# Quantum geometry of the null cone

Wolfgang<EMAIL_ADDRESS>Institute for Quantum Gravity,
Theoretical Physics III, Department of Physics
Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91052
Erlangen, Germany
30 January 2024

###### Abstract

We present a non-perturbative quantization of gravitational null initial data.
Our starting point is the characteristic null initial problem for tetradic
gravity with a parity-odd Holst term in the bulk. After a basic review about
the resulting Carrollian boundary field theory, we introduce a specific class
of impulsive radiative data. This class is defined for a specific choice of
relational clock. The clock is chosen in such a way that the shear of the null
boundary follows the profile of a step function. The angular dependence is
arbitrary. Next, we solve the residual constraints, which are the Raychaudhuri
equation and a Carrollian transport equation for an $SL(2,\mathbb{R})$
holonomy. We show that the resulting submanifold in phase space is symplectic.
Along each null generator, we end up with a simple mechanical system. The
quantization of this system is straightforward. Our basic strategy is to start
from an auxiliary Hilbert space with constraints. The physical Hilbert space
is the kernel of a constraint, which is a combination of ladder operators. The
constraint and its hermitian conjugate are second-class. Solving the
constraint amounts to imposing a simple recursion relation for physical
states. On the resulting physical Hilbert space, the $SL(2,\mathbb{R})$
Casimir is a Dirac observable. This observable determines the spectrum of the
two radiative modes. The area at the initial and final cross sections are
Dirac observables as well. They have a discrete spectrum, which agrees with
earlier results on this topic.

###### Contents

  1. 1 Introduction
  2. 2 Boundary field theory and null symplectic structure
  3. 3 Twistor quantisation of the unimodular group
  4. 4 Quantum impulsive data on the null cone
  5. 5 Summary and Interpretation of Results

##  1 Introduction

If gravity is quantum, we will find ourselves in a quantum spacetime geometry.
What does it mean to live in such a world? What are the operators and
eigenvalues that characterize the quantum nature of space and time? One of the
main difficulties to answer these questions is the issue of general
covariance. In a background invariant theory, geometry can be described only
relationally [1, 2, 3]. This implies the existence of vast gauge redundancies.
Complementary variables, such as the three-metric and Arnowitt–Deser–Misner
(ADM) momentum can provide only a redundant description of the physical state
[4, 5]. If we start from some initial data on a partial Cauchy surface
$\mathcal{B}$, we can always deform the surface into a new hypersurface
$\mathcal{B}^{\prime}$, while keeping fixed the corners
$\mathcal{C}=\partial\mathcal{B}$, to obtain a new but gauge equivalent
representation of the same physical state. For each gauge redundancy there is
a generator, which vanishes as a constraint. To speak about observables, we
need to impose the constraints and find the quantum numbers that characterize
the physical, i.e. gauge invariant, data. There are different strategies to
make this possible. One viewpoint is to attack the problem face-on. This is
the canonical approach, in which one solves the constraints on a suitable
kinematical Hilbert space [6, 7, 8, 9]. In here, following up earlier research
on the subject [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29], we will argue for a slightly different viewpoint. Roughly
speaking, the idea is as follows. Suppose, we had a quantum theory of gravity
where there is a representation of the hypersurface deformation algebra.
Consider a kinematical state $\rho_{\mathcal{B}}$ on a partial Cauchy surface
$\mathcal{B}$, e.g. a three-ball with boundary $S^{2}$. Physical states are
then equivalence classes $[\rho_{\mathcal{B}}]$ under hypersurface
deformations. If we now take a diffeomorphism and deform $\mathcal{B}$ into a
different three-surface $\mathcal{B}^{\prime}$, such that the corner
$\partial\mathcal{B}=\partial\mathcal{B}^{\prime}$ is held fixed and
$\mathcal{B}^{\prime}$ lies in the future Cauchy development of $\mathcal{B}$,
we obtain a new and gauge equivalent state
$\rho_{\mathcal{B}^{\prime}}\sim\rho_{\mathcal{B}}$. If we take this
deformation to its natural extreme, we obtain a state that lies at the
boundary of the future Cauchy development of $\mathcal{B}$, which is a
lightlike three-surface $\mathcal{N}$. If we now reverse the logic, we can
start out from a quantum state at the null boundary $\mathcal{N}$ and take it
as the unique representative (modulo residual diffeomorphisms) that defines
the entire equivalence class $[\rho_{\mathcal{B}}]=:\rho_{\mathcal{N}}$. There
are still residual gauge symmetries, but they are rather mild. This is so,
because there is more structure available. A null surface is ruled by its
light rays, points on different null rays are spacelike separated and the
residual gauge redundancies break down to angle dependent reparametrizations
of the null generators.

Still, we may expect a preservation of difficulty. Further simplifying
assumptions may be useful. In the following, we restrict ourselves to a
certain class of impulsive null initial data [30, 31, 32, 33], in which the
shear is piecewise constant along the null generators. Each null generator,
can then only carry countably many degrees of freedom. There are infinitely
many degrees of freedom along the angular directions, but the constraints or
transport equations do not couple neighbouring light rays, which are spacelike
separated. Thus, we are dealing with a perhaps large but finite-dimensional
Hamiltonian system on each light ray. We are left with a standard quantum
mechanical problem. The practical usefulness of our approach lies also in
certain infrared ambiguities that appear in perturbative gravity. If we take
our description to null infinity, each pulse has a soft graviton component,
i.e. a spin-two excitation at $\mathcal{I}^{\pm}$, whose frequency is peaked
around $\omega=0$. The resulting Fock states are no longer normalizable with
respect to the naive $L^{2}$ inner product at $\mathcal{I}^{\pm}$, because of
gravitational memory [34, 35, 36, 37]. In our truncation, we replace a
continuous profile by a large but finite sequence of pulses, and we will show
below how each such pulse can be quantized at the full non-perturbative level.
Thus, there is an implicit infrared cutoff built into our model.

The paper splits into three main parts. In Section 2, we consider the null
boundary problem for gravity in terms of spin connection variables with a
parity violating Holst term [38] in the bulk. In addition, we will also
present some new developments on the Carrollian geometry [14, 39, 40, 41] of
the spin bundle and clarify a few previous developments on charge quantization
that we first made in [24, 42]. The third section, Section 3, provides a quick
review about the quantisation of $T^{\ast}SL(2,\mathbb{R})$ in terms of
$SL(2,\mathbb{R})$ boundary twistors. This quantization will prove useful at a
later stage of this project when we will discuss how to quantise an entire
sequence of pulses by gluing the states for the individual pulses across the
interjacent corners. The main result is developed in Section 4, which consists
of two parts. The first part is about the classical phase space. We take the
covariant pre-symplectic potential and compute the pull-back to a submanifold
in phase-space, where the shear of the null generators follows the profile of
a step function. We impose the constraints and identify the Dirac observables
at the full non-perturbative level. The second part takes the resulting
finite-dimensional state space to the quantum level. We find a representation
of the canonical Heisenberg relations and define the physical states of the
system. We introduce the physical observables and discuss their spectra for a
specific class of discrete representations of $SL(2,\mathbb{R})$. Both the
shear that characterizes the strength of the radiative data as well as the
area of the cross sections of the null surface turn out to have a discrete
spectrum. It is unclear at this point, whether the continuous series
representations of $SL(2,\mathbb{R})$ have any physical significance in our
model. It is our expectation that they have not, for reasons explained in the
paper.

Our conventions follow previous articles on this topic, $a,b,c,\dots$ are
abstract tensor indices, $\alpha,\beta,\gamma,\dots$ are internal Minkoski
indices, $A,B,C,\dots$ and $\bar{A},\bar{B},\bar{C}$ are spinor indices
($\xi_{A}=\xi^{B}\epsilon_{BA}$, $\xi^{A}=\epsilon^{AB}\xi_{B}$ for
$\epsilon_{AB}=-\epsilon_{BA}$, $\epsilon^{AC}\epsilon_{BC}=\delta^{A}_{B}$)
transforming under the spin $(\tfrac{1}{2},0)$ and $(0,\tfrac{1}{2})$
representation of local $SL(2,\mathbb{C})$ transformations. The configurations
variable in the bulk are the cotetrad $e_{\alpha}\equiv e_{\alpha a}$ and the
$\mathfrak{so}(1,3)$ spin connection
$\tensor{A}{{}^{\alpha}_{\beta}}\equiv\tensor{A}{{}^{\alpha}_{\beta a}}$. Its
curvature is
$\nabla^{2}\equiv\tensor{F}{{}^{\alpha}_{\beta}}=\mathrm{d}\tensor{A}{{}^{\alpha}_{\beta}}+\tensor{A}{{}^{\alpha}_{\gamma}}\wedge\tensor{A}{{}^{\gamma}_{\beta}}$.
The metric signature is $(-$$+$$+$$+)$.

##  2 Boundary field theory and null symplectic structure

###  2.1 Holst action in a causal domain

In four spacetime dimensions, we can add a parity breaking term to the
Hilbert–Palatini action, without changing the field equations in the interior
[38, 43, 44, 45]. This freedom, which does not exist in higher dimensions, is
responsible for a new coupling constant, namely the Barbero–Immirzi parameter
$\gamma$ (w.l.g. $\gamma>0$), which can enter the action in the bulk. When
there is no cosmological constant and all matter couplings are set to zero,
the resulting Holst action is

$S_{\mathcal{M}}[A,e]=\frac{1}{16\pi
G}\int_{\mathcal{M}}\Big{[}\operatorname{\ast}(e_{\alpha}\wedge
e_{\beta})\wedge F^{\alpha\beta}[A]-\frac{1}{\gamma}e_{\alpha}\wedge
e_{\beta}\wedge F^{\alpha\beta}[A]\Big{]},$ (1)

where $\ast(e_{\alpha}\wedge
e_{\beta})=\tfrac{1}{2}\tensor{\epsilon}{{}_{\alpha\beta}^{\alpha^{\prime}\beta^{\prime}}}e_{\alpha^{\prime}}\wedge
e_{\beta^{\prime}}$ is the Hodge dual of the Plebański two-form
$\Sigma_{\alpha\beta}=e_{\alpha}\wedge e_{\beta}$, which is a two-form that
takes values in the Lorentz bundle. On shell, both terms in the action vanish.
The first term vanishes thanks to the vacuum Einstein equations $\ast
F_{\alpha\beta}\wedge e^{\beta}=0$ and the second term vanishes thanks to the
Bianchi identity $F_{\alpha\beta}\wedge e^{\beta}=0$, which is a consequence
of the torsionless condition $\nabla
e^{\alpha}=\mathrm{d}e^{\alpha}+\tensor{A}{{}^{\alpha}_{\beta}}\wedge
e^{\beta}$.

In the following, we consider how the Barbero–Immirzi parameter effects the
null boundary problem of classical and quantum general relativity. This is to
say that we study the state space and algebra of observables derived from (1)
in a four-dimensional spacetime region $\mathcal{M}$ with a boundary
$\partial\mathcal{M}=\mathcal{B}_{+}\cup\mathcal{N}\cup\mathcal{B}_{-}$ that
contains a three-dimensional null boundary $\mathcal{N}$. The bottom and top
of the boundary $\partial\mathcal{M}$ are space-like three-balls
$\mathcal{B}_{\pm}$ that are connected to $\mathcal{N}$ at co-dimension two
boundaries (corners) $\mathcal{C}_{\pm}=\partial\mathcal{B}_{\pm}$. For
simplicity and definiteness of the problem, we assume that $\mathcal{C}_{\pm}$
have the topology of a two-sphere. In addition, the null portion of the
boundary is viewed as an abstract topological manifold
$\mathcal{N}=(-1,1)\times S^{2}$, where
$\mathcal{C}_{+}=\\{+1\\}\times[S^{2}]^{-1}$ and
$\mathcal{C}_{-}=\\{-1\\}\times S^{2}$ are topological two-spheres. Any such
null surface $\mathcal{N}$ is equipped with a projector
$\pi:\mathcal{N}\rightarrow\mathcal{C},\mathcal{N}\ni(u,q)\mapsto
q\in\mathcal{C}$, which maps every light ray in $\mathcal{N}$ into a point on
the base manifold. Accordingly, we can say that a vector field $l^{a}\in
T\mathcal{N}$ is null, if its push-forward onto the base manifold vanishes,
i.e. $\pi_{\ast}l^{a}=0$. This condition serves as a partial gauge fixing in
which the direction of the light rays that are intrinsic to $\mathcal{N}$ is
kept fixed once and for all. Apart from the boundary condition
$u|_{\mathcal{C}_{\pm}}=\pm 1$, the time parameter on the null boundary is
completely arbitrary. In particular, it is not an affine or otherwise
geometrically preferred null coordinate on $\mathcal{N}$.

At the co-dimension two corners, otherwise unphysical gauge modes become
physical [46, 47, 48, 49, 50, 51, 15, 16, 12, 28, 52, 53, 27, 54]. This is a
direct consequence of how the then necessary boundary conditions break the
local gauge symmetries of general relativity in the bulk. Take a generic
solution in the interior and consider the formal evolution from the initial
values at $\mathcal{B_{-}}$ to $\mathcal{B_{+}}$ along some vector field
$\xi^{a}$, which is null if we restrict it to $\mathcal{N}$. Since radiation
can cross the null boundary, the resulting flow equations can no longer be
Hamiltonian. The system is dissipative. Additional radiative data must be
fixed before we can integrate the field equations from $\mathcal{B_{-}}$ to
$\mathcal{B_{+}}$. Since a generic four-dimensional diffeomorphism moves the
null boundary relative to the fields in the bulk, the required boundary
conditions break diffeomorphism invariance. Once a gauge symmetry is broken,
we obtain a new additional physical mode. Since the symmetry is only broken at
the boundary, the resulting physical phase space on $\mathcal{B}_{\pm}$
consist of radiative modes inside and additional boundary modes at
$\mathcal{C}_{\pm}$. The boundary modes themselves split into edge modes
$\lambda$, which serve as reference frames [55, 56, 57] for large gauge
symmetries, and their conjugate momenta, which define the charges
$Q_{\lambda}$ at the boundary.222Small gauge symmetries, on the other hand,
have gauge parameters $\lambda$ that vanish at the corners. They are mere
gauge redundancies of the theory in the bulk. Large gauge symmetries are not.

Although the Barbero–Immirzi parameter does not enter the field equations
$\ast F_{\alpha\beta}\wedge e^{\beta}=0$ and $\nabla e^{\alpha}=0$ in the
bulk, it can have observables consequences. It alters the symplectic
structure, which determines the quantisation map. This alteration is subtle.
It does not change the classical Poisson algebra of the two radiative modes at
the null boundary [28]. It only affects the boundary modes. In this way, the
generators of boundary symmetries depend explicitly on $\gamma$. At the
classical level, this amounts to a mere relabelling of canonical charges,
which does not change the physics. Consider then the dictionary between the
abstract gauge charges $\\{Q_{\lambda}\\}$ and more standard observables
$\\{O_{\alpha}\\}$ such as mass, spin, angular momentum. If we start from the
action (1), the map between $\\{O_{\alpha}\\}$ and $\\{Q_{\lambda}\\}$ depends
on $\gamma$. Hence there is some functional dependence
$Q_{\lambda}(\gamma,\\{{O}_{\alpha}\\})$. As far as the classical observables
are concerned, this dependence amounts to a mere relabelling of the list of
boundary charges. The numerical value, that each of the physical observables
$\\{{O}_{\alpha}\\}$ has, can only depend on the metric $g_{ab}$ and its
derivatives at the boundary, i.e
${O}_{\alpha}={O}_{\alpha}[g_{ab},\nabla_{a},\mathcal{C}]$. Given a classical
solution $g_{ab}$ of the field equations, the observables $\\{O_{\alpha}\\}$
are independent of $\gamma$. At the quantum level, the situation is more
subtle. Now, the logic is reversed. We start from the algebra of observables,
schematically
$[\hat{Q}_{\lambda},\hat{Q}_{\lambda^{\prime}}]=\hat{Q}_{[\lambda,\lambda^{\prime}]}$,
and seek for a representation of the canonical commutation relations on a
Hilbert space. Given such a representation, we can predict possible
eigenvalues for the charges. It is only in a second step that we can then
invert the map $Q_{\lambda}(\gamma,\\{{O}_{\alpha}\\})$ between the charges
$\\{Q_{\lambda}\\}$ and the observables $\\{O_{\alpha}\\}$. The commutation
relations for the boundary charges are determined by their algebraic
properties $\\{\hat{Q}_{\lambda}\\}$, which can be inferred from the
symmetries they generate. Hence, there is no dependence on $\gamma$. All
eigenvalues for the charges are independent of $\gamma$. Yet, the map
$Q_{\lambda}(\gamma,\\{{O}_{\alpha}\\})$ depends on $\gamma$. Thus, the
Barbero–Immirzi parameter can enter the spectrum of physical observables
$\\{{O}_{\alpha}\\}$. Besides the boundary modes, there is also the radiative
data. Since the charges must be consistent with certain flux-balance laws for
the free radiative data, the radiative data will also be affected by the
addition of $\gamma$. A similar mechanism appears in QED in the presence of a
$\theta$-term in the action, which can alter the quantization condition
between electric and magnetic charges [58]. For each different value of
$\gamma$, we can expect, therefore, unitarily inequivalent representations of
the canonical commutation relations. Depending on the value of $\gamma$, the
quantisation of the radiative data will be different. Since $\gamma$ does not
enter the classical Poisson algebra for the radiative data, this may sound
very surprising at first. But we will see below how this mechanism can play
out in practice.

###  2.2 Boundary conditions and boundary Carollian field theory

Depending on which boundary conditions we choose, we need to add different
boundary terms to the action. In the following, we consider a class of
conformal boundary conditions along the null boundary for the Holst action (1)
in the bulk. The treatment becomes particularly transparent in terms of self-
dual (spinor-valued) two-forms. The action in the bulk (1) can be now seen as
a sum of selfdual and anti-selfdual parts,

$S_{\mathcal{M}}[A,e]=\Big{[}\frac{\mathrm{i}}{8\pi\gamma
G}(\gamma+\mathrm{i})\int_{\mathcal{M}}\Sigma_{AB}\wedge
F^{AB}\Big{]}+\mathrm{cc.},$ (2)

where
$\tensor{F}{{}^{A}_{B}}=\mathrm{d}\tensor{A}{{}^{A}_{B}}+\tensor{A}{{}^{A}_{C}}\wedge\tensor{A}{{}^{C}_{B}}$
is the curvature of the self-dual connection and $\Sigma_{AB}$ is the
Plebański two-form

$\Sigma_{AB}=-\frac{1}{2}e_{A\bar{C}}\wedge\tensor{e}{{}_{B}^{\bar{C}}},$ (3)

where $e_{AA^{\prime}}$ are the tetrads in spinor notation.

A possible boundary term for conformal boundary conditions on a null surface
for the Holst action was introduced in [25, 46]. The simplest way to
understand the corresponding boundary Lagrangian is to start out from the
pull-back to the null boundary of the Plebański two-form in the bulk. A
straightforward calculation gives

$\varphi^{\ast}_{\mathcal{N}}\Sigma_{AB}=e_{(A}\wedge\bar{m}\,\ell_{B)},$ (4)

where $\ell^{A}$ is a section of the spinor bundle over the null surface and
$e_{A}$ is a spinor-valued one-form at the boundary. The complex-valued one-
forms $m_{a}$, on the other hand, define the pull-back of the spacetime metric
to the null boundary,

$\varphi^{\ast}g_{ab}=2m_{(a}\bar{m}_{b)}.$ (5)

The wedge product of $m$ and $\bar{m}$ gives the area two-form $\varepsilon$
of the null surface,

$\varepsilon_{ab}=-2\mathrm{i}m_{[a}\bar{m}_{b]}.$ (6)

To extend the dyads $(m_{a},\bar{m}_{a})$ into a co-basis of
$T^{\ast}\mathcal{N}$, we introduce a one-form

$k=-N\mathrm{d}u,$ (7)

where $u:\mathcal{N}\rightarrow(-1,1)$ is the time-coordinate, which we
introduced earlier, see section 2.1. We do not impose any specific condition
on $u$ other that it increase monotonically along the null generators
$\gamma_{q}=\pi^{-1}(q)$, $q\in\mathcal{C}$. The scalar function $N$, on the
other hand, plays the role of a lapse function. Given such a triple
$(k_{a},m_{a},\bar{m}_{a})$, we then also have a corresponding dual basis,
which is denoted by $(\ell^{a},m^{a},\bar{m}^{a})$. It satisfies
$k_{a}\ell^{a}=-1$ and $m_{a}\bar{m}^{a}=1$. All other contractions vanish.
Therefore, $\ell^{a}$ is a null vector, i.e. $\pi_{\ast}\ell^{a}=0$.

The exterior derivatives of the one-forms $(m_{a},\bar{m}_{a})$ define spin
connection coefficients intrinsic to $\mathcal{N}$,

$\mathrm{d}m=-\Big{(}\tfrac{1}{2}\vartheta_{(\ell)}+\mathrm{i}\varphi_{(\ell)}\Big{)}k\wedge
m-\sigma_{(\ell)}\,k\wedge\bar{m}+\mathrm{i}\,\gamma\,m\wedge\bar{m},$ (8)

where $\vartheta_{(\ell)}$ is the expansion of the null generators $\ell^{a}$
and $\sigma_{(\ell)}$ is the shear. The coefficients $\varphi_{(\ell)}$ and
$\gamma$, on the other hand, can be rearranged into a $U(1)$ connection

$\Gamma=k\varphi_{(\ell)}+\gamma\bar{m}+\bar{\gamma}m.$ (9)

Consider then the components of the one-form $e_{A}$ with respect to the co-
basis $(k_{a},m_{a},\bar{m}_{a})$,

$e_{A}\wedge\bar{m}=(\ell_{A}k-k_{A}m)\wedge m.$ (10)

On the null surface, the spinors $k^{A}$ and $\ell^{A}$ are linearly
independent and satisfy $k_{A}\ell^{A}=1$. Hence they define a basis in the
spinor bundle over $\mathcal{N}$. We can now take the pull back of the
selfdual connection to $\mathcal{N}$, thereby introducing a covariant
derivative333The corresponding connection, which contains depends on both
extrinsic and intrinsic data at $\mathcal{N}$, is the null surface analogue of
the self-dual Ashtekar connection [59] on a spatial hypersurface.
$D=\varphi^{\ast}_{\mathcal{N}}\nabla$, and obtain spin rotation coefficients.
If the torsionless condition is satisfied, we find

$\displaystyle\ell_{A}D\ell^{A}$
$\displaystyle=-\frac{1}{2}\big{(}\vartheta_{(\ell)}m+\sigma_{(\ell)}\bar{m}\big{)},$
(11) $\displaystyle k_{A}D\ell^{A}$
$\displaystyle=\frac{1}{2\mathrm{i}}\big{(}\Gamma+\mathrm{i}K\big{)},$ (12)

where $\Gamma$ is the $U(1)$ connection (9) and the one-form $K$ depends on
only extrinsic data. It admits the decomposition

$K=-\kappa_{(\ell)}k+\bar{\alpha}m+\alpha\bar{m},(\ref{Kdef})$ (13)

where $\kappa_{(\ell)}$ is the affinity of the null generators $\ell^{a}$ and
the coefficient $\alpha$ satisfies

$\bar{\alpha}=-N^{-1}\bar{m}^{a}\partial_{a}N+\ell^{b}k_{A}D_{b}k^{A}.$ (14)

The components $m^{b}k_{A}D_{b}k^{A}=\tfrac{1}{2}\vartheta_{(k)}$ and
$\bar{m}^{b}k_{A}D_{b}k^{A}=\bar{\sigma}_{(k)}$ are the shear and expansion of
the transversal null direction $k^{a}\equiv\mathrm{i}k^{A}\bar{k}^{\bar{A}}$.
In the following, these two transversal components play no further role in our
analysis.

Consider then the following shifted and complexified $U(1)$ connection $A$,

$A=-\mathrm{i}\Gamma+K+\vartheta_{(\ell)}k.$ (15)

Given this connection, it is then easy to show that the boundary spinors
satisfy the following boundary equations along the null surface

$\displaystyle\bar{m}\wedge\big{(}D-\tfrac{1}{2}A\big{)}$
$\displaystyle=\tfrac{1}{2}\vartheta_{(\ell)}e^{A}\wedge\bar{m},$ (16)
$\displaystyle\big{(}D+\tfrac{1}{2}A\wedge\big{)}(e_{A}\wedge\bar{m})$
$\displaystyle=0.$ (17)

Equations (16) and (17) determine Carollian transport equations for the
boundary spinors $\ell^{A}$ and $e_{A}\operatorname{mod}\bar{m}$ along the
null generators. A boundary action, whose equations of motion are (16) and
(17) is then given by

$S_{\mathcal{N}}[e_{A},\ell^{A},\tensor{A}{{}^{A}_{B}}|m,A,\vartheta]=\frac{\mathrm{i}}{8\pi\gamma
G}(\gamma+\mathrm{i})\int_{\mathcal{N}}\Big{[}e_{A}\wedge\bar{m}\wedge(D-\tfrac{1}{2}A)\ell^{A}-\tfrac{\vartheta}{4}e_{A}\wedge
e^{A}\wedge\bar{m}\Big{]}+\mathrm{cc.},$ (18)

where $\vartheta=\bar{\vartheta}$ is real, while all other configuration
variables are complex. The action assumes the standard form of Hamiltonian
mechanics, in which the action is a difference of a symplectic term
$\theta\propto e_{A}\wedge\bar{m}\wedge(D-\tfrac{1}{2}A)\ell^{A}$ and a
Hamiltonian $h\propto\tfrac{\vartheta}{4}e_{A}\wedge e^{A}\wedge\bar{m}$,
which is quadratic in the momenta $\pi_{A}=e_{A}\operatorname{mod}(\bar{m})$.
In computing the saddle points of the action, we keep a gauge equivalence
class of boundary fields $[m,A,\vartheta]$ fixed:

$\textsl{g}=[m,A,\vartheta]/_{\sim},\qquad\delta\textsl{g}=0.$ (19)

The symmetries that define this equivalence class are

\- Vertical diffeomorphisms:

$[\varphi^{\ast}m,\varphi^{\ast}A,\vartheta\circ\varphi]\sim[m,A,\vartheta]$
for all
$\varphi\in\mathrm{Diff}(\mathcal{N}:\mathcal{N}):\pi\circ\varphi=\mathrm{id}_{\mathcal{C}}$.

\- $U(1)$ transformations:

$[\mathrm{e}^{\mathrm{i}\phi}m,A+\mathrm{i}\,\mathrm{d}\phi,\vartheta]\sim[m,A,\vartheta]$
for all $\phi:\mathcal{N}\rightarrow\mathbb{R}$.

\- Dilatations of the null normal:

$[m,A+\mathrm{d}\lambda,\mathrm{e}^{\lambda}\vartheta]\sim[m,A,\vartheta]$ for
all $\lambda:\mathcal{N}\rightarrow\mathbb{R}$.

\- Conformal rescalings:

$[\mathrm{e}^{\omega}m,A,\mathrm{e}^{\omega}\vartheta]\sim[m,A,\vartheta]$ for
all $\omega:\mathcal{N}\rightarrow\mathbb{R}$.

\- Shifts of the $U(1)_{\mathbb{C}}$ connection:

$[m,A,\vartheta]\sim[m,A+\frac{\mathrm{i}}{\gamma+\mathrm{i}}\beta+\bar{\xi}_{+}m+\xi_{-}\bar{m},\vartheta]$
for all one-forms $\beta\in\Omega^{1}(\mathcal{N}:\mathbb{R})$ and scalars
$\xi_{\pm}:\mathcal{N}\rightarrow\mathbb{C}$. This condition removes all but
just one of the $3\times 2=6$ real components of the abelian boundary
connection $A\in\Omega^{1}(\mathcal{N}:\mathbb{C})$. The residual component is
essentially the affinity $\kappa_{(\ell)}$ shifted by terms proportional to
$\vartheta_{(\ell)}$ and $\varphi_{(\ell)}$ that only depend on the intrinisic
geometry of $\mathcal{N}$.

Notice that we have fixed earlier the null directions of $\mathcal{N}$ as an
external background structure, i.e. we do not change how to embed the abstract
light rays $\pi^{-1}(q),q\in\mathcal{C}$ into the null boundary $\mathcal{N}$.
This implies that the one-form $m_{a}$ can only have two complex degrees of
freedom, since $l^{a}m_{a}=0$ for all $l^{a}\in
T\mathcal{N}:\pi_{\ast}l^{a}=0$. The triple $[m,A,\vartheta]$ is thus
specified by $4+6+1=11$ local parameters on $\mathcal{N}$. The equivalence
relation removes nine of them. The remaining two degrees of freedom
characterize the free initial data along $\mathcal{N}$.

The coupled bulk plus boundary action is the sum
$S=S_{\mathcal{M}}+S_{\mathcal{N}}$ of (2) and (18). If we now vary the action
and determine its stationary points for the conformal boundary conditions
(19), we obtain the usual Einstein equations in the bulk and additional gluing
conditions and boundary field equations. The boundary field equations (16) and
(17) transport the boundary spinors $\ell^{A}$ and $e_{A}\wedge\bar{m}$ along
the null generators. The gluing conditions (4) link them to the fields in the
bulk. If we then take variations of the action that violate the boundary
conditions $\delta\textsl{g}\neq 0$, we obtain the symplectic structure. After
some straightforward algebraic manipulations, we obtain the symplectic
structure on the null surface $\mathcal{N}$, which is now given by

$\Theta_{\mathcal{N}}=\frac{\mathrm{i}}{8\pi\gamma
G}(\gamma+\mathrm{i})\int_{\mathcal{N}}\Big{[}\bbvar{d}(k\wedge\bar{m})\wedge\ell_{A}D\ell^{A}+\mathrm{i}\varepsilon\wedge\bbvar{d}A\Big{]}+\mathrm{cc.},$
(20)

where $\bbvar{d}$ is the exterior derivative on the covariant phase
space.444N.B. A perhaps much faster way to arrive at the same result is to
start out from the symplectic potential
$\Theta_{\mathcal{B}}=\frac{\mathrm{i}}{8\pi\gamma
G}(\gamma+\mathrm{i})\int_{\mathcal{B}}\Sigma_{AB}\wedge\bbvar{d}A^{AB}+\mathrm{cc.}$
on a partial Cauchy surface $\mathcal{B}$, then take a formal limit in which
$\mathcal{B}$ is sent to a null surface, and impose, by hand, that all the
variations of the spin dyad vanish $\bbvar{d}\ell^{A}=\bbvar{d}k^{A}=0$.

Let us close this section with a few comments on the residual diffeomorphism
symmetries. The abstract null boundary
$(\mathcal{N}=(-1,1)\times\mathcal{C},\pi)$ is equipped with a projector onto
the corner $\mathcal{C}$, $\pi:\mathcal{N}\rightarrow\mathcal{C}$. For any
$q\in\mathcal{C}$, the light rays $\gamma_{q}=\pi^{-1}(q)$ provide us with a
background structure, which we keep fixed.555This can be justified
relationally. To say that two events $x$ and $x^{\prime}$ lie on the same null
ray, $\pi(x)=\pi(x^{\prime})$, is a coordinate invariant statement. Thus, any
variation of the null normal $\bbvar{d}\ell^{a}$ will be again proportional to
$\ell^{a}$, $\bbvar{d}\ell^{a}\propto\ell^{a}$. This, in turn, implies that
the diffeomorphism symmetries of the null surface are broken down to the semi-
direct product of
$\mathrm{Diff}(\mathcal{C})\ltimes\mathrm{Diff}(I)^{\mathcal{C}}$, where
$\mathrm{Diff}(I)$ are the diffeomorphisms of the intervall $(-1,1)\ni u$. Of
these, only the subgroup $\mathrm{Diff}(I)^{\mathcal{C}}$ generated by
vertical vector fields $\xi^{a}\in T\mathcal{N}:\pi_{\ast}\xi^{a}=0$ that
vanish at the two corners are redundant null directions of the pre-symplectic
two-form $\Omega_{\mathcal{N}}=\bbvar{d}\Theta_{\mathcal{N}}$. Elements of the
rotational subgroup $\mathrm{Diff}(\mathcal{C})$, on the other hand, do not
generate gauge redundancies. They are true physical symmetries, i.e. motions
on the physical phase space. The residual gauge redundancies are generated by
a constraint, namely the Raychaudhuri equation

$\displaystyle 0=\mathrm{i}$
$\displaystyle\varphi^{\ast}_{\mathcal{N}}\Big{(}F_{AB}\wedge\tensor{e}{{}^{B}_{\bar{A}}}\ell^{A}\bar{\ell}^{\bar{A}}\Big{)}=(\ell_{A}D^{2}\ell^{A})\wedge\bar{m}=$
$\displaystyle=\big{(}\mathrm{d}(\ell_{A}D\ell^{A})-D\ell_{A}\wedge
D\ell^{A}\big{)}\wedge\bar{m}=\big{(}\mathrm{d}(\ell_{A}D\ell^{A})+\mathrm{i}(\Gamma+\mathrm{i}K)\wedge\ell_{A}D\ell^{A}\big{)}\wedge\bar{m}=$
$\displaystyle=\frac{1}{2}\Big{(}\ell^{a}\partial_{a}[\vartheta_{(\ell)}]-\kappa_{(\ell)}\vartheta_{(\ell)}+\frac{1}{2}\vartheta_{(\ell)}^{2}+2\sigma_{(\ell)}\bar{\sigma}_{(\ell)}\Big{)}k\wedge
m\wedge\bar{m}.$ (21)

By keeping the null directions of $\mathcal{N}$ fixed, we reduced
$\mathrm{Diff}(\mathcal{N})$ down
$\mathrm{Diff}(\mathcal{C})\ltimes\mathrm{Diff}(I)^{\mathcal{C}}$, of which
only the normal subgroup of angle dependent reparametrizations of the null
direction (forming the little group of $\mathcal{N}$, so to say) survive as
redundant gauge symmetries (i.e. null directions of
$\Omega_{\mathcal{N}}=\bbvar{d}\Theta_{\mathcal{N}}$). The elements of
$\mathrm{Diff}(\mathcal{C})$, on the other hand, are genuine symmetries
generated by boundary charges, see e.g. [49, 60, 61, 28, 25]. At this point it
is useful to compare our situation to the Hamiltonian analysis in terms of ADM
variables, where there are four constraints on a spatial hypersurface
$\Sigma$. There are the three vector constraints that generate
$\mathrm{Diff}(\Sigma)$ and there is one additional scalar constraint that
generates the motions that take us away from $\Sigma$. On a null boundary, the
situation is different. Diffeomorphisms that take us away from the null
surface, e.g. those that are generated by vector fields
$\xi^{a}\propto\mathrm{i}\tensor{e}{{}_{A\bar{A}}^{a}}k^{A}\bar{k}^{\bar{A}}$,
are no longer gauge symmetries. In fact, they are not even Hamiltonian, and
there is a clear physical reason why this must be so. To move the null
boundary relative to the metric in the bulk, new initial data must be
specified. This data is arbitrary. There are infinitely many ways to extend a
given solution in the interior to the outside. Hence, such an outward radial
evolution can no longer be Hamiltonian. At the quantum level, we expect,
therefore, that the radial evolution will be governed by some non-unitary flow
equation, e.g. a Lindblad equation for the physical states. In this way, the
hard problem of the ADM canonical framework, namely how to solve the
Wheeler–DeWitt equation, turns into a secondary question. First, we seek for a
quantum representation of the geometry of the null cone. As we will see below,
we can already do interesting physics at that stage. The question of how to
move the null boundary and trace out, thereby, an entire four-dimensional
geometry is secondary. It is related to coarse graining and radial
renormalization [62, 63, 64].

###  2.3 $SL(2,\mathbb{R})$ holonomies on the lightcone

The gauge equivalence class of boundary data (19) provides a complete
characterization of the free radiative data on the null surface boundary. Yet,
this definition is not very practical. To compute Poisson brackets among Dirac
observables, we need a more useful parametrization. In the following, the
basic idea is to use certain group-valued data to parametrize the boundary
metric at the full non-perturbative level. Instead of the metric $q_{ab}$, we
work with dyadic one-forms $(m_{a},\bar{m}_{a})$, which are normal to the null
directions, i.e. $l^{a}m_{a}=0$ for all $l^{a}\in
T\mathcal{N}:\pi_{\ast}l^{a}=0$. The one-forms $m_{a}$ are thus characterized
by four numbers. One of them can be chosen to be a conformal factor $\Omega$
that determines the overall scale of the geometry. The other three can be
repackaged into an $SL(2,\mathbb{R})$ holonomy that determines a transport
equation between an arbitrary fiducial dyad
$({}^{(0)}m_{a},{}^{(0)}\bar{m}_{a})$ and the physical dyads
$(m_{a},\bar{m}_{a})$. To pick such a fiducial dyad, we choose coordinates
$(z,\bar{z})$ on the base manifold $\mathcal{C}\simeq S^{2}$. Next, we drag
the coordinates along the null generators using the Lie derivative, i.e.
$l^{a}\partial_{a}[z]=0$ for all $l^{a}:\pi_{\ast}\ell^{a}=0$. We can now
define the following fiducial one-forms on $\mathcal{N}$. We set

${}^{(0)}m=\frac{\sqrt{2}}{1+|z|^{2}}\mathrm{d}z.$ (22)

The corresponding fiducial area element is

$d^{2}v_{o}=-\mathrm{i}\,{}^{(0)}m\wedge{}^{(0)}\bar{m}.$ (23)

To define an $SL(2,\mathbb{R})$ group action on the dyads, we introduce an
auxiliary two-dimensional real vector space $\mathbb{V}$ and its dual space
$\mathbb{V}^{\ast}$ with complex basis vectors

$\displaystyle m^{i}\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
+\mathrm{i}\end{pmatrix},$
$\displaystyle\qquad\bar{m}^{i}\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
-\mathrm{i}\end{pmatrix},$ (24) $\displaystyle
m_{i}\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}1&\mathrm{i}\end{pmatrix},$
$\displaystyle\qquad\bar{m}_{i}\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}1&-\mathrm{i}\end{pmatrix}.$
(25)

Next, we define the following co-dyad on $\mathcal{N}$

$\tensor{e}{{}^{i}_{a}}=m^{i}\bar{m}_{a}+\mathrm{cc.},$ (26)

where $i=1,2$ are internal $SL(2,\mathbb{R})$ indices. Any such co-frame field
can be now parametrized in terms of the fiducial basis by an overall conformal
factor $\Omega$ and $SL(2,\mathbb{R})$ holonomy
$\tensor{S}{{}^{i}_{j}}:\mathcal{N}\rightarrow SL(2,\mathbb{R})$ such that

$e^{i}=\Omega\,\tensor{S}{{}^{i}_{j}}\big{(}m^{j}\,{}^{(0)}\bar{m}_{a}+\mathrm{cc.}\big{)}.$
(27)

The holonomy satisfies a simple transport equation along the null generators,

$\ell^{a}\partial_{a}S=\big{(}\varphi_{(\ell)}J+(\sigma_{(\ell)}\bar{X}+\bar{\sigma}_{(\ell)}X)\big{)}S,$
(28)

where $\sigma_{(\ell)}$ is the shear and
$\varphi_{(\ell)}=-\ell^{a}\Gamma_{a}$, for the $U(1)$ connection (9). In
here, we introduced a basis in $\mathfrak{sl}(2,\mathbb{R})$, whose matrix
elements are given by

$\tensor{J}{{}^{i}_{k}}=\mathrm{i}(\bar{m}^{i}m_{k}-m^{i}\bar{m}_{k})\equiv\tensor{\varepsilon}{{}^{i}_{k}},\quad\tensor{X}{{}^{i}_{k}}=m^{i}m_{k},\qquad\tensor{\bar{X}}{{}^{i}_{k}}=\bar{m}^{i}\bar{m}_{k}.$
(29)

These matrices satisfy standard the commutation relations

$[J,X]=-2\mathrm{i}\,X,\qquad[J,\bar{X}]=+2\mathrm{i}\,\bar{X},\qquad[X,\bar{X}]=\mathrm{i}\,J$
(30)

###  2.4 Choice of relational clock

To access the two true physical degrees of freedom of the gravitational field
on a null cone, we need to impose the Raychaudhuri equation, which is the
generator on phase space of angle dependent reparametrizations of the null
generators $\gamma_{q}=\pi^{-1}(q),q\in\mathcal{C}$. At the null surface, we
have a fiducial, but unphysical, time coordinate $u$, which is the $2+1$
Carollian version of the usual ADM time coordinate $t$. Surfaces of constant
$u$ are two-dimensional spatial sections of $\mathcal{N}$, but they do not
satisfy any particular constraint. If we now also add the coordinates
$(z,\bar{z})$, which are constant along the null generators, we have a
fiducial three-dimensional coordinate system $(u,z,\bar{z})$ on $\mathcal{N}$.
Thus, we also have a basis
$(\partial_{u}^{a},\partial^{a}_{z},\partial^{a}_{\bar{z}})$ in
$T\mathcal{N}$, where $\partial_{u}^{a}$ is null. Since $k=-N\mathrm{d}u$, see
(7), we obtain

$\ell^{a}=N^{-1}\partial^{a}_{u}.$ (31)

Dilations of $\ell^{a}$, sending $\ell^{a}$ into
$\mathrm{e}^{\lambda}\ell^{a}$ are gauge symmetries. The corresponding boost
charge $K_{\lambda}[\mathcal{C}_{\pm}]$, see e.g. [28], is given by the gauge
parameter $\lambda$ integrated against the area of the cross section

$K_{\lambda}[\mathcal{C}_{\pm}]=\frac{1}{8\pi
G}\oint_{\mathcal{C}_{\pm}}\varepsilon\,\lambda.$ (32)

Dilatations of the null normal change the affinity $\kappa_{(\ell)}$, sending
$\kappa_{(\ell)}$ into
$\mathrm{e}^{\lambda}(\kappa_{(\ell)}+\ell^{a}\partial_{a}\lambda)$.

To proceed further, we choose a clock. Taking advantage of both dilatations
and angle-dependent diffeomorphisms, we introduce a new coordinate system
$(\mathcal{U},z,\bar{z})=\mathcal{U}(u,z,\bar{z}),z,\bar{z})$ on $\mathcal{N}$
in which the Raychaudhuri equation simplifies. Our choice is666We call this
clock teleological, from greek telos, which means end, purpose or goal,
because our clock knows that it will assume the values $\pm 1$ at the future
and past endpoints of $\mathcal{N}$. It is impossible to find such a clock
from data given only in a small neighbourhood of $\mathcal{C}_{-}$, i.e. the
past corner of $\mathcal{N}$.

$\displaystyle\partial^{b}_{\mathcal{U}}\nabla_{b}\partial^{a}_{\mathcal{U}}$
$\displaystyle=-\Omega^{-1}\partial_{\mathcal{U}}\Omega\,\partial^{a}_{\mathcal{U}},$
(33) $\displaystyle\mathcal{U}\big{|}_{\mathcal{C}_{\pm}}$ $\displaystyle=\pm
1.$ (34)

With respect to this clock, the Raychaudhuri equation becomes

$\frac{\mathrm{d}^{2}}{\mathrm{d}\mathcal{U}^{2}}\Omega^{2}=-2\sigma\bar{\sigma}\Omega^{2}.$
(35)

A quick side remark: if there is no shear ($\sigma=0$), the area of the cross
sections will increase linearly in $\mathcal{U}$ and $d=\sqrt{\mathcal{U}}$
becomes a sort-of luminosity distance.

In here, the shear is obtained from the transport equation for the
$SL(2,\mathbb{R})$ holonomy.777The spin coefficients $\varphi_{(\ell)}$ and
$\sigma_{(\ell)}$ refer to the physical co-frame $(k,m,\bar{m})$, which is
dual to $(\ell^{a},m^{a},\bar{m})$. It is this co-frame that enters the
definition of the kinematical symplectic potential (20). For the spin
coefficients of the gauge-fixed co-frame $(-\mathrm{d}\mathcal{U},m,\bar{m})$,
we drop the subscript and simply write $\varphi$ and $\sigma$.

$\frac{\mathrm{d}}{\mathrm{d}\mathcal{U}}S=\Big{(}\varphi
J+\big{(}\sigma\bar{X}+\mathrm{cc.}\big{)}\Big{)}S.$ (36)

The new time variable $\mathcal{U}$, which is a partial observable on the
kinematical phase space of the null surface, can be obtained from the fiducial
and unphysical time variable $u$ ($\bbvar{d}u=0$) by

$\mathcal{U}(u,z,\bar{z})=-1+\int_{-1}^{u}\mathrm{d}u^{\prime}N(u^{\prime},z,\bar{z})\mathrm{e}^{\lambda(u,z,\bar{z})},$
(37)

where $N$ is the lapse function and the gauge parameter
$\lambda=\lambda^{(0)}+\Delta\lambda$ is the sum of

$\displaystyle\lambda^{(0)}(u,z,\bar{z})$
$\displaystyle=\int_{-1}^{u}\mathrm{d}u^{\prime}\big{(}N\kappa_{(\ell)}+\Omega^{-1}\partial_{u}\Omega\big{)}(u^{\prime},z,\bar{z}),$
(38) $\displaystyle\mathrm{e}^{-(\Delta\lambda)(z,\bar{z})}$
$\displaystyle=\frac{1}{2}\int_{-1}^{1}\mathrm{d}u\,N(u,z,\bar{z})\mathrm{e}^{\lambda^{(0)}(u^{\prime},z,\bar{z})}.$
(39)

Thus, the clock depends as a functional on the lapse function $N$, the
affinity $\kappa_{(\ell)}$ and the conformal factor $\Omega$. Each of these
fields are part of the kinematical phase space equipped with the symplectic
potential (20). Therefore, the new clock is a field-dependent variable. The
same applies to the gauge parameter $\lambda$, such that

$\bbvar{d}\mathcal{U}\neq 0,\qquad\bbvar{d}\lambda\neq 0.$ (40)

In fact, the map between $\ell^{a}$ and $\partial^{a}_{\mathcal{U}}$ is a
dilation with gauge parameter $\lambda$. Going back to (31) and (37) obtain

$\ell^{a}=\mathrm{e}^{\lambda}\partial^{a}_{\mathcal{U}},\qquad k\wedge
m\wedge\bar{m}=-\mathrm{e}^{-\lambda}\mathrm{d}\mathcal{U}\wedge{m}\wedge\bar{m}.$
(41)

###  2.5 Null symplectic structure in terms of $SL(2,\mathbb{R})$ holonomies

After having introduced a paramerization for the null direction $\ell^{a}$ and
for the co-dyad $(m,\bar{m})$, see (41) and (27), we return to the pre-
symplectic potential (20). All differentials on phase space can be absorbed
back into variations of $(\mathrm{e}^{\lambda},\Omega,\Gamma)$ and an
$SL(2,\mathbb{R})$ holonomy on the lightcone. We obtain

$\displaystyle\Theta_{\mathcal{N}}=$ $\displaystyle-\frac{1}{8\pi
G}\int_{\partial\mathcal{N}}d^{2}v_{o}\,\Omega^{2}\bbvar{d}\lambda+\frac{1}{16\pi
G}\int_{\partial\mathcal{N}}d^{2}v_{o}\bbvar{d}\Omega^{2}+$
$\displaystyle-\frac{1}{8\pi\gamma
G}\int_{\mathcal{N}}\bigg{[}\frac{1}{2}d^{2}v_{o}\,\wedge\mathrm{d}\Omega^{2}\,\operatorname{Tr}\big{(}J\bbvar{d}SS^{-1}\big{)}+$
$\displaystyle\hskip
60.00009pt+d^{2}v_{o}\wedge\mathrm{d}\mathcal{U}\,\Omega^{2}\operatorname{Tr}\Big{(}(\sigma\bar{X}+\bar{\sigma}X)(\gamma\bbvar{1}-J)\bbvar{d}SS^{-1}\Big{)}\bigg{]}+$
$\displaystyle-\frac{1}{8\pi
G}\int_{\mathcal{N}}d^{2}v_{o}\,\bbvar{d}\mathcal{U}\frac{\mathrm{d}^{2}}{\mathrm{d}\mathcal{U}^{2}}\Omega^{2}-\frac{1}{8\pi\gamma
G}\int_{\mathcal{N}}d^{2}v_{o}\,\Omega^{2}\bbvar{d}\Gamma,$ (42)

where $\Gamma=-\varphi\,\mathrm{d}\mathcal{U}+\gamma\bar{m}+\bar{\gamma}m$ is
the $U(1)$ connection (9) on $\mathcal{N}$ and $(J,X,\bar{X})$ are the
$SL(2,\mathbb{R})$ generators (29). In what follows, the two boundary terms in
the first line are irrelevant: The second term is a total derivative on phase
space. Hence, it does not contribute to the presymplectic two-form. The first
term, on the other hand, can be reabsorbed back into a redefinition of the
$U(1)$ connection.888This is possible, because the generator
$K_{\lambda}[\mathcal{C}]$ for dilations that acts on the triple
$(\ell^{a},m_{a},\bar{m}_{a})$ by $\delta_{\lambda}^{\text{\it
boost}}(\ell^{a},m_{a},\bar{m}_{a})=(\lambda\ell^{a},m_{a},\bar{m}_{a})$ and
the generator $L_{\lambda}[\mathcal{C}]$ of $U(1)$ frame rotations that acts
by
$\delta_{\lambda}^{rot}(\ell^{a},m_{a},\bar{m}_{a})=(\ell^{a},\mathrm{i}\lambda
m_{a},-\mathrm{i}\lambda\bar{m}_{a})$ are proportional, i.e.
$K_{\lambda}[\mathcal{C}]=\gamma L_{\lambda}[\mathcal{C}]$. This constraint
plays an important role in covariant approaches to loop quantum gravity [65,
66]. We can thus set them both to zero.

Next, we need to further simplify this expression and remove the residual
gauge reparametrization symmetry. To this goal, we introduce a type of
interaction picture, where we seperate the $U(1)$ part of the transport
equation (36) from the shear degrees of freedom. We set

$S=RHS_{-}=RS_{I},$ (43)

where $R$ is the $U(1)$ holonomy along the light rays that generate the null
surface. We have

$R=\mathrm{e}^{\Delta
J},\quad\Delta(\mathcal{U},z,\bar{z})=\int_{-1}^{\mathcal{U}}\mathrm{d}\mathcal{U}^{\prime}\,\varphi(\mathcal{U}^{\prime},z,\bar{z}),$
(44)

where $J$ is the complex structure introduced in (29). Using the interaction
picture, the new $SL(2,R)$ holonomy $H$ satisfies the transport equation

$\frac{\mathrm{d}}{\mathrm{d}\mathcal{U}}H=(\sigma_{I}\bar{X}+\bar{\sigma}_{I}X)H,$
(45)

for the shear in the interaction picture

$\sigma_{I}(\mathcal{U},z,\bar{z})=\sigma_{(\ell)}(\mathcal{U},z,\bar{z})\mathrm{e}^{-2\mathrm{i}\Delta(\mathcal{U},z,\bar{z})}.$
(46)

The initial conditions are

$\displaystyle H(\mathcal{U}=-1,z,\bar{z})$ $\displaystyle=\bbvar{1},$ (47)
$\displaystyle S_{I}(\mathcal{U}=-1,z,\bar{z})$
$\displaystyle=S_{-}(z,\bar{z}).$ (48)

As a shorthand notation, we will also write
$S\big{|}_{\mathcal{C}_{\pm}}=S_{\pm}$ and

$\displaystyle S_{+}(z,\bar{z})$
$\displaystyle=R_{+}H(\mathcal{U}=+1)S_{-}\equiv\mathrm{e}^{J\Delta_{+}}H(\mathcal{U}=+1)S_{-},$
(49) $\displaystyle\Delta_{+}(z,\bar{z})$
$\displaystyle=\Delta(\mathcal{U}=+1,z,\bar{z}).$ (50)

To simplify (42) and impose the Raychaudhuri equation, we perform a partial
integration in the third term of equation (42). In this way, we obtain an
exterior derivative that hits the field space Maurer–Cartan form
$\bbvar{d}SS^{-1}$. The resulting expression satisfies

$\displaystyle
d^{2}v_{o}\wedge\mathrm{d}\mathopen{}\mathclose{{}\left(\bbvar{d}SS^{-1}}\right)$
$\displaystyle=d^{2}v_{o}\wedge
S\bbvar{d}\mathopen{}\mathclose{{}\left(S^{-1}\mathrm{d}S}\right)S^{-1}=$
$\displaystyle=-d^{2}v_{o}\wedge\bbvar{d}\Gamma\,J+d^{2}v_{o}\wedge
R\,\bbvar{d}\big{(}\mathrm{d}\mathcal{U}\bar{\sigma}_{I}+\mathrm{d}\mathcal{U}\sigma_{I}\bar{X}\big{)}R^{-1}+$
$\displaystyle\quad+d^{2}v_{o}\wedge\Gamma\,R\mathopen{}\mathclose{{}\left[\bbvar{d}S_{I}S_{I}^{-1},J}\right]R^{-1}+$
$\displaystyle\quad-d^{2}v_{o}\wedge
R\mathopen{}\mathclose{{}\left[\bbvar{d}S_{I}S_{I}^{-1},\mathrm{d}S_{I}S_{I}^{-1}}\right]R^{-1}.$
(51)

Next, we insert this eqaution back into the pre-symplectic structure. We
obtain

$\displaystyle\Theta_{\mathcal{N}}$ $\displaystyle=-\frac{1}{16\pi\gamma
G}\int_{\partial{\mathcal{N}}}d^{2}v_{o}\,\Omega^{2}\,\operatorname{Tr}\big{(}J\bbvar{d}SS^{-1}\big{)}+$
$\displaystyle\quad-\frac{1}{8\pi
G}\int_{\mathcal{N}}\mathrm{d}\mathcal{U}\wedge
d^{2}v_{o}\Omega^{2}\,\operatorname{Tr}\big{(}(\sigma_{I}\bar{X}+\bar{\sigma}_{I}X)\bbvar{D}S_{I}S_{I}^{-1}\big{)}+$
$\displaystyle\quad-\frac{1}{8\pi
G}\int_{\mathcal{N}}\mathrm{d}\mathcal{U}\wedge
d^{2}v_{o}\,\bbvar{d}\mathcal{U}\Big{(}\frac{\mathrm{d}^{2}}{\mathrm{d}\mathcal{U}^{2}}\Omega^{2}+2\sigma_{I}\bar{\sigma}_{I}\Omega^{2}\Big{)}.$
(52)

The last term vanishes on-shell, i.e. if the Raychaudhuri constraint (35) is
satisfied. It also shows, not very surprisingly, that the clock time
$\mathcal{U}$ is conjugate to the constraint. Furthermore, we introduced a
field space covariant derivative $\bbvar{D}$, which differs from the usual
exterior derivative on phase space by a vertical diffeomorphism such that it
annihilates the clock variable. In other words, we now have that

$\displaystyle\bbvar{D}S_{I}$
$\displaystyle=\bbvar{d}S_{I}-\bbvar{d}\mathcal{U}\big{(}\sigma_{I}\bar{X}+\bar{\sigma}X\big{)}S_{I}=\bbvar{d}S_{I}-\bbvar{d}\mathcal{U}\frac{\mathrm{d}}{\mathrm{d}\mathcal{U}}S_{I},$
(53) $\displaystyle\bbvar{D}\Omega$
$\displaystyle=\bbvar{d}\Omega-\bbvar{d}\mathcal{U}\frac{\mathrm{d}}{\mathrm{d}\mathcal{U}}\Omega,$
(54) $\displaystyle\bbvar{D}\mathcal{U}$ $\displaystyle=0.$ (55)

Equation (52) is a local version of the standard symplectic structure of
$T^{\ast}SL(2,\mathbb{R})$. It is interesting to note that, by adding the
Barbero–Immirzi parameter to the action, new entries appear in the symplectic
structure that would vanish as $\gamma\rightarrow\infty$. In this way,
representations of $T^{\ast}SL(2,\mathbb{C})$ can now suddenly count that do
otherwise not. It is this mechanism that is responsible for the discreteness
of geometry in loop quantum gravity.

###  2.6 Variation of the $SL(2,\mathbb{R})$ holonomy

To compute the Poisson brackets among Dirac observables, we will crucially
need the functional derivative of the holonomy (45). Since, however, the
covariant field space derivative $\bbvar{D}$ and the time derivative
$\frac{\mathrm{d}}{\mathrm{d}\mathcal{U}}$ commute, the calculation is
straightforward. Going back to the defining differential equation for the
holonomy, i.e. (45) and taking a functional derivative, we obtain

$H^{-1}\frac{\mathrm{d}}{\mathrm{d}\mathcal{U}}\bbvar{D}H=H^{-1}\mathopen{}\mathclose{{}\left(\bbvar{D}\sigma_{I}+\bbvar{D}\bar{\sigma}_{I}X}\right)H-\frac{\mathrm{d}}{\mathrm{d}\mathcal{U}}\mathopen{}\mathclose{{}\left(H^{-1}}\right)\bbvar{D}H.$
(56)

Next, we multiply this equation from the left by $H(\mathcal{U})$ and
integrate the resulting expression from $\mathcal{U}^{\prime}=-1$ to
$\mathcal{U}$. Taking into account that
$\bbvar{D}H(\mathcal{U}=-1)=\bbvar{D}\bbvar{1}=0$, we see

$(\bbvar{D}H)(\mathcal{U})=\int_{-1}^{\mathcal{U}}\mathrm{d}\mathcal{U}^{\prime}H(\mathcal{U})\mathopen{}\mathclose{{}\left(H^{-1}(\bbvar{D}\sigma_{I}\bar{X}+\bbvar{D}\bar{\sigma}_{I}X)H}\right)(\mathcal{U^{\prime}}).$
(57)

##  3 Twistor quantisation of ${T^{\ast}SL(2,\mathbb{R})}$

###  3.1 Twistor parametrisation of $T^{\ast}SL(2,\mathbb{R})$

To quantise the geometry of the light cone, we now turn to the phase space
$T^{\ast}SL(2,\mathbb{R})$ and its quantisation. Since
$T^{\ast}SL(2,\mathbb{R})$ is isomorphic to $T^{\ast}SU(1,1)$ and since it is
better to work with complex numbers rather than with real numbers, we can work
with $T^{\ast}SU(1,1)$ instead. The standard symplectic structure of
$T^{\ast}SU(1,1)$ is

$\Theta_{SU(1,1)}=\operatorname{Tr}(\Pi\mathrm{d}UU^{-1}),\quad\text{where}\quad(\Pi,U)\in\mathfrak{su}(1,1)\times
SU(1,1).$ (58)

To split the momentum into its rotational and translational components, we
introduce the following matrices,

$J=\begin{pmatrix}\mathrm{i}&0\\\ 0&-\mathrm{i}\end{pmatrix},\quad
X=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix},\quad\bar{X}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}.$ (59)

Notice that all products and traces are isomorphic to (29). Angular and
translational components $(L,c,\bar{c})$ of $\Pi$ are now given by

$\Pi=LJ+(c\bar{X}+\bar{c}X).$ (60)

The group $SU(1,1)$ has a standard action on $\mathbb{C}^{2}$ that preserves
the signature $(+,-)$ inner product between spinors
$U^{A},V^{A}\in\mathbb{C}^{2}$:

$\eta_{A\bar{A}}U^{A}\bar{V}^{\bar{A}}=U^{0}\overline{{V}^{0}}-U^{1}\overline{{V}^{1}}.$
(61)

In addition, the group is unimodular. For any $\mathfrak{sl}(2,\mathbb{R})$
element $\tensor{\Pi}{{}^{A}_{B}}$, we thus find the two conditions

$\displaystyle\Pi_{AB}$ $\displaystyle=\Pi_{BA},$ (62) $\displaystyle\Pi_{AB}$
$\displaystyle=-\eta_{A\bar{A}}\eta_{B\bar{B}}\bar{\Pi}^{\bar{A}\bar{B}},$
(63)

where spinor indices are raised and lowered using the skew symmetric
$\epsilon$-spinor, e.g. $\Pi_{AB}=\epsilon_{CA}\tensor{\Pi}{{}^{C}_{B}}$,
$\mathrm{Tr}(\Pi)=\tensor{\Pi}{{}^{A}_{A}}=\epsilon^{BA}\Pi_{AB}=0$.

Any symmetric spinor of rank $n$ can be split into the symmetrised tensor
products of $n$ eigenspinors [67]. Therefore, we can always find spinors
$\pi_{A}$ and $\omega_{A}$ such that,

$\Pi_{AB}=\pi_{(A}\omega_{B)}.$ (64)

In terms of these spinors, the $SL(2,\mathbb{R})$ Casimir is simply given by

$\mathrm{Tr}(\Pi^{2})=-2\mathopen{}\mathclose{{}\left(L^{2}-c\bar{c}}\right)=-\Pi_{AB}\Pi^{AB}=\frac{1}{2}(\pi_{A}\omega^{A})^{2}$
(65)

The spinors are unique up to the discrete exchange of $\pi_{A}$ and
$\omega_{A}$ and complexified dilations mapping
$\pi_{A}\rightarrow\mathrm{e}^{z}\pi_{A}$ and
$\omega_{A}\rightarrow\mathrm{e}^{-z}\omega_{A}$ for some $z\in\mathbb{C}$. To
obtain an $\mathfrak{su}(1,1)$ Lie algebra element, we now also have to
satisfy (63). There are two cases to distinguish that correspond to the
discrete and continuous series representations of $SL(2,\mathbb{R})$. In the
first case, we have

$\exists
z\in\mathbb{C}:\mathopen{}\mathclose{{}\left(\pi^{A},\omega^{A}}\right)=\mathopen{}\mathclose{{}\left(\mathrm{e}^{z}\tensor{\eta}{{}^{A}_{\bar{A}}}\bar{\pi}^{\bar{A}},-\mathrm{e}^{-z}\tensor{\eta}{{}^{A}_{\bar{A}}}\bar{\omega}^{\bar{A}}}\right).$
(66)

We can then always rescale the spinors $(\pi^{A},\omega^{A})$ such that we
have the simplified condition

$\mathopen{}\mathclose{{}\left(\pi^{A},\omega^{A}}\right)=\mathopen{}\mathclose{{}\left(\tensor{\eta}{{}^{A}_{\bar{A}}}\bar{\pi}^{\bar{A}},-\tensor{\eta}{{}^{A}_{\bar{A}}}\bar{\omega}^{\bar{A}}}\right).$
(67)

The first case corresponds, therefore, to a configuration in which

$\pi_{A}\omega^{A}=\bar{\pi}_{\bar{A}}\bar{\omega}^{\bar{A}}\Leftrightarrow
c\bar{c}\geq L^{2}.$ (68)

In the second case, on the other hand, we have

$\exists
z\in\mathbb{C}:\mathopen{}\mathclose{{}\left(\pi^{A},\omega^{A}}\right)=\mathopen{}\mathclose{{}\left(\mathrm{e}^{z}\tensor{\eta}{{}^{A}_{\bar{A}}}\bar{\omega}^{\bar{A}},-\mathrm{e}^{-z}\tensor{\eta}{{}^{A}_{\bar{A}}}\bar{\pi}^{\bar{A}}}\right).$
(69)

Again, we can reabsorb $z$ back into the definition of the two eigenspinors of
$\Pi_{AB}$. We are then only left with two possibilities,

${}^{\pm}\pi^{A}=\pm\mathrm{i}\tensor{\eta}{{}^{A}_{\bar{A}}}{}^{\pm}\bar{\omega}^{\bar{A}}$
(70)

Hence, there are two series, which we distinguish in our notation by writing
$({}^{\pm}\pi^{A},{}^{\pm}\omega^{A})$ for either sign. The second case
corresponds, therefore, to a configuration in which

$\pi_{A}\omega^{A}=-\bar{\pi}_{\bar{A}}\bar{\omega}^{\bar{A}}\Leftrightarrow
c\bar{c}\leq L^{2}.$ (71)

In the following, we exclude the case in which the Casimir vanishes, such that
$\pi_{A}\omega^{A}\neq 0$. If this condition is satisfied, the pair
$(\pi^{A},\omega^{A})$ forms a basis in $\mathbb{C}^{2}$. This implies that
the configuration variable $\tensor{U}{{}^{A}_{B}}\in SL(2,\mathbb{R})$ is
completly characterized by how it acts on this basis. We thus define a new
pair $(\underaccent{\tilde}{\pi}^{A},\underaccent{\tilde}{\omega}^{A})$ of
spinors such that

$\omega^{A}=\tensor{U}{{}^{A}_{B}}\underaccent{\tilde}{\omega}^{B},\quad\pi^{A}=\tensor{U}{{}^{A}_{B}}\underaccent{\tilde}{\pi}^{B}.$
(72)

Since $\tensor{U}{{}^{A}_{B}}$ has determinant one, these spinors satisfy the
matching condition

$C=\pi_{A}\omega^{A}-\underaccent{\tilde}{\pi}_{A}\tilde{\omega}^{A}=0.$ (73)

Given the quadruple
$(\pi^{A},\omega^{A},\underaccent{\tilde}{\pi}^{A},\underaccent{\tilde}{\omega}^{A})$,
we can reconstruct both $U\in SU(1,1)$ and $\Pi\in\mathfrak{su}(1,1)$. We can
thus take this parametrisation and insert it back into the symplectic
potential. A short calculation gives

$\displaystyle\Theta_{SU(1,1)}=\operatorname{Tr}(\Pi\mathrm{d}UU^{-1})$
$\displaystyle=\pi_{(A}\omega_{B)}\mathrm{d}\tensor{U}{{}^{A}_{C}}U^{BC}=\pi_{A}\omega_{B}\mathrm{d}\tensor{U}{{}^{A}_{C}}U^{BC}=\pi_{A}\mathrm{d}\tensor{U}{{}^{A}_{C}}\underaccent{\tilde}{\omega}^{C}=$
$\displaystyle=\pi_{A}\mathrm{d}(\tensor{U}{{}^{A}_{C}}\underaccent{\tilde}{\omega}^{C})-\pi_{A}\tensor{U}{{}^{A}_{C}}\mathrm{d}\underaccent{\tilde}{\omega}^{C}=\pi_{A}\mathrm{d}\omega^{A}-\underaccent{\tilde}{\pi}_{A}\mathrm{d}\underaccent{\tilde}{\omega}^{A}.$
(74)

For either case $\pi_{A}\omega^{A}\gtrless 0$, we can thus embed
$\Theta_{SU(1,1)}$ into a larger phase, which is now formed by the spinors
$(\pi^{A},\omega^{A},\underaccent{\tilde}{\pi}^{A},\underaccent{\tilde}{\omega}^{A})$
subject to constraints (73) and (67) and (69). The entire construction is in
complete analogy with the spinor and twistor representation of quantum
geometry [68, 69, 70, 71, 72, 73, 74, 75, 76].

###  3.2 Twistor quantisation of $T^{\ast}SL(2,\mathbb{R})$: continuous series
representations

There are two cases to consider. In the first case, $c\bar{c}\geq L^{2}$. In
this case, the spinors satisfy the reality condition (67). The solution of
this equation is immediate to find. There are phase space variables
$(p,q)\in\mathbb{C}^{2}$ such that

$\begin{pmatrix}\pi^{0}\\\ \pi^{1}\end{pmatrix}=\begin{pmatrix}p\\\
-\bar{p}\end{pmatrix},\quad\begin{pmatrix}\omega^{0}\\\
\omega^{1}\end{pmatrix}=\begin{pmatrix}\bar{q}\\\ q\end{pmatrix},$ (75)

and equally for
$(\underaccent{\tilde}{\pi}^{A},\underaccent{\tilde}{\omega}^{A})$ in terms of
$(\underaccent{\tilde}{p},\underaccent{\tilde}{q})$. Going back to the
symplectic potential (74), we infer the canonical commutatiuon relations

$\\{p,q\\}=1,\qquad\\{\bar{p},\bar{q}\\}=1.$ (76)

Going to a position representation, we obtain wavefunctions

$\langle
q,\bar{q}|\lambda,m\rangle=\frac{1}{2\pi}(q\bar{q})^{\frac{1}{2}(\mathrm{i}\lambda-1)}\mathopen{}\mathclose{{}\left(\frac{q}{\bar{q}}}\right)^{m},$
(77)

where $\lambda\in\mathbb{R}$ and $m\in\mathbb{Z}/2$. The normalisation is
$\langle\lambda,m|\lambda^{\prime},m^{\prime}\rangle=\delta(\lambda-\lambda^{\prime})\delta_{mm^{\prime}}$.
On the corresponding Hilbert space
$L^{2}(\mathbb{C},\frac{\mathrm{i}}{2}\mathrm{d}q\,\mathrm{d}\bar{q})$, the
$SL(2,\mathbb{R})$ generators admit the representation

$\displaystyle L$
$\displaystyle=-\frac{1}{2}\mathrm{Tr}(PJ)=\frac{1}{2}\mathopen{}\mathclose{{}\left(q\partial_{q}-\bar{q}\partial_{\bar{q}}}\right),$
(78) $\displaystyle c$
$\displaystyle=\mathrm{Tr}(PX)=-\mathrm{i}\bar{q}\partial_{q},$ (79)
$\displaystyle c^{\dagger}\equiv\bar{c}$
$\displaystyle=\mathrm{Tr}(P\bar{X})=-\mathrm{i}{q}\partial_{\bar{q}},$ (80)

The operators act as

$\displaystyle L|\lambda,m\rangle$ $\displaystyle=m|\lambda,m\rangle,$ (81)
$\displaystyle c|\lambda,m\rangle$
$\displaystyle=-\frac{\mathrm{i}}{2}\mathopen{}\mathclose{{}\left(\mathrm{i}\lambda+2m-1}\right)|\lambda,m-1\rangle,$
(82) $\displaystyle\bar{c}|\lambda,m\rangle$
$\displaystyle=-\frac{\mathrm{i}}{2}\mathopen{}\mathclose{{}\left(\mathrm{i}\lambda-2m-1}\right)|\lambda,m+1\rangle,$
(83)
$\displaystyle\mathopen{}\mathclose{{}\left(L^{2}-\tfrac{1}{2}(c\bar{c}+\bar{c}c)}\right)|\lambda,m\rangle$
$\displaystyle=-\frac{1}{4}(\lambda^{2}+1)|\lambda,m\rangle.$ (84)

###  3.3 Twistor quantisation of $T^{\ast}SL(2,\mathbb{R})$: discrete series
representations

The second case happens when the $SL(2,\mathbb{R})$ Casimir satisfies
$L^{2}>c\bar{c}$. In this case, the symplectic structure (74) implies that the
coefficients of the $SL(2,\mathbb{R})$ spinors are canonical creation and
annihilation operators

$\displaystyle\begin{pmatrix}{}^{+}\omega^{0}\\\
{}^{+}\omega^{1}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}d\\\
e^{\dagger}\end{pmatrix},$
$\displaystyle\qquad\begin{pmatrix}{}^{-}\omega^{0}\\\
{}^{-}\omega^{1}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}d^{\dagger}\\\
e\end{pmatrix},$ (85) $\displaystyle\begin{pmatrix}{}^{+}\pi^{0}\\\
{}^{+}\pi^{1}\end{pmatrix}$ $\displaystyle=-\mathrm{i}\begin{pmatrix}e\\\
d^{\dagger}\end{pmatrix},$ $\displaystyle\begin{pmatrix}{}^{-}\pi^{0}\\\
{}^{-}\pi^{1}\end{pmatrix}$
$\displaystyle=\mathrm{i}\begin{pmatrix}e^{\dagger}\\\ d\end{pmatrix}.$ (86)

The fundamental canonical commutation relations are

$[d,d^{\dagger}]=[e,e^{\dagger}]=1.$ (87)

All other commutators vanish. The resulting representation of the $SL(2,R)$
generators is

$\displaystyle{}^{\pm}L$
$\displaystyle=\pm\frac{1}{2}\mathopen{}\mathclose{{}\left(d^{\dagger}d+e^{\dagger}e+1}\right),$
(88) $\displaystyle{}^{+}c$ $\displaystyle={}^{-}c^{\dagger}=-\mathrm{i}de,$
(89) $\displaystyle L^{2}-\tfrac{1}{2}(cc^{\dagger}+c^{\dagger}c)$
$\displaystyle=\frac{1}{4}\mathopen{}\mathclose{{}\left(d^{\dagger}d-e^{\dagger}e-1}\right)\mathopen{}\mathclose{{}\left(d^{\dagger}d-e^{\dagger}e+1}\right).$
(90)

##  4 Quantum impulsive data on the null cone

###  4.1 Phase space of a single pulse of radiative data

In this section, we carry out the main task of this paper. The idea is to
start from a truncation of the classical phase space, in which we only
consider a specific class of impulsive null initial data. By restricting
ourselves to a finite such puls, the infinite-dimensional gravitational bulk
and boundary phase space is reduced to a finite-dimensional mechanical system
with constraints. This truncation provides a great simplification. At the end
of the section, we can take the analysis to the quantum level and solve the
constraints at the full non-perturbative level.

The first step is to restrict the phase space equipped with the null
symplectic potential (52) to a single pulse. A large but finite sequences of
such pulses, corresponding to a sequence of null slabs
$(\mathcal{N}_{1},\mathcal{N}_{2},\dots)$ can approximate to arbitrarily good
precision any smooth shear $\sigma_{(\ell)}$ with compact support on
$\mathcal{K}=\mathcal{N}_{1}\cup\mathcal{N}_{2}\cup\dots$.999As a Cauchy
sequence with respect to an auxiliary inner product on $\mathcal{K}$, e.g.
$L^{2}(\mathcal{K},d^{2}v_{o}\wedge k)$. At the quantum level, we will then
have a corresponding sequence of Hilbert spaces
$(\mathcal{H}_{\mathcal{N}_{1}},\mathcal{H}_{\mathcal{N}_{2}},\dots)$. The
physical Hilbert space for the entire boundary data will be then built from
the tensor product $\mathcal{H}_{\text{\it
kin}}=\mathcal{H}_{\mathcal{N}_{1}}\otimes\mathcal{H}_{\mathcal{N}_{2}}\otimes\dots$
by tracing over intermediate edge states at the two-dimensional corners
$\mathcal{C}_{i}=\partial\mathcal{N}_{i}\cap\partial\mathcal{N}_{i+1}$.

In each pulse, we shall restrict ourselves to configurations with constant
shear. This creates a conceptual problem. The shear $\sigma_{(\ell)}$ of a
null generator $\ell^{a}$ of $\mathcal{N}$ is a component of the spin
connection, which is a one-form in the bulk. A component of a connection does
not define a scalar. Thus, our statement can only be true in a very specific
gauge. At this point, we are free to choose whatever gauge we like, in
particular a gauge in which the Raychaudhuri equation, which is the generator
for angle dependent time reparametrizations
$\mathcal{U}\rightarrow\mathcal{U}^{\prime}(\mathcal{U},z,\bar{z})$,
$\mathcal{U}^{\prime}(\pm 1,z,\bar{z})=\pm 1$, simplifies. In addition, there
are further gauge symmetries we can use. There is a $U(1)$ gauge freedom and
the freedom to rescale $\mathcal{\ell}^{a}$. All three gauge transformations
are required to define the shear in the interaction picture, as we saw in (46)
above. It is the shear in this gauge that we restrict to be constant along the
null generators. Our requirement can be thus summarized as

$\frac{\mathrm{d}}{\mathrm{d}\mathcal{U}}\sigma_{I}=0,\qquad\sigma_{I}(\mathcal{U},z,\bar{z})=:\sigma_{I}(z,\bar{z}).$
(91)

For constant such shear, we can immediately integrate the Raychaudhuri
equation (35). We obtain

$\Omega^{2}=(E_{+}+E_{-})\frac{\sin(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}{\sin(2\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}\cos(\sqrt{2\sigma_{I}\bar{\sigma}_{I}}\mathcal{U})+(E_{+}-E_{-})\frac{\cos(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}{\sin(2\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}\sin(\sqrt{2\sigma_{I}\bar{\sigma}_{I}}\mathcal{U}),$
(92)

where $E_{\pm}$ are the initial conditions

$E_{\pm}(z,\bar{z})=\Omega^{2}(\mathcal{U}=\pm 1,z,\bar{z}).$ (93)

At this point, we may wish to restrict the norm of the shear to a finite
interval $0\leq|\sigma_{I}|<\frac{\pi}{2\sqrt{2}}$. However, no such
restriction seems necessary in our final result. In any case, at null
infinity, we are well within this regime. During its finite duration, the
pulse will have the asymptotic form
$\sigma_{I}(z,\bar{z})=\frac{\dot{\sigma}^{(0)}(z,\bar{z})}{r}+\mathcal{O}(r^{-2})$.
See also  footnote 11 for a similar remark, which will be made more precise
elsewhere.

We then also have two integrals to solve. They are

$\displaystyle\int_{-1}^{1}\mathrm{d}\mathcal{U}\,\Omega^{2}(\mathcal{U},z,\bar{z})$
$\displaystyle=+(E_{+}+E_{-})\frac{\tan(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}{\sqrt{2\sigma_{I}\bar{\sigma}_{I}}},$
(94)
$\displaystyle\int_{-1}^{1}\mathrm{d}\mathcal{U}\,\mathcal{U}\,\Omega^{2}(\mathcal{U},z,\bar{z})$
$\displaystyle=-(E_{+}-E_{-})\frac{1}{\sin(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}\frac{\mathrm{d}}{\mathrm{d}\rho}\mathopen{}\mathclose{{}\left[\frac{\sin(\rho)}{\rho}}\right]\bigg{|}_{\rho=\sqrt{2\sigma_{I}\bar{\sigma}_{I}}}.$
(95)

These integrals will prove useful when evaluating the pre-symplectic potential
on-shell, i.e. when the Raychaudhuri equation is satisfied.

Next, we need to speak about the $SL(2,\mathbb{R})$ holonomy (45). Since the
shear is now constant along the null generators, we can forget about the time-
ordering, and immediately integrate the defining differential equation (45).
We obtain,

$H=\operatorname{ch}\mathopen{}\mathclose{{}\left(\sqrt{\sigma_{I}\bar{\sigma}_{I}}(\mathcal{U}+1)}\right)\bbvar{1}+\frac{1}{\sqrt{\sigma_{I}\bar{\sigma}_{I}}}\mathopen{}\mathclose{{}\left(\bar{\sigma}_{I}X+{\sigma}_{I}\bar{X}}\right)\operatorname{sh}\mathopen{}\mathclose{{}\left(\sqrt{\sigma_{I}\bar{\sigma}_{I}}(\mathcal{U}+1)}\right),$
(96)

where we again supressed all angular dependence, i.e. $H\equiv
H(\mathcal{U},z,\bar{z})$.

We can now turn to our main task in this section and compute the physical
phase space of the system. The strategy is simple. We insert the solutions for
the $SL(2,\mathbb{R})$ holonomy $H(\mathcal{U},z,\bar{z})$ and the conformal
factor $\Omega^{2}(\mathcal{U},z,\bar{z})$, see (96) and (92), back into the
pre-symplectic structure (52), thereby calculating the pull-back from the
kinematical phase space to the physical phase space of the radiative pulse. In
doing so, we have to take into account also the boundary conditions (43) and
(49) in the interaction picture.

In what follows, we split the calculation into two steps. First of all, we
consider the term that contains the bulk integral along $\mathcal{N}$. Taking
into account the variation of the holonomy (57), we obtain

$\displaystyle\int_{\mathcal{N}}d^{2}v_{o}\wedge\mathrm{d}\mathcal{U}\,\Omega^{2}\operatorname{Tr}\mathopen{}\mathclose{{}\left((\sigma_{I}\bar{X}+\bar{\sigma}_{I}X)\bbvar{D}S_{I}S_{I}^{-1}}\right)=$
(97)
$\displaystyle=\int_{-1}^{1}\mathrm{d}\mathcal{U}\int_{-1}^{\mathcal{U}}\mathrm{d}\mathcal{U}^{\prime}\int_{\mathcal{C}}d^{2}v_{o}\,\Omega^{2}(\mathcal{U})\operatorname{Tr}\mathopen{}\mathclose{{}\left((\sigma_{I}\bar{X}+\mathrm{cc.})H(\mathcal{U},\mathcal{U}^{\prime})\mathopen{}\mathclose{{}\left(\bbvar{D}\sigma_{I}\bar{X}+\mathrm{cc.}}\right)H^{-1}(\mathcal{U},\mathcal{U}^{\prime})}\right)+$
$\displaystyle\quad+\int_{\mathcal{N}}d^{2}v_{o}\wedge\mathrm{d}\mathcal{U}\,\Omega^{2}(\mathcal{U})\operatorname{Tr}\mathopen{}\mathclose{{}\left((\sigma_{I}\bar{X}+\mathrm{cc.})H(\mathcal{U})(\bbvar{D}S_{-}S_{-}^{-1})H^{-1}(\mathcal{U})}\right),$
(98)

where
$H(\mathcal{U},\mathcal{U}^{\prime}):=H(\mathcal{U})H^{-1}(\mathcal{U}^{\prime})$
and all variables have an implicit angular dependence dropped from our
notation for brevity, e.g.
$\Omega^{2}(\mathcal{U})\equiv\Omega^{2}(\mathcal{U},z,\bar{z})$. Since,
however, the shear is assumed to be constant for the entire duration of the
pulse, see (91), the holonomies commute with the connection, i.e.
$[H(\mathcal{U}),\sigma_{I}\bar{X}+\mathrm{cc.}]=0$. This simplifies the
integrals a lot. In fact,

$\displaystyle\int_{\mathcal{N}}d^{2}v_{o}\wedge\mathrm{d}\mathcal{U}\,\Omega^{2}\operatorname{Tr}\mathopen{}\mathclose{{}\left((\sigma_{I}\bar{X}+\bar{\sigma}_{I}X)\bbvar{D}S_{I}S_{I}^{-1}}\right)=\int_{-1}^{1}\mathrm{d}\mathcal{U}\int_{\mathcal{C}}d^{2}v_{o}\,\Omega^{2}(\mathcal{U})(1+\mathcal{U})\bbvar{D}(\sigma_{I}\bar{\sigma}_{I})+$
$\displaystyle\quad+\int_{\mathcal{N}}d^{2}v_{o}\wedge\mathrm{d}\mathcal{U}\,\Omega^{2}(\mathcal{U})\operatorname{Tr}\mathopen{}\mathclose{{}\left((\sigma_{I}\bar{X}+\mathrm{cc.})(\bbvar{D}S_{-}S_{-}^{-1})}\right)=$
$\displaystyle=-\frac{1}{2}\int_{\mathcal{C}}d^{2}v_{o}\,(E_{+}-E_{-})\frac{1}{\sin(\sqrt{2\sigma_{I}\bar{\sigma}_{I})}}\frac{\mathrm{d}}{\mathrm{d}\rho}\mathopen{}\mathclose{{}\left[\frac{\sin\rho}{\rho}}\right]\bigg{|}_{\rho=\sqrt{2\sigma_{I}\bar{\sigma}_{I}}}\bbvar{D}(2\sigma_{I}\bar{\sigma}_{I})+$
$\displaystyle\quad+\frac{1}{2}\int_{\mathcal{C}}d^{2}v_{o}\,(E_{+}+E_{-})\frac{\tan(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}{\sqrt{2\sigma_{I}\bar{\sigma}_{I})}}\bbvar{D}(2\sigma_{I}\bar{\sigma}_{I})+$
$\displaystyle\quad+\int_{\mathcal{C}}d^{2}v_{o}\,(E_{+}+E_{-})\frac{\tan(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}{\sqrt{2\sigma_{I}\bar{\sigma}_{I}}}\operatorname{Tr}\mathopen{}\mathclose{{}\left((\sigma_{I}\bar{X}+\mathrm{cc.})(\bbvar{D}S_{-}S_{-}^{-1})}\right)=$
$\displaystyle=-\int_{\mathcal{C}}d^{2}v_{o}\,(E_{+}-E_{-})\bbvar{D}\mathopen{}\mathclose{{}\left[\ln\mathopen{}\mathclose{{}\left(\frac{\sin(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}{\sqrt{2\sigma_{I}\bar{\sigma}_{I}}}}\right)}\right]+$
$\displaystyle\quad-\int_{\mathcal{C}}d^{2}v_{o}\,(E_{+}+E_{-})\bbvar{D}\mathopen{}\mathclose{{}\left[\ln\mathopen{}\mathclose{{}\left(\cos(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}\right)}\right]+$
$\displaystyle\quad+\int_{\mathcal{C}}d^{2}v_{o}\,(E_{+}+E_{-})\frac{\tan(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}{\sqrt{2\sigma_{I}\bar{\sigma}_{I}}}\operatorname{Tr}\mathopen{}\mathclose{{}\left((\sigma_{I}\bar{X}+\mathrm{cc.})(\bbvar{D}S_{-}S_{-}^{-1})}\right).$
(99)

In our next step, we consider the boundary term that appears in the first line
of (52). Since $S_{+}=R(\mathcal{U}=1)H(\mathcal{U}=1)S_{-}$, see (49), we
have

$\displaystyle\operatorname{Tr}\mathopen{}\mathclose{{}\left(J\bbvar{D}S_{+}S_{+}^{-1}}\right)$
$\displaystyle=-2\bbvar{D}\Delta_{+}+\operatorname{Tr}\mathopen{}\mathclose{{}\left(J(\bbvar{D}H)(\mathcal{U}=+1)H^{-1}(\mathcal{U}=+1)}\right)+$
$\displaystyle\quad+\operatorname{Tr}\mathopen{}\mathclose{{}\left(H^{-1}(\mathcal{U}=+1)JH(\mathcal{U}=+1)\bbvar{D}S_{-}S_{-}^{-1}}\right).$
(100)

To calculate the second term in this equation, we recall that
$H(\mathcal{U})=\exp((1+\mathcal{U})(\sigma_{I}\bar{X}+\mathrm{cc.}))$. Since
the complex structure $J$ acts on the translational $SL(2,\mathbb{R})$
generators by $JX=-XJ$ and $J\bar{X}=-\bar{X}J$, see (59), we have
$JH(\mathcal{U})=H^{-1}(\mathcal{U})J$. Going back to (96), we obtain

$\displaystyle\operatorname{Tr}\mathopen{}\mathclose{{}\left(H^{-1}(\mathcal{U}=1)JH(\mathcal{U}=1)\bbvar{D}S_{-}S_{-}^{-1}}\right)=\operatorname{Tr}\mathopen{}\mathclose{{}\left(H^{-1}(\mathcal{U}=3)J\bbvar{D}S_{-}S_{-}^{-1}}\right)=$
$\displaystyle=\operatorname{ch}\mathopen{}\mathclose{{}\left(4\sqrt{\sigma_{I}\bar{\sigma}_{I}}}\right)\operatorname{Tr}\mathopen{}\mathclose{{}\left(J\bbvar{D}S_{-}S_{-}^{-1}}\right)-\mathrm{i}\frac{\operatorname{sh}(4\sqrt{\sigma_{I}\bar{\sigma}_{I}}}{\sqrt{\sigma_{I}\bar{\sigma}_{I}}}\operatorname{Tr}\mathopen{}\mathclose{{}\left((\sigma_{I}\bar{X}-\bar{\sigma}_{I}X)\bbvar{D}S_{-}S_{-}^{-1}}\right).$
(101)

Next, we also need the Maurer–Cartan form $\bbvar{D}HH^{-1}$, which can be
inferred from (96). A short calculation gives

$\displaystyle(\bbvar{D}H)H^{-1}\big{|}_{\mathcal{U}=+1}$
$\displaystyle=\frac{\mathrm{i}}{2}\frac{1}{\sqrt{\sigma_{I}\bar{\sigma}_{I}}}\mathopen{}\mathclose{{}\left(\sigma_{I}\bar{X}-\bar{\sigma}_{I}X}\right)\operatorname{sh}(4\sqrt{\sigma_{I}\bar{\sigma}_{I}})\bbvar{D}\phi_{I}+$
$\displaystyle\quad-J\operatorname{sh}^{2}(2\sqrt{\sigma_{I}\bar{\sigma}_{I}})\bbvar{D}\phi_{I}+\frac{2}{\sqrt{\sigma_{I}\bar{\sigma}_{I}}}\mathopen{}\mathclose{{}\left(\sigma_{I}\bar{X}+\bar{\sigma}_{I}X}\right)\bbvar{D}(\sigma_{I}\bar{\sigma}_{I}),$
(102)

where $\phi_{I}\equiv\phi_{I}(z,\bar{z})$ denotes the $U(1)$ angle

$\sigma_{I}=|\sigma_{I}|\mathrm{e}^{\mathrm{i}\phi_{I}}.$ (103)

Finally, we insert the results of the first step (99), and the second step,
which are collected in (100), (101), (102), back into the pre-symplectic
potential, which turns, thereby, into the symplectic potential for a single
pulse of radiation,

$\displaystyle\Theta^{\text{\it pulse}}_{\mathcal{N}}$
$\displaystyle=\frac{1}{8\pi\gamma
G}\int_{\mathcal{C}}d^{2}v_{o}E_{+}\bbvar{D}\Delta_{+}+\frac{1}{16\pi\gamma
G}\int_{\mathcal{C}}d^{2}v_{o}E_{+}\mathopen{}\mathclose{{}\left(1-\operatorname{ch}(4\sqrt{\sigma_{I}\bar{\sigma}_{I}})}\right)\bbvar{D}\phi_{I}+$
$\displaystyle\quad+\frac{1}{8\pi
G}\int_{\mathcal{C}}d^{2}v_{o}(E_{+}-E_{-})\bbvar{D}\mathopen{}\mathclose{{}\left[\ln\mathopen{}\mathclose{{}\left(\frac{\sin(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}{\sqrt{2\sigma_{I}\bar{\sigma}_{I}}}}\right)}\right]+$
$\displaystyle\quad+\frac{1}{8\pi
G}\int_{\mathcal{C}}d^{2}v_{o}(E_{+}+E_{-})\bbvar{D}\mathopen{}\mathclose{{}\left[\ln\mathopen{}\mathclose{{}\left(\cos(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}\right)}\right]+\int_{\mathcal{C}}d^{2}v_{o}\mathrm{Tr}\mathopen{}\mathclose{{}\left(\Pi^{\prime}\bbvar{D}S_{-}S_{-}^{-1}}\right),$
(104)

where we introduced the $SL(2,\mathbb{R})$ momentum variable101010The
introduction of primed variables should be clear in a moment.

$\Pi^{\prime}=JL+c^{\prime}\bar{X}+\bar{c}^{\prime}X.$ (105)

The $SL(2,\mathbb{R})$ momentum splits into the following generators on phase
space

$\displaystyle L$ $\displaystyle=\frac{1}{16\pi\gamma
G}\mathopen{}\mathclose{{}\left(E_{-}-\operatorname{ch}(4\sqrt{\sigma_{I}\bar{\sigma}_{I}})E_{+}}\right),$
(106) $\displaystyle c^{\prime}$ $\displaystyle=-\frac{1}{8\pi
G}\mathopen{}\mathclose{{}\left(\frac{1}{\sqrt{2}}(E_{+}+E_{-})\tan(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})-\frac{\mathrm{i}}{2\gamma}\operatorname{sh}(4\sqrt{\sigma_{I}\bar{\sigma}_{I}})E_{+}}\right)\mathrm{e}^{\mathrm{i}\phi_{I}}.$
(107)

The third and forth term in equation (104) are redundant. To simplify the
expression further, we absorb them back into a redefinition of the canonical
variables. Consider $U(1)$ rotation with yet unspecified angle dependent gauge
parameter $\psi=\psi(z,\bar{z})$, define

$U_{-}=\mathrm{e}^{\psi J}S_{-},$ (108)

We can then also set

$\Pi=\mathrm{e}^{\psi J}\Pi^{\prime}\mathrm{e}^{-\psi
J}=JL+c\bar{X}+\bar{c}X,$ (109)

where

$c=-\frac{1}{8\pi
G}\mathopen{}\mathclose{{}\left(\frac{1}{\sqrt{2}}(E_{+}+E_{-})\tan(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})-\frac{\mathrm{i}}{2\gamma}\operatorname{sh}(4\sqrt{\sigma_{I}\bar{\sigma}_{I}})E_{+}}\right)\mathrm{e}^{\mathrm{i}(\phi_{I}+2\psi)}.$
(110)

The $SL(2,\mathbb{R})$ symplectic potential transforms according to

$\operatorname{Tr}\mathopen{}\mathclose{{}\left(\Pi^{\prime}\bbvar{D}S_{-}S_{-}^{-1}}\right)=2L\bbvar{D}\psi+\operatorname{Tr}\mathopen{}\mathclose{{}\left(\Pi\bbvar{D}U_{-}U_{-}^{-1}}\right).$
(111)

If we now compare this equation with (104), it is clear that we can cancel the
third and forth terms in the expression for the symplectic potential (104) by
making the following choice111111Notice that in standard Bondi coordintes at
asymptotic null infinity $\mathcal{I}^{\pm}$, the asymptotic time translation
can be chosen such that it agrees to leading order in an $1/r$ expansion with
our $\mathcal{U}$ coordinate. The shear of such a pulse will then scale within
its support as
$\sigma_{I}(z,\bar{z})=\dot{\sigma}^{(0)}(z,\bar{z})/r+\mathcal{O}(r^{-2})$,
hence $\psi=\mathcal{O}(r^{-1})$ as $r\rightarrow\infty$. This is very good,
because otherwise our canonical transformation would be inadequate for
establishing a connection to perturbative gravity at $\mathcal{I}^{\pm}$.

$\psi=\gamma\ln\mathopen{}\mathclose{{}\left[\frac{\tan(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}{\sqrt{2\sigma_{I}\bar{\sigma}_{I}}}}\right],$
(112)

In fact, we obtain

$\displaystyle\Theta^{\text{\it pulse}}_{\mathcal{N}}$
$\displaystyle=+\frac{1}{16\pi\gamma
G}\int_{\mathcal{C}}d^{2}v_{o}E_{+}\mathopen{}\mathclose{{}\left(\operatorname{ch}\mathopen{}\mathclose{{}\left(4\sqrt{\sigma_{I}\bar{\sigma}_{I}}}\right)+1}\right)\bbvar{D}\mathopen{}\mathclose{{}\left[\Delta_{+}+2\gamma\ln\mathopen{}\mathclose{{}\left(\cos(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}\right)}\right]+$
$\displaystyle\quad-\frac{1}{16\pi\gamma
G}\int_{\mathcal{C}}d^{2}v_{o}E_{+}\mathopen{}\mathclose{{}\left(\operatorname{ch}\mathopen{}\mathclose{{}\left(4\sqrt{\sigma_{I}\bar{\sigma}_{I}}}\right)-1}\right)\bbvar{D}\mathopen{}\mathclose{{}\left[\Delta_{+}+\phi_{I}+2\gamma\ln\mathopen{}\mathclose{{}\left(\frac{\sin(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}{\sqrt{2\sigma_{I}\bar{\sigma}_{I}}}}\right)}\right]+$
$\displaystyle\quad+\int_{\mathcal{C}}d^{2}v_{o}\mathrm{Tr}\mathopen{}\mathclose{{}\left(\Pi\bbvar{D}U_{-}U_{-}^{-1}}\right).$
(113)

We are not completely finished yet. The variables in the first two lines are
not independent from the variables in the third line. To identify the physical
phase, we enlarge the phase space and impose a constraint, which is second
class, but not very difficult to solve. Consider first the following angle
dependent oscillator type variables

$\displaystyle a$ $\displaystyle=\frac{1}{\sqrt{8\pi\gamma
G}}\sqrt{E_{+}}\operatorname{ch}\mathopen{}\mathclose{{}\left(2\sqrt{\sigma_{I}\bar{\sigma}_{I}}}\right)\mathrm{e}^{-\mathrm{i}\mathopen{}\mathclose{{}\left(\Delta_{+}+2\gamma\ln\mathopen{}\mathclose{{}\left(\cos(\sqrt{2\sigma_{I}\bar{\sigma}_{I}})}\right)}\right)},$
(114) $\displaystyle b$ $\displaystyle=\frac{1}{\sqrt{8\pi\gamma
G}}\sqrt{E_{+}}\operatorname{sh}\mathopen{}\mathclose{{}\left(2\sqrt{\sigma_{I}\bar{\sigma}_{I}}}\right)\mathrm{e}^{+\mathrm{i}\mathopen{}\mathclose{{}\left(\Delta_{+}+\phi_{I}+2\gamma\ln\mathopen{}\mathclose{{}\left(\frac{\sin\mathopen{}\mathclose{{}\left(\sqrt{2\sigma_{I}\bar{\sigma}_{I}}}\right)}{\sqrt{2\sigma_{I}\bar{\sigma}_{I}}}}\right)}\right)}.$
(115)

On the enlarged phase space, these oscillators satisfy standard commutation
relations for creation and annihilation operators on the cross section,

$\\{a(z,\bar{z}),\bar{a}(z^{\prime},\bar{z}^{\prime})\\}=\\{b(z,\bar{z}),\bar{b}(z^{\prime},\bar{z}^{\prime})\\}=\mathrm{i}\,\delta^{(2)}_{\mathcal{C}}(z,\bar{z}|z^{\prime},\bar{z}^{\prime}).$
(116)

In addition, we have the constraint

$a\bar{a}-b\bar{b}=\frac{E_{+}}{8\pi\gamma G}>0.$ (117)

We then immediately also find

$\displaystyle E_{+}+E_{-}$ $\displaystyle=16\pi\gamma G(L+a\bar{a}),$ (118)
$\displaystyle E_{-}-E_{+}$ $\displaystyle=16\pi\gamma G(L+b\bar{b}),$ (119)
$\displaystyle\sqrt{\sigma_{I}\bar{\sigma}_{I}}$
$\displaystyle=\frac{1}{4}\ln\mathopen{}\mathclose{{}\left(\frac{\sqrt{\bar{a}a}+\sqrt{\bar{b}b}}{\sqrt{\bar{a}a}-\sqrt{\bar{b}b}}}\right).$
(120)

We thus see that the initial and final area densities $E_{\pm}$ and the shear
of the pulse can be all expressed in terms of harmonic oscillator variables.
Yet, these oscillators by themselves are not Dirac osbervables. Going back to
equation (110), in which we defined the $SL(2,\mathbb{R})$ translation
generator $c$, which is a ladder operator in the quantum theory, we obtain a
constraint on the now enlarged phase space. First of all, we note

$ab=\frac{1}{16\pi\gamma
G}E_{+}\operatorname{sh}\mathopen{}\mathclose{{}\left(4\sqrt{\sigma_{I}\bar{\sigma}_{I}}}\right)\mathrm{e}^{\mathrm{i}\phi_{I}+2\mathrm{i}\psi}.$
(121)

A short calculation gives the constraint

$\boxed{c\,\bar{a}\,\bar{b}=-\gamma\mathopen{}\mathclose{{}\left(L+\bar{a}a}\right)\sqrt{2\,\bar{a}a\,\bar{b}b}\tan\mathopen{}\mathclose{{}\left(\frac{1}{2\sqrt{2}}\ln\mathopen{}\mathclose{{}\left(\frac{\sqrt{\bar{a}a}+\sqrt{\bar{b}b}}{\sqrt{\bar{a}a}-\sqrt{\bar{b}b}}}\right)}\right)+\mathrm{i}\,\bar{a}a\,\bar{b}b.}$
(122)

The Poisson commutation relations for $T^{\ast}SL(2,\mathbb{R})$ tell us that
this constraint, which commutes with the $SL(2,\mathbb{R})$ Casimir, forms a
second class system with its complex conjugate: the right hand side clearly
commutes with its complex conjugate, the left hand side does not. At the
quantum level, we can only impose one of them, the other will map any physical
state into its orthogonal complement. In fact, the constraint (122) imposes a
simple recursion relation among the coefficients in a standard oscillator
basis expansion of physical states. Furthermore, if we restrict ourselves to
the discrete series representations of $SL(2,\mathbb{R})$, the constraint can
be solved with remarkable ease. This will be the subject of the next and final
subsection.

###  4.2 Solving the residual second-class constraint at the quantum level

In this section, we show how to solve the constraint (122) in the specific
case of the unitary discrete series representations of $SL(2,\mathbb{R})$, in
which $L>0$. Following earlier results on loop quantum gravity and isolated
horizons [77, 6, 78, 79, 80, 81, 8], we use a polymer quantization, in which
all geometry is excited at a finite number of topological defects at the null
surface. This is achieved by introducing a tessellation of the cross section
into a number of plaquettes $\\{\square_{i}\\}$,
$\bigcup_{i}\square_{i}=\mathcal{C}_{i}$. In each plaquette, we define
oscillators and $SL(2,\mathbb{R})$ generators

$\displaystyle a_{i}=\int_{\square_{i}}d^{2}v_{o}\,a,\quad
b_{i}=\int_{\square_{i}}d^{2}v_{o}\,b,\quad
c_{i}=\int_{\square_{i}}d^{2}v_{o}\,c,\quad
L_{i}=\int_{\square_{i}}d^{2}v_{o}\,L.$ (123)

The only non-vanishing commutators are

$\displaystyle[a_{i},a^{\dagger}_{j}]$
$\displaystyle=[b_{i},b^{\dagger}_{j}]=\delta_{ij},$ (124)
$\displaystyle[L_{i},c_{j}]$
$\displaystyle=-\delta_{ij}c_{j},\quad[L_{i},c_{j}^{\dagger}]=+\delta_{ij}c_{j}^{\dagger},$
(125) $\displaystyle[c_{i},c_{j}^{\dagger}]$
$\displaystyle=2\delta_{ij}L_{i}.$ (126)

Consider then the following basis states for a single plaquette

$|N,n_{a},n_{b},n_{c}\rangle,$ (127)

where $n_{a}$ and $n_{b}$ are the occupation numbers of the two oscillators,
whereas $n_{c}$ is the magnetic quantum number, i.e. the eigenvalue of
$L$.121212To be precise, there is also a second such magnetic quantum number,
which represents the second pair of oscillators
$(\underaccent{\tilde}{\pi}^{A},\underaccent{\tilde}{\omega}^{A})$ that we
introduced in (72). As far as the solution of the constraint (122) is
concerned, the corresponding magnetic quantum number
$\underaccent{\tilde}{n}_{c}$ is a mere spectator, because the oscillators
$(\underaccent{\tilde}{\pi}^{A},\underaccent{\tilde}{\omega}^{A})$ commute
with the constraint. Hence it is an additional Dirac observable, that encodes
the $SL(2,\mathbb{R})$ shape degrees of freedom of the intrinsic metric
$q_{ab}$ on the null cone. To satisfy the constraint (117), we restrict
ourselves to quantum numbers $n_{a}$ and $n_{b}$ such that

$n_{a}>n_{b}.$ (128)

In addition, $N$ determines the eigenvalue of the Casimir operator
$\hat{Q}=L^{2}-\frac{1}{2}(cc^{\dagger}+c^{\dagger}c)$. For the discrete
representations of $SL(2,\mathbb{R})$, we have $2|n_{c}|=N,N+1,\dots$ for
$N=1,2,\dots$ and

$\hat{Q}|N,n_{a},n_{b},n_{c}\rangle=\frac{1}{4}N(N-2)|N,n_{+},n_{b},n_{c}\rangle,\quad$
(129)

Clearly, $N$ and $n_{+}=n_{a}-n_{b}$ are Dirac observables, because the
corresponding operators $\hat{Q}$ and $a^{\dagger}a-b^{\dagger}b$ commute with
the constraint (122). For the same reason, any shift operator that changes the
values of $N$ and $n_{+}$ will be a Dirac observable as well. On the other
hand, neither $L$ nor $b^{\dagger}b$ and $a^{\dagger}a$ can be Dirac
observables. However, the sum $n_{-}=2L+a^{\dagger}a+b^{\dagger}b$ is and so
are the shift operators that change it. Consider, for example, a case in which
$n_{-}-n_{+}=2(N+m)>0$ (i.e. $E_{-}-E_{+}>0$). We can now immediately find a
unique superposition of states

$|N,n_{+},0,N+m\rangle,|N,n_{+}+1,1,N+m-1\rangle,\dots,|N,n_{+}+m,m,N\rangle$
(130)

such that the constraint (122) is satisfied. In here, we chose an operator
ordering such that

$\displaystyle
c\,{a}^{\dagger}{b}^{\dagger}=-\gamma\mathopen{}\mathclose{{}\left(L+{a}^{\dagger}a}\right)\sqrt{2\,{a}^{\dagger}a\,{b}^{\dagger}b}\tan\mathopen{}\mathclose{{}\left(\frac{1}{2\sqrt{2}}\ln\mathopen{}\mathclose{{}\left(\frac{\sqrt{{a}^{\dagger}a}+\sqrt{{b}^{\dagger}b}}{\sqrt{{a}^{\dagger}a}-\sqrt{{b}^{\dagger}b}}}\right)}\right)+\mathrm{i}\,{a}^{\dagger}a\,{b}^{\dagger}b$
(131)

If we try to build physical states in which $n_{-}-n_{+}<0$ and $L<0$, the
recursion relations never terminate. The corresponding physical states contain
infinitely many kinematical state vectors (127). At this stage it is an open
question whether such states are normalizable kets or should be rather seen as
distributions on the kinematical Hilbert space, i.e. elements of the algebraic
dual of the kinematical Hilbert space.

In defining the constraint (131), there are operator ordering ambiguities, but
they are mild. The only condition is that the right hand side of the
constraint (131) annihilates the kinematical state
$|N,n_{a},n_{b}=0,n_{c}\rangle$. What is the physical interpretation of the
quantum numbers for the Dirac observables we found? A short calculation gives

$\displaystyle L+a^{\dagger}a\sim\frac{E_{+}+E_{-}}{16\pi\gamma G},$ (132)
$\displaystyle L+b^{\dagger}b\sim\frac{E_{-}-E_{+}}{16\pi\gamma G},$ (133)
$\displaystyle\hat{Q}\sim\frac{1}{(8\pi
G)^{2}}\mathopen{}\mathclose{{}\left[\frac{1}{4\gamma^{2}}(E_{+}-E_{-})^{2}-\frac{1}{\gamma^{2}}E_{+}E_{-}\operatorname{sh}^{2}(2\sqrt{\sigma\bar{\sigma}})-\frac{1}{2}(E_{+}+E_{-})^{2}\tan^{2}(\sqrt{2\sigma\bar{\sigma}})}\right],$
(134)

where $\sim$ has to be understood as equality in the semiclassical limit. In
here, $\gamma$ is the Barbero–Immirzi parameter, $E_{+}$ ($E_{-}$) is the
final (initial) area density at $\mathcal{C}_{\pm}$, and
$|\sigma|=|\sigma_{I}|$ is the norm of the shear in the chosen gauge, see
(33).

A few final remarks. Throughout this research, we restricted ourselves to
discrete but unitary representations of $SL(2,\mathbb{R})$. We do not know at
this stage whether it is possible to find solutions for the continuous series
representations on the Hilbert space as well. More work is needed, but we
believe that there are no normalizable states in the kernel of (131), for
which $\hat{Q}=L^{2}-\tfrac{1}{2}(c^{\dagger}c+cc^{\dagger})<0$. In any case,
states in which $\hat{Q}<0$ will have very little to do with gravity at low
energy. This can be argued as follows. If we take a pulse of radiation of
finite duration at future (past) null infinity and insert our parametrization
back into (134), we find that $\hat{Q}$ is positive until it reaches a
critical value, which corresponds to a Bondi news flux of the order of the
Planck power $P_{\text{\it Planck}}\approx 1/G$. We will explain this point
more carefully elsewhere, but there seems to be a cutoff for the flux. What is
also interesting is that for any of our states created from (130), the
operator $L+b^{\dagger}b$ is always positive definite. This follows
immediately from the representation theory of $SL(2,\mathbb{R})$.
Geometrically, $L+b^{\dagger}b=(E_{-}-E_{+})/(16\pi\gamma G)$, which is the
difference between the eigenvalues of area at the final cross section
$\mathcal{C}_{+}$ and the initial cross section $\mathcal{C}_{-}$. Thus,
$E_{-}\geq E_{+}$, which is a version of the quasi-local second law at the
full non-perturbative quantum level. Throughout this entire work, we assumed
$E_{\pm}>0$. This is perhaps a limitation. Negative values of the area density
describe the opposite orientation, i.e. an expanding rather than contracting
null surface. In this case, all creation and annihilation operators are simply
exchanged, sending $(a,b,c,L)$ into
$(a^{\dagger},b^{\dagger},c^{\dagger},-L)$.

##  5 Summary and Interpretation of Results

To summarize, we established a non-perturbative quantization of impulsive
radiative data on the null cone. At the two cross sections, where the pulses
start or terminate, the area density is a Dirac observable. Its spectrum is
discrete. The entire quantization is based on the representation theory of
$SL(2,\mathbb{R})$. For the discrete unitary representations, the strength of
the shear degrees of freedom has a discrete spectrum as well. The spectrum is
complicated, but it can be inferred, in principle, from (134). At the end of
the previous section, we made a basic conjecture. What makes the discrete
series representations stand out in our model is that the recursion relations
(131) terminate after finitely many steps. For the continuous series
representations no such mechanism is available. Instead, the coefficients
oscillate violently and it seems hard to find finite norm states that lie in
the kernel of (131). Similar arguments suggest a universal cutoff for the
total radiated power as measured at a cross section of future null infinity,
but more work is required to make this second statement precise. If true, it
would suggest a fundamental UV cutoff due to the fundamental quantum
discreteness of geometry.

The results where established in three steps: First, we studied the null
boundary problem for general relativity with a parity odd Holst term added to
the action in the bulk. The strength of this term relative to the usual
Einstein–Hilbert Lagrangian is determined by the Barbero–Immirzi parameter
$\gamma$. Second, we made a truncation. We performed a gauge fixing and solved
the residual constraints, namely the Raychaudhuri constraint and the transport
equation for the $SL(2,\mathbb{R})$ holonomy, for specific impulsive data. By
inserting the solution back into the pre-symplectic potential, we equipped the
submanifold of impulsive radiative data with a natural symplectic structure
inherited from the vast phase space of general relativity. Third, we
established a quantum representation of the canonical commutation relations.
For each null generator, we found three quantum numbers that label physical
states: the area of the initial and final cross section and the
$SL(2,\mathbb{R})$ Casimir, which is a combination of the area of the cross
sections and the shear of the null generators, see (134).

When $\gamma\rightarrow\infty$, the action returns the usual Hilbert–Palatini
action for general relativity in the bulk. The addition of the Holst term does
not change the classical field equations. However, it affects the algebra of
boundary charges. This has observable consequences. The Barbero–Immirzi
parameter enters the spectra for the area density and shear degrees of
freedom. At the quantum level, the limit $\gamma\rightarrow\infty$ becomes
singular.

In the previous section, we studied a very peculiar family of null geometries
in which $(E_{-}-E_{+})/(16\pi\gamma G)>0$ and $L>0$, representing an
infalling quantum null cone. Clearly, more work is needed to better understand
the entire solution space of the recursion relations (131). There are numerous
more issues that we leave open for future research. For example, it would be
very nice to connect our results to the quantum geometry of isolated horizons
[79]. This would be a good test to link the description at the null boundary
to a spin network and twisted geometry representation in the bulk [82, 6,
70].1 At this stage, perhaps the most important open technical problem is to
take the tensor product of many such quantum impulsive null initial data and
trace over intermediate edge states to form quantum states for realistic
initial data. A related issue is to check whether the gluing procedure is
cylindrical consistent, i.e. stable under refinement [63, 8]. A particular
important problem is to define a family of coherent states to take the limit
to an asymptotic boundary. In this way, it will be possible to link asymptotic
scattering amplitudes at low energies to the full theory at finite distance.

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