Dimension of Space-time
Bergfinnur Durhuus1 , Thordur Jonsson2 and John F Wheater3
Why quantum gravity?
For ninety years our understanding of gravitational physics has been based on the
general theory of relativity which accurately describes many phenomena occuring at
very different distance scales: from the gravitational red-shift of light observed in the
laboratory experiment of Pound and Rebka; through the precession of the perihelion
of Mercury; to the evolution of the universe and the cosmic microwave background.
More recently discrepancies have been found in large distance physics and at present
these are the subject of intense activity. We do not know for sure what will have to
be modified; whether general relativity itself or just the parameters of the theory. On
the other hand if general relativity is used to evolve our observations of the universe
back in time we reach the conclusion that there was a Big Bang and that in the era
immediately afterwards the universe was incredibly small – in fact much smaller than
an atom. This leads to another sort of problem; things the size of atoms cannot be
understood properly without using quantum mechanics so we also expect that the
extrapolation back to the Big Bang using general relativity, which is a classical field
theory, will fail at some point. In other words, in order to understand fully the Big
Bang we expect that it is necessary to have a theory of gravitation which is fully
From the theoretical point of view there are other motivations for pursuing a
quantum theory of gravity. Ever since classical electromagnetism and quantum mechanics were combined into quantum electrodynamics theorists have believed that
there must be some way of doing the same with the theory of gravity. That belief has
only been strengthened by the successes of electroweak theory and of QCD, the theory of the strong interactions. However gravity has proved extraordinarily resistant
to consistent quantization and even today, after many decades of effort, research on
the topic continues.
Models of quantum gravity
Many different ways of formulating quantum gravity have been investigated; some,
in particular string theory, have attracted more attention than others but none is
entirely satisfactory. This article describes some of the work being done by the ENRAGE Research Training Network within the discretised approach which is explained
Mathematics Institute, University of Copenhagen
Science Institute, University of Iceland
Department of Physics, University of Oxford
in more detail below . ENRAGE also works on causal set theory , and other
researchers study loop quantum gravity . We should note that the bibliography
in this article is neither intended to be exhaustive nor to record the historical development of the subject; many relevant references can be found in the bibliographies
of the works cited here.
In classical general relativity the dynamical degree of freedom is the metric tensor
gµν . The metric satisfies Einstein’s equations that relate it to the energy-momentum
tensor describing the matter which is the source of gravitation. So for any given
set of consistent initial conditions there is just one unique evolution of the metric
in time, just as a particle follows a particular path in classical physics. When the
particle motion is quantised the single classical trajectory from point A at t = 0 to
B at time t is replaced by an amplitude that the particle starting at A ends up at
B. This amplitude is given by a sum over the set P of all possible paths from (A, 0)
to (B, t)
hB, t| A, t = 0i =
where Sp is the action for the path p. Extending this naively to the gravitational
case we expect the metric to evolve from gµν
at t = 0 to gµν
at t with amplitude
hgB , t| gA , t = 0i =
where G is the set of all possible metrics satisfying g = gA at t = 0 and g = gB at
t, and Sg is the action (the natural choice for which is the Einstein action which in
two dimensions consists of the cosmological constant term alone). Note that we have
made a number of assumptions here concerning the consistent definition of t.
To evaluate the amplitude (2) requires a systematic way of describing the set G.
The discretized random surface is one way of doing this . For simplicity consider
a two-dimensional manifold with euclidean metric (so this is not really gravity which
should have a lorentzian metric) and spherical topology. Disregarding the metric for
a moment such a manifold can be triangulated with N ≥ 4 triangles; the idea is
that by taking N → ∞ in an appropriate way we can recover a continuum space.
The trick is then to associate a triangulation with a metric in the following way: we
suppose that all triangles are equilateral of side a and define the geodesic distance
between any two points as La where L is the number of links in the shortest path
connecting them. Then every distinct triangulation T leads to a distinct metric and
the vacuum amplitude is given by
where T is the set of all distinct triangulations of the sphere and we have now set
~ = 1. As well as the metric we can identify the Ricci scalar R in this construction.
It is simply proportional to the deficit angle for parallel transport around a closed
loop divided by a2 ; since all the triangles are equilateral the deficit angle at site i
is simply given by (6 − ni ) π3 where ni is the number of triangles which have i as a
vertex. Many objects of interest have been calculated in this particular model which
is often known as ‘two-dimensional euclidean quantum gravity’.
The idea can be extended to space-times of three and four dimensions and to
lorentzian space-times; unfortunately it becomes progressively more difficult with
increasing dimension to obtain analytic results in these models although much has
been learned by doing numerical simulations [4, 5]. Our interest is in obtaining
analytic results so the rest of this article concerns low dimensional models.
A moment’s thought will show that many of the triangulations contributing to T
are highly irregular. So although they are locally two-dimensional they might have
global properties that are quite different, as do fractal graphs, and on the average the
properties of T may show little resemblance to a smooth flat two-dimensional space.
These differences can be characterised to some extent through ‘dimensions’ which
probe different aspects of the space; usually these quantities are defined so that they
all give the same result for smooth flat spaces (eg 2 for R 2 ). We will be concerned
with just two of them: the Hausdorff dimension dh and the spectral dimension ds .
The Hausdorff dimension dh is a measure of the volume growth. Let V0 (r) be the
number of points within a distance r (defined as in the previous section) of the point
hV0 (r)iµ ∼ const r dh .
Note that this definition leads to dh = 1 for a line, dh = 2 for R2 etc. In the case
of the triangulations of the sphere it was proved by Chassaing and Schaeffer  and
by Angel and Schramm [7, 8] that dh = 4, a result that had been known, albeit less
rigorously, to physicists for many years . This value requires some explanation.
The point is that the structure of the typical graph with an infinite number of points
is highly branched – it consists of an infinite tube whose diameter goes roughly like
r 2 with many finite outgrowths branching out of it:
As was proved by Angel most of the volume is in the outgrowths and this is what
makes dh = 4. This branching structure rather suggests that it might be worthwhile
studying simpler sets of graphs with an infinite spine such as trees:
These sets of graphs are simple enough that, as well as dh , the spectral dimension
ds can also be computed exactly. They still have a metric defined by the geodesic
distance between points and they still have scalar curvature defined by the deficit
angle: vertices of order 3 have R < 0, vertices of order 2 have R = 0 and vertices of
order 1 have R > 0.
The spectral dimension ds is a measure of how likely it is that a random walk
returns to the point of origin. Walks on graphs are defined so that at each time
step the walker chooses to hop to one of the σ neighbouring points with uniform
probability σ −1 . After a long time t has elapsed the probability that the walker is
back at the starting point is given by
It is known that for spaces with scalar curvature R ≥ 0 everywhere ds = dh . So
for example we have the famous √1t return probability for random walk on a line.
However spaces with branching, such as combs and trees, necessarily have R < 0 at
the branching points and ds can be different from dh . The work we have been doing
recently has been to determine ds for some of these models with branching.
The spectral dimension of several ensembles of combs is investigated in detail in 
where proofs of all the statements made here can be found. In that paper we made a
particular point of finding methods that were as elementary as possible and avoided
the use of heavy mathematical machinery.
The important point to appreciate is that, from the quantum gravity motivation
where we started, we are not interested in the properties of any one graph but in the
average properties of the whole ensemble. The models are characterised by the assumption that the length ` of each tooth is distributed identically and independently
with probability π(`).
If π(`) = 0 for ` > `0 , all teeth are of finite length and it turns out that they
do not affect the large t behaviour so ds = 1. This can be understood by noting
that although there is negative curvature at the branching points the compensating
regions of positive curvature at the ends of the teeth are nearby; so averaging over
a finite region produces zero net curvature – on large enough scales the combs just
look like a line. At the other extreme π(∞) = 1; now all teeth are infinite and the
positive curvature is pushed off to infinity. A simple calculation shows that ds = 32 ;
clearly dh = 2 so we see the effect of the branching in reducing ds below dh . In fact
to get ds = 32 it does not matter that every tooth is infinite, just that teeth have a
finite probability of being infinite. To interpolate between these regimes it is useful
to consider π(`) = b`−a , a > 1; the smaller a the longer the typical tooth and so the
larger we might expect the shift away from ds = 1 to be. Our results are shown in
the following table:
Random tooth spacing
Random tooth length a ≥ 2
Random tooth length 1 < a < 2 3 − a
It turns out that the crucial point is whether h`iµ is finite; if it is, ie a > 2, then
the teeth are short enough that ds = 1, but if it diverges then ds depends on a. The
last column is shown in the table because it is known that ds ≥ d2d
for any graph
in much more general circumstances and therefore it ought to apply to the comb
ensemble, as indeed it does.
Earlier we suggested looking at trees as an example of spaces which share some of
the characteristics of the triangulations but which are substantially simpler. In fact
certain tree ensembles are particularly interesting to us because they occur naturally
when conformal matter fields of central charge c > 1 are coupled to the 2D euclidean
quantum gravity model. The back reaction is so strong that the space collapses to
one made up of tubes whose diameter is O(a) linked up in a tree-like structure:
The finite tube diameter can have no effect on the long time properties of a random
walk so we replace the tube structures by pure trees with a branching law for the
vertices inherited from the original structures. The ensemble of trees obtained in
this way is called the generic tree ensemble. It is defined by the condition that the
variance of the distribution of vertex order is finite; all such ensembles have the same
dh and ds regardless of the details of the distribution. The typical infinite tree consists
of an infinite spine with finite trees branching off [10, 13]. So these trees have similar
properties to the triangulations but are much easier to analyse. It has been known
for a long time that they have dh = 2 and a simple proof that the ensemble average
has ds = 34 is given in  – note that this saturates the lower bound 2dh /(dh + 1).
The trees have much more negative curvature than the combs; almost every vertex
has negative curvature so perhaps it is not surprising that ds is as low as possible. For
some choices of the vertex distribution these trees can also be regarded as percolation
clusters at criticality in high dimension and in this context have been investigated
extensively by probability theorists who have proved the stronger result that almost
all members of these ensembles individually have ds = 43 [11, 14].
The actual results we have discussed apply mostly to toy problems rather than to the
quantum gravity models. So now the interesting question is whether the techniques
employed so far can be generalised to these. The main difference is that the graphs
involved now contain loops rather than just being pure branching objects; of course
the presence of loops per se has no effect on dh but the extra connectivity can
drastically affect the spectral dimension.
 J. Ambjørn, B. Durhuus and T. Jonsson, Quantum geometry: a statistical field
theory approach, Cambridge University Press, Cambridge (1997)
 F. Dowker, Causal sets and the deep structure of space-time, gr-qc/0508109
 L. Smolin, An invitation to loop quantum gravity, hep-th/0408048
 J. Ambjørn, J. Jurkiewicz and R. Loll, Spectral dimension of the universe,
Phys. Rev. Lett. 95 (2005) 171301, hep-th/0505113
 J. Ambjørn, J. Jurkiewicz and R. Loll, Reconstructing the universe, Phys. Rev. D
72 (2005) 064014, hep-th/0505154
 P. Chassaing and G. Schaeffer, Random planar lattices and integrated superbrownian excursion, Probability Theory and Related Fields, 128 (2004) 161-212,
 O. Angel and O. Schramm, Uniform infinite planar triangulations, Comm.
Math. Phys. 241 (2003) 191-213, math.PR/0207153
 O. Angel, Growth and percolation on the uniform infinite planar triangulation,
 J. Ambjørn and Y. Watabiki, Scaling in quantum gravity, Nucl. Phys. B445
 B. Durhuus, Probabilistic aspects of infinite trees and surfaces, Act. Phys. Pol.
34 (2003) 4795-4811
 M. T. Barlow and T. Kumagai, Random walk on the incipient infinite cluster
on trees, math.PR/0503118
 B. Durhuus, T. Jonsson and J. Wheater, Random walks on combs, J. Phys. A39
 B. Durhuus, T. Jonsson and J. Wheater, The spectral dimension of generic trees,
 M. T. Barlow, A. A. Járai, T. Kumagai and G. Slade, Random walk on
the incipient infinite cluster for oriented percolation in high dimensions,