**1. Introduction**

Background independence is a hallmark of general relativity that has revolutionised our conception of space and time. The picture of physical reality it paints is that of an impartial dynamical interplay between matter and gravitational fields. Spacetime is no longer a passive stage on which matter performs; it is an equally active performer in itself. Coordinates are gauge, thus losing their physical status of non-relativistic settings. In particular, the notion of time is modified drastically. It is no longer an absolute, global, external parameter uniquely encoding the full dynamics. It is instead a gauge parameter associated with a Hamiltonian constraint.

On the other hand, the well-established fields of quantum statistical mechanics and thermodynamics have been of immense use in the physical sciences. From early applications to heat engines and study of gases, to modern day uses in condensed matter systems and quantum optics, these powerful frameworks have greatly expanded our knowledge of physical systems. However, a complete extension of them to a background independent setting, such as for a gravitational field, remains an open issue [1–3]. The biggest challenge is the absence of an absolute notion of time, and thus of energy, which is essential to any statistical and thermodynamical consideration. This issue is particularly exacerbated in the context of defining statistical equilibrium, for the natural reason that the standard concepts of equilibrium and time are tightly linked. In other words, the constrained dynamics of a background independent system lacks a non-vanishing Hamiltonian in general, which makes formulating (equilibrium) statistical mechanics and thermodynamics, an especially thorny problem. This is a foundational issue, and tackling it is important and interesting in its own right, and even more so because it could provide deep insights into the nature of (quantum) gravitational systems. This paper is devoted to addressing precisely these points.

The importance of addressing these issues is further intensified in light of the deep interplay between thermodynamics, gravity and the quantum theory, first uncovered for black holes. The laws of black hole mechanics [4] were a glimpse into a curious intermingling of thermodynamics and classical gravity, even if originally only at a formal level of analogy. The discovery of black hole entropy and radiation [5–7] further brought quantum mechanics into the mix. This directly led to a multitude of new conceptual insights along with many puzzling questions which continue to be investigated still after decades. The content of the discovery, namely that a black hole must be assigned physical entropy and that it scales with the area of its horizon in Planck units, has birthed several distinct lines of thoughts, in turn leading to different (even if related) lines of investigations, such as thermodynamics of gravity, analogue gravity and holography. Moreover, early attempts at understanding the physical origin of this entropy [8] made evident the relevance of quantum entanglement, thus also contributing to the current prolific interest in fascinating connections between quantum information theory and gravitational physics.

This discovery further hinted at a quantum microstructure underlying a classical spacetime. This perspective is shared, to varying degrees of details, by various approaches to quantum gravity such as loop quantum gravity (and related spin foams and group field theories), string theory and AdS/CFT, simplicial gravity and causal set theory to name a few. Specifically within discrete non-perturbative approaches, spacetime is replaced by more fundamental entities that are discrete, quantum, and pre-geometric in the sense that no notion of smooth metric geometry and spacetime manifold exists yet. The collective dynamics of such quanta of geometry, governed by some theory of quantum gravity is then hypothesised to give rise to an emergen<sup>t</sup> spacetime, corresponding to certain phases of the full theory. This would essentially entail identifying suitable procedures to extract a classical continuum from a quantum discretuum, and to reconstruct general relativistic gravitational dynamics coupled with matter (likely with quantum corrections). This emergence in quantum gravity is akin to that in condensed matter systems in which also coarse-grained macroscopic (thermodynamic) properties of the physical systems are extracted from the microscopic (statistical and) dynamical theories of the constituent atoms. In this sense our universe can be understood as an unusual condensed matter system, brought into the existing smooth geometric form by a phase transition of a quantum gravity system of pre-geometric 'atoms' of space; in particular, as a condensate [9].

This brings our motivations full circle, and to the core of this article: to illustrate, the potential of and preliminary evidence for, a rewarding exchange between a background independent generalisation of statistical mechanics and discrete quantum gravity; and show that ideas from the former are vital to investigate statistical mechanics and thermodynamics of quantum gravity, and that its considerations in the latter could in turn provide valuable insights into the former.

These are the two facets of interest to us here. In Section 2, we discuss a potential background independent extension of equilibrium statistical mechanics, giving a succinct ye<sup>t</sup> complete discussion of past works in Section 2.1.1, and subsequently focussing on a new thermodynamical characterisation for background independent equilibrium in Section 2.1.2, which is based on a constrained maximisation of information entropy. In Section 2.2, we detail further crucial properties of this characterisation, while placing it within a bigger context of the issue of background independent statistical equilibrium, also in comparison with the previous proposals. Section 2.3 is more exploratory, remarking on exciting new connections between the thermodynamical characterisation and the thermal time hypothesis, wherein information entropy and observer dependence are seen to play instrumental roles. In Section 2.4, we discuss several aspects of a generalised thermodynamics based on the generalised equilibrium statistical mechanics derived above, including statements of the zeroth and first laws. Section 3 is devoted to statistical mechanical considerations of candidate quantum gravity degrees of freedom of combinatorial and algebraic type. After clarifying the framework for many-body mechanics of such atoms of space in Section 3.1, we give an overview of examples in Section 3.2, thus illustrating the applicability of the generalised statistical framework in quantum gravity. The one case for which we give a slightly more detailed account is that of deriving a generic covariant group field theory as an effective statistical field theory starting from a particular class of quantum Gibbs states of the underlying microscopic system. Finally, we conclude and offer some outlook.

### **2. Background Independent Equilibrium Statistical Mechanics**

Covariant statistical mechanics [1–3] broadly aims at addressing the foundational issue of defining a suitable statistical framework for constrained systems. This issue, especially in the context of gravity, was brought to the fore in a seminal work [1], and developed subsequently in [2,3,10,11]. Valuable insights from these studies on spacetime relativistic systems [1–3,11–13] have also formed the conceptual backbone of first applications to discrete quantum gravity [14–16]. In this section, we present extensions of equilibrium statistical mechanics to background independent<sup>1</sup> systems, laying out different proposals for a generalised statistical equilibrium, but emphasising on one in particular, and based on which further aspects of a generalised thermodynamics are considered. The aim here is thus to address the fundamental problem of formulating these frameworks in settings where the conspicuous absence of time and energy is particularly tricky.

Section 2.1 discusses background independent characterisations of equilibrium Gibbs states, of the general form *e*<sup>−</sup> ∑*a βa*O*a* . In Section 2.1.1, we touch upon various proposals for equilibrium put forward in past studies on spacetime covariant systems [1,3,11,17,18]. From Section 2.1.2 onwards, we focus on Jaynes' information-theoretic characterisation [19,20] for equilibrium. This was first suggested as a viable proposal for background independent equilibrium, and illustrated with an explicit example in the context of quantum gravity, in [14]. Using the terminology of [14], we call this a 'thermodynamical' characterisation of equilibrium, to contrast with the customary Kubo–Martin–Schwinger (KMS) [21] 'dynamical' characterisation2.

We devote Section 2.2 to discussing various aspects of the thermodynamical characterisation, including highlighting many of its favourable features, also compared to the other proposals. In fact, we point out how this characterisation can comfortably accommodate the other proposals for Gibbs equilibrium.

Further, as is evident shortly, the thermodynamical characterisation hints at the idea that entropy is a central player, which has been a recurring theme across modern theoretical physics. In Section 2.3, we present a tentative discussion on some of these aspects. In particular, we notice compelling new relations between the thermodynamical characterisation and the thermal time hypothesis, which further seem to hint at intriguing relations between entropy, observer dependence and thermodynamical time. We further propose to use the thermodynamical characterisation as a constructive criterion of choice for the thermal time hypothesis.

Finally, in Section 2.4, we define the basic thermodynamic quantities which can be derived immediately from a generalised equilibrium state, without requiring any additional physical and/or interpretational inputs. We clarify the issue of extracting a single common temperature for the full system from a set of several of them, and end with the zeroth and first laws of a generalised thermodynamics.

<sup>1</sup> In the original works mentioned above, the framework is usually referred to as covariant or general relativistic statistical mechanics. However, we choose to call it background independent statistical mechanics as our applications to quantum gravity are evident of the fact that the main ideas and structures are general enough to be used in radically background independent systems devoid of any spacetime manifold or associated geometric structures.

<sup>2</sup> For a more detailed discussion of the comparison between these two characterisations, we refer the reader to [14]. The main idea is that the various proposals for generalised Gibbs equilibrium can be divided into these two categories. Which characterisation one chooses to use in a given situation depends on the information/description of the system that one has at hand. For instance, if the description includes a one-parameter flow of physical interest, then using the dynamical characterisation, i.e., satisfying the KMS condition with respect to it, will define equilibrium with respect to it. The procedures defining these two categories can thus be seen as 'recipes' for constructing a Gibbs state, and which one is more suitable depends on our knowledge of the system.

### *2.1. Generalised Equilibrium*

Equilibrium states are a cornerstone of statistical mechanics, which in turn is the theoretical basis for thermodynamics. They are vital in the description of macroscopic systems with a large number of microscopic constituents. In particular, Gibbs states *e*<sup>−</sup>*β<sup>E</sup>* have a vast applicability across a broad range of fields such as condensed matter physics, quantum information and tensor networks, and (quantum) gravity, to name a few. They are special, being the unique class of states in finite systems satisfying the KMS condition3. Furthermore, usual coarse-graining techniques also rely on the definition of Gibbs measures. In treatments closer to hydrodynamics, one often considers the full (non-equilibrium) system as being composed of many interacting subsystems, each considered to be at local equilibrium. While in the context of renormalisation group flow treatments, each phase at a given scale, for a given set of coupling parameters is also naturally understood to be at equilibrium, each described by (an inequivalent) Gibbs measure.

Given this physical interest in Gibbs states, the question then is how to define them for background independent systems. The following are different proposals, all relying on different principles originating in standard non-relativistic statistical mechanics, extended to a relativistic setting.

### 2.1.1. Past Proposals

The first proposal [1,12] was based on the idea of statistical independence of arbitrary (small, but macroscopic) subsystems of the full system. The notion of equilibrium is taken to be characterised by the factorisation property of the state, *ρ*12 = *ρ*1*ρ*2, for any two subsystems 1 and 2; and the full system is at equilibrium if any one of its subsystems satisfies this property with all the rest. We notice that the property of statistical independence is related to an assumption of weak interactions [22].

This same dilute gas assumption is integral also to the Boltzmann method of statistical mechanics. It characterises equilibrium as the most probable distribution, that is one with maximum entropy4. This method is used in [11] to study a gas of constrained particles5.

The work in [3] puts forward a physical characterisation for an equilibrium state. The suggestion is that *ρ* (itself a well-defined state on the physical, reduced state space) is said to be a physical Gibbs state if its modular Hamiltonian *h* = − ln *ρ*, is a well-defined function on the physical state space; and, is such that there exists a (local) clock function *<sup>T</sup>*(**x**) on the extended state space (with its conjugate momentum *pT*(**x**)), such that the (pull-back) of *h* is proportional to (the negative of) *pT*. Importantly, when this is the case the modular flow ('thermal time', see Section 2.2) is a geometric (foliation) flow in spacetime, in which sense *ρ* is said to be 'physical'. Notice that the built-in strategy here is to define KMS equilibrium in a deparameterised system (thus it is an example of using the dynamical characterisation), since it basically identifies a state's modular Hamiltonian with a (local) clock Hamiltonian on the base spacetime manifold.

Another strategy [17] is based on the use of the ergodic principle and introduction of clock subsystems to define (clock) time averages. Again, this characterisation, similar to a couple of the previous ones, relies on the validity of a postulate, even if traditionally a fundamental one.

Finally, the proposal of [18] interestingly characterises equilibrium by a vanishing information flow between interacting histories. The notion of information used is that of Shannon (entropy), *I* = ln *N*, where *N* is the number of microstates traversed in a given history during interaction.

<sup>3</sup> The algebraic KMS condition [21] is well known to provide a comprehensive characterisation of statistical equilibrium in systems of arbitrary sizes, as long as there exists a well-defined one-parameter dynamical group of automorphisms of the system. This latter point, of the required existence of a preferred time evolution of the system, is exactly the missing ingredient in our case, thus limiting its applicability.

<sup>4</sup> Even though this method relies on maximising the entropy similar to the thermodynamical characterisation, it is more restrictive than the latter, as is made clear in Section 2.2.

<sup>5</sup> We remark that except for this one work, all other studies in spacetime covariant statistical mechanics are carried out from the Gibbs ensemble point of view.

Equilibrium between two histories 1 and 2 is encoded in a vanishing information flow, *δI* = *I*2 − *I*1 = 0. This characterisation of equilibrium is evidently information-theoretic, even if relying on an assumption of interactions. Moreover, it is much closer to our thermodynamical characterisation, because the condition of vanishing *δI* is nothing but an optimisation of information entropy.

These different proposals, along with the thermal time hypothesis [1,2], have led to some remarkable results, such as recovering the Tolman–Ehrenfest effect [13,23], relativistic Jüttner distribution [23] and Unruh effect [24]. However, they all assume the validity of one or more principles, postulates or assumptions about the system. Moreover, none (at least presently) seems to be general enough, as with the proposal below, to be implemented in a full quantum gravity setup, while also accommodating within it the rest of the proposals.
