11institutetext: Carlo Rovelli 22institutetext: Aix-Marseille University,
Université de Toulon, CPT-CNRS; The Perimeter Institute; and
The University of Western Ontario, _Department of Philosophy_ and _Rotman
Institute of Philosophy_
22email<EMAIL_ADDRESS>
Francesca Vidotto 33institutetext: The University of Western Ontario
_Department of Physics and Astronomy, Department of Philosophy, Rotman
Institute of Philosophy_
33email<EMAIL_ADDRESS>
The University of Western Ontario is located in the traditional lands of
Anishinaabek, Haudenosaunee, Lūnaapéewak and Attawandaron peoples.
This is a preprint of the chapter: “Philosophical Foundations of Loop Quantum
Gravity”, to appear in the “Handbook of Quantum Gravity”, edited by Cosimo
Bambi, Leonardo Modesto and Ilya Shapiro, 2023, Springer, reproduced with
permission of Springer.

# Philosophical Foundations of
Loop Quantum Gravity

Carlo Rovelli and Francesca Vidotto

###### Abstract

Understanding the quantum aspects of gravity is not only a matter of equations
and experiments. Gravity is intimately connected with the structure of space
and time, and understanding quantum gravity requires us to find a conceptual
structure appropriate to make sense of the quantum aspects of space and time.
In the course of the last decades, an extensive discussion on this problem has
led to a clear conceptual picture, that provides a coherent conceptual
foundation of today’s Loop Quantum Gravity. We review this foundation,
addressing issues such as the sense in which space and time are _emergent_ ,
the notion of _locality_ , the role of truncation that enables physical
computations on _finite graphs_ , the _problem of time_ , and the
characterization of the _observable quantities in quantum gravity_.

##  1 Introduction

What is a quantum spacetime? Are space and time emergent notions? If so, what
do they emerge from, and in which sense? Is there a fundamental ontology, from
which conventional space and time emerge? Does a quantum theory of gravity
require a specific time variable, as the Schrödinger equation does? If not,
what is the connection between the common notion of time and quantum gravity?
How is evolution described, in the absence of a canonical time or a fixed
background spacetime structure? Which empirically observable quantities are
well defined in a quantum spacetime? How do we compute their behavior? Are
they local is some sense?

These questions have been discussed widely and at length in the quantum
gravity literature, and routinely confuse those entering the field anew. Here
we address them, showing how a coherent conceptual framework for a quantum
theory of gravity can be cleanly defined.

We give a basic discussion of the notions of ‘space’ and ‘time’. This is
essential because substantial confusion derives from mixing up different ways
in which these notions are used. We discuss observability in general
relativistic physics and in quantum mechanics. Observability in quantum
gravity is subtle precisely because it combines the conceptual subtleties of
general relativity with those in quantum mechanics. We discuss the notion of
‘emergence’ and the so called ‘problem of time’, the precise role of the
_finite_ combinatorial structures (graphs and two-complexes) that enter
concrete calculations, their relation to locality and the reason they are
physically relevant. Our focus is on the conceptual structure of Quantum
Gravity in the Loop formulation (LQG) Rovelli:2004fk ; Rovelli2015 ;
Thiemann:2001yy ; GambiniPullinBook .

##  2 Two distinct notions of space

In its simplest usage, _space_ is the structure determined by a relation of
contiguity between physical entities. We use this notion when we say “I am in
London”, or ‘the electron has reached the detector”. In these cases we
_spatially locate_ entities (our self, London, an electron, a detector) with
respect to one another. This notion of space does not involve metric
quantities (distances, areas, volumes…) and refers to a _relation_ between
entities. This is a _relational_ notion of space.

In Newtonian physics, a different notion of space is employed. “Space”, in
this more specific usage, is a _container_ in which things, or observations,
are located, it is an entity assumed to exists by itself, independently from
the objects or the dynamical degrees of freedom. In this sense, “space” can
also be empty. Dynamical objects, or observations, are located “in space”,
namely they are located with respect to it. Space, in this sense, can have a
geometrical structure, which in Newtonian physics is described by three-
dimensional euclidean geometry. This notion of container space, formalized by
Newton (anticipated by ancient atomism), has played a fundamental role in the
development of physics. We employ it for instance when we use the Newton
equations to describe the dynamical of particles as (the evolution in time of)
their location in $\mathbb{R}^{3}$.111In the philosophical literature there
are other, distinct, discussions regarding relational aspects of space. One is
about the relational aspects of geometry. Another discussion is the confusion
between the velocity being relative versus acceleration being relative. A
third one regards the possibility of a relational reading of Newtonian
mechanics (in terms of reference systems). Here we are not referring to these
issues. The relation we refer to is simple contiguity, which does not require
any metric connotation. From the perspective of quantum gravity, Newtonian
space is better understood as special configuration of the gravitational
field: an entity. All these discussions can be traced to the famous
distinction at the beginning of the _Principia_ Newton1934 , and to Leibniz’s
relationalism: “As for my own opinion, I have said more than once, that I hold
space to be something merely relative, as time is, that I hold it to be an
order of coexistences, as time is an order of successions”. Third Paper,
paragraph 4; G VII.363 pg 25–26 Alexander ). Mixing up these various
discussions is an endless source of confusion.

The distinction between these two notions of space (relational and Newtonian)
is essential to get clarity in quantum gravity. In brief: the first extends to
the full quantum gravity regime, the second does not Rovelli2020b . The reason
the relational notion of space survives in quantum gravity is that contiguity
–and therefore localization– can be defined with respect to Newtonian or
classical general relativistic spacetime, but can also be defined with respect
to other entities, even in the absence of such Newtonian or classical general
relativistic spacetime. On the other hand, the reason the Newtonian
“container” notion of space needs to be abandoned, is that Newtonian space is
recognised on physical grounds to be an approximate description of a
particular configuration of a quantum field.

In its Newtonian version, the second of these notions of space _emerges_ ,
from the fundamental theory within a series of approximations:

  1. 1.

Newtonian space emerges from Minkowski space in the low relative-velocity
approximation (formally, this is a $c\\!\to\\!\infty$ limit, where $c$ is the
speed of light).

  2. 2.

Minkowski space emerges from a general relativistic (pseudo-)Riemannian
geometry at scales small with respect to the curvature radius (formally, it
can be identified with the tangent space at a point).

  3. 3.

A (pseudo-)Riemannian geometry emerges from the quantum geometry defined by
the LQG states and their dynamics, in a suitable classical limit (formally,
this is a $\hbar\\!\to\\!0$ limit).

The relevant notion of “emergence” here is a weak one, common in physics: in
some contexts, but not always, the physical system admits a convenient and
effective approximate description in terms of an “emergent” theory. The
emergent theory (here: General Relativity) is self-standing and autonomous,
and utilizes its own proper notions (here: relativistic spacetime). These can
be related to notions of the underpinning theory: they approximate certain
(not all) particular configurations of those.

Notice that the Newtonian intuition of the existence of space as an entity is
not contradicted neither by general relativity nor by LQG. Newtonian space is
simply better understood in these theories as the approximate description of a
—classical, or, respectively, quantum— dynamical field. With respect to this
field, location can be defined relationally, as it can be defined relationally
with respect to anything else.

In LQG, the spinnetwork states form a basis of states for the quantum
gravitational field. (Later on we shall be more precise about the meaning of
“state” in this context.) A spinnetwork state $|\Gamma,j_{\ell},v_{n}\rangle$
is determined by a labelled (abstract, not-embedded) graph $\Gamma$ with links
$\ell$ labelled by spins $j_{\ell}$ and nodes $n$ labelled by $SU(2)$
intertwiners $v_{n}$. These are a basis of $SU(2)$ invariant tensors in
$H_{n}=\otimes_{\ell\in n}V_{j_{\ell}}$, where the tensor product is over the
links $\ell$ adjacent to the node $n$ and $V_{j}$ is the Hilbert space of the
spin-$j$ representation of $SU(2)$. The nodes of the graph describe elementary
quantum excitations or “elementary quanta” of the gravitational field. (The
interpretation and the physical reasons for these discrete structures is
discussed later on, in Section 9.)

The links of the graph define a notion of contiguity between two nodes linked
by the link. The notion of contiguity between the elementary quanta determine
a spatial structure, in the sense of relational space. Therefore space, in the
relational sense, is present at the foundations of the theory. In other words,
the quanta represented by the nodes of the graph are spatially located with
respect to one another. Notice that they are _not_ located –in any sense– into
an external container space. In a slogan, they are not _in_ space; rather,
they themselves, with their contiguity relations, _make up_ a relational
space.

When matter is present Rovelli:2004fk ; Rovelli2015 ; Thiemann:2001yy , its
degrees of freedom are defined as labels on the same graphs $\Gamma$ as the
gravitational field states, therefore the notion of contiguity is equally
defined with respect to other quanta as well. That is, quanta of space and
other elementary quantum field excitations are all located with respect to one
another, defining a relational spatial structure.

Confusion between these two different meanings of “space” is at the root of a
common misunderstanding of non-perturbative quantum gravity. Some authors, for
instance, suggest that physics is inconsistent unless it is formulated in
terms of observables “located in space” Maudlin2007  taking for granted that a
notion of space as a container is necessary for understanding science. This
assumption has no base.

We locate our observations with respect to one another (both spatially and
temporally), but the idea that there should be a container space within which
observations are located is only a useful theoretical construct well utilized
by Newton, not a requirement for intelligibility. For millennia, before
Newton, humankind found the world perfectly conceivable in terms of relative
localization and not in terms of localization “in space”. In fact, even
“location” in everyday usage usually refers to adjacency to some material
object, for instance a location on Earth, and not a location with respect to
an abstract unobservable entity such as Newtonian space. To claim that science
is unintelligible without a container space is to fail to understand the
possibility of a conceptualization of the world different from the Newtonian
one. (See also the discussion in Huggett2013a ),

##  3 The emergence of the continuous metric space

Let us consider the third of the above approximations in some detail. The way
a continuous metric geometry emerges from the spinnetwork states is similar to
the way a continuous electromagnetic field emerges from the photon states of
quantum electrodynamics. A continuous (intrinsic) 3d Riemannian geometry $g$
can be approximated arbitrarily well by a 3d Regge triangulation formed by
flat tetrahedra $\tau_{n}$ connected by triangles $t_{\ell}$. The (two
skeleton of the) dual of this triangulation defines a graph $\Gamma$. Its
geometry can be captured by the variables

$G^{\ell\ell^{\prime}}_{n}=A_{\ell}A_{\ell^{\prime}}\
\vec{n}_{\ell}\cdot\vec{n}_{\ell^{\prime}}$ (1)

where $A_{\ell}$ are the areas and $\vec{n}_{\ell}$ the unit normals to the
face of the tetrahedron $\tau_{n}$. The operators

$\hat{G}^{\ell\ell^{\prime}}_{n}|\Gamma,j_{\ell},v_{n}\rangle=(\hbar
G)^{2}|\Gamma,j_{\ell},\vec{E}^{\ell}_{n}\cdot\vec{E}^{\ell^{\prime}}_{n}v_{n}\rangle$
(2)

where the vector operators $\vec{E}^{\ell}_{n}$ are $SU(2)$ generators on the
$V_{j_{\ell}}$ tensor component of $H_{n}$, are defined on the Hilbert space
${\cal H}_{\Gamma}$ spanned by the spinnetwork states on $\Gamma$. It can be
shown that there are ‘intrinsic’ coherent states $|\psi_{g}\rangle\in{{{\cal
H}_{\Gamma}}}$ such that

$\langle\psi_{g}|\hat{G}^{\ell\ell^{\prime}}_{n}|\psi_{g}\rangle=G^{\ell\ell^{\prime}}_{n}+O(\hbar)$
(3)

and the variance of these operators goes to zero with $\hbar$ Thiemann:2002vj
; Livine:2007mr ; Bianchi2011 .

Similarly, the _extrinsic_ geometry of a Riemannian 3d space embedded in a 4d
spacetime can be approximated by the an extrinsic geometry $k$ of a
triangulated 3d space. This is captured by the 4d dihedral angles
$\theta_{\ell}$ between normals to the tetrahedra; ${{\cal H}_{\Gamma}}$
carries a corresponding operator $\hat{\theta}_{\ell}$ Bianchi:2009ky ;
Freidel:2010aq  and it can be shown that there are (minimally spread)
‘extrinsic’ coherent states $|\psi_{g,k}\rangle\in{{{\cal H}_{\Gamma}}}$ that
satisfy the last equation as well as

$\langle\psi_{g,k}|\hat{\theta}_{\ell}|\psi_{g,k}\rangle=\theta_{\ell}+\mathcal{O}(\hbar).$
(4)

In simpler words: there are quantum states of LQG that approximate Riemannian
intrinsic and extrinsic geometries in the classical limit, in the usual sense
in which states of any quantum theory approximate configurations of its
classical limit.

Furthermore, there is some evidence (see below) that the LQG transition
amplitudes define a dynamics that approximates pseudo-Riemannian geometries
that solve the Einstein equation, arbitrary well. In this sense, the pseudo-
Riemannian spacetime of classical general relativity can ‘emerge’ from quantum
gravity: it is the standard sense in which classical trajectories ‘emerge’
from a quantum theory.

Various aspects of the Newtonian notion of space are lost when moving to more
general frameworks: the special relativistic framework loses the notion of a
preferred space foliation which is present in Newtonian spacetime; the general
relativistic framework loses the notion of a metric structure independent from
the dynamical degrees of freedom; the full quantum gravitational framework
lose the notion of continuous physical space. These notions are useful within
approximations, but are not appropriate to describe nature in a full quantum
gravitational regime. This is not a problem because they are not needed to
have a coherent and intelligible conceptual picture of reality.

On the other hand, the relational notion of space defined by contiguity of
dynamical entities, which is the familiar one we use when talking about space
in our everyday life, remains well defined in quantum gravity. Hence, certain
aspect of the intuitive notion of space (continuity, space as a container,
Riemannian geometry…) are _emergent_ , others (relational space) are not.
Those that emerge, emerge from the dynamics of the spinnetworks (states of the
_quantum_ gravitational field), in a way which is to a large extent similar to
the way a continuous electromagnetic field emerge from discrete photons
(states of the _quantum_ electromagnetic field).

In a full gravitational regime, the metric structure of spacetime displays the
typical quantum features. The most prominent of these are:

  1. (i)

The granularity implied by the discrete spectrum of the
$\hat{G}^{\ell\ell^{\prime}}_{n}$ operatorsRovelli:1994ge . This is the most
distinct feature and the key result of LQG Vidotto:2013jia . (On a possible
discreteness of time see Rovelli2015c  and on a possibility of actually
measuring it see Christodoulou2018b .)

  2. (ii)

The fact that the geometry can be in quantum superposition of states with
definite geometrical properties, with the usual characteristic phenomena such
as interference and entanglement. On the possibility of testing this
phenomenon (already predicted by non-relativistic quantum gravity), see
Christodoulou2018c .

  3. (iii)

The short scale fuzziness due to the fact that the various operators defining
geometry do not commute. Even the $\hat{G}^{\ell\ell^{\prime}}_{n}$ operators
do not all commute with one another Ashtekar1998 , hence cannot be
diagonalized together: this fact determines the quantum fuzziness of the
(intrinsic) 3d geometry at short scale.

We close this section addressing two issues raised in the philosophy
literature Huggett2013a . The first regards the interpretation of the states
that are superpositions of spinnetworks with different graphs. How is
contiguity well defined if there is more than one graph, and two graphs define
a different notion of contiguity? The answer to this question is in the
overall structure of the LQG Hilbert space. A Hilbert space ${\cal
H}_{\Gamma^{\prime}}$ spanned by the spinnetworks with given abstract graph
$\Gamma^{\prime}$ is a (proper) subspace of any Hilbert space ${\cal
H}_{\Gamma}$ where $\Gamma^{\prime}$ is a sub-graph of $\Gamma$. Any
specification of a superposition of states with different graphs
$\Gamma^{\prime}$ and $\Gamma^{\prime\prime}$ must be written as a state in a
state space ${\cal H}_{\Gamma}$ where both $\Gamma^{\prime}$ and
$\Gamma^{\prime\prime}$ are subgraphs of $\Gamma$. (See Section 8 below.)
Unless otherwise specified, the quantum superposition between two states with
different graphs must be interpret as a state in the Hilbert space ${\cal
H}_{\Gamma}$ where $\Gamma$ is the formed by the two _disconnected_ components
$\Gamma^{\prime}$ and $\Gamma^{\prime\prime}$. This resolves any ambiguity.

The second issue raised in Huggett2013a  is the observation that the notion of
adjacency defined by the graph may not match the one implicitly defined by an
averaged large geometry. Hence there may be two distinct notions of contiguity
in the theory: the one defined by the graph (that underpins the dynamics of
the spin networks) and the one defined by the emergent smooth geometry. This
is correct. The same happens in classical general relativity. A wormhole
smaller than the scale of observation can connect macroscopically distant
regions of spacetime. A microscopic notion of contiguity does not need to
match the macroscopic one. There can be “wild” geometries in classical general
relativity, where a similar mismatch happens, and this does not jeopardize
intelligibility.222In addition, regular geometries can be discretized in terms
of “wild” triangulations. See later. Similarly, there are “wild” states in
LQG. We do not know the physical relevance (if any) of either.

To clarify why two notions of adjacency do not represent a problem, it is
useful to ask what is the physical meaning of adjacency. The answer should not
be searched in a Kantian a priori condition for intelligibility, but in what
we have learned from experience about the world around us. The adjacency
relation that we experience is rooted in the fact that physical interactions
are local. Because of this, we only directly affect –and we only directly
receive information from– adjacent entities. In other words, the basic spatial
structure of the world is determined by what directly affects what. The
dynamics of loop quantum gravity is local on the graph (both in the
Hamiltonian and in the covariant formulations of LQG). This is why the
locality relation that we experience derives from the locality defined by the
graph structure of the states.

##  4 Observability in gravitational physics

The conceptual structure of general relativity (GR) is subtle and has confused
all relativists (including Einstein) for a long time. For this reason, decades
have been necessary before getting clarity on question like the nature of the
Schwarzschild singularity, or whether gravitational waves are physical or
gauge artifacts.

Much of the confusion stems from the fact that the theory is written in terms
of spacetime coordinates $x$ and $t$, but the physical meaning of these is
totally different from the physical meaning of the spacetime coordinates with
the same name used in special relativity and in non-relativistic physics. The
spacetime coordinates $X$ and $T$ in non relativistic and special relativistic
physics have _metric_ meaning: the spacetime coordinates $x$ and $t$ in
general relativistic physics do not have metric meaning. That is, in special
relativity for a particle to have position $X$ means to be at a physical
distance $X$ from the axis of some established physical reference frame. This
distance can be measured with a rod, a laser, or anything else. For an event
to have coordinate $T$ means to happen when a clock has measured a time lapse
$T$. Not so in GR: in GR, if a particle has position $x$ this does not mean
that the particle is at a physical distance $x$ from something. For an event
to have coordinate $t$ does not mean that a clock has measured a time lapse
$t$ from some initial time. Distance (measured in any of the above manners)
and clock readings are rather given by integrals involving the gravitational
field, such as

$T=\int_{\gamma}\sqrt{g_{ab}dx^{a}dx^{b}},$ (5)

where $g_{ab}(x)$ is the gravitational field. The value of these integrals
does not change if the coordinates are changed to new coordinates. The fact
that the coordinates have such a dramatically different meaning in the two
contexts raises continuous confusion.

Related to this is a persistent confusion about the connection between the
theory and the physical measurable quantities, usually called “observables”.
The reason of the confusion is the following. The Einstein equations are
invariant under arbitrary changes of the coordinates $x$ and $t$. It follows
that in general a quantity that depends on the coordinates $x$ and $t$ cannot
be predicted by the theory.

There are three alternative ways of interpreting this fact and using the
theory, all three equally valid (see Rovelli:1990ph ).

  1. 1.

The first is to only consider observables that are invariant under coordinate
transformations. These are predicted by the Einstein equations, once a
solution is specified. For instance, the minimal distance between the Earth
and the Moon during the current month, as measured by the return (proper-)time
on Earth of a laser signal bounced off the moon, is a quantity that does not
depend on the coordinates chosen. All quantities measured in relativistic
observational gravity can be interpreted in this manner.

  2. 2.

The second option is to interpret the coordinates as labels of concrete
reference objects whose dynamics is determined by the theory. This is a gauge
fixing of the coordinate choice, and as such it promotes the coordinates to
quantities that can be actually determined physically. This procedure is
commonly followed for instance in cosmology (in the homogeneous
approximation), where coordinates are attached to galaxies and the proper time
on these. In this language, the Einstein equations are gauge fixed on
particular coordinate choices.

  3. 3.

The third option is again to interpret the coordinates as labels of concrete
reference objects but to disregard the dynamical laws governing these
reference objects. The under-determination in the evolution equations can then
be interpreted as the result of disregarding these dynamical laws, namely
choosing physical reference systems that move arbitrarily in spacetime.

The three options are all viable, and ultimately equivalent. They refer to
different sets of variables. While the first and the second refer to the
physics of the dynamical degrees of freedom included in the theory, with
nothing else interacting, the third refers to the the physics of the dynamical
degrees of freedom of the theory interacting with other degrees of freedom.

In other words, the gauge degrees of freedom of general relativity can
alternatively be:

  1. 1.

interpreted as unphysical, namely just as a redundancy of the mathematics; or

  2. 2.

gauge fixed; or

  3. 3.

interpreted as relational degrees of freedom describing the coupling with an
external (arbitrarily moving) physical system, used as physical reference
system Rovelli2014a .

In all interpretations, spacetime localization is only relative. In the first
case, objects and events in the theory are localized with respect to one
another (the Earth and the laser pulse). In the second, they are localized
with respect to the chosen reference system (for instance, the galaxies and
their clocks). In the third, they are localized with respect to the external
arbitrarily moving reference system. Much of the conceptual confusion about
the the observables of general relativity and about the interpretation of
”spacetime points” comes from mixing up these three cases.

A consideration is important for what follows. Are observables in general
relativity _local_ in some sense? Let us consider two examples taken from
actual applications of the theory, where physical quantities are concretely
measured by experimentalists and astronomers, and compared to the theory. As a
first example, consider a detection of a gravitational wave pulse by a
gravitational interferometer. This can be thought as a curvature measurement
in a location defined by components of the detector. It is _local_ in the
sense that it only involves what happens in the location of the detector. A
second example is a typical measurements in the analysis of the general
relativistic dynamics of the Solar System. Astronomers measure the physical
distance between a given point on Earth and a given point on Venus, at a
specific time determined by some event of Earth, and defined as half the
forward-backward travel time of a laser pulse that bounces off Venus, where
the travel time is in terms of the Earth proper time. The measured quantity is
_local_ in the sense that it only involves what happens in the Solar System.
Now, say we interpret the theory according to the first option above. Are the
actual measured diffeomorphism invariant variables _local_ , in the sense that
they can be expressed as local functions of the coordinates in the dynamical
system formed by all the entities involved? Of course they are not, because no
diffeomorphism invariant quantity is local in this sense. These examples
should lead us to caution: one often reads that the absence of “local”
observables represents a major obstacle in interpreting general relativity and
quantum gravity. This is certainly not the case in the classical theory, as
shown by these examples. We shall come back on this in the quantum context.

Notice that if we adopt the third reading of general covariance, the
interpretation of what is measured in the two examples above simplifies
dramatically. Take the case of the detection of the gravitational wave pulse.
In the first interpretation, we consider the coupled dynamical system formed
by the gravitational field and the interferometer, and the measured quantities
is a highly non local function of the basic variables. In the third
interpretation, instead, we can think that the system under consideration is
just the gravitational field, and view the laboratory containing the detector
as an external (“reference”) system with which the gravitational field is
interacting. The quantity measured by the detector is a _local_ function of
the metric, in the location determined by the detector. The full
diffeomorphism invariance of the pure gravity dynamics, in other words, is
physically broken by the detector itself being located somewhere.

The two interpretations are equally correct, and both have advantages. As we
shall see, the second opens up an interesting window of opportunity in the
quantum context, in relation to the necessity of considering a Heisenberg cut
in quantum measurements.

Mixing up three interpretations above is also the source of the confusion in
the discussion about the meaning of spacetime points in general relativity
(and the “hole argument” Norton2019 ) which has been going on in philosophy of
science Roberts2020 . The discussion is confused by the fact that in the first
interpretation there is no physical definition of points independently from
the degrees of freedom _of the theory_ , but in the third there is such
definition (because the points, individuated by the coordinates, are defined
relationally with respect to the external arbitrarily moving reference
system.)

The third of the above interpretations is the reason for the strong
(irresistible, for some) intuitive appeal of the reality of a manifold
independent from the value of the gravitational field defined over it. The
points of the manifolds are possibilities for coupling other degrees of
freedom. Once we include all degrees of freedom, the manifold is dispensable
(as many relativists like to repeat Earman ). The same is true for the graph
of a spinnetwork and the two-complex of a spinfoam, if these are considered
independently from their labeling.

This discussion, more broadly, also sheds light on the general interpretation
of gauge invariance: gauge is more than mathematical redundancy, because the
gauge degrees of freedom capture ways a physical system can couple with other
physical systems. This is because (gauge invariant) couplings can couple to
gauge variant variables of a component-system. A discussion on this fact is in
Rovelli2014a .

##  5 General relativistic evolution

Physics describes processes, namely how things happen, or how they “change”.
To do so, general relativistic physics employs a more subtle notion of
evolution than Newtonian physics.

In Newtonian physics, evolution is described by writing equations that govern
how physical variables change _in time_. In general relativistic physics,
dynamical processes are described by writing equations that govern how
physical variables (including those characterising clocks) change _with
respect to one another_ Rovelli1994b ; GambiniPullin ; Giesel:2006uj ;
Domagala:2010bm . A characteristic example of utilization of this relative
notion of evolution is Loop Quantum Cosmology Agullo2013a  where the dynamics
of the universe is often coded in the relative evolution between the
cosmological scale factor and the value of a homogeneous scalar field.

More precisely, in Newtonian physics we use dynamical variables $A,B,...$ plus
a ‘special’ (preferred, canonical,…) time variable $T$. We call ‘clocks’ the
measuring devices that best track this variable. The time variable $T$ is used
as the independent variable of the evolution, and we write equations of motion
for the functions $A(T),B(T),...$. A motion can equally be represented as a
line in the space of the variables $(T,A,B,...)$, defined implicitly
(‘covariantly’) by functions of the form $f(T,A,B,...)=0$.

In a general relativistic physics, evolution is described by writing equations
that govern how physical variables (including those measured by clocks) change
_with respect to one another_. This is because there is no single canonical
time variable. Different ‘clocks’ determine distinct measurable variables
$T_{n},n=1,2,...$. Accordingly, we define evolution by giving relations
between all variables including the clocks. A motion is therefore described by
a line in the space spanned by all variables $(T_{n},A,B,..).$, defined
covariantly by functions of the form $f(T_{n},A,B,...)=0$. This line can be
parametrized by an arbitrary label $t$: this is the relativistic time
coordinate, which should not be confused with the readings $T_{n}$ of clocks.

The above is easily generalized to field theory. In a 4d (non-general-
relativistic) field theory, evolution is described by equations for fields
$A(X,T),B(X,T),...$ that depend on spacetime coordinates $(X,T)$. These
coordinates represent distances measured by rods and time intervals measured
by clocks. A motion can equally be represented as a 4d surface in the space
spanned by the variables $(X,T,A,B,...)$. In a general relativistic field
theory like general relativity, on the other hand, evolution is described by
writing equations that govern how physical variables (including distances
measured by rods and time intervals measured by clocks) change _with respect
to one another_. Evolution is given by relations between all variables
including clock variables $T_{n}$ and distance variable $X_{n}$. A motion is a
4d surface in the space of all variables $(X_{n},T_{n},A,B,...)$. This surface
can be parametrized by arbitrary 4d labels $(x,t)$: these are the relativistic
spacetime coordinates.

The fact that there is no preferred time variable in relativistic
gravitational physics can be seen for instance by noticing that in general two
clocks measure different (_proper_) times between the same couple of events,
depending on their location, speed, etcetera. Consider the following example:
launch a clock $C_{1}$ upward at some moment and catch it back when it falls
at a second moment. In the meanwhile, hold a second clock in your hands. The
two clocks will measure two different times $T_{1}$ and $T_{2}<T_{1}$ between
the launch and the catch. Which one is _the real_ time variable? The answer is
that there is no “real” time : both times are physical times, and can be taken
as independent variables.

The way evolution is treated in general relativistic physics is reflected in
the Hamiltonian structure of the dynamical theory. The general structure of
Newtonian physics is given by a Hamiltonian $H$ on a phase space $\Gamma$. The
phase space is a symplectic manifold. In a symplectic manifold, a function $H$
generates a flow: the physical motions are the orbits generated by the flow of
$H$ in $\Gamma$.

General relativistic physics requires a generalization of this structure,
which we sketch here (for more details, see Rovelli:2004fk ). The
generalization is given by a constraint $C$ on an extended (symplectic) phase
space $\Gamma_{ex}$. The symplectic form on $\Gamma_{ex}$ induces a pre-
symplectic form on the constraints surface $C=0$. The motions are the lines
(surfaces in field theory) on the constraint surfaces whose tangents are null
directions of the pre-symplectic form. The general (finite dimensional) case
reduces to the Newtonian case when $\Gamma_{ex}$ is the Cartesian product of
$\Gamma$ and a space with canonically conjugate coordinates $(T,p_{T})$, and
$C=H+p_{T}$, as can be easily verified.

The quantities like $T,A,B,...$, that include dependent as well as independent
dynamical variables are called _partial observables_ Rovelli2002b . These are
quantities that can be measured but cannot be individually predicted even with
full knowledge of the motion (as they include the independent variables of the
evolution). What the theory predicts is not the value of individual partial
observables, but, rather, relations among them. For instance, for a harmonic
oscillator with a single degree of freedom, its position $X$ and the time $T$
are both partial observables, and once the motion is known (amplitude and
phase are known), the theory predicts the value of $X$ for any given $T$, or
the possible values of $T$ for any given $X$.

The _physical phase space_ is the space of the motions. It is again a
symplectic space, and in the case of a Newtonian theory is isomorphic to the
familiar space of the initial data, but not canonically so, because the
isomorphism is determined by a value of $T$. Each point of the physical phase
space determines a relation between partial observables.

The motions can be parametrized. That is, the surfaces defined by the
functions $f(A_{n})$, where $A_{n}$ coordinatize the space of the partial
observables of the theory, can be written as functions $A_{n}(x)$ of arbitrary
coordinates $x$. The spacetime coordinates used in general relativity are
these parameters.

In quantum theory, the partial observables are represented by self adjoint
operators on an _extended_ Hilbert space ${\cal H}_{ex}$ and the dynamics is
given by a constraint operator (or set of operators) $C$ defined on ${\cal
H}_{ex}$. The transition amplitudes that define the quantum dynamics
amplitudes are given by

$W(a,b)=\langle a|P|b\rangle$ (6)

where $|a\rangle$ and $|b\rangle$ are eigenstates of (a complete set of
commuting) observables defined on ${\cal H}_{ex}$. $P$ is the projector of the
kernel of $C$ if this is a proper subspace of ${\cal H}_{ex}$. If zero is in
the continuum spectrum, $P$ can equally be defined using distributional
techniques, see for instance Marolf1994 ; Marolf1995 ; Rovelli:2004fk .

The transition amplitudes (19) define the quantum dynamics of a general
covariant quantum field theory.

To compute probabilities from the amplitudes we must remember that $H_{ex}$
describe partial observables, which include the independent variables. We can
therefore only assign probabilities to some components (say, $a_{1}$) of
$a=(a_{1},a_{0})$ at given value of others ($a_{0}$). That is, probabilities
are well defined when

$\sum_{a_{1}}|W(a_{1},a_{0},b)|^{2}=1.$ (7)

If the set of variables $a_{o}$ include a variable $t$ such that the dynamics
is symmetric under a translation in $t$, then the Hilbert space carries a
unitary representation of the group $\mathbb{R}$. If there is no variable with
this property, then there is no unitarity in this sense in the theory, but
this does not mean that probabilities are ill defined. For an enlightening
simple example of a well defined quantum system without unitarity in this
sense, see Colosi:2003si . For a discussion on the definition of probability
in the general case, see Oeckl:2005bv .

##  6 Observability in quantum physics

Classical physics assumes that all properties of a system are always sharply
defined. Not so quantum physics. Properties are given by eigenvalues of
observables and quantum mechanics only assigns properties to a system in the
context of an interaction with another system. The boundary between the two is
called the _Heisenberg cut_ VonNeumann1955 . For instance, in the Copenhagen
interpretation properties are actualized in measurements; in the relational
interpretation, they are always relative to a second system; in the many-world
interpretations, they depend on the branching and are related to Everett’s
relative states determined by a split due to a Heisenberg cut.333The
terminology “Heisenberg cut” is characteristic of the Copenhagen
interpretation, but we use it more generally to denote the separation between
systems which is needed in order to have actual values of variable also in
interpretations such as Many Worlds and Relational.

For simplicity, we use the language of the Copenhagen interpretation, but what
we say can be easily translated in the language of different interpretations.
In the Copenhagen interpretation we mentally distinguish the system from the
context, separated by the Heisenberg cut and treat the context classically.
The observables of the system take a value at a measurement, which is an
interaction between the system and the context. The theory predicts the
probability of one or the other value to be actualized in this interaction
(called _measurement_), given that other values of observables were actualized
in a previous interaction (called _preparation_). The cut can be moved outward
without changing the predictions. Denoting $a$ the set of the observables’
values actualized in the preparation and $b$ the set of the observables’
values actualized in the measurement, the conditional probabilities predicted
by the theory are given by

$P(b|a)=|W(b,a)|^{2}$ (8)

where the transition amplitude is given by

$W(b;a)=\langle b|a\rangle;$ (9)

here $|a\rangle$ and $|b\rangle$ are the relevant normalized eigenstates of
the operators corresponding to the relevant observables. (In the relational
interpretation the cut defines relational observables; in the Many World
interpretation, the cut separates two subsystems that defines the branching,
withing which variables have determined values…)

If a time $(t^{\prime}-t)$ different from zero lapses between the two
measurements and the hamiltonian is $H$, the transition probabilities are

$W(b,t^{\prime};a,t)=\langle b|e^{-\frac{i}{\hbar}(t^{\prime}-t)H}|a\rangle;$
(10)

In the general relativistic case, the time variable is included among the
partial observables, and we write the amplitude above as

$W(b,t^{\prime};a,t)=\langle b,t^{\prime}|P|a,t\rangle.$ (11)

Additionally, we can consider a ‘boundary’ Hilbert space $H_{b}=H_{in}\otimes
H_{out}$ namely the tensor product of the in and out state spaces, and express
the dynamics as a single (possibly generalised) bra on this boundary state
space

$W(b,t^{\prime};a,t)=\langle W|b,t^{\prime};a,t\rangle.$ (12)

For details on the formalism, see Rovelli:2004fk .

##  7 Observability in quantum gravity

To understand observability in quantum gravity we have to combine our
understanding of observability in general relativity with our understanding of
observability in quantum theory. From the second, we learn that what the
theory can predict is the probability of one or the other property of the
gravitational field to be actualized in interactions, across a Heisenberg cut,
with a context that can be treated as classical. The natural and simple key
for this to work is to identify the Heisenberg cut with the boundary of a four
dimensional spacetime region $\cal R$ Conrady2003b . It is not difficult to
see that any realistic observation in relativistic gravitation can be
expressed in this form.

It is particularly convenient to take $\cal R$ to be compact, and bounded by a
3d surface $\Sigma$ formed by the union of a past and a future spacelike
surfaces $\Sigma_{-}$ and $\Sigma_{+}$ joined along a two sphere. The theory
is naturally expressed in the time gauge on these surfaces. This setting
permits us to interpret the LQG transition amplitudes as transition amplitudes
from $\Sigma_{-}$ to $\Sigma_{+}$. Quantum states on $\Sigma_{-}$ and
$\Sigma_{+}$ represent quantum geometries on these surfaces. These are
interpreted as interactions between the gravitational field on $\cal R$ and
the rest of the universe, across the Heisenberg cut defined by $\Sigma$.

This setting works very well in the two contexts where we expect quantum
gravitational phenomena to be non negligible: early cosmology and around black
hole singularities. (See below.)

A preparation and (a complete) measurement at the Heisenberg cut, namely at
$\Sigma$, determines the eigenvalues of a complete set of (_partial_)
observables, and a corresponding eigenstate $|\Psi\rangle\in{\cal H}_{b}$.
(Semiclassical coherent stares are more convenient in some applications.) The
dynamics is then given by a single bra $\langle W|$ on ${\cal H}_{b}$. This is
the covariant version of the transition amplitude

$\langle W|\psi\rangle=\langle\Psi_{+}|P|\Psi_{-}\rangle$ (13)

where $P$ is discussed in Section 5, the boundary Hilbert space is discussed
in the last Section and $|\Psi_{\pm}\rangle$ are quantum states of the
geometry of, respectively, $\Sigma_{\pm}$.

Formally, if the measured quantities correspond to the 3d (intrinsic) geometry
$g$ on $\Sigma$ and we call the corresponding eigenstate $\Psi_{g}$ we may
write

$\langle W|\Psi_{g}\rangle=\int_{\partial g_{4}=g}Dg_{4}\
e^{-\frac{i}{\hbar}\int\sqrt{-g_{4}}R[g_{4}]},$ (14)

where the (ill defined) functional integration of the exponent of the Einstein
Hilbert action is over the 4d geometries $g_{4}$ on $\cal R$ bounded by $g$ on
$\Sigma$, as originally suggested by John Wheeler and Charles Misner
Misner:1957fk . The spinfoam formalism can be viewed as a way to transform the
last formula into something well defined and computable, within arbitrary
truncations.

In a semi-classical regime we expect the classical dynamics of general
relativity to be recovered from the approximation

$\langle W|\Psi_{g}\rangle\sim\sum_{n}e^{-\frac{i}{\hbar}S_{n}[g]},$ (15)

where the sum is over the different solutions $g_{4}[g]$ of Einstein equations
on $\cal R$ that induce the 3-metric $g$ on $\Sigma$ and

$S_{n}[g]=\int_{\cal R}\sqrt{-g_{4}[g]}R[g_{4}[g]]$ (16)

is the corresponding Hamilton function of general relativity Rovelli:2004fk ,
namely the value of the action on a solution with given boundary data. As well
known, full knowledge of the Hamilton function is essentially equivalent to
knowledge of the solution of the equations of motion. (To see how this still
works in the generally covariant case, see Rovelli:2004fk .)

From the conceptual point of view that concerns us here, we notice that the
observables of quantum gravity can be chosen to be sitting on the Heisenberg
cut $\Sigma$ and be partial observables. Importantly, they do not need to be
fully gauge invariant, because they represent interactions between the quantum
system studied and (‘the measurement apparatus on’) the boundary of the
classical context. What the theory provides, then, are transition amplitudes
between partial observables, and these give the physical predictions, as
illustrated above.

An alternative way to derive gauge invariant predictions from quantum gravity
is to write fully gauge invariant observables on the phase space of the
theory. If we had a sufficiently rich family of such gauge invariant
observables, these could be used to compute transition amplitudes directly,
because fully gauge invariant observables in general relativity are like
operators in the Heisenberg picture. The scalar product between their
eigenstates would directly give transition amplitudes, exhausting all relevant
dynamical information. Many authors have explored this path Bergmann:1960wb ;
Bergmann:1961wa ; Page:1983uc ; Rovelli1991 ; Perez:2001gj ; Rovelli:2001my ;
Dittrich:2004cb ; Dittrich:2005kc ; Dittrich:2006ee ; Giddings2006 ;
Giesel:2009jp ; Kaminski:2009qb ; Donnelly2016a ; Donnelly2016b ; Duch2014 ;
Duch2015 ; Bodendorfer2015 ; Duch2016 . This strategy involves an infamously
hard task, as discussed in detail by many of the authors cited, from a variety
of points of view: Dirac observables are highly non-local and obey a non-local
algebra in a general relativistic setting. The difficulty has been addressed
in various manners, such as expanding around flat space, or coupling suitable
matter fields to use as references, or using nonlocal dressings of fields, or
using geodesics from infinity to define bulk localization, or considering the
particular class of Dirac observables called the ‘evolving constants of the
motion’ RovelliQM3 . Here we shall not review those intriguing possibilities,
for which we refer to the authors cited.

The boundary strategy for computing predictions in quantum gravity described
above, instead, circumvents the infamous difficulty of writing fully gauge
invariant observables (or “Dirac observable”) in general relativity as
explicit functions of the theory’s phase space. Explicit knowledge of these
functions is not necessary to extract information from a quantum theory of
gravity. In fact, it is not necessary to extract information from the
classical theory of gravity either. This is evident form the fact that we do
not know how to write such quantities on the phase space of general
relativity, and yet in general relativistic physics, astrophysics, and
cosmology, we can routinely do general relativistic observations,
measurements, and predictions, as illustrated in Section 4. Clearly there are
ways of extracting physically meaningful information from a general
relativistic theory even without being capable of writing its Dirac
observables.

In other words, to make predictions about the behaviour of a covariant system
it is not necessary to know explicit its physical Hilbert space and the
operators which leave the physical Hilbert space invariant. Knowledge of the
extended (or kinematical) Hilbert space ${\cal H}_{ex}$ and the transition
amplitudes is sufficient. For example, consider a particle in two dimensions
described in a covariant manner with states $|x,t\rangle\in{\cal H}_{ex}$. The
full theory can be expressed in terms of the transition amplitudes

$W(x,t;x^{\prime},t^{\prime})\equiv\langle
x,t|P|x^{\prime},t^{\prime}\rangle,$ (17)

where $P$ is an operator on ${\cal H}_{ex}$, without reference to a physical
Hilbert space of solutions of the “Wheeler-deWitt equation” whose solutions
define the physical Hilbert space. If we can deparametrize the theory, the
explicit form of the transition functions can be written in the well known
form

$W(x,t;x^{\prime},t^{\prime})\equiv\langle
x|e^{-\frac{i}{\hbar}H(t-t^{\prime})}|x^{\prime}\rangle,$ (18)

but this expression may be ill defined in the general case, while (17) remains
well defined. Intuitively, and in the cases where these equations are well
defined,

$W(x,t;x^{\prime},t^{\prime})\sim\langle
x,t|\delta(C)|x^{\prime},t^{\prime}\rangle\sim\int_{(x^{\prime},t^{\prime})\to(x,t)}DX\
e^{{\frac{i}{\hbar}S}[X]}$ (19)

where $C$ is the Hamiltonian constraint, $S$ the action and the integral is a
Feynman integral over paths $X$. In the covariant formulation of LQG, the
(truncated, see below) quantities $W$ are defined by the spinfoam amplitude
and are functions of boundary states representing 3-geometries, as in (14).
There is no need to ”deparametrize” or finding Dirac observables, to study
quantum gravitational processes. Quantum transition amplitudes, like
predictions in classical general relativity, can be formulated and computed
using only partial observables, working with gauge dependent quantities and
exploiting the third of the three interpretations of general covariance listed
in the Section 4.

The relational interpretation of quantum theory Rovelli:1995fv ; Laudisa2017
is a natural setting when quantum gravity is formulated in this manner. The
relational structure of space and time merge naturally and beautifully with
the relational structure of quantum theory: the Heisenberg cut is identified
with spacetime partitions. Notice that in the Copenhagen version, we still
need an outside classical spacetime. Not so in the relational interpretation,
where what matters in only that there is am (relational) boundary between two
systems, without any presupposed geometry on this boundary.

However, the problem of quantum gravity and the problem of the interpretation
of quantum mechanics are distinct and to a large extent independent.

A concrete example of utilization of LQG transition amplitudes is given by the
calculations of what happens at the end of the evaporation of a black hole.
Most of spacetime can be treated classically, because quantum gravitational
effects are negligible. Not so the high curvature region surrounding the
classical (unphysical, because of quantum gravity) singularity and the horizon
near the end of the evaporation. This is the compact quantum region $\cal R$.
A 3d surface surrounding it can be chosen and the LQG quantum transition
amplitudes describing what can happen at the end of the evaporation can be
explicitly studied: see Bianchi2018c ; Rovelli2014ps ; DAmbrosio2021 ;
Christodoulou2018d ; Soltani2021 . (This is possible because a _classical_
exact solution for the exterior exists Haggard2014 .)

Another concrete example is the use of this covariant formalism to study the
big bang Bianchi:2010zs ; Gozzini2019 . In this case the surface $\Sigma$ can
be taken to be a single spacelike surface with the topology of a 3-sphere
after the big bang, to describe the transition from nothing to a cosmological
space (as in the Hartle-Hawking scenario), or, alternatively, as two
disconnected spacelike surfaces with the topology of a 3-sphere, to describe a
Big Bounce, as in Loop Quantum Cosmology Vidotto:2011qa . In this cosmological
context, the average value of the spins can be taken as the independent
variable (a discretized version of the cosmological scale factor), in terms of
which the dynamics of the fluctuations of the rest of the geometry evolves.

##  8 Truncation, finite graphs and finite spinfoams

The bra $\langle W|$ that gives the dynamics is defined in covariant LQG in
terms of spinfoam amplitudes, order by order in a suitable sequence of
truncations that represent increasingly fine approximations. See Rovelli2015
for a detailed technical introduction of these. Here we only discuss the
conceptual structure of the theory.

A general remark is important here, to dispel a recurring conceptual
confusion: the idea that a quantum theory, and in particular a quantum theory
of gravity, could only describe _elementary_ components of nature. There is no
reason to expect so, and this is not the right way of viewing and using
quantum theory. Quantum theory is not a theory about the elementary components
of reality: it is a theory about the quantum behaviour of _any_ physical
variable, irrespectively on whether this is elementary or composite. The
angular momentum of a molecule, for instance, is quantized and can be used in
computing transition amplitudes in the dynamics of the molecule, independently
from the internal quarks’s structure of the molecule.

In the same manner, there is no reason to see the quantization of some aspects
of the metric field as a description of elementary components: any variable,
at any scale, behaves quantum mechanically and LQG is the description of
quantum properties of gravitational degrees of freedom, at any relevant scale.
A state does not represent a “thing”: it represents the outcome of an
interaction in which certain observables take on certain values. No
measurement delivers an infinite amount of information. Any measurement
captures a _finite_ number of degrees of freedom only. An effective
theoretical description of a given phenomenon needs only to refer to the
degrees of freedom that are relevant for that phenomenon.

This is also true in quantum field theory. In standard quantum field theory
calculations are always performed on finite lattices and finite graphs. For
instance, in lattice QCD hadron’s masses are computed on lattices of _finite_
size that are large enough to include the hadron and fine enough to see the
quarks’ wavelengths, but no more. Analogously, collisions in QED and in the
electroweak theory are computed order by order in a perturbation expansion: at
each order there is a maximum number of (real and virtual) particles involved,
and therefore, again, only a finite number of degrees of freedom involved. For
this reason, both the lattices concretely used in QFT calculations and the
(Feynman) graphs in QFT perturbation theory are finite (that is, they have a
finite number of vertices).

The same holds in loop quantum gravity: concrete calculations involve spin
networks and spinfoams with finite graphs and finite two-complexes. The
calculation is expected to provide results that approximate the physical
behavior of phenomena where the corresponding degrees of freedom play a role.
When describing a phenomenon (like the end of the evaporation of a black hole,
or the possible bounce of the primordial universe), we have to single out the
degrees of freedom that may play a relevant role in the corresponding
dynamics, and describe the process in terms of these, not in terms of
everything Borja:2011di ; Vidotto2017 .

Suggestions that calculations on finite graphs and finite spinfoams are
unreliable are therefore conceptually ill-founded.

A measurement of the geometry that captures a finite number of degrees of
freedom can be modelled as follows. Given a 3d metric space with geometry $g$,
consider a simplicial decomposition of the space and call $A_{\ell}$ the areas
of the 2-simplices $\ell$, and $\vec{n}_{\ell}\cdot\vec{n}_{\ell^{\prime}}$
the angle between two vectors normal to two 2-simplices $\ell$, and
$\ell^{\prime}$ bounding the same 3-simplex $n$, in an arbitrary point of
$\ell$ and $\ell^{\prime}$, parallel transported to an arbitrary internal
point of $n$. Then equation (1) defines a family of quantities that measures
the geometry $g$ at some scale. These quantities do not commute in quantum
theory. A (smaller) set of commuting quantities is given by the areas
$A_{\ell}$ of the two simplices and the volumes $v_{n}$ of the three
simplices. A volume operator $V_{n}$ is defined on ${{\cal H}_{\Gamma}}$,
where $\Gamma$ is the graph dual to the cellular decomposition. The operators
$(A_{\ell},V_{n})$ form a commuting set of operators in ${{\cal H}_{\Gamma}}$
which is maximal up to some signs that we disregard here for simplicity. These
operators have discrete spectrum Rovelli:1994ge . Let
$|\Gamma,j_{\ell},v_{n}\rangle$ be a basis in ${{\cal H}_{\Gamma}}$ that
diagonalizes this set. The states $|\Gamma,j_{\ell},v_{n}\rangle$ can be
interpreted to represent the outcomes $(j_{\ell},v_{n})$ of these
measurements, in the standard sense of quantum theory.

This does _not_ mean that a state like $|\Gamma,j_{\ell},v_{n}\rangle$ gives a
complete description of reality in a certain spacetime region. It _only_
refers to a subset of degrees of freedom measured. The theory that describe
_these_ degrees of freedom is a good description of reality to the extent the
dynamics of these degrees of freedom is not too affected by others.

In the covariant formulation, the LQG transitions amplitudes are defined in
terms of a sequence of 4d truncations, after fixing a relevant family of
boundary states (say in ${{\cal H}_{\Gamma}}$). Each truncation is defined by
the choice of a 2-complex $\cal C$ having $\Gamma$ as boundary. The spin foam
amplitudes define a bra $\langle W_{\cal C}|$ on ${\cal R}_{\Gamma}$ Engle2007
; Freidel:2007py ; Kaminski:2009fm ; Rovelli2015 . The theory is well defined
if refining the 2-complex the amplitude converges. Numerical calculations give
some partial positive indications that this can be the case exist, see for
instance Frisoni2022 .

##  9 Physical discreteness

It is important not to confuse the discreteness introduced by the various
truncations used to define the theory (the graph of the spinnetworks, the two-
complex of the spinfoams) with the physical Planck scale discreteness
predicted by LQG. The first is only a theoretical tool, analog to the lattice
of lattice QCD. The second (absent in QCD) is a hard physical prediction of
the theory, and the most characteristic feature of LQG. It is the analog of
the discreteness of the spectra of the energy of the Hydrogen atom or non
relativistic harmonic oscillator, or the discreteness of photons. It is
derived in the theory form the spectral analysis of the operators describing
the geometry Rovelli:1994ge . It is compatible with the local Lorentz
invariance of the theory Rovelli2003 . It is this physical discreteness which
is responsible for the ultraviolet finiteness of LQG and for the resolution of
the singularities of general relativity Ashtekar:2008ay ; Rovelli2013d .

The expression of the transition amplitudes as a spinfoam sum have much in
common with a standard lattice discretization of a Feynman sum over histories
Conrady:2008ea , like the one that defines lattice QCD. However, there is a
crucial difference Rovelli2011b : in theories like lattice QCD the full
quantum theory is recovered by sending the number of lattice sites to infinity
as well as the lattice spacing to zero. Because of the underpinning
diffeomorphism invariance, only the first of these limits (that is: refining
the two complex) is required in LQG. See a detailed discussion in Rovelli2011b
.

Here “limit” must be understood in the sense of potential, not actual. What we
do in physics is to compute transition amplitudes within approximations. This
is always done within a finite truncation, as we do in standard perturbative
QFT and lattice QCD. Therefore the theory can be formally defined by the
continuous limit, but the actual theory to be used is always at arbitrary but
finite truncation.444This observation may either be seen as a simple pragmatic
consideration, or, perhaps, as a way to question the physical relevance of the
actual limit theory Vidotto:2013oxa .

A different question is how we recover classical general relativity. For this,
we have to take both the continuum limit and a “classical” limit: namely look
at scales large with respect to the Planck scale. This is usually implemented
as a large spin limit. For a while, the LQG literature contained the wrong
expectation (giving rise to an apparent “flatness problem”) that the classical
limit could be taken before, and independently from, the continuum limit. This
is not the case: the two limits must be taken together, see Han2014a ;
Rovelli2015 ; Han2017 ; Asante2020 ; Engle2022 .555Specifically: at fixed
triangulation, the LQG amplitudes approximate sufficiently well the dynamics
of discretized general relativity (Regge theory) only if the triangulation is
sufficiently fine.

##  10 Three distinct notions of time

In moving from non-relativistic physics to quantum gravity, the notion of time
undergoes alterations similar to the notion of space. However, the notion of
time is more subtle than the notion of space, raising further issues.

In the case of space, we observed that clarity is obtained by distinguishing
the common relational notion of space, according to which objects are
spatially located with respect to one another –a notion still in play in
quantum gravity– from the Newtonian notion of space, as a continuous metric
manifold with an Euclidean geometry, which emerges only in approximations.

In the case of time, the same distinction holds. In everyday life we use a
relational notion of time. Time is just a counting of happenings in
successions: for example the succession of days and years. It is a fact of
nature that there are such successions of events. Newtonian physics, on the
other hand, postulates the existence of a physical time that is independent
from any succession of events, and has a rich structure: it has a metric
structure, it is the same all over the universe, defining a global
simultaneity, and so on. As well known, many features of such Newtonian time
are approximations: they do not describe correctly the actual temporal
structure of reality. There is no single canonical clock variable in the
universe, and no global simultaneity, except in dramatic approximations like
homogeneity and isotropy in cosmology.

The absence of a single preferred time variable and the fact that no single
variable has all the features typical of Newtonian time is just a fact of
nature, and is the reason for the generalization of mechanics illustrate in
section 5. The formal structure illustrated in that section permits to define
both the classical and the quantum dynamics coherently without having to
specify a preferred time variable. As observed, non relativistic physics
describes evolution as change of the variables in time, relativistic physics
describes evolution as change of the variables with respect to one another.

In the literature, a big deal had been made about the alleged existence of a
‘problem of time’ due to this absence of a canonical time. The confusion in
this issue stems from a mixing up two distinct questons:

  1. 1.

The first question is to understand how to describe dynamical evolution in a
relativistic setting when there is no canonical time variable.

  2. 2.

The second problem is raised by our strong feeling that time “flows” in a
sense that makes it different from any other physical variable. In a Newtonian
theory, we identify the flow of time with the change in the canonical variable
$T$ of the Newtonian formalism. But we cannot do so in a theory formulated in
a way that does not select any time variable as special.

The first of these two problems can be solved classically, as illustrated in
Section 5: a dynamical theory does not require a specific time variable to be
defined. All predictions by a general relativistic theory can be obtained
without specifying a canonical, or ‘special’, time variable. So, this question
can be consistently answered. The quantities predicted by the theory are
values of some variables when other variables have given values.

A corresponding quantum formalism can also be defined, as shown in Section 5.
Alleged “quantum” solutions of this same problem, such as the Page-Wootter
construction Page:1983uc  are nothing else than this same solution expressed
in the quantum domain.

The second of the above problems, on the other hand, is rooted on a conceptual
misunderstanding. We do experience a flow of time, of course. To understand
this experience we should look at our experience as it is in reality, and not
assume that our experience reaches out directly to the deep and general
structure of reality.

What we experience is due to the specific and complex situation in which we
are. Not only our experience is in the Newtonian limit (so that we
misinterpret aspects of this limit for universal features of nature), but it
is also strongly marked by the fact that we access a small subset of the
degrees of freedom of Nature, hence we experience macroscopic coarse grained
variables that happen to have thermodynamic properties. In particular, we
happen to live in a universe with a strong entropy gradient, where the
behaviour of these macroscopic observables has a marked irreversible
character. (We do not know why. An hypothesis for the reason of this is in
Rovelli:2015 , but this is irrelevant here.) From the perspective of the
fundamental theory, this fact is accidental.

A direct consequence of this fact is that our local present has abundant
traces of the past Rovelli2020 , and past low entropy allows macroscopic
histories to branch Rovelli2020a ; Rovelli1956 . These facts determine the
epistemic and the agential arrows of time, both aligned with the entropy
gradient. These phenomena, not any preferred fundamental temporal variable,
are principally responsible for the phenomenology of our experiential time.
For an ample discussion of all this, see Rovelli2018 . All this is very
interesting, of course, but has nothing to do with quantum gravity.

(Incidentally: a consistent thermodynamic and statistical theory of the
classical gravitational field is still missing, let alone for the quantum one.
This is one of the reasons of the confusion surrounding issues like black hole
entropy. For hints and clumsy attempts in this directions, see for instance
Rovelli2002c ; Haggard2013 ; Chirco2016 .)

The notions of time that need to be distinguished in order to get clarity in
quantum gravity are therefore three:

  1. 1.

Relational time is the notion that allows us to say that two local events
happen in a direct succession next to one another. This is the analog of
relational space. This notion remains true in quantum gravity: we compute
transition amplitudes for successions of local events.

  2. 2.

Newtonian time is a quantity that is well defined only under numerous
approximations are taken. Various features of Newtonian time are lost one
after the other as approximations are undone.

  3. 3.

Experienced time includes a rich phenomenology that depends on the specific
environment around us, especially the irreversibility due to the entropy
gradient, and on the functioning of our brain and its functioning in terms of
deliberations Ismael2022 ; Price2023 .

It is perhaps clarifying to distinguish a generic notion of “change” from the
specific concept of “time”. By “change” we may mean the most generic aspect of
temporal contingency, in the following sense. We experience in the world that
things can be in a certain way “sometime” and different “some other time”.
This is a notion which is local (not global across the universe), not
necessarily oriented, and does not require a single time variable to be
described. That is, we describe the world in terms of a certain number of
quantities: $(A,B,C,...)$ and the functional dependencies between these: we
can compile lists of observations
$(A_{1},B_{1},C_{1},...)...(A_{2},B_{2},C_{2},...)...$ giving us the values of
these quantities “changed”. Physics gives us equations that constrain these
changes. For instance the change in an oscillator is described by the two
partial observables $(X,T)$ and their relation $f(X,T)=X-A\sin(\omega
T-\phi)=0$, where $A$ and $\phi$ are constants. In this particular case, we
can recognize “$T$” as what we usually call time, but in general relativity
there is no such easy recognition, and in general none of the variables
$(A,B,C,...)$ have all the qualities we ascribe to time in Newtonian physics.
By “time” we indicate a particular variable among those describing the world,
that has a particular list of properties (for instance it is monotonic along
the change) and in the approximate description of the world obtained in the
non-quantum, non-relativistic limit is associated with the quantity measured
by our clocks and with our experiential time: the sense of passing we have in
our brain. So, change and time are different. The first is part of the
conceptualization in quantum gravity, the second is not.

##  11 Conclusion

A large number of important technical issues are open in quantum gravity666For
instance, the infrared “bubble” divergences Riello2013 ; Frisoni2021 ;
Dona2022 ; Han2021a ., not to mention the persistent lack of direct empirical
support. But LQG has at its disposal not only a powerful mathematical
formalism that represents a tentative theory of gravity, but also a coherent
conceptual picture within which a possible understanding of quantum spacetime
can be framed.

The relational notions of space and time that are familiar from our common
experience remain useful in quantum gravity: events can be “next” to one
another spatially and “next” to one another temporally. Not so the structure
of a general relativistic spacetime, which only emerges in approximations. A
general covariant formalism for dynamics is well defined and clear: it is
based on the notion of partial observables: quantities that can be measured
but in general cannot be predicted by themselves. The dynamical theory gives
the correlations between these, both in the classical and quantum domains. The
observable quantities in quantum gravity are the same as those of general
relativity: in principle, any measurement in relativistic gravitational
physics is also a measurement in quantum gravity. (Any measurement in
relativistic gravitational physics can be represented as performed across a 3d
surface in the form described above.)

The entire theory has a strong relational character: localization in space and
time is relational. Measurements imply relations between spacetime regions.
Evolution is given as relative evolution. Quantum states are interpreted as
relative states in the relational interpretation of quantum theory. (The
entropy gradient as well could be a perspectival phenomenon Rovelli:2015 .)
This deeply relational aspect of reality, that comes both from general
relativity and from quantum mechanics, and that merges so naturally in quantum
gravity, is perhaps the deepest insight that quantum gravity is offering into
the nature or reality Vidotto:2022aoz .

Acknowledgments –  Sincere thanks to Emily Adlam and Pascal Rodriguez for
their extensive and thoughtful comments, and to Muxin Han, Jared Wogan and
Sofie Reid for comments on the draft of the manuscript.

This work was supported by the John Templeton Foundation Grant No.62312
_“Quantum Information Structure of Spacetime”_ (QISS). FV’s work is supported
by the Canada Research Chairs Program and by the Natural Science and
Engineering Council of Canada (NSERC) through the Discovery Grant _”Loop
Quantum Gravity: from Computation to Phenomenology”_.

## References

  * (1) C. Rovelli, Quantum Gravity.  Cambridge University Press, 2004.
  * (2) C. Rovelli and F. Vidotto, Covariant loop quantum gravity: An elementary introduction to quantum gravity and spinfoam theory.  Cambridge Univeristity Press, 2015.
  * (3) T. Thiemann, “Introduction to modern canonical quantum general relativity,” 0110034.
  * (4) R. Gambini and J. Pullin, Loops, Knots, Gauge Theory and Quantum Gravity.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1996.
  * (5) I. Newton, Scholium to the Definitions in Philosophiae Naturalis Principia Mathematica, Bk. 1 (1689); trans. Andrew Motte (1729), rev. Florian Cajori.  University of California Press, Berkeley, 1934.
  * (6) H. Alexander, ed., The Leibniz-Clarke Correspondence.  Manchester University Press, Manchester.
  * (7) C. Rovelli, “Space and Time in Loop Quantum Gravity,” in Beyond Spacetime, pp. 117–132.  2020\.  1802.02382.
  * (8) T. Maudlin, “Completeness, supervenience, and ontology. : Mathematical and Theoretical,” Journal of Physics A: Mathematical and Theoretical 40 (2007) 3151–3171.
  * (9) N. Huggett and C. Wüthrich, “Emergent spacetime and empirical (in)coherence,” Studies in History and Philosophy of Science Part B - Studies in History and Philosophy of Modern Physics 44 (aug, 2013) 276–285, 1206.6290.
  * (10) T. Thiemann, “Complexifier coherent states for quantum general relativity,” Class. Quant. Grav. 23 (2006) 2063–2118, 0206037.
  * (11) E. R. Livine and S. Speziale, “Physical boundary state for the quantum tetrahedron,” Class. Quant. Grav. 25 (2008) 85003, 0711.2455.
  * (12) E. Bianchi, P. Doná, and S. Speziale, “Polyhedra in loop quantum gravity,” Physical Review D - Particles, Fields, Gravitation and Cosmology 83 (2011), no. 4, 044035, 1009.3402.
  * (13) E. Bianchi, E. Magliaro, and C. Perini, “Coherent spin-networks,” Phys. Rev. D82 (2010) 24012, 0912.4054.
  * (14) L. Freidel and S. Speziale, “Twisted geometries: A geometric parametrisation of SU(2) phase space,” Phys.Rev. D82 (2010) 84040, 1001.2748.
  * (15) C. Rovelli and L. Smolin, “Discreteness of area and volume in quantum gravity,” Nucl. Phys. B442 (1995) 593–622, 9411005.
  * (16) F. Vidotto, “Atomism and Relationalism as guiding principles for Quantum Gravity,” 1309.1403.
  * (17) C. Rovelli and F. Vidotto, “Compact phase space, cosmological constant, and discrete time,” Physical Review D - Particles, Fields, Gravitation and Cosmology (2015) 1502.0278.
  * (18) M. Christodoulou and C. Rovelli, “On the possibility of experimental detection of the discreteness of time,” 1812.01542v1.
  * (19) M. Christodoulou and C. Rovelli, “On the possibility of laboratory evidence for quantum superposition of geometries,” Physics Letters B 792, (2018), no. 10, 64–68, 1808.05842.
  * (20) A. Ashtekar, A. Corichi, and J. A. Zapata, “Quantum theory of geometry: III. Non-commutativity of Riemannian structures,” Classical and Quantum Gravity 15 (jun, 1998) 2955–2972, 9806041.
  * (21) C. Rovelli, “What is observable in classical and quantum gravity?,” Class. Quant. Grav. 8 (1991) 297–316.
  * (22) C. Rovelli, “Why Gauge?,” Foundations of Physics 44 (2014), no. 1, 91–104, 1308.5599.
  * (23) J. D. Norton, “The Hole Argument,” The Stanford Encyclopedia of Philosophy (Summer 2019 Edition) (2019).
  * (24) B. W. Roberts and J. O. Weatherall, “New Perspectives on the Hole Argument,” mar, 2020.
  * (25) J. Earman, World enough and space-time : absolute versus relational theories of space and time.  MIT Press, Cambridge, 1989.
  * (26) C. Rovelli and L. Smolin, “The physical Hamiltonian in nonperturbative quantum gravity,” Physical Review Letters 72 (1994), no. 4, 446–449, 9308002.
  * (27) R. Gambini and J. Pullin, “A rigorous solution of the quantum Einstein equations,” Phys. Rev. D 54 (1996) 5935–5938.
  * (28) K. Giesel and T. Thiemann, “Algebraic quantum gravity (AQG): I. Conceptual setup,” Class. Quantum Grav. 24 (2007) 2465–2497.
  * (29) M. Domagala, K. Giesel, W. Kaminski, and J. Lewandowski, “Gravity quantized: Loop Quantum Gravity with a Scalar Field,” Phys.Rev. D82 (2010) 104038, 1009.2445.
  * (30) I. Agullo and A. Corichi, “Loop Quantum Cosmology,” 1302.3833.
  * (31) C. Rovelli, “Partial observables,” Physical Review D 65 (2002), no. 12, 0110035.
  * (32) D. Marolf, “Almost ideal clocks in quantum cosmology: A brief derivation of time,” Classical and Quantum Gravity 12 (dec, 1995) 2469–2486, 9412016.
  * (33) D. Marolf, “Quantum observables and recollapsing dynamics,” Classical and Quantum Gravity 12 (apr, 1995) 1199–1220, 9404053.
  * (34) D. Colosi and C. Rovelli, “A simple background-independent Hamiltonian quantum model,” Phys. Rev. D68 (2003) 104008, 0306059.
  * (35) R. Oeckl, “General boundary quantum field theory: Foundations and probability interpretation,” Adv. Theor. Math. Phys. 12 (2008) 319–352, 0509122.
  * (36) J. von Neumann, Mathematical Foundations of Quantum Mechanics,.  Princeton University Press, Princeton, 1955.
  * (37) F. Conrady, L. Doplicher, R. Oeckl, C. Rovelli, and M. Testa, “Minkowski vacuum in background independent quantum gravity,” Physical Review D - Particles, Fields, Gravitation and Cosmology 69 (jul, 2004) 7, 0307118.
  * (38) C. W. Misner, “Feynman quantization of general relativity,” Rev Mod Phys 29 (1957) 497.
  * (39) P. G. Bergmann and A. B. Komar, “Poisson brackets between locally defined observables in general relativity,” Phys. Rev. Lett. 4 (1960) 432–433.
  * (40) P. G. Bergmann, “Observables in General Relativity,” Rev. Mod. Phys. 33 (1961) 510–514.
  * (41) D. N. Page and W. K. Wootters, “Evolution without evolution: Dynamics described by stationary observables,” Phys.Rev. D27 (1983) 2885\.
  * (42) C. Rovelli, “Quantum reference systems,” Classical and Quantum Gravity 8 (1991), no. 2, 317–331.
  * (43) A. Perez and C. Rovelli, “Observables in quantum gravity,” 0104034.
  * (44) C. Rovelli, “GPS observables in general relativity,” Phys. Rev. D65 (2002) 44017, 0110003.
  * (45) B. Dittrich, “Partial and complete observables for Hamiltonian constrained systems,” Gen. Rel. Grav. 39 (2007) 1891–1927, 0411013.
  * (46) B. Dittrich, “Partial and Complete Observables for Canonical General Relativity,” Class. Quant. Grav. 23 (2006) 6155–6184, 0507106.
  * (47) B. Dittrich and J. Tambornino, “A perturbative approach to Dirac observables and their space-time algebra,” Class. Quant. Grav. 24 (2007) 757–784, 0610060.
  * (48) S. B. Giddings, D. Marolf, and J. B. Hartle, “Observables in effective gravity,” Physical Review D 74 (2006), no. 6, 64018, 0512200.
  * (49) K. Giesel, J. Tambornino, and T. Thiemann, “LTB spacetimes in terms of Dirac observables,” Class. Quant. Grav. 27 (2010) 105013, 0906.0569.
  * (50) W. Kaminski, J. Lewandowski, and T. Pawlowski, “Quantum constraints, Dirac observables and evolution: group averaging versus Schroedinger picture in LQC,” Class. Quant. Grav. 26 (2009) 245016, 0907.4322.
  * (51) W. Donnelly and S. B. Giddings, “Observables, gravitational dressing, and obstructions to locality and subsystems,” Physical Review D 94 (jul, 2016) 1607.01025.
  * (52) W. Donnelly and S. B. Giddings, “Diffeomorphism-invariant observables and their nonlocal algebra,” Physical Review D 93 (jul, 2016) 1507.07921.
  * (53) P. Duch, W. Kamiński, J. Lewandowski, and J. Świezewski, “Observables for general relativity related to geometry,” Journal of High Energy Physics 2014 (mar, 2014) 1403.8062v2.
  * (54) P. Duch, W. Kamiński, J. Lewandowski, and J. Świeżewski, “Erratum to: Observables for general relativity related to geometry,” mar, 2015.
  * (55) N. Bodendorfer, P. Duch, J. Lewandowski, and J. Świeżewski, “The algebra of observables in Gaußian normal spacetime coordinates,” Journal of High Energy Physics 2016 (oct, 2016) 1–18, 1510.04154v2.
  * (56) P. Duch, J. Lewandowski, and J. Świezewski, “Observer’s observables. Residual diffeomorphisms,” Classical and Quantum Gravity 34 (oct, 2017) 1610.03294v1.
  * (57) C. Rovelli, “Quantum evolving constants,” Phys. Rev. D 44 (1991), no. 4, 1339–1341.
  * (58) C. Rovelli, “Relational Quantum Mechanics,” Int. J. Theor. Phys. 35 (1996) 1637, 9609002.
  * (59) F. Laudisa and C. Rovelli, “Relational Quantum Mechanics,” Stanford encyclopedia of philosophy (2017).
  * (60) E. Bianchi, M. Christodoulou, F. D’Ambrosio, H. M. Haggard, and C. Rovelli, “White holes as remnants: A surprising scenario for the end of a black hole,” Classical and Quantum Gravity 35 (2018) 225003, 1802.04264.
  * (61) C. Rovelli and F. Vidotto, “Planck stars,” Int. J. Mod. Phys. D 23 (2014) 1442026, 1401.6562.
  * (62) F. D’Ambrosio, M. Christodoulou, P. Martin-Dussaud, C. Rovelli, and F. Soltani, “The End of a Black Hole’s Evaporation - Part I,” Phys. Rev. D 103 (2021) 106014, 2009.05016.
  * (63) M. Christodoulou and F. D’Ambrosio, “Characteristic time scales for the geometry transition of a black hole to a white hole from spinfoams,” 1801.03027.
  * (64) F. Soltani, C. Rovelli, and P. Martin-Dussaud, “End of a black hole’s evaporation. II.,” Physical Review D 104 (may, 2021) 106014, 2105.06876.
  * (65) H. M. Haggard and C. Rovelli, “Quantum-gravity effects outside the horizon spark black to white hole tunneling,” Physical Review D 92 (2015), no. 10, 104020, 1407.0989.
  * (66) E. Bianchi, C. Rovelli, and F. Vidotto, “Towards Spinfoam Cosmology,” Phys. Rev. D82 (2010) 84035, 1003.3483.
  * (67) F. Gozzini and F. Vidotto, “Primordial Fluctuations From Quantum Gravity,” Frontiers in Astronomy and Space Sciences 7 (jun, 2021) 1906.02211.
  * (68) F. Vidotto, “Many-nodes/many-links spinfoam: the homogeneous and isotropic case,” Class. Quant Grav. 28 (2011), no. 245005, 1107.2633.
  * (69) Enrique F. Borja, Iñaki Garay, and Francesca Vidotto, “Learning about quantum gravity with a couple of nodes,” SIGMA 7 (2011).
  * (70) F. Vidotto, “Relational quantum cosmology,” in The Philosophy of Cosmology, pp. 297–316.  aug, 2017.  1508.05543.
  * (71) J. Engle, R. Pereira, and C. Rovelli, “Flipped spinfoam vertex and loop gravity,” arXiv.org gr-qc (2007).
  * (72) L. Freidel and K. Krasnov, “A New Spin Foam Model for 4d Gravity,” Class. Quant. Grav. 25 (2008) 125018, 0708.1595.
  * (73) W. Kaminski, M. Kisielowski, and J. Lewandowski, “Spin-Foams for All Loop Quantum Gravity,” Class. Quant. Grav. 27 (2010) 95006, 0909.0939.
  * (74) P. Frisoni, F. Gozzini, and F. Vidotto, “Markov Chain Monte Carlo methods for graph refinement in covariant Loop Quantum Gravity,” 2207.02881.
  * (75) C. Rovelli and S. Speziale, “Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction,” Physical Review D 67 (2003), no. 6,.
  * (76) A. Ashtekar, “Singularity Resolution in Loop Quantum Cosmology: A Brief Overview,” J. Phys. Conf. Ser. 189 (2009) 12003, 0812.4703.
  * (77) C. Rovelli and F. Vidotto, “Evidence for Maximal Acceleration and Singularity Resolution in Covariant Loop Quantum Gravity,” Physical Review Letters 111 (2013), no. 9, 091303, 1307.3228.
  * (78) F. Conrady and L. Freidel, “Path integral representation of spin foam models of 4d gravity,” Class. Quant. Grav. 25 (2008) 245010, 0806.4640.
  * (79) C. Rovelli, “Discretizing parametrized systems: the magic of Ditt-invariance,” 1107.2310.
  * (80) F. Vidotto, “Infinities as a measure of our ignorance,”  5, 2013.  1305.2358.
  * (81) M. Han, “On spinfoam models in large spin regime,” Classical and Quantum Gravity 31 (apr, 2014) 1304.5627.
  * (82) M. Han, “Einstein equation from covariant loop quantum gravity in semiclassical continuum limit,” Physical Review D 96 (2017), no. 2, 1705.09030.
  * (83) S. K. Asante, B. Dittrich, and H. M. Haggard, “Discrete gravity dynamics from effective spin foams,” 2011.14468.
  * (84) J. Engle and C. Rovelli, “The accidental flatness constraint does not mean a wrong classical limit,” Class. Quant Grav. (2022) 117001, 2111.03166.
  * (85) C. Rovelli, “Is Time’s Arrow Perspectival?,” in The Philosophy of Cosmology, K. Chamcham, J. Silk, J. Barrow, and S. Saunders, eds., pp. 285–296.  Cambridge University Press, 2016.  1505.01125.
  * (86) C. Rovelli, “Memory and entropy,” Entropy 2022 24 (2022) 1022, 2003.06687.
  * (87) C. Rovelli, “Agency in Physics,” in Experience, abstraction and the scientific image of the world. Franco Angeli editore, 2021.  2007.05300.
  * (88) C. Rovelli, “Back to Reichenbach,” http://philsci-archive.pitt.edu/20148/ (2021).
  * (89) C. Rovelli, The Order of Time.  Riverhead, New York, 2018.
  * (90) C. Rovelli, “A note on the foundation of relativistic mechanics. I: Relativistic observables and relativistic states,” in Proceedings of the 15th SIGRAV Conference on General Relativity and Gravitational Physics.  Rome, 2002.  0111037.
  * (91) H. M. Haggard and C. Rovelli, “Death and resurrection of the zeroth principle of thermodynamics,” Physical Review D - Particles, Fields, Gravitation and Cosmology 87 (2013), no. 8, 1302.0724.
  * (92) G. Chirco, T. Josset, and C. Rovelli, “Statistical mechanics of reparametrization-invariant systems. It takes three to tango.,” Classical and Quantum Gravity 33 (2016), no. 4, 1503.08725.
  * (93) J. Ismael, “The Open Universe: Totality, Self-reference and Time,” Australasian Philosophical Review (2022) to appear.
  * (94) H. Price, “Time for Pragmatism,” in Neo-pragmatism, J. Gert, ed.  Oxford University Press, 2023.
  * (95) A. Riello, “Self-Energy of the Lorentzian EPRL-FK Spin Foam Model of Quantum Gravity,” Physical Review D - Particles, Fields, Gravitation and Cosmology 88 (feb, 2013) 1302.1781v2.
  * (96) P. Frisoni, F. Gozzini, and F. Vidotto, “Numerical analysis of the self-energy in covariant loop quantum gravity,” Physical Review D 105 (dec, 2022) 2112.14781v2.
  * (97) P. Donà, P. Frisoni, and E. Wilson-Ewing, “Radiative corrections to the Lorentzian Engle-Pereira-Rovelli-Livine spin foam propagator,” Physical Review D 106 (jun, 2022) 2206.14755v3.
  * (98) M. Han, “Four-dimensional spinfoam quantum gravity with a cosmological constant: Finiteness and semiclassical limit,” Physical Review D 104 (aug, 2021) 2109.00034.
  * (99) F. Vidotto, The relational ontology of contemporary physics.  1, 2022.  2201.00907.