Causality and Quantum Spacetime
By Jerzy Jurkiewicz, Renate Loll, and Jan Ambjørn
Scientific American, June
Edited by Andy Ross
Quantum theory and Einstein's general theory of relativity are famously at
loggerheads. Physicists have long tried to reconcile them in a theory of
quantum gravity, with only limited success.
A new approach introduces
no exotic components but rather provides a novel way to apply existing laws
to individual motes of spacetime. The motes fall into place of their own
accord, like molecules in a crystal.
This approach shows how
four-dimensional spacetime as we know it can emerge dynamically from more
basic ingredients. It also suggests that spacetime shades from a smooth
arena to a funky fractal on small scales.
The Self-Organized de Sitter Universe
By J. Ambjørn, J.
Jurkiewicz, R. Loll
arXiv:0806.0397v1 [gr-qc] 2 Jun 2008
Edited by Andy Ross
We propose a theory of quantum gravity which formulates the quantum theory
as a nonperturbative path integral, where each spacetime history appears
with the weight exp (iS), with S the Einstein-Hilbert action of the
corresponding causal geometry. The path integral is diffeomorphism-invariant
(only geometries appear) and background-independent. The theory can be
investigated by computer simulations, which show that a de Sitter universe
emerges on large scales. This emergence is of an entropic, self-organizing
nature, with the weight of the Einstein-Hilbert action playing a minor role.
Also the quantum fluctuations around this de Sitter universe can be studied
quantitatively and remain small until one gets close to the Planck scale.
The structures found to describe Planck-scale gravity are reminiscent of
certain aspects of condensed-matter systems.
De Sitter Goes
Willem de Sitter did not realize in 1917 that his
new cosmological solution to Einstein’s field equations would one day become
an integral part of our description of the universe.
In a de Sitter
universe, the distance between any two points grows exponentially as proper
time advances, with the expansion rate determined by the size of the
cosmological constant. After the discovery of the accelerated expansion of
our universe, we believe that the vacuum solution describes its inescapable
fate in the far future, with all stars and galaxies apart from our own local
galaxy cluster gradually fading from view. Besides providing a description
of the universe at late times, de Sitter space also figures as a simplified
model of the very early universe, as it undergoes rapid inflation after the
It would have been impossible to anticipate that a de
Sitter universe would one day be reconstructed from nothing but quantum
A quantum ensemble of essentially structureless,
microscopic constituents, interacting according to simple local rules
dictated by gravity, causality and quantum theory, can produce a quantum
universe, which on large scales matches perfectly a classical 4D de Sitter
universe. The derivation of this result, obtained in the context of a
candidate theory for quantum gravity based on causal dynamical
triangulations, is remarkable for a number of reasons:
— It is background
independent: no preferred classical background metric is put into the
construction at any stage.
— It is nonperturbative: the path integral,
the sum over histories, is dominated by spacetimes that are highly singular
and nonclassical on short scales.
— It is minimalist: no new fundamental
objects or symmetry principles need to be postulated.
— It comes with a
reality check: the quantum superposition is not merely a formal quantity,
but can be evaluated explicitly with the help of Monte Carlo simulations.
— It is robust: many of the details of the intermediate regularization
needed to make the path integral mathematically well defined do not affect
the final result in the continuum limit.
Putting a New Spin
on Quantum Gravity
How does our derivation of a classical
limit from a nonperturbative model of quantum gravity succeed in producing a
ground state of quantum geometry, which is not only a classical spacetime on
large scales, but in the absence of matter a physically realistic solution
of the Einstein equations?
Can the underlying theory answer
longstanding questions in quantum gravity about the true degrees of freedom
of spacetime at the Planck scale, and whether a smooth, classical spacetime
can emerge from microscopic, wild quantum fluctuations?
our new formulation of quantum gravity yields important insights into how to
think about gravity in the regime of ultra-short distances, usually captured
by the heuristic notion of a spacetime foam.
A fruitful approach is
that of viewing quantum gravity through the eyes of a condensed-matter
theorist, while paying close attention to key features of classical general
relativity, like the need for coordinate invariance and a causal structure.
Think of quantum gravity as a strongly coupled system of a very
large number of microscopic constituents, which is largely inaccessible to
analytic methods. This is a common situation in many complex systems of
theoretical interest in physics, biology and elsewhere. Powerful
computational methods enable us to derive quantitative results. Their
application relies on an intermediate discretization of the space of
spacetime geometries, in the spirit of lattice spin systems or lattice QCD,
but one that is coordinate-free and uses dynamical instead of fixed
background lattices. If a well-defined continuum limit of the path integral
exists as the discretization cut-off (or lattice mesh) is sent to zero, it
will result in a fundamental theory valid on all scales.
DIY Quantum Universe
There are straightforward construction
rules for the spacetimes contributing to the regularized version of the path
— Represent them as inequivalent piecewise flat manifolds
(triangulations) with a global proper-time structure, glued from
four-dimensional triangular building blocks in a way that avoids causal
singularities, like those associated with topology change.
— Next, set
up a Monte Carlo simulation based on a Wick-rotated version of the path
integral and measure interesting quantum observables.
To verify that the
quantum superposition created by the computer behaves like a de Sitter
universe, first convince yourself that it behaves like a 4D entity on large
scales, then measure the expectation value of its spatial volume as a
function of time. One finds a universal curve, independent of the spacetime
volume. Translating this into a continuum language, and fixing one
undetermined constant, the ratio between the time coming from the discrete
triangulation and the proper time of the continuum formulation, this is seen
to fit the shape of the de Sitter spacetime almost perfectly. We substitute
it for t to give Euclidean de Sitter space (a four-sphere) matching the
computer simulations performed for the Wick-rotated, Euclideanized path
Complexity Versus Simplicity
miraculous emergence of a (semi-)classical solution from quantum theory can
be illustrated by comparing the relevant (Euclidean) actions. The bare
action of the path integral is a straightforward discretization of the
Einstein-Hilbert action and shares its unboundedness from below.
Despite the fact that our basic building blocks and interaction rules are
simple, it is quite impossible to determine their combined dynamics
Here we are dealing with a case of self-organization, a
process where a system of a large number of microscopic constituents with
certain properties and mutual interactions exhibits collective behavior
giving rise to a new, coherent structure on a macroscopic scale. In our
case, we recover a de Sitter universe, a maximally symmetric space, even
though no symmetry assumptions were put into the path integral and our
slicing of proper time might have broken spacetime covariance. There clearly
is much to be learned from this novel way of looking at quantum gravity.
The Universe from Scratch
By J. Ambjørn,
J. Jurkiewicz, R. Loll
arXiv:hep-th/0509010v3 14 Oct 2006
Edited by Andy Ross
A piece of empty space that seems completely smooth and structureless has an
intricate microstructure. The laws of quantum theory tell us that looking at
spacetime at ever smaller scales requires ever larger energies, and this
will alter spacetime itself by curving it. But we lack a theory of quantum
gravity to give us a detailed and quantitative description of the highly
curved and fluctuating geometry of spacetime at the Planck scale. This
article outlines the approach of causal dynamical triangulations and its
achievements so far.
Searching for the quanta of spacetime
believe that probing the structure of space and time at distances far below
those currently accessible by our most powerful accelerators would reveal a
rich geometric fabric, where spacetime itself never stands still but instead
fluctuates wildly. One of the biggest challenges for physicists today is to
identify these fundamental excitations of spacetime geometry and understand
how their interaction gives rise to macroscopic spacetime.
Contemporary physics offers two main reasons to expect that as we resolve
the fabric of spacetime with an imaginary microscope at ever smaller scales,
spacetime will turn from an immutable stage into the actor itself:
Heisenberg's uncertainty relations specify that probing spacetime at very
short distances is accompanied by large quantum fluctuations in energy and
momentum. The shorter the distance, the larger the energy-momentum
— Einstein's theory of general relativity predicts that
these energy fluctuations, like any form of energy, will deform the geometry
of the spacetime, imparting a curvature that is detectable through the
bending of light rays and particle trajectories.
Together, these ideas
lead to the prediction that the quantum structure of space and time at the
Planck scale must be highly curved and dynamical.
We aim to find a
consistent description of this dynamical microstructure within a theory of
quantum gravity that unifies quantum theory and general relativity. Our
research program investigates causal nonperturbative quantum gravity and has
the name causal dynamical triangulations (CDT).
Our approach has produced
a number of results that mark it as a serious contender for a theory of
quantum gravity. There is evidence that the theory has a good classical
limit. It reproduces Einstein's classical theory at sufficiently large
scales. When one zooms out from the scale of the quantum fluctuations, one
rediscovers the smooth 4D spacetime of general relativity. And there are
indications of what the quantum structure of spacetime may be at the Planck
Why quantum gravity is special
Quantum gravity describes the
dynamics of spacetime. The degrees of freedom of a spacetime in classical
general relativity can be described by the spacetime metric, which is a
local field variable that determines the values of distance and angle
measurements in spacetime, and hence how spacetime is bent and curved
locally. Classical spacetime is determined by solving the Einstein
equations, subject to boundary conditions and a particular mass
distribution. From a quantum gravity point of view, one would like to
formulate a quantum analog of Einstein's equations, with quantum spacetime
as a solution.
Quantum field theory describes the dynamics of
elementary particles and their interactions on a fixed spacetime background,
usually the flat 4D Minkowski space of special relativity. Since at short
distances the gravitational forces are so much weaker than the others, it is
usually an excellent approximation to treat the gravitational degrees of
freedom as frozen in and non-dynamical. The geometric structure of the
Minkowski metric is part of the immutable background structure for quantum
However, quantum gravity aims to explain physical
situations that cannot generally be described in terms of linear
fluctuations of the metric field around Minkowski space or some other fixed
background metric. We aim to describe empty spacetime at very short
distances of the order of the Planck scale, 10—35 m, and the extreme and
ultradense state of the very young universe.
In quantum gravity. one
has to modify standard quantization techniques that rely on the presence of
a fixed metric background structure. Gravity must ultimately be quantized in
a way that is independent of any particular background metric and does not
simply describe the dynamics of linear perturbations around some fixed
There is no experimental or observational data
to guide the search for the correct theory of quantum gravity. We take a
rather conservative approach and adapt a set of well known physical
principles and tools to the situation of a dynamical geometry. The
principles and tools are quantum-mechanical superposition, causality,
triangulation of geometry, and elements of the theory of critical phenomena.
There is still no theory of quantum gravity that is both reasonably
complete and internally consistent mathematically. We are still looking for
a theory that is sufficiently complete to make at least some predictions
about the quantum behavior of spacetime.
The dynamical principle
The most important theoretical tool in the CDT approach
is Feynman's principle of superposing quantum amplitudes, the famous path
integral, applied to spacetime geometries. Its basic idea is to obtain a
solution to the quantum dynamics of a physical system by taking a
superposition of all possible configurations of the system, where each
configuration contributes a complex weight exp(iS) to the path integral,
which depends on the classical action S, which in turn integrates the
system's Lagrangian L over a given time interval.
For the case of a
nonrelativistic particle moving in a potential, the configurations are
continuous trajectories x(t) describing the particle's position as a
function of time t, running from an initial ti to a final tf in an interval
tif. Superposing the associated quantum amplitudes exp(iS[x(t)]), one
obtains a solution to the Schrödinger equation of the particle. The
individual paths x(t) appearing in the path integral are mostly not
physically feasible trajectories, but virtual paths, or just curves one can
draw between fixed initial and final points xi and xf:
(1) G(xi, xf, tif)
= the sum or integral over paths from xi to xf of exp(iS[x(t)])
physics of the particle is encoded in the superposition of all these virtual
paths. To extract the physical properties, one evaluates suitable quantum
operators on the ensemble of paths contributing to the path integral (1).
For example, one may compute expectation values for the position or the
energy of the particle, together with their quantum fluctuations. The
propagator (1) allows us to retrieve the classical behavior of the particle
in a limit, but it describes the full quantum dynamics of the system.
Analogously, a path integral for gravity is a superposition of all
virtual paths our universe can follow as time unfolds. These paths are
simply the different configurations for the metric field variables. A single
path is now no longer an assignment of just three coordinates (x1, x2, x3)
to each moment t in time, but rather the assignment to each t of a whole
array of numbers (the components g(x1, x2, x3, t) of the metric tensor g(x))
for each spatial point (x1, x2, x3). This is because gravity is a field
theory with infinitely many degrees of freedom. The path integral for
gravity can be written as:
(2) G(gi, gf, tif) = the sum or integral over
spacetimes g from gi to gf of exp(iS[g(x, t)])
Here S is now the
classical gravitational action associated with a spacetime metric g, with
initial and final boundary conditions gi and gf separated by a time interval
As in the particle case, the individual spacetime configurations
interpolated between the initial and final spatial geometries are not all
feasible classical spacetimes, but are much more general objects. The path
integral (2) is a superposition of all possible ways to curve an empty
spacetime. The collective behavior of the virtual spacetimes contributing to
the gravitational propagator (2) should tell us what quantum spacetime is.
To extract this geometric information, we evaluate quantum operators on the
ensemble of geometries contributing to the path integral. Defining the
gravitational path integral and extracting physical information from it is
CDT gives a precise prescription of how the path
integral should be computed and how the class of virtual paths should be
chosen. In addition, it provides technical tools to extract information
about the quantum geometry by the principle of quantum superposition. The
prescription is novel in two main ways:
— It is nonperturbative, in the
sense that the integrated geometries can have very large curvature
fluctuations at very small scales and thus be arbitrarily far away from any
classical spacetime, so no particular spacetime geometry is distinguished at
— It constrains the causal structure of the integrated
geometries, in contrast to previous Euclidean path integral approaches to
Representing spacetime geometry in CDT
define the precise class of spacetime geometries, labeled by the metric
tensor g, over which we take the sum or integral. As elsewhere in quantum
field theory, unless one chooses a careful regularization for the path
integral, it will be wildly divergent and hence mathematically useless.
Regularizing means making the path integral finite by introducing certain
cutoff parameters for the contributing configurations. These parameters are
later removed in a controlled manner.
The regularized spacetimes we
use are called piecewise flat geometries. Recall that the dynamical degrees
of freedom of a geometry are the ways in which it is locally curved.
Piecewise flat geometries are spaces that are flat everywhere apart from
small subspaces where curvature is said to be concentrated. This discretizes
curvature and vastly reduces the different number of ways spacetime can be
curved. We use a triangulated space called a Regge geometry. It can be
thought of as a space glued together from elementary pieces called
simplexes, which are higher-dimensional generalizations of triangles. Each
simplex is flat by definition. Local curvature only appears along
lower-dimensional interfaces when they are glued together.
be visualized most easily in the 2D case. Take a set of identical little
flat equilateral triangles and start gluing them pairwise together at their
edges. Points where several edges meet are called vertexes. We can make a
piece of flat space by arranging the triangles in a regular pattern so that
exactly six triangles and edges meet at each vertex. But there are many more
ways to create curved spaces by the same gluing procedure. Whenever the
number of triangles meeting at a vertex is smaller or larger than six, this
vertex will have a positive or negative curvature. By curvature we mean the
intrinsic curvature of the 2D surface that can be detected from within the
The story in higher dimensions is the same, except that the
2D triangles (or 2-simplexes) are now other flat simplexes (3-simplexes in
3D, 4-simplexes in 4D, and so on). Generally, the simplexes in dimension d
are glued together pairwise along their (d — 1)-dimensional faces, and their
curvature is concentrated at the (d — 2)-dimensional intersections of these
The Regge calculus was originally designed to approximate
smooth classical spacetimes by such piecewise flat, triangulated spaces.
This is a useful way of describing a spacetime for two reasons:
— We can
characterize a finite piece of spacetime completely by the geodesic
invariant edge length of the simplexes and the way they are glued together.
— Because we need no coordinate system for the simplexes, this
formulation avoids the redundant coordinates of Einstein gravity described
in terms of field variables g(x).
The use of triangulated spacetimes
differs in classical and quantum applications. In the classical case, the
aim is to approximate a smooth spacetime as well as possible. This can be
achieved by choosing a sequence of triangulations, where in each step of the
sequence the triangulation is finer than before and therefore converges to
the smooth manifold in the limit. Such an approximation can be very good
when the edge lengths become much smaller than the scale at which the smooth
spacetime is curved.
By contrast, in the quantum case, the aim is to
represent the path integral as well as possible, or rather to define it,
since there is currently no other way to do the computation. Here the
integral does not represent a single classical geometry but a quantum
superposition, where in general the individual spacetimes are highly
There is no precise mathematical principle to
guide this construction. Although we hope that the path integral provides an
ergodic sampling of the space of geometries, in practice we are constrained
by the need to define and regularize the path integral mathematically and
obtain a sensible classical limit.
The short-distance cutoff a is an
important part of our regularization of the spacetime geometries in the
gravitational propagator. We take the limit as a tends to zero in our search
for the continuum limit of the path integral over the regularized
geometries. We need to do this to get a final theory that does not depend on
the arbitrary details that went into the regularized model, which was only
an intermediate step in the construction. The method of using lattice
spacing a and letting a tend to zero (while renormalizing the coupling
constants as functions of a) is borrowed from the theory of critical
phenomena and is intended to ensure that the end result does not depend on
the details of the regularization. Still, this does not guarantee that we
get a viable theory.
The ensemble of virtual spacetime geometries in CDT
Given the regularized triangulated geometries, we now need to decide what
ensemble of such objects to include in the sum over geometries in (2). Here
we invoke causality.
The integration is not performed over Lorentzian
spacetimes but over Euclidean spaces. Classically, Euclidean spacetimes are
bizarre and unphysical entities, in which moving back and forth in time is
just as easy as moving back and forth in space. The reason for using them
instead of Lorentzian spacetimes of the correct physical signature is mainly
technical: in the Euclidean case, the weights exp(iS) are no longer complex
but real numbers, which simplifies a discussion of the convergence
properties of the path integral, and also makes Monte Carlo simulations
possible. However, there is no obvious relation between nonperturbative path
integrals for Lorentzian and Euclidean geometries. Indeed, causal
dynamically triangulated gravity in dimensions 2, 3 and 4 provides hard
evidence that the two path integrals are inequivalent and have completely
We can now write a regularized version of the
gravitational propagator as:
(3) G(Ti, Tf, t) = the sum over all
triangulations T from Ti to Tf of exp(iS[T])
Here T denotes a
triangulated spacetime, glued from 4-simplexes, and Ti and Tf are the
spatially triangulated bounding geometries (glued from 3-simplexes).
The gravitational action for a piecewise flat spacetime T can be schematized
(4) S(T) = (a constant times the curvature of T) + (another constant
times the volume of T)
There is a prescription for computing the
curvature and volume of a given triangulation T in terms of the edge length
and how the simplexes are glued together. The coupling constant for the
curvature is (minus the inverse of) Newton's gravitational constant, and the
constant for the volume is the cosmological constant (which may account for
the dark energy in our universe).
All the simplexes used in DT are
equilateral, and the discrete summation (3) is over inequivalent ways of
gluing the simplexes together. We need a further restriction. Consider the
number of distinct gluings of N simplexes, for a particular set of gluing
rules. Clearly, this number will grow with N, but the important question is
whether it will grow exponentially as a function of N or
super-exponentially. In the latter case, the path integral would be too
divergent to lead to a fundamental theory of gravity.
For this reason,
we cannot include a sum over topologies in the path integral. We need to fix
the topology of the spacetimes in the summation. Typically, we choose a 4D
sphere or torus. In principle, summing over topologies is possible using the
path integral formulation, but this possibility is highly impractical. From
a Euclidean point of view, we see no further natural restrictions we can
impose on the geometries.
A direct analytical evaluation of the path
integral is formidably difficult. But statistical mechanics and the theory
of critical phenomena offer a set of powerful numerical tools. We adapt
these tools to the case where the individual configurations are curved
geometries rather than spin or field configurations on a fixed background
space or lattice. We use Monte Carlo methods to simulate the ensemble of
spacetimes underlying the path integral and generate a random walk in the
space of all configurations according to a probability distribution defined
by (3). Computationally, this procedure can only be implemented on a finite
space of geometric configurations, usually by performing the simulations on
the ensemble of triangulations of a fixed discrete volume N. One repeats the
simulation for various values of N and tries to extrapolate the results to
the limit as N approaches infinity. This technique is known as finite-size
If all goes well, the quantum superposition (3) of
geometries will reproduce a classical spacetime in the classical limit. But
at small scales, the geometry should show highly nonclassical behavior
dominated by quantum fluctuations.
It turns out that the quantum
geometry generated by Euclidean DT can be in either one of two phases.
Either the geometry is completely crumpled or it is totally polymerized,
that is, degenerated into thin branching threads. These structures persist
also at large scales, and as a result the DT path integral appears to have
no meaningful classical limit, and therefore is useless for a theory of
Causal DT gives 4D
The starting point of CDT was the
hypothesis that this failure may have to do with the unphysical Euclidean
nature of the construction, and that one may be able to do better by
encoding the causal structure of Lorentzian spacetimes explicitly in the
choice of simplexes and gluing rules. We first tested the viability of the
CDT causal quantization program in lower dimensions. Superpositions like (3)
were defined by considering spacetimes glued from 2D or 3D building blocks.
This gave toy models that shared some but not all properties of the full CDT
path integral. We studied these and showed that the causal, Lorentzian path
integral in all cases gave different results from the corresponding
Euclidean path integral.
We now describe the first encouraging result
for CDT. Previous quantization attempts had failed to generate a spacetime
that can be said to be 4D on sufficiently large scales.
It may come
as a surprise that a superposition of locally 4D geometries can give
anything that is not again 4D. After all, we obtained our geometric building
blocks by cutting out small pieces from a 4D flat space. However, Euclidean
dynamically triangulated models show that the dimension can comes out as
other than 4, and indeed this seems to happen generically. The crumpled and
polymeric phases of the Euclidean model mentioned above have Hausdorff
dimensions of infinity and 2 respectively.
Roughly speaking, the
Hausdorff dimension is obtained by comparing the typical linear size r of a
convex subspace of a given space (such as its diameter) with its volume
V(r). If the leading behavior is that V(r) scales with the d-th power of r,
the space is said to have the Hausdorff dimension d.
To obtain an
effectively infinite-dimensional space by gluing N simplexes with edge
length a, one can define a sequence of triangulations whose volume goes to
infinity as N approaches infinity by choosing a gluing for each fixed N such
that every simplex shares a given vertex. Thus, no matter how large N gets,
all the simplexes of the triangulated space crowd around a single point.
This is compatible with the gluing rules, but gives rise to a space whose
dimensionality diverges because its linear size always stays at the cutoff
Conversely, one can get an effectively 1D space by gluing
the 4D building blocks into a long thin tube. So as N approaches infinity
and a approaches zero, three directions stay at the cutoff length a, and the
geometry only grows along the fourth direction.
Such spaces with
exotic dimensionality are obtained as limiting cases of regular simplicial
manifolds. Whether the gravitational path integral is dominated by
geometries of this nature in the continuum limit is a dynamical question
that cannot be decided a priori.
The answer depends on the relative
weight of energy and entropy. This results from the Boltzmann weight of a
given geometry (which in turn is a function of the values of the coupling
constants) and the number of geometries with any given Boltzmann weight. An
exotic geometry may have a very large Boltzmann weight and be energetically
favored, but there may be few such objects in the ensemble relative to the
more normal geometries, such that their contribution vanishes in the limit.
As we have seen, in Euclidean DT models for quantum gravity, the state sums
are dominated by exotic geometries that are either maximally crumpled or
like branched polymers, depending on the coupling constants.
Dimensionality becomes a dynamical quantity because the nonperturbative
gravitational path integral contains highly nonclassical geometries that are
curved at the cutoff scale a. Geometries with such unruly short-scale
behavior can dominate the path integral as a approaches zero.
short-scale picture of geometry that arises in CDT is completely
non-classical. A piece of classical spacetime, no matter how curved it is,
will always start looking like a piece of flat spacetime when the observed
scale becomes much smaller than the characteristic scale of the curvature.
By contrast, a typical quantum spacetime generated by our nonperturbative
path integral construction will never begin to look flat, no matter how fine
the resolution. However, we still do not know which microstructures can be
generated by various prescriptions for setting up the gravitational path
We wish to recover classical geometry at sufficiently large
scales. First, the quantum geometry must be 4D at large distances. A path
integral that does not pass this test does not qualify as a theory of
quantum gravity. We want a path integral that allows large short-scale
fluctuations in curvature but still allows a reasonable classical limit.
The CDT approach is an example of such a geometry. In this approach, we
impose certain causal rules on the simplexes. The rules make explicit
reference to the Lorentzian structure of the individual geometries
contributing to the path integral. The new physical insight here is that
causality at sub-Planckian scales may be responsible for the 4D nature of
The causality conditions in the CDT approach say that
each spacetime appearing in the sum over geometries should be a geometric
object that can be obtained by evolving a purely spatial geometry in time,
without changing its spatial topology. An example of a forbidden spacetime
is one where an initially connected space splits into two or more
components, or conversely where several components of a space merge.
Spacetimes with wormholes are also forbidden in the sum over geometries.
These geometries are pathological from a classical point of view.
Imagine a 3D space that undergoes branching as time progresses. Initially
the space is in one piece, so that any point in the space can be reached
from any other point along a continuous path. At some moment in time, the
space splits into two parts, which then remain cut off from each other.
Classically, something goes wrong with the light cone structure. The
assignment of light cones to spacetime points cannot be smooth, since there
is at least one point in spacetime (the branching point) where it is
undefined which way a light ray from the past goes into the future. Since
the light cones define the causal structure of spacetime, this is an example
of a geometry where causality is violated.
The classical Einstein
equations cannot describe spacetimes with such changing topologies. The
absence of branching points (and merge points) from a Lorentzian geometry is
invariant under diffeomorphisms because different notions of time always
share the same overall arrow of time. To introduce branching points and
their cutoff regions, one would need to reverse the time flow in entire open
regions of spacetime, which cannot be done by an allowed coordinate
transformation. In the Euclidean theory, which has no distinguished arrow of
time, one cannot talk about the absence or presence of branching points in a
In CDT, we use the Lorentzian structure of
contributing geometries explicitly and exclude all spacetimes with topology
changes and therefore acausal behavior. This constraint on the path integral
histories cannot be derived from the classical considerations of causality.
The individual path integral geometries are not smooth classical objects, so
there is no obvious reason to forbid any particular quantum fluctuations of
the geometry, including those that include topology changes.
theoretically possible that a quantum superposition of acausal spacetimes
leads to a quantum spacetime where causality is somehow restored
dynamically, at least in the macroscopic limit. However, this is not
confirmed in the Euclidean version of DT, which has no causality
restrictions but is unable to reconstruct a 4D space. Conversely, the fact
that individual path integral geometries in CDT are causal is not sufficient
to guarantee that the quantum geometry it generates is too. This is still
What is the quantum spacetime generated by CDT?
emergence of classical geometry is an important test in quantum gravity. The
dimensionality of spacetime is only one of many quantum observables one may
use to check the ground state geometry generated by CDT at various length
Although CDT histories come with a notion of proper time,
they do not otherwise have a natural coordinate system. Even if we
introduced coordinate systems on the individual triangulated spacetimes,
there is no way to mark the same point simultaneously in all of them.
Individual points do not have any physical significance in empty space. We
are thus forced to phrase any question about local curvature properties,
say, in terms of quantities that are meaningful in the context of a
diffeomorphism-invariant theory. For example, we can do so in terms of
n-point correlation functions, where the location of each of the n points
has been averaged over spacetime.
The correlation function that has
been studied up to now in CDT measures the correlation between the volumes
V(t) of spatial slices (slices of constant time t) some fixed proper-time
distance td apart, that is, a suitably normalized version of the expectation
(5) <V(0) V(td)> = the sum over t from 0 to te of <V(t) V(t + td)>
Here the ensemble average is taken over simplicial spacetimes with time
extension te and with fixed 4-volume N. One piece of evidence that spacetime
is 4D at large distances is the following fact. In order to map the
expectation values given by (5) on top of each other for different values of
the spacetime volume N, the time distance has to be rescaled by a power p,
where p is the DH-th root of N, and DH is the cosmological Hausdorff
dimension. It turns out that DH = 4 within measuring accuracy. So what we
might call continuum time scales correctly with the spacetime volume. Such
scaling is not given a priori in the presence of large geometric quantum
fluctuations, even though the simplexes are 4D at the cutoff scale.
Another result about the large-scale geometry of the quantum spacetime
dynamically generated by CDT makes contact with quantum cosmology. Almost
every aspect of our current standard model of cosmology is based on a
radical truncation of Einstein's theory to a single global degree of
freedom, the scale factor a(t), which describes the scale of the universe as
a function of time t. This truncation is motivated by assuming that the
universe is homogeneous and isotropic at the largest scales. We can try to
extract information about the quantum behavior of the universe by quantizing
the classically truncated system.
With a quantum geometry
construction that is not truncated, we can ask what it predicts for the
dynamics of the scale factor. We can extract an effective action for the
scale factor from CDT by integrating out all other degrees of freedom in the
full quantum theory. The resulting action takes the same functional form as
the standard action of a minisuperspace cosmology for a closed universe, up
to an overall sign. The collective effect of the local gravitational
excitations seems to result in the same kind of contribution as that coming
from the scale factor itself, but with the opposite sign. The consequences
of this result for quantum cosmology are currently being explored.
However, computer simulations reveal that the semiclassical approximation is
no longer an adequate description of the observed behavior of the scale
factor when the latter becomes small. This is an indicator for new
quantum-gravitational effects appearing at short distances. This is where
new quantum physics will appear.
We also have some first insights
into the microstructure of quantum spacetime. The evidence comes from yet
another way of probing the effective dimensionality of spacetime. The idea
is to define a diffusion process (equivalently, a random walk) on the
triangulated geometries in the path integral over spacetimes, and to deduce
geometric information about the underlying quantum spacetime from the
behavior of the diffusion as a function of the diffusion time. The beauty of
this procedure is its wide applicability, since diffusion processes can be
defined not just on smooth manifolds but on much more general spaces, such
as our triangulations and even on fractal structures.
interested in the spectral dimension, which is the effective dimension of
the carrier space seen by the diffusion process. It can be extracted from
the return probability P(sigma), which measures the probability of a random
walk to have returned to its origin after diffusion time sigma (or sigma
evolution steps in a discrete implementation). For diffusion on a flat
d-dimensional manifold, we have an exact relation. For general spaces, we
define the spectral dimension to depend on sigma: small values of sigma
probe the small-distance properties of the underlying space, and large
values its large-distance geometry. The spectral dimension extracted for the
quantum geometry of CDT is a twofold average over the starting point of the
diffusion process (which is initially peaked at a given simplex) and over
all geometries contributing to the path integral.
show a scale dependence for the spacetime dimension. At large distances it
approaches the value 4 asymptotically. But as we probe the geometry at ever
shorter distances, this dimension decreases continuously to an extrapolated
value of 2 within measuring accuracy. Such a scale dependence has never
before been observed in statistical models of quantum gravity and is a clear
indication that spacetime behaves highly nonclassically at short distances
close to the Planck length. Further investigations suggest a fractal
Conclusions and outlook
This article has offered an
overview of some fundamental issues addressed by quantum gravity. It has
described our attempt to arrive at a consistent quantum theory of gravity by
the use of causal dynamical triangulations (CDT). This approach has yielded
a number of hard results concerning the emergence of classical geometry from
a Feynman-type superposition of spacetimes.
More features of the
classical theory still need to be established, such as the presence of
attractive gravitational forces obeying Newton's law. However, the new
physics lies beyond the classical approximation. Here the challenge is to
extract more detailed information about the short-scale structure of quantum
spacetime and to uncover physical consequences that may be detectable.
The paradigm of spacetime beginning to emerge from CDT is that of a
scale-invariant, fractal, and effectively lower-dimensional structure at the
Planck scale, which only at a larger scale acquires the well known classical
features of geometry.
Jan Ambjørn is a member of
the Royal Danish Academy and a professor at the Niels Bohr Institute in
Copenhagen and at Utrecht University in the Netherlands.
Jerzy Jurkiewicz is head of the department of the theory of complex
systems at the Institute of Physics at the Jagiellonian University in
Krakow. His many past positions include one at the Niels Bohr Institute in
Renate Loll is a professor at Utrecht University, where
she heads one of the largest groups for quantum gravity research in Europe.
Previously she worked at the Max Planck Institute for Gravitational Physics
in Golm, Germany, where she held a Heisenberg Fellowship.
AR Whew! This is wonderful. I
was thrilled by the causal dynamical approach when I first heard of it in
2004 and now I see more clearly why it's worth getting excited about. This
is the best news in quantum gravity since the early days of string theory.
Even better for me, because the constructive and dynamical approach appeals
to me philosophically.
Note added 2010-08-10:
Ambjørn et al. have reacted
to a recent proposal by Petr Horava to break the Lorentz symmetry to build a
new theory of quantum gravity.