On the non-perturbative bulk Hilbert space of JT gravity

Luca V. Iliesiu1,2, Adam Levine3, Henry W. Lin2, Henry Maxfield2, Márk Mezei4

1 Department of Physics, University of California, Berkeley, CA 94720, USA

2 Stanford Institute for Theoretical Physics, Stanford University, Stanford,
CA 94305, USA

3 Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA

4 Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2
6GG, UK

Abstract

What is the bulk Hilbert space of quantum gravity? In this paper, we resolve
this problem in $2d$ JT gravity, both with and without matter, providing the
first example of an explicit definition of a non-perturbative Hilbert space
specified in terms of metric variables. The states are wavefunctions of the
length and matter state, but with a non-trivial and highly degenerate inner
product. We explicitly identify the null states, and discuss their importance
for defining operators non-perturbatively. To highlight the power of the
formalism we developed, we study the non-perturbative effects for two bulk
linear operators that may serve as proxies for the experience of an observer
falling into a two-sided black hole: one captures the length of an Einstein-
Rosen bridge and the other captures the center-of-mass collision energy
between two particles falling from opposite sides. We track the behavior of
these operators up to times of order $e^{S_{\text{BH}}}$, at which point the
wavefunction spreads to the complete set of eigenstates of these operators. If
these observables are indeed good proxies for the experience of an infalling
observer, our results indicate an $O(1)$ probability of detecting a firewall
at late times that is self-averaging and universal.

###### Contents

  1. 1 Introduction & summary
    1. 1.1 The bulk Hilbert space on geodesics
    2. 1.2 Defining bulk operators
    3. 1.3 Case study I: length operators
    4. 1.4 Case study II: center-of-mass collision energy
    5. 1.5 Outline
  2. 2 The non-perturbative geodesic Hilbert space
    1. 2.1 Perturbative Hilbert space of pure JT: review
    2. 2.2 Defining the non-perturbative theory
    3. 2.3 Relating geodesic and asymptotic states
    4. 2.4 An alternative ensemble perspective
    5. 2.5 The Hamiltonian is unmodified
  3. 3 Null states: redundancies in the bulk Hilbert space
  4. 4 Defining non-perturbative bulk operators
    1. 4.1 Challenges and ambiguities
    2. 4.2 A class of length operators
    3. 4.3 Path integral interpretation
  5. 5 Case-study I: a length operator
    1. 5.1 The Hamiltonian in the length eigenvector basis
    2. 5.2 Evolution of the wormhole length
    3. 5.3 The wormhole velocity
  6. 6 Generalization to JT with matter
    1. 6.1 Perturbative JT with matter
      1. 6.1.1 Geodesic and asymptotic states
      2. 6.1.2 QFT in AdS2 and $\mathfrak{sl}(2)$ representations
      3. 6.1.3 The half-disk and Hamiltonian eigenstates
    2. 6.2 The non-perturbative definition of the theory
    3. 6.3 The non-perturbative inner product
    4. 6.4 Null states & redundancies
  7. 7 Case study II: a center-of-mass collision energy operator
    1. 7.1 Semi-classical approximation
    2. 7.2 Relation to the Casimir operator
    3. 7.3 Perturbative probability of detecting a fixed Casimir
    4. 7.4 A violation of charge conservation for the Casimir operator
    5. 7.5 Non-perturbative definition of the Casimir
  8. 8 Discussion
    1. 8.1 Firewalls
    2. 8.2 The conditions underlying our construction
    3. 8.3 A non-perturbative Wheeler-DeWitt equation
    4. 8.4 An enlarged bulk Hilbert space with baby universes
    5. 8.5 Relation to non-isometric codes
    6. 8.6 Embedding into other microscopic models
  9. A The Hilbert space and eigenfunctions in the boundary particle formalism
    1. A.1 Hilbert space from the boundary particle formalism
    2. A.2 One-particle Hilbert space
    3. A.3 Evaluating the overlap of energy eigenfunctions
  10. B Deriving the Casimir distribution from the Gravitational OPE
  11. C Details on the numerics
    1. C.1 Rescaling from the semi-circle to the Schwarzian spectrum
    2. C.2 Length and velocities
  12. D The dimension of the Hilbert space

##  1 Introduction & summary

In recent years, it has become increasingly apparent that non-perturbative
gravitational effects have remarkable explanatory power. Highlights include
quantifying the information content of Hawking radiation [1, 2, 3, 4], aspects
of spectral statistics of black hole microstates [5, 6, 7, 8, 9], and counting
states of supersymmetric black holes using the supergravity path integral [10,
11, 12]. These calculations can all be described in the framework of general
relativity as an effective quantum theory, without using microscopic ideas
such as string theory or holographic duality. Einstein’s geometric theory of
dynamical spacetime and quantum mechanics thus have far more unity and scope
than previously envisaged.

However, the progress described above involves quantities that can be defined
asymptotically, in a region where gravity is absent or weak. Concretely, the
calculations involve boundary conditions that fix the asymptotic spacetime
geometry and fields (for example, at the boundary of an asymptotically AdS
spacetime), even if there are large quantum fluctuations in the interior. This
becomes an important limitation for addressing more inherently ‘bulk’
questions in contexts like the black hole interior (specifically, the
experience of an infalling observer at late times) or cosmological spacetimes.
We would therefore like a better understanding of non-perturbative
gravitational effects in bulk language, with states described in terms of
wavefunctions of spatial metrics and matter fields. In particular, such a
language seems necessary to properly define bulk linear operators and describe
their physics.

In this paper, we make progress towards this goal in the simple two-
dimensional theory of JT gravity (coupled to matter). Specifically, we take
the existing definition of the perturbative bulk Hilbert space defined on
geodesic slices and describe how it is modified by non-perturbative effects.
We then use this construction to study bulk operators non-perturbatively.

A prominent idea about the Hilbert space of gravity is that of redundancy, or
null states. Historically, effective quantum gravity has been regarded as
inconsistent due to the black hole information paradox; this is underpinned by
the fact that a black hole can apparently have far too many interior states
(as defined by quantizing fluctuations around a classical background
spacetime, taking perturbative gravitational constraints into account). A
possible resolution is that the geometric description of the Hilbert space is
perfectly consistent but that apparently different states are not, in fact,
linearly independent. States that are orthogonal in perturbation theory can
obtain small overlaps from non-perturbative effects; if these overlaps are
finely tuned, the resulting matrix of inner products can be degenerate, with
many ‘null states’ of zero norm. We will see this idea implemented very
concretely in our study of JT gravity, obtaining explicit expressions for the
degenerate inner product and null state wavefunctions. This phenomenon has
important consequences for operators on the Hilbert space and for the
associated observables in any theory of quantum gravity.

In the remainder of the introduction, we summarise our main results before
outlining the structure of the paper.

###  1.1 The bulk Hilbert space on geodesics

A simplifying feature of JT gravity is that there is a tunable parameter
$S_{0}$ which suppresses non-perturbative effects without altering
perturbative gravity. In the limit $S_{0}\to\infty$, we recover a self-
consistent ‘perturbative’ theory, which includes all-loop quantum fluctuations
but not topology-changing processes. We begin by describing the Hilbert space
of this theory (which we call $\mathcal{H}_{0}$), before addressing the non-
perturbative effects. Here, we restrict to pure JT gravity, leaving the
generalisation to include matter to the main text.

We are interested in the ‘two-sided’ Hilbert space, with two asymptotic
spatial boundaries. We can describe states in $\mathcal{H}_{0}$ as
wavefunctions $\psi(\ell)$ of the renormalized geodesic length $\ell$ between
these boundaries, with the usual $L^{2}$ inner product (denoted
$\langle\cdot|\cdot\rangle_{0}$ to distinguish it from the full inner product
introduced later):

$\mathcal{H}_{0}\simeq L^{2}(\mathbb{R}),\qquad|\psi\rangle=\int
d\ell\,\psi(\ell)|\ell\rangle,\qquad\langle\psi|\psi\rangle_{0}=\int|\psi(\ell)|^{2}d\ell\qquad(S_{0}\to\infty).$
(1.1)

We get this simple local inner product because every allowed geometry in this
$S_{0}\to\infty$ theory (which must have the topology of a disc) has a unique
geodesic between given boundary points. Time evolution is generated by a
simple Hamiltonian $H$ acting on the wavefunction $\psi(\ell)$, of a non-
relativistic particle in an exponential potential [13, 14]:

$H=H_{L}=H_{R}=-\tfrac{1}{2}\partial_{\ell}^{2}+2e^{-\ell}.$ (1.2)

This evolves either on the left or right boundary, since the left and right
Hamiltonians $H_{L,R}$ are equal for pure JT gravity (this no longer holds
with matter). This operator has a continuous spectrum (the positive reals
$E>0$), so there is an infinite-dimensional space of states in any window of
energies; this is a version of the information problem.

What happens when $S_{0}$ is finite, and we include non-perturbative effects?
A complication is that we no longer have a unique geodesic on every geometry,
so we might expect that a Hilbert space in terms of geodesic lengths no longer
makes sense.111To avoid confusion, note that this is different from the baby
universe Hilbert space [15].  Nonetheless, we can continue to define bulk
‘basis’ states $|\ell\rangle$ by geodesic boundary conditions in the path
integral. However, the inner product $\langle\ell^{\prime}|\ell\rangle$
becomes non-trivial, computed by the path integral over all geometries bounded
by a pair of geodesics of lengths $\ell,\ell^{\prime}$; this idea was
articulated before in [16]. Our first main result is an expression for this
inner product in terms of the microscopic data of the theory (which in AdS/CFT
language we can think of as the dual quantum mechanics). In pure JT gravity,
this data is simply the discrete spectrum $E_{i}$ ($i=0,1,2,\ldots$) of the
Hamiltonian in the complete non-perturbative theory. For a superposition
$|\psi\rangle=\int d\ell\,\psi(\ell)|\ell\rangle$ of geodesic states as above,
we find

$\langle\psi|\psi\rangle=e^{-S_{0}}\sum_{i}\left|\int\phi_{E_{i}}(\ell)\psi(\ell)\,d\ell\right|^{2},$
(1.3)

where $\phi_{E}(\ell)$ are eigenfunctions of the Schrödinger Hamiltonian (1.2)
(given explicitly in (2.2)). Thus, for example, the inner product between two
states with fixed length is given by

$\displaystyle\langle\ell|\ell^{\prime}\rangle=e^{-S_{0}}\sum_{i}\phi_{E_{i}}(\ell)\phi_{E_{i}}(\ell^{\prime})\,.$
(1.4)

Normally, JT gravity is regarded as being dual to an ensemble of quantum
systems with a random Hamiltonian, so we should explain what we mean by the
spectrum $E_{i}$. There are two possible (but equivalent) perspectives we can
take. First, we could take our definition of the non-perturbative theory to
include not only sums over topologies, but also additional effects which ‘fix
the member of the ensemble’ (for example, additional asymptotic boundaries to
fix a single ‘$\alpha$-state’ of baby universes, a non-perturbatively small
non-local interaction or eigenbranes). This leaves us with some specific non-
perturbative spectrum (and a factorizing theory), the precise nature of which
will depend on the details. Alternatively, we can think of the matrix of inner
products $\langle\ell^{\prime}|\ell\rangle$ as a random variable, with $E_{i}$
the eigenvalues of a random matrix. In that case, we will demonstrate the
equality between random variables (1.3) holds for all moments of the inner
product to all orders in a genus expansion, which is explicitly computable.
Specifically, the average of this random variable takes the form,

$\displaystyle\overline{\innerproduct{\ell^{\prime}}{\ell}}\quad=\quad\delta(\ell-\ell^{\prime})+\quad\leavevmode\hbox
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O(e^{-4S_{0}})\,.$ (1.5)

In either case, we use only some simple properties of the bulk theory and not
the full details of its non-perturbative definition. This is particularly
useful once we add matter since, in that case, the precise definition of the
non-perturbative corrections is more ambiguous due to divergences on
spacetimes with nontrivial topology.

The modified norm (1.3) is highly degenerate. For any wavefunction
$\psi(\ell)$, we can construct the ‘transform’ $\widehat{\psi}(E)=\int
d\ell\,\phi_{E}(\ell)\psi(\ell)$, and the norm depends only on the values
$\widehat{\psi}(E_{i})$ at the discrete energies giving the non-perturbative
spectrum. In particular, if $\widehat{\psi}(E_{i})=0$ for all $i$, then we
have a null state (and there will be many such states for which the function
$\psi(\ell)$ is nonzero). Nonetheless, (1.3) is positive semi-definite. Note
that we reserve the term ‘wavefunction’ for the function $\psi(\ell)$ in the
decomposition of a state in terms of geodesic states; because the inner
product is modified and states $\ket{\ell}$ with different $\ell$ are no
longer orthogonal, this is _not_ equal to the overlap
$\langle\ell|\psi\rangle$ (though this does have an alternative interpretation
as a non-perturbative ‘Wheeler-DeWitt wavefunction’, discussed in section
8.3).

Despite the modified norm, the Hamiltonian (1.2) we used for the perturbative
theory continues to apply in the exact theory. More precisely, acting with the
differential operator (1.1) on a wavefunction $\psi(\ell)$ for the state
$|\psi\rangle$ produces a wavefunction for $H|\psi\rangle$. Nonetheless, due
to the redundancy of the geodesic length description, there are many other
ways to represent the same operator.

###  1.2 Defining bulk operators

Now that we have a description of the bulk Hilbert space $\mathcal{H}$, we can
begin to discuss operators. Since the underlying vector space of wavefunctions
$\psi(\ell)$ is unchanged by the non-perturbative effects, one might think
that an operator on the perturbative Hilbert space $\mathcal{H}_{0}$
straightforwardly gives us an operator on $\mathcal{H}$. However, there is an
important subtlety: a generic operator on $\mathcal{H}_{0}$ will not map null
states to null states, so does not give a well-defined operator on
$\mathcal{H}$. (A second problem is that a Hermitian operator on
$\mathcal{H}_{0}$ will not typically be Hermitian with respect to the modified
inner product.) Given this, the non-perturbative definition of a
perturbatively-defined bulk operator is inherently ambiguous.

Take the simple example of a ‘length operator’ $\widehat{\ell}$ for which
geodesic states are eigenstates:
$\widehat{\ell}|\ell\rangle=\ell|\ell\rangle$. This is the usual position
operator on the perturbative Hilbert space (1.1), but it is ill-defined with
the non-perturbative inner product (1.3) for the reasons explained above.
However, for any invertible function $\mathcal{F}(\ell)$ of the length, we can
make an alternative definition that does not suffer from the same problems:

$\displaystyle\widehat{\ell_{\mathcal{F}}}:=\mathcal{F}^{-1}\left(\int
d\ell\,\mathcal{F}(\ell)|\ell\rangle\langle\ell|\right).$ (1.6)

With a trivial inner product, geodesic states $|\ell\rangle$ would be
eigenstates of this operator with eigenvalue $\ell$, but this is not true once
$\langle\ell^{\prime}|\ell\rangle$ is non-diagonal. We get a different
operator for different functions $\mathcal{F}$, so there is an ambiguity in
defining operators non-perturbatively. This ambiguity can also be seen at a
path integral level when summing over geometries with higher topologies. As
we’ve already emphasized, higher-topology AdS2 geometries have multiple
geodesics between any two boundary points. Thus, when defining the geodesic
length operator there is an ambiguity in which geodesic we choose (or the
relative weight we lend to each geodesic). The relation between the Hilbert
space ambiguity and the geometric ambiguity can be best seen by looking at the
operator $\mathcal{F}(\widehat{\ell_{\mathcal{F}}})$, which has the same
eigenfunctions as and related eigenvalues to $\widehat{\ell_{\mathcal{F}}}$.
By using (1.3), we shall show that matrix elements of this operator have a
path integral interpretation as a weighted sum of $\mathcal{F}(\ell)$ over all
possible geodesics:

$\displaystyle\bra{\psi^{\prime}}\mathcal{F}(\widehat{\ell_{\mathcal{F}}})\ket{\psi}=\int\frac{Dg_{\mu\nu}D\Phi}{\text{Diffs}}\left(\sum_{\gamma}\mathcal{F}(\ell_{\gamma})\right)e^{-I_{\text{JT}}}\,,$
(1.7)

where the boundary conditions are determined by the states $|\psi\rangle$ and
$|\psi^{\prime}\rangle$, and the sum runs over all non self-intersecting
geodesic slices $\gamma$ between specified boundary points. In other words,
for each relative weight of geodesics, there is an associated length operator
that we can define at the bulk Hilbert space level.

To highlight the usefulness of our formalism, we shall focus on two examples
of bulk operators, both of which shall serve as useful probes for the interior
of two-sided black holes.

###  1.3 Case study I: length operators

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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{20.62395pt}{27.74565pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ Figure 1:  Two-sided
near-extremal black hole setup in which we shall study the collision energy
between an observer and a particle, infalling from opposite sides. In the
left-most diagram, an infalling
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}W$ particle
encounters a collision almost immediately after crossing the horizon. By
applying a boost to the middle figure, we get the right-most diagram. We see
that ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}W}$
does not experience any collision with
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}V$ until
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}W}$ gets
extremely close to the future inner horizon (which is expected to be singular
in any realistic near-extremal black hole.) In other words, while there is
still a high-energy collision in the growing wormhole, the $W$ particle enjoys
a potentially large amount of proper time in the interior of the black hole
before the collision. In this sense, a growing wormhole is safer than a
shrinking wormhole.

aStudying the behavior of length operators non-perturbatively can help us
understand whether an infalling observer sees an energetic particle when
crossing the horizon of a black hole, a firewall [17, 18]. Consider the setup
presented in figure 1. An observer falls into a two-sided black hole from the
right at point $W$, and a particle is sent in from the left at point $V$. The
existence of a firewall is correlated with whether the wormhole between $V$
and $W$ is growing or shrinking [19, 20, 21]. A shrinking wormhole leads to an
exponential blue-shift factor which will lead to a violent collision between
an infalling observer and the particle sent in from the opposite side. The
bulk state, in this case, is usually referred to as a white hole. A growing
wormhole leads to an exponential red-shift factor which makes the collision
peaceful. The bulk state, in this case, is a black hole. Thus, the probability
distribution for the ”velocities” of the wormhole length can offer a proxy for
the probability of hitting a firewall. Perturbatively, if we fix the energies
on both sides to be close to $E$, this velocity operator should only have two
possible eigenvalues ($+\sqrt{2E}$ and $-\sqrt{2E}$), the probability of
seeing a negative eigenvalue approximating that of seeing a firewall. While at
early times, we expect the state of the black hole to have approximately zero
probability of detecting a shrinking wormhole, at very late times, the black
hole could be in a complicated linear combination of velocity eigenstates. At
such times, what is the probability of detecting a shrinking wormhole?

To understand such probabilities we should thus start by having a well-defined
length operator that acts on the bulk Hilbert space discussed above. A length
operator $\widehat{\ell}_{\Delta}$ that we will study in detail will be
obtained by taking $\mathcal{F}(\ell)=e^{-\Delta\ell}$ in (1.6). This operator
can be viewed as computing the length from the matrix elements of the bi-local
operator $O_{L}O_{R}$, each operator of scaling dimension $\Delta$ being
inserted on opposite sides of the black hole. In other words, this length
operator is directly defined in terms of simple boundary operators. As we
shall explain, the matrix elements of this length operator can be explicitly
computed for any member of the JT ensemble, and from this, given a
wavefunction, we can compute the probability of detecting any eigenvalue of
the length operator or any eigenvalue of its associated velocity
operator.222We will take the velocity operator to be defined as
$\widehat{v}_{\Delta}=i[H,\ell_{\Delta}]$, where $H=H_{L}=H_{R}$ is the ADM
Hamiltonian operator on either boundary. By numerically sampling from the
ensemble dual to JT gravity, we can obtain probability distributions for the
observed lengths and velocities that capture all possible corrections in
$e^{-S_{0}}$.

Starting with a two-sided black hole in a microcanonical window,333With
$H_{L}=H_{R}$. we find that the length of the wormhole between V and W starts
by growing linearly with time, after which it plateaus at a time and value
$~{}e^{S_{\text{BH}}}$.444This result is consistent with previous proposals
for the length operator in the literature, for example [22, 21]. However, as
far as we are aware, the Hilbert space interpretation for all length operators
discussed in the past was unclear.  Consequently, the expectation value of the
associated wormhole velocity vanishes at late times. To better understand this
result, we compute the probability distribution of detecting any given
wormhole velocity. We find that the probability of detecting a negative
velocity is $1/2$ at late times. At first sight, this suggests that if the
velocity operator is a good proxy for firewalls, the probability of
encountering a firewall at late times is also $1/2$. This probability is self-
averaging and universal: it is independent of the choice of ensemble and of
the value of $\Delta$ that we choose in order to define
$\widehat{\ell}_{\Delta}$. This is consistent with the findings of [21], who
computed the probability for the wormhole to shrink to first non-trivial order
in $e^{-S_{0}}$. However, we find that an $O(1)$ fraction of this probability
is concentrated on eigenvalues of the velocity operator that do not have a
good perturbative description, i.e. they are not close to $\pm\sqrt{2E}$. In
other words, a more refined description of our results is that there are equal
$O(1)$ probabilities of detecting a black hole/white hole state, and there is
an $O(1)$ probability of detecting a state that does not have any good
perturbative description; we will refer to such states as gray holes.555This
is slightly different than what is known as the gray hole scenario put forward
in [19]. There, the state at late times was believed to be in a linear
combination of black hole states and white hole states. Here, we see the
appearance of a third type of states that do not have a good semi-classical
description, i.e. a gray hole state.  This suggests that even if we start with
a perturbatively well-defined firewall operator, whose eigenvalues are $\pm 1$
for detecting or not detecting a firewall, its non-perturbative definition
will have some eigenvalues that could be very different from $\pm 1$ and
associated eigenstates that do not have a good perturbative description.

###  1.4 Case study II: center-of-mass collision energy

A second operator that serves as a more direct proxy for the experience of an
infalling observer is the center-of-mass collision energy between the observer
and the infalling particle from the opposite side, as in figure 1. We will
thus define and study the operator associated with this energy. In our AdS2
geometry, the CM collision energy can be obtained by studying the Casimir
operator associated with the $\mathfrak{sl}(2,\mathbb{R})$ isometry of the
spacetime [23, 24]. While the perturbative properties of the length operator
were previously understood, the collision energy operator or the related
Casimir operator has not been previously studied in the perturbative regime.
For the two-particle setup we are considering in figure 1, the eigenvalues of
such an operator are quantized. This operator is also conserved at the
perturbative level: thus, we can analyze the two-particle wavefunction in the
collision energy basis is equal on all time slices past the insertion points
of $V$ and $W$. We will thus analyze this two-particle wavefunction in the
Casimir basis on some slice to the future of $V$ and $W$ and track its
behavior as we change the time at which $V$ and $W$ are inserted.
Perturbatively, from this Casimir operator, we find that the likeliest
collision energy that an observer would detect grows exponentially with time,
and the wavefunction spread also grows exponentially with time.

Because it quickly spreads with time, the wavefunction in the collision basis
sees non-perturbative corrections at relatively early times, $t\sim S_{0}$.
The origin of such corrections comes from wormhole contributions in which the
conservation of the Casimir is violated, a phenomenon similar to the violation
of global symmetry charge conservation discussed in [25, 26, 27]. Even at such
early times, an observer should see that the eigenvalues of the Casimir no
longer have the quantized spacing predicted by the perturbative calculation.
Instead, we show that at large times the behavior of the Casimir operator
mimics that of an exponential of the length operator that we discussed above.
Thus, the same ambiguities that appeared in the non-perturbative definition of
the length reappear when studying the behavior of the collision energy at late
times. We thus expect that the universal behavior seen for the length operator
to also be present for the collision energy operator, whose expectation value
should thus plateau at a time and values $\sim e^{S_{0}}$.

###  1.5 Outline

The outline of the paper is as follows. In section 2, we review the
perturbative properties of JT gravity, after which we derive the non-
perturbative modification to the inner product and explore its effect on
Hamiltonian evolution. In section 3, we explicitly derive all null-states that
exist in our construction that need to be quotiented out from our vector space
in order to have a well-defined Hilbert space. In section 4, we explain in
further detail why there is an ambiguity in the non-perturbative definition of
any bulk operator both within the Hilbert space and at the path integral
level. In section 5, we focus on a specific choice of wormhole length and
velocity operators and study them numerically in great detail. In section 6,
we generalize our construction of a non-perturbative bulk Hilbert space to JT
gravity with matter. We then use this construction in section 7 to explicitly
study the CM collision energy operator both at a perturbative and non-
perturbative level. Finally, in section 8, we discuss several consequences and
interpretations of our results.
Note: During the development of these results, we became aware of the upcoming
work of [28], who present related results (to all orders in $e^{-S_{0}}$) for
the probability of detecting a firewall.

##  2 The non-perturbative geodesic Hilbert space

###  2.1 Perturbative Hilbert space of pure JT: review

We begin by reviewing the relevant results of quantization of pure JT gravity
on spacetimes of disc topology, which can be obtained from many different
perspectives [23, 29, 30, 31, 32, 14, 33, 34].

We study the Hilbert space with two asymptotic boundaries, defined by
Dirichlet boundary conditions for the proper length of the boundary
$L=\beta/\epsilon$ and for the dilaton
$\Phi|_{\partial}=\frac{\Phi_{b}}{\epsilon}$, where we take the cutoff
$\epsilon\to 0$. The value of $\phi_{b}$ simply sets the units of energy and
time, and we work in units with $\phi_{b}=1$. The physical time $t$ on this
boundary is defined (according to the usual holographic prescription) as
$\epsilon$ times the proper time. The theory has a single degree of freedom,
which we can take to be the renormalized geodesic length $\ell$ between the
two boundaries: $\ell$ equals the proper length minus $2\log\epsilon^{-1}$,
which gives a finite (but not necessarily positive) quantity when we take
$\epsilon\to 0$.

The resulting ‘perturbative Hilbert space’ $\mathcal{H}_{0}=L^{2}(\mathbb{R})$
consists of wavefunctions of the length, with the usual inner product as given
in (1.1). We can define states directly in this description using geodesic
boundary conditions, giving $\delta$-function states of definite length
$|\ell\rangle$, normalised so that
$\langle\ell^{\prime}|\ell\rangle_{0}=\delta(\ell-\ell^{\prime})$. Time
evolution on the left or right side is generated by a Hamiltonian $H_{L}$ or
$H_{R}$; these operators are in fact equal for pure JT, and act on
wavefunctions $\psi(\ell)$ as

$H=H_{L}=H_{R}=-\frac{1}{2}\partial_{\ell}^{2}+2e^{-\ell}.$ (2.1)

The Hamiltonian is diagonalised by scattering states $|E\rangle$ for $E>0$,
with wavefunctions

$\phi_{E}(\ell)=\langle\ell|E\rangle_{0}=4K_{i\sqrt{8E}}\left(4e^{-\ell/2}\right).$
(2.2)

We have defined these states with a convenient normalisation, which is singled
out (up to an overall constant) by the fact that the wavefunction becomes
independent of $E$ when we take $\ell\to-\infty$. (Note also that the Bessel
function is real.) This gives the following inner products and completeness
relation on $\mathcal{H}_{0}$:

$\langle
E^{\prime}|E\rangle_{0}=\frac{\delta(E-E^{\prime})}{\rho_{0}(E)}\,,\qquad\int_{0}^{\infty}dE\,\rho_{0}(E)|E\rangle\langle
E|=\mathbb{1}_{0}\,.$ (2.3)

Here we have defined

$\rho_{0}(E)=\frac{1}{4\pi^{2}}\sinh\left(2\pi\sqrt{2E}\right)\,$ (2.4)

which (up to a factor of $e^{S_{0}}$) is familiar as the density of states
coming from the disk in JT: that is $Z_{0}(\beta)=e^{S_{0}}\int
dE\,\rho_{0}(E)e^{-\beta E}$, where the left-hand side is defined as a path
integral over Euclidean spacetimes with disk topology and renormalized
boundary length $\beta$.

This disk partition function can also be interpreted as an inner product
$Z_{0}(\tau+\tau^{\prime})=\langle\tau^{\prime}|\tau\rangle_{0}$ (computed at
infinite $S_{0}$) by cutting it into two half-disks, with $|\tau\rangle$ the
two-sided state defined by an asymptotic Euclidean boundary of renormalized
length $\tau>0$. These states will be useful for us because they have a simple
definition as thermofield double (TFD) states in the non-perturbative theory
(defined by a dual boundary quantum mechanics as described in a moment). We
can write $|\tau\rangle$ in terms of the energy eigenstates above, with

$|\tau\rangle=e^{S_{0}/2}\int dE\,\rho_{0}(E)e^{-\tau E}|E\rangle;$ (2.5)

this is fixed (up to an $E$-dependent phase) by the requirement that the
overlaps give $Z_{0}$. We can also write $|\tau\rangle$ as a superposition of
geodesic states,

$\begin{gathered}|\tau\rangle=e^{S_{0}/2}\int
d\ell\,Z_{{\put(4.0,4.0){\oval(8.0,8.0)[b]}\put(0.0,4.0){\line(1,0){8.0}}\phantom{\subset\,}}}(\tau;\ell)|\ell\rangle,\\\
Z_{\put(4.0,4.0){\oval(8.0,8.0)[b]}\put(0.0,4.0){\line(1,0){8.0}}\phantom{\subset\,}}(\tau;\ell)=\int
dE\,\rho_{0}(E)e^{-\tau E}\phi_{E}(\ell).\end{gathered}$ (2.6)

This expression is obtained by taking the inner product with $|\ell\rangle$
using (2.2), but can also be computed directly from a path-integral on the
half-disk geometry bounded by an asymptotic boundary and a geodesic boundary
[31]:

$\displaystyle
e^{S_{0}/2}Z_{{\put(4.0,4.0){\oval(8.0,8.0)[b]}\put(0.0,4.0){\line(1,0){8.0}}\phantom{\subset\,}}}(\tau;\ell)=\langle\ell|\tau\rangle_{0}=\leavevmode\hbox
to125.19pt{\vbox to119.9pt{\pgfpicture\makeatletter\hbox{\hskip
62.59598pt\lower-62.59598pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{-62.59598pt}{2.84544pt}\pgfsys@moveto{-62.59598pt}{2.84544pt}\pgfsys@lineto{-62.59598pt}{-62.59598pt}\pgfsys@lineto{62.59598pt}{-62.59598pt}\pgfsys@lineto{62.59598pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{62.59598pt}{-62.59598pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{56.90552pt}{31.42845pt}{31.42845pt}{56.90552pt}{0.0pt}{56.90552pt}\pgfsys@curveto{-31.42845pt}{56.90552pt}{-56.90552pt}{31.42845pt}{-56.90552pt}{0.0pt}\pgfsys@curveto{-56.90552pt}{-31.42845pt}{-31.42845pt}{-56.90552pt}{0.0pt}{-56.90552pt}\pgfsys@curveto{31.42845pt}{-56.90552pt}{56.90552pt}{-31.42845pt}{56.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{}\pgfsys@moveto{-56.90552pt}{2.84544pt}\pgfsys@lineto{56.90552pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.08334pt}{-7.83199pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$\ell$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.19516pt}{-51.9451pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\tau$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (2.7)

###  2.2 Defining the non-perturbative theory

To study a similar geodesic Hilbert space in the non-perturbative theory, we
must first give a precise definition of what we mean by that theory.
Foreshadowing the key idea, we will identify a relation between geodesic
states $|\ell\rangle$ and the asymptotic ‘TFD’ states $|\tau\rangle$ which is
independent of the full details of the bulk theory. Using this, we can define
the inner products $\langle\ell^{\prime}|\ell\rangle$ (and matrix elements
like $\langle\ell^{\prime}|e^{-\tau H}|\ell\rangle$) by first rewriting them
in terms of asymptotically-defined inner products
$\langle\tau^{\prime}|\tau\rangle$. We can write these overlaps as $Z(\beta)$,
which is defined by an asymptotic boundary condition with a Euclidean thermal
circle of length $\beta=\tau+\tau^{\prime}$. So, it is sufficient to define
the theory via the amplitudes with asymptotic circular boundaries $Z(\beta)$.
While this definition of the theory involves Euclidean quantities, with the
answer in hand, we can nonetheless apply it to real-time evolution and
Lorentzian questions.

As sketched in section 1.1, there are two possible (closely related)
perspectives. The first is to directly take the definition and interpretation
of JT gravity given in [7]. With that definition, the theory computes
amplitudes $Z_{n}(\beta_{1},\ldots,\beta_{n})$ with $n$ disconnected circular
boundaries (order-by-order in a genus expansion). $Z_{n}$ is then interpreted
as an $n$th moment $\overline{Z(\beta_{1})\cdots Z(\beta_{n})}$ of random
variables $Z(\beta)$, with $\overline{\phantom{I}\cdot\phantom{I}}$ denoting
the average with respect to some probability distribution. In this
distribution, $Z(\beta)$ is the partition function $\Tr(e^{-\beta H})$ of a
random Hamiltonian $H$, with discrete (random) spectrum
$\\{E_{i}\\}_{i=0}^{\infty}$. We can describe this as an ensemble of quantum
mechanical theories dual to pure JT. Directly importing this to the geodesic
states, we will end up calculating statistics
$\overline{\langle\ell_{1}^{\prime}|\ell_{1}\rangle\cdots\langle\ell_{n}^{\prime}|\ell_{n}\rangle}$,
where $\langle\ell^{\prime}|\ell\rangle$ is interpreted a random inner
product.

For the second perspective, we can ask what the inner product
$\langle\ell^{\prime}|\ell\rangle$ looks like if we select a specific member
of the ensemble. In practical terms, we can sample a single random matrix $H$
from the probability distribution, and use the resulting specific spectrum
$\\{E_{i}\\}_{i=0}^{\infty}$ to define the theory. Then, an $n$ boundary
amplitude factorizes into a product of $n$ partition functions
$Z(\beta)=\sum_{i}e^{-\beta E_{i}}$ (so we need only ever consider a single
asymptotic boundary of spacetime). It is reasonable to expect the result for a
typical spectrum to be indicative of the situation in a theory with a
conventional (non-ensemble) holographic dual. From the bulk perspective, this
means that we are choosing a specific $\alpha$-state of closed ‘baby’
universes [35, 36, 15]. We can arrange this by modifying the boundary
conditions, perhaps by adding a carefully arranged set of additional
asymptotic boundaries [15], or similarly by additional ‘eigenbrane’ boundary
conditions on which spacetime can terminate [37], though we will not need to
be very specific about the details. In any case, with this perspective, we
take a single boundary dual quantum mechanics and compute the associated
single matrix of inner products $\langle\ell^{\prime}|\ell\rangle$.

We will mostly write with this latter perspective in mind, though they are
essentially equivalent since the ensemble can always be recovered by thinking
of $\\{E_{i}\\}$ as random variables rather than specific numbers.

We should comment on how the two-sided Hilbert space of JT is related to the
Hilbert space $\mathcal{H}_{\partial}$ of this boundary dual quantum
mechanics. Since we have two asymptotic spatial boundaries, we expect to have
two copies of the dual system $\mathcal{H}_{R}\simeq\mathcal{H}_{\partial}$
and $\mathcal{H}_{L}\simeq\mathcal{H}_{\partial}^{*}$ (the complex conjugate
Hilbert space) living on the right and left boundaries respectively, and thus
a total Hilbert space $\mathcal{H}\simeq\mathcal{H}_{L}\otimes\mathcal{H}_{R}$
with a basis $|i_{L},i_{R}\rangle$ of eigenstates of left and right
Hamiltonians $H_{L},H_{R}$. While this is indeed the case when we consider JT
gravity coupled to matter, it fails in pure JT since all states have
$H_{L}=H_{R}$. This is clear from the definition of the bulk theory through
the amplitudes as described above, since the TFD states $|\tau\rangle$ (which
satisfy $H_{L}=H_{R}$ automatically) span all possible boundary conditions
creating states of $\mathcal{H}$. Thus, $\mathcal{H}$ consists only of the
‘diagonal’ subspace spanned by states $|i,i\rangle$ with $i_{L}=i_{R}=i$
(which we will simply write as $|i\rangle$).

Finally, we note that there is a third perspective, which is natural from the
gravitational point of view, but gives an inequivalent construction. Once we
include fluctuations of topology, it is no longer clear that we can describe a
two-boundary bulk Hilbert space by only a single geodesic length, because
there are processes which produce closed ‘baby’ universes. If we keep track of
these closed components, we arrive at an inherently different Hilbert space,
which will be explored in future work (previewed in section 8.4). The ensemble
is recovered by ‘tracing out’ the closed universe sector, while the single
dual emerges by placing the closed universes in an $\alpha$ state.

###  2.3 Relating geodesic and asymptotic states

With the preliminaries out the way, we can now proceed to compute the full
inner product $\langle\ell^{\prime}|\ell\rangle$, defined as the path integral
over all Euclidean spacetimes bounded by two geodesics of lengths
$\ell,\ell^{\prime}$. To do this, we first consider computing the inner
product between thermofield double states $\langle\tau^{\prime}|\tau\rangle$,
computed by the path integral $Z(\tau+\tau^{\prime})$ with boundary conditions
of an asymptotic thermal circle. We observe that for pure JT gravity, in every
spacetime contributing to this path integral, there is a unique geodesic
homotopic to each segment of the asymptotic boundary. Furthermore, the region
between the boundary and this geodesic is a simple half-disk, namely the
geometry pictured in (2.7). We discuss the generality of this result and the
underlying assumptions in section 8.2. Now, we are free to cut along these
geodesics and divide the path integral into three parts: two half-discs and an
integral over spacetimes bounded by two geodesics, which defines the length-
basis inner product $\langle\ell^{\prime}|\ell\rangle$:

$Z(\tau+\tau^{\prime})=e^{S_{0}}\int d\ell
d\ell^{\prime}\,Z_{\put(4.0,4.0){\oval(8.0,8.0)[b]}\put(0.0,4.0){\line(1,0){8.0}}\phantom{\subset\,}}(\tau;\ell)Z_{\put(4.0,4.0){\oval(8.0,8.0)[b]}\put(0.0,4.0){\line(1,0){8.0}}\phantom{\subset\,}}(\tau^{\prime};\ell^{\prime})\langle\ell^{\prime}|\ell\rangle.$
(2.8)

This formula will be enough to compute $\langle\ell^{\prime}|\ell\rangle$, see
Figure 2.

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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ Figure 2: On the
left, we show a sample contribution to the disk partition function. It
includes a handle disk. By stripping off the half-disk wavefunctions, we
extract a contribution to the inner product
$\innerproduct{\ell}{\ell^{\prime}}$.

We are defining $Z(\beta)$ non-perturbatively as the partition function of a
dual quantum mechanics, so we can rewrite it as a sum over the spectrum
$\\{E_{n}\\}$ of the Hamiltonian. Using this as well as (2.6) for the half-
disk path integral, we can write (2.8) as

$\sum_{i=0}^{\infty}e^{-(\tau+\tau^{\prime})E_{i}}=e^{S_{0}}\int d\ell
d\ell^{\prime}dE\rho_{0}(E)dE^{\prime}\rho_{0}(E^{\prime})e^{-(\tau
E+\tau^{\prime}E^{\prime})}\phi_{E}(\ell)\phi_{E^{\prime}}(\ell^{\prime})\langle\ell^{\prime}|\ell\rangle.$
(2.9)

Now, for this to hold for all $\tau,\tau^{\prime}$ we we must have

$\sum_{i=0}^{\infty}\delta(E-E_{i})\delta(E^{\prime}-E_{i})=e^{S_{0}}\rho_{0}(E)\rho_{0}(E^{\prime})\int
d\ell
d\ell^{\prime}\phi_{E}(\ell)\phi_{E^{\prime}}(\ell^{\prime})\langle\ell^{\prime}|\ell\rangle.$
(2.10)

Finally, we can invert this using $\int
dE\rho_{0}(E)\phi_{E}(\ell)\phi_{E}(\ell^{\prime})=\delta(\ell-\ell^{\prime})$
(the completeness relation for the states $|E\rangle$ in
$\mathcal{H}_{0}$)666The fact that this integral runs only over $E>0$ might
raise suspicions that something goes wrong when the spectrum contains negative
energies. In fact, while the orthogonality of $\phi_{E}(\ell)$ states only
makes sense for $E>0$, the integral $\int
d\ell\phi_{E}(\ell)Z_{\put(4.0,4.0){\oval(8.0,8.0)[b]}\put(0.0,4.0){\line(1,0){8.0}}\phantom{\subset\,}}(\tau,\ell)$
still converges to $e^{-\tau E}$ for $E<0$. So while the intermediate step
(2.10) does not make sense because the order of integration cannot be
exchanged, the final result is still correct even when $E_{i}<0$ for some
states. to obtain our final result for the non-perturbative geodesic inner
product:

$\langle\ell^{\prime}|\ell\rangle=e^{-S_{0}}\sum_{i=0}^{\infty}\phi_{E_{i}}(\ell)\phi_{E_{i}}(\ell^{\prime}).$
(2.11)

Note that if we approximate the sum $\sum_{i}$ by an integral $\int
dE\,e^{S_{0}}\rho(E)$ with the disk density of states, we recover
$\langle\ell^{\prime}|\ell\rangle_{0}=\delta(\ell-\ell^{\prime})$. This means
that the inner product of wavefunctions with broad, coherent energy resolution
will be very close to the perturbative result. Nonetheless, for some purposes
(2.11) is a dramatic departure from $\langle\ell^{\prime}|\ell\rangle_{0}$.

We can express this result more directly in terms of the states themselves: we
have learned that the relation (2.6) expressing the thermofield double state
in terms of the geodesic states $|\tau\rangle=\int
d\ell\,Z_{\put(4.0,4.0){\oval(8.0,8.0)[b]}\put(0.0,4.0){\line(1,0){8.0}}\phantom{\subset\,}}(\tau;\ell)|\ell\rangle$
continues to hold exactly non-perturbatively! We verify this by computing the
overlap of $|\tau\rangle$ with any state (for which it’s sufficient to take
another thermofield double, since these span the Hilbert space), and cutting
the resulting path integral along the geodesic homotopic to the boundary. In
this form, it’s also straightforward to write the geodesic states directly in
terms of an orthonormal basis $|i\rangle$ of energy eigenstates ($\langle
j|i\rangle=\delta_{i,j}$), using the decomposition
$|\tau\rangle=\sum_{n}e^{-\tau E_{i}}|i\rangle$ of the thermofield double:

$\langle i|\tau\rangle=e^{-\tau E_{i}}\implies\langle
i|\ell\rangle=e^{-S_{0}/2}\phi_{E_{i}}(\ell)\qquad(i=0,1,2,\ldots).$ (2.12)

###  2.4 An alternative ensemble perspective

To reiterate the argument and to make the non-perturbative path integral
somewhat more concrete, it is illuminating to restate things in the context of
the JT ensemble [6], rather than a single specific dual spectrum as above. For
this, consider the connected contribution to the product of $n$ inner
products, defined by a path integral over connected geometries with $n$ pairs
of geodesic boundaries. Such a product has a genus expansion:

$\overline{\langle\ell_{1}^{\prime}|\ell_{1}\rangle\cdots\langle\ell_{n}^{\prime}|\ell_{n}\rangle}_{\text{conn.}}=e^{(1-2g-n)S_{0}}\sum_{g=0}^{\infty}\eta_{n,g}(\ell_{1},\ldots,\ell_{n};\ell_{1}^{\prime},\ldots,\ell_{n}^{\prime}),$
(2.13)

where
$\eta_{n,g}(\ell_{1},\ldots,\ell_{n};\ell_{1}^{\prime},\ldots,\ell_{n}^{\prime})$
is computed by summing over all genus $g$ geometries with $n$ boundaries, each
consisting of a pair of geodesics of renormalized lengths
$\ell,\,\ell^{\prime}$. The $n=1,g=0$ contribution gives the perturbative
result $\delta(\ell-\ell^{\prime})$, while other terms give non-perturbative
corrections suppressed by powers of $e^{-S_{0}}$. For example, the following
geometries can be used to compute $\eta_{1,1}$ and $\eta_{2,0}$ respectively
which give the leading non-perturbative corrections to
$\overline{\innerproduct{\ell^{\prime}}{\ell}}$ and
$\overline{\innerproduct{\ell^{\prime}_{1}}{\ell_{1}}\innerproduct{\ell^{\prime}_{2}}{\ell_{2}}}_{\text{conn.}}$:777Here,
we are assuming the sum over geometries solely includes orientable surfaces. A
similar $e^{-S_{0}}$ expansion would apply to the unorientable case.

$\displaystyle\overline{\innerproduct{\ell^{\prime}}{\ell}}\quad=\quad\delta(\ell-\ell^{\prime})+\quad\leavevmode\hbox
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}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{74.68849pt}\pgfsys@curveto{-4.26784pt}{81.09041pt}{-4.26784pt}{89.62611pt}{0.0pt}{96.02806pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{}\pgfsys@moveto{-2.13408pt}{72.5544pt}\pgfsys@curveto{5.33488pt}{81.09041pt}{5.33488pt}{89.62611pt}{-2.13408pt}{98.16214pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad+\quad
O(e^{-4S_{0}})$ (2.14)
$\displaystyle\overline{\innerproduct{\ell^{\prime}_{1}}{\ell_{1}}\innerproduct{\ell^{\prime}_{2}}{\ell_{2}}}_{\text{conn.}}\quad=\quad\leavevmode\hbox
to171.52pt{\vbox to127.61pt{\pgfpicture\makeatletter\hbox{\hskip
85.75827pt\lower 13.63213pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-35.49269pt}{18.76958pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ell_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-45.53247pt}{60.67093pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ell^{\prime}_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-35.49269pt}{82.78828pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ell_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-45.53247pt}{124.68964pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ell^{\prime}_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{}}{{}}{}{}{}{}{}
{}{{}{}{}}{}{} {}{{}{}{}}{}{} {}{{}{}{}}{}{}{}{}{}{}{}{}{} {}{{}{}{}}{}{}
{}{{}{}{}}{}{} {}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\definecolor[named]{pgffillcolor}{rgb}{0.9,0.9,1}\pgfsys@color@rgb@fill{0.9}{0.9}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{85.35828pt}{46.94704pt}\pgfsys@curveto{85.35828pt}{46.94704pt}{54.52242pt}{46.82399pt}{42.67914pt}{42.67888pt}\pgfsys@curveto{30.83585pt}{38.53375pt}{11.84328pt}{17.07178pt}{0.0pt}{17.07178pt}\pgfsys@curveto{-11.84328pt}{17.07178pt}{-30.83585pt}{38.53375pt}{-42.67914pt}{42.67888pt}\pgfsys@curveto{-54.52242pt}{46.82399pt}{-85.35828pt}{46.94704pt}{-85.35828pt}{46.94704pt}\pgfsys@lineto{-85.35828pt}{46.94704pt}\pgfsys@curveto{-85.35828pt}{46.94704pt}{-54.52242pt}{47.07008pt}{-42.67914pt}{51.2152pt}\pgfsys@curveto{-30.83585pt}{55.36032pt}{-11.84328pt}{76.8223pt}{0.0pt}{76.8223pt}\pgfsys@curveto{11.84328pt}{76.8223pt}{30.83585pt}{55.36032pt}{42.67914pt}{51.2152pt}\pgfsys@curveto{54.52242pt}{47.07008pt}{85.35828pt}{46.94704pt}{85.35828pt}{46.94704pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{{}}{}{}{}{}{}
{}{{}{}{}}{}{} {}{{}{}{}}{}{} {}{{}{}{}}{}{}{}{}{}{}{}{}{} {}{{}{}{}}{}{}
{}{{}{}{}}{}{} {}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\definecolor[named]{pgffillcolor}{rgb}{0.9,0.9,1}\pgfsys@color@rgb@fill{0.9}{0.9}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{85.35828pt}{110.96574pt}\pgfsys@curveto{85.35828pt}{110.96574pt}{54.52242pt}{110.8427pt}{42.67914pt}{106.69759pt}\pgfsys@curveto{30.83585pt}{102.55246pt}{11.84328pt}{81.09048pt}{0.0pt}{81.09048pt}\pgfsys@curveto{-11.84328pt}{81.09048pt}{-30.83585pt}{102.55246pt}{-42.67914pt}{106.69759pt}\pgfsys@curveto{-54.52242pt}{110.8427pt}{-85.35828pt}{110.96574pt}{-85.35828pt}{110.96574pt}\pgfsys@lineto{-85.35828pt}{110.96574pt}\pgfsys@curveto{-85.35828pt}{110.96574pt}{-54.52242pt}{111.08879pt}{-42.67914pt}{115.2339pt}\pgfsys@curveto{-30.83585pt}{119.37903pt}{-11.84328pt}{140.841pt}{0.0pt}{140.841pt}\pgfsys@curveto{11.84328pt}{140.841pt}{30.83585pt}{119.37903pt}{42.67914pt}{115.2339pt}\pgfsys@curveto{54.52242pt}{111.08879pt}{85.35828pt}{110.96574pt}{85.35828pt}{110.96574pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.85358pt}\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\definecolor[named]{pgffillcolor}{rgb}{0.9,0.9,1}\pgfsys@color@rgb@fill{0.9}{0.9}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{-16.00467pt}{46.94704pt}\pgfsys@curveto{-9.54314pt}{71.06183pt}{-9.54314pt}{86.85094pt}{-16.00467pt}{110.96574pt}\pgfsys@curveto{-9.76332pt}{121.77611pt}{9.76332pt}{121.77611pt}{16.00467pt}{110.96574pt}\pgfsys@curveto{9.54314pt}{86.85094pt}{9.54314pt}{71.06183pt}{16.00467pt}{46.94704pt}\pgfsys@curveto{9.76332pt}{36.13667pt}{-9.76332pt}{36.13667pt}{-16.00467pt}{46.94704pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{{}}{}{}{}{}{}
{}{{}{}{}}{}{} {}{{}{}{}}{}{} {}{{}{}{}}{}{}{}{}{}{}{}{}{} {}{{}{}{}}{}{}
{}{{}{}{}}{}{} {}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{}\pgfsys@moveto{85.35828pt}{110.96574pt}\pgfsys@curveto{85.35828pt}{110.96574pt}{54.52242pt}{110.8427pt}{42.67914pt}{106.69759pt}\pgfsys@curveto{30.83585pt}{102.55246pt}{11.84328pt}{81.09048pt}{0.0pt}{81.09048pt}\pgfsys@curveto{-11.84328pt}{81.09048pt}{-30.83585pt}{102.55246pt}{-42.67914pt}{106.69759pt}\pgfsys@curveto{-54.52242pt}{110.8427pt}{-85.35828pt}{110.96574pt}{-85.35828pt}{110.96574pt}\pgfsys@lineto{-85.35828pt}{110.96574pt}\pgfsys@curveto{-85.35828pt}{110.96574pt}{-54.52242pt}{111.08879pt}{-42.67914pt}{115.2339pt}\pgfsys@curveto{-30.83585pt}{119.37903pt}{-11.84328pt}{140.841pt}{0.0pt}{140.841pt}\pgfsys@curveto{11.84328pt}{140.841pt}{30.83585pt}{119.37903pt}{42.67914pt}{115.2339pt}\pgfsys@curveto{54.52242pt}{111.08879pt}{85.35828pt}{110.96574pt}{85.35828pt}{110.96574pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{{}}{}{}{}{}{}
{}{{}{}{}}{}{} {}{{}{}{}}{}{} {}{{}{}{}}{}{}{}{}{}{}{}{}{} {}{{}{}{}}{}{}
{}{{}{}{}}{}{} {}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{}\pgfsys@moveto{85.35828pt}{46.94704pt}\pgfsys@curveto{85.35828pt}{46.94704pt}{54.52242pt}{46.82399pt}{42.67914pt}{42.67888pt}\pgfsys@curveto{30.83585pt}{38.53375pt}{11.84328pt}{17.07178pt}{0.0pt}{17.07178pt}\pgfsys@curveto{-11.84328pt}{17.07178pt}{-30.83585pt}{38.53375pt}{-42.67914pt}{42.67888pt}\pgfsys@curveto{-54.52242pt}{46.82399pt}{-85.35828pt}{46.94704pt}{-85.35828pt}{46.94704pt}\pgfsys@lineto{-85.35828pt}{46.94704pt}\pgfsys@curveto{-85.35828pt}{46.94704pt}{-54.52242pt}{47.07008pt}{-42.67914pt}{51.2152pt}\pgfsys@curveto{-30.83585pt}{55.36032pt}{-11.84328pt}{76.8223pt}{0.0pt}{76.8223pt}\pgfsys@curveto{11.84328pt}{76.8223pt}{30.83585pt}{55.36032pt}{42.67914pt}{51.2152pt}\pgfsys@curveto{54.52242pt}{47.07008pt}{85.35828pt}{46.94704pt}{85.35828pt}{46.94704pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad+\quad
O(e^{-2S_{0}})\,.$ (2.15)

Now, just as in the argument above, we can relate the quantities $\eta_{n,g}$
to the usual $n$ boundary genus $g$ amplitudes
$Z_{n,g}(\beta_{1},\ldots,\beta_{n})$ which compute the terms in the genus
expansion of the cumulants of the partition function, by slicing along $2n$
geodesics. Explicitly,

$Z_{n,g}(\tau_{1}+\tau_{1}^{\prime},\ldots,\tau_{n}+\tau_{n}^{\prime})=\int\left(\prod_{k=1}^{n}d\ell_{k}d\ell_{k}^{\prime}Z_{\put(4.0,4.0){\oval(8.0,8.0)[b]}\put(0.0,4.0){\line(1,0){8.0}}\phantom{\subset\,}}(\tau_{k};\ell_{k})Z_{\put(4.0,4.0){\oval(8.0,8.0)[b]}\put(0.0,4.0){\line(1,0){8.0}}\phantom{\subset\,}}(\tau^{\prime}_{k};\ell^{\prime}_{k})\right)\eta_{n,g}(\ell_{1},\ldots,\ell_{n};\ell_{1}^{\prime},\ldots,\ell_{n}^{\prime}).$
(2.16)

Here, we have the advantage that we can be very concrete about the geometries
involved and the geodesic slicing since the path integral at fixed $n,g$
involves only a finite-dimensional integral over a moduli space of hyperbolic
surfaces. For each such surface, one can easily verify the crucial fact that
there is a unique geodesic homotopic to a given segment of boundary (in
particular, the analysis reduces to a ‘trumpet’ region lying between each
asymptotic boundary and a closed geodesic of length $b$).

Using (2.16),
$\eta_{n,g}(\ell_{1},\ldots,\ell_{n};\ell_{1}^{\prime},\ldots,\ell_{n}^{\prime})$
can be explicitly computed. Just like
$Z_{n,g}(\tau_{1}+\tau_{1}^{\prime},\ldots,\tau_{n}+\tau_{n}^{\prime})$ can be
computed by using trumpet geometries glued to hyperbolic surfaces with
arbitrary genus and geodesic boundaries, so too can the same algorithm be
applied to
$\eta_{n,g}(\ell_{1},\ldots,\ell_{n};\ell_{1}^{\prime},\ldots,\ell_{n}^{\prime})$.
The only new ingredient we require is a modified trumpet,

$\displaystyle\psi(\ell,\ell^{\prime};b)\equiv\leavevmode\hbox
to114.61pt{\vbox to72.32pt{\pgfpicture\makeatletter\hbox{\hskip
57.30551pt\lower-36.1602pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{}{}{}{} {}{{}{}{}}{}{}
{}{{}{}{}}{}{} {}{{}{}{}}{}{}{}{}{}{}{}{}{} {}{{}{}{}}{}{} {}{{}{}{}}{}{}
{}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.3}\pgfsys@invoke{
}{}\pgfsys@moveto{-56.90552pt}{0.0pt}\pgfsys@curveto{-56.90552pt}{0.0pt}{-36.34828pt}{0.08203pt}{-28.45276pt}{2.84544pt}\pgfsys@curveto{-20.55724pt}{5.60886pt}{-7.89552pt}{19.91684pt}{0.0pt}{19.91684pt}\pgfsys@curveto{7.89552pt}{19.91684pt}{20.55724pt}{5.60886pt}{28.45276pt}{2.84544pt}\pgfsys@curveto{36.34828pt}{0.08203pt}{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@curveto{56.90552pt}{0.0pt}{36.34828pt}{-0.08203pt}{28.45276pt}{-2.84544pt}\pgfsys@curveto{20.55724pt}{-5.60886pt}{7.89552pt}{-19.91684pt}{0.0pt}{-19.91684pt}\pgfsys@curveto{-7.89552pt}{-19.91684pt}{-20.55724pt}{-5.60886pt}{-28.45276pt}{-2.84544pt}\pgfsys@curveto{-36.34828pt}{-0.08203pt}{-56.90552pt}{0.0pt}{-56.90552pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.48334pt}{-31.02275pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ell_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ell_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.14583pt}{-3.47221pt}\pgfsys@invoke{
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}\hbox{{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ $\displaystyle=\int
dE\rho_{\text{Trumpet}}(E,b)\phi_{E}(\ell_{1})\phi_{E}(\ell_{2})$
$\displaystyle=4K_{0}\left(4\sqrt{e^{-\ell_{2}}+e^{-\ell_{1}}+2e^{-\left(\ell_{2}+\ell_{1}\right)/2}\cosh\frac{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}{2}}\right)\,,$
(2.17)

where the trumpet density of states is given by
$\rho_{\text{Trumpet}}(E,b)=\frac{\cos(b\sqrt{E})}{\pi\sqrt{E}}$. From this,
we can obtain any connected contribution to
$\eta_{n,g}(\ell_{1},\ldots,\ell_{n};\ell_{1}^{\prime},\ldots,\ell_{n}^{\prime})$
by gluing such trumpets to hyperbolic surfaces with $n$ geodesic boundaries,

$\displaystyle\eta_{n,g}(\ell_{1},\ldots,\ell_{n};\ell_{1}^{\prime},\ldots,\ell_{n}^{\prime})_{\text{conn.}}=\int\prod_{i=1}^{n}b_{i}\,db_{i}\,\psi(\ell_{1},\ell_{1}^{\prime};b_{1})\cdots\psi(\ell_{n},\ell_{n}^{\prime};b_{n})V_{g,n}(b_{1},\cdots,b_{n})\,,$
(2.18)

where $V_{g,n}$ is the volume of the moduli space of genus $g$ hyperbolic
surfaces with geodesic boundaries of specified lengths.

As a concrete example, let us compute the leading order connected correction
to
$\overline{\innerproduct{\ell^{\prime}_{1}}{\ell_{1}}\innerproduct{\ell^{\prime}_{2}}{\ell_{2}}}$
shown in (2.15). Using $V_{0,2}(b_{1},b_{2})=\delta(b_{1}-b_{2})/b_{2}$, the
geometry appearing in (2.15) gives

$\displaystyle\overline{\innerproduct{\ell^{\prime}_{1}}{\ell_{1}}\innerproduct{\ell^{\prime}_{2}}{\ell_{2}}}_{\text{conn.}}=$
$\displaystyle
16\int{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}\,\mathrm{d}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}\,{K_{0}\left(4\sqrt{e^{-\ell_{1}^{\prime}}+e^{-\ell_{1}}+2e^{-\left(\ell_{1}^{\prime}+\ell_{1}\right)/2}\cosh\frac{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}{2}}\right)K_{0}\left(4\sqrt{e^{-\ell_{2}}+e^{-\ell_{2}^{\prime}}+2e^{-\left(\ell_{2}+\ell_{2}^{\prime}\right)/2}\cosh\frac{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}{2}}\right)}.$
(2.19)

By summing over $g$ and performing the integrals over the geodesic lengths
$b_{i}$, the average product of inner-products can be simply expressed in
terms of the spectral density correlator
$\overline{\rho(E_{1})\dots\rho(E_{n})}$ in the JT ensemble, which is well
known in the literature. From this, (2.13) can be conveniently rewritten as

$\displaystyle\overline{\langle\ell_{1}^{\prime}|\ell_{1}\rangle\cdots\langle\ell_{n}^{\prime}|\ell_{n}\rangle}=$
$\displaystyle\int\mathrm{d}E_{1}\cdots\mathrm{d}E_{n}\,\overline{\rho(E_{1})\cdots\rho(E_{n})}\phi_{E_{1}}(\ell_{1})\phi_{E_{1}}(\ell_{1}^{\prime})\cdots\phi_{E_{n}}(\ell_{n})\phi_{E_{n}}(\ell_{n}^{\prime})\,.$
(2.20)

Since this holds for any $n$, from (2.20) we can deduce what the individual
inner products are in each member of the JT ensemble with energies
$\\{E_{i}\\}$. Namely, (2.20) implies

$\langle\ell^{\prime}|\ell\rangle=e^{-S_{0}}\int
dE\rho(E)\phi_{E}(\ell)\phi_{E}(\ell^{\prime}),$ (2.21)

where $\langle\ell^{\prime}|\ell\rangle$ and $\rho(E)$ are interpreted as
random variables. More precisely, our argument shows that this equality holds
for all moments to all orders in the genus expansion. Rewriting the density
$\rho(E)=\sum_{i}\delta(E-E_{i})$ in terms of a discrete spectrum, this is the
same as (2.11).

We note that it is tricky to write a direct formula for geodesic states in
terms of energy eigenstates analogous to (2.12) in the ensemble language
(unless we simply interpret the energies $E_{i}$ in (2.12) as random
variables). This is due to a conflict between continuum and discrete
normalized energy eigenfunctions. For similar reasons, it is subtle to attempt
to argue directly for the relationship between energy eigenstates and geodesic
states, which is why we choose to go via the thermofield double states where
such issues do not arise.

###  2.5 The Hamiltonian is unmodified

From the decomposition of $|\ell\rangle$ into energy eigenstates, it is
immediate that the Schrödinger operator (2.1) continues to generate time
translations on superpositions of geodesic states non-perturbatively. More
precisely, we mean that

$|\psi\rangle=\int d\ell\,\psi(\ell)|\ell\rangle\implies H|\psi\rangle=\int
d\ell\left(-\tfrac{1}{2}\psi^{\prime\prime}(\ell)+2e^{-\ell}\psi(\ell)\right)|\ell\rangle$
(2.22)

for sufficiently nice wavefunctions $\psi$ (in particular, decaying at large
$\ell$). To verify this, take the inner product of each side with an energy
eigenstate $\langle n|$, and use the fact that $\langle n|\ell\rangle$ is an
eigenstate of the differential operator. Some care is needed with this and
similar statements because the wavefunction $\psi(\ell)$ is not equal to
$\langle\ell|\psi\rangle$ non-perturbatively, and (as we will explain in the
next section) it is not even uniqely determined from the state $|\psi\rangle$.

This argument is a little abstract, but we can understand it intuitively with
a more direct derivation from the path integral. To do this, we can generalize
the above calculation of the inner product to compute a matrix element
$\langle\ell^{\prime}|e^{-\tau H_{L}}|\ell\rangle$, which means including
insertion of a piece of Euclidean boundary on the left side (the right side
would give the same answer in the end since $H_{L}=H_{R}$). Now, for a given
geometry that appears in the path integral, consider continuously deforming
the past boundary geodesic (defining $|\ell\rangle$) by moving its right
endpoint forward in Euclidean time while keeping it as a geodesic. Under the
same conditions as above, this process will sweep out a ‘wedge’ bounded by
geodesics of length $\ell,\ell^{\prime\prime}$ and the asymptotic boundary
segment of length $\tau$. The path integral of this wedge computes the matrix
elements of the time-evolution operator in the perturbative theory
$\langle\ell^{\prime\prime}|e^{-\tau H_{L}}|\ell\rangle_{0}$. The non-
perturbative matrix element is recovered by multiplying by
$\langle\ell^{\prime}|\ell^{\prime\prime}\rangle$ and integrating over
$\ell^{\prime\prime}$. But the time dependence all appears in the piece which
is independent of the non-perturbative corrections (analogous to
$Z_{\put(4.0,4.0){\oval(8.0,8.0)[b]}\put(0.0,4.0){\line(1,0){8.0}}\phantom{\subset\,}}$
above), which gives us the above (unmodified perturbative) representation for
the Hamiltonian. It is essential for this result that in every possible
spacetime and for every possible initial geodesic, by moving an endpoint, we
can obtain a continuous family of geodesics that never encounter interesting
topology or other non-perturbative features:888See discussion in section 8.

$\displaystyle\langle\ell^{\prime}|e^{-H_{L}{\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}\tau}}|{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\ell}\rangle=\int
d\ell^{\prime\prime}\,\langle\ell^{\prime}|\ell^{\prime\prime}\rangle\,\langle\ell^{\prime\prime}|e^{-H_{L}{\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}\tau}}|{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\ell}\rangle_{0}\
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