Causal Dynamical Triangulations:
Gateway to Nonperturbative Quantum Gravity

J. Ambjørn$\,{}^{a,b}$ and R. Loll$\,{}^{a}$

a Institute for Mathematics, Astrophysics and Particle Physics, Radboud
University
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands.
email<EMAIL_ADDRESS>

b The Niels Bohr Institute, Copenhagen University
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark.
email<EMAIL_ADDRESS>

Abstract

A powerful strategy to treat quantum field theories beyond perturbation theory
is by putting them on a lattice. However, the dynamical and symmetry structure
of general relativity have for a long time stood in the way of a well-defined
lattice formulation of quantum gravity. These issues are resolved by using
Causal Dynamical Triangulations (CDT) to implement a nonperturbative,
background-independent path integral for Lorentzian quantum gravity on
dynamical lattices. We describe the essential ingredients of this formulation,
and how it has allowed us to move away from formal considerations in quantum
gravity to extracting quantitative results on the spectra of diffeomorphism-
invariant quantum observables, describing physics near the Planck scale. Key
results to date are the emergence of a de Sitter-like quantum universe and the
discovery of an anomalous spectral dimension at short distances.

###### Contents

  1. 1 Introduction
  2. 2 The need for a nonperturbative path integral
  3. 3 Simplicial building blocks and setting up the path integral
    1. 3.1 Euclidean case
    2. 3.2 Lorentzian case
    3. 3.3 Including matter fields
  4. 4 CDT as a lattice and continuum field theory
    1. 4.1 Phase diagram and phase transitions
    2. 4.2 The quantum de Sitter universe
    3. 4.3 The CDT effective action
  5. 5 Observables
    1. 5.1 Dimensions
    2. 5.2 Curvatures

##  1 Introduction

To outsiders and practitioners alike, quantum gravity can at times appear as a
subject whose aims, progression, relevance and perhaps very existence are
forever elusive. Our search for a fundamental theory of quantum gravity is
hampered by the extreme weakness of the gravitational interactions, which
implies that large quantum effects are naturally associated with extreme
energy scales. As a consequence, there are currently no experiments or
observations to guide our theory-building, or help us discriminate between
fruitful ideas and idle speculation.

Another sense in which gravity is fundamentally different is the fact that its
dynamical field is spacetime itself, in contrast with the nongravitational
fields of the standard model of particle physics, which propagate on a fixed,
inert background spacetime. This has long given rise to speculations that the
dynamical nature of spacetime in gravity may ultimately be incompatible with
the basic tenets of quantum field theory, and that the construction of a
theory of quantum gravity therefore requires radically new ingredients and
physical principles.

By contrast, the theory and methodology presented here, subsumed under the
name Causal Dynamical Triangulations or CDT for short, shows that there is a
conceptually much simpler way to understand quantum gravity beyond
perturbation theory. It uses purely quantum field-theoretic concepts and a
minimal set of ingredients and free parameters, but nevertheless produces
robust results that are neither trivial nor obvious. Its quantum dynamics is
defined in terms of the time-honoured gravitational path integral or sum over
histories

$Z=\\!\int\\!{\cal D}[g]\;\mathrm{e}^{\,iS^{\mathrm{grav}}[g]},$ (1)

where the superposition is taken over all spacetime geometries $[g]$, each
with a complex weight depending on its gravitational action
$S^{\mathrm{grav}}[g]$. As usual in quantum field theory, the formal continuum
path integral (1) is ill-defined and highly divergent. It needs nontrivial
input to transform it from a mere statement of intent to a mathematically
well-defined description from which physical predictions can be derived

The first key point regarding this ansatz is that CDT provides a precise
prescription for regularizing (1) in a manifestly diffeomorphism-invariant and
background-independent way, which also incorporates the Lorentzian signature
of spacetime. The spacetime configurations in the path integral are
regularized in terms of dynamical lattices made from small triangular
Minkowskian building blocks, from which the formulation takes its name. To
obtain a theory of quantum gravity one still needs to perform a continuum
limit, where the ultraviolet (short-distance) lattice regulator is removed and
coupling constants are suitably renormalized.

Figure 1: Quantum-fluctuating spacetime foam, from a 2D toy model of quantum
gravity. (Courtesy of T. Budd)

While the continuum limit of a two-dimensional toy version of the path
integral (1) can be found analytically [15], unsurprisingly for a strongly
interacting quantum field theory in four dimensions this is of course not
possible for quantum gravity proper. The second key point is that a
nonperturbative evaluation of the CDT-regularized path integral and of
suitable observables is nevertheless possible numerically, by using powerful
Monte Carlo methods. Although the idea of lattice quantum gravity as the
gravitational analogue of lattice QCD has been around for a long time, as
described in [24], due to the dynamical nature of spacetime geometry and its
invariance structure it has taken much longer to adapt the very successful
arsenal of lattice methods in quantum and statistical field theory to the case
of gravity, culminating in today’s lattice gravity à la CDT.

The ability to perform numerical experiments allows us to quantitatively
explore the dynamical content of the gravitational path integral near the
Planck scale. In the absence of actual experiments, this provides a unique and
valuable reality check on how gravity behaves on ultrashort length scales,
where spacetime is supposed to become a quantum spacetime foam (cf. Fig. 1).
It also serves as a concrete blueprint for the type of universal results we
can expect quantum gravity to deliver, given the nature of diffeomorphism-
invariant observables and the structural lack of detailed analytical control
in a nonperturbative Planckian regime. In what follows, we will highlight key
structural properties and results obtained from using Causal Dynamical
Triangulations; further technical details can be found in the major reviews by
[10, 25].

##  2 The need for a nonperturbative path integral

The action that gives rise to the classical Einstein equations,

$S[g_{\mu\nu},\phi]=\frac{1}{16\pi G}\int d^{4}x\sqrt{|\det
g|}\,(R-2\Lambda)+\int d^{4}x\sqrt{|\det g|}\ {\cal L}_{m}(g_{\mu\nu},\phi),$
(2)

consists of a gravitational part, depending only on the Lorentzian metric
field $g_{\mu\nu}$, the so-called Einstein-Hilbert action, and a matter part
with Lagrangian ${\cal L}_{m}$, which includes any matter fields $\phi$
coupled to gravity. In eq. (2), $G$ is the gravitational and $\Lambda$ the
cosmological coupling constant. With the standard choice of units $\hbar=c=1$,
Newton’s constant $G$ has negative mass dimension $-2$. This implies that a
quantum theory of gravity that is based on a perturbative expansion in $G$
around flat spacetime will not be a renormalizable quantum field theory.

To make sense of the gravitational path integral (1), one needs to define it
nonperturbatively. For nongravitational field theories on flat spacetime,
lattice quantum field theories provide such a definition. The steps are as
follows: Wick-rotate from flat Minkowskian spacetime to flat Euclidean space
by analytically continuing to imaginary time, which will usually result in a
well-defined Euclidean action. Replace the continuum by a lattice, such that
its coordinates $x$ are replaced by discrete lattice point $x_{i}$, and the
fields $\phi(x)$ are represented by lattice fields $\phi(x_{i})$. Then find a
suitable lattice action for the fields $\phi(x_{i})$ that optimally
approximates the continuum action. Starting from a lattice with a finite
number of lattice points, the lattice-regularized path integral will be a
finite-dimensional, well-defined integral. One can then study the limit in
which the number of lattice points goes to infinity and the distance between
lattice points to zero. If such a limit can be shown to exist, one has a
nonperturbative definition of the Euclidean path integral and thus of a
corresponding Euclidean quantum field theory. To get back to Minkowskian
spacetime one can appeal to the Osterwalder-Schrader reconstruction theorem,
which ensures that this continuation is possible under rather general
conditions [28]. Lattice field theories have been very useful in addressing
nonperturbative aspects of renormalizable quantum field theories. When it
comes to nonrenormalizable quantum field theories, they are virtually the only
tool we have at our disposal to try to define them in a rigorous way.

In view of this state of affairs, putting gravity on the lattice seems like an
obvious strategy to make sense of the gravitational path integral (1), which
involves an integration over geometries. However, one is immediately
confronted with a number of problems. The program outlined above fails already
at the first step: in the absence of a Minkowskian background geometry and a
preferred notion of time, there is no obvious analogue of the imaginary-time
prescription underlying the standard Wick rotation, or known way to associate
a general Lorentzian metric with a Riemannian one. A popular way to sidestep
the lack of a rotation to Euclidean signature is to start from a different set
of field configurations, namely the space of four-dimensional Riemannian
metrics $g^{\rm eu}_{\mu\nu}$. These are unphysical, in the sense of lacking a
causal structure or notion of time. The associated Euclidean path integral

$Z^{\rm eu}=\\!\int\\!{\cal D}[g^{\rm
eu}]\;\mathrm{e}^{\,-S^{\mathrm{grav}}[g^{\rm eu}]}$ (3)

defines what is usually called Euclidean quantum gravity, and has a priori no
relation with its Lorentzian counterpart (1). If we nevertheless use it as
input for the lattice theory, we immediately face the problem that the
Euclidean Einstein-Hilbert action $S^{grav}[g^{\rm eu}]$ is unbounded below,
compounding the ill-defined nature of (3). Setting this issue aside still
leaves the challenge of finding a lattice discretization which constitutes a
good approximation of the space of all geometries and properly takes the
diffeomorphism-invariance of gravity into account. Even if we manage to
appropriately represent the gravitational lattice fields and action, compute
the path integral and take the limit of infinite lattice size and vanishing
lattice spacing $a$, it is not obvious how to get back to a quantum field
theory of Lorentzian geometries without an Osterwalder-Schrader theorem.

Remarkably, the above steps can be accomplished when one starts from the
Lorentzian path integral (1), discretizes the space of geometries rather than
the metrics and rotates these to Euclidean signature, before performing the
path integral. This turns out to yield different results from putting
Euclidean quantum gravity on the lattice and has taken nonperturbative quantum
gravity to a new level, in developments that will be summarized below.

##  3 Simplicial building blocks and setting up the path integral

###  3.1 Euclidean case

To illustrate the key construction principles of CDT and how they address some
of the issues raised in Sec. 2, we first discuss the technically slightly
easier case of Euclidean dynamical triangulations, abbreviated DT or EDT, the
Euclidean-signature precursor of CDT lattice gravity. On the one hand, it
implements the idea of discretizing geometries rather than metrics, without
ever appealing to a coordinate system. On the other hand, it implements the
idea of random geometry, where the configuration space of the path integral
consists of geometries that can be assembled in all possible ways from a small
number of elementary building-block types, subject to a set of elementary
gluing rules [6].

The building blocks of EDT in $d\\!=\\!4$ are four-dimensional equilateral
flat simplices, whose flat interior geometry is determined completely by the
geodesic length of their one-dimensional links or edges. Path integral
configurations are obtained by gluing these four-simplices pairwise along
their three-dimensional faces, subject to simplicial-manifold conditions,
either with or without restrictions on the overall topology. A closed
equilateral triangulation $T$ is a particular case of a piecewise linear
manifold, and a Riemannian version of classical Regge calculus [29] can be
used to assign to it a simplicial approximation

$S^{\rm eu}(T)=-\kappa_{2}N_{2}(T)+\kappa_{4}N_{4}(T)$ (4)

of the Einstein-Hilbert action, where $N_{k}(T)$ denotes the number of
$k$-dimensional simplices in $T$. Assigning a physical, dimensionful length
$a$ to the links of the triangulations, the dimensionless lattice coupling
constants $\kappa_{2}$ and $\kappa_{4}$ in (4) can be expressed in terms of
$a$ and the previously introduced continuum couplings $G$ and $\Lambda$. The
exceedingly simple form of (4) comes from the fact that volumes in equilateral
simplicial manifolds come in discrete units, as do the deficit angles
associated with the two-dimensional subsimplices, which are a measure of
intrinsic curvature (cf. Sec. 5.2). The Euclidean path integral can then be
written as

$Z^{\rm eu}(\kappa_{2},\kappa_{4})=\sum_{T}\tfrac{1}{C(T)}\
\mbox{e}^{\,\kappa_{2}N_{2}(T)-\kappa_{4}N_{4}(T)}=\sum_{N_{2},N_{4}}{\cal
N}(N_{2},N_{4})\;\mbox{e}^{\,\kappa_{2}N_{2}-\kappa_{4}N_{4}},$ (5)

where ${\cal N}(N_{2},N_{4})$ is the number of triangulations containing
$N_{4}$ four-simplices and $N_{2}$ triangles, and $C(T)$ is the size of the
automorphism group of $T$. This illustrates that the partition function (5) is
purely combinatorial. If we restrict the discrete volume $N_{4}$ to be finite,
$Z^{\rm eu}(\kappa_{2},\kappa_{4})$ is well defined and finite, because there
is only a finite number of configurations contributing to the right-hand side.
If the restriction of finite $N_{4}$ is lifted and arbitrary topologies for
$T$ are allowed, one can show that the growth of ${\cal N}(N_{2},N_{4})$ with
$N_{4}$ is such that $Z^{\rm eu}(\kappa_{2},\kappa_{4})$ is ill-defined for
any choice of the coupling constants $\kappa_{2}$ and $\kappa_{4}$. One
therefore cannot study the limit as $N_{4}\\!\to\\!\infty$. The
superexponential growth of ${\cal N}(N_{2},N_{4})$ can be traced to the
proliferation of triangulations with a complicated topology. On the basis of
numerical evidence, it is believed that ${\cal N}(N_{2},N_{4})$ only grows
exponentially with $N_{4}$ if the topology of the piecewise linear manifolds
in the sum (5) is fixed.

Assuming that this conjecture is correct, let us from now on fix the topology
of the contributing geometries. Then there exists a so-called critical
coupling $\kappa_{4}^{c}(\kappa_{2})$ such that $Z^{\rm
eu}(\kappa_{2},\kappa_{4})$ is well-defined for
$\kappa_{4}\\!>\\!\kappa_{4}^{c}(\kappa_{2})$ and we can study the
large-$N_{4}$ limit for given $\kappa_{2}$ by fine-tuning $\kappa_{4}$ to
$\kappa_{4}^{c}(\kappa_{2})$ from above. Using the path integral $Z^{\rm
eu}(\kappa_{2},\kappa_{4})$ to calculate the average number $\langle
N_{4}\rangle$, we want to fine-tune in such a way that $\langle
V_{4}\rangle\propto\langle N_{4}\rangle a^{4}$ can be interpreted as the
average continuum volume of the universe as $a\to 0$.

Note that the choice of building blocks has tamed the problem of the unbounded
Euclidean action mentioned in Sec. 2. The latter can be traced to the
conformal mode of the metric, whose kinetic term enters the action with a
negative sign. It means that ever more rapid oscillations of this mode can
make the action arbitrarily large and negative. Using identical building
blocks imposes a uniform limit on how rapidly the geometry can oscillate, set
by the size of the simplicial building block, which implies that the
regularized path integral has no conformal divergence. An analysis of the
phase diagram of the system shows that the suppression of the unboundedness of
the action persists also in the continuum limit $N_{4}\\!\to\\!\infty$,
$a\\!\to\\!0$, as long as the bare coupling constant $\kappa_{4}$ is chosen
large enough, $\kappa_{4}>\kappa^{c}_{4}(\kappa_{2})$.

To summarize, the regularized Euclidean path integral (5) can be viewed as the
partition function of a statistical system of piecewise linear geometries
constructed from elementary, triangular building blocks, where the name
dynamical triangulations emphasizes the fact that unlike in standard lattice
field theory, the lattice is not fixed but itself plays the role of a
statistical field. One can then proceed to use methods from the theory of
critical phenomena to study the system’s infinite-volume limit. This can be
done analytically for the analogue in $d\\!=\\!2$ of the path integral (5),
reproducing the results of an evaluation of the Euclidean path integral in the
continuum (see [6] for a review).

However, it has not been possible to find interesting continuum limits of the
EDT path integral (5) in $d\\!=\\!4$. For small $\kappa_{2}$, typical
triangulations are completely crumpled, in the sense that their four-simplices
are clustered around a few lattice vertices of extremely large order, while
triangulations at large $\kappa_{2}$ degenerate into so-called branched
polymers or trees. A phase transition at $\kappa_{2}=\kappa_{2}^{c}$ separates
these two phase space regions of degenerate geometry, but unfortunately it is
a first-order phase transition. As a result, there is no smooth transition
between the two phases and no obvious candidate for taking a continuum limit.
Efforts to locate a second-order transition in Euclidean quantum gravity by
using a more complicated action than (4) are ongoing [16].

###  3.2 Lorentzian case

Quite apart from these unresolved issues of the nonperturbative Euclidean path
integral, our primary interest is of course physical, Lorentzian quantum
gravity, obtained from the original path integral (1) over Lorentzian
geometries. The method of causal dynamical triangulations employs a Lorentzian
analogue of the equilateral building blocks of EDT, while ensuring the
presence of a well-defined analytic continuation of the complex path integral
to a real partition function that can be investigated with the help of Monte
Carlo simulations. A choice of four-simplices which realizes these objectives
is shown in Fig. 2. The interior geometry of the simplices is that of flat
Minkowski space, and their edges come in two different types, namely space-
and timelike, with two different squared edge lengths

$\ell^{2}_{s}=a^{2},\qquad\ell^{2}_{t}=-\alpha a^{2},\quad\alpha>0,$ (6)

where $a$ again denotes the lattice spacing (and short-distance regulator) and
$\alpha$ is a constant. Unlike in the Riemannian situation of Sec. 3.1, each
four-simplex now has a well-defined causal (or lightcone) structure.

The other ingredient that must be specified to implement the regularized path
integral are the gluing rules for these building blocks. The gluing should
again yield a simplicial manifold. In addition, only faces of the same type,
with respect to the space- and timelike character of their edges, can be glued
together. The most important difference with respect to the Euclidean case is
that each triangulation must have a well-behaved causal structure, in the
sense of obeying a lattice version of global hyperbolicity. This means not
only a fixed topology of spacetime, but a topology that is the direct product
$M\\!=\\![0,1]\times\Sigma$ of a time direction and a fixed spatial topology
$\Sigma$. This is implemented by giving each four-dimensional triangulation
$T$ a sliced structure, such that $T$ is given by a sequence of three-
dimensional spatial triangulations $\Sigma({t_{i}})$, labelled sequentially by
an integer $t_{i}$ representing a discrete notion of proper time. A more
detailed discussion of the nature and physical interpretation of this notion
of time is given in [25].

Note furthermore that the presence of a notion of time is essential to proving
unitarity of the continuum theory, by showing that the transfer matrix with
respect to this time obeys reflection positivity. This is nontrivial to
achieve in lattice gravity and systems of random geometry in particular, as
reviewed by [24], but has been shown in CDT (see [12, 10] for definitions and
further discussion).

Figure 2: CDT building blocks of type (4,1) and (3,2) interpolating between
spatial slices $\Sigma(t_{i})$ and $\Sigma(t_{i+1})$; spacelike edges in blue
and timelike ones in red.

Note that a priori there are no hard physical arguments for what kind of
configurations should be included in the nonperturbative gravitational path
integral, and folklore from nongravitational theories may not be applicable.
The choice to impose strong conditions on the causal structure of individual
path integral histories is in the spirit of what has been suggested by [31] in
a continuum context. The relevant hurdles that must be cleared by any
alternative prescription is to show that the regularized path integral exists
and is well-defined, and that an appropriate continuum limit can be taken,
leading to interesting physical results and reproducing classical features in
an appropriate classical limit. Lattice gravity based on CDT passes these
nontrivial tests and at the same time is the only known cure to terminal
pathologies like the geometric crumpling and polymerization seen in EDT.

Each slice of constant integer time is a three-dimensional equilateral
Euclidean triangulation with edges of length $a$, in accordance with (6). The
spacetime between two subsequent slices $\Sigma(t_{i})$ and $\Sigma(t_{i+1})$
consists of a layer of four-simplices of type $(m,5\\!-\\!m)$, $m=1,\ldots,4$,
which are defined as sharing a $(m\\!-\\!1)$-simplex with $\Sigma(t_{i})$ and
a $(4\\!-\\!m)$-simplex with $\Sigma(t_{i+1})$. Fig. 2 shows the four-
simplices of type (4,1) and (3,2); those of type (1,4) and (2,3) can be
obtained by time reflection. As indicated, the links connecting neighbouring
spatial slices are timelike, while all others are spacelike.

The simplicial form $S(T)$ of the Einstein-Hilbert action obtained from Regge
calculus is again simple and linear in the counting variables
$N_{4}^{(m,5-m)}(T)$ of the various four-simplex types, as well as the number
$N_{0}(T)$ of vertices. For a closed $T$, whose time direction has been
compactified for simplicity, the action is

$S(T)=\tilde{k}_{0}N_{0}(T)+\tilde{k}_{1}\Big{(}N_{4}^{(1,4)}(T)+N_{4}^{(4,1)}(T)\Big{)}+\tilde{k}_{2}\Big{(}N_{4}^{(2,3)}(T)+N_{4}^{(3,2)}(T)\Big{)},$
(7)

where the coupling constants $\tilde{k}_{i}$ depend linearly on the lattice
couplings $\kappa_{2},\kappa_{4}$ introduced earlier, and nonlinearly on the
parameter $\alpha$ (see [12, 10] for details). The regularized version of the
Lorentzian path integral (1) now reads

$Z(\kappa_{2},\kappa_{4};\alpha)=\sum_{T}\tfrac{1}{C(T)}\ \mbox{e}^{\,iS(T)},$
(8)

where the summation is over causal triangulations assembled according to the
gluing rules discussed above. For a finite upper bound on the discrete volume
$N_{4}$, (8) is finite and well defined.

An added bonus of this construction is that for each Lorentzian triangulation
with length assignments (6) we can perform an analytic continuation in the
lower half of the complex $\alpha$-plane such that $\alpha$ changes from
positive to negative values. This turns all timelike edges into spacelike
ones, and the corresponding piecewise flat geometry becomes Euclidean. For a
certain range of $\alpha$, including $\alpha\\!=\\!1$, we obtain the usual
relation between the Euclidean and Lorentzian actions, namely

$S(T,-\alpha\\!-\\!i\epsilon)=:iS_{E}(T,\alpha),\qquad\ell^{2}_{t}=\alpha
a^{2}>0,$ (9)

where the symbol $S_{E}$ denotes the Euclidean action after the analytic
continuation. It implies that we can compute the path integral using the real
Boltzmann weights $\mbox{e}^{\,-S_{E}(T)}$ instead of the complex weights
$\mbox{e}^{\\.{i}S(T)}$, and after performing the sum make the analytic
continuation $-\alpha\to\alpha$ back to the Lorentzian path integral. Note
that the set of piecewise flat Euclidean geometries used in the construction
of this analytically continued path integral is a strict subset of the
Euclidean geometries considered in EDT, which will be seen to lead to
different and much more interesting results. To summarize,

$Z(\kappa_{2},\kappa_{4};\alpha)\to
Z(\kappa_{2},\kappa_{4};-\alpha)=Z_{E}(\kappa_{2},\kappa_{4};\alpha)=\sum_{T}\tfrac{1}{C(T)}\
\mbox{e}^{\,-S_{E}(T,\alpha)},$ (10)

where

$S_{E}(T,\alpha)=\hat{k}_{0}N_{0}(T)+\hat{k}_{1}\Big{(}N_{4}^{(1,4)}(T)+N_{4}^{(4,1)}(T)\Big{)}+\hat{k}_{2}\Big{(}N_{4}^{(2,3)}(T)+N_{4}^{(3,2)}(T)\Big{)},$
(11)

with $\hat{k}_{a}(\alpha)\\!:=\\!-i\tilde{k}_{a}(-\alpha)$. Like in the EDT
case discussed in Sec. 3.1, the gravitational path integral $Z_{E}$ has the
form of a statistical partition function of random geometries. A two-
dimensional version has been solved analytically by [15], leading to results
that are inequivalent to those of the Euclidean two-dimensional path integral,
with a different universal behaviour of its observables, and a natural
analytical continuation of the obtained continuum propagator back to
Lorentzian signature. Although these are mere toy models of the four-
dimensional theories, they highlight the inequivalence between Lorentzian and
Riemannian metric signature in nonperturbative quantum gravity.

The analysis of the four-dimensional Lorentzian path integral in terms of a
CDT lattice formulation and its results to date are the subject of the
following sections. A crucial tool for extracting quantitative results beyond
perturbation theory are Markov chain Monte Carlo methods. Their application to
systems of dynamical geometry presents specific challenges, whose resolution
has been described elsewhere, see e.g. [10].

###  3.3 Including matter fields

We have so far considered the lattice formulation of the gravitational,
Einstein-Hilbert part of the action (2), but the inclusion of matter in this
formalism is straightforward. To illustrate the point, let us discuss how to
incorporate a scalar field $\phi(x)$ with continuum (Euclidean) action

$S_{m}[\phi,g_{\mu\nu}]=\int_{M}d^{4}x\sqrt{\det
g}\;\Big{(}g^{\mu\nu}\partial_{\mu}\phi(x)\partial_{\nu}\phi(x)+V(\phi)\Big{)}.$
(12)

The simplest CDT lattice implementation of this action on a triangulation $T$
is obtained by placing the scalar fields at the vertices $x_{i}$ of the
lattice dual to $T$ (equivalently, at the centres of the four-simplices of
$T$) and using as lattice derivative the difference $\phi(x_{i})-\phi(x_{j})$
between scalar fields at neighbouring pairs of such vertices. For a given $T$
this leads to the lattice action

$S_{m}[\phi,T]=\sum_{(x_{i},x_{j})}\big{(}\phi(x_{i})-\phi(x_{j})\big{)}^{2}+\sum_{x_{i}}V(\phi(x_{i})),$
(13)

where $(x_{i},x_{j})$ denotes pairs of neighbouring vertices, i.e. links in
the dual lattice. To obtain a combined gravity-matter path integral, one
integrates for each triangulation $T$ over all matter field configurations
$\\{\phi(x_{i})\\}$, weighted by $\mbox{e}^{-S_{m}[\phi,T]}$, and then perform
the sum over all geometries. On both EDT and CDT lattices one can also
incorporate gauge fields [2, 18], as well as fermions. The latter can be
included because local frames can be introduced on individual four-simplices
in a straightforward way as shown explicitly by [11] in their analysis of
gravitational Wilson loops.

##  4 CDT as a lattice and continuum field theory

Having specified the ingredients which turn the formal Lorentzian path
integral (1) into a properly regularized expression, based on a statistical
system of piecewise linear geometries with a well-defined causal structure, we
can tap into standard concepts and techniques from the theory of critical
phenomena to understand how this lattice system can give rise to a continuum
field theory.

The first thing to stress is that the chosen building blocks do not have a
fundamental status, and do not define a fundamental physical length scale. On
the contrary, the link length $a$ plays the role of a quantum field-theoretic
UV cutoff. To define an interacting continuum theory of quantum gravity it
must be taken to zero. Taking a Wilsonian point of view on the renormalization
of lattice field theories, we should look for second-order phase transitions
of the lattice-regularized model. Such transitions are associated with
divergent correlation lengths of the lattice fields, which implies that the
physics related to these correlation lengths will exhibit universality: it
will to a large degree be independent of the arbitrary discretization choices
made when setting up the lattice theory, and many different lattice theories
can lead to the same continuum quantum field theory, if it can be shown to
exist.

Explicit universality has been observed in two-dimensional toy versions of the
path integral, and is expected to be present in four dimensions also. The two-
dimensional models can be solved analytically in either signature, as has
already been mentioned, and provide beautiful illustrations of Wilson’s ideas.
For example, one can vastly extend the space of configurations and coupling
constants by including not only triangles, but also squares, pentagons, etc.
with different weights. The enlarged phase space contains a critical surface
of finite co-dimension where one can take the cutoff $a\\!\to\\!0$ such that a
suitable correlation length diverges, leading to what one can view as
continuum theories of two-dimensional quantum gravity, see [6, 1] for details.
The important message from these exactly soluble examples is that the lattice
formalism of dynamical triangulations provides a natural regularization of
diffeomorphism-invariant, backgound-independent quantum theories of gravity
and geometry, without any need to question the foundational principles of
quantum field theory.

###  4.1 Phase diagram and phase transitions

Turning next to the evaluation and analysis of the gravitational path integral
formulated in terms of CDT, we follow the strategy of looking for second-order
phase transitions in the phase space spanned by the bare coupling constants.
The Wick-rotated action $S_{E}$ in (10) and (11) depends on the three
parameters $\kappa_{2},\kappa_{4}$ and $\alpha$. One can debate to what extent
$\alpha$ can be viewed as a coupling constant, since it was introduced merely
as a finite scale-factor between space- and timelike link lengths. However,
the results reported below suggest that in a Wilsonian spirit it is more
appropriate to view the resulting $\alpha$-dependence as a generalization of
the Regge action for $\alpha\\!=\\!1$, which classically is an equally valid
discretization of the Einstein-Hilbert action. The conventional
parametrization of the action (11) is given by

$S_{E}(T;k_{0},k_{4},\Delta)=-\big{(}k_{0}\,\\!+\\!\,6\Delta\big{)}N_{0}(T)+k_{4}N_{4}(T)+\Delta\big{(}N_{4}^{(4,1)}(T)\\!+\\!N_{4}^{(1,4)}(T)\big{)},$
(14)

where

$N_{4}(T):=N_{4}^{(1,4)}(T)\\!+\\!N_{4}^{(4,1)}(T)\\!+\\!N_{4}^{(3,2)}(T)\\!+\\!N_{4}^{(2,3)}(T)$
(15)

and the coupling constants $k_{0}$, $k_{4}$ and $\Delta$ are specific
functions of $\kappa_{2},\kappa_{4}$ and $\alpha$, where we again refer to
[10] for details. In particular, the asymmetry parameter $\Delta$ satisfies
$\Delta\\!=\\!0$ for $\alpha\\!=\\!1$.

Since the corresponding four-dimensional path integral or partition function
$Z_{E}(k_{0},k_{4},\Delta)$ cannot be calculated analytically, the phase
diagram of the theory is explored with the help of Monte Carlo simulations.
For the results reported below the topology chosen for the spatial slices was
spherical, $\Sigma\\!\cong\\!S^{3}$. There are also simulations for toroidal
slices of topology $T^{3}$, which have been reviewed by [5].

The phase structure for the spherical case has the following features.
Analogous to what was reported in Sec. 3.1 for the coupling $\kappa_{4}$ in
EDT, there is a transition from a finite to an infinite expectation value
$\langle N_{4}\rangle$ of the four-volume at a critical value
$k_{4}^{c}(k_{0},\Delta)$ of $k_{4}$, and the partition function is not
defined for $k_{4}<k_{4}^{c}(k_{0},\Delta)$. As a consequence, there is a two-
dimensional critical surface parametrized by $k_{0}$ and $\Delta$. Since we
are interested in the behaviour for large $\langle N_{4}\rangle$, the
simulations should stay as close to this surface as possible. Note that in
itself, the infinite-volume or thermodynamic limit $\langle
N_{4}\rangle\to\infty$ does not necessarily imply interesting continuum
physics. For that, we need to scan the two-dimensional coupling-constant plane
spanned by $(k_{0},\Delta)$ for the presence of phase transitions of second
order. The result of this analysis is shown in Fig. 3.

Figure 3: Phase diagram of CDT lattice gravity.

Of the four phases found, the de Sitter phase $C_{\rm dS}$ has our main
interest, as will become clear below. Two of the other phases can be thought
of as Lorentzian analogues of the degenerate phases of EDT, namely of the
branched-polymer phase dominated by the conformal mode (phase $A$), and of the
crumpled phase (phase $B$). The bifurcation phase $C_{\rm b}$ is characterized
by the emergence of a distinguished, string-like structure consisting of
vertices of high order, indicating a breakdown of the conjectured homogeneity
of geometry in the de Sitter phase [4]. While the physical relevance of the
$C_{\rm b}$-phase is still being explored, there is strong evidence that
interesting physical behaviour is present in the $C_{\rm dS}$-phase, regarding
both infrared and ultraviolet properties. This also means that the phase
transition lines bordering the de Sitter phase are of particular interest. The
$C_{\rm dS}$-$C_{\rm b}$ transition line has been identified as second-order
[19], and the $C_{\rm dS}$-$B$ transition line is another promising candidate
under investigation. Points along these transition lines are prime candidates
in the search for UV fixed points at which one may be able to find a continuum
theory that is well-behaved in the UV. Before reporting on progress in
locating such a fixed point, we will first describe some of the infrared
physical properties that have been found in the de Sitter phase.

###  4.2 The quantum de Sitter universe

One of the remarkable features of the gravitational path integral in terms of
CDT is the emergence of a macroscopically four-dimensional quantum spacetime
whose large-scale properties are compatible with those of a classical de
Sitter universe. We start by reviewing the evidence that the dynamically
generated quantum spacetime is de Sitter-shaped, as shown by [8, 9].

The numerical simulations take place inside the de Sitter phase and for
computer-technical reasons are performed such that the discrete four-volume
$N_{4}$ of the geometries fluctuates around a given value $\bar{N}_{4}$.
Running Monte Carlo simulations at different and increasing volumes
$\bar{N}_{4}$ then allows for the use of powerful finite-size scaling methods
when analyzing the data. In addition, the computational set-up involves a
fixed number $t_{\rm tot}$ of time steps, but it turns out that the precise
choice is unimportant as long as $t_{\rm tot}$ is sufficiently large.

In the simulations one can easily monitor the number $N_{3}(i)$ of three-
simplices in a three-dimensional spatial triangulation at a given time
$t_{i}$. The function $N_{3}(i)$ is called the shape or volume profile of the
spacetime. We can measure the expectation value $\langle
N_{3}(i)\rangle_{\bar{N}_{4}}$ and the correlator $\langle
N_{3}(i)N_{3}(j)\rangle_{\bar{N}_{4}}$, as well as the corresponding
fluctuations of $N_{3}(i)$. The outcome is illustrated in Fig. 4.

Figure 4: Volume profile and fluctuations as a function of time $i$. Within
the range of nonvanishing three-volume, the measured volume profile $\langle
N_{3}(i)\rangle_{\bar{N}_{4}}$ at $\bar{N}_{4}=360.000$, extrapolated to
continuous time (red curve) and the $\cos^{3}$-profile on the right-hand side
of eq. (16) (black curve) are indistinguishable at the given resolution. Blue
bars indicate the size of fluctuations $\Delta N_{3}(i)$ around the average.

The volume profile consists of an extended part of nonvanishing macroscopic
volume and a stalk, where the three-volume is as small as allowed by the
regularity condition imposed (i.e. $N_{3}(i)\\!=\\!5$) to prevent the universe
from pinching off. In other words, spatial volume in the stalk is zero from a
continuum point of view.

The beautiful result is that the overall shape observed for the nonvanishing
part of the volume profile is described perfectly by the functional expression

$\langle
N_{3}(i)\rangle_{\bar{N}_{4}}=c\,\bar{N}_{4}\;\frac{3}{4}\,\frac{1}{\omega\bar{N}_{4}^{1/4}}\,\cos^{3}\bigg{(}\frac{i}{\omega\bar{N}_{4}^{1/4}}\bigg{)},$
(16)

and the correlator of the three-volume fluctuations by

$\langle\delta N_{3}(i)\delta
N_{3}(j)\rangle_{\bar{N}_{4}}=\bar{N}_{4}\;F\bigg{(}\frac{i}{\omega\bar{N}_{4}^{1/4}},\frac{j}{\omega\bar{N}_{4}^{1/4}}\bigg{)},\quad\delta
N_{3}(i):=N_{3}(i)\\!-\\!\langle N_{3}(i)\rangle,$ (17)

where $c$ and $F(0,0)$ are of order 1, and $c$ and $\omega$ are functions of
$k_{0}$ and $\Delta$. Eq. (16) should be compared to the formula for the
volume $V_{3}(t)$ of the three-sphere located at a geodesic distance $t$ from
the north pole of a round four-sphere, which is given by

$V_{3}(t)=V_{4}\;\frac{3}{4}\;\frac{1}{\omega_{0}V_{4}^{1/4}}\;\cos^{3}\bigg{(}\frac{t}{\omega_{0}V_{4}^{1/4}}\bigg{)},\quad\omega_{0}^{4}=\frac{3}{8\pi^{3}}.$
(18)

One concludes that by a suitable finite rescaling of the lattice distance
units of space and time, the volume profile generated in the lattice
simulations matches precisely that of the (Euclideanized) classical de Sitter
solution to general relativity, anywhere in phase $C_{\rm dS}$. Defining
$(\Delta N_{3}(i)_{\bar{N}_{4}})^{2}\\!:=\\!\langle\delta N_{3}(i)\delta
N_{3}(i)\rangle_{\bar{N}_{4}}$ and assuming one can write $V_{3}(t)\propto
N_{3}(i)a^{3}$, we have

$\frac{\Delta V_{3}(t)}{V_{3}(t)}\Big{|}_{V_{4}}=\frac{\Delta
N_{3}(i)_{\bar{N}_{4}}}{\langle N_{3}(i)\rangle_{\bar{N}_{4}}}.$ (19)

The right-hand side of this relation can be obtained from the Monte Carlo
results for (16) and (17), and has been found to agree well with the left-hand
side, which can be computed using semiclassical perturbation theory around the
classical de Sitter solution. Note that the height of the curve in Fig. 4
scales like $\bar{N}_{4}^{3/4}$ and its width like $\bar{N}_{4}^{1/4}$,
whereas the fluctuations $\Delta N_{3}(i)$ according to (17) only scale like
$\bar{N}_{4}^{1/2}$. It means that the relative size of the fluctuations
scales to zero with increasing four-volume.

We conclude that CDT lattice gravity, without putting in any background
geometry, dynamically generates a quantum spacetime whose shape is that of a
semiclassical de Sitter space, including quantum fluctuations, which is an
unprecedented result in nonperturbative quantum gravity.

###  4.3 The CDT effective action

From the knowledge of the measurements of the volume profile $\langle
N_{3}(i)\rangle$ and the correlator $\langle\delta N_{3}(i)\delta
N_{3}(j)\rangle$ one can determine an effective action for $N_{3}(i)$ that
reproduces these measurements. This reconstruction can proceed in a number of
ways, but the final result is amazingly simple, see [10, 7] for details.
Considering only the physics of the region of nonvanishing three-volume, one
obtains

$S_{\rm eff}(\bar{N}_{4})=\frac{1}{\Gamma}\
\sum_{i}\left(\frac{\big{(}N_{3}(i\\!+\\!1)-N_{3}(i)\big{)}^{2}}{N_{3}({i})\big{)}}+\delta\,N_{3}^{1/3}(i)\right).$
(20)

The quantities $\Gamma$ and $\delta$ depend on $k_{0}$, $\Delta$ and
$\bar{N}_{4}$, and their dependence on $\bar{N}_{4}$ becomes stronger as the
second-order $C_{\rm dS}$-$C_{\rm b}$ phase transition is approached. For
fixed $(k_{0},\Delta)$, the $\bar{N}_{4}$-dependence disappears for
sufficiently large $\bar{N}_{4}$. Introducing $s_{i}\\!=\\!i/\bar{N}_{4}$ and
$n_{3}(s_{i})\\!=\\!N_{3}(i)/\bar{N}^{3/4}_{4}$ and replacing the summation
over $i$ by an integral over $s$ for large $\bar{N}_{4}$, one can rewrite (20)
as

$S_{\rm eff}(\bar{N}_{4})=\frac{\sqrt{\bar{N}_{4}}}{\Gamma}\int
ds\;\left(\frac{\dot{n}_{3}^{2}(s)}{n_{3}(s)}+\delta\,n_{3}^{1/3}(s)\right).$
(21)

This is precisely the minisuperspace action of [20] if one makes the
identifications

$\frac{\sqrt{\bar{N}_{4}}}{\Gamma}=\frac{\sqrt{V_{4}}}{24\pi
G},\qquad\delta=\delta_{0}\equiv\frac{3}{8\pi^{2}}.$ (22)

Although the measured $\delta$ is not equal to $\delta_{0}$, a rescaling of
the relative length assignments of space- and timelike links allows us to
obtain such an agreement for any values of $k_{0}$ and $\Delta$ in the de
Sitter phase $C_{\rm dS}$, as stated earlier.

Hartle and Hawking resorted to a minisuperspace reduction of gravity to study
some aspects of quantum gravity, and the unboundedness problem of the
Euclidean action in particular. As mentioned in Sec. 3.2 above, CDT provides a
nonperturbative solution to this problem. Moreover, by integrating out all
degrees of freedom except for the three-volume $V_{3}(t)$ in CDT, one is
effectively deriving the Hartle-Hawking minisuperspace action from full
quantum gravity. It would be very interesting to determine corrections to the
effective action (20), but the accuracy of the numerical data does not
currently allow this.

The above analysis suggests the presence of a semiclassical de Sitter-like
universe, unless we are too close to the phase boundaries. We can therefore
use relation (22) to obtain an estimate of the lattice spacing $a$ in terms of
physical length units, assuming $V_{4}\propto\bar{N}_{4}\,a^{4}$. This yields
$a\\!=\\!\tilde{c}\sqrt{G}$, where $\tilde{c}$ is of order 1. In other words,
$a$ is of the order of the Planck length, but not much smaller, and the
typical linear size of a quantum spacetime simulated in the computer is in the
range of 10—20 Planck length.

Returning to our earlier discussion about the phase structure, we are
interested in identifying and approaching a possible UV fixed point. Using
various renormalization group techniques, such a fixed point has been found in
continuum quantum gravity [30]. We have seen that in CDT lattice gravity any
putative fixed point will likely be located on one of the boundaries of the de
Sitter phase $C_{\rm dS}$. The effective action (21) can be useful in relating
the two formulations, since it can be compared to the effective actions
calculated in the renormalization group analysis. The quantity $\Gamma$
measured in CDT changes with $k_{0}$ and $\Delta$, and close to the boundary
also with $\bar{N}_{4}$, while the quantities $G$ and $V_{4}$ on the right-
hand side of eq. (22), calculated with the help of the renormalization group,
changes as a function of the renormalization group scale. Close to the UV
fixed point the latter can be related to the UV cutoff of the lattice theory,
given by the lattice spacing $a$. Work currently in progress indicates that a
putative UV lattice fixed point may be located at one of the two triple points
of the CDT phase diagram.

##  5 Observables

The physical content of quantum gravity is encoded in its geometric
observables, which in the context of CDT lattice gravity are given by
operationally well-defined, diffeomorphism-invariant quantum operators, whose
eigenvalues are computable and finite in a continuum limit. Observables in
pure gravity are nonlocal, and in the absence of a reference system – which
could take the form of coupled matter fields or (asymptotic) boundaries with a
fixed metric structure – are typically given as spacetime integrals or
averages of quasi-local scalar quantities.

What should be kept in mind regarding the type of observable accessible in a
nonperturbative and strongly quantum-fluctuating regime is the fact that in a
continuum limit, typical geometric configurations in the path integral are
continuous, but nowhere differentiable. In particular, they are not smooth,
and even finite piecewise flat triangulations do not possess local coordinate
systems that extend over more than pairs of adjacent simplices, due to their
singular curvature structure. There is therefore no useful notion of tensor
calculus, and consequently no meaningful way to talk about a quantum metric
$\hat{g}_{\mu\nu}$ or its expectation value – disregarding for the moment its
noninvariant behaviour under diffeomorphisms – not even in a coarse-grained or
approximate sense. Even when considering local scalar quantities, say, by
trying to implement local invariants of the Riemann tensor in terms of naïve
finite-difference expressions on the lattice, they will typically be highly
divergent in a continuum limit. Indeed, a well-defined renormalized notion of
Ricci curvature, constructed from quasi-local distance and volume data, has
only recently been implemented, see Sec. 5.2 below.

These properties are not flaws or peculiarities of this particular formulation
of nonperturbative quantum gravity, but reflect the highly nonclassical nature
of quantum spacetime in a near-Planckian regime. Nevertheless, a well-defined
metric structure is still present, which allows us to measure distances and
volumes, and is crucial in the construction of quantum observables. In fact,
these rods and clocks of quantum geometry are close in spirit to how we
explore the properties of classical gravitating systems in our universe with
the help astrophysical measurements.

Secs. 4.2 and 4.3 above illustrated the global nature of the observable
studied, in this case the total spatial volume $V_{3}(t)$ as a function of
proper time $t$, with the latter defined invariantly as the distance from the
beginning of the universe at vanishing spatial volume. The robust character of
this observable clearly contributes to the fact that we can match its
expectation value with great accuracy to a classical de Sitter volume profile,
and that quantum fluctuations are relatively small in size. For a general
quantum observable, there is no a priori guarantee that a semiclassical limit
will be observable in the scale window that can be accessed computationally.
Even more interesting from a quantum-gravitational point of view, we may
observe genuine quantum signatures, characteristic deviations from an expected
classical behaviour that need not be small, and may not be visible in any
perturbative approach. An important example is the spectral dimension of
spacetime. It is one of the selected quantum observables we discuss below.
Further examples and more comprehensive treatments of observables can be found
in [10, 25, 26].

###  5.1 Dimensions

In classical gravity, the dimension of spacetime is given by its topological
dimension, which is nondynamical and fixed to 4. There are various ways of
characterizing the dimension of a nonsmooth, nonclassical metric space, which
on such a space may not coincide with the topological dimension or with each
other. A powerful and popular method to quantify the behaviour of such a
geometry is by its spectral and Hausdorff dimension, defined in the form of
scaling exponents that characterize the growth of volumes (of a diffusion
cloud, of a geodesic ball) as function of a diffusion time and a radial
distance respectively.

The relevance of these fractal dimensions as observables in quantum gravity is
two-fold, as necessary conditions for the existence of a classical limit and
by exhibiting true quantum features. On a smooth manifold, the fractal
dimensions are equal to the topological dimension, but this is generally not
true for quantum geometries, like those obtained from a nonperturbative
gravitational path integral. In particular, even when the microscopic building
blocks used to assemble the path integral histories are four-dimensional, the
resulting quantum superposition will in general not be four-dimensional, on
any scale. This somewhat surprising fact was one of the lessons of Euclidean
quantum gravity à la EDT, and took a while to recognize. Noncanonical values
of fractal dimensions on Planckian scales can be perfectly acceptable as
quantum signatures, that is, as expressions of the nonclassical character of
the geometry. However, the absence of four-dimensionality on large scales
signals the absence of a well-defined classical limit, making the quantum
gravity candidate theory ineligible.

Spacetime-averaged fractal dimensions can be defined and evaluated on the
ensemble of random geometries underlying the CDT approach, with interesting
results. Starting with a continuum analysis of the spectral dimension, let
$\Delta_{g}$ denote the Laplacian on a smooth $d$-dimensional Riemannian
manifold $M$ with metric $g_{\mu\nu}(x)$. Diffusion on $M$ is described by the
equation

$\frac{\partial}{\partial\sigma}K_{g}(x,y;\sigma)=\Delta_{g}(x)K_{g}(x,y;\sigma),$
(23)

whose solution $K_{g}(x,y;\sigma)$, the so-called heat kernel, is interpreted
as the probability that a particle starting at the point $x$ diffuses
(performs a random walk) to the point $y$ in diffusion time $\sigma$. For
equal arguments, $K_{g}(x,x;\sigma)$ is the return probability at $x$. For a
finite manifold volume $V$ the average return probability $K_{g}(\sigma)$ can
be written as

$K_{g}(\sigma)=\frac{1}{V}\int_{M}d^{d}x\sqrt{g(x)}\;K_{g}(x,x;\sigma)=\frac{1}{(4\pi\sigma)^{D_{s}/2}}\sum_{r=0}^{\infty}A_{r}\sigma^{r},\quad
A_{0}=1,$ (24)

where the coefficients $A_{r}$ can be expressed as integrals over local
invariants, like suitable contractions of powers of the curvature tensor. In
eq. (24), $D_{s}$ is the spectral dimension, which for a smooth manifold $M$
satisfies $D_{s}=d$. Diffusion processes can also be defined on more general,
nonsmooth metric spaces by using an appropriate implementation of the
diffusion equation (23), allowing us to extract a spectral dimension.

Since $K_{g}(\sigma)$ is invariant under diffeomorphisms, it makes sense to
consider its quantum average $\langle K_{g}(\sigma)\rangle$ in a theory of
quantum gravity, including on the configurations of CDT lattice gravity. The
expectation value $\langle K_{g}(\sigma)\rangle_{V_{4}}$ at constant volume
$V_{4}$ has been measured in the de Sitter phase of CDT, using Monte Carlo
simulations [14, 13]. The extracted spectral dimension as a function of the
diffusion time $\sigma$ is shown in Fig. 5. The fact that $D_{s}$ depends
nontrivially on $\sigma$ came as a surprise. Since some power of $\sigma$ will
be a measure of the linear size of the region explored by the diffusion
process, small values of $\sigma$ correspond to short-distance excursions from
the starting point $x$, characterizing the short-scale dimension of quantum
spacetime. Similarly, large diffusion times $\sigma$ are associated with long-
distance properties.

Figure 5: Spectral dimension $D_{s}$ as a function of the diffusion time
$\sigma$, measured for a spacetime volume $N_{4}\\!=\\!360.000$. The averaged
measurement data lie along the central curve, together with a superimposed
best fit $D_{s}(\sigma)\\!=\\!4.02-119/(54+\sigma)$, and the two outer curves
represent error bars.

In flat space, only the first term on the right-hand side of (24) will
contribute, as long as $\sigma$ is not too large, such that the measurements
are not subject to finite-size effects. If typical spacetime geometries were
fractal and self-similar up to some scale, one would still expect the first
term in the expansion to dominate, leading to an initial plateau of the curve
for $D_{s}$, possibly at a value $D_{s}\not=4$. Instead, we observe a
nontrivial, monotonically increasing spectral dimension. Reassuringly, the
asymptotic value for large $\sigma$, extrapolated from the measurements, is
compatible with 4, the correct classical limit.

Equally important and even more intriguing is the observation of a genuine
quantum signature for the short-scale dimension, which extrapolated from the
data is given by $D_{s}(\sigma\rightarrow 0)=1.80\pm 0.25$, a phenomenon
dubbed dynamical dimensional reduction. This value, which may be interpreted
as evidence that quantum spacetime is effectively two-dimensional in the
ultraviolet (UV), is remarkable since it could indicate a nonperturbative
resolution of quantum gravity’s nonrenormalizability. Since the initial
discovery of this effect, there have been numerous attempts to compute the
spectral dimension in other approaches to quantum gravity, which have largely
found a similar behaviour. It suggests that the short-distance dimensional
reduction could be a universal feature of quantum gravity, as advocated by
[17].

It is worth emphasizing that the spectral dimension for $\sigma\rightarrow 0$
is a rare instance of a nonperturbative gauge-invariant observable that is
simply a number, which can be calculated and compared across approaches.
Although this result has not been related explicitly to phenomenology, it
represents genuine progress in a field that used to be characterized by a
complete absence of computable observables beyond perturbation theory.

Another way to characterize a metric space $M$ is by its Hausdorff dimension
$d_{H}$, which we will take to mean the growth rate of the volume $V(B_{r})$
of geodesic balls $B_{r}$ as a function of their radius $r$ and averaged over
$M$,

$\bar{V}(B_{r})\propto r^{d_{H}}.$ (25)

All of the required ingredients exist on the triangulations we are interested
in. This includes a notion of geodesic link distance between lattice vertices
$x$ and $y$, defined as the number of one-dimensional links (lattice edges) in
the shortest path of consecutive links connecting $x$ and $y$. The volume
$V(B_{r})$ is then given by the number of vertices with link distance smaller
or equal to $r$ from a given vertex. The expectation value of the Hausdorff
dimension in the de Sitter phase of CDT was measured by [14] on geometries
with volumes of up to $N_{4}=180.000$ and found to be $d_{H}=4.01\pm 0.05$
over a wide range of scales. We conclude that this particular observable
exhibits classical behaviour throughout, in the sense of expectation values.

###  5.2 Curvatures

The central role of curvature in classical general relativity raises the
question of whether something similar could be true in quantum gravity also.
Since the Riemannian curvature is a complicated rank-4 tensor, which depends
on second derivatives of the metric and encodes a large amount of local
curvature data, one may immediately anticipate difficulties in representing it
in the quantum theory. The challenge beyond perturbation theory is to
construct diffeomorphism-invariant curvature operators that are appropriately
regularized and renormalized. Recalling the nowhere-differentiable character
of the quantum geometry in CDT, it is unclear a priori whether a meaningful
notion of curvature can be defined in a nonperturbative regime at all. As we
will see below, this has now been realized.

Figure 6: Each of the triangles meeting at the vertex $v$ contributes an angle
$\alpha_{i}$. Cutting the two-dimensional triangulation open as indicated and
flattening it out on a straight surface reveals a deficit angle $\delta_{v}$.

Before presenting this new notion of quantum curvature, let us review some
well-known facts about finite piecewise flat manifolds and the singular
curvature assignments they carry. A $d$-dimensional simplicial manifold, like
the ones occurring in EDT or CDT, can be thought of as a nonsmooth metric
space of Euclidean or Lorentzian signature, whose individual triangular
building blocks are flat by construction. This implies that curvature can only
reside along lower-dimensional (sub-)simplices, which turn out to be the
$(d-2)$-dimensional simples, so-called hinges, as already mentioned in Sec.
3.1. The presence of curvature at such a hinge is exhibited by parallel-
transporting a vector around the hinge along a small closed loop $\gamma$ in a
plane perpendicular to the hinge. This is most easily visualized in a two-
dimensional triangulation, where the hinges are the (zero-dimensional)
vertices. The deficit angle $\delta_{v}$ at a vertex $v$ is the difference
between $2\pi$ and the sum of the angles of the triangles meeting at $v$ (Fig.
6). It is a direct measure of the intrinsic, Gaussian curvature located at
$v$, and is also equal to the rotation angle that appears in the holonomy of
the Levi-Civita connection $\Gamma$ along $\gamma$,

$U_{\gamma}(\Gamma):=\mathrm{P}\exp\Big{(}-\oint_{\gamma}\Gamma\Big{)},$ (26)

which describes how a tangent vector is rotated during its parallel transport
along $\gamma$, where P denotes path ordering. The construction of holonomies
(26) and their associated Wilson loops in CDT in $d\\!=\\!4$ is described in
[11].

[29] has given a prescription for the total scalar curvature of a simplicial
manifold $M$ in terms of a weighted sum over all deficit angles in $M$. For an
equilateral EDT configuration, this expression – together with the
cosmological-constant term – gives rise to the form (4) of the Einstein-
Hilbert action, and similarly in CDT for the Lorentzian version (7) of the
action. However, for reasons recapitulated in [26], it turns out that the
integrated scalar curvature à la Regge diverges in the continuum limit and is
therefore not a good curvature observable in four-dimensional EDT or CDT
quantum gravity.

An alternative to the deficit-angle curvature, which is better behaved in the
UV, has been discovered recently. This so-called quantum Ricci curvature
relies purely on local distance and volume measurements and not on the
availability of tensor calculus. It can nevertheless also be applied to smooth
Riemannian manifolds, where in the limit of infinitesimal distances it
reproduces the local Ricci curvature $Ric(v,v)\\!=\\!R_{\mu\nu}v^{\mu}v^{\nu}$
associated with a normal vector $v^{\mu}$ [21, 22]. However, its key strength
is the applicability to a much larger class of nonsmooth metric spaces,
including simplicial manifolds.

The main ingredient in its construction is the average sphere distance
$\bar{d}(S^{\delta}_{p},S^{\delta}_{p^{\prime}})$ between two spheres of
geodesic radius $\delta$, whose centres $p$ and $p^{\prime}$ are also a
distance $\delta$ apart. On a smooth four-dimensional Riemannian manifold
$(M,g_{\mu\nu})$ it is given by

$\bar{d}(S^{\delta}_{p},S^{\delta}_{p^{\prime}}):=\frac{1}{vol(S_{p}^{\delta})}\frac{1}{vol(S^{\delta}_{p^{\prime}})}\int_{S^{\delta}_{p}}d^{3}\\!q\,\sqrt{{\det
h}}\int_{S^{\delta}_{p^{\prime}}}d^{3}\\!q^{\prime}\sqrt{\det h^{\prime}}\
d_{g}(q,q^{\prime}),$ (27)

where $d_{g}(q,q^{\prime})$ denotes the geodesic distance of the points $q$
and $q^{\prime}$, and $h$ and $h^{\prime}$ are the metrics induced on the two
spheres (Fig. 7). The quasi-local quantum Ricci curvature $K_{q}$ at scale
$\delta$ is then defined in terms of the quotient of the average sphere
distance (27) and the centre distance as

$\bar{d}(S_{p},S_{p^{\prime}})/\delta=c_{q}(1-K_{q}(p,p^{\prime})),\;\;\;\delta=d_{g}(p,p^{\prime}),$
(28)

where $c_{q}$ is a nonuniversal $\delta$-independent constant with
$0\\!<\\!c_{q}\\!<\\!3$, depending on the type and dimension of the metric
space, and $K_{q}$ captures the nonconstant remainder.

Figure 7: Comparing the distance
$\bar{d}(S^{\delta}_{p},S^{\delta}_{p^{\prime}})$ of two $\delta$-spheres
$S_{p}^{\delta}$ and $S^{\delta}_{p^{\prime}}$ with the distance $\delta$ of
their centres $p$ and $p^{\prime}$.

The simplest way of constructing a curvature observable for a given $M$ is by
integrating the quotient (28) over all pairs $(p,p^{\prime})$ of points at
fixed distance $\delta$. Important for the implementation in quantum gravity
is that the sphere distance (27) is equally well defined on CDT geometries. It
allows for the evaluation of the expectation value
$\langle\bar{d}(\delta)/\delta\rangle$ as a function of the geodesic length
scale $\delta$, the so-called curvature profile. The curvature profile in CDT
quantum gravity in $d\\!=\\!4$ in a near-Planckian regime has been measured
and found to be compatible with that of a classical de Sitter space in [23].
This confirms the viability of the quantum Ricci curvature in nonperturbative
quantum gravity and the de Sitter-nature of the dynamically generated quantum
spacetime, both highly nontrivial results.

The availability of a notion of Ricci curvature which is well-defined in a
nonperturbative regime and can in principle pick up direction-dependence (via
the relative position of the two spheres) is significant. It opens the door to
investigating new quantum observables that will lead to a more fine-grained
understanding of the properties of quantum spacetime. Examples are
diffeomorphism-invariant measures of homogeneity [27] and correlation
functions [3]. This is important from a physical point of view since the de
Sitter-like quantum universe provides a model for the very early universe, and
moreover has been derived from first principles. Its predictions should be
compared with the ad-hoc assumptions made in standard early-universe cosmology
regarding the homogeneous and isotropic nature of spacetime and the spectrum
of the quantum fluctuations that act as seeds of structure formation. A re-
examination of these assumptions from the perspective of nonperturbative
quantum gravity may give us new insights into whether or not they are
justified.

Acknowledgement. We are most grateful to the numerous collaborators and
colleagues who contributed to the results reported here and elsewhere.

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To control for potential cohort effects for firm exit times, we include exit-date fixed effects interacted with existing time-verying fixed effects, that is, let $\alpha'_{k(j)t}$ ($\alpha'_{k(ij)t}$) be defined as the interaction of $\alpha_{k(j)t}$ ($\alpha_{k(ij)t}$) with an indicator variable that takes value 1 if firm $j$ last appears in our sample at time period $t$, and zero otherwise. We replace $\alpha_{k(j)t}$ with $\alpha'_{k(j)t}$ in (<ref>) and (<ref>), and we replace $\alpha_{k(ij)t}$ with $\alpha'_{k(ij)t}$ in (<ref>) and (<ref>).

§.§.§ Firm size restrictions

We present four different samples of firms, defined by number of workers at EPSW announcement. Recall that our baseline sample includes firms with between 6 and 13 workers at announcement, accounting respectively for percentiles 39 and 78 of the firm size distribution at announcement, resulting in a total of 3,201 firms.
Column V presents a “doughnut hole” sample which excludes all firms with either 9 or 10 workers at announcement from our baseline sample, resulting in a total of 2,789 firms. This sample is to account for the mechanical increase in likelihood that the excluded firms' treatment status does not match whether they are bound by EPSW at any given point in time. Column VI presents a narrower sample which includes firms with between 7 and 12 workers at announcement and excludes all others,  accounting for percentiles 48 and 75 respectively in the firm size distribution at announcement, resulting in a total of 2,424 firms. Column VII presents a wider sample which includes firms with between 5 and 14 workers at announcement and excludes all others, accounting for percentiles 28 and 80 respectively in the firm size distribution at announcement, resulting in a total of 3,898 firms.

Our results reanalyze the specifications in (<ref>), (<ref>), (<ref>), and (<ref>) with these alternative samples.

§.§ Effect of EPSW on the overall gender wage gap

In <Ref>, we estimate the effect of EPSW on the wage gap in male majority versus female majority labor markets. Here, we estimate the effect of EPSW across all labor markets, which can be interpreted as the average effect of EPSW on the total wage gap in Chile.

We consider a panel in which an observation is a worker-firm-month.
We estimate triple difference models of the following form:

\begin{align}
\ln w_{ijt} = &~ \alpha_i + \omega_{it} +\alpha_j + \alpha_{k(ij)t} +  \gamma_{1} (above10_j \times post_t) + \gamma_2 (male_i \times post_t) \notag\\
& + \gamma_3 (above10_j \times male_i) + \beta^{gap} (above10_j \times male_i \times post_t) + X_{ijt}\Lambda + \epsilon_{ijt} \label{emp_eq3}
\end{align}

where $X_{ijt}$ is a vector of firm-month level controls and worker-firm-month level controls. The firm-month level controls are the share of workers younger than the median age at the industry-region, share of workers with tertiary education, and the share of workers that have long-term contracts. The worker-firm-month level controls are the number of months at the firm and an indicator for whether the worker's  earnings reach the top-coding threshold.
$\alpha_{k(ij)t}$ are time fixed effects for workers in set $k(ij)$, where $k(ij)$ is a set of workers including $i$ and a comparison group of workers to $i$ employed at a comparison group of firms to $j$. Workers comparison groups are defined by equivalence across three binary dimensions at time $t$ at firm $j$: an indicator for tertiary education, an indicator for long-term versus fixed-term contract, and an indicator for being above median age in the particular industry-region in which firm $j$ operates. Across specifications, we consider finer and finer firm comparison groups defined by firm industry and region.
Our coefficient of interest is $\beta^{gap}$ and we interpret it as the effect of the policy on the (percentage) wage gap between male and female workers. Positive values indicate a relative increase in male wages.

To understand more about the dynamic effects of EPSW, we estimate the following triple difference model year by year, where we omit the year before policy announcement as the reference period, so that the set of years included is $\mathcal{T}=\{2005,2006,2007,2009,...,2013\}$. By construction, each year (indexed by $\tau$) corresponds to twelve time periods (indexed by $t$).

\begin{align}
\ln w_{ijt} = & ~\alpha_i + \omega_{it} + \alpha_j + \alpha_{k(ij)t} + \sum_{\tau\in \mathcal{T}}^{} \gamma_{1\tau} (above10_j \times year_\tau)  \notag\\
& + \sum_{\tau\in \mathcal{T}}^{} \gamma_{2\tau} (male_i \times year_\tau) + \gamma_3 (above10_j \times male_i)  \notag\\
& + \sum_{\tau\in \mathcal{T}}^{} \beta^{gap}_{\tau} D_{ijt} + X_{ijt}\Lambda + \epsilon_{ijt} \label{emp_eq4}
\end{align}

where $D_{ijt}$ is an indicator that equals 1 in time period $t$ if firm $j$ employs at least 10 long-term workers at the time of policy announcement and $i$ is male, and zero otherwise. $\beta^{gap}_\tau$ is the average difference in log wages between men and women in treated versus control firms in year $\tau$ (relative to 2008).

Our identifying assumption is that parallel trends hold between treated and control firms, that is, $\mathbb{E}[\epsilon_{ijt}\cdot D_{ijt}]=0$ for all $t$ [Olden and Møen, 2022]. This strategy builds in a partial falsification test, in that we expect coefficient estimates of $\beta^{gap}_\tau$ to be zero for $\tau<2009$.

Tables <ref> and  <ref> present estimates from (<ref>). Table <ref> presents specifications detailed in <Ref> and Table <ref> presents specifications with various firm comparison groups $k(j)$. In all specifications EPSW increases the gender wage gap  (in favor of men), however, the results of some specifications are not statistically  significant at conventional levels. Additionally, we present
estimates from (<ref>) to support our parallel trends assumption. The mean pre-treatment estimates are small and statistically insignificant, and we cannot reject an  F-test that all pre-treatment coefficients are jointly zero.

Effect of EPSW on Gender Wage Gap

1c(I)   1c(II)   1c(III)   1c(IV)   1c(V)   1c(VI)   1c(VII)

1cBaseline   1cNo firm   1cNo   1cUnbalanced   1cDoughnut   1cNarrower    1cWider

1c   1cFEs   1ccontrols   1csample   1chole   1cband   1cband

($\hat \beta^{gap}$) Treated $\times$ Male $\times$ Post     0.0257*        0.0236*         0.0264*       0.0140               0.0319*       0.0170         0.0079

(0.0133)            (0.0128)               (0.0134)       (0.0106)            (0.0179)            (0.0142)       (0.0113)

Mean Pre-Treatment                              -0.0051           -0.0025               -0.0048       -0.0146                -0.0019             -0.0200        -0.0132

(0.0124)             (0.0118)                (0.0126)        (0.0107)                 (0.0164)              (0.0142)         (0.0116)

No Pre-Trend p-value                           0.9041             0.8934                0.8984        0.3330                 0.8190              0.5739         0.5602

Number of Firms                                           3,201              3,201                3,201        6,120                 2,789                2,424          3,898

Number of Observations                                    3,405,855          3,405,855            3,405,855         5,427,234         2,902,147            2,687,403      4,051,757

Fixed effects

Firm                                    Yes     No     Yes     Yes     Yes     Yes     Yes

Worker                                     Yes    Yes     Yes     Yes     Yes     Yes     Yes

Worker age                                        Yes     Yes     Yes     Yes     Yes     Yes     Yes

Month $\times$ industry $\times$ county                    Yes     Yes     Yes     Yes     Yes     Yes     Yes

$\times$  tertiary education

$\times$ contract type $\times$ worker age

Firm-month level controls                  Yes     Yes     No     Yes     Yes     Yes     Yes

Worker-month level controls                Yes     Yes     No     Yes     Yes     Yes     Yes

Notes: The unit of analysis is the worker-firm-month and the dependent variable is the natural logarithm of the worker's wage at the firm in a given month.
Column I presents estimated coefficients  $\hat \beta^{gap}$ for our baseline regression specification presented in (<ref>). Time-varying fixed effects are defined as the intersection of: the firm's industry, the firm's county, an indicator for worker tertiary education, an indicator for worker contract type, an indicator for a worker being above median age in the industry-region.
Firm-month controls included are: the share of workers younger than the median age in the industry-region, the share of workers with tertiary education, and the share of workers that have long-term contracts. Worker-firm-month levels controls included are the number of months at the firm and an indicator for reaching the earnings truncation threshold.
The mean pre-treatment effect is the mean of $\hat \beta^{gap}_\tau$ for $\tau\in\{2005, 2006, 2007\}$ calculated from (<ref>), and the no pre-trend p-value is derived from a joint $F$-test that  $\beta^{gap}_\tau=0$ for all $\tau\in\{2005, 2006, 2007\}$.
Columns II-VII present the analogous information for alternative sample selections and empirical specifications as described in <Ref>. Throughout, standard errors in parentheses are two-way clustered at the firm and month levels. *, **, and *** indicate statistical significance at the 10%, 5%, and 1% levels, respectively.

Effect of EPSW on Gender Wage Gap–Alternative Time Trends

1c(I)   1c(II)   1c(III)   1c(IV)   1c(V)   1c(VI)

($\hat \beta^{gap}$) Treated $\times$ Male $\times$ Post          0.0269*           0.0298**          0.0234*           0.0262**          0.0264*           0.0257*

(0.0146)                (0.0143)                (0.0133)                (0.0131)                (0.0134)                (0.0133)

Mean Pre-Treatment       -0.0024               -0.0055               -0.0027               -0.0066               -0.0048               -0.0051

(0.0139)                 (0.0134)                 (0.0131)                 (0.0128)                 (0.0126)                 (0.0124)

No Pre-Trend p-value         0.9759                  0.8678                  0.9842                  0.9107                  0.8984                  0.9041

Number of Firms                  3,201                    3,201                    3,201                    3,201                    3,201                    3,201

Number of Observations        3,559,172                 3,558,635                 3,535,936                 3,535,362                 3,405,855                 3,405,855

Fixed effects

Firm                                    Yes     Yes     Yes     Yes     Yes     Yes

Worker                                     Yes     Yes     Yes     Yes     Yes     Yes

Worker age                                        Yes     Yes     Yes     Yes     Yes     Yes

Month $\times$ tertiary education                    Yes      No     No     No     No     No

$\times$ contract type $\times$ worker age

Month $\times$ industry $\times$ tertiary education                  No     Yes     No     Yes     No     No

$\times$ contract type $\times$ worker age

Month $\times$ county
$\times$ tertiary education           No      No     Yes     Yes     No     No

$\times$ contract type $\times$ worker age

Month $\times$ industry $\times$ county           No      No     No     No     Yes     Yes

$\times$ tertiary education $\times$ contract type

$\times$ worker age

Firm-month level controls                  No      No      No      No     No     Yes

Worker-month level controls                No      No      No      No     No     Yes

Notes: This table displays estimated coefficients  for the regression described in (<ref>). In particular, we present estimates of $\hat \beta^{gap}$. The unit of the panel is the worker-firm-month and the dependent variable is the natural logarithm of monthly earnings. Among firms in our sample that employ both male and female workers at EPSW announcement, the within-firm wage gap in favor of men, averaged across firms, is 35.8%. Each column presents a time-varying fixed effect for a different comparison group of firms $k(ij)$ for each worker $i$ and each firm $j$. All columns include worker comparison groups defined by equivalence across three binary dimensions at time $t$ at firm $j$: an indicator for tertiary education, an indicator for long-term versus fixed-term contract, and an indicator for being above median age in the particular industry-region in which firm $j$ operates. Columns differentially include firm comparison groups. In column I, all firms are included in the comparison group. In column II firms with the same industry code are included in the comparison group. In column III firms in the same geographic county are included in the comparison group. In column IV, there are two comparison groups: firms in the same geographic county and firms with the same industry code. In column V, the comparison group includes firms that are both in the same county and have the same industry code. Column VI adds the firm-month level controls (the share of workers younger than the median age in the industry-region, the share of workers with tertiary education, and the share of workers that have long-term contracts) and worker-firm-month level controls (number of months at the firm and an indicator for reaching the earnings truncation threshold) to the model in column V. The mean pre-treatment effect is the mean of $\hat \beta^{gap}_\tau$ for $\tau\in\{2005, 2006, 2007\}$ calculated from (<ref>), and the no pre-trend p-value is derived from a joint $F$-test that  $\beta^{gap}_\tau=0$ for all $\tau\in\{2005, 2006, 2007\}$. Throughout, standard errors in parentheses are two-way clustered at the firm and month levels. *, **, and *** indicate statistical significance at the 10%, 5%, and 1% levels, respectively.

Figure <ref> displays the estimated coefficients of interest from (<ref>).
Prior to EPSW, the coefficient of interest is statistically indistinguishable from 0 in all periods. In the first year of EPSW, the wage gap rises by 2.02 percentage points in the treated group compared to the control group (p-value = 0.102) and rises to 2.52 percentage points (p-value = 0.172) by year five of EPSW.

Dynamic Path of EPSW's Effect on Gender Wage Gap

Notes: This figure displays estimated coefficients $\hat \beta^{gap}_\tau$ for the regression described in (<ref>). The specification corresponds to that in column I of Table <ref>. Bars depict 95% confidence intervals.

§.§ Details on data samples

We use administrative data from the Chilean unemployment insurance system from January 2005 to December 2013. In our data, an observation is a worker-firm-month. We observe two stratified (by firm size) random samples—a 1% sample and a 3% sample—of firms, and for each sampled firm we observe the entire monthly working history of every worker that was every employed by the sampled firm, regardless of whether the worker remains at the sampled firm or not. Therefore, we observe some workers during time periods they are employed by other, non-sampled firms. We do not directly observe which firms are sampled. For that reason, our dataset includes “incidental firms” for which we do not necessarily observe the entire workforce at any given moment in time. This happens, for example, if a worker from a sampled firm switched to a non-sampled firm.

Because we do not observe the entire workforce for these incidental firms, we do not observe the size (number of workers) for these firms. This naturally leads to a potential concern with our size-based empirical strategy. To address this potential concern, we attempt to filter out the incidental firms. Our filtering approach is built on the notion that, due to the sampling procedure, we anticipate incidental firms to have high variance in the number (and presence) of workers across time periods.

Our filtering procedure does the following. 1) For each of the 1% and 3% samples, we compute for each firm the first and the last month it is observed. We drop firms that employ no workers for some months in between these two dates in any of the two samples, i.e. if the firm has “holes” in its employment history. 2) For firms in our sample, we compute the monthly average number of workers in each firm in our data and we drop firms that have an average of fewer than 4 in any of the two samples. 3) For each of the firms present in both the 1% and 3% samples, we compute the average number of workers of each firm across time, and we drop firms that have different average numbers of workers in the two samples. Descriptive statistics for the set of firms and workers left after the filtering are presented in column I of Table <ref>.