JT Gravity on a Finite Lorentzian Strip: Time dependent Quantum Gravity
Amplitudes

J. A. Rosabal<EMAIL_ADDRESS>
Departamento de Electromagnetismo y Electrónica, Universidad de Murcia, Campus
de Espinardo, 30100 Murcia, Spain.

We formulate JT quantum gravity on a finite Lorentzian strip. Due to the
spatial boundaries of the strip, it is possible to define left and right
proper times. With respect to these times we compute non-perturbatively the
quantum gravity (QG) time dependent transition amplitude. Lagrangian and
Hamiltonian formulations are presented. Special attention is paid to the four
corner terms (Hayward terms) in the action that are needed in order to have a
well defined variational problem. From a detailed analysis of the gravity
boundary condition on the spatial boundary, we find that while the lapse and
the shift functions are independent Lagrange multipliers on the bulk, on the
spatial boundary, these two are related. This fact leads to an algebraic
equation of motion for a particular degree of freedom that is conveniently
introduced on the spatial boundaries whose solution can be plugged back into
the action allowing to fully determine the time dependent transition
amplitude. The final result suggests that time evolution is non-unitary for
most of the boundary conditions. Interestingly enough, unitary could be
recovered when spatial $\text{AdS}_{2}$ boundary conditions are imposed. Other
wave functions for other topologies obtained from the strip by gluing its
spatial boundaries are also presented. Remarkably these do not exhibit any
non-unitary evolution behavior.

###### Contents

  1. 1 Introduction
  2. 2 JT Action, Surface and Junction Terms
  3. 3 Boundary Conditions, JT Lagrangian and Hamiltonian Actions and, Surface and Junctions Terms
    1. 3.1 JT Lagrangian Action and Boundary Terms
    2. 3.2 JT Hamiltonian Action and Boundary Terms
  4. 4 JT Quantum Gravity on the Strip
    1. 4.1 Bulk Calculation
  5. 5 Other topologies
  6. 6 Conclusions
  7. A Cylinder propagator

##  1 Introduction

Studying time evolution in QG is particularly challenging. The main reason is
due to the lack of time evolution itself. A possible resolution to incorporate
time is to explore the theory on manifolds with spatial boundaries. Over them
some of the components of the metric are fixed, and a proper time over the
spatial boundaries can be defined.

Here we focus on JT gravity Teitelboim:1983ux ; Jackiw:1984je  over a strip
with Lorentzian signature metrics. Our main motivation to choose a strip with
Lorentzian metrics is to have an arena where time dependent non-perturbative
QG amplitudes can be computed. In this work we will always refer to the
Lorentzian time just by time.

As we are interested in computing QG amplitudes the strip should be bounded in
time too, Fig. 1. Due to the four corners where the boundary is non-
differentiable, studying QG on this manifold is harder than on its counterpart
manifolds. Nonetheless, the non-smoothness of the corners can be handled by
the inclusion of the Hayward term in the action Hayward:1993my ;
Hayward:1992ix .

We compute the time dependent amplitude directly in the Lorentzian manifold in
a non perturbative way. Although we do not include matter in this work, it can
be regarded as a first step towards attacking the information paradox problem
from a different perspective. Rather than computing entropies, we focus on the
time dependent amplitudes to answer the question: is evolution in QG unitary?
After all, entropies in QG could give some hints about unitarity, but a
definitive answer about (non-)unitary evolution will come from the amplitudes
and their behavior along time (Hawking:1976ra, ). Our final result suggests
that evolution is non-unitary for the strip with generic boundary conditions.
However, there is a case where unitary evolution could be achieved. It is when
when $\text{AdS}_{2}$, spatial boundary conditions are imposed
Maldacena:2019cbz .

Intriguingly, other wave functions related to other topologies obtained from
the strip by gluing the time-like boundaries do not exhibit this non-unitary
behavior. In fact, after tracing over the degrees of freedom defined on the
spatial boundaries all references to the Lorenzian time disappears, as it
should be for the QG wave function on the resulting geometries. This part of
the work can also be seen as a complement to Iliesiu:2020zld  and
Griguolo:2021wgy  where similar results have been found but using different
methods to the ones presented here.

The paper is organized as follows. In the next section we define JT gravity on
a Lorentzian strip. In section 3 we discuss generic boundary conditions on
this manifold.

Although classically there could not exist solutions for generic boundary
conditions, at the quantum level the story is different. Quantically, any
boundary function is a good candidate to appear in the wave function. In this
section, we also introduce the Lagrangian and the Hamiltonian form of the
action. Special attention is paid to the Hayward terms, which are crucial
Hayward:1993my ; Hayward:1992ix  to properly define the boundary action and
the degrees of freedom over the boundaries.

In section 4 we start the quantum exploration of JT gravity on the Lorentzian
strip. We pose the problem for the transition amplitude we are interested in
and show how to solve the gravitational path integral on this manifold. It is
important to note that because we are assuming generic boundary conditions, in
general there could not exists classical solutions for the metric. This means
that integrating out first the dilaton field to solve the path integral is not
an option here.

To solve the path integral we propose a canonical transformation inspired by
the solution of the QG constraints Henneaux:1985nw ; Louis-Martinez:1993bge .
Although we do not use the conventional methods to solve a path integral, in
the end we are able to fully find the amplitude because we alternate between
the path integral and the canonical approach. Interestingly enough the time
dependent amplitude indicates that the evolution is non-unitary for most of
the boundary conditions.

In section 5 we show how to get other wave functions that are associated to
other manifolds with different topologies. We get these manifolds after gluing
the time-like boundaries of the strip. These wave functions do not reflect any
non-unitary evolution. Conclusion are presented in section 6.

##  2 JT Action, Surface and Junction Terms

Let M, be a sufficiently well behaved 2-dimensional finite strip, where a time
function t, which foliates M, into a set of constant t, space-like slices
$\Sigma_{\text{t}}$, can be defined. The boundaries of M, consists of the
initial and final space-like slices $\Sigma_{i}$, and $\Sigma_{f}$, see Fig.
1, as well as two time-like boundaries $\text{B}_{\textbf{L}}$,
$\text{B}_{\text{R}}$. For every $\Sigma_{\text{t}}$. The intersections
$\Sigma_{i}\cap\text{B}_{\text{L},\text{R}}$, and
$\Sigma_{f}\cap\text{B}_{\text{L},\text{R}}$, are called the junctions and are
denoted by $\text{J}_{i,}$, and $\text{J}_{f,}$, with an extra sub-index L, R
depending on the location on the spatial boundaries. For the strip in two
dimension the four junctions are just points.

Figure 1: The $(1+1)$ strip manifold M, the space-like boundaries
$\Sigma_{i}$, and $\Sigma_{f}$, the time-like boundaries $\text{B}_{L}$ and
$\text{B}_{R}$, and the junctions $\text{J}_{i,L}$, $\text{J}_{i,R}$, and
$\text{J}_{f,L}$, $\text{J}_{f,R}$. $\Sigma_{t}$, a particular constant time
slice and $\text{B}_{t,L}$ and $\text{B}_{t,R}$ the intersection
$\Sigma_{t}\cap\text{B}_{\text{L},\text{R}}$.

The appropriate form of the Einstein Hilbert or JT action to make quantum
gravity on manifolds with non-smooth boundaries Fig. 1 includes boundary, as
well as, junction contributions. When we fix only the induced metric and the
dilaton field on the boundaries the JT action reads as

$\displaystyle\text{S}=$
$\displaystyle\int\limits_{\text{M}}\text{d}^{2}x\sqrt{-\text{g}}\
\phi\big{(}\text{R}-\lambda\big{)}-2\sum\limits_{\text{A}}\epsilon_{\text{A}}\int\limits_{\text{A}}\text{d}y_{\text{A}}\sqrt{|\text{h}|}\
\phi\text{K}$ (1)
$\displaystyle+2\phi\eta\Big{|}_{\text{J}_{i,\text{R}}}^{\text{J}_{f,\text{R}}}-2\phi\eta\Big{|}_{\text{J}_{i,\text{L}}}^{\text{J}_{f,\text{L}}}.$

Where h, is fixed over $\Sigma_{i}$, $\Sigma_{f}$, $\text{B}_{\text{L}}$ and
$\text{B}_{\text{R}}$. K, is the extrinsic curvature of the boundaries. The
index A, runs over the boundaries; and $\epsilon_{\text{A}}$, is defined as

$\epsilon_{\text{A}}=\begin{cases}+1,&\text{if}\ \text{A}=\text{B}_{\text{L}}\
,\text{B}_{\text{R}}\\\ -1,&\text{if}\ \text{A}=\Sigma_{i}\ ,\Sigma_{f}\
.\end{cases}$

The scalar function $\eta$, is defined as

$\eta=\text{arcsinh}\big{(}\hat{\text{n}}^{(\Sigma)}_{|_{\text{B}}}\cdot\hat{\text{n}}^{(\text{B})}\big{)},$
(2)

where $\hat{\text{n}}^{(\Sigma)}_{|_{\text{B}}}$, represents the two-
dimensional unit normal to $\Sigma_{t}$, evaluated on
$\Sigma_{t}\cap\text{B}$, and $\hat{\text{n}}^{(\text{B})}$, is the two-
dimensional unit normal to B. We would like to stress that $\eta$, is defined
over the whole time-like boundaries $\text{B}_{\text{L},\text{R}}$, including
their boundaries, i.e., the junctions $\text{J}_{i}$, and $\text{J}_{f}$, and
$\eta\in(-\infty,+\infty)$.

The variation of (1) is given by

$\displaystyle\delta\text{S}$
$\displaystyle=\int\limits_{\text{M}}\text{d}^{2}x\sqrt{-\text{g}}\Big{[}\big{(}\text{R}-\lambda)\delta\phi+\big{(}-\nabla_{\mu}\nabla_{\nu}\
\phi+\Box\ \phi\ \text{g}_{\mu\nu}+\frac{\lambda}{2}\phi\
\text{g}_{\mu\nu}\big{)}\delta\text{g}^{\mu\nu}\Big{]}$ (3)
$\displaystyle-\sum\limits_{\text{A}}\epsilon_{\text{A}}\int\limits_{\text{B}_{\text{A}}}\text{d}y_{\text{A}}\sqrt{|\text{h}|}\Big{[}\phi\Big{(}\text{K}_{ab}-\text{K}\
\text{h}_{ab}+\text{n}_{(\text{A})}^{\sigma}\partial_{\sigma}\ \phi\
\text{h}_{ab}\Big{)}\delta\text{h}^{ab}+2\text{K}\ \delta\phi\Big{]}.$

After imposing $\delta\text{h}_{ab}=\delta\phi=0$, over the boundaries, we get
the vacuum JT’s equations of motion. Note that no term including $\eta$,
appears in (3).

##  3 Boundary Conditions, JT Lagrangian and Hamiltonian Actions and, Surface
and Junctions Terms

In this section, we introduce the ADM decomposition of the two-metric. Without
loss of the generality, we introduce a coordinate system over
$\Sigma_{\text{t}}$, such that the spatial boundaries are located at a
constant value of a coordinate that we call r. We perform a detailed analysis
of the boundary conditions, to find the relation the lapse N, and the shift
$\text{N}^{\text{r}}$, functions satisfy over the spatial boundaries. This
relation evidences that while N, and $\text{N}^{\text{r}}$, are independent
over M, over $\text{B}_{\text{L},\text{R}}$, they are not.

In the Lagrangian formulation, this boundary relation between N, and
$\text{N}^{\text{r}}$, is taken into account during the variation process of
(1). However, we show that in the Hamiltonian formulation, this is not the
case. Therefore, to take into account the variations of N, and
$\text{N}^{\text{r}}$, over the spatial boundary it is convenient to introduce
a new field to parameterize the boundary condition.

We present the Lagrangian and the Hamiltonian form of the JT gravity action in
the ADM formalism and the boundary equation of motion associated with the new
field. We discuss that while the variation of (1) does not lead to any
boundary equation, the variation of the Hamiltonian action does lead to a
boundary equation.

This boundary equation of motion is of algebraic type. This fact allows us to
solve this equation in terms of the others boundary degrees of freedom and
plug the solution back in the action. This treatment of the boundary action
can also be found in Lau:1995fr ; Kummer:1996si  but in a different context.
In these two references, it can also be found a more detailed analysis of some
of the issues discussed in this section but written in a different notation.

The ADM metric is given by

$\text{ds}^{2}=-\text{N}^{2}\text{dt}^{2}+\Lambda^{2}(\text{dr}+\text{N}^{\text{r}}\text{dt})^{2}.$
(4)

On a constant time slice $\Sigma_{\text{t}}$, the induced metric takes the
form

$\text{ds}^{2}_{|_{\text{t}}}=\Lambda^{2}\text{dr}^{2},$ (5)

and, $\sqrt{|\text{h}|}=\Lambda$. In particular on $\Sigma_{(i,f)}$,
$\text{ds}^{2}_{|_{\text{t}_{(i,f)}}}=\Lambda^{2}_{(i,f)}\text{dr}^{2}$, with
$\Lambda_{(i,f)}$, given functions of r. On the spatial boundaries the metric
reads as

$\text{ds}^{2}_{|_{\text{r}_{\text{cons}}}}=-\Big{(}\text{N}^{2}-(\Lambda\text{N}^{\text{r}})^{2}\Big{)}\text{dt}^{2}=-\overline{\text{N}}^{2}\text{dt}^{2},$
(6)

and, $\sqrt{|\text{h}|}=\overline{\text{N}}$.

From (6) we see that none of the degree of freedom N, $\text{N}^{\text{r}}$,
or $\Lambda$, are fully specified over the spatial boundaries, instead what is
fully specified is the combination

$\text{N}^{2}-(\Lambda\text{N}^{\text{r}})^{2}=\overline{\text{N}}^{2},$ (7)

Where $\overline{\text{N}}$, is a given function of $t$.

For the sake of completeness let us present the unit normal vectors to
$\Sigma_{t}$, and $\text{B}_{\text{L},\text{R}}$. Their expressions will be
helpful for the subsequent discussions. In the coordinates $(t,\text{r})$, one
can define the functions

$\displaystyle\text{F}^{(\Sigma)}$ $\displaystyle=$ $\displaystyle t_{0}-t,$
$\displaystyle\text{F}^{(\text{B})}$ $\displaystyle=$
$\displaystyle\text{r}_{0}-\text{r}.$ (8)

Such that a slice of constant time $t$, or constant r, are specified by the
relations $\text{F}^{(\Sigma)}=\text{F}^{(\text{B})}=0$.

The unit normal vectors are defined as

$\displaystyle\hat{\text{n}}_{\mu}^{(\Sigma)}$ $\displaystyle=$
$\displaystyle\frac{\partial_{\mu}\text{F}^{(\Sigma)}}{(-\text{g}^{\alpha\beta}\partial_{\alpha}\text{F}^{(\Sigma)}\partial_{\beta}\text{F}^{(\Sigma)})^{\frac{1}{2}}}=\text{N}(-1,0),$
(9) $\displaystyle\hat{\text{n}}^{(\text{B})}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\partial_{\mu}\text{F}^{(\text{B})}}{(\text{g}^{\alpha\beta}\partial_{\alpha}\text{F}^{(\text{B})}\partial_{\beta}\text{F}^{(\text{B})})^{\frac{1}{2}}}=\frac{(0,-1)}{(\Lambda^{-2}-\text{N}^{-2}\text{N}^{\text{r}}\text{N}^{\text{r}})^{\frac{1}{2}}}.$

The scalar product between both unit normal vectors over B, is

$(\text{g}^{\mu\nu}\hat{\text{n}}_{\mu}^{(\Sigma)}\hat{\text{n}}_{\nu}^{(\text{B})})_{|_{\text{r}_{\text{cons}}}}=\frac{\Lambda\text{N}^{\text{r}}}{(\text{N}^{2}-(\Lambda\text{N}^{\text{r}})^{2})^{\frac{1}{2}}}=\frac{\Lambda\text{N}^{\text{r}}}{\overline{\text{N}}}.$
(10)

Now, (2) can be written as

$\eta=\text{arcsinh}\big{(}\frac{\Lambda\text{N}^{\text{r}}}{\overline{\text{N}}}\big{)}.$
(11)

Let us go back to discuss the boundary conditions expressed in the coordinates
$(t,\text{r})$. We would like to stress that we can not impose separately
conditions on each functions appearing in (7).

Using (11) and (7) we note that

$\displaystyle\Lambda\text{N}^{\text{r}}_{|_{\text{r}_{\text{cons}}}}$
$\displaystyle=$ $\displaystyle\overline{\text{N}}\ \text{sinh}(\eta),$ (12)
$\displaystyle\text{N}_{|_{\text{r}_{\text{cons}}}}$ $\displaystyle=$
$\displaystyle\overline{\text{N}}\ \text{cosh}(\eta).$

Previous expressions indicate we can use $\eta$, to parameterize 111Although
(7) can be parameterized in infinitely many ways, we find convenient to use
(12) because it is the optimum one to carry out the subsequent calculations.
the degrees of freedom over the spatial boundaries.

###  3.1 JT Lagrangian Action and Boundary Terms

In this section we introduce the Lagrangian ADM form of the JT gravity action,
paying special attention to the boundary contribution. In two dimension the
Ricci scalar can be written as

$\text{R}=-2\nabla_{\mu}\Big{(}\text{K}\hat{\text{n}}^{\mu}+\hat{\text{n}}^{\nu}\nabla_{\nu}\hat{\text{n}}^{\mu}\Big{)},$
(13)

Using (13), from (1) we arrive at the JT Lagrangian form of the action,

$\text{S}=2\int\limits_{\text{M}}\text{d}^{2}x\Big{[}\Lambda(\partial_{t}\phi-\text{N}^{\text{r}}\partial_{\text{r}}\phi)\text{K}+\Lambda^{-1}\partial_{\text{r}}\phi\partial_{\text{r}}\text{N}-\frac{\lambda}{2}\text{N}\Lambda\phi\Big{]}+2\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\text{d}t\
\partial_{t}\phi\ \eta,$ (14)

where
$\text{K}=-\Lambda^{-1}\text{N}^{-1}\Big{(}\partial_{t}\Lambda-\partial_{\text{r}}(\Lambda\text{N}^{\text{r}})\Big{)}$,
and

$\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}=\int\limits_{\text{B}_{\text{R}}}-\int\limits_{\text{B}_{\text{L}}},$
(15)

in the interval $[t_{i},t_{f}]$. To get this action we have proceed in a
similar way as in Hawking:1996ww ; Rosabal:2021fao , of course we have taken
into account the particularities of 2d gravity. Note the unusual boundary term
in (14). It is needed to have a well defined variation process and it will be
essential in the Hamiltonian formulation. As $\phi$, is given over the
boundaries, in particular $\partial_{t}\phi$, would be fixed over the spatial
boundary too.

###  3.2 JT Hamiltonian Action and Boundary Terms

Let us introduce the JT Hamiltonian. From (14) we can compute the canonically
conjugate momenta

$\displaystyle\text{P}_{\Lambda}$ $\displaystyle=$
$\displaystyle-2\text{N}^{-1}\Big{(}\partial_{t}\phi-\text{N}^{\text{r}}\partial_{r}\phi\Big{)}=-2\hat{\text{n}}^{\mu}\partial_{\mu}\phi,$
(16) $\displaystyle\text{P}_{\phi}$ $\displaystyle=$
$\displaystyle-2\text{N}^{-1}\Big{(}\partial_{t}\Lambda-\partial_{\text{r}}(\Lambda\text{N}^{\text{r}})\Big{)}=-2\Lambda\
\text{K}.$

After some manipulations, we arrive at

$\displaystyle\text{S}=$
$\displaystyle\text{S}_{\text{M}}+\text{S}_{\text{B}},$ (17)
$\displaystyle{}=$
$\displaystyle\int\limits_{t_{i}}^{t_{f}}\text{d}t\int\limits_{\text{-r}_{0}}^{\text{r}_{0}}\text{dr}\Big{[}\text{P}_{\phi}\partial_{t}\phi+\text{P}_{\Lambda}\partial_{t}\Lambda-\text{N}\cal{H}-\text{N}^{\text{r}}\cal{H}_{\text{r}}\Big{]}$
$\displaystyle+\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\text{d}t\Big{(}\overline{\text{N}}\
\text{P}_{\Lambda}\
\text{sinh}(\eta)-2\overline{\text{N}}\Lambda^{-1}\partial_{\text{r}}\phi\
\text{cosh}(\eta)+2\partial_{t}\phi\ \eta\Big{)},$

where we have used (12) in the boundary action of (17), and

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\text{P}_{\phi}\text{P}_{\Lambda}+2\partial_{\text{r}}\big{(}\Lambda^{-1}\partial_{\text{r}}\phi\big{)}+\lambda\
\Lambda\ \phi,$ (18) $\displaystyle{\cal H}_{\text{r}}$ $\displaystyle=$
$\displaystyle\partial_{\text{r}}\phi\text{P}_{\phi}-\Lambda\partial_{\text{r}}\text{P}_{\Lambda}.$

In the Hamiltonian formulation N, and $\text{N}^{\text{r}}$, play the role of
Lagrange multipliers Hartle:1983ai . Their equations of motion only enforce
the Hamiltonian and the momentum constraint ${\cal H}={\cal H}_{\text{r}}=0$,
on M.

Variations of (17) with respect to N, $\text{N}^{\text{r}}$, $\Lambda,\ \phi$,
$\text{P}_{\Lambda}$, and $\text{P}_{\phi}$, over M, lead to an equivalent
systems of equations of motion to that obtained from the Lagrangian (14).
However, for the action written in the ADM Hamiltonian form extra care is
needed with the variations over $\text{B}_{\text{L},\text{R}}$. Note that
while the variations of the Lagrange multipliers N, and $\text{N}^{\text{r}}$,
are independent over M, over the boundaries $\text{B}_{\text{L},\text{R}}$,
they are related each other through relation (7). Therefore, the variation of
N, and $\text{N}^{\text{r}}$, over $\text{B}_{\text{L},\text{R}}$, are not
independent.

To take care of the variations of N, and $\text{N}^{\text{r}}$, over
$\text{B}_{\text{L},\text{R}}$, it is convenient to parameterize the solutions
of (7), using (12), and regard $\eta$, as an independent degree of freedom
defined over $\text{B}_{\text{L},\text{R}}$. This is the reason why the
boundary action in (17) has been explicitly written in terms of $\eta$, and
the others boundary degrees of freedom, instead of N, $\text{N}^{\text{r}}$.

The boundary equations of motion, associated to $\eta$, on
$\text{B}_{\text{L},\text{R}}$, read as

$\overline{\text{N}}\ \text{P}_{\Lambda}\
\text{cosh}(\eta)-2\overline{\text{N}}\Lambda^{-1}\partial_{\text{r}}\phi\
\text{sinh}(\eta)+2\partial_{t}\phi=0.$ (19)

Equation (19) is contained in the first equation in (16). This can be checked
by plugging (12) in (19). In other words, equation (19) can be viewed as the
restriction to the time-like boundaries of the first equation in (16).
Previous statement ensures there are no contradictions between the Lagrangian
and the Hamiltonian formulations. Nonetheless, we emphasize that (19) and (16)
come from the variation of two different and unrelated degrees of freedom.
While the former comes from the variation of $\eta$, defined only over the
spatial boundaries, the latter, in the Hamiltonian formulation, comes from the
variation of $\text{P}_{\Lambda}$, over M, including its boundaries.

##  4 JT Quantum Gravity on the Strip

In this section we explore quantum JT gravity on the finite Lorentzian strip.
Our final goal is to compute the time dependent transition amplitude. To this
end we will use both, the canonical and the path integral formulation of
gravity.

Before pose the problem we want to make an important remark. Note that
integrating out first the field $\phi$, is not a clear option to solve the
path integral in this manifold. We are not sure whether a classical solution
satisfying the boundary conditions (5) and (6) for generic $\Lambda_{i,f}$,
and for generic $\overline{\text{N}}_{\text{L},\text{R}}$, does exist. For
sure when $\Lambda_{i,f}$, and $\overline{\text{N}}_{\text{L},\text{R}}$, are
restricted to one of the well known cases, the solution will exist and
integrating out first the field $\phi$, becomes an option again.

Let us now pose the problem. We are interested in computing the transition
amplitude between an initial configuration $(\phi,\Lambda)_{i}$, at some
initial time $t_{i}$, and a final one $(\phi,\Lambda)_{f}$, at some final time
$t_{f}$.

In the path integral formulation this transition amplitude is represented by

$\displaystyle\Psi\Big{[}(\phi,\Lambda)_{f},(\phi,\Lambda)_{i}\
;(\phi,\overline{\text{N}})_{\text{R}},(\phi,\overline{\text{N}})_{\text{L}}\Big{]}$
$\displaystyle=\int\text{D}\big{[}\text{N},\text{N}^{\text{r}},\phi,\Lambda,\text{P}_{\phi},\text{P}_{\Lambda}\big{]}_{\big{|}_{\text{M}}}$
$\displaystyle\ \ \
\int\text{D}\big{[}\text{N},\text{N}^{\text{r}},\Lambda,\text{P}_{\Lambda}\big{]}_{\big{|}_{\text{B}_{\text{L},\text{R}}}}$
$\displaystyle\delta\big{[}\text{N}^{2}-(\Lambda\text{N}^{\text{r}})^{2}-\overline{\text{N}}^{2}\big{]}_{\big{|}_{\text{B}_{\text{L},\text{R}}}}\text{e}^{\text{i}\text{S}}.$
(20)

Conveniently we have split the measure into the bulk and the boundaries
contribution. The functional Dirac delta enforces the boundary condition (7),
over the time-like boundaries, and

$\displaystyle\text{S}=$
$\displaystyle\int\limits_{t_{i}}^{t_{f}}\text{d}t\int\limits_{\text{-r}_{0}}^{\text{r}_{0}}\text{dr}\Big{[}\text{P}_{\phi}\partial_{t}\phi+\text{P}_{\Lambda}\partial_{t}\Lambda-\text{N}\cal{H}-\text{N}^{\text{r}}\cal{H}_{\text{r}}\Big{]}$
(21)
$\displaystyle+\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\text{d}t\Big{(}\Lambda\text{N}^{\text{r}}\
\text{P}_{\Lambda}-2\text{N}\Lambda^{-1}\partial_{\text{r}}\phi+2\partial_{t}\phi\
\text{arcsinh}\big{(}\frac{\Lambda\text{N}^{\text{r}}}{\overline{\text{N}}}\big{)}\Big{)}.$

We do not specified the measure of the path integral, because in the end we
will not solve it explicitly. After performing the change of variables over
the spatial boundaries only

$\displaystyle\Lambda\text{N}^{\text{r}}$ $\displaystyle=$ $\displaystyle{\cal
R}\ \text{sinh}(\eta),$ (22) N $\displaystyle=$ $\displaystyle{\cal R}\
\text{cosh}(\eta),$

the transition amplitude can be written as

$\displaystyle\Psi\Big{[}(\phi,\Lambda)_{f},(\phi,\Lambda)_{i}\
;(\phi,\overline{\text{N}})_{\text{R}},(\phi,\overline{\text{N}})_{\text{L}}\Big{]}$
$\displaystyle=\int\text{D}\big{[}\text{N},\text{N}^{\text{r}},\phi,\Lambda,\text{P}_{\phi},\text{P}_{\Lambda}\big{]}_{\big{|}_{\text{M}}}$
$\displaystyle\ \
\int\text{D}\big{[}\eta,\Lambda,\text{P}_{\Lambda}\big{]}_{\big{|}_{\text{B}_{\text{L},\text{R}}}}\text{e}^{\text{i}\text{S}},$
(23)

where S, is as in (17). Given the form of the boundary action
$\text{S}_{\text{B}}$, in principle we could point wise integrate $\eta$, to
get Hankel or modified Bessel functions of the second kind. However, we do not
know how to handle the infinity product of functions we get. Instead of
integrating $\eta$, we will take advantage of the fact that its equation of
motion is of algebraic type. In this case we are allowed to solve for $\eta$,
in terms of the others boundary degrees of freedom and plug the solution in
the action Lau:1995fr ; Kummer:1996si . We stress that for algebraic equations
of motion this is not a semi-classical procedure, so we still are in a non-
perturbative regimen. This procedure is completely equivalent to integrating
in $\eta$.

The solution to (19), is

$\displaystyle\eta=$
$\displaystyle\text{ln}\Big{(}\frac{(2\overline{\text{N}}^{-1}\partial_{t}\phi)+\sqrt{(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+(2\overline{\text{N}}^{-1}\partial_{t}\phi)^{2}-\text{P}_{\Lambda}^{2}}}{(2\Lambda^{-1}\partial_{\text{r}}\phi)-\text{P}_{\Lambda}}\Big{)}.$
(24)

Plugging it in the boundary action we get

$\displaystyle\text{S}_{\text{B}}$
$\displaystyle=\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\text{d}t\overline{\text{N}}\Big{[}-\sqrt{(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+(2\overline{\text{N}}^{-1}\partial_{t}\phi)^{2}-\text{P}_{\Lambda}^{2}}$
(25)
$\displaystyle+(2\overline{\text{N}}^{-1}\partial_{t}\phi)\text{ln}\Big{(}\frac{(2\overline{\text{N}}^{-1}\partial_{t}\phi)+\sqrt{(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+(2\overline{\text{N}}^{-1}\partial_{t}\phi)^{2}-\text{P}_{\Lambda}^{2}}}{(2\Lambda^{-1}\partial_{\text{r}}\phi)-\text{P}_{\Lambda}}\Big{)}\Big{]}.$

Before moving to the bulk calculation we would like to make a further
assumption which simplifies the next calculations. It will be implemented in
two steps. First, we will assume that over the spatial boundaries $\phi$, is
constant. This assumption translates into the conditions

$\partial_{t}\phi_{\big{|}_{\text{B}_{\text{L}},\text{B}_{\text{R}}}}=0.$ (26)

Then, the transition amplitude (4) reduces to

$\displaystyle\Psi\Big{[}(\phi,\Lambda)_{f},(\phi,\Lambda)_{i}\
;(\phi,\overline{\text{N}})_{\text{R}},(\phi,\overline{\text{N}})_{\text{L}}\Big{]}$
$\displaystyle=\int\text{D}\big{[}\text{N},\text{N}^{\text{r}},\phi,\Lambda,\text{P}_{\phi},\text{P}_{\Lambda}\big{]}_{\big{|}_{\text{M}}}$
(27) $\displaystyle\ \ \
{}\int\text{D}\big{[}\Lambda,\text{P}_{\Lambda}\big{]}_{\big{|}_{\text{B}_{\text{L},\text{R}}}}\text{e}^{\text{i}\text{S}},$

with S, given by

S
$\displaystyle=\int\limits_{t_{i}}^{t_{f}}\text{d}t\int\limits_{\text{-r}_{0}}^{\text{r}_{0}}\text{dr}\Big{[}\text{P}_{\phi}\partial_{t}\phi+\text{P}_{\Lambda}\partial_{t}\Lambda-\text{N}\cal{H}-\text{N}^{\text{r}}\cal{H}_{\text{r}}\Big{]}$
(28)
$\displaystyle-\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\text{d}t\
\overline{\text{N}}\sqrt{(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}-\text{P}_{\Lambda}^{2}}.$

Note that now, $\phi_{\text{R}}$, and $\phi_{\text{L}}$ are constants. In
addition by consistency

$\displaystyle\phi_{i}(-\text{r}_{0})=\phi_{f}(-\text{r}_{0})$
$\displaystyle=$ $\displaystyle\phi_{\text{R}},$
$\displaystyle\phi_{i}(+\text{r}_{0})=\phi_{f}(+\text{r}_{0})$
$\displaystyle=$ $\displaystyle\phi_{\text{L}}.$ (29)

###  4.1 Bulk Calculation

In order to show how to solve the full path integral (27) we find instructive
first to look at the classical solution to the constraint ${\cal H}={\cal
H}_{\text{r}}=0$, Henneaux:1985nw ; Louis-Martinez:1993bge .

This is

$\displaystyle\text{P}_{\Lambda}$ $\displaystyle=$
$\displaystyle\Big{(}\text{C}(t)+(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+2\lambda\phi^{2}\Big{)}^{\frac{1}{2}},$
(30) $\displaystyle\text{P}_{\phi}$ $\displaystyle=$ $\displaystyle
2\frac{\partial_{\text{r}}(2\Lambda^{-1}\partial_{\text{r}}\phi)+\lambda\Lambda\phi}{\Big{(}\text{C(t)}+(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+2\lambda\phi^{2}\Big{)}^{\frac{1}{2}}},$

where $\text{C}(t)$, is an arbitrary function of $t$. If we were performing
this calculation over an space-time without spatial boundaries, for instance a
cylinder, it would be enough to make the substitution
$\text{P}_{\Lambda}=-\text{i}\frac{\delta}{\delta\Lambda}$, and
$\text{P}_{\phi}=-\text{i}\frac{\delta}{\delta\phi}$, in (30) and solve for
$\Psi$, the corresponding functional equations, i.e.,

$\displaystyle-\text{i}\frac{\delta}{\delta\Lambda}\Psi$ $\displaystyle=$
$\displaystyle\Big{(}\text{C(t)}+(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+2\lambda\phi^{2}\Big{)}^{\frac{1}{2}}\Psi,$
(31) $\displaystyle-\text{i}\frac{\delta}{\delta\phi}\Psi$ $\displaystyle=$
$\displaystyle
2\frac{\partial_{\text{r}}(2\Lambda^{-1}\partial_{\text{r}}\phi)+\lambda\Lambda\phi}{\Big{(}\text{C(t)}+(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+2\lambda\phi^{2}\Big{)}^{\frac{1}{2}}}\Psi.$

In fact, these equations are equivalent to the Wheeler-DeWitt equation and the
momentum constraint, with a particular factor ordering. In this case we would
find that

$\Psi=\text{exp}\Big{[}\text{i}\
\Omega^{\prime}[\Lambda,\phi;\text{C}]\Big{]},$ (32)

with

$\Omega^{\prime}[\Lambda,\phi;\text{C}]=\Omega[\Lambda,\phi;\text{C}]+\text{G}(\text{C}),$
(33)

and

$\displaystyle\Omega[\Lambda,\phi;\text{C}]=$
$\displaystyle\int\limits_{\Sigma_{f}}\text{dr}\
\Lambda\Big{[}\sqrt{\text{C}+(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+2\lambda\phi^{2}}$
(34)
$\displaystyle+(2\Lambda^{-1}\partial_{\text{r}}\phi)\text{ln}\Big{(}\frac{(2\Lambda^{-1}\partial_{\text{r}}\phi)+\sqrt{\text{C}+(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+2\lambda\phi^{2}}}{(2\Lambda^{-1}\partial_{\text{r}}\phi)-\sqrt{\text{C}+(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+2\lambda\phi^{2}}}\Big{)}\Big{]},$

where $\text{G}(\text{C})$, is an arbitrary function of the constant
$\text{C}(t_{f})$.

For the transition amplitude we are interested in, the former procedure does
not apply as it stands because in the action (28) there is a boundary
contribution that does not decouple from the bulk path integral.

The crucial observation here is that we can regard the solution (30) as an
off-shell change of variables in the path integral. In fact it is a canonical
transformation whose generating functional is given by
$\Omega[\Lambda,\phi;\text{C}]$, (34), with no extra boundary terms. A similar
transformation has appeared in the literature before but in a different
context within Einstein QG in four dimensions Kuchar:1994zk .

After performing the canonical transformation (30) to (28) we get (note that
the transformation (30) maps ${\cal H}$, and ${\cal H}_{\text{r}}$, to zero)

S
$\displaystyle=\int\limits_{t_{i}}^{t_{f}}\text{d}t\int\limits_{\text{-r}_{0}}^{\text{r}_{0}}\text{dr}\Big{[}\text{P}_{\text{C}}\partial_{t}\text{C}+\tilde{\text{P}}_{\phi}\partial_{t}\phi-{\cal
K}[\text{C},\phi,\text{P}_{\text{C}},\tilde{\text{P}}_{\phi}]\Big{]}$
$\displaystyle+\int\limits_{t_{i}}^{t_{f}}\text{d}t\frac{\text{d}}{\text{d}t}\Omega[\Lambda,\phi;\text{C}]-\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\text{d}t\
\overline{\text{N}}\sqrt{-\text{C}-2\lambda\phi^{2}}.$

where

$\displaystyle\text{P}_{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{\delta}{\delta\Lambda}\Omega[\Lambda,\phi;\text{C}],$ (35)
$\displaystyle\text{P}_{\phi}$ $\displaystyle=$
$\displaystyle\tilde{\text{P}}_{\phi}+\frac{\delta}{\delta\phi}\Omega[\Lambda,\phi;\text{C}],$
$\displaystyle\text{P}_{\text{C}}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\text{C}}\Omega[\Lambda,\phi;\text{C}],$

and the new Hamiltonian

${\cal
K}[\text{C},\phi,\text{P}_{\text{C}},\tilde{\text{P}}_{\phi}]=-\frac{1}{2}\text{N}\tilde{\text{P}}_{\phi}\sqrt{\text{C}+(2\Lambda^{-1}\partial_{\text{r}}\phi)^{2}+2\lambda\phi^{2}}+\text{N}^{\text{r}}\partial_{\text{r}}\phi\tilde{\text{P}}_{\phi}.$
(36)

In the new action the Lagrange multipliers impose the same constraint,
$\tilde{\text{P}}_{\phi}=0$. In the path integral this means that after
integration in N the action will further reduce to

$\text{S}=\int\limits_{t_{i}}^{t_{f}}\text{d}t\ \Pi_{\text{C}}\
\partial_{t}\text{C}+\Omega[\Lambda,\phi;\text{C}]\Big{|}_{t_{i}}^{t_{f}}-\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\text{d}t\
\overline{\text{N}}\sqrt{-\text{C}-2\lambda\phi^{2}}$ (37)

where $\Pi_{\text{C}}(t)=\int\limits_{\text{-r}_{0}}^{\text{r}_{0}}\text{dr}\
\text{P}_{\text{C}}(t)$, and the transition amplitude

$\displaystyle\Psi\Big{[}(\phi,\Lambda)_{f},(\phi,\Lambda)_{i}\
;(\phi,\overline{\text{N}})_{\text{R}},(\phi,\overline{\text{N}})_{\text{L}}\Big{]}$
$\displaystyle=\text{e}^{\text{i}\Omega[\Lambda,\phi;\text{C}]\Big{|}_{t_{i}}^{t_{f}}}\int\text{D}\text{\text{C}}\
\text{D}\Pi_{\text{C}}\ $ $\displaystyle\ \
\text{exp}\Big{[}\text{i}\int\limits_{t_{i}}^{t_{f}}\text{d}t\ \Pi_{\text{C}}\
\partial_{t}\text{C}+$ (38)
$\displaystyle-\text{i}\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\text{d}t\
\overline{\text{N}}\sqrt{-\text{C}-2\lambda\phi^{2}}\Big{]},$

Note that (37) is complex for most of the values of C, and $\lambda$.

To further proceed it is convenient to perform the $\Pi_{\text{C}}$, integral
in (4.1). This integration gives rise a functional Dirac delta
$\delta[\partial_{t}\text{C}]$. It imposes the condition
$\text{C}=\text{const}$, while removes all the C integrals. In the end we get

$\displaystyle\Psi\Big{[}(\phi,\Lambda)_{f},(\phi,\Lambda)_{i}\
;(\phi,\overline{\text{N}})_{\text{R}},(\phi,\overline{\text{N}})_{\text{L}}\Big{]}$
$\displaystyle=\chi(\text{C})\
\text{e}^{\text{i}\Omega[\Lambda,\phi;\text{C}]\Big{|}_{t_{i}}^{t_{f}}}$
$\displaystyle\text{exp}\Big{[}{\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\text{d}t\
\overline{\text{N}}\sqrt{\text{C}+2\lambda\phi^{2}}}\Big{]}.$ (39)

Where $\chi(\text{C})$, is coming from the fact that the new Hamiltonian
$\text{h}[\text{C},t]$,

$\text{h}[\text{C},t]=\big{(}\overline{\text{N}}_{\text{R}}(t)-\overline{\text{N}}_{\text{L}}(t)\big{)}\sqrt{-\text{C}-2\lambda\phi^{2}},$
(40)

in the new action is a function of C, 222To better understand the origin of
$\chi(\text{C})$, it is convenient to look at the quantum mechanical problem
(4.1) à la Schrödinger. For this case the Schrödinger equation reads
$\text{i}\partial_{t_{f}}\psi(\text{C},t_{f})=\text{h}[\text{C},t_{f}]\
\psi(\text{C},t_{f})=\big{(}\overline{\text{N}}_{\text{R}}(t_{f})-\overline{\text{N}}_{\text{L}}(t_{f})\big{)}\sqrt{-\text{C}-2\lambda\phi^{2}}\
\psi(\text{C},t_{f}).$ (41) It is straightforward to see that we can separate
variables $\psi(\text{C},t_{f})=\chi(\text{C})\text{f}(t_{f})$, to finally get
$\psi(\text{C},t_{f})=\chi(\text{C})\text{exp}\Big{[}{\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\text{d}t\
\overline{\text{N}}\sqrt{\text{C}+2\lambda\phi^{2}}\Big{]}}.$ (42) . This
function $\chi(\text{C})$, can be fixed by demanding initial conditions for
the time dependent transition amplitude (4.1). Since C, and $\phi$, are
constant we can easily introduce the invariant proper time $\tau$,

$\tau=\int\limits_{t_{i}}^{t_{f}}\text{d}t\ \overline{\text{N}}.$ (43)

In this basis the transition amplitude reads

$\displaystyle\Psi\Big{[}\phi,\Lambda_{f},\Lambda_{i}\
;\tau_{\text{R}},\tau_{\text{L}}\Big{]}$
$\displaystyle=\int\limits_{0}^{\infty}\text{dC}\ \chi(\text{C})\
\text{e}^{\text{i}\Omega[\Lambda,\phi;\text{C}]\Big{|}_{t_{i}}^{t_{f}}}$
$\displaystyle\text{exp}\Big{[}\sqrt{\text{C}+2\lambda\phi^{2}}(\tau_{\text{R}}-\tau_{\text{L}})\Big{]}.$
(44)

where, one more integral has been included (can be included) to fulfill the
initial conditions. When $t_{f}\rightarrow t_{i}$, $\tau=0$, and the initial
conditions are expressed as

$\displaystyle\Psi\Big{[}\phi,\Lambda_{f},\Lambda_{i}\ ;0,0\Big{]}$
$\displaystyle=\int\limits_{0}^{\infty}\text{dC}\ \chi(\text{C})\
\text{e}^{\text{i}\Omega[\Lambda,\phi;\text{C}]\Big{|}_{t_{i}}^{t_{f}}}=\Psi_{0}\Big{[}\phi,\Lambda_{f},\Lambda_{i}\Big{]},$
(45)

with $\Psi_{0}\Big{[}\phi,\Lambda_{f},\Lambda_{i}\Big{]}$, a given functional.
Note that when $t_{f}\rightarrow t_{i}$, not necessarily
$\Lambda_{f}\rightarrow\Lambda_{i}$. Solving the equation (45) for
$\chi(\text{C})$, complete determines the time dependent state.

Now, we implement the second part of the assumption we implemented in previous
section. Assuming that $\phi$, is constant over the space-like boundaries
implies

$\partial_{\text{r}}\phi_{\big{|}_{\Sigma_{i},\Sigma_{f}}}=0.$ (46)

Then, introducing the invariant proper length l,

$\text{l}=\int\limits_{-\text{r}_{0}}^{\text{r}_{0}}\text{dr}\ \Lambda,$ (47)

we have

$\Psi\Big{[}\phi,\text{l}_{f},\text{l}_{i}\
;\tau_{\text{R}},\tau_{\text{L}}\Big{]}=\int\limits_{0}^{\infty}\text{dC}\
\chi(\text{C})\text{exp}\Big{[}\sqrt{\text{C}+2\lambda\phi^{2}}\Big{(}\text{i}(\text{l}_{f}-\text{l}_{i})+(\tau_{\text{R}}-\tau_{\text{L}})\Big{)}\Big{]},$
(48)

Note, that we can rewrite the transition amplitude as

$\Psi\Big{[}\phi,\text{l}_{f},\text{l}_{i}\
;\tau_{\text{R}},\tau_{\text{L}}\Big{]}=\int\limits_{0}^{\infty}\text{dC}\
\chi(\text{C})\text{exp}\Big{[}\sqrt{\text{C}+2\lambda\phi^{2}}\ \text{i}\
\text{L}\Big{]},$ (49)

where $\text{L}=\oint\text{ds}$, is taken clockwise around the boundary

$\text{L}=\int\limits_{\text{r}_{0}}^{\text{-r}_{0}}\text{ds}_{\big{|}_{\Sigma_{i}}}+\int\limits_{t_{i}}^{t_{f}}\text{ds}_{\big{|}_{\text{B}_{\text{L}}}}+\int\limits_{\text{-r}_{0}}^{\text{r}_{0}}\text{ds}_{\big{|}_{\Sigma_{f}}}+\int\limits_{t_{f}}^{t_{i}}\text{ds}_{\big{|}_{\text{B}_{\text{R}}}}.$
(50)

From (43) it is straightforward to see that (48) is invariant under the
corner-preserving reparametrizations. Additionally we can see that (49)
satisfies a sort of reduced Wheeler-DeWitt equation with complex variable
$\text{L}=(\text{l}_{f}-\text{l}_{i})-\text{i}(\tau_{\text{R}}-\tau_{\text{L}})$,

$\Big{(}\partial_{\text{L}}(\text{L}^{-1}\partial_{\phi})+2\lambda\phi\Big{)}\Psi[\phi,\text{L}]=0.$
(51)

We stress that we are not developing a mini-superspace model of JT gravity.
The appearance of (51), which is similar to the equation one gets in some
mini-superspace models, see for instance Iliesiu:2020zld  (for
$\text{L}\in\mathbb{R}$), is a consequence of the restriction of the dilaton
field to constant values over the boundaries. Nonetheless (51), could be
helpful for determining some of the amplitudes.

Having the time dependent transition amplitude (48) some comments are in
order. To fully determine
$\Psi\Big{[}\phi,\text{l}_{f},\text{l}_{i},\tau_{\text{R}},\tau_{\text{L}}\Big{]}$,
we need to specify $\chi(\text{C})$, which for the amplitudes we have computed
it is not interpreted as a density of states. In general, and unlike the
calculation of the partition function, it could take negative values, or it
could even be a complex function. It is fixed by demanding some boundary
conditions (initial conditions) (45) for the amplitude. It is worth to remark
here that finding $\chi(\text{C})$, from (45), in general, could be involved
without knowing explicitly the product between states 333If there exists a
functional measure $\text{M}[\phi_{f},\Lambda_{f}]$, such that
$\int\text{D}\phi_{f}\text{D}\Lambda_{f}\ \text{M}[\phi_{f},\Lambda_{f}]\
\text{exp}\Big{[}\text{i}\Big{(}\Omega[\Lambda_{t_{f}},\phi_{t_{f}};\text{C}]-\Omega[\Lambda_{t_{f}},\phi_{t_{f}};\text{C}^{\prime}]\Big{)}\Big{]}=\text{f}(\text{C})\delta\big{(}\text{C}-\text{C}^{\prime}\big{)},$
(52) The product (52) can be considered as the inner product in the Hilbert
space..

Note that on the one hand, (48) contains a growing (decreasing) time dependent
exponential factor. This is an indication that, for some states the
probability will not be conserved. In other words, for a generic
$\chi(\text{C})$, evolution seems to be non-unitary, independently of the sign
of $\lambda$. Although there are states that behead unitary. For instance,
when $\chi(\text{C})=\delta(\text{C}-\text{C}_{0})$, depending of the sign of
$\text{C}_{0}+2\lambda\phi^{2}$, the probability could be conserved or not.
Nonetheless these states, despite some of them behave unitary, describe
unrealistic scenarios.

On the other hand there is a possibility where evolution could be unitary. For
that to happen we need to have $\sqrt{\text{C}+2\lambda\phi^{2}}<0$, for all
values of C. Note that this happens only if

$\lambda<0,\ \ \text{and}\ \
\phi_{\large{|}_{\text{B}_{(\text{R},\text{L})}}}=\frac{1}{\epsilon},$ (53)

with $\epsilon\rightarrow 0$. These conditions can be recognized with the
$\text{AdS}_{2}$, boundary conditions Maldacena:2019cbz . In this case we can
take the metric near the spatial boundaries, located at $\sigma=0$, and
$\sigma=\pi$, to be

$\text{ds}^{2}=\frac{-\text{d}t^{2}+\text{d}\sigma^{2}}{\text{sin}^{2}(\sigma)}.$
(54)

For this metric the boundary condition for $\overline{\text{N}}$, rewrites as

$\overline{\text{N}}_{(\text{R},\text{L})}=\frac{\overline{\text{N}}_{0\
(\text{R},\text{L})}}{\epsilon}.$ (55)

In addition we can not regard $\phi$, as a constant over the four boundaries.
In fact now we need to have a varying dilaton over the initial and the final
slices, as well as a varying $\Lambda$. Note that if the dilaton were constant
over these slices then by continuity of the function (29) it would be infinity
over them. The issue is that with a constant dilaton over $\Sigma_{i}$, and
$\Sigma_{f}$, the limit $\epsilon\rightarrow 0$, of the action in (27) is not
well defined. In addition there not exists counterterm that can fix this
limit. Consistency among boundaries conditions (29) implies that

$\lim_{\sigma\to\text{B}}\phi_{(i,f)}=\infty,$ (56)

for the varying dilaton over $\Sigma_{i}$, and $\Sigma_{f}$.

Starting from (4.1), it is straightforward to see that

$\displaystyle\Psi\Big{[}(\phi,\Lambda)_{f},(\phi,\Lambda)_{i}\
;(\infty,\infty)_{\text{R}},(\infty,\infty)_{\text{L}}\Big{]}$
$\displaystyle=\int\limits_{0}^{\infty}\text{dC}\ \chi(\text{C})\
\text{e}^{\text{i}\Omega[\Lambda,\phi;\text{C}]\Big{|}_{t_{i}}^{t_{f}}}$
$\displaystyle\text{exp}\Big{[}-\text{i}\frac{\text{C}}{2\sqrt{2\lambda}}(\tau_{\text{R}}-\tau_{\text{L}})\Big{]}.$
(57)

Where we have removed the infinity contribution
$\frac{\sqrt{2\lambda}}{\epsilon^{2}}(\tau_{\text{R}}-\tau_{\text{L}})$. From
this term one can read of the counterterm needed in the action (1) to make it
well defined for these particular boundary conditions. It is given by

$\text{S}_{\text{counterterm}}=-\sqrt{2\lambda}\int\limits_{(\text{B}_{\text{R}}-\text{B}_{\text{L}})}\phi\
\text{ds}.$ (58)

Now it is clear to see that with the proper product between states (52) and,
because of the appearance of the complex unit in front of the proper time in
the exponent of (4.1) this amplitude might evolve unitary 444If (52) holds,
the probability does not dependent on time and it is given by
$\text{P}(\tau)=\int\limits_{0}^{\infty}\text{dC}\
\chi(\text{C})\chi^{*}(\text{C})\ \text{f}(\text{C})=\text{P}(0).$ (59) .

##  5 Other topologies

In what follows we will see how by gluing the two spatial boundaries we can
make contact with some known results. To glue these two boundaries we can not
consider constant functions over the boundaries.

Starting from (21) or (25) we immediately see that once we identify the
functions defined over $\text{B}_{\text{R}}$, with the ones over
$\text{B}_{\text{L}}$, their contribution cancel each other. So, after tracing
over the boundary degrees of freedom we will get a multiplicative infinity
constant that can be ignored.

From this point on we can proceed similarly to previous section to get

$\displaystyle\Psi\Big{[}(\phi,\Lambda)_{f},(\phi,\Lambda)_{i}\Big{]}$
$\displaystyle=\int\limits_{0}^{\infty}\text{dC}\ \chi(\text{C})\
\text{e}^{\text{i}\Omega[\Lambda,\phi;\text{C}]\Big{|}_{t_{i}}^{t_{f}}},$ (60)

defined over a cylinder. Now for simplicity we can restrict the boundaries
functions to be constants. In this case we have

$\Psi\Big{[}(\phi,l)_{f},(\phi,l)_{i}\Big{]}=\int\limits_{0}^{\infty}\text{dC}\
\chi(\text{C})\text{exp}\Big{[}\text{i}\sqrt{\text{C}+2\lambda\phi_{f}^{2}}\
\text{l}_{f}-\text{i}\sqrt{\text{C}+2\lambda\phi_{i}^{2}}\
\text{l}_{i}\Big{]}.$ (61)

In principle there should exists a function $\chi{\text{(C)}}$, such that from
(61) we get the finite cutoff Lorentzian version of the cylinder propagator,
see appendix A. However, here we have not been able to fully find this
function, see (Iliesiu:2020zld, ) for a discussion about this topic.

In addition, when we set to zero the length of the initial segment/circle,
depending at which stage of the calculation we shrink it, the strip/cylinder
might reduce to a disk with a conical defect Iliesiu:2020zld ; Mertens:2019tcm
or a cap. Now, we can get the corresponding amplitudes from (61). After
setting $\text{l}_{i}=0$, we can see that we do not need to specify
$\phi_{i}$, any more. As it should be for a these geometries. In the end we
have

$\Psi[\phi,l]=\int\limits_{0}^{\infty}\text{dC}\
\chi(\text{C})\text{exp}\Big{[}\text{i}\sqrt{\text{C}+2\lambda\phi^{2}}\
\text{l}\Big{]}.$ (62)

Now, with

$\chi(\text{C})=\text{sinh}\big{(}2\pi\sqrt{\text{C}}\big{)},$ (63)

for instance, we can make contact with the Lorentzian version of the finite
cutoff JT wave function for $\lambda>0$, i.e., the $\text{dS}_{2}$, version of
it,

$\Psi[\phi,l]=\frac{\phi^{2}\
\text{l}}{(\text{l}^{2}-4\pi^{2})}\text{H}_{2}^{(1)}\Big{(}\sqrt{2\lambda}\
|\phi|(\text{l}^{2}-4\pi^{2})^{\frac{1}{2}}\Big{)}.$ (64)

Although (62) seems problematic when $\lambda<0$, the analytic extension of
(64) to these values of $\lambda$ is well defined. In this case we get what
could be interpreted as the Lorentzian version of the finite cutoff JT wave
function for $\text{AdS}_{2}$,

$\Psi[\phi,l]=\frac{\phi^{2}\
\text{l}}{(\text{l}^{2}-4\pi^{2})}\text{K}_{2}\Big{(}\sqrt{2|\lambda|}\
|\phi|(\text{l}^{2}-4\pi^{2})^{\frac{1}{2}}\Big{)}.$ (65)

Wave functions (64) and (65) can be contrasted with their respective
counterparts found in (Iliesiu:2020zld, ).

Now, we would like to study (49) on the strip for the choice (63). The
analytically extended (to all values of $\tau_{\text{R}}-\tau_{\text{L}}$ )
new function with $\lambda>0$, is given by (64) where
$\text{l}\rightarrow\text{L}$. This function is regular for all values of
$(\phi,\text{l}_{f},\text{l}_{i},\tau_{\text{R}},\tau_{\text{L}})$, except
when both time-like boundaries have the same boundary condition,
$\text{N}_{\text{R}}-\text{N}_{\text{L}}=0$ (or
$\tau_{\text{R}}-\tau_{\text{L}}=0$), and $|\text{l}_{f}-\text{l}_{i}|=2\pi$.

##  6 Conclusions

In this work we have studied JT quantum gravity on a finite strip with
Lorentzian metrics. This manifold provides a natural setting for computing
time dependent amplitudes.

We have noted that independent of the boundary condition
$\overline{\text{N}}$, (7) the three degrees of freedom
$(\text{N},\text{N}^{\text{r}},\Lambda)$, that are independent in the bulk,
get entangled at the time-like boundaries. In the Lagrangian formulation
relation (7) is taken into account during the variation process. In the
Hamiltonian formulation, where $(\text{N},\text{N}^{\text{r}})$, play the role
of Lagrange multipliers (17), the variation with respect to these fields must
be taken carefully.

Although one could consider $(\text{N},\text{N}^{\text{r}})$, as the degrees
of freedom all the way until the spatial boundaries, we have shown that it is
more convenient to use the Hayward angle $\eta$, (2) to parameterize
$(\text{N},\text{N}^{\text{r}})$, (12) over the time-like boundaries. In this
way, varying $\eta$, is equivalent to varying $(\text{N},\text{N}^{\text{r}})$
over the spatial boundaries while the condition (7) is respected.

The equation associated to $\eta$, (19) is the restriction to the time-like
boundaries of the first equation in (16). This is a desirable fact because it
indicates that there are not contradiction between the Lagrangian and the
Hamiltonian formulations. In the former one never encounters a boundary
equation of motion, see section 2. It is important to note, however, that from
the action in the Hamiltonian form (17), these two equations come from two
different and independent degrees of freedom. One comes from the variation of
$\eta$, while the other comes from the variation of $\text{P}_{\Lambda}$.

In section 4 we start the exploration of JT quantum gravity. We pose the
gravitational path integral for the amplitude we are interested in (4), and
solve it in two parts.

In the first part we focus on the time-like boundaries degrees of freedom,
specifically $\eta$. Although we know how to point wise solve the integrals
for $\eta$, leading to Hankel or modified Bessel functions of the second kind
555 From (4) and (17) we can see that the $\eta$, integrals are the integral
representation of the Hankel functions., we do not know how to handle the
infinite product we get. Certainly it would be nice if we could compare the
result of this infinite product with the result obtained here by just
following the observation that the equation for $\eta$, is of algebraic type.

From this observation, instead of solving the integrals we solve the classical
equation of motion associated to $\eta$, (19) in terms of the other boundary
degrees of freedom and plug the solution back into the action. It might look
here we are proceeding in a semi-classical way but it is not the case. We
still are performing this calculation in a non-perturbative fashion. This is
an standard procedure for this kind of actions. For instance, in string theory
thanks to this procedure we can go from the Polyakov to the Nambu-Goto action.

In the second part we focus on the bulk degrees of freedom. Here to fully find
the amplitude we alternate between the path integral and the canonical
approach. As it is clear at this stage, we can not integrate the dilaton field
in order to solve the path integral. Instead, and based on the classical and
the quantum solutions of the constraints (30)-(34), we propose a canonical
transformation (35) whose generating functional is given in (34). After some
simplifications, we finally arrive at the time dependent QG amplitude (49).
Surprisingly, (49) indicates that the evolution could be non-unitary for most
of the boundary conditions. More surprising perhaps is the fact that for the
particular case of $\text{AdS}_{2}$ boundary conditions unitarity is recovered
(4.1).

As a complement and to make contact with some results found in Iliesiu:2020zld
and Griguolo:2021wgy , or at least contrast with some of the results in these
references, we present the section 5. Here we explore the cylinder and the cap
manifolds as spaces obtained from gluing the time-like boundaries of the
strip. We find the propagator on the cylinder (67) and the $\text{dS}_{2}$,
and $\text{AdS}_{2}$, Lorentzian versions of the cap wave functions (64) and
(65). Remarkably, in these wave functions there are not references to any time
thus there are not indications that evolution could be non-unitary. In the end
we discuss a clear example of non-unitary evolution on the strip for the
particular choice (63) of the function $\chi(\text{C})$.

As explained in Iliesiu:2020zld  finite cutoff JT gravity with
$\text{AdS}_{2}$, boundary conditions can be interpreted as a
$\text{T}\overline{\text{T}}$, deformation of the Schwarzian action. This was
checked in Euclidean JT gravity. Applying similar reasoning here we should
expect that an holographic interpretation for what we have done here could be
given in terms of a $\text{T}\overline{\text{T}}$ deformed Schwarzian theory
with Lorentzian signature. This connection will be explored elsewhere.

## Acknowledgments

We are grateful to Jose J. Fernandez-Melgarejo and Javier Molina-Vilaplana for
comments and support. We would also like to thanks Pablo Bueno, Roberto
Emparan and Arpan Bhattacharyya for discussions, comments and suggestions and
to the member of the gravitation group of ICREA at the Barcelona University
for the hospitality and the interesting discussion during the presentation of
this work.

This work is supported by Next Generation EU through the Maria Zambrano grant
from the Spanish Ministry of Universities under the Plan de Recuperación,
Transformación y Resiliencia, and by the Spanish Ministerio de Innovación y
Ciencia, CARM Fundación Séneca and Universidad de Murcia under the grant
PID2021-125700NA-C22, 21257/PI/19.

##  Appendix A Cylinder propagator

In a pure speculative sense we would like to make some comments on the
cylinder propagator. From a third quantization point of view this object
satisfies the equation Iliesiu:2020zld

$\Big{(}\partial_{\text{l}_{f}}(\text{l}_{f}^{-1}\partial_{\phi})+2\lambda\phi\Big{)}\Psi\Big{[}(\phi,l)_{f},(\phi,l)_{i}\Big{]}=\frac{\delta(\text{l}_{f}-\text{l}_{i})\delta(\phi_{f}-\phi_{i})}{\text{l}_{f}}.$
(66)

We have noted that for the choice
$\chi(\text{C})\propto\frac{\text{d}}{\text{dC}}(\frac{1}{\sqrt{\text{C}}})$,
we can get this propagator from (61)

$\Psi\Big{[}(\phi,l)_{f},(\phi,l)_{i}\Big{]}\propto\text{l}_{f}\text{H}_{0}^{(1)}\Big{(}\sqrt{2\lambda}\sqrt{(\text{l}_{f}^{2}-\text{l}_{i}^{2})(\phi_{f}^{2}-\phi_{i}^{2})}\Big{)},$
(67)

in the cases $(\textbf{l}_{i}=0,\phi_{i}\neq 0)$ and $(\textbf{l}_{i}\neq 0,\
\phi_{i}=0)$. To solve this integral for these cases we first regularize the
integral i.e.,
$\int\limits_{0}^{\infty}\rightarrow\int\limits_{\epsilon}^{\infty}$. Then
after integration by parts, with the substitution
$\text{C}=2\lambda\phi^{2}\text{sinh}^{2}(s)$, with $\lambda>0$ the integral
can be straightforwardly solved. As a final step we remove all the divergent
terms and take the limit $\epsilon\rightarrow 0$.

Note that with the change of variables

$\displaystyle x$ $\displaystyle=$ $\displaystyle\text{l}^{2}+\phi^{2},$
$\displaystyle y$ $\displaystyle=$ $\displaystyle\text{l}^{2}-\phi^{2},$

We can bring the Helmholtz equation in Minkowski space with a Dirac delta
source, whose solution is the Green function

$\text{H}_{0}^{(1)}\Big{(}\sqrt{2\lambda}\sqrt{(x-x^{\prime})^{2}-(y-y^{\prime})^{2}}\Big{)},$
(68)

into (66).

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