Thermodynamics of quantum spacetime histories

We show that the simplicity constraints, which define the dynamics of spin foam models, imply, and are implied by, the first law of thermodynamics, when the latter is applied to causal diamonds in the quantum spacetime. This result reveals an intimate connection between the holographic nature of gravity, as reflected by the Bekenstein entropy, and the fact that general relativity and other gravitational theories can be understood as constrained topological field theories. To state and derive this correspondence we describe causal diamonds in the causal structure of spin foam histories and generalize arguments given for the near horizon region of black holes by Frodden, Gosh and Perez [Phys. Rev. D 87, 121503 (2013); Phys. Rev. D 89, 084069 (2014); Phys. Rev. Lett. 107, 241301 (2011); Phys. Rev. Lett.108, 169901(E) (2012).] and Bianchi [arXiv:1204.5122.]. This allows us to apply a recent argument of Jacobson [Phys. Rev. Lett. 116, 201101 (2016).] to show that if a spin foam history has a semiclassical limit described in terms of a smooth metric geometry, that geometry satisfies the Einstein equations. These results suggest also a proposal for a quantum equivalence principle.

Figure 1

A causal diamond defined by two events, $f$ and $e$. $W$ is the corner, or waist, bounding a space like three-disk, $S$.

Figure 2

A tetrahedron and its labeling.

Figure 3

The quantum near horizon region $\mathcal{R}$ constructed in a spin foam by a series of generalized boosts, here generated by $1\to 4$ moves, which are here illustrated in this two dimensional figure by $1\to 2$ moves.

Figure 4

A generalized boost realized in $1+1$ dimensional dual Pachner moves.

Figure 5

Two $3+1$ dimensional boost moves carried out in succession. Each fixes the black triangle which common to three tetrahedra, in blue, purple and red. Two $1\to 4$ moves are carried out in succession, which evolve from the blue to the purple to the red tetrahedra.

Figure 6

Two $3+1$ dimensional boost moves in the last figure. Each is a $1\to 4$ move represented by a four simplex that fixes the black triangle.