# Statistical properties of the two dimensional Feshbach–Villars oscillator
(FVO) in the Rotating Cosmic String space-time

Abdelmalek Bouzenada<EMAIL_ADDRESS>;
<EMAIL_ADDRESS>Laboratoire de Physique Appliquée et Théorique
Université Larbi-Tébessi-, Tébessa, Algeria  Abdelmalek Boumali
<EMAIL_ADDRESS>Laboratoire de Physique Appliquée et Théorique
Université Larbi-Tébessi-, Tébessa, Algeria

(September 3, 2024)

###### Abstract

This paper is concerned with an investigation of the quantum mechanical
dynamics of massive, spinless relativistic Klein-Gordon particle in the space-
time generated by a rotating cosmic string. The equations of motion are found
by the use of the first-order Feshbach-Villars formulation of the Klein-Gordon
equation. The wave-functions and the associated energies were deduced ( both
in the free and in the interaction case). Following that, the partition
function was approximated using Zeta function representation. Moreover, by
considering the energy spectrum of the system in question, the thermal
properties are presented. We examine the behavior of these properties as a
function of the physical parameters of the model such as rotation, curvature,
and quantum numbers. Therefore, the impact of the topological defect on the
quantum system under investigation is discussed.

Klein-Gordon oscillator, Feshbach–Villars equations, topological defects,
cosmic string space-time, partition function, thermal properties.

###### pacs:

04.62.+v; 04.40.−b; 04.20.Gz; 04.20.Jb; 04.20.−q; 03.65.Pm; 03.50.−z;
03.65.Ge; 03.65.−w; 05.70.Ce

##  I Introduction

It is of general interest to understand the influence of the gravitational
field on the dynamics of quantum mechanical systems. Einstein’s theory of
general relativity (GR) (key-1, ), on the one hand, provides a persuasive
description of gravity as a geometric property of space-time. In particular,
it demonstrates that the classical gravitational field is a manifestation of
the curvature of space-time. It has, for example, successfully predicted the
existence of gravitational waves (key-2, ) and black holes(key-3, ). Quantum
mechanics (QM), on the other hand, is the framework for describing the
behavior of particles in the microscopic scale (key-4, ). It is an extremely
successful theory (typically quantum field theory) in explaining how the tiny
particles interact and how three of the four fundamental forces of nature: the
weak, strong, and electromagnetic interactions are emerged (key-5, ). However,
the attempts to have a unified theory which can reconcile general relativity
and quantum mechanics i.e. a theory of quantum gravity, is still suffering
from several obstacles and technical issues that cannot being encountered, at
least until the recent time (key-6, ; key-7, ) .

One basic method for formulating a theory in which the interface between
gravity and relativistic quantum mechanics arises, is by generalizing the
aspects of the relativistic dynamics of particles in flat Minkowski space to
an arbitrary curved background geometry (key-8, ; key-9, ) , and therefore one
can extend this formulation to establish a broad picture on how the
gravitational field affects on the relativistic particles at quantum level. In
this way, it is possible to adapt the approach to deal with different models
where the notion of curvature appears, and hence incorporating more
predictions on the values of macroscopic observables which are required to
make relevant experimental verification of certain phenomenological
consequences, specifically in astrophysics and cosmology. In addition,
understanding the thermodynamic behavior of relativistic particles where
gravitational effects have to be taken into account (key-10, ; key-11, ;
key-12, ), and analyzing the associated features i.e, the fundamental
statistical quantities, would offer the possibility of obtaining useful and
essential results in the context of describing the quantum behavior of
gravity.

Over the past few decades, topological defects ( domain walls, cosmic strings,
monopoles, and textures ) have been tackled intensively as a subject of
research and still remain one of the most active fields in condensed matter
physics, cosmology, astrophysics and elementary particle models. It is
believed that these structures raised as a consequence of the Kibble mechanism
(key-13, ; key-14, ; key-15, ) where the defects are formed in symmetry-
breaking phase transitions during the cooling of the early universe (key-16, ;
key-17, ). The particular defect we are interested in, is cosmic strings (for
more reviews we refer the reader to (key-18, )). These objects ( static or
rotating) can produce observable effects. For example, they provide a
promising scheme for seeding galaxy formation and the gravitational lensing
effects. Moreover, by studying cosmic strings and their properties, we can
learn much more about particle physics at very high energies in different
scenarios. In addition, the possibility that cosmic strings could behave like
superconducting wires has been raised with intriguing consequences in modern
physics.

It has been known for many years now that the harmonic oscillator (HO) is
considered as an essential tool in many disciplines of theoretical physics
(key-19, ). It is a well studied exactly solvable model which can be used to
analyze various complex problems in the framework of quantum mechanics
(key-20, ). Furthermore the relativistic generalization of the quantum
harmonic oscillator yields an effective model in explaining very diverse
aspects of molecular, atomic and nuclear interactions. Indeed, the feature of
having a complete set of exact analytical solutions when dealing with such
model can give rise to significantly different explanations of many
mathematical and physical phenomena, and hence related applications can be
achieved via the underlying formulation.

It is now accepted that the behavior of several relativistic quantum systems
depends crucially on the so-called Dirac oscillator (DO). As pointed out by
Itô et al. (key-21, ) in earlier developments of spin-1/2 particle dynamics
with linear trajectory. They showed that the non-relativistic limit of this
system leads to the ordinary harmonic oscillator with a strong spin-orbit
coupling term. Actually, according to Moshinsky and Szczepaniak (key-22, ) the
above mentioned DO could be obtained from the free Dirac equation by
introducing an external linear potential through a minimal substitution of the
momentum operator $\hat{p}\longrightarrow\hat{p}-im\omega\beta\hat{r}$. It is
interesting to point out that besides the theoretical focus on studying the
DO, valuable insights can be obtained by considering the physical
interpretation which is certainly essential in understanding many relevant
applications.

Inspired by DO, an analogous formalism was introduced for the case of bosonic
particles and hence it was named a Klein-Gordon oscillator (KGO) (key-23, ;
key-24, ). The covariant form of this model in curved space-time and within
different configurations, has recently been an active field of investigation
by several authors. There are numerous contributions on the subject of the
relativistic quantum motions of scalar and vector particles under
gravitational effects produced by different curved space-time geometries, for
example, the problem of the interaction between KGO coupled harmonically with
topological defects in Kaluza-Klein theory is studied in Ref. (key-25, ). The
relativistic quantum dynamics of spin-0 particles in a rotating cosmic string
space-time with a scalar and vector potentials of Coulomb-type has been
studied in Ref. (key-26, ). Furthermore, rotating effects on the scalar field
in the cosmic string space-time, in the space-time with space-like dislocation
and in the space-time with a spiral dislocation have been investigated in Ref.
(key-27, ). Recently, the authors of Ref. (key-28, ) have analyzed the KGO in
a cosmic string space-time and studied the effects stemming from the rotating
frame and non-commutativity in momentum space. In addition, the KGO subjected
to a magnetic quantum flux in the presence of a Cornell-type scalar and
Coulomb-type vector potentials in a rotating cosmic string space-time have
been examined in Ref. (key-29, ).

Attempts to investigate the relativistic spin-0, spin-1 bosons and spin-1/2
fermions wave functions and their time evolution have been pursued by various
authors (key-30, ; key-31, ; key-32, ) making use of the Hamiltonian form i.e,
having Schrodinger’s equations. The so-called Feshbach-Villars (FV) equations
(key-33, ) are of particular interest in this respect. These equations were
initially constructed in the purpose of permitting a relativistic single
particle interpretation of the second order KG equation. For the later case,
FV equations originate from splitting the KG wave function into two components
in order to obtain an equation with first order time derivative. In recent
decades, a number of papers have been produced with the aim of exploring the
relativistic dynamical properties of single particles and solving their wave
equations by adopting the FV scheme (e.g, Refs. (key-34, ; key-35, ; key-36, ;
key-37, ; key-38, ; key-39, ; key-40, ) and other related references cited
therein).

In a quest to map out a description of the dynamics of spinless, massive
bosons in a space-time with a topological defect, namely the cosmic string
geometry, we employ the methods of the FV transformation to derive FV
equations of motion in a static and rotating cosmic string backgrounds. The
main purpose of this paper is , firstly to solve the derived FV equations and
obtain the associated spectra, and secondly, to investigate the thermodynamic
properties of these systems and understand how the geometry of the space-time
affects on the observable quantities.

The structure of this paper is as follows. In the next section we derive the
FV equations for scalar boson in Minkowski and static Cosmic string space-time
considering both the free and the interaction case. We introduce the KG
oscillator in a Hamiltonian form, then we solve the obtained equations to
deduce the eigenstates and the energy levels. In section 3 we furthermore
construct the previous equations in the case where the geometry of the cosmic
string is characterized by a rotation, the same procedure of Section 2 is used
in this section. In section 4 we study the statistical properties of the
system in question and we discuss the influence of the topological defect on
these quantities. We give our conclusions in section 5. Throughout the paper,
we will always use natural units $\hbar=c=G=K_{B}=1$ , and our metric
convention is $(+,-,-,-$).

##  II The FV Representation of Spin-0 Particle in Minkowski Space-time

###  II.1 Klein-Gordon Equation in Schrödinger Form

In this section, we deal with the relativistic quantum description of spin-0
particle propagating in Minkowski space-time with the metric tensor
$\eta_{\mu\nu}=\text{diag}\left(1,-1,-1,-1\right)$. For a scalar massive
particle $\Phi$ with mass $m>0$, the standard covariant KG equation reads
(key-41, ; key-42, )

$\left(\eta^{\mu\nu}D_{\mu}D_{\nu}+m^{2}\right)\Phi(x,t)=0,$ (1)

where $D_{\mu}=i\left(p_{\mu}-eA_{\mu}\right)$ denotes the minimally coupled
covariant derivative. $p_{\mu}=\left(E,-p_{i}\right)$ is the canonical four
momentum and $A_{\mu}=\left(A_{0},-A_{i}\right)$ is the electromagnetic four
potential, respectively. $e$ is the magnitude of the particle charge.

At this place, it is worth emphasizing that Eq. ((1)) can be rewritten in a
Hamiltonian form with the time first derivative i.e, a Schrodinger’s equation

$\mathcal{H}\Phi\left(x,t\right)=i\,\frac{\partial}{\partial t}\Phi(x,t).$ (2)

The Hamiltonian $\mathcal{H}$ can be defined by the use of the FV
linearization procedure i.e, transforming Eq. (1) to a first order in time
differential equation. We introduce the two component wave function (key-43, ;
key-44, ),

$\Phi(x,t)=\left(\begin{array}[]{c}\phi_{1}(x,t)\\\
\phi_{2}(x,t)\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1+\frac{i}{m}\mathcal{D}\\\
1-\frac{i}{m}\mathcal{D}\end{array}\right)\psi(x,t),$ (3)

here $\psi(x,t)$ obeys the KG wave equation, and we have defined $\mathcal{D}$
such that

$\mathcal{D}=\frac{\partial}{\partial t}+ieA_{0}(x),$ (4)

the above transformation ((3)) consists in introducing wave functions
satisfying the conditions

$\psi=\phi_{1}+\phi_{2},\qquad
i\mathcal{D}\psi=m\left(\phi_{1}-\phi_{2}\right).$ (5)

For our subsequent review, it is convenient to write

$\displaystyle\phi_{1}$
$\displaystyle=\frac{1}{2m}\left[m+i\frac{\partial}{\partial
t}-eA_{0}\right]\psi$ (6) $\displaystyle\phi_{2}$
$\displaystyle=\frac{1}{2m}\left[m-i\frac{\partial}{\partial
t}+eA_{0}\right]\psi,$

Equivalently, Eq. ((1)) becomes

$\displaystyle\left[i\frac{\partial}{\partial
t}-eA_{0}\right]\left(\phi_{1}+\phi_{2}\right)$
$\displaystyle=m\left(\phi_{1}-\phi_{2}\right)$ (7)
$\displaystyle\left[i\frac{\partial}{\partial
t}-eA_{0}\right]\left(\phi_{1}-\phi_{2}\right)$
$\displaystyle=\left[\frac{\left(p_{i}-eA_{i}\right)^{2}}{m}+m\right]\left(\phi_{1}+\phi_{2}\right),$

Addition and subtraction of these two equations lead to the system of coupled
differential equations of first order in time,

$\displaystyle\frac{\left(p_{i}-eA_{i}\right)^{2}}{2m}\left(\phi_{1}+\phi_{2}\right)+\left(m+eA_{0}\right)\phi_{1}$
$\displaystyle=i\frac{\partial\phi_{1}}{\partial t}$ (8)
$\displaystyle\frac{-\left(p_{i}-eA_{i}\right)^{2}}{2m}\left(\phi_{1}+\phi_{2}\right)-\left(m-eA_{0}\right)\phi_{2}$
$\displaystyle=i\frac{\partial\phi_{2}}{\partial t},$

Using Eqs. ((8)), the FV Hamiltonian of a scalar particle in the presence of
the electromagnetic interaction may be written as

$\mathcal{H}_{KG}=\left(\tau_{z}+i\tau_{y}\right)\frac{\left(p_{i}-eA_{i}\right)^{2}}{2m}+m\tau_{z}+eA_{0}(x),$
(9)

where $\tau_{i}\,\left(i=x,y,z\right)$ are the conventional $2\times 2$ Pauli
matrices given by

$\tau_{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\tau_{y}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\tau_{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (10)

It is noteworthy that the Hamiltonian ((9)) fulfills the generalized hermicity
condition 111The Hamiltonian $\mathcal{H}$ is said to be pseudo-Hermitian if
there is an invertible, Hermitian, linear operator $\beta$ such that
$\mathcal{H}^{\dagger}=\beta\mathcal{H}\beta^{-1}$ (key-45, ),

$\mathcal{H}_{KG}=\tau_{z}\mathcal{H}_{KG}^{\dagger}\tau_{z},\qquad\mathcal{H}_{KG}^{\dagger}=\tau_{z}\mathcal{H}_{KG}\tau_{z}.$
(11)

For the free particle propagation i.e, no interaction is considered
$\left(A_{\mu}=0\right),$ the one dimensional FV Hamiltonian simplifies to

$\mathcal{H}_{0}=\left(\tau_{z}+i\tau_{y}\right)\frac{p_{x}^{2}}{2m}+m\tau_{z},$
(12)

The solutions to the free Hamiltonian (which is independent of time) are just
stationary states. Assuming a solution of the form (key-36, ),

$\Phi\left(x,t\right)=\Phi\left(x\right)e^{-iEt}=\left(\begin{array}[]{c}\phi_{1}\left(x\right)\\\
\phi_{2}\left(x\right)\end{array}\right)e^{-iEt},$ (13)

with $E$ being the energy of the system. Thus, Eq. ((2)) can then be written
as

$\mathcal{H}_{0}\Phi\left(x\right)=E\Phi\left(x\right),$ (14)

which is the one-dimensional FV equation of the free relativistic spin-0
particle and it is carried out with the aim of having an alternative
Schrodinger’s to KG equation. In what follows, the above method will be used
to find the solutions of wave equations in curved space-time, namely, the
cosmic string.

###  II.2 KGO in Schrödinger Form

For the next discussion, it is preferable to analyze the KGO in Minkowski
space-time exploiting the method described previously. In order to examine the
KGO in the FV representation, we start with the minimal substitution
$p_{\mu}\longrightarrow p_{\mu}+im\omega x_{\mu}$ of the momentum operator
(key-46, ). Here $\omega$ is the oscillation frequency and $x_{\mu}=(0,x,0,0)$
. Thus, by generalizing the momentum operator, the one dimensional FV
Hamiltonian ((12)) becomes

$\mathcal{H}_{KGO}=\frac{1}{2m}\left(\tau_{z}+i\tau_{y}\right)\left(P_{x}-im\omega
x\right)\left(P_{x}+im\omega x\right)+m\tau_{z},$ (15)

In this way, one obtains the Schrödinger formulation of KGO in one dimension
by substituting the Ansatz ((13)) into Eq. ((14)) by means of Eq. ((15)),

$\displaystyle\frac{1}{2m}\left[\frac{d^{2}}{dx^{2}}-m^{2}\omega^{2}x^{2}+m\omega\right]\left(\phi_{1}+\phi_{2}\right)+m\phi_{1}$
$\displaystyle=E\phi_{1}$ (16)
$\displaystyle-\frac{1}{2m}\left[\frac{d^{2}}{dx^{2}}-m^{2}\omega^{2}x^{2}+m\omega\right]\left(\phi_{1}+\phi_{2}\right)-m\phi_{2}$
$\displaystyle=E\phi_{2},$

after performing a calculation similar to the one was done in the previous KG
equation case, then it follows that

$\left[\frac{d^{2}}{dx^{2}}-m^{2}\omega^{2}x^{2}+\left(E^{2}-m^{2}+m\omega\right)\right]\psi\left(x\right)=0,$
(17)

or,

$\left[\frac{d^{2}}{dx^{2}}-\lambda_{1}x^{2}+\lambda_{2}\right]\psi\left(x\right)=0$
(18)

where we have set,

$\lambda_{1}=m^{2}\omega^{2},\qquad\lambda_{2}=E^{2}-m^{2}+m\omega.$ (19)

Eq. ((18)) is a second order differential equation for the field $\psi$
describing the KGO dynamics in one dimensional Minkowski space-time. The
solution of the above equation can be found in literature, and a quantization
condition of the energy is followed from that solution, providing

$\frac{E^{2}-m^{2}+m\omega}{2m}=\left(n+\frac{1}{2}\right)\omega,\qquad
n=0,1,2,\cdots.$ (20)

After arranging and simplifying the condition ((20)), we find the following
expression of the energy spectrum

$E=\pm\sqrt{2m\omega n+m^{2}},$ (21)

Eq.((21)) presents the relativistic energy spectrum of KGO in Minkowski space-
time.

##  III The FV Representation of Spin-0 Particle in Cosmic String Space-time

The purpose of this section is to study the KGO in the background geometry of
a cosmic string with the use of the FV scheme. It is well known that the
generally covariant relativistic wave equations of a scalar particle in a
Riemannian space-time defined by the metric tensor $g_{\mu\nu}$ , can be
obtained by reformulating the KG equation such that ( see, e.g, the textbooks
(key-8, ; key-9, ) )

$\left(\square+m^{2}-\xi R\right)\Phi(x,t)=0,$ (22)

where $\square$ is the Laplace-Beltrami operator given by

$\square=g^{\mu\nu}D_{\mu}D_{\nu}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\right),$
(23)

$\xi$ is a real dimensionless coupling constant and $R$ is the Ricci scalar
curvature defined by $R=g^{\mu\nu}R_{\mu\nu}$ where $R_{\mu\nu}$ is the Ricci
curvature tensor. $g^{\mu\nu}$ is the inverse metric tensor and
$g=\det\left(g^{\mu\nu}\right)$.

We now would like to study the quantum dynamics of spin-0 particles in the
space-time induced by a (2+1)-dimensional static cosmic string and set up the
corresponding FV formulation.

###  III.1 FV formulation of KGO in static cosmic string space-time

Before we study the KGO in the Hamiltonian representation, let us first derive
the KG wave equation for the free relativistic scalar particle propagating in
the cosmic string space-time that is assumed to be static and cylindrical
symmetric.

The general expression for a (3+1)-dimensional cosmic string metric is defined
by the line element (key-47, ; key-48, )

$\displaystyle ds^{2}$ $\displaystyle=g_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle=dt^{2}-dr^{2}-\alpha^{2}r^{2}d\varphi^{2}-dz,$ (24)

in cylindrical coordinates 222Note that this metric is an exact solution to
Einstein’s field equations for $0\leq\mu<1/4,$ and by setting
$\varphi^{\prime}=\alpha\varphi$, then it represents a flat conical exterior
space with angle deficit $\delta\phi=8\pi\mu$.. Here $-\infty\leq
t\leq+\infty$, $r\geq 0$, $0\leq\varphi\leq 2\pi$, $-\infty\leq z\leq+\infty,$
and $\alpha\in[0,1[\>$ is the angular parameter which determines the angular
deficit $\delta\varphi=2\pi(1-\alpha)$, and it is related to the linear mass
density $\mu$ of the string by $\alpha=1-4\mu$.

For the sake of simplicity in treating our quantum mechanical problem, let us
work in a lower dimensional space in which $z=\text{cont}$, and because there
is no structure in $z$-direction, we can suppress it (key-49, ; key-50, ).
Thus, the metric of a static cosmic string with cylindrical symmetry has a
$(2+1)$-dimensional form 333Since this metric is Lorentz-invariant under
boosts in the $(t,z)$ plane (key-18, ; key-48, ), and by the virtue of
rotational symmetry along the $z$-axis, it is reasonable to assume that the
theory is invariant in the 2-dimentional $(r,\varphi)$ surface.

$ds^{2}=dt^{2}-dr^{2}-\alpha^{2}r^{2}d\varphi^{2}$ (25)

where the components of the metric and the inverse metric tensors are,
respectively,

$g_{\mu\nu}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\ 0&0&-\left(\alpha
r\right)^{2}\end{array}\right),\quad
g^{\mu\nu}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\
0&0&\frac{-1}{\left(\alpha r\right)^{2}}\end{array}\right)$ (26)

Its is worthy to mention that the subject of spinless massive particles in the
geometry generated by a static cosmic string background has been discussed in
several papers ( see, e.g, (key-25, ; key-51, ))

In what follows, we shall adopt the procedure presented in Refs. (key-52, ;
key-53, ) to derive the FV form of KG wave equation in curved manifolds. We
use the generalized Feshbach–Villars transformation (GFVT) 444An equivalent
transformation was proposed earlier in Ref.(key-54, ) which is appropriate for
describing both massive and massless particles. In the GFVT, the components of
the wave function $\Phi$ are given by (key-52, )

$\psi=\phi_{1}+\phi_{2},\qquad
i\tilde{\mathcal{D}}\psi=\mathcal{N}\left(\phi_{1}-\phi_{2}\right),$ (27)

where $\mathcal{N}$ is an arbitrary nonzero real parameter, and we have
defined $\tilde{\mathcal{D}}=\frac{\partial}{\partial t}+\mathcal{Y},$ with

$\mathcal{Y}=\frac{1}{2g^{00}\sqrt{-g}}\left\\{\partial_{i},\sqrt{-g}g^{0i}\right\\},$
(28)

The curly bracket in Eq. ((28)) denotes the anti-commutator. For the above
mentioned transformation, the Hamiltonian reads

$\mathcal{H}_{GFVT}=\tau_{z}\left(\frac{\mathcal{N}^{2}+\mathcal{T}}{2\mathcal{N}}\right)+i\tau_{y}\left(\frac{-\mathcal{N}^{2}+\mathcal{T}}{2\mathcal{N}}\right)-i\mathcal{Y},$
(29)

with

$\mathcal{T}=\frac{1}{g^{00}\sqrt{-g}}\partial_{i}\sqrt{-g}g^{ij}\partial_{j}+\frac{m^{2}-\xi
R}{g^{00}}-\mathcal{Y}^{2},$ (30)

here and below $(i,j=1,2,3)$. We note that for $\mathcal{N}=m$, the original
FV transformations are satisfied.

Now, considering the metric (25), it is easy to find that $R=0$, in other
words, the space-time is locally flat (there is no local gravity), and hence
the coupling term is vanishing555The case $\xi=0$ is refereed to as minimal
coupling. However, for massless theory, $\xi$ takes the value 1/6 (in 4
dimensions). Then, in this later case, the equations of motion are conformally
invariant..

A straightforward calculation leads to $\mathcal{Y}=0,$ then we obtain

$\mathcal{T}=-\frac{d^{2}}{dr^{2}}-\frac{1}{r}\frac{d}{dr}-\frac{1}{\alpha^{2}r^{2}}\frac{d^{2}}{d\varphi^{2}}+m^{2},$
(31)

Using these results to find the Hamiltonian (29), then because of the time and
angular independence in the metric (25), one can assume a solution of the form
666we seek solutions that are cylindrical symmetric, i.e. solutions that have
a rotational symmetry in the $(x,y)$-plane and do not explicitely depend on
$z$.,

$\Phi(t,r,\varphi)=\Phi(r)e^{-i\left(Et-j\varphi\right)},$ (32)

where $j=0,\pm 1,\pm 2,..$ are the eigenvalues of the $z$component of the
angular momentum operator. Therefore, it follows that the KG equation (22) may
be written equivalently to the following two coupled equations

$\displaystyle\left(\mathcal{N}^{2}+\mathcal{T}\right)\phi_{1}+\left(-\mathcal{N}^{2}+\mathcal{T}\right)\phi_{2}$
$\displaystyle=2\mathcal{N}E\phi_{1}$
$\displaystyle-\left(\mathcal{N}^{2}+\mathcal{T}\right)\phi_{2}-\left(-\mathcal{N}^{2}+\mathcal{T}\right)\phi_{1}$
$\displaystyle=2\mathcal{N}E\phi_{2},$ (33)

The sum and the difference of the two last equations give a second order
differential equation for the field $\psi$. Thus, the radial equation is
written as follows

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{\zeta^{2}}{r^{2}}+\kappa\right]\psi\left(r\right)=0,$
(34)

where we have set

$\zeta=\frac{j}{\alpha},\qquad\kappa=\sqrt{E^{2}-m^{2}}.$ (35)

We can observe that Eq. ((34)) is a Bessel equation and its general solution
is defined by (key-55, )

$\psi\left(r\right)=A\,J_{|\zeta|}\left(\kappa
r\right)+B\,Y_{|\zeta|}\left(\kappa r\right),$ (36)

where $J_{|\zeta|}\left(\kappa r\right)$ and $Y_{|\zeta|}\left(\kappa
r\right)$ are the Bessel functions of order $\zeta$ and of the first and the
second kind, respectively. Here $A$ and $B$ are arbitrary constants. We notice
that at the origin when $\zeta=0$, the function $J_{|\zeta|}\left(\kappa
r\right)\neq 0$. However, $Y_{|\zeta|}\left(\kappa r\right)$ is always
divergent at the origin. In this case, we will consider only
$J_{|\zeta|}\left(\kappa r\right)$ when $\zeta\neq 0$. Hence, we write the
solution to Eq. ((34)) as follows

$\psi\left(r\right)=A\,J_{\frac{|j|}{\alpha}}\left(\sqrt{E^{2}-m^{2}}\,r\right),$
(37)

using this solution, we can now write the complete two-components wave-
function of the spinless massive KG particle in the space-time of a static
cosmic string

$\psi\left(t,r,\varphi\right)=|C|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)e^{-i\left(Et-j\varphi\right)}\,J_{\frac{|j|}{\alpha}}\left(\sqrt{E^{2}-m^{2}}\,r\right),$
(38)

The constant $|C|$ can be obtained by the appropriate normalization condition
associated with the KG equation ( e.g, see Ref. (key-56, ; key-57, )),
however, it is fortunate that non determining the normalization constants
throughout this manuscript does not affect the final results.

We turn now to the particular case where we want to extend the GFVT for the
KGO. In general, we need to perform a substitution of the momentum operator in
Eq. ((22)). Consequently, it is possible to rewrite Eq. ((30)) as follows

$\mathcal{T}=\frac{1}{\sqrt{-g}}\left(\frac{\partial}{\partial r}-m\omega
r\right)\left(-\sqrt{-g}\right)\left(\frac{\partial}{\partial r}+m\omega
r\right)-\frac{1}{\alpha^{2}r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}+m^{2},$
(39)

Similarly, a straightforward calculation based on the procedure that was
carried out in the above discussion, one can obtain the following differential
equation

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}+m^{2}\omega^{2}r^{2}-\frac{\sigma^{2}}{r^{2}}+\delta\right]\psi\left(r\right)=0,$
(40)

with

$\sigma^{2}=\left(\frac{j}{\alpha}\right)^{2},\qquad\delta=E^{2}-m^{2}+2m\omega.$
(41)

Eq. ((40)) is the KGO for spin-0 particle in the space-time of a static cosmic
string. To obtain the solution of this equation, we first propose a
transformation of the radial coordinate

$\chi=m\omega r^{2},$ (42)

substituting the expression for $\chi$ into Eq. ((40)), we obtain

$\left[\frac{d^{2}}{d\chi^{2}}+\frac{1}{\chi}\frac{\partial}{d\chi}-\frac{\sigma^{2}}{4\chi^{2}}+\frac{\delta}{4m\omega\chi}-\frac{1}{4}\right]\psi\left(\chi\right)=0.$
(43)

Now, if we study the asymptotic behavior of the wave function at the origin
and infinity, and since we are looking for regular solutions, we may suppose a
solution that has the form

$\psi\left(\chi\right)=\chi^{\frac{\left|\sigma\right|}{2}}e^{-\frac{\chi}{2}}F\left(\chi\right),$
(44)

As before, this can be substituted back into Eq. ((43)), then we have

$\chi\frac{d^{2}F\left(\chi\right)}{d\chi^{2}}+\left(|\sigma|+1-\chi\right)\frac{dF\left(\chi\right)}{d\chi}-\left(\frac{|\sigma|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2}\right)F\left(\chi\right)=0,$
(45)

This is the confluent hyper-geometric equation (key-58, ) whose solutions are
expressed in terms of the confluent hyper-geometric function type

$F\left(\chi\right)=_{1}F_{1}\left(\frac{\left|\sigma\right|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},+1,\chi\right),$
(46)

We should note that the solution ((46)) must be a polynomial function of
degree $n$. However, taking $n\rightarrow\infty$ imposes a divergence issue.
We can have a finite polynomial only if the factor of the last term in Eq.
((45)) is a negative integer, meaning,

$\frac{\left|\sigma\right|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2}=-n\qquad,n=0,1,2,\cdots.$
(47)

(a) $j=1.\,a=0.1,\,\omega=m=1$ Figure 1: Energy of KGO in rotating cosmic
string as a function of the quantum number $n$ for different values of
$\alpha$ in (a) and of $a$ in (b).

Exploiting this result, and by the insertion of the parameters ((41)), we can
obtain the quantified energy spectrum of KGO in the static cosmic string
space-time, hence,

$E\left(n,\alpha,j\right)=\pm\sqrt{4m\omega
n+\frac{2m\omega\left|j\right|}{\alpha}+m^{2}},$ (48)

We can observe that the energy depends explicitly on the angular deficit
$\alpha$. In other words, the curvature of the space-time which is influenced
by the topological defect i.e, the cosmic string through the deficit parameter
will affects on the relativistic dynamics of the scalar particle by generating
a gravitational field due to the presence of the wedge angle. Fig. (1) shows
the energy levels of the KGO as a function of the quantum number $n$ in a
static cosmic string space characterized by a wedge parameter $\alpha$ which
takes different values.

The corresponding wave function is given by

$\psi\left(t,r,\varphi\right)=|\tilde{C}|\left(m\omega
r^{2}\right)^{\frac{\left|j\right|}{2\alpha}}e^{-\frac{m\omega
r^{2}}{2}}{}_{1}F_{1}\left(\frac{\left|j\right|}{2\alpha}-\frac{\delta}{4m\omega}+\frac{1}{2},\frac{|j|}{\alpha}+1,m\omega
r^{2}\right),$ (49)

Then the general eigenfunctions are expressed as

$\psi\left(t,r,\varphi\right)=|\tilde{C}|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)\left(m\omega
r^{2}\right)^{\frac{\left|j\right|}{2\alpha}}e^{-\frac{m\omega
r^{2}}{2}}e^{-i\left(Et-j\varphi\right)}{}_{1}F_{1}\left(\frac{\left|j\right|}{2\alpha}-\frac{\delta}{4m\omega}+\frac{1}{2},\frac{|j|}{\alpha}+1,m\omega
r^{2}\right),$ (50)

where $|\tilde{C}|$ is the normalization constant.

(a) $n=0,\,\alpha=0.5$ (b) $n=1,\,\alpha=0.5$ (c) $n=2,\,\alpha=0.5$ (d)
$n=3,\,\alpha=0.5$

Figure 2: Plots of the density of KGO in static cosmic string as a function of
the radial distance $r$ for different values of $n$ with the parameters
$\alpha=0.5$ and $j=1.$

Having the complete wave-function above, we are in position to calculate the
density $\rho_{{}_{KGO}}$ corresponds to the KGO given by (key-33, )

$\rho_{{}_{KGO}}=\psi^{*}\tau_{z}\psi$ (51)

In Fig. (2) we plot the density of KGO in static cosmic string as a function
of the radial distance $r.$ Obviously, we can see that the density is affected
by the choice of the variable $\mathcal{N}$.

###  III.2 FV formulation of KGO in rotating cosmic string space-time(41)

In this section we shall analyze the KGO in the background geometry of a
rotating cosmic string in three dimensions. Similarly to the case studied in
Sec. III, the equations of motion of a scalar particle can be achieved by
considering the GFVT. Several authors studied the quantum dynamics of
relativistic particles in the space-time of a cosmic string with rotational
effects, and many models were considered in this context. For instance, in a
previous paper by Mazur (key-59, ), the quantum mechanical properties of
massive (or massless) particles in the gravitational field of spinning cosmic
strings were discussed. He showed that the energy should be quantized in the
presence of non-zero angular momentum of the string. Later, Gerbert and Jackiw
(key-60, ) have presented solutions for the KG and Dirac equations in the
(2+1)-dimensional space-time created by a massive point particle, with
arbitrary angular momentum. In Ref. (key-61, ), the vacuum expectation value
of the stress-energy tensor for a massless scalar field conformally coupled to
gravitation was discussed. The authors of Ref. (key-62, ) have examined the
behavior of a quantum test particle satisfying the Klein-Gordon equation in a
space-time of a spinning cosmic string. Additionally, it was shown in Ref.
(key-63, ) that rotating cosmic string solution of $U(1)$ scalar field theory
with a cylindrical symmetric energy density can be characterized as an extrema
of the field’s energy for given angular and linear momenta. Moreover,
topological and geometrical phases due to gravitational field of a cosmic
string that has mass and angular momentum were investigated in Ref. (key-64,
).

Recently, the subject of gravitational effects of rotating cosmic strings has
attracted considerable interest in connection with the dynamics of
relativistic quantum particles and their properties. For example, vacuum
fluctuations for a massless scalar field around a spinning cosmic string were
studied in Ref. (key-65, ) applying a re-normalization method. In like manner,
vacuum polarization of a scalar field in the gravitational background of a
spinning cosmic string was investigated in Ref. (key-66, ). Furthermore, the
authors of Ref. (key-67, ) have analyzed the Landau levels of a spinless
massive particle in the spacetime of a rotating cosmic string by means of a
fully relativistic approach. Wang et al. (key-68, ) have treated the model of
the KGO coupled to a uniform magnetic field in the background of the rotating
cosmic string. Also, the problem of a spinless relativistic particle subjected
to a uniform magnetic field in the spinning cosmic string space-time was
addressed in Ref.(key-69, ). Besides, the relativistic quantum dynamics of a
KG scalar field subjected to a Cornell potential in spinning cosmic string
space-time was presented in Ref. (key-70, ). In addition, the relativistic
scalar charged particle in a rotating cosmic string space-time with Cornell-
type potential and Aharonov-Bohm effect was analyzed in Ref. (key-29, ). Other
publications in which rotating cosmic strings have been considered to study
the quantum properties of relativistic systems could be mentioned in this
respect (e.g, (key-71, ; key-72, ; key-73, ; key-74, ; key-75, ; key-76, ;
key-77, ; key-78, ) and related references therein). Aside from investigating
bosonic particles in that specific space-time, in the literature one can find
various discussions concerning the effects of the topology and geometry of
this space-time on the quantum dynamics of fermionic particles and their
behaviour ( see for instance Refs. (key-79, ; key-80, ; key-81, ; key-82, ;
key-83, ; key-84, ; key-85, )).

In the next discussion, we extend the problem studied in Sec. III to a more
general space-time with non-zero angular momentum. We consider a massive,
relativistic spin-0 particle whose wave-function is denoted by $\Psi$ and
satisfies the KG equation 22 in the space-time induced by a (2+1)-dimensional
stationary rotating cosmic string which is described by the following line
element (key-86, ; key-87, ; key-59, ).

$ds^{2}=dt^{2}+2a\,dt\,d\phi-
dr^{2}-\left(\alpha^{2}r^{2}-a^{2}\right)d\varphi^{2},$ (52)

always the cylindrical coordinates $(t,r,\varphi)$ are with usual ranges, and
$\alpha=1-4\mu$ is the wedge parameter with $\alpha<1.$ Here, the rotation
parameter $a=4J$ has units of distance with $J$ stands for the angular
momentum of the string.

The covariant and contravariant components of the metric tensor are

$g_{\mu\nu}=\left(\begin{array}[]{ccc}1&0&a\\\ 0&-1&0\\\
a&0&-\left(\alpha^{2}r^{2}-a^{2}\right)\end{array}\right),\quad
g^{\mu\nu}=\left(\begin{array}[]{ccc}1-\frac{a^{2}}{\alpha^{2}r^{2}}&0&\frac{a}{\alpha^{2}r^{2}}\\\
0&-1&0\\\
\frac{a}{\alpha^{2}r^{2}}&0&-\frac{1}{\alpha^{2}r^{2}}\end{array}\right).$
(53)

To study the relativistic quantum motion of a scalar boson interacting with
the gravitational field of the background geometry defined by the metric 52,
we need to write the generic KG equation in the curved space in question,
given by

$\left(\square+m^{2}-\xi R\right)\Psi(t,r,\varphi)=0,$ (54)

where $\Psi$ is a two-components vector

$\Psi(x)=\left(\begin{array}[]{c}\phi_{1}\left(x\right)\\\
\phi_{2}\left(x\right)\end{array}\right),\qquad x\equiv(t,r,\varphi).$ (55)

satisfying the condition

$\psi=\phi_{1}+\phi_{2},\qquad
i\mathcal{D}^{\prime}\psi=\mathcal{N}\left(\phi_{1}-\phi_{2}\right),$ (56)

with $\mathcal{D}^{\prime}=\frac{\partial}{\partial t}+\mathcal{Y^{\prime}}.$

The next step is to develop the approach used in Sec III for the case of
rotating cosmic strings. We adopt the GFVT to derive the equations of motion
for this problem, then we solve them to obtain the wave functions and the
energy spectra. Following Ref. (key-52, ; key-53, ) , the identification of
the Hamiltonian leads to the idea of employing a two-components formulation of
the KG type fields. In order to rewrite Eq. (54) in the Hamiltonian form, one
requires to introduce new definitions for the quantities presented in Eqs.
((29)), ((30)) and ((28)). In the case under consideration, we have

$\mathcal{H}_{GFVT}^{\prime}=\tau_{z}\left(\frac{\mathcal{N}^{2}+\mathcal{T^{\prime}}}{2\mathcal{N}}\right)+i\tau_{y}\left(\frac{-\mathcal{N}^{2}+\mathcal{T^{\prime}}}{2\mathcal{N}}\right)-i\mathcal{Y}^{\prime},$
(57) $\displaystyle\mathcal{T}^{\prime}$
$\displaystyle=\partial_{i}\frac{G^{ij}}{g^{00}}\partial_{j}+\frac{m^{2}-\xi
R}{g^{00}}+\frac{1}{\mathcal{F}}\nabla_{i}\left(\sqrt{-g}G^{ij}\right)\nabla_{j}\left(\frac{1}{\mathcal{F}}\right)+\sqrt{\frac{\sqrt{-g}}{g^{00}}}G^{ij}\nabla_{i}\nabla_{j}\left(\frac{1}{\mathcal{F}}\right)+\frac{1}{4\mathcal{F}^{4}}\left[\nabla_{i}\left(\mathcal{U}^{i}\right)\right]^{2}$
$\displaystyle\qquad-\frac{1}{2\mathcal{F}^{2}}\nabla_{i}\left(\frac{g^{0i}}{g^{00}}\right)\nabla_{j}\left(\mathcal{U}^{i}\right)-\frac{g^{0i}}{2g^{00}\mathcal{F}^{2}}\nabla_{i}\nabla_{j}\left(\mathcal{U}^{i}\right),$
(58)
$\mathcal{Y}^{\prime}=\frac{1}{2}\left\\{\partial_{i},\sqrt{-g}\frac{g^{0i}}{g^{00}}\right\\},$
(59)

where

$G^{ij}=g^{ij}-\frac{g^{0i}g^{0j}}{g^{00}},\qquad\mathcal{F}=\sqrt{g^{00}\sqrt{-g}},\qquad\mathcal{U}^{i}=\sqrt{-g}g^{0i}.$
(60)

It has been shown in Ref. (key-53, ) that the transformations ((57)), ((58))
and (59) with the definitions ((60)) are exact and covers any inertial and
gravitational fields. It should be noted that these exact transformations
ensures to obtain the block-diagonal form of the Hamiltonian
$\mathcal{H}_{GFVT}$ which does not depend on the parameter $\mathcal{N}.$ For
the geometry ((53)), the eigenvalues of the operator $\mathcal{Y}^{\prime}$
are given by

$\mathcal{Y}^{\prime}\Psi\left(t,r,\varphi\right)=\frac{1}{2}\left\\{\partial_{2},\sqrt{-g}\frac{g^{02}}{g^{00}}\right\\}\Psi\left(t,r,\varphi\right)=\frac{g^{02}}{g^{00}}\frac{\partial}{\partial\varphi}\Psi\left(t,r,\varphi\right),$
(61)

where the field $\Psi$ obeys the non-unitary transformation
$\Psi\equiv\Phi^{\prime}=\mathcal{F}\Phi$ which permits to obtain pseudo-
Hermitian Hamiltonian
$\mathcal{H}_{GFVT}^{\prime}=\mathcal{F}\mathcal{H}_{GFVT}^{\prime}\mathcal{F}^{-1},$
$\mathcal{H}_{GFVT}^{\prime}=\tau_{z}\left(\mathcal{H}_{GFVT}^{\prime}\right)^{\dagger}\tau_{z}.$

After some mathematical calculations, Eq. ((58)) takes the form

$\mathcal{T}^{\prime}=-\frac{1}{\mathcal{F}}\left[\left(\frac{\partial}{\partial
r}\right)\left(\sqrt{-g}\frac{\partial}{\partial
r}\right)+\frac{\sqrt{-g}}{\alpha^{2}r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}\right]\frac{1}{\mathcal{F}}+\frac{m^{2}}{g^{00}}-\left(\frac{g^{02}}{g^{00}}\frac{\partial}{\partial\varphi}\right)^{2},$
(62)

where for the metric ((52)) the Ricci scalar vanishes i.e, $R=0$.

Using the two components fields (55) and the Hamiltonian ((57)), we can
express the field equation ((54)) as the Schrodinger’s equation

$\mathcal{H}_{GFVT}^{\prime}\Psi^{\prime}(x)=i\,\frac{d}{dt}\Psi^{\prime}(x),\qquad
x\equiv(t,r,\varphi).$ (63)

To solve this eigenvalue problem, let us consider for the wave function the
Ansatz below

$\Psi(x)=\mathcal{F}\Phi\left(x\right)=\mathcal{F}\left(\begin{array}[]{c}\phi_{1}\left(r\right)\\\
\phi_{2}\left(r\right)\end{array}\right)e^{-\left(iEt-j\varphi\right)},$ (64)

substituting ((64)) into Eq. ((63)), we find the following coupled
differential equations

$\displaystyle\mathcal{N}^{2}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)+\mathcal{T}^{\prime}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)+2\mathcal{N}\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\phi_{1}$
$\displaystyle=2\mathcal{N}E\mathcal{F}\phi_{1}$ (65)
$\displaystyle\mathcal{N}^{2}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)-\mathcal{T}^{\prime}\mathcal{F}\left(\phi_{1}+\phi_{2}\right)+2\mathcal{N}\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\phi_{2}$
$\displaystyle=2\mathcal{N}E\mathcal{F}\phi_{2},$

We can add and subtract the two equations of ((65)) to get, respectively,

$\begin{aligned}
\mathcal{N}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)+\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\left(\phi_{1}+\phi_{2}\right)&=E\mathcal{F}\left(\phi_{1}+\phi_{2}\right)\end{aligned},$
(66)

and

$-\mathcal{T}^{\prime}\mathcal{F}\left(\phi_{1}+\phi_{2}\right)+\mathcal{N}\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\left(\phi_{1}-\phi_{2}\right)=\mathcal{N}E\mathcal{F}\left(\phi_{1}-\phi_{2}\right),$
(67)

where we have used the relation

$\phi_{1}=\frac{1}{\mathcal{N}}\left(E-\frac{g^{02}}{g^{00}}j\right)\phi_{2},$
(68)

After simple algebraic manipulations we arrive at the following second order
differential equation for the radial function $\psi(r)$

$\left[\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
r}\left(\sqrt{-g}\frac{\partial}{\partial
r}\right)-\frac{\left(aE+j\right)^{2}}{\alpha^{2}r^{2}}+E^{2}-m^{2}\right]\psi\left(r\right)=0,$
(69)

setting
$\varsigma^{2}=\frac{\left(aE+j\right)^{2}}{\alpha^{2}},\gamma=E^{2}-m^{2},$
yields

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{\varsigma^{2}}{r^{2}}+\gamma\right]\varphi_{s}\left(r\right)=0,$
(70)

Eq. ((70)) is expressed as a Bessel differential equation, its solutions can
be written in terms of Bessel function of first kind as

$\psi\left(r\right)=A^{\prime}\,J_{\frac{aE+j}{\alpha}}\left(\sqrt{E^{2}-m^{2}}\,r\right),$
(71)

where $A^{\prime}$ is an integration constant.

The complete eigenstates are given by

$\psi\left(t,r,\varphi\right)=|C^{\prime}|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)e^{-i\left(Et-j\varphi\right)}\,J_{\frac{aE+j}{\alpha}}\left(\sqrt{E^{2}-m^{2}}\,r\right),$
(72)

From now on we proceed to study the Klein Gordon oscillator in a rotating
cosmic string space-time . Firstly, we start by considering a scalar quantum
particle embedded in the background gravitational field of the space-time
described by the metric ((52)). In this way, we shall introduce a replacement
of the momentum operator $p_{i}\longrightarrow p_{i}+im\omega x_{i}$ where
$p_{i}=i\nabla_{i}$ in Eq. ((58)). Then, we have

$\mathcal{T}^{\prime}=-\frac{1}{\mathcal{F}}\left[\left(\frac{\partial}{\partial
r}-m\omega r\right)\left(\sqrt{-g}\left(\frac{\partial}{\partial r}+m\omega
r\right)\right)+\frac{\sqrt{-g}}{\alpha^{2}r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}\right]\frac{1}{\mathcal{F}}+\frac{m^{2}}{g^{00}}-\left(\frac{g^{02}}{g^{00}}\frac{\partial}{\partial\varphi}\right)^{2},$
(73)

Based on the previous analyses, we shall apply the GFVT for the case of KGO in
the concerned space through the same steps used before. Inserting Eq. ((73))
and Eq. ((61)) into the Hamiltonian ((57)), then assuming the solution ((64))
gives two coupled differential equations similar to those of Eq. ((65)) but
with different value of $\mathcal{T}^{\prime}$.

Manipulating exactly the same steps before, we obtain the following radial
equation

$\left[\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}-m^{2}\omega^{2}r^{2}-\frac{\vartheta^{2}}{r^{2}}+\delta\right]\psi\left(r\right)=0,$
(74)

where we have defined

$\vartheta^{2}=\left(\frac{aE+j}{\alpha}\right)^{2},\qquad\delta=E^{2}-m^{2}+2m\omega.$
(75)

To solve Eq. ((74)), we introduce a new dimensionless variable
$\mathcal{R}=m\omega r^{2}$, and by substitution into Eq. (74) , the resulting
equation reads

$\left[\frac{d^{2}}{d\mathcal{R}^{2}}+\frac{1}{\mathcal{R}}\frac{d}{d\mathcal{R}}-\frac{\vartheta^{2}}{4\mathcal{R}^{2}}-\frac{1}{4}+\frac{\delta}{4m\omega\mathcal{R}}\right]\psi\left(\mathcal{R}\right)=0,$
(76)

To eliminate the term $d\psi(\mathcal{R})\big{/}d\mathcal{R},$ we consider the
following change of variable

$\psi(\mathcal{R})=\mathcal{R}^{-\frac{1}{2}}\Xi\left(\mathcal{R}\right),$
(77)

Eq. ((76)) then becomes

$\frac{d^{2}\Xi\left(\mathcal{R}\right)}{d\mathcal{R}^{2}}+\left[-\frac{1}{4}+\frac{\delta}{4m\omega\mathcal{R}}+\frac{\frac{1}{4}-\left(\frac{\vartheta}{2}\right)^{2}}{\mathcal{R}^{2}}\right]\Xi\left(\mathcal{R}\right)=0,$
(78)

and it has the form of the Whittaker differential equation (key-55, ). The
general solution of this equation, which is regular at the origin, is given by

$\Xi\left(\mathcal{R}\right)=|C|\mathcal{R}^{-\tfrac{1}{2}}M\left(\frac{\delta}{4m\omega},\frac{|\vartheta|}{2},\mathcal{R}\right),$
(79)

where $|C|$ is an arbitrary constant and
$M\left(\frac{\delta}{4m\omega},\frac{|\vartheta|}{2},\mathcal{R}\right)$ is
the Whittaker M-function defined via the confluent hyper-geometric functions
as

$M\left(\frac{\delta}{4m\omega},\frac{|\vartheta|}{2},\mathcal{R}\right)=e^{-\frac{\mathcal{R}}{2}}\mathcal{R}^{\frac{\left|\vartheta\right|+1}{2}}{}_{1}F_{1}\left(\frac{|\vartheta|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},|\vartheta|+1,\mathcal{R}\right),$
(80)

Using the definition ((80)), the final expression of the wave-function of the
spinless KGO propagating in the rotating cosmic string background can be
represented as

$\psi\left(t,r,\varphi\right)=|\mathtt{C}|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)\left(m\omega
r^{2}\right)^{\frac{\left|\vartheta\right|}{2}}e^{-\frac{m\omega}{2}r^{2}}e^{-i\left(Et-j\varphi\right)}{}_{1}F_{1}\left(\frac{|\vartheta|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},|\vartheta|+1,m\omega
r^{2}\right),$ (81)

where the parameters $\vartheta$ and $\delta$ are defined in Eq. ((75)).

Again, The asymptotic behavior of the confluent hyper-geometric function
implies that

$\frac{|\vartheta|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2}=-n,$ (82)

(a) $j=1.\,a=0.1,\,\omega=m=1$ (b) $j=1.\,\alpha=0.5,\,\omega=m=1$

Figure 3: Energy of KGO in rotating cosmic string as a function of the quantum
number $n$ for different values of $\alpha$ in (a) and of $a$ in (b).

Hence, after inserting $\vartheta$ and $\delta$ and by solving Eq. ((82)) for
$E$, we obtain the energy levels for our scalar particle

$E^{\pm}(n,\alpha,a,j)=\frac{am\omega}{\alpha}\pm\sqrt{4m\omega
n+\left(\frac{am\omega}{\alpha}\right)^{2}+\frac{2m\omega|j|}{\alpha}+m^{2},}$
(83)

(a) $n=0,\,a=0.1,\,\alpha=0.5$ (b) $n=1,\,a=0.1,\,\alpha=0.5$ (c)
$n=2,\,a=0.1,\,\alpha=0.5$ (d) $n=3,\,a=0.1,\,\alpha=0.5$

Figure 4: Plots of the density of KGO in rotating cosmic string as a function
of the radial distance $r$ for different values of $n$ with the parameters
$a=0.5$ and $j=1.$

(a) $\mathcal{N}=m,\,a=0.1$ (b) $\mathcal{N}=m,\,\alpha=0.5$

Figure 5: Plots of the density of KGO in rotating cosmic string as a function
of the radial distance $r$ for different values of $\alpha$ and $a$ with the
parameters $m=\omega=1$ and $j=1.$

As an interesting side result, we have an expression for the energy in which
we see all the parameters that characterize the background geometry. Indeed,
the energy spectrum does not depend only on the angular deficit parameter
$\alpha$, but also on the rotation parameter $a$. We expect that the external
gravitational field due to a rotating cosmic string affects on the
relativistic dynamics of the quantum particle, essentially, since it has non-
zero curvature concentrated along the $z$-axis and its angular momentum
appears throughout the parameter $a.$ By comparing the energy ((83)) with the
energy of KGO in a static cosmic string ((48)), we notice that
$E\left(n,\alpha,a,j\right)$ is shifted by the amount $am\omega\big{/}\alpha$
and enlarged by a conical factor. We plot the energy eigenvalues as a function
of the quantum number $n$ with fixed parameters $\omega=j=1$ and for different
values of the variable $\alpha$ and $a$ shown in Fig. (3).

Taking this result into the consideration, we can find the density for the FV-
KGO in the rotating cosmic string space as a function of the radial distance
$r$ and fixed parameter $\alpha$. Fig. (4)shows that the density is affected
when choosing the $\mathcal{N}$ variable for different situations. Now, the
case where $\mathcal{N}=m$ that corresponds with the case of KGO is presented
in the figure. (5). This figure shows that density is strongly dependent with
the parameter of rotation $a$.

##  IV Thermal properties of FVO in Rotating Cosmic String

This section is devoted to look into the thermal properties which may arise
throughout our study of the KGO in the space-time produced by a rotating
cosmic string. Having already obtained the energy spectrum of KGO in that
particular space, we are now in a position to present the thermodynamics of
the model encountered in Sec. (III.2) by calculating the partition function
which provides all the physical information of our quantum system. To better
illustrate our results, we present several figures in the next discussion.

###  IV.1 Partition function

We are interested in determining the partition function of KGO interacting
with the gravitational field induced by rotating cosmic string. Consequently,
the statistical quantities such as the free energy $F,$the mean energy $U,$the
entropy $S$and the specific heat $C_{v}$, associated with the relativistic
spinless particle in that space-time can be obtained from the partition
function.

The canonical ensemble partition function at finite temperature $T$ is given
by

$Z=\stackrel{{\scriptstyle[}}{{n}}=0]{\infty}{\sum}e^{-\beta E_{n}},$ (84)

where $E_{n}$ denotes the corresponding energy eigenvalues and
$\beta=\frac{1}{K_{B}T}$ with $K_{B}$ being the Boltzmann constant. Given the
spectrum (83) and by considering only the positive energies i.e,
$E_{n}=E_{n}^{+}$, we can express the partition function as

$Z=\stackrel{{\scriptstyle[}}{{n}}=0]{\infty}{\sum}e^{-\beta\left(\mathsf{Y}+\sqrt{\mathsf{A}n+\mathsf{B}}\right)}=e^{-\beta\mathsf{Y}}\stackrel{{\scriptstyle[}}{{n}}=0]{\infty}{\sum}e^{-\beta\sqrt{\mathsf{A}n+\mathsf{B}}},$
(85)

where

$\mathsf{Y}=\frac{am\omega}{\alpha},\qquad\mathsf{A}=4m\omega,\qquad\mathsf{B}=\left(\frac{am\omega}{\alpha}\right)^{2}+\frac{2m\omega|j|}{\alpha}+m^{2}$

In what follows, we closely follow the methods outlined in Ref.(key-88, ) and
revised in (key-89, ) with the intention of approximating the infinite sum
apparent in Eq. (84). For this purpose, we shall begin with rewriting the
series representation of the partition function $Z$ in terms of the Zeta
function. Hence we have

$\stackrel{{\scriptstyle[}}{{n}}=0]{\infty}{\sum}e^{-\beta\sqrt{\mathsf{A}n+\mathsf{B}}}=e^{-\widetilde{\beta}}+\stackrel{{\scriptstyle[}}{{n}}^{\prime}=0]{\infty}{\sum}e^{-\tilde{\beta}\sqrt{\mathsf{D}}\sqrt{\frac{1}{\mathsf{D}}+\left(n^{\prime}+1\right)}},\qquad
n=n^{\prime}+1.$ (86)

where we have introduced the notation

$\tilde{\beta}=\beta\sqrt{\mathsf{B}},\qquad\mathsf{D=\frac{A}{B}.}$

Using the Cahen-Mellin integral (key-90, ; key-91, )

$e^{-\mathit{z}}=\frac{1}{2\pi
i}\stackrel{{\scriptstyle[}}{{\mathfrak{}}}{c}-i\infty]{\mathfrak{c}+i\infty}{\int}\Gamma\left(s\right)z^{-s}ds,\qquad\left(|\textrm{arg}z|<\frac{\pi}{2};z\neq
0\right).$

where the path of integration is the vertical line $\Re(s)=\mathfrak{c}$, with
$\mathfrak{c}\in\mathbb{R}^{*}$, lying to the right of all the poles of
$\Gamma(s),$ the term
$\stackrel{{\scriptstyle[}}{{n}}^{\prime}=0]{\infty}{\sum}e^{-\tilde{\beta}\sqrt{\mathsf{D}}\sqrt{\frac{1}{\mathsf{D}}+\left(n^{\prime}+1\right)}}$
can be expressed as

$\stackrel{{\scriptstyle[}}{{n}}^{\prime}=0]{\infty}{\sum}e^{-\tilde{\beta}\sqrt{\mathsf{D}}\sqrt{\frac{1}{\mathsf{D}}+\left(n^{\prime}+1\right)}}=\frac{1}{2\pi
i}\stackrel{{\scriptstyle[}}{{\mathfrak{}}}{c}-i\infty]{\mathfrak{c}+i\infty}{\int}\left(\tilde{\beta}\sqrt{\mathsf{D}}\right)^{-s}\Gamma\left(s\right)\zeta_{H}\left(\frac{s}{2},\frac{1}{\mathsf{D}}+1\right),$
(87)

Figure 6: Contour Integration of the Mellin transformation of the problems in
question (key-89, ; key-91, )

$\Gamma(s)$ and $\zeta_{H}\left(\frac{s}{2},\frac{1}{\mathsf{D}}+1\right)$
refers to Euler Gamma and Hurwitz zeta functions, receptively. We note that
the integral converges only when $s\big{/}2>1$ requires $\mathfrak{c}>2.$ As
we need to work out the integral (87), it is clear that the integrand has
poles at $s=-k;(k=0,1,2,\cdots)$ of $\Gamma(s),$with residue
$(-1)^{k}\big{/}k!$ (see Figure. (6)). In addition, we have a simple pole at
$s=2$ of $\zeta_{H}\left(\frac{s}{2},\frac{1}{\mathsf{D}}+1\right)$ with
residue equals to $2.$

A straightforward evaluation of the residues at the poles $\\{s=0,2,-k\\}$
leads to the following expression of the partition function

$Z(\tilde{\beta})=e^{-\tilde{\beta}\mathsf{YB}^{\nicefrac{{-1}}{{2}}}}\left[-\frac{1}{2}+\frac{1}{\mathsf{D}}\left(\frac{2}{\widetilde{\beta}^{2}}-1\right)+e^{-\widetilde{\beta}}+\stackrel{{\scriptstyle[}}{{n}}=1]{\infty}{\sum}\frac{\left(-\widetilde{\beta}\sqrt{\mathsf{D}}\right)}{n!}^{n}\zeta_{H}\left(-\frac{n}{2},1+\frac{1}{\mathsf{D}}\right)\right],$
(88)

where we have used the identities

$\zeta_{H}\left(0,p\right)=\frac{1}{2}-p.$ (89)

###  IV.2 Thermal properties

Having estimated the partition function of KGO in rotating cosmic string, we
are allowed to compute the main thermodynamic quantities,

$F=-\frac{\sqrt{B}}{\tilde{\beta}}\ln\left(Z\right),U=-\sqrt{B}\frac{d\ln\left(Z\right)}{d\tilde{\beta}},$
(90)
$S=\ln\left(Z\right)-\tilde{\beta}\frac{d\ln\left(Z\right)}{d\tilde{\beta}},\,C_{V}=\tilde{\beta}{}^{2}\frac{d^{2}\ln\left(Z\right)}{d\tilde{\beta}^{2}}.$

The series representations of the partition function $Z$ in terms of the
$\zeta_{H}$-function and the Boltzmann representation introduced in Eq. ((88))
can be evaluated numerically considering only a finite number of terms in the
series.

Fig. (7) shows the different thermal quantities of KGO in (2+1)-dimensional
rotating cosmic string for $a=0.1$ derived from the partition function keeping
$n_{max}=1000$ terms of the series in Eq. ((88)). Following the figure, all
thermal quantities are plotted versus a reduced temperature $\tau$: here from
the curves of the numerical entropy function, no abrupt change, around
$\tau_{0}$ has been identified in the curves of specific heat. This means that
the curvature, observed in the specific heat curve does not exhibit or
indicate an existence of a phase transition around a $\tau_{0}$ temperature.
In addition, the effect of the parameter $\alpha$ for fixed $a$ is well
observed in the curves of specific heat. These curves increased when $\alpha$
decreases.

Fig(8) shows the same quantities but for fixed $\alpha=0.5$ and different
values of $a.$ As we can see, all the curves of the specific heat coincide a
round of $\tau_{0}$ temperature. So, the effect of the rotation on the thermal
properties of our oscillator has no effect.

(a) Free energy $F$ versus the reduced temperature $\tau$ (b) Total energy $U$
versus the reduced temperature $\tau$ (c) Entropy $S$ versus the reduced
temperature $\tau$ (d) Specific heat $C_{V}$ versus the reduced temperature
$\tau$

Figure 7: Statistical functions of KGO in (2+1)-dimensional rotating cosmic
string versus $\tilde{\beta}$ for $a=0.1$ and fixed parameters $j=m=\omega=1.$

(a) Free energy $F$ versus the reduced temperature $\tau$ (b) Total energy $U$
versus the reduced temperature $\tau$ (c) Entropy $S$ versus the reduced
temperature $\tau$ (d) Specific heat $C_{V}$ versus the reduced temperature
$\tau$

Figure 8: Statistical functions of KGO in (2+1)-dimensional rotating cosmic
string versus $\tilde{\beta}$ for $\alpha=0.5$ and fixed parameters
$j=m=\omega=1.$

##  V Conclusion

The aim of the present work is to study the relativistic dynamics of spinless
quantum particle via the Feshbach-Villars representation of two models,
namely, the interaction of KGO with the gravitational field generated by the
background geometry of : a) static cosmic strings and b) rotating cosmic
strings. Starting from the revising the FV formulation of scalar fields in
Minkowski space-time, we derived the corresponding formulations in two
different curved manifolds. We obtained the exact solutions of both systems
and we presented the quantized energy spectra which depend on the parameters
that characterize the space-time topology. It is not surprising to find that
the wave-function of our quantum system are expressed in terms of the
confluent hyper-geometric functions for both static and rotating cosmic
strings, since the former can be described throughout the previous one by
choosing appropriate coordinate transformation. In our above discussion, we
made use of the generalized Feshbach-Villars transformations in order to get
relevant observables. Such transformations have been shown to be exact and
covers any inertial and gravitational fields. In the last section, we have
examined the thermal properties of the system in question by computing the
partition function in terms of the Hurwitz zeta function. We have noticed that
these thermal quantities are affected by the geometrical and topological
parameters of the background geometry.

##  VI Funding

  * •

Authors report that no funding is available.

##  VII Author Contribution

  * •

Abdelmalek Buzenada concentrates on programming the system’s thermal
properties.

  * •

Abdelmalek Boumali reviews, writes and completes the main text of the
manuscript

  * •

All authors reviewed the manuscript

##  VIII Data Availability Statemen

  * •

The datasets used and/or analysed during the current study available from the
corresponding author on reasonable request.

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# Unsupervised Classification of Voiced Speech and Pitch Tracking Using
Forward-Backward Kalman Filtering

Benedikt Boenninghoff 1, Robert M. Nickel 2, Steffen Zeiler 1, Dorothea
Kolossa 1

###### Abstract

The detection of voiced speech, the estimation of the fundamental frequency
and the tracking of pitch values over time are crucial subtasks for a variety
of speech processing techniques. Many different algorithms have been developed
for each of the three subtasks. We present a new algorithm that integrates the
three subtasks into a single procedure. The algorithm can be applied to pre-
recorded speech utterances in the presence of considerable amounts of
background noise. We combine a collection of standard metrics, such as the
zero-crossing rate for example, to formulate an unsupervised voicing
classifier. The estimation of pitch values is accomplished with a hybrid
autocorre- lation-based technique. We propose a forward-backward Kalman filter
to smooth the estimated pitch contour. In experiments we are able to show that
the proposed method compares favorably with current, state-of-the-art pitch
detection algorithms.

##  1 Introduction

Human speech can roughly be divided into voiced and unvoiced sounds. Unvoiced
sounds are produced via a non-periodic/turbulent airflow along the vocal tract
and the lips/ teeth. Voiced speech sounds are produced by the vibration of the
vocal chords, where the airflow is interrupted periodically. The inverse of
the duration between two interruption epochs is called the fundamental
frequency or $F_{0}$ [1].

An accurate tracking of $F_{0}$ over time is still considered a major
technical challenge. In general, speech is not perfectly periodic and the
correct detection of voiced speech sounds as well as the accurate estimation
of the pitch value is difficult, particularly when speech is recorded in a
noisy environment.

When we estimate the pitch we usually do not only have a single voiced speech
frame to work with but an entire spoken utterance. The task, then, is to track
the pitch value throughout all voiced segments of the utterance. This can be
divided into three subtasks: (a) detection of voiced sounds, (b) pitch
estimation and (c) pitch tracking.

(a) Different methods have been developed to distinguish between voiced and
unvoiced or silent speech segments [2, 3]. The decision can be made before the
pitch estimation algorithm is applied or it can be coupled with the result of
the pitch estimation [4, 5, 6].

(b) Various parametric as well as non-parametric pitch estimation methods have
been studied, using properties of both time and frequency domain
characteristics [6]. The algorithms differ in terms of their signal model and
the application of statistical tools to detect repeating patterns within a
signal. The autocorrelation function is a wide- spread tool to detect pitch
within a signal in the time domain. Autocorrelation functions are applied in
Praat [7] and Yin [8]. In the frequency domain, the cepstrum meth- od [9] and
the harmonic periodic spectrum [10] can be used to estimate maximum pitch
values. Hybrid methods such as Pefac [11] and BaNa [12] use properties in both
domains. In [13] multi-pitch estimation algorithms embedded in a statistical
framework are developed based on a harmonic constraint. It is assumed that the
signal is composed of complex sinusoids. In [14] a maximum likelihood (ML)
pitch estimator is developed using a real-valued sinusoid-based signal model.
Halcyon [15] presents an algorithm for instantaneous pitch estimation that
decomposes the signal into narrow-band components and calculates fine pitch
values based on the estimated instantaneous parameters.

(c) There exist several methods to find the correct pitch track for a speech
sentence. BaNa uses the Viterbi algorithm to find the best path of the given
pitch candidates for each frame with respect to a cost function. In [14] two
approaches based on hidden Markov models and Kalman filters are proposed to
track the ML pitch estimate. A neural network based pitch tracking method is
presented in [16], where Pefac is used to generate feature vectors and again
the Viterbi algorithm is applied to detect the optimal pitch sequence.

Even neglecting voiced speech classification and/or the training phase, some
of the existing algorithms need long computation times. The goal of this paper
is to combine the three subtasks from above to formulate a computationally
efficient voiced speech classifier and pitch tracking algorithm. The algorithm
works reliably on speech utterances that have been pre-recorded in a noisy
environment. A very important aspect is that no training phase is needed. To
that end, we have devised a method that autonomously detects voiced frames
without a training phase. The problem is solved by combining different
approaches from [2, 3] to create a feature vector for unsupervised k-means
clustering and subsequent linear discriminant analysis.

Using the resulting voiced speech segments, we can estimate the corresponding
pitch values. Here, we use a hybrid method based on the spectro-temporal
autocorrelation function (SpTe-ACF) as proposed in [3]. All estimated pitch
values are then input to the proposed forward-backward Kalman filter (FB-KF)
to obtain a smoothed pitch track for the entire speech utterance.

##  2 Classification of Voiced Speech

Voiced and unvoiced speech segments have different characteristics regarding
their periodicity, frequency content, spectral tilt, etc. which can be used
for their distinction.

###  2.1 Feature Extraction

Let $x_{k}(n)$ with $0\leq n\leq N-1$ be the $k$-th frame of length $N$. We
combine several voiced speech classification methods to construct a feature
vector $\boldsymbol{v}(k)=[v_{1}(k),\ldots,v_{D}(k)]^{T}$, where each entry is
the normalized result of one classification method. Since some methods do not
work if there is neither noise nor speech present in a frame, only speech
frames with enough power are considered. To detect silent frames, we compute
$P_{x}(k)=\frac{1}{N}\sum_{n=0}^{N-1}x_{k}^{2}(n)$. For
$P_{x}(k)<\delta_{P_{x}}$, our entire feature vector is set to
$\boldsymbol{v}(k)=[-1,\ldots,-1]^{T}$. All computed classification methods
are normalized in such a way that they are restricted within the interval
$[-1,1]$, where a value close to one is assigned as a voiced frame [3].

In the following, we combine five different methods, namely the periodic
similarity measure, zero-crossing rate, spectrum tilt, pre-emphasized energy
ratio and low-band to full-band ratio as described in [3].

###  2.2 Unsupervised Decision Making

The next task is to find a linear weighting vector
$\boldsymbol{w}^{T}\boldsymbol{v}(k)=\sum_{d=1}^{D}w_{d}v_{d}(k)$ such that we
can assign $\boldsymbol{v}(k)$ to one class $C_{i}$ with
$i=\left\\{0,1\right\\}$, where $C_{0}$ denotes the set of unvoiced frames and
$C_{1}$ is the set of voiced frames. The class labels for all feature vectors
can be obtained using k-means clustering. The algorithm is initialized with
means $\boldsymbol{\mu}_{0}=[-1\ldots-1]^{T}$ and
$\boldsymbol{\mu}_{1}=[1\ldots 1]^{T}$ so that the class allocation is unique.
Having the class labels, we can apply a linear discriminant analysis to obtain
the optimal weighting vector. For two classes it is simplified to [17]

$\displaystyle\boldsymbol{w}\propto\boldsymbol{S}_{w}^{-1}\big{[}\boldsymbol{m}_{0}-\boldsymbol{m}_{1}\big{]},$
(1)

where the local mean vectors $\boldsymbol{m}_{i}$ with
$i=\left\\{0,1\right\\}$ result from the last round of k-means clustering and
$\boldsymbol{S}_{w}^{-1}$ denotes the inverse of the within-class scatter
matrix. As the $D$-dimensional scatter matrix is symmetric, there are at most
$D(D+1)/2$ entries to compute. In our simulations, utterances with a length of
at least $1$ second are sufficient to be able to compute the inverse of the
within-class scatter matrix, provided that each class is sufficiently
represented. Now we can use the linear classifier of Eq. (1) for the decision
making,

$\displaystyle\boldsymbol{w}^{T}\boldsymbol{v}(k)\underset{\mathrm{unvoiced}}{\overset{\mathrm{voiced}}{\gtrless}}0.$
(2)

Since class allocation is unique, the decision boundaries can be learned ad-
hoc for each sentence separately, which provides two advantages: First, no
training phase is needed. Second, the instantaneous adjustment of the
classifier considers the current speaker as well as the noise type.

(a) Noise-free speech sentence.

(b) Voiced-speech classification.

(c) Voiced-speech classification after SpTe-ACF.

Figure 1: Speech signal with additive car noise at $0$ dB SNR and classified
voiced frames.

Fig. 1 (a) presents an example for the voiced-speech classification and the
corresponding original noise-free signal waveform. Fig. 1 (b) shows the
decision for the same signal waveform with additive car noise. Applying Eq.
(2), we can observe two different types of errors:

In some frames the speech is masked by the additive noise so that the speech
frame is wrongly marked as unvoiced. The second error type occurs if the noise
also includes periodic elements. As a result, the speech frame may be
misclassified as voiced. We can recover some of the second type of error by
post-processing, which will be explained below. The final voiced speech
decision including post-processing is illustrated in Fig. 1 (c).

##  3 Pitch Tracking

The SpTe-ACF combines the normalized time- and frequency-domain
autocorrelations to obtain an estimate for the current pitch value given a
voiced speech frame. Subsequent to the SpTe-ACF, we apply the FB-KF to fit the
entire pitch track.

###  3.1 Spectro-Temporal Autocorrelation

In post-processing, we first change the decision for those single unvoiced
frames that lie between two voiced frames.

Then, as described in [3], we compute the time-domain autocorrelation
$R_{t}(k,l)$ and the frequency-domain autocorrelation $R_{s}(k,l)$ for
$N_{\mathrm{low}}\leq l\leq N_{\mathrm{high}}$. The lower and upper limits for
the pitch search are chosen to be
$N_{\mathrm{low}}=\max\big{\\{}1,\big{\lfloor}F_{s}/F_{\mathrm{max}}\big{\rfloor}\big{\\}}$
and
$N_{\mathrm{high}}=\min\big{\\{}\big{\lceil}F_{s}/F_{\mathrm{min}}\big{\rceil},N\big{\\}}$,
where $\lfloor\leavevmode\nobreak\ \rfloor$ and $\lceil\leavevmode\nobreak\
\rceil$ denote downward and upward rounding, respectively, $F_{s}$ is the
sampling frequency and $F_{\mathrm{min}}$, $F_{\mathrm{max}}$ define the
minimum and maximum values of $F_{0}$. Now, both methods are combined as
follows

$\displaystyle R_{st}(k,l)=\alpha_{R}R_{t}(k,l)+(1-\alpha_{R})R_{s}(k,l),$ (3)

to obtain the SpTe-ACF. The weighting factor in Eq. (3) is chosen as
$\alpha_{R}=0.5$. The observation for the Kalman filter is then given by

$\displaystyle\qquad\qquad N(k)=\arg\max_{l}\big{\\{}R_{st}(k,l)\big{\\}}.$
(4)

After applying the pitch estimation algorithm to all voiced frames, we change
the voicing decision of all frames where the estimated pitch values lie on the
boundary of the search interval, i.e.
$N(k)\in\left\\{N_{\mathrm{low}},N_{\mathrm{high}}\right\\}$, to unvoiced.
Finally, we change the decision for all single voiced frames that lie between
two unvoiced frames to obtain contiguous segments.

###  3.2 Kalman Filter

The Kalman filter is based on tracking $N_{0}(k)=\frac{F_{s}}{F_{0}(k)}$.
Finding the correct pitch track of a speech signal is equivalent to tracking
an unknown time-varying state variable using noisy observations. To describe
the transition between consecutive states $N_{0}(k)$ and $N_{0}(k-1)$ we can
assume a simple first-order Markov model [18] to formulate the state equation
$N_{0}(k)=N_{0}(k-1)+\Delta N_{0}(k-1)$, where $\Delta
N_{0}(k)\sim\mathscr{N}(0,\sigma^{2}_{\Delta_{0}})$ is the system noise. The
observation equation can be defined as $N(k)=N_{0}(k)+\Delta N(k)$, where
$\Delta N(k)\sim\mathscr{N}(0,\sigma^{2}_{\Delta}(k))$ is the observation
noise. The prediction step of our scalar Kalman filter is given by [19]

$\displaystyle\hat{N}_{0}(k|k-1)$ $\displaystyle=\hat{N}_{0}(k-1|k-1),$
$\displaystyle\hat{P}_{0}(k|k-1)$
$\displaystyle=\hat{P}_{0}(k-1|k-1)+\sigma^{2}_{\Delta_{0}}$

and the correction step can be written as

$\displaystyle g(k)$
$\displaystyle=\frac{\hat{P}_{0}(k|k-1)}{\hat{P}_{0}(k|k-1)+\sigma^{2}_{\Delta}(k)},$
$\displaystyle\hat{N}_{0}(k|k)$
$\displaystyle=\hat{N}_{0}(k|k-1)+g(k)\big{[}N(k)-\hat{N}_{0}(k|k-1)\big{]},$
$\displaystyle\hat{P}(k|k)$
$\displaystyle=\big{[}1-g(k)\big{]}\hat{P}_{0}(k|k-1).$

In this context, the variance of the system noise is fixed and chosen
a-priori. We need to estimate, however, the frame-dependent uncertainty of our
observation $\sigma^{2}_{\Delta}(k)$ in each iteration step. Similarly to [14]
the instantaneous variance is obtained by the ML estimate based on the last
$L$ values of Eq. (4),

$\displaystyle\hat{\sigma}^{2}_{\Delta}(k)=\frac{1}{L}\sum_{i=k}^{k-L+1}{\big{[}N(i)-\hat{\mu}_{N}(i)\big{]}^{2}},$
(5)

where $\hat{\mu}_{N}(k)$ denotes the estimated mean value of our observation,
which can be computed recursively by

$\displaystyle\hat{\mu}_{N}(k)$ $\displaystyle=\alpha
N(k)+(1-\alpha)\hat{\mu}_{N}(k-1).$ (6)

The Kalman filter is applied twice, forward and backward to obtain the
respective estimated pitch values $\hat{N}_{0,f}(k|k)$ and
$\hat{N}_{0,b}(k|k)$.

###  3.3 Forward-Backward Pitch Tracking

The fusion of $\hat{N}_{0,f}(k|k)$ and $\hat{N}_{0,b}(k|k)$ follows by
estimating the maximum of the log-likelihood, which is defined as

$\displaystyle\log\mathscr{L}(N_{0}(k))=\log\big{\\{}p(\hat{N}_{0,f}(k|k),\hat{N}_{0,b}(k|k)|N_{0}(k))\big{\\}}.$
(7)

For the joint distribution we assume that $\hat{N}_{0,f}$ and $\hat{N}_{0,b}$
are conditionally independent given the true state $N_{0}$,

$\displaystyle p(\hat{N}_{0,f}(k|k),\hat{N}_{0,b}(k|k)|N_{0}(k$
$\displaystyle))$ $\displaystyle=p(\hat{N}_{0,f}(k|k)|N_{0}(k))$
$\displaystyle\leavevmode\nobreak\ p(\hat{N}_{0,b}(k|k)|N_{0}(k)).$ (8)

If we further assume that $\hat{N}_{0,f}$ and $\hat{N}_{0,b}$ in Eq. (3.3) are
Gaussian distributed,

$\displaystyle p(\hat{N}_{0,f}(k|k)|N_{0}(k))$
$\displaystyle\sim\mathscr{N}(N_{0}(k),\sigma^{2}_{\Delta,f}(k)),$
$\displaystyle p(\hat{N}_{0,b}(k|k)|N_{0}(k))$
$\displaystyle\sim\mathscr{N}(N_{0}(k),\sigma^{2}_{\Delta,b}(k)),$

then the ML estimate of Eq. (7) is given by

$\displaystyle\hat{N}_{0}(k)$
$\displaystyle=\arg\max_{N_{0}(k)}\big{\\{}\log\mathscr{L}(N_{0}(k))\big{\\}}$
$\displaystyle=\frac{\hat{\sigma}^{2}_{\Delta,b}(k)\hat{N}_{0,f}(k|k)+\hat{\sigma}^{2}_{\Delta,f}(k)\hat{N}_{0,b}(k|k)}{\hat{\sigma}^{2}_{\Delta,f}(k)+\hat{\sigma}^{2}_{\Delta,b}(k)}.$
(9)

Finally, the relation of the estimated fundamental frequency and Eq. (3.3) is
given by

$\displaystyle\hat{F}_{0}(k)=\frac{F_{s}}{\hat{N}_{0}(k)}.$ (10)

(a) Speech waveform of male speaker.

(b) Forward Kalman filtering results with GT (black).

(c) Backward Kalman filtering results with GT (black).

(d) FB-Kalman pitch estimates plotted on spectrum.

Figure 2: Variance-based averaging of the results of forward and backward
Kalman filters.

Plots (b)-(d) of Fig. 2 illustrate the advantage of the FB-KF for the speech
waveform in Fig. 2 (a). Starting with Fig. 2 (b), we see the curves of
$\hat{N}_{0,f}(k|k)\pm 10\leavevmode\nobreak\ \hat{\sigma}_{\Delta,f}(k)$ and
the groundtruth (GT), which is computed as described in Section 4. We can
observe that at the beginning or after speech pauses, the uncertainty of the
observation increases. The increase follows from the change of $F_{0}$ after
speech pauses. Similarly, we observe the same results in the backward variance
in Fig. 2 (c). Applying Eq. (3.3) and (10) leads to greater reliability at the
beginning and end of speech segments which can be seen in Fig. 2 (d). See, for
example, the behavior around 0.7 s.

(a) Noise-free speech waveform.

(b) SpTe-ACF and FB-KF results plotted on noise-free spectrum.

Figure 3: SpTe-ACF and FB-Kalman for $0$ dB factory noise plotted on the clean
spectrum.

In addition, Fig. 3 shows the pitch tracking results, before and after
applying the FB-KF to SpTe-ACF in a noisy environment. The clear speech
waveform of Fig. 3 (a) is superimposed with factory noise at $0$ dB. As a
result, some pitch estimates of the SpTe-ACF differ from the plotted clear
spectrum in the background. The FB-KF is able to correct the measurement
errors here, as well.

##  4 Simulation Results

In [12, 6, 15] recorded speech signals from the Keele, LDC, CSTR and PTDB-TUG
databases are used to compare different state-of-the-art unsupervised pitch
estimation algorithms. Here, we test the FB-KF based SpTe-ACF with the TIMIT
database to find optimal parameters in terms of Gross Pitch Error (GPE) and
(Mean Fine Pitch Error) MFPE. First we count all frames, where the relative
distance between the estimated pitch value and the reference pitch value is
higher than 10 %,

$\displaystyle\mathrm{error}(k)=\begin{cases}1&\text{if}\leavevmode\nobreak\
\frac{|F_{0}(k)-\hat{F}_{0}(k)|}{F_{0}(k)}>0.1\\\
0&\text{otherwise}.\end{cases}$

Then, the GPE ratio is defined as
$\mathrm{GPE}=\frac{1}{K}\sum_{k=1}^{K}\mathrm{error}(k)$. In the following,
we only estimate the pitch values for the frames classified as voiced and
calculate the corresponding GPE. For all voiced frames without GPE, the MFPE
is defined as

$\displaystyle\mathrm{MFPE}=\frac{1}{N_{e}}\sum_{k=1}^{N_{e}}\frac{|F_{0}(k)-\hat{F}_{0}(k)|}{F_{0}(k)},$

where $N_{e}$ denotes the number of frames without GPE [20]. We use Praat to
determine the groundtruth $F_{0}(k)$. The searching interval is defined as
$F_{\mathrm{min}}=$ 60 Hz and $F_{\mathrm{max}}=$ 460 Hz and the power
threshold is determined as
$\delta_{P_{x}}=0.3\cdot\mathrm{E}\left\\{P_{x}(k)\right\\}$. The goal is to
find a set of parameters $\\{L,\alpha,\sigma^{2}_{\Delta_{0}}\\}$ which solves
the trade-off between minimizing the GPE and minimizing the MFPE. The test
signals show good results for both metrics if we choose $L=8$, $\alpha=0.95$,
$\sigma^{2}_{\Delta_{0}}=0.06$. If we vary the set of parameters, i.e. $L=6$,
$\alpha=0.9$, $\sigma^{2}_{\Delta_{0}}=0.01$, GPE rate decreases, but this
setting also results in an increase of MFPE rate.

We use the ARCTIC [21] database for evaluation, where the signals are sampled
at $F_{s}$ = 16 kHz. We compare the proposed method with Yin, Pefac, BaNa and
Halcyon which are applied with their default parameters. First we need to
calculate the GT as a reference for the GPE. A speech frame is marked to be
voiced, when the ratio of the distance between the maximum and minimum pitch
estimates with respect to the minimum value of all algorithms is smaller than
10 %. The GT value is then computed as the average of all pitch estimates.
Otherwise the frame is considered as unvoiced. As a consequence, there are
some ARCTIC signals where only single frames within the entire sentence
remain. Therefore, we apply the unsupervised voiced speech classification
method as described in Section 2 and consider only those signals, where at
least 60 % of voiced frames are still available.

(a) GPE, average over all noise types.

(b) MFPE, average over all noise types.

Figure 4: Simulation results for $4$ speakers, $50$ signals, $8$ noise types
and $6$ different SNR values.

Fig. 4 presents the resulting GPE rates for different types of additive noise
[22] and for a range of SNR values. Regarding the curves in Fig. 4 (a) for
different algorithms, it can be seen, that the FB-KF based SpTe-ACF achieves
the highest overall average detection rate for SNR values up to 0 dB. In
addition, Fig. 4 (b) highlights the MFPE rates for a range of SNR values.
Again, the FB-KF based SpTe-ACF achieves a good performance for SNR values up
to 0 dB. In summary, we observe an acceptable trade-off between GPE and MFPE
for the chosen set of parameters.

| Yin | Pefac | BaNa | Halcyon | SpTe-ACF & FB-KF
---|---|---|---|---|---
avg. time | 0.03 s | 0.64 s | 4.16 s | 3.64 s | 0.26 s
Table 1: Average computation time in seconds for all algorithms implemented in
Matlab. The average length of all speech signals is $3.07$ s.

##  5 Conclusions

We have presented a forward-backward Kalman filtering technique for pitch
tracking in noisy environments. The procedure includes a sentence-based method
for unsupervised voiced-unvoiced classification. The combination of the
results of the forward and backward KF can capture and correct outliers in the
pitch estimate. The success of the algorithm is highly dependent on a good
performance of the voiced-unvoiced classification stage. If the observation
input for the Kalman filter contains too many outliers then the Kalman filter
is drifting. Simulation results based on the ARCTIC database, however, show
that the SpTe-ACF based FB-KF achieves a good pitch estimation accuracy in
noisy environments for SNR values down to $0$ dB. In addition, the
computational effort for pitch estimation and tracking is significantly lower
than that for BaNa and Halcyon. Table 1 provides a very coarse comparison of
computational complexity for some of the considered algorithms based on MATLAB
execution times.

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