# REEXAMINATION OF THE HIERARCHY PROBLEM
FROM A NOVEL GEOMETRIC PERSPECTIVE

HIROFUMI SAKUMA† IZUMI OJIMA§<EMAIL_ADDRESS><EMAIL_ADDRESS>Research Origin for Dressed Photon,
Yokohama City, Kanagawa 221-0022, Japan 111Bldg.1-1F, 3-13-19 Moriya-cho,
Kanagawa-ku, Yokohama, Kanagawa, 221-0022 Japan
KAZUYA OKAMURA  College of Engineering
Chubu University Center for Mathematical Science and Artificial Intelligence
1200 Matsumoto-cho, Kasugai-shi, Aichi 487-8501, Japan
<EMAIL_ADDRESS>
((Day Month Year); (Day Month Year))

###### Abstract

A lucid interpretation of the longstanding hierarchy problem is proposed based
on the unconventional model of the universe recently proposed by the authors.
Our heuristic cosmological model is developed by considering Penrose and
Petit’s original ideas as the Weyl curvature hypothesis, conformal cyclic
cosmology, and the twin universe model. The uniqueness of our model lies in
its incorporation of dark energy and matter, and its single key parameter,
adjusted by observational data, is the radius ($R_{dS}$) of a four-dimensional
($4D$) hypersphere called de Sitter space. We presuppose that our $4D$
universe originated from the spontaneous conformal symmetry (SCS) breaking of
a light field with a null distance. We show that in this SCS breaking state,
the energy–momentum tensor of the space-like electromagnetic field, whose
existence is inevitable for quantum electromagnetic field interactions
(Greenberg–Robinson theorem), becomes isomorphic to the divergence-free
Einstein tensor in the general theory of relativity. Furthermore, we reveal
the $R_{dS}$ dependency of the $4D$ gravitational field. Based on these
findings, we show an intriguing relation between the magnitude of the
gravitational coupling constant and $R_{dS}$. A solution to the hierarchy
problem is derived by assuming that $R_{dS}$ depends on the “newly defined
cosmological time”.

###### keywords:

hierarchy problem; coupling constant; dark energy; dark matter; conformal
symmetry breaking, algebraic QFT, micro–macro duality, extra dimension.

##  1 Introduction

This study aims to shed new light on the unresolved hierarchy problem in
particle physics. The problem stems from the incredible difference existing
between, for instance, the magnitudes of electromagnetic and gravitational
coupling constants. This inspired the investigations of Dirac[1, 2], Milne[3],
and other researchers’ [4, 5, 6, 7], wherein they refute the hypothesis on the
invariance property of physical constants, particularly of gravitational
constant $G$. To understand the problem, the possibility of simultaneous
temporal change of all physical constants was examined first by Troitskii[8]
and explored further by Petit[9, 10]. At the end of the 20th century, an
entirely different approach to explaining the hierarchy problem was presented
by Randoll and Sundrum[11], who introduced a notion of warped extra dimensions
based on the brane dynamics in superstring theory. Notably, our model is
distinct from the abovementioned two in the sense that it incorporates dark
energy and matter as crucial components. However, our model inherits important
characteristics from those preceding researches. For instance, it addresses
the “temporal”evolution of the magnitude of the cosmological constant
associated with the extra expansion of the universe in the fifth dimension
($D$) perpendicular to our $4D$ universe.

The elucidation of new ideas is frequently accompanied by the dissemination of
important knowledge. Although such knowledge might be new or old, it is often
not widespread in the relevant community. In our case, we have three pieces of
such important knowledge: two of which, the Greenberg–Robinson (GR)
theorem[12, 13] and the micro–macro duality (MMD) theory of Ojima[14, 15], are
considered old information, whereas the Clebsch dual space-like
electromagnetic (CDSE) theory[16, 17, 18] built on the two preceding pieces of
knowledge is considered new knowledge. As demonstrated by the Schrödinger’s
cat, the prevailing understanding of the invisible microscopic quantum world
and its connection to the visible macroscopic world is unsatisfactory. The MMD
theory, built on the basic philosophy of the Araki–Haag–Kastler formulation of
the algebraic quantum field theory (QFT), is an ambitious scheme aiming to
provide a solution to the Schrödinger’s cat problem in quantum physics.
Notably, herein, the term “quantum physics” denotes relativistic quantum
physics unless otherwise stated.

In the field of quantum physics, the prevailing knowledge is quantum mechanics
(QM) with finite degrees of freedom. The knowledge required for this study is
not that on QM but on QFT with infinite degrees of freedom, whose dynamical
behaviors are considerably more complex than those of QM. Moreover, the
crucial difference between QM and QFT lies in the number of sectors, where a
sector is defined as the dynamical domain in which the principle of
superposition on the quantum states holds good. For a QM system, only one
sector exists (Stone–von Neumann theorem[19]), whereas for a system described
by QFT, multiple unitarily inequivalent sectors exist (initially highlighted
by van Hove in 1952). Initially, the abovementioned van Hove phenomena were
considered as the pathological characteristics of a QFT-based system with
infinite degrees of freedom. However, later, the existence of such multiple
sectors was determined to be the source of dynamical richness with which the
visible macroscopic world can emerge from the invisible quantum microscopic
world.

Ojima’s MMD theory was developed from the Doplicher–Haag–Roberts (DHR)’s
original sector theory[20, 21], which attempted to formulate the QFT based
solely on the maximal set of observables. In addition, Ojima developed the
theory further by generalizing it in such a way that it could cope with the
important physical mechanism of (spontaneous) symmetry breaking of given
quantum fields. Notably, in the MMD theory, the original sector notion based
on irreducible representations is revised into a generalized one with factor
representation. In the revised variant, the factor representation centers
feature attributes of either trivial or nontrivial, corresponding to
microquantum or macroclassical factor representations, respectively. In
addition, these generalized sectors are mutually disjoint (separated by the
absence of intertwiners), which is a stronger notion than unitary
inequivalence. Thus, the MMD theory became a comprehensive QFT, which can
consistently connect the quantum world, where the algebra of physical
quantities obeys the noncommutative law, to the classical world, where the
corresponding algebra is replaced by the one with commutative law. Thus, the
MMD theory embodies the mathematically rigorous quantum–classical
correspondence. A plain explanation of the essence of the MMD theory without
resorting to sophisticated mathematical representations has been provided by
Sakuma et al.[22].

Clarifying the mechanism of nonlinear field interactions would pose the
biggest dynamical challenge in QFT studies. As is well known, the essential
features of Fock spaces, such as the positive energy spectra in the state
vector space generated by repeated applications of the creation operators on
the Fock vacuum $|0\rangle$ (under the cyclicity assumption), are derived from
the following Eq. (1):

$\phi^{(f)}(x^{0},\tilde{x})=\int\frac{d^{3}\tilde{k}}{\sqrt{(2\pi)^{3}2E_{k}}}[a(\tilde{k})\exp{(-ik_{\nu}x^{\nu})}+a^{\dagger}(\tilde{k})\exp{(ik_{\nu}x^{\nu})}],$
(1)

where ($a^{\dagger}(\tilde{k})$, $a(\tilde{l})$) and ($\tilde{x}$ and
$\tilde{k}$), respectively, denote a pair of creation–annihilation operators
and of three vectors comprising spatial components of $x^{\mu}$ and $k_{\nu}$,
with $E_{k}$ defined by $E_{k}:=\sqrt{(\tilde{k})^{2}+(m_{0})^{2}}$. Although
the $\phi^{f}$ field thus constructed embodies the wave–particle duality of a
quantum system, the $\phi^{f}$ field and the associated 4-momentum $p^{\mu}$
are, respectively, restricted by the following linear Klein–Gordon (KG)
equation and the on-shell condition of the form

$[\hbar^{2}\partial^{\nu}\partial_{\nu}+(m_{0}c)^{2}]\phi^{(f)}=0,\;\;\;\;p_{\nu}p^{\nu}=(m_{0}c)^{2}\geq
0,$ (2)

where the sign convention of $(+,-,-,-)$ is used for the Lorentzian metric
tensor. Regarding these expressions, however, with the linearity of Eq. (2)
overlooked, the abovementioned characteristic features of Fock spaces are
frequently misinterpreted as the universal structure to be found in
interacting multiparticle systems. Accordingly, $|0\rangle$ becomes as
mysterious as the creation of everything from emptiness.

In the quest to clarify quantum field interactions, the GR theorem stands as a
noteworthy forerunner. In fact, it provides a clear-cut mathematical criterion
for distinguishing nonlinear quantum field interactions from the time
evolution of the free modes ($\phi^{(f)}$) with the second quantization
described in Eq. (1). The GR theorem states that if the Fourier transform
($\varphi(p)$) of a given quantum field ($\varphi(x)$) does not contain an
off-shell space-like momentum ($p_{\mu}$) satisfying $p_{\nu}p^{\nu}<0$, then
$\varphi(x)$ is a generalized free field. We believe that this theorem
highlights the missing factor in the presently incomplete QFT. In the
appendix, we provide a sketchy overview of the theorem derived based on
Wightman axioms. In the field of elementary particle physics, such off-shell
quantities are conventionally dismissed as nonphysical ghosts or extremely
short-lived transitory entities. Consequently, the wide physical science
communities have not come to an important research conclusion on off-shell
quantities in general, despite their importance.

In conventional quantum electrodynamics (QED), the longitudinal mode of a
field is eliminated as a nonphysical mode, although, classically, the Coulomb
mode plays an important role in electromagnetic field interactions. This well-
known fact highlights the limitation of the prevailing QED theory in affording
a satisfactory explanation of the consistent connection between the quantum
world and classical one. As the first step to a solution, based on the GR
theorem and MMD theory, the first author successfully extended a free
electromagnetic field into a space-like momentum domain [23]. As this
extension was achieved by introducing Clebsch variables [24] used in the study
of the Hamiltonian structure of barotropic fluid, we call the extended
electromagnetic field the Clebsch dual (CD) field. Notably, energy–momentum
tensor of the field assumes the same form as Einstein’s tensor in the theory
of general relativity. Namely, the CD field bears the characteristics of
spacetime itself in the sense of general relativity and can be shown to be a
promising candidate for understanding the enigmatic dark energy field [17]. As
the CD field plays an important role in our discussion of the hierarchy
problem, we will recapitulate its necessary points in Section 2.

As stated at the beginning of this section, we aim to present a novel view of
the hierarchy problem in Section 3. To do that, we think it is inevitable to
explore the different ideas published recently in our series of papers[17, 25,
26]. Accordingly, considering that majority of readers would be unfamiliar
with those ideas, in Section 2, we provide a bird’s eye view of the grand
design of our research. The overview encompasses all individual aspects
necessary in the pursuit of what we call “off-shell science”, originating from
dressed photon (DP) studies in the field of nanophotonics. Thereafter, in
Section 4, we highlight an intriguing finding on the meaning of Maldacena
duality[27] in superstring theory, which applies to the early stages of our
new model of the universe. Finally, the summary and conclusions are presented
in Section 5.

##  2 Overview of our ongoing research on reformulating the quantum field
theory

As stated in the preceding section, one of the focal points of our research on
the extended light field is off-shell quantum fields involving quantum field
interactions. The existence of off-shell (superluminal wave-) momenta implies,
as in the case of quantum spin entanglement, an instantaneous connectedness of
the entire universe. This naturally leads us to some cosmological problems
that are not necessarily within the scope of this study. As cosmology remains
a vast, unknown frontier in the physical sciences, in dealing with mysterious
cosmological problems, we adopt a simple and effective approach, the Occam’s
razor principle.

In line with studying the extended light field, we use a unique hypothesis.
The hypothesis is similar to the conformal cyclic cosmology (CCC) concept
proposed by Penrose[28], whose characteristics were explained in Sakuma and
Ojima[25], and hereinafter, it will be referred to as CCC of the second kind
(CCC-2nd). According to it, as in the case of the creation and annihilation of
the matter and antimatter pair, the twin universes as metric spacetimes are
birthed by a certain kind of spontaneous conformal symmetry breaking of a
light field with null distance ($ds^{2}=0$). The twin universes thus created
are divided by an event horizon intrinsically embedded in the de Sitter space
structure, caused by the most dominant component of dark energy. Eons later,
the twins meet at the event horizon to return to the original light field, and
this cycle repeats forever. In our cosmological hypothesis, the flatness,
isotropy, and horizon problems are resolved, respectively, by the observed
existence ratio of dark energy to matter, the Weyl curvature hypothesis
proposed by Penrose[29], and the existence of a superluminous off-shell
electromagnetic field.

In this section, we first explain the emergence of a space-like domain known
as de Sitter space mentioned above, which is necessary for quantum field
interactions. The emergence mechanism of the familiar time-like domain to
which not only the classical but also partial quantum fields belong must be
quite different from that of the space-like domain. Second, we provide a clear
explanation of the mechanism based on our new interpretation of conformal
gravity.

###  2.1 Conformal symmetry breaking and emergence of spacetime

Concerning the symmetry breaking of quantum fields, the MMD theory states that
the symmetric space ($G/H$) studied by Cartan[30], which has various desirable
properties, emerges in the generalized sector classifying space if the
symmetry of a given system, described by a Lie group ($G$), is broken up into
its subgroup, $H$, remaining unbroken. A concrete example of this case will be
explained in this subsection concerning the emergence of spacetime, not as a
purely mathematical concept having nothing to do with physics, but as a
physical entity. Recall that $4D$ spacetime occupies a special position in
differential topology. A most notable example would be the one highlighted by
Donaldson [31]. In the context of our present research on QFT, the following
properties of $4D$ spacetime are deemed important:
(i) the free Maxwell equation on electromagnetism is scale free only in $4D$
spacetime and (ii) the Weyl (conformal) curvature can be defined in spacetimes
with dimensions larger than or equal to 4.

If we combine the statement of the MMD theory on an emerging symmetric space
($G/H$) with property (i), i.e., the free Maxwell field that can play the role
of $H$, then one of the assertions of our cosmological theory (CCC-2nd)
proposing a $4D$ spacetime theory is restated as follows: the universe as a
symmetric space ($G/H$) emerges as the conformal symmetry breaking of the $H$
field with $H$ being an important remaining unbroken subgroup. In our series
of papers cited in Section 1, we show how space- and time-like domains of
$G/H$ emerge from $H$. Next, we briefly recapitulate the main points of those
emergence processes, starting from the simple space-like case.

####  2.1.1 Emergence of the space-like domain

The energy–momentum tensor ($T_{\mu}^{\;\;\nu}$) of the free Maxwell field
assumes the form of $T_{\mu}^{\;\;\nu}=F_{\mu\sigma}F^{\nu\sigma}$ with
$F_{\mu\sigma}=\nabla_{\mu}A_{\sigma}-\nabla_{\sigma}A_{\mu}$, where
$\nabla_{\mu}$ denotes the covariant derivative and other notations are
conventional. As is well known, this free field is characterized by a couple
of conditions:

$F_{\nu\sigma}F^{\nu\sigma}=0\;\;\;\;\textnormal{and}\;\;\;\;F_{01}F_{23}+F_{02}F_{31}+F_{03}F_{12}=0.$
(3)

In extending this null field to the space-like momentum domain, focusing on
the properties of the Riemann curvature tensor ($R_{\alpha\beta\gamma\delta}$)
and the mixed form of the Ricci tensor ($R_{\mu}^{\;\;\nu}$), as shown below,
is quite helpful:

$\displaystyle R_{\beta\alpha\gamma\delta}$ $\displaystyle=$ $\displaystyle-
R_{\alpha\beta\gamma\delta},\;\;R_{\alpha\beta\delta\gamma}=-R_{\alpha\beta\gamma\delta},\;\;R_{\gamma\delta\alpha\beta}=R_{\alpha\beta\gamma\delta},$
(4) $\displaystyle R_{\alpha\beta\gamma\delta}$ $\displaystyle+$
$\displaystyle
R_{\alpha\gamma\delta\beta}+R_{\alpha\delta\beta\gamma}=0,\;\;R_{\mu}^{\;\;\nu}:=R_{\mu\sigma}^{\;\;\;\;\nu\sigma}.$
(5)

Here, the first equation in Eq. (5) is called the first Bianchi identity,
which is closely related to the second equation in Eq. (3), namely, the
orthogonality of electric and magnetic fields.

In relativistic fluid dynamics, if the vorticity tensor field
($\omega_{\mu\nu}$) satisfies the second equation in Eq. (3), then the fluid
is classified as a barotropic (isentropic) fluid (cf. [26]). This observation
motivates us to use Clebsch parameterization (CP) to represent the
electromagnetic vector potential ($U_{\mu}$) in space-like momentum domains.
We can achieve this because the rotational part of the barotropic fluid
velocity ($u_{\mu}$) can be represented by two Clebsch parameters, $\lambda$
and $\phi$, to give $u_{\mu}=\lambda\nabla_{\mu}\phi$ [24]. Notably, $\lambda$
and $\phi$ serve as the canonically conjugate variables for the Hamiltonian
structure of the barotropic fluid motions. In our case, the two gradient
vectors, $C_{\mu}:=\nabla_{\mu}\phi$ and $L_{\mu}:=\nabla_{\mu}\lambda$, with
the orthogonality condition, $C^{\nu}L_{\nu}=0$, provide a dynamical basis on
which electric and magnetic fields can be successfully introduced into the
space-like momentum domain.

To understand the meaning of CP, we first explain the light-like case to
compare it with the conventional quantization of the light field. In the
second part on the pure space-like case, we will show that the energy–momentum
tensor ($\hat{T}_{\mu}^{\;\;\nu}$) in the extended space-like domain becomes
isomorphic to the Einstein tensor in the theory of general relativity. This
will reveal that the extended electromagnetic field automatically possesses
the property of spacetime through which quantum electromagnetic field
interactions occur (GR theorem). Namely, this spacetime, including the light-
like case, plays the role of “virtual photons” in the conventional QED.

Light-like case: $U^{\nu}(U_{\nu})^{*}=0$, where $(\cdot)^{*}$ denotes the
complex conjugate of $(\cdot)$.
For Clebsch variables $\lambda$ and $\phi$ satisfying

$\displaystyle\nabla^{\nu}\nabla_{\nu}\lambda-(\kappa_{0})^{2}\lambda$
$\displaystyle=$ $\displaystyle
0,\;\;L_{\mu}=\nabla_{\mu}\lambda,\;\;\;\;\nabla^{\nu}\nabla_{\nu}\phi=0,\;\;C_{\mu}=\nabla_{\mu}\phi,$
(6) $\displaystyle C^{\nu}(L_{\nu})^{*}$ $\displaystyle=$ $\displaystyle
0,\;\;U_{\mu}:=\lambda C_{\mu},$ (7)

where $(\kappa_{0})^{-1}$ is DP constant $l_{dp}\approx 50$ nm, determined by
Ohtsu[17], we have

$\displaystyle
U^{\nu}\nabla_{\nu}U_{\mu}=0,\;\;S_{\mu\nu}:=\nabla_{\mu}U_{\nu}-\nabla_{\nu}U_{\mu}=L_{\mu}C_{\nu}-L_{\nu}C_{\mu},$
(8) $\displaystyle\hat{T}_{\mu}^{\;\;\nu}=S_{\mu\sigma}S^{\nu\sigma}=\rho
C_{\mu}C^{\nu},\;\;\rho:=L^{\nu}(L_{\nu})^{*}<0,\;\;\nabla_{\nu}\hat{T}_{\mu}^{\;\;\nu}=0.$
(9)

Notice that $\hat{T}_{\mu}^{\;\;\nu}$ has a dual wave
($S_{\mu\sigma}S^{\nu\sigma}$) and particle ($\rho C_{\mu}C^{\nu}$)
representation. The latter is consistent with the conventional quantization of
QED where this mode is eliminated from the physically meaningful domain as the
particle density ($\rho$) in Eq. (9) becomes negative. As referenced in the
introduction, in re-examining the Nakanishi–Lautrup formalism of the abelian
gauge theory, Ojima[32] highlighted that the nonparticle form corresponding to
($S_{\mu\sigma}S^{\nu\sigma}$) in our model plays substantial physical roles,
which is consistent with the GR theorem.

Space-like case: $U^{\nu}(U_{\nu})^{*}<0$.
In this case, we modify Eq. (6) as follows:

$\displaystyle\nabla^{\nu}\nabla_{\nu}\lambda-(\kappa_{0})^{2}\lambda$
$\displaystyle=$ $\displaystyle
0,\;\;\nabla^{\nu}\nabla_{\nu}\phi-(\kappa_{0})^{2}\phi=0,\;\;C^{\nu}(L_{\nu})^{*}=0,$
(10) $\displaystyle U_{\mu}:$ $\displaystyle=$ $\displaystyle(\lambda
C_{\mu}-\phi L_{\mu})/2.$ (11)

By doing so, the associated energy–momentum tensor ($\hat{T}_{\mu}^{\;\;\nu}$)
now becomes

$\hat{T}_{\mu}^{\;\;\nu}=S_{\mu\sigma}S^{\nu\sigma}-\frac{1}{2}g_{\mu}^{\;\;\nu}S_{\sigma\tau}S^{\sigma\tau}.$
(12)

Considering the definition of the Ricci tensor ($R_{\mu}^{\;\;\nu}$) in Eq.
(5), we see that Eq. (12) is isomorphic to the Einstein tensor
($G_{\mu}^{\;\;\nu}=R_{\mu}^{\;\;\nu}-g_{\mu}^{\;\;\nu}R/2$).

As Dirac’s equation ($(i\gamma^{\nu}\partial_{\nu}+m)\Psi=0$) is the square
root of the time-like KG equation
($(\partial^{\nu}\partial_{\nu}+m^{2})\Psi=0$), the quantum field
($\Psi_{(M)}$) that satisfies the square root of the space-like KG equation
($(\partial^{\nu}\partial_{\nu}-(\kappa_{0})^{2})\Psi=0$) is shown to be the
electrically neutral Majorana fermionic field
({$(\gamma^{\nu}_{(M)}\partial_{\nu}-\kappa_{0})\Psi_{(M)}=0$}). As the space-
like $\Psi_{(M)}$ represents a nonparticle mode, we denote it as the Majorana
fermionic field instead of the Majorana fermion. Owing to Pauli’s exclusion
principle, $\Psi_{(M)}$ with a half-integer spin of 1/2 cannot occupy the same
state. A possible configuration where a couple of $\Psi_{(M)}$s form a bosonic
$S_{\mu\nu}$ field can be identified using the Pauli–Lubanski 4-vector
($W_{\mu}$), which describes the spin states of moving particles.

$W_{\mu}=\frac{1}{2}\epsilon_{\mu\nu\lambda\sigma}M^{\nu\lambda}p^{\sigma},$
(13)

where $\epsilon_{\mu\nu\lambda\sigma}$ denotes the $4D$ totally antisymmetric
Levi–Civita tensor, $M^{\nu\lambda}$ and $p^{\sigma}$ are angular and linear
momenta, respectively. As Eq. (13) is rewritten as

$\displaystyle\left(\begin{array}[]{l}W_{0}\\\ W_{1}\\\ W_{2}\\\
W_{3}\end{array}\right)=\left(\begin{array}[]{cccc}0&M^{23}&M^{31}&M^{12}\\\
-M^{23}&0&M^{03}&-M^{02}\\\ -M^{31}&-M^{03}&0&M^{01}\\\
-M^{12}&M^{02}&-M^{01}&0\end{array}\right)\left(\begin{array}[]{r}p^{0}\\\
p^{1}\\\ p^{2}\\\ p^{3}\end{array}\right),$ (26)

we can see that two different fields, $(M^{\mu\nu},p^{\mu}$) and
($N^{\mu\nu},q^{\mu}$), which satisfy the orthogonality condition
($p^{\nu}q_{\nu}=0$, corresponding to $C^{\nu}(L_{\nu})^{*}=0$ in Eq. (10)),
can share the same $W_{\mu}$ and hence combine to form a spin 1 bosonic field,
$S_{\mu\nu}$.

The CDSE field ($S_{\mu\nu}$) has another noteworthy geometric feature, which
is particularly important for its application in cosmology. A plane wave of
the form, $\psi=\hat{\psi}_{c}\exp{[i(k_{\nu}x^{\nu})]}$, satisfies the
equation of a couple of $\lambda$ and $\phi$ fields in Eq. (10) with
$\partial^{\nu}\psi(\partial_{\nu}\psi)^{*}=-(\kappa_{0})^{2}[\hat{\psi}_{c}(\hat{\psi}_{c})^{*}].$
Therefore, for vector $L_{\mu}$, we have

$L^{\nu}(L_{\nu})^{*}=-(\kappa_{0})^{2}[\hat{\lambda}_{c}(\hat{\lambda}_{c})^{*}]=const.<0,$
(27)

whose structure resembles that of the tangent vector on de Sitter space.
Notably, the importance of this space in the context of spacetime quantization
was first noted by Snyder[33]. A pseudo-hypersphere with radius $R_{dS}$
embedded in $R^{5}$ is called de Sitter space, which is represented as

$(\eta_{0})^{2}-(\eta_{1})^{2}-(\eta_{2})^{2}-(\eta_{3})^{2}-(\eta_{4})^{2}=-(R_{dS})^{2}=-3(\Lambda_{dS})^{-1}.$
(28)

Here, $3(\Lambda_{dS})^{-1}:=(R_{dS})^{2}$ is a frequently employed
alternative expression of $R_{dS}$. Snyder showed that the spacetime
($x^{\mu}$) together with the 4-momentum ($p_{\mu}$), defined as

$p_{\mu}=\frac{\hbar}{l_{p}}\frac{\eta_{\mu}}{\eta_{4}},\;\;(0\leq\mu\leq
3),\;\;p^{\nu}p_{\nu}=\left(\frac{\hbar}{l_{p}}\right)^{2}\left[1-\left(\frac{R_{dS}}{\eta_{4}}\right)^{2}\right]<0,$
(29)

can be quantized without breaking Lorentz’s invariance, where $l_{p}$ denotes
Planck’s length. We have already seen that an isomorphism exists between the
Einstein tensor ($G_{\mu}^{\;\;\nu}$) and ($\hat{T}_{\mu}^{\;\;\nu}$) in Eq.
(12) whose r.h.s. can be quantized with the Majorana field. Thus, we
conjecture that the spacetime quantization scheme of Snyder is consistent with
the quantization of $\hat{T}_{\mu}^{\;\;\nu}$. Eq. (27 and 29) show that
$L_{\mu}$ and $p_{\mu}$ are on the submanifold of de Sitter space,
parameterized by $\eta_{4}=const.$

As nonlinear electromagnetic field interactions are ubiquitous, the incessant
occurrence of instantaneous excitation–deexcitation processes in the
$S_{\mu\nu}$ field must be prevalent in the universe. In such processes, we
can show that a unique state exists, $|M3\rangle_{g}$, of $S_{\mu\nu}$, which
behaves as if it is “the ground state”of $S_{\mu\nu}$ in the sense that
$|M3\rangle_{g}$ is occupied by $S_{\mu\nu}$ at every moment. To observe this
phenomenon, we consider Eq. (13) again. As the spatial dimension of our
spacetime is three, the maximum number of space-like momentum vectors
satisfying the orthogonality condition (10): ($p^{\nu}q_{\nu}=0$), is also
three. That is, for three different sets of ($M^{\mu\nu},\;p^{\mu}$),
($N^{\mu\nu},\;q^{\mu}$), and ($L^{\mu\nu},\;r^{\mu}$) with
$p^{\nu}q_{\nu}=0;\;p^{\nu}r_{\nu}=0;\;q^{\nu}r_{\nu}=0$, the following unique
spin vector configuration exists:

$2W_{\mu}=\epsilon_{\mu\nu\lambda\sigma}M^{\nu\lambda}p^{\sigma}=\epsilon_{\mu\nu\lambda\sigma}N^{\nu\lambda}q^{\sigma}=\epsilon_{\mu\nu\lambda\sigma}L^{\nu\lambda}r^{\sigma}.$
(30)

This indicates the existence of a compound state of the Majorana fermionic
field with spin 3/2, called the Rarita–Schwinger state ($|M3\rangle_{g}$).
Notice that a bosonic field, $S_{\mu\nu}$, can be constructed by any pair in
the set of {($M^{\mu\nu},\;p^{\mu}$), ($N^{\mu\nu},\;q^{\mu}$), and
($L^{\mu\nu},\;r^{\mu}$)} mentioned above. In our scenario of the
electromagnetic field interactions in which the ubiquitous field interactions
induce the incessant occurrence of excitation–deexcitation processes, the
$|M3\rangle_{g}$ state behaves like “the ground state”of $S_{\mu\nu}$ in the
sense that any vacant configuration of $S_{\mu\nu}$, if it exists, must be
reoccupied in a moment. This makes $|M3\rangle_{g}$ a constantly occupied
state from a macroscopic timescale.

The key question regarding $|M3\rangle_{g}$ is whether it is an observable
quantity in either a direct or an indirect fashion as $S_{\mu\nu}$, as a
space-like quantity, is not observable in general. First, we can say that,
within the framework of relativistic QFT, any observable, without exception,
associated with the given internal symmetry is the invariant under the action
of the transformation group materializing the symmetry under consideration.
The second helpful observation is that, as was highlighted after Eq. (29),
Lorentz’s invariance still holds for Snyder’s quantum version of the
isomorphism between Einstein tensor $G_{\mu}^{\;\;\nu}$ and the Majorana
version of $\hat{T}_{\mu}^{\;\;\nu}$ given in Eq. (12). This is in addition to
the fact that Lorentz’s invariance, indicating an external symmetry, is
related to the internal ones through supersymmetry. Based on these
observations, we assume that
{$|M3\rangle_{g}:=\Sigma_{(i=1)}^{(3)}\hat{T}_{\nu}^{\;\;\nu}(x^{i})\rightarrow
R$}, as the invariant of the general coordinate transformations, is an
observable as it is indirectly related to the actual observable quantity,
i.e., the expansion rate of the universe.

The simplest model of dark energy is expressed as a cosmological term,
$\Lambda_{(de)}g_{\mu\nu}$, with $\Lambda_{(de)}<0$ in the following Einstein
field equation, with the sign convention $R>0$ for a matter-dominated closed
universe:

$R_{\mu}^{\;\;\nu}-\frac{R}{2}g_{\mu}^{\;\;\nu}+\Lambda_{(de)}g_{\mu}^{\;\;\nu}=-\frac{8\pi
G}{c^{4}}T_{\mu}^{\;\;\nu}.$ (31)

From the discussions developed thus far, we can identify observable
$\Lambda_{(de)}$ as

$\Lambda_{(de)}=|M3\rangle_{g}=\Sigma_{(i=1)}^{(3)}\hat{T}_{\nu}^{\;\;\nu}(x^{i})<0.$
(32)

Recall that the key parameter ($\kappa_{0}$) in Eq. (6) for determining
Clebsch variables $\lambda$ and $\phi$ is the DP constant, which has been
experimentally determined. Using the experimental value of
$(\kappa_{0}^{-1})\approx 50$ nm, we obtain $\Lambda_{(de)}=-2.47\times
10^{-53}m^{-2}$. Conversely, the value of $\Lambda_{obs}$ derived by Planck
satellite observations [34] is estimated as $\Lambda_{(obs)}\approx-3.7\times
10^{-53}m^{-2}$.

####  2.1.2 Emergence of the time-like domain

As shown above, the CP used to develop our dark energy model as a space-like
electromagnetic field is an analytic approach that sheds light on the
characteristics of barotropic (isentropic) fluid motions. Next, we show that
the knowledge on fluid dynamics is again quite helpful in identifying a
promising dark matter model, which can be derived by considering baroclinic
(nonisentropic) fluid motions. For clarity, we begin with the definitions of
barotropic and baroclinic fluids. For simplicity, let us consider the
nonrelativistic equation of motion of ideal gas flows:

$\partial_{t}v_{\mu}+v^{\nu}\partial_{\nu}v_{\mu}=-\partial_{\mu}p/\rho=-C_{p}\partial_{\mu}T+T\partial_{\mu}s,$
(33)

where $v_{\mu}$ denotes the $3D$ velocity field; $p$, $\rho$, $T$, and $s$
denote pressure, density, absolute temperature, and specific entropy,
respectively; and $C_{p}$ denotes specific heat at constant pressure. A given
fluid motion is barotropic if $\partial_{\mu}s=0$ (namely isentropic);
otherwise, it is baroclinic. A particularly important conserved quantity for
the general baroclinic case is Ertel’s potential vorticity ($Q$[35]), which
has the following form:

$Q:=\frac{1}{\rho}(\vec{\nabla}\times\vec{v})\cdot\vec{\nabla}s,\;\;\Longrightarrow\;\;\partial_{t}Q+v^{\nu}\partial_{\nu}Q=0.$
(34)

Notice that $Q$ can be adopted to label the fluid particles defined in the
Lagrangian specification of the flow field. Alternatively, $Q$ may play the
role of the “physical coordinates”of the space in which a given baroclinic
fluid system undergoes time evolution.

To clarify the underlying mechanism of the interpretation of dark matter in
terms of $Q$, Sakuma et al.[26] examined the novel relativistic representation
($\Omega_{T}$) of $Q$. They found that under the assumption of a low energy
limit within which the conservation of the fluid particle number ($n$) having
the form of $\nabla_{\nu}(nu^{\nu})=0$ holds,

$\displaystyle\Omega_{T}:=\Omega/T,\;\;\Omega:=\omega_{01}\omega_{23}+\omega_{02}\omega_{31}+\omega_{03}\omega_{12},$
(35)
$\displaystyle\Omega_{T}u^{\mu}=\;\nabla_{\nu}[^{*}(\omega^{\mu\nu})(\sigma/n)],\;\;\Longrightarrow\;\;\nabla_{\nu}(\Omega_{T}u^{\nu})=0.$
(36)

Here, $u^{\mu}$, $\omega_{\mu\nu}$, ${}^{*}\omega^{\mu\nu}$, $T$, $\sigma$,
and $n$ are the nondimensional 4-velocity vector, relativistic vorticity
tensor, Hodge’s dual of $\omega^{\mu\nu}$, absolute temperature, specific
entropy, and particle number per unit volume, respectively. Furthermore, the
current ($\Omega_{T}u^{\mu}$) is identified as gravitational entropy current.
In additional, the conservation law given in the second equation in Eq. (36)
shows that the first equation may be considered as “Maxwell’s equation” in
which $\Omega_{T}u^{\mu}$ plays the role of the “electric current”. They
further derived the following important relations valid among metric tensor
$g^{\mu\nu}$, Weyl curvature tensor $W^{\alpha\beta\gamma\delta}$, and
relativistic vorticity tensor $\omega^{\mu\nu}$ under the assumption of
$W^{2}\Omega^{2}\neq 0$

$\displaystyle
g^{\mu\nu}=\frac{W^{\mu\alpha\beta\gamma}W^{\nu}_{\;\;\alpha\beta\gamma}}{W^{2}/4}$
$\displaystyle=$
$\displaystyle\frac{{}^{*}{\omega}^{\mu\sigma}(^{*}{\omega}^{\kappa\lambda})\omega^{\nu}_{\;\;\sigma}\omega_{\kappa\lambda}}{(^{*}{\omega}^{\kappa\lambda}\omega_{\lambda\kappa})^{2}/4}=\frac{{}^{*}{\omega}^{\mu\sigma}(^{*}{\omega}^{\kappa\lambda})\omega^{\nu}_{\;\;\sigma}\omega_{\kappa\lambda}}{(4\Omega)^{2}/4},$
(37) $\displaystyle W^{2}:$ $\displaystyle=$ $\displaystyle
W^{\alpha\beta\gamma\delta}W_{\alpha\beta\gamma\delta},\;\;\Omega:=\;^{*}{\omega}^{\kappa\lambda}\omega_{\lambda\kappa}.$

These suggest the existence of the minimum values of $(W_{0})^{2}$ and
$(\Omega_{0})^{2}$. In general, $g^{\mu\nu}$ is a purely mathematical concept
as it is dependent on the choice of the coordinates. However, Eq. (37) assigns
a unique physical characteristic to $g^{\mu\nu}$ such that the emergent
symmetric space ($G/H$) in our cosmological theory, as a metric space, assumes
the form of a certain kind of spin-network to which the skew-symmetric
properties of Eq. (4) are reflected. In relativistic dynamics, mass
distribution is directly related to the distribution of scalar curvature
$R_{\nu}^{\;\;\nu}$. By contrast, from Eq. (37), we observe that $W^{2}$ is
directly correlated with $(4\Omega)^{2}$ and that
$\Omega_{T}u^{\mu}=(\Omega/T)u^{\mu}$ behaves like a conserved density
current. As the pure Weyl curvature represents the vacuum of relativistic
dynamics, a relatively strong current ($\Omega_{T}u^{\mu}$) in a cosmological
environment with a negligible value of $R_{\nu}^{\;\;\nu}$ compared with
$\Omega_{T}$ would behave like the invisible density current, which is the
model of dark matter in our theory. Using Aoki et al.’s[36] recent research
outcomes on relativistic conserved charges and entropy current, Sakuma et
al.[26] also showed that the entropy current $\Omega_{T}u^{\mu}$ is
dynamically associated with an energy–momentum tensor of the form, $\lambda
g_{\mu\nu}$, with $\lambda$ being a positive constant, where $g_{\mu\nu}$ is
to be interpreted by Eq. (37).

Thus, a series of the above analyses on $\Omega_{T}u^{\mu}$ suggests that the
energy–momentum tensor of dark matter assumes the form of
$\Lambda_{(dm)}g_{\mu\nu}$, where constant $\Lambda_{(dm)}$ is to be
determined by observations. The consensus ranges of the estimated percentage
of dark energy and dark matter are (68%–76%; mean = 72%) and (20%–28%; mean =
24%), respectively. Thus, the following equation:

$\Lambda_{(dm)}\approx\hat{\Lambda}_{(de)}:=-\Lambda_{(de)}/3=\Lambda_{dS}/3=1/(R_{dS})^{2},$
(38)

is an estimate consistent with the observation, where $R_{dS}$ and
$\Lambda_{dS}/3$ denote the radius of de Sitter space and its alternative
expression defined in Eq. (28).

Notably, in our cosmological scenario of CCC-2nd, $\hat{T}_{\mu}^{\;\;\nu}$ in
Eq. (12), shown to be isomorphic to the space-like Einstein tensor
($G_{\mu}^{\;\;\nu}$) and $\Lambda_{(dm)}g^{\mu\nu}$ with $g^{\mu\nu}$ given
by Eq. (37), provide the physical space- and time-like spacetimes,
respectively. These spacetimes emerged from the spontaneous symmetry breaking
of the primordial light field with null distance ($ds^{2}=0$). To consolidate
our CCC-2nd scenario here, we show that a unique vector boson exists in our
model, which plays the role of the Nambu–Goldstone boson (NGB) associated with
the spontaneous symmetry-breaking process.

To start, we consider again the space-like CP of $U^{\mu}$ explained in Eq.
(10, 11). The field strength ($S_{\mu\nu}$) defined by the curl of $U_{\mu}$
assumes the same form as the one in the light-like case given in the second
equation in Eq. (8). Contrarily, the null geodesic equation
($U^{\nu}\nabla_{\nu}U_{\mu}=0$) in Eq. (8) is replaced by

$U^{\nu}\nabla_{\nu}U_{\mu}=-S_{\mu\nu}U^{\nu}+\nabla_{\nu}(U^{\nu}U_{\nu}/2)=0;\;\;U^{\nu}U_{\nu}<0.$
(39)

In relativistic fluid dynamics, the magnitude of $U^{\mu}$, defined by
$V:=U^{\nu}U_{\nu}/2$, can be normalized as $V=1$[37]. As the equations on
$\lambda$ and $\phi$ in Eq. (10) are linear, we can introduce a similar
normalization for $L_{\mu}=\nabla_{\mu}\lambda$ and
$C_{\mu}=\nabla_{\mu}\phi$. The natural normalization would be

$L^{\nu}L_{\nu}=-(m_{\lambda})^{2}\lambda^{2},\;\;C^{\nu}C_{\nu}=-(m_{\phi})^{2}\phi^{2},$
(40)

where the wavenumber vector, $k^{\mu}$, of the respective plane-wave solutions
satisfies $k^{\nu}k_{\nu}=-(m_{\lambda})^{2}$ and
$k^{\nu}k_{\nu}=-(m_{\phi})^{2}$. For the space-like CP of $U^{\mu}$ discussed
thus far, we have

$(m_{\lambda})^{2}=(m_{\phi})^{2}=(\kappa_{0})^{2}.$ (41)

Using the orthogonality condition, $C^{\nu}(L_{\nu})^{*}=0$, in Eq. (10) and
Eq. (40), a couple of important characteristics of space-like $U^{\mu}$ fields
can readily be expressed as

$\nabla_{\nu}U^{\nu}=0,\;\;\;V=-\frac{1}{8}\lambda^{2}\phi^{2}[(m_{\lambda})^{2}+(m_{\phi})^{2}].$
(42)

Now, revisiting Eq. (10), we consider a different case in which we only
replace the second equation,
$\nabla^{\nu}\nabla_{\nu}\phi-(\kappa_{0})^{2}\phi=0$, with
$\nabla^{\nu}\nabla_{\nu}\phi+(\kappa_{0})^{2}\phi=0$. With this change, Eq.
(41) changes into $(m_{\lambda})^{2}=(\kappa_{0})^{2}=-(m_{\phi})^{2}$.
Consequently, $V$ in Eq. (42) vanishes, and $U^{\mu}$ becomes a null vector.
Notably, the form of $S_{\mu\nu}U^{\nu}$ in Eq. (39) is similar to that of
Lorentz force, $F_{\mu\nu}(ev^{\nu})$, where $F_{\mu\nu}$ and $ev^{\nu}$
denote the background electromagnetic field and an electric current with
charge $e$, respectively. A direct expression of $-S_{\mu\nu}U^{\nu}$ is

$\displaystyle-S_{\mu\nu}U^{\nu}$ $\displaystyle=$
$\displaystyle-(L_{\mu}C_{\nu}-L_{\nu}C_{\mu})(\lambda C^{\nu}-\phi
L^{\nu})/2$ (43) $\displaystyle=$
$\displaystyle-\lambda\phi(\kappa_{0})^{2}(\phi L_{\mu}-\lambda
C_{\mu})/2=\lambda\phi(\kappa_{0})^{2}U_{\mu}.$

Conversely, as we have $\nabla_{\nu}U^{\nu}=-\lambda\phi(\kappa_{0})^{2}$, in
this case, Eq. (43) becomes

$S_{\mu\nu}U^{\nu}=-\lambda\phi(\kappa_{0})^{2}U_{\mu}=(\nabla_{\nu}U^{\nu})U_{\mu}\neq
0.$ (44)

Eq. (44) expresses that the null vector $U^{\mu}$ with a nonvanishing
irrotational part is a vector field for which $\kappa_{0}$ and $i\kappa_{0}$
play the role of the field source as in the case of $\pm e$ in
electromagnetism. Furthermore, the field source behaves as if it is an
“electrically charged virtual photon” responsible for nonlinear
electromagnetic field interactions. In our CCC-2nd scenario, the emergence of
a couple of these sources ($\kappa_{0}$ and $i\kappa_{0}$) is interpreted as
the consequence of the spontaneous symmetry breaking of the light vector field
whose temporal and spatial components exist in a balanced manner. Thus, we
believe that the vector field, $U^{\mu}$, with the abovementioned
characteristics plays the role of NGB in our cosmological scenario. Moreover,
it plays substantial roles in the nonlinear field interactions between the
quantum generalized sectors defined by Ojima[14], existing on not only time-
like but also space-like domains, and the classical generalized sectors on
time-like domains. In the subsequent section, we show that the DP constant
($l_{dp}\approx 50$ nm), defined as the inverse of $\kappa_{0}$, gives the
scale of the Heisenberg cut dividing the whole universe into micro-quantum and
macro-classical worlds (cf. Eq. (54)).

Interestingly, we can unravel many conceptual aspects of QFT from such a
specific theory as the description of the behaviors of DPs. Here, the
concentration of the momentum support of the dressed photon field in the
space-like domain ($p^{2}=p_{\mu}^{\;\;}p^{\mu}<0$) plays the most important
role, which is in sharp contrast to the time-like momenta owing to the
particle-like excitations. This contrast between the space- and time-like
momenta should not be superficially considered. However, the important factor
is the medium or environment constructed by the space-like momenta, in
contrast to the particle-like motions associated with the time-like momenta.
From this viewpoint, the distribution of different sizes of space-like momenta
($p^{2}<0$) describes the difference among different media, which constitutes
the categorical background of algebraic QFT constructed by arrows of space-
like momenta. Therefore, this kind of interpretation of space-like momenta
associated with dressed photons exhibits the categorical essence of QFT.

Along this line, we promote new research on the relevance of Kan extentions in
the QFT of dressed photons. This is applicable if the four terms constituting
MMD are recombined into the three terms of micro-dynamical systems (comprising
micro-dynamics acting on the micro-algebra) and macro-states, together with a
macro-classifying space constructed by the classifying parameters of macro-
states.

##  3 Approach to the hierarchy problem

The Einstein field equation incorporating our dark energy and matter model is
expressed as follows:

$\displaystyle
R_{\mu}^{\;\;\nu}-\frac{R}{2}g_{\mu}^{\;\;\nu}+\Lambda_{(dm)}g_{\mu}^{\;\;\nu}$
$\displaystyle=$ $\displaystyle-\frac{8\pi
G}{c^{4}}[T_{\mu}^{\;\;\nu}+\\{\Lambda_{(de)}g_{\mu}^{\;\;\nu}\\}],$ (45)
$\displaystyle\Lambda_{(dm)}$ $\displaystyle=$
$\displaystyle-\Lambda_{(de)}/3+\epsilon>0.$

Here, our present concern is the dark matter field,
$\Lambda_{(dm)}g_{\mu}^{\;\;\nu}$, as the main source of the gravitational
field. As the first step, by comparing Coulomb’s law with the universal law of
gravitation, we obtain

$F_{e}=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r^{2}},\;\;\;F_{g}=G\frac{m_{1}m_{2}}{r^{2}}.$
(46)

Let us examine the $F_{e}/F_{g}$ ratio, where the notations used in Eq. (46)
are conventional. By doing so, it would be natural to choose the fundamental
electric charge ($e$) for $q_{1}=q_{2}$. However, for $m_{1}=m_{2}$, we
encounter a serious challenge as we cannot single out “the fundamental mass
charge” like $e$ in the case of $F_{e}$. To overcome this problem, recall
first that, for our dark energy model, a unique state, $|M3\rangle_{g}$,
exists, which behaves like “the ground state” of $S_{\mu\nu}$. Through Eq.
(32) and Eq. (38), this “ground state” is directly related to
$\Lambda_{(dm)}$. By contrast, in Eq. (37), we highlight the existence of the
minimum value of $W^{2}$, i.e., $(W_{0})^{2}\neq 0$, associated with the
conformal symmetry breaking of the light field ($H$). Thus, we naturally
assume that $\Lambda_{(dm)}=|W_{0}|$. Notably, in our previous studies where
we did not consider the possibility of the “temporal change” of
$\Lambda_{(dm)}$ in the dark matter field ($\Lambda_{(dm)}g_{\mu\nu}$), we
simply asserted $\Lambda_{(dm)}=W_{0}=const$. However, in the present
discussion where we consider the “temporal change” of $\Lambda_{(dm)}$, we
assume that $W_{0}$ is a certain positive constant that satisfies

$W_{0}:=Min\\{\Lambda_{(dm)}\\}.$ (47)

Under this new hypothesis, we can regard $\Lambda_{(dm)}$ as the fundamental
mass of the gravitational field; the justification will be provided shortly.

As the dimension of $\Lambda_{(dm)}$ is $(\textnormal{length})^{-2}$, we
introduce $m_{\lambda}$ having the dimension of mass, corresponding to
$\Lambda_{(dm)}$. Substituting $e$ and $m_{\lambda}$ into $F_{e}$ and $F_{g}$
in Eq. (46), respectively, we obtain

$\displaystyle\frac{F_{e}}{F_{g}}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi\epsilon_{0}}\frac{1}{G(m_{\lambda})^{2}}=\frac{e^{2}}{4\pi\epsilon_{0}c\hbar}\frac{c\hbar}{G(m_{\lambda})^{2}}=\alpha\left(\frac{m_{p}}{m_{\lambda}}\right)^{2},$
(48) $\displaystyle\alpha$ $\displaystyle:=$
$\displaystyle\frac{e^{2}}{4\pi\epsilon_{0}c\hbar},\;\;\;m_{p}:=\sqrt{\frac{c\hbar}{G}},$
(49)

where $\alpha$ and $m_{p}$ are the coupling constant of the electromagnetic
field and Planck’s mass. For $\Lambda_{(dm)}$, using Eq. (38), namely,
$\Lambda_{(dm)}\approx-\Lambda_{(de)/3}$, and using the concrete expression of
Eq. (32) obtained in reference [17], we have

$\Lambda_{(dm)}\approx\frac{4\pi
Gh}{c^{3}}\frac{(\kappa_{0})^{2}}{\epsilon}=\frac{8\pi^{2}G\hbar}{c^{3}}\frac{(\kappa_{0})^{2}}{\epsilon}=8\pi^{2}(l_{p})^{2}\frac{1}{\epsilon(l_{dp})^{2}};\;\;l_{dp}:=(\kappa_{0})^{-1}.$
(50)

Here, $l_{p}$ and $l_{dp}$ are the Planck length and DP length defined as the
inverse of the DP constant in Eq. (6); $\epsilon$ denotes a dimension
adjusting coefficient of unit length squared. Two reasons exist for the
appearance of $\epsilon$ in the expression of $\Lambda_{(dm)}$. First, the
quantity, $\Lambda_{(de)}=\Sigma_{(i=1)}^{(3)}\hat{T}_{\nu}^{\;\;\nu}(x^{i})$,
in Eq. (32) is related to the “radiation pressure” of the $S_{\mu\nu}$ field.
Second, the calculation of such a quantity is required to make the energy
quantization of the light-like $S_{\mu\nu}$ field consistent with $E=h\nu$ for
the usual radiation field. As we have introduced $m_{\lambda}$ as the
elemental mass corresponding to $\Lambda_{(dm)}$, we can determine it by the
following formal identification using Einstein’s field equation:

$\Lambda_{(dm)}g_{\mu}^{\;\;\nu}=\frac{8\pi
G}{c^{4}}[(\rho_{\lambda}c^{2})u_{\mu}u^{\nu}],\;\;\Rightarrow\;\;\Lambda_{(dm)}(l_{\epsilon})^{3}=\frac{8\pi
G}{c^{2}}m_{\lambda},\;\;m_{\lambda}=\rho_{\lambda}(l_{\epsilon})^{3},$ (51)

where $l_{\epsilon}$ denotes unit length. Therefore, using Eq. (48), Eq. (49),
Eq. (50), and Eq. (51), we finally obtain

$\frac{F_{e}}{F_{g}}=\frac{\alpha}{\pi^{2}}\frac{(l_{dp})^{4}}{(l_{p})^{2}}\frac{1}{\epsilon}.$
(52)

Substituting $\alpha=7.3\times 10^{-3}$, $\pi^{2}=9.9$, $l_{p}=1.6\times
10^{-35}$m, $l_{dp}\approx 5.0\times 10^{-8}$m, and
$\epsilon=1$$\textnormal{m}^{2}$ into Eq. (52), we obtain

$\frac{F_{e}}{F_{g}}=1.7\times 10^{37},$ (53)

which appears to be consistent with the conventional rough estimates obtained
without using $\Lambda_{(dm)}$.

Having derived Eq. (52), now we examine the consequence of the “temporal
change” of $R_{dS}(=1/\sqrt{\Lambda_{(dm)}})$ applicable to $F_{e}/F_{g}$ in
Eq. (52). In the introduction, we cited previous studies on (cosmological)
time-dependent physical constants. Suppose that $\Lambda_{(dm)}$ is such a
time-dependent quantity, then $\Lambda_{(dm)}g_{\mu}^{\;\;\nu}$ in Eq. (45)
ceases to be a divergence-free term. Notice, however, that as $\Lambda_{(dm)}$
is directly related to the radius ($R_{dS}$, in Eq. (38)) of de Sitter space
(Eq. (28)), the partial derivative of $\Lambda_{(dm)}$ with respect to the
cosmological coordinate ($x^{\mu}$) vanishes. This is because the shape of the
isotropic universe is given by a de Sitter space and
$\nabla_{\mu}R_{dS}\;(0\leq\mu\leq 3)$ is on the tangent hyperplane of
$R_{dS}=const.$, on which the gradient of the local $4D$ spacetime coordinate
($x^{\mu}$) exists. Thus, in this sense, $\Lambda_{(dm)}g_{\mu}^{\;\;\nu}$
remains a divergence-free term, although the radius can either expand or
shrink in the fifth dimension perpendicular to the gradient of the $4D$
coordinates ($x^{\mu}$). The “temporal change” of $\Lambda_{(dm)}$ that we
consider is the change in the magnitudes of the dark energy and matter fields.

For simplicity, in our discussion, except for $\Lambda_{(de)}$ and the related
$\Lambda_{(dm)}$, we assume that all physical constants are fixed quantities.
From Eq. (38, 50), we readily obtain

$l_{dp}\approx\frac{2\sqrt{2}\pi}{\sqrt{\epsilon}}\frac{l_{p}}{\sqrt{\Lambda_{(dm)}}}=\frac{2\sqrt{2}\pi}{\sqrt{\epsilon}}l_{p}R_{dS},$
(54)

which implies that the DP length ($l_{dp}$) affords the geometric mean of the
smallest Planck length ($l_{p}$) and the largest scale of our universe
($R_{dS}$). Furthermore, under the assumption that $l_{p}=const.$, $l_{dp}$
becomes proportional to $R_{dS}$. Thus, we can choose $l_{dp}$ as the unique
geometrical parameter of our cosmological model. From the viewpoint of the
unification of four forces, examining the case where we have $F_{e}/F_{g}=1$
is interesting. A simple calculation shows that

$\displaystyle\textnormal{the present value}:\;\;\frac{F_{e}}{F_{g}}$
$\displaystyle=$ $\displaystyle 1.7\times 10^{37};\;\;l_{dp}\approx 5.0\times
10^{-8}\textnormal{m}$ (55) $\displaystyle\textnormal{the unification
value}:\;\;\frac{F_{e}}{F_{g}}$ $\displaystyle=$ $\displaystyle
1;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;l_{dp}\approx 2.4\times
10^{-17}\textnormal{m}.$ (56)

Thus, using Eq. (50), we observe that the unification value of
$\Lambda_{(dm)}(u)$ is ($4.4\times 10^{18}$) times larger than the present
value of $\Lambda_{(dm)}(p)$.

We focus on dark energy and matter mainly because of their extreme
predominance over material substances. This implies that, in thermodynamic
terminology, the dark matter field ($\Lambda_{(dm)}g_{\mu\nu}$) with positive
$\Lambda_{(dm)}$ resulting from Weyl curvature and dark energy field
($\Lambda_{(de)}g_{\mu\nu}$) with negative $\Lambda_{(de)}g_{\mu\nu}$
resulting from Ricci curvature would work as high- and low-temperature
reservoirs, respectively, for the gravity-driven temporal evolution of such
material systems as stars, galaxies, clusters of galaxies, and the large-scale
structure of the cosmos. Moreover, in this cosmic thermodynamical system,
$\Omega_{T}u^{\mu}$, defined in Eq. (36), gives the gravitational entropy
flow. As the energy density of the two thermal reservoirs are finite, the
initial “temperature difference” between them, measured by
$\Lambda_{(dm)}(u)-\Lambda_{(de)}(u)\approx
4\Lambda_{(dm)}(u)/3=4[R_{dS}(u)]^{-2}/3\;$, would decrease with the temporal
evolution of material systems. This result can be observed as the extra
expansion (increase of $R_{dS}$) of our universe in the fifth dimension,
directly related to the temporal increase in the ratio of the previously
discussed coupling constants, $F_{e}/F_{g}\propto(R_{dS})^{2}$. Although the
dynamics we have discussed are unrelated to superstring theories, it is
interesting to bring our attention to Witten’s noteworthy remark[38] made at
Strings ‘95: “eleven-dimensional supergravity arises as a low energy limit of
the ten-dimensional Type IIA superstring.” This appears to be qualitatively
similar to our present situation, in which our $4D$ universe undergoes an
extra expansion into the surrounding fifth-dimensional space. The expansion
starts from the initial high-energy state of $F_{e}/F_{g}\approx 1$ with a
negligible magnitude of $W^{2}$ in Eq. (37) to low energy states having large
values of $W^{2}$ as the measure of conformal gravity.

##  4 Implication of Maldacena (AdS/CFT) duality

In subsection 2.1.1, we saw that the CDSE field ($S_{\mu\nu}$) is closely
related to de Sitter space having the well-known scale-free property. Recall
first that spinor is an irreducible representation of the universal covering
group, $SU(2)$ of $SO(3)$. For the $4D$ spacetime case, in which we have the
Lorentz transformation group ($SO(1,3)$), the Lorentzian spinor ($SL(2,C)$)
corresponds to $SU(2)$ in the case of $SO(3)$. When we further extend
$SO(1,3)$ into the $4D$ conformal transformation group ($SO(2,4)$), $SL(2,C)$
is extended into $SU(2,2)$, which operates on Penrose’s twistor in the $4D$
complex spacetime. As in the case of the above extension of spinor, we can
also consider a similar extension of the electromagnetic field ($F_{\mu\nu}$)
as the $U(1)$ gauge field. We believe that the CP (explained in subsection
2.1.1) applied to extend $F_{\mu\nu}$ into a space-like momentum domain is
what is required for such an extension of conformal transformation, which is
closely related to the important notion of modular form.

The fact that the emergence of de Sitter space through CP is an inevitable
consequence of the extension of $SL(2,C)$ can be readily verified from the
following properties of the Lorentzian spinor, $\Psi(V^{\mu})$:

$\displaystyle\qquad\Psi(V^{\mu})=V^{\mu(\mu)^{\prime}}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}V^{00^{\prime}}&V^{01^{\prime}}\\\
V^{10^{\prime}}&V^{11^{\prime}}\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}V^{0}+V^{3}&V^{1}+iV^{2}\\\
V^{1}-iV^{2}&V^{0}-V^{3}\end{array}\right],$ (61) $\displaystyle
det\Psi(V^{\mu})$ $\displaystyle=$
$\displaystyle\frac{1}{2}[(V^{0})^{2}-(V^{1})^{2}-(V^{2})^{2}-(V^{3})^{2}].$
(62)

For the space-like vector, $V^{\mu}$, $det\Psi(V^{\mu})$ becomes negative so
that Eq. (62) becomes isomorphic to the second equation in Eq. (29). As Eq.
(28) and Eq. (29) are connected by a one-to-one correspondence through
parameter $\eta_{4}$ and the latter is isomorphic to Eq. (62), we thus see
that de Sitter space and $\Psi(V^{\mu})$ share the same symmetry.

Now, we compare the following forms of $g^{\mu\nu}$:

$\displaystyle g^{\mu\nu}_{(I)}$ $\displaystyle=$
$\displaystyle\frac{-1}{\Lambda_{(I)}}\left(R_{(I)}^{\mu\nu}-\frac{R_{(I)}}{2}g^{\mu\nu}_{(I)}\right),\;\;(W^{\alpha\beta\gamma\delta}=0),\;(I=1\;\textnormal{or}\;2),$
(63) $\displaystyle g^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{W^{\mu\alpha\beta\gamma}W^{\nu}_{\;\;\alpha\beta\gamma}}{W^{2}/4}\;\;(W^{2}\neq
0),$ (64)

where Eq. (63) represents either de Sitter space with
$\Lambda_{(1)}=\Lambda_{(de)}<0$ or anti de Sitter space (AdS) with
$\Lambda_{(2)}=\Lambda_{(dm)}>0$, depending on the index ($I$).

In our CCC-2nd hypothesis, the isotropy of an early universe is explained by
the small amplitude of $W^{2}$ (the Weyl curvature hypothesis proposed by
Penrose[29]), which motivates us to examine the possibility that the following
limit exists:

$\displaystyle
g^{\mu\nu}=\frac{W^{\mu\alpha\beta\gamma}W^{\nu}_{\;\;\alpha\beta\gamma}}{W^{2}/4}\;\longrightarrow\;g^{\mu\nu}_{(2)}=\frac{-1}{\Lambda_{(2)}}\left(R_{(2)}^{\mu\nu}-\frac{R_{(2)}}{2}g^{\mu\nu}_{(2)}\right),\;\textnormal{as}\;\;W^{2}\;\rightarrow\;0.$
(65)

Within the framework of our CCC-2nd hypothesis, this gives a diffeomorphism
defined as the time reversal of a given cosmological time development. As
$g^{\mu\nu}$ on the l.h.s. gives the gravitational field, whereas the r.h.s.
represents AdS, Eq. (65) can be regarded as the “Maldacena (AdS/CFT)
duality”[27] in CCC-2nd.

##  5 Summary and conclusion

Using our recently developed unconventional model incorporating dark energy
and matter, we re-examined the hierarchy problem. As the model itself is not
well known, we began our discussion by recapitulating the key concepts used in
our new theory to elucidate an elusive phenomenon called DP in the field of
nanophotonics. This phenomenon unexpectedly inspired the novel perspective on
cosmological problems addressed in this study. The concepts include the GR
theorem for nonlinear quantum field interactions, Ojima’s MMD theory developed
from DHR’s original sector theory in QFT, and CDSE field developed by the
first author, together with the unconventional introduction of conformal
gravity explained in subsection 2.1.

In addition, as an outstanding feature of our cosmological theory, we use the
basic view that our universe is undergoing an infinite cycle of birth and
death as in the case of CCC. However, a crucial difference exists between CCC
and our theory (CCC-2nd), namely, the twin structure of a universe arising
from the intrinsic property of de Sitter space. Our reason for supporting such
a view is twofold. First, we believe that the basic principle of the universe
is not complicated but simple. The creation and annihilation mechanisms of the
matter and antimatter pair through the intervention of a light field observed
in laboratories can also be applied naturally to the case of a twin universe
configuration. Once we accept this conceptually simple view on the creation of
the universe, we need not worry about complicated parameter tuning processes
used in the widely prevailing theory of cosmic inflation, which favors the
creation of everything from emptiness.

Second, as mentioned in the introductory section, the idea of the creation of
everything from emptiness appears to be conceived from either the
misinterpretation or the extended interpretation of Fock vacuum $|0\rangle$,
which is, as far as QFT is concerned, introduced within the framework of
linear dynamics on free modes. In our cosmological theory, as Eq. (38) shows,
the universe expands, keeping a quasi-equilibrium between $\Lambda_{(dm)}$ and
$\hat{\Lambda}_{(de)}$, which behave as “high and low temperature reservoirs”,
respectively, for the temporal evolutions of systems in the universe. Thus,
the essential global aspect of the expanding universe in a quasi-equilibrium
state should be described by the Tomita–Takesaki theory[39] as a thermodynamic
Kubo–Martin–Schwinger (KMS) state with infinite degrees of freedom. As the KMS
state is a mixed one, its corresponding Gel’fand–Naimark–Segal representation
is reducible. Therefore, for $\mathcal{M}$, defined as a von Neumann algebra
on Hilbert space $\mathfrak{H}$, there exists its commutant,
$\mathcal{M}^{\prime}$, which satisfies the following inversion relation:

$\displaystyle
J\mathcal{M}J=\mathcal{M}^{\prime},\;\;\;e^{itH}\mathcal{M}e^{-itH}=\mathcal{M},\;\;\;J^{2}=1,$
(66) $\displaystyle JHJ=-H,$ (67)

where $H$ and $J$ denote the Hamiltonian and antiunitary operators called
modular conjugation, respectively. Notice that the spectrum of the Hamiltonian
is symmetric with respect to its sign, which indicates the existence of states
with negative energy. We believe that the result of the Tomita–Takesaki theory
applies to the case of a twin universe configuration and to the thermodynamics
of the observed cosmic background radiation whose energy-spectrum distribution
is given by the black body radiation.

Eq. (52) stands as the central result in our study, showing the clear causal
relation between the existence of noncompactified extra dimension and
$F_{e}/F_{g}$, in which the DP length ($l_{dp}$) plays the key role in
determining the value of $F_{e}/F_{g}$. Through Eq. (54), we see that $l_{dp}$
divides the universe into two domains. Furthermore, its present value, given
in Eq. (55), suggests that $l_{dp}$ affords the scale of the Heisenberg cut
dividing the micro-quantum and the macro-classical worlds.

As the final remark on our study, we wish to make a few brief comments on the
innovative observational outcomes of the James Webb Space Telescope (JWST).
One of the quite unexpected findings of the JWST Advanced Deep Extragalactic
Survey is that the events considered to have occurred in the very early stages
of the universe actually occurred much earlier than expected. We have shown
that the magnitudes of $\Lambda_{(dm)}$ and $\Lambda_{(de)}$ given in Eq. (38)
in the early stage of the universe were considerably greater than those in the
present one. Thus, we conjecture that the standard $\Lambda$CDM model used in
cosmological simulations is not suitable for simulating the temporal
development of the early universe. Eq. (45) shows that the isotropic expansion
owing to $\Lambda_{(de)}g_{\mu\nu}$ on the r.h.s. manifests itself through a
conformally flat Ricci curvature. Conversely, the local gravitational
attraction owing to $\Lambda_{(dm)}g_{\mu\nu}$ on the l.h.s. is highly
nonisotropic because of the intrinsic property of the Weyl curvature. Data
from CEERS survey by JWST show that most of the very early galaxies assumed
elongated shapes similar to noodles or surfboards, which appears to be
consistent with our conjecture on the gravitational effect arising from the
nonisotropic Weyl curvature. Presumably, the observed large-scale structure of
the present universe comprising galaxy filaments and voids is also the
manifestation of this nonisotropic Weyl curvature effect.

Regarding the cosmic age problem relating to the very early formation of
primordial galaxies, in his 2023 paper, Gupta[40] proposed an intriguing
resolution derived from the outcome of a hybrid cosmological model called
CCC(covarying coupling constants) + TL(tired light). The idea of CCC is
similar to that of Dirac’s hypothesis on varying physical constants, such as
gravitational constant ($G$) and speed of light ($c$). Gupta introduced the
CCC model as an extended version of the $\Lambda$CDM model with a variant
cosmological constant. In that sense, our model with a varying
$\Lambda_{(dm)}(\approx-\Lambda_{(de)}/3)$ is similar to the CCC model.
However, a crucial difference exists between these two models. In Gupta’s
model, parameter $\alpha$, called the strength of the coupling constant’s
variation, plays a substantial role, whereas in our model, all usual physical
constants, except for $\Lambda$, are assumed to be nonvariant. We adopt this
assumption because it is the simplest one. Therefore, the question of whether
physical constants are unchanging or not remains unanswered. Qualitatively,
much larger value of $\Lambda_{(dm)}$ in our model of the very early universe
and much larger value of gravitational constant $G$ in Gupta’s model would
have a similar impact on the formation of early galaxies. The most important
characteristics of our model is the possibility of the universe expansion in
fifth dimensional direction, which reveals that our universe as a $4D$
Riemannian manifold is not a closed entity but is open to a higher dimensional
realm.

## Acknowledgments

All of the authors express their sincere appreciation for the distinguished
leadership of M. Ohtsu, Prof. Emer. of the University of Tokyo, who has been
and is still driving unique “off-shell science project” originated from his
lifelong dressed photon research in the field of nanophotonics.

##  Appendix A Greenberg-Robinson theorem

The essence of the theorem is that off-shell fields is never compatible with
the on-shell condition characterizing (generalized) free fields in the special
relativistic situation. To precisely state the theorem, we briefly present the
Wightman system of axioms. In this appendix, we use the natural system of
units, the unit system such that $c=1$ and $\hbar=1$.

The following family of four axioms is referred to as the Wightman system of
axioms:

[1. Field operator]

For a neutral field, a field operator $\phi(x)$, $x\in\mathbb{R}^{4}$, is
defined by an operator on a Hilbert space $\mathcal{H}$.

[2. The covariance condition]

A (projective) unitary representation ($U$) of the Poincaré group exists,
$\mathcal{P}_{+}^{\uparrow}=\mathbb{R}^{4}\rtimes\mathcal{L}_{+}^{\uparrow}$
on $\mathcal{H}$ such that

$U(a,L)\phi(x)U(a,L)^{\ast}=\phi(Lx+a)$ (68)

for all
$(a,L)\in\mathcal{P}_{+}^{\uparrow}=\mathbb{R}^{4}\rtimes\mathcal{L}_{+}^{\uparrow}$.

[3. The causality condition]

$\phi(x)$, $x\in\mathbb{R}^{4}$, satisfy the local commutativity, that is,

$[\phi(x),\phi(y)]=\phi(x)\phi(y)-\phi(y)\phi(x)=0$ (69)

for all pairs $(x,y)$ of mutually space-like points.

[4. Vacuum state and the spectrum condition]

A unit vector, $\Omega$ of $\mathcal{H}$, called the vacuum vector and a
spectral measure ($E$) of $\mathbb{R}^{4}$ on $\mathcal{H}$ exist, satisfying
the following two conditions:

  1. 4.1

$\Omega$ satisfies $U(a,L)\Omega=\Omega$ for all
$(a,L)\in\mathcal{P}_{+}^{\uparrow}$. $E$ satisfies

$U(a,I)=\int_{\overline{V_{+}}}e^{ia\cdot p}\;dE(p)$ (70)

for all $a\in\mathbb{R}^{4}$, where
$\overline{V_{+}}=\\{x=(x^{\mu})\in\mathbb{R}^{4}\,|\,x^{2}=x_{\mu}x^{\mu}\geq
0\;\text{and}\;x^{0}\geq 0\\}$ and $a\cdot p=a^{\mu}p_{\mu}$ ($\Omega$ then
satisfies $E(\\{0\\})\Omega=\Omega$).

  2. 4.2

$\Omega$ is cyclic for $\mathcal{P}(\mathbb{R}^{4})$, the polynomial algebra
over $\mathbb{C}$ generated by $\\{\phi(x)\,|\,x\in\mathbb{R}^{4}\\}$, i.e.,
$\mathcal{H}=\overline{\mathcal{P}(\mathbb{R}^{4})\Omega}=\overline{\\{X\Omega\,|\,X\in\mathcal{P}(\mathbb{R}^{4})\\}}$.

Strictly, every field operator is defined as an operator-valued distribution.
However, we omit the detail (see [41] for example). Moreover, no field
operator is defined at each point $x$ of the Minkowski space, so that
$\phi(x)$ is simply a symbolic notation.

A neutral field $\phi(x)$ is irreducible if it satisfies the following
condition: If $B$ is a bounded linear operator on $\mathcal{H}$ such that

$\langle\Omega|B\phi(x_{1})\phi(x_{2})\cdots\phi(x_{m})\Omega\rangle=\langle\phi(x_{m})^{\ast}\phi(x_{m-1})^{\ast}\cdots\phi(x_{1})^{\ast}\Omega|B\Omega\rangle$
(71)

for every natural number $m$ and points $x_{1},\cdots,x_{m}$ of the Minkowski
space, then it is of the form $B=kI$, where $k\in\mathbb{C}$ and $I$ is the
identity operator on $\mathcal{H}$.

The Fourier transform $\hat{\phi}(p)$ of a field operator $\phi(x)$ is defined
by

$\hat{\phi}(p)=\dfrac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{4}}e^{-ip\cdot
x}\phi(x)\;d^{4}x.$ (72)

A field $\phi(x)$ is called a generalized free field if it satisfies the
commutation relation

$[\phi(x),\phi(y)]=iP(x-y)I$ (73)

for all $x,y\in\mathbb{R}^{4}$. The function $P(x)$ is called the function
defined by

$P(x)=\int_{0}^{\infty}D_{\sqrt{\xi}}(x)\;d\sigma(\xi),$ (74)

where $\sigma$ is a measure on $\mathbb{R}_{\geq
0}=\\{x\in\mathbb{R}\,|\,x\geq 0\\}$,

$D_{m}(x)=\dfrac{1}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\sin(p^{0}x^{0})e^{-i{\bm
p}\cdot{\bm x}}\dfrac{d{\bm p}}{p^{0}}$ (75)

and $p^{0}=\sqrt{{\bm p}^{2}+m^{2}}$. The free Klein-Gordon field $\varphi(x)$
with square mass $m^{2}(>0)$, a typical example, satisfies the commutation
relation $[\varphi(x),\varphi(y)]=iD_{m}(x-y)I$ for all
$x,y\in\mathbb{R}^{4}$.

######  Theorem A.1 (Greenberg-Robinson [42, 43]).

Let $\phi(x)$ be an irreducible neutral field. If an open subset exists, $O$,
of $\mathbb{R}^{4}\backslash\overline{V_{\pm}}$ such that $\hat{\phi}(p)=0$
for all $p\in O$, where
$\overline{V_{\pm}}=\\{x\in\mathbb{R}^{4}\,|\,x^{2}=x_{\mu}x^{\mu}\geq 0\\}$,
then $\phi(x)$ is a generalized free field.

The contraposition of this theorem implies that

If an irreducible neutral field $\phi(x)$ is not a generalized free field, it
holds that $\hat{\phi}(x)\neq 0$ for all $p\in\mathbb{R}^{4}$.

This fact indicates that the involvement of off-shell momenta is specific to
interacting quantum fields and is never observed in (generalized) free fields
at the four-dimensional Minkowski space.

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