Carlo Rovelli

Baptiste Le Biha

Space and Time in Loop Quantum Gravity

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I. INTRODUCTION

Newton's success sharpened our understanding of the nature of space and time in the XVII century. Einstein's special and general relativity improved this understanding in the XX century. Quantum gravity is expected to take a step further, deepening our understanding of space and time, by grasping of the implications for space and time of the quantum nature of the physical world.

The best way to see what happens to space and time when their quantum traits cannot be disregarded is to look how this actually happens in a concrete theory of quantum gravity. Loop Quantum Gravity (LQG) [START_REF] Rovelli | Quantum Gravity[END_REF][START_REF] Rovelli | Covariant Loop Quantum Gravity[END_REF][START_REF] Thiemann | Modern Canonical Quantum General Relativity[END_REF][START_REF] Ashtekar | Introduction to Loop Quantum Gravity[END_REF][START_REF] Gambini | Loops, Knots, Gauge Theories and Quantum Gravity[END_REF][START_REF] Gambini | Introduction to loop quantum gravity[END_REF][START_REF] Perez | The Spin-Foam Approach to Quantum Gravity[END_REF] is among the few current theories sufficiently developed to provide a complete and clear-cut answer to this question.

Here I discuss the role(s) that space and time play in LQG and the version of these notions required to make sense of a quantum gravitational world. For a detailed discussion, see the first part of the book [START_REF] Rovelli | Quantum Gravity[END_REF]. A brief summary of the structure of LQG is given in the Appendix, for the reader unfamiliar with this theory.

II. SPACE

Confusion about the nature of space -even more so for time-originates from failing to recognise that these are stratified, multi-layered concepts. They are charged with a multiplicity of attributes and there is no agreement on a terminology to designate spacial or temporal notions lacking same of these attributes. When we say 'space' or 'time' we indicate different things in different contexts.

The only route to clarify the role of space and time in quantum gravity is to ask what we mean in general when we say 'space' or 'time' [START_REF] Van Fraassen | An introduction to the philosophy if time and space[END_REF]. There are distinct answers to this question; each defines a different notion of 'space' or 'time'. Let's disentangle them. I start with space, and move to time, which is more complex, later on.

Relational space: 'Space' is the relation we use when we locate things. We talk about space when we ask "Where is Andorra?" and answer "Between Spain and France". Location is established in relation to something else (Andorra is located by Spain and France). Used in this sense 'space' is a relation between things. It does not require metric connotations. It is the notion of space Aristoteles refers to in his Physics, Descartes founds on 'contiguity', and so on. In mathematics it is studied by topology. This is a very general notion of space, equally present in ancient, Cartesian, Newtonian, and relativistic physics.

This notion of space is equally present in LQG. In LQG, in fact, we can say that something is in a certain location with respect to something else. A particle can be at the same location as a certain quantum of gravity. We can also say that two quanta are adjacent. The network of adjacency of the elementary quanta of the gravitational field is captured by the graph of a spin network (see Appendix). The links of the graph are the elementary adjacency relations. Spin networks describe relative spacial arrangements of dynamical entities: the elementary quanta.

Newtonian space: In the XVII century, in the Principia, Newton introduced a distinction between two notions of space [START_REF] Newton | Scholium to the Definitions in Philosophiae Naturalis Principia Mathematica[END_REF]. The first, which he called the "common" one, is the one illustrated in the previous item. The second, which he called the "true" one, is what has been later called Newtonian space. Newtonian space is not a relation between objects: it is assumed by Newton to exist also in the absence of objects. It is an entity with no dynamics, with a metric structure: that of a 3d Euclidean manifold. It is postulated by Newton on the basis of suggestions from ancient Democritean physics, and is essential for his theoretical construction. 1 Special relativity modifies this ontology only marginally, merging Newtonian space and time into Minkowski's spacetime.

In quantum gravity, Minkowski spacetime and hence Newtonian space appear only as an approximations, as we shall see below. They have no role at all in the foundation of the theory.

General relativistic space: Our understanding of the actual physical nature of Newtonian space (and Minkowski spacetime) underwent a radical sharpening with the discovery of General Relativity (GR). The empirical success of GR -slowly cumulated for a century and recently booming-adds much credibility to the effectiveness of this step.

What GR shows is that Newtonian space is indeed an entity as Newton postulated, but is not nondynamical as Newton assumed. It is a dynamical entity, very much akin to the electromagnetic field: a gravitational field. Therefore in GR there are two distinct spacial notions. The first is the simple fact that dynamical entities (all entities in the theory are dynamical) are localized with respect to one another ("This black hole is inside this globular cluster"). The second is a left-over habit from Newtonian logic: the habit of calling 'space' (or 'spacetime') one particular dynamical entity: the gravitational field. There is nothing wrong in doing so, provided that the substantial difference between these three notions of space (order of localization, Newtonian non-dynamical space, gravitational field) is clear.

LQG treats space (in this sense) precisely as GR does: a dynamical entity that behaves as Newtonian space in a certain approximation. However, in LQG this dynamical entity has the usual additional properties of quantum entities. These are three: (i) Granularity. The quantum electromagnetic field has granular properties: photons. For the same reason, the quantum gravitational field has granular properties: the elementary quanta represented by the nodes of a spin network. Photon states form a basis in the Hilbert state of quantum electromagnetism like spin network states form a basis in the Hilbert space of LQG. (ii) Indeterminism. The dynamics of the 'quanta of space' (like that of photons) is probabilistic. (iii) Relationalism. Quantum gravity inherits all features of quantum mechanics including the weirdest. Quantum theory (in its most common interpretation) describes interactions among systems where properties become actual. So happens in LQG to the gravitational field: the theory describes how it interacts with other systems (and with itself) and how its properties become actual in interactions. More on this after we discuss time.

III. TIME

The case with time is parallel to space, but with some additional levels of complexity [START_REF] Callender | The Oxford handbook of philosophy of time[END_REF].

Relational time: 'Time' is the relation we use when we locate events. We are talking about time when we ask "When shall we meet?" and answer "In three days". Location of events is given with respect to something else. (We shall meet after three sunrises.) Used in this sense time is a relation between events. This is the notion Aristoteles refers to in his Physics 2 , and so on. It is a very general notion of time, equally present in ancient, Cartesian, Newtonian, and relativistic physics.

When used in this wide sense, 'time' is definitely present in LQG. In LQG we can say that something happens when something else happens. For instance, a particle is emitted when two quanta of gravity join. Also, we can say that two events are temporally adjacent. A network of temporal adjacency of elementary processes of the gravitational field is captured by the spinfoams (see Appendix).

Newtonian time: In the Principia, Newton distinguished two notions of time. The first, which he called the "common" one, is the one in the previous item. The second, which he called the "true" one, is what has been later called Newtonian time. Newtonian time is assumed to be "flowing uniformly", even when nothing happens, with no influence from events, and to have a metric structure: we can say when two time intervals have equal duration. Special relativity modifies the Newtonian ontology only marginally, merging Newtonian space and time into Minkowski spacetime.

In LQG (Minkowsky spacetime and hence) Newtonian time appears only as an approximation. It has no role at all in the foundation of the theory.

General relativistic time:

What GR has shown is that Newtonian time is indeed (part of) an entity as Newton postulated, but this entity is not nondynamical as Newton assumed. Rather, it is an aspect of a dynamical field, the gravitational field.

What the reading T of a common clock tracks, for instance, is a function of the gravitational field g µν ,

T = g µν dx µ dx ν . (1)
In GR, therefore, there are two distinct kinds of temporal notions. The first is the simple fact that all events are localized with respect to one another ("This gravity wave has been emitted when the two neutron stars have merged", "The binary pulsar emits seven hundred pulses during an orbit"). The second is a left-over habit from Newtonian logic: the habit of calling 'time' (in 'spacetime') aspects of one specific dynamical entity: the gravitational field. Again, there is nothing wrong in doing so, 2 The famous definition is: Time is ἀριθμός κινήσ εως κατὰ τὸ πρότερον καὶ ῎ υστερον "The number of change with respect to before and after" (Physics, IV, 219 b 2; see also 232 b 22-23) [START_REF]Physics[END_REF].

provided that the difference between three notions of time (relative order of events, Newtonian non dynamical time, the gravitational field) are clear.

LQG treats time (in this sense) as GR does: there is no preferred clock time, but many clock times measured by different clocks. In addition, however, clock times undergo standard quantum fluctuations like any other dynamical variable. There can be quantum superpositions between different values of the same clock time variable T .

Our common intuition about time is profoundly marked by natural phenomena that are not generally present in fundamental physics. Unless we disentangle these from the aspects of time described above, confusion reigns (I have extensively discussed the multiple aspects of temporality in the recent book [START_REF] Rovelli | The Order of Time[END_REF]). These fall into two classes:

Irreversible time: When dealing with many degrees of freedom we recur to statistical and thermodynamical notions. In an environment with an entropy gradient there are irreversible phenomena. The existence of traces of the past versus the absence of traces of the future, or the apparent asymmetry of causation and agency, are consequences of the entropy gradient (of what else?). Our common intuition about time is profoundly marked by these phenomena. We do not know why was entropy as low in the past universe [START_REF] Earman | The Past Hypothesis: Not Even False[END_REF]. (A possibility is that this is a perspectival effect due to the way the physical system to which we belong couples with the rest of the universe [START_REF] Rovelli | Is Time's Arrow Perspectival?[END_REF].) Whatever the origin of the entropic gradient, it is a fact that all irreversible phenomena of our experience can be traced to (some version) of it [START_REF] Reichenbach | The philosophy of space and time[END_REF][START_REF] Albert | Time and Change[END_REF][START_REF] Price | Time's Arrow[END_REF]. This has nothing to do with the role of time in classical or quantum mechanics, in relativistic physics or in quantum gravity. There is no compelling reason to confuse these phenomena with issues of time in quantum gravity.

Accordingly, nothing refers to 'causation, 'irreversibility' or similar, in LQG. LQG describes physical happening, the way it happens, its probabilistic relations, the microphysics, not the statistics of many degrees of freedom, entropy gradients or related irreversible phenomena.

To address these, and understand the source of the the features that make a time variable 'special', we need a general covariant quantum statistical mechanics. Key steps in this direction exist (see [START_REF] Connes | Von Neumann algebra automorphisms and time thermodynamics relation in general covariant quantum theories[END_REF][START_REF] Rovelli | Statistical mechanics of gravity and the thermodynamical origin of time[END_REF] on thermal time, and [START_REF] Chirco | Statistical mechanics of reparametrization-invariant systems. It takes three to tango[END_REF] and references therein) but are incomplete. They have no direct bearing on LQG.

Experiential time:

The second class of phenomena that profoundly affects our intuition of time are those following from the fact that our brain is a machine that (because of the entropy gradient) remembers the past and works constantly to antici-pate the future [START_REF] Buonomano | Your Brain Is A Time Machine: The Nueroscience and Physics of Time[END_REF]. This working of our brain gives us a distinctive feeling about time: this is the feeling we call "flow", or the "clearing" that is is our experiential time [START_REF] Heidegger | Gesamtausgabe[END_REF]. This depends on the working of our brain, not on fundamental physics [START_REF] James | The Principles of Psychology[END_REF]. It is a mistake to search something pertaining to our feelings uniquely in fundamental physics. It would be like asking fundamental physics to directly justify the fact that a red frequency is more vivid to our eyes than a green one: a question asked the wrong chapter of science.

Accordingly, nothing refers to "flowing", "passage" or the similar in LQG. LQG describes physical happening [START_REF] Dorato | Rovelli' s relational quantum mechanics, monism and quantum becoming[END_REF], the way they happen, their probabilistic relations, not idiosyncrasies of our brain (or our culture [START_REF] Everett | Don't Sleep, There Are Snakes[END_REF]).

IV. PRESENTISM OR BLOCK UNIVERSE?

A FALSE ALTERNATIVE.

An ongoing discussion on the nature of time is framed as an alternative between presentism and block universe (or eternalism). This is a false alternative. Let me get rid of this confusion before continuing.

Presentism is the idea of identifying what is real with what is present now, everywhere in the universe. Special relativity and GR make clear that an objective notion of 'present' defined all over the universe is not in the physical world. Hence there can be no objective universal distinction between past, present and future. Presentism is seriously questioned by this discovery, because to hold it we have to base it on a notion of present that lacks observable ground, and this is unpalatable. A common response states that (i) we must therefore identify what is real with the ensemble of all events of the universe, including past and future ones [START_REF] Putnam | Time and Physical Geometry[END_REF], and (ii) this implies that, since future and past are equally real, the passage of time is illusory, and there is no becoming in nature [START_REF] Mctaggart | The Unreality of Time[END_REF].

The argument is wrong. (i) is just a grammatical choice about how we decide to use the ambiguous adjective "real", it has no content [START_REF] Austin | Sense and Sensibilia[END_REF][START_REF] Quine | On What There Is[END_REF]. (ii) is mistaken because it treats time too rigidly, failing to realise that time can behave differently from our experience, and still deserve to be called time.

The absence of a preferred objective present does not imply that temporality and becoming are illusions. Events happen, and this we call 'becoming', but their temporal relations form a structure richer than we previously thought. We have to adapt our notion of becoming to this discovery, not discard it.

There are temporal relations, but these are local and not global; more precisely, there is a temporal ordering but it is a partial ordering, and not a complete one. The universe is an ensemble of processes that happen, and these are not organised in a unique global order. In the classical theory, they are organised in a nontrivial geom-etry. In the quantum theory, in possibly more complex patterns.

The expression "real now here" can still be used to denote an ensemble of events that sit on the portion of a common simultaneity surface for a group of observers in slow relative motion; the region it pertains to must be small enough for the effects of the finite speed of light to be smaller than the available time resolution. When these conditions are not met, the expression "real now" simply makes no sense.

Therefore the discovery of relativity does not imply that becoming or temporality are meaningless or illusory: it implies that they behave in a more subtle manner than in our pre-relativistic intuition. The best language for describing the universe remains a language of happening and becoming, not a language of being. Even more so when we fold quantum theory in.

LQG describes reality in terms of processes. The amplitudes of the theory determine probabilities for processes to happen. This is a language of becoming, not being. In a process, variables change value. The quantum states of the theory code the possible set of values that are transformed into each other in processes.

In simple words, the now is replaced by here and now, not by a frozen eternity.

Temporality in the sense of becoming is at the roots of the language of LQG. But in LQG there is no preferred time variable, as I discuss in the next section.

V. "ABSENCE OF TIME" AND RELATIVE EVOLUTION: TIME IS NOT FORZEN

What is missing in LQG is not becoming. It is a (preferred) time variable.

Let me start by reviewing the (different) roles of the coordinates in Newtonian physics and GR. Newtonian space is a 3d Euclidean space and Newtonian time is a uniform 1d metric line. Euclidean space admits families of Cartesian coordinates X and the time line carries a natural (affine) metric coordinate T . These quantities are tracked by standard rods and clocks. Rods and clocks are not strictly needed for localisation in time and space, because anything can be used for relative localisation, but they are convenient in the presence of a rigid background metric structure such as the Newtonian, or the special relativistic one.

Rods and clocks are also useful in GR, but far less central. Einstein relayed on rods and clocks in the early days of the theory, but later realized that this was a mistake and repeatedly de-emphasized their role at the foundation of his theory. In fact, he cautioned against giving excessive weight to the fact that the gravitational field defines a geometry [START_REF] Lehmkuhl | Why Einstein did not believe that general relativity geometrizes gravity[END_REF]. He regarded this fact as a convenient mathematical feature and a useful tool to connect the theory to the geometry of newtonian space [START_REF] Einstein | Geometrie und Erfahrung[END_REF], but the essential about GR is not that it describes gravitation as a manifestation of a Riemannian spacetime geom-etry; it is that it provides a field theoretical description of gravitation [START_REF] Einstein | The Meaning of Relativity[END_REF].

GR's general coordinates x, t are devoid of metrical meaning, unrelated to rods and clocks, and arbiltrarilly assigned to events. This is imposed by the fact that the dynamics of rods and clocks is determined by interaction with the gravitational field. Therefore the general relativistic coordinates do not have the direct physical interpretation of Newtonian and special relativistic coordinates. To compare the theory to reality we have to find coordinate-invariant quantities. This generates some technical complication but is never particularly hard in realistic applications. But the relativistic t coordinates should not be confused with intuitive time, nor with clock time. Clock time is computed in the theory by the proper time (1) along a worldline. The reason is that this quantity counts, say, the oscillations of a mechanism following the worldline. Contrary to what often wrongly stated, this is not a postulate of the theory: it is a consequences of the equations of motion of the mechanism.

Given two events in spacetime, the clock time separation between them depends on the worldline of the clock. Therefore there is no single meaning to the time separation between two events. This does not make the notion of time inconsistent: it reveals it to be richer than our naive intuition. It is a fact that two clocks separated and then taken back together in general do not indicate the same time. Accord of clocks is an approximative phenomenon due to the peculiar environment in which we conduct our usual business.

Due to the discrepancy between clocks, it makes no sense to interpret dynamics as evolution with respect to one particular clock, as Newton wanted. 3 Accordingly, the dynamics of GR is not expressed in terms of evolution in a single clock time variable; it is expressed in terms of relative evolution between observable quantities (a detailed discussion is in Chapter 3 of [START_REF] Rovelli | Quantum Gravity[END_REF]). This fact makes it possible to get rid of the t variable all-together, and express the dynamical evolution directly in terms of the relative evolution of dynamical variables (Chapter 3 of [START_REF] Rovelli | Quantum Gravity[END_REF]). Thus, special clocks or preferred spacial or temporal variables are not needed in relativistic physics.

A formulation of classical GR that does not employ the time variable t at all is the Hamilton-Jacobi formulation [START_REF] Peres | On Cauchy's problem in General Relativity[END_REF]. It is expressed uniquely in terms of the three metric q ab of a spacelike surfaces and defined by two equations

D a δS[q] q ab = 0, G abcd δS[q] q ab δS[q] q cd + det q R[q] = 0 (2)
where G abcd = q ac q bd + q ad q bc -q ab q cd and R is the Ricci scalar of q. Notice the absence of any temporal coordinate t. In principle, knowing the solutions of these equations is equivalent to solving the Einstein equations.

Here S[q] is the Hamilton-Jacobi function of GR. When q is the 3-metric of the boundary of a compact region R of an Einstein space, S[q] can be taken to be the action of a solution of the field equations in this region. It is the quantity connected to the LQG amplitudes as in (A1). Absence of a time variable does not mean that "time is frozen" or that the theory does not describe dynamics, as unfortunately is still heard.

Equations ( 2) provide indeed an equivalent formulation of standard GR and can describe the solar system dynamics, black holes, gravitational waves and any other dynamical process, where things become, without any need of an independent t variable. In these phenomena many physical variables change together, and no preferred clock or parameter is needed to track change.

The same happens in LQG. The quantum versions of (2) formally determine the transition amplitudes between quantum states of the gravitational field. These can be coupled to matter and clocks. Variables change together and no preferred clock variable is used in the theory.

It is in this weak sense that it is sometimes said that "time does not exist" at the fundamental level in quantum gravity. This expression means that there is no time variable in the fundamental equations. It does not mean that there is no change in nature. The theory indeed is formulated in terms of probability amplitudes for processes.

VI. QUANTUM THEORY WITHOUT SCHR ÖDINGER EQUATION

Quantum mechanics requires some cosmetic adaptations in order to deal with the way general relativistic physics treats becoming. General relativistic physics describes becoming as evolution of variables that change together, any of them can be used to track change. No preferred time variable is singled out. Quantum mechanics instead is commonly formulated in terms of a preferred independent clock variable T . Evolution in T is expressed either in the form of Schrödinger equation

i ∂ψ ∂T = Hψ (3)
or as a dynamical equation for the variables

dA dT = i [A, H], (4)
where H is the Hamiltonian operator. Neither of these equations is adapt to describe relativistic relative evolution. The extension of quantum theory to the relativistic evolution is however not very hard, and has been developed by many authors, starting from Dirac. See for instance Chapter 5 of [START_REF] Rovelli | Quantum Gravity[END_REF], or [START_REF] Rovelli | Covariant Loop Quantum Gravity[END_REF] or, on a slightly different perspective, the extensive work of Jim Hartle [START_REF] Hartle | Spacetime quantum mechanics and the quantum mechanics of spacetime[END_REF] on this topic.

Like classical mechanics, quantum mechanics can be phrased as a theory of the probabilistic relations between the values of variables evolving together, rather than variables evolving with respect to a single time parameter. The Schrödinger equation is then replaced by a Wheeler-DeWitt equation

Cψ = 0, (5)
or as a dynamical equation for the variables

[A, C] = 0 (6)
for a suitable Wheeler-DeWitt operator C. Again: these equations do not mean that time is frozen or there is no dynamics. They mean that the dynamics is expressed as joint evolution between variables, instead than evolution with respected to a single special variable. Formally: the → 0 limit of equation ( 5) is the second equation in [START_REF] Rovelli | Covariant Loop Quantum Gravity[END_REF]; given boundary values, ( 5) is formally solved by the transition amplitudes W ; these can be expressed as a path integral over fields in the region and in the → 0 limit W ∼ e i S , where S is a solution to [START_REF] Rovelli | Covariant Loop Quantum Gravity[END_REF]. These are formal manipulations. LQG provides a finite and well defined expressions for W , at any order in a truncation in the number of degrees of freedom. Quantum theory does not describe how things 'are'. It describe quantum events that happen when systems interact [START_REF] Rovelli | Space is blue and birds fly through it[END_REF]. We mentally separate a 'quantum system', for a certain time interval, from the rest of the world, and describe the way this interact with its surroundings. This peculiar conceptual structure at the foundations of quantum theory takes a surprising twist in quantum gravity.

In quantum gravity we identify the process of the 'quantum system' with a finite spacetime region. This yields a remarkable dictionary between the relational structure of quantum theory and the relational structure of relativistic spacetime: quantum transition ↔ 4d spacetime region initial and final states ↔ 3d boundaries interaction ('measurement') ↔ continguity Thus the quantum states of LQG sit naturally on 3d boundaries of 4d regions (see Figure 1) [START_REF] Oeckl | A 'general boundary' formulation for quantum mechanics and quantum gravity[END_REF]. The quantum amplitudes are associated to what happens inside the regions. Intuitively, they can be understood as path integral over all possible internal geometries, at fixed boundary data. For each set of boundary data, the theory gives an amplitude, that determines the probability for this process to happen, with respect to other processes.

Remarkably: the net of quantum interactions between systems is the same thing as the net of adjacent spacetime regions. 2. Spacetime is a name given to the gravitational field in classical GR. In LQG there is a gravitational field, but it is not a continuous metric manifold. It is a quantum field with the usual quantum properties of discreteness, indeterminism and quantum relationality.

3. Space and time can refer to preferred variables used to locate things or to track change, in particular reading of meters and clocks. In LQG, rods and clocks and their (quantum) behaviour can in principle be described, but play no role in the foundation of the theory. The equations of the theory do not have preferred spacial or temporal variables.

4. Thermal, causal, "flowing" aspects of temporality are ground on chapters of science distinct from the elementary quantum mechanics of reality. They may involve thermal time, perspectival phenomena, statistics, brain structures, or else.

5. The universe described by quantum gravity is not flowing along a single time variable, nor organised into a smooth Einsteinian geometry. It is a network of quantum processes, related to one another, each of which obeys probabilistic laws that the theory captures. The net of quantum interactions between systems is identified with the net of adjacent spacetime regions.

These are the roles of space and time in Loop Quantum Gravity. Much confusion about these notions in quantum gravity is confusion between these different meanings of space and time.

Appendix A: Loop Quantum Gravity in a nutshell

As any quantum theory, LQG can be defined by a Hilbert space, an algebra of operators and a family of transition amplitudes. The Hilbert space H of the theory admits a basis called the spin network basis, whose states |Γ, j l , v n are labelled by a (abstract, combinatorial) graph Γ, a discrete quantum number j l for each link l of the graph, and a discrete quantum number v n for each node n of the graph. The nodes of the graph are interpreted as elementary 'quanta of gravity' or 'quanta of space', whose adjacency is determined by the links, see Figure 2. These quanta do not live on some space: rather, they themselves build up physical space. The volume of these quanta is discrete and determined by v n . The area of the surfaces separating two nodes is also discrete, and determined by j l . The elementary quanta of space do not have a sharp metrical geometry (volume and areas are not sufficient to determine geometry), but in the limit of large quantum numbers there are states in H that approximate 3d geometries arbitrarily well, in the same sense in which linear combinations of photon states approximate a classical electromagnetic field. The spin network states are eigenstates of operators A l and V l in the operator algebra of the theory, respectively associated to nodes and links of the graph. In the classical limit these operators become functions of the Einstein's gravitational field g µν , determined by the standard relativistic formulas for area and volume. For instance, V (R) = R √ det q, for the volume of a 3d spacial region R, where q is the 3-metric induced on R.

In the covariant formalism (see [START_REF] Rovelli | Covariant Loop Quantum Gravity[END_REF]), transition amplitudes are defined order by order in a truncation on the number of degrees of freedom. At each order, a transition amplitude is determined by a spinfoam: a combinatorial structure C defined by elementary faces joining on edges in turn joining on vertices (in turn, labeled by quantum numbers on faces and edges), as in Figure 3. A spinfoam can be viewed as the Feynman graph of a history of a spin network; equivalently, as a (dual) dis-crete 4d geometry: a vertex corresponds to an elementary 4d region, an elementary process. The boundary of a spinfoam is a spin network. The theory associates an amplitude W C (Γ, j l , v n ) (a complex number) to spinfoams. These are ultraviolet finite. Several theorems relate them to the action (more precisely the Hamilton function S) of GR, in the limit of large quantum numbers. This is the expected formal relation between the quantum dynam-ics, expressed in terms of transition amplitudes W and its classical limit, expressed in terms of the action S:

W ∼ e i S , (A1)

where W and S are both functions of the boundary data. This concludes the sketch of the formal structure of (covariant) LQG. Notice that nowhere in the basic equations of the theory a time coordinate t or a space coordinate x show up.

Figure 1 .

 1 Figure 1. A compact spacetime region is identified with a quantum transition. The states of LQG sit on its boundary.

Figure 2 .

 2 Figure 2. The graph of a spin network and an intuitive image of the quanta of space it represents.

Figure 3 .

 3 Figure 3. Spinfoam: the time evolution of a spin network.

During the XIX century, certain awkward aspects of this Newtonian hypostasis led to the development of the notion of 'physical reference system': the idea that Newtonian space captures the properties of preferred systems of bodies not subjected to forces. This is correct but already presupposes the essential ingredient: a fixed metric space, permitting to locate things with respect to distant references bodies. Thus the notion of reference system does not add much to the novelty of the Newtonian ontology.

Given two clocks that measure different time intervals between two events, it make no sense to ask which of the two is 'true time': the theory simply allows us to compute the way each changes with respect to the other.


Eugenio Bianchi

Hal M Haggard

Carlo Rovelli

The boundary is mixed

We show that Oeckl's boundary formalism incorporates quantum statistical mechanics naturally, and we formulate general-covariant quantum statistical mechanics in this language. We illustrate the formalism by showing how it accounts for the Unruh effect. We observe that the distinction between pure and mixed states weakens in the general covariant context, and surmise that local gravitational processes are indivisibly statistical with no possible quantal versus probabilistic distinction.

I. INTRODUCTION

Quantum field theory and quantum statistical mechanics provide a framework within which most of current fundamental physics can be understood. In their usual formulation, however, they are not at ease in dealing with gravitational physics. The difficulty stems from general covariance and the peculiar way in which general relativistic theories deal with time evolution. A quantum statistical theory including gravity requires a generalized formulation of quantum and statistical mechanics.

A key tool in this direction which has proved effective in quantum gravity, is Oeckl's idea [START_REF] Oeckl | A 'general boundary' formulation for quantum mechanics and quantum gravity[END_REF][START_REF] Oeckl | General boundary quantum field theory: Foundations and probability interpretation[END_REF] of using a boundary formalism, reviewed below. This formalism combines the advantages of an S-matrix transition-amplitude language with the possibility of defining the theory without referring to asymptotic regions. It is a language adapted to general covariant theories, where "bulk" observables are notoriously tricky, because it can treat dependent and independent variables on the same footing. This formalism allows a general covariant definition of transition amplitudes, n-point functions and in particular the graviton propagator [START_REF] Rovelli | Graviton propagator from backgroundindependent quantum gravity[END_REF][START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF]. These are defined on compact spacetime regions-the dependence on the boundary metric data makes general covariance explicit and circumvents the difficulties (e.g. [START_REF] Arkani-Hamed | A Measure of de Sitter entropy and eternal inflation[END_REF]) usually associated to the definition of these quantities in a general covariant theory.

In the boundary formalism, the focus is moved from "states", which describe a system at some given time, to "processes", which describe what happens to a local system during a finite time-span. For a conventional nonrelativistic system, the quantum space of the processes, B (for "boundary"), is simply the tensor product of the initial and final Hilbert state spaces. Tensor states in B represent processes with given initial and final states.

What about the vectors in B that are not of the tensor form? Remarkably, it turns out that mixed statistical quantum states are naturally represented by these non-tensor states [START_REF] Bianchi | Talk at the 2012 Marcel Grossmann meeting[END_REF]. Here we formalize this observation, showing how statistical expectation values are expressed in this language. This opens the way to a systematic treatment of general-covariant quantum statistical mechanics, a problem still wide open.

The structure of this paper is as follows: In Section II, we start from conventional non-relativistic mechanics and move "upward" towards more covariance: we construct the formal structures that define the boundary formalism, characterize physical states and operators, define the dynamics through amplitudes, and show how statistical states and equilibrium states can be treated. In Section III, we adapt the boundary formalism to a general covariant language by including the independent evolution parameter (the "time" partial observable) into the configuration space. This is the step that permits the generalization to general covariant systems. Once these structures are clear, in Section IV we take them as fundamental, and show that they retain their meaning also in the more general cases where the system is genuinely general relativistic. In Section V we apply the formalism to the Unruh effect and in Section VI we draw some tentative conclusions regarding quantum gravity.

These point towards the idea that any local gravitational process is statistical.

II. NON-RELATIVISTIC FORMALISM

A. Mechanics Consider a Hamiltonian system with configuration space C. Call x ∈ C a generic point in C. The corresponding quantum system is defined by a Hilbert space H and a Hamiltonian operator H. We indicate by A, B, ... ∈ A the self-adjoint operators representing observables. In the Schrödinger representation, which diagonalizes configuration variables, a state ψ is represented by the functions ψ(x) = x|ψ , where |x is a (possibly generalized) eigenvector of a family of observables that coordinatizes C (we use the Dirac notation also for generalized states, as Dirac did). States evolve in time by ψ t = e -iHt ψ o . For convenience we call H t the Hilbert space isomorphic to H, thought of as the space of states at time t.

Fix a time t and consider the non-relativistic boundary space

B t = H 0 ⊗ H * t , (1)
arXiv:1306.5206v1 [gr-qc] 21 Jun 2013

where the star indicates the dual space. This space can be interpreted as the space of all (kinematical) processes.

The state Ψ = ψ ⊗ φ * ∈ B t represents the process that takes the initial state ψ into the final state φ in a time t. For instance, if ψ and φ are eigenstates of operators corresponding to given eigenvalues, then Ψ represents a process where these eigenvalues have been measured at initial and final time.

In the Schrödinger representation, vectors in B t have the form ψ(x, x ) = x, x |ψ . The state |x, x ≡ |x ⊗ x | represents the process that takes the system from x to x in a time t. The interpretation of the states in B t which are not of the tensor form is our main concern in this paper and is discussed below.

There are two notable structures on the space B t .

(a) A linear function W t on B t , which completely codes the dynamics. This is defined by its action

W t (ψ ⊗ φ * ) := φ|e -iHt |ψ (2)
on tensor states, and extended by linearity to the entire space. This function codes the dynamics because its value on any tensor state ψ ⊗ φ * gives the probability amplitude of the corresponding process that transforms the state ψ into the state φ. Notice that the expression of W t in the Schrödinger basis reads

W t (x, x ) = x |e -iHt |x , (3)
which is precisely the Schrödinger-equation propagator, and can be represented formally as a Feynman path integral from x to x in a time t, and, of course, it codes the dynamics of the theory.

(b) There is a nonlinear map σ that sends H into B t , given by

σ : ψ → ψ ⊗ (e -iHt ψ) * . (4)
Boundary states in the image of σ represent processes that have probability amplitude equal to one, as can be easily verified using [START_REF] Oeckl | General boundary quantum field theory: Foundations and probability interpretation[END_REF] and ( 4). The process σ(ψ) is the one induced by the initial state ψ.

In general, we shall call any vector Ψ ∈ B t that satisfies

W t (Ψ) = 1 (5)
a "physical boundary state."

These are the basic structures of the boundary formalism in the case of a non-relativistic system.

B. Statistical mechanics

The last equation of the previous section is linear, hence a linear combination of solutions is also a solution. But linear combinations of tensor states are not tensor states. What do the solutions of (5) which are not of the tensor form represent? Consider a statistical state ρ. By this we mean here a trace class operator in H that can be mixed or pure. An operator in H is naturally identified with a vector in H ⊗ H * , of course. In particular, let |n be an orthogonal basis that diagonalizes ρ, then

ρ = n c n |n n|. (6)
The corresponding element in B 0 is

ρ = n c n |n ⊗ n| (7)
and we will from now on identify the two quantities. That is, below we often write states in H ⊗ H * , as operators in H. The numbers c n in (6) are the statistical weights. They satisfy

n c n = 1 (8)
because of the trace condition on ρ, which expresses the fact that probabilities add up to one. Thus the state ρ can be seen as an element of B 0 . Consider the corresponding element of B t , defined by

ρ t := n c n |n n|e iHt . (9)
It is immediate to see that

W t (ρ t ) = 1. ( 10
)
Therefore we have found the physical meaning of the other (normalized) solutions of [START_REF] Arkani-Hamed | A Measure of de Sitter entropy and eternal inflation[END_REF]. They represent statistical states. Notice that these are expressed as vectors in the boundary Hilbert space B t . (See also [START_REF] Oeckl | A positive formalism for quantum theory in the general boundary formulation[END_REF].)

The expectation value of the observable A in the statistical state ρ is

A = Tr[Aρ], (11)
the correlation between two observables is

AB = Tr[ABρ], (12)
and the time dependent correlation is

A(t)B(0) = Tr[e iHt Ae -iHt Bρ], (13)
of which the two previous expressions are special cases. These quantities can be expressed in the simple form

A(t)B(0) = W t ( (B ⊗ A) ρ t ) (14)
because

W t ((B ⊗ A)ρ t ) = Tr[e -iHt Bρ t A] = Tr[e -iHt Bρe iHt A],
here the placement of ρ t within the trace reflects the fact that its left factor is in the initial space and its right factor is in the final space (and A does not need a dagger because it is self-adjoint). Therefore the boundary formalism permits a direct reformulation of quantum statistical mechanics in terms of general boundary states, boundary operators and the W t amplitude. Consider states of Gibbs's form ρ = N e -βH . The corresponding state in B t is

ρ t = N n e -βEn e iEnt |n n| = N e iH(t+iβ) ( 15
)
where |n is the energy eigenbasis and N = N (β), determined by the normalization, is the inverse of the partition function. A straightforward calculation shows that for these states the correlations ( 14) satisfy the KMS condition

A(t)B(0) = B(-t -iβ)A(0) (16)
which is the mark of an equilibrium state. Thus Gibbs states are the equilibrium states.

C. L1 and L2 norms: physical states and pure states

The two classes of solutions illustrated in the previous two subsections (pure states and statistical states) exhaust all solutions of the physical boundary state condition when B t decomposes as a tensor product of two Hilbert spaces:

B t = H 0 ⊗ H * t . (17)
This can be shown as follows. Consider an orthonormal basis |n in H 0 . Due to the unitarity of the time evolution, the vectors (e -iHt |n ) * form a basis of H * t . Therefore any state in B t can be written in the form

Ψ = nn c nn |n ⊗ (e -iHt |n ) * . ( 18
)
The physical states satisfy

W |Ψ = nn c nn n |e iHt e -iHt |n = n c nn = 1, ( 19
)
therefore they correspond precisely to the operators

ρ = nn c nn |n n | ( 20
)
in H 0 , satisfying the condition Trρ = 1 [START_REF] Wightman | Quantum field theory in terms of vacuum expectation values[END_REF] which is to say: they are the statistical states. In particular, they are pure states if they are projection operators,

ρ 2 = ρ.
Observe that in general a statistical state in B t is not a normalized state in this space. Rather, its L 2 norm satisfies

|Ψ| 2 = nn |c nn | 2 ≤ 1 (22)
where the equality holds only if the state is pure. This is easy to see in a basis that diagonalizes ρ, because the trace condition implies that all eigenvalues are equal or smaller than 1 and sum to 1. Thus there is a simple characterization of physical states and pure states: the first have the "L 1 " norm [START_REF] Streater | PCT, Spin and Statistics, and All That[END_REF] equal to unity. The second have also the "L 2 " norm |Ψ| 2 equal to unity.

III. RELATIVISTIC FORMALISM A. Relativistic mechanics

Let us now take a step towards the relativistic formalism where the time variable is treated on the same footing as the configuration variables.

With this aim, consider again the same system as before and define the extended configuration space E = C × R. Call (x, t) ∈ E a generic point in E. Let Γ ex = T * E be the corresponding extended phase space and C = p t + H the Hamiltonian constraint, where p t is the momentum conjugate to t. The corresponding quantum system is characterized by the extended Hilbert space K and a Wheeler-deWitt operator C [START_REF] Rovelli | Quantum Gravity[END_REF].

Indicate by A, B, ... ∈ A the self-adjoint operators representing partial observables [START_REF] Rovelli | Partial observables[END_REF] defined in K. In the Schrödinger representation that diagonalizes extended configuration variables, states are given by functions ψ(x, t) = x, t|ψ . The physical states are the solutions of the Wheeler-deWitt equation Cψ = 0, which here is just the Schrödinger equation. Physical states are the (generalized) vectors ψ(x, t) in K that are solutions of the Schrödinger equation.

The space H formed by the physical states that are solutions of the Schrödinger equation is clearly in oneto-one correspondence with the space H 0 of the states at time t = 0. Therefore there is a linear map that sends H 0 into (a suitable completion of) K, simply defined by sending the state ψ(x) into the solution ψ(x, t) of the Schrödinger equation such that ψ(x, 0) = ψ(x). Vice versa, there is a (generalized) projection P from (a dense subspace of) K to H, that sends a state ψ(x, t) to a solution of the Schrödinger equation. This can be formally obtained from the spectral decomposition of C, or, more simply, by

(P ψ)(x, t) = dx dt W (t-t ) (x, x ) ψ(x , t ). ( 23
)
Now, without fixing a time, the relativistic boundary state space is defined by

B = K ⊗ K * . ( 24
)
Notice the absence of the t-label subscript. In the Schrödinger representation, vectors in B have the form ψ(x, t, x , t ) = x, t, x , t |ψ . This space can again be interpreted as the space of all (kinematical) processes, where now the boundary measurement of the clock time t is treated on the same footing as the other partial observables. Thus for instance |x, t, x , t ≡ |x, t ⊗ x , t | represents the process that takes the system from the configuration x at time t to the configuration x at time t .

The two structures considered above simplify on the space B.

(a) The dynamics is completely coded by a linear function W (no t label!) on B. This is defined extending by linearity

W (φ * ⊗ ψ) := φ|P |ψ . ( 25
)
Its expression in the Schrödinger basis reads

W (x, t, x , t ) = x, t|P |x , t = x|e iH(t-t ) |x , (26)
which is once again nothing but the Schrödingerequation propagator, now seen as a function of initial and final extended configuration variables. The variable t is not treated as an independent evolution parameter, but rather is treated on equal footing with the other partial observables. The operator P can still be represented as a suitable Feynman path integral in the extended configuration space, from the point (x, t) to the point (x , t ).

(b) Second, there is again a nonlinear map σ that sends K into B, now simply given by

σ : ψ → ψ ⊗ ψ * . ( 27
)
States in the image of this map are "physical", namely represent processes that have probability amplitude equal to one, only if ψ satisfies the Schrödinger equation. In this case, a straightforward calculation verifies that

W (Ψ) = 1. ( 28
)
As before, we call "physical" any state in B solving this equation.

B. Relativistic statistical mechanics

As before, linear combinations of physical states represent statistical states. A general relativistic statistical state is a statistical superposition of solutions of the equations of motion [START_REF] Rovelli | Statistical mechanics of gravity and the thermodynamical origin of time[END_REF]. 1 Consider again the state (6) in this 1 A concrete example is illustrated in [START_REF] Rovelli | The Statistical state of the universe[END_REF].

language: if ψ n is the full time-dependent solution of the Schrödinger equation corresponding to the initial state |n , we can now represente the state (6) in B simply by

ρ = n c n ψ n ψ * n . (29)
Explicitly, in the Schrödinger basis

ρ(x, t, x , t ) = n c n ψ n (x, t) ψ n (x , t ). ( 30
)
The equilibrium statistical state at inverse temperature β is given by ρ(x, t, x , t ) = N n e iEn(t-t +iβ) ψ n (x) ψ n (x ).

= N e iH(t-t +iβ) .

(

) 31
where ψ n (x) are the energy eigenfunctions.

The correlation functions between partial observables are now given simply by

AB = W ((A ⊗ B) ρ). ( 32
)
Notice the complete absence of the time label t in the formalism. Any temporal dependence is folded into the boundary data. (However, see the next section for a generalization of the KMS property and equilibrium.) This completes the construction of the boundary formalism for a relativistic system. We now have at our disposal the full language and we can "throw away the ladder," keep only the structure constructed, and extend it to far more arbitrary systems, including relativistic gravity.

IV. GENERAL BOUNDARY

We now generalize the boundary formalism to genuinely (general) relativistic systems that do not have a non-relativistic formulation.

A quantum system is defined by the triple (B, A, W ). The Hilbert space B is interpreted as the boundary state space, not necessarily of the tensor form. A is an algebra of self-adjoint operators on B. The elements A, B, ... ∈ A represent partial observables, namely quantities to which we can imagine associating measurement apparatuses, but whose outcome is not necessarily predictable (think for instance of a clock). The linear map W on B defines the dynamics.

Vectors Ψ ∈ B represent processes. If Ψ is an eigenstate of the operator A ∈ A with eigenvalue a, it represents a process where the corresponding boundary observable has value a. The quantity

W (Ψ) = W |Ψ (33)
is the amplitude of the process. Its modulus square (suitably normalized) determines the relative probability of distinct processes [START_REF] Rovelli | Quantum Gravity[END_REF]. A physical process is a vector in B that has amplitude equal to one, namely satisfies

W |Ψ = 1. ( 34
)
The expectation value of an operator A ∈ A on a physical process Ψ is

A = W |A|Ψ . ( 35
)
If a tensor structure in B is not given, then there is no a priori distinction between pure and mixed states. The distinction between quantum incertitude and statistical incertitude acquires meaning only if we can distinguish past and future parts of the boundary [START_REF] Smolin | On the nature of quantum fluctuations and their relation to gravitation and the principle of inertia[END_REF][START_REF] Smolin | Quantum gravity and the statistical interpretation of quantum mechanics[END_REF].

So far, there is no notion of time flow in the theory. The theory predicts correlations between boundary observables. However, as pointed out in [START_REF] Connes | Von Neumann algebra automorphisms and time thermodynamics relation in general covariant quantum theories[END_REF], a generic state Ψ on the algebra of local observables of a region defines a flow α τ on the observable algebra by the Tomita theorem [START_REF] Connes | Von Neumann algebra automorphisms and time thermodynamics relation in general covariant quantum theories[END_REF], and the state Ψ satisfies the KMS condition for this flow

A(τ )B(0) = B(-τ -iβ)A(0) , (36)
where A(τ ) = α τ (A). It will be interesting to compare the flow generated in this manner with the flow generated by a statistical state within the boundary Hilbert space. If a flow is given a priori, the KMS states for this flow are equilibrium states for this flow.

In a general relativistic theory including gravity, no flow is given a priori, but we can still distinguish physical equilibrium states as follows: an equilibrium state is a state that defines a mean geometry and whose Tomita flow is given by a timelike Killing vector of this geometry: see [START_REF] Rovelli | General relativistic statistical mechanics[END_REF].

V. UNRUH EFFECT

As an example application of the formalism, we describe the Unruh effect [START_REF] Unruh | Notes on black hole evaporation[END_REF] in this language. Other treatments with a focus on the general boundary formalism are [START_REF] Colosi | The Unruh Effect in General Boundary Quantum Field Theory[END_REF][START_REF] Banisch | Vacuum states on timelike hypersurfaces in quantum field theory[END_REF]. Consider a partition of Minkowski space into two regions M and M separated by the two surfaces

Σ 0 : {t = 0, x ≥ 0}, Σ η : {t = ηx, x ≥ 0}. ( 37
)
The region M is a wedge of angular opening η and M is its complement (Figure 1). Consider a Lorentz invariant quantum field theory on Minkowski space, say satisfying the Wigtmam axioms [START_REF] Streater | PCT, Spin and Statistics, and All That[END_REF]: in particular, energy is positive-definite and there is a single Poincaré-invariant state, the vacuum |0 . How is the vacuum described in the boundary language?

In general, a boundary state φ b on ∂M = Σ = Σ 0 ∪ Σ η is a vector in the Hilbert space B = H 0 ⊗ H * η , where H 0 and H η are Hilbert spaces associated to the states on Σ 0 FIG. 1. The wedge M in Minkowski space.

and Σ η respectively. The conventional Hilbert space H associated to the t = 0 surface is the tensor product of two Hilbert spaces H = H L ⊗ H R that describe the degrees of freedom to the left or right of the origin. We can identify H R and H 0 since they carry the same observables: the field operators on Σ 0 . Because the theory is Lorentz-invariant, H carries a representation of the Lorentz group. The self-adjoint boost generator K in the t, x plane does not mix the two factors H L and H R . If we call k its eigenvalues and |k, α L,R , its eigenstates in the two factors with α labeling the distinct degenerate levels of k, then it is a well known result [START_REF] Bisognano | On the Duality Condition for Quantum Fields[END_REF]

that 0|k, α L = e -πk k, α| R (38)
which we can write in the form

|0 = dk dα e -πk |k, α L ⊗ |k, α R . (39)
Tracing over H L gives the density matrix in H R

ρ 0 = Tr L |0 0| = e -2πK (40)
which determines the result of any vacuum measurement, and therefore any measurment [START_REF] Wightman | Quantum field theory in terms of vacuum expectation values[END_REF], performed on Σ 0 . The evolution operator W η in the angle η, associated to the wedge, sends Σ 0 to Σ η and is

W η = e -iηK . (41)
These two quantities give immediately the boundary expression of the vacuum on Σ:

ρ η = ρ 0 e iηK = e i(η+2πi)K (42)
This is the vacuum in the boundary formalism. It is a KMS state at temperature 1/2π with respect to the flow generated by K in η. For an observer moving with constant proper acceleration a along the hyperboloid of points with constant distance from the origin, this flow is proportional to proper time s

s = η/a. (43)
And therefore the vacuum is a KMS state, namely a thermal state, at the Unruh temperature (restoring )

T = a 2π . (44)
This is the manner in which the Unruh effect is naturally described in the boundary language. Notice that no reference to accelerated observers or special basis in Hilbert space is needed to identify the thermal character of the vacuum on the η-wedge.

An interesting remark is that the expectation values of operators on Σ can be equally computed using the region M which is complementary to the wedge M . Let us first do this for η = 0. In this case, the insertion of the empty region M cannot alter the value of the observables, and therefore it is reasonable to take the boundary state we associate to it to be the unit operator.

ρ = 1l (45)
And therefore

ρη = e -iηK . (46)
For consistency, we have then that the evolution operator associated to M must be Wη = e i(η+2πi)K .

Therefore the evolution operator and the boundary state simply swap their roles when going from a region to its complement. 2 Notice that there exists a geometrical transformation that rotates Σ 0 into Σ η , obtained by rotating it clockwise, rather than anti clockwise. This rotation is not implemented by a proper Lorentz transformation, because the Lorentz group rotates Σ 0 at most only up to the light cone t = -x. But it can nevertheless be realized by extending a Lorentz transformation x = cosh(η)x + sinh(η)t t = sinh(η)x + cosh(η)t (48) to a complex parameter iη x = cosh(iη)x + sinh(iη)t = cos(η)x + i sin(η)t t = sinh(iη)x + cosh(iη)t = i sin(η)x + cos(η)t. (49)

For a small η = , this transformation rotates the positive x axis infinitesimally into the complex t plane. The Lorentz group acts on the expectation values of the theory, and in particular on the expectation values of products of its local observables. Since the n-point functions 2 This can be intuitively understood in terms of path integrals: the evolution operator is the path integral on the interior of a spacetime region, at fixed boundary values; the boundary state can be viewed as the path integral on the exterior of the region.

In the case under consideration, the vacuum is singled out by the boundary values of the field at infinity. For a detailed discussion, see [22].

of a quantum field theory where the energy is positive can be continued analytically for complex times (Theorem 3.5, pg. 114 in [START_REF] Streater | PCT, Spin and Statistics, and All That[END_REF]), this action is well defined on expectation values. In particular, we can rotate (t, x) infinitesimally into the complex t plane, and then rotate around the real t, x plane, passing below the light cone x = ±t in complex space. In other words, by adding a small complex rotation into imaginary time, we can rotate a space-like half-line into a timelike one [START_REF] Gibbons | Action integrals and partition functions in quantum gravity[END_REF][START_REF] Bianchi | Entropy of Non-Extremal Black Holes from Loop Gravity[END_REF]. A full rotation is implemented by U (2πi), giving (47). Finally, observe that the vacuum is the unique Poincaré invariant state in the theory. This implies that if a state is Poincaré invariant then it is thermal at the Unruh temperature on the boundary of the wedge. This is clearly a reflection of correlations with physics beyond the edge of the wedge.

Since vacuum expectation values determine all local measurable quantum-field-theory observables, this implies that the boundary state is unavoidably mixed. In essence the available field operators are insufficient to purify the state. This can be seen physically as follows: in principle, we can project the state onto a pure state on Σ 0 , breaking Poincaré invariance by singling out the origin, but to do so we need a complete measurement of field values for x > 0 and therefore an infinite number of measurements, which would move the state out of its folium [START_REF] Haag | Local Quantum Physics[END_REF]. We continue these considerations in the next section.

VI. RELATION WITH GRAVITY AND THERMALITY OF GRAVITATIONAL STATES

So far, gravity has played no direct role in our considerations. The construction above, however, is motivated by general relativity, because the boundary formalism is not needed as long as we deal with a quantum field theory on a fixed geometry, but becomes crucial in quantum gravity, where it allows us to circumvent the difficulties raised by diffeomorphism invariance in the quantum context.

In quantum gravity we can study probability amplitudes for local processes by associating boundary states to a finite portion of spacetime, and including the quantum dynamics of spacetime itself in the process. Therefore the boundary state includes the information about the geometry of the region itself.

The general structure of statistical mechanics of relativistic quantum geometry has been explored in [START_REF] Rovelli | General relativistic statistical mechanics[END_REF], where equilibrium states are characterized as those whose Tomita flow is a Killing vector of the mean geometry. Up until now it hasn't been possible to identify the statistical states in the general boundary formalism and so this strategy hasn't been available in this more covariant context. With a boundary notion of statistical states this becomes possible. It becomes possible, in particular, to check if given boundary data allow for a mean geometry that interpolates them. In quantum gravity we are interested in spacelike boundary states where initial and final data can be given, therefore a typical spacetime region will have the lens shape depicted in Figure 2. Past and future components of the boundary will meet on wedge-like two-dimensional "corner" regions. Now, say we assume that a quantum version of the equivalence principle holds, for which the local physics at the corner is locally Lorentz invariant. Then the result of the previous section indicates that the boundary state of the lens region will be mixed. Any such boundary state in quantum gravity is a mixed state. (Other arguments for the thermality of local spacetime processes are in [START_REF] Martinetti | Diamonds's temperature: Unruh effect for bounded trajectories and thermal time hypothesis[END_REF].) The dynamics at the corner is governed by the corner terms of the action [START_REF] Carlip | The Off-shell black hole[END_REF][START_REF] Bianchi | Horizon energy as the boost boundary term in general relativity and loop gravity[END_REF], which can indeed be seen as responsible for the thermalization [START_REF] Massar | How the change in horizon area drives black hole evaporation[END_REF][START_REF] Jacobson | Horizon entropy[END_REF].

Up to this point we have emphasized the mixed state character of the boundary states in order to make a clear connection with the standard quantum formalism. However, note that from the perspective of the fully covariant general boundary formalism (see section IV) there is always a single boundary Hilbert space B that can be made bipartite in many different manners. From this point of view it is more natural to call these boundary states nonseparable. Then, local gravitational states are entangled states. This was first appreciated in the context of the examples treated in [22], which was an inspiration for the present work.

Recently Bianchi and Myers have conjectured that in a theory of quantum gravity, for any sufficiently large region corresponding to a smooth background spacetime, the entanglement entropy between the degrees of freedom describing the given region with those describing its complement are given by the Bekenstein-Hawking entropy [START_REF] Bianchi | On the Architecture of Spacetime Geometry[END_REF]. The Bianchi-Myers conjecture and the considerations above result in a compelling picture supporting a quantum version of the equivalence principle.

Both the mixing of the state near a corner and the Bianchi-Myers conjecture can be seen as manifestations of the fact that by restricting the region of interest to a finite spatial region we are tracing over the correlations between this region and the exterior, and therefore we are necessarily dealing with a state which is not pure. If, as we expect, the boundary formalism is crucial for extracting physical amplitudes from quantum gravity, all this appears to imply that the notion of pure state is irrelevant in local quantum gravitational physics and therefore statistical fluctuations cannot be disentangled from quantum fluctuations in quantum gravity [START_REF] Smolin | On the nature of quantum fluctuations and their relation to gravitation and the principle of inertia[END_REF][START_REF] Smolin | Quantum gravity and the statistical interpretation of quantum mechanics[END_REF].

FIG. 2 .

 2 FIG. 2. Lens shaped spacetime region with spacelike boundaries and corners (filled circles).

EB acknowledges support from a Banting Postdoctoral Fellowship from NSERC. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. HMH acknowledges support from the National Science Foundation (NSF) International Research Fellowship Program (IRFP) under Grant No. OISE-1159218.