# Vacuum effects induced by a plate in de Sitter spacetime in the presence of
a cosmic string

W. Oliveira dos Santos1 and E. R. Bezerra de Mello1
2Departamento de Física, Universidade Federal da Paraíba
58.059-970, Caixa Postal 5.008, João Pessoa, PB, Brazil E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we investigate the vacuum expectation values of the field
squared and the energy-momentum tensor associated to a charged massive scalar
quantum field in a $(1+D)$-dimensional de Sitter spacetime induced by a plate
(flat boundary) and a carrying-magnetic-flux cosmic string. In our analysis we
admit that the flat boundary is perpendicular to the string, and the scalar
field obeys the Robin boundary condition on the plate. In order to do develop
this analysis, we obtain the complete set of normalized positive-energy
solution of the Klein-Gordon equation compatible with the model setup. Having
obtained these bosonic modes, we construct the corresponding Wightman
function. The latter is given by the sum of two terms: one associated with the
boundary-free spacetime, and the other induced by the flat boundary. Although
we have imposed the Robin boundary condition on the field, we apply our
formalism considering specifically the Dirichlet and Neumann boundary
conditions. The corresponding parts have opposite signs. Because the analysis
of bosonic vacuum polarization in boundary-free de Sitter space and in
presence of a cosmic string, in some sense, has been developed in the
literature, here we are mainly interested in the calculations of the effects
induced by the boundary. In this way, closed expressions for the corresponding
expectation values are provided, as well as their asymptotic behavior in
different limiting regions. We show that the conical topology due to the
cosmic string enhances the boundary induced vacuum polarization effects for
both field squared and the energy-momentum tensor, compared to the case of a
boundary in pure de Sitter spacetime. Moreover, the presence of cosmic string
and boundary induce non-zero stress along the direction normal to the
boundary. The corresponding vacuum force acting on the boundary is also
investigated.

Keywords: cosmic string, magnetic flux, de Sitter spacetime, flat boundaries

##  1 Introduction

De Sitter (dS) spacetime is one of the most interesting solutions allowed by
the Theory of General Relativity. This curved maximally symmetric spacetime
allows that several physical problem in Quantum Field Theory can be exactly
solvable; moreover, the importance of these theoretical analysis increased by
the appearance of the inflationary cosmology scenario [1]. In great number of
inflationary models, approximated dS spacetime is employed to solve relevant
problems in standard Cosmology. During an inflationary epoch, quantum
fluctuations in the inflaton field introduce inhomogeneities and may affect
the transition toward the true vacuum. These fluctuations play important role
in the generation of cosmic structures from inflation. More recently
astronomical observations of high redshift supernovae, galaxy clusters and
cosmic microwave background [2, 3, 4] indicate that at the present epoch, the
Universe is accelerating and can be well approximated by a world with a
positive cosmological constant.

Cosmic strings are linear topologically stable gravitational defects which may
have been created in the early Universe after Planck time by a vacuum phase
transition [5, 6]. The gravitational field produced by a cosmic string may be
approximated by a planar angle deficit in the surface orthogonal to the
string. The simplest theoretical model which describes a straight and
infinitely long cosmic string is given by a Dirac-delta type distribution for
the energy-momentum tensor along the linear defect. This object can also be
described by coupling the energy-momentum tensor associated with the Maxwell-
Higgs system [7] with the Einstein equations [8, 9].

Many of the high-energy theories of fundamental physics are formulated in
higher dimensional spacetimes. Although topological defects have been first
analyzed in four-dimensional spacetime [5, 6], they have been considered in
the context of braneworld as well. In this scenario the defects live in a
$n-$dimensional submanifold embedded in a $(4+n)$-dimensional Universe. The
cosmic string case corresponds to two additional extra dimensions. In this
context the gravitational effect of global strings has been considered in [10,
11] as responsible for the compactification from six to four spacetime
dimensions, naturally producing the observed hierarchy between electroweak and
gravitational forces. The analysis of quantum effects for various spin fields
on the background of dS spacetime have been discussed by several authors (see,
for instance, [12]-[31] and references therein). Also for the cosmic string
spacetime these analyses have considered in [32]-[38] and [39]-[43], for
scalar and fermionic fields, respectively.

Another type of vacuum polarization is due to the presence of boundaries. This
is the Casimir effect. In this sense, the analysis of the Casimir effect of
the scalar, vector and fermionic fields in the idealized cosmic string
spacetime, obeying boundary conditions have been analyzed in [44], [45, 46]
and [47, 48, 49], respectively. Considering a flat boundary orthogonal to the
string, the vacuum effects associated with scalar and fermionic fields have
been investigated in [50, 51].

The analysis of the vacuum polarization associated with a real bosonic field
in dS spacetime in the presence of a cosmic string has been developed in [52].
The fermionic vacuum polarization in the same setup model has been considered
in [53]. Our objective in this paper is to extend the analysis of bosonic
quantum vacuum considering charged field in this background and the presence
of a flat boundary, throughout the paper also called plate.

The plan of the paper is the following. In the next section, we present the
setup model under consideration, and the complete set of normalized positive
energy solutions of the Klein-Gordon equation that obeys Robin boundary
conditions on a plate orthogonal to the string. Having obtained the complete
set of bosonic modes, in section 3 we construct the positive frequency
Wightman function. We show that this function is given by the sum of two
distinct contributions: the first one associated with the $(1+D)-$dimensional
dS spacetime in the presence of a cosmic string without boundary, and the
other induced by the boundary condition satisfied by the scalar field. Both
parts are presented in a closed form. Although we have obtained the complete
set of bosonic modes satisfying Robin boundary condition on the plate, in
section 4 we calculate the vacuum expectation value (VEV) of the field
squared, considering as direct applications of the formalism, the Dirichlet
and Neumann boundary conditions, separately. As we will see this VEV is given
in terms of a pure dS spacetime contribution, plus a term induced by the
magnetic-flux-carrying cosmic string, and finally a contribution induced by
the flat boundary orthogonal to the string. Moreover, in this section we
explicitly detail the asymptotic behavior of the boundary induced part in
different regions. To complete the analysis we present plots of the VEV of the
field squared as function of relevant physical variables considering different
values of the parameter associated with the conical structure generated by the
cosmic string. In section 5 we calculate the VEV of the energy-momentum tensor
induced by the boundary considering directly the Dirichlet and Neumann
conditions. We analyze specifically the energy-density and its asymptotic
expressions for different regions. Also in this section we present some plots
for the energy-density as function of some physical relevant variables. We
finalize section 5 by analyzing the vacuum force acting on the boundary.
Finally the main results of the paper are summarized in section 6.

##  2 Background geometry and matter field content

The main objective of this section is to obtain the complete set of positive
energy solution of the Klein-Gordon equation, in a $(1+D)$-dimensional dS
spacetime in the presence of an infinitely long straight cosmic string,
obeying the Robin boundary condition on a flat boundary orthogonal to the
string. In order to do that, we write the corresponding line element in
cylindrical coordinate [52, 53]:

$ds^{2}=dt^{2}-e^{2t/a}\left(dr^{2}+r^{2}d\phi^{2}+dz^{2}+\sum_{i=1}^{D-3}dx_{i}^{2}\right)\
,$ (2.1)

where $r\geq 0$ and $\phi\in[0,2\pi/q]$. The parameter $q\geq 1$ encodes the
conical geometry. The coordinates $(t,z,x_{i})\in(-\infty,\infty)$, and $a$
stands for the length scale of dS spacetime and it is related with the
cosmological constant, $\Lambda$, and the curvature scalar, $R$, by the
following relations:

$\Lambda=\frac{D(D-1)}{2a^{2}},\quad R=\frac{D(D+1)}{a^{2}}\ .$ (2.2)

Besides the synchronous time coordinate above, for convenience of the
discussion we adopt the conformal time, $\tau$, defined as $\tau=-ae^{-t/a}$
with $\tau\in(-\infty,0]$. By doing so, we get

$ds^{2}=\left(\frac{a}{\tau}\right)^{2}\left(d\tau^{2}-dr^{2}-r^{2}d\phi^{2}-dz^{2}-\sum_{i=1}^{D-3}dx_{i}^{2}\right)\
.$ (2.3)

Note that the line element inside brackets corresponds to the metric tensor
associated with an idealized cosmic string in Minkowski spacetime.

In this paper, we want to analyze the vacuum effects associated to a
propagating charged scalar field in the dS spacetime in the presence of a
magnetic-carrying-flux cosmic string and a flat boundary perpendicular to the
string. For this purpose we consider the following Klein-Gordon field
equation:

$(g^{\mu\nu}\mathcal{D}_{\mu}\mathcal{D}_{\nu}+m^{2}+\xi R)\varphi(x)=0\ ,$
(2.4)

where $\mathcal{D}_{\mu}=\nabla_{\mu}+ieA_{\mu}$ with $m$ being the mass of
the field. Moreover, in the equation above we have introduced a non minimal
coupling between the background curvature and the scalar field through the
term $\xi R$, with $\xi$ being the curvature coupling constant and $R$ the
Ricci scalar. The magnetic flux along the string axis is implemented through
the gauge field $A_{\mu}=A_{\phi}\delta_{\mu}^{\phi}$, with a constant
$A_{\phi}=-q\Phi/2\pi$, being $\Phi$ the magnetic flux along the string.

For consideration of the single flat boundary we impose that solutions of the
Klein-Gordon equation (2.4) obey the Robin boundary conditions given by

$(1+\beta n^{\mu}\nabla_{\mu})\varphi(x)\Big{|}_{z=0}=0\ ,$ (2.5)

where $\beta$ is a constant coefficient. In particular, for $\beta=0$ and
$\beta=\infty$, the Robin boundary conditions are reduced to the Dirichlet and
Neumann boundary conditions, respectively. In both sides of the plate one has
$n^{\mu}=\delta_{z}^{\mu}$. According to the notation above, the flat boundary
is located at $z=0$. Moreover, note that in our setup problem the cosmic
string is perpendicular to the boundary.

###  2.1 Bosonic modes

In this section our goal is to find the complete set of normalized positive
energy solutions of the Klein-Gordon equation (2.4) that satisfies (2.5).

In the geometry of the spacetime under consideration (2.3) and with the gauge
field $A_{\mu}=A_{\phi}\delta_{\mu}^{\phi}$, the Klein-Gordon equation reduces
to

$\displaystyle\Bigg{[}\frac{\partial^{2}}{\partial\tau^{2}}+\frac{(1-D)}{\tau}\frac{\partial}{\partial\tau}+\frac{D(D+1)\xi+(ma)^{2}}{\tau^{2}}-\frac{\partial^{2}}{\partial
r^{2}}-\frac{1}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\left(\frac{\partial}{\partial\phi}+ieA_{\phi}\right)^{2}$
(2.6) $\displaystyle-$ $\displaystyle\frac{\partial^{2}}{\partial
z^{2}}-\sum_{i=1}^{D-3}\frac{\partial^{2}}{\partial(x^{i})^{2}}\Bigg{]}\varphi(x)=0\
.$

The equation is completely separable and in accordance to the cylindrical
symmetries. We propose the following Ansatz:

$\varphi(x)=f(\tau)R(r)h(z)e^{iqn\phi+i\vec{k}\cdot\vec{x}_{||}}\ ,$ (2.7)

where $\vec{x}_{||}$ represents the coordinates along the $(D-3)$ extra
dimensions and $\vec{k}$ the corresponding momenta. The unknown function
$h(z)$ will be specified by the Robin boundary conditions obeyed by the scalar
field on the plate placed at $z=0$.

Taking the Ansatz (2.7) into (2.6) and admitting that

$\frac{\partial^{2}h(z)}{\partial z^{2}}=-k_{z}^{2}h(z)\ ,$ (2.8)

we get the following differential equations for the functions $f(\tau)$ and
$R(r)$:

$\Bigg{[}\frac{\partial^{2}}{\partial\tau^{2}}+\frac{(1-D)}{\tau}\frac{\partial}{\partial\tau}+\frac{D(D+1)\xi+(ma)^{2}}{\tau^{2}}+\lambda^{2}\Bigg{]}f(\tau)=0\
,$ (2.9)

and

$\Bigg{[}\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}-\frac{q^{2}(n+\alpha)^{2}}{r^{2}}+p^{2}\Bigg{]}R(r)=0\ ,$ (2.10)

with

$\lambda=\sqrt{p^{2}+k_{z}^{2}+\vec{k}^{2}}$ (2.11)

and the notation

$\alpha=\frac{eA_{\phi}}{q}=-\frac{\Phi}{\Phi_{0}}\ ,$ (2.12)

being $\Phi_{0}=2\pi/e$ the quantum flux.

The solution for (2.10) regular at $r=0$ is given by

$R(r)=J_{q|n+\alpha|}(pr)\ ,$ (2.13)

being $J_{\mu}(x)$ the Bessel function of first kind [54]. The solution for
the time-dependent equation is given by the linear combination of Hankel
functions:

$f(\tau)=\eta^{D/2}(c_{1}H_{\nu}^{(1)}(\lambda\eta)+c_{2}H_{\nu}^{(2)}(\lambda\eta))\
,$ (2.14)

with the order given by

$\nu=\sqrt{D^{2}/4-m^{2}a^{2}-\xi D(D+1)}\ .$ (2.15)

Moreover, in (2.14) we have introduced the variable $\eta=|\tau|$ and
$H_{\nu}^{(l)}(x)$, $l=1,\ 2$, represents the Hankel function [54]. Different
choices of the coefficients $c_{1,2}$ in (2.14) lead to different choices of
the vacuum state. In this paper we consider the Bunch-Davies vacuum,
corresponding to the choice $c_{2}=0$.

As to the solution of the $z$-dependent equation, obeying the Robin boundary
conditions (2.5) on the flat boundary at $z=0$, is given by

$h(z)=\cos[k_{z}z+\alpha_{1}(k_{z})]\ ,$ (2.16)

with the notation

$e^{2i\alpha_{1}(x)}=\frac{i\beta x-1}{i\beta x+1}\ .$ (2.17)

Finally, combining (2.13), (2.14) and (2.16), the mode functions that obey the
Klein-Gordon equation (2.4) and also satisfy the Robin boundary conditions
(2.5) on the plate is given by

$\varphi_{\sigma}(x)=C_{\sigma}\eta^{D/2}H_{\nu}^{(1)}(\lambda\eta)J_{q|n+\alpha|}(pr)\cos[k_{z}z+\alpha_{1}(k_{z})]e^{iqn\phi+i\vec{k}\cdot\vec{x}_{||}}\
,$ (2.18)

where $\sigma=\\{\lambda,p,n,k_{z},\vec{k}\\}$ is the set of quantum numbers
specifying each mode of the field. The coefficient $C_{\sigma}$ is determined
by the orthonormalization condition

$-i\int
d^{D-1}x\int_{0}^{\infty}dz\sqrt{|g|}g^{00}[\varphi_{\sigma}(x)\partial_{\tau}\varphi_{\sigma^{\prime}}^{\ast}(x)-\varphi_{\sigma^{\prime}}^{\ast}(x)\partial_{\tau}\varphi_{\sigma}(x)]=\delta_{\sigma,\sigma^{\prime}}\
,$ (2.19)

where the integral is evaluated over the spatial hypersurface $\tau=$ const,
and $\delta_{\sigma,\sigma^{\prime}}$ represents the Kronecker-delta for
discrete indices and Dirac-delta function for continuous ones. For the mode
functions in (2.18), the normalization condition leads to

$|C_{\sigma}|^{2}=\frac{qpe^{i(\nu-\nu^{\ast})\pi/2}}{2(2\pi)^{D-2}a^{D-1}}\
.$ (2.20)

##  3 Wightman function

Our aim in this paper is to investigate the vacuum polarization effects
emerging from the background setup described in the previous section. For this
purpose we will make use of the Wightman function. This function is of special
interest since its form is useful in calculations of vacuum expectation values
of physical observables that depend on bilinear field operators. In
particular, the properties of the vacuum can be described by the Wightman
function, $W(x,x^{\prime})=\langle
0|\hat{\varphi}(x)\hat{\varphi}^{\dagger}(x^{\prime})|0\rangle$, where
$|0\rangle$ stand for the vacuum state. For evaluation of this function, we
will adopt the mode sum formula technique by which the Wightman function takes
the form:

$W(x,x^{\prime})=\sum_{\sigma}\varphi_{\sigma}(x)\varphi_{\sigma}^{\ast}(x^{\prime})\
.$ (3.1)

where $\sum_{\sigma}$ stands for summations over the set of discrete and
integrals over continuos quantum numbers,
$\sigma=\\{\lambda,p,n,k_{z},\vec{k}\\}$.

Taking (2.18), along with the coefficient (2.20), into (3.1), we obtain

$\displaystyle W(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{8q(\eta\eta^{\prime})^{D/2}}{(2\pi)^{D}a^{D-1}}\sum_{n=-\infty}^{\infty}e^{inq\Delta\phi}\int_{0}^{\infty}dppJ_{q|n+\alpha|}(pr)J_{q|n+\alpha|}(pr^{\prime})\int
d\vec{k}e^{i\vec{k}\cdot\Delta\vec{x}_{||}}$ (3.2) $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dk_{z}K_{\nu}(e^{-\pi
i/2}\eta\lambda)K_{\nu}(e^{\pi
i/2}\eta^{\prime}\lambda)\cos[k_{z}z+\alpha_{1}(k_{z})]\cos[k_{z}z^{\prime}+\alpha_{1}(k_{z})]\
,$

where $\Delta\phi=\phi-\phi^{\prime}$ and
$\Delta\vec{x}_{\parallel}=\vec{x}_{\parallel}-\vec{x}_{\parallel}^{\prime}$
and $K_{\nu}(x)$, represents the Macdonald function [55]. Moreover, to obtain
the expression above we have introduced the notation
$\lambda=\sqrt{p^{2}+k_{z}^{2}+\vec{k}^{2}}$ and used the relation [55]

$e^{i(\nu-\nu^{\ast})\pi/2}H_{\nu}^{(1)}(\lambda\eta)\big{[}H_{\nu}^{(1)}(\lambda\eta^{\prime})\big{]}^{\ast}=\frac{4}{\pi^{2}}K_{\nu}(-i\lambda\eta)K_{\nu}(i\lambda\eta^{\prime})\
.$ (3.3)

In order to integrate over the quantum numbers $p$ and $\vec{k}$, we can make
use of the following identity [56]:

$K_{\nu}(a)K_{\nu}(b)=\frac{1}{2}\int_{-\infty}^{\infty}dy\int_{0}^{\infty}dww^{-1}\exp[-2\nu
y-abw^{-1}\cosh(2y)-w/2-(a^{2}+b^{2})/2w]\ .$ (3.4)

Taking this expression into (3.2), we get

$\displaystyle W(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{4q(\eta\eta^{\prime})^{D/2}}{(2\pi)^{D}a^{D-1}}\int_{-\infty}^{\infty}dy\int_{0}^{\infty}dww^{-1}e^{-2\nu
y-w/2}$ (3.5) $\displaystyle\times$
$\displaystyle\sum_{n=-\infty}^{\infty}e^{inq\Delta\phi}\int dppe^{-\sigma
p^{2}/2w}J_{q|n+\alpha|}(pr)J_{q|n+\alpha|}(pr^{\prime})\int
d\vec{k}e^{i\vec{k}\cdot\Delta\vec{x}_{||}-\sigma\vec{k}^{2}/2w}$
$\displaystyle\times$ $\displaystyle\int_{0}^{\infty}dk_{z}e^{-\sigma
k_{z}^{2}/2w}\cos[k_{z}z+\alpha_{1}(k_{z})]\cos[k_{z}z^{\prime}+\alpha_{1}(k_{z})]\
,$

with the notation

$\varepsilon=2\eta\eta^{\prime}\cosh(2y)-\eta^{2}-\eta^{\prime 2}\ .$ (3.6)

The integrations over $p$ and $\vec{k}$ can be performed with the help of the
formulas given in [54]. By doing this, we obtain

$\displaystyle W(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{4q(\eta\eta^{\prime})^{D/2}}{(2\pi)^{(D+3)/2}a^{D-1}}\int_{-\infty}^{\infty}dy\int_{0}^{\infty}dww^{-1}\left(\frac{w}{\varepsilon}\right)^{(D-1)/2}e^{-2\nu
y-w/2-(r^{2}+r^{\prime 2}+\Delta\vec{x}_{||}^{2})w/(2\varepsilon)}$ (3.7)
$\displaystyle\times$
$\displaystyle\sum_{n=-\infty}^{\infty}e^{inq\Delta\phi}I_{q|n+\alpha|}\left(\frac{rr^{\prime}w}{\sigma}\right)\int_{0}^{\infty}dk_{z}e^{-\varepsilon
k_{z}^{2}/2w}\cos[k_{z}z+\alpha_{1}(k_{z})]$ $\displaystyle\times$
$\displaystyle\cos[k_{z}z^{\prime}+\alpha_{1}(k_{z})]\ .$

Introducing a new variable of integration, $\chi=w/2\varepsilon$, we can write

$\displaystyle W(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(\eta\eta^{\prime})^{D/2}}{\pi^{\frac{D+3}{2}}a^{D-1}}\int_{0}^{\infty}d\chi\chi^{(D-3)/2}e^{-(r^{2}+r^{\prime
2}+\Delta\vec{x}_{||}^{2})\chi}\int_{-\infty}^{\infty}dye^{-2\nu
y-\sigma\chi}$ (3.8) $\displaystyle\times$
$\displaystyle\sum_{n=-\infty}^{\infty}e^{inq\Delta\phi}I_{q|n+\alpha|}\left(2rr^{\prime}\chi\right)\int_{0}^{\infty}dk_{z}e^{-k_{z}^{2}/4\chi}\cos[k_{z}z+\alpha_{1}(k_{z})]$
$\displaystyle\times$ $\displaystyle\cos[k_{z}z^{\prime}+\alpha_{1}(k_{z})]\
.$

Now we can integrate over $y$. By taking into account (3.6) and using the
formula [52]

$\int_{-\infty}^{\infty}dye^{-2\nu
y-2x\eta\eta^{\prime}\cosh(2y)}=K_{\nu}(2\eta\eta^{\prime}x)\ ,$ (3.9)

we can write (3.8) as

$\displaystyle W(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(\eta\eta^{\prime})^{D/2}}{\pi^{\frac{D+3}{2}}a^{D-1}}\int_{0}^{\infty}d\chi\chi^{(D-3)/2}e^{-(r^{2}+r^{\prime
2}+\Delta\vec{x}_{||}^{2}-\eta^{2}-\eta^{\prime 2})\chi}$ (3.10)
$\displaystyle\times$
$\displaystyle\sum_{n=-\infty}^{\infty}e^{inq\Delta\phi}I_{q|n+\alpha|}\left(2rr^{\prime}\chi\right)K_{\nu}(2\eta\eta^{\prime}\chi)\int_{0}^{\infty}dk_{z}e^{-k_{z}^{2}/4\chi}\cos[k_{z}z+\alpha_{1}(k_{z})]$
$\displaystyle\times$ $\displaystyle\cos[k_{z}z^{\prime}+\alpha_{1}(k_{z})]\
.$

We now turn to the integration over $k_{z}$. In this sense, we use the
identity

$2\cos[k_{z}z+\alpha_{1}(k_{z})]\cos[k_{z}z^{\prime}+\alpha_{1}(k_{z})]=\cos[k_{z}(z-z^{\prime})]+\cos[k_{z}(z+z^{\prime})+2\alpha_{1}(k_{z})]\
,$ (3.11)

allowing us to write the Wightman function as

$W(x,x^{\prime})=W_{\rm{cs}}^{\rm{dS}}(x,x^{\prime})+W_{\rm{pl}}(x,x^{\prime})\
.$ (3.12)

The first term on the r.h.s. of the above expression is the Wightman function
due to the cosmic string with its magnetic flux in dS spacetime without
boundary. Note that in the absence of the magnetic flux, this term coincides
with the one investigated in Ref. [52]. Explicitly, this term reads

$\displaystyle W_{\rm{cs}}^{\rm{dS}}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(\eta\eta^{\prime})^{D/2}}{2\pi^{\frac{D+3}{2}}a^{D-1}}\int_{0}^{\infty}d\chi\chi^{(D-3)/2}e^{-(r^{2}+r^{\prime
2}+\Delta\vec{x}_{||}^{2}-\eta^{2}-\eta^{\prime
2})\chi}K_{\nu}(2\eta\eta^{\prime}\chi)$ (3.13) $\displaystyle\times$
$\displaystyle\sum_{n=-\infty}^{\infty}e^{inq\Delta\phi}I_{q|n+\alpha|}\left(2rr^{\prime}\chi\right)\int_{0}^{\infty}dk_{z}e^{-k_{z}^{2}/4\chi}\cos[k_{z}(z-z^{\prime})]\
.$

As to the second term, it is the Wightman function induced by the plate.111The
Wightman function associated to a single plate in dS spacetime was studied in
[57]. It is given by

$\displaystyle W_{\rm{pl}}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(\eta\eta^{\prime})^{D/2}}{2\pi^{\frac{D+3}{2}}a^{D-1}}\int_{0}^{\infty}d\chi\chi^{(D-3)/2}e^{-(r^{2}+r^{\prime
2}+\Delta\vec{x}_{||}^{2}-\eta^{2}-\eta^{\prime
2})\chi}K_{\nu}(2\eta\eta^{\prime}\chi)$ (3.14) $\displaystyle\times$
$\displaystyle\sum_{n=-\infty}^{\infty}e^{inq\Delta\phi}I_{q|n+\alpha|}\left(2rr^{\prime}\chi\right)\int_{0}^{\infty}dk_{z}e^{-k_{z}^{2}/4\chi}\cos[k_{z}(z+z^{\prime})+2\alpha_{1}(k_{z})]\
.$

From this point, let us develop each term separately, starting with the
string-induced term (3.13). The integral over $k_{z}$ can be easily performed.
After integration, (3.13) reads

$\displaystyle W_{\rm{cs}}^{\rm{dS}}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(\eta\eta^{\prime})^{D/2}}{2\pi^{\frac{D}{2}+1}a^{D-1}}\int_{0}^{\infty}d\chi\chi^{D/2-1}e^{-(r^{2}+r^{\prime
2}+\Delta\vec{x}_{||}^{2}+\Delta z^{2}-\eta^{2}-\eta^{\prime
2})\chi}K_{\nu}(2\eta\eta^{\prime}\chi)$ (3.15) $\displaystyle\times$
$\displaystyle\sum_{n=-\infty}^{\infty}e^{inq\Delta\phi}I_{q|n+\alpha|}\left(2rr^{\prime}\chi\right)\
,$

where $\Delta z=z-z^{\prime}$.

The parameter $\alpha$ in Eq. (2.12) can be written in the form

$\alpha=n_{0}+\alpha_{0},\ \textrm{with}\ |\alpha_{0}|<\frac{1}{2}\ ,$ (3.16)

being $n_{0}$ an integer number. This allow us to sum over the quantum number
$n$ in (3.15) by using the formula obtained in [58]:

$\displaystyle\sum_{n=-\infty}^{\infty}e^{iqn\Delta\phi}I_{q|n+\alpha|}(x)=\frac{1}{q}\sum_{k}e^{x\cos(2\pi
k/q-\Delta\phi)}e^{i\alpha_{0}(2\pi k-q\Delta\phi)}$ (3.17) $\displaystyle-$
$\displaystyle\frac{e^{-iqn_{0}\Delta\phi}}{2\pi i}\sum_{j=\pm 1}je^{ji\pi
q|\alpha_{0}|}\int_{0}^{\infty}dy\frac{\cosh{[qy(1-|\alpha_{0}|)]}-\cosh{(|\alpha_{0}|qy)e^{-iq(\Delta\phi+j\pi)}}}{e^{x\cosh{(y)}}\big{[}\cosh{(qy)}-\cos{(q(\Delta\phi+j\pi))}\big{]}}\
,$

where

$-\frac{q}{2}+\frac{\Delta\phi}{\Phi_{0}}\leq
k\leq\frac{q}{2}+\frac{\Delta\phi}{\Phi_{0}}\ .$ (3.18)

The substitution of (3.17) into (3.15), allow us to integrate over $\chi$ with
the help of the formula given below [54]

$\int_{0}^{\infty}dxx^{\mu-1}e^{-vx}K_{\nu}(x)=\frac{\sqrt{\pi}2^{\nu}\Gamma(\mu-\nu)\Gamma(\mu+\nu)}{\Gamma(\mu+1/2)(v+1)^{\mu+\nu}}F\Bigg{(}\mu+\nu,\nu+\frac{1}{2};\mu+1/2;\frac{v-1}{v+1}\Bigg{)}\
,$ (3.19)

where $F(a,b;c,z)$ represents the hypergeometric function [54]. Thus, the
Wightman function due to the string, in closed form, is given by

$\displaystyle W_{\rm{cs}}^{\rm{dS}}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2(2\pi)^{\frac{D+1}{2}}a^{D-1}}\Biggr{\\{}\sum_{k}e^{i\alpha_{0}(2\pi
k-q\Delta\phi)}F_{\nu}^{D/2}(u_{k})$ (3.20) $\displaystyle-$
$\displaystyle\frac{e^{-iqn_{0}\Delta\phi}}{2\pi i}\sum_{j=\pm 1}je^{ji\pi
q|\alpha_{0}|}\int_{0}^{\infty}dy\frac{\cosh{[qy(1-|\alpha_{0}|)]}-\cosh{(|\alpha_{0}|qy)e^{-iq(\Delta\phi+j\pi)}}}{\cosh{(qy)}-\cos{(q(\Delta\phi+j\pi))}}$
$\displaystyle\times$ $\displaystyle F_{\nu}^{D/2}(u_{y})\Biggl{\\}}\ ,$

where we have introduced the function

$F_{\nu}^{D/2}(v)=\frac{2^{\nu+1/2}\Gamma(D/2-\nu)\Gamma(D/2+\nu)}{\Gamma(D/2+1/2)(v+1)^{D/2+\nu}}F\Bigg{(}\frac{D}{2}+\nu,\nu+\frac{1}{2};\frac{D+1}{2};\frac{v-1}{v+1}\Bigg{)}\
,$ (3.21)

and the variables

$\displaystyle u_{k}$ $\displaystyle=$ $\displaystyle-1+\frac{r^{2}+r^{\prime
2}-2rr^{\prime}\cos(\Delta\phi+2\pi k/q)+\Delta\vec{x}_{||}^{2}+\Delta
z^{2}-(\eta-\eta^{\prime})^{2}}{2\eta\eta^{\prime}}\ ,$ $\displaystyle u_{y}$
$\displaystyle=$ $\displaystyle-1+\frac{r^{2}+r^{\prime
2}+2rr^{\prime}\cosh(y/2)+\Delta\vec{x}_{||}^{2}+\Delta
z^{2}-(\eta-\eta^{\prime})^{2}}{2\eta\eta^{\prime}}\ .$ (3.22)

Now we turn to the boundary-induced term. Since the integral over $k_{z}$
unfortunately can be performed for an arbitrary value of the parameter
$\beta$, we will study the particular cases of the Dirichlet and Neumann
boundary conditions. According to (2.17), for Dirichlet BC ($\beta=0$) we have
$\alpha_{1}(k_{z})=\pi/2$ and for Neumann BC ($\beta\rightarrow\infty$),
$\alpha_{1}(k_{z})=0$. Therefore, for these boundary conditions, the Wightman
function due to the plate is

$\displaystyle W_{\rm{pl}}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\delta_{(\rm{J})}\frac{q(\eta\eta^{\prime})^{D/2}}{2\pi^{\frac{D+3}{2}}a^{D-1}}\int_{0}^{\infty}d\chi\chi^{(D-3)/2}e^{-(r^{2}+r^{\prime
2}+\Delta\vec{x}_{||}^{2}-\eta^{2}-\eta^{\prime
2})\chi}K_{\nu}(2\eta\eta^{\prime}\chi)$ (3.23) $\displaystyle\times$
$\displaystyle\sum_{n=-\infty}^{\infty}e^{inq\Delta\phi}I_{q|n+\alpha|}\left(2rr^{\prime}\chi\right)\int_{0}^{\infty}dk_{z}e^{-k_{z}^{2}/4\chi}\cos[k_{z}(z+z^{\prime})]\
,$

where $\rm{J}=D$ for Dirichlet BC, $\delta_{(\rm{J})}=1$, and $\rm{J}=N$ for
Neumann BC, $\delta_{(\rm{J})}=-1$. Now integrating over $k_{z}$, using the
formula (3.17) for the sum over $n$ and subsequently integrating over the
variable $\chi$, we arrive to

$\displaystyle W_{\rm{pl}}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\delta_{(\rm{J})}}{2(2\pi)^{\frac{D+1}{2}}a^{D-1}}\Biggr{\\{}\sum_{k}e^{i\alpha_{0}(2\pi
k-q\Delta\phi)}F_{\nu}^{D/2}(v_{k})$ (3.24) $\displaystyle-$
$\displaystyle\frac{e^{-iqn_{0}\Delta\phi}}{2\pi i}\sum_{j=\pm 1}je^{ji\pi
q|\alpha_{0}|}\int_{0}^{\infty}dy\frac{\cosh{[qy(1-|\alpha_{0}|)]}-\cosh{(|\alpha_{0}|qy)e^{-iq(\Delta\phi+j\pi)}}}{\cosh{(qy)}-\cos{(q(\Delta\phi+j\pi))}}$
$\displaystyle\times$ $\displaystyle F_{\nu}^{D/2}(v_{y})\Biggl{\\}}\ ,$

with the arguments

$\displaystyle v_{k}$ $\displaystyle=$ $\displaystyle-1+\frac{r^{2}+r^{\prime
2}-2rr^{\prime}\cos(\Delta\phi+2\pi
k/q)+\Delta\vec{x}_{||}^{2}+(z+z^{\prime})^{2}-(\eta-\eta^{\prime})^{2}}{2\eta\eta^{\prime}}\
,$ $\displaystyle v_{y}$ $\displaystyle=$
$\displaystyle-1+\frac{r^{2}+r^{\prime
2}+2rr^{\prime}\cosh(y/2)+\Delta\vec{x}_{||}^{2}+(z+z^{\prime})^{2}-(\eta-\eta^{\prime})^{2}}{2\eta\eta^{\prime}}\
.$ (3.25)

##  4 Field squared

In this section we want to investigate the VEV of the field squared. This
quantity is in the calculation of the energy-momentum tensor as we will see.
Formally it is defined by evaluating the Wightman at the coincidence limit:

$\langle|\varphi|^{2}\rangle=\lim_{x\rightarrow x^{\prime}}W(x,x^{\prime})\ ,$
(4.1)

where $|\varphi|^{2}$ is here understood as
$\hat{\varphi}(x)\hat{\varphi}^{\dagger}(x)$. According to (3.12), we
decompose this VEV in the form

$\langle|\varphi|^{2}\rangle=\langle|\varphi|^{2}\rangle_{\rm{dS}}+\langle|\varphi|^{2}\rangle_{\rm{cs}}+\langle|\varphi|^{2}\rangle_{\rm{pl}}\
.$ (4.2)

The contribution $\langle|\varphi|^{2}\rangle_{\rm{dS}}$ comes from the term
$k=0$ in (3.20), and it is induced in the pure de Sitter spacetime. This term
is divergent and independent of the spacetime coordinates. This contribution
has been already investigated in [59, 60, 61]. Specifically for $D=3$ this
renormalized VEV is given by

$\displaystyle\langle|\varphi|^{2}\rangle_{\rm{dS}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}a^{2}}\left\\{\left(\frac{m^{2}a^{2}}{2}+6\xi-1\right)\left[\Psi\left(\frac{3}{2}+\nu\right)+\Psi\left(\frac{3}{2}-\nu\right)-2\ln(ma)\right]\right.$
(4.3) $\displaystyle-$
$\displaystyle\left.\frac{(6\xi-1)^{2}}{m^{2}a^{2}}+\frac{1}{30m^{2}a^{2}}-6\xi+\frac{2}{3}\right\\}\
,$

where $\Psi(x)$ is the logarithimic derivative of the gamma-function.

The second contribution, $\langle|\varphi|^{2}\rangle_{\rm{cs}}$, is induced
by the magnetic-flux-carrying cosmic string in the dS spacetime. Because the
presence of the cosmic string does not introduce additional curvature, this
contribution is finite for points with $r\neq 0$.222This contribution
considering real bosonic field has been analyzed in Ref. [52]. Explicitly, the
VEV of the field squared induced by the cosmic string, taking into account the
interaction of the bosonic field with the magnetic flux, reads:

$\displaystyle\langle|\varphi|^{2}\rangle_{\rm{cs}}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{\frac{D+1}{2}}a^{D-1}}\Biggr{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2\pi
k\alpha_{0})F_{\nu}^{D/2}(2r_{p}^{2}s_{k}^{2}-1)$ (4.4) $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{h(q,\alpha_{0},y)}{\cosh{(qy)}-\cos{(q\pi)}}F_{\nu}^{D/2}(2r_{p}^{2}c_{y}^{2}-1)\Biggl{]}\
,$

where $[q/2]$ represents the integer part of $q/2$, and the prime on the sign
of the summation over $k$ means that for even values of $q$, the term $k=q/2$
should be taken with the coefficient $1/2$. Moreover, we have defined the
function

$h(q,\alpha_{0},y)=\cosh[qy(1-|\alpha_{0}|)]\sin(q\pi|\alpha_{0}|)+\cosh(qy|\alpha_{0}|)\sin[q\pi(1-|\alpha_{0}|)]\
,$ (4.5)

with

$\displaystyle s_{k}=\sin(\pi k/q)$ $\displaystyle c_{y}=\cosh(y/2)\ .$ (4.6)

In (4.4) , $r_{p}=r/\eta$ and $z_{p}=z/\eta$ are the proper distances from the
string and the plate, respectively, in unities of the dS spacetime curvature,
$a$.

As to the last term in the r.h.s. of (4.2), it can be yet decomposed in two
contributions:

$\langle|\varphi|^{2}\rangle_{\rm{pl}}=\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(0)}+\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0})}\
.$ (4.7)

The first contribution on the r.h.s. of the above expression is the term
induced purely by the plate in the dS spacetime and has been analyzed in [57].
Explicitly, for Dirichlet or Neumann boundary conditions, it is given by

$\displaystyle\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{\delta_{(\rm{J})}}{(2\pi)^{\frac{D+1}{2}}a^{D-1}}F_{\nu}^{D/2}(2z_{p}^{2}-1)\
.$ (4.8)

The second contribution, induced by the magnetic-flux-carrying cosmic string
and the plate, is a new one and is given by

$\displaystyle\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$
$\displaystyle=$
$\displaystyle\frac{\delta_{(\rm{J})}}{(2\pi)^{\frac{D+1}{2}}a^{D-1}}\Biggl{[}\sideset{}{{}^{\prime}}{\sum}_{k=0}^{[q/2]}\cos(2\pi
k\alpha_{0})F_{\nu}^{D/2}(w_{k})$ (4.9) $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{h(q,\alpha_{0},y)}{\cosh{(qy)}-\cos{(q\pi)}}F_{\nu}^{D/2}(w_{y})\Biggr{]}\
,$

where we have introduced the notations

$\displaystyle w_{k}$ $\displaystyle=$ $\displaystyle
2r_{p}^{2}s_{k}^{2}+2z_{p}^{2}-1$ $\displaystyle w_{y}$ $\displaystyle=$
$\displaystyle 2r_{p}^{2}c_{y}^{2}+2z_{p}^{2}-1\ .$ (4.10)

Let us now investigate the behavior of the field squared for some special and
asymptotic cases. We would like to remind that in this paper, our goal is the
study of the new contribution induced by the presence of the cosmic string and
the plate. For a conformal massless quantum scalar field, we have $\nu=1/2$
according to (2.15), and in this case we have the relation

$F_{1/2}^{D/2}(x)=\frac{\Gamma(D/2-1/2)}{(1+x)^{(D-1)/2}}\ ,$ (4.11)

which replaced in (4.9), yields

$\displaystyle\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$
$\displaystyle=$
$\displaystyle\delta_{(\rm{J})}\frac{2\Gamma(D/2-1/2)}{(4\pi)^{\frac{D+1}{2}}}\Big{(}\frac{\eta}{a}\Big{)}^{D-1}\Biggr{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{\cos(2\pi
k\alpha_{0})}{(r^{2}s_{k}^{2}+z^{2})^{(D-1)/2}}$ (4.12) $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{h(q,\alpha_{0},y)}{[\cosh{(qy)}-\cos{(q\pi)}](r^{2}c_{y}^{2}+z^{2})^{(D-1)/2}}\Biggl{]}\
.$

This result is conformally related to the VEV of field squared induced in the
Minkowski bulk with a plate placed at $z=0$ and a cosmic string with a
magnetic flux, for both Dirichlet and Neumann boundary conditions (see Ref.
[62]).

We now analyze the behaviour of
$\langle|\varphi^{2}|\rangle_{\rm{pl}}^{(q,\alpha_{0})}$ for points close and
far from the string. For points outside of the plate, $z\neq 0$, the VEV of
the field squared on the string is finite and can be obtained directly by
putting $r=0$ in (4.9). In the opposite limit, i.e., for points far from
string we need the asymptotic expression for large arguments of the function
$F_{\nu}^{D/2}(x)$. In the leading term, we have the approximate formula:

$F_{\nu}^{D/2}(x)\approx\frac{2^{\nu-1/2}}{\sqrt{\pi}}\Gamma(\nu)\Gamma(D/2-\nu)x^{-D/2+\nu}\
.$ (4.13)

Therefore, for distant points from the string, $r\gg\eta,z$, we can use the
above asymptotic formula in (4.9), yielding

$\displaystyle\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$
$\displaystyle\approx$
$\displaystyle\delta_{(\rm{J})}\frac{2^{2\nu-D-1}\Gamma(\nu)\Gamma(D/2-\nu)}{\pi^{\frac{D}{2}+1}a^{D-1}r_{p}^{D-2\nu}}\Biggr{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{\cos(2\pi
k\alpha_{0})}{s_{k}^{D-2\nu}}$ (4.14) $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{c_{y}^{-(D-2\nu)}h(q,\alpha_{0},y)}{\cosh{(qy)}-\cos{(q\pi)}}\Biggl{]}\
.$

We now turn to the behaviour of the field squared induced by the string and
plate, regarding the limiting regions along the $z$-axis. On the surface of
the plate, $z=0$, $\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$ is
finite for points outside the string, $r\neq 0$. On the other hand, far from
the plate location, $z\gg\eta,r$, we can once again make use of (4.13),
obtaining

$\displaystyle\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$
$\displaystyle\approx$
$\displaystyle\delta_{(\rm{J})}\frac{2^{2\nu-D-1}\Gamma(\nu)\Gamma(D/2-\nu)}{\pi^{\frac{D}{2}+1}a^{D-1}z_{p}^{D-2\nu}}\Biggr{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2\pi
k\alpha_{0})$ (4.15) $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{h(q,\alpha_{0},y)}{\cosh{(qy)}-\cos{(q\pi)}}\Biggl{]}\
.$

Finally, we consider the Minkowskian limit, $a\rightarrow\infty$, with fixed
value of $t$. In this limit, the geometry under consideration is reduced to
the geometry of a cosmic string in the background of $(D+1)$-dimensional
Minkowski spacetime. For the analysis of this limit, the representation for
the field squared given in (4.9) is not convenient. Thus, for this purpose, we
return to the Wightman function representation induced by the plate given in
(3.23). For the coordinate $\tau$ in the arguments of the modified Bessel
function one has $\eta\approx|t-a|$. Also in this limit we have $\nu\gg 1$,
which is imaginary and, according to (2.15), is $\nu\approx ima$. By making
use of the uniform asymptotic expansion for the Macdonald function of
imaginary order given in [63], replacing it in (3.23) and subsequently taking
the coincidence limit, after some intermediate steps, we get

$\displaystyle\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0}),(\rm{M})}$
$\displaystyle=$
$\displaystyle\delta_{(\rm{J})}\frac{2m^{D-1}}{(2\pi)^{\frac{D+1}{2}}}\Biggl{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2\pi
k\alpha_{0})\frac{K_{\frac{D-1}{2}}\Big{(}2m\sqrt{r^{2}s_{k}^{2}+z^{2}}\Big{)}}{\Big{(}2m\sqrt{r^{2}s_{k}^{2}+z^{2}}\Big{)}^{(D-1)/2}}$
(4.16) $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{h(q,\alpha_{0},y)}{\cosh{(qy)}-\cos{(q\pi)}}\frac{K_{\frac{D-1}{2}}\Big{(}2m\sqrt{r^{2}c_{y}^{2}+z^{2}}\Big{)}}{\Big{(}2m\sqrt{r^{2}c_{y}^{2}+z^{2}}\Big{)}^{(D-1)/2}}\Biggr{]}\
,$

with the notation

$f_{\nu}(x)=\frac{K_{\nu}(x)}{x^{\nu}}\ .$ (4.17)

Note that this result exactly coincides with the corresponding VEV of the
field squared calculated in the $(D+1)$-dimensional Minkowskian bulk [62].

In Fig.1 we plot the behavior the field quared induced by the plate,
$\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$, as function of
$r/\eta$ for $D=3$, considering minimum coupling, $\xi=0$, and fixed
parameters $z/\eta=0.5$ and $\alpha_{0}=0.25$, for different values of the
parameter $q$. In the left panel we adopt $ma=1$ (real $\nu$) and in the right
panel $ma=2$ (imaginary $\nu$). We note from these graphs that the VEV is
finite on the cosmic string and goes to zero far from it. From these plots we
also note that the effect of the deficit angle due to the string is to enhance
the intensity of the VEV of the field squared. Moreover, comparison between
the panel shows that the intensity decreases with the mass.

Figure 1: The VEV of the complex scalar field squared induced by the plate for
$D=3$, Eq. (4.9), is plotted in terms of the proper distance from the cosmic
string in units of $a$, $r/\eta$, for $z/\eta=0.5$ $\alpha_{0}=0.25$, $\xi=0$
and $q=1$, 2 and 3. The plot on the left is for $ma=1$, while the plot on the
right is for $ma=2$. The numbers near the curves are the values of $q$.

In Fig. 2, we present the behavior of (4.9) as function of proper distance
from the plate in units of $a$, $z/\eta$, for $D=3$ and fixed values
$r/\eta=0.75$, $\alpha_{0}=0.25$ and $\xi=0$, considering different values of
$q$. In the left panel we assume $ma=1$, and in the right panel $ma=2$. These
plots shows that the VEV of the field squared is finite on the plate and
vanishes far from it.

Figure 2: The VEV of the field squared induced by the plate, Eq. (4.9), is
plotted for $D=3$ as function of $z/\eta$, for fixed $\xi=0$,
$\alpha_{0}=0.25$ and $r/\eta=0.75$. The numbers near the curves correspond to
the values of deficit angle parameter, $q$. The left panel corresponds to
$ma=1$ and the right panel to $ma=2$. Numbers near the curves are the values
of $q$.

##  5 Energy-momentum tensor

We now turn to the calculation of the VEV of the energy-momentum tensor in the
setup under consideration. For this end, we use the formula [64]

$\langle T_{\mu\nu}\rangle=\lim_{x^{\prime}\rightarrow
x}(D_{\mu}D_{\nu^{\prime}}^{\dagger}+D_{\mu^{\prime}}^{\dagger}D_{\nu})W(x,x^{\prime})-2[\xi
R_{\mu\nu}+\xi\nabla_{\mu}\nabla_{\nu}-(\xi-1/4)g_{\mu\nu}\nabla_{\alpha}\nabla^{\alpha}]]\langle|\varphi|^{2}\rangle\
,$ (5.1)

where $R_{\mu\nu}=Dg_{\mu\nu}/a^{2}$ is the Ricci tensor for the dS spacetime
and $D_{\mu}=\nabla_{\mu}+ieA_{\mu}$. Similarly to the VEV of the field
squared, the VEV of the energy-momentum tensor can be decomposed as

$\langle T_{\mu\nu}\rangle=\langle T_{\mu\nu}\rangle_{\rm{dS}}+\langle
T_{\mu\nu}\rangle_{\rm{cs}}+\langle T_{\mu\nu}\rangle_{\rm{pl}}\ .$ (5.2)

The first contribution in the r.h.s. of the above equation, $\langle
T_{\mu\nu}\rangle_{\rm{dS}}$, is induced in the pure dS spacetime and is a
coordinate independent quantity, due to the maximal symmetry of dS spacetime
and the vacuum state under consideration. Additionally, this contribution is
divergent and some renormalization scheme is needed to provide a finite
result. This procedure has been considered in [59, 60, 61]. The second term on
the r.h.s., $\langle T_{\mu\nu}\rangle_{\rm{cs}}$, is induced by the cosmic
string and the magnetic flux along its core and it is finite away from the
string.333For the case of real scalar field, this VEV has been calculated in
[52]. Here we decided do not exhibit this contribution in order do not extend
this paper too long, since our propose is to investigate the energy-momentum
tensor induced by the plate only, $\langle T_{\mu\nu}\rangle_{\rm{pl}}$. The
latter can be decomposed in two contributions:

$\langle T_{\mu\nu}\rangle_{\rm{pl}}=\langle
T_{\mu\nu}\rangle_{\rm{pl}}^{(0)}+\langle
T_{\mu\nu}\rangle_{\rm{pl}}^{(q,\alpha_{0})}\ ,$ (5.3)

where the first contribution on the r.h.s. is induced purely by the plate in
dS spacetime and has been analyzed in [57]. The second term is a new
contribution and is induced by the string and the plate. This contribution is
finite away from the string or from the plate location, as will see below. In
what follows our analysis is constrained to the latter contribution.

The covariant d’Alembertian acting in (4.9) provides

$\displaystyle\Box\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$
$\displaystyle=$
$\displaystyle-\frac{4\delta_{(\rm{J})}}{(2\pi)^{\frac{D+1}{2}}a^{D+1}}\Biggl{\\{}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2\pi
k/q)\Bigg{[}\Big{[}1+2\gamma^{2}-(D+2)(s_{k}^{2}r_{p}^{2}+z_{p}^{2})\Big{]}$
(5.4) $\displaystyle\times$ $\displaystyle\frac{\partial}{\partial
y}F_{\nu}^{D/2}(y)+4\Big{[}(s_{k}^{4}r_{p}^{2}+z_{p}^{2})-(s_{k}^{2}r_{p}^{2}+z_{p}^{2})^{2}\Big{]}\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\Bigg{]}\Bigg{|}_{y=w_{k}}$ $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{h(q,\alpha_{0},y)}{\cosh{(qy)}-\cos{(q\pi)}}\Bigg{[}\Big{[}1+2\gamma^{2}-(D+2)(c_{y}^{2}r_{p}^{2}+z_{p}^{2})\Big{]}$
$\displaystyle\times$ $\displaystyle\frac{\partial}{\partial
y}F_{\nu}^{D/2}(y)+4\Big{[}(c_{y}^{4}r_{p}^{2}+z_{p}^{2})-(c_{y}^{2}r_{p}^{2}+z_{p}^{2})^{2}\Big{]}\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\Bigg{]}\Bigg{|}_{y=w_{y}}\Biggr{\\}}\ ,$

with $w_{k}$ and $w_{y}$ been given by (4.10).

For the geometry under consideration, only the $\nabla_{\eta}\nabla_{r}$,
$\nabla_{r}\nabla_{z}$, $\nabla_{\eta}\nabla_{z}$ and
$\nabla_{\mu}\nabla_{\mu}$ differential operators contribute when acting on
the VEV of the field squared. The remaining contributions come from the
electromagnetic covariant derivatives acting on the Wightman function. As to
the azimuthal component, it is more convenient to act the
$D_{\phi}D_{\phi^{\prime}}^{\dagger}$ operator in (3.23), and subsequently
take the coincidence limit in all coordinates, including the angular one.
Following this procedure, we obtain:

$\mathcal{I}(q,\alpha,y)=\sum_{n=-\infty}^{\infty}q^{2}(n+\alpha)^{2}I_{q|n+\alpha|}(y)\
,$ (5.5)

where $y=2rr^{\prime}\chi$. This sum can be developed by using the
differential equation obeyed by the modified Bessel function. Then we get,

$\mathcal{I}(q,\alpha,y)=\bigg{(}y^{2}\frac{d^{2}}{dy^{2}}+y\frac{d}{dy}-y^{2}\bigg{)}\sum_{n=-\infty}^{\infty}I_{q|n+\alpha|}(y)\
,$ (5.6)

where this last sum is given by [65]

$\sum_{n=-\infty}^{\infty}I_{q|n+\alpha|}(y)=\frac{e^{y}}{q}+\frac{2}{q}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2\pi
k\alpha_{0})e^{y\cos(2\pi
k/q)}-\frac{1}{\pi}\int_{0}^{\infty}dy\frac{e^{-y\cosh(y)}f(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}\
.$ (5.7)

Note that the first term on the r.h.s. is induced purely by the plate in dS
spacetime and accounts for the contribution $\langle
T_{\nu}^{\mu}\rangle_{\rm{pl}}^{(0)}$ only and, therefore, is not taken into
account in further analysis.

The boundary induced contribution in the VEV of the energy-momentum tensor is
calculated by making use of the corresponding parts in the Wightman function
and VEV of the field squared. After long but straightforward steps, we get (no
summation over $\mu$)

$\displaystyle\langle T_{\mu}^{\mu}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$
$\displaystyle=$
$\displaystyle-\frac{2\delta_{(\rm{J})}}{(2\pi)^{\frac{D+1}{2}}a^{D+1}}\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2\pi
k\alpha_{0})G_{\mu}^{\mu}(s_{k},r_{p},z_{p},w_{k})$ (5.8) $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{h(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}G_{\mu}^{\mu}(c_{y},r_{p},z_{p},w_{y})\Bigg{]}\
,$

with the functions

$\displaystyle G_{0}^{0}(\gamma,u,v,y)$ $\displaystyle=$ $\displaystyle D\xi
F_{\nu}^{D/2}(y)+\Big{[}-1+\xi_{1}(1+2\gamma^{2})-D\xi_{1}(\gamma^{2}u^{2}+v^{2})\Big{]}$
$\displaystyle\times$ $\displaystyle\frac{\partial}{\partial
y}F_{\nu}^{D/2}(y)+4\xi_{1}(\gamma^{4}u^{2}+v^{2})\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\ ,$ $\displaystyle G_{1}^{1}(\gamma,u,v,y)$
$\displaystyle=$ $\displaystyle D\xi
F_{\nu}^{D/2}(y)+\Bigl{\\{}-2+4\xi(1+\gamma^{2})+\big{[}1-(D+1)\xi_{1}\big{]}(\gamma^{2}u^{2}+v^{2})\Bigr{\\}}$
$\displaystyle\times$ $\displaystyle\frac{\partial}{\partial
y}F_{\nu}^{D/2}(y)+4\xi_{1}\Big{[}v^{2}-(\gamma^{2}u^{2}+z^{2})^{2}\Big{]}\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\ ,$ $\displaystyle G_{2}^{2}(\gamma,u,v,y)$
$\displaystyle=$ $\displaystyle D\xi
F_{\nu}^{D/2}(y)+\Bigl{\\{}-1+\gamma^{2}+\xi_{1}(1+\gamma^{2})+\big{[}1-(D+1)\xi_{1}\big{]}(\gamma^{2}u^{2}+v^{2})\Bigr{\\}}$
$\displaystyle\times$ $\displaystyle\frac{\partial}{\partial
y}F_{\nu}^{D/2}(y)+4\left\\{(4\xi\gamma^{2}-1)\gamma^{2}u^{2}+\xi_{1}\Big{[}v^{2}-(\gamma^{2}u^{2}+z^{2})^{2}\Big{]}\right\\}\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\ ,$ $\displaystyle G_{3}^{3}(\gamma,u,v,y)$
$\displaystyle=$ $\displaystyle D\xi
F_{\nu}^{D/2}(y)+\Bigl{\\{}2\xi_{1}\gamma^{2}+\big{[}1-(D+1)\xi_{1}\big{]}(\gamma^{2}u^{2}+v^{2})\Bigr{\\}}$
$\displaystyle\times$ $\displaystyle\frac{\partial}{\partial
y}F_{\nu}^{D/2}(y)+4\xi_{1}\Big{[}\gamma^{4}u^{2}-(\gamma^{2}u^{2}+z^{2})^{2}\Big{]}\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\ ,$ $\displaystyle G_{i}^{i}(\gamma,u,v,y)$
$\displaystyle=$ $\displaystyle D\xi
F_{\nu}^{D/2}(y)+\Bigl{\\{}-1+\xi_{1}(1+2\gamma^{2})+\big{[}1-(D+1)\xi_{1}\big{]}(\gamma^{2}u^{2}+v^{2})\Bigr{\\}}$
(5.9) $\displaystyle\times$ $\displaystyle\frac{\partial}{\partial
y}F_{\nu}^{D/2}(y)+4\xi_{1}\Big{[}(\gamma^{4}u^{2}+v^{2})-(\gamma^{2}u^{2}+z^{2})^{2}\Big{]}\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\ ,$

with $\xi_{1}=4\xi-1$. In the notation of the equation above, the indexes 0,
1, 2, 3 correspond to the coordinates $\eta$, $r$, $\phi$, $z$ and $i=4,...,D$
correspond to the extra dimensions ones. Additionally, we have three off-
diagonal components,

$\displaystyle G_{0}^{1}(\gamma,u,v,y)$ $\displaystyle=$ $\displaystyle
u\gamma^{2}\Bigg{[}(\xi_{1}-1)\frac{\partial}{\partial
y}F_{\nu}^{D/2}(y)+4\xi_{1}(u^{2}\gamma^{2}+v^{2})\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\Bigg{]}\ ,$ (5.10) $\displaystyle
G_{1}^{3}(\gamma,u,v,y)$ $\displaystyle=$
$\displaystyle-4\xi_{1}uv\gamma^{2}\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\ ,$ (5.11) $\displaystyle G_{0}^{3}(\gamma,u,v,y)$
$\displaystyle=$ $\displaystyle v\Bigg{[}(\xi_{1}-1)\frac{\partial}{\partial
y}F_{\nu}^{D/2}(y)+4\xi_{1}(u^{2}\gamma^{2}+v^{2})\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\Bigg{]}\ .$ (5.12)

Note that in (5.8) the term $k=0$ is not included. As explained in the
previous section, this is the boundary-induced term in the absence of the
string and magnetic flux, which has been investigated in [57]. Therefore, we
are not considering the analysis of that term.

At this point we want to say that the above VEV of the energy-momentum tensor
satisfies the trace relation:

$\langle
T_{\mu}^{\mu}\rangle=2[D(\xi-\xi_{c})\nabla_{\mu}\nabla^{\mu}\langle|\varphi|^{2}\rangle+m^{2}\langle|\varphi|^{2}\rangle]\
.$ (5.13)

In particular, this is zero for a conformal massless quantum scalar field.
Moreover, due to the off-diagonal components, from the conservation condition,
$\nabla_{\mu}\langle T_{\nu}^{\mu}\rangle=0$, three non-trivial differential
equations take place:

$\displaystyle\partial_{\eta}\langle T_{0}^{0}\rangle+\frac{1}{\eta}[\langle
T_{\mu}^{\mu}\rangle-(D+1)\langle
T_{0}^{0}\rangle]+\frac{1}{r}\partial_{r}(r\langle
T_{0}^{1}\rangle)+\partial_{z}\langle T_{0}^{3}\rangle=0\ ,$
$\displaystyle\frac{1}{r}\partial_{r}(r\langle
T_{1}^{1}\rangle)+\partial_{\eta}\langle T_{1}^{0}\rangle+\partial_{z}\langle
T_{1}^{3}\rangle-\frac{1}{r}\langle T_{2}^{2}\rangle-\frac{(D+1)}{\eta}\langle
T_{1}^{0}\rangle=0\ ,$ $\displaystyle\partial_{z}\langle
T_{3}^{3}\rangle+\frac{1}{r}\partial_{r}(r\langle
T_{3}^{1}\rangle)+\partial_{\eta}\langle
T_{3}^{0}\rangle-\frac{(D+1)}{\eta}\langle T_{3}^{0}\rangle=0\ .$

It can also be checked that these relations are satisfied by the components of
the VEV of the energy-momentum tensor above.

Let us now investigate some special and asymptotic cases of the energy
density. We start analysing the conformal massless scalar field case. In this
situation we have $\nu=1/2$. Using the corresponding formula for the function
$F_{1/2}^{D/2}(x)$ given in (4.11), after some intermediate steps, we have:

$\displaystyle\langle T_{0}^{0}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$
$\displaystyle=$
$\displaystyle-\delta_{(\rm{J})}\frac{2\Gamma(D/2+1/2)}{(4\pi)^{\frac{D+1}{2}}D}\left(\frac{\eta}{a}\right)^{D+1}\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2\pi
k\alpha_{0})s_{k}^{2}\frac{[2-(D-1)(s_{k}^{2}-1)]r^{2}+2z^{2}}{(r^{2}s_{k}^{2}+z^{2})^{(D+3)/2}}$
(5.15) $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{c_{y}^{2}h(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}\frac{[2-(D-1)(c_{y}^{2}-1)]r^{2}+2z^{2}}{(r^{2}c_{y}^{2}+z^{2})^{(D+3)/2}}\Bigg{]}\
.$

We turn now to the study of the asymptotic cases. Near the string,
$r\ll\eta,z$, for $z\neq 0$ and $q|\alpha_{0}|>1$, the energy density is
finite and we can directly put $r=0$. On the other hand, for $q|\alpha_{0}|<1$
the energy density on the string diverges as $\langle
T_{0}^{0}\rangle_{\rm{pl}}^{(q,\alpha_{0})}\sim 1/r_{p}^{2(1-q|\alpha_{0}|)}$.
Moreover, far from the string, $r\gg\eta,z$, we proceed in the same way as we
did with the VEV of the field squared. Thus, to the leading order term, we get

$\displaystyle\langle T_{0}^{0}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$
$\displaystyle\approx$
$\displaystyle-\delta_{(\rm{J})}\frac{2^{2\nu-D-2}D[1-\xi_{1}(D-2\nu-1)]\Gamma(\nu)\Gamma(D/2-\nu)}{\pi^{\frac{D}{2}+1}a^{D+1}r_{p}^{D-2\nu}}$
(5.16) $\displaystyle\times$
$\displaystyle\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{\cos(2\pi
k\alpha_{0})}{s_{k}^{D-2\nu}}-\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{c_{y}^{-(D-2\nu)}h(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}\Bigg{]}\
.$

On the plate, $z=0$, the VEV of the energy density induced by the string and
plate is finite for points outside, $r\neq 0$. In the opposite limit,
$z\gg\eta,r$, the corresponding asymptotic expression, to the leading order,
is

$\displaystyle\langle T_{0}^{0}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$
$\displaystyle\approx$
$\displaystyle-\delta_{(\rm{J})}\frac{2^{2\nu-D-2}D[1-\xi_{1}(D-2\nu-1)]\Gamma(\nu)\Gamma(D/2-\nu)}{\pi^{\frac{D}{2}+1}a^{D+1}z_{p}^{D-2\nu}}$
(5.17) $\displaystyle\times$
$\displaystyle\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2\pi
k\alpha_{0})-\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{h(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}\Bigg{]}\
.$

In the Minkowskian limit, $a\rightarrow\infty$, we follow the same procedure
described in the previous section. Therefore, in this limit the energy density
is given by:

$\displaystyle\langle T_{0}^{0}\rangle_{\rm{pl}}^{(q,\alpha_{0}),(\rm{M})}$
$\displaystyle=$
$\displaystyle\delta_{(\rm{J})}\frac{2m^{D-1}}{(2\pi)^{\frac{D+1}{2}}}\Biggl{\\{}\sum_{k=1}^{[q/2]}\cos(2\pi
k\alpha_{0})\Biggl{[}\Big{(}1-\xi_{1}(1+2\gamma^{2})\Big{)}f_{\frac{D+1}{2}}\left(2m\sqrt{s_{k}^{2}r^{2}+z^{2}}\right)$
(5.18) $\displaystyle+$ $\displaystyle
4\xi_{1}f_{\frac{D+3}{2}}\left(2m\sqrt{s_{k}^{2}r^{2}+z^{2}}\right)\Biggr{]}-\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{h(q,\alpha_{0},y)}{\cosh{(qy)}-\cos{(q\pi)}}\Biggl{[}\Big{(}1-\xi_{1}(1+2\gamma^{2})\Big{)}$
$\displaystyle\times$ $\displaystyle
f_{\frac{D+1}{2}}\left(2m\sqrt{c_{y}^{2}r^{2}+z^{2}}\right)+4\xi_{1}f_{\frac{D+3}{2}}\left(2m\sqrt{c_{y}^{2}r^{2}+z^{2}}\right)\Biggr{]}\Biggr{\\}}\
,$

with the function $f_{\nu}(x)$ defined in (4.17). Note that the expression
above exactly coincides with the VEV calculated in $(D+1)$-dimensional
Minkowkian bulk [62].

In Fig. 3 we exhibit the behavior the energy density induced by the plate as
function of $r/\eta$ for $D=3$, considering minimum coupling, $\xi=0$, and
fixed parameters $z/\eta=0.5$ and $\alpha_{0}=0.45$, for different values of
the parameter $q$. In the left panel we adopt $ma=1$ (real $\nu$) and in the
right panel $ma=2$ (imaginary $\nu$). As we can see from these plots, the
energy density is divergent on the string for $q|\alpha_{0}|<1$ (curve with
$q=1.5$) and is finite for $q|\alpha_{0}|>1$ (curve with $q=3.5$). This is in
accordance to our asymptotic analysis for points near the string. Moreover,
similarly to the field squared, the comparison between the plots shows that
the intensity of the energy density decreases with the mass.

Figure 3: The energy density induced by the plate for $D=3$ is plotted in
terms of the proper distance from the cosmic string, $r/\eta$, for
$z/\eta=0.5$ $\alpha_{0}=0.45$, $\xi=0$ and $q=1$, 2 and 3. The plot on the
left is for $ma=1$, while the plot on the right is for $ma=2$. The numbers
near the curves are the values of $q$.

In Fig. 4 we present the behavior of $\langle
T_{0}^{0}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$ as function of proper distance
from the plate in units of $a$, $z/\eta$, for $D=3$ fixed values
$r/\eta=0.75$, $\alpha_{0}=0.25$ and $\xi=0$, considering different values of
$q$. In the left panel we assume $ma=1$, and in the right panel $ma=2$. These
plots shows that the VEV of the energy density is finite on the plate and
vanishes far from it.

Figure 4: The VEV of the energy density induced by the plate is plotted for
$D=3$ as function of $z/\eta$, for fixed $\xi=0$, $\alpha_{0}=0.25$ and
$r/\eta=0.75$. The numbers near the curves correspond to the values of deficit
angle parameter, $q$. The left panel corresponds to $ma=1$ and the right panel
to $ma=2$. Numbers near the curves are the values of $q$.

The normal vacuum force acting on the plate is another interesting quantity
present in the energy-momentum tensor given by (5.8). This quantity is given
by the component $\langle T_{3}^{3}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$ at
$z=0$ and is finite for $r\neq 0$:

$\displaystyle\langle
T_{3}^{3}\rangle_{\rm{pl}}^{(q,\alpha_{0})}\Big{|}_{z=0}$ $\displaystyle=$
$\displaystyle-\frac{2\delta_{(\rm{J})}}{(2\pi)^{\frac{D+1}{2}}a^{D+1}}\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2\pi
k\alpha_{0})G_{z}(s_{k},2s_{k}^{2}r_{p}^{2}-1)$ (5.19) $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{h(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}G_{z}(c_{y},2c_{y}^{2}r_{p}^{2}-1)\Bigg{]}\
,$

with

$\displaystyle G_{z}(u,\gamma,y)$ $\displaystyle=$ $\displaystyle D\xi
F_{\nu}^{D/2}(y)+\Bigl{\\{}2\xi_{1}\gamma^{2}+\big{[}1-(D+1)\xi_{1}\big{]}\gamma^{2}u^{2}\Bigr{\\}}$
(5.20) $\displaystyle\times$ $\displaystyle\frac{\partial}{\partial
y}F_{\nu}^{D/2}(y)+4\xi_{1}\gamma^{4}u^{2}(1-u^{2})\frac{\partial^{2}}{\partial
y^{2}}F_{\nu}^{D/2}(y)\ .$

We see that even in the case of $q=1$, there is a pressure on the flat plate
caused by the integral term, which is a consequence of the interaction between
the field and the magnetic flux running along the string’s core.

For a conformally coupled massless field, the vacuum effective pressure on the
plate, $\langle T_{3}^{3}\rangle_{\rm{pl}}^{(q,\alpha_{0})}\Big{|}_{z=0}$, can
be even further simplified:

$\displaystyle\langle
T_{3}^{3}\rangle_{\rm{pl}}^{(q,\alpha_{0})}\Big{|}_{z=0}$ $\displaystyle=$
$\displaystyle\delta_{(\rm{J})}\frac{(D-1)^{2}}{(4\pi)^{\frac{D+1}{2}}D}\left(\frac{\eta}{ar}\right)^{D+1}\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{\cos(2\pi
k\alpha_{0})}{s_{k}^{D-1}}$ (5.21) $\displaystyle-$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{c_{y}^{-(D-1)}h(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}\Bigg{]}\
.$

Additionally, in the absence of magnetic flux, $\alpha_{0}=0$, and integer
$q$, the second term on the r.h.s. of the above equation is absent, and for
$D=3$, we have a simple expression,

$\langle
T_{3}^{3}\rangle_{\rm{pl}}^{(q,\alpha_{0})}\Big{|}_{z=0}=\frac{\delta_{(\rm{J})}}{36\pi^{2}}\left(\frac{\eta}{ar}\right)^{4}(q^{2}-1)\
,$ (5.22)

which is conformally related with the one calculated in the Minkowskian bulk
[66].

In Fig. 5 the behavior of the vacuum effective pressure on the plate as
function of the distance from the string is exhibited for a minimally coupled
field, considering $D=3$ and $\alpha_{0}=0.25$. In the left panel we consider
$ma=1$ and in the right one, $ma=2$. The numbers near the curves are values of
the angle deficit, $q$.

Figure 5: The vacuum normal force on the plate for a minimal coupling and
$D=3$ is plotted in terms $r/\eta$, considering $\alpha_{0}=0.25$ and $q=1$, 2
and 3. The plot on the left is for $ma=1$, while the plot on the right is for
$ma=2$. The numbers near the curves are the values of $q$.

##  6 Conclusion

In this paper we have investigated the effects on the bosonic vacuum induced
by a flat boundary in a $(1+D)-$dimensional dS background in the presence of
cosmic string orthogonal to the plate. Specifically we have calculated the
VEVs of the field squared and the energy-momentum tensor. In order to do that
we had first to obtain the complete set of normalized positive frequency
solutions of the Klein-Gordon equation compatible with the boundary condition
imposed on the field at the plate, located at $z=0$. Having obtained this set
of solutions, we constructed the Wightman function given by the sum over these
modes. This function was expressed as the sum of a contribution associated
with the dS space in the presence of a cosmic string without boundary, plus a
contribution induced by the plate. Unfortunately we could not obtain the
latter in a closed form for arbitrary values of the $\beta$ parameter present
in (2.5), so we decided to consider the field obeying Dirichlet and Neumann
boundary conditions separately in our analysis. The corresponding formulas for
the VEVs of the field squared and the energy-momentum tensor differ only by an
overall sign. Because the analysis of the VEVs associated with real scalar
fields in higher-dimensional dS in the presence of a cosmic string spacetime
have been developed in the literature, in this paper we we focused our
investigation on the contribution induced by the boundary.

In section 4, we have developed the calculation of the VEV of the field
squared induced by the flat boundary, $\langle|\varphi|^{2}\rangle_{\rm{pl}}$.
There we have shown that this quantity is decomposed in two parts, the first,
$\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(0)}$, corresponds to the contribution
induced by the plate in absence of the cosmic string, and the second induced
by the cosmic string considering the interaction of the charged bosonic field
with the magnetic flux,
$\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$. Because the
contribution induced purely by the plate has been investigated in the
literature [57], we focus only on the second part. In this way we have
investigated this observable in some asymptotic regions of the space. For
$r\gg\eta,\ z$, we found (4.14); on the other hand for $z\gg\eta,\ r$, the
asymptotic expression is given by (4.15). We can see that both results go to
zero with the inverse of the corresponding proper distance at the power
$D-2\nu$. In the Minkowskian limit, i.e, considering $a\to\infty$ and fixing
the value of $t$, the corresponding VEV was provided in (4.16). Moreover, two
figures exhibit the behavior of
$\langle|\varphi|^{2}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$, as function of
$r/\eta$ and $z/\eta$ for different values of the parameter $q$ and $ma$. They
are Figures (1) and (2), respectively.

In section 5, we have calculated all components of the VEV of the energy-
momentum tensor, $\langle T_{\nu}^{\mu}\rangle_{\rm{pl}}$, induced by the flat
plate in dS spacetime in the presence of a cosmic string. We have obtained,
besides diagonal components, three off-diagonal ones.444In the case of absence
of a flat boundary anlyzed in [52], only one off-diagonal component takes
place. As in the case of VEV of the field squared, this observable is
decomposed in a part induced by the boundary only in dS, $\langle
T_{\nu}^{\mu}\rangle_{\rm{pl}}^{(0)}$, plus the part induced by the presence
of the cosmic string, $\langle
T_{\nu}^{\mu}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$. Here in this section we
analyze only the part induced in the presence of the string. In fact this
contribution takes into account, besides the presence of the flat plate in dS,
the interaction of the charged scalar field with the magnetic flux running
along the string’s core and with the non-trivial topology of the space. As to
the magnetic interaction, it provides corrections in the two terms in (5.8).
The first one associated with the sum over the $k$, and the other associated
with the integral. Moreover, we observe that both corrections depend only on
the fractional part of the ratio of the magnetic flux by the quantum one,
$\alpha_{0}$. So the correction corresponds to a Aharonov-Bohm effect. In this
section we have proved that the VEV of the energy-momentum tensor satisfies
the trace relation, Eq. (5.13), and the conservation condition, Eq. (5).
Special attention has been given to the energy density, $\langle
T_{0}^{0}\rangle_{\rm{pl}}^{(q,\alpha_{0})}$. We have investigated its
behavior in some limiting cases. For a massless field conformally coupled with
the geometry this VEV is given by Eq. (5.15). Moreover, the energy density is
finite on the string for $z\neq 0$, and far from it, $r\gg\eta,z$, its
asymptotic behavior is given by (5.16). On the surface of the plate, $z=0$,
the energy density is finite for $r\neq 0$, and in the opposite limit is given
by (5.17). As we can see in both cases, the energy-density decays with the
inverse of the corresponding proper distance. The Minkowskian limit of the
energy-density is given by (5.18). In order to exhibit the behavior of the
energy density as function of $r/\eta$ and $z/\eta$, the proper distances from
the string and the plate, respectively, measured in the units of dS curvature
radius, $a$, we provide, for $D=3$, the figures 3 and 4, considering different
values of $q$ and $ma$. As our final remark we present the pressure on the
flat boundary. This quantity is given by the evaluation of the $(3-3)$
component of the energy-momentum tensor evaluated at $z=0$. The corresponding
expression is given in (5.19). For conformally coupled massless field, the
pressure is given by (5.21). Considering the absence of magnetic flux and for
integer $q$, the pressure reads, (5.22). The numerical behaviors of the
pressure, considering $D=3$, and different values of $q$ and $ma$, as function
of $r/\eta$ are given in the plots of figure 5.

As our final remark, we would like to say that the present work may has an
important role in the future investigation of vacuum polarization effects in
dS spacetime [67]. In fact, it is of our particular interest to analyze the
Casimir effect induced by a cosmic string orthogonal to two plates in this
background geometry.

## Acknowledgments

W.O.S is supported under grant 2022/2008, Paraíba State Research Foundation
(FAPESQ). E.R.B.M is partially supported by CNPq under Grant no
301.783/2019-3.

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# Unsupervised Video Person Re-identification via Noise and Hard frame Aware
Clustering

###### Abstract

Unsupervised video-based person re-identification (re-ID) methods extract
richer features from video tracklets than image-based ones. The state-of-the-
art methods utilize clustering to obtain pseudo-labels and train the models
iteratively. However, they underestimate the influence of two kinds of frames
in the tracklet: 1) noise frames caused by detection errors or heavy
occlusions exist in the tracklet, which may be allocated with unreliable
labels during clustering; 2) the tracklet also contains hard frames caused by
pose changes or partial occlusions, which are difficult to distinguish but
informative. This paper proposes a Noise and Hard frame Aware Clustering
(NHAC) method. NHAC consists of a graph trimming module and a node re-sampling
module. The graph trimming module obtains stable graphs by removing noise
frame nodes to improve the clustering accuracy. The node re-sampling module
enhances the training of hard frame nodes to learn rich tracklet information.
Experiments conducted on two video-based datasets demonstrate the
effectiveness of the proposed NHAC under the unsupervised re-ID setting.

Index Terms—  re-ID, unsupervised, clustering

##  1 Introduction

Compared with image-based person re-identification (re-ID), video-based person
re-ID has gained increasing attention due to the rich spatial-temporal
information that exists in video tracklets [1]. However, most video-based
person re-ID methods are conducted in a supervised manner, requiring intensive
manual labeling [2]. To be scalable in real-world applications, recent studies
focusing on the unsupervised video-based person re-ID [3] are broadly divided
into two categories.

Fig. 1: Illustration of the impact of noise and hard frames. (a) shows the
feature space of integral and clean tracklet features, respectively. Due to
the noise frames, the features of tracklets $a$ and $b$ have a high
similarity, and they will be wrongly merged during clustering. When noise
frames are trimmed, the features of clean tracklets $a^{\prime}$ and
$b^{\prime}$ are able to be easily distinguished. (b) shows the feature space
of frames in the tracklet. The hard frames are more scattered than easy frames
and do not have the same high aggregation frame feature spaces as easy frames.

1) Association-based: methods utilize tracklet association to mine labels from
the unlabeled tracklets. Some methods obtain labels under the assumption of
long tracklet or spatio-temporal topology [4, 5, 6]. For example, Ye et al.
[5] employed a static strategy to associate similar tracklet data for further
training. Liu et al. [6] carried out a reciprocal nearest neighbor search for
negative sample mining to improve the accuracy of tracklet association.
However, these methods still require several pedestrian tracklet labels. To
avoid the usage of person identity labeling, other methods explore the
tracklet association under different cameras [7, 8]. Li et al. [7] and Chen et
al. [8] utilized intra-camera and cross-camera anchors to enhance tracklet
association learning. Yet, these methods may face great challenges when
camera-related information is not provided.

2) Clustering-based: methods utilize clustering to perform unsupervised
learning. These methods generally include two main steps: i) clustering is
conducted on the features extracted from the person representation model, then
pseudo labels are assigned to each sample according to their clustering
results; ii) the model is retrained with samples and their pseudo labels. Lin
et al. [9] proposed a Bottom-Up Clustering (BUC) method to enhance the intra-
cluster tightness and inter-cluster well-separation. Ding et al. [10] proposed
a Dispersion Based Clustering (DBC) method to build high-quality clusters.
Recently, Wu et al. [11] took full advantage of the intrinsic tracklet
appearance information to formulate a novel Tracklet Self-Supervised Learning
(TSSL) method.

However, as shown in Figure 1, current clustering-based methods underestimate
the influence of two kinds of frames in the tracklet. 1) Noise frames: the
tracklet may contain noise frames caused by detection errors or heavy
occlusions. These noise frames introduce certain bias to the tracklet feature
(the tracklet feature is usually represented by average pooling to multiple
frame features). As a result, different pedestrians may share a high
similarity illustrated in Figure 1 (a), $a$ and $b$ are two biased tracklets
with different IDs but are easily merged during clustering. 2) Hard frames:
there are hard frames in the tracklet caused by pose changes or partial
occlusions of the pedestrian. These hard frames are difficult to distinguish
but contain rich information. As shown in Figure 1 (b), the feature
distribution of hard frames is more scattered than that of easy frames in the
frame feature spaces. As a result, the features within the same tracklets are
not sufficiently close, and the margins between different tracklets are not
adequately large.

Aiming to address these problems, this paper proposes a Noise and Hard frame
Aware Clustering (NHAC) method to optimize clustering accuracy and feature
embedding space for unsupervised video-based person re-ID. To suppress the
feature bias caused by noise nodes, a graph trimming module is presented to
improve the accuracy of tracklet similarity. The graph trimming module
utilizes the diversities between features to identify and remove noise nodes.
Then, a stable tracklet graph structure is obtained to make the
characteristics of the tracklet more accurate. To learn the rich information
from hard frame nodes, a node re-sampling module is designed to learn abundant
pedestrian tracklet information and obtain a discriminative feature space.
This module explores the dispersion of tracklets to identify hard nodes and to
learn the rich information of tracklets between the input data and feature
embedding levels. Our contributions in this paper are summarized in two folds:

  * •

We propose a noise and hard frame aware clustering method for unsupervised
video-based person re-ID. It is built on the application-specific
characteristics existing in the video re-ID task.

  * •

We design a graph trimming module to avoid feature bias caused by noise frames
and design a node re-sampling module to enhance the training of hard frames.

##  2 Methodology

The proposed NHAC consists of three main parts: 1) A clustering framework is
utilized to obtain a reliable cluster structure. 2) The graph trimming module
improves the clustering accuracy by removing the noise frames node. 3) The
node re-sampling module enhances the training of tracklet hard frames node to
learn abundant pedestrian tracklet information. Figure 2 illustrates the
overview of NHAC.

Fig. 2: The framework of our method. The CNN model is initially trained with
the initial pseudo labels. In each cycle, tracklet feature embeddings are
firstly extracted by the graph trimming module; after clustering, new training
data are obtained through the node re-sampling module; then, the network is
re-trained with new pseudo labels and input data.

###  2.1 The Clustering Framework

Clustering is an effective strategy for unsupervised learning. However, it is
non-trivial to straightforwardly utilize clustering for person re-ID, since
person re-ID is a fine-grained recognition task with large intra-class
differences whilst subtle inter-class variations properties. To obtain a
suitable cluster structure for person re-ID, we employ the BUC clustering-
based unsupervised re-ID method proposed by Lin et al. [9]. This method
iteratively updates the CNN by two steps, i.e. model training and cluster
merging.

Given an unlabeled training set $\chi=\\{{X_{1}},{X_{2}},\ldots,{X_{N}}\\}$ of
$N$ tracklets, each tarcklet contains $L$ frames, i.e.
${X_{i}}=\\{x_{i}^{j}\\}_{j=1}^{L}$. To explore the clustering of tracklet, we
deploy a feature embedding model ${f_{\theta}}(\cdot)$ (where $\theta$ is the
network parameters). During the training, we randomly select $M$ frames in the
tracklet to extract features, and then average pooling these features. It is
formulated as $\boldsymbol{v}_{i}=\rm{avgpool}({f_{\theta}}(X_{i}^{M}))$.
Since we do not have ground truth labels in the initial phase, we allocate
each tracklet to a different cluster, so each tracklet has a pseudo label. The
tarcklet $X_{i}$ belongs to the $c$-th cluster probability is defined as:

$p(c|X_{i},V)=\frac{{\exp(\boldsymbol{V}_{c}^{\rm{T}}\boldsymbol{v}_{i}/\tau)}}{{\sum\nolimits_{j=1}^{C}{\exp(\boldsymbol{V}_{j}^{\rm{T}}\boldsymbol{v}_{i}/\tau)}}},$
(1)

where $\boldsymbol{V}\in{R^{{n_{\theta}}\times C}}$ is the lookup table that
stores the centroid feature of each cluster, $n_{\theta}$ is the dimension of
the feature, $C$ is the number of clusters at the present stage,
${\boldsymbol{V}_{j}}$ is the $j$-th column of $\boldsymbol{V}$ and $\tau$ is
a temperature parameter. At the initial training stage, $C$ is the number of
tarcklets and at the following stages, $C$ will gradually decrease. During the
training, we calculate cosine similarities between feature
${\boldsymbol{v}_{i}}$ with all the centroid feature of each cluster by
${\boldsymbol{V}^{\rm{T}}}\cdot{\boldsymbol{v}_{i}}$. Then we update the
$y_{i}$-th column of table $\boldsymbol{V}$ by
${\boldsymbol{V}_{{y_{i}}}}=\frac{1}{2}({\boldsymbol{V}_{{y_{i}}}}+{\boldsymbol{v}_{i}})$.
Finally, the loss function is formulated as:

${\cal L}{\rm{=-log(p(}}{{y}_{i}}{|}{{X}_{i}}{,\boldsymbol{V}))}.$ (2)

The cluster merging is vital in each iteration of the clustering. Following
Lin et al. [9], the distance between two clusters is represented by the
minimum pairwise distance. The trained model ${f_{\theta}}(\cdot)$ is used to
extract tracklet features $\beta=\\{{F_{1}},{F_{2}},\ldots,F_{N}\\}$. The
distance between cluster $A$ and $B$ is formulated as:
${D}(A,B)=\mathop{\min}\limits_{{a}\in A,{b}\in B}d({F_{a}},{F_{b}})$.
${F_{a}},{F_{b}}$ are the features of the tracklets in clusters $A$ and $B$
respectively. $d({F_{a}}$, ${F_{b}})$ is defined as the euclidean distance
between the feature embeddings of two tracklets. In order to make the merged
clusters more reliable, the method is based on a hierarchical clustering
algorithm. In each iteration of clustering, this method selects the nearest
top-$m$ clusters to merge, where $m=N*mp$, and $mp\in(0,1)$ denotes the
merging percentage of clustering. Then, according to the result of cluster
merging to assign pseudo labels to each tracklet. In the end, this framework
iteratively updates the CNN to enhance model performance through the above two
steps.

###  2.2 Graph Trimming Module

In video-based person re-ID, each tracklet is usually automatically acquired
by the detector. However, the tracklet may contain noise frames caused by
detection errors. In addition, heavy occlusions of pedestrians may also
generate noise frames. Because the tracklet features are usually represented
by average pooling to transform multiple frame vectors into a single feature
vector, noise frames may easily bias the tracklet feature. Thus, we design a
graph trimming module to reduce the impact of noise frames.

For a single tracklet, we treat each frame as a node, and multiple nodes
constitute a tracklet graph. Compared with other nodes in the tracklet graph,
we find a large discrepancy in the noise nodes. Specifically, for each
tracklet ${X_{i}}=\\{x_{i}^{j}\\}_{j=1}^{L}$, the features of all nodes are
extracted as $\hat{F_{i}}=\\{f_{i}^{j}\\}_{j=1}^{L}$. Then we use average
pooling to get the graph central node feature
${F_{i}}=\mathrm{avgpool}(\hat{F_{i}})$. We calculate each node with central
node cosine similarity $s_{i}^{j}=\cos(f_{i}^{j},{F_{i}})$. The cosine
similarity between $\hat{F_{i}}$ and $F_{i}$ is expressed as
${S_{i}}=\\{s_{i}^{1},s_{i}^{2},\ldots,s_{i}^{L}\\}$. Due to the large bias of
noise nodes, intuitively, we can trim some noise nodes by setting a fixed
threshold according to the similarity.

However, this method presents two problems. 1) There are variations between
different datasets, and their corresponding threshold should be different. 2)
Different tracklets of the same dataset have different lengths and qualities,
it is easy to make the judgment of noise nodes inaccurate if the threshold is
fixed. In order to trim accurately more noise nodes, we set the dynamic
threshold by calculating the variance of the graph. Specifically, the dynamic
threshold ${q_{i}}$ is expressed as:

${q_{i}}={{\sum\nolimits_{j=1}^{L}{u_{i}^{j}}}\mathord{\left/{\vphantom{{\sum\nolimits_{j=1}^{L}{u_{i}^{j}}}{(L*\delta)}}}\right.\kern-1.2pt}{(L*\delta)}},$
(3)

where $L$ is the number of nodes in the tracklet graph, and $\delta$ is the
parameter that controls the degree of noise relaxation.
$u_{i}^{j}={(1-s_{i}^{j})^{2}}$ is the square of the inverse cosine similarity
between the current node and the center node. For node $x_{i}^{j}$, when
$u_{i}^{j}>{q_{i}}$, it is judged as a noise node and needs to be trimmed. For
each tracklet, after graph trimming node, the reserved features are expressed
as
$\hat{F_{i^{{}^{\prime}}}}=\\{f_{i}^{j}|u_{i}^{j}<q_{i},j\in\\{1,2,\ldots,L\\}\\}$
and the trimmed tracklet feature is expressed as
${F_{i^{{}^{\prime}}}}=\mathrm{avgpool}({\hat{F_{i^{{}^{\prime}}}}})$.
Finally, through the graph trimming module, the distance between cluster $A$
and $B$ is formulated as
${D}(A,B)=\mathop{\min}\limits_{{{a^{{}^{\prime}}}}\in
A,{{b^{{}^{\prime}}}}\in B}d({F_{a^{{}^{\prime}}}},{F_{b^{{}^{\prime}}}})$.

Table 1: Comparisons with state-of-the-arts. “One”, “Camera”, and “None”
denote one-example annotation, camera annotation, and no extra annotation,
respectively. ”-” denotes that the results are not provided in the original
paper. $\rm{1^{st}/2^{nd}}$ best results are in bold$\rm{/}$underline.

Methods | Venue | Annotation | DukeMTMC-VideoReID | MARS
---|---|---|---|---
Rank-1 | Rank-5 | Rank-10 | mAP | Rank-1 | Rank-5 | Rank-10 | mAP
DGM+IDE [5] | ICCV’17 | One | 42.3 | 57.9 | 68.3 | 33.6 | 36.8 | 54.0 | - | 16.8
Stepwise [6] | ICCV’17 | One | 56.2 | 70.3 | 79.2 | 46.7 | 41.2 | 55.5 | - | 19.6
RACE [12] | ECCV’18 | One | - | - | - | - | 43.2 | 57.1 | 62.1 | 24.5
EUG [4] | CVPR’18 | One | 72.7 | 84.1 | - | 63.2 | 62.6 | 74.9 | - | 42.4
DAL [8] | BMVC’18 | Camera | - | - | - | - | 49.3 | 65.9 | 72.2 | 23
UTAL [7] | TPAMI’20 | Camera | - | - | - | - | 49.9 | 66.4 | - | 35.2
OIM [13] | CVPR’17 | None | 51.1 | 70.5 | 76.2 | 43.8 | 33.7 | 48.1 | 54.8 | 13.5
BUC [9] | AAAI’19 | None | 74.8 | 86.8 | 89.7 | 66.7 | 55.1 | 68.3 | 72.8 | 29.4
DBC [10] | BMVC’19 | None | 75.6 | 88.5 | 91.0 | 67.4 | 58.5 | 70.1 | 73.5 | 31.7
TSSL [11] | AAAI’20 | None | 73.9 | - | - | 64.6 | 56.3 | - | - | 30.5
Ours | - | None | 82.8 | 92.7 | 95.6 | 76.0 | 61.8 | 75.3 | 79.9 | 40.1

###  2.3 Node Re-sampling Module

In order to make the model learn rich information from hard nodes of the
tracklet during training, we propose a node re-sampling module to improve the
discriminability of the tracklet feature space.

For node $\\{x_{i}^{j}\\}_{j=1}^{L}$ in tracklet ${X_{i}}$, the less
information will be learned if it is easier to be distinguished. To find nodes
with rich information, the predicted probability of the current node ID is
usually utilized for judgment. However, it is impossible to judge the
correctness of the pseudo label of the current tracklet for unsupervised
tasks. For each node in the tracklet, the closer it is to the central node,
the less information is obtained by the model training. Similar to the graph
trimming module, for each tracklet, we calculate the cosine similarity between
each node with the central node, i.e.
${S_{i}}=\\{s_{i}^{1},s_{i}^{2},\ldots,s_{i}^{L}\\}$, and calculate its mean,
i.e.
${\bar{s_{i}}}={{\sum\nolimits_{j=1}^{L}{s_{i}^{j}}}\mathord{\left/{\vphantom{{\sum\nolimits_{j=1}^{L}{s_{i}^{j}}}L}}\right.\kern-1.2pt}L}$.
According to the value ${\bar{s_{i}}}$, the node is divided into two sets
${g_{i}},{b_{i}}$. When $s_{i}^{j}>\bar{s_{i}}$, the node is easily
distinguished, i.e.
$g_{i}=\\{x_{i}^{j}|s_{i}^{j}>\bar{s_{i}},j\in\\{1,2,\ldots,L\\}\\}$;
otherwise the node is hard to be distinguished, i.e.
$b_{i}=\\{x_{i}^{j}|s_{i}^{j}<\bar{s_{i}},j\in\\{1,2,\ldots,L\\}\\}$. We found
that the number of ${g_{i}}$ nodes in a tracklet is usually more than the
number of ${b_{i}}$ nodes. In order to learn more information from hard nodes
during training, the set ${b_{i}}$ is oversampled to the length of set
${g_{i}}$, i.e.
$b_{i}^{*}=\mathrm{oversampling}{({b_{i}})_{\mathrm{len}({g_{i}})}}$. Finally,
we get a new set ${E_{i}}$ through oversampling, $M$ frame nodes in set
$E_{i}$ is randomly selected during training.

${E_{i}}=\left\\{{\begin{array}[]{*{20}{l}}{[{g_{i}},b_{i}^{*}],\qquad\rm{if}\
\mathrm{len}({g_{i}})>\mathrm{len}({b_{i}})}\\\
{[{g_{i}},{b_{i}}],\qquad\rm{others}}\end{array}}\right\\}.$ (4)

In addition, the undersampling and its combination with oversampling also
enhance the learning of hard nodes. 1) The undersampling method randomly
selects $n$ nodes in $g_{i}$ to obtain set $g_{i}^{*}$, where $n$ is the
length of $b_{i}$. Then $g_{i}^{*}$ is combined with $b_{i}$ to get a new
train set. However, random selection is uncertain and may reduce the diversity
of nodes available for training. 2) The oversampling and undersampling union
method combines the sets $b_{i}^{*}$ and $g_{i}^{*}$ to get a new train set.
Although this method improves the learning of hard nodes, it may lead the
model to overfit the minority samples. In experiments, the effectiveness of
the oversampling method will also be demonstrated in Section 3.4.

Although the model has learned rich information from hard nodes in the
training data, the distribution of the feature space still needs to be
improved. To directly optimize feature space, triplet loss is utilized in this
work which needs to generate batches of positive and negative pairs during
training. The characteristics of the video tracklet are used to design
tracklet triplet loss. In the same tracklet, the labels of different nodes
should be consistent. Therefore, the positive pairs are easy to be formed
between nodes in the same tracklet. During training, $M$ frames are randomly
selected to get set ${X_{i}^{M}}$ in each tracklet, which is divided into $K$
parts with length $M/K$. Since these $K$ parts have the same label, the anchor
$x_{i}$ and the positive pairs $x_{i,p}^{*}$ are easy to be obtained. For
negative pairs, pseudo labels are used to generate negative pairs
$x_{i,n}^{*}$. The proposed tracklet triplet loss is expressed as:

${{\cal
L}_{t}}=\max(0,\alpha+D({v_{{x_{i}}}},{v_{x_{i,p}^{*}}})-D({v_{{x_{i}}}},{v_{{x_{i,n}^{*}}}})),$
(5)

where $\alpha$ denotes a margin,
$\\{{v_{x_{i}}},v_{x_{i,p}^{*}},{v_{x_{i,n}^{*}}}\\}$ are the feature vectors
of $\\{{x_{i}},x_{i,p}^{*},{x_{i,n}^{*}}\\}$ respectively, $x_{i,p}^{*}$ is
positive tracklet and ${x_{i,n}^{*}}$ is negative tracklet.

##  3 Experiments

###  3.1 Datasets

We conduct extensive experiments on the DukeMTMC-VideoReID [4] dataset and the
MARS [14] dataset to evaluate the proposed NHAC. On both the DukeMTMC-
VideoReID and the MARS, each tracklet is treated as an individual sample in
the model training. Note that, our method does not utilize any annotation
information for model initialization or training. We adopt Rank-$k$ and mean
average precision (mAP) to evaluate our method.

###  3.2 Implementation Details

We adopt ResNet-50 as the CNN backbone to conduct all the experiments and
initialized it using the ImageNet pre-trained model. During training, the
number of training epochs in the first stage, the batch size, and the dropout
rate were set to be 20, 16, and 0.5 respectively. The parameters $mp$, $K$,
and $\delta$ in Eq.(3) were set to 0.05, 2, and 0.5 respectively. $\alpha$ in
Eq.(5) is set to 0.3. We set $\tau$ = 0.1 following [13] and select $M=16$
frames as the input for each tracklet. In the cluster merging and final
evaluation, average pooling was used to aggregate the frame-level features
into a tracklet representation. We adopt stochastic gradient descent (SGD)
with a momentum of 0.9 to optimize the model. The learning rate is initialized
to 0.1 and set to 0.01 after 15 epochs. We fixed the first three residual
blocks of ResNet-50 to save GPU memory and to boost iterations. The proposed
method was implemented on Pytorch and trained with NVIDIA TITAN Xp GPU.

###  3.3 Comparison with the State-of-the-Arts

In Table 1, we compare our method with the state-of-the-art algorithms on the
two large-scale video-based datasets. On the DukeMTMC-VideoReID dataset, our
NHAC achieves 76.0% in mAP and 82.8% in Rank-1, improving the state-of-the-art
performance by 8.6 points and 7.2 points, respectively. Compared to the BUC
method, our method achieves 9.3 and 8.0 points of improvement on mAP accuracy
and Rank-1, respectively.

On the MARS dataset, we achieve 40.1% in mAP and 61.8% in Rank-1, which beats
the state-of-the-art cluster-based algorithms by 8.4 and 3.3 points,
respectively. Compared to BUC, we achieve 10.7 and 6.7 points of improvement
in mAP accuracy and Rank-1. Compared with the association-based approaches,
our method outperforms most existing state-of-the-art methods except EUG. As
stated in [4], EUG initializes its model with a small number of tracklet
labels. Therefore, EUG is not fully unsupervised. Compared with UTAL and DAL
which utilize camera annotations, our method gives the best performance.

Table 2: Evaluating the components of NHAC on the DukeMTMC-VideoReID and the
MARS datasets.

Components | DukeMTMC | MARS
---|---|---
Rank-1 | mAP | Rank-1 | mAP
Baseline | 74.5 | 66.8 | 56.1 | 31.6
NHAC w/o NRM | 76.5 | 69.3 | 60.0 | 35.4
NHAC w/o GTM | 81.5 | 74.9 | 59.9 | 37.1
NHAC | 82.8 | 76.0 | 61.8 | 40.1

|  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |
easy | noise | hard
Fig. 3: Examples of easy, noise and hard nodes in tracklets.

###  3.4 Diagnostic Studies

To evaluate the effectiveness of our method, we analyze the impact of the two
main modules (graph trimming, node re-sampling), hyper-parameter $\delta$, re-
sampling criterion, and robustness.

Graph Trimming Module. The Graph Trimming Module (GTM) assigns more accurate
pseudo labels during clustering. As shown in Table 2 (see NHAC w/o NRM), we
show the impact of the GTM on the clustering algorithm. On the DukeMTMC-
VideoReID dataset, the module is 2.0% and 2.5% higher than baseline in Rank-1
and mAP, respectively. On the MARS dataset, the improvements are 3.9% for
Rank-1 and 3.8% for mAP. The improvement is more obvious for GTM on the MARS
dataset. Because the overall quality of the tracklets in the MARS dataset is
higher and the noise frames are more distinct compared to other frames, they
can be accurately cropped to achieve good performance. Figure 3 illustrates
some examples of noise nodes in the tracklets.

Node Re-sampling Module. We evaluate the effectiveness of our Node Re-sampling
Module (NRM) by comparing it to our baseline as shown in Table 2 (see NHAC w/o
GTM). NRM helps the model to learn from hard frames at both the input data and
feature embedding levels. Therefore, the performance is significantly improved
by 7.0% Rank-1 and 8.1% mAP on the DukeMTMC-VideoReID, while improving 3.8%
Rank-1 and 5.5% mAP on the MARS. Note that, on the DukeMTMC-VideoReID dataset,
the improvement of this module is more significant. The main reason for this
is that on DukeMTMC-VideoReID dataset has a more salient hard frame variance
and longer tracklet length compared to MARS (around twice of MARS), which
makes the model learn richer information. Some examples of hard nodes are
shown in Figure 3.

---
(a) NHAC w/o NRM | (b) NHAC
Fig. 4: Analysis of the parameter $\delta$ on the DukeMTMC-VideoReID dataset.
(a) NHAC w/o NRM: only have the Graph Trimming Module (GTM); (b) NHAC: the
method with GTM and NRM modules.

Hyper-parameter $\delta$. Hyper-parameter $\delta$ controls the degree of
identifying noise relaxation, which is helpful to improve the accuracy of
trimming noise nodes. We evaluate different values for $\delta$ in Figure 4.
We evaluate the performance of NHAC w/o NRM and NHAC, respectively, as the
value of $\delta$ varies from 0.1 to 0.9. For NHAC w/o NRM, Figure 4 shows
that Rank-1 and mAP achieve their highest value when $\delta$=0.6 and
$\delta$=0.5, respectively. For NHAC, it consists of modules GTM and NRM. The
best Rank-1 and mAP are obtained when $\delta$=0.5. The reason is that when
$\delta$ is too small, the degree of identifying noise is relaxing, which
makes the module difficult to trim noise nodes. When $\delta$ is too large, a
large number of non-noise nodes will be trimmed, which results in insufficient
diversity of tracklet features.

Table 3: Comparison of different re-sampling criteria on the DukeMTMC-
VideoReID dataset.

Criterion | Rank-1 | Rank-5 | Rank-10 | mAP
---|---|---|---|---
Over | 82.8 | 92.7 | 95.6 | 76.0
Under | 81.9 | 92.0 | 95.2 | 75.1
Over+Under | 80.9 | 92.3 | 94.0 | 73.5

Re-sampling Criterion. Table 3 illustrates the results of three re-sampling
criteria. The oversampling method (Over) achieves the best result with the
Rank-1 = 82.8% and mAP = 76.0%. For the undersampling method (Under), we
observe a slightly lower performance with the Rank-1 = 81.9% and mAP = 75.1%.
When using the oversampling and undersampling union method (Over+Under), we
observe the Rank-1 and mAP accuracy of 80.9% and 73.5%, respectively. The main
reason may lie in the different percentages of easy and hard frames in each
tracklet, where the oversampling and undersampling union method may lead the
model to overfit the minority samples.

---
Fig. 5: The Rank-1 and mAP performances with the different iterations on
DukeMTMC-VideoReID dataset.
---
Fig. 6: Percentage of hard and noise nodes in the MARS and DukeMTMC-VideoReID
datasets.

Robustness. Figure 5 illustrates the performance change over clustering
iterations in the DukeMTMC-VideoReID dataset. Throughout the iteration of our
method, the Rank-1 accuracy gradually rises from 44.0% to 82.8%; while the mAP
accuracy gradually rises from 39.8% to 76.0%. After the $14^{th}$ iteration,
the model performance stops rising and begins to drop. Compared with the DBC
and BUC methods, we observe that our method performs fall behind each other
before the $6^{th}$ iteration but diverged afterward, with our outperforming
other methods by a relatively large margin. Since our method mines for less
varying but more reliable tracklet mergers at the initial stage, this helps
our method to merge tracklets across cameras more accurately in the remaining
phases. Note that, from the $10^{th}$ iteration to the $17^{th}$ iteration,
our method always gives the best performance, i.e. with a mAP accuracy higher
than 67.4% and a Rank-1 accuracy higher than 75.6%, which demonstrates the
robustness of our proposed method.

##  4 Conclusion

We present a Noise and Hard frame Aware Clustering (NHAC) method to handling
unsupervised video-based person re-ID tasks. It optimizes both clustering and
training steps for the existence of noise and hard frames in tracklets. The
graph trimming module suppresses the tracklet feature bias caused by noise
nodes. The node re-sampling module enhances the model’s learning of rich
information from hard nodes.

Discussion: Figure 6 shows that MARS has a higher percentage of hard nodes and
fewer noise nodes compared to DukeMTMC-VideoReID. Since the tracklets in MARS
have a higher quality, the improvement of our method on MARS is not so
significant as that on DukeMTMC-VideoReID.

##  5 Acknowledgement

This work was supported by the Natural Science Foundation of China (U1803262,
61602349, 61440016, 61801335).

## References

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