# How the mass of a scalar field influences Resonance Casimir-Polder
interaction in Schwarzschild spacetime

Arpan Chatterjee<EMAIL_ADDRESS>Saptarshi Saha
<EMAIL_ADDRESS>Chiranjeeb Singha<EMAIL_ADDRESS>Department of Physical Sciences, Indian Institute of Science Education and
Research Kolkata, Mohanpur - 741 246, WB, India

###### Abstract

_Resonance Casimir-Polder interaction(RCPI)_ occurs in nature when one or more
atoms are in their excited states and the exchange of real photon is involved
between them due to vacuum fluctuations of the quantum fields. In recent
times, many attempts have been made to show that the curved spacetime such as
the _de-Sitter spacetime_ can be separated from a _thermal Minkowski
spacetime_ using _RCPI_. Motivated from these ideas, here we study the _RCPI_
between two atoms that interact with a massive scalar field in Schwarzschild
spacetime provided the atoms are placed in the near-horizon region.
Subsequently, we use the tool of the _open quantum system_ and calculate the
Lamb shift of the atomic energy level of the entangled states. We show that
the behavior of _RCPI_ modifies depending on the mass of the scalar field. In
the high mass limit, the interaction becomes short-range and eventually
disappears beyond a characteristic length scale of $1/m$, where $m$ is the
mass of the scalar field.

###### pacs:

04.62.+v, 04.60.Pp

##  I Introduction

The phenomena of _Casimir Polder interaction (CPI)_ occurs between an atom and
a conducting plate due to vacuum fluctuation of the quantized fields [1, 2,
3]. It has major applications from the field of quantum electrodynamics (QED)
to the signature of gravity [4, 5, 6]. _CPI_ mimics van der Waals force in
non-relativistic regime where a dispersive force is acting in few nanometer
length scale [7, 8]. In physical chemistry it has also huge implications in
atomic adsorption and desorption in the thermally excited surfaces [9].
Experimental realization exists in the case of the motion of Bose-Einstein
condensate (BEC) under surface _CPI_ [10]. The other successful efforts have
been made to probe more complicated contexts like detecting spacetime
curvature [11, 6, 12], Unruh effect [13, 14, 15], Hawking radiation of a black
hole and checking thermal and nonthermal scaling in a black hole spacetime
[16, 17]. One can show that the background spacetime and the relativistic
motion of interacting systems can modify the _CPI_ [13, 14, 15, 11, 6, 12, 17,
16, 18, 19, 20, 21, 22].

_CPI_ generates correlations between multiple quantum systems [23, 24, 25].
Two uncorrelated atom also can be entangled if they both share common
environment [26, 27]. Recently Lindblad master equation approach is [28]
successfully used to explain the interatomic correlations at Schwarzschild
spacetime where vacuum fluctuation of the quantum field takes the vital role
to create the entanglement between them. This interatomic correlations are the
source of _RCPI_ [6, 29]. Subsequently it has been shown that an uniformly
accelerated atom behaves like a system in a thermal bath (_Thermalization
theorem_) [30]. Therefore, _RCPI_ is exactly equal to the second order shift
(Lamb shift) [31] due to system-field interaction Hamiltonian [32, 33, 6].

In this paper following the recent works, [6, 29], we theoretically
investigate the _RCPI_ between two atoms that interact with a massive scalar
field in Schwarzschild spacetime. Two atoms are initially uncorrelated and
interact individually with the scalar field. The cross-terms of the two
individual system-field coupling Hamiltonian in the second-order of the
quantum master equation will give rise to the entanglement between the atoms
[27]. Here we show how the interatomic correlations between the atoms depend
on the mass of the scalar field by calculating the Lamb shift due to system-
field coupling.

We organize the paper in the following way. In section-II we give a general
description of _open quantum system_ approach-GKSL (Gorini-Kossakowski-
Sudarshan-Lindblad) formalism [34, 35, 36] for the two-atom system then
discuss the common environment effect and finally compute the shifts of the
energy level of the entangled states due to Lamb shift Hamiltonian. In
section- III we review the form of the Schwarzschild metric in the near
horizon and find the expression of the two-point correlation function for a
massive scalar field. Recently it has been shown that, for Schwarzschild
spacetime, beyond a characteristic length scale which is proportional to the
inverse of the surface gravity $\kappa$, the _RCPI_ between two entangled
atoms is characterized by a $1/L^{2}$ power-law provided the atoms are located
close to the horizon [29]. However, the length scale limit beyond the
characteristic value is not compatible with the local flatness of the
spacetime. A massless scalar field has been considered in the ref. [29] to
calculate _RCPI_. In section-IV we compute _RCPI_ between two atoms that
interact with a massive scalar field and point out its dependence on the mass
term of the scalar field. Subsequently, we show that the _RCPI_ becomes short-
range and eventually disappears beyond a characteristic length scale of $1/m$,
$m$ being the mass of the scalar field.

##  II Dynamics of a two-atom system

We consider a system of two-atom weakly interacts with a common environment
but they do not have any mutual interaction. Here we choose the environment to
be a quantized massive scalar field. This is a general description of the
_open quantum system_ [37], therefore, we follow the formalism of the quantum
master equation [31]. The Hamiltonian of the full system can be expressed as,

$\displaystyle\mathcal{H}=\mathcal{H}_{\text{\tiny
S}}^{0}+\mathcal{H}_{\text{\tiny L}}^{0}+\mathcal{H}_{\text{\tiny SL}}~{}.$
(1)

We assume the atoms have the same Zeeman levels and throughout the paper, we
use the natural units, $\hbar=c=1$. The free Hamiltonian of the system can be
written as $\mathcal{H}_{\text{\tiny S}}^{0}=\mathcal{H}_{\text{\tiny
S}}^{01}+\mathcal{H}_{\text{\tiny
S}}^{02}=\frac{1}{2}\omega_{0}\sigma_{3}^{(1)}+\frac{1}{2}\omega_{0}\sigma_{3}^{(2)}$
(The superscript in the Pauli matrices represent the atom number). Here
$\sigma_{i}$ are the Pauli matrices, $\omega_{0}$ is the frequency of Zeeman
levels and we defined $|g\rangle,|e\rangle$ as the corresponding ground and
excited state of the atoms. The free Hamiltonian of the scalar field
$\mathcal{H}_{\text{\tiny L}}^{0}$ can be written as [38],

$\displaystyle\mathcal{H}_{\text{\tiny
L}}^{0}=\int\frac{d^{3}k}{(2\pi)^{3}}\omega_{k}a^{\dagger}(k)a(k)~{},$ (2)

where $\omega_{k}=\sqrt{k^{2}+m^{2}}$, the frequency of the scalar field. Here
m is the mass of the scalar field and $a^{\dagger},a$ are the creation and
annihilation operator of the quantized field respectively. The coupling
Hamiltonian between the atom-scalar field can be defined as [28],

$\displaystyle\mathcal{H}_{\text{\tiny
SL}}=\lambda\sum\limits_{\mu=0}^{3}\big{[}\sigma_{\mu}^{(1)}\otimes\phi(x_{1})+\sigma_{\mu}^{(2)}\otimes\phi(x_{2})\big{]}~{},$
(3)

where $\lambda$ is the coupling constant and $\phi$ represents the scalar
field. Initially, the atoms and the scalar field are separated from each
other. Therefore, the initial density matrix can be written as
$\rho(0)=\rho_{s}(0)\otimes|0\rangle\langle 0|$ (“Born-Markov” approximation
[37]). Here $|0\rangle$ is the vacuum state of the massive scalar field and
$\rho_{s}(0)$ is the initial density matrix of the system. For a closed
system, “Von-Neumann-Liouville” equation is used to describe the full
dynamics,

$\displaystyle\frac{d\rho(t)}{dt}$ $\displaystyle=$
$\displaystyle-i[\mathcal{H}_{\text{\tiny S}}^{0}+\mathcal{H}_{\text{\tiny
L}}^{0}+\mathcal{H}_{\text{\tiny SL}},\rho(t)].$ (4)

Here, $\rho(t)$ is the total density matrices. Starting from the equation (4)
we perturbatively expand the time evolution operator
($U(t,t_{0})=\mathcal{T}\exp\\{-i\int^{t}_{t_{0}}dt^{\prime}(\mathcal{H}_{\text{\tiny
S}}^{0}+\mathcal{H}_{\text{\tiny L}}^{0}+\mathcal{H}_{\text{\tiny SL}})\\}$)
to the second order of coupling Hamiltonian [31] and taking trace over field
degrees of freedom, we get the quantum master equation, which is also known as
GKSL equation [34, 35, 36]. The quantum master equation in the lab-frame with
a proper-time $\tau$ can be expressed as [31],

$\displaystyle\frac{d\rho_{s}(\tau)}{d\tau}=-i[\mathcal{H}_{\text{\tiny
S}}^{0}+\mathcal{H}_{lamb},\rho_{s}(\tau)]+\mathcal{L}\big{(}\rho_{s}(\tau)\big{)}~{}.$
(5)

Here, $\mathcal{H}_{lamb}$ is the Lamb-shift Hamiltonian that leads to the
renormalization of Zeeman Hamiltonian and
$\mathcal{L}\big{(}\rho_{s}(\tau)\big{)}$ is the dissipator of the master-
equation which can be written as [28],

$\displaystyle\mathcal{L}\big{(}\rho_{s}\big{)}$ $\displaystyle=$
$\displaystyle\sum\limits_{a,b=1}^{2}\sum\limits_{j,k=1}^{3}\gamma^{ab}_{jk}\Big{(}\sigma_{b}^{k}\rho_{s}\sigma_{a}^{j}-\frac{1}{2}\\{\sigma_{a}^{j}\sigma_{b}^{k},\rho_{s}\\}\Big{)}~{},$
(6) $\displaystyle\mathcal{H}_{lamb}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\sum\limits_{a,b=1}^{2}\sum\limits_{j,k=1}^{3}\mathcal{S}_{jk}^{ab}\sigma_{a}^{j}\sigma_{b}^{k}~{},$
(7)

where $\mathcal{S}_{jk}^{ab}$ and $\gamma^{ab}_{jk}$ depend on the Fourier
transforms of the two-point correlation functions of the scalar field. The
shift terms and dissipative terms are coming from second-order terms of atom-
field coupling Hamiltonian and they are _Kramer-kronig_ pair to each-other
[37]. As we are interested in the shift between the energy levels of the two-
atom system, our main focus is to demonstrate the Lamb-shift term and not to
think about decay terms. However, the decay terms are responsible for taking
the two-atom system to the equilibrium configuration, which is the
_Bekenstein-Hawking temperature_ [30] of the massive scalar field. Lamb shift
can be calculated from the Hilbert transforms of the Fourier transforms of the
two-point functions which are shown below,

$\displaystyle\mathcal{K}^{ab}(\omega_{0})=\frac{\mathcal{P}}{\pi
i}\int\limits^{\infty}_{-\infty}d\omega\frac{\mathcal{G}^{ab}(\omega)}{\omega-\omega_{0}}~{}.$
(8)

Here $\mathcal{P}$ denotes the principal value. $\mathcal{G}^{ab}(\omega)$ are
the Fourier transforms of the two-point correlation functions of the scalar
field. The Fourier transforms of the two-point functions are given by,

$\displaystyle\mathcal{{G}}^{ab}(\omega)$ $\displaystyle=$
$\displaystyle\int\limits^{\infty}_{-\infty}d\Delta\tau~{}e^{i\omega\Delta\tau}~{}{G}^{ab}(\Delta\tau)~{},$
(9) $\displaystyle\text{where,}\,\,\,\,G^{ab}(\Delta\tau)$ $\displaystyle=$
$\displaystyle\langle{\Phi}(\tau,\mathbf{x}_{a}){\Phi}(\tau^{\prime},\mathbf{x}_{b})\rangle~{},$
(10)

and $\Delta\tau=(\tau-\tau^{\prime})$. The exact form of
$\mathcal{S}^{ab}_{jk}$ can be written as [6],

$\displaystyle\mathcal{S}^{ab}_{jk}=A^{ab}\delta_{jk}-iB^{ab}\epsilon_{jkl}\delta_{3l}-A^{ab}\delta_{3j}\delta_{3k}~{},$
(11)

where the terms $A^{ab}$ and $B^{ab}$ are given by,

$\displaystyle A^{ab}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4}\left[\mathcal{K}^{ab}(\omega_{0})+\mathcal{K}^{ab}(-\omega_{0})\right],$
(12) $\displaystyle B^{ab}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4}\left[\mathcal{K}^{ab}(\omega_{0})-\mathcal{K}^{ab}(-\omega_{0})\right].$
(13)

###  II.1 Entanglement between two atoms through a massive scalar field

Two atoms are initially uncorrelated but due to interaction with the massive
scalar field, the energy levels become correlated. So, there is a formation of
field-induced entanglement between the atoms which depends on the two-point
correlation functions [27]. The two-point functions can be computed along the
trajectories of the atoms thus they depend on the spacetime background. Now we
aim to calculate the expectation value of $\mathcal{H}_{lamb}$ in the
entangled states to investigate the effect of interatomic correlations between
two atoms in Schwarzschild spacetime. Here we take the symmetric and anti-
symmetric state of the dipolar-coupled Hamiltonian for simplicity because
other states are unchanged due to interatomic correlations [39]. The symmetric
and anti-symmetric states of two atom-system are given by,

$\displaystyle|E\rangle=\frac{|e_{1}\rangle|g_{2}\rangle+|g_{1}\rangle|e_{2}\rangle}{\sqrt{2}}~{};\quad|A\rangle=\frac{|e_{1}\rangle|g_{2}\rangle-|g_{1}\rangle|e_{2}\rangle}{\sqrt{2}}~{}.$

One can find the expressions for energy level shifts as [6],

$\displaystyle\delta E_{S_{LS}}=\langle E|\mathcal{H}_{lamb}|E\rangle$
$\displaystyle=-\frac{i}{2}\left[\sum^{3}_{j=1}\left(\mathcal{S}^{12}_{jj}+\mathcal{S}^{21}_{jj}+\mathcal{S}^{11}_{jj}+\mathcal{S}^{22}_{jj}\right)-2(\mathcal{S}^{12}_{33}+\mathcal{S}^{21}_{33})\right],$
$\displaystyle\delta E_{A_{LS}}=\langle A|\mathcal{H}_{lamb}|A\rangle$
$\displaystyle=\frac{i}{2}\left[\sum^{3}_{j=1}\left(\mathcal{S}^{12}_{jj}+\mathcal{S}^{21}_{jj}-\mathcal{S}^{11}_{jj}-\mathcal{S}^{22}_{jj}\right)\right].$
(14)

Here $\delta E_{S_{LS}}$ stands for the second-order energy level shift of the
symmetric state and $\delta E_{A_{LS}}$ stands for corresponding energy level
shift of the antisymmetric state. In the next section we compute these shifts
of the energy level of the entangled states for a massive scalar field in
Schwarzschild spacetime when the atoms are located close to the horizon.

##  III Schwarzschild metric in near horizon region

Here we consider (3+1) dimensional Schwarzschild spacetime to compute the
_RCPI_ between two entangled atoms interact with a massive scalar field when
the atoms are located close to the horizon. The Schwarzschild spacetime is
described by the line element,

$d{s}^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}~{},$ (15)

where, $f(r)=\left(1-r_{s}/r\right)$ and $r_{s}=2GM$ is the Schwarzschild
radius related to the line element. A proper distance $l$ from the horizon to
a radial distance $r$ is defined by the formula [40],

$\displaystyle
l=\sqrt{r(r-r_{s})}+r_{s}\sinh^{-1}(\sqrt{\frac{r}{r_{s}}-1})~{}.$ (16)

In terms of $l$, the Schwarzschild metric (15) becomes

$d{s}^{2}=-f(r)dt^{2}+dl^{2}+r^{2}(l)d\theta^{2}+r^{2}(l)\sin^{2}\theta
d\phi^{2}~{},$ (17)

where, $f(r)=\left(1-r_{s}/r(l)\right)$. Now near the horizon, where
$r=r_{s}+\delta$, $l=2\sqrt{r_{s}\delta}$ and on a small angular region which
is around $\theta\sim 0$, the new coordinates have been defined as [29],

$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
l\cosh\frac{t}{2r_{s}}~{};~{}~{}T=l\sinh\frac{t}{2r_{s}}~{};$ $\displaystyle
X_{2}$ $\displaystyle=$ $\displaystyle
r_{s}\theta\cos\phi~{};~{}~{}X_{3}=r_{s}\theta\sin\phi~{}.$ (18)

Using this coordinates (III), the line element (15) becomes [29],

$d{s}^{2}=-dT^{2}+dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2}~{},$ (19)

which expresses the Minkowski spacetime [40, 41, 42, 43, 29].

###  III.1 Two point correlation function for the massive scalar field

In the position space, using the coordinates of the inertial metric (19), the
two-point function for a massive scalar field of mass $m$ can be expressed as
[38],

$\displaystyle G(x,x^{\prime})$ $\displaystyle\equiv$ $\displaystyle\langle
0|\hat{\Phi}(x)\hat{\Phi}(x^{\prime})|0\rangle=\langle
0|\hat{\Phi}(T,\mathbf{x})\hat{\Phi}(T^{\prime},\mathbf{x}^{\prime})|0\rangle$
(20) $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{\frac{3}{2}}}\delta(k^{2}-m^{2})e^{-ik(x-x^{\prime})}~{}.$

After integrating the above expression, the two-point function for the scalar
field can be written as [44],

$\displaystyle
G(x-x^{\prime})=-\frac{im}{4\pi^{2}}\frac{K_{1}(im\sqrt{(T-T^{\prime}-i\epsilon)^{2}-(\mathbf{x}-\mathbf{x}^{\prime})^{2}})}{\sqrt{(T-T^{\prime})^{2}-(\mathbf{x}-\mathbf{x}^{\prime})^{2}}}~{}.$

Here $K_{1}$ is the modified Bessel function of the second kind,
$(\mathbf{x}-\mathbf{x}^{\prime})^{2}=(X_{1}-X_{1}^{\prime})^{2}+(X_{2}-X_{2}^{\prime})^{2}+(X_{3}-X_{3}^{\prime})^{2}$
and $\epsilon$ is a small, positive parameter which is introduced to evaluate
two-point function. In the limit $mR<<1$ the function behaves like the two-
point correlation function (III.1) for a massless scalar field case [45]. In
this limit, the expression of two-point function is shown below,

$\displaystyle G(x-x^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi^{2}R^{2}}~{}.$ (22)

On the other hand in the high mass limit, $mR>>1$, the correlation function
(III.1) is given by [45],

$\displaystyle G(x-x^{\prime})$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
R}\Big{(}\frac{im}{2\pi R}\Big{)}^{\frac{1}{2}}e^{-imR}~{},$
$\displaystyle\text{where,}\qquad R$ $\displaystyle=$
$\displaystyle\sqrt{(T-T^{\prime})^{2}-(\mathbf{x}-\mathbf{x}^{\prime})^{2}}~{}.$
(23)

Using equation (III) $R$ can be written as [29],

$\displaystyle R=2l\sinh\frac{(\tau-\tau^{\prime})}{2l}~{}.$ (24)

Now to avoid divergence at $(\tau-\tau^{\prime}=0),R$ is transformed by the
following transformation ($\tau$ is the proper time),

$\displaystyle R=2l\sinh\frac{(\tau-\tau^{\prime})}{2l}+i\epsilon~{}.$ (25)

Here $1/l$ behaves as the acceleration of the system and $\epsilon$ is a small
positive quantity. So,

$\displaystyle|G(x-x^{\prime})|$ $\displaystyle=$
$\displaystyle\Big{(}\frac{m}{32\pi^{3}|R|^{3}}\Big{)}^{\frac{1}{2}}e^{-m\epsilon}~{},$
$\displaystyle\text{where}\qquad|R|^{2}$ $\displaystyle=$
$\displaystyle\epsilon^{2}+\Big{(}2l\sinh\frac{(\tau-\tau^{\prime})}{2l}\Big{)}^{2}.$
(26)

Using Lebesgue’s bounded convergence theorem the Fourier transform of
correlation function goes to zero for the large mass limit [45, 30], which
also indicates in high mass limit _RCPI_ may disappear.

##  IV _RCPI_ for the two-atom system

Two static atom placed in two different position
$\to(r,\theta,\phi)\,\,\text{and}\,\,(r,\theta^{\prime},\phi)$ which are close
to the horizon. Here $\theta$ and $\theta^{\prime}$ are assumed to be small.
The response function for a detector (atom) per unit time $(T_{0})$ is given
by [44],

$\displaystyle\frac{\mathcal{F}(\omega)}{T_{0}}$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{\infty}\frac{dk\,k^{2}}{2\pi\sqrt{k^{2}+m^{2}}}|\beta_{k}|^{2}\delta(\omega-\sqrt{k^{2}+m^{2}})~{}.$
(27)

From the analogy of _thermalization theorem_ [30] $|\beta_{k}^{2}|$ is the
number of particle with mode $k$ of the isotropic bath with temperature
$\mathcal{T_{H}}=\frac{1}{2\pi l}$ in the static space time. Following the
above discussion, the Fourier transforms of the two-point correlation
functions (10) for these two spacetime points can be written as,

$\displaystyle\mathcal{G}^{11}(\omega)$ $\displaystyle=$
$\displaystyle\mathcal{G}^{22}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\frac{\Omega(\omega,m)}{1-e^{-2\pi
l\Omega(\omega,m)}}~{},$ $\displaystyle\mathcal{G}^{12}(\omega)$
$\displaystyle=$ $\displaystyle\mathcal{G}^{21}(\omega)$ (28) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\frac{\Omega(\omega,m)}{1-e^{-2\pi
l\Omega(\omega,m)}}g(\Omega(\omega,m),L/2)~{}.$

Here we define $\Omega(\omega,m)=\sqrt{\omega^{2}-m^{2}}\,\,\theta(\omega-m)$.
For $m=0$ the expression exactly matches with the ref. [29]. From the
mathematical point of view $\omega$ is replaced with
$\sqrt{\omega^{2}-m^{2}}\,\,\theta(\omega-m)$ for this case. We also define
$g\big{(}\Omega(\omega,m),z\big{)}=\frac{\sin[2l\,\Omega(\omega,m)\sinh^{-1}(z/l)]}{2z\,\Omega(\omega,m)\sqrt{1+z^{2}/l^{2}}}$
and $L$ denotes the proper distance between $(r,\theta,\phi)$ and
$(r,\theta^{\prime},\phi)$. Calculating the Hilbert transforms (8) of the
above functions (28) the coefficients (12,13) can be written as,

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}\mathcal{P}}{8\pi^{2}i}\int^{\infty}_{-\infty}d\omega\,\Omega(\omega,m)\left(\frac{1}{\omega-\omega_{0}}+\frac{1}{\omega+\omega_{0}}\right)\times$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\frac{1}{1-e^{-2\pi
l\Omega(\omega,m)}}~{},$ $\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}\mathcal{P}}{8\pi^{2}i}\int^{\infty}_{-\infty}d\omega\,\Omega(\omega,m)\left(\frac{1}{\omega-\omega_{0}}-\frac{1}{\omega+\omega_{0}}\right)\times$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\frac{1}{1-e^{-2\pi
l\Omega(\omega,m)}}~{},$ $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}\mathcal{P}}{8\pi^{2}i}\int^{\infty}_{-\infty}d\omega\,\Omega(\omega,m)\left(\frac{1}{\omega-\omega_{0}}+\frac{1}{\omega+\omega_{0}}\right)\times$
$\displaystyle\qquad\qquad\frac{1}{1-e^{-2\pi
l\Omega(\omega,m)}}g(\Omega(\omega,m),L/2)~{},$ $\displaystyle B_{2}$
$\displaystyle=$
$\displaystyle\frac{\lambda^{2}\mathcal{P}}{8\pi^{2}i}\int^{\infty}_{-\infty}d\omega\,\Omega(\omega,m)\left(\frac{1}{\omega-\omega_{0}}-\frac{1}{\omega+\omega_{0}}\right)\times$
(29) $\displaystyle\qquad\qquad\frac{1}{1-e^{-2\pi
l\Omega(\omega,m)}}g(\Omega(\omega,m),L/2)~{}.$

Here,
$A^{11}=A^{22}=A_{1},~{}A^{12}=A^{21}=A_{2},~{}B^{11}=B^{22}=B_{1},~{}B^{12}=B^{21}=B_{2}$.
Put the coefficients (29) in the following equations (11),

$\displaystyle\mathcal{S}^{11}_{jk}=\mathcal{S}^{22}_{jk}=A_{1}\delta_{jk}-iB_{1}\epsilon_{jkl}\delta_{3l}-A_{1}\delta_{3j}\delta_{3k}~{},$
$\displaystyle\mathcal{S}^{12}_{jk}=\mathcal{S}^{21}_{jk}=A_{2}\delta_{jk}-iB_{2}\epsilon_{jkl}\delta_{3l}-A_{2}\delta_{3j}\delta_{3k}~{},$
(30)

the shifts of the energy level of the symmetric state and the anti-symmetric
state (14) of the two-atom system are given by,

$\displaystyle\delta E_{S_{LS}}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{4\pi^{2}}\int^{\infty}_{0}d\omega\,\Omega(\omega,m)\left(\frac{1}{\omega-\omega_{0}}+\frac{1}{\omega+\omega_{0}}\right)\times$
$\displaystyle\qquad\qquad\qquad\left[g(\Omega(\omega,m),L/2)+1\right]~{},$
$\displaystyle\delta E_{A_{LS}}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4\pi^{2}}\int^{\infty}_{0}d\omega\,\Omega(\omega,m)\left(\frac{1}{\omega-\omega_{0}}+\frac{1}{\omega+\omega_{0}}\right)\times$
(31)
$\displaystyle\qquad\qquad\qquad\left[g(\Omega(\omega,m),L/2)-1\right]~{}.$

These shifts depend on the proper length $L$. To calculate the Casimir-Polder
force between the atoms we need to take a derivative with respect to $L$ [6].
So, we neglect the terms which do not depend on $L$. From the above
discussion, the Lamb shift of the symmetric and anti-symmetric state of the
two-atom system is then given by,

$\displaystyle\delta E_{S}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{4\pi^{2}}\int\limits_{0}^{\infty}d\omega\,\Omega(\omega,m)\Big{(}\frac{1}{\omega-\omega_{0}}+\frac{1}{\omega+\omega_{0}}\Big{)}\times$
$\displaystyle\qquad\qquad\qquad g(\Omega(\omega,m),L/2)~{},$
$\displaystyle\delta E_{A}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4\pi^{2}}\int\limits_{0}^{\infty}d\omega\,\Omega(\omega,m)\Big{(}\frac{1}{\omega-\omega_{0}}+\frac{1}{\omega+\omega_{0}}\Big{)}\times$
(32) $\displaystyle\qquad\qquad\qquad g(\Omega(\omega,m),L/2)~{}.$

After substituting $\omega^{2}-m^{2}=z^{2}$, the integral form look like,

$\displaystyle\mathcal{I}=\mp\frac{\lambda^{2}}{2\pi^{2}}\int\limits_{0}^{\infty}dz\frac{z}{z^{2}+m^{2}-\omega_{0}^{2}}\,\eta\sin(\alpha
z)~{},$ (33)

where $\eta=\frac{1}{2L\sqrt{1+\frac{L^{2}}{4l^{2}}}}$ and
$\alpha=2l\sinh^{-1}(\frac{L}{2l})$. Previously from the description of the
response function per unit time of a scalar field (28), the term
$\theta(\omega-m)$ suggests that momentum cannot be imaginary but now in the
expression of the integrand (33) there is no restriction of theta function on
$(m^{2}-\omega_{0}^{2})$, it can be either positive or negative or zero.

Figure 1:  Dependence of energy shifts _w.r.t_ mass of the scalar field for
$\omega_{0}^{2}>m^{2}$. (a) for $L>>l$ we take $L=1000$, $l=1$,
$\omega_{0}=10$ and $m$ is varying from 1 to 10 (b) for $l>>L$ we take $L=1$,
$l=1000$, $\omega_{0}=100$ and $m$ is varying from 1 to 100. Here
$m,~{}L,~{}l,~{}\omega_{0}$ are in arbitrary units.  Figure 2:  Dependence of
energy shifts _w.r.t_ mass of the scalar field for $\omega_{0}^{2}<m^{2}$. (a)
for $L>>l$ we take $L=1000$, $l=1$, $\omega_{0}=1$ and $m$ is varying from 1
to 1.1 (b) for $l>>L$ we take $L=1$, $l=1000$, $\omega_{0}=1$ and $m$ is
varying from 1 to 10. Here $m,~{}L,~{}l,~{}\omega_{0}$ are in arbitrary units.

###  IV.1 Case-I, $\omega_{0}^{2}>m^{2}$

For case-I, the poles of the integrand (33) lie on the real-line. Therefore,
the expressions for Lamb shift of two atoms are given by,

$\displaystyle\delta E_{S}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{4\pi
L\sqrt{1+\frac{L^{2}}{4l^{2}}}}\cos(2\sqrt{\omega_{0}^{2}-m^{2}}\,\,l\sinh^{-1}\big{(}L/2l\big{)})~{},$
$\displaystyle\delta E_{A}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4\pi
L\sqrt{1+\frac{L^{2}}{4l^{2}}}}\cos(2\sqrt{\omega_{0}^{2}-m^{2}}\,\,l\sinh^{-1}\big{(}L/2l\big{)}).$

We also note that there exists a characteristic length scale $l$, which up to
the leading order approximation is equal to
$\frac{\sqrt{1-\frac{r_{s}}{r}}}{\kappa}$, where $\kappa=\frac{1}{2r_{s}}$ is
the surface gravity [29]. To analyze the behavior of the _RCPI_ , here we also
consider both the limits of the proper distance between the atoms which are
either larger or smaller than the characteristic length scale. From equation
(IV.1), it is shown that in the limit $L>>l$ (or
$\Delta\theta>>2\sqrt{\frac{\delta}{r_{s}}}$) [29], the expressions are,

$\displaystyle\delta E_{S}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}~{}l}{2\pi
L^{2}}\cos(2\sqrt{\omega_{0}^{2}-m^{2}}\,\,l\log(L/l))~{},$
$\displaystyle\delta E_{A}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}~{}l}{2\pi
L^{2}}\cos(2\sqrt{\omega_{0}^{2}-m^{2}}\,\,l\log(L/l))~{},$ (35)

and in the limit $L<<l$ (or $\Delta\theta<<2\sqrt{\frac{\delta}{r_{s}}}$),

$\displaystyle\delta E_{S}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{4\pi
L}\cos(\sqrt{\omega_{0}^{2}-m^{2}}\,\,L)~{},$ $\displaystyle\delta E_{A}$
$\displaystyle=$ $\displaystyle\frac{\lambda^{2}}{4\pi
L}\cos(\sqrt{\omega_{0}^{2}-m^{2}}\,\,L)~{}.$ (36)

The schematic diagrams for both the limits are shown in FIG. 1.

###  IV.2 Case-II, $\omega_{0}^{2}<m^{2}$

For case-II, the poles in the integrand (33) lie on the imaginary axis, the
expressions for _RCPI_ are given by,

$\displaystyle\delta E_{S}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{4\pi
L\sqrt{1+\frac{L^{2}}{4l^{2}}}}e^{-2\sqrt{m^{2}-\omega_{0}^{2}}\,\,l\sinh^{-1}\big{(}L/2l\big{)}}~{},$
$\displaystyle\delta E_{A}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4\pi
L\sqrt{1+\frac{L^{2}}{4l^{2}}}}e^{-2\sqrt{m^{2}-\omega_{0}^{2}}\,\,l\sinh^{-1}\big{(}L/2l\big{)}}~{}.$
(37)

The expressions have an exponential decaying term, which indicates the same
kind of result of the two-point correlation function at $mR>>1$. Therefore,
the correlation function and RCPI vanish in the large mass limit. From
equation (IV.1), in the limit $L>>l$ (or
$\Delta\theta>>2\sqrt{\frac{\delta}{r_{s}}}$), the expressions are given by,

$\displaystyle\delta E_{S}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}~{}l}{2\pi
L^{2}}e^{-2\sqrt{m^{2}-\omega_{0}^{2}}\,\,l\log(L/l)}~{},$
$\displaystyle\delta E_{A}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}~{}l}{2\pi
L^{2}}e^{-2\sqrt{m^{2}-\omega_{0}^{2}}\,\,l\log(L/l)}~{},$ (38)

and in the limit $L<<l$ (or $\Delta\theta<<2\sqrt{\frac{\delta}{r_{s}}}$),

$\displaystyle\delta E_{S}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{4\pi
L}e^{-\sqrt{m^{2}-\omega_{0}^{2}}\,\,L}~{},$ $\displaystyle\delta E_{A}$
$\displaystyle=$ $\displaystyle\frac{\lambda^{2}}{4\pi
L}e^{-\sqrt{m^{2}-\omega_{0}^{2}}\,\,L}~{}.$ (39)

The schematic diagrams for both the limits are shown in FIG. 2 .

###  IV.3 Case-III, $\omega_{0}^{2}=m^{2}$

For case-III, the pole of the integrand (33) is at the origin, $z=0$. The
expressions for _RCPI_ are shown below,

$\displaystyle\delta E_{S}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{4\pi L\sqrt{1+\frac{L^{2}}{4l^{2}}}},$
$\displaystyle\delta E_{A}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4\pi L\sqrt{1+\frac{L^{2}}{4l^{2}}}}.$ (40)

From equation (IV.2), in the limit $L>>l$ (or
$\Delta\theta>>2\sqrt{\frac{\delta}{r_{s}}}$), the expressions are given by,

$\displaystyle\delta E_{S}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}~{}l}{2\pi L^{2}},$ $\displaystyle\delta
E_{A}$ $\displaystyle=$ $\displaystyle\frac{\lambda^{2}~{}l}{2\pi L^{2}},$
(41)

and in the limit $L<<l$ (or $\Delta\theta<<2\sqrt{\frac{\delta}{r_{s}}}$),

$\displaystyle\delta E_{S}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{4\pi L},$ $\displaystyle\delta E_{A}$
$\displaystyle=$ $\displaystyle\frac{\lambda^{2}}{4\pi L}~{}.$ (42)

Both the FIG. 1 and FIG. 2 merge at the same point for $\omega_{0}=m$. We also
check analytically the pair of equations, (IV.1,IV.2) are equal with the
expression (IV.3) and similarly for (IV.1,IV.2) are equal with (IV.3).
However, The length scale limit beyond a characteristic value ($L>>l$) for the
three cases (IV.1,IV.2,IV.3) is not compatible with the local flatness of the
spacetime [29].

##  V Discussions

From the analogy of _equivalence principle_ we can say an accelerating
observer mimics an observer in a gravitational field and from _thermalization
theorem_ we find a connection between an accelerating atom and a quantum
system connected to a thermal bath. Therefore, we use the quantum master
equation approach to describe the dynamics of an accelerating two-atom system
interacting with a massive scalar field. We find the expressions for the
second-order shift term for system-field coupling Hamiltonian. In the presence
of vacuum fluctuations of the field, the effective shift terms are responsible
for the interatomic correlations between the atoms. Here we calculate the
_RCPI_ for the Schwarzschild spacetime in the near-horizon region. From the
recent studies, the _RCPI_ using a massless scalar field beyond a
characteristic length scale follows a $1/L^{2}$ power-law behavior [29]. The
length-scale is equivalent to the inverse of the surface gravity $\kappa$ in
Schwarzschild spacetime. However, the limit is not compatible with the locally
flat space-time. For the massive scalar field, we also find the same kind of
behavior. With this characteristic length scale, we also introduce a mass
dependence of _RCPI_ , which serves another independent length scale in the
dynamics. The key point we have addressed here is both the limits
($\omega_{0}^{2}>m^{2},\omega_{0}^{2}<m^{2}$) merge to a constant-value which
is independent of the mass of scalar field and the frequency of Zeeman levels.
There is a competition between the frequency of Zeeman levels and the mass of
the scalar field when they are equal the energy-shifts become maximum. This
particular value exactly shows the behavior of a two-atom system whose
frequency of Zeeman levels is zero in a massless scalar field. In summary, we
have shown that in the low mass limit it exactly follows the behavior reported
in ref. [29] with the frequency modulated by a mass term and in the high mass
limit, the behavior of _RCPI_ becomes short-range and eventually disappears
beyond a characteristic length scale of $1/m$. So, the atoms become
uncorrelated in the large mass limit.

###### Acknowledgements.

We sincerely thank Rangeet Bhattacharyya, Golam Mortuza Hossain, Sumanta
Chakraborty, Siddharth Lal, Bibhas Ranjan Majhi, Ashmita Das, Arnab
Chakrabarti and Chandramouli Chowdhury for many useful discussions. AC and SS
thank University Grant Commission (UGC) for a junior research fellowship.

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$\lambda\in E_{n}$,}\\\
\mathfrak{sl}(W^{\lambda}_{\overline{0}})\oplus\mathfrak{sl}(W^{\lambda}_{\overline{1}})&\text{if
$\lambda\in F_{n}$,}\end{cases}$

where $\mathfrak{sl}(W_{\lambda})$ denotes the diagonally embedded copy of
$\mathfrak{sl}(W_{\lambda})$ in
$\mathfrak{gl}(W^{\lambda}_{\overline{0}})\oplus\mathfrak{gl}(W^{\lambda}_{\overline{1}})$,
as in (4.3.3). The homogeneous subspaces of $W^{\lambda}$ are submodules for
the action of $\mathbb{C}A_{n}$, and hence also for the action of
$\mathfrak{D}$; if $\lambda\in E_{n}$ these submodules are both isomorphic to
the irreducible $|\mathbb{C}A_{n}|$-module $S^{\lambda}$, while if $\lambda\in
F_{n}$ they are isomorphic to the irreducible $|\mathbb{C}A_{n}|$-modules
$S^{\lambda^{+}}$ and $S^{\lambda^{-}}$.

For $\lambda\in\overline{\mathcal{P}}(n)$, let
$\rho_{\lambda}:\mathfrak{D}\to\operatorname{End}(W^{\lambda})_{\overline{0}}$
be the $\mathfrak{D}$-module structure map, let
$\mathfrak{D}_{\lambda}=\ker(\rho_{\lambda})$, and let
$\mathfrak{D}^{\lambda}$ be the orthogonal complement of
$\mathfrak{D}_{\lambda}$ with respect to the Killing form on $\mathfrak{D}$.
Then $\mathfrak{D}^{\lambda}$ is an ideal in $\mathfrak{D}$,
$\mathfrak{D}=\mathfrak{D}^{\lambda}\oplus\mathfrak{D}_{\lambda}$ as a Lie
algebra, and $\rho_{\lambda}$ induces a Lie algebra isomorphism
$\mathfrak{D}^{\lambda}\cong W^{\lambda}(\mathfrak{D})$. Thus
$\mathfrak{D}^{\lambda}$ is a simple ideal in $\mathfrak{D}$ (if $\lambda\in
E_{n}$), or is uniquely expressible as a direct sum of two simple ideals in
$\mathfrak{D}$ (if $\lambda\in F_{n}$).

######  Lemma 4.7.1.

lemma:Dlambda-Dmu-intersection Let $n\geq 5$, and let
$\lambda,\mu\in\overline{\mathcal{P}}(n)$. If
$\mathfrak{D}^{\lambda}\cap\mathfrak{D}^{\mu}\neq 0$, then $\lambda=\mu$.

###### Proof.

Let $\mathfrak{a}=\mathfrak{D}^{\lambda}\cap\mathfrak{D}^{\mu}$, and suppose
$\mathfrak{a}\neq 0$. There are several cases to consider:

  1. (1)

$\lambda,\mu\in E_{n}$. Then
$\mathfrak{D}^{\lambda}=\mathfrak{a}=\mathfrak{D}^{\mu}$, and the maps
$\rho_{\lambda}$ and $\rho_{\mu}$ induce isomorphisms $\mathfrak{a}\cong
W^{\lambda}(\mathfrak{D})$ and $\mathfrak{a}\cong W^{\mu}(\mathfrak{D})$.

  2. (2)

$\lambda\in E_{n}$ and $\mu\in F_{n}$. Then
$\mathfrak{D}^{\lambda}=\mathfrak{a}$, and $\mathfrak{a}$ is one of the two
simple ideals that comprise $\mathfrak{D}^{\mu}$. The map $\rho_{\lambda}$
induces an isomorphism $\mathfrak{a}\cong W^{\lambda}(\mathfrak{D})$, while
the map $\rho_{\mu}$ maps $\mathfrak{a}$ isomorphically into (precisely) one
of the summands $\mathfrak{sl}(W^{\mu}_{\overline{0}})$ or
$\mathfrak{sl}(W^{\mu}_{\overline{1}})$ of $W^{\mu}(\mathfrak{D})$. (There is
also the symmetric case $\lambda\in F_{n}$ and $\mu\in E_{n}$, which we omit.)

  3. (3)

$\lambda,\mu\in F_{n}$ and $\mathfrak{a}$ is a simple ideal. Then
$\mathfrak{a}$ is one of the two simple ideals that comprise
$\mathfrak{D}^{\lambda}$, and also for $\mathfrak{D}^{\mu}$. The map
$\rho_{\lambda}$ sends $\mathfrak{a}$ isomorphically into (precisely) one of
the summands $\mathfrak{sl}(W^{\lambda}_{\overline{0}})$ or
$\mathfrak{sl}(W^{\lambda}_{\overline{1}})$ of $W^{\lambda}(\mathfrak{D})$,
and similarly for $\rho_{\mu}$.

  4. (4)

$\lambda,\mu\in F_{n}$ and
$\mathfrak{D}^{\lambda}=\mathfrak{a}=\mathfrak{D}^{\mu}$. Then the maps
$\rho_{\lambda}$ and $\rho_{\mu}$ induce isomorphisms $\mathfrak{a}\cong
W^{\lambda}(\mathfrak{D})$ and $\mathfrak{a}\cong W^{\mu}(\mathfrak{D})$.

In each of the four cases, one must have $\dim(W^{\lambda})=\dim(W^{\mu})$.
Then fixing homogeneous bases for $W^{\lambda}$ and $W^{\mu}$, we can make the
identifications $W^{\lambda}=\mathbb{C}^{m|m}=W^{\mu}$ for some
$m\in\mathbb{N}$, and we can interpret $\rho_{\lambda}$ and $\rho_{\mu}$ as
Lie algebra homomorphisms

$\mathfrak{D}\to\operatorname{End}(\mathbb{C}^{m|m})_{\overline{0}}=\operatorname{End}(\mathbb{C}^{m|0})\oplus\operatorname{End}(\mathbb{C}^{0|m})=\operatorname{End}(\mathbb{C}^{m})\oplus\operatorname{End}(\mathbb{C}^{m}).$

First consider case (1), in which $\lambda,\mu\in E_{n}$. Then the images of
$\rho_{\lambda}$ and $\rho_{\mu}$ are each equal to the diagonal copy of
$\mathfrak{sl}(\mathbb{C}^{m})$ in
$\operatorname{End}(\mathbb{C}^{m|m})_{\overline{0}}$, and the composite
induced map

$\mathfrak{sl}(\mathbb{C}^{m})\xrightarrow{\rho_{\lambda}^{-1}}\mathfrak{a}\xrightarrow{\rho_{\mu}}\mathfrak{sl}(\mathbb{C}^{m})$

is a Lie algebra automorphism. Lie algebra automorphisms of
$\mathfrak{sl}(\mathbb{C}^{m})$ come in two forms:

  1. (1)

$X\mapsto gXg^{-1}$ for some $g\in GL(\mathbb{C}^{m})$, or

  2. (2)

$X\mapsto-(gX^{t}g^{-1})$ for some $g\in GL(\mathbb{C}^{m})$, where $X^{t}$
denotes the transpose of $X$.

If $\rho_{\mu}\circ\rho_{\lambda}^{-1}$ is of the first form, then the
$\mathfrak{D}$-module structures on $\mathbb{C}^{m|m}$ afforded by
$\rho_{\lambda}$ and $\rho_{\mu}$ are isomorphic, i.e., $W^{\lambda}\cong
W^{\mu}$ as $\mathfrak{D}$-modules. If $\rho_{\mu}\circ\rho_{\lambda}^{-1}$ is
of the second form, then the $\mathfrak{D}$-module structure on
$\mathbb{C}^{m|m}$ afforded by $\rho_{\lambda}$ is isomorphic to the dual of
the $\mathfrak{D}$-module structure afforded by $\rho_{\mu}$, i.e.,
$W^{\lambda}\cong(W^{\mu})^{*}$ as $\mathfrak{D}$-modules. Applying module-
iso-equivalent and remark:self-dual, this implies that $W^{\lambda}\cong
W^{\mu}$ as $\mathbb{C}S_{n}$-supermodules, and hence $\lambda=\mu$.

The reasoning for the other cases proceeds similarly. For example, in cases
(2) and (3), one deduces that one of the $\mathfrak{D}$-module composition
factors in $W^{\mu}$ is isomorphic to one (resp. both, if $\lambda\in E_{n}$)
of the $\mathfrak{D}$-module composition factors (or their duals) in
$W^{\lambda}$. In case (4), one deduces that (both of) the
$\mathfrak{D}$-module composition factors $W^{\mu}$ are isomorphic to the
$\mathfrak{D}$-module composition factors in $W^{\lambda}$, perhaps up to
duals and parity change. In any case, lemma:submodule-equivalent and
prop:irred-equivalent then imply that as $|\mathbb{C}A_{n}|$-modules,
$W^{\lambda}$ and $W^{\mu}$ have irreducible constituents in common, which is
only possible if $\lambda=\mu$. (In particular, cases (2) and (3) are
impossible.) ∎

######  Proposition 4.7.2.

prop:D(gzero)-structure Let $n\geq 5$. Then
$\mathfrak{D}=\bigoplus_{\lambda\in\overline{\mathcal{P}}(n)}\mathfrak{D}^{\lambda}$.

###### Proof.

The sum $\sum_{\lambda\in\overline{\mathcal{P}}(n)}\mathfrak{D}^{\lambda}$ is
a direct sum as a consequence of lemma:Dlambda-Dmu-intersection, and the sum
is equal to all of $\mathfrak{D}$ as a consequence of the module structure map
$\mathfrak{D}\to\bigoplus_{\lambda\in\overline{\mathcal{P}}(n)}\operatorname{End}(W^{\lambda})$,
$\sigma\mapsto\bigoplus_{\lambda\in\overline{\mathcal{P}}(n)}W^{\lambda}(\sigma)$,
being faithful. ∎

######  Theorem 4.7.3.

theorem:finaltheorem Let $n\geq 2$. Then
$\mathfrak{g}_{n}=\mathfrak{D}(\mathbb{C}S_{n})+\mathbb{C}\cdot T_{n}$.

###### Proof.

The theorem is true for $n\in\left\\{2,3,4,5\right\\}$ by lemma:gn-equality,
so we may assume that $n\geq 6$. We observed previously in (4.1.7) that
$\mathfrak{g}_{n}\subseteq\mathfrak{D}(\mathbb{C}S_{n})+\mathbb{C}\cdot T_{n}$
and that $T_{n}\in\mathfrak{g}_{n}$, so we just need to show that
$\mathfrak{D}(\mathbb{C}S_{n})\subseteq\mathfrak{g}_{n}$. Henceforward in this
proof, we will let $\mathfrak{g}=\mathfrak{g}_{n}$, and we will identify
$\mathbb{C}S_{n}$ with its image under the superalgebra isomorphism of
cor:CSn-as-superalgebra.

First we will show that
$\mathfrak{D}(\mathbb{C}S_{n})_{\overline{1}}\subseteq\mathfrak{g}$. By
(4.1.4), one has

$\mathfrak{D}(\mathbb{C}S_{n})_{\overline{1}}=\Big{[}\bigoplus_{\lambda\in
E_{n}}\mathfrak{sq}(W^{\lambda})_{\overline{1}}\Big{]}\oplus\Big{[}\bigoplus_{\lambda\in
F_{n}}\mathfrak{sl}(W^{\lambda})_{\overline{1}}\Big{]},$

and by prop:D(gzero)-structure, one has, with notation as in (4.7.1),

$\displaystyle\mathfrak{D}(\mathfrak{g}_{\overline{0}})$
$\displaystyle=\Big{[}\bigoplus_{\lambda\in
E_{n}}\mathfrak{sl}(W_{\lambda})\Big{]}\oplus\Big{[}\bigoplus_{\lambda\in
F_{n}}\mathfrak{sl}(W^{\lambda}_{\overline{0}})\oplus\mathfrak{sl}(W^{\lambda}_{\overline{1}})\Big{]}.$

Then $\mathfrak{D}(\mathbb{C}S_{n})_{\overline{1}}$ is a direct sum of
pairwise non-isomorphic irreducible
$\mathfrak{D}(\mathfrak{g}_{\overline{0}})$-modules. Since
$\mathfrak{g}_{\overline{1}}$ is a
$\mathfrak{D}(\mathfrak{g}_{\overline{0}})$-submodule of
$\mathfrak{D}(\mathbb{C}S_{n})_{\overline{1}}+\mathbb{C}\cdot T_{n}$, it must
contain some subset of the irreducible summands in
$\mathfrak{D}(\mathbb{C}S_{n})_{\overline{1}}$. Using theorem:Wlambda(gn), we
see that each of these summands is contained in the image of the corresponding
projection map $W^{\lambda}:\mathfrak{g}\to\operatorname{End}(W^{\lambda})$,
and hence must have been contained in $\mathfrak{g}_{\overline{1}}$. Thus
$\mathfrak{D}(\mathbb{C}S_{n})_{\overline{1}}\subseteq\mathfrak{g}$.

Now applying lemma:lower-bound-dim and rem:generated-by-odd, we deduce that
$\mathfrak{sl}(W^{\lambda})\subseteq\mathfrak{g}$ for each $\lambda\in F_{n}$,
and we deduce that $\mathfrak{sq}(W^{\lambda})\subseteq\mathfrak{g}$ for all
$\lambda\in E_{n}$ with the exception of
$\lambda\in\left\\{(n),(1^{n})\right\\}$. For the sake of definiteness,
suppose $\lambda=(n)\in E_{n}$. For this partition, one has
$\mathfrak{sq}(W^{(n)})=\mathbb{C}\operatorname{id}_{W^{(n)}}$. If $\tau\in
S_{n}$ is any transposition, then
$1_{\mathbb{C}S_{n}}=\frac{1}{2}[\tau,\tau]\in\mathfrak{g}$. But under the
isomorphism of cor:CSn-as-superalgebra, one has
$1_{\mathbb{C}S_{n}}=\sum_{\lambda\in\overline{\mathcal{P}}(n)}\operatorname{id}_{W^{\lambda}}$,
so

$\operatorname{id}_{W^{(n)}}=1_{\mathbb{C}S_{n}}-\Big{(}\sum_{\begin{subarray}{c}\lambda\in\overline{\mathcal{P}}(n)\\\
\lambda\neq(n)\end{subarray}}\operatorname{id}_{W^{\lambda}}\Big{)}\in\mathfrak{g}.$

Thus $\mathfrak{sq}(W^{(n)})\subseteq\mathfrak{g}$, and hence
$\mathfrak{D}(\mathbb{C}S_{n})\subseteq\mathfrak{g}$. ∎

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