0}\;\sum\limits_{\text{Euclidean}\atop\text{triang.}\,T}\frac{1}{C(T)}\,{\rm
e}^{-S^{\text{R}}[T]},$ (31)

appeared in 1991 [214, 213], and were followed by a boost of activity (see
[101] for an overview of results).252525The measure in the path integral (31)
is “democratic”, in the sense that each triangulation $T$ carries the same
weight, up to the factor $C(T)$, denoting the size of the automorphism group
of $T$, a natural number almost always equal to 1. They found evidence of a
two-phase structure, consisting of a crumpled and a branched-polymer phase,
with a third, ”crinkled” phase appearing when a nontrivial measure term is
included in the path integral [215]. However, neither during the 1990s nor in
recent re-examinations of the Euclidean version of DT quantum gravity evidence
of a second-order phase transition has been found [216, 217, 218], which makes
it difficult to attach a standard continuum interpretation to this model,
apart from the challenges of interpreting it in Lorentzian terms or
establishing its unitarity, given the absence of reflection positivity.

## Q27: What are causal dynamical triangulations?

Causal dynamical triangulations (CDT) were introduced as an attempt to make
nonperturbative quantum gravity more physical (i.e. to allow for notions of
time and causal structure, unlike in Euclidean models) and come up with
quantitative evidence for nontrivial continuum limits [16, 219]. One can
simply think of CDT as a “Lorentzian variant” of Euclidean DT, but this
understates the challenge one usually faces in quantum gravity, namely, to
describe a quantum theory of Lorentzian spacetimes while maintaining access to
powerful computational tools that are needed to extract the physical
properties of these theories, which typically requires a real, Euclidean
version of the path integral. In CDT, both aspects are realized due to the
presence of a well-defined “Wick rotation” [220, 100], an analytic
continuation in one of the geometric parameters of the model, which maps the
regularized Lorentzian path integral

$Z^{\text{CDT}}=\lim_{a\rightarrow
0}\;\sum\limits_{\text{Lorentzian}\atop\text{triang.}\,T}\frac{1}{C(T)}\,{\rm
e}^{iS^{\text{R}}[T]}$ (32)

to a real partition function which can be investigated with numerical
approximation methods. Note that this partition function is different from the
Euclidean path integral (31), because the sum is over a different ensemble of
triangulations, which – importantly – leads to inequivalent outcomes.

The CDT path integral addresses all difficulties raised in Q24. All elements
of its construction are defined on a regularized version of the space
${\mathcal{G}}(M)$ of geometries, and the path integral has no residual gauge
invariance. Unless the bare inverse Newton constant is chosen too large, the
conformal divergence (28) coming from the gravitational action is entropically
suppressed: configurations that maximize the kinetic term exist, but are
relatively rare in the nonperturbative ensemble in the continuum limit [221,
191]. The regularized geometries carry a natural discrete time parameter, with
respect to which reflection positivity has been shown [220], and which on
sufficiently large scales turns out to be proportional to cosmological proper
time [222, 223]. The presence of an explicit time parameter is the simplest
way to enforce global hyperbolicity, which is an important feature that
distinguishes the CDT from the DT ensemble. However, explicit studies in
$D\\!=\\!3$ suggest that a time foliation is not strictly necessary to achieve
this [224, 225].262626Note that, unlike in the continuum, e.g. in the
$(3\\!+\\!1)$-decomposition of canonical gravity (cf. Q19) or in Hořava-
Lifshitz gravity [226], a choice of time here is not related to any breaking
of diffeomorphism invariance, since this invariance is already manifest. Also,
it is not a priori associated with any physical notion of time, see [100] for
further discussion.

Like DT, also CDT quantum gravity is amenable to Markov chain Monte Carlo
methods and has already been studied extensively (see [73, 100, 227] for
reviews). The model has a nontrivial phase structure and two lines of second-
order phase transitions [228, 229, 230, 231, 232], which provide candidates
for critical points where a continuum theory may be defined. For
topology272727For technical convenience, time in the computer simulations is
often compactified. $M\\!=\\!S^{1}\\!\times\\!S^{3}$, at least one of the
observed phases, the so-called de Sitter phase, features an emergent four-
dimensional geometry [233, 234], which is a highly nontrivial result as will
be explained in Q28. In the sense of expectation values, the large-scale shape
of the emergent geometry is that of a classical de Sitter space [222, 223,
235]. Likewise, its large-scale curvature properties are compatible with those
of a de Sitter space [236], despite being extremely quantum-fluctuating on
short scales.

There is a proof-of-concept study of renormalization group flows in the CDT
phase space [23], but mainly for technical reasons it has not yet been
possible to make definite statements about these flows near the second-order
phase transitions. This is needed to better understand the nature of the UV
completion and whether or not the asymptotic safety scenario (cf. Q23) can be
corroborated. Irrespectively, CDT appears to be a valid calculational
framework that makes it possible to measure quantum observables within an
order of magnitude of the Planck scale (see Q28, Q29), uncovering “Planckian
fingerprints” [237, 223] as well as extracting classical limits [238]. This
constitutes considerable progress in a field where traditionally there were no
operationally well-defined, approach-independent quantities to be computed.
One central task of nonperturbative quantum gravity is to uncover more of such
diffeomorphism-invariant observables. Lastly, it should be noted that (C)DT
quantum gravity theories have very few free parameters and a large degree of
universality. This means they can potentially be falsified, a highly desirable
property for any candidate quantum gravity theory to have. For this, it is not
necessarily required to develop testable phenomenology that probes the deep
quantum regime

## Q28: How can the dimension of quantum spacetime not be 4?

One of the most challenging and exciting tasks in nonperturbative quantum
gravity is to understand the nature of quantum geometry near the Planck scale
and to develop appropriate notions (read “quantum observables”) and
approximation tools to explore this terra incognita. Here, the word “geometry”
should be understood in a generalized sense, since we do not know a priori
which attributes of classical spacetime geometry, if any, will continue to be
meaningful in a Planckian regime.

Fortunately, as already mentioned in Q14, many important lessons have been
learned about quantum-geometric systems over the last decades, primarily in
the context of random geometry, which includes models of (causal) dynamical
triangulations in various dimensions, but also studies of causal set theory
[62] and tensor models [239, 240]. For systems of random geometry the logic is
straightforward: elementary building blocks of geometry are assembled into
metric spaces, which are then made to interact according to the rules of
quantum field theory, i.e. superposed in a path integral, where each geometry
appears with an amplitude weight that depends on its gravitational action, as
in (31) and (32).

The reason why such simple, non-exotic ingredients can lead to dynamical
systems with nontrivial and sometimes very unexpected properties has to do
with their nonperturbative collective behaviour, which at the level of their
partition functions is governed by a complex interplay between energy and
entropy, and with the fact that their construction involves a nontrivial
limiting process. In fact, one of the unexpected and unwelcome lessons one has
learned is that in $D\\!\geq\\!3$, systems of Riemannian random geometries,
made from $D$-dimensional building blocks, generically exhibit geometrically
degenerate behaviour that does not allow for an interpretation in terms of
extended, $D$-dimensional geometry on any scale above the cutoff scale.

How is it possible that a superposition of spacetimes, each one assembled from
four-dimensional simplices, generates a quantum spacetime that is not four-
dimensional? One way in which this can happen is that in a particular region
of phase space the four-simplices preferably arrange themselves into a
structure that at the cutoff scale $a$ is a four-dimensional manifold (by
construction), but when viewed from a larger scale282828any finite physical
scale, after making the lattice infinitely fine, i.e. in the limit
$a\rightarrow 0$ looks like a branched polymer. The latter is a well-known
statistical system, characterized by a Hausdorff dimension of 2 and a spectral
dimension of 4/3 on all scales. In other words, the dimension of the
microscopic building blocks does not determine the dimension of the emergent
“quantum spacetime”!

Unfortunately, the properties of branched polymers are incompatible with the
existence of a classical limit that reproduces the extended spacetimes of
general relativity. Over time, it has been understood that the occurrence of
this pathological behaviour is generic, and is not only present in DT quantum
gravity, where it does not even depend on the dimensionality of the simplices
[70, 241, 71], but also beyond [242]. Polymerization is not the only type of
degeneracy afflicting ensembles of Euclidean random geometry beyond
$D\\!=\\!2$ and potentially standing in the way of a large-scale gravity
interpretation [243, 244]. However, requiring a well-defined causal structure
to be present for path integral histories, as is done in CDT quantum gravity,
gives rise to a phase structure which is not entirely dominated by such
degeneracies, but allows for the presence of behaviour compatible with four-
dimensionality on large scales, as we will describe below.

The above discussion highlights the important role of dimensionalities as
diffeomorphism-invariant observables characterizing spacetimes beyond
classicality. While superstring theory has taught us not to take spacetime
dimension 4 for granted microscopically, the dynamical mechanism at work here
is quite different and directly related to the presence of large quantum
fluctuations on short scales and the absence of any a priori preferred
classical background. As is illustrated by the example of the branched-polymer
behaviour, in nonperturbative quantum gravity these dimensions will
generically not assume their canonical, classical value of 4, and may not even
be an integer, which implies that they are not associated with any classical
space. The latter is perfectly acceptable in a deep quantum regime, but not in
the limit of large, classical distances.

Another nonperturbative and qualitatively new phenomenon is that the dimension
of spacetime can be scale-dependent. This was first discovered in CDT quantum
gravity [237] for the spectral dimension $D_{S}$, which associates a dimension
with the behaviour of a diffusion process. On a compact manifold $M$ of volume
$V(M)$, one considers the probability $P(x,x,\sigma)$ of a random walker to
return to a point $x\\!\in\\!M$ after a diffusion time $\sigma$. To obtain a
diffeomorphism-invariant quantity, one integrates $P$ over the manifold to
obtain the average return probability

$\mathcal{R}_{V}(\sigma)=\frac{1}{V(M)}\int\\!\\!d^{4}x\ \sqrt{\det
g}\,P(x,x,\sigma)\sim\frac{1}{\sigma^{D_{S}/2}},$ (33)

from which the spectral dimension $D_{S}$ is extracted as (minus two times)
the power of the leading term in a small-$\sigma$ expansion [73]. When
evaluating (33) on the ensemble of CDT configurations in $D\\!=\\!4$, one
finds that the expectation value $\langle D_{S}\rangle$ on large scales is
given by $4.02\pm 0.1$, i.e. compatible with 4, signalling a correct classical
limit, but near the Planck scale drops smoothly to $1.80\pm 0.25$, which is
compatible with the value 2 within measurement accuracy [237, 234]. The scale
in this context is set by $\sigma$, because the typical maximal linear
distance away from the point $x$ reached by the random walker is of the order
of $\sqrt{\sigma}$.

Following the discovery of this dynamical dimensional reduction near the
Planck scale, many arguments and computations have corroborated this
phenomenon from the perspective of other quantum gravity approaches (see e.g.
[245, 246, 247, 248, 249]). What makes the short-scale spectral dimension such
an attractive quantity is the fact that it can be computed across a whole
range of candidate theories, in a field largely devoid of any computable
“numbers”. It has been suggested [250] that dimensional reduction may be a
universal feature of nonperturbative quantum gravity292929see [100] for some
cautionary remarks, which means that $D_{S}\\!=\\!2$ could have a status
similar to the famous formula $S_{\text{BH}}\\!=\\!A/4$ for the entropy of a
black hole of area $A$, which by contrast is a prediction of semi-classical
gravity. Despite the fact that it is currently difficult to derive
phenomenological consequences of a Planck-scale dimensional reduction in a
stringent manner, the spectral dimension is already being used as a benchmark
for comparing different quantum theories of gravity.

## Q29: What (other) quantum observables are there?

When introducing observables in pure gravity in Q6, we explained that they are
necessarily nonlocal in nature, as a consequence of background independence.
The same holds at the level of the quantum theory. For example, as we saw in
eq. (33), the extraction of the spectral dimension involves a spacetime
average. For this or any other observable ${\mathcal{O}}[g]$, its expectation
value $\langle{\mathcal{O}}\rangle$ in the nonperturbative quantum theory is
obtained by taking an additional ensemble average,

$\langle{\mathcal{O}}\rangle=\frac{1}{Z}\int\limits_{{\cal G}(M)}\\!\\!{\cal
D}[g]\,{\mathcal{O}}[g]\,{\rm e}^{iS^{\rm EH}[g]},$ (34)

where for simplicity we are using a formal continuum language, and $Z$ was
defined in eq. (27) above. Although it may appear natural to use observables
${\mathcal{O}}[g]$ which are spacetime integrals of scalars constructed from
the Riemann tensor and its covariant derivatives, this is in general not
feasible, both technically and as a matter of principle. In a continuum
formulation, each instance of the Riemann tensor contains double derivatives
of the metric at the same point, which must be regularized and renormalized
while respecting diffeomorphism invariance, a highly nontrivial requirement.
Taming of the associated infinities may not be possible, and may introduce
ambiguities or an uncontrolled gauge dependence. In discrete or piecewise flat
nonperturbative formulations, tensor calculus is generally not available and
derivatives are not defined.

Due to these difficulties, diffeomorphism-invariant $n$-point
functions303030where the locations of the $n$ points are integrated over,
subject to fixed pairwise geodesic distances of curvature scalars are for the
most part beyond reach in four dimensions, although they would in theory
provide for a rich set of observables. The quantum observables that have been
constructed and studied so far rely on notions of geodesic distance and
volumes, which do not require the presence of a smooth geometric structure,
and have a relatively simple functional form in terms of these basic
quantities. Of course, the need for a proper renormalization of physical
observables persists. Other considerations regarding feasibility have to do
with computational and numerical limitations. For models of lattice gravity,
these relate e.g. to system size, computing power and the efficiency of the
Monte Carlo algorithms.

Given these multiple requirements, it is rather nontrivial that a number of
quantum observables have nevertheless been identified and investigated:

$\bullet$

A notion of dimension different from the spectral dimension of Q28 is the
(local) Hausdorff dimension $d_{H}$, which was already mentioned in Q17. It is
determined from the scaling behaviour of the volume of a geodesic ball
${\mathcal{B}}_{R}(x)$ of geodesic radius $R$ centred at $x\\!\in\\!M$. To
obtain a diffeomorphism-invariant quantity, the ball volume
$\text{vol}({\mathcal{B}}_{R}(x))$ is averaged over $M$, and $d_{H}$ is
defined by the leading power of its expectation value

$\langle\,\overline{\text{vol}({\mathcal{B}}_{R})}\,\rangle\sim R^{d_{H}}$
(35)

for small $R$. A large-scale or cosmological Hausdorff dimension can be
extracted by relating the average linear size of a spacetime to its total
volume, for different volumes. In the de Sitter phase of CDT quantum gravity
it was determined to be 4 within measuring accuracy [234, 73], signalling a
good classical limit.

$\bullet$

A geometric observable of a global nature is the volume profile or shape
$V_{3}(\tau)$ of the spacetime, which is given by the three-volume of spatial
slices as a function of (cosmological) proper time $\tau$. The latter is a
geometric concept, obtained by measuring the geodesic distance from an initial
surface or an initial singularity, emulating a corresponding classical
construction [251]. In CDT quantum gravity with spacetime topology
$M\\!=\\!S^{1}\\!\times\\!S^{3}$, the expectation value $\langle
V_{3}(\tau)\rangle$ can be mapped onto the volume profile of a (Euclidean) de
Sitter space with great accuracy [222, 223, 235], while for spacetime topology
$M\\!=\\!S^{1}\\!\times\\!T^{3}$ the volume profile is constant in $\tau$
[252]. Remarkably, both nonperturbative models therefore reproduce the scale
factor of a solution of a classical cosmological minisuperspace model in the
sense of expectation values [100, 253].313131Note that the three-volume
satisfies $V_{3}(\tau)\\!\propto\\!a^{3}(\tau)$, where $a(\tau)$ is the scale
factor of cosmology mentioned in Q17. Moreover, the dynamics of the quantum
fluctuations $\delta V_{3}(\tau)$ of the three-volume around its mean measured
in simulations is consistent with a semiclassical analysis of the
corresponding quantum fluctuations in the continuum [223, 73]. The quantum
effective action for the scale factor can also be studied in asymptotic
safety, with consistent findings [254].

$\bullet$

The most recent ingredient in constructing quantum observables is a notion of
Ricci curvature applicable to nonsmooth metric spaces. This quantum Ricci
curvature, inspired by the Ollivier-Ricci curvature in pure mathematics [255],
is extracted by comparing the distance between two geodesic spheres with the
distance of their respective centres [256, 257]. The simplest observable one
can construct from it is the so-called curvature profile $\bar{K}_{\delta}$
[258], the spacetime average of a quantity $K_{\delta}(x)$, which has the
interpretation of a Ricci scalar coarse-grained over a neighbourhood of
diameter $\delta$ at the point $x$. Evaluating its expectation value
$\langle\,\bar{K}_{\delta}\,\rangle$ gives rise to a scale-dependent,
renormalized Ricci curvature that turns out to be well defined in a Planckian
regime. It has been used in CDT quantum gravity to show that the averaged
scalar curvature of its emergent quantum geometry is compatible with that of a
classical de Sitter space. This further strengthens the evidence for the
existence of a classical limit of this particular quantum gravity theory, and
paves the way for more elaborate curvature observables, which also use the
directional character of the quantum Ricci curvature [236].

These examples illustrate the nature and scope of the type of quantum
observable one has access to in nonperturbative quantum gravity. The fact that
these quantities are very different from those one typically studies in
classical general relativity is not a bug, but a feature of Planckian physics.
This is a regime where most questions one asks classically are simply not
operationally well defined, because of the absence of a classical background
and any preferred coordinate system, and the presence of large quantum
fluctuations.

The entropy of a black hole is a case in point: to make sense of this notion
in the nonperturbative quantum theory would in the first instance require us
to develop a diffeomorphism-invariant characterization of what constitutes a
black hole in this context. This is not at all straightforward, since the
usual treatments use a semiclassical setting of quantum fields on a
(quasi-)classical background geometry and do not involve the full,
nonperturbative dynamics of the gravitational field itself. By the same token,
there are no obvious, renormalized Planckian analogues of event
horizons323232see [259] for exploratory considerations of how to construct a
“quantum horizon finder” in CDT, areas and entropy. Secondly, the question is
whether quantum black holes, in whichever way defined, can be shown to arise
dynamically in quantum gravity, in a regime we have computational access to.
This may depend e.g. on the choice of boundary conditions for the path
integral, but at the present stage little is known about this in any
nonperturbative approach. For a related perspective on black holes and their
entropy in the asymptotic safety scenario, see [179].

The ongoing challenge of quantum gravity is to improve our calculational
abilities to systematically explore existing observables and to identify new
ones. When working in lattice approaches, it should be noted that by far not
every coordinate- or labeling-invariant quantity one can define on finite
lattices (e.g. by counting lattice substructures of a certain type) will lead
to an interesting observable in the continuum limit. Many such quantities will
be discretization artefacts, in the sense of not exhibiting interesting
scaling behaviour in a finite-size scaling analysis [199, 98] as the lattice
size increases. This also implies that they will not have a large-scale
classical limit.

Lastly, let us comment on the nonlocal nature of purely gravitational
observables. It is sometimes suggested to use relational observables of the
type “scalar quantity $X$ at the point where scalar quantity $Y$ assumes the
value 5.8”. This localizes the quantity $X$ in the point(s) where
$Y\\!=\\!5.8$, but does not make the corresponding quantum observable local in
an operational sense, since each configuration in the sum over geometries has
to be scanned globally to find these points. In addition, such observables
tend to be functionally involved, which also hinders their implementation in a
Planckian regime. A different way forward to close the gap between (semi-)
classical and nonperturbative quantum gravity is to compute the nonlocal,
integrated observables that are used in the quantum theory (like the curvature
and volume profiles mentioned above) on selected classical curved manifolds.
This seems particularly suggestive for the early universe, which is still
(relatively) small, and where one would like to compare nonperturbative
findings with continuum results obtained on a fixed cosmological background.

## Q30: What is the future of quantum gravity?

An overall picture emerges from the panorama of questions and answers we have
presented in these lecture notes. After 35 years of excursions into the world
of extended fundamental objects, quantum gravity has entered a post-loop and
post-string era. We are witnessing a renaissance of quantum field
theory333333see [260] for a recent example in a post-string spirit, with the
benefit of computational techniques that have been developed during roughly
the same period, to deal with the dynamical nature of gravity and to explore
the nonperturbative sector of the quantum theory. Covariant, path-integral
methods are being employed almost universally.

In pursuit of constructing a nonperturbative quantum gravity theory valid on
all scales, without relying on the presence of asymptotically flat or
stationary regions of spacetime, we examined which types of tools can get us
there. Given the state of the art in understanding interacting quantum field
theories in four dimensions beyond perturbation theory, analytical methods
alone are highly unlikely to provide the answer. In the absence of guidance
from experiment, numerical tools are therefore essential to develop and test
our candidate theories of quantum gravity quantitatively.

Progress in quantum gravity will depend on the continued development of
nonperturbative computational methods, hand in hand with theoretical and
analytical considerations. Approaches that lend themselves to a numerical
treatment have a clear advantage over others. We introduced the most promising
lattice gravity approaches currently on the market, which are based on the
powerful idea of working directly with spaces of geometries, avoiding the use
of coordinates. The aim of both lattice formulations and continuum treatments
in the context of asymptotic safety is to investigate how a nonperturbative
renormalization à la Wilson can be realized in quantum gravity [22, 179],
ideally complementing each other. Concrete, related research goals are to
further strengthen the evidence for an ultraviolet fixed point, and to explore
the universal and nonuniversal features of physics along renormalization group
trajectories running into the fixed point, using suitable observables.
Diffeomorphism-invariant observables are absolutely essential to progress;
however, we must keep in mind that quantities which can be meaningfully
studied in a Planckian regime tend to differ from their classical counterparts
(cf. Q29).

Tackling these issues is not at all trivial, but the fact that there is a
blueprint for how to proceed, which is not tied to any single approach, is a
sign of the growing maturity of the field. For example, any candidate theory
of quantum gravity with a sufficient degree of uniqueness, and whose
computational capabilities beyond perturbation theory are sufficiently
developed may provide further evidence for or against the currently favoured
high-energy scenario of asymptotic safety.

We have also highlighted why quantum gravity and the structure of its
nonperturbative ground state at the Planck scale can be very nontrivial,
despite the absence of any exotic ingredients and the adherence to the
principles of quantum field theory, albeit in a background-independent
context. This is illustrated by two-dimensional toy models of quantum (random)
geometry, which have a rigorous mathematical underpinning, and exhibit highly
nonclassical behaviour (cf. Q17, Q18). In full, four-dimensional quantum
gravity, the scale-dependence of the spectral dimension (see Q28) has given us
a first glimpse of the rich world of dynamical quantum geometry, where many
other nonperturbative surprises may be awaiting us. Not long ago we had little
idea that these structures existed and at the same time could have a well-
defined classical limit, since we lacked some theoretical insights343434We are
referring to the introduction of a causal structure à la CDT in lattice
gravity. In general, the role of the metric signature and the status of
unitarity in nonperturbative quantum gravity need to be understood better
[100, 185, 179]. and the appropriate computational tools to explore either
aspect.

This situation has improved significantly: although by far not a complete set,
we now have a number of quantum observables at our disposal (Q29), which allow
us to understand selected (near-)Planckian properties of quantum gravity in
quantitative terms. Their analysis provides both potential benchmarks for
comparing different approaches and stepping stones for further development,
most importantly, to determine to what extent any microscopic findings can
lead to physical consequences on more macroscopic scales. A particularly
fruitful area of research may be the very early universe, where one can look
for a dynamical, quantum-gravitational origin of “spacetime as we know it” and
for purely gravitational sources of structure formation. These are challenging
and ambitious goals, which promise a high reward and – if nature is kind to us
– may be just within the reach of a theory of quantum gravity.

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