Comments on wormholes and factorization

Phil Saad1, Stephen H. Shenker2, and Shunyu Yao2

1School of Natural Sciences,

Institute for Advanced Study, Princeton, NJ 08540

2Stanford Institute for Theoretical Physics,

Stanford University, Stanford, CA 94305

Abstract

> In AdS/CFT partition functions of decoupled copies of the CFT factorize. In
> bulk computations of such quantities contributions from spacetime wormholes
> which link separate asymptotic boundaries threaten to spoil this property,
> leading to a “factorization puzzle.” Certain simple models like JT gravity
> have wormholes, but bulk computations in them correspond to averages over an
> ensemble of boundary systems. These averages need not factorize. We can
> formulate a toy version of the factorization puzzle in such models by
> focusing on a specific member of the ensemble where partition functions will
> again factorize.
>
> As Coleman and Giddings-Strominger pointed out in the 1980s, fixed members
> of ensembles are described in the bulk by “$\alpha$-states“ in a many-
> universe Hilbert space. In this paper we analyze in detail the bulk
> mechanism for factorization in such $\alpha$-states in the topological model
> introduced by Marolf and Maxfield (the “MM model”) and in JT gravity. In
> these models geometric calculations in $\alpha$ states are poorly
> controlled. We circumvent this complication by working in approximate
> $\alpha$ states where bulk calculations just involve the simplest
> topologies: disks and cylinders.
>
> One of our main results is an effective description of the factorization
> mechanism. In this effective description the many-universe contributions
> from the full $\alpha$ state are replaced by a small number of effective
> boundaries. Our motivation in constructing this effective description, and
> more generally in studying these simple ensemble models, is that the lessons
> learned might have wider applicability. In fact the effective description
> lines up with a recent discussion of the SYK model with fixed couplings [1].
> We conclude with some discussion about the possible applicability of this
> effective model in more general contexts.

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 Summary of the paper
  2. 2 Periodic orbits
  3. 3 The disk-and-cylinder approximation
    1. 3.1 Introduction to section
    2. 3.2 The MM and CGS model
    3. 3.3 States of Closed Universes
    4. 3.4 Factorization in the CGS model
    5. 3.5 The CGS model with “species”
    6. 3.6 An effective description with random boundaries
  4. 4 Approximate $\alpha$ states in the MM model
    1. 4.1 Review of the MM model
    2. 4.2 Approximate $\alpha$ states in the MM model
    3. 4.3  CGS approximation errors in an approximate $\alpha$ state
  5. 5 JT gravity
    1. 5.1 Review of JT gravity
    2. 5.2 JT in the disk-and-cylinder approximation and an effective model
    3. 5.3 Approximate $\alpha$ states in JT gravity
  6. 6 Discussion
    1. 6.1 SYK model
    2. 6.2 Some open questions
  7. A The diagonal sum is self-averaging
  8. B Explicit calculations in the CGS model with species
  9. C The autocorrelation function in JT gravity
  10. D Bound on the disk-and-cylinder approximation in JT gravity
  11. E Correlators of many copies of $\widehat{Y}_{E}(T)$
  12. F Errors from the $3$-holed sphere in JT gravity
  13. G Formal discussion about factorization in the No-Boundary state

##  1 Introduction

###  1.1 Overview

Gauge/gravity duality currently provides the only known complete definition of
quantum gravity. In this framework bulk quantum gravity, in certain
situations, has a dual description as a well defined non-gravitational
boundary quantum system. Many parts of the dictionary between bulk and
boundary quantities have been established, but our understanding of the bulk
is still incomplete.

There are certain sharp tensions between our current knowledge of the bulk and
the well-defined boundary description whose resolution would certainly deepen
our understanding of quantum gravity. The one we will focus on in this paper
is the puzzle of factorization, first described in this context in [2, 3].

Consider two decoupled boundary CFTs, called L and R, with corresponding
partition functions $Z_{L}$ and $Z_{R}$. The boundary description immediately
shows that the partition function of the combined system $Z_{LR}$ factorizes –
that is, it is equal to $Z_{L}Z_{R}$. But a bulk calculation seems to include
gravitational configurations connecting the two boundaries, spacetime
wormholes are the paradigmatic example. Naively such configurations spoil
factorization. The problem is to understand how a complete bulk description
resolves this apparent contradiction.

Figure 1: Here we picture a spacetime wormhole which contributes to
$Z_{L}Z_{R}$.

This question has become more pressing recently because spacetime wormholes
have been argued to be useful in explaining a number of nonperturbative
properties of quantum black holes in the context gauge/gravity duality. A
first example is the “eternal traversable wormhole“ [4]; Another example is
the “ramp” region in the long-time behavior of the spectral form factor [5, 6]
and of correlation functions [7] that is a signature of random matrix
statistics in the quantum chaotic boundary system. Another example is the
replica wormhole calculation of the Page curve [8, 9] and of squared matrix
elements [10] in models of evaporating black holes. Some of these situations,
in particular the spectral form factor and squared matrix elements, are
described by decoupled boundary systems and so the wormhole explanation gives
rise to a factorization puzzle.

The most complete calculations of these quantities have been done in systems
like the SYK model [11, 12, 13] and its low energy limit, JT gravity [14, 15,
16, 17, 18]. These systems are dual to ensembles of boundary quantum systems
[6, 19]. In the SYK model one averages over an ensemble of Hamiltonians with
different values of the couplings between the fermions. In JT gravity one
averages over Hamiltonian matrices drawn from a certain random matrix
ensemble. Then instead of computing genuine products of partition functions,
one considers averages of products of partition functions. Denoting the
average over the ensemble by $\langle\cdot\rangle$, the average over L and R
systems $\langle Z_{L}Z_{R}\rangle$ in general will not factorize into the
product $\langle Z_{L}\rangle\langle Z_{R}\rangle$. In the the bulk these
averaged products typically receive contributions from wormholes, which
accurately calculate the deviation from factorization. There is no puzzle
here.111There has been a substantial amount of additional recent work on
wormholes and ensembles. Examples include [20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].

The connection between wormholes and ensembles is actually an old one, going
back to the ideas of Coleman [42] and Giddings-Strominger [43] in the 1980s.
These ideas have recently been reformulated in the AdS/CFT context, with
important extensions, by Marolf and Maxfield [44] in work we will discuss
extensively below.

But the paradigmatic examples of gauge/gravity duality consist of bulk
gravitational systems dual to single boundary theories, not ensembles. Bulk
spacetime wormhole configurations joining decoupled boundaries may well exist
in these systems [3, 45, 25, 46, 32, 47, 48]. How could this be compatible
with factorization?

Perhaps the wormholes are actually absent, or don’t contribute for some
reason. But on general grounds, especially the universality of random matrix
behavior in such quantum chaotic systems, we expect that decoupled non-
averaged systems display certain features reminiscent of the wormhole effects.
An example we will focus on is the ramp in the spectral form factor, the
product of analytically continued partition functions
$Z_{L}(\beta+iT)Z_{R}(\beta+iT)$. This product must, of necessity, factorize.
How can this be compatible with the wormhole signal?

For a chaotic non-averaged system, including the boundary theories in standard
examples of gauge/gravity duality, one expects, again based on random matrix
universality, that the spectral form factor is a noisy function for
sufficiently late times, including the ramp region [49]. This noise is
typically the same order as the signal. For systems described by ensembles,
like SYK and JT gravity, one expects similar noisy behavior for a fixed draw
from the ensemble. In suitably rich ensembles, like those in SYK and JT
gravity, this noise can largely be removed by ensemble averaging; the
resulting averaged spectral form factor has a smooth ramp rather than an
erratic one, giving the signal well described by wormholes. These large noisy
fluctuations can be quantified by studying ensemble averaged statistics like
the variance, higher moments, and time autocorrelation functions. Such
statistics can also be computed in the bulk using wormholes.222This is the
case as long as we restrict attention to finite order moments, and times short
compared to the plateau time.

So while in systems like SYK and JT gravity, the wormhole does not give a good
approximation for a fixed draw from the ensemble, the wormhole has a clear
role in computing the ensemble averages. However, in non-averaged systems the
wormhole cannot be isolated this way, and due to the large fluctuations it
does not give a uniformly good approximation to the answer. So one might
expect that wormholes are not relevant here.

But random matrix universality also indicates that the autocorrelation time of
the noise in the spectral form factor is short, of order the inverse size of
the energy window considered [49, 50],333This autocorrelation time will be
derived in the JT system in appendix C. and so performing a time average along
the ramp leads to a smooth curve which matches the wormhole result. So it
seems appealing to entertain the possibility that wormholes play some kind of
role in the bulk description of non-averaged systems.444Of course this might
not be the case – these features might have a completely different origin.
Note that this time averaging correlates the L and R observables so there is
no factorization puzzle.555The Renyi entropies computed by replica wormholes
[9, 8] are other examples of correlated observables which do not raise a
factorization issue, and where noise is suppressed.

The goal of this paper is to explore, compare and contrast some scenarios for
how wormholes and factorization can coexist in non-averaged gauge/gravity
duals. We currently do not have enough control in any standard holographic
system to address these subtle questions. So we must of necessity return to
simple models, in particular those described by ensembles. Here, as mentioned
above, we can study a version of the factorization puzzle by looking at just
one element of the ensemble. Computations for a single ensemble element will
include contributions from wormholes, but nevertheless still manage to produce
a factorized answer. Our hope, and at this point it is only a hope, is that
the mechanisms we examine might give some insight into how standard non-
averaged holographic systems behave.

Specifically, we will examine the Marolf-Maxfield (MM) model [44] where a
fixed element of the ensemble is referred to as an $\alpha$ state [42, 43]).
We will then turn to JT gravity where a single element of the ensemble
corresponds to a particular random matrix drawn from the appropriate
distribution[51]. To set some context we will begin by discussing a simple
analogy – the periodic orbit theory of semiclassical quantum chaos [52];

We will also make some brief remarks comparing these models to another system,
the SYK model with a single instance of the random fermion couplings. We have
recently presented an analysis of this system in another paper, written
jointly with Douglas Stanford [1].

In each model we will endeavor to study the simplest situation where the
factorization puzzle arises. In JT gravity we will only study the ramp region
of the spectral form factor and ignore the plateau. This amounts to ignoring
the discreteness of the random matrix eigenvalue spectrum, and treating it as
a continuous density.666See [53] for progress in controlling effects related
to the discreteness of the spectrum. This allows us to focus on the simplest
topologies which we will call the disk (the Euclidean black hole) and the
cylinder (the wormhole). Similarly in the MM model we will ignore the
discreteness of the spectrum of $\alpha$ states and consider sharply peaked
but smooth approximate $\alpha$ states. Again only disks and cylinders will
contribute.

###  1.2 Summary of the paper

We now give a brief summary of this rather lengthy paper and its main results.

In Section 2, Periodic orbits. we briefly review the computation of the
spectral form factor in the periodic orbit theory of semiclassical quantum
chaos. In these systems, one uses the Gutzwiller trace formula to express the
spectral form factor as a double-sum over periodic orbits, one for each copy
of the system. The off-diagonal summands have erratic phases, but the diagonal
terms are smooth. As a result, averaging removes the off-diagonal terms, with
the diagonal terms giving the resulting “ramp”. Making an analogy between this
diagonal orbit sum and the wormhole, we are led to ask what is the analog of
the off-diagonal structure in gravity.

In Section 3. The disk-and-cylinder approximation, we study a toy model of
gravity with wormholes, which we call the Coleman-Giddings-Strominger (CGS)
model. In this model, the analogs of partition functions are computed by sums
over surfaces with topologies limited to disks and cylinders.777A similar
model has recently been discussed in this context in [36]. We view the
simplest version of this model as an approximation to the MM model, and a more
complicated variant of this model as an approximation to JT gravity. We leave
the justification of this approximation to sections 4 and 5. The general goal
of this section is to describe the simple picture of factorization which
applies in the disk-and-cylinder approximation, and introduce an effective
description based on this simple structure.

The main points of Section 3 are as follows:

  * •

Using the framework of Marolf and Maxfield [44], building on [42, 43], we view
the path integral for this model as computing correlation functions of
boundary operators $\widehat{Z}_{I}$ in a state $|\psi\rangle$ of closed
universes. Eigenstates $|\psi_{\alpha}\rangle$ of the boundary operators
correspond to fixed members of an ensemble of theories. Then our interest is
in studying correlation functions in a eigenstate, or “$\alpha$ state” [43,
42, 44].

  * •

While in general an $\alpha$ state $|\psi_{\alpha}\rangle$ is a superposition
of states with any number of closed universes, correlation functions of $n$
boundary operators depend solely on the components of a state
$|\psi^{2}_{\alpha}\rangle$ (which is simply related to
$|\psi_{\alpha}\rangle$) with up to $n$ universes. So rather than depending on
the whole many-universe structure of the closed-universe wavefunction, an
$n$-point function only depends on an $n$-universe state.

  * •

In an $\alpha$ state, the $n$-universe components of
$|\psi^{2}_{\alpha}\rangle$ needed to compute an $n$-point function are simply
determined by the one-universe component. This follows from the fact that an
$n$-point function in an $\alpha$ state factorizes into a product of one-point
functions. In many ways this one-universe component provides a simpler
description of a fixed member of the ensemble than the many-universe $\alpha$
state.

  * •

In this model, studying the factorization problem amounts to comparing a two-
point function in an $\alpha$ state with the product of one-point functions.
In our formalism, this involves comparing the two-universe component of
$|\psi^{2}_{\alpha}\rangle$ with the product of one-universe components.
Schematically,

$\displaystyle\langle\psi_{\alpha}|\widehat{Z}_{I}\widehat{Z}_{J}|\psi_{\alpha}\rangle$
$\displaystyle\supset\text{Cylinder}+\text{Two-universe component}$ (1.1)
$\displaystyle\langle\psi_{\alpha}|\widehat{Z}_{I}|\psi_{\alpha}\rangle\langle\psi_{\alpha}|\widehat{Z}_{J}|\psi_{\alpha}\rangle$
$\displaystyle\supset\big{(}\text{One-universe
component}\big{)}\cdot\big{(}\text{One-universe component}\big{)}$ (1.2)

In appropriate cases, we can use the analogy with periodic orbit theory and
view the cylinder as a diagonal sum in some basis of one-universe states.
Expressed in that basis, the one-universe component of
$|\psi^{2}_{\alpha}\rangle$ is a sum over one-universe states with random
coefficients, and the product of one-point functions
$\langle\psi_{\alpha}|\widehat{Z}_{I}|\psi_{\alpha}\rangle\langle\psi_{\alpha}|\widehat{Z}_{J}|\psi_{\alpha}\rangle$
is a double sum. Then the two-universe component of
$|\psi^{2}_{\alpha}\rangle$, which is added to the “diagonal” cylinder,
behaves like an off-diagonal sum. This off-diagonal character has a geometric
origin in the many-universe description, resulting from a geometric “exclusion
effect.”

  * •

We can capture the behavior of the CGS model in an $\alpha$ state with an
effective model, in which the many-universe $\alpha$ state is traded for a
random “$\Psi$” boundary condition. This $\Psi$ boundary condition is
essentially an effective description of the one-universe component of
$|\psi^{2}_{\alpha}\rangle$. “Broken cylinders” with one $\widehat{Z}_{I}$
boundary and one $\Psi$ boundary summarize the full many-universe structure of
the $\alpha$ state. As we explain in the Discussion, these broken cylinders
are closely related to the “half-wormholes” found in [1].

Upon averaging, pairs of broken cylinders are glued together along their
$\Psi$ boundaries to form the usual cylinders. While this effective
description naturally captures some aspects of the full theory, it does not
have an “exclusion effect” to enforce the off-diagonal character of pairs of
$\Psi$ boundaries. Instead one must implement this by hand with an ad-hoc
“exclusion rule.”

Figure 2: Averaging glues pairs of broken cylinders together along their
$\Psi$ boundaries to produce full cylinders.

The rest of the paper is devoted to applying the simple picture found in the
CGS model to the MM model and JT gravity.

In Section 4, Approximate $\alpha$ states in the MM model, we study the MM
model. We show that the disk-and-cylinder approximation of the MM model, which
is equivalent to the CGS model from the previous section, is valid for
appropriate correlation functions in approximate $\alpha$ states. Establishing
the validity of this approximation then allows us to apply the ideas of
Section 3 to this model.

In Section 5 JT gravity, we study JT gravity. The main idea of the section is
that the disk-and-cylinder approximation is valid for appropriate correlation
functions in approximate $\alpha$ states, and thus the ideas from Section 3
may also be applied to JT gravity. However, we postpone the justification of
this approximation to the end of the section, and instead begin by assuming
the validity of the disk-and-cylinder approximation and exploring the results.
In particular, we focus on the effective description of JT gravity with $\Psi$
boundaries, which is especially vivid in this context. In this model, the
broken cylinders are naturally constructed by bisecting the cylinder along a
circular geodesic, and giving this new boundary a random $\Psi$ boundary
condition. This boundary condition can be described by a random function
$\Psi(b)$, where $b$ is the length of the boundary. In this effective
description, we see how both ensemble-averaging and time-averaging can glue
the $\Psi(b)$ boundaries together to form wormholes.

We end Section 5 by studying approximate $\alpha$ states in JT gravity. Exact
$\alpha$ states in JT gravity correspond to fixing a discrete set of energy
levels in the boundary theory, which may be described in the bulk using
“Eigenbranes” [51, 54]. However, the physics on timescales less than the
plateau time888For simple observables. Correlation functions with
exponentially many operator insertions may be sensitive to the discreteness of
the boundary spectrum. is not sensitive to the discreteness of the boundary
spectrum. In the averaged theory, physics on these timescales can be described
in the bulk with wormholes, and in the boundary as Gaussian fluctuations of a
coarse-grained, continuous approximation to the boundary spectrum. Then to
construct approximate $\alpha$ states, rather than following an approach which
directly mirrors the approach in Section 4, which would involve approximately
fixing a discrete set of energy levels, we instead approximately fix a coarse-
grained, continuous version of the density of states. In these approximate
$\alpha$ states, we argue that the disk-and-cylinder approximation is valid at
least until a timescale determined by the error specified in the
approximation. This error is quantified by the small variance in the
approximate $\alpha$ state. Specifying an exponentially small variance, this
timescale can be made exponentially long in the entropy, $\sim e^{cS}$, with
$c<1$.

We begin the Discussion by comparing our findings in this paper with recent
results about the SYK model published in a joint paper with Douglas Stanford
[1]. In that work we analyzed the SYK model for a fixed choice of boundary
couplings, which is analogous to studying gravity in an exact $\alpha$ state.
As we briefly explain in this section, the results from that work are
reminiscent of our findings in this paper. In the remainder of the Discussion,
we raise several questions and give brief comments on each.

##  2 Periodic orbits

Here we briefly review an idea, originally due to Berry [55], that explains
the ramp in the spectral form factor for few-body quantum chaotic systems in
the semiclassical approximation.999For a detailed review see [52]. It suggests
a picture of how wormholes and factorization can be reconciled [6]. This
picture gives a reference example that we will compare and contrast to other
model systems in subsequent sections.

Consider a chaotic classical mechanical system, like a billiard ball moving on
a suitably shaped billiard table. In a semiclassical limit we can write a
Minkowski time partition function $Z(it)={\rm Tr}[e^{-iHt}]$ using a suitable
formulation of the Feynman path integral called the Gutzwiller trace
formula.101010Strictly speaking this trace should be restricted to an energy
window as in the quantity $Y(it)$ defined subsequently. Schematically we have

$Z(iT)={\rm Tr}[e^{-iHT}]=\sum_{a}D_{a}e^{iS_{a}}$ (2.1)

where $a$ runs over classical periodic orbits, $S_{a}$ is the classical action
of the orbit, and $D_{a}$ is the one loop fluctuation determinant. The
spectral form factor is given by the product of partition functions

$Z_{L}(iT)Z_{R}(-iT)=\sum_{ab}D_{a}D_{b}e^{i(S_{a}-S_{b})}$ (2.2)

In this non-averaged system this quantity manifestly factorizes, as reflected
in the double sum.

But this double sum simplifies if we do various kinds of averaging. For
example we could average over a window of times $T$, or we could average over
an ensemble of related chaotic Hamiltonians. The time averaged case does not
factorize because the observable is correlated, the ensemble case doesn’t
because the Hamiltonians are.

At large $T$ the actions $S_{a}$ are large and the phases rapidly oscillate.
For large, but not too large, times the only terms in (2.2) that survive
averaging are the “diagonal” ones $a=b$, up to a relative time translation.
There are exponentially many such orbits, compensated for by exponentially
small one-loop factors $D_{a}$. The relative time translation mode gives a
factor of $T$, explaining the linear behavior of the ramp in the spectral form
factor. This is Berry’s “diagonal approximation” [55]. It is clear that the
passage from the double sum to the diagonal single sum destroys factorization,
as expected in an averaged system.

All of the above is a “boundary” quantum mechanical analysis, but it is
appealing to consider the diagonal sum as an analog of the bulk wormhole
configuration, where the L and R factors are linked by the diagonal
correlation pattern. This analogy suggests a way to restore the factorization
in a non-averaged system. We keep the wormhole but need to add back in the
bulk analog of the off-diagonal terms that restore the manifestly factorizing
double sum in (2.2).

Note that in this analogy the wormhole is an emergent concept, it plays no
role in the fundamental description of the system. Note also that just two
copies of the boundary Hilbert space are involved here, and hence just two
sums. To study correlators with more $Z$’s, like
$\langle(Z_{L}(iT)Z_{R}(-iT))^{k}\rangle$, $2k$ sums would be required.

##  3 The disk-and-cylinder approximation

###  3.1 Introduction to section

In this section we begin our discussion of simple bulk models that display a
factorization problem. We focus here on a very interesting model introduced by
Marolf and Maxfield (the MM model) [44] that illustrates many of the basic
issues. The model is a reformulation and extension of the old ideas of Coleman
[42] and Giddings-Strominger [43] to the AdS/CFT context. The only degree of
freedom in this model is the topology of the bulk manifold and contains
arbitrarily many one dimensional asymptotic AdS type boundaries connected by
2-manifolds of arbitrary genus. We will focus on an approximation that amounts
to limiting the 2-manifold topologies to disks and cylinders (spacetime
wormholes). This simpler model is close to the one studied by Coleman and
Giddings-Strominger and we refer to it as the CGS model. The MM and CGS models
are reviewed in The MM and CGS model.

These models describe states in a “many-universe” Hilbert space which is
reviewed in States of closed universes. There are certain states in this
Hilbert space called $\alpha$ states in which partition functions $Z$ assume a
definite value – eigenstates of the ${\widehat{Z}}$ operator. Expectation
values of products of $Z$’s manifestly factorize in such states. The analysis
of such exact $\alpha$ states in the full MM model is subtle, so to simplify
our analysis we will study approximate $\alpha$ states in which expectation
values of $Z$ products approximately factorize, so there still is a sharp
factorization puzzle. This amounts to focusing on the simple disk and cylinder
topologies of the CGS model. Factorization in the CGS model and its “multi-
species” extension is discussed in Factorization in the CGS model and
Factorization in the CGS model with species. We postpone the analysis of the
errors in approximating the full MM model in approximate $\alpha$ states by
the CGS model until The MM model and show that some of the subtle properties
of the MM model, null states and discrete $\alpha$ spectrum, do not play a
role in resolving the approximate factorization puzzle. The analysis of errors
in approximating JT gravity by the CGS model with species is postponed to JT
gravity.

We should emphasize that in these models the spacetime wormhole connecting
different boundaries is a fundamental part of the description of the system.
This is in contrast to the periodic orbit picture discussed in Section 2 where
the wormhole is emergent. Nonetheless in an (approximate) $\alpha$ state
factorization is (approximately) restored. Roughly speaking in an
(approximate) $\alpha$ state additional contributions corresponding to the
“off-diagonal” terms in the periodic orbit story are present, allowing the
entire result to be written in the factorized double sum form. We make this
analogy more precise by describing these contributions in terms of a random
single-universe wavefunction.

We end with An effective description with random boundaries, in which we
introduce an effective description of the CGS model in an $\alpha$ state. In
this effective description, the contributions from the many-universe $\alpha$
state are replaced with contributions from a few “effective” boundaries, with
a random boundary condition. An unusual feature of this effective description
is that we must modify the conventional sum over geometries to include an
“exclusion rule”, which instructs us to remove either the cylinders connecting
partition function boundaries, or the corresponding “diagonal” contribution of
pairs of cylinders with effective boundaries. This exclusion rule has a
geometric origin in the full theory, but must be put in by hand in the
effective model.

###  3.2 The MM and CGS model

To set the context we first review the elementary parts of the MM model
introduced in [44]. This model is defined by a Euclidean path integral which
computes quantities $Z_{n}$. $Z_{n}$ is computed by summing over (oriented)
two-dimensional spacetimes $M$ with $n$ circular boundaries. The surface $M$
may have any number of disconnected components.

Figure 3: Here we show an example contribution to $Z_{n}$ (for $n\geq 4)$.
Each spacetime contributing to $Z_{n}$ has $n$ circular boundaries, and may
have any number of connected components. The lower set of dots represent the
$n-4$ boundaries not explicitly drawn.

The path integral for $Z_{n}$ is analogous to the AdS gravity path integral
which computes a product of $n$ partition functions. The $n$ circular
boundaries are analogous to the asymptotically AdS boundaries. However, in
this model the action depends solely on the topology of the spacetime, so the
“partition functions” $Z_{n}$ computed in this model do not depend on a
temperature.

The Euclidean action in this model is simply proportional to the Euler
character of the spacetime $M$, $I(M)=-S_{0}\chi(M)$. We think of $S_{0}$ as a
large number analogous to the black hole entropy.111111Note that our
conventions differ from those used in [44] by a shift of one unit of $\chi$.
The choice used in [44] gives integer values for “partition functions”
appropriate to a boundary Hilbert space trace. We choose to use the
geometrically more natural normalization. None of the lessons we draw are
affected by this. Then the path integral reduces to a sum over topologies,
weighed by the action $\exp[-I(M)]$ and a measure factor $\mu(M)$,

$Z_{n}=\sum_{\text{Surfaces }M}\mu(M)e^{-I(M)}.$ (3.1)

The measure factor is defined to treat disconnected spacetimes without
boundary as indistinguishable. If $M$ has $m_{g}$ connected components of
genus $g$ and no boundary,

$\mu(M)=\prod_{g}\frac{1}{m_{g}!}.$ (3.2)

The measure factor $\mu(M)$ will not play a large role in our discussion.
$\mu(M)$ is not equal to one only for spacetimes without boundaries; the
contributions of these spacetimes factor out of all quantities we compute and
are normalized away.

For practice, we will briefly describe the computations of $Z_{0}$, $Z_{1}$,
and $Z_{2}$.

To compute $Z_{0}$ we sum over all oriented surfaces with no boundaries. We
can view this sum as a sum over the numbers $m_{g}$ of connected components
with genus $g$. The Euler character of a surface $M$ with $m_{g}$ connected
components of genus $g$ is simply the sum of the the Euler characters of the
connected components

$\chi(M)=\sum_{g=0}^{\infty}m_{g}(2-2g).$ (3.3)

Then,

$Z_{0}=\prod_{g=0}^{\infty}\bigg{[}\sum_{m_{g}=0}^{\infty}\frac{1}{m_{g}!}e^{m_{g}S_{0}(2-2g)}\bigg{]}=\prod_{g=0}^{\infty}\exp\bigg{[}e^{S_{0}(2-2g)}\bigg{]}=\exp\bigg{[}\frac{e^{2S_{0}}}{1-e^{-2S_{0}}}\bigg{]}.$
(3.4)

Figure 4: Here we illustrate some examples of contributions to $Z_{0}$,
labeled by the number of connected components for each genus, $m_{g}$.

To compute $Z_{1}$ we sum over surfaces with one circular boundary. We can
view this sum as a sum over the numbers $m_{g}$ of connected components with
no boundary and genus $g$, and over the genus $g_{1}$ of the connected
component with a boundary.

Because the Euler character is a sum of the Euler characters of the connected
components, the path integral factorizes into the product of the sum over
surfaces with no boundary $Z_{0}$, and the sum over connected surfaces with
one boundary.

$Z_{1}=Z_{0}\times\sum_{g_{1}=0}^{\infty}e^{S_{0}(2-2g_{1}-1)}.$ (3.5)

The second factor in the above formula is the sum over connected surfaces with
one boundary, which are classified by their genus $g_{1}$. The Euler character
of a connected surface with $k$ boundaries and genus $g_{1}$ is
$\chi=2-2g_{1}-n$; in the present case we have $n=1$.

Focusing on this factor, and defining a normalized path integral
$\tilde{Z}_{1}$,

$\tilde{Z}_{1}\equiv\frac{Z_{1}}{Z_{0}}=e^{S_{0}}+e^{-S_{0}}+\dots=\frac{e^{S_{0}}}{1-e^{-2S_{0}}}.$
(3.6)

Figure 5: Here we have pictured the leading contributions to $\tilde{Z}_{1}$,
which is given by a sum over spacetimes with one connected component and a
single circular boundary. We have labeled the contributions by their Euler
character.

Finally, we compute $Z_{2}$, the sum over surfaces with two circular
boundaries. Again, as in the computation of $Z_{1}$, the contributions of
connected components with no boundaries factorizes out, leaving a sum over
connected surfaces with boundaries. This will happen for any $Z_{n}$.

In the case of two boundaries, the components with boundaries may have one or
two boundaries. The sum over these surfaces can be split into the sum over
surfaces with two connected components with genus $g_{1}$ and $g_{2}$, each
with one boundary, and the sum over surfaces with one connected component with
genus $g_{12}$ and two boundaries.

$\tilde{Z}_{2}\equiv\frac{Z_{2}}{Z_{0}}=\bigg{(}\sum_{g_{1}=0}^{\infty}e^{S_{0}(2-2g_{1}-1)}\bigg{)}\bigg{(}\sum_{g_{2}=0}^{\infty}e^{S_{0}(2-2g_{2}-1)}\bigg{)}+\sum_{g_{12}=0}^{\infty}e^{S_{0}(2-2g_{12}-2)}.$
(3.7)

Figure 6: Here we have pictured the leading contributions to $\tilde{Z}_{2}$,
which is given by a sum over spacetimes with two circular boundaries,
including only spacetimes where each connected component has at least one of
these two boundaries. The cylinder contribution, with $\chi=0$, is an example
of a spacetime wormhole, connecting two separate boundaries.

At large $S_{0}$, complicated topologies provide small contributions to the
$Z_{n}$ for $n\ll e^{S_{0}}$. For example, we look at the calculation of
$\tilde{Z}_{2}-\tilde{Z}_{1}^{2}$. This quantity diagnoses the failure of
factorization of the “partition functions” in this model, which will be of
central interest to us.

$\tilde{Z}_{2}-\tilde{Z}_{1}^{2}$ is given by a sum over surfaces with one
connected component with two circular boundaries. The cylinder gives a
contribution of one, while the cylinder with a handle added gives a
contribution of $e^{-2S_{0}}$.

$\tilde{Z}_{2}-\tilde{Z}_{1}^{2}=1+e^{-2S_{0}}+\dots$ (3.8)

Figure 7: The leading fully connected contribution to $\tilde{Z}_{k}$, which
includes only contributions from spacetimes with a single connected component,
is given by the $k$-holed sphere. This contribution has Euler character
$-k+2$, so the connected correlators for $k>2$ are exponentially small, $\sim
e^{-(k-2)S_{0}}$.

More general “connected correlators” such as
$\tilde{Z}_{4,c}\equiv\tilde{Z}_{4}-2\tilde{Z}_{2}^{2}-2\tilde{Z}_{2}\tilde{Z}_{1}^{2}-\tilde{Z_{1}}^{4}$,
will be small when $S_{0}$ is large. For $k\ll e^{S_{0}}$, $\tilde{Z}_{k,c}$,
the ‘connected k-boundary correlator’, will be given to leading order by the
contribution of the spacetime with the topology of a sphere with $k$ holes.
This has an Euler character of $\chi=2-k$, so

$\tilde{Z}_{k,c}\approx e^{-(k-2)S_{0}},\hskip 20.0ptk\ll e^{S_{0}}.$ (3.9)

Neglecting these exponentially small terms, the $\tilde{Z}_{k}$ behave
approximately like moments of a Gaussian distributed random variable. This
interpretation will be made precise in the later sections.

Motivated by this Gaussian behavior, we define a modification of the MM model,
which we dub the Coleman-Giddings-Strominger (CGS) model. This model is
obtained from the MM model by restricting the sum over topologies to only
include spacetimes whose connected components have the topology of the disk or
the cylinder.

In this CGS model the $\tilde{Z}_{n}$ behave exactly like the moments of a
Gaussian random variable with mean $\tilde{Z}_{1}$ and variance Cyl,

$\displaystyle\tilde{Z}_{1}$ $\displaystyle=\text{Disk}=e^{S_{0}}$ (3.10)
$\displaystyle\tilde{Z}_{2}$
$\displaystyle=\text{Disk}^{2}+\text{Cyl}=e^{2S_{0}}+1$ (3.11)
$\displaystyle\tilde{Z}_{2,c}$ $\displaystyle=\text{Cyl}=1$ (3.12)
$\displaystyle\tilde{Z}_{k,c}$ $\displaystyle=0,\hskip 20.0ptk>2.$ (3.13)

Here the subscript “c” denotes the connected contribution to $\tilde{Z}_{k}$.

The CGS model lacks some of the interesting features of the MM model, which
are linked to the contributions from higher-genus surfaces. However, this
model has its own factorization problem, since $\tilde{Z}_{2,c}=\text{Cyl}\neq
0$. We will focus on this issue.

In Section 4 we will describe in more detail how and when the MM model can be
reduced to the CGS model in computations related to factorization. However,
for now we will focus solely on the CGS model.

###  3.3 States of Closed Universes

Marolf and Maxfield [44] also introduced a Hilbert space structure associated
to spacetimes with boundaries, interpreting quantities like the $Z_{k}$ as
amplitudes in a quantum theory of closed universes. The Gaussian behavior of
the $\tilde{Z}_{n}$ at large $S_{0}$ discussed in the previous section can be
interpreted in this way as expectation values of a quantum harmonic oscillator
position operator $\widehat{Z}$ in a certain state of closed universes. In
this section we briefly review the discussion in [44] of the quantum mechanics
of closed universes. The main points that we will explain are:

  * •

The Hilbert space of closed universes is spanned by the states
$\widehat{Z}^{k}|NB\rangle$. The No-Boundary state, $|NB\rangle$, is a state
of no closed universes. $\widehat{Z}$ is a Hermitian operator which creates
and destroys closed universes with the spatial topology of a circle.

  * •

The inner product between states $\widehat{Z}^{k}|NB\rangle$,
$\widehat{Z}^{k^{\prime}}|NB\rangle$ is defined to be equal to the path
integral $\tilde{Z}_{k+k^{\prime}}$.121212Normalizing the No-Boundary state
correponds to dividing out by the sum over spacetimes without boundaries,
$Z_{0}$.

  * •

The operators $\widehat{Z}$ are observables. By expressing a general state
$|\psi\rangle$ as a superposition of the eigenvectors $|Z_{\alpha}\rangle$ of
$\widehat{Z}$, $|\psi\rangle=\sum_{\alpha}\psi_{\alpha}|Z_{\alpha}\rangle$, we
can relate correlation functions of the operator $\widehat{Z}$ to moments of a
variable $Z$ in a classical probability distribution. The wavefunction
$\psi_{\alpha}$ determines the classical probability distribution
$|\psi_{\alpha}|^{2}$.

  * •

The path integrals $\tilde{Z}_{n}$ compute expectation values of $\widehat{Z}$
in the No-Boundary state, which has a Gaussian wavefunction. The variance of
the corresponding probability distribution is given by the cylinder
contribution to the path integral.

  * •

Computing correlation functions of “observables,” that is $\widehat{Z}$s, in
more general states involves path integrals with two classes of circular
boundaries - boundaries associated with the observables, and boundaries
associated with the state of the closed universes.

####  3.3.1 The Hilbert space of closed universes: Z basis

To describe the Hilbert space of closed universes, we start by introducing the
“No-Boundary” state $|NB\rangle$.131313The No-Boundary state is sometimes
referred to as the Hartle-Hawking state. We can think of $|NB\rangle$ as the
state of no closed universes. There are complications in defining an
appropriate notion of closed universe number, since closed universes can
nucleate, annihilate, join and split. However, we will use a certain notion of
closed universe number appropriate in the CGS model and the “perturbative”
limit of the MM model, where we ignore joining and splitting effects. The No-
Boundary state is the zero closed universe state with this definition of
closed universe number.

We will discuss the rest of the Hilbert space in three ways. First we describe
an overcomplete basis of states which is closely related to the path integral
description of the model; the “Z basis”. Then we will discuss an orthonormal
basis which is appropriate for describing the CGS model and some limits of the
MM model, the “Number“ or “N” basis. Finally we describe the eigenbasis of the
observable $\widehat{Z}$, the “$\alpha$ basis”. This basis naturally connects
the physics of closed universes to ensemble averaging.

To obtain the Z basis, we act on the No-Boundary state $|NB\rangle$ with the
Hermitian operator $\widehat{Z}$ raised to a non-negative integer power $k$.
We may define the action of this operator, and the states
$\widehat{Z}^{k}|NB\rangle\equiv|Z^{k}\rangle$, by giving a formula for their
inner product, $\langle NB|\widehat{Z}^{k^{\prime}}\widehat{Z}^{k}|NB\rangle$.
The path integral gives a natural definition of these inner products.

$\langle
NB|\widehat{Z}^{k^{\prime}}\widehat{Z}^{k}|NB\rangle=\tilde{Z}_{k+k^{\prime}}\leavevmode\nobreak\
.$ (3.14)

Figure 8: Here we depict an example contribution to the overlap between states
$|Z^{k}\rangle$ and $|Z^{k^{\prime}}\rangle$.

With this definition, we can compute general overlaps of normalized states
$|\psi\rangle$, $|\psi^{\prime}\rangle$ and matrix elements of products of the
operator $\widehat{Z}$ in these states by expressing the states as
superpositions of the $\widehat{Z}^{k}|NB\rangle$ to write these quantities as
sums over path integrals $\tilde{Z}_{k}$,141414In the MM model,
nonperturbative effects in the topological expansion for $\tilde{Z}_{k}$
render this inner product positive semidefinite; the proper Hilbert space is
then obtained by modding out by null states [44]. As a consequence, the $Z$
basis expression for a state $|\psi\rangle$ is non-unique. Subtleties related
to null states will not be important in this section, but we will discuss them
briefly in Sections 4 and 5.

$\displaystyle\langle\psi^{\prime}|\widehat{Z}^{n}|\psi\rangle$
$\displaystyle=\bigg{(}\sum_{k^{\prime}}(\psi^{\prime}_{k^{\prime}})^{*}\langle
NB|\widehat{Z}^{k^{\prime}}\bigg{)}\widehat{Z}^{n}\bigg{(}\sum_{k}\psi_{k}\widehat{Z}^{k}|NB\rangle\bigg{)}$
(3.15)
$\displaystyle=\sum_{k,k^{\prime}}(\psi^{\prime}_{k^{\prime}})^{*}\psi_{k}\;\;\langle
NB|\widehat{Z}^{k^{\prime}+k+n}|NB\rangle$ (3.16)
$\displaystyle=\sum_{k,k^{\prime}}(\psi^{\prime}_{k^{\prime}})^{*}\psi_{k}\;\tilde{Z}_{k+k^{\prime}+n}.$
(3.17)

Figure 9: In representing the path integral computing these inner products
between normalized states, we draw diagrams like the above example. Normalized
states $|\psi\rangle$ and $\langle\psi^{\prime}|$ are denoted with boundaries
behind vertical purple lines, while operator insertions in correlation
functions are denoted by boundaries inserted between the bra and ket.

It is clear that in this inner product the operator $\widehat{Z}$ that is
Hermitian. We will often denote expectation values
$\langle\psi|\widehat{O}|\psi\rangle$ by $\langle\widehat{O}\rangle_{\psi}$ .

####  3.3.2 The Hilbert space of closed universes: N basis

To describe the next basis of interest, the Number or N basis, we begin by
discussing the state of a single closed universe.

In the CGS or MM models, the states of closed universes are rather simple.
First we consider a single closed universe.151515When we refer to a single
closed universe we are referring to a closed one dimensional manifold,
topologically a circle. There are no degrees of freedom characterizing a
single closed universe, so there is only one state associated to a single
closed universe. We can try to create a state with a single closed universe by
acting with the operator $\widehat{Z}$ on the No-Boundary state. However, we
can see that $\widehat{Z}|NB\rangle\equiv|Z\rangle$ might not be a good
definition of a single-universe state, as the overlap with the No-Boundary
state is nonzero. In the CGS model, the overlap between the normalized states
is161616Here, and in the remainder of this paper, $|NB\rangle$ represents the
normalized No-Boundary state.

$\frac{\langle NB|Z\rangle}{\sqrt{\langle
Z|Z\rangle}}=\frac{\text{Disk}}{\sqrt{\text{Disk}^{2}+\text{Cyl}}}=1-\mathcal{O}(e^{-2S_{0}}).$
(3.18)

The closed universe created by $\widehat{Z}$ has a large amplitude to
disappear via the disk diagram.

Figure 10: Here we have pictured the disk contribution to the overlap between
$|Z\rangle$ and the No-Boundary state. The purple lines denote that the bra
and ket in the figure are normalized.

The disk acts like a tadpole in field theory. In that context we subtract the
tadpole to define a field which creates a one-particle state.

Here, the one-closed universe state is naturally defined as

$|1\rangle\propto(\widehat{Z}-\text{Disk})|NB\rangle.$ (3.19)

Figure 11: We can represent the state $|Z\rangle$ as disk plus a single
universe state, as shown above. The disk describes the amplitude for the
closed universe created by $\widehat{Z}$ to annihilate, and thus gives the
overlap between $|Z\rangle$ and the No-Boundary state. The cylinder describes
the amplitude for the closed universe created by $\widehat{Z}$ to end up in
the state $|1\rangle$.

The operator $\widehat{Z}-\text{Disk}$ can create and destroy single closed
universes, so it is natural to express it as a sum of creation and
annihilation operators,

$\displaystyle\widehat{Z}-\text{Disk}$ $\displaystyle=a+a^{\dagger}$ (3.20)
$\displaystyle[a,a^{\dagger}]$ $\displaystyle=\text{Cyl}=1$ (3.21)
$\displaystyle a|NB\rangle$ $\displaystyle=0$ (3.22) $\displaystyle
a^{\dagger}|NB\rangle$ $\displaystyle=|1\rangle.$ (3.23)

In general, we define the (unnormalized!) $n$-(closed) universe state as
$|n\rangle=(a^{\dagger})^{n}|NB\rangle$. These span the Hilbert space of a
harmonic oscillator in the orthonormal N basis, and we see that
$\widehat{Z}-\text{Disk}$ is just the conventional position operator.

It is instructive to use creation and annihilation operators to describe the
state $\widehat{Z}^{2}|NB\rangle$. By normal ordering the operator
$\widehat{Z}^{2}$, we find

$\displaystyle\widehat{Z}^{2}|NB\rangle$
$\displaystyle=\big{(}\text{Disk}^{2}+2\;\text{Disk}(a+a^{\dagger})+(a+a^{\dagger})^{2}\big{)}|NB\rangle$
(3.24)
$\displaystyle=\big{(}\text{Disk}^{2}+2\;\text{Disk}(a+a^{\dagger})+a^{2}+(a^{\dagger})^{2}+2a^{\dagger}a+\text{Cyl}\big{)}|NB\rangle$
(3.25)
$\displaystyle=(\text{Disk}^{2}+\text{Cyl})|NB\rangle+2\;\text{Disk}|1\rangle+|2\rangle.$
(3.26)

Figure 12: Here we have depicted the state $|Z^{2}\rangle$ in the N basis. The
first two terms, with two disks and a cylinder, describe the component with
zero closed universes; the disks describe the amplitude for the closed
universes created by $\widehat{Z}$ to separately annihilate, and the cylinder
describes the amplitude for them to annihilate with each other. The remaining
terms describe the components with one or two universes, in which either one
or zero of the closed universes are annihilated via the disk spacetime.

Here we can see pictorially why the normal ordering is related to the cylinder
topology.

The N basis is particularly useful because states of different closed universe
number are orthogonal171717The normalization of these states is chosen to
simplify later formulas.

$\langle n|m\rangle=\text{Cyl}^{n}\;n!\;\delta_{nm}=n!\;\delta_{nm}.$ (3.27)

This is in contrast with the states $\widehat{Z}^{n}|NB\rangle$ or even
$(\widehat{Z}-\text{Disk})^{n}|NB\rangle$ which are highly non-orthogonal due
to tadpoles (the disk) and normal ordering (the cylinder).

In the CGS model, the number eigenstates form an orthogonal basis of the
Hilbert space, and the Hilbert space is identical to that of a harmonic
oscillator. However the MM model includes surfaces of arbitrary topology which
give nonzero overlaps between states of different universe number. For
example, the three holed sphere gives an overlap between the one-universe and
two-universe states. Furthermore, nonperturbative effects in the topological
expansion drastically change the structure of the Hilbert space – there are
now a large number of null states. Marolf and Maxfield demonstrated a dramatic
consequence of such nonperturbative effects. They change the spectrum of the
$\widehat{Z}$ operator from a continuous one – the values of the harmonic
oscillator position – to a discrete one in the full model.

In Section 4 we will return to the full MM model and address these issues. We
will find that while these effects play a role in the computations of
correlators in exact $\widehat{Z}$ eigenstates, there are many approximate
eigenstates for which they are unimportant. In these states, the CGS model
serves as a good approximation to the full MM model. The situation will be
similar in JT gravity, discussed in Section 5. There, a suitable
generalization of the CGS model serves as a good approximation in computations
in certain approximate eigenstates, for which spacetimes with topologies other
than the disk and cylinder can be ignored.

We now turn to a formal discussion of such eigenstates.

####  3.3.3 The Hilbert space of closed universes: $\alpha$ basis

We now introduce the $\alpha$ basis, the eigenbasis of the observable
$\widehat{Z}$. We label the eigenstates of $\widehat{Z}$ as $|\alpha\rangle$,
with real eigenvalues $Z_{\alpha}$,

$\widehat{Z}|\alpha\rangle=Z_{\alpha}|\alpha\rangle.$ (3.28)

$\widehat{Z}$ may or may not have a continuous spectrum, with delta function
normalized eigenstates: In the CGS model $\widehat{Z}$ has a continuous
spectrum; while in the MM model $\widehat{Z}$ has a discrete spectrum.

It is interesting to consider the expression for expectation values of powers
of $\widehat{Z}$ in a general state $|\psi\rangle$ when expressed in the
orthonormal $|\alpha\rangle$ basis.

$\displaystyle\langle\widehat{Z}^{n}\rangle_{\psi}\equiv\langle\psi|\widehat{Z}^{n}|\psi\rangle$
$\displaystyle=\sum_{\alpha,\beta}\psi^{*}_{\beta}\psi_{\alpha}\langle\beta|\widehat{Z}^{n}|\alpha\rangle$
(3.29)
$\displaystyle=\sum_{\alpha,\beta}\psi^{*}_{\beta}\psi_{\alpha}\langle\beta|\alpha\rangle
Z_{\alpha}^{n}$ (3.30)
$\displaystyle=\sum_{\alpha}|\psi_{\alpha}|^{2}Z_{\alpha}^{n}.$ (3.31)

Here we have assumed that the spectrum of $\widehat{Z}$ is discrete, though
the generalization to the case with a continuous spectrum is clear.

The last line in equation (3.29) allows us to describe quantum expectation
values as expectation values in a classical ensemble of systems with
probability distribution given by
$P^{(\psi)}(Z_{\alpha})=|\psi_{\alpha}|^{2}$.

We can summarize the relationship between the closed universe states and
ensemble averages as follows:181818The relationship between the physical
states $|\psi\rangle$ and distributions $P^{(\psi)}(Z_{\alpha})$ is not one-
to-one; states which differ only by the phases of their components
$\psi_{\alpha}$ map to the same distribution. In general, correlation
functions of physically relevant operators are invariant under
$|\psi\rangle\rightarrow e^{i\theta(\widehat{Z})}|\psi\rangle$.

Eigenvectors $|\alpha\rangle$ of $\widehat{Z}$
$\displaystyle\longleftrightarrow\;\text{Elements $\alpha$ of an ensemble of
dual boundary theories.}$ (3.32) States $|\psi\rangle$ of closed universes
$\displaystyle\longleftrightarrow\;\text{Probability distributions
$P^{(\psi)}(Z_{\alpha})=|\psi_{\alpha}|^{2}$ over the ensemble.}$ (3.33)
Eigenvalues $Z_{\alpha}$ of $\widehat{Z}$
$\displaystyle\longleftrightarrow\;\text{Partition function $Z_{\alpha}$ of
the member $\alpha$ of the ensemble.}$ (3.34) Correlation functions
$\langle\widehat{Z}^{n}\rangle_{\psi}$
$\displaystyle\longleftrightarrow\;\text{Ensemble averages $\langle
Z^{n}\rangle_{P}=\sum_{\alpha}P^{(\psi)}(Z_{\alpha})\;Z_{\alpha}^{n}$}.$
(3.35)

In particular, we can consider expectation values in the No-Boundary state
$|NB\rangle$. These describe a probability distribution

$P^{(NB)}(Z_{\alpha})=|\langle\alpha|NB\rangle|^{2}.$ (3.36)

In the CGS model, the spectrum of $\widehat{Z}$ is continuous, and we can just
label the eigenvalues by a continuous real parameter $z$. Then the probability
distribution corresponding to the No-Boundary state is Gaussian; $|NB\rangle$
is just the (shifted) harmonic oscillator ground state:

$P^{(NB)}(z)dz\propto\exp\bigg{[}-\frac{(z-\text{Disk})^{2}}{2\;\text{Cyl}}\bigg{]}dz.$
(3.37)

In more general theories, we expect that the No-Boundary state can often be
well approximated by a Gaussian state. This is true, for example, in the MM
model when $S_{0}$ is large, and is a good approximation for many observables
in JT gravity at large entropy. The variance of the Gaussian wavefunction is
described by spacetimes with the topology of a cylinder which connects two
asymptotic boundaries.

To connect to AdS/CFT we interpret $\langle\widehat{Z}\rangle_{NB}$ as the
partition function with one boundary computed using the Euclidean black hole
spacetime. In examples such as the MM and CGS models and JT gravity the
apparent lack of factorization of $\langle\widehat{Z}\widehat{Z}\rangle_{NB}$
is resolved by interpreting this expectation value as an ensemble average of
quantum mechanical boundary systems. Such ensemble averages need not factorize
since the averaging over parameters in the boundary Hamiltonian correlates the
two $Z$’s. The failure of factorization, measured by the connected correlator
$\langle\widehat{Z}^{2}\rangle_{NB,c}$, is the nonzero variance of the
partition function in this ensemble.

Of course this immediately suggests a way to calculate a product of partition
functions which does factorize; don’t compute in the No-Boundary state, but in
an $\alpha$ state $|\alpha\rangle$. The probability distribution for this
state has no variance, and correlation functions clearly factorize.

$\langle\alpha|\widehat{Z}\widehat{Z}|\alpha\rangle=Z_{\alpha}Z_{\alpha}.$
(3.38)

In examples like the MM model and JT gravity, there are many $\alpha$ states
$|\alpha\rangle$, and thus many members of an ensemble of boundary duals to
focus on. This mechanism for factorization seems incompatible with many
conventional examples of AdS/CFT, where there are no hints of an ensemble of
boundary duals, and thus no sign of the existence of multiple distinct
$\alpha$ states in the bulk dual [44, 56]. However, one might hope that by
studying this mechanism for factorization one might learn lessons that are
useful for understanding factorization in conventional AdS/CFT.

####  3.3.4 Reformulating correlation functions as overlaps

In the CGS and MM models, a general state $|\psi\rangle$ of the closed
universes can be expressed in terms of a function191919In the full MM model,
different choices of $\psi(\widehat{Z})$ may create states which may differ
only by a null state. $\psi(\widehat{Z})$ of the boundary operator
$\widehat{Z}$ acting on the No-Boundary state,

$|\psi\rangle=\psi(\widehat{Z})|NB\rangle.$ (3.39)

We will always choose $\psi(\widehat{Z})$ so that $|\psi\rangle$ is
normalized. Consider the $n$-point function

$\displaystyle\langle\psi|\widehat{Z}^{n}|\psi\rangle$ $\displaystyle=\langle
NB|\psi(\widehat{Z})^{*}\widehat{Z}^{n}\psi(\widehat{Z})|NB\rangle$ (3.40)
$\displaystyle=\langle
NB|\widehat{Z}^{n}\;\;|\psi(\widehat{Z})|^{2}|NB\rangle.$ (3.41)

Defining the (unnormalized) state
$|\psi^{2}\rangle\equiv|\psi(\widehat{Z})|^{2}|NB\rangle\rightarrow\psi(\widehat{Z})^{2}|NB\rangle$,202020Note
that the state $|\psi^{2}\rangle$, as well as correlation functions computed
with it, only depend on the modulus of the function $\psi(\widehat{Z})$. We
can then take $\psi(\widehat{Z})$ to be real without loss of generality. we
can view the $n$-point function as an overlap of two states

$\langle\widehat{Z}^{n}\rangle_{\psi}=\langle Z^{n}|\psi^{2}\rangle.$ (3.42)

Essentially, $|\psi^{2}\rangle$ combines the closed universes from the bra and
the ket into one state. This reformulation will play an important role in our
discussion of factorization.212121Had we considered a correlation function of
operators which do not commute with $\widehat{Z}$, we would not have been able
to express the correlation function as an overlap with $|\psi^{2}\rangle$.
However, there is no clear gravity interpretation of such correlation
functions so we will not consider them.,

Figure 13: In this figure, we show how to view an expectation value as an
overlap of two states. The operator insertion $\widehat{Z}^{n}$ in the
correlation function on the LHS is then viewed as creating a state
$|Z^{n}\rangle$ on the RHS. The red line on the RHS, associated with the ket,
denotes that the state is not normalized, but is instead a state
$|\psi^{2}\rangle$ for $|\psi\rangle$ normalized. Then the RHS depicts an
overlap between two unnormalized states.

Viewing correlation functions as overlaps with the state $|\psi^{2}\rangle$ is
particularly useful when states with different numbers of closed universes are
orthogonal, as is the case in the CGS model. The state $|Z^{n}\rangle$ is a
superposition of number eigenstates $|k\rangle$ with a maximum of $k=n$
universes. For example, in (3.24) we found that $|Z^{2}\rangle$ has components
with up to two universes.

Then, by inserting a complete set of states in the N-basis, the overlap
$\langle Z^{n}|\psi^{2}\rangle$ only involves the number basis components of
$|\psi^{2}\rangle$ with up to $n$ universes

$\langle Z^{n}|\psi^{2}\rangle=\sum_{k=0}^{n}\frac{1}{\text{Cyl}^{k}k!}\langle
Z^{n}|k\rangle\langle k|\psi^{2}\rangle.$ (3.43)

The factors of $1/\text{Cyl}$ come from the normalization of the resolution of
the identity. $\widehat{1}=\sum_{k=0}^{\infty}\frac{1}{\langle
k|k\rangle}|k\rangle\langle k|$, where $\langle k|k\rangle=\text{Cyl}^{k}k!$.
For the case $n=0$, we are computing the norm of $|\psi\rangle$, which is
equal to the overlap of $|\psi^{2}\rangle$ with the No-Boundary state. This is
equal to the norm of $|\psi\rangle$, which is set to one.

The one-point function of $\widehat{Z}$ involves only one nontrivial component
of $|\psi^{2}\rangle$, the one-universe component

$\langle\widehat{Z}\rangle_{\psi}=\langle Z|\psi^{2}\rangle=\langle
Z|NB\rangle+\frac{1}{\text{Cyl}}\langle Z|1\rangle\langle 1|\psi^{2}\rangle.$
(3.44)

Physically, the factor of $1/\text{Cyl}$ from the resolution of the identity
tells us that when we glue two cylinders together, we multiply their weights
and divide by Cyl so that normalized cylinders are glued together to make
normalized cylinders:
$\frac{1}{\text{Cyl}}\text{Cyl}\times\text{Cyl}=\text{Cyl}$.

Pictorially, we represent this equation in Figure 14.

Figure 14: Here we have depicted the computation of the one-point function
$\langle\widehat{Z}\rangle_{\psi}$, viewed as the overlap $\langle
Z|\psi^{2}\rangle$. On the LHS we see a boundary corresponding to the bra
$\langle Z|$, and a red line denoting the ket $|\psi^{2}\rangle$. In the first
term on the RHS of the first line, we picture the contribution where the
boundary for the state $\langle Z|$ is capped off with a disk. This
contribution also includes a sum over spacetimes ending on the
$|\psi^{2}\rangle$ boundaries, which are denoted by dashed red lines. The
entire contribution, which gives the overlap $\langle NB|\psi^{2}\rangle=1$,
is circled with a red dotted line. In the second term, we picture the
contribution for which the boundary for the state $\langle Z|$ connects to the
$|\psi^{2}\rangle$ boundaries. In the top line we have depicted this
contribution along the lines of (3.44), as a product of the amplitude $\langle
Z|1\rangle$ and the one-universe component $\langle 1|\psi^{2}\rangle$. The
one-universe component is computed by the spacetimes within the dotted red
line. In the second line we view this contribution more directly as an overlap
with the one-universe component of $|\psi^{2}\rangle$. The red circular line
denotes a boundary corresponding to the unnormalized state $\langle
1|\psi^{2}\rangle|1\rangle$, rather than the normalized state $|1\rangle$.

The full, many-universe state $|\psi^{2}\rangle$ is prepared by including any
number of $\widehat{Z}$ boundaries in the path integral. However, the one-
point function only involves the zero and one-universe components.

The overlap $\langle Z|NB\rangle$ is equal to the one-point function of
$\widehat{Z}$ in the No-Boundary state, given by the disk. We can think of
this as the contribution from nucleating a single closed universe from the
closed universe vacuum.

The overlap $\langle Z|1\rangle=1$ is the amplitude between a single closed
universe, in the normalized state $|1\rangle$, and the unnormalized state
$|Z\rangle$. In the computation of the one-point function, we must multiply
this amplitude by the one-universe component of $|\psi^{2}\rangle$.

The amplitude $\langle Z|1\rangle$ is computed by the cylinder topology, while
the one-universe component $\langle 1|\psi^{2}\rangle$ is computed by summing
over many disconnected spacetimes with boundaries created by the operator
$\psi(\widehat{Z})^{2}$. The product $\frac{1}{\text{Cyl}}\langle
Z|1\rangle\langle 1|\psi^{2}\rangle$ then represents the contribution to the
overlap $\langle Z|\psi^{2}\rangle$ from the $\widehat{Z}$ boundary absorbing
a single closed universe created by $\psi(\widehat{Z})^{2}$. Pictorially, we
represent this with the cylinder in Figure 14, with one boundary corresponding
to the state $|Z\rangle$, and the other red boundary corresponding to the one-
universe component of $|\psi^{2}\rangle$.

Figure 15: Here we have pictured an example contribution to the one-universe
component of $|\psi^{2}\rangle$.

In the CGS model, there are no contributions to the overlap $\langle
Z|\psi^{2}\rangle$ in which the $\widehat{Z}$ boundary absorbs two or more
closed universes created by $\psi(\widehat{Z})^{2}$. These contributions would
involve spacetimes with connected components with three or more boundaries,
such as the three-holed sphere, and these spacetimes are not present in the
CGS model.

The two-point function of $\widehat{Z}$ involves only one additional component
of the many-universe state $|\psi^{2}\rangle$, the two-universe component.

$\langle\widehat{Z}^{2}\rangle_{\psi}=\langle Z^{2}|\psi^{2}\rangle=\langle
Z^{2}|NB\rangle+\frac{1}{\text{Cyl}}\langle Z^{2}|1\rangle\langle
1|\psi^{2}\rangle+\frac{1}{2\;\text{Cyl}^{2}}\langle Z^{2}|2\rangle\langle
2|\psi^{2}\rangle.$ (3.45)

Pictorially, we represent this in Figure 16

Figure 16: Here we have pictured the contributions to the RHS of (3.45). The
pair of disks and the cylinder correspond to $\langle Z^{2}|NB\rangle$, the
disk and cylinder with a red circular boundary corresponds to
$\frac{1}{\text{Cyl}}\langle Z^{2}\rangle|1\rangle\langle 1|\psi^{2}\rangle$,
and the cylinders with linked red boundaries corresponds to
$\frac{1}{2\;\text{Cyl}^{2}}\langle Z^{2}|2\rangle\langle 2|\psi^{2}\rangle$.
The linked boundaries represent the two-universe component $\langle
2|\psi^{2}\rangle$; this is distinct from a pair of unlinked red circular
boundaries, which would denote two copies of the one-universe component
$\langle 1|\psi^{2}\rangle$.

Some of the contributions to the two-point function are simply products of
contributions to the one-point function. However, the two-point function
involves the cylinder connecting the two $\widehat{Z}$ boundaries (included in
the overlap $\langle Z^{2}|NB\rangle$), as well as the amplitude
$\frac{1}{2\;\text{Cyl}^{2}}\langle Z^{2}|2\rangle=1$ for the two
$\widehat{Z}$ boundaries to each absorb two closed universes in a normalized
state, multiplied by the two-universe component $\langle 2|\psi^{2}\rangle$.
In Figure 16, we have represented this second contribution with two cylinders
with linked red boundaries. The pair of linked red boundaries represents the
two-universe state $\langle 2|\psi^{2}\rangle$. Here the link differentiates
this state from two copies of the one-universe state $\langle
1|\psi^{2}\rangle$. As with the one-universe component, this two-universe
component is computed by summing over many disconnected spacetimes with
boundaries created by the operator $\psi(\widehat{Z})^{2}$.

###  3.4 Factorization in the CGS model

In the special case that $|\psi\rangle$ is an eigenstate of $\widehat{Z}$ (an
$\alpha$ state) with eigenvalue $z$, correlation functions should factorize.
In particular, the two-point function should be equal to the square of the
one-point function,

$\langle\widehat{Z}^{2}\rangle_{z}=\langle\widehat{Z}\rangle_{z}^{2}=z^{2}.$
(3.46)

The subscript $z$ denotes the $\alpha$ state
$|z\rangle\equiv|\psi_{z}\rangle$.

When viewing the correlation functions as overlaps of the form $\langle
Z^{n}|\psi^{2}_{z}\rangle$, factorization implies a particularly simple
relationship between the components $\langle k|\psi^{2}_{z}\rangle$ with
different universe number. This simple relationship will be the basis of an
effective description we describe in Section 3.6.

Terms in the two-point function only involving the disk topology manifestly
factorize into contributions from the square of the one-point function. So to
isolate the nontrivial contributions we consider correlation functions of
$\widehat{Z}-\text{Disk}$ and compare our expressions for

$\langle\widehat{Z}-\text{Disk}\rangle_{z}^{2}=\langle
1|\psi_{z}^{2}\rangle^{2}.$ (3.47)

and

$\langle(\widehat{Z}-\text{Disk}))^{2}\rangle_{z}=\text{Cyl}+\langle
2|\psi_{z}^{2}\rangle.$ (3.48)

in terms of the components of $|\psi^{2}\rangle$, using equations (3.44) and
(3.45) with $|\psi\rangle=|\psi_{z}\rangle$. The fact that
$\langle\widehat{Z}\rangle_{z}=z$ allows us to identify $\langle
1|\psi_{z}^{2}\rangle=z-\text{Disk}$. Since $z$ is Gaussian distributed in the
No-Boundary state/ensemble, the one-universe component $\langle
1|\psi_{z}^{2}\rangle$ is as well (but here with zero mean). Denoting
averaging $z$ over the No-Boundary ensemble with $\mathbb{E}[\dots]_{NB}$,

$\mathbb{E}\big{[}\langle 1|\psi_{z}^{2}\rangle\big{]}_{NB}=0,\hskip
20.0pt\mathbb{E}\big{[}\big{(}\langle
1|\psi_{z}^{2}\rangle\big{)}^{2}\big{]}_{NB}=\text{Cyl}=1.$ (3.49)

Equation (3.47) tells us that the one-universe component $\langle
1|\psi^{2}_{z}\rangle$ contains all the information about the $\alpha$ state.
We can see this indirectly by using the fact that all other correlation
functions in the $\alpha$ state must factorize into products of this one-point
function, which is determined by $\langle 1|\psi^{2}_{z}\rangle$, or more
directly by noting that the identification $\langle
1|\psi_{z}^{2}\rangle=z-\text{Disk}$ fixes the $\alpha$ state $|z\rangle$ in
terms of $\langle 1|\psi^{2}_{z}\rangle$. We can then view $\langle
1|\psi_{z}^{2}\rangle$ as the parameter describing the set of eigenstates, or
equivalently the random variable describing the ensemble of boundary theories;
rather than describing the theory in an $\alpha$ state in terms of the many-
universe $\alpha$ state $|\psi_{z}\rangle$, we can simply use the random one-
universe wavefunction $\langle 1|\psi^{2}_{z}\rangle$. This reduction in the
number of universes/boundaries, as well as the simple Gaussian statistics of
this one-universe component, are the key simplifications in our description of
factorization.

As the one-universe component $\langle 1|\psi^{2}_{z}\rangle$ contains all the
information about the $\alpha$ state, the remaining components must be
redundant variables. Using (3.43) to generalize (3.48), an $n$-point function
can be expressed in terms of the components $\langle k|\psi^{2}_{z}\rangle$
with $k\leq n$. On the other hand, we can write the $k$-point function as a
product of one-point functions, which only depend on the one-universe
component. Factorization of the $k$-point function then implies identities
that relate the $k$-universe components to the one-universe component.

For example, the equality between the square of the one-point function (3.47)
and the two-point function (3.48) implies that the two-universe component is
equal to the square of the one-universe component, minus the cylinder,

$\langle 2|\psi_{z}^{2}\rangle=\langle 1|\psi_{z}^{2}\rangle^{2}-\text{Cyl}.$
(3.50)

Figure 17: The two-universe wavefunction $\langle 2|\psi^{2}_{z}\rangle$ is
related to the one-universe wavefunction $\langle 1|\psi^{2}_{z}\rangle$
through this identity. This identity ensures that the two-point function in an
$\alpha$ state factorizes.

We briefly note that using (3.49), we see that the two-universe component
averages to zero, $\mathbb{E}\big{[}\langle 2|\psi^{2}_{z}\rangle\big{]}=0$.

Now we take a moment to compare this computation to the computation of the
spectral form factor in periodic orbit theory (2.2). The rough analogy we will
make is to think of the one-universe component $\langle 1|\psi_{z}^{2}\rangle$
as the sum over orbits describing $Z(iT)$ or $Z(-iT)$, and the cylinder as the
diagonal part of the double-sum over orbits. This second comparison is
motivated by the fact that the cylinder and diagonal sum both describe the
ensemble average.

The two-universe component $\langle 2|\psi_{z}^{2}\rangle$ then plays the role
of the off-diagonal part of the double-sum over orbits; this is what we need
to add to the cylinder (analogous to the diagonal sum) that computes the
averaged answer in order to find an answer that factorizes.

Of course, this analogy is very coarse, especially since in this simple model
the one-universe and two-universe components of $|\psi_{z}^{2}\rangle$ are not
given by any natural sum. However, in more complicated models with a richer
closed universe Hilbert space, such as the models we consider in the remainder
of this paper, this analogy will be more appropriate.

So far we have just considered the factorization of the two-point function
into the square of the one-point function. As these correlation functions
depend only on the one and two-universe components of $|\psi^{2}_{z}\rangle$,
we found no constraints on the higher universe number components. To find the
analogous relationship between the $n$-universe component $\langle
n|\psi^{2}_{z}\rangle$ and the one-universe component, we demand that the
$n$-point function factorize. This tells us that that the $n$-universe
component is equal to the $n$th power of the one-universe component, minus
corrections from replacing pairs of the one-universe component with the
cylinder. For example, in the case $n=3$,

$\langle 3|\psi_{z}^{2}\rangle=\langle
1|\psi_{z}^{2}\rangle^{3}-3\;\text{Cyl}\;\langle 1|\psi_{z}^{2}\rangle.$
(3.51)

Figure 18: The three-universe wavefunction $\langle 3|\psi^{2}_{z}\rangle$,
depicted by the three linked circles, is related the the one-universe
wavefunction $\langle 1|\psi^{2}_{z}\rangle$ through this identity, which we
may derive by demanding that the three-point function in an $\alpha$ state
factorizes. Similar identities describing the $n$-universe components of
$|\psi^{2}_{z}\rangle$ can be derived by demanding that the $n$-point function
factorizes.

Through these relations we can see that the full many-universe $\alpha$ state
can be simply described in terms of a one-universe state, which has simple
Gaussian statistics. Though we can think of this wavefunction as computed by a
sum involving spacetimes with many boundaries and many connected components,
along the lines of Figure 15, we may also instead think of independently
defining this wavefunction through its statistics (3.49), without recourse to
the many-universe description, and defining the remaining components of
$|\psi^{2}_{z}\rangle$ through these recursion relations. We will exploit this
point of view later, in Section 3.6.

####  3.4.1 Explicit calculation

It is useful to check that the relations (3.50) and (3.51) for
$|\psi_{z}^{2}\rangle$ are obeyed using the explicit expression for the
eigenstate $|z\rangle$,

$|z\rangle\propto\delta(\widehat{Z}-z)|NB\rangle.$ (3.52)

We then identify $\psi_{z}(\widehat{Z})\propto\delta(\widehat{Z}-z)$.

This expression is difficult to work with directly; instead we express the
delta function as an integral,

$\psi_{z}(\widehat{Z})\propto\int_{-\infty}^{\infty}dp\;e^{ip(\widehat{Z}-z)}.$
(3.53)

This expression leads to an expression for the eigenstate $|z\rangle$ in the
“spacetime D-brane” basis introduced in [44]. The state
$|p\rangle=e^{ip\widehat{Z}}|NB\rangle$ is thought of as a spacetime D-brane
with a parameter $p$, since the computation of correlation functions in these
states mirrors Polchinski’s perturbative computation of string theory
amplitudes in the presence of a D-brane [57].222222These are close analogs of
the D-branes considered in [6] which are also created by the exponential of a
closed universe operator. In that case the operator being exponentiated is the
Laplace transform of the partition function. In the closely related minimal
string theory, versions of these states describe D-branes of FZZT type [58,
59]. In the CGS model, we can view the brane state $|p\rangle$ as a coherent
state of closed universes. Using the expression
$\widehat{Z}=\text{Disk}+a+a^{\dagger}$ and normal ordering the exponential
$e^{ip\widehat{Z}}$ using the BCH formula we find

$e^{ip\widehat{Z}}|NB\rangle=e^{ip\text{Disk}}e^{-\text{Cyl}\frac{p^{2}}{2}}e^{ipa^{\dagger}}|NB\rangle.$
(3.54)

In order to verify that the $n$-universe components of $|\psi_{z}^{2}\rangle$
for $|\psi\rangle=|z\rangle$ obey the relations (3.50) and (3.51), we must
normal order the operator
$\psi_{z}(\widehat{Z})^{2}\propto\delta(\widehat{Z}-z)^{2}$. To do this, it is
helpful to regulate the delta function integral by adding a small Gaussian
convergence factor,

$\psi_{z,\Delta}(\widehat{Z})\propto\int_{-\infty}^{\infty}dp\;e^{-\Delta^{2}p^{2}+ip(\widehat{Z}-z)}.$
(3.55)

This operator creates a sharp Gaussian state $|z,\Delta\rangle$, peaked around
$z$ with a small width proportional to $\Delta$.

$|z\rangle_{\Delta}\propto\int_{-\infty}^{\infty}dp\;e^{-\Delta^{2}p^{2}+ip(\widehat{Z}-z)}|NB\rangle\propto
e^{-\frac{(\widehat{Z}-z)}{4\Delta^{2}}}|NB\rangle.$ (3.56)

We compute $\psi_{z,\Delta}(\widehat{Z})^{2}$ by expressing the product of two
gaussian integrals over $p$ and $p^{\prime}$ as the product of two gaussian
integrals in the sum and difference variables $p_{+}=p+p^{\prime}$ and
$p_{-}=p-p^{\prime}$.

$\psi_{z,\Delta}(\widehat{Z})^{2}\propto\int_{-\infty}^{\infty}dpdp^{\prime}e^{-\Delta^{2}(p^{2}+p^{\prime
2})+i(p+p^{\prime})(\widehat{Z}-z)}=\bigg{[}\int_{-\infty}^{\infty}dp_{-}e^{-\frac{\Delta^{2}p_{-}^{2}}{2}}\bigg{]}\int_{-\infty}^{\infty}dp_{+}e^{-\frac{\Delta^{2}p_{+}^{2}}{2}+ip_{+}(\widehat{Z}-z)}.$
(3.57)

The integral over $p_{-}$ doesn’t depend on $\widehat{Z}$, it simply
contributes a normalization factor. $\psi_{z,\Delta}(\widehat{Z})^{2}$ is also
a superposition of exponential brane operators $e^{ip_{+}\widehat{Z}}$, so we
can simply normal order using the BCH formula. Normal ordering introduces a
Gaussian term $e^{-\text{Cyl}\frac{p_{+}^{2}}{2}}$ so we can safely take the
limit $\Delta\rightarrow 0$ to find

$|\psi_{z}^{2}\rangle\propto\int_{-\infty}^{\infty}dp_{+}\;e^{-\text{Cyl}\frac{p_{+}^{2}}{2}+ip+(\text{Disk}-z)}e^{ip_{+}a^{\dagger}}|NB\rangle.$
(3.58)

Expanding the exponential $e^{ip_{+}a^{\dagger}}$, we find

$|\psi_{z}^{2}\rangle\propto\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}dp_{+}\;e^{-\text{Cyl}\frac{p_{+}^{2}}{2}+ip_{+}(\text{Disk}-z)}\frac{(ip_{+})^{k}}{k!}|k\rangle.$
(3.59)

The constant of proportionality is fixed by the condition that $\langle
NB|\psi_{z}^{2}\rangle=1$.

To take the $n$-universe component we use the overlap $\langle
n|k\rangle=\text{Cyl}^{n}n!\delta_{nk}$. Then the $n$-universe component is
given by the $n$’th moment of $ip_{+}\text{Cyl}$ in this Gaussian. We can
rescale our variables $p_{+}\rightarrow p_{+}^{\prime}=p_{+}\text{Cyl}$ so
that the Gaussian has a center $ip_{+}^{\prime}=z-\text{Disk}$ and variance
Cyl.

The one-universe component is given by the center of the Gaussian.

$\langle 1|\psi_{z}^{2}\rangle=z-\text{Disk},$ (3.60)

and the two-universe component is proportional to the two-point function of
$ip_{+}^{\prime}$,

$\displaystyle\langle 2|\psi_{z}^{2}\rangle$
$\displaystyle=(z-\text{Disk})^{2}-\text{Cyl}$ (3.61) $\displaystyle=\langle
1|\psi_{z}^{2}\rangle^{2}-\text{Cyl}.$ (3.62)

The general pattern describing the $n$-universe components is precisely
described by the Gaussian integral (3.59), with Wick contractions between the
$n$ factors of $p_{+}^{\prime}$ contributing the cylinder corrections in
(3.50) and (3.51).

As an aside, we will mention that the use of the delta function (or
regularized delta function) to describe the eigenstate $|z\rangle$ is related
to how one would naturally compute correlation functions from the ensemble-
average point of view discussed in Section 3.3.3. In this point of view, we
think about correlation functions in a state $|\psi\rangle$ as ensemble
averages over the eigenvalue $z^{\prime}$ of $\widehat{Z}$ with a distribution
$P^{(\psi)}(z^{\prime})$

$\langle\widehat{Z}^{n}\rangle_{\psi}=\int
dz^{\prime}\;P^{(\psi)}(z^{\prime})\;z^{\prime n}.$ (3.63)

Choosing $|\psi\rangle$ to be an eigenstate means that we choose
$P^{(\psi)}(z^{\prime})$ to focus on a single element $z^{\prime}=z$ of the
ensemble; in other words we choose
$P^{(\psi)}(z^{\prime})=\delta(z-z^{\prime})$. If we multiply and divide this
distribution by the No-Boundary distribution, then correlation functions in
the eigenstate $|z\rangle$ can be described as correlation functions in the
No-Boundary state with an operator proportional to $\delta(\widehat{Z}-z)$
inserted. Comparing with our prescription for computing a correlation function
in the state $|\psi\rangle$ by inserting the operator $\psi(\widehat{Z})^{2}$
in the No-Boundary state, we can identify
$\psi_{z}(\widehat{Z})^{2}\propto\delta(\widehat{Z}-z)$. To compare this with
our earlier identification of
$\psi_{z}(\widehat{Z})\propto\delta(\widehat{Z}-z)$, it is useful to use the
regulated integral expression for the delta function; the square of the delta
function is itself a delta function, up to an (infinite) normalization.

###  3.5 The CGS model with “species”

To summarize the results of the previous section: in the CGS model, $n$-point
correlation functions in a state $|\psi\rangle$ are simple functions of the
$k$-universe components of the state $|\psi^{2}\rangle$ with $k\leq n$. The
contributions of the $k$-universe components describe the absorption of $k$
closed universes by the $n$ $\widehat{Z}$ operators in the correlation
function. In the case that $|\psi\rangle$ is an eigenstate $|z\rangle$, an
$n$-point correlation function factorizes into the $n$’th power of the one-
point function; after subtracting off the disks, this is simply the $n$’th
power of the one-universe component $\langle 1|\psi^{2}\rangle$. In order for
this to happen, the various contributions to the $n$-point function from
cylinders and $k$-universe components $\langle k|\psi^{2}\rangle$ with $k\geq
2$ must sum up to powers of the one-universe component $\langle
1|\psi^{2}\rangle$. The basic way in which this happens is illustrated by the
two-point function; the two-universe component plus the cylinder is equal to
the square of the one-universe component. This is roughly analogous to the
periodic orbit story from Section 2, with the cylinder playing the role of a
diagonal sum over orbits, and the two-universe component playing the role of
the off-diagonal sum.

The CGS model is very simple; there is only a single one-universe state,
$|1\rangle$, created by the operator $\widehat{Z}$. This model will serve as
an approximation to the MM model, as we will demonstrate in Section 4.
However, in order to approximate more complicated theories of gravity we must
generalize the CGS model so that a single closed universe can have many
possible states. The appropriate generalizations of the relations (3.50) and
(3.51) describing the components of $|\psi^{2}\rangle$ for $|\psi\rangle$ an
eigenstate sharpens the analogy with the periodic orbit story from Section 2.
Because this generalization has many internal states of a single closed
universe, we refer to this model as the “CGS model with species”.

To motivate this generalization, we begin by discussing some more general
aspects of the quantum mechanics of closed universes with a theory with a
negative cosmological constant, following [44]. To construct the Hilbert space
of closed universes, we mirror the procedure used in Section 3.3, with the
addition of a larger set of closed universe operators
$\widehat{Z}\rightarrow\widehat{Z}[\mathcal{J}]$. $\widehat{Z}[\mathcal{J}]$
is an operator which creates an asymptotically AdS boundary in the path
integral (which we can think of as periodic in complexified time), with
boundary conditions labeled by $\mathcal{J}$.232323It is not necessary that
these operators create states on asymptotic boundaries. Later in Section 5 we
will work with states of finite size closed universes.,242424In three or more
dimensions, we could also consider operators which create closed universes
with more general spatial topologies. $\mathcal{J}$ may represent a fixed
metric, as in the special case of the partition function
$\widehat{Z}[\mathcal{J}]\rightarrow\widehat{Z}(\beta)$. More generally
$\mathcal{J}$ denotes the boundary conditions of bulk fields $\phi$ (including
the metric) at an asymptotically AdS boundary.

Following the procedure reviewed in Section 3.3, we define the operators
$\widehat{Z}[\mathcal{J}]$ by their matrix elements in the No-Boundary state,
and we define these matrix elements via the gravity path integral with the
corresponding boundary conditions:

$\langle
NB|\widehat{Z}[\mathcal{J}_{1}]\dots\widehat{Z}[\mathcal{J}_{n}]|NB\rangle=\int_{\phi\rightarrow\mathcal{J}}D\phi\;e^{-I[\phi]}.$
(3.64)

We can then roughly define the Hilbert space as the span of states created by
acting on the No-Boundary state with the operators
$\widehat{Z}[\mathcal{J}]$.252525See [44] for a more precise and detailed
discussion.

This definition of the matrix elements makes clear that the
$\widehat{Z}[\mathcal{J}]$ commute. Hermitian conjugation of the
$\widehat{Z}[\mathcal{J}]$ is defined by
$\widehat{Z}[\mathcal{J}]^{\dagger}=\widehat{Z}[\mathcal{J}^{*}]$, with
$\mathcal{J}^{*}$ the CPT conjugation of the boundary conditions
$\mathcal{J}$. For simplicity, in the remainder of this section we will
restrict our attention to Hermitian operators.262626The operators
$\widehat{Z}(\beta+iT)$, and related fixed-energy operators we will consider
in later sections are not Hermitian, but we can take their real and imaginary
parts to form Hermitian operators. Then the formulas from this section can be
applied directly.

In this more general setting, $\alpha$ states now diagonalize all of the
$\widehat{Z}[\mathcal{J}]$ for different sources $\mathcal{J}$,

$\widehat{Z}[\mathcal{J}]|\alpha\rangle=Z_{\alpha}[\mathcal{J}]|\alpha\rangle.$
(3.65)

Expectation values of products of operators $\widehat{Z}(\beta)$ in a state
$|\psi\rangle$ are then equal to ensemble averages of products of partition
functions with a distribution described by the components of $|\psi\rangle$,
$P^{(\psi)}_{\alpha}=|\langle\alpha|\psi\rangle|^{2}$,

$\langle\widehat{Z}[\mathcal{J}_{1}]\dots\widehat{Z}[\mathcal{J}_{n}]\rangle_{\psi}=\sum_{\alpha}P^{(\psi)}_{\alpha}\;Z_{\alpha}[\mathcal{J}_{1}]\dots
Z_{\alpha}[\mathcal{J}_{1}].$ (3.66)

In the case that we take $\widehat{Z}[\mathcal{J}]$ to be a partition function
$\widehat{Z}(\beta)$, the eigenvalues $Z_{\alpha}(\beta)$ describe the
partition function of an element of a boundary ensemble. As will be the case
when we study JT gravity in Section 5, we can then think of elements of the
boundary ensemble as being labeled by the set of energy eigenvalues of the
boundary Hamiltonian.

####  3.5.1 Introduction to the CGS model with “species”

We now define a generalization of the CGS model as a theory in which a set of
boundary operators $\widehat{Z}_{I}$, which can be thought of as linear
superpositions of operators $\widehat{Z}[\mathcal{J}]$, are independent
Gaussian variables in the No-Boundary state.

$\displaystyle\langle\widehat{Z}_{I}\rangle_{NB}$
$\displaystyle=\text{Disk}_{I}$ (3.67)
$\displaystyle\langle\widehat{Z}_{I}\widehat{Z}_{J}\rangle_{NB,c}$
$\displaystyle=\text{Cyl}\;\delta_{IJ}=\delta_{IJ}$ (3.68)
$\displaystyle\langle\widehat{Z}_{I_{1}}\dots\widehat{Z}_{I_{n}}\rangle_{NB,c}$
$\displaystyle=0,\hskip 20.0ptk>2.$ (3.69)

In the remainder of this section, we will not explicitly write factors of Cyl,
which is set equal to one here.

In general the operators $\widehat{Z}[\mathcal{J}]$ may have a more
complicated inner product,
$\langle\widehat{Z}[\mathcal{J}]\widehat{Z}[\mathcal{J}^{\prime}]\rangle_{NB,c}=\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime})$,
which is computed by the path integral with two boundaries. For example,
partition function operators $\widehat{Z}(\beta)$ in JT gravity have a two-
point function which is not diagonal in $\beta$. In that case we can simply
diagonalize the correlation matrix
$\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime})$ and rescale the operators to
find independent, unit-variance variables $\widehat{Z}_{I}$. For simplicity we
will assume that there are finitely many independent Gaussian variables
$\widehat{Z}_{I}$.

Figure 19: The connected two-point function
$\langle\widehat{Z}[\mathcal{J}]\widehat{Z}[\mathcal{J}^{\prime}]\rangle_{NB,c}=\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime})$
is computed by the path integral over cylindrical spacetimes with
$\mathcal{J}$ boundary conditions on one boundary and $\mathcal{J}^{\prime}$
boundary conditions on the other.

In our applications we will use this version of the CGS model to approximate
JT gravity. In that case the natural operators $\widehat{Z}[\mathcal{J}]$ are
analytically continued partition functions $\widehat{Z}(\beta+iT)$ (or their
fixed energy counterparts $\widehat{Y}_{E}(T)$, to be defined in Section 5).
However, it is possible that this version of the CGS model may be relevant for
describing higher dimensional theories of gravity as well, in cases where
higher dimensional spacetime wormholes give approximately Gaussian correlators
of operators $\widehat{Z}[\mathcal{J}]$ in the No-Boundary state.272727For
example, in higher dimensional theories of gravity, the double cone [5], when
it dominates, describes Gaussian correlation functions of
$\widehat{Y}_{E}(T)$. Other calculations in higher dimensions include [25, 24,
32, 47].

####  3.5.2 $n$-universe states in the CGS model with species

The Gaussian correlators (3.67) prompt us to think of the operators
$\widehat{Z}_{I}$ as the shifted position operators of many independent
harmonic oscillators. The $\widehat{Z}_{I}$ can be expressed in terms of
creation and annihilation operators

$\widehat{Z}_{I}=\text{Disk}_{I}+a^{\dagger}_{I}+a_{I}.$ (3.70)

The $a^{\dagger}_{I}$ and $a_{I}$ satisfy

$\displaystyle[a_{I},a^{\dagger}_{J}]$ $\displaystyle=\delta_{IJ}$ (3.71)
$\displaystyle a_{I}|NB\rangle$ $\displaystyle=0.$ (3.72)

The $a^{\dagger}_{I}$ then create orthonormal one-universe states,

$|I\rangle\equiv a^{\dagger}_{I}|NB\rangle,\hskip 20.0pt\langle
I|J\rangle=\delta_{IJ},$ (3.73)

which form a basis of the one-universe Hilbert space. Note that

$|Z_{I}\rangle\equiv\widehat{Z}_{I}|NB\rangle=\text{Disk}_{I}|NB\rangle+|I\rangle.$
(3.74)

The full closed universe Hilbert space is spanned by $n$-universe states
labeled by the indices $I_{1}\dots I_{n}$ up to reordering,

$|\\{I_{1}\dots I_{n}\\}\rangle=a^{\dagger}_{I_{1}}\dots
a^{\dagger}_{I_{n}}|NB\rangle.$ (3.75)

With a slight abuse of terminology, we will also refer to this basis as the “N
basis”. The states $|\\{I_{1}\dots I_{n}\\}\rangle$ with different sets of
species $\\{I_{1}\dots I_{n}\\}$ are orthogonal,

$\langle\\{J_{1}\dots J_{m}\\}|\\{I_{1}\dots I_{n}\\}\rangle=C_{\\{I_{1}\dots
I_{n}\\}}\delta_{\\{J_{1}\dots J_{m}\\},\\{I_{1}\dots I_{n}\\}},$ (3.76)

where $C_{\\{I_{1}\dots I_{n}\\}}$ can be determined from normal ordering, and
is equal to a product of factorials of the multiplicites of each species
appearing in the set $\\{I_{1}\dots I_{n}\\}$.

A general state $|\psi\rangle$ can be described by its components
$\psi_{\\{I_{1}\dots I_{n}\\}}$,

$|\psi\rangle=\psi_{0}|NB\rangle+\sum_{n=1}^{\infty}\sum_{\\{I_{1}\dots
I_{n}\\}}\frac{\psi_{\\{I_{1}\dots I_{n}\\}}}{C_{\\{I_{1}\dots
I_{n}\\}}}|\\{I_{1}\dots I_{n}\\}\rangle.$ (3.77)

The components $\psi_{\\{I_{1}\dots I_{n}\\}}=\langle\\{I_{1}\dots
I_{n}\\}|\psi\rangle$ can be thought of as a symmetric tensor.

####  3.5.3 Correlation functions

Now we will discuss the computations of the correlators
$\langle\widehat{Z}_{I_{1}}\dots\widehat{Z}_{I_{n}}\rangle_{\psi}$ in a
general state $|\psi\rangle$ before focusing on the case that $|\psi\rangle$
is a joint eigenstate of all of the $\widehat{Z}_{I}$.

We again approach this by viewing correlation functions in a state
$|\psi\rangle=\psi(\widehat{Z}_{I})|NB\rangle$ as an overlap between the state
$|\psi^{2}\rangle=\psi(\widehat{Z}_{I})^{2}|NB\rangle$282828As in the single-
species version of the CGS model, we can take $\psi(\widehat{Z}_{I})$ to be
real. and a state $|Z_{I_{1}}\dots
Z_{I_{n}}\rangle=\widehat{Z}_{I_{1}}\dots\widehat{Z}_{I_{n}}|NB\rangle$
created by the operators inserted in the correlation function. We then insert
a complete set of states in the basis $|\\{I_{1}\dots I_{k}\\}\rangle$. The
simplification that we take advantage of is that the states $|Z_{I_{1}}\dots
Z_{I_{n}}\rangle$ contain only up to $n$ universes. Denoting the projector
onto the $k$-universe subspace as

$\widehat{P}_{k}=\sum_{\\{J_{1}\dots J_{k}\\}}\frac{1}{C_{\\{J_{1}\dots
J_{k}\\}}}|\\{J_{1}\dots J_{k}\\}\rangle\langle\\{J_{1}\dots J_{k}\\}|,$
(3.78)

we may express a correlation function as a sum over overlaps in the N basis,

$\displaystyle\langle Z_{I_{1}}\dots$ $\displaystyle
Z_{I_{n}}|\psi^{2}\rangle=\langle\widehat{Z}_{I_{1}}\dots\widehat{Z}_{I_{n}}|NB\rangle+\sum_{k=1}^{\infty}\langle
Z_{I_{1}}\dots Z_{I_{n}}|\widehat{P}_{k}|\psi^{2}\rangle$ (3.79)
$\displaystyle=\langle Z_{I_{1}}\dots
Z_{I_{n}}|NB\rangle+\sum_{k=1}^{n}\sum_{\\{J_{1}\dots J_{k}\\}}\frac{\langle
Z_{I_{1}}\dots Z_{I_{n}}|\\{J_{1}\dots J_{k}\\}\rangle}{C_{\\{J_{1}\dots
J_{k}\\}}}\langle\\{J_{1}\dots J_{k}\\}|\psi^{2}\rangle.$ (3.80)

Here we used the fact that if $|\psi\rangle$ is normalized, $\langle
NB|\psi^{2}\rangle=1$. $\langle Z_{I_{1}}\dots Z_{I_{n}}|NB\rangle$ is equal
to the correlation function in the No-Boundary state, and is a sum of products
of $\text{Disk}_{I}$ and the cylinder $\delta_{IJ}$.

Now we focus on the one and two-point functions. Using (3.74), the one-point
function is simply equal to the disk plus the cylinder, with one boundary of
the cylinder in the state $\widehat{P}_{1}|\psi^{2}\rangle$, where
$\widehat{P}_{1}$ is the projector onto the one-universe sector of the Hilbert
space. Using the orthonormality of the states $|I\rangle,\;|J\rangle$, this
cylinder is just equal to the component $\langle I|\psi^{2}\rangle$,

$\displaystyle\langle Z_{I}\rangle_{\psi}$
$\displaystyle=\text{Disk}_{I}+\sum_{J}\langle Z_{I}|J\rangle\langle
J|\psi^{2}\rangle$ (3.81) $\displaystyle=\text{Disk}_{I}+\langle
I|\psi^{2}\rangle.$ (3.82)

Subtracting off the disks for simplicity, as in (3.74), the two-point function
of operators $\widehat{Z}_{I}$ and $\widehat{Z}_{J}$ is a sum of the cylinder
between the two operators, which is equal to $\delta_{IJ}$, and a pair of
cylinders with two boundaries in the two-universe state
$\widehat{P}_{2}|\psi^{2}\rangle$,

$\displaystyle\langle(\widehat{Z}_{I}-\text{Disk}_{I})(\widehat{Z}_{J}-\text{Disk}_{J})\rangle_{\psi}$
$\displaystyle=\delta_{IJ}+\sum_{\\{KL\\}}\frac{\langle
Z_{I}Z_{J}|\\{KL\\}\rangle}{C_{\\{KL\\}}}\langle\\{KL\\}|\psi^{2}\rangle$
(3.83) $\displaystyle=\delta_{IJ}+\langle\\{IJ\\}|\psi^{2}\rangle.$ (3.84)

Here we used the fact that $\langle
Z_{I}Z_{J}|\\{KL\\}\rangle=C_{\\{KL\\}}\delta_{\\{KL\\},\\{IJ\\}}$.

In the applications considered in this paper, most correlation functions of
interest are correlation functions of non-independent operators
$\widehat{Z}[\mathcal{J}]$, such as the analytically continued partition
function $\widehat{Z}(\beta+iT)$. These operators can be expressed as linear
superpositions of the independent operators $\widehat{Z}_{I}$. The relation
between these two operators is captured by the overlap $\langle
Z[\mathcal{J}]|Z_{I}\rangle$ or $\langle Z[\mathcal{J}]|I\rangle$. This second
overlap subtracts off the disk contributions and is computed by the cylinder
with a $Z[\mathcal{J}]$ boundary and a boundary in the state $|I\rangle$.

Subtracting off the disks, we can then express a correlator of the more
general $Z[\mathcal{J}]$ by “gluing” the $\langle Z[\mathcal{J}]|I\rangle$
cylinders to the correlators of disk-subtracted orthonormal $\widehat{Z}_{I}$.

For the one-point function of $\widehat{Z}[\mathcal{J}]$, the one-universe
wavefunction $\langle I|\psi^{2}\rangle$ is glued to the cylinder $\langle
Z[\mathcal{J}]|I\rangle$,

$\langle\widehat{Z}[\mathcal{J}]\rangle_{\psi}=\text{Disk}(\mathcal{J})+\sum_{I}\langle
Z[\mathcal{J}]|I\rangle\langle I|\psi^{2}\rangle.$ (3.85)

Pictorally, we may represent this in a way similar to Figure 14. In this case
the red circular boundary represents the one-universe wavefunction
$\widehat{P}_{1}|\psi^{2}\rangle$, which contains single universes of many
species.

The disk-subtracted two-point function of more general operators
$\widehat{Z}[\mathcal{J}]$, $\widehat{Z}[\mathcal{J}^{\prime}]$, is obtained
by gluing the cylinders $\langle Z[\mathcal{J}]|I\rangle$ to (3.83),

$\langle(\widehat{Z}[\mathcal{J}]-\text{Disk}(\mathcal{J}))(\widehat{Z}[\mathcal{J}^{\prime}]-\text{Disk}(\mathcal{J}^{\prime}))\rangle_{\psi}=\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime})+\sum_{IJ}\langle
Z[\mathcal{J}]|I\rangle\langle
Z[\mathcal{J}^{\prime}]|J\rangle\langle\\{IJ\\}|\psi^{2}\rangle.$ (3.86)

In the above equation the cylinder
$\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime})$ comes from gluing two $\langle
Z[\mathcal{J}]|I\rangle$ cylinders to the $\langle I|J\rangle=\delta_{IJ}$
cylinder,

$\displaystyle\sum_{IJ}\langle Z[\mathcal{J}]|I\rangle\langle
Z[\mathcal{J}^{\prime}]|J\rangle\delta_{IJ}$ $\displaystyle=\sum_{I}\langle
Z[\mathcal{J}]|I\rangle\langle I|Z[\mathcal{J}^{\prime}]\rangle$ (3.87)
$\displaystyle=\langle\widehat{Z}[\mathcal{J}]\widehat{Z}[\mathcal{J}^{\prime}]\rangle_{NB,c}\equiv\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime}).$
(3.88)

In the first line we used the fact that the $\widehat{Z}[\mathcal{J}]$ are
Hermitian, and we can choose the overlaps $\langle Z[\mathcal{J}]|I\rangle$ to
be real, and in the second line we used that the states
$|Z[\mathcal{J}]\rangle$ are orthogonal to the states with $k\geq 2$
universes, so that, subtracting off contributions from disks by considering
the connected correlator, the projector
$\widehat{P}_{1}=\sum_{I}|I\rangle\langle I|$ can be replaced with the
identity.

Figure 20: We can split the path integal over cylinders with
$\widehat{Z}[\mathcal{J}]$ and $\widehat{Z}[\mathcal{J}]$ boundaries by
inserting a complete set of intermediate one-universe states.

####  3.5.4 Factorization in the CGS model with species

In order to study factorization we now consider the case in which
$|\psi\rangle=|\psi_{\\{z_{I}\\}}\rangle$ is an eigenstate of the
$\widehat{Z}_{I}$,

$\widehat{Z}_{I}|\psi_{\\{z_{I}\\}}\rangle=z_{I}|\psi_{\\{z_{I}\\}}\rangle.$
(3.89)

The eigenvalues $z_{I}$ can be thought of as random variables labeling the
ensemble of boundary theories. In the probability distribution provided by the
No-Boundary state these variables are independently Gaussian with unit
variance and a mean given by the disk.

Using equation (3.81), we can identify $\langle
I|\psi^{2}_{\\{z_{I}\\}}\rangle=z_{I}-\text{Disk}_{I}$, so the one-universe
components $\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle$ can equivalently be
thought of as the random Gaussian variables characterizing the ensemble. These
variables have mean zero, and are independently distributed with unit
variance. Denoting averaging the $z_{I}$ with the No-Boundary distribution by
$\mathbb{E}[\dots]_{NB}$,

$\mathbb{E}\big{[}\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle\big{]}_{NB}=0,\hskip
20.0pt\mathbb{E}\big{[}\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle\langle
J|\psi^{2}_{\\{z_{I}\\}}\rangle\big{]}_{NB}=\delta_{IJ}.$ (3.90)

A single draw from this ensemble is the one-universe wavefunction
$\widehat{P}_{1}|\psi^{2}_{\\{z_{I}\\}}\rangle$ for a specific choice of
$\\{z_{I}\\}$. As in the single-species version of the CGS model, this one-
universe wavefunction contains all of the information about the full many-
universe $\alpha$ state. The remaining components are related to the one-
universe component by identities which generalize (3.50) and (3.51), which we
will derive by demanding that correlation functions factorize.

To understand these identities intuitively, it will be useful to make an
analogy between the expressions for correlation functions in this model as
sums over the components of $|\psi^{2}_{\\{z_{I}\\}}\rangle$, and the
expressions from Section 2 for products of $Z(iT)$ in periodic orbit theory as
sums over orbits.

Equation (3.85) expresses the one point function of an operator
$\widehat{Z}[\mathcal{J}]$ in an $\alpha$ state as a sum over single closed
universe states $I$. Typically, for an $\alpha$ state there will be many
nonzero terms in this sum. We will make an analogy between this sum and the
sum over orbits describing $Z(iT)$ in Section 2,

$\sum_{a}D_{a}e^{iS_{a}}\hskip 10.0pt\sim\hskip 10.0pt\sum_{I}\langle
Z[\mathcal{J}]|I\rangle\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle.$ (3.91)

We should note that in this analogy we are not literally identifying the sum
over closed universe states $I$ with an orbit sum for some system describing
the CGS model. Instead we are trying to make a formal analogy between two sums
over many random variables, the random phases $e^{iS_{a}}$ and the random
compontents $\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle$.292929 We refer to
Section 5.2 for some attempts to make a more dynamical analogy. We also
emphasize that the sum $\sum_{I}$ over closed universe states is not
equivalent to the sum over energies $E_{n}$ computing $Z(iT)$ in the boundary
theory.303030For example, in the application to JT gravity in Section 5, we
will see that the ramp in the spectral form factor is given by a diagonal sum
in the $|I\rangle$ basis, while the ramp is not given by a diagonal sum over
boundary energies.

In the periodic orbit story, the spectral form factor $Z(iT)Z(iT)^{*}$ is
given by a double sum over orbits, which is decomposed into the diagonal and
off-diagonal parts. The diagonal sum gives the ensemble average of the
spectral form factor. In the gravity computation, this average is given by the
cylinder topology, so we will think of the diagonal sum over orbits as
analogous to the cylinder,

$\sum_{a}|D_{a}|^{2}\hskip 10.0pt\sim\hskip
10.0pt\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime}).$ (3.92)

Since we are making an analogy between the orbit sum and the sum over closed
universes $I$, we should find that the diagonal part of the double sum over
$I,J$ describing the product of one-point functions of operators
$\widehat{Z}[\mathcal{J}],\;\widehat{Z}[\mathcal{J}^{\prime}]$ to also be
approximately equal to the cylinder. In the case that the dimension $d_{1}$ of
the single closed universe Hilbert space is large, then for typical operators
$\widehat{Z}[\mathcal{J}]$ and a typical $\alpha$ state this is indeed the
case,

$\displaystyle\sum_{I=1}^{d_{1}}\langle Z[\mathcal{J}]|I\rangle\langle
Z[\mathcal{J}^{\prime}]|I\rangle\;\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle^{2}$
$\displaystyle=\sum_{I=1}^{d_{1}}\langle Z[\mathcal{J}]|I\rangle\langle
I|Z[\mathcal{J}^{\prime}]\rangle\bigg{(}1+\mathcal{O}\bigg{(}\sqrt{\frac{1}{d_{1}}}\bigg{)}\bigg{)}$
(3.93)
$\displaystyle=\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime})+\mathcal{O}\bigg{(}\sqrt{\frac{1}{d_{1}}}\bigg{)}.$
(3.94)

In the first equality we used the reality of the matrix elements and the
results of Appendix A, assuming that the $\widehat{Z}[\mathcal{J}]|I\rangle$
are “smooth”. This will be true for many natural observables which do not
diagonalize the cylinder, including the observables we will study in JT
gravity in Section 5. In these cases, essentially what happens is that the
fluctuations of the random $\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle^{2}$
around their average value add incoherently, so for large $d_{1}$ we can
replace them by the average $\langle
I|\psi^{2}_{\\{z_{I}\\}}\rangle^{2}\rightarrow 1$ for typical choices of
$\\{z_{I}\\}$.

Figure 21: For appropriate choices of operators $\widehat{Z}[\mathcal{J}]$,
$\widehat{Z}[\mathcal{J}^{\prime}]$, the diagonal part of the product of
cylinders $\langle
Z[\mathcal{J}]|\widehat{P}_{1}|\psi^{2}_{\\{z_{I}\\}}\rangle\langle
Z[\mathcal{J}^{\prime}]|\widehat{P}_{1}|\psi^{2}_{\\{z_{I}\\}}\rangle$ is
approximately equal to the cylinder
$\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime})$ at large $d_{1}$ for typical
choices of $\\{z_{I}\\}$.

We will refer to this as an approximate “diagonal = cylinder” identity. This
further sharpens our analogy between the orbit sum and the sum over $I$;
ensemble averages can be computed by restricting to diagonal sums. Though this
“diagonal = cylinder” identity only holds in certain cases, the diagonal and
off-diagonal structure exhibited in these cases serves as a useful heuristic
for understanding the identities relating the components of
$|\psi^{2}_{\\{z_{I}\\}}\rangle$. We will use these terms to describe our
formulas, keeping quotation marks around the words “diagonal” and “off-
diagonal” to remind the reader of our heuristic use of these terms.

With this analogy in mind we examine the two-point function in an eigenstate,
and demand that it factorize into the product of one-point functions. First we
will start with the two-point function of the $\widehat{Z}_{I}$ with the disks
subtracted (3.83).

$\langle(\widehat{Z}_{I}-\text{Disk}_{I})(\widehat{Z}_{J}-\text{Disk}_{J})\rangle_{\\{z_{I}\\}}=\delta_{IJ}+\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle.$
(3.95)

Using (3.81) and demanding that this two-point function factorizes,
$\langle(\widehat{Z}_{I}-\text{Disk}_{I})(\widehat{Z}_{J}-\text{Disk}_{J})\rangle_{\\{z_{I}\\}}=\langle(\widehat{Z}_{I}-\text{Disk}_{I})\rangle_{\\{z_{I}\\}}\langle(\widehat{Z}_{J}-\text{Disk}_{J})\rangle_{\\{z_{I}\\}}$,
we see that the one-universe components and the two-universe components of
$|\psi^{2}_{\\{z_{I}\\}}\rangle$ must be related by

$\boxed{\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle=\langle
I|\psi^{2}_{\\{z_{I}\\}}\rangle\langle
J|\psi^{2}_{\\{z_{I}\\}}\rangle-\delta_{IJ}.}$ (3.96)

This equation is the many-species generalization of equation (3.50),
containing the essential requirement for factorization. However, in this case
it is relationship between symmetric matrices with indices $I$ and $J$, with
the matrix $\delta_{IJ}$ interpreted as the cylinder $\langle I|J\rangle$.

Looking at the off-diagonal components, $I\neq J$, we find

$\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle=\langle
I|\psi^{2}_{\\{z_{I}\\}}\rangle\langle J|\psi^{2}_{\\{z_{I}\\}}\rangle,\hskip
20.0ptI\neq J.$ (3.97)

This equation is a nontrivial relation between components of
$|\psi^{2}_{\\{z_{I}\\}}\rangle$ with different universe number; the off-
diagonal components of the two-universe wavefunction factorize into products
of the one-universe component. While we arrived at this relation in a simple
way by demanding that the two-point functions factorize, we will see in
Section 3.5.6 and Appendix B that this relation is complicated in terms of the
geometric representation of the states $|\psi^{2}_{\\{z_{I}\\}}\rangle$.

While the off-diagonal components of
$\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle$ factorize, the fact that the
cylinder is diagonal in the $I,J$ indices picks out the diagonal, $I=J$,
components of $\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle$ as special.
These components are related by

$\langle\\{II\\}|\psi^{2}_{\\{z_{I}\\}}\rangle=\langle
I|\psi^{2}_{\\{z_{I}\\}}\rangle^{2}-1.$ (3.98)

The factor of one subtracted on the RHS corresponds to the cylinder $1/\langle
1|1\rangle$. Using (3.90), the average of the RHS is zero, indicating that on
average, the diagonal part of the product of one-universe components
compensates for the cylinder so that the two-universe component is off-
diagonal on average. The fluctuations of
$\langle\\{II\\}|\psi^{2}_{\\{z_{I}\\}}\rangle$ around their average value of
zero are nonzero. However, the fluctuations are independent for different $I$.
In applying this formula to correlators of operators
$\widehat{Z}[\mathcal{J}]$, we will find sums over these diagonal components,
which then add incoherently. Using the results in Appendix A, we will see that
in appropriate correlators, we will be able to ignore the diagonal components
and treat them as if they were zero. In this situation, the
“diagonal=cylinder” identity allows us to heuristically view the two-universe
components as purely “off-diagonal”.

This fits with our analogy to periodic orbits; in order for the two-point
function to factorize, we must add an “off-diagonal” sum to the (diagonal)
cylinder, with the off-diagonal summands being simply a product of summands
for the one-point functions.

To see this in more detail, we apply equation (3.96) to the two-point
functions of more general operators $\widehat{Z}[\mathcal{J}]$. Gluing the
cylinders $\langle Z[\mathcal{J}]|I\rangle$ to (3.96), we find

$\displaystyle\sum_{I,J=1}^{d_{1}}\langle Z[\mathcal{J}]|I\rangle\langle
Z[\mathcal{J}^{\prime}]|J\rangle\;\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle$
$\displaystyle=$ (3.99) $\displaystyle\bigg{(}\sum_{I=1}^{d_{1}}\langle
Z[\mathcal{J}]|I\rangle\langle
I|\psi^{2}_{\\{z_{I}\\}}\rangle\bigg{)}\bigg{(}\sum_{J=1}^{d_{1}}\langle
Z[\mathcal{J}^{\prime}]|J\rangle\langle
J|\psi^{2}_{\\{z_{I}\\}}\rangle\bigg{)}$
$\displaystyle-\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime}).$ (3.100)

Figure 22: The identity (3.96) implies the pictured relationship between
contributions to two-point functions and one-point functions in an $\alpha$
state. The linked red boundaries denote the two-universe component of
$|\psi^{2}_{\\{z_{I}\\}}\rangle$, while the unlinked red boundaries denote the
one-universe component.

This equation is precise, independent of $d_{1}$. However, at large $d_{1}$ we
can use the “diagonal=cylinder” identity to connect more sharply with the
periodic orbits story and allow us to simplify the description of
factorization.

The (disk subtracted) two-point function is the sum of the cylinder
$\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime})$ and the contribution of the
two-universe component $\sum_{IJ}\langle Z[\mathcal{J}]|I\rangle\langle
Z[\mathcal{J}^{\prime}]|J\rangle\;\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle$.
This second contribution is a double-sum over closed universe states $I$ and
$J$. Since we are identifying the diagonal part of the orbit sum with the
cylinder, we should identify the off-diagonal part of the sum with the
contribution of the two-universe component.

Along with (3.93), (3.99) tells us that this identification is sound. The
diagonal part of the double-sum in (3.99) is small for large $d_{1}$, so the
double-sum is essentially off-diagonal.

Figure 23: When the “diagonal=cylinder” identity applies, the identity
relating contributions to the two-point function and one-point function
simplies; the contribution on the LHS to the two-point function is
approximately equal to the off-diagonal part of the contribution of the
$\alpha$ state to the product of one-point functions. In the computation of
the two-point function, the “off-diagonal” LHS is added to the “diagonal”
cylinder, yielding a factorizing double-sum.

To summarize, we can see the factorization of the two-point function by
denoting
$(\text{1-universe})_{I}\equiv\langle\widehat{Z}[\mathcal{J}]|I\rangle\langle
I|\psi^{2}_{\\{z_{I}\\}}\rangle$ and
$(\text{2-universe})_{IJ}\equiv\langle\widehat{Z}[\mathcal{J}]\widehat{Z}[\mathcal{J^{\prime}}]|I\rangle\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle$,
and schematically writing

$\displaystyle\langle(\widehat{Z}-\text{Disk})^{2}\rangle_{\\{z_{I}\\}}$
$\displaystyle=\hskip 35.0pt\text{Cyl}\hskip 39.0pt+\hskip
10.0pt\sum_{IJ}(\text{2-universe})_{IJ}$ (3.101)
$\displaystyle\approx\sum_{I=J}(\text{1-universe})_{I}^{2}\hskip 10.0pt+\hskip
10.0pt\sum_{I\neq J}(\text{1-universe})_{I}(\text{1-universe})_{J}$ (3.102)
$\displaystyle=\sum_{IJ}(\text{1-universe})_{I}(\text{1-universe})_{J}$
(3.103)
$\displaystyle=\bigg{(}\sum_{I}(\text{1-universe})_{I}\bigg{)}^{2}=\big{(}\langle\widehat{Z}-\text{Disk}\rangle_{\\{z_{I}\\}}\big{)}^{2}.$
(3.104)

We emphasize that though at large $d_{1}$ the calculation approximately has
this “diagonal + off-diagonal” structure, giving a simple picture of
factorization, (3.96) allows the two-point function to factorize even at small
$d_{1}$.

The “off-diagonal” structure of the two-universe wavefunction
$\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle$ described by (3.96) has a
geometrical origin. To see this, compare the computations of the product of
one-universe components $\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle\langle
J|\psi^{2}_{\\{z_{I}\\}}\rangle$ and a two-universe wavefunction
$\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle$.

$\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle$ is computed by the sum over
spacetimes with a boundary in the state $|I\rangle$ and boundaries in the
state $|\psi^{2}_{\\{z_{I}\\}}\rangle$. The $|I\rangle$ boundary can connect
to just one of the $|\psi^{2}_{\\{z_{I}\\}}\rangle$ boundaries at a time with
a cylinder, and the component $|I\rangle$ is equal to a sum over the
$|\psi^{2}_{\\{z_{I}\\}}\rangle$ boundaries the $|I\rangle$ boundary connects
to. The product of components $\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle\langle
J|\psi^{2}_{\\{z_{I}\\}}\rangle$ can then be thought of as a double-sum over
the $|\psi^{2}_{\\{z_{I}\\}}\rangle$ boundaries that the $|I\rangle$ and
$|J\rangle$ boundaries connect to.

Figure 24: Here we compare the computation of the product of one-universe
components $\langle I|\psi^{2}\rangle\langle J|\psi^{2}\rangle$ with the two-
universe component $\langle\\{IJ\\}|\psi^{2}\rangle$. In the computation of
the one-universe component, the $|I\rangle$ boundary connects to a boundary in
$|\psi^{2}\rangle$ labeled by $a$ with a cylinder and the $|J\rangle$ boundary
connects to $b$. For fixed $a,b$, we sum over the spacetimes connecting to the
remaining boundaries in $|\psi^{2}\rangle$. Then we sum over the choices
$a,b$. In the computation of the two-universe component, we similarly sum over
spacetimes for which the $|I\rangle$ and $|J\rangle$ boundaries connect to
boundaries in $|\psi^{2}\rangle$, labeled by $a,b$ with cylinders. However, in
this case the $|\psi^{2}\rangle$ boundaries are shared, and there are no
contributions from $a=b$; these “diagonal” terms are “excluded”. In an
$\alpha$ state $|\psi_{\\{z_{I}\\}}\rangle$, the state is tuned so that this
exclusion results in the diagonal/off-diagonal structure in $I,J$, (3.96).

The two-universe component $\langle\\{IJ\\}|\psi^{2}_{\\{z_{I}\\}}\rangle$ is
given by a similar computation, where we sum over spacetimes with an
$|I\rangle$ boundary and a $|J\rangle$ boundary, but now these boundaries
“share” the $|\psi^{2}_{\\{z_{I}\\}}\rangle$ boundaries. When we sum over the
$|\psi^{2}_{\\{z_{I}\\}}\rangle$ boundaries that $|I\rangle$ and $|J\rangle$
connect to with cylinders, we do not include the diagonal term in the sum,
since the $|I\rangle$ and $|J\rangle$ boundaries cannot connect to the same
boundary with a cylinder simultaneously.313131If we had included more
complicated topologies, as in the MM model, there could be such a connection,
but it would involve a more complicated topology like the three-holed sphere.
This would be suppressed because of its Euler character.

We will refer to this phenomenon as the “exclusion effect” and is illustrated
in Figure 24.

This geometrical picture is rough, and does not explain other aspects of the
relation between the one-universe and two-universe components of
$|\psi^{2}_{\\{z_{I}\\}}\rangle$. For example, it is clear geometrically why
the two-universe component is generally not equal to the product of one-
universe components. However, the fact that the two-universe component
factorizes precisely into the product of one-universe components is a result
of a fine-tuning in the many-universe state.323232This property is more
clearly understood using the explicit description of
$|\psi^{2}_{\\{z_{I}\\}}\rangle$ as a regulated delta-function in 3.5.6.
However, identifying the geometric origin of the difference between the linked
and unlinked pair of red boundaries in 22 and 23 (the two-universe component
of $|\psi^{2}_{\\{z_{I}\\}}\rangle$ and the product of one-universe
components) will be important in our effective description of factorization in
Section 3.6. In Appendix G, we will also use this exclusion effect in a more
general context to emphasize the importance of the “sharing” of the $\alpha$
state boundaries in allowing correlation functions to factorize.

####  3.5.5 Factorization of higher-point correlators

As an aside, we briefly discuss the factorization of higher point correlators.

To generalize equation (3.96) to higher point correlators, it is useful to
first express this equation in a basis-independent way. Viewing the multi-
universe states $|\\{I_{1}\dots I_{n}\\}\rangle$ as (symmetrized) tensor
products of single-universe states, and introducing the maximally entangled
state of two universes $|MAX\rangle=\sum_{I}|I\rangle\otimes|I\rangle$, we can
express (3.96) in a basis independent way as

$\widehat{P}_{2}|\psi^{2}_{\\{z_{I}\\}}\rangle=(\widehat{P}_{1}|\psi^{2}_{\\{z_{I}\\}}\rangle)^{\otimes
2}-|MAX\rangle.$ (3.105)

The projection operators onto $n$-universe states $\widehat{P}_{n}$ were
defined in equation (3.78). Geometrically, the state $|MAX\rangle$ corresponds
to the state of two universe produced by the cylinder spacetime, and $\langle
Z[\mathcal{J}]Z[\mathcal{J}^{\prime}]|MAX\rangle=\text{Cyl}(\mathcal{J},\mathcal{J}^{\prime})$.

We can think of the lack of factorization of correlators in the No-Boundary
state as follows: The cylinder corresponds to the pair creation of two closed
universes in the maximally entangled state, and the contribution of the
cylinder to correlation functions describes the absorption of these entangled
universes by the operators in the correlator. The entanglement between
different closed universes then spoils factorization.

In order for correlators to factorize, the closed universes absorbed by the
operators in the correlator must not be entangled. We can think of this many-
universe state as the sum of an $n$-universe component of
$|\psi^{2}_{\\{z_{I}\\}}\rangle$ and the state of pairwise-entangled universes
created by cylinders. Then the $n$-universe component of
$|\psi^{2}_{\\{z_{I}\\}}\rangle$ must be a product state,
$(\widehat{P}_{1}|\psi^{2}_{\\{z_{I}\\}}\rangle)^{\otimes n}$, minus
corrections which cancel the pairwise-entangled cylinder states, somewhat as
in (3.51). For example,

$\widehat{P}_{3}|\psi^{2}_{\\{z_{I}\\}}\rangle=(\widehat{P}_{1}|\psi^{2}_{\\{z_{I}\\}}\rangle)^{\otimes
3}-{3\choose
2}\mathcal{S}\big{[}(\widehat{P}_{1}|\psi^{2}_{\\{z_{I}\\}}\rangle)\otimes|MAX\rangle\big{]}.$
(3.106)

where the operator $\mathcal{S}[\dots]$ symmetrizes its argument. This can be
derived by demanding that the three-point function, which probes the three-
universe component of the many-universe state, factorizes.

####  3.5.6 Explicit representation of the $\alpha$ state

We now briefly discuss the computation of correlators using the explicit
(regularized) delta function representation of an $\alpha$-state, generalizing
the results of Section 3.96 to the many-species case.

We can represent $\psi(\widehat{Z}_{I})$ for an approximate eigenstate of the
$\widehat{Z}_{I}$ with eigenvalues $z_{I}$ as a product of regularized delta
functions,

$\psi_{\\{z_{I}\\},\Delta}(\widehat{Z}_{I})\propto\int_{-\infty}^{\infty}dp_{I}\;e^{-\Delta^{2}\sum_{I}p_{I}^{2}+i\sum_{I}p_{I}(\widehat{Z}_{I}-z_{I})}.$
(3.107)

We have chosen the same width for each of the Gaussian regularizations of the
delta functions. Squaring this to obtain
$\psi_{\\{z_{I}\\},\Delta}(\widehat{Z}_{I})^{2}$, we can change variables to
the sum and difference variables of the $p_{I}$ and $p_{I}^{\prime}$ and
integrate out the difference variables to find an expression for
$\psi_{\\{z_{I}\\},\Delta}(\widehat{Z}_{I})^{2}$ as a product of Gaussians
integrals in the sum variables. This is just many copies of the single delta
function expression (3.59) for $\psi_{z}(\widehat{Z})^{2}$ in the one-species
model.

Now we can normal order the exponential $e^{i\sum_{I}\widehat{Z}_{I}}$ to find
an expression for the alpha state in the N-basis $|I_{1}\dots I_{k}\rangle$,

$\psi_{\\{z_{I}\\},\Delta}(\widehat{Z}_{I})^{2}|NB\rangle\propto\int_{-\infty}^{\infty}dp_{+I}e^{\frac{\Delta+1}{2}\sum_{I}p_{+I}^{2}+i\sum_{I}p_{+I}(\text{Disk}_{I}-z_{I})}\sum_{k=0}^{\infty}\sum_{\\{I_{1}\dots
I_{k}\\}}\frac{(ip_{+I_{1}})\dots(ip_{+I_{k}})}{C_{\\{I_{1}\dots
I_{k}\\}}}|\\{I_{1}\dots I_{k}\\}\rangle.$ (3.108)

We can safely take the limit $\Delta\rightarrow 0$, so that the Gaussian
integrals have a width determined by the cylinder, which is equal to one.

The components $\langle\\{I_{1}\dots I_{k}\\}|\psi^{2}_{\\{z_{I}\\}}\rangle$
are then given by moments of a multi-dimensional Gaussian integral. The Wick
contractions describing these moments exactly match the pattern
(3.105),(3.106), which we expect to describe these components, with the
identification $\langle
I|\psi^{2}_{\\{z_{I}\\}}\rangle=z_{I}-\text{Disk}_{I}$.

In Appendix B, we discuss the explicit computation of the factorization of the
two-universe components into products of one-universe components in more
detail.

###  3.6 An effective description with random boundaries

In the CGS model, as well as in the MM model and JT gravity, an $\alpha$ state
is a complicated many-universe state. The full description of factorization in
these states depends crucially on the many-universe structure, with identities
such as (3.96) resulting from the “exclusion effect” in a fine-tuned many-
universe state. On the other hand, if we take the identities such as (3.96) as
given, the many-universe structure of the theory does not play a large role;
correlation functions can be described simply in terms of the one-universe
state $\widehat{P}_{1}|\psi^{2}_{\alpha}\rangle$. The components of this one-
universe state have simple statistics, they are independent Gaussian
variables, so the ensemble of boundary theories is also more simply described
in terms of this one-universe state.

Furthermore, these relations often simplify when applied to correlators at
large dimension $d_{1}$ of the one-universe Hilbert space, when the
$n$-universe components in a non-orthormal basis described by operators
$\widehat{Z}[\mathcal{J}]$ have an “off-diagonal” structure. In this case, the
description of correlators in terms of the one-universe wavefunction
$\widehat{P}_{1}|\psi^{2}_{\alpha}\rangle$ becomes especially simple,
resembling the periodic orbits story. The cylinder behaves like a “diagonal”
sum, the two-universe component behaves like an “off-diagonal” sum, and they
add to a factorizing double-sum.

In contrast to the periodic orbits story, though, the (diagonal) cylinder is a
fundamental part of the model. We add the two-universe component to the
cylinder, which is then required to have an “off-diagonal” structure, as
pictured in Figure 23. The “exclusion effect” underlying this relation has a
fundamentally many-universe explanation. It is the essential way in which the
many-universe character of the theory enters into the calculation of
correlators in an $\alpha$ state.

In this section we introduce an effective model meant to describe the CGS
model in an $\alpha$ state, but not in a more general state of the closed
universes. This effective description takes advantage of this simple
description of the CGS model in an $\alpha$ state in terms of one-universe
states $\widehat{P}_{1}|\psi^{2}_{\alpha}\rangle$. In the effective model, we
replace the contributions from the random many-universe $\alpha$ state with
contributions from a small number of random “$\Psi$” boundaries; to compute a
product of $n$ partition functions, we need only $n$ $\Psi$ boundaries.

For a given computation, there are different choices of effective description
which give the same result. These choices correspond in the full theory to the
choice of computing an $n$-point function using a manifestly factorized
description or not,
$\langle\widehat{Z}[\mathcal{J}]^{n}\rangle_{\alpha}=\langle\widehat{Z}[\mathcal{J}]\rangle_{\alpha}^{n}$.

In one version of the effective model, we only allow “unlinked” $\Psi$
boundaries in the path integral. Each unlinked $\Psi$ boundary comes with a
random boundary condition, fixed to be the same for each boundary. A single
$\Psi$ boundary replaces the contribution of one-universe states
$\widehat{P}_{1}|\psi^{2}_{\alpha}\rangle$. For example, the contribution of
one $\Psi$ boundary to a partition function is a “broken cylinder”\- a
cylinder with a $\widehat{Z}[\mathcal{J}]$ boundary, and a $\Psi$ boundary.
This broken cylinder replaces the contribution of the one-universe components
$\langle I|\psi^{2}_{\\{z_{I}\\}}\rangle$, (3.92), in the full model. In this
version of the effective model, we disallow, or “exclude”, cylinders that
connect partition function boundaries. Then this version of the effective
model corresponds to the manifestly factorized computation of correlation
functions in the full theory.

In the other version of the effective model, we allow cylinders connecting
partition functions, but introduce “linked” $\Psi$ boundaries rather than
unlinked boundaries. As in the other version of the effective model, we sum
over spacetimes with different numbers of circular $\Psi$ boundaries; however,
linked $\Psi$ boundaries come with a different boundary condition. Rather than
giving each $\Psi$ boundary the same random boundary condition, the boundary
condition is “shared” between the linked $\Psi$ boundaries. Roughly, the
linked $\Psi$ boundary condition is like a product of unlinked boundary
conditions, with the “diagonal” contributions “excluded”. These linked $\Psi$
boundaries then replace the contributions of the $n$-universe components
$\widehat{P}_{n}|\psi^{2}\rangle$, and this version of the effective
description corresponds to the not-manifestly-factorized computation of
correlators in the full theory.

This distinction between linked and unlinked $\Psi$ boundaries is a
consequence of abandoning the many-universe description of the full theory. In
the effective model, we have a choice between using unlinked $\Psi$
boundaries, but excluding the full cylinders, or using linked $\Psi$
boundaries with the diagonal components excluded. This ad-hoc “exclusion
rule”, which instructs us to exclude either the cylinder, or the “diagonal”
piece of the linked $\Psi$ boundaries (related by a “diagonal=cylinder
identity”), is put in by hand in this effective model, but has a natural
geometrical origin in the full theory. For example, the “off-diagonal”
structure of the two-universe components of $|\psi^{2}_{\alpha}\rangle$
results from the geometric exclusion preventing two cylinders from ending on
the same $\alpha$ state boundary.

This effective description will turn out to be useful for describing JT
gravity, discussed in Section 5. The effective description of JT gravity in an
$\alpha$ state has similar advantages and disadvantages to the effective
description of the CGS model; in particular the “exclusion rule” must be
explained by appealing to the full many-universe description of JT gravity.
However, JT gravity could possibly serve as a laboratory for exploring some
generalizations in which the random $\Psi$ boundaries are replaced by
pseudorandom boundary conditions, with the pseudorandomness resulting from
chaotic bulk dynamics.

This effective description is similar to the description of the SYK model for
a fixed choice of couplings. However, in the SYK model the distinction between
different versions of the effective model, with linked and unlinked $\Psi$
boundaries, is explained in a simple, but nongeometrical way in the full
theory, as discussed in [1]. We briefly review these ideas in the Discussion.

####  3.6.1 Definition of the effective model

Now we describe this effective description in more detail. This effective
description is meant to capture the behavior of correlation functions of the
CGS model in an $\alpha$ state, but without using the apparatus of the many-
closed-universe quantum mechanics. Since we will no longer be making use of
many-universe states, we will no longer think about the observables we are
computing as correlation functions. In this effective description, partition
functions always factorize, so we can think of them as numbers,333333We could
also think of the partition functions computed in the effective description as
correlation functions in the No-Boundary state of the bulk theory with $\Psi$
boundaries. We will comment on this perspective in the Appendix G.

$\langle\widehat{Z}[\mathcal{J}_{1}]\dots\widehat{Z}[\mathcal{J}_{n}]\rangle_{\alpha}\longrightarrow
Z^{(\Psi)}[\mathcal{J}_{1}]\dots Z^{(\Psi)}[\mathcal{J}_{n}].$ (3.109)

Here $Z^{(\Psi)}[\mathcal{J}]$ is a fixed number. The symbol $\Psi$ denotes
the choice of the element of the ensemble of boundary theories to which the
$\alpha$ state corresponds, so we should identify
$Z^{(\Psi)}[\mathcal{J}]=\langle\widehat{Z}[\mathcal{J}]\rangle_{\alpha}$. We
are using the symbol $\Psi$ instead of $\alpha$ to label members of the
ensemble in order to emphasize that we do not think of $\Psi$ as described by
some many-universe $\alpha$ state.

Now we give a prescription for computing the observables
$Z^{(\Psi)}[\mathcal{J}_{1}]\dots Z^{(\Psi)}[\mathcal{J}_{n}]$ using this
effective description. First, we discuss the version of the effective model
with no cylinders, and only unlinked $\Psi$ boundaries.

$Z^{(\Psi)}[\mathcal{J}]$ is computed by a path integral over spacetimes with
each connected component has a boundary with a $\mathcal{J}$ boundary
condition; we do not include spacetimes disconnected from these boundaries. In
the sum over geometries, we also freely include any number of “$\Psi$
boundaries”.343434The rules for including $\Psi$ boundaries in the path
integral are similar to the rules for including EOW brane loops in a
modification of the MM model [44]. Restricting our attention to the case that
the full theory is a two-dimensional theory of gravity, these are circular
boundaries with an identical random boundary condition labeled by $\Psi$, as
pictured in Figure 25.353535In this effective model we relax the earlier
condition that we consider states of closed universes only on asymptotic
boundaries, so that we may think of the $\Psi$ boundary as living on a “bulk”
circular boundary.. We refer to such configurations as “broken
cylinders.”363636As we explain in the Discussion, these broken cylinders are
closely related to the “half-wormholes” found in [1].

Figure 25: Here we have pictured the contribution to $Z^{(\Psi)}[\mathcal{J}]$
from one $\Psi$ boundary, denoted by the wiggly red boundary. This “broken