Week 9

作者: 悟空金月饺子 | 来源:发表于2021-05-13 13:50 被阅读0次

整理了一下之前的想法，不过还是有很多想不通。

When we talk about wormholes what do we realy mean? Mainly there are two types of wormholes: the replica wormhole and the Euclidean wormhole.
The replica wormhole is the contribution to the quantities like $\text{tr}(\rho_R^n)$ , so we may think of that it is assocaited with some density matrix?(Q: what kind of density matrix? ) They play the role of solving information paradox. (quantum correlation vs. classical correlation?)

However, the Euclidean wormhole is the saddle point approximation of the partition function of the gravity theory (w/ or w/out source? Q: what is the interpretaion from the point of view of field theory side?). They play the role of explained the ramp of the spectral form factor. (a dephasing or decoherence channel: correlation vs. non-correlation? Pure states become mixed. Aonther possible way to understand it is perhaps we do not have a direct gravitional way to describe the pure state, but because of the dephaing or decoherence of a chaotic system, the pure state is well approximated by a mixed state which we have a gravitional description.) They are responsible for the non-factorization.

What is the relation between them? A possible way to understand the relation is that we also assume that the Euclidean wormhole is related to some density matrix, then we put the two kinds of wormholes on the same footing.

(Q: Is dephasing same as ensemble average?)

\section{Model I: JT with EOW}
[1911.11977]

The Euclidean action of the system is
$I=I_{JT}+\mu \int_{\text{brane}}ds,$
where the integral is along the worldline of the EOW brane. We assume that the EOW brane has a very large number k of possible internal states which describe the interior partners of the early Hawking radiation so we can also think of them as the black states. Including the Hawking radiation outside of the horizon, the state of the whole system is

$|\psi\rangle=\frac{1}{\sqrt{k}}\sum_{i=1}^k |\psi_i\rangle |i\rangle_R$

To compute the entropy of system $R$ , we consider the reduced density matrix

$\sum_k \langle\psi_k| (\frac{1}{k}\sum_{i,j}|i\rangle\langle j|\otimes |\psi_i\rangle \langle \psi_j|)|\psi_k\rangle=\rho_R$
$\rho_R=\frac{1}{k}\sum_{i,j=1}^{k}|j\rangle \langle i|_R \langle \psi_i|\psi_j\rangle_B=\frac{1}{k}\sum_{i}|i\rangle \langle i|+\dots$
On the right hand side, the first term describes a maximally mixed state, so there is only classical correlation. However the additional terms denoted by $\dots$ should capture the quantum coherence (off-diagonal part of the density matrix if they exist). Because we have assumed that the states $|\psi\rangle$ are orthogonal, so there is no quantum correlation in $\rho_R$ . But if we try to compute the purity i.e. the second Renyi entropy of $\rho_R$ which is supposed to be

$\rho_R^2=\frac{1}{k^2}\sum_{i,j,k}|i\rangle \langle k| |k\rangle \langle j|\langle \psi_i|\psi_k\rangle \langle \psi_k|\psi_j\rangle$
$\text{tr}(\rho_R^2)=\frac{1}{k^2}\sum_{i,j}|\langle \psi_i|\psi_j\rangle|^2$

while from the gravitional path integral we indeed found the replica wormhole contribution

$\text{tr}(\rho_R^2)=k^{-1}+e^{-S_0}$

This result is puzzling considering

$\rho_R^2=\frac{1}{k}\rho_R$

It suggests that there is quantum correlation in $\rho_R$ but to 'see' it we need to consider the quantity like $\rho_R^2$ . The gravitional path integral is not literally computing quantum amplitudes $\langle \psi_i|\psi_j\rangle$ , but instead computing some coarse-grained version, where we average over some microscopic information.

The lack of factorization of the amplitude is familiar from the connection between Euclidean wormholes and disorder.

There are many questions can be asked. Maybe the most urgent two are where does the quantum correlation come from and in a non-averaged situation where factorizatoin must hold what is one supposed to do with the wormholes?

In particular we know we can not just eliminate the wormhole contributions because it indeeds explain the ramp of the spectral form factor. But for the un-averaged theory, the wormhole is not whole solution, there is also quantum noise which gives rise to erratic fluctuation of the spectral form factor and the fluctuation is almost in the same order or the smooth wormhole contribution.

One possible answer to the factorization puzzle is that one does not eliminate the paired diagonal terms corresponding to the wormhole instead one adds back in all the other unpaired off-diagonal terms. And the off-diagnoal terms would correspond to the quantum noise.

\section{Model II, MM topological model}
It is a gravitional theory which is dual ensembel average of boundary theory.
Instead of products of partition function $Z(\beta)$ , we compute correlation functions of a boundary operator $\hat{Z}$ in states of closed universe.
$\langle NB|\hat{Z}^3|NB\rangle$
or include state boundaries
$\langle \psi|\hat{Z}^3\rangle$
where $\psi$ is some function of $Z$ .

In eigenstates of $\hat{Z}$ , correlation function factorize because it is just a number. So the eigenstate corresponds to a single memeber of an ensemble of boundary theories.

In an exact eigenstate, there includes surfaces with many boundaries and high genus and there surfaces are not suppressed order by order in the topological expansion. However we can consider the approximate Gaussian eigenstates where we ingore the effects from the discretness of the spectrum of $\hat{Z}$ , and correspond to only keeping disks and cylinders, i.e. Goleman-Giddlings-Strominger model.

In this approxiamtion, $\hat{Z}$ is like the coordinate of a harmonic oscillator

$\hat{Z}=a+a^\dagger,\quad a|NB\rangle=0$

So $|\hat{Z}^k\rangle$ is a superposition of n-universe state with at most $n=k$
It is conveneint to think about the correlation functions as overlaps states

$\langle \hat{Z}^k\rangle_\psi=\langle Z^k|\psi^2\rangle=\sum_{n=0}^k \langle Z^k|n\rangle \langle n|\psi^2\rangle$
For $|\psi\rangle$ an to be an eigenstate, factorization requires

$\langle 1|\psi^2\rangle^2=\langle 2|\psi^2\rangle+cylinder$

To model JT gravity, we may think there are many $\hat{Z}$ such as $Z(\beta+i T)$ for different T. In order to relate to the JT gravity calculation more closely we introduce a basis $|b_i\rangle$ corresponding to circular spatial slices with length b and zero extrinsic curvature

$\langle b|b'\rangle=\frac{1}{b}\delta(b-b')$

Then factorization requires

$\langle b_1|\psi^2\rangle \langle b_2|\psi^2\rangle=\langle b_1b_2|\psi^2\rangle+\frac{1}{b_1}\delta(b_1-b_2).$

So the second term is the wormhole piece and the first term will correspond to the off-diagonal noise.

Week 9

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