Causality and Quantum Spacetime

By Jerzy Jurkiewicz, Renate Loll, and Jan Ambjørn
Scientific American, June 2008

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 Quantum

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 big bang.

It would have been impossible to anticipate that a de Sitter universe would one day be reconstructed from nothing but quantum fluctuations.

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?

We believe 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.

The DIY Quantum Universe

There are straightforward construction rules for the spacetimes contributing to the regularized version of the path integral:
— 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 integral.

Complexity Versus Simplicity

This 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 analytically.

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

We 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 uncertainty.

— 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 scale.

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 field theories.

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 background spacetime.

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 underlying CDT

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)])

The 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 tif.

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 very difficult.

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 the outset.

— It constrains the causal structure of the integrated geometries, in contrast to previous Euclidean path integral approaches to quantum gravity.

Representing spacetime geometry in CDT

We now 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.

This can 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 surface.

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 faces.

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 nonclassical objects.

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 different properties.

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 as:

(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 scaling.

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 quantum gravity.

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 length a.

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.

The 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 integral.

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 our universe.

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 coordinate-invariant way.

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.

It is 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 unknown.

What is the quantum spacetime generated by CDT?

The 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 scales.

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 value:

(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.

We are 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.

Our measurements 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 microstructure.

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 Copenhagen.

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.