Dressed Photon Constant as a Key Parameter for the Conformal Cyclic Cosmology of Twin Universes

As an important follow-up report on the latest study of the first author (H.S.) on an off-shell 1 quantum field causing a dressed photon and dark energy, we further discuss a couple of intriguing 2 subjects based on our new theory. One is the dressed photon constant. If we use it, in addition to h̄ and c, 3 as the third component of natural units, then it is defined as the geometric mean of the smallest and the 4 largest lengths: Planck length and that relating to the cosmological constant. Interestingly, this length (≈ 5 50 nanometers) seems to give a rough measure of the Heisenberg cut for electromagnetic phenomena. 6 The other is a new perspective on cosmology that combines two original notions, i.e., twin universes and 7 conformal cyclic cosmology, proposed respectively by Petit and Penrose, into one novel picture where 8 universes expand self-similarly. We show the possibility that twin universes having a dual structure 9 of [matter with (dark energy & matter)] vs. corresponding anti-entities, separated by an event horizon 10 embedded in the geometric structure of de Sitter space, undergo endless cyclic processes of birth and 11 death, as in the case of the pair creation and annihilation of elementary particles through the intervention 12 of a conformal light field. 13

First, the physical interpretations of QFT described by the interacting Heisenberg fields ϕ H are realized by the notion of on-shell particles contained in ϕ H with the 4-mometum p µ given by Eq. (1): p 2 := η µν p µ p ν := p ν p ν = (m 0 c) 2 ≥ 0, µ, ν = 0, 1, 2, 3, where we adopt the sign convention (+1, −1, −1, −1) for the Minkowski metric η given by The physical meaning of the asymptotic fields φ as (as = in or out) can be seen in their role in a scattering 43 process formed by the in-fields φ in 1 (p 1 ), · · ·, φ in m (p m ) with momenta p 1 , · · ·, p m converging from the 44 remote past to the scattering center and by the out-fields φ out 1 (q 1 ), · · ·, φ out m (q n ) with momenta q 1 , · · ·, q n carrying the above momentum spectrum as an observable quantity can be easily realized as a free field 48 obtained by the so-called second quantization, as shown below. Owing to its linearity, the asymptotic field 49 φ as is governed by the well-known Klein-Gordon (KG) equation (2). 50 In the simplest case of a scalar field φ as , the first quantization p µ → ih∂ µ applied to (1) realizes the KG equation: [h 2 ∂ ν ∂ ν + (m 0 c) 2 ]φ as = 0, where the operand φ as determined by the second quantization becomes a quantum field φ as describing a multiparticle system given by space generated by repeated applications of the creation operators on the Fock vacuum |0 (under the 58 cyclicity assumption) are misinterpreted as the universal structure to be found in interacting multiparticle 59 systems. Accordingly, |0 becomes as mysterious as the creation of everything from emptiness. We return 60 to this point in section 4 on cosmology. 61 The mutual relations among the Poincaré group P, Heisenberg field ϕ H , asymptotic field φ as and 62 momentum spectrum (p µ ) can be clearly visualized by means of the quadrality scheme to describe the The asymptotic field φ as given by (3)  Greenberg-Robinson theorem [4,5], states that if the Fourier transformation ϕ(p) of a given quantum field φ as (x) 69 does not contain an off-shell spacelike momentum p µ with p ν p ν < 0 (cf. Eq. (1)), then φ as (x) is a generalized free 70 field. A caveat to be made here is that a spacelike momentum field does not necessarily mean the presence 71 of a tachyonic field representing particle-like localized energy field moving with superluminous velocity, 72 which breaks the Einstein causality. This localized field is known to be unstable such that the existing 73 spacelike momentum fields take naturally nonlocal wavy forms. Another crucial piece of knowledge 74 necessary to understand the enigmatic DP phenomena is the important property of quantum fields with 75 infinite degrees of freedom. Unlike a quantum mechanical system with finite degrees of freedom, for which 76 we have only one sector with unitary time evolution governing a given system, there exist, for quantum 77 fields with infinite degrees of freedom, multiple sectors [6] that are mutually disjoint (i.e., separated by the 78 absence of intertwiners), stronger than unitary inequivalence. Regarding the unitary equivalence, Haag's 79 theorem [7] states that any quantum field satisfying Poincaré covariance is a free field if it is connected to a free 80 field by a unitary transformation. According to this no-go theorem, it is meaningless to replace the interacting 81 Heisenberg field with a unitarily transformed free field obtained from the interaction term represented by 82 the well-known Dyson S-matrix. In this way, the essential part of our common knowledge cultivated in 83 quantum mechanical systems with finite degrees of freedom is invalidated in relativistic QFT.

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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,  and the latest S3O) derived an extended field covering the spacelike momentum domain by applying a mathematical technique called Clebsch parameterization to electromagnetic 4-vector potential A µ . The extension of the field was To confirm what is stated above, let us consider Maxwell's equation (5) and the associated 140 energy-momentum tensor (7), together with its divergence (8), as follows: If the Lorentz gauge condition ∂ ν A ν = 0 is imposed, additionally or formally, to the above Maxwell's equation, then Eq. (5) reduces to ∂ ν ∂ ν A µ = 0, according to which the free Maxwell's equation can be identified in the sense of j µ = 0. Apart from this conventional method, however, another possibility to find the free equation begins with without assuming ∂ ν A ν = 0. In this case, (5) tells us that we have a nontrivial (∂ µ φ = 0) balance equation The first equation in (10) can be justified in two steps: First, from (5) and (8), we see that the conservation law of ∂ ν T ν µ = 0 is satisfied when j ν = 0 in the usual free case (8). In the case of (10), however, we use the if F µν ⊥ ∂ ν φ with ∂ µ ∂ µ φ = 0. This expression indicates that the longitudinally propagating vector ∂ ν φ is 142 physical in the sense that it satisfies the energy-momentum conservation.

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In the second step of the physical justification of (10), we consider (9) in terms of α µ and χ given in (6), which becomes is expressed in terms of the gradient of the vector potentialφ, namely, (v 1 = ∂ 1φ , v 2 = ∂ 2φ ); on the other 149 hand, the incompressibility of the fluid makes its motion nondivergent such that (v 1 , v 2 ) is alternatively 150 expressed as (v 1 = −∂ 2ψ , v 2 = ∂ 1ψ ), whereψ denotes a stream function. Equating these two, we obtain can say that the vector field ∂ µ φ is the physical mode that represents a longitudinally propagating electric 157 field.

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The orthogonality condition (11) is mathematically equivalent to the relativistic hydrodynamic parameters (λ, φ) whose two degrees of freedom are equal to those of ( E, M) in electromagnetic waves.

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Thus, in case [I] of the semi-spacelike CD field, the electromagnetic vector potential U µ is parameterized as where κ 0 is a constant determined by DP experiments. If we introduce two gradient vectors L µ := ∂ µ λ and C µ := ∂ µ φ, then the skew-symmetric field strength S µν can be represented by a simple bivector of the form 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 S µν : (P f (S)) 2 = Det(S µν ), and the barotropic fluid motions governed by the equation of motion ω µν (wu ν ) = 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 C µ would naturally be expected to satisfy from which we obtain In deriving (18), we utilized the fact that C ν ∂ ν C µ = 0. For (18), the following orthogonality condition in the CD field can be imposed as an additional condition, which turns out later to be an important equation.

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To see in what sense (19) is consistent with (15), we consider a null geodesic field (U ν U ν = 0): which is expected to satisfy an extended light field. Using (13) and (15), we readily obtain which vanishes by the orthogonality condition (19). The importance of (19) in the CD field formulation is that L µ must be a spacelike vector, because L µ satisfying (19) is either C µ 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 C µ and L µ , we can show the form of the extended Maxwell's equation: The energy-momentum tensorT ν µ of the lightlike CD field can be derived easily from the conventional 169 one with the following form: T ν µ = −F µσ F νσ . Considering the sign change of the energy at the boundary 170 between the timelike and spacelike domains, we define the tensor as The negative density ρ corresponds to the negative norm of the longitudinal modes in the QED, which 172 makes this mode unphysical in the conventional interpretation. However, we believe that the usage of the 173 term 'unphysical' in this context is inappropriate because if we regard the CD field as virtual photons, 174 then the former is physical in the sense that the latter, as the mediator of the electromagnetic force, is 175 physical though it is invisible. As the argument regarding the reference point of the gravitational potential 176 energy shows, the decision regarding whether a given quantity under consideration is physical depends 177 essentially on the physical setting of our problem; hence, the Clebsch duality relation between F µν and 178 S µν should not be viewed as the duality between physical and unphysical aspects but instead as the 179 duality between the positive and negative sides of the light-cone p 2 = 0, the latter of which is, as we will 180 see in section 3 on cosmology, often closely related to the invisibility of a given quantity. Actually, the 181 "state-dependent" physicality of the longitudinal photons was already pointed out by Ojima [17], who 182 stated that while the longitudinal photons or unphysical Goldstone bosons in the Higgs mechanism are 183 eliminated from the physical space of states in the usual formulation, this statement applies to the above 184 modes only in their particle forms. In their nonparticle forms, the former appear physically as infrared 185 Coulomb tails, and the latter, as the so-called "macroscopic wave functions" arising from the Cooper 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 ∂ ν φ = 0 given by the second equation 192 in (10) to allow the following extended vector potential U µ , which is advected by itself along a geodesic: The form of S µν given by the first equation in (15) remains unchanged in (24). Note that the condition can certainly be considered a gauge fixing condition, but at the same time, Corresponding to the above extension, the energy-momentum tensor satisfying the conservation law Note thatŜ αβγδ defined above has the same skew-symmetric properties as those of the Riemann tensor 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 = hν. Since the CD field variable L µ has the dimension of length, we introduce a certain quantized elemental length l dp whose inverse is κ 0 , namely, the discretization of ρ leads to which can be considered an energy quantization of the CD field. Recall that the Dirac equation of the form can be regarded as the "square root" of the timelike KG equation (∂ ν ∂ ν + m 2 )Ψ = 0. Hence, the Dirac equation for the spacelike KG equation 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 ν (M) ∂ ν + m)Ψ = 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 W 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, where M µν and p ν denote the angular and linear momenta of a given Majorana field, respectively, while 211 N µν and q ν are the corresponding momenta of the other, of which the linear momentum q ν is perpendicular 212 to p ν . We believe that this configuration (30) gives a quantum mechanical justification for the orthogonality 213 condition (19) and (25) of the CD field.

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For a plane wave solution (λ =λ c exp[i(k ν x ν )]) to the spacelike KG equation (14), following commutation relations are derived. For the definitions of p µ ,p µ andx µ , we have where l p denotes the Planck length and ξ µ takes a value of −1 when µ = 0 and 1 when µ = 0, from which where ijk is Edington's epsilon and L i and M i are angular momentum vectors generated respectively by 223 (spatial-spatial) and (spatial-temporal) rotations. Snyder further showed that the "Lorentz transformation" 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 [19]. Through those experiments, the maximum size of the DP that can be considered as l dp introduced in (27) was estimated to be 50 nanometer < l dp = (κ 0 ) −1 < 70 nanometer.
As emphasized in the introduction, we do not yet know a reliable QFT that can deal with the off-shell 234 properties of the field playing an important role in the DP generating mechanism. Thus, we need to resort 235 to a certain kind of simplified argument to bring in the experimental outcome to CD field theory. In the 236 following, we give such a simplified argument. In the first paragraph of the introduction, we mentioned 237 that the existence of point-like singularities, similar to the pointed end of a fiber probe or impurities with 238 extremely tiny size scattered across a given background material, is the crucial element for generating DPs. 239 We can safely say that field interactions in which these singularities come into play should be so serious 240 that the involvement of the spacelike momenta predicted by the Greenberg-Robinson theorem will be 241 crucial in these cases compared with those without singularities.

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As a heuristic example, let us consider a simple wave propagation: certain background field. One may regard it as a wave, say, in the atmosphere. When the wave exists in a 244 uniform background, it propagates such that it satisfies (∂ ν ∂ ν + k 2 )ψ = 0, with k 2 := (k 0 ) 2 − (k 1 ) 2 , which 245 can be regarded as a "unitary" time evolution of a free mode in the timelike sector. If the background field 246 becomes nonuniform but its degree of nonuniformity is rather smooth, then though its way of propagation 247 is deformed to some extent, we can describe the deformed propagation pattern by employing perturbative (valid in the domain x 1 ≥ 0) respectively corresponding to the abovementioned properties of (i) and (ii).

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Needless to say, this example due to the atmospheric dynamics could be transferred to situations involving 262 interactions among elementary particles, where a "severe interaction" would evoke these changes on the 263 interacting Heisenberg fields to which on-shell field theory cannot be applied. We believe that this simple 264 toy model gives an intuitive explanation of the essential features of severe field interactions involving a 265 certain kind of discontinuity and why spacelike momentum modes are necessary to describe these field 266 interactions. 267 Aharonov et al.
[20] conducted an advanced analysis of the response behavior of the spacelike KG equation perturbed by a point-like delta function δ(x 0 )δ(x 1 ), 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 λ(x 0 , r) = exp(±k 0 x 0 )R(r), where R(r) satisfies  (19) and (25), the orthogonal configuration must be broken 272 down by the perturbation, and the timelike pair will turn, respectively, into λ(x 0 , r) = exp(±k 0 x 0 )R(r), 273 namely, particle and antiparticle pairs, since an electrically neutral antiparticle can be considered a particle 274 traveling backward in time. The excited field is nonpropagating in nature; thus, a pair of particle and 275 antiparticle fields will be combined into either an "electric" field with spin 0 or a "magnetic" field with 276 spin 1 [21]. We believe that the DP is generated through this pair annihilation of the Majorana field. Since 277 the DP field is basically electromagnetic, once it is generated, its behavior in a uniform environment can be

On dark energy and dark matter 290
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 p ν and q ν given in (30) gives us a hint to solve this problem concerning the ground state. For spacetime with 3 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 This compound state with a resultant spin 3/2 is called a Rarita-Schwinger state, which we denote by 291 |M3 g . The important characteristic of |M3 g is that the CD vector boson field can be excited from any Therefore, we can say that |M3 g exists not as a momentary virtual state but as a stable invisible off-shell state. In the following, we show that |M3 g exerts on the universe a cosmological effect identified as dark 297 energy.

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To investigate the property of |M3 g , let us consider plane wave solutions λ and φ for the spacelike case of U ν U ν < 0, in which λ = N λλc exp(ik ν x ν ) and φ = N φφc exp(ik ν x ν ), with k ν k ν = −(κ 0 ) 2 , wherê λ c andφ c denote elemental amplitudes of the respective fields and N λ and N 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 where N is a combined number N := N λ N φ and we can identifyλ c asλ c = (κ 0 ) −2 , sinceλ 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 ofT ν ν (1), denoted as |T ν ν (1)|: where (•) * denotes the complex conjugate of (•). The right-hand side of (39) can be evaluated by the lightlike case of the CD field (23), in which we haveT µν = ρC µ C ν . For the lightlike case, we have Next, we consider a case in which the k µ vector of φ is parallel to the x 1 direction and consider a rectangular parallelepiped V spanned by the vectors (1/k 1 , 1, 1). For k 0 = ν 0 /c, where c and ν 0 denote the light velocity and the frequency of the φ field, the volume integral ofT 0 0 /(−N 2 ) over V as the energy per quantum is where denotes the unit length squared. Equating (41) with E = hν 0 , we obtain 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 (S 23 , S 02 ), (S 31 , S 03 ) and (S 12 , S 01 ). The energy-momentum tensorT ν µ (3) derived by the superposition of these fields becomeŝ In deriving (43), we set S 23 = S 31 = S 12 = σ and S 01 = S 02 = S 03 = τ. We note thatT 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 0 0 (3) = −3σ 2 < 0). Dark energy (with positive energy density) * T ν µ (3) having exactly the same trace as that of the anti-dark energyT ν µ (3) can be introduced by the Hodge dual exchange between (σ, τ) and (iτ, iσ) in (43), which becomes * T ν 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 equation (34), the isomorphism betweenT µν and G 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 Λg µν , then the Einstein field equation with the sign convention of R µν = R σ µνσ together with the metric convention of (+1, −1, −1, −1) becomes where Λ becomes negative for an expanding universe. Before proceeding further, we note that * T 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, since * T 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 |M3 g with spin 3/2. The crucial problem in our analysis therefore is whether we can find observable quantities in * T 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 de g ν ν defined by is observable as the invariant of the general coordinate transformation, which is consistent with the built-in 299 Lorentz invariance of Snyder's momentum space on which the CD field is constructed. Thus, the validity 300 of our new model on dark energy can be checked by comparing the following two models: where Λ obs denotes the value obtained by Planck satellite observations. (In S3O, Λ obs in the above equation |M3 g seems to be a promising candidate model for dark energy.

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In the above arguments on the dark energy model, the physical meaning of the "real"cosmological term Λg µν should be revised, because it does not correspond in our model to dark energy. We believe that one of the intriguing possibilities is that Λ dm g µν 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 g µν in terms of the Weyl (conformal) curvature tensor W αβγδ as long as its magnitude does not vanish, namely, as shown by straightforward calculations [27]. 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 2 along the radial direction in the field of W αβγδ in the well-known spherically symmetric Schwarzschild outer solution of a given star suggests that the local maxima of W 2 would behave as "particles"or that its existence tends to correlate with the created matter field. Therefore, T µν , defined asT 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 ofT µν will further accelerate the deviation of spacetime from the FRW metric and hence serve as the fostering mechanism of galaxy formation. (In equation (30) of S3O, the aboveT 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 : the theoretical justification of which is given in the next section, where conformal cyclic cosmology (CCC)
Equation (53) reveals that if we choosel dp := l dp /2 √ π as the third component of a natural unit in which we setl dp = 1, thenl dp gives the geometric mean of the smallest scale l p and the largest one of l dm in that natural unit system. By rewriting the second equation in (46) as we can use this equation to estimate the DP constant l † dp solely by the fundamental physical constants G, h, and c together with the observed cosmological constant Λ obs in place of the above Λ de . Directly from the second equation in (54), we obtain l † dp ≈ 40.0 nm, Experiments : 50 nm < l dp < 70 nm . (55)

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De Sitter space, described by the Einstein field equation yields a familiar solution given by where the constant R 0 serves as the coefficient of the time-dependent scale factor. In the use of (50), this solution can be simplified by taking R 0 = l p into At the end of section 3, the simultaneous conformal symmetry breaking in electromagnetic and gravitational fields was mentioned. We now explain what this means. Recall that the energy-momentum tensorT ν µ of the spacelike (U ν U ν < 0) CD field is given in subsection 2.1 by (26), which is isomorphic to the Einstein tensor G ν µ . The same quantityT ν µ also emerges from the lightlike case of U ν U ν = 0 by replacing ∂ ν ∂ ν φ = 0 with [∂ ν ∂ ν − (κ 0 ) 2 ]φ = 0, which can be regarded as the breaking of both symmetries, i.e., conformal and gauge (cf.(10)). Therefore, this conformal symmetry breaking from the lightlike to the spacelike CD field can be seen as responsible simultaneously for the breaking from ds 2 = 0 to nonzero ds 2 in (58) through (53). A well-known remarkable characteristic of the solution (58) is that it is transformed into a stationary solution by the following variable changes: where D is defined either by 1 > D := 1 − Λ dm (r ) 2 > 0 (case I) or by 1 > D := Λ dm (r ) 2 − 1 > 0 (case II). Note that the metric (59) is similar in form to the Schwarzschild metric given below, for which an event horizon exists at r = α, while that in (59) exists at r = √ 1/Λ dm .  In case I of the stationary metric (59), we have r = 0 by the synchronization t = t of t and t owing 315 to (60). If t is adjusted as t = Θt, (Θ > 1), then we see that r moves from 0 to 1/ (Λ dm ) as t moves 316 from 0 to +∞. Similarly, in case II, we see that r moves from 2/(Λ dm ) to 1/ (Λ dm ) as t moves from 317 0 to +∞. This dual structure, illustrated in Fig. 1, clearly shows that by taking t = 0 as the origin of 318 time from which twin Big Bang universes evolve, they will meet at the event horizon in (59) an eon later 319 (t = ∞). To the best of our knowledge, the concept of twin universes with matter vs. antimatter duality 320 was first discussed by Petit [28]. We believe that his cosmological model fits exactly into the configuration 321 illustrated in Fig. 1, which tells us that (Λ dm ) −1 is a genuine characteristic length scale of our universe.

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This justifies the fact that Λ dm defined in (50) is the cosmological constant that appears in the form of (49).

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The forward and backward time evolutions of twin universes correspond, respectively, to positive and 324 negative field operators of the 4-momentum, while the existence of twin universes naturally explains the 325 reason why one-sided energy spectra at the level of state vector space works for many practical situations 326 in each universe. If the birth of these twin universes was brought about by conformal symmetry breaking 327 of certain light fields in which the duality between "matter (with positive energy) and antimatter (with 328 negative energy)" works as the separation rule of the twin structure, then the twin pair will return to the 329 original light fields when they meet at the event horizon. The next Big Bangs of the twin pair will occur at 330 certain locations on this event horizon distant from each other by √ 2/Λ dm .

331
According to the arguments developed thus far, we can say that the original conformal light field is 332 composed of light fields with the following duality structures: where the symbol * denotes the Hodge duality explained in the derivation of (44). Although (62) and (63) can be considered as light and antilight fields, they can coexist as free modes without interacting with each other, unlike the case of matter and antimatter interactions. Since all of these fields are trace free, the associated Ricci scalar curvature is zero. Equation (26) tells us that the Riemann curvature associated with these light fields takes the form R λρµν = F λρ F µν (= S λρ S µν ). In addition to R ν ν = 0, we can readily show R µν R µν = 0 using (23). Under the former condition R ν ν = 0, the Weyl tensor W λρµν assumes the form thus, by direct calculations using the latter condition of R µν R µν = 0, we obtain W 2 = 0. Hence, for light fields (62) and (63), we have The second equation in (65) is related to Penrose's Weyl curvature hypothesis [29].

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In modern cosmology, cosmic inflation theory was introduced to explain the observed highly tuned 335 initial condition of the Big Bang, in which the notion of "false vacua" plays a key role in explaining the 336 tremendous exponential expansion of space. In the introduction, however, we pointed out that the notion 337 of the vacuum state in conventional QFT is highly biased by the one in Fock space, which may be called

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When we take into account the remarkable abundance ratios of invisible dark energy and dark matter 362 in comparison to the negligible one of ordinary visible matter, the time evolution of visible material 363 subsystems in the universe, for instance, galaxy cluster formations, may be compared to the "heat engines" 364 working between invisible "heat reservoirs" with higher and lower temperature, which respectively 365 correspond to dark matter with positive energy and negative dark energy. If we denote the space averaged 366 W 2 by W 2 | ave. , then due to the property of universal gravitation, it will increase with the passage of time 367 and hence may be related to the gravitational entropy of the visible subsystem in the universe. From this 368 viewpoint, the effect of the gravitational field, including that of dark matter, modeled as Λ dm g µν in our 369 theory, can be interpreted by a certain model of thermodynamics. Actually, attempts at this have already 370 been made, for instance, by [31,32].
As the final remarks on CCC, first, we note that the conformal symmetry of free Maxwell's equation 372 holds well only in four dimensions, which may explain why the dimensions of spacetime in which we live 373 are four. Second, the first author would appreciate if his philosophical preference of helical evolution to 374 cyclic motion is reflected in CCC. His speculative "Book of Genesis" on CCC is as follows: