*3.2. Applications*

We briefly sketch below some examples of applying the above framework.

A couple of examples for a classical system are studied in [15]. In the process of applying the thermodynamical characterisation, these cases introduce a statistical, effective manner of imposing a given (set of) first class constraint(s), that is *C* = 0, instead of the exact, strong way *C* = 0. In one case, the condition of closure of a classical *d*-polyhedron is relaxed in this statistical manner, while in the other the boundary gluing constraints amongs<sup>t</sup> the polyhedral atoms of space are relaxed in this way to describe fluctuating twisted geometries. Brief summaries of these follow.

In the first example, starting from the extended state space Γex = *I S*2 *AI* of intrinsic geometries of a *d*-polyhedron with face areas {*AI*}*<sup>I</sup>*=1,...,*d*, closure is implemented via the following su(2)<sup>∗</sup>-valued function on Γex,

$$J = \sum\_{I=1}^{d} x\_I \tag{28}$$

which is the momentum map associated with the diagonal action of *SU*(2). Satisfying closure exactly is to have *J* = 0. Then, applying the thermodynamic characterisation to the scalar component functions of *J*, that is requiring *Ja* = 0 (*a* = 1, 2, <sup>3</sup>), gives a Gibbs distribution on Γex of the form *e*<sup>−</sup> ∑*a βa Ja* with a vector-valued temperature (*βa*) ∈ su(2). Thus, we have a thermal state for a classical polyhedron that is fluctuating in terms of its closure, with the fluctuations controlled by the parameter *β*. In fact, this state generalises Souriau's Gibbs states [26,27] to the case of Lie group (Hamiltonian) actions associated with first class constraints.

In the other example, the set of half-link gluing (or face-sharing) conditions for a boundary graph are statistically relaxed. It is known that an oriented (closed) boundary graph *γ*, with *M* nodes and *L* links, labelled with (*g*, *x*) ∈ *T*∗(*SU*(2)) variables admits a notion of discrete (closed) twisted geometry [48]. Twisted geometries are a generalisation of the more rigid Regge geometries, wherein the shapes of the shared faces are left arbitrary and only their areas are constrained to match. From the present constructive many-body viewpoint, one can understand these states instead as a result of satisfying a set of *SU*(2)- and su(2)<sup>∗</sup>-valued gluing conditions (denoted, respectively, by {*C*} and {*D*}) on an initially disconnected system of several labelled open nodes. That is, starting from a system of *M* number of labelled open nodes, one ends up with a twisted geometric configuration if the set of gluing constraints on the holonomy and flux variables corresponding to a given *γ*, {*<sup>C</sup>*,*<sup>a</sup>*(*gng*<sup>−</sup><sup>1</sup> *m* ) = 0, *<sup>D</sup>*,*<sup>a</sup>*(*xn* − *xm*) = <sup>0</sup>}*<sup>γ</sup>*, are satisfied strongly (component-wise). Here, = 1, 2, ..., *L* labels a full link, *a* = 1, 2, 3 is *SU*(2) component index, and subscripts *n* refer to the half-link (belonging to the full link) of node *n*. We can then choose instead to impose these constraints weakly by requiring only its statistical averages in a state to vanish. This gives a *γ*-dependent state on Γ*M*, written formally as

$$\rho\_{\{\gamma,a,\emptyset\}} \propto e^{-\sum\_{\ell} \sum\_{a} a\_{\ell,a} \mathbb{C}\_{\ell,a} + \oint\_{\ell,a} D\_{\ell,a}} \equiv e^{-G\_{\gamma}(a,\emptyset)} \tag{29}$$

where *α*, *β* ∈ R3*<sup>L</sup>* are generalised inverse temperatures characterising this fluctuating twisted geometric configuration. In fact, one can generalise this state to a probabilistic superposition of such internally fluctuating twisted geometries for an *N* particle system (thus, defined on Γ*N*), which includes

contributions from distinct graphs, each composed of a possibly variable number of nodes *M*. A state of this kind can formally be written as,

$$\rho\_N = \frac{1}{Z\_N(M\_{\text{max}}, \lambda\_{\gamma^t}, a\_\prime \beta)} e^{-\sum\_{M=2}^{M\_{\text{max}}} \sum\_{\{\gamma\}\_M} \frac{1}{\lambda\_{\text{var}}(\gamma)} \lambda\_\gamma \sum\_{i\_1 \neq \dots \neq i\_M = 1}^{N} G\_\gamma(\vec{g}\_{i\_1} \vec{x}\_{i\_1 \wedge \dots \wedge \vec{g}\_{i\_M}} \vec{x}\_{i\_M} \vec{x}\_{i\_M} \mu\_\gamma\theta)} \tag{30}$$

where *i* is the particle index, and *M*max ≤ *N*. The value of *M*max and the set {*γ*}*M* for a fixed *M* are model-building choices. The first sum over *M* includes contributions from all admissible (depending on the model, determined by *M*max) different *M*-particle subgroups of the full *N* particle system, with the gluing combinatorics of various different boundary graphs with *M* nodes. The second sum is a sum over all admissible boundary graphs *γ*, with a given fixed number of nodes *M*. Furthermore, the third sum takes into account all *M*-particle subgroup gluings (according to a given fixed *γ*) of the full *N* particle system. We note that the state in Equation (30) is a further generalisation of that presented in [15]; specifically, the latter is a special case of the former for the case of a single term *M* = *M*max = *N* in the first sum. Further allowing for the system size to vary, that is considering a variable *N*, gives the most general configuration, with a set of coupling parameters linked directly to the underlying microscopic model,

$$Z(M\_{\text{max},\prime}\lambda\_{\gamma\prime}\mathfrak{a},\beta) = \sum\_{N\geq 0} e^{\mu N} Z\_N(M\_{\text{max},\prime}\lambda\_{\gamma\prime}\mathfrak{a},\beta) \,. \tag{31}$$

A physically more interesting example is considered in [14], which defines a thermal state with respect to a spatial volume operator,

$$
\hat{\rho} = \frac{1}{Z} e^{-\beta \hat{\mathcal{V}}} \tag{32}
$$

where Vˆ = *dg v*(*g*)*ϕ*ˆ†(*g*)*ϕ*<sup>ˆ</sup>(*g*) is a positive, self-adjoint operator on H*<sup>F</sup>*, and the state is a well-defined density operator on the same. In fact, with a grand-canonical extension of it, this system can be shown to naturally support Bose–Einstein condensation to a low-spin phase [14]. Clearly, this state encodes thermal fluctuations in the volume observable, which is especially an important one in the context of cosmology. In fact, the rapidly developing field of condensate cosmology [54] for atoms of space of the kind considered here, is based on modelling the underlying system as a condensate, and subsequently extracting effective physics from it. These are certainly crucial steps in the direction of obtaining cosmological physics from quantum gravity [9]. It is equally crucial to enrich further the microscopic quantum gravity description itself, and extract effective physics for these new cases. One such important case is to consider thermal fluctuations of the gravitational field at early times, during which our universe is expected to be in a quantum gravity regime. That is, to consider *thermal* quantum gravity condensates using the frameworks laid out in this article (as opposed to the zero temperature condensates that have been used till now), and subsequently derive effective physics from them. This case would then directly reflect thermal fluctuations of gravity as being of a proper quantum gravity origin. This is investigated in [28].

We end this section by making a direct link to the definition of group field theories using the above framework. Group field theories (GFT) [37–39] are non-local field theories defined over (copies of) a Lie group. Most widely studied (Euclidean) models are for real or complex scalar fields, over copies of *SU*(2), *Spin*(4) or *SO*(4). For instance, a complex scalar GFT over *SU*(2) is defined by a partition function of the following general form,

$$Z\_{\rm GFT} = \int \left[ D\mu(\varphi, \Phi) \right] \, e^{-S\_{\rm GFT}[\varphi, \Phi]} \tag{33}$$

where *μ* is a functional measure which in general is ill-defined, and *S*GFT is the GFT action of the form (for commonly encountered models),

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

$$S\_{\rm GFT} = \int\_{G} d\mathfrak{g}\_{1} \int\_{G} d\mathfrak{g}\_{2} \, K(\mathfrak{g}\_{1}, \mathfrak{g}\_{2}) \mathfrak{g}(\mathfrak{g}\_{1}) \mathfrak{g}(\mathfrak{g}\_{2}) + \int\_{G} d\mathfrak{g}\_{1} \int\_{G} d\mathfrak{g}\_{2} \dots V(\mathfrak{g}\_{1}, \mathfrak{g}\_{2}, \dots) f(\mathfrak{g}, \mathfrak{g}) \tag{34}$$

where *g* ∈ *G*, and the kernel *V* is generically non-local, which convolutes the arguments of several *ϕ* and *ϕ*¯ fields (written here in terms of a single function *f*). It defines the interaction vertex of the dynamics by enforcing the combinatorics of its corresponding (unique, via the inverse of the bulk map) boundary graph.

*Z*GFT defines the covariant dynamics of the GFT model encoded in *S*GFT. Below we outline a way to derive such covariant dynamics from a suitable quantum statistical equilibrium description of a system of quanta of space defined previously in Section 3.1. The following technique of using field coherent states is the same as in [15,29], but with the crucial difference that here we do not claim to define, or aim to achieve any correspondence (even if formal) between a canonical dynamics (in terms of a projector operator) and a covariant dynamics (in terms of a functional integral). Here we simply show a quantum statistical basis for the covariant dynamics of a GFT, and in the process, reinterpret the standard form of the GFT partition function in Equation (33) as that of an effective statistical field theory arising from a coarse-graining and further approximations of the underlying statistical quantum gravity system.

We saw in Section 3.1 that the dynamics of the polyhedral atoms of space is encoded in the choices of propagators and interaction vertices, which can be written in terms of kinetic and vertex operators in the Fock description. In our present considerations with a single type of atom (*SU*(2)-labelled 4-valent node), let us then consider the following generic kinetic and vertex operators,

$$\hat{\mathbb{K}} = \int\_{SL(2)^8} [d\mathfrak{g}] \, \, \hat{\mathfrak{g}}^{\dagger}(\vec{\mathfrak{g}}\_1) K(\vec{\mathfrak{g}}\_1, \vec{\mathfrak{g}}\_2) \phi(\vec{\mathfrak{g}}\_2) \quad , \quad \hat{\mathbb{V}} = \int\_{SL(2)^{4\mathcal{N}}} [d\mathfrak{g}] \, \, V\_{\uparrow}(\vec{\mathfrak{g}}\_1, ..., \vec{\mathfrak{g}}\_N) \hat{f}(\phi, \phi^{\dagger}) \tag{35}$$

where *N* > 2 is the number of 4-valent nodes in the boundary graph *γ*, and ˆ *f* is a function of the ladder operators with all terms of a single degree *N*. For example, when *N* = 3, this function could be ˆ *f* = *<sup>λ</sup>*1*ϕ*<sup>ˆ</sup>*ϕ*<sup>ˆ</sup>*ϕ*ˆ† + *<sup>λ</sup>*2*ϕ*ˆ†*ϕ*<sup>ˆ</sup>*ϕ*ˆ†. As shown above, in principle, a generic model can include several distinct vertex operators. Even though what we have considered here is the simple of case of having only one, the argumen<sup>t</sup> can be extended directly to the general case.

Operators Kˆ and Vˆ have well-defined actions on the Fock space H*<sup>F</sup>*. Using the thermodynamical characterisation then, we can consider the formal constraints<sup>16</sup> Kˆ = constant and Vˆ = constant, to write down a generalised Gibbs state on H*<sup>F</sup>*,

$$\hat{\rho}\_{\{\hat{\mathbb{B}}\_{\mathbf{a}}\}} = \frac{1}{Z\_{\{\hat{\mathbb{B}}\_{\mathbf{a}}\}}} e^{-\beta\_1 \mathbb{R} - \beta\_2 \mathbb{V}} \tag{36}$$

where *a* = 1, 2 and the partition function<sup>17</sup> is,

$$Z\_{\{\beta\_a\}} = \text{Tr}\_{\mathcal{H}\_F}(e^{-\beta\_1 \triangle - \beta\_2 \triangle})\,. \tag{37}$$

An effective field theory can then be extracted from the above by using a basis of coherent states on H*F* [15,29,55]. Field coherent states give a continuous representation on H*F* where the parameter labelling each state is a wave (test) function [55]. For the Fock description mentioned in Section 3.1, the coherent states are

$$<\langle \psi \rangle = e^{\phi^\dagger(\psi) - \phi(\psi)} \left| 0 \right\rangle \tag{38}$$

<sup>16</sup> A proper interpretation of these constraints is left for future work.

<sup>17</sup> This partition function will in general be ill-defined as expected. One reason is the operator norm unboundedness of the ladder operators.

where |0 is the Fock vacuum (satisfying *ϕ*<sup>ˆ</sup>(*g*)|<sup>0</sup> = 0 for all *g*), *ϕ*<sup>ˆ</sup>(*ψ*) = *SU*(2)<sup>4</sup> *ψ*¯*ϕ*<sup>ˆ</sup> and its adjoint are smeared operators, and *ψ* ∈ H. The set of all such states provides an over-complete basis for H*<sup>F</sup>*. The most useful property of these states is that they are eigenstates of the annihilation operator,

$$
\langle \Phi(\vec{\g}) | \Psi \rangle = \psi(\vec{\g}) \left| \Psi \right\rangle \,. \tag{39}
$$

The trace in the partition function in Equation (37) can then be evaluated in this basis,

$$Z\_{\{\beta\_{\pi}\}} = \int [D\mu(\psi, \vec{\psi})] \,\, \langle \psi \vert \, e^{-\beta\_1 \hat{\mathbb{K}} - \beta\_2 \hat{\mathbb{V}}} \, \vert \psi \rangle \tag{40}$$

where *μ* here is the coherent state measure [55]. The integrand can be treated and simplified along the lines presented in [15] (to which we refer for details), to ge<sup>t</sup> an effective partition function,

$$Z\_0 = \int \left[ D\mu(\psi, \bar{\psi}) \right] e^{-\beta\_1 \mathcal{K}[\psi, \psi] - \beta\_2 V[\psi, \psi]} = Z\_{\{\beta\_x\}} - Z\_{\mathcal{O}(\hbar)} \tag{41}$$

where subscript 0 indicates that we have neglected higher order terms, collected inside *<sup>Z</sup>*O(*h*¯), resulting from normal orderings of the exponent in *<sup>Z</sup>*{*βa*}, and the functions in the exponent are *K* = *ψ*| : Kˆ : |*ψ* and *V* = *ψ*| : Vˆ : |*ψ* . It is then evident that *Z*0 has the precise form of a generic GFT partition function. It thus *defines* a group field theory as an effective statistical field theory, that is

$$\mathbf{Z}\_{\text{CFT}} := \mathbf{Z}\_0 \,. \tag{42}$$

From this perspective, it is clear that the generalised inverse temperatures (which are basically the intensive parameters conjugate to the energies in the generalised thermodynamics setting of Section 2.4) *are* the coupling parameters defining the effective model, thus characterising the phases of the emergen<sup>t</sup> statistical group field theory, as would be expected. Moreover, from this purely statistical standpoint, we can understand the GFT action more appropriately as Landau–Ginzburg free energy (or effective "Hamiltonian', in the sense that it encodes the effective dynamics), instead of a Euclidean action, which might imply having Wick rotated a Lorentzian measure, even in an absence of any such notions as is the case presently. Lastly, deriving in this way the covariant definition of a group field theory, based entirely on the framework presented in Section 3.1, strengthens the statement that a group field theory is a field theory of combinatorial and algebraic quanta of space [38,39].

### **4. Conclusions and Outlook**

We have presented an extension of equilibrium statistical mechanics for background independent systems, based on a collection of results and insights from old and new studies. While various proposals for a background independent notion of statistical equilibrium have been summarised, one in particular, based on the constrained maximisation of information entropy, has been stressed upon. We have argued in favour of its potential by highlighting its many unique and valuable features. We have remarked on interesting new connections with the thermal time hypothesis, in particular suggesting to use this particular characterisation of equilibrium as a criterion of choice for the application of the hypothesis. Subsequently, aspects of a generalised framework for thermodynamics have been investigated, including defining the essential thermodynamic potentials, and discussing generalised zeroth and first laws.

We have then considered the statistical mechanics of a candidate quantum gravity system, composed of many atoms of space. The choice of (possibly different types of) these quanta is inspired directly from boundary structures in loop quantum gravity, spin foam and group field theory approaches. They are combinatorial building blocks (or boundary patches) of graphs, labelled with suitable algebraic data encoding discrete geometric information, with their constrained many-body dynamics dictated by bulk bondings between interaction vertices and amplitude functions. Generic statistical states can then be defined on a many-body state space, and generalised Gibbs states can be defined using the thermodynamical characterisation [14]. Finally, we have given an overview of applications in quantum gravity [14–16,28]. In particular, we have derived the covariant definition of group field theories as a coarse-graining using coherent states of a class of generalised Gibbs states of the underlying system with respect to dynamics-encoding kinetic and vertex operators; and in this way reinterpreted the GFT partition function as an effective statistical field theory partition function, extracted from an underlying statistical quantum gravity system.

More investigations along these directions will certainly be worthwhile. For example, the thermodynamical characterisation could be applied in a spacetime setting, such as for stationary black holes with respect to the mass, charge and angular momentum observables, to explore further its physical implications. The black hole setting could also help unfold how the selection of a single preferred temperature can occur starting from a generalised Gibbs measure. Moreover, it could offer insights into relations with the thermal time hypothesis, and help better understand some of our more intuitive reasonings presented in Section 2.3, and similarly for generalised thermodynamics. It requires further development, particularly for the first and second laws. For instance, in the first law as presented above, the additional possible work contributions need to be identified and understood, particularly in the context of background independence. For these, and other thermodynamical aspects, we could benefit from Souriau's generalisation of Lie group thermodynamics [26,27].

There are many avenues to explore also in the context of statistical mechanics and thermodynamics of quantum gravity. In the former, for example, it would be interesting to study potential black hole quantum gravity states [56]. In general, it is important to be able to identify suitable observables to characterise an equilibrium state of physically relevant cases. On the cosmological side, for instance, those phases of the complete quantum gravity system which admit a cosmological interpretation will be expected to have certain symmetries whose associated generators could then be suitable candidates for the generalised energies. Another interesting cosmological aspect to consider is that of inhomogeneities induced by early time volume thermal fluctuations of quantum gravity origin, possibly from an application of the volume Gibbs state [14] (or a suitable modification of it) recalled above. The latter aspect of investigating thermodynamics of quantum gravity would certainly benefit from confrontation with studies on thermodynamics of spacetime in semiclassical settings. We may also need to consider explicitly the quantum nature of the degrees of freedom, and use insights from the field of quantum thermodynamics [57], which itself has fascinating links to quantum information [58].

### **Funding:** This research received no external funding.

**Acknowledgments:** Many thanks are due to Daniele Oriti for valuable discussions and comments on the manuscript. Special thanks are due also to Goffredo Chirco and Mehdi Assanioussi for insightful discussions. The generous hospitality of the Visiting Graduate Fellowship program at Perimeter Institute, where part of this work was carried out, is gratefully acknowledged. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade.

**Conflicts of Interest:** The author declares no conflict of interest.
