**1. Introduction**

Application studies of quantum theory in nanosciences have continued to accomplish a variety of spectacular modern technological achievements. The technology involving the dressed photon (DP) phenomena is one such achievement that makes the impossible possible. While a reliable theory has not ye<sup>t</sup> been established to explain the characteristic behaviors of DPs, a comprehensive review of DP studies, including the impossibility of understanding DP phenomena within the conventional framework of Maxwell's equation, was given by Ohtsu [1], together with a series of associated intriguing technologies and the status of theoretical attempts to understand DPs up to 2017. The research on the DP phenomena is now being pursued more actively than ever before both experimentally and theoretically. The most important point on the DP, clarified through decades-long investigations, is that the DP field is not a simple variant of the light field such as evanescent light, which is essentially a free mode, but involves largely transmuted and locally condensed (within an area smaller than several tens of nanometers) electromagnetic field energy achieved through light–matter field interactions involving point-like singularities, which seem to be a key factor for DP generation. The peculiarity of the DP field compared with the free light field is concisely summarized in Section 1 of the latest paper on DPs by Sakuma et al. [2] (S3O hereafter), where a new theory is proposed, focusing on the aspects of quantum field interactions thus far neglected.

The real reason for the unsuccessful attempts at a full-fledged theory of DPs seems to be related to the fact that a DP is not a free mode, but is the outcome of light–matter field interactions, the complexity of which makes constructing a simple mathematical model

**Citation:** Sakuma, H.; Ojima, I. On the Dressed Photon Constant and Its Implication for a Novel Perspective on Cosmology. *Symmetry* **2021**, *13*, 593. https://doi.org/10.3390/ sym13040593

Academic Editor: Ignatios Antoniadis

Received: 2 March 2021 Accepted: 1 April 2021 Published: 2 April 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

1

difficult. In fact, contrary to the above-mentioned remarkable technological successes of quantum theory, the current stage of development of quantum field theory (QFT) is far from a firmly established one, such as the theory of Newtonian mechanics. From this viewpoint, a major stumbling block might be the lack of mathematical support for interacting quantum field models satisfying the covariance under the Poincaré group P in 4-dimensional Minkowski spacetime (defined as the crossed product P := R<sup>4</sup> - L of the Lorentz group L acting on the 4-dimensional Minkowski spacetime R4). While the main subject here is the DP system, to be described as a subsystem of relativistic 4-dimensional QFT, a survey of the basic structure of the 4-dimensional QFT itself would be useful for our purpose of discussing the various aspects of the DP system.

First, the physical interpretations of QFT described by the interacting Heisenberg fields *ϕH* are realized by the notion of on-shell particles contained in *ϕH* with the 4-mometum *pμ* given by Equation (1):

$$p^2 := \eta\_{\mu\nu} p^\mu p^\nu := p\_\nu p^\nu = (m\_0 \mathfrak{c})^2 \ge 0, \quad \mu, \nu = 0, 1, 2, 3,\tag{1}$$

where we adopt the sign convention (+1, −1, −1, −<sup>1</sup>) for the Minkowski metric *η* given by

$$
\eta\_{\mu\nu} = \begin{pmatrix} +1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}.
$$

The physical meaning of the asymptotic fields *φas* (*as* = *in* or *out*) can be seen in their role in a scattering process formed by the in-fields *φin*1 (*p*1), ···, *φinm* (*pm*) with momenta *p*1, ···, *pm* converging from the remote past to the scattering center and by the out-fields *φout* 1 (*q*1), ···, *φout m* (*qn*) with momenta *q*1, ···, *qn* diverging from the scattering center to the remote future. In contrast with the interacting Heisenberg field *ϕ<sup>H</sup>*, which causes and controls the above scattering process behind the scenes, the asymptotic field *φas* carrying the above momentum spectrum as an observable quantity can be easily realized as a free field obtained by the so-called second quantization, as shown below. Owing to its linearity, the asymptotic field *φas* is governed by the well-known Klein–Gordon (KG) Equation (2). ¯

In the simplest case of a scalar field *φas*, the first quantization *pμ* → *i h∂μ* applied to (1) realizes the KG equation:

[

$$
\hbar^2 \partial^\nu \partial\_\nu + (m\_0 c)^2] \phi^{\text{as}} = 0,\tag{2}
$$

where the operand *φas* determined by the second quantization becomes a quantum field *φas* describing a multi-particle system given by

$$\phi^{\rm as}(\mathbf{x}^{0}, \tilde{\mathbf{x}}) = \int \frac{d^{3}k}{\sqrt{(2\pi)^{3}2E\_{k}}} [a(\tilde{k})\exp\left(-ik\_{\nu}\mathbf{x}^{\nu}\right) + a^{\dagger}(\tilde{k})\exp\left(ik\_{\nu}\mathbf{x}^{\nu}\right)].\tag{3}$$

Here, (*a*†(˜*k*), *a*(˜*l*)) and (*x*˜ and ˜*k*), respectively, denote a pair of creation-annihilation operators and of 3-vectors consisting of spatial components of *xμ* and *k<sup>ν</sup>*, with *Ek* defined by *Ek* := (˜*k*)<sup>2</sup> + (*<sup>m</sup>*0)2. A familiar Fock space is constructed on the basis of (3) and of the vacuum state vector |0 satisfying *a*|0 = 0, according to which a positive energy spectrum is selected in the state vector space. While the field *φas* thus constructed embodies the *wave–particle duality* of a quantum system, it still lives in the realm of linearity due to the linear KG Equation (2). With the restriction due to this linearity (or the on-shell property (1)) overlooked, however, 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 (under the cyclicity assumption) are misinterpreted as the universal structure to be found in interacting multiparticle systems. Accordingly, |0 becomes as mysterious as the creation of everything from emptiness. We return to this point in Section 4 on cosmology.

The mutual relations among the Poincaré group P, Heisenberg field *ϕ<sup>H</sup>*, asymptotic field *φas*, and momentum spectrum (*pμ*) can be clearly visualized by means of the quadrality scheme to describe the duality relation between Micro and Macro (Micro-Macro duality based on the quadrality scheme [3]):

**Remark 1.** *In the specific example of scattering process with asymptotic completeness, the original quadrality scheme of micro–macro duality can be seen in the above relations among the dynamics* P *acting on the algebras of interacting Heisenberg fields ϕH and of their asymptotic fields φas and the spectrum of energy-momentum pμ. It gives a unified categorical description of the system of interacting quantum fields in terms of quantum and classical systems, both of which are characterized dynamically by their non-commutative and commutative algebras. As our new ideas on quantum field theory of the dressed photons depends heavily on this quadrality scheme, it will be convenient to explain here its minimal essential points to those who are familiar only with quantum mechanics with finite degrees of freedom.*

*The scheme is a theoretical framework consisting of a couple of different dualities that are interweaved to describe the theoretically phenomena under consideration: among the four basic ingredients in the scheme, Dynamics and the Algebra* X *of physical quantities belong to the micro side of the quantum system, while the remaining two elements—States (and their representations) and Spectrum— belong to the macro side. To visualize the invisible quantum micro system, we need to exert certain action E* : A→X *on the microscopic quantum system* X *from the macro side* A*. The response of the acted micro side to the acting macro side is to be given by F* : A←X *, according to which we have an adjoint pair of functors* A *<sup>F</sup>*(*x*) - *<sup>E</sup>*(*a*) X *; (x* ∈ X *and a* ∈ A*). In this way, we see that the basic structure of the quantum theory is mathematically formulated by the so-called "adjunction" in category theory, which can be understood as the precise mathematical form of "duality"*A X *(one of the weaker forms of equivalence), where* X *and* A*, respectively, denote unknown mathematical object belonging to micro system and known object (as the familiar vocabulary) in the classical macro system and symbol denotes natural equivalence.*

*As we see in the above diagram, the abscissa axis represents the duality between the algebra* X *of quantum variables and its states with Gel'fand–Naimark–Segal (GNS) representations realized in a Hilbert space. Central problematic issues we have in considering quantum systems with infinite degrees of freedom would be those on unitary nonequivalence and the uniqueness of irreducible decomposition, which are usually regarded as a pathological aspect of systems with infinite degrees of freedom. However, omitting the details of extensive researches so far done on the generalized sector problem, we can briefly summarize the main conclusions of them as follows. A system with infinite degrees of freedom can be represented with multiple sectors where a sector is defined by a factor representation with trivial center containing only scalar multiples of the identity, which generalizes the notion of irreducible representations with trivial commutants. Here, disjointness means the absence of intertwiners, as the refined notion of unitary nonequivalence adapted to the situations with infinite degrees of freedom. By this kind of generalization, we also have the change in the classification of representation, that is to say, an irreducible representation is to be replaced by a factorial representation which has a self-evident center playing the role of a commutative (classical) order parameter. Thus, we show that macroscopic order parameters emerge naturally from the disjoint representations appearing in the micro systems and the spectrum of those order parameters gives the classification space for describing a variety of configurations the micro system would take.* The duality relation illustrated in the ordinate axis, that is,

[*Dyn* - *Spec*] expresses the duality between invariability and variability of coupled micro and macro systems.

The asymptotic fields *φas* given by (3) are placed in this scheme in duality relation with the interacting Heisenberg fields *ϕ<sup>H</sup>*, where *φas* itself consist only of linear free modes without anything to do with nonlinear field interactions having the off-shell property. Because the clear-cut mathematical criterion to distinguish nonlinear field interactions from the free time evolution of noninteracting modes, known as the Greenberg–Robinson theorem [4,5], states that *if the Fourier transform ϕ*(*p*) *of a given quantum field φas*(*x*) *does not contain an off-shell spacelike momentum pμ with pν pν* < 0 *(cf. Equation (1)), then φas*(*x*) *is a generalized free field*. A caveat to be made here is that a spacelike momentum field does not necessarily mean the presence of a tachyonic field representing particle-like *localized energy field* moving with superluminous velocity, which violates the Einstein causality. This localized field is known to be unstable such that the existing spacelike momentum fields take naturally simple wavy forms. Another crucial piece of knowledge necessary to understand the enigmatic DP phenomena is the important property of quantum fields with *infinite degrees of freedom*, referred to in the above remark. As is well known, we have only one sector in the familiar case of quantum mechanical systems with *finite degrees of freedom* which are governed by unitary time evolution (the Stone–von Neumann theorem [6]). In sharp contrast to this situation, quantum fields with infinite degrees of freedom have multiple sectors [3,7], which are mutually disjoint (i.e., separated by the absence of intertwiners), stronger than unitary inequivalence. Regarding the unitary equivalence, Haag's theorem [8] states that *any quantum field satisfying Poincaré covariance is a free field if it is connected to a free field by a unitary transformation*. According to this no-go theorem, it is meaningless to consider that an interacting Heisenberg field can be realized through a unitary transformation of a free field by means of the well-known Dyson S-matrix involving the interaction term. In this way, the essential part of our common knowledge cultivated in quantum mechanical systems with finite degrees of freedom is invalidated in relativistic QFT.

The notions of spacelike momentum field and the existence of multiple sectors must be quite foreign for many who are unfamiliar with quantum systems with infinite degrees of freedom, so that it is worthwhile to give a simple heuristic example. Let us consider a simple wave propagation, *ψ* = exp *<sup>i</sup>*(*k*0*x*<sup>0</sup> − *<sup>k</sup>*1*x*<sup>1</sup>), in a certain background field. One may regard it as a wave, say, in the atmosphere. When the wave exists in a uniform background, it propagates such that it satisfies (*∂ν∂ν* + *k*<sup>2</sup>)*ψ* = 0, with *k*2 := (*k*0)<sup>2</sup> − (*k*1)2, which may be compared to a "unitary" time evolution of a free mode in the timelike sector. If the background field becomes nonuniform but its degree of nonuniformity is rather smooth, then though its way of propagation is deformed to some extent, we can describe the deformed propagation pattern by employing perturbative methods, and the solution still remains in the timelike sector mentioned above. As an extreme case of severe interactions with the environmental field for which the perturbative method is break down, we can consider a frontal instability of the atmosphere in which the front is defined as a line of discontinuity of the temperature and velocity fields. A wavelike perturbation with small amplitude put into this frontal zone, due to hydrodynamic shear instability, can no longer keep its wavy form, and its amplitude starts to either (i) grow or to (ii) damp exponentially in a region that is narrow in the traverse direction. In view of such situations that QFT is basically a theory involving complex numbers and that the frequency and wave number of a given wavelike field represent the energy and momentum, the abrupt change in the energy and momentum brought about by a certain kind of discontinuity of the field can be represented in the simplest crude model by a discrete jump of (*k*0, *k*1) into (±*il*0, −*il*1) with *l*2 := (*l*0)<sup>2</sup> − (*l*1)<sup>2</sup> > 0. Note that with this abrupt change, (*∂ν∂ν* + *k*<sup>2</sup>)*ψ* = 0 becomes (*∂ν∂ν* − *l*<sup>2</sup>)*ψ* = 0, namely, the wave dynamics shifts abruptly from a timelike sector to a spacelike one with the properties exp(∓*l*0*x*<sup>0</sup>) and exp(−*l*1*x*<sup>1</sup>) (valid in the domain *x*1 ≥ 0), respectively, corresponding to the above-mentioned properties of (i) and (ii). Needless

to say, this example, due to the atmospheric dynamics, could be transferred to situations involving interactions among elementary particles, where a "severe interaction" would evoke these changes on the interacting Heisenberg fields to which on-shell field theory cannot be applied. We believe that this simple toy model gives an intuitive explanation of the essential features of severe field interactions involving a certain kind of discontinuity and why spacelike momentum modes are necessary to describe these field interactions. We will further discuss this problem in Section 2.2 on DP model.

Now, going back to the general argumen<sup>t</sup> on QFT, notice that the above two theorems in axiomatic QFT for relativistic quantum fields, especially the first one, justify our investigation into the existence of a spacelike momentum domain, *in the sense of a different sector*, with which the conventional Maxwell's equation is to be augmented for a complete description of electromagnetic field interactions. A helpful hint regarding an appropriate form of the spacelike momentum can be found in the longitudinal Coulomb mode or the virtual photon, which behaves as a carrier of electromagnetic force. In their series of papers, Sakuma et al. (and the latest S3O [9–12]) derived an extended field covering the spacelike momentum domain by applying a mathematical technique called *Clebsch parameterization* to electromagnetic 4-vector potential *<sup>A</sup>μ*. The extension of the field was accomplished in two steps: (I) semi-spacelike and (II) spacelike extensions. To avoid confusion, here we replace the common notation *<sup>A</sup>μ* for a 4-vector potential with *<sup>U</sup>μ*. In step (I), *<sup>U</sup>μ* satisfies

$$[\left(\partial^{\nu}\partial\_{\nu} - (\kappa\_0)^2\right)l]\_{\mu} = 0, \quad ll\_{\nu}l l^{\nu} = 0,\tag{4}$$

where *κ*0 is an important constant, to be identified as the DP constant. At first glance, one may consider this to be the wrong equation, as a null (massless) condition *UνU<sup>ν</sup>* = 0 seems to be incompatible with the first equation in (4). As shown in the next section, however, it is indeed correct. The reason why it looks bizarre is because it corresponds to a longitudinally propagating electromagnetic wave of which the quantum version is eliminated as unphysical in the conventional interpretation. We believe that this bizarre mode, massless in the sense of *UνU<sup>ν</sup>* = 0, corresponds qualitatively to an *invisible virtual photon, i.e., a U*(1) *gauge boson*, and in step (II), this field is extended further to the case of a genuine spacelike field satisfying *UνU<sup>ν</sup>* < 0. As we will touch upon in Section 2.2, the formulation of steps (I) and (II) is generalized to cover the case of a curved spacetime. As the first equation in (4) can be considered a dual form of the timelike Proca equation, i.e., [*∂ν∂ν* + ( *<sup>m</sup>*0)<sup>2</sup>]*<sup>A</sup>μ* = 0, we call it the Clebsch dual (CD) field and denote its skew-symmetric field strength by *<sup>S</sup>μν* := *∂μ<sup>U</sup>ν* − *∂νUμ*.

As the source-free Maxwell's equation is conformally invariant, the derivation of an augmented Maxwell field can be viewed mathematically as a conformal extension of the electromagnetic field *<sup>F</sup>μν*. From this viewpoint, note that the derivation of the CD field is conceptually similar to the notion of a twistor introduced by Penrose [13], and in this sense, the essence of our new proposal on cosmology has a closer connection to the conformal cyclic cosmology (CCC) proposed by Penrose [14] than the antipodal twin universe model of Petit [15]. To see this, let us consider the rotation group *SO*(3) acting on three-dimensional vectors. For *SO*(3), the universal covering group *SU*(2) exists, which is locally isomorphic to *SO*(3) and in relation to which a spinor is defined as its irreducible representation. Extending this context to the Lorentz group *SO*(1, 3) in four-dimensional spacetime, *SL*(2, *C*) arises as the universal covering group corresponding to *SU*(2). If we further extend *SO*(1, 3) to a four-dimensional conformal group, then *SO*(1, 3) and *SL*(2, *C*) are extended, respectively, to *SO*(2, 4) and *SU*(2, <sup>2</sup>), and Penrose's twistor appears as an element of the complex four-dimensional space on which *SU*(2, 2) acts. As a parallel argument, we can consider the case of a conformal extension of the electromagnetic field *<sup>F</sup>μν* that acts on the spinor as a *U*(1) gauge field. CD field *<sup>S</sup>μν*, introduced as the spacelike extension of *<sup>F</sup>μν*, is thus also regarded as a conformal extension of *<sup>F</sup>μν*. As has been shown in S3O, we believe that this fact explains why the CD field plays an important role in the dark energy dynamics of the self-similarly (conformally) expanding universe described as a de Sitter space, in sharp contrast to the simple-minded intuition that the mutual relations between the DP and cosmological phenomena are irrelevant owing to their extremely large scale difference.

This paper is organized as follows. To discuss the theme addressed in the title, we first need prior knowledge on the CD field, which is a very new concept, and on several important conclusions on cosmology reported in S3O. We reserve Sections 2 and 3 for the purpose of recapitulating the minimal required knowledge in a simple way. Then, in Section 4, we discuss the main topics of this paper, namely, the dressed photon constant and a perspective on the possible relation between our novel cosmology and the CCC.
