can be understood. Second, considering two most interesting cases of random and holographic fluctuations of the background space, we find that it is random fluctuations that gives running dimension resulting in modification of Newton's inverse square law in a perfect agreement with the modification coming from one-loop gravitational radiative corrections.


Introduction
Recently a profound quantum gravitational effect of dimension reduction of space -time near the Planck scale was discovered in the framework of two different approaches to quantum gravity [1].As thus far there is no final picture of quantum gravity and the assault on it goes along several ways, it is very desirable to derive this result from an underlaying principle that is common for all approaches.Most likely such a fundamental principle seems to be finite resolution of space -time, like quantum mechanics implies finite resolution of phase space.In this or another way, quantum gravity strongly indicates the finite resolution of space -time, that is, space -time uncertainty.Space -time uncertainty is common for all approaches to quantum gravity be it: space -time uncertainty relations in string theory [2,3]; noncommutative space -time approach [4]; loop quantum gravity [5]; or space-time uncertainty relations coming from a simple Gedankenexperiments of space -time measurement [6].Well known entropy bounds emerging via the merging of quantum theory and general relativity also imply finite space -time resolution [7].The combination of quantum theory and general relativity in one or another way manifests that the conventional notion of distance breaks down the latest at the Planck scale l P ≃ 10 −33 cm [8].Indeed, this statement can be understood in a very simple physical terms.(In what follows we will assume system of units = c = k B = 1).Namely, posing a question to what maximal precision can one mark a point in space by placing there a test particle, one notices that in the framework of quantum field theory the quantum takes up at least a volume, δx 3 , defined by its Compton wavelength δx 1/m.To not collapse into a black hole, general relativity insists the quantum on taking up a finite amount of room defined by its gravitational radius δx l 2 P m.Combining together both quantum mechanical and general relativistic requirements one finds δx max(m −1 , l 2 P m) .
From this equation one sees that a quantum occupies at least the volume ∼ l 3 P .Since our understanding of time is tightly related to the periodic motion along some length scale, this result implies in general an impossibility of spacetime distance measurement to a better accuracy than ∼ l P .Therefore, the point in space -time can not be marked (measured) to a better accuracy than ∼ l 4 P .It is tantamount to say that the space -time point undergoes fluctuations of the order of ∼ l 4 P , we refer the reader to a very readable paper of Alden Mead [8] for his discussion regarding the status of a fundamental (minimum) length l P , as this conceptual standpoint was unanimous in almost all subsequent papers albeit many authors apparently did not know this paper.Over the space -time region l 4 these local fluctuations add up in this or another way that results in four volume fluctuation of l 4 .In view of the fact how the local fluctuations of space -time add up over the macroscopic scale (l ≫ l P ), different scenarios come into play.Most interesting in quantum gravity are random and holographic fluctuations.From the very outset let us notice that the length scale l we are interested in is a horizon distance l H .If the local fluctuations, ł P , are of random nature then over the length scale l H they add up as δl H = (l H /l p ) 1/2 l P .In the holographic case, the local fluctuations, ł P , add up over the length scale l H in such a way to ensure the black hole entropy bound on the horizon region δl H = (l H /l P ) 1/3 l P .Throughout this paper we will consider these two cases separately.Taking note of finiteness of the space-time resolution in quantum gravity, one immediately faces the question what operational meaning can be given to the space -time dimension.The fundamental to the generalized mathematical treatment of dimension for a set under consideration is an idea of measurement at scale ǫ, for each ǫ we measure a set in a way that ignores irregularities of size less than ǫ, and we see how this measurement behaves as ǫ → 0. For more details we refer the reader to a very readable book of Falconer [9].As Falconer notices in the introduction of his book , "A glance at a recent physics literature shows the variety of natural objects that are described as fractals − cloud boundaries, topographical surfaces, coastlines, turbulence in fluids, and so on.None of these are actual fractals − their fractal features disappear if they are viewed at sufficiently small scales."However, this naive expectation is impeded by quantum gravity.

Box -counting dimension
Because of quantum gravity the dimension of space -time appears to depend on the size of region, it is somewhat smaller than 4 and monotonically increases with increasing of size of the region [1].We can account for this effect in a simple and physically clear way that allows us to write simple analytic expressions for space -time dimension running.In what follows we will use a box -counting dimension [9].Box -counting dimension is one of the most widely used dimension largely due to its ease of mathematical calculation and empirical estimation.A major disadvantage of the Hausdorff dimension [10] is that in many cases it is hard to calculate or to estimate by computational methods.Except of some "pathological" cases that have no physical interest, the Hausdorff dimension is equivalent to the boxcounting dimension [11].Let us consider a set F that is understood to be a subset of four dimensional Euclidean space R 4 , and let l 4 be a smallest box containing this set, F ⊆ l 4 .The mathematical concept of dimension tells us that for estimating the dimension of F we have to cover it by ǫ 4 cells and counting the minimal number of such cells, N(ǫ), we can determine the dimension, d ≡ dim(F ) as a limit d = d(ǫ → 0), where n d(ǫ) = N and n = l/ǫ.For more details see [9].This definition can be written in a more familiar form as Certainly, in the case when F = l 4 , by taking the limit d(ǫ → 0) we get the dimension to be 4. From the fact that we are talking about the dimension of a set embedded into the four dimensional space, F ⊂ R 4 , it automatically follows that its dimension can not be greater than 4, d ≤ 4. Hence, for a fractal F uniformly filling the box l 4 we have the reduction of its volume in comparison with the four dimensional value that would be Introducing δN = n(ǫ) 4 − N(ǫ), the reduction of dimension ε = 4 − d can be written as (2) Quantum gravity, whatever the particular approach is, shows up a finite space -time resolution.The local fluctuations, ǫ = l P , add up over the length scale l resulting in fluctuation δl(l).Respectively, for the region l 4 we have the deviation (fluctuation) from the four dimensional value of volume of the order δV = δl(l) 4 .One naturally finds that this fluctuation of volume has to account for the reduction of dimension1 .It is worth noticing that albeit locally (that is, at each point) the space -time undergoes fluctuations of the order ∼ l P , for the fluctuations add up over the length scale l to δl(l), the region l 4 effectively looks as being made of cells δl(l) 4 that immediately prompts the rate of volume fluctuation.

Dimension running/reduction of space -time in the case of random fluctuations
In the case when local fluctuations of space -time are of random nature we expect the Poison fluctuation of volume l 4 of the order δV = l 4 /l 4 P l 4 P [15].Simply, this value of δV can be understood in the operational sense that in measuring of volume l 4 with the precision l 4 P one naturally expects the error l 4 P to take on ± sign with equal probability at each step of measurement that leads to the summation of error with the factor l 4 /l 4 P .The same can be said in the way that the local fluctuations l P take on ± sign along the length scale l with equal likelihood that results in amplification factor √ l/l P over this length scale, see for a detailed discussion [13].Respectively, from Eq.( 2) one gets ( This equation gives the running of dimension with respect to the size of region l.

Dimension running/reduction of space -time in the case of holographic fluctuations
Considering a weakly gravitating system in asymptotically flat space-time, the Bekenstein entropy bound tells us that the maximum number of bits that can be stored inside the region l 3 with the energy E can not exceed [7] S E l .
We will typically ignore the numerical factors of order unity and will make an effort to keep the equations as simple as possible in order to not obscure the underlying physical concepts.Maximum number of bits is set respectively by E max ∼ l/l2 P above which the gravitational collapse of this energy into a black hole will take place Taking note of this fact, that the maximum amount of information available to an observer within the cosmological horizon is given by Eq.( 5) with l = l H , one finds the maximal space-time resolution over the horizon scale to be 2 Thus in the holographic case the four volume V = l 4 H undergoes fluctuation of the order δV = δl 4 H ≃ l 8/3 P l 4/3 H that with respect to the Eq.( 2) yields It is curious to notice that if one assumes the holographic fluctuations to pertain to the space only but not to the time, that is, if we use three volume instead of the four one in Eq.( 7), the dimension will coincide with ( 3).This convergence of results seems intriguing, so one could simply argue the use of three volume instead of the four one because the entropy bound Eq.( 4) has to do immediately with the spatial region, but it is certainly a bit subtle question needing further scrutiny.

QFT reasoning for understanding of space -time dimension reduction in light of quantum gravity
It is an old well known idea that the melding of quantum theory and gravity typically indicates the presence of an inherent UV cutoff.In view of the above discussion, the emergence of such an intrinsic UV scale can be understood in a simple physical way that the background metric fluctuations does not allow QFT to operate with a better precision than the background space resolution.That is, if we have a characteristic IR scale l, then the UV cutoff, Λ, is naturally bounded by the fluctuation δl(l), Λ 1/δl(l).In its turn, the presence of IR scale is well motivated by the existence of a cosmological horizon, l l H . Thus knowing a particular IR scale, the presence of corresponding UV cutoff tells us that the Feynman diagrams pertaining to this theory become finite.This result in terms of dimensional regularization inevitably favors the dwindling of dimension.

Discussion
First of all let us discuss the validity region for Eqs.(3,7).From the above discussion one simply infers that the validity condition is ε ≪ 1.That is, the discussion is valid as long as four volume fluctuation δV satisfies δV ≪ V = l 4 .How far in the early cosmology can we use Eqs.(3,7) ?Say, for the length scale ∼ 1/10 16 GeV corresponding to the GUT, the ε random that is larger than ε holographic gives ε random ∼ 10 −6 , so that the validity condition is satisfied with good accuracy.Recalling that the inflation energy scale, E in f lation , is bounded from above by (non) observation of tensor fluctuations of the cosmic microwave background radiation (relict gravitational wave background) [17], with the current limit being E in f lation 10 16 GeV [18], one infers that even during the inflation stage we can safely use the Eqs.(3,7).
It is important to decide between the Eqs.(3,7) for both of them can not be true.In the low energy regime (≪ m P ) general relativity can be successfully treated as an effective quantum field theory [19].So that it is possible to unambiguously compute quantum effects due to graviton loops, as long as the momentum of the particles in the loops is cut off at some scale ≪ m P .The results are independent of the structure of any ultraviolet completion, and therefore constitute genuine low energy predictions of any quantum theory of gravity.Following this way of reasoning it has been possible to compute one-loop quantum correction to the Newtonian potential [20] V Let us compare this result with the modification of the Newton's law due to dimension running Eqs.(3,7).Modification of the Newton's law due to dimension reduction can be estimated without too much trouble by writing it in the Planck units One easily finds Substituting the Eq.( 3) which appears to be in perfect agreement with the Eq.( 8).While the Eq.( 7) gives obviously incorrect result.Thus the model of random fluctuations appears to be favored over the holographic one.It is interesting to notice that dark energy models based on the space-time uncertainty relations also manifest the random fluctuations to be more likely [12].Curiously enough, the uncertainty relation δl = (l P l) 1/2 is favored over the Károlyházy uncertainty relation, δl = l 2/3 P l 1/3 , even in the framework of Gedankenexperiments for space-time measurement [13].It is somewhat disappointing that hitherto we do not know how to work at a fundamental level with the theories having dynamical dimension.Nevertheless, some attempts to study the cosmology with a variable space dimension have been already made in literature, see for instance [21].No doubt it would be very interesting to take a close look at the cosmology with the running dimension in order to identify the corresponding experimental signatures.Besides the early cosmology, for a phenomenological study of quantum -gravitational reduction/running of spacetime dimension, the QFT effects measured with a high precision call for attention for one can estimate in a systematic way the corresponding quantum corrections.Such an investigation for studying the influence of the dimension running on the running of gauge couplings has been done in our paper [22].In an upcoming paper [23] we studied the corrections to the hydrogen spectrum due to dimension reduction.A few experimental signatures of this kind can be found in [14,24].