Statistical equilibrium of tetrahedra from maximum entropy principle

Abstract

Discrete formulations of (quantum) gravity in four spacetime dimensions build space out of tetrahedra. We investigate a statistical mechanical system of tetrahedra from a many-body point of view based on non-local, combinatorial gluing constraints that are modelled as multi-particle interactions. We focus on Gibbs equilibrium states, constructed using Jaynes' principle of constrained maximisation of entropy, which has been shown recently to play an important role in characterising equilibrium in background independent systems. We apply this principle first to classical systems of many tetrahedra using different examples of geometrically motivated constraints. Then for a system of quantum tetrahedra, we show that the quantum statistical partition function of a Gibbs state with respect to some constraint operator can be reinterpreted as a partition function for a quantum field theory of tetrahedra, taking the form of a group field theory. CONTENTS

Thermal representations in group field theory: squeezed vacua and quantum gravity condensates

“…Similarly for the matter part, let us consider a basis of complex-valued smooth functions T α (φ) in L 2 (φ), labelled by a discrete index α, satisfying orthonormality 5 For more functional rigour, the test functions may be defined on the dense subspace of smooth functions C ∞ (.) ⊂ L 2 (.).…”

“…where β ℓ are generalized inverse temperatures conjugate to a given set of observables O ℓ [5,6,20]. This state is a result of maximising the information entropy, − ln ρ ρ , under the set of constraints O ℓ ρ = constant and 1 ρ = 1.…”

“…We can describe the same structures from a many-body perspective [19], and treat as more fundamental an in-teracting system of many such quanta. This viewpoint enables us to import formal techniques from standard many-body physics for macroscopic systems, by treating a quantum polyhedron or an open spin network node as a single atom or particle of interest [5,6,20,21]. This further leads to a modelling of an extended region of discrete quantum space (a spin network) as a multi-particle state with a large number of such quanta, and a region of dynamical quantum spacetime (a spin foam) as an interaction process.…”

Thermal Quantum Spacetime

“…This brings us to the proposal of characterising a generalised Gibbs state based on a constrained maximisation of information (Shannon or von Neumann) entropy [14][15][16], along the lines advocated by Jaynes [19,20] purely from the perspective of evidential statistical inference. Jaynes' approach is fundamentally different from other more traditional ones of statistical physics.…”

“…7 Given this, then ρ {βa} is clearly KMS with respect to the flow X ρ ∼ ∂/∂t (orÛ ρ (t) ∼ e iĥt ) generated by its modular Hamiltonian h = a β a O a . In particular, ρ {βa} is not stationary with respect to the individual flows X a generated by O a , unless they satisfy [X a , X a ] = 0 for all a, a [15].…”