*2.1. Clebsch Dual Field*

As mentioned above, the CD field can be regarded as a field of longitudinal electromagnetic waves. To understand this, we first note that a serious misunderstanding regarding the longitudinally propagating wave modes has persisted. In the physical science communities, this misunderstanding has been prevailing and left untouched, but it cannot be overlooked in the present context. As a matter of fact, one frequently encounters this statement in standard textbooks on electromagnetism, which asserts that electromagnetic waves are not longitudinal but transversal. This concept seems, however, to be a superfluous reaction to the assertion in "advanced" quantum electrodynamics (QED), where longitudinal modes are eliminated as unphysical. In the classical theory of electromagnetism, however, the longitudinally propagating modes have been proved unmistakably to exist in *a light beam with finite width*, both theoretically in [16] and experimentally in [17]. In these papers, the existence of longitudinal modes is shown without using the electromagnetic 4-vector potential *<sup>A</sup>μ*. Here, the significance of introducing the CD field can be seen in the following two aspects:


To confirm what is stated above, let us consider Maxwell's Equation (5) and the associated energy-momentum tensor (7), together with its divergence (8):

$$
\partial^{\nu} F\_{\mu \nu} = \quad \partial^{\nu} (\partial\_{\mu} A\_{\nu} - \partial\_{\nu} A\_{\mu}) = \left[ -\partial^{\nu} \partial\_{\nu} A\_{\mu} + \partial\_{\mu} (\partial^{\nu} A\_{\nu}) \right] = j\_{\mu \nu} \tag{5}
$$

$$A\_{\mu} \quad = \quad a\_{\mu} + \partial\_{\mu} \chi\_{\nu} \quad (\partial\_{\nu} a^{\nu} = 0, \ \phi := \partial\_{\nu} A^{\nu} = \partial\_{\nu} \partial^{\nu} \chi). \tag{6}$$

$$\left(T\_{\mu}^{\ v}\right)\_{\mu} = \left. -F\_{\mu\tau}F^{\upsilon\tau} + \frac{1}{4}\eta\_{\mu}^{\upsilon}F\_{\tau\tau}F^{\upsilon\tau} \right.\left. \left(F\_{\tau\tau}F^{\upsilon\tau} = 0\right) \text{ for free wave modes} \right)\_{\prime} \tag{7}$$

$$
\partial\_\nu T^\nu\_\mu \quad = \quad \partial\_\nu (-F\_{\mu\sigma} F^{\nu\sigma}) = F\_{\mu\nu} \partial\_\sigma F^{\nu\sigma} = F\_{\mu\nu} \mathbf{j}^\nu. \tag{8}
$$

If the Lorentz gauge condition *∂νAν* = 0 is imposed, additionally or *formally*, to the above Maxwell's equation, then Equation (5) reduces to *∂ν∂νAμ* = 0, according to which the free Maxwell's equation can be identified in the sense of *jμ* = 0. Apart from this conventional method, however, another possibility to find the free equation begins with

$$
\partial^{\nu}\partial\_{\nu}A\_{\mu} = 0,\tag{9}
$$

without assuming *∂νAν* = 0. In this case, (5) tells us that we have a nontrivial (*∂μφ* = 0) balance equation

$$
\partial^{\nu}F\_{\mu\nu} = \partial\_{\mu}\phi\_{\prime} \quad \to \quad \partial^{\mu}\partial\_{\mu}\phi = \partial^{\mu}\partial^{\nu}F\_{\mu\nu} = 0. \tag{10}
$$

The first equation in (10) can be justified in two steps: First, from (5) and (8), we see that the conservation law of *∂νT ν μ* = 0 is satisfied when *j ν* = 0 in the usual free case (8). In the case of (10), however, we use the expression *∂νT νμ* = *<sup>F</sup>μν∂σFνσ* in (8) and *∂νFμν* = *∂μ∂ν<sup>A</sup>ν* in (5), which leads to

$$
\partial\_\nu T\_\mu{}^\nu = F\_{\mu\nu} \partial^\nu \phi = 0,\tag{11}
$$

if *<sup>F</sup>μν* ⊥ *∂νφ* with *∂μ∂μφ* = 0. This expression indicates that the longitudinally propagating vector *∂νφ* is physical in the sense that it satisfies the energy-momentum conservation.

In the second step of the physical justification of (10), we consider (9) in terms of *αμ* and *χ* given in (6), which becomes

$$
\partial^\nu \partial\_\nu a\_\mu^{(h)} = 0, \quad \partial^\nu \partial\_\nu a\_\mu^{(i)} + \partial^\nu \partial\_\nu (\partial\_\mu \chi) = 0,\tag{12}
$$

with homogeneous and inhomogeneous solutions, i.e., *α*(*h*) *μ* and *α*(*i*) *μ* , respectively, for a given *χ* satisfying the second equation in (10). *α*(*h*) *μ* obviously represents a transverse mode, and the second equation gives a balance between the rotational and irrotational modes. The existence of this balance is well documented in the hydrodynamic literature explaining the mathematical description of the irrotational motion of a two-dimensional incompressible fluid. Due to the irrotationality of the motion, the velocity vector (*<sup>v</sup>*1, *v*2) is expressed in terms of the gradient of the vector potential *φ*ˆ, namely, (*<sup>v</sup>*1 = *<sup>∂</sup>*1*φ*ˆ, *v*2 = *<sup>∂</sup>*2*φ*ˆ); on the other hand, the incompressibility of the fluid makes its motion nondivergent such that (*<sup>v</sup>*1, *v*2) is alternatively expressed as (*<sup>v</sup>*1 = −*∂*2*ψ*ˆ, *v*2 = *<sup>∂</sup>*1*ψ*ˆ), where *ψ*ˆ denotes a stream function. Equating these two, we obtain *<sup>∂</sup>*1*φ*<sup>ˆ</sup> = −*∂*2*ψ*ˆ, *<sup>∂</sup>*2*φ*<sup>ˆ</sup> = *<sup>∂</sup>*1*ψ*ˆ, showing that *φ*ˆ and *ψ*ˆ satisfy the Cauchy–Riemann relation in complex analysis. This heuristic example serves as a helpful reference in proving that *a null vector current ∂μφ propagating along the x*1*axis perpendicular to <sup>F</sup>μνcan be reinterpreted as the current of the longitudinal* (*x*<sup>1</sup>*-directed) electric field, of which a detailed explanation is given in [10].* As referred to at the beginning of this subsection, the existence of this longitudinally propagating electric field was actually reported in [16,17]. Thus, we can say that the vector field *∂μφ* is the physical mode that represents a longitudinally propagating electric field.

The orthogonality condition (11) is mathematically equivalent to the relativistic hydrodynamic equation of motion of a barotropic (isentropic) fluid [18]: *ωμν*(*wu<sup>ν</sup>*) = 0, where *ωμν* := *∂μ*(*wuν*) − *∂ν*(*wuμ*), *<sup>u</sup>ν*, and *w* are the vorticity tensor, 4-velocity, and proper enthalpy density of the fluid, respectively. This observation suggests that the unknown form of the 4-vector potential *<sup>U</sup>μ* can be clarified through the Clebsch parameterization [19] because the Clebsch parameterization is used to study the Hamiltonian structure of the above-mentioned barotropic fluid motion in terms of a couple of canonically conjugate scalar parameters (*<sup>λ</sup>*, *φ*) whose two degrees of freedom are equal to those of (-*E*, *M*- ) in electromagnetic waves. Thus, in case (I) of the semi-spacelike CD field, the electromagnetic vector potential *<sup>U</sup>μ* is parameterized as

$$\mathcal{U}\_{\mathbb{H}} = \, \, \lambda \partial\_{\mathbb{H}} \phi \,, \, \left( \phi = \partial\_{\mathbb{V}} A^{\mathbb{V}} \right) \text{ which satisfies } \partial^{\mathbb{V}} \partial\_{\mathbb{V}} \phi = 0 \text{)} \,, \tag{13}$$

$$
\partial^\nu \partial\_\mathcal{V} \lambda - (\kappa\_0)^2 \lambda \quad = \quad 0,\tag{14}
$$

where *κ*0 is a constant determined by DP experiments. If we introduce two gradient vectors—*Lμ* := *∂μλ* and *Cμ* := *∂μφ*, then the skew-symmetric field strength *<sup>S</sup>μν* can be represented by a simple bivector of the form

$$S\_{\mu\nu} = L\_{\mu}\mathbb{C}\_{\nu} - L\_{\nu}\mathbb{C}\_{\mu\nu} \quad \rightarrow \quad Pf(S) := S\_{01}\mathbb{S}\_{23} + S\_{02}\mathbb{S}\_{31} + S\_{03}\mathbb{S}\_{12} = 0,\tag{15}$$

which shows that, as in the case of - *E* and *H* - of an electromagnetic wave, the "electric" and "magnetic" fields of the CD field also satisfy the above orthogonality condition. *P f*(*S*) in (15) is the Pfaffian of the skew-symmetric matrix *<sup>S</sup>μν* : (*P f*(*S*))<sup>2</sup> = *Det*(*<sup>S</sup>μν*) and the barotropic fluid motions governed by the equation of motion *ωμν*(*wu<sup>ν</sup>*) = 0 are characterized by the condition that the Pfaffian vanishes. Another important property of an electromagnetic

wave is that - *E* and *H*- are advected along a null Poynting vector. In the CD model now under consideration, a null vector *Cμ* would naturally be expected to satisfy

$$
\mathbb{C}^\upsilon \partial\_\upsilon L\_\mu = 0,\tag{16}
$$

from which we obtain

$$L^{\mu}(\mathbb{C}^{\upsilon}\partial\_{\upsilon}L\_{\mu})\ \ =\ \ 0,\ \ \rightarrow \ \ \ \mathbb{C}^{\upsilon}\partial\_{\upsilon}(L^{\mu}L\_{\mu})=0,\tag{17}$$

$$\mathbb{C}^{\mu}(\mathbb{C}^{\nu}\partial\_{\boldsymbol{\nu}}L\_{\mu})\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; C^{\nu}\partial\_{\boldsymbol{\nu}}(\mathbb{C}^{\mu}L\_{\mu}) = 0. \tag{18}$$

In deriving (18), we utilized the fact that *Cν∂νCμ* = 0. For (18), the following orthogonality condition in the CD field

$$L\_V \mathbb{C}^V = 0 \tag{19}$$

can be imposed as an additional condition, which turns out later to be an important equation.

To see in what sense (19) is consistent with (15), we consider a null geodesic field (*UνU<sup>ν</sup>* = 0):

$$
\mathcal{U}I^{\nu}\partial\_{\nu}\mathcal{U}\_{\mu} = \mathcal{U}^{\nu}(\partial\_{\nu}\mathcal{U}\_{\mu} - \partial\_{\mu}\mathcal{U}\_{\nu}) = 0,\tag{20}
$$

which is expected to satisfy an extended light field. Using (13) and (15), we readily obtain

$$
\Delta U^{\nu} \partial\_{\nu} U\_{\mu} = -S\_{\mu \nu} (\lambda \mathcal{C}^{\nu}) = (\mathbb{C}\_{\mu} L\_{\nu} - L\_{\mu} \mathbb{C}\_{\nu}) (\lambda \mathcal{C}^{\nu}) = (L\_{\nu} \mathcal{C}^{\nu}) \lambda \mathcal{C}\_{\mu \nu} \tag{21}
$$

which vanishes by the orthogonality condition (19). The importance of (19) in the CD field formulation is that *Lμ* must be a spacelike vector, because *Lμ* satisfying (19) is either *Cμ* or a spacelike vector, which explains why the *λ* field introduced in the CD formulation satisfies the spacelike KG equation given in (14). Using the relations derived above between *Cμ* and *<sup>L</sup>μ*, we can show the form of the extended Maxwell's equation:

$$
\partial^\nu \mathcal{S}\_{\nu\mu} = (\mathbf{x}\_0)^2 \mathcal{U}\_\mu \quad \Longleftrightarrow \quad [\partial^\nu \partial\_\nu - (\mathbf{x}\_0)^2] \mathcal{U}\_\mu = 0, \quad \text{(with } \partial\_\nu \mathcal{U}^\nu = 0). \tag{22}
$$

The energy-momentum tensor *T*ˆ *ν μ* of the lightlike CD field can be derived easily from the conventional one with the following form: *T ν μ* = <sup>−</sup>*FμσFνσ*. Considering the sign change of the energy at the boundary between the timelike and spacelike domains, we define the tensor as

$$\begin{aligned} \hat{\mathcal{T}}\_{\mu\nu} &: \quad = \mathcal{S}\_{\mu\sigma} \mathcal{S}\_{\nu}{}^{\sigma} = (L\_{\mu} \mathbb{C}\_{\sigma} - \mathbb{C}\_{\mu} L\_{\sigma}) (L\_{\nu} \mathbb{C}^{\sigma} - \mathbb{C}\_{\nu} L^{\sigma}) \\ &= \quad (L\_{\sigma} L^{\sigma}) \mathbb{C}\_{\mu} \mathbb{C}\_{\nu} = \rho \mathbb{C}\_{\mu} \mathbb{C}\_{\nu}, \ \rho := L\_{\sigma} L^{\sigma} < 0. \end{aligned} \tag{23}$$

The negative density *ρ* corresponds to the negative norm of the longitudinal modes in the QED, which makes this mode unphysical in the conventional interpretation. However, we believe that the usage of the term "unphysical" in this context is inappropriate, because if we regard the CD field as virtual photons, then the former is physical in the sense that the latter, as the mediator of the electromagnetic force, is physical though it is invisible. As the argumen<sup>t</sup> regarding the reference point of the gravitational potential energy shows, the decision regarding whether a given quantity under consideration is physical depends essentially on the physical setting of our problem; therefore, the Clebsch duality relation between *<sup>F</sup>μν* and *<sup>S</sup>μν* should not be viewed as the duality between physical and unphysical aspects but instead as the duality between the positive and negative sides of the light-cone *p*2 = 0, the latter of which is, as we will see in Section 3 on cosmology, often closely related to the invisibility of a given quantity. Actually, the "state-dependent" physicality of the longitudinal photons was already pointed out by Ojima [20], who stated that while the longitudinal photons or unphysical Goldstone bosons in the Higgs mechanism are eliminated from the physical space of states in the usual formulation, this statement applies to the above modes only in their particle forms. In their non-particle forms, the former appear physically as infrared Coulomb tails, and the latter, as the so-called "macroscopic

wave functions" arising from the Cooper pairs, both of which play essential physical roles. The CD formulation based on the Greenberg–Robinson theorem has revealed that the momenta of the non-particle forms in the above statement are invisible non-localized spacelike ones. Thus, regarding the negativity of *ρ*, we point out that it can be likened to the simple fact that the complexified time coordinate *ict* in Minkowski space is invisible, though it is an important element without which we cannot describe a given dynamical system in a satisfactory way.

In step (II) of the CD field formulation, we relax the condition *∂ν∂νφ* = 0 given by the second equation in (10) to allow the following extended vector potential *<sup>U</sup>μ*, which is advected by itself along a geodesic:

$$\begin{split} \mathcal{U}\_{\mu} := \frac{1}{2} (\lambda \mathbb{C}\_{\mu} - \phi \mathbb{L}\_{\mu}), \implies \mathcal{U}^{\nu} \partial\_{\nu} \mathcal{U}\_{\mu} = \begin{aligned} -\mathbb{S}\_{\mu \nu} \mathcal{U}^{\nu} + \frac{1}{2} \partial\_{\mu} (\mathcal{U}^{\nu} \mathcal{U}\_{\nu}) &= 0, \\ \mathcal{U}\_{\nu} \mathcal{U}^{\nu} &< 0, \end{aligned} \tag{24}$$

$$
\partial^\nu \partial\_\nu \lambda - (\kappa\_0)^2 \lambda = 0, \ \partial^\nu \partial\_\nu \phi - (\kappa\_0)^2 \phi \quad = \quad 0, \ \mathcal{C}^\nu L\_\nu = 0. \tag{25}
$$

The form of *<sup>S</sup>μν*, given by the first equation in (15), remains unchanged in (24). Note that the condition *∂ν∂νφ* = 0 (*φ* = *∂νAν*) can certainly be considered a gauge fixing condition, but at the same time, the second equation in (10) can be interpreted as a special gauge condition where gauge invariance is represented by the charge conservation due to *∂μ∂νFμν* = 0, while *∂μφ* is not a usual timelike electric current.

In the extended Maxwell's equation given in (22), an electrically neutral current (*<sup>κ</sup>*0)<sup>2</sup>*Uμ* = (*<sup>κ</sup>*0)<sup>2</sup>(*λ∂μφ*) behaves exactly like *jμ* in the original Maxwell's equation, which shows that the constant *κ*0 serves as a fundamental unit, such as the electric charge. Therefore, violation of condition (10) causes gauge symmetry breaking, according to which the CD field extended in step (II) suffers from breakdown of both the gauge symmetry and conformal symmetry in the sense of *UνU<sup>ν</sup>* = 0.

Corresponding to the above extension, the energy-momentum tensor satisfying the conservation law of *∂νT*<sup>ˆ</sup> *νμ* = 0 is redefined as

$$\begin{array}{rcl} \mathcal{T}\_{\mu\nu} & = & \mathcal{S}\_{\mu\sigma\nu}{}^{\sigma} - \frac{1}{2} \mathcal{S}\_{a\beta}{}^{a\beta} \eta\_{\mu\nu\prime} & \mathcal{S}\_{a\beta\gamma\delta} := \mathcal{S}\_{a\beta} \mathcal{S}\_{\gamma\delta\prime} \\ \iff & \mathcal{G}\_{\mu\nu} := \mathcal{R}\_{\mu\nu} - \mathcal{R} \mathcal{g}\_{\mu\nu}/2. \end{array} \tag{26}$$

Note that *S* ˆ *αβγδ* defined above has the same skew-symmetric properties as those of the Riemann tensor *<sup>R</sup>αβγδ*, including the first Bianchi identity, *<sup>S</sup>α*[*βγδ*] = 0 (equivalent to the second equation in (15)), which is valid as *<sup>S</sup>μν* is a bivector field given by the first equation in (15). Thus, *T* ˆ *μν* given in (26) becomes isomorphic to the Einstein tensor *<sup>G</sup>μν* := *<sup>R</sup>μν* − *Rgμν*/2, where the Ricci tensor *<sup>R</sup>μν* := *<sup>R</sup>σμνσ*.

### *2.2. Quantization of the CD Field and DP Model*

Going back to (23), we note that it is isomorphic to the energy-momentum tensor of freely moving fluid particles. The *ρ* field for an actual fluid will be discretized if the kinetic theory of molecules is taken into account. When the light field is quantized, this form will obey Planck's quantization of light energy *E* = *h<sup>ν</sup>*. As the CD field variable *Lμ* has the dimension of length, we introduce a certain quantized elemental length *ldp* whose inverse is *κ*0, namely, the discretization of *ρ* leads to

$$\kappa\_0 := (l\_{dp})^{-1}\text{.}\tag{27}$$

which can be considered an energy quantization of the CD field. Recall that the Dirac equation of the form

$$(i\gamma^{\nu}\partial\_{\nu} + m)\Psi = 0\tag{28}$$

can be regarded as the "square root" of the timelike KG equation (*∂ν∂ν* + *m*<sup>2</sup>)<sup>Ψ</sup> = 0. Therefore, the Dirac equation for the spacelike KG equation (*∂ν∂ν* − (*<sup>κ</sup>*0)<sup>2</sup>)<sup>Ψ</sup> = 0 must be

$$i(\gamma^\nu \partial\_\nu + \kappa\_0)\Psi = 0.\tag{29}$$

On the other hand, an electrically neutral Majorana representation exists for (28), in which all the *γ* matrices become purely imaginary such that these matrices have the form (*γ<sup>ν</sup>*(*M*)*∂ν* + *m*)<sup>Ψ</sup> = 0, which is identical to (29). The Majorana field is fermionic with a half-integer spin 1/2; thus, the same (momentum) state cannot be occupied by two fields according to Pauli's exclusion principle. Note that by using the Pauli–Lubanski vector *<sup>W</sup>μ* to describe the spin polarization of moving particles, we can find a specific orthogonal momentum configuration of a pair of Majorana fields whose resultant spin becomes 1, namely,

$$M\_{\mu\nu}p^{\nu} = N\_{\mu\nu}q^{\nu} = \mathcal{W}\_{\mu\nu} \tag{30}$$

where *<sup>M</sup>μν* and *pν* denote the angular and linear momenta of a given Majorana field, respectively, while *<sup>N</sup>μν* and *qν* are the corresponding momenta of the other, of which the linear momentum *qν* is perpendicular to *p<sup>ν</sup>*. We believe that this configuration (30) gives a quantum mechanical justification for the orthogonality condition (19) and (25) of the CD field.

For a plane wave solution (*λ* = *λ* ˆ *c* exp[*i*(*kνx<sup>ν</sup>*)]) to the spacelike KG equation (14), *Lν* = *∂νλ* satisfies

$$L^\nu L\_\nu^\* = - (\kappa\_0)^2 (\mathbb{X}\_c \mathbb{X}\_c^\*) = \text{const.} < 0,\tag{31}$$

which shows that the momentum vector *Lμ* lies in a submanifold of the Lorentzian manifold, called de Sitter space in cosmology, which is a pseudo-hypersphere with a certain constant radius embedded in *R*5. Quite independent of the cosmological arguments on de Sitter space, Snyder [21] discussed the unique role of this space in spacetime quantization. He showed that with the introduction of the hypothetical momentum 5-vector *η<sup>μ</sup>*(<sup>0</sup> ≤ *μ* ≤ 4) in *R*<sup>5</sup> constrained to lie on the de Sitter space, i.e., *ηνη*<sup>∗</sup>*ν* = <sup>−</sup>(*ηc*)<sup>2</sup> = *const*., the following commutation relations are derived. For the definitions of *pμ*, *p*<sup>ˆ</sup>*μ*, and *<sup>x</sup>*<sup>ˆ</sup>*μ*, we have

$$\begin{array}{rcl} p\_{\mu} &:& \quad = \frac{\hbar}{l\_{p}} \frac{\eta\_{\mu}}{\eta\_{4}}, \quad \not p\_{\mu} := \ -\frac{i\hbar}{l\_{p}\eta\_{4}} \frac{\partial}{\partial \eta\_{\mu}}, \quad \not \mathfrak{k}^{\mu} := i\mathfrak{l}\_{p} \left(\eta\_{4}\frac{\partial}{\partial \eta\_{\mu}} - \mathfrak{J}\_{\mu}\eta\_{\mu}\frac{\partial}{\partial \eta\_{4}}\right); \\ \not \mathfrak{k}(0 \le \leftarrow \mu \le 3), \end{array} \tag{32}$$

where *lp* denotes the Planck length, and *ξμ* takes a value of −1 when *μ* = 0 and 1 when *μ* = 0, from which we obtain

$$\begin{aligned} \left[\mathfrak{k}^{\mu}, \mathfrak{p}\_{\mu}\right] &= \quad i\bar{l}\left[1 + \mathfrak{f}\_{\mu}^{\mathrm{T}} \left(\frac{l\_{p}}{\bar{l}\mathfrak{t}}\right)^{2} (p\_{\mu})^{2}\right], \\\left[\mathfrak{k}^{\mu}, \mathfrak{p}\_{\nu}\right] &= \quad \left[\mathfrak{k}^{\nu}, \mathfrak{p}\_{\mu}\right] = i\bar{l}\mathfrak{l} \left(\frac{l\_{p}}{\bar{l}\mathfrak{t}}\right)^{2} p\_{\mu} p\_{\nu} \quad 0 \le (\mu, \nu) \le 3, \end{aligned} \tag{33}$$

$$\left[\mathfrak{X}^{i},\mathfrak{X}^{j}\right]\_{\cdot} = \begin{array}{c} i(l\_{p})^{2} \\ \frac{1}{\hbar} \end{array} \mathfrak{c}\_{ijk} L\_{k\prime} \quad \left[\mathfrak{X}^{0},\mathfrak{X}^{i}\right] = \frac{i(l\_{p})^{2}}{\hbar} M\_{\hat{\mathbf{i}}} ; \quad 1 \le (i,j,k) \le 3,\tag{34}$$

where *ijk* is Eddington's epsilon, and *Li* and *Mi* are angular momentum vectors generated, respectively, by (spatial-spatial) and (spatial-temporal) rotations. *Snyder further showed that the "Lorentz transformation" in his spacelike momentum space* {*η<sup>μ</sup>*}, (0 ≤ *μ* ≤ 3) *naturally induces the Lorentz transformation in the usual spacetime* {*x<sup>μ</sup>*}. *Thus, the energy-momentum tensor T* ˆ *μν of the CD field given in (26) can be regarded as the one constructed on this Snyder's momentum "spacetime" ημ with Lorentz invariance as in the case of <sup>R</sup>μν*, *also constructed on the spacetime xμ with Lorentz invariance, which becomes a very important property in the discussion of dark energy in the next section.* In [12] and S3O, we showed that, by virtue of the bivector property of *<sup>S</sup>μν* given in (15), the form of *T* ˆ *μν* can be extended to a curved spacetime. Thus, the intriguing isomorphism between *<sup>T</sup>*<sup>ˆ</sup>*μν* and *<sup>G</sup>μν* in (26) seems to sugges<sup>t</sup> an important consequence: the quantization of the CD field attained by the above commutation relations may also be applied to the quantization of the gravitational field. The research pursuing this goal can be found, for instance, Girelli [22] and Glikman [23].

Now, we move on to a new DP model. Although the constant *κ*0 plays a crucial role in formulating the CD field, its value clearly cannot be determined solely by theoretical arguments. We already explained in S3O how the value of the DP constant *κ*0 was estimated by the extensive DP experiments by Ohtsu, who utilized the photochemical vapor deposition and autonomous etching techniques [24]. Through those experiments, the maximum size of the DP that can be considered as *ldp*introduced in (27) was estimated to be

$$50\text{ nanometer} \quad < l\_{dp} = \left(\kappa\_0\right)^{-1} < \text{ 70 nanometer.}\tag{35}$$

As emphasized in the introduction, we do not ye<sup>t</sup> know a reliable QFT that can deal with the off-shell properties of the field playing an important role in the DP generating mechanism. Thus, we need to resort to a certain kind of simplified argumen<sup>t</sup> to bring in the experimental outcome to CD field theory. In the following, we give such a simplified argument. In the first paragraph of the introduction, we mentioned that the existence of point-like singularities, similar to the pointed end of a fiber probe or impurities with extremely tiny size scattered across a given background material, is the crucial element for generating DPs. We can safely say that field interactions in which these singularities come into play should be so serious that the involvement of the spacelike momenta predicted by the Greenberg–Robinson theorem will be crucial in these cases compared with those without singularities.

Remember that, in the introductory Section 1, we have touched upon a heuristic toy model with which we show the intervention of spacelike momentum in the field interactions. Aharonov et al. [25] conducted an advanced analysis of the response behavior of the spacelike KG equation perturbed by a point-like delta function *<sup>δ</sup>*(*x*<sup>0</sup>)*δ*(*x*<sup>1</sup>), in which the above essential aspect was incorporated. They showed that the solutions excited by this point-like disturbance consist of two different types: the stable spacelike mode and the unstable timelike mode. The unstable timelike mode excited from the spacelike KG Equation (14) with spherical symmetry has the form *<sup>λ</sup>*(*x*0,*r*) = exp(±*k*0*x*<sup>0</sup>)*R*(*r*), where *<sup>R</sup>*(*r*) satisfies

$$R'' + \frac{2}{r}R' - (\hat{\kappa}\_I)^2 R = 0, \quad (\hat{\kappa}\_I)^2 := (k\_0)^2 - (\kappa\_0)^2 > 0,\tag{36}$$

according to which *<sup>R</sup>*(*r*) is the Yukawa potential of *<sup>R</sup>*(*r*) = exp(−*κ*<sup>ˆ</sup>*r<sup>r</sup>*)/*<sup>r</sup>*. For a Majorana field, as with the quantum version of the *λ* field, the energy in terms of *k*0 is discretized by *κ*0, as shown in (27). Thus, the nonzero minimum *Min*[*κ*<sup>ˆ</sup>*r*] in the Yukawa potential is *κ*0, which gives the maximum size of the localized DP to be compared with the experimental result (35). Although the CD field consists of a pair of Majorana fields satisfying the orthogonality conditions (19) and (25), the orthogonal configuration must be broken down by the perturbation, and the timelike pair will turn, respectively, into *<sup>λ</sup>*(*x*0,*r*) = exp(±*k*0*x*<sup>0</sup>)*R*(*r*), namely, particle and antiparticle pairs, as an electrically neutral antiparticle can be considered a particle traveling backward in time. The excited field is non-propagating in nature; thus, a pair of particle and antiparticle fields will be combined into either an "electric" field with spin 0 or a "magnetic" field with spin 1 [26]. We believe that the DP is generated through this pair annihilation of the Majorana field. As the DP field is basically electromagnetic, once it is generated, its behavior in a uniform environment can be described by the Proca equation of the form *∂ν∂νAμ* + (*<sup>κ</sup>*0)<sup>2</sup>*Aμ* = 0. From the viewpoint of nanophotonical engineering, however, what really matters is the control of the DP energy flows driven by the existence of point-like sources and sinks. In the above argument, we showed that the energy of incident photons working as the triggering cause of *<sup>δ</sup>*(*x*<sup>0</sup>) at the singular point eventually turns into the energy of the DP. At the present stage, we do not have clear knowledge of the sink mechanisms, but the research on DP energy flow with source–sink-type

driving forces is pursued actively by employing a certain class of quantum walk models [27–29]. Intuitively, however, we can expect that some kind of *ζ*-function enters here as the carrier to convey the above singularity waves, which explains the observation of *ζ*-function singularities in the quantum walks. Moreover, the parallelism between *ζ*-functions and partition functions (the latter appearing in statistical mechanics) explains the relevance of Tomita–Takesaki modular duality [30] to the basis of the conformal symmetry discussed below.

### **3. On Dark Energy and Dark Matter**

In our discussion so far, we have developed a new concept of a CD field carrying spacelike momentum modes, which are required for electromagnetic field interactions. In comparison to the conventional QFT, the CD field can be compared with invisible virtual photons that can be excited from the vacuum (|0 = 0), regarded as the ground state of a one-sided energy spectrum within the bound of the uncertainty principle. Apparently, simply employing this excitation scenario is problematic because the concept of the CD field contradicts the vacuum state mentioned above. We believe that the orthogonal relation between a pair of momentum vectors *pν* and *qν* given in (30) gives us a hint to solve this problem concerning the ground state. For spacetime with three spatial dimensions, as shown below, the maximum number of Majorana fermion fields as the limited capacity of spacetime is also three, of which the configuration is shown by

$$M\_{\mu\nu}p^{\nu} = N\_{\mu\nu}q^{\nu} = L\_{\mu\nu}r^{\nu} = \mathcal{W}\_{\mu}.\tag{37}$$

This compound state with a resultant spin 3/2 is called a Rarita–Schwinger state, which we denote by |*<sup>M</sup>*3*<sup>g</sup>*. The role of the vector |*<sup>M</sup>*3*g* is to give the GNS cyclic vector of a mixed state which is disjoint from the vacuum state whose cyclic vector is given by |0 [31]. The important characteristic of |*<sup>M</sup>*3*g* is that the CD vector boson field can be excited from any of the three different pairs, which propagates along one of the (*x*1, *x*2, *x*3) directions. In view of the universality of electromagnetic interactions, the incessant occurrence of excitation–de-excitation cycles between |*<sup>M</sup>*3*g* and non-ground states makes |*<sup>M</sup>*3*g* a fully occupied state in the macroscopic time scale. Therefore, we can say that |*<sup>M</sup>*3*g* exists not as a momentary virtual state, but also as a stable invisible off-shell state. In the following, we show that |*<sup>M</sup>*3*g* exerts on the universe a cosmological effect identified as dark energy.

To investigate the property of |*<sup>M</sup>*3*<sup>g</sup>*, let us consider plane wave solutions *λ* and *φ* for the spacelike case of *UνU<sup>ν</sup>* < 0, in which *λ* = *<sup>N</sup>λλ*<sup>ˆ</sup> *c* exp(*ikνx<sup>ν</sup>*) and *φ* = *<sup>N</sup>φφ*<sup>ˆ</sup>*c* exp(*ikνx<sup>ν</sup>*), with *kνk<sup>ν</sup>* = −(*<sup>κ</sup>*0)2, where *λ*ˆ *c* and *φ*ˆ*c* denote elemental amplitudes of the respective fields, and *Nλ* and *<sup>N</sup>φ* are the numbers of the respective modes. As Equation (15) shows, *λ* and *φ* always appear in the form of a product; thus, we may rewrite these two expressions as

$$
\lambda = N(\kappa\_0)^{-2} \exp\left(ik\_\nu \mathbf{x}^\nu\right), \quad \phi = \hat{\phi}\_c \exp\left(ik\_\nu \mathbf{x}^\nu\right), \tag{38}
$$

where *N* is a combined number *N* := *<sup>N</sup>λNφ*, and we can identify *λ*ˆ *c* as *λ*ˆ *c* = (*<sup>κ</sup>*0)−2, as *λ*ˆ *c* has the dimension of (*length*)2. By substituting these into the first equation in (26) and setting *N* = 1, we obtain the absolute value of *T*ˆ *ν ν* (1), denoted as |*T*ˆ *ν ν* (1)|:

$$|\hat{T}\_{\nu}^{\,\,\nu}(1)| = -2[\hat{\phi}\_{\varepsilon}(\hat{\phi}\_{\varepsilon})^\*] < 0,\tag{39}$$

where (•)<sup>∗</sup> denotes the complex conjugate of (•). The right-hand side of (39) can be evaluated by the light-like case of the CD field (23), in which we have *T*ˆ*μν* = *ρCμC<sup>ν</sup>*. For the light-like case, we have *φ* = *φ*ˆ*c* exp(*ikνx<sup>ν</sup>*), *kνk<sup>ν</sup>* = 0 and *λ* = *<sup>N</sup>*(*<sup>κ</sup>*0)−<sup>2</sup> exp(*ilνx<sup>ν</sup>*), *lνl ν* = −(*<sup>κ</sup>*0)2, from which we have

$$(\mathbb{C}\_{\mu})^\* \mathbb{C}^\vee = k\_{\mu} k^\nu \hat{\phi}\_{\varepsilon} (\hat{\phi}\_{\varepsilon})^\*, \quad \rho = -N^2 (\kappa\_0)^{-2}. \tag{40}$$

Next, we consider a case in which the *kμ* vector of *φ* is parallel to the *x*1 direction and consider a rectangular parallelepiped *V* spanned by the vectors (1/*k*1, 1, <sup>1</sup>). For *k*0 = *<sup>ν</sup>*0/*<sup>c</sup>*, where *c* and *ν*0 denote the light velocity and the frequency of the *φ* field, the volume integral of *T*ˆ 0 0 /(− *N*<sup>2</sup>) over *V* as the energy per quantum is

$$\frac{1}{\left(-N^{2}\right)}\int\_{V}\mathcal{T}\_{0}^{0}dx^{1}dx^{2}dx^{3} = (\kappa\_{0})^{-2}\epsilon[\phi\_{c}(\phi\_{c})^{\*}]\frac{\nu\_{0}}{c},\tag{41}$$

where denotes the unit length squared. Equating (41) with *E* = *h<sup>ν</sup>*0, we obtain

$$\ln c(\kappa\_0)^2 = \varepsilon[\hat{\phi}\_\varepsilon(\hat{\phi}\_\varepsilon)^\*], \quad \varepsilon = 1 \,\text{(meter)}^2. \tag{42}$$

As stated after (37), we need three fields propagating along the *x*1, *x*2, and *x*3 directions to achieve isotropic radiation of the CD field. These three fields are given by (*<sup>S</sup>*23, *<sup>S</sup>*02), (*<sup>S</sup>*31, *<sup>S</sup>*03), and (*<sup>S</sup>*12, *<sup>S</sup>*01). The energy-momentum tensor *T*ˆ *ν μ* (3) derived by the superposition of these fields becomes

$$
\mathcal{T}^{\nu}\_{\mu}(3) = \begin{pmatrix}
\tau\sigma & 2\tau^2 - \sigma^2 & 0 & 0 \\
\tau\sigma & 0 & 2\tau^2 - \sigma^2 & 0 \\
\tau\sigma & 0 & 0 & 2\tau^2 - \sigma^2
\end{pmatrix}.\tag{43}
$$

In deriving (43), we set *S*23 = *S*31 = *S*12 = *σ* and *S*01 = *S*02 = *S*03 = *τ*. We note that *T*ˆ *ν μ* (3) can be regarded as the energy-momentum tensor of the anti-dark energy (dark energy with a negative energy density, that is, *T*ˆ 0 0 (3) = −3*σ*<sup>2</sup> < 0). Dark energy (with positive energy density) ∗*T*<sup>ˆ</sup> *ν μ* (3) having exactly the same trace as that of the anti-dark energy *T*ˆ *ν μ* (3) can be introduced by the Hodge dual exchange between (*<sup>σ</sup>*, *τ*) and (*<sup>i</sup><sup>τ</sup>*, *<sup>i</sup>σ*) in (43), which becomes

$${}^{\nu}\hat{\mathcal{T}}\_{\mu}^{\;\;\nu}(3) = \begin{pmatrix} 3\tau^2 & \tau\sigma & \tau\sigma & \tau\sigma \\ -\tau\sigma & -2\sigma^2 + \tau^2 & 0 & 0 \\ -\tau\sigma & 0 & -2\sigma^2 + \tau^2 & 0 \\ -\tau\sigma & 0 & 0 & -2\sigma^2 + \tau^2 \end{pmatrix}. \tag{44}$$

At this point, we recall the important remark on the validity of extending our discussion, which started from Minkowski space, to the case of a curved spacetime. As already pointed out in the explanation of Snyder space written in italics below in Equation (34), the isomorphism between *<sup>T</sup>*<sup>ˆ</sup>*μν* and *<sup>G</sup>μν* given in (26) can be extended to a curved spacetime by virtue of the bivector property of (15). If the dark energy is modeled by a cosmological term of <sup>Λ</sup>*gμν*, then the Einstein field equation with the sign convention of *<sup>R</sup>μν* = *Rσμνσ* together with the metric convention of (+1, −1, −1, −<sup>1</sup>) becomes

$$R^{\nu}\_{\mu} - \frac{R}{2} \mathbf{g}^{\nu}\_{\mu} + \Lambda \mathbf{g}^{\nu}\_{\mu} = -\frac{8\pi G}{c^4} T^{\nu}\_{\mu}{}^{\nu} \tag{45}$$

where Λ becomes negative for an expanding universe. Before proceeding further, we note that ∗*T*<sup>ˆ</sup> *ν μ* (3) is not a quantity that directly fits into the conventional cosmological analysis utilizing the isotropic spacetime structure assumed by Weyl's hypothesis on the cosmological principle. First, as ∗*T*<sup>ˆ</sup> *ν μ* (3) is spacelike in nature, it cannot be reduced to a diagonalized matrix form. Second, it is the energy-momentum tensor of fermionic |*<sup>M</sup>*3*g* with spin 3/2. The crucial problem in our analysis therefore is whether we can find observable quantities in ∗*T*<sup>ˆ</sup> *ν μ* (3). Because the relevant criterion for singling out an observable quantity may depend on the situation, we have no choice but to make a good guess. The fact that seems to work as "the guiding principle" is that within the framework of relativistic QFT, any observable without exception associated with a given internal symmetry is invariant under the action of a transformation group materializing the symmetry under consideration. By extending this knowledge on the internal symmetry to the external (spacetime) one, we assume that the trace Λ*deg νν* defined by

$$
\Lambda\_{\rm d\varepsilon} g\_{\nu}^{\;V} := -\frac{8\pi G}{c^4} \, ^\*\hat{T}\_{\nu} (\mathfrak{d}) > 0, \quad \rightarrow \quad \Lambda\_{\rm d\varepsilon} = \frac{12\pi G h}{c^3 \epsilon} (\kappa\_0)^2 \tag{46}
$$

is observable as the invariant of the general coordinate transformation, which is consistent with the built-in Lorentz invariance of Snyder's momentum space on which the CD field is constructed. Thus, the validity of our new model on dark energy can be checked by comparing the following two models:

$$\begin{aligned} R\_{\mu}^{\;\nu} - \frac{R}{2} \mathbf{g}\_{\mu}^{\;\nu} - \Lambda\_{obs} \mathbf{g}\_{\mu}^{\;\nu} &= \quad - \frac{8\pi G}{c^4} T\_{\mu}^{\;\nu} \\ R\_{\mu}^{\;\nu} - \frac{R}{2} \mathbf{g}\_{\mu}^{\;\nu} &= \quad - \frac{8\pi G}{c^4} T\_{\mu}^{\;\nu} + \Lambda\_{d\epsilon} \mathbf{g}\_{\mu}^{\;\nu} \end{aligned} \tag{47}$$

where Λ*obs* denotes the value obtained by Planck satellite observations. (In S3O, Λ*obs* in the above Equation (47) appeared with the wrong sign in the corresponding Equation (25), which should be corrected.) Using (39), ∗*T* ˆ *ν ν* (3) = 3*T*<sup>ˆ</sup> *νν* (1), and (42), we obtain Λ*de* ≈ 2.47 × 10−<sup>53</sup> m<sup>−</sup><sup>2</sup> and Λ*obs* ≈ 3.7 × 10−<sup>53</sup> m<sup>−</sup><sup>2</sup> [32]. Thus, |*<sup>M</sup>*3*g* seems to be a promising candidate model for dark energy.

In the above arguments on the dark energy model, the physical meaning of the "real" cosmological term <sup>Λ</sup>*gμν* should be revised, because it does not correspond in our model to dark energy. We believe that one of the intriguing possibilities is that <sup>Λ</sup>*dmgμν* with Λ*dm* > 0 (valid in our sign convention) represents dark matter. The main reason for this is due to a simple fact that we can represent the metric tensor *gμν* in terms of the Weyl (conformal) curvature tensor *<sup>W</sup>αβγδ* as long as its magnitude does not vanish, namely,

$$\mathcal{g}\_{\mu\nu} = \frac{4}{W^2} \mathcal{W}\_{\mu a \beta \gamma} \mathcal{W}\_{\nu}{}^{a\beta\gamma}, \quad \mathcal{W}^2 := \mathcal{W}\_{a\beta\gamma\delta} \mathcal{W}^{a\beta\gamma\delta} \neq 0,\tag{48}$$

as shown by straightforward calculations [33]. Recall that Weyl curvature represents the deviation of spacetime from the conformally flat Friedmann–Robertson–Walker (FRW) metric for an isotropic universe. In addition, the monotonic decrease in *W*<sup>2</sup> along the radial direction in the field of *<sup>W</sup>αβγδ* in the well-known spherically symmetric Schwarzschild outer solution of a given star suggests that the local maxima of *W*<sup>2</sup> would behave as "particles" or that its existence tends to correlate with the created matter field. Therefore, *T* ˜ *μν*, defined as

$$
\bar{T}\_{\mu\nu} := \Lambda\_{dm} g\_{\mu\nu}, \quad \Lambda\_{dm} > 0, \text{ g\u0} > 0,\tag{49}
$$

to be put on the left-hand side of (45), gives an energy-momentum tensor of this pseudomatter field as a candidate for dark matter. The existence of *<sup>T</sup>*˜*μν* will further accelerate the deviation of spacetime from the FRW metric and thus serve as the fostering mechanism of galaxy formation. (In Equation (30) of S3O, the above *T* ˜ *μν* was defined with negative Λ*dm*, which is a second error related to the first error of +Λ*obs* in (47)). In determining the magnitude of Λ*dm*, we first refer to the observational fact that the estimated abundance ratio of dark energy to dark matter is 3 : 1. AS Λ*de* = −<sup>∗</sup>*T* ˆ *ν ν* (3) = −3*T*<sup>ˆ</sup> *νν* (1), we have

$$
\Lambda\_{dm} = -\hat{T}\_\nu^\nu(1) = \frac{\Lambda\_{dc}}{3},
\tag{50}
$$

the theoretical justification of which is given in the next section. Notice that the constant *T* ˆ *ν ν* (1) appearing first in (39) is a quantity belonging to the off-shell electromagnetic field discussed in Section 2.1 in which spacelike CD field is introduced by the conformal symmetry breaking (CSB) of light-like CD field. Although we already alluded to the importance of CSB in our previous paper (S3O), our discussion on it in the context of cosmological dynamics remains quite vague. In the subsequent section covering the main theme of this paper, we will show that the new notion of CSB which applies simultaneously to

electromagnetic as well as gravitational fields will play an important role in connecting our novel cosmological model to the preceding intriguing CCC proposed by Penrose [14,34].
