# Thermal Quantum Spacetime

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## Abstract

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## 1. Introduction

## 2. Background Independent Equilibrium Statistical Mechanics

#### 2.1. Generalised Equilibrium

#### 2.1.1. Past Proposals

#### 2.1.2. Thermodynamical Characterisation

#### 2.2. Remarks

- There are two key features of this characterisation. The first is the use of evidential (or epistemic, or Bayesian) probabilities, thus taking into account the given evidence $\left\{{U}_{a}\right\}$; and second is a preference for the least-biased (or most “honest”) distribution out of all the different ones compatible with the given evidence. It is not enough to arbitrarily choose any that is compatible with the prior data. An aware observer must also take into account their own ignorance, or lack of knowledge honestly, by maximising the information entropy.
- This notion of equilibrium is inherently observer-dependent because of its use of the macrostate thermodynamic description of the system, which in itself is observer-dependent due to having to choose a coarse-graining, that is the set of macroscopic observables.
- Given a generalised Gibbs state, the question arises as to which flow it is stationary with respect to. Any density distribution or operator satisfies the KMS condition (which implies stationarity) with respect to its own modular flow. In fact, by the Tomita–Takesaki theorem [21], any faithful algebraic state over a von Neumann algebra is KMS with respect to its own one-parameter modular (Tomita) flow.7 Given this, then ${\rho}_{\left\{{\beta}_{a}\right\}}$ is clearly KMS with respect to the flow ${X}_{\rho}\sim \partial /\partial t$ (or ${\stackrel{\u02c6}{U}}_{\rho}\left(t\right)\sim {e}^{i\stackrel{\u02c6}{h}t}$) generated by its modular Hamiltonian $h={\sum}_{a}{\beta}_{a}{\mathcal{O}}_{a}$. In particular, ${\rho}_{\left\{{\beta}_{a}\right\}}$ is not stationary with respect to the individual flows ${X}_{a}$ generated by ${\mathcal{O}}_{a}$, unless they satisfy $[{X}_{a},{X}_{{a}^{\prime}}]=0$ for all $a,{a}^{\prime}$ [15]. In fact, this last property shows that the proposal in [1,12] based on statistical independence (that is, $[{X}_{{\rho}_{1}},{X}_{{\rho}_{2}}]=0$) can be understood as a special case of this one, when the state is defined for a pair of observables that are defined on mutually exclusive subspaces of the state space. In this case, their respective flows will automatically commute and the state will be said to satisfy statistical independence.
- To be clear, the use of the “most probable” characterisation for equilibrium is not new in itself. It was used by Boltzmann in the late 19th century, and utilised (also within a Boltzmann interpretation of statistical mechanics) in a constrained system in [11]. The fact that equilibrium configurations maximise the system’s entropy is also not new: it was well known already in the time of Gibbs8. The novelty here is: in the revival of Jaynes’ perspective, of deriving 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.
- The procedure is versatile, being applicable to a wide array of cases (both classical and quantum), relying only on a sufficiently well-defined mathematical description in terms of a state space, along with a set of observables with dynamically constant averages ${U}_{a}$ defining a suitable macrostate of the system9.
- Evidently, this manner of defining equilibrium statistical mechanics (and from it, thermodynamics) does not lend any fundamental status to energy, nor does it rely on selecting a single, special (energy) observable out of the full set $\left\{{\mathcal{O}}_{a}\right\}$. It can thus be crucial in settings where concepts of time and energy are dubious at the least, or not defined at all as in non-perturbative quantum gravity.
- It has a technical advantage of not needing any (one-parameter) symmetry (sub-)groups of the system to be defined a priori, unlike the dynamical characterisation based on the standard KMS condition.
- It is independent of any additional physical assumptions, hypotheses or principles that are common to standard statistical physics, and, in the present context, to the other proposals of generalised equilibrium recalled in Section 2.1. Some examples of these extra ingredients (not required in the thermodynamical characterisation) that we have already encountered are ergodicity, weak interactions, statistical independence, and often a combination of them.
- It is independent of any physical interpretations attached (or not!) to the quantities and setup involved. This further amplifies its appeal for use in quantum gravity where the geometrical (and physical) meanings of the quantities involved may not necessarily be clear from the start.
- One of the main features (which helps accommodate the other proposals as special cases of this one) is the generality in the choice of observables ${\mathcal{O}}_{a}$ allowed naturally by this characterisation. In principle, they need only be mathematically well-defined in the given description of the system (regardless of whether it is kinematic i.e., working at the extended state space level, or dynamic, i.e., working with the physical state space), satisfying convexity properties so that the resultant Gibbs state is normalisable. More standard choices include a Hamiltonian in a non-relativistic system, a clock Hamiltonian in a deparameterised system [3,14], and generators of kinematic symmetries such as rotations, or more generally of one-parameter subgroups of Lie group actions [26,27]. Some of the more unconventional choices include geometric observables such as volume [14,28], (component functions of the) momentum map associated with geometric closure of classical polyhedra [15,16], half-link gluing (or face-sharing) constraints of discrete gravity [15], a projector in group field theory [15,29], and generic gauge-invariant observables (not necessarily symmetry generators) [11]. We refer to [14] for a more detailed discussion.

#### 2.3. Relation to Thermal Time Hypothesis

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

## 3. Equilibrium Statistical Mechanics in Quantum Gravity

#### 3.1. Framework

#### 3.1.1. Atoms of Quantum Space and Kinematics

#### 3.1.2. Interacting Quantum Spacetime and Dynamics

#### 3.1.3. Generalised Equilibrium States

#### 3.2. Applications

## 4. Conclusions and Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Rovelli, C. Statistical mechanics of gravity and the thermodynamical origin of time. Class. Quantum Gravity
**1993**, 10, 1549–1566. [Google Scholar] [CrossRef] - Connes, A.; Rovelli, C. Von Neumann algebra automorphisms and time thermodynamics relation in general covariant quantum theories. Class. Quantum Gravity
**1994**, 11, 2899–2918. [Google Scholar] [CrossRef] - Rovelli, C. General relativistic statistical mechanics. Phys. Rev.
**2013**, 87, 084055. [Google Scholar] [CrossRef] [Green Version] - Bardeen, J.M.; Carter, B.; Hawking, S.W. The Four laws of black hole mechanics. Commun. Math. Phys.
**1973**, 31, 161–170. [Google Scholar] [CrossRef] - Bekenstein, J.D. Black holes and the second law. Lett. Nuovo Cimento
**1972**, 4, 737–740. [Google Scholar] [CrossRef] - Bekenstein, J.D. Black holes and entropy. Phys. Rev.
**1973**, 7, 2333–2346. [Google Scholar] [CrossRef] - Hawking, S.W. Particle Creation by Black Holes. Commun. Math. Phys.
**1975**, 43, 199–220. [Google Scholar] [CrossRef] - Bombelli, L.; Koul, R.K.; Lee, J.; Sorkin, R.D. A Quantum Source of Entropy for Black Holes. Phys. Rev. D
**1986**, 34, 373–383. [Google Scholar] [CrossRef] - Oriti, D. The universe as a quantum gravity condensate. C. R. Phys.
**2017**, 18, 235–245. [Google Scholar] [CrossRef] - Rovelli, C. The Statistical state of the universe. Class. Quantum Gravity
**1993**, 10, 1567. [Google Scholar] [CrossRef] - Montesinos, M.; Rovelli, C. Statistical mechanics of generally covariant quantum theories: A Boltzmann-like approach. Class. Quantum Gravity
**2001**, 18, 555–569. [Google Scholar] [CrossRef] - Chirco, G.; Haggard, H.M.; Rovelli, C. Coupling and thermal equilibrium in general-covariant systems. Phys. Rev.
**2013**, 88, 084027. [Google Scholar] [CrossRef] [Green Version] - Rovelli, C.; Smerlak, M. Thermal time and the Tolman-Ehrenfest effect: Temperature as the “speed of time”. Class. Quantum Gravity
**2011**, 28, 075007. [Google Scholar] [CrossRef] - Kotecha, I.; Oriti, D. Statistical Equilibrium in Quantum Gravity: Gibbs states in Group Field Theory. New J. Phys.
**2018**, 20, 073009. [Google Scholar] [CrossRef] - Chirco, G.; Kotecha, I.; Oriti, D. Statistical equilibrium of tetrahedra from maximum entropy principle. Phys. Rev.
**2019**, 99, 086011. [Google Scholar] [CrossRef] [Green Version] - Chirco, G.; Kotecha, I. Generalized Gibbs Ensembles in Discrete Quantum Gravity. In Geometric Science of Information 2019; Nielsen, F., Barbaresco, F., Eds.; Springer: Cham, Switzerland, 2019. [Google Scholar]
- Chirco, G.; Josset, T.; Rovelli, C. Statistical mechanics of reparametrization-invariant systems. It takes three to tango. Class. Quantum Gravity
**2016**, 33, 045005. [Google Scholar] [CrossRef] - Haggard, H.M.; Rovelli, C. Death and resurrection of the zeroth principle of thermodynamics. Phys. Rev.
**2013**, 87, 084001. [Google Scholar] [Green Version] - Jaynes, E.T. Information Theory and Statistical Mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Jaynes, E.T. Information Theory and Statistical Mechanics. II. Phys. Rev.
**1957**, 108, 171–190. [Google Scholar] [CrossRef] - Bratteli, O.; Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics—I, II; Springer: Berlin/Heidelberg, Germany, 1987. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Statistical Physics, Part 1; Volume 5 of Course of Theoretical Physics; Butterworth-Heinemann: Oxford, UK, 1980. [Google Scholar]
- Chirco, G.; Josset, T. Statistical mechanics of covariant systems with multi-fingered time. arXiv
**2016**, arXiv:1606.04444. [Google Scholar] - Martinetti, P.; Rovelli, C. Diamonds’s temperature: Unruh effect for bounded trajectories and thermal time hypothesis. Class. Quantum Gravity
**2003**, 20, 4919–4932. [Google Scholar] [CrossRef] - Haggard, H.M. Gibbsing spacetime: A group field theory approach to equilibrium in quantum gravity. New J. Phys.
**2018**, 20, 071001. [Google Scholar] [CrossRef] - Souriau, J.-M. Structure des Systemes Dynamiques; Dunod: Paris, France, 1969. [Google Scholar]
- Marle, C.-M. From tools in symplectic and poisson geometry to J.-M. Souriau’s theories of statistical mechanics and thermodynamics. Entropy
**2016**, 18, 370. [Google Scholar] [CrossRef] - Assanioussi, M.; Kotecha, I. Thermal quantum gravity condensates and group field theory cosmology. In progress.
- Oriti, D. Group field theory as the 2nd quantization of Loop Quantum Gravity. Class. Quantum Gravity
**2016**, 33, 085005. [Google Scholar] [CrossRef] - Rovelli, C. “Forget time”. Found. Phys.
**2011**, 41, 1475–1490. [Google Scholar] [CrossRef] - Rovelli, C. Is Time’s Arrow Perspectival? In Simon Saunders; Silk, J., Barrow, J.D., Chamcham, K., Eds.; The Philosophy of Cosmology: Cambridge, UK, 2017; pp. 285–296. [Google Scholar]
- Jaynes, E.T. The Gibbs Paradox. In Maximum Entropy and Bayesian Methods; Smith, C.R., Erickson, G.J., Neudorfer, P.O., Eds.; Springer: Dordrecht, The Netherlands, 1992; pp. 1–21. [Google Scholar]
- Araki, H. Relative Entropy of States of Von Neumann Algebras. Publ. Res. Inst. Math. Sci. Kyoto
**1976**, 1976, 809–833. [Google Scholar] - Oriti, D. Spacetime as a quantum many-body system. arXiv
**2017**, arXiv:1710.02807. [Google Scholar] - Oriti, D.; Ryan, J.P.; Thurigen, J. Group field theories for all loop quantum gravity. New J. Phys.
**2015**, 17, 023042. [Google Scholar] [CrossRef] [Green Version] - Michael, P. Reisenberger and Carlo Rovelli. Space-time as a Feynman diagram: The Connection formulation. Class. Quantum Gravity
**2001**, 18, 121–140. [Google Scholar] - Freidel, L. Group field theory: An Overview. Int. J. Theory Phys.
**2005**, 44, 1769–1783. [Google Scholar] [CrossRef] - Oriti, D. The group field theory approach to quantum gravity. In Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter; Oriti, D., Ed.; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Oriti, D. The microscopic dynamics of quantum space as a group field theory. arXiv
**2011**, arXiv:1110.5606. [Google Scholar] - Bianchi, E.; Dona, P.; Speziale, S. Polyhedra in loop quantum gravity. Phys. Rev. D
**2011**, 83, 044035. [Google Scholar] [CrossRef] - Bodendorfer, N. An elementary introduction to loop quantum gravity. arXiv
**2016**, arXiv:1607.05129. [Google Scholar] - Perez, A. The Spin Foam Approach to Quantum Gravity. Living Rev. Relativ.
**2013**, 16, 3. [Google Scholar] [CrossRef] - Hamber, H.W. Quantum Gravity on the Lattice. Gen. Relativ. Gravit.
**2009**, 41, 817–876. [Google Scholar] [CrossRef] - Finocchiaro, M.; Oriti, D. Spin foam models and the Duflo map. arXiv
**2018**, arXiv:1812.03550. [Google Scholar] - Regge, T. General relativity without coordinates. Nuovo Cimento
**1961**, 19, 558–571. [Google Scholar] [CrossRef] - Regge, T.; Williams, R.M. Discrete structures in gravity. J. Math. Phys.
**2000**, 41, 3964–3984. [Google Scholar] [CrossRef] [Green Version] - Dittrich, B.; Speziale, S. Area-angle variables for general relativity. New J. Phys.
**2008**. [Google Scholar] [CrossRef] - Freidel, L.; Speziale, S. Twisted geometries: A geometric parametrisation of SU(2) phase space. Phys. Rev. D
**2010**, 82, 084040. [Google Scholar] [CrossRef] - Rovelli, C.; Speziale, S. On the geometry of loop quantum gravity on a graph. Phys. Rev.
**2010**, 82, 044018. [Google Scholar] [CrossRef] - Dittrich, B.; Ryan, J.P. Phase space descriptions for simplicial 4d geometries. Class. Quantum Gravity
**2011**, 28, 065006. [Google Scholar] [CrossRef] - Barbieri, A. Quantum tetrahedra and simplicial spin networks. Nucl. Phys.
**1998**, 518, 714–728. [Google Scholar] [CrossRef] [Green Version] - Baez, J.C.L.; Barrett, J.W. The Quantum tetrahedron in three-dimensions and four-dimensions. Adv. Theor. Math. Phys.
**1999**, 3, 815–850. [Google Scholar] [CrossRef] - Kegeles, A.; Oriti, D.; Tomlin, C. Inequivalent coherent state representations in group field theory. Class. Quantum Gravity
**2018**, 35, 125011. [Google Scholar] [CrossRef] - Pithis, A.G.; Sakellariadou, M. Group field theory condensate cosmology: An appetizer. Universe
**2019**, 5, 147. [Google Scholar] [CrossRef] - Klauder, J.; Skagerstam, B. Coherent States; World Scientific: Singapore, 1985. [Google Scholar]
- Oriti, D.; Pranzetti, D.; Sindoni, L. Black Holes as Quantum Gravity Condensates. Phys. Rev.
**2018**, 97, 066017. [Google Scholar] [CrossRef] - Vinjanampathy, S.; Anders, J. Quantum thermodynamics. Contemp. Phys.
**2016**, 57, 545–579. [Google Scholar] [CrossRef] [Green Version] - Goold, J.; Huber, M.; Riera, A.; del Rio, L.; Skrzypczyk, P. The role of quantum information in thermodynamics—A topical review. J. Phys. A Math. Theor.
**2016**, 49, 143001. [Google Scholar] [CrossRef]

1. | 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. |

2. | 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. |

3. | 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. |

4. | 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. |

5. | 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. |

6. | Local, in the sense of being observer-dependent (see Section 2.2). |

7. | |

8. | 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. |

9. | 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. |

10. | By this we mean that the term $\langle d{\mathcal{O}}_{a}\rangle $, based on the same observables defining the generalised energies ${U}_{a}$, 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 $d{W}_{a}$. 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. |

11. | Thus, not necessarily embedded into any continuum spatial manifold. |

12. | |

13. | 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 $\overline{v}$; link or full link ↔ boundary edge connecting two multivalent vertices ${\overline{v}}_{1},{\overline{v}}_{2}$; half-link ↔ boundary edge connecting a multivalent vertex $\overline{v}$ and a bivalent vertex $\stackrel{\u02c6}{v}$. 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. |

14. | 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. |

15. | This would generically be true for any second quantised operator [29]. |

16. | A proper interpretation of these constraints is left for future work. |

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

**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,\dots $), and bisected by one bivalent node (shown here in green).

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Kotecha, I.
Thermal Quantum Spacetime. *Universe* **2019**, *5*, 187.
https://doi.org/10.3390/universe5080187

**AMA Style**

Kotecha I.
Thermal Quantum Spacetime. *Universe*. 2019; 5(8):187.
https://doi.org/10.3390/universe5080187

**Chicago/Turabian Style**

Kotecha, Isha.
2019. "Thermal Quantum Spacetime" *Universe* 5, no. 8: 187.
https://doi.org/10.3390/universe5080187