---
abstract: 'Quantum-gravity renders the space-time dimension to depend on the size of region; it monotonically increases with the size of region and asymptotically approaches four for large distances. This effect was discovered in 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 \[hep-th/0508202\]. However, along these approaches the interpretation and the physical meaning of the effective change of dimension at shorter scales is not clear. Without invoking particular models in this essay we show 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. In this way we derive a simple analytic expression of space-time dimension running, which implies the modification of Newton’s inverse square law in a perfect agreement with the modification coming from one-loop gravitational radiative corrections.'
address:
- 'Andronikashvili Institute of Physics, 6 Tamarashvili St., Tbilisi 0177, Georgia'
- |
    Faculty of Physics and Mathematics, Chavchavadze State University,\
    32 Chavchavadze Ave., Tbilisi 0179, Georgia
author:
- Michael Maziashvili
title: ' Quantum-gravitational running/reduction of space-time dimension '
---

Quantum gravity ,Effective dimension of Space-time.

04.60.-m ,

 {#section .unnumbered}

A new door for a versatile study of the quantum gravity phenomenology is open by a profound quantum gravity effect of space-time dimension running/reduction discovered recently in two different approaches to quantum gravity [@AJLLR]. Two important questions arise in this regard. First we have to understand what the physical meaning of this effect is, and second, we should find an expression of dimension running, which will have more or less universal form irrespective to the particular approaches to quantum gravity. Both of these questions are left open and call for attention. These are the questions we address in this essay. Before addressing the questions we should be clear about the concept of gravitational running of dimension. We start from four dimensional Minkowskian space-time in which the matter fields leave but gravity is switched off. When gravity enteres the game the efective dimension of space-time (what observer can measure) appears to depend on the size of region, it becomes smaller than four at small scales and monotonically increases with increasing the size of region [@AJLLR]. In order to get good understanding of this statement let us recall what the dimension is [@Falconer].

Let us consider a subset $\mathcal{F}$ of four dimensional Minkowskian space. To determine the dimension of $\mathcal{F}$ first we have to introduce Euclidean metric on it, that is, to consider it as a subset of Euclidean space $\mathbb{R}^4$. It is straitforward to Euclideanize the initial metric that makes $\mathcal{F}$ a subset of $\mathbb{R}^4$. Then let $l^4$ be a smallest box containing this set, $\mathcal{F} \subseteq l^4$. For estimating the dimension of $\mathcal{F}$ we have to cover it by $\epsilon^4$ cells and counting the minimal number of such cells, $N(\epsilon)$, we determine the dimension, $d \equiv \dim(\mathcal{F})$ as a limit $d = d(\epsilon \rightarrow 0)$, where $n^{d(\epsilon)} = N $ and $n = l/\epsilon$ [@Falconer]. This definition is referred to as a box-counting dimension and can be written in a more familiar form as $$d \,=\, \lim\limits_{\epsilon \rightarrow 0} {\ln N(\epsilon) \over
\ln {l \over \epsilon}}~.$$ Certainly, in the case when $\mathcal{F} = l^4$, by taking the limit $d(\epsilon \rightarrow 0)$ 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, $\mathcal{F} \subset
\mathbb{R}^4$, it automatically follows that its dimension can not be greater than $4$, $d \le 4$. We see that the volume of a fractal $\mathcal{F}$ uniformly filling the box $l^4$ is reduced in comparison with the four dimensional value $l^4$ $$\label{volumefluct} V(\mathcal{F}) \,=\, \lim\limits_{\epsilon \rightarrow 0} N(\epsilon)\epsilon^4
\,=\, \lim\limits_{\epsilon \rightarrow
0}n(\epsilon)^{d(\epsilon)}\epsilon^4 \,\leq \, l^4 ~.$$

So, in a generic mathematical treatment of dimension we measure a set in a way that ignores irregularities of size less than $\epsilon$, and then we see how this measurement behaves as $\epsilon \rightarrow 0$. 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. Such a description is motivated by the fact that in any real measurement we use a finite resolution $\epsilon$. In this case the dimension takes the form $(l/\epsilon)^{d(\epsilon)} = N(\epsilon) $, which after introducing $\varepsilon(\epsilon) = 4 - d(\epsilon)\,,~~\delta N =
n(\epsilon)^4 - N(\epsilon)$ can be written as $$\label{opdim} \varepsilon(\epsilon) \,=\, - {\ln
\left(1 - {\delta N(\epsilon) \over n(\epsilon)^4 } \right)\over
\ln n(\epsilon) } \, \approx \, {1 \over \ln n(\epsilon)}
\,{\delta N(\epsilon) \over n(\epsilon)^4}~.$$ However, it is widely held belief that none of these objects are actual fractals $-$ that their fractal features disappear if they are viewed at sufficiently small scales. But this naive expectation is impeded by quantum gravity. Quantum gravity strongly indicates the finite resolution of space-time. A glance at a recent quantum gravity literature obviously shows that the finite resolution of space-time is common for all approaches to quantum gravity: space-time uncertainty relations in string theory [@String1; @String2]; noncommutative space-time approach [@noncommutative]; loop quantum gravity [@Loop]; or space-time uncertainty relations coming from a simple [Gedankenexperiments]{} of space-time measurement [@Gedankenexperiments]. Well known entropy bounds emerging via the merging of quantum theory and general relativity also imply finite space-time resolution [@entropybounds]. 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 $l_P \simeq
10^{-33}$cm [@minimumlength].

So, in quantum gravity the resolution of space-time is set by the Planck length $\epsilon = l_P$. The local fluctuations $\sim l_P$ add up over the length scale $l$ to $\delta
l = (l_Pl)^{1/2}$ [@camelia; @mazia]. Respectively, for the region $l^4$ we get the volume fluctuation of the order $\delta V = \delta l^4$, or in other words, in quantum gravity we expect the Poison fluctuation of volume $l^4$ of the order $ \delta V =
(l^2/l_P^2)\, l_P^4 $ [@Sorkin]. In view of Eq.(\[volumefluct\]) one naturally finds that this volume fluctuation has to account for the reduction of dimension.[^1] Thus, using the Eq.(\[opdim\]) one gets $n = l/l_P\,,~\delta N = l^2/l_P^2$, $$\label{rundim} \varepsilon (l) \,=\,{1 \over \ln {l\over l_P}}\,\, 
{\delta V \over l^4}  \,=\, {1 \over \ln {l\over l_P}}\,
\left({l_P \over l}\right)^2~.$$ This equation (which was obtained on general grounds) gives the running of dimension with respect to the size of region $l$. Having found this equation, the question of its reliability naturally occurs. For this reason we consider the following particular example.

In the low energy regime ($\ll m_P$) general relativity can be successfully treated as an effective quantum field theory [@Donoghue]. 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 $\ll m_P$. 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 [@Khriplovich]

$$\label{oneloop}V(r) = -l_P^2{m_1\,m_2 \over r}\left[1 + \frac{41}{10\pi}  \left({l_P \over r}\right)^2 \right]
~.$$

Let us compare this result with the modification of the Newton’s law due to dimension running Eq.(\[rundim\]).

Modification of the Newton’s law due to dimension reduction can be estimated without too much trouble by writing it in the Planck units

$$-l_Pm_1\,m_2 \,{ l_P \over r} \equiv  -l_Pm_1\,m_2 \,{ 1 \over \xi} ~.$$ One easily finds

$${ 1 \over \xi^{1 - \varepsilon}}  = {e^{\varepsilon\ln\xi} \over \xi} =  {1 \over \xi} \left(1 +  \varepsilon\ln\xi +
\ldots~\right)~.$$ Substituting the Eq.(\[rundim\]) $$\varepsilon = {1 \over \xi^2\ln\xi} ~,$$ we get

$$V(r) = -l_P^2{m_1\,m_2 \over r}\left[1 +  \left({l_P \over r}\right)^2 \right] ~,$$ which appears to be in perfect agreement with the Eq.(\[oneloop\]).

 Acknowledgments  {#acknowledgments .unnumbered}
=================

Author acknowledges useful discussions with Zurab Silagadze. The work was supported in part by the *CRDF/GRDF* grant.

[00]{}

J. Ambjorn, J. Jurkiewicz and R. Loll, Phys. Rev. Lett.  [**95**]{}, 171301 (2005) \[arXiv: hep-th/0505113\];

O. Lauscher and M. Reuter, JHEP [**0510**]{}, 050 (2005) \[arXiv: hep-th/0508202\];

R. Loll, Class. Quant. Grav.  [**25**]{}, 114006 (2008) \[arXiv: 0711.0273 \[gr-qc\]\].

K. Falconer, [*Fractal geometry: Mathematical foundations and applications*]{} (second edition, Wiley, 2003).

G. Veneziano, Europhys. Lett. 2 (1986) 199;

D. J. Gross and P. F. Mende, Nucl. Phys. B303 (1988) 407;

D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B216 (1989) 41;

K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B234 (1990) 276;

R. Guida, K. Konishi and P. Provero, Mod. Phys. Lett. A6 (1991) 1487.

M. Li and T. Yoneya, Phys. Rev. Lett. 78 (1997) 1219, hep-th/9611072;

T. Yoneya, Prog. Theor. Phys. 103 (2000) 1081, hep-th/0004074;

T. Yoneya, Prog. Theor. Phys. Suppl. 171 (2007) 87.

D. Ahluwalia, Phys. Lett. B339 (1994) 301, gr-qc/9308007;

M. Maggiore, Phys. Lett. B304 (1993) 65, hep-th/9301067;

M. Maggiore, Phys. Rev. D49 (1994) 5182, hep-th/9305163;

M. Maggiore, Phys. Lett. B319 (1993) 83, hep-th/9309034;

S. Doplicher, K. Fredenhagen and J. E. Roberts, Commun. Math. Phys. 172 (1995) 187, hep-th/0303037;

N. Seiberg and E. Witten, JHEP 9909 (1999) 032, hep-th/9908142.

A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 21, (2004) R53;

C. Rovelli, *Quantum Gravity*, (Cambridge University Press, Cambridge 2004);

T. Thiemann, *Introduction to Modern Canonical Quantum General Relativity*, (Cambridge University Press, Cambridge, 2007).

C. Alden Mead, Phys. Rev. 135 (1964) B849;

C. Alden Mead, Phys. Rev. 143 (1966) 990;

T. Padmanabhan, Class. Quant. Grav. 4 (1987) L107;

G. Amelino-Camelia, Mod. Phys. Lett. A9 (1994) 3415, gr-qc/9603014;

F. Scardigli, Phys. Lett. B452 (1999) 39, hep-th/9904025;

R. Adler and D. Santiago, Mod. Phys. Lett. A14 (1999) 1371, gr-qc/9904026;

F. Károlyházy, Nuovo Cim. A42 (1966) 390;

F. Károlyházy, A. Frenkel and B. Lukács, in *Physics as Natural Philosophy* (Eds. A. Shimony and H. Feschbach, MIT Press, Cambridge, MA, 1982);

F. Károlyházy, A. Frenkel and B. Lukács, in *Quantum Concepts in Space and Time* (Eds. R. Penrose and C. J. Isham, Clarendon Press, Oxford, 1986);

Y. J. Ng and H. van Dam, Mod. Phys. Lett. A9 (1994) 335;

N. Sasakura, Prog. Theor. Phys. 102 (1999) 169, hep-th/9903146;

M. Maziashvili, Int. J. Mod. Phys. D16 (2007) 1531, gr-qc/0612110.

J. D. Bekenstein, Phys. Rev. D23 (1981) 287;

J. D. Bekenstein, Phys. Rev. D49 (1994) 1912, gr-qc/9307035;

R. Bousso, JHEP 9907 (1999) 004, hep-th/9905177;

E. E. Flanagan, D. Marolf and R. M. Wald, Phys. Rev. D62 (2000) 084035, hep-th/9908070;

R. Bousso, Rev. Mod. Phys. 74 (2002) 825, hep-th/0203101;

R. Bousso, E. E. Flanagan and D. Marolf, Phys. Rev. D68 (2003) 064001, hep-th/0305149.

L. J. Garay, Int. J. Mod. Phys. A10 (1995) 145, gr-qc/9403008.

G. Amelino-Camelia, Lect. Notes Phys. 541 (2000) 1, gr-qc/9910089.

M. Maziashvili, Int. J. Mod. Phys.  A [**23**]{}, 1747 (2008) \[arXiv: 0709.0898 \[gr-qc\]\]. R. D. Sorkin, Int. J. Theor. Phys.  [**36**]{}, 2759 (1997) \[arXiv: gr-qc/9706002\]; R. D. Sorkin, AIP Conf. Proc.  [**957**]{}, 142 (2007) \[arXiv: 0710.1675 \[gr-qc\]\].

K. Svozil and A. Zeilinger, Int. J. Mod. Phys.  A [**1**]{}, 971 (1986); A. Zeilinger and K. Svozil, Phys. Rev. Lett.  [**54**]{}, 2553 (1985).

J. F. Donoghue, Phys. Rev. Lett.  [**72**]{}, 2996 (1994) \[arXiv:gr-qc/9310024\]; Phys. Rev.  D [**50**]{}, 3874 (1994) \[arXiv:gr-qc/9405057\].

C. P. Burgess, Living Rev. Rel.  [**7**]{}, 5 (2004) \[arXiv:gr-qc/0311082\].

I. B. Khriplovich and G. G. Kirilin, J. Exp. Theor. Phys.  [**95**]{}, 981 (2002) \[Zh. Eksp. Teor. Fiz.  [**95**]{}, 1139 (2002)\] \[arXiv: gr-qc/0207118\];

N. E. J. Bjerrum-Bohr, J. F. Donoghue and B. R. Holstein, Phys. Rev.  D [**67**]{}, 084033 (2003) \[Erratum-ibid.  D [**71**]{}, 069903 (2005)\] \[arXiv: hep-th/0211072\];

A. Akhundov and A. Shiekh, Electron. J. Theor. Phys.  [**17**]{}, 1 (2008) \[arXiv: gr-qc/0611091\];

S. Faller, Phys. Rev.  D [**77**]{}, 124039 (2008) \[arXiv: 0708.1701 \[hep-th\]\];

I. B. Khriplovich and G. G. Kirilin, J. Exp. Theor. Phys.  [**98**]{}, 1063 (2004) \[Zh. Eksp. Teor. Fiz.  [**125**]{}, 1219 (2004)\] \[arXiv: gr-qc/0402018\].

[^1]: This suggestion has been made in [@mazia], though the rate of volume fluctuation was overestimated in this paper. Let us also notice that the necessity of operational definition of dimension because of quantum mechanical uncertainties (not quantum-gravitational !) was first stressed in [@SZ].
