*2.2. Remarks*


<sup>6</sup> Local, in the sense of being observer-dependent (see Section 2.2).

<sup>7</sup> This is also the main ingredient of the thermal time hypothesis [1,2], which we return to below.

<sup>8</sup> However, as Jaynes points out in [19], these properties were relegated to side remarks in the past, not really considered to be fundamental to the theory or to the justifications for the methods of statistical mechanics.

equilibrium statistical mechanics in terms of evidential probabilities, solely as a problem of statistical inference without depending on the validity of any further conjectures, physical assumptions or interpretations; and in the suggestion that it is general enough to apply to genuinely background independent systems, including quantum gravity. Below, we list some of these more valuable features.


In Section 3.2, we outline some examples of using this characterisation in quantum gravity, while a detailed investigation of its consequences in particular for covariant systems on a spacetime manifold is left to future studies.

<sup>9</sup> In fact, in hindsight, we could already have anticipated a possible equilibrium description in terms of these constants, whose existence is assumed from the start.

### *2.3. Relation to Thermal Time Hypothesis*

This section outlines a couple of new intriguing connections between the thermodynamical characterisation and the thermal time hypothesis, which we think are worthwhile to be explored further. Thermal time hypothesis [1,2] states that the (geometric) modular flow of the (physical, equilibrium) statistical state that an observer happens to be in is the time that they experience. It thus argues for a thermodynamical origin of time [30].

What is this state? Pragmatically, the state of a macroscopic system is that which an observer is able to observe and assigns to the system. It is not an absolute property since one can never know everything there is to know about the system. In other words, the state that the observer "happens to be in" is the state that they are able to detect. This leads us to sugges<sup>t</sup> that the thermodynamical characterisation can provide a suitable *criterion of choice* for the thermal time hypothesis.

What we mean by this is the following. Consider a macroscopic system that is observed to be in a particular macrostate in terms of a set of (constant) observable averages. The thermodynamical characterisation then provides the least biased choice for the underlying (equilibrium) statistical state. Given this state then, the thermal time hypothesis would imply that the (physical) time experienced by this observer is the (geometric) modular flow of the observed state.

Jaynes [19,20] turned the usual logic of statistical mechanics upside-down to stress on entropy and the observed macrostate as the starting point, to define equilibrium statistical mechanics in its entirety *from* it (and importantly, a further background independent generalisation, as shown above). Rovelli [1], later with Connes [2], turned the usual logic of the definition of time upside-down to stress on the choice of a statistical state as the starting point to identify a suitable time flow *from* it. The suggestion here is to merge the two and ge<sup>t</sup> an operational way of implementing the thermal time hypothesis.

It is interesting to see that the crucial property of observer-dependence of relativistic time arises as a natural consequence of our suggestion, directly because of the observer-dependence of any state defined using the thermodynamical characterisation. This way, thermodynamical time is intrinsically "perspectival" [31] or "anthropomorphic" [32].

To be clear, this criterion of choice will not single out a preferred state, by the very fact that it is inherently observer-dependent. It is thus compatible with the basic philosophy of the thermal time hypothesis, namely that there is no preferred physical time.

Presently the above suggestion is rather conjectural, and certainly much work remains to be done to understand it better, and explore its potential consequences for physical systems. Here, it may be helpful to realise that the thermal time hypothesis can be sensed to be intimately related with (special and general) relativistic systems, and so might the thermodynamical characterisation when considered in this context. Thus, for instance, Rindler spacetime or stationary black holes might offer suitable settings to begin investigating these aspects in more detail.

The second connection that we observe is much less direct, and is via information entropy. The generator of the thermal time flow [1], − ln *ρ*, can immediately be observed to be related to Shannon entropy in Equation (2). Moreover, in the general algebraic (quantum) field theoretic setting, the generator is the log of the modular operator Δ of von Neumann algebra theory [2]. A modification of it, the relative modular operator, is known to be an algebraic measure of relative entropy [33], which in fact has seen a recent revival in the context of quantum information and gravity. This is a remarkable feature in our opinion, which compels us to look for deeper insights it may have to offer, in further studies.

### *2.4. Generalised Thermodynamic Potentials, Zeroth and First Laws*

Traditional thermodynamics is the study of energy and entropy exchanges. However, what is a suitable generalisation of it for background independent systems? This, as with the question of a generalised equilibrium statistical mechanics which we have considered until now, is still open. In the following, we offer some insights gained from preceding discussions, including identifying certain thermodynamic potentials, and generalised zeroth and first laws.

Thermodynamic potentials are vital, particularly in characterising the different phases of the system. The most important one is the partition function *<sup>Z</sup>*{*βa*}, or equivalently the free energy

$$\Phi(\{\beta\_a\}) := -\ln Z\_{\{\beta\_a\}}\,. \tag{5}$$

It encodes complete information about the system from which other thermodynamic quantities can be derived in principle. Notice that the standard definition of a free energy *F* comes with an additional factor of a (single, global) temperature, that is we normally have Φ = *βF*. However, for now, Φ is the more suitable quantity to define and not *F* since we do not (yet) have a single common temperature for the full system. We return to this point below.

Next is the thermodynamic entropy (which by use of the thermodynamical characterisation has been identified with information entropy), which is straightforwardly

$$S(\left\{\mathcal{U}\_a\right\}) = \sum\_a \beta\_a \mathcal{U}\_a - \Phi \tag{6}$$

for generalised Gibbs states of the form in Equation (3). Notice again the lack of a single *β* scaling the whole equation at this more general level of equilibrium.

By varying *S* such that the variations *dUa* and *d*O*a* are independent [19], a set of generalised heats can be defined

$$dS = \sum\_{a} \beta\_{a} (d\mathcal{U}\_{a} - \langle d\mathcal{O}\_{a} \rangle) =: \sum\_{a} \beta\_{a} \, d\mathcal{Q}\_{a} \tag{7}$$

and, from it (at least part of the10) work done on the system *dWa* [15], can be identified

$$d\mathcal{W}\_{\mathfrak{a}} := \left< d\mathcal{O}\_{\mathfrak{a}} \right> = \frac{1}{\beta\_{\mathfrak{a}}} \int\_{\Gamma} d\lambda \, \frac{\delta\Phi}{\delta\mathcal{O}\_{\mathfrak{a}}} \, d\mathcal{O}\_{\mathfrak{a}} \,. \tag{8}$$

From the setup of the thermodynamical characterisation presented in Section 2.1.2, we can immediately identify *Ua* as generalised "energies". Jaynes' procedure allows these quantities to *democratically* play the role of generalised energies. None had to be selected as being *the* energy in order to define equilibrium. This a priori democratic status of the several conserved quantities can be broken most easily by preferring one over the others. In turn, if its modular flow can be associated with a physical evolution parameter (relational or not), then this observable can play the role of a dynamical Hamiltonian.

Thermodynamic conjugates to these energies are several generalised inverse temperatures *β<sup>a</sup>*. By construction, each *βa* is the periodicity in the flow of O*<sup>a</sup>*, in addition to being the Lagrange multiplier for the *a*th constraint in Equation (1). Moreover, these same constraints can determine *β<sup>a</sup>*, by inverting the equations

$$\frac{\partial \Phi}{\partial \beta\_a} = \mathcal{U}\_a;$$

or equivalently from

$$\frac{\partial S}{\partial \mathcal{U}\_a} = \beta\_a \,. \tag{10}$$

<sup>10</sup> By this we mean that the term *d*O*a* , based on the *same* observables defining the generalised energies *Ua*, can be seen as reflecting some work done on the system. However, naturally, we do not expect or claim that this is all the work that is/can be performed on the system by external agencies. In other words, there could be other work contributions, in addition to the terms *dWa*. A better understanding of work terms in this background independent setup, will also contribute to a better understanding of the generalised first law presented below.

In general, {*βa*} is a multi-variable inverse temperature. In the special case when O*a* are component functions of a dual vector, then - *β* ≡ (*βa*) is a vector-valued temperature. For example, this is the case when O ≡ {O - *a*} are dual Lie algebra-valued momentum maps associated with Hamiltonian actions of Lie groups, as introduced by Souriau [26,27], and appearing in the context of classical polyhedra in [15].

As shown above, a generalised equilibrium is characterised by several inverse temperatures, but an identification of a single common temperature for the full system is of obvious interest. This can be done as follows [12,15]. A state of the form in Equation (3), with modular Hamiltonian

$$h = \sum\_{a} \beta\_{a} \mathcal{O}\_{a} \tag{11}$$

generates a modular flow (with respect to which it is at equilibrium), parameterised by

$$t = \sum\_{a} \frac{t\_a}{\beta\_a} \tag{12}$$

where *ta* are the flow parameters of O*<sup>a</sup>*. The strategy now is to reparameterise the same trajectory by a rescaling of *t*,

$$
\pi := \mathfrak{t}/\beta \tag{13}
$$

for a real-valued *β*. It is clear that *τ* parameterises the modular flow of a rescaled modular hamiltonian ˜ *h* = *βh*, associated with the state

$$\bar{\rho}\_{\notin} = \frac{1}{\mathbb{Z}\_{\notin}} e^{-\hat{\mathsf{H}}} = \frac{1}{\mathbb{Z}\_{\notin}} e^{-\beta \hat{\mathsf{H}}} \tag{14}$$

characterised now by a single inverse temperature *β*.

In fact, this state can be understood as satisfying the thermodynamical characterisation for a single constraint

$$
\langle h \rangle = \text{constant} \tag{15}
$$

instead of several of them as in Equation (1). Clearly, this rescaling is not a trivial move. It corresponds to the case of a weaker, single constraint which by nature corresponds to a different physical situation wherein there is exchange of information between the different observables (so that they can thermalise to a single *β*). This can happen for instance when one observable is special (e.g., the Hamiltonian) and the rest are functionally related to it (e.g., the volume or number of particles). Whether such a determination of a single temperature can be brought about by a more physically meaningful technique is left to future work. Having said that, it will not change the general layout of the two cases as outlined above.

One immediate consequence of extracting a single *β* is regarding the free energy, which can now be written in the familiar form as

$$
\Phi = \beta F.\tag{16}
$$

This is most directly seen from the expression for the entropy,

$$\mathcal{S} = - \langle \ln \bar{\rho}\_{\beta} \rangle\_{\beta\_{\beta}} = \beta \sum\_{a} \beta\_{a} \mathcal{U}\_{a} + \ln Z \quad \Leftrightarrow \quad \mathcal{F} = \mathcal{U} - \beta^{-1} \mathcal{S} \tag{17}$$

where *U* ˜ = ∑*a βaU* ˜ *a* is a total energy, and tildes mean that the quantities are associated with the state *ρ*˜*β*. Notice that the above equation clearly identifies a single conjugate variable to entropy, the temperature *β*−1.

It is important to remark that, in the above method to ge<sup>t</sup> a single *β*, we still do not need to choose a special observable, say O, out of the given set of O*<sup>a</sup>*. If one were to do this, i.e., select O as a dynamical energy (so that by extension the other O*a* are functions of this one), then by standard arguments, the rest of the Lagrange multipliers will be proportional to *β*, which in turn would then be the common inverse temperature for the full system. The point is that this latter instance is a special case of the former.

*Universe* **2019**, *5*, 187

We end this section with zeroth and first laws of generalised thermodynamics. The crux of the zeroth law is a definition of equilibrium. Standard statement refers to a thermalisation resulting in a single temperature being shared by any two systems in thermal contact. This can be extended by the statement that at equilibrium, all inverse temperatures *βa* are equalised. This is in exact analogy with all intensive thermodynamic parameters, such as the chemical potential, being equal at equilibrium.

The standard first law is basically a statement about conservation of energy. In the generalised equilibrium case corresponding to the set of individual constraints in Equation (1), the first law is satisfied *a*th-energy-wise,

$$d\mathcal{U}\_{\mathfrak{a}} = d\mathcal{Q}\_{\mathfrak{a}} + d\mathcal{W}\_{\mathfrak{a}} \,. \tag{18}$$

The fact that the law holds *a*-energy-wise is not surprising because the separate constraints in Equation (1) for each *a* mean that observables O*a* do not exchange any information amongs<sup>t</sup> themselves. If they did, then their Lagrange multipliers would no longer be mutually independent and we would automatically reduce to the special case of having a single *β* after thermalisation.

On the other hand, for the case with a single *β*, variation of the entropy in Equation (17) gives

$$d\tilde{S} = \beta \sum\_{a} \beta\_{a} (d\mathcal{U}\_{a} - \langle d\mathcal{O}\_{a} \rangle) =: \beta d\tilde{\mathcal{Q}}\tag{19}$$

giving a first law with a more familiar form, in terms of total energy, total heat and total work variations

$$d\hat{d}\hat{l} = d\tilde{Q} + d\tilde{W}.\tag{20}$$

As before, in the even more special case where *β* is conjugate to a single preferred energy, then this reduces to the traditional first law. We leave the verification of the second law for the generalised entropy to future work. Further, the quantities introduced above and the consequences of this setup also need to be investigated in greater detail.

### **3. Equilibrium Statistical Mechanics in Quantum Gravity**

Emergence of spacetime is the outstanding open problem in quantum gravity that is being addressed from several directions. One such is based on modelling quantum spacetime as a many-body system [34], which further complements the view of a classical spacetime as an effective macroscopic thermodynamic system. This formal suggestion allows one to treat extended regions of quantum spacetime as built out of discrete building blocks whose dynamics is dictated by non-local, combinatorial and algebraic mechanical models. Based on this mechanics, a formal statistical mechanics of the quanta of space can be studied [14,15]. Statistical mixtures of quantum gravity states are better suited to describe generic boundary configurations with a large number of quanta. This is in the sense that given a region of space with certain known macroscopic properties, a more reasonable modelling of its underlying quantum gravity description would be in in terms of a mixed state rather than a pure state, essentially because we cannot hope to know precisely all microscopic details to prefer one particular microstate. A simple example is having a region with a fixed spatial volume and wanting to estimate the underlying quantum gravity (statistical) state [11,14].

In addition to the issue of emergence, investigating the statistical mechanics and thermodynamics of quantum gravity systems would be expected to contribute towards untangling the puzzling knot between thermodynamics, gravity and the quantum theory, especially when applied to more physical settings, such as cosmology [28].

In the rest of this article, we use results from the previous sections to outline a framework for equilibrium statistical mechanics for candidate quanta of geometry (along the lines presented in [14,15], but generalising further to a richer combinatorics based on [35]), and within it give an overview of some concrete examples. In particular, we show that a group field theory can be understood as an effective statistical field theory derived from a coarse-graining of a generalised Gibbs configuration of the underlying quanta. In addition to providing an explicit quantum statistical

basis for group field theories, it further reinforces their status as being field theories for quanta of geometry [36–39]. As expected, we see that even though the many-body viewpoint makes certain techniques available that are almost analogous to standard treatments, there are several non-trivialities such as that of background independence, and physical (possible pre-geometric and effective geometric) interpretations of the statistical and thermodynamic quantities involved.
