Typicality in spin-network states of quantum geometry

Abstract

In this letter we extend the so-called typicality approach, originally formulated in statistical mechanics contexts, to SU(2) invariant spin network states. Our results do not depend on the physical interpretation of the spin-network, however they are mainly motivated by the fact that spin-network states can describe states of quantum geometry, providing a gauge-invariant basis for the kinematical Hilbert space of several background independent approaches to quantum gravity. The first result is, by itself, the existence of a regime in which we show the emergence of a typical state. We interpret this as the prove that, in that regime there are certain (local) properties of quantum geometry which are "universal". Such set of properties is heralded by the typical state, of which we give the explicit form. This is our second result. In the end, we study some interesting properties of the typical state, proving that the area-law for the entropy of a surface must be satisfied at the local level, up to logarithmic corrections which we are able to bound

Hidden messenger from quantum geometry: Towards information conservation in quantum gravity

“…So in order to describe black holes, we should restrict ourselves to the regime where the entropy of b obeys the area law even though it can be expressed as ln Ω eff for some effective dimension Ω eff (cf. also the second paper of [11]). In the typicality regime, either the entropy (15) or (16) follows the area law: S J 0 ≫N ≫1 is extensive in the number of area patches and respectively the 2J S ∼ E = 2A in S N ≫J 0 ≫1 are quantized areas.…”

“…On the other hand, the large N regime is of interest for quantum gravity. In this case the system's von Neumann entropy is shown in [11] to be…”

“…With these in mind, let us turn to the typicality in spin network states [11]. Consider the total Hilbert space of N edges each spin of which is bounded by a constant J ≫ 1.…”

“…The entropy of the remaining system b is therefore maximal and proportional to ln D k−r (| J| d , J S − J d ), which means that the angular momentum basis of the intertwiner Hilbert space provides a complete description of the configurations of a "black hole". We can then take the weight in ρ d as the deviation from the canonical weight near that in ρ S and repeat the approximation of the dimension function D performed in [11] to extract the form e ∆S . Also notice that we are dealing with the spin network states describing arbitrary quantum geometries instead of those of black holes.…”

Hamiltonian flows of Lorentzian polyhedra: Kapovich-Millson phase space and SU(1, 1) intertwiners

“…This provides a natural measure on the space of Lorentzian polyhedra with space-like normals and opens the door to the analysis of concentration of measure phenomena on GL N (R) and typicality as in the Euclidean case [8,34,35].…”