*3.1. Framework*

The candidate atoms of space considered here are geometric (quantum) *d*-polyhedra (with *d* faces), or equivalently open *d*-valent nodes with its half-links dressed by the appropriate algebraic data [40]. This choice is motivated strongly by loop quantum gravity [41], spin foam [42], group field theory [36–39] and lattice quantum gravity [43] approaches in the context of 4d models. Extended discrete space and spacetime can be built out of these fundamental atoms or "particles", via kinematical compositions (or boundary gluings) and dynamical interactions (or bulk bondings), respectively. In this sense, the perspective innate to a many-body quantum spacetime is a constructive one, which is naturally also extended to the statistical mechanics based on this mechanics.

Two types of data specify a mechanical model, combinatorial and algebraic. States and processes of a model are supported on combinatorial structures, here abstract<sup>11</sup> graphs and 2-complexes, respectively; and algebraic dressings of these structures adds discrete geometric information. Thus, different choices of combinatorics and algebraic data gives different mechanical models. For instance, the simplest spin foam models (and their associated group field theories) for 4d gravity are based on: boundary combinatorics based on a 4-valent node (or a tetrahedron), bulk combinatorics based on a 4-simplex interaction vertex, and algebraic (or group representation) data of *SU*(2) labelling the boundary 4-valent graphs and bulk 2-complexes.

Clearly, this is not the only choice, in fact far from it. The vast richness of possible combinatorics, compatible with our constructive point of view, is comprehensively illustrated in [35]12. The various choices for variables to label the discrete structures with (so that they may encode a suitable notion of discrete geometry, which notion depending exactly on the variables chosen and constraints imposed on them) have been an important subject of study, starting all the way from Regge [45–50]. Accommodation of these various different choices is ye<sup>t</sup> another appeal of the constructive many-body viewpoint and this framework. After clarifying further some of these aspects in the following, we choose to work with simplicial combinatorics and *SU*(2) holonomy-flux data for the subsequent examples.

### 3.1.1. Atoms of Quantum Space and Kinematics

In the following, we make use of some of the combinatorial structures defined in [35]. However we are content with introducing them in a more intuitive manner, and not recalling the rigorous definitions as that would not be particularly valuable for the present discussion. The interested reader can refer to [35] for details13.

<sup>11</sup> Thus, not necessarily embedded into any continuum spatial manifold.

<sup>12</sup> In fact, the work in [35] is phrased in a language closer to the group field theory approach, but the structures are general enough to apply elsewhere, such as in spin foams, as evidenced in [44].

<sup>13</sup> For clarity, we note that the terminology used here is slightly different from that in [35]. Specifically, the dictionary between here ↔ there is: combinatorial atom or particle ↔ boundary patch; interaction/bulk vertex ↔ spin foam atom; boundary node ↔ boundary multivalent vertex *v*¯; link or full link ↔ boundary edge connecting two multivalent vertices *v*¯1, *v*¯2; half-link ↔ boundary edge connecting a multivalent vertex *v*¯ and a bivalent vertex *v*ˆ. This minor difference is mainly due to a minor difference in the purpose for the same combinatorial structures. Here, we are in a setup where the accessible states are boundary states, for which a statistical mechanics is defined; and the case of interacting dynamics is considered as defining a suitable (amplitude) functional over the the boundary state space. On the other hand, the perspective in [35] is more in a spin foam constructive setting, so that modelling the 2-complexes as built out of fundamental spin foam atoms is more natural there.

The primary objects of interest to us are boundary patches, which we take as the combinatorial atoms of space. To put simply, a boundary patch is the most basic unit of a boundary graph, in the sense that the set of all boundary patches generates the set of all connected bisected boundary graphs. A bisected boundary graph is simply a directed boundary graph with each of its full links bisected into a pair of half-links, glued at the bivalent nodes (see Figure 1). Different kinds of atoms of space are then the different, inequivalent boundary patches (dressed further with suitable data), and the choice of combinatorics basically boils down to a choice of the set of admissible boundary patches. Moreover, a model with multiple inequivalent boundary patches can be treated akin to a statistical system with multiple species of atoms.

The most general types of boundary graphs are those with nodes of arbitrary valence, and including loops. A common and natural restriction is to consider loopless structures, as they can be associated with combinatorial polyhedral complexes [35]. As the name suggests, loopless boundary patches are those with no loops, i.e., each half-link is bounded on one end by a unique bivalent node (and on the other by the common, multivalent central node). A loopless patch is thus uniquely specified by the number of incident half-links (or equivalently, by the number of bivalent nodes bounding the central node). A *d*-patch, with *d* number of incident half-links, is simply a *d*-valent node. Importantly for us, it is the combinatorial atom that supports (quantum) geometric states of a *d*-polyhedron [40,51,52]. A further common restriction is to consider graphs with nodes of a single, fixed valence, that is to consider *d*-regular loopless structures.

Let us take an example. Consider the boundary graph of a 4-simplex as shown in Figure 1. The fundamental atom or boundary patch is a 4-valent node. This graph can be constructed starting from five open 4-valent nodes (denoted *m*, *n*, ..., *q*), and gluing the half-links, or equivalently the faces of the dual tetrahedra, pair-wise, with the non-local combinatorics of a complete graph on five 4-valent nodes. The result is ten bisected full links, bounded by five nodes. It is important to note here that a key ingredient of constructing extended boundary states from the atoms are precisely the half-link gluing, or face-sharing conditions on the algebraic data decorating the patches. For instance, in the case of standard loop quantum gravity holonomy-flux variables of *<sup>T</sup>*<sup>∗</sup>(*SU*(2)), the face-sharing gluing constraints are area matching [48], thus lending a notion of discrete classical twisted geometry to the graph. This is much weaker than a Regge geometry, which could have been obtained for the same variables if instead the so-called shape-matching conditions [47] are imposed on the pair-wise gluing of faces/half-links. Thus, kinematic composition (boundary gluings) that creates boundary states depends on two crucial ingredients, the combinatorial structure of the resultant boundary graph, and face-sharing gluing conditions on the algebraic data.

From here on, we restrict ourselves to a single boundary patch for simplicity, a (gauge-invariant) 4-valent node dressed with *SU*(2) data, i.e., a quantised tetrahedron [40,51]. However, it should be clear from the brief discussion above (and the extensive study in [35]) that a direct generalisation of the present (statistical) mechanical framework is possible also for these more enhanced combinatorial structures.

The phase space of a single classical tetrahedron, encoding both intrinsic and extrinsic degrees of freedom (along with an arbitrary orientation in R3) is

$$
\Gamma = T^\* \left( S\mathcal{U}(2)^4 / S\mathcal{U}(2) \right) \tag{21}
$$

where the quotient by *SU*(2) imposes geometric closure of the tetrahedron. The choice of domain space is basically the choice of algebraic data. For instance, in Euclidean 4d settings a more apt choice would be the group *Spin*(4), and *SL*(2, C) for Lorentzian settings. Then, states of a system of *N* tetrahedra belong to Γ*N* = <sup>Γ</sup><sup>×</sup>*N*, and observables would be smooth (real-valued) functions defined on Γ*N* [14,15].

The quantum counterparts are,

$$\mathcal{H} = L^2 (S\mathcal{U}(2)^4 / S\mathcal{U}(2))\tag{22}$$

for the single-particle Hilbert space, and H*N* = H<sup>⊗</sup>*<sup>N</sup>* for an *<sup>N</sup>*-particle system. In the quantum setting, we can go a step further and construct a Fock space based on the above single-particle Hilbert space,

$$\mathcal{H}\_F = \bigoplus\_{N \ge 0} \text{sym}\,\mathcal{H}\_N \tag{23}$$

where the symmetrisation of *<sup>N</sup>*-particle spaces implements a choice of bosonic statistics for the quanta, mirroring the graph automorphism of node exchanges. One choice for the algebra of operators on H*F* is the von Neumann algebra of bounded linear operators. A more common choice though is the larger \*-algebra generated by ladder operators *ϕ*ˆ, *ϕ*ˆ†, which generate the full H*F* by acting on a cyclic Fock vacuum, and satisfy a commutation relations algebra

$$\left[\phi(\vec{g}\_1), \phi^\dagger(\vec{g}\_2)\right] = \int\_{SL(2)} dh \prod\_{I=1}^4 \delta(g\_{1I} h g\_{2I}^{-1})\tag{24}$$

where *g* ≡ (*gI*) ∈ *SU*(2)<sup>4</sup> and the integral on the right ensures *SU*(2) gauge invariance. In fact, this is the Fock representation of an algebraic bosonic group field theory defined by a Weyl algebra [14,29,53].

**Figure 1.** Bisected boundary graph of a 4-simplex, as a result of non-local pair-wise gluing of half-links. Each full link is bounded by two 4-valent nodes (denoted here by *m*, *n*, ...), and bisected by one bivalent node (shown here in green).

### 3.1.2. Interacting Quantum Spacetime and Dynamics

Coming now to dynamics, the key ingredients here are the specifications of propagators and admissible interaction vertices, including both their combinatorics, and functional dependences on the algebraic data, i.e., their amplitudes.

The combinatorics of propagators and interaction vertices can be packaged neatly within two maps defined in [35], the bonding map and the bulk map, respectively. A bonding map is defined between two bondable boundary patches. Two patches are bondable if they have the same number of nodes and links. Then, a bonding map between two bondable patches identifies each of their nodes and links, under the compatibility condition that if a bounding bivalent node in one patch is identified with a particular one in another, then their respective half-links (attaching them to their respective central nodes) are also identified with each other. Thus, a bonding map basically bonds two bulk vertices via (parts of) their boundary graphs to form a process (with boundary). This is simply a bulk edge, or propagator.

The set of interaction vertices can themselves be defined by a bulk map. This map augments the set of constituent elements (multivalent nodes, bivalent nodes, and half-links connecting the two) of *any* bisected boundary graph, by one new vertex (the bulk vertex), a set of links joining each of the original boundary nodes to this vertex, and a set of two-dimensional faces bounded by a triple of the bulk vertex, a multivalent boundary node and a bivalent boundary node. The resulting structure is an interaction vertex with the given boundary graph14. The complete dynamics is then given by the chosen combinatorics, supplemented with amplitude functions that reflect the dependence on the algebraic data.

The interaction vertices can in fact be described by vertex operators on the Fock space in terms of the ladder operators. An example vertex operator, corresponding to the 4-simplex boundary graph shown in Figure 1, is

$$\vec{\nabla}\_{\mathsf{4sim}} = \int\_{S1L(2)^{20}} [d\underline{\chi}] \, \phi^{\dagger}(\vec{\chi}\_1) \phi^{\dagger}(\vec{\chi}\_2) \, V\_{\mathsf{4sim}}(\vec{\chi}\_1, \dots, \vec{\chi}\_5) \phi(\vec{\chi}\_3) \phi(\vec{\chi}\_4) \phi(\vec{\chi}\_5) \tag{25}$$

where the interaction kernel *V*4sim = *<sup>V</sup>*4sim({*gijg*<sup>−</sup><sup>1</sup> *ji* }*<sup>i</sup>*<*j*) (for *i*, *j* = 1, ..., 5) encodes the combinatorics of the boundary graph. There are of course other vertex operators associated with the *same* graph (that is with the same kernel), but including different combinations of creation and annihilation operators15.

Thus, a definition of kinematics entails: defining the state space, which includes specifying the combinatorics (choosing the set of allowed boundary patches, which generate the admissible boundary graphs), and the algebraic data (choosing variables to characterise the discrete geometric states supported on the boundary graphs); and defining the algebra of observables acting on the state space. A definition of dynamics entails: specifying the propagator and bulk vertex combinatorics and amplitudes. Together, they specify the many-body mechanics.

### 3.1.3. Generalised Equilibrium States

Outlined below is a generalised equilibrium statistical mechanics for these systems [14,15], along the lines laid out in Section 2. For a system of many classical tetrahedra (in general, polyhedra), a statistical state *ρN* can be formally defined on the state space Γ*N*. If it satisfies the thermodynamical characterisation with respect to a set of functions on Γ*N*, then it will be an equilibrium state. Further, a configuration with a varying number of tetrahedra can be described by a grand-canonical type state [15] of the form

$$Z = \sum\_{N \ge 0} \mathfrak{e}^{\mu N} Z\_N \tag{26}$$

where *Z N* = Γ*N dλ ρ N*, and *μ* is a chemical potential. Similarly, for a system of many quantum tetrahedra, a generic statistical state *ρ*ˆ is a density operator on H*F*; and generalised equilibrium states with a varying number of quanta are

$$Z = \text{Tr}\_{\mathcal{H}\_F} \left( \varepsilon^{-\sum\_a \beta\_a \mathcal{O}\_a + \mu \hat{N}} \right) \tag{27}$$

<sup>14</sup> An interesting aspect is that the bulk map is one-to-one, so that for every distinct bisected boundary graph, there is a unique interaction vertex which can be defined from it.

<sup>15</sup> This would generically be true for any second quantised operator [29].

where *N*ˆ = *dg ϕ*ˆ†(*g*)*ϕ*<sup>ˆ</sup>(*g*) is the number operator on H*<sup>F</sup>*. Operators of natural interest here are the ones encoding the dynamics, i.e. vertex (and kinetic) operators (see Section 3.2 below). Such grand-canonical type boundary states are important because one would expect quantum gravity dynamics to not be number conserving in general [15,29]. In addition, naturally, in both cases, what the precise content of equilibrium is depends crucially on which observables O*a* are used to define the state. As pointed out in Section 2.2, and exemplified in the cases below in Section 3.2, there are many choices and types of observables one could consider in principle. Which ones are the relevant ones in a given situation is in fact a crucial part of the problem.
