Wormhole calculus, replicas, and entropies

TL;DR

The paper investigates how spacetime wormholes and baby universes influence entropy and correlation calculations that use the replica method. It combines the wormhole calculus with standard quantum rules to derive restricted patterns of replica connections, showing that only nearest-neighbor wormholes arise and that baby universes induce an ensemble of couplings with probability distribution |psi(α)|^2. Repeated experiments in the parent universe effectively collapse this ensemble, yielding a definite coupling in the large-experiment limit, while maintaining unitary evolution for later processes. The work challenges the common assumption of summing over all replica wormhole topologies and clarifies how topology-changing effects could be compatible with unitary black hole evolution, though it also highlights puzzles, such as the pinwheel geometries, that require further justification of underlying amplitudes.

Abstract

We investigate contributions of spacetime wormholes, describing baby universe emission and absorption, to calculations of entropies and correlation functions, for example those based on the replica method. We find that the rules of the "wormhole calculus," developed in the 1980s, together with standard quantum mechanical prescriptions for computing entropies and correlators, imply definite rules for $\textit{limited}$ patterns of connection between replica factors in simple calculations. These results stand in contrast with assumptions that all topologies connecting replicas should be summed over, and call into question the explanation for the latter. In a "free" approximation baby universes introduce probability distributions for coupling constants, and we review and extend arguments that successive experiments in a "parent" universe increasingly precisely fix such couplings, resulting in ultimately pure evolution. Once this has happened, the nontrivial question remains of how topology-changing effects can modify the standard description of black hole information loss.

Figures (7)

• Figure 1: To compute the amplitude to go from an initial state of the parent universe, plus some number of BUs, to a similar final state, we integrate over all intermediate geometries. In this figure we sketch one particular geometry contributing to the transition amplitude between four and three BUs.
• Figure 2: Shown is a sketch of the geometry used in a replica method calculation of the $n$th Renyi entropy of the density matrix \ref{['totdens']}. Time runs upwards (downwards) in the lower (upper) copies. This calculation produces only wormholes that connect different replicas in the pattern ${\bar{1}}-2$, ${\bar{2}}-3$, $\cdots$, ${\bar{n}} - 1$. We also indicate how the parent universes are identified at time $T$. The wormhole joining at $1$ is emitted from $\bar{n}$ while the one emitted from $\bar{3}$ joins $4$, etc. Wormholes connecting $1-1$, $\bar{1}-\bar{1}$, $2-2$, etc. are present, but not shown. The right panel shows a rearrangement of the diagrams making the purity of \ref{['totdens']} manifest.
• Figure 3: Shown is a sketch of the replica method calculation of the $n$th Renyi entropy of the reduced density matrix $\rho_p = {\rm Tr}_{\rm BU} \,\rho$. Time runs upwards (downwards) in the lower (upper) copies. Here there are no wormholes connecting different replicas, and the connections have the pattern ${\bar{1}}-1$, ${\bar{2}}-2$, $\cdots$, ${\bar{n}} - n$. We also indicate how the parent universes are identified. Wormholes connecting $1-1$, $\bar{1}-\bar{1}$, $2-2$, etc. are present, but not shown.
• Figure 4: In this diagram we show the in-in calculation of a two point function (represented by the black dots). The bottom (top) represent the time evolution upwards (downwards) creating the bra (ket) at time $T$ along the dashed line, in a diagram like those described in GiSl3. The state is glued at the dashed line including the operator insertions and the rules of QM would require us to include wormholes between them. We depict one of these wormholes.
• Figure 5: (a) Penrose diagram of a Schwarschild black hole. We indicate two Cauchy slices in the two-sided spacetime, $\Sigma_1$ and $\Sigma_2$. (b) Euclidean no-boundary preparation of the two-sided state on the surface $\Sigma_2$.