Quantum flux operators for Carrollian

diffeomorphism in general dimensions

Ang<EMAIL_ADDRESS>Wen-Bin<EMAIL_ADDRESS>Jiang
<EMAIL_ADDRESS>& Run-Ze<EMAIL_ADDRESS>

School of Physics, Huazhong University of Science and Technology,
Wuhan, Hubei 430074, China

We construct Carrollian scalar field theories in general dimensions, mainly
focusing on the boundaries of Minkowski and Rindler spacetime, whose quantum
flux operators form a faithful representation of Carrollian diffeomorphism up
to a central charge, respectively. At future/past null infinity, the fluxes
are physically observable and encode rich information of the radiation. The
central charge may be regularized to be finite by spectral zeta function or
heat kernel method on the unit sphere. For the theory at the Rindler horizon,
the effective central charge is proportional to the area of the bifurcation
surface after regularization. Moreover, the zero mode of supertranslation is
identified as the modular Hamiltonian, linking Carrollian diffeomorphism to
quantum information theory. Our results may hold for general null
hypersurfaces and provide new insight in the study of the Carrollian field
theory, asymptotic symmetry group and entanglement entropy.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Scalar field theory at future null infinity
    1. 3.1 Null infinity
    2. 3.2 Bulk theory
    3. 3.3 Boundary theory
    4. 3.4 Flux operators for Carrollian diffeomorphism
    5. 3.5 Commutators and regularization of central charge
  4. 4 Scalar field theory on Rindler horizon
    1. 4.1 Rindler horizon
    2. 4.2 From bulk to boundary
    3. 4.3 Flux operators for Carrollian diffeomorphism
    4. 4.4 Relation to Modular Hamiltonian
    5. 4.5 Observables in Unruh effect
  5. 5 Null hypersurfaces
  6. 6 Conclusion and discussion
  7. A Unit sphere $S^{m}$
    1. A.1 Spherical coordinates
    2. A.2 Spherical harmonics
    3. A.3 Addition theorem
    4. A.4 Conformal Killing vectors
  8. B Mode expansion
    1. B.1 Null infinity
    2. B.2 Null hypersurface $\mathcal{H}^{-}$
    3. B.3 Rindler horizon $\mathcal{H}^{--}$
  9. C Zeta-function regularizations
    1. C.1 Definitions
    2. C.2 Regularization

##  1 Introduction

Null hypersurfaces play a central role in the causal structure of spacetime.
They are important in many physical systems, such as future/past null infinity
of asymptotically flat spacetime and black hole horizons. The notable feature
of null hypersurfaces is that the metric is degenerate. Recently, the null
geometry has been studied in the context of Carrollian manifolds [1, 2, 3, 4,
5].

The Carrollian diffeomorphism preserving the Carroll structure [6] could
provide a natural separation between space and time [7]. More interestingly,
it has been shown in a series of papers [6, 8, 9] that one may construct
quantum flux operators that generate Carrollian diffeomorphism up to anomalous
terms at future null infinity. The operators are defined through the flux
densities which are radiated to future null infinity and they encode rich
information of the time and angle dependencies about the radiation. The
operators are totally defined at future null infinity and could form a
representation of Carrollian diffeomorphism. Rather interestingly, the
radiation fluxes are physically observables. Therefore, it indicates that all
the Carrollian diffeomorphism should be regarded as large gauge
transformations, extending the BMS groups [10, 11, 12, 13, 14, 15, 16].

However, there are still two remaining problems. In higher dimensions, there
are consistent fall-off conditions that prevent the supertranslations from
being asymptotic symmetries [17, 18]. On the other hand, the Weinberg soft
theorem is believed to be valid in all dimensions [19, 20, 21]. There exist
interesting discussions on the extended symmetry in higher dimensions [22, 23,
24]. Moreover, there are various null hypersurfaces (not necessarily null
infinity) on which one could define “localized charges” [25, 26, 27]. One of
the most intriguing example is the Killing horizon of black holes [28, 29, 30,
31, 32]. From the intrinsic perspective of Carrollian manifold, there are no
essential differences among various null hypersurfaces [33, 34, 35].
Therefore, one could define field theory and find quantum operators that
represent Carrollian diffeomorphism for any null hypersurfaces in general
dimensions.

In this paper, we will study the quantum flux operators associated with
Carrollian diffeomorphism in general dimensions. We will use two typical null
hypersurfaces, the null infinity and the Rindler horizon, to present the
results. More explicitly, we reduce the bulk massless scalar theory in
Minkowski spacetime to future null infinity and construct the radiation
fluxes. The flux operators form a faithful representation of the Carrollian
diffeomorphism up to a central charge which may be regularized through the
spectral zeta function on the unit sphere. The result is also consistent with
the heat kernel method. We will also study the scalar field theory in the
Rindler wedge and reduce it to the null boundary, a Killing horizon associated
with a Lorentz boost in Minkowski spacetime. Since the Killing horizon is not
at infinity, we do not impose any artificial fall-off condition except that
the field is finite at the null hypersurface. The field theory can be massive
since any particles can cross through the null hypersurface. Similar to the
case of null infinity, the quantum flux operators form a representation of
Carrollian diffeomorphism at the Killing horizon. The supertranslation flux
operators form a Virasoro algebra with a divergent central charge. We use heat
kernel method and Pauli-Villars method to regularize the central charge. They
lead to the same conclusion that there is an effective central charge which is
proportional to area of the bifurcation surface of the Killing horizons. Using
the conservation law of stress tensor and Stokes’ theorem, we identify the
zero mode of supertranslation as the modular Hamiltonian of the Rindler wedge.
The result indicates that Carrollian diffeomorphism may be related to quantum
information theory. We try to calculate the expectation values of the quantum
flux operators for the vacuum and excited states. The relative fluxes, which
are defined as the difference of the expectation values between two quantum
states, may provide new observables in Unruh effect.

The structure of the paper is as follows. In section 2, we will introduce the
basic ingredients on the Carrollian manifold. In the following two sections,
we will construct quantum flux operators on future null infinity and Rindler
horizon and discuss their various properties. We extend the results to general
null hypersurfaces in section 5 and conclude in section 6. Technical details
are relegated to three appendices.

Conventions and notations

We will use the following conventions in this paper. Lorentz manifolds are
denoted by $\mathbb{M},\mathbb{N}$, etc., with signature $(-,+,+,\cdots,+)$.
Tensors on Lorentz manifolds are denoted by lowercase Greek indices
$\mu,\nu,\cdots$. Timelike hypersurfaces are denoted by
$\mathfrak{H},\mathfrak{I}$, and so on. A null hypersurface is a Carrollian
manifold which is denoted as $\mathcal{H},\mathcal{I}$, etc.. Usually they can
be factorized as a product such as

$\mathcal{H}=\textbf{R}\times S$ (1.1)

where $\bf R$ is the retarded/advanced time direction and $S$ is a Riemannian
manifold. Tensors on Carrollian manifolds are labeled by lowercase Roman
indices $a,b,c$, etc. from the first half of the alphabet. Riemannian
manifolds are denoted by uppercase Roman letters. Tensors on Riemannian
manifolds are indexed by uppercase Roman indices $A,B,C$, etc..

##  2 Preliminaries

In this section, we will review the minimal ingredients on Carrollian
diffeomorphism which are relevant to later discussion. A Carrollian manifold
$\mathcal{C}$ with topology ${\bf R}\times S$ has a degenerate metric

$ds^{2}\equiv\bm{h}=h_{AB}d\theta^{A}d\theta^{B},\quad A,B=1,2,\cdots,d-2$
(2.1)

as well as a null vector

$\displaystyle\bm{\chi}=\partial_{u}.$ (2.2)

The metric $\bm{h}$ also defines the geometry of the $d-2$ dimensional
manifold $S$, which is assumed to be Riemannian with signature
$(+,+,\cdots,+)$. The $d-2$ coordinates $\theta^{A},A=1,2,\cdots,d-2$ may also
be collected as $\Omega=(\theta^{1},\theta^{2},\cdots,\theta^{d-2})$. The null
vector $\bm{\chi}$ generates the time direction of the Carrollian manifold. We
have used the coordinate $u$ to denote the time for this manifold. The volume
form for the manifold $S$ is denoted by

$d\Omega=\sqrt{\det h}d\theta^{1}\wedge\cdots\wedge d\theta^{d-2}$ (2.3)

and the unique volume $\bm{\epsilon}$ for the Carrollian manifold
$\mathcal{C}$ is given by

$i_{\bm{\chi}}\bm{\epsilon}=d\Omega\quad\Rightarrow\quad\bm{\epsilon}=du\wedge
d\Omega=\sqrt{\det h}du\wedge d\theta^{1}\wedge\cdots\wedge d\theta^{d-2}.$
(2.4)

For brevity, we will also write it as

$\bm{\epsilon}=dud\Omega.$ (2.5)

Carrollian diffeomorphism is generated by the vectors $\bm{\xi}$

$\mathcal{L}_{\bm{\xi}}\bm{\chi}=\mu\bm{\chi}.$ (2.6)

The general solution of the equation (2.6) is

$\displaystyle\bm{\xi}_{f,Y}=f(u,\Omega)\partial_{u}+Y^{A}(\Omega)\partial_{A}$
(2.7)

where $\bm{\xi}_{f}=f(u,\Omega)\partial_{u}$ generates general
supertranslations (GSTs) and $\bm{\xi}_{Y}=Y^{A}(\Omega)\partial_{A}$
generates special superrotations (SSRs) within the terminology of [6]. One may
also consider general superrotations (GSRs) which is generated by
$Y^{A}(u,\Omega)\partial_{A}$. However, this turns out to break the null
structure of the Carrollian manifold and we will not focus on it in this work.
The supertranslations and superrotations are borrowed from BMS
transformations. Nevertheless, in general situations, they are not related to
the original BMS transformations which are only defined at future/past null
infinity. The vectors (2.7) form an infinite dimensional Lie algebra

$\displaystyle\
[\bm{\xi}_{f_{1},Y_{1}},\bm{\xi}_{f_{2},Y_{2}}]=\bm{\xi}_{f_{1}\dot{f}_{2}-f_{2}\dot{f}_{1}+Y_{1}^{A}\partial_{A}f_{2}-Y_{2}^{A}\partial_{A}f_{1},[Y_{1},Y_{2}]}$
(2.8)

where $\dot{f}_{i}\equiv\partial_{u}f_{i},i=1,2$. In particular, for pure
supertranslations, the corresponding Lie algebra is

$\displaystyle\
[\bm{\xi}_{f_{1}},\bm{\xi}_{f_{2}}]=\bm{\xi}_{f_{1}\dot{f}_{2}-f_{2}\dot{f}_{1}}$
(2.9)

which is called T-Witt algebra in [36]. For pure superrotations, the Lie
algebra is

$\ [\bm{\xi}_{Y_{1}},\bm{\xi}_{Y_{2}}]=\bm{\xi}_{[Y_{1},Y_{2}]}$ (2.10)

where $\bm{\xi}_{Y}$ generates diffeomorphism of the manifold $S$. In [6, 8,
9], the authors find that the Carrollian diffeomorphism generated by
$\bm{\xi}$ may be realized by quantum flux operators $Q_{\bm{\xi}}$ whose
algebra is exactly (2.8) up to anomalous quantum corrections

$\displaystyle[Q_{\bm{\xi}_{1}},Q_{\bm{\xi}_{2}}]=iQ_{[\bm{\xi}_{1},\bm{\xi}_{2}]}+K_{\bm{\xi}_{1},\bm{\xi}_{2}}.$
(2.11)

We will use the conventions

$\mathcal{T}_{f}=Q_{\bm{\xi}_{f}}\quad\text{and}\quad\mathcal{M}_{Y}=Q_{\bm{\xi}_{Y}}$
(2.12)

to distinguish supertranslations and superrotations. The operators
$Q_{\bm{\xi}}$ and the anomalous corrections $K_{\bm{\xi}_{1},\bm{\xi}_{2}}$
may depend on the field $F$ defined on the Carrollian manifold in general

$Q_{\bm{\xi}}=Q_{\bm{\xi}}[F],\quad
K_{\bm{\xi}_{1},\bm{\xi}_{2}}=K_{\bm{\xi}_{1},\bm{\xi}_{2}}[F].$ (2.13)

For real scalar theory, the unique anomalous term is a central charge of
T-Witt algebra

$K_{\bm{\xi}_{f_{1}},\bm{\xi}_{f_{2}}}=C_{T}(f_{1},f_{2})=-\frac{i}{48\pi}\delta^{(d-2)}(0)\mathcal{I}_{f_{1}\dddot{f_{2}}-f_{2}\dddot{f_{1}}}$
(2.14)

such that the algebra becomes a Virasoro algebra with more transverse indices

$[\mathcal{T}_{f_{1}},\mathcal{T}_{f_{2}}]=\mathcal{T}_{f_{1}\dot{f}_{2}-f_{2}\dot{f}_{1}}+C_{T}(f_{1},f_{2}).$
(2.15)

The identity operator in (2.14) is defined as

$\displaystyle\mathcal{I}_{f}=\int dud\Omega f(u,\Omega),$ (2.16)

while the Dirac delta function with argument equalling to zero

$\delta^{(d-2)}(0)=\delta^{(d-2)}(\Omega-\Omega^{\prime})|_{\Omega^{\prime}=\Omega}$
(2.17)

is proportional to the density of states in the transverse direction, e.g. the
density of states on the manifold $S$. Since the spectrum of the field in the
transverse direction is continuous in general, the Dirac delta function
$\delta^{(d-2)}(0)$ is divergent. Note that we have already written the
results in $d$ dimensions. In the vector and gravitational theory, there are
additional anomalous terms which may be interpreted as helicity flux operators
whose explicit forms can be found in [8, 9]. The helicity flux operators
generate super-duality transformations (angle dependent duality
transformations) which are essential for the field theories with nonzero
spins.

To find the operators $Q_{\bm{\xi}}$, one may embed the Carrollian manifold
into $d$ dimensional spacetime $\mathbb{M}$ where the bulk theory is well
defined. The stress tensor $T_{\mu\nu}$ of the bulk theory is conserved such
that one may integrate the current

$j_{\bm{\xi}}^{\mu}=T^{\mu}_{\ \nu}\xi^{\nu}$ (2.18)

over the Carrollian manifold $\mathcal{C}$ to obtain

$Q_{\bm{\xi}}=\int_{\mathcal{C}}(d^{d-1}x)_{\mu}j_{\bm{\xi}}^{\mu}.$ (2.19)

The vector $\bm{\xi}$ may be chosen as the generator of Carrollian
diffeomorphism (2.7). If $\bm{\xi}$ is a Killing vector when extended into the
bulk spacetime $\mathbb{M}$, the operator $Q_{\bm{\xi}}$ is the corresponding
charge which crosses the Carrollian manifold $\mathcal{C}$ during the whole
time. Therefore, $Q_{\bm{\xi}}$ could be interpreted as a leaky flux from bulk
to boundary.

The above results have been mainly found at null infinity in four dimensional
flat spacetime. In this paper, we will extend it to general null hypersurfaces
and general dimensions.

##  3 Scalar field theory at future null infinity

In this section, we will construct various scalar field theories at future
null infinity ($\mathcal{I}^{+}$) from bulk reduction in general dimensions.
The overall framework is to embed $\mathcal{I}^{+}$ into Minkowski spacetime
$\mathbb{R}^{1,d-1}$ and reduce the bulk theory from $\mathbb{R}^{1,d-1}$ to
$\mathcal{I}^{+}$.

###  3.1 Null infinity

The future/past null infinity ($\mathcal{I}^{+}/\mathcal{I}^{-}$) is a
Carrollian manifold in asymptotically flat spacetime. In this work, we will
discuss $\mathcal{I}^{+}$ for Minkowski spacetime $\mathbb{R}^{1,d-1}$ which
can be described in Cartesian coordinates $x^{\mu}=(t,x^{i})$

$\displaystyle ds^{2}=-dt^{2}+dx^{i}dx^{i}=\eta_{\mu\nu}dx^{\mu}dx^{\nu},$
(3.1)

where $\mu=0,1,\cdots,d-1$ denotes the components of spacetime coordinates and
$i=1,2,\cdots,d-1$ labels the components of spatial coordinates. We also use
the retarded coordinate system $(u,r,\theta^{A})$ and write the metric of
$\mathbb{R}^{1,d-1}$ as

$\displaystyle
ds_{\mathbb{R}^{1,d-1}}^{2}=-du^{2}-2dudr+r^{2}\gamma_{AB}d\theta^{A}d\theta^{B},\quad
A,B=1,2,\cdots,d-2.$ (3.2)

The future null infinity $\mathcal{I}^{+}$ is a $d-1$ dimensional Carrollian
manifold

$\displaystyle\mathcal{I}^{+}={\bf R}\times S^{d-2}$ (3.3)

with a degenerate metric

$\displaystyle
ds_{\mathcal{I}^{+}}^{2}\equiv\bm{\gamma}=\gamma_{AB}d\theta^{A}d\theta^{B}=ds^{2}_{S^{d-2}}.$
(3.4)

The spherical coordinates $\theta^{A}=(\theta_{1},\cdots,\theta_{d-2})$ are
used to describe the unit sphere $S^{d-2}$ whose metric $ds^{2}_{S^{d-2}}$ can
be found in Appendix A. We will also use the notation
$\Omega=(\theta_{1},\cdots,\theta_{d-2})$ to denote the spherical coordinates
in the context. The covariant derivative $\nabla_{A}$ is adapted to the metric
$\gamma_{AB}$, while the covariant derivative $\nabla_{\mu}$ is adapted to the
Minkowski metric in Cartesian frame. The integral measure on $\mathcal{I}^{+}$
is abbreviated as

$\displaystyle\int
dud\Omega\equiv\int_{-\infty}^{\infty}du\int_{S^{d-2}}d\Omega,$ (3.5)

where the integral measure on $S^{d-2}$ is

$\displaystyle\int
d\Omega\equiv\int_{S^{d-2}}d\Omega=[\prod_{j=2}^{d-2}\int_{0}^{\pi}\sin^{j-1}\theta_{j}d\theta_{j}]\int_{0}^{2\pi}d\theta_{1}.$
(3.6)

Besides the metric (3.4), there is also a distinguished null vector

$\bm{\chi}=\partial_{u}$ (3.7)

which is to generate the retarded time direction.

To obtain the metric of the Carrollian manifold (3.4) from the bulk metric,
one may choose a cutoff

$\displaystyle r=R$ (3.8)

such that the induced metric on the hypersurface

$\displaystyle\mathfrak{H}_{R}=\\{p\in\mathbb{R}^{1,d-1}|\
p=(u,r,\theta_{1},\cdots,\theta_{d-2})\ \text{with}\ r=R\\}$ (3.9)

is

$\displaystyle
ds^{2}=-du^{2}+R^{2}\gamma_{AB}d\theta^{A}d\theta^{B}=R^{2}(-\frac{du^{2}}{R^{2}}+\gamma_{AB}d\theta^{A}d\theta^{B}).$
(3.10)

We use a Weyl scaling to remove the conformal factor and take the limit

$\displaystyle R\to\infty$ (3.11)

while keeping the retarded time $u$ finite such that (3.10) becomes the metric
of the Carrollian manifold $\mathcal{I}^{+}$. We define the limit

$\displaystyle\lim{}_{+}=\lim_{r\to\infty,\ u\ \text{finite}}$ (3.12)

to send the quantities on $\mathfrak{H}_{r}$ to $\mathcal{I}^{+}$. Similarly,
taking the limit below

$\displaystyle\lim{}_{-}=\lim_{r\to\infty,\ v\ \text{finite}}$ (3.13)

sends the quantities on $\mathfrak{H}_{r}$ to $\mathcal{I}^{-}$ where $v$ is
the advanced time $v=t+r$.

###  3.2 Bulk theory

We may consider a general scalar theory with the action

$\displaystyle S[\Phi]=\int
d^{d}x\sqrt{-g}[-\frac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi-V(\Phi)].$
(3.14)

Only massless particles can arrive at $\mathcal{I}^{+}$, and therefore we
consider massless theory with the potential

$V(\Phi)=\sum_{k=3}^{\infty}\frac{\lambda_{k}}{k}\Phi^{k}.$ (3.15)

We impose the fall-off condition

$\Phi(t,\bm{x})=\frac{\Sigma(u,\Omega)}{r^{\Delta}}+\frac{\Sigma^{(2)}(u,\Omega)}{r^{\Delta+1}}+\cdots$
(3.16)

for the bulk scalar field. The constant $\Delta$ is the scaling dimension of
the bulk field and is related to the dimension $d$ through

$\Delta=\frac{d-2}{2}.$ (3.17)

Now we consider the equation of motion (EOM)

$\displaystyle\partial_{\mu}\partial^{\mu}\Phi-V^{\prime}(\Phi)=\partial_{\mu}\partial^{\mu}\Phi-\sum_{k=3}^{\infty}{\lambda_{k}}\Phi^{k-1}=0.$
(3.18)

For the first term, one can show that

$\displaystyle\partial_{\mu}\partial^{\mu}=-2\partial_{u}\partial_{r}-\frac{2\Delta}{r}\partial_{u}+\partial_{r}^{2}+\frac{2\Delta}{r}\partial_{r}+\frac{1}{r^{2}}\nabla_{A}\nabla^{A},$
(3.19)

which leads to

$\displaystyle\partial_{\mu}\partial^{\mu}\Phi=$
$\displaystyle\bigl{(}-2\partial_{u}\partial_{r}-\frac{2\Delta}{r}\partial_{u}+\partial_{r}^{2}+\frac{2\Delta}{r}\partial_{r}+\frac{1}{r^{2}}\nabla_{A}\nabla^{A}\bigr{)}\bigl{[}\frac{\Sigma(u,\Omega)}{r^{\Delta}}+\sum_{k\geq
2}^{\infty}\frac{\Sigma^{(k)}(u,\Omega)}{r^{\Delta+k-1}}\bigr{]}$
$\displaystyle=$ $\displaystyle\ \sum_{k\geq
2}^{\infty}\frac{1}{r^{\Delta+k}}\Big{[}2(k-1)\dot{\Sigma}^{(k)}+(\Delta+k-2)(k-1-\Delta)\Sigma^{(k-1)}+\nabla_{A}\nabla^{A}\Sigma^{(k-1)}\Big{]}.$
(3.20)

For the potential term, we find

$\displaystyle V^{\prime}(\Phi)=\sum_{k\geq
2}^{\infty}\frac{1}{r^{\Delta+k}}\sum_{n=3}^{\infty}{\lambda_{n}}\sum_{k_{1},\cdots,k_{n-1}\geq
1}^{\sum_{j=1}^{n-1}k_{j}=\Delta+k-(n-1)(\Delta-1)}\Sigma^{(k_{1})}\cdots\Sigma^{(k_{n-1})}.$
(3.21)

Note that it does not contribute to EOM for $k=0$. Moreover, for $k=1$,
potential also has no contribution unless $\Delta=1$ (corresponding to
dimension 4). If $\Delta=1$, the leading EOM is $\lambda_{3}\Sigma^{2}=0$,
which results in $\lambda_{3}=0$. Otherwise, the leading EOM is trivial.
Nevertheless, we write out the EOM at order $\mathcal{O}({1}/{r^{\Delta+k}})$

$\displaystyle 0=$ $\displaystyle
2(k-1)\dot{\Sigma}^{(k)}+(\Delta+k-2)(k-\Delta-1)\Sigma^{(k-1)}+\nabla_{A}\nabla^{A}\Sigma^{(k-1)}$
$\displaystyle-\sum_{n=3}^{\infty}{\lambda_{n}}\sum_{k_{1},\cdots,k_{n-1}\geq
1}^{k_{1}+\cdots+k_{n-1}=\Delta+k-(n-1)(\Delta-1)}\Sigma^{(k_{1})}\cdots\Sigma^{(k_{n-1})}.$
(3.22)

In odd dimensions, $\Delta$ is a half integer. Therefore, only the terms with
$n=\text{even}$ have contributions in the second line. In general dimensions,
the leading coefficients $\Sigma$ encode the dynamical propagating degree of
freedom. All the subleading coefficients $\Sigma^{(k)},\ k\geq 2$ are
determined by the EOM up to initial data $\Sigma^{(k)}(u_{0},\Omega),\ k\geq
2$ where $u_{0}$ is the initial time on $\mathcal{C}$.

###  3.3 Boundary theory

The pre-symplectic form of the theory is

$\displaystyle\bm{\Theta}(\delta\Phi;\Phi)=-\int_{\mathfrak{H}_{r}}(d^{d-1}x)_{\mu}\partial^{\mu}\Phi\delta\Phi$
(3.23)

where

$\displaystyle(d^{d-1}x)_{\mu}=\frac{1}{(d-1)!}\epsilon_{\mu\nu_{1}\cdots\nu_{d-1}}dx^{\nu_{1}}\wedge\cdots\wedge
dx^{\nu_{d-1}}=-r^{d-2}m_{\mu}dud\Omega.$ (3.24)

The normal co-vector $m_{\mu}$ of the hypersurface $\mathfrak{H}_{r}$ is

$dr=n_{i}dx^{i}\quad\Rightarrow\quad
m_{\mu}=\frac{1}{2}(n_{\mu}+\bar{n}_{\mu})$ (3.25)

where the null vectors $n_{\mu}$ and $\bar{n}_{\mu}$ are defined as

$n^{\mu}=(1,n^{i}),\quad\bar{n}^{\mu}=(-1,n^{i})$ (3.26)

with $n^{i}={x^{i}}/{r},i=1,2,\cdots,d-1$ being the normal vector of $S^{d-2}$
embedded into $\mathbb{R}^{d-1}$. The symplectic form for the theory on
$\mathfrak{H}_{r}$ is

$\bm{\Omega}^{\mathfrak{H}_{r}}(\delta\Phi;\delta\Phi;\Phi)=-\int_{\mathfrak{H}_{r}}(d^{d-1}x)_{\mu}\partial^{\mu}\delta\Phi\wedge\delta\Phi.$
(3.27)

With the fall-off condition (3.16), the symplectic form becomes

$\displaystyle\bm{\Omega}(\delta\Sigma;\delta\Sigma;\Sigma)=\lim{}_{+}\bm{\Omega}^{\mathfrak{H}_{r}}(\delta\Phi;\delta\Phi;\Phi)=\int
dud\Omega\delta\Sigma\wedge\delta\dot{\Sigma}$ (3.28)

at $\mathcal{I}^{+}$. This is identified as the symplectic form of the
Carrollian scalar field theory at $\mathcal{I}^{+}$. We may find the following
commutators for the field $\Sigma$

$\displaystyle[\Sigma(u,\Omega),\Sigma(u^{\prime},\Omega^{\prime})]$
$\displaystyle=\frac{i}{2}\alpha(u-u^{\prime})\delta^{(d-2)}(\Omega-\Omega^{\prime}),$
(3.29a)
$\displaystyle[\Sigma(u,\Omega),\dot{\Sigma}(u^{\prime},\Omega^{\prime})]$
$\displaystyle=\frac{i}{2}\delta(u-u^{\prime})\delta^{(d-2)}(\Omega-\Omega^{\prime}),$
(3.29b)
$\displaystyle[\dot{\Sigma}(u,\Omega),\dot{\Sigma}(u^{\prime},\Omega^{\prime})]$
$\displaystyle=\frac{i}{2}\delta^{\prime}(u-u^{\prime})\delta^{(d-2)}(\Omega-\Omega^{\prime}),$
(3.29c)

where the function $\alpha(u-u^{\prime})$ is

$\displaystyle\alpha(u-u^{\prime})=\frac{1}{2}[\theta(u^{\prime}-u)-\theta(u-u^{\prime})]$
(3.30)

with $\theta(x)$ being the Heaviside step function

$\theta(x)=\left\\{\begin{array}[]{cc}1,&\ x>0\\\ 0,&\ x<0.\end{array}\right.$
(3.31)

The field $\Sigma$ may be expanded in a set of complete basis of
$\mathcal{I}^{+}$

$\displaystyle\Sigma(u,\Omega)=\int_{0}^{\infty}\frac{d\omega}{\sqrt{4\pi\omega}}[a_{\omega,\bm{\ell}}e^{-i\omega
u}Y_{\bm{\ell}}(\Omega)+a_{\omega,\bm{\ell}}^{\dagger}e^{i\omega
u}Y^{*}_{\bm{\ell}}(\Omega)]$ (3.32)

with the commutators

$\displaystyle[a_{\omega,\bm{\ell}},a_{\omega^{\prime},\bm{\ell}^{\prime}}]=[a_{\omega,\bm{\ell}}^{\dagger},a_{\omega^{\prime},\bm{\ell}^{\prime}}^{\dagger}]=0,$
(3.33a)
$\displaystyle[a_{\omega,\bm{\ell}},a_{\omega^{\prime},\bm{\ell}^{\prime}}^{\dagger}]=\delta(\omega-\omega^{\prime})\delta_{\bm{\ell},\bm{\ell}^{\prime}}.$
(3.33b)

In the expansion (3.32), the functions $Y_{\bm{\ell}}(\Omega)$ are spherical
harmonic functions on $S^{d-2}$. Readers may find more details in Appendix
A.2. We have checked the mode expansion (3.32) using canonical quantization
method in Appendix B.1. With the canonical quantization, we may define the
vacuum state $|0\rangle$ through

$a_{\omega,\bm{\ell}}|0\rangle=0.$ (3.34)

Therefore, we find the following correlators

$\displaystyle\langle
0|\Sigma(u,\Omega)\Sigma(u^{\prime},\Omega^{\prime})|0\rangle$
$\displaystyle=\beta(u-u^{\prime})\delta^{(d-2)}(\Omega-\Omega^{\prime}),$
(3.35a) $\displaystyle\langle
0|\Sigma(u,\Omega)\dot{\Sigma}(u^{\prime},\Omega^{\prime})|0\rangle$
$\displaystyle=\frac{1}{4\pi(u-u^{\prime}-i\epsilon)}\delta^{(d-2)}(\Omega-\Omega^{\prime}),$
(3.35b) $\displaystyle\langle
0|\dot{\Sigma}(u,\Omega)\Sigma(u^{\prime},\Omega^{\prime})|0\rangle$
$\displaystyle=-\frac{1}{4\pi(u-u^{\prime}-i\epsilon)}\delta^{(d-2)}(\Omega-\Omega^{\prime}),$
(3.35c) $\displaystyle\langle
0|\dot{\Sigma}(u,\Omega)\dot{\Sigma}(u^{\prime},\Omega^{\prime})|0\rangle$
$\displaystyle=-\frac{1}{4\pi(u-u^{\prime}-i\epsilon)^{2}}\delta^{(d-2)}(\Omega-\Omega^{\prime}).$
(3.35d)

###  3.4 Flux operators for Carrollian diffeomorphism

In this section, we will define flux operators for Carrollian diffeomorphism
in general dimensions. For any conserved current $j^{\mu}$ in the bulk, we may
construct the corresponding flux radiated to $\mathcal{I}^{+}$ through the
limit

$\mathcal{F}[j]=\lim{}_{+}\int_{\mathfrak{H}_{r}}(d^{d-1}x)_{\mu}j^{\mu}=-\lim{}_{+}\int_{\mathfrak{H}_{r}}dud\Omega
r^{d-2}m_{\mu}j^{\mu}.$ (3.36)

The stress tensor for the theory (3.14) is

$T_{\mu\nu}=\partial_{\mu}\Phi\partial_{\nu}\Phi+g_{\mu\nu}\mathcal{L}$ (3.37)

where $\mathcal{L}$ is the Lagrangian of the theory. Note that we have the
expansion

$\displaystyle\Phi=\sum_{k\geq
1}\frac{\Sigma^{(k)}(u,\Omega)}{r^{\Delta+k-1}},\quad\Sigma^{(1)}=\Sigma$
(3.38)

for the field $\Phi$ which leads to

$\displaystyle\partial_{\mu}\Phi=$ $\displaystyle\sum_{k\geq
1}\frac{1}{r^{\Delta+k-1}}\big{[}{-n_{\mu}\dot{\Sigma}^{(k)}-\frac{1}{2}(\Delta+k-2)(n_{\mu}+\bar{n}_{\mu})\Sigma^{(k-1)}-Y_{\mu}^{A}\nabla_{A}\Sigma^{(k-1)}}\big{]},$
(3.39)

where we have assumed $\Sigma^{(0)}=0$. The vector $Y_{\mu}^{A}$ is the
gradient of the null vector $n_{\mu}$

$Y_{\mu}^{A}=-\nabla^{A}n_{\mu}.$ (3.40)

Similarly, we have

$\displaystyle\partial_{\mu}\Phi\partial_{\nu}\Phi=\sum_{k_{1},k_{2}\geq 1}$
$\displaystyle\frac{1}{r^{2\Delta+k_{1}+k_{2}-2}}\big{[}(n_{\mu}\dot{\Sigma}^{(k_{1})}+(\Delta+k_{1}-2)m_{\mu}\Sigma^{(k_{1}-1)}+Y_{\mu}^{A}\nabla_{A}\Sigma^{(k_{1}-1)})$
$\displaystyle\times(n_{\nu}\dot{\Sigma}^{(k_{2})}+(\Delta+k_{2}-2)m_{\mu}\Sigma^{(k_{2}-1)}+Y_{\nu}^{A}\nabla_{A}\Sigma^{(k_{2}-1)})\big{]}.$
(3.41)

The stress tensor has the following fall-off behaviour

$\displaystyle
T_{\mu\nu}=\frac{t_{\mu\nu}^{(2\Delta)}}{r^{2\Delta}}+\frac{t_{\mu\nu}^{(2\Delta+1)}}{r^{2\Delta+1}}+\cdots$
(3.42)

where

$\displaystyle t^{(2\Delta)}_{\mu\nu}=n_{\mu}n_{\nu}\dot{\Sigma}^{2},$ (3.43)
$\displaystyle
t^{(2\Delta+1)}_{\mu\nu}=(n_{\mu}\dot{\Sigma}^{(2)}+{\Delta}\Sigma
m_{\mu}+Y_{\mu}^{A}\nabla_{A}\Sigma)n_{\nu}\dot{\Sigma}+(\mu\leftrightarrow\nu)-\eta_{\mu\nu}\Delta\dot{\Sigma}\Sigma.$
(3.44)

For any Killing vector $\bm{\xi}$ in $\mathbb{R}^{1,d-1}$, we may find a
conserved current

$\displaystyle j^{\mu}_{\bm{\xi}}=T^{\mu\nu}\xi_{\nu}.$ (3.45)

Therefore, the flux corresponding to the Killing vector $\bm{\xi}$ is

$\displaystyle\mathcal{F}_{\bm{\xi}}\equiv\mathcal{F}[j_{\bm{\xi}}]=-\lim{}_{+}\int_{\mathcal{H}_{r}}dud\Omega
r^{d-2}m_{\mu}T^{\mu\nu}\xi_{\nu}.$ (3.46)

  1. 1.

For a translation generated by
$\bm{\xi}_{\text{trans.}}=c^{\mu}\partial_{\mu}$, the energy-momentum flux is

$\displaystyle\mathcal{F}_{\text{trans.}}=-\int dud\Omega
c^{\mu}n_{\mu}\dot{\Sigma}^{2}.$ (3.47)

  2. 2.

For a Lorentz rotation generated by
$\bm{\xi}_{\text{LT}}=\omega^{\mu\nu}(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})$,
the angular momentum or center of mass flux is

$\displaystyle\mathcal{F}_{\text{LT}}=\int
dud\Omega[\frac{u}{d-2}\omega^{\mu\nu}\nabla_{A}Y^{A}_{\mu\nu}\dot{\Sigma}^{2}+\omega^{\mu\nu}Y_{\mu\nu}^{A}\dot{\Sigma}\nabla_{A}\Sigma],$
(3.48)

where $Y_{\mu\nu}^{A}$ is the conformal Killing vectors on $S^{d-2}$. Readers
can find more details about the derivation in Appendix A.4.

From the fluxes (3.47) and (3.48) we may define the energy density flux
operator

$\displaystyle T(u,\Omega)=:\dot{\Sigma}^{2}:$ (3.49)

and angular momentum density flux operator

$M_{A}(u,\Omega)=\frac{1}{2}\left(:\dot{\Sigma}\nabla_{A}\Sigma-\Sigma\nabla_{A}\dot{\Sigma}:\right).$
(3.50)

We have used normal ordering $:\cdots:$ in the definition. The angular
momentum density flux operator $M_{A}(u,\Omega)$ is only fixed up to a
constant $\lambda$

$M_{A}(u,\Omega;\lambda)=\frac{1}{2}\left(:\lambda\dot{\Sigma}\nabla_{A}\Sigma-(1-\lambda)\Sigma\nabla_{A}\dot{\Sigma}:\right)$
(3.51)

since they lead to the same fluxes in (3.48) after integration by parts.
Following [6], we fix $\lambda=\frac{1}{2}$ using the orthogonality condition

$\langle 0|T(u,\Omega)M_{A}(u^{\prime},\Omega^{\prime})|0\rangle=0.$ (3.52)

Note the Poincaré group is a subgroup of Carrollian diffeomorphism for
$\mathcal{I}^{+}$. We may exploit (2.19) and extend the fluxes to general
Carrollian diffeomorphism. After some algebra, we find the following extended
flux operators

$\displaystyle\mathcal{T}_{f}$ $\displaystyle=\int dud\Omega
f(u,\Omega)T(u,\Omega),$ (3.53a) $\displaystyle\mathcal{M}_{Y}$
$\displaystyle=\int dud\Omega Y^{A}(\Omega)M_{A}(u,\Omega)$ (3.53b)

for Carrollian diffeomorphism. The time and angle dependence of the density
flux operators may be transformed to the following smeared operators

$\displaystyle\bar{\mathcal{T}}_{f}$ $\displaystyle=\int dud\Omega
f(u,\Omega)T(u,\Omega),$ (3.54a) $\displaystyle\bar{\mathcal{M}}_{Y}$
$\displaystyle=\int dud\Omega Y^{A}(u,\Omega)M_{A}(u,\Omega).$ (3.54b)

We add a bar in the definition to distinguish them with the one from
Carrollian diffeomorphism. The operator $\bar{\mathcal{T}}_{f}$ is equal to
$\mathcal{T}_{f}$ exactly for any smooth function $f=f(u,\Omega)$. On the
other hand, the operator $\bar{\mathcal{M}}_{Y}$ matches with
$\mathcal{M}_{Y}$ only for time independent vectors $Y^{A}=Y^{A}(\Omega)$. As
has been noticed, for general time dependent vectors $Y^{A}=Y^{A}(u,\Omega)$,
the corresponding operator generates non-local terms which may relate to the
violation of the null structure under GSRs.

There is a rather different way to obtain the same flux operators. From
Hamiltonian theory, we learn that the Hamiltonian corresponds to a general
vector $\bm{\xi}$ may be obtained from the Hamilton equation

$\delta
H_{\bm{\xi}}=i_{\bm{\xi}}\bm{\Omega}(\delta\Sigma;\delta\Sigma;\Sigma)$ (3.55)

where $i_{\bm{\xi}}$ is the interior product along the direction $\bm{\xi}$ in
phase space. The variation of the boundary scalar field $\Sigma$ can be
obtained from bulk diffeomorphism

$\displaystyle\delta_{\bm{\xi}}\Phi=\xi^{\lambda}\partial_{\lambda}\Phi\quad\Rightarrow\quad\delta_{\bm{\xi}}\Sigma=\lim{}_{+}r^{\Delta}\delta_{\bm{\xi}}\Phi.$
(3.56)

In Minkowski spacetime $\mathbb{R}^{1,d-1}$, the generator of a global
translation along the direction $c^{\mu}$ is

$\bm{\xi}_{\text{trans.}}=c^{\mu}\partial_{\mu}=c^{\mu}[-n_{\mu}\partial_{u}+m_{\mu}\partial_{r}-\frac{1}{r}Y_{\mu}^{A}\partial_{A}].$
(3.57)

We have inserted a constant vector $c^{\mu}$ into the expression to denote the
direction of the translation, and transformed to retarded coordinates at the
last step. Therefore, the boundary field $\Sigma$ is transformed to

$\displaystyle\delta_{\bm{\xi}_{\text{trans.}}}\Sigma=-c^{\mu}n_{\mu}\dot{\Sigma}.$
(3.58)

We observe that the generator of the translation $\bm{\xi}_{\text{trans.}}$
reduces to

$\displaystyle-c^{\mu}n_{\mu}\partial_{u}$ (3.59)

at $\mathcal{I}^{+}$. Obviously, it is the Carrollian diffeomorphism generated
by

$\bm{\xi}_{f}=f(u,\Omega)\partial_{u}$ (3.60)

with

$\displaystyle f(u,\Omega)=-c^{\mu}n_{\mu}.$ (3.61)

It is natural to assume that the transformation of the boundary field $\Sigma$
under a GST is

$\delta_{f}\Sigma(u,\Omega)=f(u,\Omega)\dot{\Sigma}.$ (3.62)

Note this is also the transformation of a boundary scalar field under the
Carrollian diffeomorphism generated by $\bm{\xi}_{f}$

$\delta_{f}\Sigma(u,\Omega)=\bm{\xi}_{f}(\Sigma)=\mathcal{L}_{\bm{\xi}_{f}}\Sigma=f(u,\Omega)\dot{\Sigma}.$
(3.63)

In this derivation, there is no need to assume $\bm{\xi}$ being a Killing
vector or an asymptotic Killing vector near $\mathcal{I}^{+}$. There would be
a corresponding Hamiltonian $H_{f}$ related to the GST through the Hamilton
equation (3.55). The infinitesimal variation of the Hamiltonian is

$\displaystyle\delta
H_{f}=i_{\bm{\xi}_{f}}\bm{\Omega}(\delta\Sigma;\delta\Sigma;\Sigma)$ (3.64)

which is integrable. The integrable flux is

$\displaystyle H_{f}=\int dud\Omega f(u,\Omega)\dot{\Sigma}^{2}(u,\Omega).$
(3.65)

This is the classical version of the operator $\mathcal{T}_{f}$. Similarly,
the generator of a Lorentz transformation parameterized by an antisymmetric
constant tensor $\omega^{\mu\nu}$ is

$\displaystyle\bm{\xi}_{\text{LT}}$ $\displaystyle=$
$\displaystyle\omega^{\mu\nu}(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})$
(3.66) $\displaystyle=$
$\displaystyle\omega^{\mu\nu}[-\frac{1}{2}un_{\mu\nu}\partial_{u}+\frac{1}{2}(u+r)n_{\mu\nu}\partial_{r}+(Y_{\mu\nu}^{A}-\frac{u}{r}\bar{m}_{\mu\nu}^{A})\partial_{A}]$

where the antisymmetric tensors $n_{\mu\nu}$ and $\bar{m}_{\mu\nu}$ are
defined in Appendix A.4. The boundary field $\Sigma$ is transformed as

$\displaystyle\delta_{\bm{\xi}_{\text{LT}}}\Sigma=\omega^{\mu\nu}[\frac{u}{d-2}\nabla_{A}Y_{\mu\nu}^{A}\dot{\Sigma}+Y_{\mu\nu}^{A}\nabla_{A}\Sigma+\frac{1}{2}\nabla_{A}Y_{\mu\nu}^{A}\Sigma]$
(3.67)

under Lorentz transformation. We note that the generator (3.66) reduces to

$\displaystyle\omega^{\mu\nu}[\frac{u}{d-2}\nabla_{A}Y^{A}_{\mu\nu}\partial_{u}+Y_{\mu\nu}^{A}\partial_{A}]$
(3.68)

at $\mathcal{I}^{+}$. This is a Carrollian diffeomorphism

$\displaystyle\bm{\xi}_{f,Y}=f(u,\Omega)\partial_{u}+Y^{A}(\Omega)\partial_{A}$
(3.69)

with

$\displaystyle f=\frac{u}{d-2}\omega^{\mu\nu}\nabla\cdot
Y_{\mu\nu}\quad\text{and}\quad Y^{A}=\omega^{\mu\nu}Y^{A}_{\mu\nu}.$ (3.70)

Therefore, it is natural to assume the variation of $\Sigma$ under SSR to be

$\displaystyle\delta_{\bm{\xi}_{Y}}\Sigma\equiv\Delta(Y;\Sigma;u,\Omega)=Y^{A}\nabla_{A}\Sigma+\frac{1}{2}\nabla_{A}Y^{A}\Sigma$
(3.71)

which could be reduced to (3.67) for Lorentz transformations555We need to
subtract a supertranslation which is generated by $\bm{\xi}_{f}$ with
$f=\frac{u}{d-2}\nabla\cdot Y$.. Note the transformation law (3.71) is
independent of the dimension $d$. However, one should notice that it is not
equal to the following transformation

$\bm{\xi}_{Y}(\Sigma)=Y^{A}\nabla_{A}\Sigma$ (3.72)

using boundary Lie derivative. To understand this transformation, we integrate
the infinitesimal transformation (3.71) and find the finite transformation

$\Sigma(u,\Omega)\to\Sigma^{\prime}(u,\Omega^{\prime})=\Big{|}\frac{\partial\theta^{\prime}}{\partial\theta}\Big{|}^{-1/2}\Sigma(u,\Omega)\,\,,$
(3.73)

where $|\frac{\partial\theta^{\prime}}{\partial\theta}|$ is the Jacobian under
finite superrotation

$(u,\theta^{A})\to(u,\theta^{\prime A}).$ (3.74)

Thus, the field $\Sigma$ is a scalar density under superrotations with weight
$w=\frac{1}{2}$ due to the result (3.73). In contrast, $\Sigma$ is a scalar
under general supertranslations. In general, we may define a scalar density
$\Lambda(u,\Omega)$ of weight $w$ under superrotation (3.74) by the following
transformation

$\Lambda(u,\Omega)\to\Lambda^{\prime}(u,\Omega^{\prime})=\Big{|}\frac{\partial\theta^{\prime}}{\partial\theta}\Big{|}^{-w}\Lambda(u,\Omega).$
(3.75)

Infinitesimally, this is

$\delta_{Y}\Lambda(u,\Omega)=Y^{A}(\Omega)\nabla_{A}\Lambda(u,\Omega)+w\nabla_{A}Y^{A}\Lambda(u,\Omega).$
(3.76)

Interestingly, we could find the integrable flux for a SSR labeled by
$Y^{A}(\Omega)$

$\displaystyle H_{Y}=\int dud\Omega\dot{\Sigma}\Delta(Y;\Sigma;u,\Omega).$
(3.77)

In quantum theory, the operators should be normal ordered and we find the same
flux operator

$\mathcal{M}_{Y}=\int dud\Omega:\dot{\Sigma}\Delta(Y;\Sigma;u,\Omega):\ =\int
dud\Omega Y^{A}(\Omega)M_{A}(u,\Omega).$ (3.78)

Before we close this subsection, we will shortly discuss the integrability
problem for general scalar densities $\Lambda(u,\Omega)$. Assuming the
boundary scalar theory is defined through the symplectic form

$\bm{\Omega}(\delta\Lambda;\delta\Lambda;\Lambda)=\int
dud\Omega\delta\Lambda\wedge\delta\dot{\Lambda}$ (3.79)

and the superrotation variation of the field $\Lambda$ is given by (3.76)
where $w$ is free. The corresponding Hamiltonian is

$\displaystyle\delta H_{Y}=\delta\int
dud\Omega\dot{\Lambda}(Y^{A}\nabla_{A}\Lambda+w\nabla_{A}Y^{A}\Lambda)+(1-2w)\int
dud\Omega\nabla_{A}Y^{A}\ \dot{\Lambda}\delta\Lambda.$ (3.80)

The first part of above equation on the right hand side is integrable while
the second part is not except for $w=\frac{1}{2}$.

###  3.5 Commutators and regularization of central charge

By computing the commutators

$\displaystyle[\mathcal{T}_{f},\Sigma(u,\Omega)]$
$\displaystyle=-if(u,\Omega)\dot{\Sigma}(u,\Omega),$ (3.81a)
$\displaystyle[\mathcal{M}_{Y},\Sigma(u,\Omega)]$
$\displaystyle=-i\Delta(Y;\Sigma;u,\Omega),$ (3.81b)

we confirm that $\mathcal{T}_{f}$ and $\mathcal{M}_{Y}$ are generators for the
Carrollian diffeomorphism given that $\Sigma$ is a scalar density of weight
$\frac{1}{2}$ under superrotations. More explicitly, $\mathcal{T}_{f}$
generates GSTs while $\mathcal{M}_{Y}$ generates SSRs. They should form Lie
algebra (2.8) up to central extension terms due to quantum corrections.
Indeed, we find

$\displaystyle\ [\mathcal{T}_{f_{1}},\mathcal{T}_{f_{2}}]$
$\displaystyle=C_{T}(f_{1},f_{2})+i\mathcal{T}_{f_{1}\dot{f}_{2}-f_{2}\dot{f}_{1}},$
(3.82a) $\displaystyle\ [\mathcal{T}_{f},\mathcal{M}_{Y}]$
$\displaystyle=-i\mathcal{T}_{Y^{A}\nabla_{A}f},$ (3.82b) $\displaystyle\
[\mathcal{M}_{Y},\mathcal{M}_{Z}]$ $\displaystyle=i\mathcal{M}_{[Y,Z]},$
(3.82c)

where the central charge is

$C_{T}(f_{1},f_{2})=-\frac{i}{48\pi}\delta^{(d-2)}(0)\mathcal{I}_{f_{1}\dddot{f_{2}}-f_{2}\dddot{f_{1}}}.$
(3.83)

The Dirac delta function on $S^{d-2}$ with the argument equalling to zero is
divergent666Readers may find more details in A.3.

$\displaystyle\delta^{(d-2)}(0)$ $\displaystyle=$
$\displaystyle\delta^{(d-2)}(\Omega-\Omega^{\prime})|_{\Omega^{\prime}=\Omega}=\sum_{\bm{\ell}}Y_{\bm{\ell}}^{*}(\Omega)Y_{\bm{\ell}}(\Omega)=\frac{\sum_{\bm{\ell}}1}{I(0)}$
(3.84)

where $\sum_{\bm{\ell}}1$ counts the number of all possible eigenstates on
$S^{d-2}$ and $I(0)$ is the area of the unit sphere. Therefore, Dirac delta
function $\delta^{(d-2)}(0)$ counts the number of states on $S^{d-2}$. Now we
will try to regularize the Dirac delta function. Notice that the eigenstates
on $S^{d-2}$ obey the Laplace equation

$\Delta_{S^{d-2}}Y_{\bm{\ell}}(\Omega)=-\ell_{m}(\ell_{m}+d-3))Y_{\bm{\ell}}(\Omega),$
(3.85)

where $Y_{\bm{\ell}}(\Omega)$ is the spherical harmonic function and
$\Delta_{S^{d-2}}=\nabla^{A}\nabla_{A}$ is the Laplace operator on $S^{d-2}$.
The Laplace operator is a natural elliptic operator which may be used to
define the spectral zeta function777Spectral zeta function has been used by
Hawking [37] to regularize the partition function of quantum fields in curved
spacetime. on $S^{d-2}$

$\zeta_{S^{d-2}}(s)\equiv\text{tr}^{\prime}(-\Delta_{S^{d-2}})^{-s}$ (3.86)

where $\text{tr}^{\prime}$ is to sum over all possible eigenstates except the
one with $\ell_{m}=0$. The eigenstates with zero eigenvalue is excluded
otherwise the summation would be divergent for any $\text{Re}(s)>0$. Using the
degeneracy of the eigenstates and the eigenvalue of the spherical harmonics,
we find

$\zeta_{S^{d-2}}(s)=\sum_{\ell=1}^{\infty}g(\ell,d-2)\left(\ell(\ell+d-3)\right)^{-s}.$
(3.87)

The spectral zeta function $\zeta_{S^{d-2}}(s)$ is convergent for sufficient
large $s$ and can be analytically continued to the complex plane except for a
single pole. Since the degeneracy $g(\ell,d-2)$ is a polynomial of $\ell$ of
degree $d-3$, we may always reduce $\zeta_{S^{d-2}}(s)$ to a finite summation
of the functions

$\displaystyle f(s;a,b,c)=\sum_{\ell=1}^{\infty}\ell^{-s+b}(\ell+a)^{-s+c}$
(3.88)

where $a,b,c$ are integers. In [38, 39], this function has been treated
carefully and expressed as a series of Riemann zeta functions (see Appendix
C). The number of eigenstates on $S^{d-2}$ may be regularized to

$\displaystyle\\#\,\text{of
eigenstates}=\sum_{\bm{\ell}}1=1+\zeta_{S^{d-2}}(0).$ (3.89)

In odd dimensions, the regularization method leads to 0. In even dimensions,
the results have been listed in table 1.

dimension $d$ | 4 | 6 | 8 | 10
---|---|---|---|---
$\\#\,$ of eigenstates | $\frac{1}{3}$ | $\frac{29}{90}$ | $\frac{1139}{3780}$ | $\frac{32377}{113400}$
Table 1: Regularization of the number of eigenstates for even dimensions

We may also define a heat kernel888For a review on heat kernel method, we
consult [40]. associated with the Laplacian

$\displaystyle K_{S^{d-2}}(\sigma)=\text{tr}\ e^{\sigma\Delta_{S^{d-2}}}$
(3.90)

where tr is sum over all possible eigenstates, including the one with
$\ell_{m}=0$. Therefore,

$\displaystyle
K_{S^{d-2}}(\sigma)=1+\sum_{\ell=1}^{\infty}g(\ell,d-2)e^{-\sigma\ell(\ell+d-3))}.$
(3.91)

The summation is convergent for any $\text{Re}(\sigma)>0$. It turns out that
the heat kernel has the pole structure near $\sigma=0$

$\displaystyle
K_{S^{d-2}}(\sigma)=\sum_{k=0}^{\infty}a_{k}(d)\sigma^{k-\frac{d-2}{2}}.$
(3.92)

There are only finite number of divergent terms in the expansion and we may
extract the constant term by regularization. Interestingly, the constant term
is always zero for odd dimensions. When $d$ is even, the asymptotic expansion
of the heat kernel can be found in the Appendix of [41]. The results matches
with the ones from spectral zeta function regularization.

##  4 Scalar field theory on Rindler horizon

###  4.1 Rindler horizon

In Minkowski spacetime $\mathbb{R}^{1,d-1}$, the Killing horizon associated
with the generator of Lorentz boost along the direction of $x^{d-1}$ is the
set of points where

$\displaystyle\bm{X}^{2}=0\quad\text{and}\quad\bm{X}\not=0\quad\text{with}\quad\bm{X}=x_{d-1}\partial_{t}+t\partial_{d-1}.$
(4.1)

The solution of null Killing vector condition $\bm{X}^{2}=0$ is the union of
the following two hypersurfaces

$\displaystyle\mathcal{H}^{-}(\bm{X})$ $\displaystyle=\\{t=-x^{d-1}\\},$
(4.2a) $\displaystyle\mathcal{H}^{+}(\bm{X})$ $\displaystyle=\\{t=x^{d-1}\\},$
(4.2b)

whose intersection is the bifurcation surface $B(\bm{X})$ with $\bm{X}=0$ and

$\displaystyle B(\bm{X})=\\{t=x^{d-1}=0\\}.$ (4.3)

This leads to the four Killing horizons associated with $\bm{X}$

$\displaystyle\mathcal{H}^{--}(\bm{X})$ $\displaystyle=\\{t=-x^{d-1},t<0\\},$
(4.4a) $\displaystyle\mathcal{H}^{-+}(\bm{X})$
$\displaystyle=\\{t=-x^{d-1},t>0\\},$ (4.4b)
$\displaystyle\mathcal{H}^{+-}(\bm{X})$ $\displaystyle=\\{t=x^{d-1},t<0\\},$
(4.4c) $\displaystyle\mathcal{H}^{++}(\bm{X})$
$\displaystyle=\\{t=x^{d-1},t>0\\}.$ (4.4d)

The above description is showed in figure 1.

$\mathcal{H}^{++}$$B$$\mathcal{H}^{--}$$\mathcal{H}^{-+}$$B$$\mathcal{H}^{+-}$right
Rindler wedgeleft Rindler wedge$t$$x^{d-1}$ Figure 1: Killing horizons and
Rindler wedge. There are four Killing horizons associated with the Lorentz
boost vector $\bm{X}$. The left and right Rindler wedge are separated by the
null hypersurfaces $\mathcal{H}^{\pm}$. There is a bifurcation surface $B$
which is the entangling surface between left and right region.

Without loss of generality, we will study the Carrollian field theory on the
Killing horizon $\mathcal{H}^{--}(\bm{X})$. This is one of the null boundary
of the Rindler spacetime which is also called Rindler horizon. We will write
it as $\mathcal{H}^{--}$ and omit $\bm{X}$ in the following. We note that it
is also possible to define Carrollian field theories on the null hypersurface
$\mathcal{H}^{-}=\mathcal{H}^{--}\cup\mathcal{H}^{-+}\cup B$. The interested
reader may find more details in Appendix B.3.

Rindler spacetime is a patch of Minkowski spacetime which may be obtained from
the coordinate transformation

$t=\rho\sinh\tau,\quad x^{d-1}=\rho\cosh\tau,\quad x_{\perp}^{A}=x^{A},\quad
A=1,2,\cdots,d-2.$ (4.5)

The Rindler time has an imaginary periodic $2\pi$

$\tau\sim\tau+2\pi i,$ (4.6)

whose inverse is the temperature of the thermal bath in Unruh effect [42]. The
metric of Rindler spacetime is

$ds_{\text{Rindler}}^{2}=-\rho^{2}d\tau^{2}+d\rho^{2}+\delta_{AB}dx_{\perp}^{A}dx_{\perp}^{B},\quad
0<\rho<\infty,\quad-\infty<\tau<\infty,\quad-\infty<x_{\perp}^{A}<\infty.$
(4.7)

We may define a new coordinate

$\lambda=\log\rho,\quad-\infty<\lambda<\infty,$ (4.8)

and the metric becomes

$ds_{\text{Rindler}}^{2}=e^{2\lambda}(-d\tau^{2}+d\lambda^{2})+\delta_{AB}dx_{\perp}^{A}dx_{\perp}^{B}.$
(4.9)

With the retarded time in Rindler spacetime

$u=\tau-\lambda,\quad-\infty<u<\infty,$ (4.10)

the metric of the Rindler spacetime may be written in retarded coordinates
$(u,\lambda,x_{\perp}^{A})$ as

$ds_{\text{Rindler}}^{2}=e^{2\lambda}(-du^{2}-2dud\lambda)+\delta_{AB}dx_{\perp}^{A}dx_{\perp}^{B}=-\rho^{2}du^{2}-2\rho
dud\rho+\delta_{AB}dx_{\perp}^{A}dx_{\perp}^{B}.$ (4.11)

To obtain the metric of the Carrollian manifold $\mathcal{H}^{--}$ from bulk
metric, one may choose a cutoff

$\displaystyle\rho=\epsilon$ (4.12)

such that the induced metric on the hypersurface

$\displaystyle\mathfrak{H}_{\epsilon}=\\{p\in\text{Rindler spacetime}|\
p=(u,\rho,x_{\perp}^{1},\cdots,x_{\perp}^{d-2})\ \text{with}\
\rho=\epsilon\\}$ (4.13)

is

$ds^{2}=-\epsilon^{2}du^{2}+\delta_{AB}dx_{\perp}^{A}dx_{\perp}^{B}.$ (4.14)

The Killing horizon $\mathcal{H}^{--}$ can be parameterized by $d-1$
coordinates $(u,x_{\perp}^{A})$ whose metric may be obtained by taking the
limit $\epsilon\to 0$ for (4.14)

$ds^{2}_{\mathcal{H}^{--}}=\delta_{AB}dx_{\perp}^{A}dx_{\perp}^{B}.$ (4.15)

Therefore, the Killing horizon $\mathcal{H}^{--}$ is a Carrollian manifold
with a degenerate metric (4.15) as well as a null vector
$\bm{\chi}=\partial_{u}$. The integration measure on $\mathcal{H}^{--}$ is

$\displaystyle\int dud\bm{x}_{\perp}$ (4.16)

while in this case we have

$\int d\bm{x}_{\perp}=\prod_{j=1}^{d-2}\int_{-\infty}^{\infty}dx_{\perp}^{j}.$
(4.17)

Similarly, we may define an advanced coordinate

$v=\tau+\lambda,$ (4.18)

and the metric of the Rindler spacetime in advanced coordinates
$(v,\lambda,\theta^{A})$ is

$ds^{2}_{\text{Rindler}}=e^{2\lambda}(-dv^{2}+2dvd\lambda)+\delta_{AB}dx_{\perp}^{A}dx_{\perp}^{B}=-\rho^{2}dv^{2}+2\rho
dvd\rho+\delta_{AB}dx_{\perp}^{A}dx_{\perp}^{B}.$ (4.19)

By taking the limit $\lambda\to-\infty$ while keeping the advanced time $v$
finite, we find the metric of the Carrollian manifold $\mathcal{H}^{++}$

$ds^{2}_{\mathcal{H}^{++}}=\delta_{AB}dx_{\perp}^{A}dx_{\perp}^{B}.$ (4.20)

The null vector of the Carrollian manifold $\mathcal{H}^{++}$ is
$\partial_{v}$. We will define the limits $\lim{}_{-}$ and $\lim{}_{+}$ to
reduce the quantities from bulk hypersurface $\mathfrak{H}_{\epsilon}$ to
$\mathcal{H}^{--}$ and $\mathcal{H}^{++}$, respectively

$\displaystyle\lim{}_{-}$ $\displaystyle=\lim_{\lambda\to-\infty,\ u\
\text{finite}}=\lim_{\rho\to 0,\ u\ \text{finite}},$ (4.21a)
$\displaystyle\lim{}_{+}$ $\displaystyle=\lim_{\lambda\to-\infty,\ v\
\text{finite}}=\lim_{\rho\to 0,\ v\ \text{finite}}.$ (4.21b)

One can see these two limits in figure 2. An attentive reader may have the
confusion that why we should reduce the theory from Rindler spacetime to
$\mathcal{H}^{--}$. It seems equivalent to reduce the theory straightforwardly
from the theory in Minkowski spacetime. Indeed, one could do so, as has been
shown in Appendix B.2, and what one actually gets is the theory defined on
$\mathcal{H}^{-}$. One should integrate out degrees of freedom on
$\mathcal{H}^{-+}\cup B$ to find the final results.

$i^{+}$$\mathcal{I}^{+}$$i^{0}$$\mathcal{I}^{-}$$i^{-}$const. $r$ slices
$\mathfrak{H}_{r}$$r$ increase,while $u$ finitewhile $v$ finite$r$
increase,$u$$v$ (a)

$\mathcal{H}^{++}$$B$$\mathcal{H}^{--}$const. $\rho$ slices
$\mathfrak{H}_{\rho}$$\rho$ decrease,while $u$ finite$\rho$ decrease,while $v$
finite$t$$x^{d-1}$

(b)

Figure 2: The figure (a) on the left hand side is the Penrose diagram of
Minkowski spacetime $\mathbb{R}^{1,d-1}$. We may reduce bulk quantities to
future/past null infinity $\mathcal{I}^{+}/\mathcal{I}^{-}$ by choosing a
series of constant $r$ slices $\mathfrak{H}_{r}$ and then taking corresponding
limits. Figure (b) is the right Rindler wedge, which is a part of Minkowski
spacetime. The null boundary of Rindler spacetime is denoted as
$\mathcal{H}^{--}$ and $\mathcal{H}^{++}$. We may define Carrollian field
theory by reducing the bulk theory to the null hypersurfaces. In the figure,
we have plotted constant $\rho$ slices which are hyperbolic curves in the
spacetime diagram.

###  4.2 From bulk to boundary

In this section, we will construct Carrollian scalar field theory at the
Killing horizon $\mathcal{H}^{--}$ from bulk reduction in general dimensions.
We still consider a general scalar theory with the action (3.14). The
potential could include a mass term since massive particles can also arrive at
$\mathcal{H}^{--}$.

$V(\Phi)=\sum_{k=2}^{\infty}\frac{\lambda_{k}}{k}\Phi^{k},\quad\lambda_{2}=m^{2}.$
(4.22)

By embedding the theory in Minkowski spacetime, there is no need to impose any
artificial boundary condition at $\mathcal{H}^{--}$ since they should be
determined by the whole system. Instead, a scalar field should be finite at
any position of Minkowski spacetime. Therefore, we just acquire the finiteness
of $\Phi$ at $\mathcal{H}^{--}$. Using Taylor expansion, we assume the series
expansion of $\Phi$ near $\mathcal{H}^{--}$

$\Phi(t,\bm{x})=\Sigma(u,\bm{x}_{\perp})+\sum_{k=1}^{\infty}\Sigma^{(k)}(u,\bm{x}_{\perp})\rho^{k}$
(4.23)

for the bulk scalar field. The pre-symplectic form of the theory is

$\displaystyle\bm{\Theta}(\delta\Phi;\Phi)=-\int_{\mathfrak{H}_{\rho}}(d^{d-1}x)_{\mu}\partial^{\mu}\Phi\delta\Phi$
(4.24)

where

$\displaystyle(d^{d-1}x)_{\mu}=\frac{1}{(d-1)!}\epsilon_{\mu\nu_{1}\cdots\nu_{d-1}}dx^{\nu_{1}}\wedge\cdots\wedge
dx^{\nu_{d-1}}=-\rho\hat{m}_{\mu}dud\bm{x}_{\perp}.$ (4.25)

The normal covector $\hat{m}_{\mu}$ of the hypersurface $\mathfrak{H}_{\rho}$
is $d\rho$ which implies

$\hat{m}_{\rho}=1,\quad\hat{m}_{u}=\hat{m}_{A}=0,$ (4.26)

and we have normalized it to 1

$\hat{m}\cdot\hat{m}=1.$ (4.27)

The symplectic form of the theory is

$\bm{\Omega}(\delta\Phi;\delta\Phi;\Phi)=-\int_{\mathfrak{H}_{\rho}}(d^{d-1}x)_{\mu}\partial^{\mu}\delta\Phi\wedge\delta\Phi.$
(4.28)

With the fall-off condition (4.23), the symplectic form becomes999Similar to
the constraint equations (3.22) at null infinity, we may also use the
differential operator
$\displaystyle\partial_{\mu}\partial^{\mu}=\partial_{\rho}^{2}+\rho^{-1}\partial_{\rho}-2\rho^{-1}\partial_{u}\partial_{\rho}+\partial_{A}\partial^{A},$
(4.29) and reduce the bulk EOM to
$\displaystyle\Sigma^{(1)}-2\dot{\Sigma}^{(1)}=0,$ (4.30a)
$\displaystyle(k+2)^{2}\Sigma^{(k+2)}-2(k+2)\dot{\Sigma}^{(k+2)}+\nabla_{A}\nabla^{A}\Sigma^{(k)}-\sum_{n=2}^{\infty}\sum_{k_{1},\cdots,k_{n-1}\geq
0}^{\sum_{j=1}^{n-1}k_{j}=k}\lambda_{n}\Sigma^{(k_{1})}\cdots\Sigma^{(k_{n-1})}=0,\quad
k\geq 0.$ (4.30b) Therefore, only the leading mode $\Sigma$ is free from
dynamical equations. All the subleading modes $\Sigma^{(k)},\ k\geq 1$ are
determined by the equations up to initial data.

$\displaystyle\bm{\Omega}(\delta\Sigma;\delta\Sigma;\Sigma)=\int
dud\bm{x}_{\perp}\delta\Sigma\wedge\delta\dot{\Sigma}$ (4.31)

at $\mathcal{H}^{--}$. This is identified as the symplectic form of the
Carrollian scalar field theory at $\mathcal{H}^{--}$. Interestingly, this is
formally the same as the one at $\mathcal{I}^{+}$ though the Carrollian
manifold and the boundary field $\Sigma$ are not identical. We may find the
following commutators for the field $\Sigma$

$\displaystyle[\Sigma(u,\bm{x}_{\perp}),\Sigma(u^{\prime},\bm{x}_{\perp}^{\prime})]$
$\displaystyle=\frac{i}{2}\alpha(u-u^{\prime})\delta^{(d-2)}(\bm{x}_{\perp}-\bm{x}_{\perp}^{\prime}),$
(4.32a)
$\displaystyle[\Sigma(u,\bm{x}_{\perp}),\dot{\Sigma}(u^{\prime},\bm{x}_{\perp}^{\prime})]$
$\displaystyle=\frac{i}{2}\delta(u-u^{\prime})\delta^{(d-2)}(\bm{x}_{\perp}-\bm{x}_{\perp}^{\prime}),$
(4.32b)
$\displaystyle[\dot{\Sigma}(u,\bm{x}_{\perp}),\dot{\Sigma}(u^{\prime},\bm{x}_{\perp}^{\prime})]$
$\displaystyle=\frac{i}{2}\delta^{\prime}(u-u^{\prime})\delta^{(d-2)}(\bm{x}_{\perp}-\bm{x}_{\perp}^{\prime}).$
(4.32c)

The field $\Sigma$ may be expanded in a set of complete basis of
$\mathcal{H}^{--}$

$\displaystyle\Sigma(u,\bm{x}_{\perp})=\int_{0}^{\infty}\frac{d\omega}{\sqrt{4\pi\omega}}\int\frac{d\bm{k}_{\perp}}{\sqrt{(2\pi)^{d-2}}}[a^{\rm
R}_{\omega,\bm{k}_{\perp}}e^{-i\omega
u+i\bm{k}_{\perp}\cdot\bm{x}_{\perp}}+a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}e^{i\omega
u-i\bm{k}_{\perp}\cdot\bm{x}_{\perp}}]$ (4.33)

with the commutators

$\displaystyle[a^{\rm R}_{\omega,\bm{k}_{\perp}},a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}]=[a_{\omega,\bm{k}_{\perp}}^{\rm
R\dagger},a_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}^{\rm R\dagger}]=0,$
(4.34a) $\displaystyle[a^{\rm
R}_{\omega,\bm{k}_{\perp}},a_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}^{\rm
R\dagger}]=\delta(\omega-\omega^{\prime})\delta^{(d-2)}(\bm{k}_{\perp}-\bm{k}_{\perp}^{\prime}).$
(4.34b)

In the expansion (4.33), the functions $e^{-i\omega
u+i\bm{k}\cdot\bm{x}_{\perp}}$ and their complex conjugates form a set of
complete basis of $\mathcal{H}^{--}$. We have also checked the mode expansion
(4.33) using canonical quantization method in Appendix B.3. With the canonical
quantization, we may define the vacuum state $|0\rangle_{\text{R}}$ through

$a^{\rm R}_{\omega,\bm{\ell}}|0\rangle_{\text{R}}=0.$ (4.35)

We have added a subscript R to denote the vacuum state of Rindler spacetime.
It should be distinguished with the vacuum state of Minkowski spacetime
$|0\rangle$ defined in previous section. Therefore, we find the following
correlators

$\displaystyle{}_{\text{R}}\langle
0|\Sigma(u,\bm{x}_{\perp})\Sigma(u^{\prime},\bm{x}_{\perp}^{\prime})|0\rangle_{\text{R}}$
$\displaystyle=\beta(u-u^{\prime})\delta^{(d-2)}(\bm{x}_{\perp}-\bm{x}_{\perp}^{\prime}),$
(4.36a) $\displaystyle{}_{\text{R}}\langle
0|\Sigma(u,\bm{x}_{\perp})\dot{\Sigma}(u^{\prime},\bm{x}_{\perp}^{\prime})|0\rangle_{\text{R}}$
$\displaystyle=\frac{1}{4\pi(u-u^{\prime}-i\epsilon)}\delta^{(d-2)}(\bm{x}_{\perp}-\bm{x}_{\perp}^{\prime}),$
(4.36b) $\displaystyle{}_{\text{R}}\langle
0|\dot{\Sigma}(u,\bm{x}_{\perp})\Sigma(u^{\prime},\bm{x}_{\perp}^{\prime})|0\rangle_{\text{R}}$
$\displaystyle=-\frac{1}{4\pi(u-u^{\prime}-i\epsilon)}\delta^{(d-2)}(\bm{x}_{\perp}-\bm{x}_{\perp}^{\prime}),$
(4.36c) $\displaystyle{}_{\text{R}}\langle
0|\dot{\Sigma}(u,\bm{x}_{\perp})\dot{\Sigma}(u^{\prime},\bm{x}_{\perp}^{\prime})|0\rangle_{\text{R}}$
$\displaystyle=-\frac{1}{4\pi(u-u^{\prime}-i\epsilon)^{2}}\delta^{(d-2)}(\bm{x}_{\perp}-\bm{x}_{\perp}^{\prime}).$
(4.36d)

###  4.3 Flux operators for Carrollian diffeomorphism

In this section, we will apply the formula (2.19) to the Killing horizon and
calculate the flux operators for Carrollian diffeomorphism. To understand the
physical meaning of the flux operators, we may study the flux operators
associated with Killing vectors at first. Rindler spacetime is locally flat
which has $d(d+1)/2$ Killing vectors inherited from Minkowski spacetime.
Therefore, we may use these generators to discuss the symmetries of Rindler
spacetime. The Killing vectors are listed below which are expressed in terms
of the quantities in retarded coordinates $(u,\rho,\bm{x}_{\perp})$

$\displaystyle\partial_{t}=e^{u}\partial_{u}-\frac{1}{2}(\rho
e^{u}-\rho^{-1}e^{-u})\partial_{\rho},$ (4.37a)
$\displaystyle\partial_{d-1}=-e^{u}\partial_{u}+\cosh\tau\partial_{\rho}=-e^{u}\partial_{u}+\frac{1}{2}(\rho
e^{u}+\rho^{-1}e^{-u})\partial_{\rho},$ (4.37b)
$\displaystyle\partial_{A}=\frac{\partial}{\partial x_{\perp}^{A}},$ (4.37c)
$\displaystyle t\partial_{d-1}+x_{d-1}\partial_{t}=\partial_{u},$ (4.37d)
$\displaystyle
t\partial_{A}+x_{A}\partial_{t}=e^{u}(x_{\perp})_{A}\partial_{u}-\frac{1}{2}(x_{\perp})_{A}(e^{u}\rho-e^{-u}\rho^{-1})\partial_{\rho}+\frac{1}{2}(e^{u}\rho^{2}-e^{-u})\partial_{A},$
(4.37e) $\displaystyle
x_{d-1}\partial_{A}-x_{A}\partial_{d-1}=-e^{u}(x_{\perp})_{A}\partial_{u}+\frac{1}{2}(x_{\perp})_{A}(e^{u}\rho+e^{-u}\rho^{-1})\partial_{\rho}+\frac{1}{2}(e^{u}\rho^{2}+e^{-u})\partial_{A},$
(4.37f) $\displaystyle
x_{A}\partial_{B}-x_{B}\partial_{A}=(x_{\perp})_{A}\partial_{B}-(x_{\perp})_{B}\partial_{A}.$
(4.37g)

In the light cone coordinates

$\displaystyle x^{\pm}=t\pm x^{d-1}\quad\Leftrightarrow\quad
x_{\pm}=-\frac{1}{2}x^{\mp},$ (4.38)

we find the following generators

$\displaystyle\bm{\xi}_{-}$
$\displaystyle=\partial_{-}=\frac{1}{2}(\partial_{t}-\partial_{d-1})=e^{u}\partial_{u}-\frac{1}{2}\rho
e^{u}\partial_{\rho}\sim e^{u}\partial_{u},$ (4.39a)
$\displaystyle\bm{\xi}_{+}$
$\displaystyle=\partial_{+}=\frac{1}{2}(\partial_{t}+\partial_{d-1})=\frac{e^{-u}}{2\rho}\partial_{\rho},$
(4.39b) $\displaystyle\bm{\xi}_{A}$ $\displaystyle=\partial_{A},$ (4.39c)
$\displaystyle\bm{\xi}_{+-}$
$\displaystyle=2(x_{+}\partial_{-}-x_{-}\partial_{+})=\partial_{u},$ (4.39d)
$\displaystyle\bm{\xi}_{+A}$
$\displaystyle=(x_{+}\partial_{A}-x_{A}\partial_{+})=\frac{1}{2}e^{-u}\partial_{A}-x_{A}\frac{e^{-u}}{2\rho}\partial_{\rho},$
(4.39e) $\displaystyle\bm{\xi}_{-A}$
$\displaystyle=(x_{-}\partial_{A}-x_{A}\partial_{-})=-x_{A}e^{u}\partial_{u}+\frac{1}{2}\rho
x_{A}e^{u}\partial_{\rho}-\frac{1}{2}\rho^{2}e^{u}\partial_{A}\sim-
x_{A}e^{u}\partial_{u},$ (4.39f) $\displaystyle\bm{\xi}_{AB}$
$\displaystyle=x_{A}\partial_{B}-x_{B}\partial_{A}.$ (4.39g)

The vectors $\bm{\xi}_{+}$ and $\bm{\xi}_{+A}$ blow up near $\mathcal{H}^{--}$
and they do not generate Carrollian diffeomorphism. To be more precise, we
note the field $\Phi$ transforms along the direction of the Killing vectors as

$\delta_{\bm{\xi}}\Phi=\xi^{\mu}\partial_{\mu}\Phi.$ (4.40)

Note that the variation of the field $\Phi$ along the direction of
$\bm{\xi}_{+}$ or $\bm{\xi}_{+A}$ blows up and violates the fall-off condition
(4.23). Therefore, we should exclude the generators $\bm{\xi}_{+}$ and
$\bm{\xi}_{+A}$. On the other hand, the variations of the field $\Phi$ along
the directions
$\bm{\xi}_{-},\bm{\xi}_{A},\bm{\xi}_{+-},\bm{\xi}_{-A},\bm{\xi}_{AB}$ preserve
the fall-off conditions and we find the following variations for the boundary
field $\Sigma$

$\displaystyle\delta_{\bm{\xi}_{-}}\Sigma$ $\displaystyle=e^{u}\dot{\Sigma},$
(4.41a) $\displaystyle\delta_{b^{A}\bm{\xi}_{A}}\Sigma$
$\displaystyle=b^{A}\partial_{A}\Sigma,$ (4.41b)
$\displaystyle\delta_{\bm{\xi}_{+-}}\Sigma$ $\displaystyle=\dot{\Sigma},$
(4.41c) $\displaystyle\delta_{c^{A}\bm{\xi}_{-A}}\Sigma$
$\displaystyle=-c^{A}x_{A}e^{u}\dot{\Sigma},$ (4.41d)
$\displaystyle\delta_{\omega^{AB}\bm{\xi}_{AB}}\Sigma$
$\displaystyle=\omega^{AB}(x_{A}\partial_{B}-x_{B}\partial_{A})\Sigma.$
(4.41e)

We have inserted constant vectors $b^{A},c^{A}$ or antisymmetric tensor
$\omega^{AB}$ to balance the indices. From the explicit form of the
generators, we conclude that
$\bm{\xi}_{-},\bm{\xi}_{A},\bm{\xi}_{+-},\bm{\xi}_{-A},\bm{\xi}_{AB}$ could
reduce to special choice of Carrollian diffeomorphism at the boundary
$\mathcal{H}^{--}$

$\displaystyle\bm{\xi}_{-}$ $\displaystyle=\bm{\xi}_{f,Y}\quad\text{with}\quad
f=e^{u},\ Y^{A}=0,$ (4.42a) $\displaystyle b^{A}\bm{\xi}_{A}$
$\displaystyle=\bm{\xi}_{f,Y}\quad\text{with}\quad f=0,\ Y^{A}=b^{A},$ (4.42b)
$\displaystyle\bm{\xi}_{+-}$
$\displaystyle=\bm{\xi}_{f,Y}\quad\text{with}\quad f=1,\ Y^{A}=0,$ (4.42c)
$\displaystyle c^{A}\bm{\xi}_{-A}$
$\displaystyle=\bm{\xi}_{f,Y}\quad\text{with}\quad f=-c^{A}x_{A}e^{u},\
Y^{A}=0,$ (4.42d) $\displaystyle\omega^{AB}\bm{\xi}_{AB}$
$\displaystyle=\bm{\xi}_{f,Y}\quad\text{with}\quad f=0,\
Y^{A}=2\omega^{CA}x_{C}.$ (4.42e)

All the transformations (4.41) could be unified by

$\displaystyle\delta_{\bm{\xi}_{f,Y}}\Sigma=f(u,\bm{x}_{\perp})\dot{\Sigma}+Y^{A}(\bm{x}_{\perp})\partial_{A}\Sigma+w\nabla_{A}Y^{A}\
\Sigma$ (4.43)

with the special choice of the $f$ and $Y^{A}$ listed in (4.42). Note that the
last term is always zero due to the vanishing of the term $\nabla_{A}Y^{A}$
for all the choices of $Y^{A}$ in (4.42). Therefore, The constant $w$ is free
at this moment. Interestingly, the function $f(u,\bm{x}_{\perp})$ is
automatically time dependent from (4.42a) and (4.42d). This fact strongly
indicates that one should consider time dependent supertranslations. Another
noticeable fact is that the translation along Rindler time is a Lorentz boost
in Minkowski spacetime. Therefore, the corresponding flux should be
interpreted as center of mass flux in Minkowski spacetime. Similarly, a
superrotation flux with $Y^{A}=b^{A}$ should be interpreted as a momentum flux
in Minkowski spacetime. Nevertheless, we can construct the leaky fluxes across
$\mathcal{H}^{--}$ for the Killing vectors (4.42) by

$\displaystyle\mathcal{F}_{\bm{\xi}}\equiv\mathcal{F}[j_{\bm{\xi}}]=-\lim{}_{-}\int_{\mathfrak{H}_{\rho}}(d^{d-1}x)_{\mu}j^{\mu}_{\bm{\xi}}=-\lim{}_{-}\int_{\mathfrak{H}_{\rho}}dud\bm{x}_{\perp}\
\rho T^{\rho}_{\ \mu}\xi^{\mu},$ (4.44)

where

$\displaystyle j^{\mu}_{\bm{\xi}}=T^{\mu\nu}\xi_{\nu}$ (4.45)

and $T^{\mu\nu}$ is the stress tensor given by (3.37) and $\bm{\xi}$ is a
Killing vector of Rindler spacetime. Given the fall-off conditions (4.23), we
find

$\displaystyle\partial_{u}\Phi$
$\displaystyle=\sum_{k=0}^{\infty}\dot{\Sigma}^{(k)}\rho^{k},$ (4.46a)
$\displaystyle\partial_{\rho}\Phi$
$\displaystyle=\sum_{k=0}^{\infty}(k+1)\Sigma^{(k+1)}\rho^{k},$ (4.46b)
$\displaystyle\partial_{A}\Phi$
$\displaystyle=\sum_{k=0}^{\infty}\partial_{A}\Sigma^{(k)}\rho^{k},$ (4.46c)
$\displaystyle\partial^{u}\Phi$
$\displaystyle=-\rho^{-1}\partial_{\rho}\Phi=-\sum_{k=0}^{\infty}(k+1)\Sigma^{(k+1)}\rho^{k-1},$
(4.46d) $\displaystyle\partial^{\rho}\Phi$
$\displaystyle=-\rho^{-1}\partial_{u}\Phi+\partial_{\rho}\Phi=-\sum_{k=0}^{\infty}\dot{\Sigma}^{(k)}\rho^{k-1}+\sum_{k=0}^{\infty}(k+1)\Sigma^{(k+1)}\rho^{k},$
(4.46e) $\displaystyle\partial^{A}\Phi$
$\displaystyle=\sum_{k=0}^{\infty}\partial^{A}\Sigma^{(k)}\rho^{k}$ (4.46f)

and

$\displaystyle\partial^{\rho}\Phi\partial_{u}\Phi$
$\displaystyle=-\rho^{-1}\dot{\Sigma}^{2}+\cdots,$ (4.47a)
$\displaystyle\partial^{\rho}\Phi\partial_{\rho}\Phi$
$\displaystyle=-\rho^{-1}\dot{\Sigma}\Sigma^{(1)}+\cdots,$ (4.47b)
$\displaystyle\partial^{\rho}\Phi\partial_{A}\Phi$
$\displaystyle=-\rho^{-1}\dot{\Sigma}\nabla_{A}\Sigma+\cdots,$ (4.47c)
$\displaystyle\partial_{\mu}\Phi\partial^{\mu}\Phi$
$\displaystyle=-2\rho^{-1}\Sigma^{(1)}\dot{\Sigma}+\cdots.$ (4.47d)

To compute (4.44), we need the following fall-off behaviour of stress tensor

$\displaystyle T^{\rho}_{\ u}$
$\displaystyle=-\rho^{-1}\dot{\Sigma}^{2}+\mathcal{O}(\rho^{0}),$ (4.48a)
$\displaystyle T^{\rho}_{\ \rho}$
$\displaystyle=t^{\rho}_{\rho}+\mathcal{O}(\rho),$ (4.48b) $\displaystyle
T^{\rho}_{\ A}$
$\displaystyle=-\rho^{-1}\dot{\Sigma}\nabla_{A}\Sigma+\mathcal{O}(\rho^{0}).$
(4.48c)

where

$\displaystyle
t^{\rho}_{\rho}=\frac{1}{2}(\Sigma^{(1)})^{2}-\frac{1}{2}\partial_{A}\Sigma\partial^{A}\Sigma-\sum_{k=2}^{\infty}\frac{\lambda_{k}}{k}\Sigma^{k}.$
(4.49)

Therefore, we find the fluxes corresponding to the Killing vectors (4.39)

$\displaystyle\mathcal{F}_{\bm{\xi}_{-}}$ $\displaystyle=\int
dud\bm{x}_{\perp}e^{u}\dot{\Sigma}^{2},$ (4.50a)
$\displaystyle\mathcal{F}_{\bm{\xi}_{+}}$ $\displaystyle=-\frac{1}{2}\int
dud\bm{x}_{\perp}e^{-u}t^{\rho}_{\rho},$ (4.50b)
$\displaystyle\mathcal{F}_{b^{A}\bm{\xi}_{A}}$ $\displaystyle=\int
dud\bm{x}_{\perp}b^{A}\dot{\Sigma}\nabla^{A}\Sigma,$ (4.50c)
$\displaystyle\mathcal{F}_{\bm{\xi}_{+-}}$ $\displaystyle=\int
dud\bm{x}_{\perp}\dot{\Sigma}^{2},$ (4.50d)
$\displaystyle\mathcal{F}_{\tilde{c}^{A}\bm{\xi}_{+A}}$
$\displaystyle=\frac{1}{2}\int
dud\bm{x}_{\perp}e^{-u}\tilde{c}^{A}[\dot{\Sigma}\nabla_{A}\Sigma+x_{A}t^{\rho}_{\rho}],$
(4.50e) $\displaystyle\mathcal{F}_{c^{A}\bm{\xi}_{-A}}$ $\displaystyle=\int
dud\bm{x}_{\perp}c^{A}x_{A}e^{u}\dot{\Sigma}^{2},$ (4.50f)
$\displaystyle\mathcal{F}_{\omega^{AB}\bm{\xi}_{AB}}$ $\displaystyle=\int
dud\bm{x}_{\perp}2\omega^{CA}x_{C}\dot{\Sigma}\nabla_{A}\Sigma.$ (4.50g)

The fluxes
$\mathcal{F}_{\bm{\xi}_{-}},\mathcal{F}_{\bm{\xi}_{+-}},\mathcal{F}_{c^{A}\bm{\xi}_{-A}}$
are exactly the previous $\mathcal{T}_{f}$ with $f=e^{u},\ 1$ and
$c^{A}x_{A}e^{u}$, respectively. Similarly, the flux
$\mathcal{F}_{b^{A}\bm{\xi}_{A}},\mathcal{F}_{\omega^{AB}\bm{\xi}_{AB}}$ are
exactly the previous $\mathcal{M}_{Y}$ with $Y^{A}=b^{A}$ and
$2\omega^{CA}x_{C}$ respectively. On the other hand, the fluxes
$\mathcal{F}_{\bm{\xi}_{+}}$ and $\mathcal{F}_{\tilde{c}^{A}\bm{\xi}_{+A}}$
are finite but not related to Carrollian diffeomorphism.

Now we can construct the flux operators associated with Carrollian
diffeomorphism. For any GST, the flux is

$\displaystyle\mathcal{F}_{\bm{\xi}_{f}}=\int
dud\bm{x}_{\perp}f(u,\bm{x}_{\perp})\dot{\Sigma}^{2}(u,\bm{x}_{\perp}).$
(4.51)

We may uplift to operators using normal ordering

$\mathcal{T}_{f}=\int
dud\bm{x}_{\perp}f(u,\bm{x}_{\perp}):\dot{\Sigma}^{2}(u,\bm{x}_{\perp}):.$
(4.52)

Similarly, we can find the flux associated with any SSR

$\mathcal{F}_{\bm{\xi}_{Y}}=\int
dud\bm{x}_{\perp}Y^{A}(\bm{x}_{\perp})\dot{\Sigma}\nabla_{A}\Sigma.$ (4.53)

The corresponding flux operator is

$\displaystyle\mathcal{M}_{Y}=\int
dud\bm{x}_{\perp}Y^{A}(\bm{x}_{\perp})M_{A}(u,\bm{x}_{\perp}),$ (4.54)

where we have chosen

$M_{A}(u,\bm{x}_{\perp})=\frac{1}{2}:(\dot{\Sigma}\nabla_{A}\Sigma-\Sigma\nabla_{A}\dot{\Sigma}):.$
(4.55)

We should mention that one could also use Hamilton equation (3.55) to obtain
the same results for Carrollian diffeomorphism. For a GST generated by

$\bm{\xi}_{f}=f(u,\bm{x}_{\perp})\partial_{u}\quad\text{with}\quad\delta_{f}\Sigma(u,\bm{x}_{\perp})=f(u,\bm{x}_{\perp})\dot{\Sigma}(u,\bm{x}_{\perp}),$
(4.56)

the corresponding Hamiltonian is exactly the flux operator (4.52) after normal
ordering. For a SSR generated by

$\bm{\xi}_{Y}=Y^{A}(\bm{x}_{\perp})\partial_{A}\quad\text{with}\quad\delta_{Y}\Sigma(u,\bm{x}_{\perp})=Y^{A}(\bm{x}_{\perp})\nabla_{A}\Sigma(u,\bm{x}_{\perp})+w\nabla_{A}Y^{A}(\bm{x}_{\perp})\
\Sigma(u,\bm{x}_{\perp}),$ (4.57)

we find an integrable flux only for $w=\frac{1}{2}$ which is again exactly the
flux operator $\eqref{fluxsr}$. Note the constant $w$ is absent from the
variation

$\bm{\xi}_{Y}(\Sigma)=Y^{A}\nabla_{A}\Sigma,$ (4.58)

we already meet with this problem at future null infinity. In that case, we
can fix the value $w=\frac{1}{2}$ by considering Lorentz transformations in
the bulk. However, one cannot fix it in the Killing horizon $\mathcal{H}^{--}$
as has been shown in (4.43). Nevertheless, the value $w$ should be
$\frac{1}{2}$ due to the following reasons.

  1. 1.

The Hamiltonian for superrotation is integrable only for $w=\frac{1}{2}$.

  2. 2.

The Dirac delta function
$\delta^{(d-2)}(\bm{x}_{\perp}-\bm{x}_{\perp}^{\prime})$ is a scalar density
of weight $1$. Therefore, the consistency for the commutator (4.32) has
already fixed $w=\frac{1}{2}$.

It turns out that the commutators between $\mathcal{T}_{f}$ and
$\mathcal{M}_{Y}$ are exactly the same as (3.82). The central charge (3.83) is
also divergent. However, in this case, we find

$\displaystyle\delta^{(d-2)}(\bm{0})$ $\displaystyle=$
$\displaystyle\delta^{(d-2)}(\bm{x}_{\perp}-\bm{x}_{\perp}^{\prime})|_{\bm{x}_{\perp}^{\prime}=\bm{x}_{\perp}}=\frac{1}{(2\pi)^{d-2}}\int
d\bm{k}_{\perp}e^{i\bm{k}_{\perp}(\bm{x}_{\perp}-\bm{x}_{\perp}^{\prime})}|_{\bm{x}_{\perp}^{\prime}=\bm{x}_{\perp}}=\frac{1}{(2\pi)^{d-2}}\int
d\bm{k}_{\perp}.$

It shows that the central charge also counts the number of eigenstates in the
transverse direction. To regularize it, we introduce a UV cutoff $2\pi/a$
where $a$ has the dimension of a microscopic length scale. In lattice theory,
$a$ is the distance of adjacent lattice points. Therefore, the Dirac function
becomes

$\displaystyle\delta^{(d-2)}(\bm{0})=\frac{1}{(2\pi)^{d-2}}\int_{0}^{2\pi/a}d\bm{k}_{\perp}=\frac{1}{a^{d-2}}=\frac{1}{V_{a}}.$
(4.59)

At the last step, we have used the fact that $a^{d-2}$ is the volume of the
smallest cell of the lattice,

$V_{a}=a^{d-2}.$ (4.60)

Therefore, the central charge should be proportional to the density of states
in the transverse direction, which has exactly the same physical meaning with
the one at $\mathcal{I}^{+}$. We may also use the heat kernel method to define

$\displaystyle K(\sigma)=\text{tr}\ e^{\sigma\Delta}$ (4.61)

with $\Delta$ being the Laplacian operator in the transverse direction. The
spectrum of the Laplacian operator is $-\bm{k}_{\perp}^{2}$ which is
continuous. Then the heat kernel becomes

$\displaystyle K(\sigma)=\frac{1}{(2\pi)^{d-2}}\int
d^{d-2}\bm{k}_{\perp}e^{-\sigma\bm{k}_{\perp}^{2}}=\frac{1}{(4\pi\sigma)^{(d-2)/2}}.$
(4.62)

As $\sigma\to 0$, the heat kernel approaches the Dirac delta function
$\delta^{(d-2)}(\bm{0})$. Since the dimension of $\sigma$ is length square, we
may introduce a UV cutoff $a$ with dimension length and identify

$\sigma=\frac{a^{2}}{4\pi}.$ (4.63)

Combining (4.62) and (4.63), we find the same result as (4.59).

Now we choose the basis for the supertranslation function

$f_{\omega,\bm{k}_{\perp}}(u,\bm{x}_{\perp})=e^{-i\omega
u}e^{i\bm{k}_{\perp}\cdot\bm{x}_{\perp}},$ (4.64)

and define the Fourier transformation of the flux operator

$\mathcal{T}_{\omega,\bm{k}_{\perp}}=\int
dud\bm{x}_{\perp}f_{\omega,\bm{k}_{\perp}}(u,\bm{x}_{\perp}):\dot{\Sigma}^{2}:$
(4.65)

Then we find the Virasoro algebra

$\displaystyle[\mathcal{T}_{\omega,\bm{k}_{\perp}},\mathcal{T}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}]=(\omega^{\prime}-\omega)\mathcal{T}_{\omega+\omega^{\prime},\bm{k}_{\perp}+\bm{k}_{\perp}^{\prime}}-\frac{\omega^{3}}{24\pi}\times(2\pi)^{d-1}\delta^{(d-2)}(\bm{0})\delta(\omega+\omega^{\prime})\delta^{(d-2)}(\bm{k}_{\perp}+\bm{k}_{\perp}^{\prime}).$
(4.66)

The Dirac delta function
$\delta^{(d-2)}(\bm{k}_{\perp}+\bm{k}_{\perp}^{\prime})$ is zero for
$\bm{k}_{\perp}^{\prime}\not=-\bm{k}_{\perp}$. The divergence for the central
charge only appears for $\bm{k}_{\perp}^{\prime}=-\bm{k}_{\perp}$. In this
case, we find

$\displaystyle(2\pi)^{d-2}\delta^{(d-2)}(\bm{k}_{\perp}+\bm{k}_{\perp}^{\prime})|_{\bm{k}_{\perp}^{\prime}=-\bm{k}_{\perp}}=\int
d\bm{x}_{\perp}=L^{d-2}=\text{Area}(B)$ (4.67)

where we have introduced an IR cutoff $L$ which is the large length scale in
the transverse direction. At the last step, we have switched the $L^{d-2}$ to
the area of the bifurcation surface $B$101010This regularization is similar to
restrict the particles in a box [43].. Combining with the regularization
(4.59), we find the Virasoro algebra

$\displaystyle\
[\mathcal{T}_{\omega,\bm{k}_{\perp}},\mathcal{T}_{\omega^{\prime},-\bm{k}_{\perp}}]=(\omega^{\prime}-\omega)\mathcal{T}_{\omega+\omega^{\prime},0}-\frac{\omega^{3}}{12}\times\frac{\text{Area}(B)}{a^{d-2}}\delta(\omega+\omega^{\prime}).$
(4.68)

The quantity

$\bar{c}=\frac{\text{Area}(B)}{a^{d-2}}$ (4.69)

is exactly the number of states in the transverse directions. We add a bar
over “c” to distinguish it with the central charge $c=\delta^{(d-2)}(\bm{0})$
which is the density of states in the transverse directions. We will call it
an effective central charge.

###  4.4 Relation to Modular Hamiltonian

We have defined flux operators associated with supertranslations and
superrotations for the Rindler wedge. In Rindler wedge, the fundamental
quantity is the modular Hamiltonian [44]

$H_{\text{modular}}=2\pi\int_{V}d^{d-1}\bm{x}x^{d-1}T_{tt}(t=0,\bm{x})$ (4.70)

where $T_{tt}(t=0,\bm{x})$ is the $tt$ component of the stress tensor on the
surface $V$ which is defined by

$V=\\{p=(t,x^{1},\cdots,x^{d-2})|t=0,\ x^{d-1}>0\\}.$ (4.71)

The modular Hamiltonian (4.70) is defined for general relativistic field
theories and plays a central role in the context of geometric entanglement
entropy. As a subregion of Minkowski spacetime, $V$ defines the Rindler wedge
whose null boundaries are exactly $\mathcal{H}^{--}$ and $\mathcal{H}^{++}$.
There could be relations between the flux operators associated with Carrollian
diffeomorphism and modular Hamiltonian. We will explore this problem in the
following.

$V$$B$$\mathcal{H}^{--}$$\mathcal{H}^{++}$$\bm{\zeta}$$\bm{\xi}^{(0)}$$\bm{\xi}$$\bm{\xi}^{(+)}$$t$$x^{d-1}$
Figure 3: The flow generated by the vector field $\bm{\zeta}$ in Rindler
wedge. The vector field reduces to the generator of Carrollian diffeomorphism
$\bm{\xi}$ on $\mathcal{H}^{--}$ and $\bm{\xi}^{(+)}$ on $\mathcal{H}^{++}$.
It crosses the Cauchy surfaces and reduces to the vector field
$\bm{\xi}^{(0)}$ on $V$.

We will derive a general formula for the flux operators at first. In figure 3,
we draw a vector field $\bm{\zeta}$ in the Rindler wedge. The vector
$\bm{\zeta}$ reduces to the generator of Carrollian diffeomorphism
$\bm{\xi}/\bm{\xi}^{(+)}$ on $\mathcal{H}^{--}/\mathcal{H}^{++}$

$\displaystyle\bm{\zeta}=\bm{\xi}+\mathcal{O}(\rho),\quad\bm{\zeta}=\bm{\xi}^{(+)}+\mathcal{O}(\rho).$
(4.72)

It will crosses $V$ and defines a vector $\bm{\xi}^{(0)}$ on this spacelike
hypersurface

$\displaystyle\bm{\xi}^{(0)}=\bm{\zeta}|_{V}.$ (4.73)

From the conservation of stress tensor, we can find the following identity
from Stokes’ theorem

$\displaystyle\int_{\text{bulk}}d^{d}xT^{\mu\nu}\nabla_{\mu}\zeta_{\nu}=\int_{\mathcal{H}^{++}}(d^{d-1}x)_{\mu}T^{\mu\nu}\zeta_{\nu}-\int_{\mathcal{H}^{--}}(d^{d-1}x)_{\mu}T^{\mu\nu}\zeta_{\nu},$
(4.74)

where the domain of integration on the left hand side is the Rindler wedge
which is denoted as “bulk”. We have dropped the terms at infinity since the
modes are suppressed exponentially in Rindler spacetime which is checked in
Appendix B.3. Since the stress tensor is symmetric

$T_{\mu\nu}=T_{\nu\mu}$ (4.75)

and the vector $\bm{\zeta}$ generates a diffeomorphism in the bulk with

$\delta_{\bm{\zeta}}g_{\mu\nu}=\nabla_{\mu}\zeta_{\nu}+\nabla_{\nu}\zeta_{\mu},$
(4.76)

we may recap (4.74) as

$\displaystyle\frac{1}{2}\int_{\text{bulk}}d^{d}xT^{\mu\nu}\delta_{\bm{\zeta}}g_{\mu\nu}=Q_{\bm{\xi}}-Q_{\bm{\xi}^{(+)}}$
(4.77)

where we have used the definition of flux operators associated with a
Carrollian diffeomorphism generated by $\bm{\xi}$ or $\bm{\xi}^{(+)}$.
Interestingly, for the Killing vectors $\bm{\zeta}$, the flux operators should
be equal to each other

$Q_{\bm{\xi}}=Q_{\bm{\xi}^{(+)}},\quad\bm{\zeta}\ \text{is a Killing vector}.$
(4.78)

The derivation could be extended to other bulk region. For example, when we
choose the shaded region which is bounded by $\mathcal{H}^{--}$ and $V$ in
figure 3, the identity becomes

$\displaystyle\frac{1}{2}\int_{\text{shaded
region}}d^{d}xT^{\mu\nu}\delta_{\bm{\zeta}}g_{\mu\nu}=Q_{\bm{\xi}}-Q_{\bm{\xi}^{(0)}}$
(4.79)

where we have defined a “charge” associated with $\bm{\xi}^{(0)}$ on $V$

$\displaystyle
Q_{\bm{\xi}^{(0)}}=\int(d^{d-1}x)_{\mu}T^{\mu\nu}\xi^{(0)}_{\nu}.$ (4.80)

Again, for a Killing vector $\bm{\zeta}$, we can identify the flux operators
on $\mathcal{H}^{--}$ with the charges on $V$

$\displaystyle Q_{\bm{\xi}}=Q_{\bm{\xi}^{(0)}},\quad\bm{\zeta}\ \text{is a
Killing vector}.$ (4.81)

The modular Hamiltonian is generated by the modular flow $\partial_{\tau}$,
which is exactly the Killing vector
$\bm{\xi}_{-}=x_{d-1}\partial_{t}+t\partial_{d-1}$. In retarded coordinates,
we have found

$\bm{\xi}_{-}=\partial_{u}$ (4.82)

which generates the supertranslation with

$f(u,\bm{x}_{\perp})=1.$ (4.83)

On $V$, the Killing vector $\partial_{\tau}$ becomes

$\bm{\xi}^{(0)}=x_{d-1}\partial_{t},\quad t=0.$ (4.84)

Therefore, we can identify the modular Hamiltonian as the flux operator
$\mathcal{T}_{f=1}$ in Rindler wedge

$\displaystyle\mathcal{T}_{f=1}=Q_{\bm{\xi}^{(0)}}=\int_{V}(d^{d-1}x)_{\mu}T^{\mu}_{\
\nu}\xi^{(0)\nu}=\int_{V}d^{d-1}\bm{x}x^{d-1}T_{tt}(t=0,\bm{x})=\frac{1}{2\pi}H_{\text{modular}}.$
(4.85)

This identity strongly indicates that the flux operators $\mathcal{T}_{f}$ and
$\mathcal{M}_{Y}$ are physically important. We may regard them as an extension
of modular Hamiltonian for subsystems. The reduced density matrix for a
Rindler wedge becomes

$\rho_{\text{Rindler}}=e^{-H_{\text{modular}}}=e^{-2\pi\mathcal{T}_{f=1}}.$
(4.86)

As a by product, we list the one-to-one correspondence between the operators
on $\mathcal{H}^{--}$ and $V$ for the Killing vectors (4.42).

Killing vectors $\bm{\zeta}$ | Flux operators $(f,Y^{A})$ | Operators in the subregion
---|---|---
$\frac{1}{2}(\partial_{t}-\partial_{d-1})$ | $(e^{u},0)$ | $\int_{V}d^{d-1}\bm{x}(T_{tt}-T_{t,d-1})$
$b^{A}\partial_{A}$ | $(0,b^{A})$ | $b^{A}\int_{V}d^{d-1}\bm{x}T_{tA}$
$2(x_{+}\partial_{-}-\partial_{-}\partial_{+})$ | $(1,0)$ | $H_{\text{modular}}$
$c^{A}(x_{-}\partial_{A}-x_{A}\partial_{-})$ | $(-c^{A}x_{A}e^{u},0)$ | $\frac{c^{A}}{2}\int_{V}d^{d-1}\bm{x}(x_{A}T_{t,d-1}-x_{d-1}T_{tA})$
$\omega^{AB}(x_{A}\partial_{B}-x_{B}\partial_{A})$ | $(0,\omega^{CA}x_{C})$ | $\omega^{CA}\int_{V}d^{d-1}\bm{x}T_{tA}x_{C}$
Table 2: Correspondence between operators on $\mathcal{H}^{--}$ and $V$.

###  4.5 Observables in Unruh effect

The Rindler wedge is the spacetime which is observed by an accelerated
observer with constant acceleration $a=1$. The Unruh effect says that the
accelerated observer could detect a thermal bath in the Minkowski vacuum
$|0\rangle$. From mode expansion which is reviewed in Appendix B.3, the vacuum
defined in $\mathcal{H}^{--}$ is the Rindler vacuum $|0\rangle_{\rm R}$ which
is not $|0\rangle$. In this section, we will compute the expectation value of
the flux operators $\mathcal{T}_{f}$ and $\mathcal{M}_{Y}$ in the Minkowski
vacuum $|0\rangle$ and excited states to propose that they are physical
observables for the accelerated observer.

The starting point is the identification of the modular Hamiltonian and the
zero mode of supertranslation flux operator (4.85). In quantum information
theory, considering a subsystem $R$ which is obtained by tracing out the
degrees of freedom of its complement $\bar{R}$, we may find the reduced
density matrix $\mathfrak{\rho}_{R}$ for the subsystem

$\mathfrak{\rho}_{R}=\text{tr}_{\bar{R}}\mathfrak{\rho}$ (4.87)

where $\mathfrak{\rho}$ is the density matrix of the whole system
$R\cup\bar{R}$. The reduced density matrix may be rewritten as an exponential
of the modular Hamiltonian

$\mathfrak{\rho}_{R}=e^{-H_{\text{modular}}}$ (4.88)

which is given by (4.85) for the Rindler wedge. The entanglement entropy for
the subsystem is

$S_{R}=-\text{tr}_{R}\mathfrak{\rho}_{R}\log\mathfrak{\rho}_{R}=\text{tr}_{R}\mathfrak{\rho}_{R}H_{\text{modular}}.$
(4.89)

Switching to the whole system with the relation (4.87), we conclude that the
entanglement entropy is the expectation value of modular Hamiltonian. Now for
the Rindler wedge, we may replace the modular Hamiltonian to the
supertranslation flux operator with $f=1$

$S_{R}=2\pi\text{tr}_{R}(\mathfrak{\rho}_{R}\mathcal{T}_{f=1})=2\pi\text{tr}(\mathfrak{\rho}\mathcal{T}_{f=1}).$
(4.90)

Now we may extend the modular Hamiltonian to general flux operators
$\mathcal{T}_{f}$ and $\mathcal{M}_{Y}$, it is natural to define the following
expectation values

$\displaystyle\mathfrak{T}_{f}(\mathfrak{\rho})\equiv\text{tr}_{R}(\mathfrak{\rho}_{R}\mathcal{T}_{f})=\text{tr}(\mathfrak{\rho}\mathcal{T}_{f}),\qquad\mathfrak{M}_{Y}(\mathfrak{\rho})\equiv\text{tr}_{R}(\mathfrak{\rho}_{R}\mathcal{M}_{Y})=\text{tr}(\mathfrak{\rho}\mathcal{M}_{Y}).$
(4.91)

They are direct extension of entanglement entropy for Rindler wedge. These are
also the fluxes across the Killing horizon observed by the accelerated
observer. When the system is in the vacuum state $|0\rangle$, the density
matrix is

$\rho_{0}=|0\rangle\langle 0|.$ (4.92)

Sometimes, we compare the fluxes for different states. Given two states
characterized by density matrices $\mathfrak{\rho}_{1}$ and
$\mathfrak{\rho}_{2}$, we may define the following relative fluxes

$\displaystyle\Delta\mathfrak{T}_{f}(\mathfrak{\rho}_{1}|\mathfrak{\rho}_{2})$
$\displaystyle=\mathfrak{T}_{f}(\mathfrak{\rho}_{1})-\mathfrak{T}_{f}(\mathfrak{\rho}_{2})=\text{tr}[(\mathfrak{\rho}_{1}-\mathfrak{\rho}_{2})\mathcal{T}_{f}],$
(4.93a)
$\displaystyle\Delta\mathfrak{M}_{Y}(\mathfrak{\rho}_{1}|\mathfrak{\rho}_{2})$
$\displaystyle=\mathfrak{M}_{Y}(\mathfrak{\rho}_{1})-\mathfrak{M}_{Y}(\mathfrak{\rho}_{2})=\text{tr}[(\mathfrak{\rho}_{1}-\mathfrak{\rho}_{2})\mathcal{M}_{Y}].$
(4.93b)

In the following, we will calculate the quantities (4.91) in vacuum and
excited states and find the relative fluxes according to (4.93).

####  4.5.1 Minkowski vacuum

We use the mode expansion to find the operators in momentum space

$\displaystyle\mathcal{T}_{f}=$
$\displaystyle-\frac{1}{2}\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\omega^{\prime}\sqrt{\omega\omega^{\prime}}\int
d\bm{k}_{\perp}\int
d\bm{k}_{\perp}^{\prime}[\mathfrak{f}(\omega+\omega^{\prime},\bm{k}_{\perp}+\bm{k}_{\perp}^{\prime})a^{\rm
R}_{\omega,\bm{k}_{\perp}}a^{\rm R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}$
$\displaystyle-2\mathfrak{f}^{*}(\omega-\omega^{\prime},\bm{k}_{\perp}-\bm{k}^{\prime}_{\perp})a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}+\mathfrak{f}^{*}(\omega+\omega^{\prime},\bm{k}_{\perp}+\bm{k}_{\perp}^{\prime})a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R\dagger}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}],$ (4.94a)
$\displaystyle\mathcal{M}_{Y}=$
$\displaystyle-\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\omega^{\prime}\int
d\bm{k}_{\perp}\int d\bm{k}_{\perp}^{\prime}a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}k_{\perp}^{A}\mathfrak{Y}^{*}_{A}(\bm{k}_{\perp}-\bm{k}_{\perp}^{\prime})$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\int_{0}^{\infty}d\omega\int
d\bm{k}_{\perp}\int d\bm{k}_{\perp}^{\prime}a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega,\bm{k}_{\perp}^{\prime}}(k_{\perp}^{A}+k_{\perp}^{{}^{\prime}A})\mathfrak{Y}^{*}_{A}(\bm{k}_{\perp}-\bm{k}_{\perp}^{\prime})$
(4.94b)

where $\mathfrak{f}(\omega,\bm{k}_{\perp})$ is the Fourier transform of the
function $f(u,\bm{x}_{\perp})$

$\mathfrak{f}(\omega,\bm{k}_{\perp})=\frac{1}{(2\pi)^{d-1}}\int
dud\bm{x}_{\perp}f(u,\bm{x}_{\perp})e^{-i\omega
u+i\bm{k}_{\perp}\cdot\bm{x}_{\perp}}$ (4.95)

and $\mathfrak{Y}_{A}(\bm{k}_{\perp})$ is the Fourier transform of
$Y_{A}(\bm{x}_{\perp})$ with

$\mathfrak{Y}_{A}(\bm{k}_{\perp})=\frac{1}{(2\pi)^{d-2}}\int
d\bm{x}_{\perp}Y_{A}(\bm{x}_{\perp})e^{i\bm{k}_{\perp}\cdot\bm{x}_{\perp}}.$
(4.96)

The operators $a^{\rm R}_{\omega,\bm{k}_{\perp}},a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}$ are related to
$b_{\bm{p}},b_{\bm{p}}^{\dagger}$ by Bogoliubov coefficients

$\displaystyle a^{\rm R}_{\omega,\bm{k}_{\perp}}$ $\displaystyle=\int
d\bm{p}(\alpha_{\omega,\bm{k}_{\perp};\bm{p}}^{*}b_{\bm{p}}-\beta_{\omega,\bm{k}_{\perp};\bm{p}}^{*}b_{\bm{p}}^{\dagger}),$
(4.97a) $\displaystyle a^{\rm R\dagger}_{\omega,\bm{k}_{\perp}}$
$\displaystyle=\int
d\bm{p}(\alpha_{\omega,\bm{k}_{\perp};\bm{p}}b^{\dagger}_{\bm{p}}-\beta_{\omega,\bm{k}_{\perp};\bm{p}}b_{\bm{p}}).$
(4.97b)

The Bogoliubov coefficients are

$\displaystyle\alpha_{\omega,\bm{k}_{\perp};\bm{p}}$
$\displaystyle=\frac{1}{2\pi}\sqrt{\frac{\omega}{E}}e^{\pi\omega/2}\Gamma(i\omega)(\frac{E+p^{d-1}}{2})^{-i\omega}\delta^{(d-2)}(\bm{k}_{\perp}-\bm{p}_{\perp}).$
(4.98a) $\displaystyle\beta_{\omega,\bm{k}_{\perp};\bm{p}}$
$\displaystyle=-\frac{1}{2\pi}\sqrt{\frac{\omega}{E}}e^{-\pi\omega/2}\Gamma(i\omega)(\frac{E+p^{d-1}}{2})^{-i\omega}\delta^{(d-2)}(\bm{k}_{\perp}+\bm{p}_{\perp}).$
(4.98b)

Readers can find the derivation in Appendix B.3. Therefore, we find the
following expectation values in Minkowski vacuum

$\displaystyle\langle 0|a^{\rm R}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle$ $\displaystyle=-\int
d\bm{p}\
\alpha^{*}_{\omega,\bm{k}_{\perp};\bm{p}}\beta^{*}_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}},$
(4.99a) $\displaystyle\langle 0|a^{\rm R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle$ $\displaystyle=\int
d\bm{p}\
\beta_{\omega,\bm{k}_{\perp};\bm{p}}\beta^{*}_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}},$
(4.99b) $\displaystyle\langle 0|a^{\rm R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R\dagger}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle$
$\displaystyle=-\int d\bm{p}\
\beta_{\omega,\bm{k}_{\perp};\bm{p}}\alpha_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}}.$
(4.99c)

Utilizing the integral identity

$\int_{-\infty}^{\infty}\frac{dx}{\sqrt{b^{2}+x^{2}}}(\sqrt{b^{2}+x^{2}}+x)^{iz}=2\pi\delta(z),$
(4.100)

we find

$\displaystyle\langle 0|a^{\rm R}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle$
$\displaystyle=\frac{1}{2\pi}e^{\pi\omega}\sqrt{\omega\omega^{\prime}}\Gamma(-i\omega)\Gamma(i\omega)\delta(\omega+\omega^{\prime})\delta^{(d-2)}(\bm{k}_{\perp}+\bm{k}_{\perp}^{\prime}),$
(4.101a) $\displaystyle\langle 0|a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle$
$\displaystyle=\frac{1}{2\pi}e^{-\pi\omega}\omega\Gamma(-i\omega)\Gamma(i\omega)\delta(\omega-\omega^{\prime})\delta^{(d-2)}(\bm{k}_{\perp}-\bm{k}_{\perp}^{\prime}),$
(4.101b) $\displaystyle\langle 0|a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R\dagger}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle$
$\displaystyle=\frac{1}{2\pi}e^{-\pi\omega}\sqrt{\omega\omega^{\prime}}\Gamma(-i\omega)\Gamma(i\omega)\delta(\omega+\omega^{\prime})\delta^{(d-2)}(\bm{k}_{\perp}+\bm{k}_{\perp}^{\prime})$
(4.101c)

In (4.94a), the integration variables $\omega,\omega^{\prime}$ are non-
negative. Therefore, the contributions from $\langle 0|a^{\rm
R}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle$ and $\langle 0|a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R\dagger}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle$ are zeros and

$\displaystyle\mathfrak{T}_{f}(\mathfrak{\rho}_{0})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\omega^{\prime}\int
d\bm{k}_{\perp}\int
d\bm{k}_{\perp}^{\prime}\sqrt{\omega\omega^{\prime}}\mathfrak{f}^{*}(\omega-\omega^{\prime},\bm{k}-\bm{k}_{\perp}^{\prime})\langle
0|a^{\rm R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle$ (4.102)
$\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int_{0}^{\infty}d\omega\int
d\bm{k}_{\perp}\mathfrak{f}^{*}(0,\bm{0})\omega^{2}e^{-\pi\omega}\Gamma(-i\omega)\Gamma(i\omega)$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\omega\frac{\omega}{e^{2\pi\omega}-1}\int
d\bm{k}_{\perp}\mathfrak{f}^{*}(0,\bm{0}).$

At the last step, we used the identity for Gamma function

$\Gamma(ix)\Gamma(-ix)=\frac{\pi}{x\sinh\pi x}.$ (4.103)

Similarly, one can obtain

$\displaystyle\mathfrak{M}_{Y}(\mathfrak{\rho}_{0})$ $\displaystyle=$
$\displaystyle-\delta(0)\int_{0}^{\infty}d\omega\frac{1}{e^{2\pi\omega}-1}\int
d\bm{k}_{\perp}k_{\perp}^{A}\mathfrak{Y}_{A}^{*}(\bm{0}).$ (4.104)

Since the integrand is odd, we conclude that the vacuum expectation value of
$\mathfrak{M}_{Y}(\mathfrak{\rho}_{0})$ vanishes. This shows that the observer
will detect a thermal bath without any time and angular independent
information. We notice that the expectation value
$\mathfrak{T}_{f}(\mathfrak{\rho}_{0})$ is divergent in Minkowski vacuum. The
integral in the transverse direction can be regularized by choosing a cutoff
in the UV region, similar to (4.59). To be precise, we notice
$\mathfrak{f}(0,\bm{0})$ is a constant and the divergent integral is exactly

$\int d\bm{k}_{\perp}=\frac{(2\pi)^{d-2}}{a^{d-2}}.$ (4.105)

The length dimension of $\bm{x}_{\perp}$ is 1 while the retarded time $u$ is
dimensionless, following from the transformations (4.5) and (4.10). Therefore
the supertranslation function $f(u,\bm{x}_{\perp})$ is dimensionless. From the
definition of $\mathfrak{f}$, we conclude that $\mathfrak{f}^{*}(0,\bm{0})$
has the dimension of area in the transverse direction. We parameterize

$\displaystyle\mathfrak{f}^{*}(0,\bm{0})=\gamma\frac{\text{Area}(B)}{(2\pi)^{d-2}}$
(4.106)

by a dimensionless constant $\gamma$ which may be expressed as

$\displaystyle\gamma=(2\pi)^{d-2}\frac{\int
dud\bm{x}_{\perp}f(u,\bm{x}_{\perp})}{\int d\bm{x}_{\perp}}.$ (4.107)

Therefore, the expectation value $\mathfrak{T}_{f}(\mathfrak{\rho}_{0})$
becomes

$\mathfrak{T}_{f}(\mathfrak{\rho}_{0})=\gamma\frac{\text{Area}(B)}{a^{d-2}}\int_{0}^{\infty}d\omega\frac{\omega}{e^{2\pi\omega}-1}.$
(4.108)

To check the consistency of the regularization, we choose $f=1$ and its
Fourier transform is

$\displaystyle\mathfrak{f}^{*}(0,\bm{0})=\frac{1}{(2\pi)^{d-1}}\int
dud\bm{x}_{\perp}=\frac{TL^{d-2}}{(2\pi)^{d-1}},$ (4.109)

where $T$ and $L$ are the IR cutoffs for the retarded time and transverse
directions, respectively. Comparing with (4.106) we read

$\gamma=\frac{T}{2\pi}.$ (4.110)

Note that $T$ is dimensionless by definition. As we have mentioned, the
expectation value $\mathfrak{T}_{f=1}(\rho_{0})$ is the entanglement entropy
for the Rindler wedge. It is well known that the entanglement entropy for a
continuous subsystem is proportional to area of the entanglement surface [45,
46, 47]. Our regularization method matches with this conclusion.

####  4.5.2 Excited states in Minkowski spacetime

Considering the one particle state with definite momentum $\bm{p}$

$|\bm{p}\rangle=b_{\bm{p}}^{\dagger}|0\rangle,$ (4.111)

we could find the following expectation values

$\displaystyle\langle\bm{p}|a_{\omega,\bm{k}_{\perp}}^{\rm R}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|\bm{p}\rangle$
$\displaystyle=-(\alpha_{\omega,\bm{k}_{\perp};\bm{p}}^{*}\beta_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}}^{*}+\beta_{\omega,\bm{k}_{\perp};\bm{p}}^{*}\alpha^{*}_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}})+\langle
0|a_{\omega,\bm{k}_{\perp}}^{\rm R}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle\langle\bm{p}|\bm{p}\rangle,$
(4.112a) $\displaystyle\langle\bm{p}|a_{\omega,\bm{k}_{\perp}}^{\rm
R\dagger}a^{\rm R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|\bm{p}\rangle$
$\displaystyle=\alpha_{\omega,\bm{k}_{\perp};\bm{p}}\alpha^{*}_{\omega^{\prime},\bm{k}^{\prime}_{\perp};\bm{p}}+\beta_{\omega,\bm{k}_{\perp};\bm{p}}\beta^{*}_{\omega^{\prime},\bm{k}^{\prime}_{\perp};\bm{p}}+\langle
0|a_{\omega,\bm{k}_{\perp}}^{\rm R\dagger}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle\langle\bm{p}|\bm{p}\rangle,$
(4.112b) $\displaystyle\langle\bm{p}|a_{\omega,\bm{k}_{\perp}}^{\rm
R\dagger}a^{\rm
R\dagger}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|\bm{p}\rangle$
$\displaystyle=-(\alpha_{\omega,\bm{k}_{\perp};\bm{p}}\beta_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}}+\beta_{\omega,\bm{k}_{\perp};\bm{p}}\alpha_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}})+\langle
0|a_{\omega,\bm{k}_{\perp}}^{\rm R\dagger}a^{\rm
R\dagger}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle\langle\bm{p}|\bm{p}\rangle.$
(4.112c)

We can easily obtain the following relative fluxes

$\displaystyle\Delta\mathfrak{T}_{f}(\mathfrak{\rho}|\mathfrak{\rho}_{0})$
$\displaystyle\equiv\frac{\langle\bm{p}|\mathcal{T}_{f}|\bm{p}\rangle}{\langle\bm{p}|\bm{p}\rangle}{-\bra{0}\mathcal{T}_{f}\ket{0}},$
(4.113a)
$\displaystyle\Delta\mathfrak{M}_{Y}(\mathfrak{\rho}|\mathfrak{\rho}_{0})$
$\displaystyle\equiv\frac{\langle\bm{p}|\mathcal{M}_{Y}|\bm{p}\rangle}{\langle\bm{p}|\bm{p}\rangle},$
(4.113b)

where

$\displaystyle\langle\bm{p}|\mathcal{T}_{f}|\bm{p}\rangle=$
$\displaystyle-\frac{1}{4\pi^{2}}\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\omega^{\prime}\frac{\omega\omega^{\prime}}{E}\cosh\frac{\pi(\omega-\omega^{\prime})}{2}\Gamma(-i\omega)\Gamma(-i\omega^{\prime})(\frac{E+p^{d-1}}{2})^{i(\omega+\omega^{\prime})}\mathfrak{f}(\omega+\omega^{\prime},\bm{0})$
$\displaystyle-\frac{1}{4\pi^{2}}\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\omega^{\prime}\frac{\omega\omega^{\prime}}{E}\cosh\frac{\pi(\omega-\omega^{\prime})}{2}\Gamma(i\omega)\Gamma(i\omega^{\prime})(\frac{E+p^{d-1}}{2})^{-i(\omega+\omega^{\prime})}\mathfrak{f}^{*}(\omega+\omega^{\prime},\bm{0})$
$\displaystyle+\frac{1}{2\pi^{2}}\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\omega^{\prime}\frac{\omega\omega^{\prime}}{E}\cosh\frac{\pi(\omega+\omega^{\prime})}{2}\Gamma(i\omega)\Gamma(-i\omega^{\prime})(\frac{E+p^{d-1}}{2})^{-i(\omega-\omega^{\prime})}\mathfrak{f}^{*}(\omega-\omega^{\prime},\bm{0}),$
(4.114a) $\displaystyle\langle\bm{p}|\mathcal{M}_{Y}|\bm{p}\rangle=$
$\displaystyle-\frac{1}{2\pi}\frac{p_{\perp}^{A}\mathfrak{Y}_{A}^{*}(0)}{E}\int_{0}^{\infty}d\omega.$
(4.114b)

Note that the expectation value of $\mathcal{M}_{Y}$ is divergent. However,
the normalization of the state $|\bm{p}\rangle$

$\displaystyle\langle\bm{p}|\bm{p}\rangle=\delta^{(d-1)}(\bm{0})$ (4.115)

is also divergent. We may consider the following wave packet

$\displaystyle|\sigma\rangle=\int d\bm{p}\sigma_{\bm{p}}|\bm{p}\rangle$
(4.116)

which is the superposition of the state with definite momentum. The
coefficients $\sigma_{\bm{p}}$ is chosen to normalize the state
$|\sigma\rangle$

$\langle\sigma|\sigma\rangle=1\quad\Rightarrow\quad\int
d\bm{p}\sigma_{\bm{p}}^{*}\sigma_{\bm{p}}=1.$ (4.117)

The density matrix becomes

$\displaystyle\mathfrak{\rho}_{\sigma}=|\sigma\rangle\langle\sigma|.$ (4.118)

Using the transition matrices

$\displaystyle\langle\bm{p}|a^{\rm R}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|\bm{p}^{\prime}\rangle$
$\displaystyle=-(\alpha^{*}_{\omega,\bm{k}_{\perp};\bm{p}^{\prime}}\beta^{*}_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}}+\alpha^{*}_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}^{\prime}}\beta^{*}_{\omega,\bm{k}_{\perp};\bm{p}})+\langle
0|a^{\rm R}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle\langle\bm{p}|\bm{p}^{\prime}\rangle,$
(4.119a) $\displaystyle\langle\bm{p}|a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|\bm{p}^{\prime}\rangle$
$\displaystyle=\alpha_{\omega,\bm{k}_{\perp};\bm{p}}\alpha^{*}_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}^{\prime}}+\beta_{\omega,\bm{k}_{\perp};\bm{p}^{\prime}}\beta^{*}_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}}+\langle
0|a^{\rm R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle\langle\bm{p}|\bm{p}^{\prime}\rangle,$
(4.119b) $\displaystyle\langle\bm{p}|a^{\rm
R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R\dagger}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|\bm{p}^{\prime}\rangle$
$\displaystyle=-(\alpha_{\omega,\bm{k}_{\perp};\bm{p}}\beta_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}^{\prime}}+\alpha_{\omega^{\prime},\bm{k}_{\perp}^{\prime};\bm{p}}\beta_{\omega,\bm{k}_{\perp};\bm{p}^{\prime}})+\langle
0|a^{\rm R\dagger}_{\omega,\bm{k}_{\perp}}a^{\rm
R\dagger}_{\omega^{\prime},\bm{k}_{\perp}^{\prime}}|0\rangle\langle\bm{p}|\bm{p}^{\prime}\rangle,$
(4.119c)

the relative fluxes are

$\displaystyle\Delta\mathfrak{T}_{f}(\mathfrak{\rho}_{\sigma}|\mathfrak{\rho}_{0})=$
$\displaystyle-\frac{1}{8\pi^{2}}\int d\bm{p}\int
d\bm{p}^{\prime}\sigma^{*}_{\bm{p}}\sigma_{\bm{p}^{\prime}}\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\omega^{\prime}\frac{\omega\omega^{\prime}}{\sqrt{EE^{\prime}}}\Gamma(-i\omega)\Gamma(-i\omega^{\prime})\mathfrak{f}(\omega+\omega^{\prime},\bm{p}_{\perp}^{\prime}-\bm{p}_{\perp})$
$\displaystyle\times\
[(\frac{E+p^{d-1}}{2})^{i\omega^{\prime}}(\frac{E^{\prime}+p^{{}^{\prime}d-1}}{2})^{i\omega}e^{\pi(\omega-\omega^{\prime})/2}+(\frac{E+p^{d-1}}{2})^{i\omega}(\frac{E^{\prime}+p^{{}^{\prime}d-1}}{2})^{i\omega^{\prime}}e^{-\pi(\omega-\omega^{\prime})/2}]$
$\displaystyle-\frac{1}{8\pi^{2}}\int d\bm{p}\int
d\bm{p}^{\prime}\sigma^{*}_{\bm{p}}\sigma_{\bm{p}^{\prime}}\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\omega^{\prime}\frac{\omega\omega^{\prime}}{\sqrt{EE^{\prime}}}\Gamma(i\omega)\Gamma(i\omega^{\prime})\mathfrak{f}^{*}(\omega+\omega^{\prime},\bm{p}_{\perp}-\bm{p}_{\perp}^{\prime})$
$\displaystyle\times[(\frac{E+p^{d-1}}{2})^{-i\omega}(\frac{E^{\prime}+p^{{}^{\prime}d-1}}{2})^{-i\omega^{\prime}}e^{\pi(\omega-\omega^{\prime})/2}+(\frac{E+p^{d-1}}{2})^{-i\omega^{\prime}}(\frac{E^{\prime}+p^{{}^{\prime}d-1}}{2})^{-i\omega}e^{-\pi(\omega-\omega^{\prime})/2}]$
$\displaystyle+\frac{1}{4\pi^{2}}\int d\bm{p}\int
d\bm{p}^{\prime}\sigma_{\bm{p}}^{*}\sigma_{\bm{p}^{\prime}}\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\omega^{\prime}\frac{\omega\omega^{\prime}}{\sqrt{EE^{\prime}}}\Gamma(i\omega)\Gamma(-i\omega^{\prime})$
$\displaystyle\times[\mathfrak{f}^{*}(\omega-\omega^{\prime},\bm{p}_{\perp}-\bm{p}_{\perp}^{\prime})(\frac{E+p^{d-1}}{2})^{-i\omega}(\frac{E^{\prime}+p^{{}^{\prime}d-1}}{2})^{i\omega^{\prime}}e^{\pi(\omega+\omega^{\prime})/2}$
$\displaystyle+\mathfrak{f}^{*}(\omega-\omega^{\prime},\bm{p}_{\perp}^{\prime}-\bm{p}_{\perp})(\frac{E+p^{d-1}}{2})^{i\omega^{\prime}}(\frac{E^{\prime}+p^{{}^{\prime}d-1}}{2})^{-i\omega}e^{-\pi(\omega+\omega^{\prime})/2}],$
(4.120a)
$\displaystyle\Delta\mathfrak{M}_{Y}(\mathfrak{\rho}_{\sigma}|\mathfrak{\rho}_{0})=$
$\displaystyle-\frac{1}{4\pi}\int d\bm{p}\int
d\bm{p}^{\prime}\sigma_{\bm{p}}^{*}\sigma_{\bm{p}^{\prime}}\int_{0}^{\infty}d\omega\frac{(p_{\perp}+p_{\perp}^{\prime})^{A}}{\sqrt{EE^{\prime}}}$
$\displaystyle\times\Big{[}\frac{\mathfrak{Y}_{A}^{*}(\bm{p}_{\perp}-\bm{p}^{\prime}_{\perp})}{1-e^{-2\pi\omega}}\left(\frac{E+p^{d-1}}{E^{\prime}+p^{{}^{\prime}d-1}}\right)^{-i\omega}-\frac{\mathfrak{Y}_{A}^{*}(\bm{p}^{\prime}_{\perp}-\bm{p}_{\perp})}{e^{2\pi\omega}-1}\left(\frac{E+p^{d-1}}{E^{\prime}+p^{{}^{\prime}d-1}}\right)^{i\omega}\Big{]}.$
(4.120b)

The relative fluxes are finite and encode the time and angle dependent
information of the excited states.

##  5 Null hypersurfaces

We consider a general null hypersurface $\mathcal{N}$ with topology
${\bf{R}}\times N$ whose metric is degenerate

$ds^{2}_{\mathcal{N}}=h_{AB}d\theta^{A}d\theta^{B}.$ (5.1)

The metric $h_{AB}$ is also the metric of the $d-2$ dimensional manifold $N$.
We need a null vector $\bm{\chi}=\partial_{u}$ to generate the time direction
of the Carrollian manifold $\mathcal{N}$. The null hypersurface may be
embedded into $d$ dimensional spacetime whose metric may be expanded near
$\mathcal{N}$ as[48, 49, 50]

$\displaystyle
ds^{2}=Kdu^{2}-2dud\rho+H_{AB}(d\theta^{A}+\Lambda^{A}du)(d\theta^{B}+\Lambda^{B}du)$
(5.2)

where $K,\Lambda^{A},H_{AB}$ depend on the coordinates $u,\rho,\theta^{A},\
A=1,2,\cdots,d-1$ and satisfy the conditions

$\displaystyle K|_{\rho=0}=0,\quad\Lambda^{A}|_{\rho=0}=0,\quad
H_{AB}|_{\rho=0}=h_{AB}.$ (5.3)

We may expand $K,\Lambda^{A},H_{AB}$ by Taylor series

$\displaystyle K(u,\rho,\theta)$
$\displaystyle=-2\kappa(u,\theta)\rho-2\sum_{k=2}^{\infty}\kappa^{(k)}(u,\theta)\rho^{k},$
(5.4a) $\displaystyle\Lambda^{A}(u,\rho,\theta)$
$\displaystyle=\lambda^{A}(u,\theta)\rho+\sum_{k=2}^{\infty}\lambda^{A(k)}(u,\theta)\rho^{k},$
(5.4b) $\displaystyle H_{AB}(u,\rho,\theta)$
$\displaystyle=h_{AB}+\sum_{k=1}^{\infty}h_{AB}^{(k)}(u,\theta)\rho^{k}$
(5.4c)

such that the near $\mathcal{N}$ metric is

$\displaystyle ds^{2}=-2\kappa\rho du^{2}-2dud\rho+2\lambda_{A}\rho
dud\theta^{A}+(h_{AB}+\rho
h^{(1)}_{AB})d\theta^{A}d\theta^{B}+\mathcal{O}(\rho^{2}),$ (5.5)

where $\lambda_{A}=\lambda^{B}h_{AB}$. The coordinate system is usually called
null Gaussian coordinates. The hypersurface $\mathcal{N}$ sits at $\rho=0$.
The components of the metric could be written out explicitly as

$\displaystyle g_{uu}$
$\displaystyle=K+H_{AB}\Lambda^{A}\Lambda^{B}=-2\kappa\rho+\mathcal{O}(\rho^{2}),$
(5.6a) $\displaystyle g_{u\rho}$ $\displaystyle=-1,\quad g_{\rho\rho}=0,\quad
g_{\rho A}=0,$ (5.6b) $\displaystyle g_{uA}$
$\displaystyle=H_{AB}\Lambda^{B}=\lambda_{A}\rho+\mathcal{O}(\rho^{2}),$
(5.6c) $\displaystyle g_{AB}$ $\displaystyle=H_{AB}=h_{AB}+\mathcal{O}(\rho).$
(5.6d)

The determinant of the metric (5.2) is

$\displaystyle-\det g=\det H=\det
h(1+\rho\,\text{tr}\,h^{(1)}+\mathcal{O}(\rho^{2}))$ (5.7)

where $\det H$ and $\det h$ are the determinants of the metric $H_{AB}$ and
$h_{AB}$, respectively. The trace of $h_{AB}^{(1)}$ is denoted as
$\text{tr}\,h^{(1)}$ with

$\displaystyle\text{tr}\,h^{(1)}=h^{AB}h_{AB}^{(1)}.$ (5.8)

The inverse metric of $g_{\mu\nu}$ is

$\displaystyle g^{uu}$ $\displaystyle=0,\quad g^{u\rho}=-1,\quad g^{uA}=0,$
(5.9a) $\displaystyle g^{\rho\rho}$
$\displaystyle=-K=2\kappa\rho+\mathcal{O}(\rho^{2}),$ (5.9b) $\displaystyle
g^{\rho A}$
$\displaystyle=-H^{AB}\Lambda_{B}=-\lambda^{A}\rho+\mathcal{O}(\rho^{2}),$
(5.9c) $\displaystyle g^{AB}$ $\displaystyle=H^{AB}=h^{AB}+\mathcal{O}(\rho).$
(5.9d)

The integral measure on $\mathcal{N}$ is abbreviated as

$\int dud\bm{\theta}\equiv\int_{-\infty}^{\infty}du\int_{N}d\bm{\theta}$
(5.10)

with

$\displaystyle\int_{N}d\bm{\theta}=\int_{N}\sqrt{\det h}d\theta^{1}\cdots
d\theta^{d-2}.$ (5.11)

The normal covector of the hypersurface $\mathcal{N}$ is

$\displaystyle\mathfrak{m}=(-K)^{-1/2}d\rho\quad\Rightarrow\quad\mathfrak{m}_{\rho}=(-K)^{-1/2}=(2\kappa\rho)^{-1/2},\quad\mathfrak{m}_{u}=\mathfrak{m}_{A}=0.$
(5.12)

We may consider a general scalar theory with the action (3.14) whose field
$\Phi$ may be expanded as

$\Phi(u,\rho,\bm{\theta})=\Sigma(u,\bm{\theta})+\rho\Sigma^{(1)}(u,\bm{\theta})+\cdots$
(5.13)

near $\mathcal{N}$. However, we may allow a mass term since the null
hypersurface $\mathcal{N}$ is placed at a finite position in spacetime which
could be approached by massive particles similar to the Rindler spacetime.
Therefore, we may expand the potential perturbatively as

$V(\Phi)=\sum_{k=2}^{\infty}\frac{\lambda_{k}}{k}\Phi^{k},$ (5.14)

where $\lambda_{2}$ is the mass square

$\lambda_{2}=m^{2}.$ (5.15)

The pre-symplectic form of the scalar theory on the null hypersurface is

$\displaystyle\bm{\Theta}(\delta\Phi;\Phi)=-\int_{\mathcal{N}}(d^{d-1}x)_{\mu}\partial^{\mu}\Phi\delta\Phi,$
(5.16)

where $(d^{d-1}x)_{\mu}$ is defined as

$\displaystyle(d^{d-1}x)_{\mu}=\frac{1}{(d-1)!}\epsilon_{\mu\nu_{1}\cdots\nu_{d-1}}dx^{\nu_{1}}\wedge\cdots\wedge
dx^{\nu_{d-1}}=-\sqrt{2\kappa\rho}\mathfrak{m}_{\mu}dud\bm{\theta}=-\delta_{\mu}^{\rho}dud\bm{\theta}.$
(5.17)

From the expansion of the field $\Phi$, we find

$\displaystyle\partial_{u}\Phi$
$\displaystyle=\sum_{k=0}^{\infty}\dot{\Sigma}^{(k)}\rho^{k}=\dot{\Sigma}+\mathcal{O}(\rho),$
(5.18a) $\displaystyle\partial_{\rho}\Phi$
$\displaystyle=\sum_{k=1}^{\infty}k\Sigma^{(k)}\rho^{k-1}=\Sigma^{(1)}+\mathcal{O}(\rho),$
(5.18b) $\displaystyle\partial_{A}\Phi$
$\displaystyle=\sum_{k=0}^{\infty}\partial_{A}\Sigma^{(k)}\rho^{k}=\partial_{A}\Sigma+\mathcal{O}(\rho).$
(5.18c)

Therefore, we obtain

$\displaystyle\partial^{u}\Phi$
$\displaystyle=g^{u\rho}\partial_{\rho}\Phi=-\Sigma^{(1)}+\mathcal{O}(\rho),$
(5.19a) $\displaystyle\partial^{\rho}\Phi$ $\displaystyle=g^{\rho
u}\partial_{u}\Phi+g^{\rho\rho}\partial_{\rho}\Phi+g^{\rho
A}\partial_{A}\Phi=-\dot{\Sigma}+\mathcal{O}(\rho),$ (5.19b)
$\displaystyle\partial^{A}\Phi$
$\displaystyle=g^{A\rho}\partial_{\rho}\Phi+g^{AB}\partial_{B}\Phi=h^{AB}\partial_{B}\Sigma+\mathcal{O}(\rho).$
(5.19c)

It follows that the symplectic form is

$\displaystyle\bm{\Omega}(\delta\Sigma;\delta\Sigma;\Sigma)$ $\displaystyle=$
$\displaystyle-\int_{\mathcal{N}}(d^{d-1}x)_{\mu}\partial^{\mu}\delta\Phi\wedge\delta\Phi$
(5.20) $\displaystyle=$ $\displaystyle\int
dud\bm{\theta}\delta\Sigma\wedge\delta\dot{\Sigma},$

which implies that the commutators between $\Sigma$ and $\dot{\Sigma}$ are

$\displaystyle[\Sigma(u,\bm{\theta}),\Sigma(u^{\prime},\bm{\theta}^{\prime})]=\frac{i}{2}\alpha(u-u^{\prime})\delta^{(d-2)}(\bm{\theta}-\bm{\theta}^{\prime}),$
(5.21a)
$\displaystyle[\Sigma(u,\bm{\theta}),\dot{\Sigma}(u^{\prime},\bm{\theta}^{\prime})]=\frac{i}{2}\delta(u-u^{\prime})\delta^{(d-2)}(\bm{\theta}-\bm{\theta}^{\prime}),$
(5.21b)
$\displaystyle[\dot{\Sigma}(u,\bm{\theta}),\dot{\Sigma}(u^{\prime},\bm{\theta}^{\prime})]=\frac{i}{2}\delta^{\prime}(u-u^{\prime})\delta^{(d-2)}(\bm{\theta}-\bm{\theta}^{\prime}).$
(5.21c)

The field $\Sigma$ may be expanded by the superposition of positive frequency
modes and negative frequency modes

$\displaystyle\Sigma(u,\bm{\theta})=\int_{0}^{\infty}\frac{d\omega}{\sqrt{4\pi\omega}}[a_{\omega}(\bm{\theta})e^{-i\omega
u}+a^{\dagger}_{\omega}(\bm{\theta})e^{i\omega u}]$ (5.22)

with the following commutators between $a_{\omega}(\bm{\theta})$ and
$a^{\dagger}_{\omega}(\bm{\theta})$

$[a_{\omega}(\bm{\theta}),a_{\omega^{\prime}}(\bm{\theta}^{\prime})]=[a^{\dagger}_{\omega}(\bm{\theta}),a^{\dagger}_{\omega^{\prime}}(\bm{\theta}^{\prime})]=0,\quad[a_{\omega}(\bm{\theta}),a^{\dagger}_{\omega^{\prime}}(\bm{\theta}^{\prime})]=\delta(\omega-\omega^{\prime})\delta^{(d-2)}(\bm{\theta}-\bm{\theta}^{\prime}).$
(5.23)

Indeed, one can check the commutation relations (5.21) are satisfied by virtue
of the commutators (5.23). Therefore, we may interpret
$a_{\omega}(\bm{\theta})$ and $a^{\dagger}_{\omega}(\bm{\theta})$ as
annihilation and creation operators. More explicitly, we may define the vacuum
state as

$\displaystyle a_{\omega}(\bm{\theta})|0\rangle_{\mathcal{N}}=0$ (5.24)

and the creation operator $a_{\omega}^{\dagger}(\bm{\theta})$ acts on the
vacuum $|0\rangle_{\mathcal{N}}$ should create a particle with frequency
$\omega$ at the position $\bm{\theta}$. We add a subscript $\mathcal{N}$ to
emphasize that the vacuum state may depend on the null hypersurface
$\mathcal{N}$. The correlators are

$\displaystyle{}_{\mathcal{N}}\langle
0|\Sigma(u,\bm{\theta})\Sigma(u^{\prime},\bm{\theta}^{\prime})|0\rangle_{\mathcal{N}}$
$\displaystyle=\beta(u-u^{\prime})\delta^{(d-2)}(\bm{\theta}-\bm{\theta}^{\prime}),$
(5.25a) $\displaystyle{}_{\mathcal{N}}\langle
0|\Sigma(u,\bm{\theta})\dot{\Sigma}(u^{\prime},\bm{\theta}^{\prime})|0\rangle_{\mathcal{N}}$
$\displaystyle=\frac{1}{4\pi(u-u^{\prime}-i\epsilon)}\delta^{(d-2)}(\bm{\theta}-\bm{\theta}^{\prime}),$
(5.25b) $\displaystyle{}_{\mathcal{N}}\langle
0|\dot{\Sigma}(u,\bm{\theta})\Sigma(u^{\prime},\bm{\theta}^{\prime})|0\rangle_{\mathcal{N}}$
$\displaystyle=-\frac{1}{4\pi(u-u^{\prime}-i\epsilon)}\delta^{(d-2)}(\bm{\theta}-\bm{\theta}^{\prime}),$
(5.25c) $\displaystyle{}_{\mathcal{N}}\langle
0|\dot{\Sigma}(u,\bm{\theta})\dot{\Sigma}(u^{\prime},\bm{\theta}^{\prime})|0\rangle_{\mathcal{N}}$
$\displaystyle=-\frac{1}{4\pi(u-u^{\prime}-i\epsilon)^{2}}\delta^{(d-2)}(\bm{\theta}-\bm{\theta}^{\prime}).$
(5.25d)

Now it is straightforward to use the formula (2.19) to find the flux operators
associated with Carrollian diffeomorphisms. The results are

$\displaystyle\mathcal{T}_{f}$ $\displaystyle=\int
dud\bm{\theta}f(u,\bm{\theta}):\dot{\Sigma}^{2}:,$ (5.26a)
$\displaystyle\mathcal{M}_{Y}$ $\displaystyle=\frac{1}{2}\int
dud\bm{\theta}Y^{A}(\bm{\theta})(:\dot{\Sigma}\nabla_{A}\Sigma-\Sigma\nabla_{A}\dot{\Sigma}:).$
(5.26b)

We can also obtain similar algebra (3.82) with a divergent central charge

$c=\delta^{(d-2)}(\bm{0})=\delta^{(d-2)}(\bm{\theta}-\bm{\theta}^{\prime})|_{\bm{\theta}^{\prime}=\bm{\theta}}.$
(5.27)

There is a set of eigenfunctions $f_{\bm{q}}(\bm{\theta})$ of the Laplace
operator on the manifold $N$

$\Delta_{N}f_{\bm{q}}(\bm{\theta})=-\lambda_{\bm{q}}f_{\bm{q}}(\bm{\theta})$
(5.28)

where $\lambda_{\bm{q}}$ is the corresponding eigenvalue. The eigenfunctions
are normalized to 1 and orthogonal to each other111111We have assumed the
manifold is compact and the eigenvalues are discrete in the following.

$\int_{N}d\bm{\theta}f_{\bm{q}}(\bm{\theta})f^{*}_{\bm{q}^{\prime}}(\bm{\theta})=\delta_{\bm{q},\bm{q}^{\prime}}.$
(5.29)

The completeness relation for the eigenfunctions is

$\sum_{\bm{q}}f_{\bm{q}}(\bm{\theta})f^{*}_{\bm{q}}(\bm{\theta}^{\prime})=\delta^{(d-2)}(\bm{\theta}-\bm{\theta}^{\prime}).$
(5.30)

Summing over all possible eigenstates in (5.29) and we obtain

$\displaystyle\text{tr}\
1=\sum_{\bm{q}}1=\sum_{\bm{q}}\sum_{\bm{q}^{\prime}}\delta_{\bm{q},\bm{q}^{\prime}}=\sum_{\bm{q}}\int_{N}d\bm{\theta}f_{\bm{q}}(\bm{\theta})f^{*}_{\bm{q}}(\bm{\theta})=\int_{N}d\bm{\theta}\delta^{(d-2)}(\bm{0}).$
(5.31)

Therefore, the central charge is

$\displaystyle
c=\delta^{(d-2)}(\bm{0})=\frac{\sum_{\bm{q}}1}{\int_{N}d\bm{\theta}}=\frac{\text{tr}\
1}{\text{Area of $N$}}=\text{density of states on $N$}.$ (5.32)

We may still use the heat kernel method [51] to regularize the central charge
for general Riemannian manifold without boundary.

##  6 Conclusion and discussion

In this work, we have constructed the quantum flux operators for scalar field
theory which form a faithful representation of Carrollian diffeomorphism for
null hypersurfaces up to a non-zero central charge. We have checked the
results mainly for two interesting null hypersurfaces and then extend them to
general null hypersurfaces. In the first case, the flux operators associated
with Carrollian diffeomorphism at future null infinity are finite in general
dimensions. Therefore, our results overcome various problems in the context of
higher dimensional asymptotic symmetry analysis and also consistent with
graviton soft theorems. Secondly, we extend our method to the Rindler horizon
and find similar flux operators which are associated with the accelerated
observer. They are natural operators for the Rindler spacetime, a subsystem in
the context of quantum information. We conclude that the zero mode of the
supertranslation flux is actually the modular Hamiltonian of the subsystem.
These results build a connection between Carrollian diffeomorphism and quantum
information theory. We could define the expectation values of the
supertranslation and superrotation flux operators associated with the
subsystem, which extend the geometric entanglement entropy. By computing the
expectation value of $\mathcal{T}_{f}$ for the vacuum state explicitly, we
could check the regularization method of the central charge using the area law
of entanglement entropy. We also define the relative fluxes between different
states which may characterize the difference between any two states, and may
be new observables in Unruh effect. Finally, we also propose to regularize the
central charge using spectral zeta function or heat kernel method. We could
find a finite result for the theory at null infinity. For the Rindler wedge,
we conclude that there is an effective central charge which is proportional to
the area of the bifurcation surface

$\bar{c}=\frac{\text{Area}(B)}{a^{d-2}}.$ (6.1)

This result is consistent with entanglement entropy by identifying the modular
Hamiltonian and the zero mode of supertranslation flux. The result shows that
the effective central charge $\bar{c}$ counts the number of microstates in the
transverse directions which is similar to the area law of black hole entropy.
However, the results depend on the UV cutoff which comes from the property of
continuous quantum field theory. There are various open questions in this
direction.

  * •

It should be able to extend our results to the theory with nonzero spins. In
[8, 9], it has been shown that more anomalous terms should be included for
these theories. It would be nice to work out the details for general null
hypersurfaces in general dimensions.

  * •

Spectral geometry of Carrollian manifold. To study the regularization of the
central charge, we compute the spectral zeta function associated with the
Carrollian manifold $\mathcal{C}=\textbf{R}\times N$

$\zeta_{\mathcal{C}}(s)=\text{tr}^{\prime}(-\Delta_{N})^{-s}$ (6.2)

where $\Delta_{N}$ is the Laplace operator of the Riemannian manifold $N$. We
read out the central charge from $1+\zeta_{\mathcal{C}}(0)$. It would be nice
to extract more physical information from the spectral zeta function for
general Carrollian manifolds. A related question is to study the heat kernel
associated with a general Carrollian manifold

$K_{\mathcal{C}}(\sigma)=\text{tr}\ e^{\sigma\Delta_{N}}.$ (6.3)

  * •

Periodicity of imaginary time. For a general subsystem or a black hole, the
imaginary time should be periodic and the inverse period is the temperature of
the system. We still do not discuss the consequence of this condition. Taking
Rindler spacetime as an example, the retarded time $u$ is periodic in
imaginary direction

$u\sim u+2\pi i.$ (6.4)

To be consistent with the periodic condition, the basis function should obey
the equation

$f_{\omega,\bm{k}_{\perp}}(u,\bm{x}_{\perp})=f_{\omega,\bm{k}_{\perp}}(u+2\pi
i,\bm{x}_{\perp})$ (6.5)

which implies

$\omega=in,\quad n=0,\pm 1,\pm 2,\cdots.$ (6.6)

We may redefine

$L_{n,\bm{k}_{\perp}}=i\mathcal{T}_{in,\bm{k}_{\perp}},$ (6.7)

then the Virasoro algebra becomes

$\displaystyle\ [L_{n,\bm{k}_{\perp}},L_{n^{\prime},\bm{k}_{\perp}^{\prime}}]$
$\displaystyle=(n-n^{\prime})L_{n+n^{\prime},\bm{0}}+\frac{\bar{c}}{12}n^{3}\delta_{n,-n^{\prime}}\quad\text{for}\quad\bm{k}_{\perp}^{\prime}=-\bm{k}_{\perp},$
(6.8a) $\displaystyle\
[L_{n,\bm{k}_{\perp}},L_{n^{\prime},\bm{k}_{\perp}^{\prime}}]$
$\displaystyle=(n-n^{\prime})L_{n+n^{\prime},\bm{k}_{\perp}+\bm{k}_{\perp}^{\prime}}\quad\text{for}\quad\bm{k}_{\perp}^{\prime}\not=-\bm{k}_{\perp}.$
(6.8b)

We find an infinite tower of two dimensional Virasoro algebras whose label $n$
is discrete, which is exactly the same as two dimensional conformal field
theory. It would be nice to explore this topic in the future.

  * •

Event horizon and Hawking radiation. Our method can be extended to event
horizon of black holes. It would be very interesting to study the Carrollian
diffeomorphism and its associated quantum flux operators at the event horizon.
A better understanding of the representation theory of the algebra may provide
the key insight on Hawking radiation.

  * •

Relation to quantum information theory. We have noticed that the quantum flux
operators defined on the null hypersurface are direct generalization of the
null modular Hamiltonians defined in [52]. For each spatial subregion, there
is a null boundary which is exactly described by a Carrollian manifold. It
would be interesting to explore the physical interpretation of the quantum
flux operators in the context of quantum information theory.

Acknowledgments. The work of J.L. is supported by NSFC Grant No. 12005069.

##  Appendix A Unit sphere $S^{m}$

In this Appendix, we will review various aspects of the unit sphere $S^{m}$
with $m=d-2$.

###  A.1 Spherical coordinates

The isometry group is $SO(m)$ the conformal group is $SO(m+1,1)$ for $S^{m}$.
We note that the conformal group is isomorphic to $SO(d-1,1)$ which is the
isometry group of Minkowski spacetime $\mathbb{R}^{1,d-1}$. The unit sphere
$S^{m}$ embedded into Minkowski spacetime $\mathbb{R}^{1,d-1}$ introduces the
coordinate transformation from Cartesian coordinates of $d$ dimensional
Minkowski spacetime to spherical coordinates
$\left(t,r,\theta^{A}\right),A=1,2,\cdots,m$ as following

$x^{1}=r\prod_{j=1}^{m}\sin\theta_{j},\quad
x^{i}=r(\prod_{j=i}^{m}\sin\theta_{j})\cos\theta_{i-1},\quad 2\leq i\leq m+1.$
(A.1)

More explicitly, we have

$\displaystyle x^{1}$
$\displaystyle=r\sin\theta_{m}\sin\theta_{m-1}\cdots\sin\theta_{3}\sin\theta_{2}\sin\theta_{1},$
(A.2) $\displaystyle x^{2}$
$\displaystyle=r\sin\theta_{m}\sin\theta_{m-1}\cdots\sin\theta_{3}\sin\theta_{2}\cos\theta_{1},$
$\displaystyle x^{3}$
$\displaystyle=r\sin\theta_{m}\sin\theta_{m-1}\cdots\sin\theta_{3}\cos\theta_{2},$
$\displaystyle\cdots$ $\displaystyle x^{m}$
$\displaystyle=r\sin\theta_{m}\cos\theta_{m-1},$ $\displaystyle x^{m+1}$
$\displaystyle=r\cos\theta_{m},$

where

$r\geq 0;\quad 0\leq\theta_{1}\leq 2\pi;\quad 0\leq\theta_{i}\leq\pi,\quad
2\leq i\leq m.$ (A.3)

The normal vector of the sphere $S^{m}$ is

$\displaystyle n^{i}=\frac{x^{i}}{r}.$ (A.4)

In spherical coordinates, the metric of Minkowksi spacetime is

$ds^{2}=-dt^{2}+dx^{2}_{i}=-dt^{2}+dr^{2}+r^{2}ds^{2}_{S^{m}},$ (A.5)

where the metric of the unit sphere $S^{m}$ is

$\displaystyle ds^{2}_{S^{m}}=\gamma_{AB}d\theta^{A}d\theta^{B},\quad
A,B=1,2,\cdots,m.$ (A.6)

The components of the metric $\gamma_{AB}$ is

$\gamma_{AB}=r^{-2}\delta_{ij}\frac{\partial
x^{i}}{\partial\theta^{A}}\frac{\partial
x^{j}}{\partial\theta^{B}}=\delta_{ij}Y_{A}^{i}Y_{B}^{j},$ (A.7)

where $Y_{A}^{i}$ is the gradient of the normal vector $n^{i}$

$\displaystyle Y^{i}_{A}=-\frac{\partial
n^{i}}{\partial\theta^{A}}=-\nabla_{A}n^{i}.$ (A.8)

The explicit metric of the sphere may be found by the following recursion
formula

$\displaystyle ds_{S^{1}}^{2}=d\theta_{1}^{2},\quad
ds_{S^{n}}^{2}=d\theta_{n}^{2}+\sin\theta_{n}^{2}ds_{S^{n-1}}^{2},\quad n\geq
2.$ (A.9)

The determinant of the metric is

$\displaystyle\sqrt{\gamma}=\prod_{j=2}^{m}\sin^{j-1}\theta_{j}.$ (A.10)

###  A.2 Spherical harmonics

The scalar spherical harmonic functions on $S^{m}$ are given by [53]

$\displaystyle
Y_{\bm{\ell}}(\Omega)=\frac{e^{i\ell_{1}\theta_{1}}}{\sqrt{2\pi}}[\prod_{j=2}^{m}c(j,\ell_{j},\ell_{m-1})(\sin\theta_{j})^{-(j-2)/2}P_{\ell_{j}+\frac{j-2}{2}}^{-(\ell_{m-1}+\frac{j-2}{2})}(\cos\theta_{j})],$
(A.11)

where the vector $\bm{\ell}$ is

$\bm{\ell}=(\ell_{1},\ell_{2},\cdots,\ell_{m})$ (A.12)

with $\ell_{j},\ j=1,2,\cdots,m$ being integers that satisfy

$\ell_{m}\geq\ell_{m-1}\geq\cdots\geq\ell_{2}\geq|\ell_{1}|.$ (A.13)

The associated Legendre function of the first kind is defined through the
hypergeometric function:

$\displaystyle
P_{\nu}^{\mu}(x)=\frac{1}{\Gamma(1-\mu)}\left(\frac{1-x}{1+x}\right)^{-\mu/2}{}_{2}F_{1}(-\nu,\nu+1;1-\mu;\frac{1-x}{2}).$
(A.14)

The coefficient $c(a_{1},a_{2},a_{3})$ is chosen as

$\displaystyle
c\left(a_{1},a_{2},a_{3}\right)=\sqrt{\frac{2a_{2}+a_{1}-1}{2}\frac{\left(a_{2}+a_{3}+a_{1}-2\right)!}{\left(a_{2}-a_{3}\right)!}}$
(A.15)

such that the spherical harmonic functions are normalized and orthogonal to
each other

$\int_{S^{m}}d\Omega
Y_{\bm{\ell}}^{*}(\Omega)Y_{\bm{\ell}^{\prime}}(\Omega)=\delta_{\bm{\ell},\bm{\ell}^{\prime}},$
(A.16)

where $Y_{\bm{\ell}}^{\ast}\left(\Omega\right)$ is the complex conjugate of
$Y_{\bm{\ell}}\left(\Omega\right)$. The completeness of spherical harmonic
functions is

$\sum_{\bm{\ell}}Y_{\bm{\ell}}^{*}(\Omega)Y_{\bm{\ell}}(\Omega^{\prime})=\delta(\Omega-\Omega^{\prime}).$
(A.17)

The harmonic functions obey the following Laplace equation on $S^{m}$

$\nabla^{A}\nabla_{A}Y_{\bm{\ell}}(\Omega)=-\ell_{m}(\ell_{m}+m-1)Y_{\bm{\ell}}(\Omega).$
(A.18)

Therefore, for any fixed quantum number $\ell_{m}=L$, the degeneracy of the
spherical harmonic functions is [54]

$\displaystyle
g(L,m)=\sum_{\bm{\ell}}^{\ell_{m}=L}1=\sum_{\ell_{m-1}=0}^{\ell_{m-1}=L}\cdots\sum_{\ell_{2}=0}^{\ell_{3}}\sum_{\ell_{1}=-\ell_{2}}^{\ell_{2}}1=\frac{(2L+m-1)(L+m-2)!}{L!(m-1)!}.$
(A.19)

The generating function of the degeneracy in general dimension is

$\displaystyle
G(x,m)\equiv\sum_{L=0}^{\infty}g(L,m)x^{L}=\frac{1+x}{(1-x)^{m}}.$ (A.20)

###  A.3 Addition theorem

As is well known, the addition theorem in four dimensions states that

$C_{\ell}^{1/2}\left(\cos\gamma\right)=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^{\ell}Y_{\ell,m}^{\ast}\left(\Omega\right)Y_{\ell,m}\left(\Omega^{\prime}\right),\quad\bm{\ell}=\left(m,\ell\right),$
(A.21)

where $\gamma$ is the intersection angle between the two directions labeled by
$\Omega$ and $\Omega^{\prime}$. The notation $C_{\ell}^{k}(x)$ is the so-
called Gegenbauer polynomial which may be defined by hypergeometric function

$C_{\ell}^{k}\left(x\right)=\frac{\Gamma\left(2k+\ell\right)}{\ell!\Gamma\left(2k\right)}\,\,_{2}F_{1}\left(-\ell,2k+\ell;\frac{1}{2}+k;\frac{1-x}{2}\right).$
(A.22)

In $4$ dimensions, $k$ takes the value $1/2$ and the intersection angle
$\gamma$ becomes $0$ when $\Omega^{\prime}=\Omega$, thus the addition theorem
reduces to

$C_{\ell}^{1/2}\left(1\right)=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^{\ell}{Y_{\ell,m}^{\ast}\left(\Omega\right)Y_{\ell,m}\left(\Omega\right)},\quad\bm{\ell}=\left(m,\ell\right),$
(A.23)

The left hand side could be reduced to the familiar Legendre polynomial of
order $\ell$. In general $d$ dimensions, we have a similar addition theorem,

$C_{{\ell_{m}}}^{(m-1)/2}\left(\cos\gamma\right)=K_{{\ell_{m}}}{\sum_{\bm{\ell}}}^{\prime}{Y_{\bm{\ell}}^{\ast}\left(\Omega\right)Y_{\bm{\ell}}\left(\Omega^{\prime}\right)},$
(A.24)

where $k$ is related to the dimension $d$ by

$k=\frac{d-3}{2}=\frac{m-1}{2}$ (A.25)

and $K_{{\ell_{m}}}$ is a constant which is not important in this work. The
summation ${\sum_{\bm{\ell}}}^{\prime}$ is to sum over all possible quantum
numbers while fixing the value of $\ell_{m}$

${\sum_{\bm{\ell}}}^{\prime}=\sum_{\ell_{m-1}=0}^{\ell_{m}}\cdots\sum_{\ell_{2}=0}^{\ell_{3}}\sum_{\ell_{1}=-\ell_{2}}^{\ell_{2}}.$
(A.26)

When $\Omega^{\prime}=\Omega$, the addition theorem becomes

$C_{{\ell_{m}}}^{(m-1)/2}\left(1\right)=K_{{\ell_{m}}}{\sum_{\bm{\ell}}}^{\prime}{Y_{\bm{\ell}}^{\ast}\left(\Omega\right)Y_{\bm{\ell}}\left(\Omega\right)},$
(A.27)

The special value $C_{{\ell_{m}}}^{(m-1)/2}\left(1\right)$ can be found from
the definition (A.22)

$C_{{\ell_{m}}}^{(m-1)/2}\left(1\right)=\frac{\Gamma(m+\ell_{m}-1)}{\ell_{m}!(m-1)!}.$
(A.28)

Integrating both sides of (A.27) over the $S^{m}$ and considering the
orthonormality (A.16) yields

$\begin{split}&\int_{S^{m}}{d\Omega}C_{{\ell_{m}}}^{(m-1)/2}\left(1\right)=K_{{\ell_{m}}}{\sum_{\bm{\ell}}}^{\prime}{\int_{S^{m}}{d\Omega}Y_{\bm{\ell}}^{\ast}\left(\Omega\right)Y_{\bm{\ell}}\left(\Omega\right)},\\\
\Rightarrow\quad&C_{\bm{\ell}}^{(m-1)/2}\left(1\right)I(0)=K_{\ell_{m}}{\sum_{\bm{\ell}}}^{\prime}{1}=K_{\ell_{m}}g(\ell_{m},m).\end{split}$
(A.29)

where $I(0)$ is the area of the unit sphere $S^{m}$:

$I\left(0\right)\equiv\int_{S^{m}}{d\Omega}=\frac{2\pi^{\left(m+1\right)/2}}{\Gamma\left[\left(m+1\right)/2\right]},$
(A.30)

Finally substituting the result above into (A.27) yields

${\delta^{(d-2)}(0)}=\sum_{\bm{\ell}}{Y_{\bm{\ell}}^{\ast}\left(\Omega\right)Y_{\bm{\ell}}\left(\Omega\right)}=\sum_{\ell_{m}=0}^{\infty}\frac{g(\ell_{m},m)}{I(0)}=\frac{\sum_{\bm{\ell}}1}{I(0)}.$
(A.31)

This is the density of states on $S^{m}$.

As a matter of fact, this conclusion is immediately derived if we take sum for
the orthogonality relation of the spherical harmonics

$\displaystyle\int_{S^{m}}d\Omega
Y_{\bm{\ell}}^{*}(\Omega)Y_{\bm{\ell}^{\prime}}(\Omega)=\delta_{\bm{\ell},\bm{\ell}^{\prime}},$
(A.32)

and notice the completeness relation

$\displaystyle\sum_{\bm{\ell}}Y_{\bm{\ell}}(\Omega)Y^{*}_{\bm{\ell}}(\Omega^{\prime})=\delta(\Omega-\Omega^{\prime}).$
(A.33)

It follows that

$\displaystyle\sum_{\bm{\ell},\bm{\ell}^{\prime}}\delta_{\bm{\ell},\bm{\ell}^{\prime}}\int_{S^{m}}d\Omega
Y_{\bm{\ell}}^{*}(\Omega)Y_{\bm{\ell}^{\prime}}(\Omega)=\sum_{\bm{\ell},\bm{\ell}^{\prime}}\delta_{\bm{\ell},\bm{\ell}^{\prime}}\quad\Rightarrow\quad\delta^{(d-2)}(0)\int_{S^{m}}d\Omega=\sum_{\bm{\ell}}1.$
(A.34)

This method may be generalized to any compact manifold where the
eigenfunctions have the similar orthogonality and completeness relations.

###  A.4 Conformal Killing vectors

The conformal group of $S^{m}$ is $SO(m+1,1)$ which is isomorphic to the
Lorentz group of $\mathbb{R}^{1,d-1}$. The conformal algebra $so(m+1,1)$ is
generated by

$\frac{(m+2)(m+1)}{2}=\frac{d(d-1)}{2}$ (A.35)

conformal Killing vectors (CKVs) of $S^{m}$. The CKVs could be found by
embedding $S^{m}$ into $\mathbb{R}^{1,d-1}$ and making use of the Lorentz
algebra of $\mathbb{R}^{1,d-1}$. We may define two null vectors
$n^{\mu},\bar{n}^{\mu}$ in Cartesian coordinates

$n^{\mu}=(1,n^{i}),\qquad\bar{n}^{\mu}=(-1,n^{i}).$ (A.36)

They satisfy the relations

$n^{2}=\bar{n}^{2}=0,\qquad n\cdot\bar{n}=2.$ (A.37)

A spacetime vector $m^{\mu}$ and timelike vector $\bar{m}^{\mu}$ are related
to $n^{\mu}$ and $\bar{n}^{\mu}$ by the relations

$m^{\mu}=\frac{1}{2}(n^{\mu}+\bar{n}^{\mu})=(0,n^{i}),\qquad\bar{m}^{\mu}=\frac{1}{2}(n^{\mu}-\bar{n}^{\mu})=(1,0).$
(A.38)

They are normalized and orthogonal to each other

$m^{2}=-\bar{m}^{2}=1,\qquad m\cdot\bar{m}=0.$ (A.39)

Hence, the Cartesian coordinates are related to the retarded coordinates by

$x^{\mu}=rn^{\mu}+u\bar{m}^{\mu},$ (A.40)

and the partial derivatives in Cartesian coordinates can be rewritten as

$\partial_{\mu}=-n_{\mu}\partial_{u}+m_{\mu}\partial_{r}-\frac{1}{r}Y_{\mu}^{A}\partial_{A},$
(A.41)

where

$Y_{\mu}^{A}=-\nabla^{A}n_{\mu}=-\nabla^{A}\bar{n}_{\mu}=-\nabla^{A}m_{\mu}.$
(A.42)

The partial derivatives $\partial_{\mu}$ can also be regarded as the
translation operator in $\mathbb{R}^{1,d-1}$. Therefore, we may also write it
as

$\displaystyle
P_{\mu}=\partial_{\mu}=-n_{\mu}\partial_{u}+m_{\mu}\partial_{r}-\frac{1}{r}Y_{\mu}^{A}\partial_{A}.$
(A.43)

We can construct the following antisymmetric tensors

$\displaystyle n_{\mu\nu}=n_{\mu}\bar{n}_{\nu}-n_{\nu}\bar{n}_{\mu},\quad
Y_{\mu\nu}^{A}=Y_{\mu}^{A}n_{\nu}-Y^{A}_{\nu}n_{\mu},\quad\bar{m}^{A}_{\mu\nu}=Y_{\mu}^{A}\bar{m}_{\nu}-Y_{\nu}^{A}\bar{m}_{\mu},$
(A.44)

in which the tensors $n_{\mu\nu}$ and $\bar{m}_{\mu\nu}^{A}$ are related to
$Y_{\mu\nu}^{A}$ through

$n_{\mu\nu}=-\frac{2}{d-2}\nabla_{A}Y^{A}_{\mu\nu},\qquad\bar{m}^{A}_{\mu\nu}=-\frac{1}{d-2}\nabla^{A}\nabla_{C}Y^{C}_{\mu\nu}.$
(A.45)

The Lorentz algebra is generated by

$\displaystyle L_{\mu\nu}=x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}.$ (A.46)

Using retarded coordinates and the relations (A.40)-(A.41), we find

$\displaystyle
L_{\mu\nu}=-\frac{1}{2}un_{\mu\nu}\partial_{u}+\frac{1}{2}(u+r)n_{\mu\nu}\partial_{r}+(Y_{\mu\nu}^{A}{+}\frac{u}{r}\bar{m}_{\mu\nu}^{A})\partial_{A}.$
(A.47)

To find the CKVs of $S^{m}$, we may pull back the generator $L_{\mu\nu}$ to
$S^{m}$ by setting $u=0$ and $r=1$. Therefore, the CKVs of $S^{m}$ are
$Y_{\mu\nu}^{A}$. More explicitly, they are denoted as $Y_{i}^{A}$ and
$Y_{ij}^{A}$ in the context

$\displaystyle Y_{i}^{A}\equiv Y_{0i}^{A},\quad Y_{ij}^{A}\equiv
Y_{i}^{A}n_{j}-Y_{j}^{A}n_{i}.$ (A.48)

The vectors $Y_{ij}^{A}$ are Killing vectors of $S^{m}$ and they form the
algebra $so(m)$. The vectors $Y_{i}^{A}$ are strictly CKVs of $S^{m}$ and they
obey the equation

$\displaystyle\nabla_{A}Y^{i}_{B}=\nabla_{B}Y^{i}_{A}=\frac{\gamma_{AB}}{d-2}\nabla_{C}Y^{iC}.$
(A.49)

In the following, we collect several important identities about the CKVs.

  1. 1.

Products among $n^{\mu},\bar{n}^{\mu},m^{\mu},\bar{m}^{\mu}$

$\displaystyle n^{2}=\bar{n}^{2}=0,\quad m^{2}=-\bar{m}^{2}=1,$ (A.50a)
$\displaystyle n\cdot\bar{n}=2,\quad n\cdot m=1,\quad
n\cdot\bar{m}=-1,\quad\bar{n}\cdot m=1,\quad\bar{n}\cdot\bar{m}=1,\quad
m\cdot\bar{m}=0.$ (A.50b)

  2. 2.

Orthogonality

$Y_{\mu}^{A}n^{\mu}=Y_{\mu}^{A}\bar{n}^{\mu}=Y_{\mu}^{A}m^{\mu}=Y_{\mu}^{A}\bar{m}^{\mu}=0.$
(A.51)

  3. 3.

Products between $Y_{\mu\nu}^{A}$ and
$n^{\mu},\bar{n}^{\mu},m^{\mu},\bar{m}^{\mu}$

$\displaystyle Y_{\mu\nu}^{A}n^{\nu}=0,\quad
Y_{\mu\nu}^{A}\bar{n}^{\nu}=2Y_{\mu}^{A},\quad
Y^{A}_{\mu\nu}m^{\nu}=Y_{\mu}^{A},\quad
Y_{\mu\nu}^{A}\bar{m}^{\nu}=-Y_{\mu}^{A}.$ (A.52)

  4. 4.

Derivatives of $n_{\mu},\bar{n}_{\mu},m_{\mu},\bar{m}_{\mu}$

$\displaystyle\nabla_{A}\nabla_{B}{n}_{\mu}=\nabla_{A}\nabla_{B}\bar{n}_{\mu}=\nabla_{A}\nabla_{B}{m}_{\mu}=-\gamma_{AB}m_{\mu},$
(A.53)
$\displaystyle\nabla^{A}\nabla_{A}n_{\mu}=\nabla^{A}\nabla_{A}\bar{n}_{\mu}=\nabla^{A}\nabla_{A}m_{\mu}=-(d-2)m_{\mu},$
(A.54)
$\displaystyle\nabla_{A}\nabla_{B}\bar{m}_{\mu}=0,\qquad\nabla^{A}\nabla_{A}\bar{m}_{\mu}=0.$
(A.55)

  5. 5.

Products between $Y_{\mu}^{A}$ and $Y_{\mu\nu}^{A}$

$\displaystyle Y_{\mu}^{A}Y^{\mu B}=\gamma^{AB},\quad Y_{\mu}^{A}Y_{\nu
A}=\eta_{\mu\nu}-\frac{1}{2}(n_{\mu}\bar{n}_{\nu}+n_{\nu}\bar{n}_{\mu}),$
(A.56a) $\displaystyle Y_{\mu\nu}^{A}Y^{\mu B}=\gamma^{AB}n_{\nu}.$ (A.56b)

  6. 6.

Derivatives of $Y_{\mu}^{A}$

$\displaystyle\nabla_{A}Y_{B}^{\mu}=\gamma_{AB}m^{\mu},\quad\nabla_{A}Y^{A\mu}=(d-2)m^{\mu},$
(A.57a) $\displaystyle Y_{\mu}^{A}\nabla_{A}Y^{\mu B}=0.$ (A.57b)

  7. 7.

Derivatives of $Y_{\mu\nu}^{A}$

$\displaystyle\nabla^{A}Y^{B}_{\mu\nu}=-\frac{1}{2}\gamma^{AB}n_{\mu\nu}+2Y^{A}_{[\mu}Y^{B}_{\nu]},\qquad\nabla_{A}Y_{\mu\nu}^{A}=-\frac{(d-2)}{2}n_{\mu\nu}.$
(A.58)

##  Appendix B Mode expansion

###  B.1 Null infinity

In Minkowski spacetime $\mathbb{R}^{1,d-1}$, the scalar field is may be
expanded in plane waves