Counting surface states in the loop quantum gravity

TL;DR

The paper investigates geometrical entropy in loop quantum gravity by treating the classical surface geometry as a macro-state and counting micro-states formed by surface spin networks. Using a simple ensemble of open surface puncture states, it constructs a partition function over puncture configurations and shows that the resulting entropy scales linearly with the surface area, $S(A) \approx \alpha' A$, with $\alpha' \approx 1.01$ in units where area is measured by $16\pi l_p^2$. This linear scaling holds for both open and closed surfaces when a strong approximation criterion is imposed, which excludes degenerate states; a complementary unordered-puncture analysis yields $S(A) \propto \sqrt{A}$. The results illustrate how geometrical entropy emerges from macro-micro correspondences in quantum geometry and hint at connections to horizon entropy in black hole thermodynamics.

Abstract

We adopt the point of view that (Riemannian) classical and (loop-based) quantum descriptions of geometry are macro- and micro-descriptions in the usual statistical mechanical sense. This gives rise to the notion of geometrical entropy, which is defined as the logarithm of the number of different quantum states which correspond to one and the same classical geometry configuration (macro-state). We apply this idea to gravitational degrees of freedom induced on an arbitrarily chosen in space 2-dimensional surface. Considering an `ensemble' of particularly simple quantum states, we show that the geometrical entropy $S(A)$ corresponding to a macro-state specified by a total area $A$ of the surface is proportional to the area $S(A)=αA$, with $α$ being approximately equal to $1/16πl_p^2$. The result holds both for case of open and closed surfaces. We discuss briefly physical motivations for our choice of the ensemble of quantum states.

Figures (6)

• Figure 1: Vertex of a 3-d spin network (a) and its intersection with the surface S (b). Edges $1,2,3$ are tangential ones.
• Figure 2: Intertwining operator assigned to the vertex $v$ of a generalized surface spin network 'remembers' what were the spins $j_4,j_5,j_6$ of the edges $4,5,6$ incident at the surface.
• Figure 3: The valence of the simplest vertex is two.
• Figure 4: The simplest surface spin network states are specified by a set of punctures on the boundary.
• Figure 5: Two sets of punctures which are considered as specifying different quantum states.