Violation of the Einstein's Equivalence Principle for a
Composite Quantum Body

A.G. Lebed

Department of Physics, University of Arizona, 1118 E.
4-th Street, Tucson, AZ 85721, USA

Recently, we have started to investigate behavior of a composite
quantum body in an external gravitational field in the framework
of General Relativity [see, for a review, A. G. Lebed, Mod. Phys.
Lett. A, 35, 2030010 (2020)]. As the simplest example, we
have considered a hydrogen atom in a weak gravitational field. Our
results are the following. The Einstein's Equivalence Principle
survives for the most of macroscopic ensembles of the atoms,
containing the stationary quantum states. On the other hand, we
have demonstrated that this principle is sometimes broken. In
particular, it is broken for the so-called Gravitational demons,
which are the coherent macroscopic ensembles of two or more
stationary quantum states in the hydrogen atoms. In the above
cited paper we have considered the Gedanken experiment, where the
gravitational field is suddenly switched on in a free from
gravitation space. In the current paper we consider the much more
realistic from experimental point of view Gedanken experiment and
come to the same conclusion about violations of the Einstein's
Equivalence Principle for the Gravitational demons.

The Einstein's Equivalence Principle of equality between inertial
and gravitational masses is a cornerstone of the theory of General
Relativity [1,2]. During the recent space mission "MICROSCOPE",
validity of the principle for ordinary condensed matter was
established with a great accuracy, $|m_i - m_g|/m_i \leq 10^{-16}
- 10^{-17}$ [3,4], whereas two other space missions, "GG" and
"STEP", are currently in their preparation stages. Nevertheless, a
validity of the Einstein's Equivalence Principle in quantum case
is not obvious and needs its theoretical and hopefully
experimental investigations. Since quantum theory of gravity has
not been developed yet, in our previous papers (see Refs. [5-8])
we have theoretically considered the so-called semi-classical
variant of General Relativity, where gravitational field is not
quantized but the matter is quantized. In particular, in the
framework of the semi-classical approach, we have shown that
active and passive gravitational masses for the simplest quantum
macroscopic ensembles - ensembles of hydrogen atoms in stationary
states - preserve the Einstein's Equivalence Principle. On the
other hand, we have also demonstrated that there exist such
unusual quantum ensembles, called the Gravitational demons, which
result in violations of the Einstein's Equivalence Principle for
both active [5,6,8] and passive [6-8] gravitational masses. Note
that the Gravitational demons are defined as the coherent
macroscopic ensembles of the quantum superpositions of two or
several stationary states. As shown by us in the above cited
papers, the expectation values of active and passive gravitational
masses in such macroscopic ensembles are not anymore related to
the expectation value of energy by the famous Einstein's formula,
$m_g \neq E/c^2$. In particular, in Refs. [6-8] we have considered
the Gedanken experiment, where the Gravitational demons are
created by laser technique in ultra-cold matter [9] in the absence
of a gravitational field and then we have switched on the field at
$t=0$. In this paper, we consider more realistic from experimental
point of view Gedanken experiment, where we create the
Gravitational demon in the presence of gravitational field and
then at $t=0$ we change the field a little bit. Our conclusions
regarding passive gravitational mass are similar to the above
discussed ones.

Let us consider for further calculations the so-called weak
gravitational field limit [1,2], which describes the Earth's
gravitational field with great accuracy in the framework of the
General Relativity spacetime,

\begin{equation}
(ds)^2(R) = -[1 + 2 \phi(R)/c^2 ](cdt)^2 + [1 - 2 \phi(R)/c^2 ]
[(dx)^2 +(dy)^2+(dz)^2 ], \ \ \ \ \phi (R)= - GM/R,
\end{equation}

where $\phi(R)$ is the Newtonian potential, $c$ is the light
velocity, $G$ is the Newtonian gravitational constant, $R$ is a
distance from a center of the Earth, and $M$ is the mass of Earth.
Note that, in accordance with the local Lorentz invariance, we can
define the following proper local space-time coordinates,

\begin{equation}
\tilde x(R) =[1-\phi(R)/c^2]  x, \ \ \ \tilde y(R)= [1-\phi(R)/c^2
] y, \ \ \ \tilde z(R)=[1-\phi(R)/c^2 ] z , \ \ \ \tilde t(R)=
[1+\phi(R)/c^2] t,
\end{equation}

where the spacetime interval has the Minkowski's form:

\begin{equation}
[d \tilde s(R)]^2 = -[c d \tilde t(R)]^2 + [d \tilde x(R)]^2 + [d
\tilde y(R)]^2 + [d \tilde z(R)]^2.
\end{equation}

[Note that the weak field approximation (1),(2) allows us below to
perform all calculations with accuracy up to the first order of
the small parameter $|\phi|/c^2 \ll 1$.] The Schrödinger
equation for electrons in a hydrogen atom in the proper local
coordinates (2) can be expressed in the approximation, where we
neglect the so-called tidal terms, as

\begin{equation}
i \hbar [\partial \Psi({\bf \tilde r},\tilde t)]/ \partial \tilde
t = \hat H_0 (\hat {  \bf \tilde p},{\bf \tilde r}) \Psi({\bf
\tilde r},\tilde t)  ,
\end{equation}

where $\hat H_0 (\hat {  \bf \tilde p},{\bf \tilde r})$ is the
electron Hamiltonian in the hydrogen atom. We stress that Eq.(4)
can be written if position of the proton is fixed by some
non-gravitational forces and if we omit all the so-called tidal
terms, which are extreemely small near the Earth.

Below, we perform the following Gedanken experiment. At $t=0$ we
create a quantum macroscopic ensemble of the coherent
superpositions of the two stationary states in the atom – $1S$
(with the corresponding wave function $\Psi_1[{\bf \tilde r(R)}]$)
and $2S$ (with the corresponding wave function  $\Psi_2[{\bf
\tilde r(R)}]$):

\begin{equation}
\tilde \Psi^1_{12}(\tilde r, \tilde t) = \exp [-i m_e c^2 \tilde
t(R) /\hbar ]
\frac{1}{\sqrt{2}} \biggl\{ \Psi_1[\tilde r(R)] \exp [-i E_1 \tilde t(R)/ \hbar]
+ \Psi_2[\tilde r(R)] \exp(i \tilde \alpha) \exp [-i E_2 \tilde
t(R)/ \hbar] \biggl\} ,
\end{equation}

where a phase difference between the stationary wave functions is
the same for each microscopic state in the macroscopic ensemble,

\begin{equation}
\tilde \alpha = const,
\end{equation}

and the wave functions of the stationary states are known to be
real and are normalized in the local proper coordinates:

\begin{equation}
\int^{+\infty}_{-\infty} [\Psi_1(\tilde r)]^2 d^3 \tilde r =
\int^{+\infty}_{-\infty} [\Psi_2(\tilde r)]^2 d^3\tilde r =1, \ \
\ \ \int^{+\infty}_{-\infty}  \Psi_1(\tilde r) \Psi_2(\tilde r)
d^3 \tilde r=0,
\end{equation}

where $m_e$ is the electron bare mass. For convenience, here we
rewrite wave function (5) in the initial spacetime coordinates,

\begin{eqnarray}
&&\Psi^1_{12}(r,t) = \exp \{ -i m_e c^2 t [1+\phi (R_1) /c^2] /
\hbar \} [1- \phi(R_1) /c^2]^{3/2}
\nonumber\\
&&\times \biggl( \Psi_1[r(1-\phi(R_1)/c^2)]  \exp \{-i E_1
t[1+\phi(R_1)/c^2] / \hbar \}  \
\nonumber\\
&&+ \Psi_2[r(1-\phi(R_1)/c^2)]  \exp(i \tilde \alpha) \exp \{-i
E_2 t[1+\phi(R_1)/c^2] / \hbar\} \biggl) / \sqrt{2} ,
\end{eqnarray}

where $\phi_1 = \phi(R_1)$ is value of the gravitational potential
(1) at initial moment of time, $t=\tilde t =0$.

Let us continue our Gedanken experiment. We suddenly change the
gravitational potential into the value $\phi_2=\phi(R_2) =
\phi(R_1) + \delta \phi$, where $|\delta \phi| \ll |\phi(R_1)|$,
and ask how the expectation value of energy of our Gravitational
demon (5),(8) is changed, provided that time of the change of the
gravitational potential is less than the characteristic time of
the electron quasi-classical motion corresponding to the wave
function (8). Then, at $t > 0$, we have the following wave

\begin{eqnarray}
&&\Psi^2_{12}(r,t) =\exp \{ -i m_e c^2 t [1+\phi (R_2) /c^2] /
\hbar \} [1- \phi(R_2) /c^2]^{3/2}
\nonumber\\
&&\times \biggl(A \ \Psi_1[r(1-\phi(R_2)/c^2)]  \exp \{-i E_1
t[1+\phi(R_2)/c^2] / \hbar \}  \
\nonumber\\
&&+ B \ \Psi_2[r(1-\phi(R_2)/c^2)]  \exp \{-i E_2
t[1+\phi(R_2)/c^2] / \hbar\} \biggl)  ,
\end{eqnarray}

where A and B are some complex numbers. Since we have a sudden
perturbation of the spacetime in our Gedanken experiment, a law of
the conservation of the number of particles can be expressed as

\begin{equation}
|\Psi^1_{12}(r,t=0)|^2  = |\Psi^2_{12}(r,t=0)|^2 .
\end{equation}

[Note that probabilities to occupy higher energy levels are
proportional to $(\delta \phi)^2/c^4$ and, thus, are not
considered in this paper.] It is important that we are not
interested in common phase of the complex numbers $A$ and $B$,
therefore, we can rewrite Eq.(10) in the following form,

\begin{equation}
\Psi^1_{12}(r,t=0) = \Psi^2_{12}(r,t=0).
\end{equation}

Taking into account Eqs.(8) and (9), we obtain

\begin{eqnarray}
&&\frac{1}{\sqrt{2}} \biggl( 1-\frac{\phi_1}{c^2}\biggl)^{3/2}
\biggl\{\Psi_1[r(1-\phi_1/c^2)] \ + \exp(i \tilde \alpha)
\Psi_2[r(1-\phi_1/c^2)]\biggl\}
\nonumber\\
&&=\biggl( 1-\frac{\phi_2}{c^2}\biggl)^{3/2} \biggl\{A \
\Psi_1[r(1-\phi_2/c^2)] \ + B \ \Psi_2[r(1-\phi_2/c^2)]\biggl\} .
\end{eqnarray}

Let us solve Eq.(12) using orthogonality conditions between the
wave functions $\biggl(
1-\frac{\phi_i}{c^2}\biggl)^{3/2}\Psi_1[r(1-\phi_i/c^2)]$ and
$\biggl( 1-\frac{\phi_i}{c^2}\biggl)^{3/2}
\Psi_2[r(1-\phi_i/c^2)]$, where $i=1,2$. As a result we find:

\begin{equation}
\frac{1}{\sqrt{2}} = A \  I_{11}+B \ I_{12}, \ \ \
\frac{\exp(i\tilde \alpha)}{\sqrt{2}} = A \ I_{21} + B \ I_{22} \
\end{equation}

\begin{equation}
I_{ij}= \biggl(1- \frac{\phi_1}{c^2} \biggl)^{3/2} \biggl(1-
\frac{\phi_2}{c^2} \biggl)^{3/2} \int^{+\infty}_{-\infty}
\Psi_i\biggl[r\biggl(1-\frac{\phi_1}{c^2} \biggl)\biggl] \
\Psi_j\biggl[r\biggl(1- \frac{\phi_2}{c^2} \biggl)\biggl] d^3r.
\end{equation}

Here, we calculate the matrix elements $I_{ij}$ by means of the
method developed in Refs. [6-8] with accuracy up to the first
orders of $|\delta \phi|/c^2 \ll 1$ and $|\phi_i|/c^2 \ll 1$.
After the lengthy but straightforward calculations, we obtain

\begin{equation}
I_{11} = I_{22}  = 1.
\end{equation}

\begin{equation}
I_{12} = -  I_{21}  = \frac{\delta \phi}{c^2} \alpha =
\frac{\delta \phi}{c^2} \int^{+\infty}_{-\infty} \Psi'_1(r) \ r \
\Psi_2(r) d^3r =  \frac{\delta \phi}{c^2} \frac{V_{12}}{E_2-E_1} ,
\  \  \ \Psi'_1 (r) = \frac{d \Psi_1(r)}{dr},
\end{equation}

where $E_1$ and $E_2$ are energies of the ground state $1S$ and
the first exited state $2S$, correspondingly;    $V_{12}$ is the
matrix element of the so-called quantum virial operator in the
hydrogen atom [6-8,10]:

\begin{equation}
V_{12}=\int^{+\infty}_{-\infty} \Psi_1(r) \hat V(r) \Psi_2(r)
d^3r, \ \ \ \hat V(r) = \frac{{\bf p}^2}{m_e}-\frac{e^2}{r'},
\end{equation}

where ${\bf p}$ is electron momentum operator, $r'$ is a distance
between electron and the fixed proton; $e$ is the electron charge.

First of all, we solve the system of linear equations (13) to find
the complex numbers A and B:

\begin{equation}
A= \frac{1}{\sqrt{2}} \biggl[ 1 - \alpha \biggl(\frac{\delta
\phi}{c^2} \biggl) \exp(i \tilde \alpha) \bigg], \ \ \ \ B=
\frac{1}{\sqrt{2}} \biggl[ \exp(i \tilde \alpha) + \alpha \biggl(
\frac{\delta \phi}{c^2}\biggl) \biggl] .
\end{equation}

Then, we make sure that they preserve the normalization conditions
with the accuracy of calculations accepted in this paper,

\begin{eqnarray}
|A|^2 + |B|^2 = \frac{1}{2} \biggl\{ \biggl[ 1 - \alpha \biggl(
\frac{\delta \phi}{c^2} \biggl) \exp(i \tilde \alpha) \biggl]
\biggl[ 1 - \alpha \biggl( \frac{\delta \phi}{c^2} \biggl) \exp(-
i \tilde \alpha) \biggl]
\nonumber\\
+\biggl[ \exp(i \tilde \alpha) + \alpha \biggl( \frac{\delta
\phi}{c^2}\biggl) \biggl]  \biggl[ \exp(- i \tilde \alpha) +
\alpha \biggl( \frac{\delta \phi}{c^2}\biggl) \biggl] \biggl\}
\approx 1 .
\end{eqnarray}

And finally, we calculate the expectation energy of the
Gravitational demon (9):

\begin{eqnarray}
\biggl< E \biggl> = \biggl[ E_1 + m_e \delta \phi +
\biggl(\frac{E_1}{c^2}  \biggl) \delta \phi \biggl] |A|^2 +
\biggl[ E_2 + m_e \delta \phi + \biggl(\frac{E_2}{c^2}  \biggl)
\delta \phi \biggl] |B|^2
\nonumber\\
= \frac{1}{2} \biggl\{ \biggl[ E_1 + m_e \delta \phi + \biggl(\frac{E_1}{c^2}  \biggl) \delta \phi \biggl] \biggl[ 1 - \alpha \biggl( \frac{\delta \phi}{c^2} \biggl) \exp(i \tilde \alpha) \biggl] \biggl[ 1 - \alpha \biggl( \frac{\delta \phi}{c^2} \biggl) \exp(- i \tilde \alpha) \biggl]
\nonumber\\
+\biggl[ E_2 + m_e \delta \phi + \biggl(\frac{E_2}{c^2}  \biggl)
\delta \phi \biggl] \biggl[ \exp(i \tilde \alpha) + \alpha \biggl(
\frac{\delta \phi}{c^2}\biggl) \biggl]  \biggl[ \exp(- i \tilde
\alpha) + \alpha \biggl( \frac{\delta \phi}{c^2}\biggl) \biggl\}.
\end{eqnarray}

As a result we obtain:

\begin{equation}
\biggl< E \biggl> = \frac{E_1 + E_2}{2} + \biggl( m_e + \frac{E_1
+ E_2}{2 c^2} + \frac{V_{12}}{c^2} \cos(\tilde \alpha) \biggl)
\delta \phi,
\end{equation}

where the expectation value of the gravitational mass per one
electron can be define as

\begin{equation}
\biggl< m_g \biggl> =  m_e + \biggl(\frac{E_1 + E_2}{2 c^2}\biggl)
+ \biggl(\frac{V_{12}}{c^2} \biggl) \cos(\tilde \alpha) ,
\end{equation}

Let us discuss Eq.(22). The first term is the bare electron mass,
the second term is the expected in relativity contribution to the
gravitational mass from the energies of electron levels (5). The
third term is the anomalous one and completely unexpected in
relativity contribution from the so-called virial term [6-8]. It
depends on the fixed difference of phases, $\tilde \alpha =
const$, in the coherent macroscopic ensemble of two stationary
states (5),(6). On the basis of Eq.(22), we can make conclusion
about the breakdown of the equivalence between expectation values
of energy and gravitational mass for the described above
Gravitational demon. We put forward a natural hypothesis that our
results are not restricted by ensembles of the hydrogen atoms but
are qualitatively true for any coherent macroscopic ensemble of
two or several stationary states of any composite quantum object.

In conclusion, we discuss in a brief some of the experimental
aspects of possible discovery of the violation of the Einstein's
Equivalence Principle. First of all, we again pay attention that
the phase difference in two quantum states in  the macroscopic
ensemble, $\tilde \alpha$, has to be fixed with good enough
accuracy (5),(6), otherwise the anomalous term in Eq.(22)
disappears after averaging over the phase. Such ensemble, in
principle, can be created by some laser technique with possibly
ultra-cold atoms or molecules in the Earth's laboratory [9]. The
second important point is that the measurements of a weight of the
ensemble has to be done very quickly, i.e. quicker than the time
scales characterizing wave functions in the quantum ensemble. Our
next steps will be to find more convenient composite quantum
objects for the above mentioned procedures.

We are thankful to Natalia N. Bagmet (Lebed), Steven Carlip,
Fulvio Melia, Pierre Meystre, Keneth Nordtvedt, Douglas Singleton,
and Vladimir E. Zakharov for useful discussions.

[1]
L.D. Landau and E.M. Lifshitz, The Classical Theory of
Fields (Butterworth-Heineman, Oxford, 2003).

[2]
C.W. Misner, K.S. Thorne, and J.A. Weeler, Gravitation
(Princeton University Press, Princeton 2017).

[3]
P. Touboul Phys. Rev. Lett. 119, 231101 (2017).

[4]
P. Touboul Class. Quantum Grav. 36, 225006 (2019).

[5]
A.G. Lebed J. Phys.: Conf. Ser. 738, 012036 (2016).

[6]
A.G. Lebed Int. J. Mod. Phys. D 28, 1930020 (2019).

[7]
A.G. Lebed Mod. Phys. Lett. A 35, 2030010 (2020).

[8]
A.G. Lebed, Editor, Breakdown of Einstein's Equivalence
Principle (World Scientific, Singapore, 2023).

[9]
P. Meystre, private communication (2019).

[10]
D. Park, Introduction to the Quantum Theory (Dover
Publications, 2005).


# Primordial monopolium as dark matter

Vicente Vento  Departamento de Física Teórica-IFIC, Universidad de Valencia-
CSIC, 46100 Burjassot (Valencia), Spain.

###### Abstract

The existence of monopoles is a characteristic signature of Kaluza-Klein
multidimensional theories. The topology of these solutions is extremely
interesting. The existence of a dipole solution, which we have associated to a
monople-anti-monopole bound state, is the leitmotiv of this investigation. The
dipole in its lowest energy state, which we here call also monopolium, is
electromagnetically inert in free space interacting only gravitationally.
Monopolium when interacting with time dependent magnetic fields acquires a
time dependent induced magnetic moment and radiates. We have analyzed the most
favorable astrophysical scenario for radiative monopolium and found that the
amount of radiation is so small that is not detectable by conventional
equipments. These findings suggest that Kaluza-Klein monopolium, if existent,
would be a candidate for a primordial dark matter constituent.

###### pacs:

04.20.Jb, 04.50+h, 04.50Gh,04.80-y

##  I introduction

The nature of dark matter is one of the most relevant problems of physics
today. Cosmological experiments have determined its abundance Aghanim:2018eyx
. From observational and numerical simulations we have some knowledge about
its distribution in galactic halos Wechsler:2018pic . Still the nature of dark
matter remains unknown. Since the favorite explanations in the past, the
weakly interacting particles, have eluded detection it is worthwhile to
consider other possibilities.

Monopole physics took a dramatic turn when ’t Hooft 'tHooft:1974qc  and
Polyakov Polyakov:1974ek  independently discovered that the SO(3) Georgi-
Glashow model Georgi:1972cj  inevitably contains monopole solutions. These
topological monopoles appear naturally in Grand Unified Theories and are
impossible to create in present day particle colliders because of their huge
GUT mass Preskill:1984gd ; Shnir:2005xx  and they are very rear in the
universe since inflation reduces their density dramatically Guth:1980zm ;
Linde:1981mu . ’t Hooft-Polyakov monopoles cannot built up dark matter because
of their incredibly strong interactions Dirac:1948um .

It has been a long time since Kaluza and Klein first investigated the idea of
unification of gravitation and gauge theories by exploring the possibility
that the number of spacetime dimensions is greater than four Kaluza:1984ws ;
Klein:1926tv . The idea behind Kaluza-Klein (KK) theories is that the world
has more than three spatial dimensions and some of them are curled up to form
a circle so small as to be unobservable. KK theories have been the subject of
revived interest in recent years since many standard model extensions in extra
dimensions yield KK theories. KK theories contain a very rich topological
structure, which includes very heavy monopoles whose mass is of the order of
the Planck mass Gross:1983hb ; Sorkin:1983ns ; Newman:1963yy ; Hawking:1976jb
; Gibbons:1979xm . But most important for the present study, they also contain
other soliton solutions in different topological sectors. In particular, the
dipole, which has the quantum numbers of a monopole-anti-monopole bound state.
This state in conventional gauge theories was called monopolium Hill:1982iq ;
Vento:2007vy ; Epele:2007ic , and we kept the name here. In conventional gauge
theories monopolium has vacuum quantum numbers and annihilates into more
elementary constituents. However, in KK theories monopolium does not belong to
the topological sector of the vacuum and therefore it is classically stable
Gross:1983hb ; Kerr:1963ud ; Gibbons:1976ue . That solitons of KK theories
could supply dark matter has been briefly discussed in the past
Overduin:1998pn . The aim of this paper is to show that monopolium, as it
appears in KK theories, if it survives inflation in sufficient numbers, could
be a candidate for a primordial constituent of dark matter, and thus its study
would open a window into the physics of more than four dimensions.

The presentation is based on the scenario discussed by Gross, Perry and Sorkin
(GPS) Gross:1983hb ; Sorkin:1983ns , which serves as a toy model, but our
results are based on properties which arise in realistic scenarios. Let us
describe them precisely. A topology where the dipole does not have vacuum
quantum numbers Newman:1963yy ; Hawking:1976jb ; Gibbons:1979xm ; Kerr:1963ud
; Gibbons:1976ue . The dipole structure, described in terms of some parameter
$d$ (distance between center of the poles), approaches the monopole-anti-
monopole structure as $d$ becomes large. This also occurs in topological
schemes with different topological sectors like in the Skyrme Model
Zahed:1986qz . The dipole can thus be interpreted as a bound state with a mass
smaller than twice the monopole mass. The parameter $d$ describes the magnetic
moment of the dipole. The ground state (smallest mass) dipole, i.e. when
$d\rightarrow 0$, has vanishingly small dipole moment and therefore it is
electromagnetically neutral and thus has only gravitational interaction. The
GPS model has all these ingredients and therefore we are going to use it as a
toy model to develop the ideas and get orders of magnitude for observables.
However, the GPS model has a peculiar way of realizing the equivalence
principle, which we will discuss in some detail, and we will show one
mechanism for avoiding it.

In section II we are going to recall some aspects of the topology of the
Kaluza-Klein theory in particular the existence of monopoles and dipoles. In
section III we analyze if the dipole survives the inflationary process. In
section IV we look for possible electromagnetic observables of monopolium and
in section V we discuss possible experimental observations. Their size allows
us to propose monopolium as a dark matter component. We end by some concluding
remarks.

##  II Kaluza-Klein monopolium

Kaluza-Klein theories Kaluza:1984ws ; Klein:1926tv  contain topological
solitons which can be identified as magnetic monopoles satisfying the Dirac
quantization condition, $eg=\frac{1}{2}$ Dirac:1948um . We next recall the
structure of the KK monopoles by whose features can be extended to models with
more compact dimensions Ezawa:1983gp ; Bais:1984xb ; Lee:1984ica ; Xu:1988zj ;
Mann:2005gk ; Cotaescu:2005jg .

The simplest topological solution in the GPS model is a magnetic monopole
which arises from the generalization to more dimensions of the Taub-NUT
solitonic solution Newman:1963yy ; Hawking:1976jb  and is described by the
following metric

$ds^{2}=-dt^{2}+V(dx^{5}+4m(1-\cos\theta)d\phi)^{2}+\frac{1}{V}(dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}),$ (1)

where $V^{-1}=1+\frac{4m}{r}$ and $r,\theta,\phi$ the spherical coordinates.
At large distances the gauge field from the GPS solution behaves as a Dirac
monopole Dirac:1948um

$A_{\phi}\sim\frac{4m(1-\cos\theta)}{r\sin\theta}$ (2)

whose inertial mass $M$ is,

$M\sim\frac{M_{Pl}}{4\sqrt{\alpha}},$ (3)

where $M_{Pl}$ is Planck’s mass and $\alpha$ the fine structure constant. Note
the non-perturbative character of this result in the inverse dependence on the
structure constant.

These KK monopoles have a characteristic, which distinguishes them from all
other types of monopoles, that makes them interesting and is the leitmotiv of
the present investigation. In ordinary Yang-Mills theories one does not find
solitons which are stable magnetic dipoles because the superposition of a
monopole and an anti-monopole has the topology of the vacuum and therefore
nothing prevents them from annihilating. The situation is quite different in
KK theories. The vacuum solution belongs to a different topological sector
than the monopole-anti-monopole solution Gibbons:1979xm . Since one cannot
evolve smoothly from one type of solution to the other the monopole-anti-
monopole pair will not annihilate classically. In the GPS scenario, KK dipoles
can be constructed by the Kerr-Schwarzschild metric, which are regular
solutions in euclidean $3+1$ dimensions, by compactifying the fourth dimension
Gross:1983hb . This provides a very interesting topological state which has no
magnetic charge and behaves like a magnetic dipole at large distances

$A_{\phi}\sim\frac{4M_{d}d\sin\theta}{r^{2}},$ (4)

Figure 1: Monopolium mass $M_{d}/M$ as a function of the distance between the
poles $d/M$.

where $M_{d}$ is the dipole mass and $d$ measures the distance between the
poles. For large $d$ this solution coincides with the product ansatz of the
monopole-antimonopole solution.

The mass of this dipole state is determined by the following equation,

$M_{d}=2\left(M-\frac{M_{d}^{2}}{\sqrt{4M_{d}^{2}+d^{2}}}\right),$ (5)

where we use the Planckian System of untits $\hbar=c=G=1$. As can be seen in
Fig. 1 its mass ranges naturally from twice the monopole mass for large $d$
when the poles are far away from each other, to the monopole mass when the
poles are strongly coupled on top of each other. Thus a pair of monopole-anti-
monopole if they get close enough to feel a strong attraction will make a
neutral dipole state which we call monopolium. For $d\sim 0$ this state is
electromagnetically inert and will only interact gravitationally. Monopolium
is classically stable and therefore may only decay quantum mechanically in a
tunnel type process being therefore very long lived, i.e. almost stable. The
model possesses other topological states of higher genus which at present are
of no interest for our discussion. This study can be generalized to higher
dimensions Ezawa:1983gp ; Bais:1984xb ; Lee:1984ica ; Xu:1988zj ; Mann:2005gk
; Cotaescu:2005jg .

In the GPS model the inertial masses of monopole and dipole are different from
the gravitational mass which is zero in a peculiar realization of the
equivalence principle Gross:1983hb . These strange gravitational properties of
the Kaluza-Klein theory are connected with the existence of a pseudo-Goldstone
massless scalar field, a dreaded dilaton, as can be seen from the equivalent
low energy theory (LET) Gross:1983hb

${\mathcal{S}}=-\frac{1}{16\pi
G}\int{d^{4}x\left(R_{4}+\frac{1}{6}g_{\mu\,\nu}\frac{\partial_{\mu}\sigma\,\partial_{\nu}\sigma}{\sigma^{2}}+\frac{1}{4}\sigma
F_{\mu\,\nu}F^{\mu\,\nu}\right)}$ (6)

where $\Phi\sim\log{\sigma}$ is the dilaton, which couples to gravity and
electromagnetism. Note that in the LET the masses of the other particles get
multiplied by a conformal factor $\bar{\mu}=\mu\,\sigma^{-1/6}$.

It has been hinted perturbatively, that in a full calculation quantum effects
might provide the dilaton with a mass Appelquist:1982zs . Given the scales
involved that mass would be at the level of the Planck mass and thus the
dilaton field can be screened out of the theory.

If we add a mass term to the above action $\frac{1}{2}M^{2}\Phi^{2}$, the
field equation for the scalar field becomes in the asymptotic limit
($r\rightarrow\infty$) of the spherical dipole metric Gross:1983hb ,

$\frac{\rho^{4}}{m^{2}}\left((1-2\,\rho)\frac{d^{2}\Phi}{d\rho^{2}}-2\,\frac{d\Phi}{d\rho}\right)-\sqrt{(1-2\,\rho)}\,M^{2}\
\Phi=0$ (7)

where $\rho=\frac{m}{r}$, $m$ being a parameter of the metric and $r$ the
spherical radial coordinate. In the asymptotic limit $\rho\rightarrow 0$ and
$M$ is planckian. In the limit where the mass is effective,
$r\rightarrow\infty$, i.e. $\rho\rightarrow 0$, Eq.(7) leads to
$\Phi\sim\exp{(-\frac{mM}{\rho})}\rightarrow 0$ and therefore
$\sigma\rightarrow 1$ and thus $\bar{\mu}=\mu$.

When the dilaton has no mass the equations of motion of a test particle of
mass $\bar{\mu}$ lead to a cancellation between the newtonian force
$\frac{1}{2}\bar{\mu}\partial_{i}(-g_{00})U^{0}U^{0}$ and
$\partial_{i}\bar{\mu}$, where $U^{\mu}=\frac{dx^{\mu}}{d\tau}$ Gross:1983hb .
This cancellation is instrumental in the vanishing of the gravitational mass.
However, once a dilaton mass is incorporated, the $\Phi$ field vanishes and
$\bar{\mu}$ becomes approximately constant and there is no cancelation.
Moreover since $M$ will be planckian the cancellation occurs for relatively
small values of $r$. Thus we expect that in any realistic KK theory where the
dilaton has been screened out the equivalence principle will be realized in
the conventional way and thus we assume that the inertial y gravitational
masses are equal, noting that the inertial mass is what enters into our
calculation. Moreover, although we will use the results of the GPS model, the
only requirements for the anlysis that follows is that the monopole has a mass
proportional to Planck’s mass and that the dipole has a mass which varies from
twice the monopole mass to a smaller mass when the deformation of the dipole
diminishes. The factors in Eq.(3) or Eq.(5) are irrelevant given the huge
numbers involved. Only the exponents count!

KK monopoles and monopolium have huge masses, larger than GUT monopoles
Preskill:1984gd ; Shnir:2005xx , and therefore they have to be created at very
early times and and their density will be strongly affected by inflation
Guth:1980zm ; Linde:1981mu . Is there any chance of having some remnants at
present?

##  III Monopolium as a dark matter candidate

Let us assume a cosmological model which expands isotropically in $3+n$
spatial dimensions. A fluctuation of the geometry during this expansion causes
$n$ dimensions to compactify. One may regard compactification as a phase
transition and use the horizon distance as an order parameter Harvey:1984fe .
This results in the production of one KK monopole per horizon distance
Zeldovich:1978wj ; Preskill:1979zi . If compactification occurs at $T\sim M$
then $d_{H}(T)\sim\frac{M_{PL}}{M^{2}}\sim\frac{16\,\alpha}{M_{Pl}}$ and in a
radiation dominated expansion the ratio of the number density of KK monopoles,
$n(T)$, to the number density of photons $n_{\gamma}(T)$ is Harvey:1984fe

$\frac{n(T)}{n_{\gamma}(T)}\sim\left(\frac{M}{M_{PL}}\right)^{3}\sim\left(\frac{1}{4\sqrt{\alpha}}\right)^{3}.$
(8)

The interactions at formation between monopole and anti-monopole and with the
charge in the plasma are dominated by the long range magnetic coupling.
Monopole and anti-monopole after diffusing through the plasma will bind
instead of annihilate Preskill:1979zi . The pair builds a highly excited
monopolium of mass approximately $M_{d}\sim 2M$ and large inter-pole distance
$d$. The strong magnetic coupling will make monopole and anti-monopole cascade
down to the ground state whose mass is $M_{d}\sim M$ and whose separation
distance $d\sim 0$ (see Fig. 1). This ground state is electromagnetically
inert, i.e. zero magnetic charge and zero magnetic dipole moment. It only is
subject to gravitational interaction. Since monopole and anti-monopole are
produced in equal numbers the number density of monopolium initially is still
$n(M)$. This is a huge number,
$n(M)\sim\left(\frac{1}{4\sqrt{\alpha}}\right)^{3}n_{\gamma}(M)\sim\;10^{107}\,m^{-3}$.
Since monopolium is basically stable its density decrease via the simplified
Boltzman’s equation without annihilation terms,

$\frac{n(T)}{T^{3}}=\frac{n(T^{\prime})}{T^{{}^{\prime}3}}\;\mbox{ for
}\;T^{\prime}<T,$ (9)

leading to a number density incompatible with standard cosmology. Thus we are
lead to inflation.

Before inflation at $T\sim M$ the density is

$n(M)\sim 10^{107}\,m^{-3}.$ (10)

Assuming for the as an approximation that all matter is made of monopolium we
get,

$n(2.7K)=\frac{\rho_{matter}(2.7K)}{M}\sim\;10^{-20}\,m^{-3},$ (11)

where $\rho_{matter}(2.7)$ is the matter density of the universe now. Let us
assume that after inflation the reheating temperature of the universe is
$T_{BB}$, then at the reheating period we get by Eq.(9) a monopolium density

$n(T_{BB})\sim n(2.7K)\left(\frac{T_{BB}}{2.7K}\right)^{3}.$ (12)

Thus from Eq.(10) and Eq.(12) a direct relation can be established between the
reheating temperature and the number of e-foldings required from inflation as
shown in Fig. 2. For example for a reheating temperature of $10^{27}K$ the
number of required e-foldings is $37$. With these properties just discussed,
monopolium in its ground state, could be an important candidate for
constituent of dark matter or in any case an exotic primordial constituent of
the universe.

Figure 2: Number of e-foldings as a function of the decimal Log of the
reheating temperature in Kelvin.

Can the de-excitation of monopolium from the excited state at which it is
formed to the ground state by emitting photons influence the present microwave
background? The time scale for de-excitation is large compared with the
inflation period and therefore this process occurs during the reheating
period. Assume the most favorable scenario that the excess mass from $2M$ to
$M$ is emitted as photons. In Fig. 3 we show the ratio of the energy density
of the radiated photons during de-excitation $\rho_{\gamma M}(T)=Mn(T)$ given
by,

$\rho_{\gamma
M}(T)=0.26\;10^{-1}\left(\frac{T}{K}\right)^{3}\frac{GeV}{m^{3}},$ (13)

to the true photon density determined by

$\rho_{\gamma}(T)=0.47\,10^{-5}\left(\frac{T}{K}\right)^{4}\frac{GeV}{m^{3}}.$
(14)

As shown in the figure the emitted photons do not contribute significantly to
reheating since they are a small fraction of the photons associated to
conventional blackbody radiation.

Figure 3: Decimal Log of the ratio of the energy density of the photons
radiated from monopolium to the conventional blackbody photon radiation as a
function on the decimal Log of the reheating temperature in Kelvin.

Our analysis leads to two possible scenarios depending on the size of the
inflationary process and subsequent reheating period. If the inflationary
process is around 37 e-foldings monopolium could be a major constituent of
dark matter. However, if inflation is more dramatic, then the density of
monopolium in the Universe will be reduced considerably, but monopolium might
still form clouds of primordial non interacting matter. The first scenario is
extremely exciting because it opens a very accessible door to extra
dimensions. The second scenario is more modest but also worth pursuing.

##  IV Signals of monopolium matter

Since monopolium is made of magnetically charged particles one might wonder
how monopolium reacts not in empty space but in the midst of the magnetic
fields that crowd the universe. Does its behavior give rise to observable
signals that discard monopolium as a dark matter particle?

Monopolium is a monopole-anti-monopole bound state and in its ground state
does not have a permanent dipole moment. However, in the vicinity of a
magnetic field it can get an induced magnetic dipole moment through its
response to the external magnetic field.

$\overrightarrow{\cal{M}}=\alpha_{M}\vec{B}=2g\,\overrightarrow{\Delta d},$
(15)

where $\overrightarrow{\cal M}$ is the magnetic dipole induced in the system,
$\vec{B}$ the magnetic field, $\alpha_{M}$ defines the magnetic (paramagnetic)
susceptibility and $\Delta d=|\overrightarrow{\Delta d}|$ is the induced
variation of the pole distance and therefore a measure of the stiffness of
monopolium.

When clouds of monopolium matter traverse magnetic fields the induced magnetic
moment will transform the space fluctuations of the magnetic field into a time
dependence that will generate electromagnetic radiation. Assume for simplicity
that the time dependence is harmonic,

${\cal{M}}={\cal{M}}_{0}\cos{\omega t}.$ (16)

In this case monopolium matter will radiate with a power proportional to the
size of the induced magnetic moment Jackson:1998nia

$W=\frac{k^{4}{\cal{M}}_{0}^{2}}{3}$ (17)

where $k$ is the modulus of the wave vector
$k=|\vec{k}|=\frac{\omega}{c}=\frac{2\pi}{\lambda}$.

Thus as a cloud of monopolium matter approaches a magnetic field the region
close to it will emit radiation while the one far away will not. That behavior
will distinguish monopolium matter from other types of inert matter. In
general the emission will not be harmonic but any emission can be formulated
as a combination of harmonics. The crucial parameter for observability is the
magnetic moment amplitude ${\cal M}_{0}$.

For a spring of constant $\kappa$ between two charged magnetic poles the
polarizability is

$\alpha_{M}=\frac{g^{2}}{\kappa}.$ (18)

as can be shown classically and quantum mechanically.

We use the Born-Oppenheimer approximation to find the spring constant of
monopolium at short distances. From Eq.(5) the $d<<M$ limit leads to

$\frac{M_{d}}{M}\sim 1+\frac{1}{16}\frac{d^{2}}{M^{2}}.$ (19)

Transforming $d$ to length dimensions becomes

$M_{d}=M+\frac{1}{16}\frac{d^{2}}{MG^{2}}$ (20)

where the masses are measured in $GeV$, $d$ in $fm$ and we use Newton’s
constant as $G=1.32\;10^{-39}\;\frac{fm}{GeV}$. The spring constant becomes
therefore $\kappa=\frac{1}{8MG^{2}}$, thus $\alpha_{M}=8Mg^{2}G^{2}$ and

${\cal M}_{0}=8Mg^{2}G^{2}B,$ (21)

where $B$ is measured in $GeV^{2}$.

Using the Dirac’s quantization condition as $\frac{eg}{\hbar c}=\frac{1}{2}$
and the fine structure constant $\alpha=\frac{e^{2}}{\hbar c}$ we get
Jackson:1998nia

$\alpha_{M}=8\left(\frac{\hbar c}{2e}\right)^{2}MG^{2}\sim
54\;MG^{2}\;fm^{3},$ (22)

thus

${\cal M}_{0}\sim 1.4\;10^{3}MG^{2}B\;fm.$ (23)

To summarize, monopolium matter which is electromagnetically inert in the
interstellar vacuum in the presence of time dependent magnetic fields emits
radiation as determined by Eqs.(17) and (23). This radiation is a signal that
differentiates monopolium clouds from other exotic matter clouds. Is this
signal big enough to be detected? If affirmative it would require the planning
of experiments to confirm their existence. If not observable, monopolium could
be a possible dark matter constituent. The aim of the next section is to
estimate the size of that emission.

##  V Astrophysical scenario

In order to answer the question posed in the previous section we analyze the
most favorable astrophysical scenario by means of a simple model. Imagine a
cloud of monopolium matter moving towards the center of a galaxy. We use a
very simple model for the magnetic core of certain type of galaxies. We assume
a basically flat galaxy, circular in shape, and which contains two regions,
the external region with radius $R$ such that $R_{G}>R>R_{C}$, where the
magnetic field is oriented in a clockwise direction, and the internal region
with radius $R$ such that $R<R_{C}$, where the magnetic field points in an
anti-clockwise direction as shown in Fig. 4. In the transition region the
field is maximal and its direction rapidly flipping.

Figure 4: Schematic picture of the magnetic field in our model galaxy. $B$ is
measured in microgauss ($\mu$G) and $x,y,z$ in kiloparsecs (Kpc).

This type of models have been studied in the literature Jaffe:2019iuk  but we
simplify them further for our calculation. Let us assume that a monopolium
cloud approaches the galaxy from the right and that the monopolium cloud is
small enough compared with the size of the galaxy ($\sim 1\,\mbox{Kpc}$) that
we can treat it as a point like object with mass $M_{C}$. Thus we reduce the
effective B field for the calculation to the one shown in Fig. 5.

Figure 5: One dimensional model of the magnetic field of a galaxy used in the
calculation as a function of distance.

We can formulate mathematically the magnetic field by

$\displaystyle B(R)=$
$\displaystyle\;\,B_{0}\;\tanh{\left(\frac{R-R_{C}}{R_{1}}\right)}\exp{\left(-\left|\frac{R-R_{C}}{R_{0}}\right|\right)}\hskip
42.67912pt$ $\displaystyle{\mbox{f}or}\;R>0,$ $\displaystyle=$ $\displaystyle-
B_{0}\;\tanh{\left(\frac{R+R_{C}}{R_{1}}\right)}\exp{\left(-\left|\frac{R+R_{C}}{R_{0}}\right|\right)}\hskip
42.67912pt$ $\displaystyle{\mbox{f}or}\;R<0,$

where $B_{0}\sim 15\,\mu G$ the largest possible field found in galaxies,
$,R_{0}=5\,\mbox{Kpc},R_{G}\sim 15\,\mbox{Kpc}$, $R_{C}\sim\,5\,\mbox{Kpc}$
and $R_{1}<<R_{0}$. We have chosen $R_{1}=1.0\,\mbox{pc}$ to plot the figure.
Let the monopolium cloud move from the right towards the center with a
velocity $v$. The moving cloud feels a time dependent magnetic field $B(t)\sim
B(R=R_{G}-vt)$, where $R_{G}$ is the point where we start measuring the time
as can be seen in Fig. 6. The most interesting region is close to $R_{C}$ as
we shall see.

The power emitted by a non harmonic force is Mirmoosa:2020pft

$W(t)\sim{\cal M}(t)\frac{d^{4}{\cal
M}(t)}{dt^{4}}\sim\alpha_{M}^{2}B(t)\frac{d^{4}B(t)}{dt^{4}}.$ (25)

where ${\cal M}(t)=\alpha_{M}B(t)$, where recall that $\alpha_{M}$ is the
magnetic polarizability of monopolium, Eq.(22). The power can be written as

$W(t)\sim\alpha_{M}^{2}B_{o}^{2}\gamma^{4}b(x)\frac{d^{4}b(x)}{dx^{4}}$ (26)

where $x=\frac{R_{G}-R_{C}-vt}{R_{1}}$, $\gamma=\frac{v}{R_{1}}$, and

$\displaystyle b(x)$ $\displaystyle=$
$\displaystyle\tanh{(x)}\exp{(-\,a\,x)}\;\;\mbox{ if }\hskip
14.22636pt0<x<\frac{R_{G}-R_{c}}{R_{1}}$ $\displaystyle=$
$\displaystyle\tanh{(x)}\exp{(+\,a\,x)}\;\;\mbox{ if
}\;\;-\frac{R_{c}}{R_{1}}<x<0$

being $a=\frac{R_{1}}{R_{0}}$.

Figure 6: Representation of the time evolution of the magnetic field seen by
the monopolium cloud.

The emission of power taking into account the magnetic field direction is
shown in Fig.7a which is the emitted wave signal. The total energy emitted in
the process at the point $R_{G}-vt$ is

$E(vt)\sim\int_{0}^{vt}dvt^{\prime}W(vt^{\prime})$ (28)

In Fig. 7b we show the energy emitted as the cloud penetrates the galaxy
noticing that most of the emission occurs precisely where the power wave is
formed around $R_{C}$ and approximately during the interval
$\sim[R_{C}-R_{1},R_{C}+R_{1}]$.

If we integrate over the whole interval from $R_{G}$ to the origin we get

$E(R_{G})\sim\alpha_{M}^{2}B_{0}^{2}\gamma^{3}{\cal
I}(R_{G},R_{C},R_{0},R_{1})$ (29)

where

${\cal
I}(R_{G},R_{C},R_{0},R_{1})=\int_{-\frac{R_{C}}{R_{1}}}^{\frac{R_{G}-R_{C}}{R_{1}}}dx\,b(x)\frac{d^{4}b(x)}{dx^{4}}.$
(30)

Note that ${\cal I}$ is a numerical integral which dependes on the shape of
the magnetic field and the structure of the galaxy as determined by the galaxy
parameters $R_{G},R_{C},R_{0},R_{1}$ but does not depend on the velocity. The
only velocity dependence is in the factor $\gamma$ in front of Eq.(29).
$R_{1}$ is the less known parameter but it is smaller than the rest of size
parameters. We use for the other size parameters realistic values as shown in
the figures $B_{0}=\,15\mu\mbox{G}$, which is the value of the highest
possible galactic magnetic fields observed in nature, $R_{G}\sim
15\,\mbox{Kpc}$, $R_{C}=5\,\mbox{Kpc}$, $R_{0}\sim 5\,\mbox{Kpc}$ and for
$R_{1}\sim 1\,\mbox{pc}$. This small value of $R_{1}$ makes the transition
from one field direction to the other very abrupt.

Figure 7: a) Emitted power taking into account the direction of the field. b)
The structure of the emitted energy as a function of the penetration distance.

Studying the dependence of the Integral on $R_{1}$ for small values of $R_{1}$
we find that it goes to a good approximation like a constant times $R_{1}$ as
shown in Fig 8, i.e. ${\cal
I}(R_{G},R_{C},R_{0},R_{1})/R_{1}\sim\mbox{constant}$. For the chosen values
of the parameters the constant factor is approximately $2$.

The emitted energy when the monopolium cloud passes across $R_{C}$,

$E\sim
2\alpha_{M}^{2}B_{0}^{2}\gamma^{3}\left(\frac{R_{1}}{\mbox{Kpc}}\right)n(T)V_{cloud}$
(31)

where $n(T)$ is the monopolium density which we parametrize by a temperature
and its determined by the relation

$n(T)=n(2.7K)\left(\frac{T}{2.7}\right)^{3}$ (32)

where $n(2.7)$ is the monopolium density today as shown in Eq.(11), and
$V_{cloud}$ is the volume of the monopolium cloud. Assuming a spherical cloud
we get for the energy emitted when the cloud passes across the region $R_{C}$

$E=\frac{8\pi}{9}\left(\frac{1}{0.197}\right)\alpha_{M}^{2}B_{0}^{2}\left(\frac{v}{c}\right)^{3}n(2.7)\left(\frac{T}{2.7}\right)^{3}\left(\frac{R_{cloud}}{R_{1}}\right)^{3}\left(\frac{R_{1}}{\mbox{Kpc}}\right)\mbox{GeV}$
(33)

where we have used the following units: $\alpha_{M}$ is in fm3, $B_{0}$ in
GeV/fm, $n(2.7)$ in fm-3 which lead to an energy in GeV. The duration in
passing the region is governed by the cloud diameter, $\Delta T=2R_{cloud}/v$.
Using a favorable case where $B_{0}\sim 15\,\mu$G, $R_{cloud}\sim
1\,\mbox{Kpc}$, $R_{1}\sim\,1\mbox{pc}$, $v=0.01$ c, the emitted power is
extremely small $P\sim 10^{-160}$ GeV/s $\sim 10^{-170}$ Joule/s at emission.
The smallness is associated with the smallness of the polarizability
$\alpha_{m}\sim 10^{-56}\,\mbox{fm}^{3}$ and also the small magnitude of the
galactic fields. Thus monopolium matter associated with Kaluza-Klein monopoles
is really dark and the fact that it is stable makes it very improbable to
detect via its decay.

Figure 8: We plot the structure of the dependence of ${\cal
I}(R_{G},R_{C},R_{0},R_{1})/R_{1}$ as a function of $R_{1}$ for small values
of this parameter

##  VI Concluding remarks

The existence of monopoles is a characteristic signature of Kaluza-Klein
multidimensional theories Gross:1983hb ; Sorkin:1983ns  based on fundamental
developments in lower dimensional theories Newman:1963yy ; Hawking:1976jb ;
Gibbons:1979xm . The topology of these theories is extremely interesting, in
particular the leitmotiv of my presentation is inspired by the existence of
the dipole solution Gross:1983hb ; Kerr:1963ud ; Gibbons:1976ue , which has
been associated to a monopole-anti-monopole bound state. In KK theories this
state is classically stable since it does not have the quantum numbers of the
vacuum and therefore is long lived. This state has been named monopolium. Its
lowest energy state, i.e. $d\sim 0$ and $M_{d}\sim M$, is electromagnetically
inert in free space. We have studied its properties and therefore suggest to
propose it as a dark matter constituent.

In order to get some consequences of this line of thought we have used the
details of the GPS model which makes the presentation simpler by dealing with
exact formulas. But the outcome would be the same accepting some very general
features: the topology Newman:1963yy ; Hawking:1976jb ; Gibbons:1979xm ;
Kerr:1963ud ; Gibbons:1976ue  ; the production mechanism by the
compactification transition Zeldovich:1978wj ; Preskill:1979zi ; Harvey:1984fe
; the naturalness of parameters in terms of $M_{Pl}$ and $G$. The only formula
used from GPS can simply substituted by $M_{d}\sim A+Bd^{2}$, where $A\sim
M_{Pl}$ and $B\sim(M_{Pl}G^{2})^{-1}$ by dimensional arguments. The only
difference with the present calculation would be numerical coefficients, which
are almost irrelevant, since what matters here are exponents.

Due to its huge mass, $M\sim M_{Pl}$, KK monopoles can only be produced in
large quantities via a compactification phase transition in the early Universe
Harvey:1984fe . Their long range magnetic force will bind the KK poles form
monopolium since monopoles and anti-monopoles are produced in equal
quantitites. We have shown that there is a good chance that monopolium
survives in detectable amounts after a conventional inflationary period and
that their photon emission arising by de-excitation from the initially highly
excited state to the ground state will not affect the cosmic microwave
background. We have analyzed the observability of monopolium by using its
electromagnetic properties induced in the presence of astronomical magnetic
fields. We have seen that its magnetic polarizability is extremely small and
thus detection is impossible even if appearing in large monopolium clouds. At
this point we cannot think of any more favorable scenario than this one. There
are objects, like magnetars Kaspi:2017fwg , with much higher magnetic fields,
but these fields are of very small extension, at the level of kms, thus
$V_{cloud}$ would be extremely small, and the emitted power even smaller than
the one we have calculated. The other scenario we would like to study in the
future is what happens with a cloud on monopolium in the proximity of a
supermassive black holes in the center of the galaxies Ghez:2008ms ;
Gillessen:2008qv , but at this point we do not see how this phenomenon would
increase observability. Thus KK monopolium seems very difficult to detect and
therefore an ideal candidate to be one of the constituents of dark matter.
Probably monopolium clouds will be only detectable by gravitational waves
arising from their interaction with black holes.

What really remains to investigate is the discovery of possible signals to
detect KK monopoles or KK monopolium. In this work we have assumed their
existence and have described some properties which lead to monopolium as being
a primordial dark matter constituent. But a proof, even a good argument of its
existence is lacking. Its huge mass makes production unfeasible in
accelerators, the small magnitude of its electromagnetic effects and its small
present day density make it difficult to detect by astronomical observatories.
Thus, the aim next is to search for properties of KK monopoles or KK
monopolium, which help in proposing experiments which might prove or disprove
its existence.

## Acknowledgement

Discussions with Gabriela Barenboim, Huner Fanchiotti, Armando Pérez, Sergio
Palomares, Santiago Noguera, and Javier Vijande are acknowledged. I would like
to thank Carlos García Canal and Marco Traini for a careful reading of the
manuscript and valuable comments and suggestions, and Marco Traini also for
sharing his profound knowledge on polarizable systems with me. Conversations
with V. Vento-Bosch were inspiring to initiate this study. This work was
supported in part by the MICINN and UE Feder under contract
FPA2016-77177-C2-1-P.

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# Uncovering Quasi-periodic Nature of Physical Systems: A Case Study of
Signalized Intersections

Suddhasattwa Das1, Shakib Mustavee2, Shaurya Agarwal2 1The author is with the
Department of Mathematics, George Masion University, Fairfax, Virginia, USA
(email: iamsuddhasattwa@gmail.com)2The authors are with the Department of
Civil Engineering, University of Central Florida, Orlando, USA (email:
smustavee@gmail.com), (email<EMAIL_ADDRESS>
###### Abstract

This paper presents a novel approach to analyze quasiperiodically driven
dynamical systems. It aims to develop a complete data-driven framework for
modeling such unknown dynamics. To achieve this, we characterize Koopman
eigenfrequencies as generating frequencies of the quasiperiodic driver of the
system. We compute true eigenfrequencies of Koopman operators by applying the
theory of Reproducing Kernel Hibert Space (RKHS) and results from ergodic
theory. We also demonstrate the decomposition of quasiperiodically driven
dynamics into two components, i) the quasiperiodic driving source with
generating frequencies and ii) the driven nonlinear dynamics. A unique aspect
of the proposed framework is that it applies to the analysis of systems where
the periodic component is either non-dominant or even absent. As a case study,
we analyze a system of nine traffic signalized intersections. The proposed
framework accurately reconstructs the measured queue lengths of the signalized
intersections and makes stable long-term predictions.

###### Index Terms:

Koopman operator, Reproducing Kernel Hibert Space, Quasiperiodic systems,
Signalized intersections

##  I Introduction

Many physical phenomena show periodicities, such as oscillations and limit
cycles, characterized by a single frequency or period. There are also many
phenomena, such as the arrival of fall foliage, the arctic ice cycle, and
planetary dynamics, in which there are multiple driving frequencies. As a
result of which the states of the system do not show exact and regular
periodicity. Such systems are called _quasiperiodic systems_. More complicated
systems exhibit chaos and a high degree of nonlinearity but having a
quasiperiodic component as a driving source. Examples are astronomical systems
[1], climate systems [2, 3], geophysical flows on periodic domains [4],
epidemics [5], and computational neuroscience [6].

The goal of this paper is to develop a robust method for reconstructing
quasiperiodically driven dynamical systems using the Koopman operator
framework. Koopman operator is originally a linear operator that describes the
spatiotemporal evolution of a nonlinear dynamical system. Projection of
observables on the eigenfunction of the Koopman operator is defined as Koopman
modes. Koopman modes can efficiently capture quasiperiodic components of a
flow and outperform proper orthogonal decomposition characterizing the
evolution on limit cycles, and tori [7]. Koopman operator has also proven to
be successful for describing highly non-periodic dynamics by decomposing the
underlying dynamics into periodic and quasiperiodic patterns [6]. In Section
II-B, we describe quasiperiodicity in terms of eigenfunctions of the Koopman
operator. However, an accurate estimation of the eigenfunctions and
eigenfrequencies has remained an elusive task. Techniques such as DMD [7, 8],
deep neural networks [9, 10], or Fourier averaging [1] are inadequate when the
dynamics has a substantial chaotic component.

Our novel approach uses a technique from _RKHS interpolation theory_ developed
in [11] to extract the true eigenfrequencies. After that, we proceed to
identify different components of the dynamics in a manner similar in structure
to [6]. This approach leads to a robust theoretical and numerical method to
reconstruct quasiperiodically driven systems having strong chaotic components
by identifying their true eigenfrequencies. As a case study, we study the
queue length dynamics on a corridor of traffic signalized intersections. Such
a system has quasiperiodic driving sources, but the measurement data also
reflects unpredictable phenomena such as accidents, road constructions, and
human factors. It makes the study more challenging.

Contributions: The contributions of this paper is as follows:

  * •

We provide an alternative mathematical formulation of the quasiperiodic
coordinates in terms of the _Koopman operator_ that correctly recovers both
the periodic and chaotic part of the dynamics.

  * •

The proposed framework is even applicable to the systems where the periodic
component is either non-dominant or even absent.

  * •

We describe the physical significance of the quasiperiodic sources obtained
from the proposed technique

Outline: Rest of the paper is arranged as follows: The dynamical systems
theory related to quasiperiodically driven systems is described in Section
II-A. We discuss the relevant concepts and techniques in Section II-B and
Section II-C. The proposed data-driven implementation is described in Section
III. Section IV performs a case study using real measurements from traffic
intersections and discusses the results.

##  II Proposed Framework

###  II-A Formulation

A quasiperiodically driven system has the form

$\begin{split}\theta_{n+1}&=\theta_{n}+\vec{\rho}\bmod\mathbb{T}^{d}\\\
x_{n+1}&=g(x_{n},\theta_{n})\end{split},$ (1)

where $f$ is some nonlinear function. The angular coordinate $\theta$ is a
point on a $d$-dimensional torus $\mathbb{T}^{d}$. $\theta$ represents the
_phase_ of a driving quasiperiodic system. The vector $\vec{\rho}$ is called
the _rotation vector_ [12, 13], it represents the angular increments at each
step for each of the coordinates of $\theta$. If the underlying system arises
from a continuous time system by taking samples at intervals $\Delta t$, then
$\vec{\rho}=\Delta t\vec{\omega}$ for some angular frequency vector
$\vec{\omega}$. The variable $x$ lies in some abstract or unknown manifold
$\mathcal{X}$. Let $\Omega:=\mathbb{T}^{d}\times\mathcal{X}$. Thus (1) is a
_one-way coupled_ or _skew-product_ dynamical system on the space $\Omega$ :

$F:\Omega\to\Omega,\quad\left(\begin{array}[]{c}\theta_{n+1}\\\
x_{n+1}\end{array}\right)=F\left(\begin{array}[]{c}\theta_{n}\\\
x_{n}\end{array}\right):=\left(\begin{array}[]{c}\theta_{n}+\vec{\omega}\\\
g(\theta_{n},x_{n})\end{array}\right).$ (2)

The space $\Omega$ is unknown, and its points are pairs of points from
$\mathbb{T}^{d}$ and $\mathcal{X}$ respectively. This abstract formulation
applies to all quasiperiodically driven systems. We further assume the
dynamics in (1) to be of the form

$\begin{split}\theta_{n+1}&=\theta_{n}+\Delta t\vec{\omega}\bmod 2\pi\\\
x_{n+1}&=g_{per}(\theta_{n})+g_{chaos}(\theta_{n},x_{n})\end{split}$ (3)

Thus the task is to find (i) the _quasiperiodicity dimension_ $d$ and the
rotation vector $\omega$; and the functions (ii) $g_{per}$ and (iii)
$g_{chaos}$. The functions and spaces above will be assumed to be unknown. The
only information about the system will be through a collection of $k$
observations / measurements, represented collectively as a map
$Y:\Omega\to^{k}$. $Y$ is possibly unknown, and possibly a low-dimensional /
partial observation of $\Omega$. It generates a sequence of $k$-dimensional
data points $\left\\{y_{n}:=Y(\theta_{n},x_{n})\;:\;n=0,1,2,\ldots\right\\}$,
where $(\theta_{n},x_{n})$ is a trajectory of the dynamics in (2) under $F$.

###  II-B Koopman operator and its spectrum

The Koopman operator $U$ is essentially a time-shift operator. It operates on
functions instead of points on the phase space. Given a function
$\phi:\Omega\to$, $U\phi$ is the function defined as

$(U\phi)(z):=\phi\left(Fz\right),\quad\forall z\in\Omega,$ (4)

where $F$ is the underlying dynamical system (2). $\phi$ can be interpreted as
a measurement or observation on the phase space $\Omega$, and $U\phi$ is the
evolution/transformation of this measurement with the dynamics.

##### Koopman eigenfrequencies

Since $U$ is unitary; its spectrum must lie on the unit circle of the complex
plane. The eigenvalues of $U$ correspond to the point spectrum, and any
eigenfunction $\zeta$ has a corresponding eigenvalue of the form
$e^{\iota\omega}$ for some $\omega\in$. Thus

$(U^{n}\zeta)(z)\stackrel{{\scriptstyle\mbox{by
def.}}}{{=}}\zeta(F^{n}z)=e^{\iota\omega n}\zeta(z),\quad\forall
n\in\mathbb{N}.$ (5)

$\omega$ is called the Koopman eigenfrequency corresponding to $\zeta$. Thus
the time-evolution of Koopman eigenfunctions is equivalent to multiplication
by $e^{\iota\omega n}$ as a function of time $n$.

$U$ always has the constant functions as eigenfunctions with eigenfrequency
$0$. $U$ may or may not have other eigenfrequencies. The collection of
eigenfunctions and (eigen)-frequencies have an algebraic structure to them.
For any two frequencies $\omega_{1},\omega_{2}$, and integers $a,b$,
$a\omega_{1}+b\omega_{2}$ is also a frequency [11]. If the system has at least
one nonzero frequency, then it has all harmonics of that frequency and thus
infinitely many frequencies. A collection of eigenfrequencies is said to be
independent if no integer linear combination is an integer. If the system has
two independent frequencies, then all its frequencies are together dense on
the real line. A collection of frequencies will be called a _basis_ or
_generating_ set of eigenfrequencies if they are independent, and all
frequencies of the system can be generated by taking integer linear
combinations of frequencies from this set. There is no unique choice of a
basis, but all bases will have the same cardinality $d$, called the
quasiperiodicity dimension $d$. $d$ is a fixed finite number if $\Omega$ is a
finite-dimensional manifold.

##### Koopman eigenfunctions and torus dynamics

Koopman eigenfunctions reveal quasiperiodic dynamics embedded in the system.
Any $k$ Koopman eigenfunctions leads to a rotation on a $k$-dimensional torus:

${\Omega}$${\Omega}$${\mathbb{T}^{k}}$${\mathbb{T}^{k}}$$\scriptstyle{\zeta_{1},\ldots,\zeta_{k}}$$\scriptstyle{F}$$\scriptstyle{\zeta_{1},\ldots,\zeta_{k}}$$\scriptstyle{R_{\vec{\omega}}}$

Here, $R_{\vec{\omega}}$ is a rotation by the vector $\vec{\omega}$ of
frequencies. If these eigenfrequencies are independent, then this map will be
surjective. Taking $d=k$ implies that the dynamics has an embedded / factor
torus rotation of the same dimension as the quasiperiodicity dimension.

This completes our examination of the quasiperiodic structure of the dynamics
(2). We next discuss some techniques from Functional Analysis for
reconstructing the quasiperiodic component and its complement.

###  II-C Kernels and integral operators

A kernel is a function $k:M\times M\to$ on some space $M$. The quantity
$k(x,y)$ is measure of similarity, closeness or distance between two points
$x,y\in M$. Kernel based methods have been used very effectively to obtain
information such as statistical manifolds [14], geometric information [15],
and dynamical information such as tracer flows [4], stable/unstable foliations
[16], Koopman spectrum [17, 11]. The techniques in this paper are based on
[11]. We shall use the _Gaussian kernel_

$k_{\epsilon}(x,y):=\exp\left(-\frac{1}{\epsilon}d(x,y)^{2}\right),$

where $\epsilon$ is called the _bandwidth parameter_ , and $d(\cdot,\cdot)$ is
some notion of metric or distance on the space.

##### Delay-coordinates

The data sequence $y_{n}$ is obtained through an observation $Y$. However $Y$
may not be a one-to-one map and its values may not correspond to unique states
in $\Omega$. We convert $Y$ into an embedding using the method of delay
coordinates [18], by incorporating $Q$ delays to get the map
$Y^{(Q)}:\Omega\to^{k(Q+1)}$ :

$Y^{(Q)}(\omega):=\left(Y(\omega),Y(F^{1}\omega),\ldots,Y(F^{Q}\omega)\right).$

Thus the delay coordinated version of each point $y_{n}$ is

$y_{n}\leftrightarrow y_{N}^{(Q)}:=\left(y_{n}y_{n+1},\ldots,y_{n+Q}\right).$

We next use the Gaussian shape function to implicitly obtain a kernel
$k_{\epsilon}:\Omega\times\Omega\to$ as follows

$k_{\epsilon}(z,z^{\prime}):=\exp\left(-\frac{1}{\epsilon}\left\|Y^{(Q)}(z)-Y^{(Q)}(z^{\prime})\right\|^{2}\right).$
(6)

Even if the two states $z,z^{\prime}$ are unknown, the left-hand side in (6)
can be computed since the right-hand side only uses the observation map $Y$.
We next modify $k_{\epsilon}$ by a process called _bistochastic normalization_
[19] to get a kernel $p_{\epsilon}$ which is symmetric, Markovian, and more
adapted to the non-uniform distribution of the data.

$\begin{split}&\deg_{R}(z):=\int
k_{\epsilon}(z,z^{\prime})d\mu(z^{\prime}),\,\deg_{L}(z):=\int\frac{k_{\epsilon}(z,z^{\prime})}{\deg_{R}(z)}d\mu(z^{\prime}).\\\
&\tilde{k}_{\epsilon}(z,z^{\prime\prime}):=\frac{k_{\epsilon}(z,z^{\prime\prime})}{\deg_{R}(z)\deg_{L}(z^{\prime\prime})^{1/2}}\\\
&p_{\epsilon}(z,z^{\prime}):=\int\tilde{k}_{\epsilon}(z,z^{\prime\prime})\tilde{k}_{\epsilon}(z^{\prime\prime},z^{\prime})d\mu(z^{\prime\prime}).\end{split}$

##### Kernel integral operator

Associated to the kernel $p_{\epsilon}:\Omega\times\Omega\to$ is the integral
operator $P_{\epsilon}$, which operates on $L^{2}(\mu)$ functions
$\phi:\Omega\to$ as

$(P_{\epsilon}\phi)(z):=\int_{\Omega}p_{\epsilon}(z,z^{\prime})\phi(z^{\prime})d\mu(z^{\prime}).$

$P_{\epsilon}$ is a _compact_ , symmetric operator on $L^{2}(\mu)$ [[,
e,g,]]DGJ_compactV_2018. Moreover $P_{\epsilon}$ has a complete basis of
eigenfunctions

$P_{\epsilon}\phi_{j}=\lambda_{j}\phi_{j},\quad j=0,1,2,\ldots,$

where the indexing is done so that the $\lambda_{j}$s are in decreasing order.
Due to the normalizations carried out, we have $\phi_{1}\equiv 1_{\Omega}$,
the constant function equal to $1$ everywhere. Moreover, the eigenvalues
satisfy$1=\lambda_{1}\geq\lambda_{2}\geq\lambda_{2}\geq\ldots>0$. Also
importantly, the $\phi_{j}$ are an orthonormal basis, i.e.,

$\langle\phi_{i},\phi_{j}\rangle_{L^{2}(\mu)}:=\int\phi_{i}^{*}(x)\phi_{j}(x)d\mu(x)=\delta_{i,j}.$

All these properties of the $\lambda_{j}$ and $\phi_{j}$ are useful for
_kernel-based learning_ , in which we recreate or extrapolate unknown
functions from some samples using these $\phi_{j}$s as a basis.

##### Kernel based learning

One of the main advantages of kernel-based approaches is that while the
$\phi_{j}$ can be approximated to any degree of accuracy by solving an
eigenvalue equation of a data-driven matrix; they can be easily extended from
vectors to a continuous function over the entire data space k(|Q|+1) and thus
on $\Omega$. The $\phi_{j}$s also happen to be left singular vectors of the
asymmetric operator $\tilde{K}_{\epsilon}$, with $\sqrt{\lambda_{j}}$ and
$\gamma_{j}$ being the associated singular values and right singular vectors.
Thus for an arbitrary point $z\in^{k(|Q|+1)}$, we have

$\begin{multlined}\phi_{j}(z)=\lambda_{j}^{-1}\int
p_{\epsilon}(z,z^{\prime})\phi_{j}(z^{\prime})d\mu(z^{\prime})\\\
=\lambda_{j}^{-1}\int\tilde{k}_{\epsilon}(z,z^{\prime})\gamma_{j}(z^{\prime})d\mu(z^{\prime}).\end{multlined}\phi_{j}(z)=\lambda_{j}^{-1}\int
p_{\epsilon}(z,z^{\prime})\phi_{j}(z^{\prime})d\mu(z^{\prime})\\\
=\lambda_{j}^{-1}\int\tilde{k}_{\epsilon}(z,z^{\prime})\gamma_{j}(z^{\prime})d\mu(z^{\prime}).$
(7)

These $\phi_{j}$ for the basis for _learning_ any function
$f:^{k(Q+1)}\to^{d}$. It is done by first computing the components of $f$
along the first $L$ $\phi_{j}$

$f_{l}:=\langle\phi_{l},f\rangle_{L^{2}(\mu)}=\int_{{}^{k(Q+1)}}\phi_{l}^{*}(z)f(z)d\mu(z),$

and then taking the sum/integral

$f(y)\approx\sum_{l=1}^{L}f_{l}\phi_{l}(y)=\int_{{}^{k(Q+1)}}p_{\epsilon}(y,z)\sum_{l=1}^{L}\frac{f_{l}}{\lambda_{l}}\phi_{l}(z)d\mu(z).$

The parameter $L$ is called the _spectral truncation_ parameter; it is the
size of the hypothesis space.

We next describe the use of the fast-Fourier transform to derive the Koopman
eigenfrequencies but on the $L$ eigenfunctions instead of the raw data.

##  III The data-driven procedure

In the data-driven approach, all of the entities described in Section II-C
have a _data-driven analog_. We begin with the invariant measure $\mu$ itself.
It will be replaced by
$\mu_{N}=\frac{1}{N}\sum_{n=1}^{N}\delta_{y_{n}^{(Q)}}$, the average of the
Dirac-delta measure on the data points $y_{n}^{(Q)}$. These are called
_sampling/empirical_ measures. Their integrals with respect to these sampling
measures are given by

$\int_{{}^{k(Q+1)}}\phi
d\mu_{N}=\frac{1}{N}\sum_{n=1}^{N}\phi\left(\delta_{y_{n}^{(Q)}}\right),$

for every continuous test function $\phi:^{k(Q+1)}\to$. The kernel integral
operators $K$, $\tilde{K}$ and $P$ will be approximated as $N\times N$
matrices $\left[K\right]$, $\left[\tilde{K}\right]$ and $\left[P\right]$ :

$\left[K\right]_{i,j}:=k_{\epsilon}\left(y_{i}^{(Q)},y_{j}^{(Q)}\right),\;\vec{d}:=\frac{1}{N}\left[K\right]\vec{1}_{N},\;.$
(8)

Next compute

$\vec{q}:=\frac{1}{N}\left[K\right]D^{-1},\quad
D:=\operatorname{diag}(\vec{d}).$ (9)

$\vec{d}$ and $\vec{q}$ are called _left and right degree vectors_
respectively. Finally compute

$\left[\tilde{K}\right]:=D^{-1}\left[K\right]Q^{-1/2},\quad
Q:=\operatorname{diag}(\vec{q}).$ (10)

The matrix $\left[P\right]:=\left[\tilde{K}\right]\left[\tilde{K}\right]^{*}$
is not computed explicitly. See [19, Algorithm 1] for more details. Compute
the top $L$ singular values $1=\sigma_{1}\geq\ldots\geq\sigma_{L}$ of
$\left[\tilde{K}\right]$ and the corresponding left eigenvectors
$\vec{\phi}_{1},\ldots,\vec{\phi}_{L}$ and right singular vectors
$\vec{\gamma}_{1},\ldots,\vec{\gamma}_{L}$. Set $\lambda_{i}:=\sigma_{i}^{2}$,
for $i=1,\ldots,L$. The set of vectors
$\left\\{\vec{\phi}_{l}\;:\;l=1,\ldots,L\right\\}$ and
$\left\\{\vec{\gamma}_{l}\;:\;l=1,\ldots,L\right\\}$ are both orthonormal
systems for $\mathbb{C}^{N}$. The eigenvectors have continuous extensions to
any $y\in^{k(Q+1)}$ as

$\begin{split}&\bar{\phi}_{l}(y):=\frac{1}{N\lambda_{l}}\vec{k}_{\epsilon}(y)^{\top}Q^{-1/2}\vec{\gamma}_{l}\\\
&\vec{k}_{\epsilon}(y):=\left(k_{\epsilon}\left(y,y_{1}^{(Q)}\right),\ldots,k_{\epsilon}\left(y,y_{N}^{(Q)}\right)\right).\end{split}$
(11)

The function $\bar{\phi}_{l}$ is continuous as the vector
$\vec{k}_{\epsilon}(y)$ is a continuous function of $y$. If $y$ in the above
equation is substituted by one of the data-points $y_{n}^{(Q)}$, then by
design $\bar{\phi}_{l}\left(y_{n}^{(Q)}\right)=\vec{\phi}_{l,n}$. Thus
$\bar{\phi}_{l}$ is indeed a continuous extension of the vector
$\vec{\phi}_{l}$. This feature of extendability and easy evaluation at
arbitrary points is one of the most powerful tools of kernel-based methods.

##### RKHS based spectral filtering

The following procedure was described in [11, Algorithm 1], and accepts as
parameters $\epsilon_{1},\epsilon_{2}>0$ and integer $L_{0}>1$. Let
$\mathcal{F}_{N}$ denote the discrete Fourier transform on vectors of length
$N$. Let $\left[\Phi\right]$ be the $N\times L$ matrix whose $l$-th column is
$\vec{\phi}_{l}$. Set
$\Lambda:=\operatorname{diag}\left(\lambda_{1},\ldots,\lambda_{L}\right)$ and
compute

$\left[\hat{\Phi}\right]:=\mathcal{F}\left[\Phi\right]\quad\left[H\right]:=\left[\hat{\Phi}\right]\Lambda^{-0.5}.$

Next, compute the RKHS-norms as

$\left[W\right]_{n,1}:=\left|\left[H\right]_{n,1}\right|,$
$\left[W\right]_{n,l+1}:=\left[W\right]_{n,l}+\left|\left[H\right]_{n,l+1}\right|,l=1,\ldots,L-1.$

Now let $J=1,\ldots,N$. Discard all the $j\in J$ for which
$\left[W\right]_{j,L_{0}}<\epsilon_{1}$. Of the remaining $j\in J$, discard
those $j$ for which
$\ln\left[W\right]_{j,L}-\ln\left[W\right]_{j,L_{0}}>\epsilon_{2}$. Compute
$\omega_{j}=\frac{2\pi j}{N\Delta t}$ for all of the remaining $j\in J$. These
frequencies $0=\omega_{1}<\omega_{2}<\ldots<\omega_{m}$ can be interpreted to
be true Koopman eigenfrequencies with substantial presence in the original
data.

##### Periodic and chaotic components

The identified frequencies $0=\omega_{1}<\omega_{2}<\ldots<\omega_{m}$ are by
no means exhaustive, they are only a finite subset of a usually infinite set
of Koopman eigenfrequencies. However, they represent those (true) frequencies
that have a significant presence in the data. The threshold $\epsilon_{1}$ is
meant to be a numerical implementation of frequencies being significant. We
next construct the periodic component $g_{per}:\to^{k}$ as

$g_{per}(t):=\Re\sum_{j=1}^{m}\left[A\right]_{j,:}e^{\iota\omega_{j}t}.$ (12)

The $m\times k$ matrix $\left[A\right]$ is the least-squares solution to

$\Re\left[F\right]\left[A\right]=\left[Y\right],$

where $Y$ is the data-matrix and $\left[F\right]$ is $N\times m$ matrix

$\left[F\right]_{n,j}:=(2-\delta_{j,1})e^{\iota n\omega_{j}}\quad 1\leq n\leq
N,\,1\leq j\leq m.$

Next, we construct the chaotic component

$g_{chaos}:^{k(Q+1)}\to^{k},\;g_{chaos}:=\sum_{l=1}^{L}\left[E\right]_{l,:}\bar{\phi}_{l},$
(13)

where $\left[Y_{non}\right]:=\left[Y\right]-\Re\left[F\right]\left[A\right]$
and $\left[E\right]:=\left[\Phi\right]^{*}\left[Y_{non}\right]$.

##### The reconstruction

We avoid the task of identifying a set of generating frequencies by directly
using the selected frequencies in the approximation

$g_{per}\left(\vec{\theta}+n\vec{\tilde{\omega}}\right)\approx\sum_{j=1}^{m}a_{j}\exp\left(\iota
n\omega_{j}\right).,\quad n=0,1,2,\ldots.$ (14)

Using this simplification in (14), and the formulas in (13) and (12), we
create the following data-driven model of the dynamics :

$\begin{split}y^{0}_{n+1}&:=g_{per}(n\Delta
t)+g_{chaos}\left(y^{0}_{n},\ldots,y^{Q}_{n}\right)\\\
y^{1}_{n+1}&:=y^{0}_{n}\\\ \vdots&=\vdots\\\
y^{Q}_{n+1}&:=y^{Q-1}_{n}\end{split}.$ (15)

Here each $y^{q}_{n}\in^{k}$, thus making the state vector
$y_{n}=\left(y^{0}_{n},\ldots,y^{Q}_{n}\right)$ a vector in k(|Q|+1). We have
thus created a standalone dynamical system k(|Q|+1) which is conjugate to the
latent dynamics.

##  IV Case Study: Signalized Intersection Corridor

###  IV-A Data Description

The case study analyzes queue length measurements from nine adaptive traffic
signals located on the Alafaya Trail (SR-434) in East Orlando, FL. The
obtained data includes the details of each movement with the time, duration,
queue length, and waiting time. It provides information on eight movements:
north left (NL), north through (NT), south left (SL), south through (ST), east
left (EL), east through (ET), west left (WL), and west through (WT). In this
study, we focus on the queue length formation of northbound through movements.
The raw data was processed and calibrated by Rahman, and et al. [20] and was
resampled at regular intervals of $\Delta t=2\text{minutes}$. This work uses
the processed data from [20].

Figure 1: Northbound queue length formation at the corridor at $i^{th}$ time

###  IV-B Problem Formulation

We formulate the signalized intersection corridor as a quasiperiodically
driven dynamical system. The underlying dynamics of the system are high
dimensional, and its governing equations are unknown. We hypothesize that the
signalized intersection corridor system obeys a dynamics of the form (2), and
the observed queue lengths are generated through some measurement function
$Y$, as described in Section II-A. Note that the measurement $Y$ is not
necessarily one-to-one. In particular, it may not be possible to connect the
$y_{i}$ with a dynamical rule of the form $y_{i+1}=\tilde{F}(y_{i})$. Rather
$y_{i}$ should be interpreted as a partial observation of the true state in
$\Omega$, in the $i$-th time frame. In Figure 1 we show the visual
representation of $y_{i}$. We use only this data to obtain a parameter-free
reconstruction of the dynamical system.

###  IV-C Computation

We arrange the data in the form of a matrix $\left[Y\right]$ with $k=9$
columns. Each column corresponds to the traffic queue length as a function of
time, at one among $9$ intersections along Alfaya Trail. We used a bandwidth
parameter of $\epsilon=0.1$ for the Gaussian kernel. We compute a total of
$L=1001$ eigenfunctions for the bistochastic kernel. We set $\epsilon_{1}=0.1$
and then choose the parameters $L_{0}=5$ and $\epsilon_{2}=2.5$ using the
heuristic approach shown in Figure 2. Lower the value of $\epsilon_{2}$, more
frequencies get filtered out and higher the probability of the identification
being correct. The two thresholds $\epsilon_{1}$ and $\epsilon_{2}$ are based
on the asymptotic behavior in two different directions [11, Theorem 1,4].
Combined, they provide a surer guarantee of identification of true
eigenfrequencies and the discarding of _spurious_ eigenvalues or _pseudo_
-spectrum.

Figure 2:  Choice of the thresholds $L_{0},\epsilon_{2}$ for the traffic
intersection data. $L_{0}$ was chosen to be $100$, as it corresponds to the
approximate location of an abrupt jump in the curve in the left panel. Given
this choice of $L_{0}$, we plotted the quantity
$\ln\left(W(n,l)/W(n,L_{0})\right)$ as a function of $n\in 0,\ldots,N/2$. We
choose $\epsilon_{2}$ to be $2.5$, which is approximately the first of the two
points of inflection of the curve in the red panel, as indicated by the
horizontal red and blue lines.

###  IV-D Results

####  IV-D1 Generating Frequencies

This work analyzed $20,000$ snapshots of queue lengths for the nine
intersections to extract Koopman frequencies, i.e., generating frequencies of
the quasiperiodic driving source of the dynamics. Figure 3 exhibits the
spectrum of Koopman eigenfrequencies. The $x$ axis denotes periods
corresponding to the frequencies. We separate longer periods from the shorter
ones and present them in two different panels for convenience of presentation.
For both the panels, the y-axis shows $\left[W\right]_{j,L_{0}}$ which we have
denoted as amplitude for the selected period indices $j$. Figure 3 shows that
dominant periods are clustered around 1 hour, 2 hours, 3 hours, 6 hours, 12
hours, 14 hours, 3.5 days, 7 days, and 14 days. The identified periods
correspond to the natural periods of the system. These periods are consistent
with the results in [21]. The authors decomposed speed measurements from an
intersection corridor via multiscale multifractal analysis (MMA). They
reported that the dominant periodicities on weekdays are 7 days, 24, 12, 8, 6,
and 3 hours while on weekends are 12 and 24 hours. In the present work, we did
not differentiate weekend data from weekday data. However, we took a different
set of observables (i.e., northbound through queue length data) and a
completely different intersection corridor system, our identified
quasiperiodic frequencies matched with that of [21]. This finding corroborates
that the proposed technique can successfully identify generating frequencies
of the quasiperiodic driving force of the intersection dynamical systems.

Figure 3:  Selected frequencies and time-periods.

####  IV-D2 Reconstruction and Prediction

We reconstruct the original data from the decomposed parts, which is shown in
Figure 4. The red curve is the output obtained from the reconstruction at each
intersection. The curves closely following each other, including the moments
when fluctuations occur. Although reconstructed dynamical models differ from
the true system, this difference is inevitable in a learning problem. However,
the reconstruction of $g_{per}$ is bounded, which guarantees that the dynamics
under (3) would remain bounded, and the deviation of the trajectories also
remain bounded. In Figure 5 we illustrate the error in reconstruction and
prediction by computing the normalized relative error

$\text{error}_{rel}^{(i)}(n):=\frac{\left|y_{n}^{(i)}-\hat{y}_{n}^{(i)}\right|}{\max_{n}\left|y_{n}^{(i)}\right|},i=1,\ldots
9.$ (16)

Here $\hat{y}_{n}^{(i)}$ denotes the output of the reconstructed system (15).

Figure 4: Dynamical reconstruction of NB queue lengths, shown for the first
$4$ intersections. Figure 5:  Dynamical reconstruction and relative prediction
error (16) of NB queue length data. Normalized error for intersections in
$n^{th}$ snapshot are indicated as $e^{1}(n),\ldots,e^{9}(n)$. The top two
plots shows a sudden and qualitative difference in the error beyond the
training period of $n=20000$. The bottom two plots focus on the prediction
performance beyond the training period. The system (15) initialized with the
state $y^{(Q)}_{t=2200}$, and iterated over the time window $n=[22000,26000]$.
An increased size of the moving average window reduces the effect of random
outliers or events and diminishes the prediction error. Number of snapshots
used to calculate moving averages in each cases is denoted as M.A and shown in
each plot.

##  V Conclusion

This work developed a data-driven framework for modeling quasi-periodic
dynamical systems using Koopman theoretic approach. The proposed approach can
handle dynamics with strong nonlinear and chaotic components. Thus, it is
applicable across domains. We performed a case study using queue length data
on a corridor of nine signalized intersections. The proposed approach
accurately identified the generating frequencies and the results for
reconstruction and prediction are encouraging. The long-term prediction error
remained bounded without exogenous inputs, unlike recurrent neural network-
based methods such as long short-term memory (LSTM). Moreover, in comparison
to deep NNs, the proposed technique is not a black-box approach. All these
advantages make it a promising candidate for future research.

## Acknowledgements

Authors would like to thank Dr. Samiul Hasan and his research group for
providing access to the intersection dataset.

Ran

pages1 Ranpages30 Ranpages4 Ranpages1 Ranpages19 Ranpages1 Ranpages8
Ranpages13 Ranpages35 Ranpages23 Ranpages72 Ranpages17 Ranpages32 Ranpages39
Ranpages38 Ranpages62 Ranpages14 Ranpages1

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