Unified field theory for electromagnetic and gravity fields with the introduction of quantized space–time and zero-point energy

In our previous papers [1,3], using only the concepts of the zero-point energy and quantized space– times, all the fields including gravity were explained. However, the previous papers had the following limitations: First, the concept of the quantized space-time must be experimentally confirmed. Second, we should clarify the meaning of the quantized Einstein’s gravity equation, which is derived in [1]. Moreover, in another paper [2], we succeeded in describing the neutrinos’ self-energy and their oscillations. However, this paper assumes the rest energy of 3-leptons in advance, which is why we needed to uncover the reason why leptons have 3-generations. As mentioned, using the concepts of the zero-point energy and quantized space–times, we derived the quantized Einstein’s gravity equation in our previous paper [1]. The paper provides an analytical solution of this equalized Einstein’s equation, which implies the conservation of angular momentum in terms of quantized space–times. Employing this solution and without the standard big bang model, a unique form of acceleration equation for the acceleration-expansion universe is derived. Moreover, the temperature of the cosmic microwave background (CMB) emission is also obtained. Further, this solution results in an analytical (not numerical) derivation of the gravity wave. Moreover, based on the configuration of quantized space–times in terms of both electric and magnetic fields, we analytically attempted to calculate every equation in terms of electromagnetic and gravity fields, using the solution of the quantized Einstein’s gravity equation. As a result of this theory, first the calculated acceleration and temperature of CMB emission agree with the measurements. Furthermore, the analytical solution of the quantized Einstein’s gravity equation resulted in all the laws of electromagnetic and gravity fields in addition to the analytically derived gravity wave, which agrees well with the recent measurements. Moreover, the calculations of the energies in the basic configuration of the quantized space–times resulted in all 3leptons’ rest energies. Considering this basic configuration is uniformly distributed everywhere in the universe, we can conclude that τ-particles or static magnetic field energy derived from the basic configuration of the quantized space–times is the identity of dark energy, which also distributes uniformly in the universe. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 July 2020 © 2020 by the author(s). Distributed under a Creative Commons CC BY license. doi:10.20944/preprints202007.0462.v1

Keywords: unified field theory, zero-point energy, quantized space-time, quantized Einstein's gravity equation, conservation of angular momentum in terms of quantized space-times 1. Introduction

Summary of the entire contents including previous works
This paper introduces concepts of quantized space-times and zero-point energy. By these concepts, we succeed in reinforcing our previously established unified particle theory [1,3] as well as giving the reason for 3-generations of leptons. Furthermore, these concept results in quantization of the Einstein's gravity equation and its analytical solution imply, in a real sense, the conservation of angular momentums in terms of quantized space-times. This solution solves various current universe problems, such as acceleration-expansion universe, dark energy, and so on. Furthermore, this solution of the quantized Einstein's gravity equation creates all the laws and equations regarding electromagnetic and gravity fields. Considering our previous works [1,3] described that electromagnetic and gravity fields were created from the zero-point energy in principle, the electromagnetic and gravity fields were related to the weak interaction, strong interaction, neutrinos, quarks, protons, and neutrons, because all these fields are also created from the zero-point energy [1,3]. Thus, we now reinforce the unified field theory on previous paper and present the basic principle that the conservation of angular momentums in terms of quantized space-times, i.e., both the zero-point energy and quantized spacetimes, creates all physics laws regarding particle physics.

Background
In our previous papers [1,3], with the zero-point energy and the concepts of quantized space-times, it is succeeded to describe all the interactions of electromagnetic interaction, gravity interaction, weak interaction, and strong interaction with no numerical or fitting methods, with the agreements of the measurements. Moreover, in another paper [2], we described neutrino self-energy and their oscillations analytically, which agrees with the measurements.
However, in these previous papers, we did not describe Regarding 3-leptons masses, we succeed in obtaining these masses from the basic configuration of the quantized space-times, which basically agrees with the measurements. This result is important because our presented concept of the quantized space-times is certified by the measurements.
Moreover, because the dark energy generally distributes uniformly, it is allowed to conclude that the energy of this configuration of the quantized space-times implies the dark energy. Furthermore, because this configuration also implies that the static magnetic field energy (GeV order), this fact can explain recent measurements [4,5] that everywhere in the universe there are static magnetic fields even in no macroscopic objects.
One of the significances of this paper is that we succeed in obtaining the analytical (not numerical) solution of the Einstein's gravity equation. This is because the introduction of quantized space-times results in the quantization of the Einstein's gravity equation, which enables us to solve this equation analytically. The resultant facts from this analytical solution are 1) The temperature for the cosmic microwave background (CMB) emission is predicted, which agrees with the measurements.
2) The acceleration-expansion of the universe can be described, which also agrees well with the recent measurements. The expansion of the universe was known in the era of Einstein, but this expansion is taking acceleration, which was reported in [6]. We claim that this fact cannot be explained according to the standard big bang model.
3) The gravitational wave is calculated analytically. Thus far, gravitational wave has been calculated only using the numerical methods. 4) With the concept of the quantized space-times and with this solution of the quantized Einstein's equation, the conservation of angular momentums of the quantized space-times creates all the laws in electromagnetism and gravity. That is, with our previous papers [1,3], the unified field theory of particle physics is now reinforced. Now let us consider the problems in the current standard big bang model. dark energy. However, this paper claims that this energy is merely the well-known particle, which obeys general gravitational law. Therefore, this paper claims that the dark energy which exhibits repulsive forces does not exist.
In short, the standard big bang model cannot describe the recent cosmology problems and is not supported by the measurements in a real sense. In particular, the above-mentioned "Li problem" is serious. Therefore a new model has been pursued by other researchers recently.

Summary of significances in the present paper
As a summary of significances in this paper, confirming the agreements with the measurements, we succeeded in confirming the existence of the basic configuration of the quantized space-times, and the quantized space-times' concept and zero-point energy concept resulted in an analytical solution of the Einstein's gravity equation, which implies the conservation of angular momentum of the quantized space-times. This solution creates all the electromagnetic and gravity fields' laws and equations. In our previous papers [1,3], the weak interaction, strong interaction and particle fields themselves are well described only with the concepts of the zero-point energy and quantized space-times. Therefore we now reach an important principle: all the physical fields and their laws are created only by the conservation of angular momentums in terms of the quantized time-spaces, i.e., the zero-point energy with the introduction of quantized space-times.
In other contribution, this paper could obtain the reason why leptons and neutrinos have 3-generations, which has been puzzled since the establishment of particle physics. Additionally, the main problems in the cosmology have been solved without the standard big bang model.

Concept of a quantized space-time
Starting the result of Dirac equation, which implies that a photon creates an electron and a positron.
where ω0, me, and c denote a constant angular frequency, the mass of an electron, and the speed of light, respectively.
This equation can be interpreted as This equation produces the minimum quantized length 0 and time t 0 in terms of a space-time: We derive a more general constant quantized space-time's length and time: We consistently assume that the above length (5) and time (6) are the minimum length and time. Thus, they cannot be divided further. As discussed later, it was found that these concepts are supported by the measurement.
In eq. (2), the left-hand side is identical for the zero-point energy in the Harmonic oscillator Hamiltonian.
As every quantum field theory argues, the first term implies AC electromagnetism. However, the second term called zero-point energy, which is neglected in the quantum field theory, is more important because of DC electromagnetism. Note that we have also described that first term photon creates the Maxwell's time-dependent equations in view of different approaches from the quantum field theory. Figure 1 shows a schematic of the basic configuration of the quantized space-times. Two quantized space-times in terms of an electric field are rotating with velocity v. Each quantized spacetime in terms of an electric field has embedded up-and down-spin electrons, respectively. Note that it is necessary to distinguish these embedded electrons from real-body electrons. In this way, a quantized space-time in terms of a magnetic field is induced. As discussed later, the energies of these two quantized space-times are commonly expressed as the zero-point energy. The forces F is the Lorentz force originating from this static magnetic field, which is identified with the attractive gravity forces F from the gravitational field. This is important because, in this scale, a gravity and a magnetic field are unified, which results in the quantized Einstein's gravity equation. An important notation that this configuration can be described by the solution of the quantized Einstein's gravity equation, as will be discussed later. First, two quantized space-times embedding electrons, which are not real bodies, have rotations. An important point is that these two embedded electrons have opposite spins with each other. That is, one has up-spin and the other has down-spin. This fact will take the importance when considering the creation of a μ-particle. The radius of the quantized time-space is , which is determined using the Dirac equation and Lorentz contraction. For details, refer to our previous paper [1].
By rotations of the embedded electrons, i.e., the rotations of the two quantized space-times, another quantized space-times in terms of a magnetic field is induced. This magnetic field accompanies a concept of flux.
That is, as shown, another quantized space-time whose radius is the same as . Therefore, it is very important to distinguish two quantized space-times as the different ones:

1) A quantized space-time accompanying an embedded electron in terms of an electric field 2) A quantized space-time induced in terms of a magnetic field
Moreover, a force F is generated as a result of the magnetic field generation, as the Lorentz force. As discussed in our various previous papers, generally when relative momentum in terms of two charged particles is zero, these two particles experience an attractive force, which stems from the Lorentz force. As shown, this attractive force F is identical to an attractive gravity force, which is related to the quantization of the Einstein's gravity equation.
where r, G, uB, uG,and uE denote the distance, the gravitational constant, magnetic field energy, gravity field energy and electric field energy.  where Rμν, Tμν, gμν and R denote the Riemann curvature tensor, the energy flux tensor, the metric tensor, and the Ricci tensor.
As a result of substituting the gravitational constant G from eq. (8), we obtain Now, it is assumed that the macroscopic tensor gμν to be approximately the Minkowski tensor gij because, for a quantized space-time, an analytical differential cannot be defined [1]. That is, it merely implies a division by λc. [1]. Similarly, the energy density is given using zero-point energy: Thus, Tμν is approximated using the Minkowski tensor: Considering the above, Einstein's gravitational equation is transformed to As mentioned, the energy of a quantized space-time regarding a magnetic field is given as the zeropoint energy.
Assuming the Ricci tensor 1/2R to be substantially smaller than the first term, we obtain Moreover, considering the initial equation (8), the ratio uB/uE in eq. (15) is obtained.
Considering this, we obtain conclusively = (

Zero-point energy in quantized space-time for gravity or magnetic field
Let us estimate the zero-point energy in terms of a magnetic or gravity field as well as in terms of an electric field concerning a quantized space-time. Here, we consider the energy level | | from the special relativity Equation (18) This energy level is located at the middle position within the band gap of the vacuum, i.e., the energy gap is 2| |.
As indicated in the following eq. (19), if v ≠ 0 in eq. (18), this energy gap 2| | is produced by the product of both the fine-structure constant α and zero-point energy.
This fact will be proved while discussing the CMB in this paper or literature [8] has already proved it.
Note that if v = 0 in eq. (18), this equation | | implies the zero-point energy in terms of an electric field related to eq. (2). Conversely, the velocity v in eq. (18) here is assumed to be critical velocity vc for an electron. That is, when largely accelerated, an electron takes the maximum velocity vc less than the speed of light c. According to our previous paper [1,2], an electron can accompany an e-neutrino and thus the e-neutrino speed is equal to the critical velocity of an electron. Therefore, in eq. (18), v = vc is substituted by 0.994c [1,2].
It is allowed to consider that this value agrees with the measurement of a τ-particle [9].
In summary, the zero-point energy related to a special relativity energy, eq. (18), where v = 0 implies the rest energy of an electron and is related to the quantized space-time in terms of an electric field.
While v in eq. (18) is the critical velocity vc for an electron, the zero-point energy in eq. (19) implies a quantized space-time in terms of magnetic or gravity field.
Related items are listed below: 1) As described later, because the τ-particle mass is equal to the energy of a quantized time-space in terms of magnetic or gravity field, we will conclude that dark energy, which distributes uniformly in the universe, is τ-particle mass or magnetic field energy of a quantized space-time.
2) As described in the later section, an important point is that the existence of energies of the magnetic field and the gravity field in a quantized space-time will be proved by the presence of a measured τparticle. Figure 2 indicates schematic how the magnetic or gravity field in a quantized space-time (i.e., the combination of two embedded electrons in two quantized space-times) is collapsed. Each quantized space-time in terms of a magnetic or gravity field has torque property whose moment corresponds to the magnetic field vector.

Three-generations of lepton 2.3.1 Collapse of the basic configuration of quantized space-times
By the work of torque property, if two magnetic field vectors of two quantized space-times in terms of a magnetic field take the same direction, a larger magnetic field is generated. On the contrary, however, if two magnetic field vectors of two quantized space-times in terms of magnetic field take reverse directions, the net magnetic field vanishes. In this way, although the combination energy of two embedded electrons in two quantized space-times is quite large [2], a quantized space-time in terms of a magnetic or gravity field collapses. This fact results in creations of τ-and μ-particles as discussed later.

Fig. 2
Schematic that two quantized space-times in terms of a magnetic or gravity field interact with each other and the way how these quantized space-times collapse.
First, a quantized space-time in terms of magnetic field has torque property whose moment corresponds to its magnetic field vector. The superposition case in this figure is that the two magnetic field vectors are maximally strengthened. Generally, in the location of the universe, at which the gravity field becomes extremely strong, this maximum superposition occurs. An important case is the cancelation with each other. In this case, the quantized space-times in terms of a magnetic field collapse, and as discussed in the body in this paper, τ-and μ-particles appear according to the energy conservation. The magnetic field energy is converted to τ-particles, while μparticle energy comes from the spin interaction of two electrons embedded in quantized space-times in terms of electric field.

Superposition
Cancel with each other

Masses of μ-and τ-particles from the basic configuration of the quantized space-times
This paper claims that the masses of 3-generation leptons stem from the abovementioned collapse of the basic configuration of quantized space-times. As a result of the collapse, three energies are generated from a collapsed quantized space-time in terms of a magnetic or gravity field. Based on Fig.   1, we can conclude the following: 1. The combination energy between two embedded electrons in quantized space-times in terms of electric field, i.e., the magnetic field (gravity field) energy in a quantized space-time is converted.
This energy corresponds to the rest energy of τ-particle.
2. Each embedded electrons in two quantized space-times, which take rotations and which induce the magnetic field energy in a quantized space-time, have interactions in terms of spins (up-and down-spin). This interaction is converted to the rest energy of μ-particle.
3. As a result of the collapse of the τ-or μ-particle, real electrons automatically appear in the collapsed mode of these particles. This fact comes from the lifetime of τ-or μ-particles.
In the Result section, the actual calculations will be conducted.

The analytical solution of the quantized Einstein's gravity equation
We now consider the quantized Einstein's gravity (QEG) equation according to our paper [1].
giving Minkowski tensor This QEG equation requires Riemann curvature tensor be a specific form, as the following reason.
1) Because is a diagonal matrix, Riemann curvature tensor is a diagonal matrix considering eq. (21).
2) The QEG equation must automatically express the Lorentz conservation, not to add this Lorentz conservation as a condition.
3) Riemann curvature tensor is a covariance tensor. Thus, this covariance tensor must be composed by direct product of covariance position vector .
where i in the fourth term denotes the imaginary unit.
(Note that the symbol × implies direct product of vectors in this paper) Therefore (26) In the QEG, eq. (21) taking the trace Tr to form the Lorentz conservation: According to this equation, the equation automatically presents the Lorentz conservation.
In the derived equation, time t should be consider a period, where ri implies radius of rotation at each location indexed by i, and we assume the cylindrical coordinates. Note that the radius ri depends on the values of variable z. Moreover, variable angular frequency ω was introduced because time t implies a period.
As described later, the derived equation implies a quantized space-time's rotation and z gives anisotropic property. We will see that this solution of the QEG well describes both electromagnetism and gravity fields in later section.
Considering Equations (21) Moreover, as time t implies a period, the equation automatically introduces unique time in terms of special relativity.

Derivation of acceleration a(r)
Let us undertake calculations for acceleration-expansion of the universe. First, let us consider the analytical solution of the quantized Einstein's equation again.
Index i is associated with both radius and zero-point energy 1 2 ħ . Note that we assume z=0 here.
Because ri should be considered macroscopic variable of radius in this section, the first term of the right-hand side (i.e., the zero-point energy having the index i) should be neglected in this section. As a result, macroscopic variable radius r appears and thus a unique angler frequency is derived as Acceleration a(r) is derived as follows: Note that considering eq. (29), ri must indicate a radius of the rotation. Thus, it is necessary to introduce the circumference rL for radius r, and the acceleration is derived as eq. (32).
In the above equation, eq. (30) was used. Moreover, T in eq. (32) implies a period.
Now we calculate a difference in velocity, Δv, considering 1 Mps. Considering the general definition of an acceleration, a deferential velocity Δv is expressed as The above result and equation (30) include macroscopic variable distance r and this movement implies rotations, which results in acceleration (32). We can take a physical picture that our entire universe is undergoing rotations but its angular frequency and its acceleration is not the conventional and basic form but includes variable distance r, which results from the solution of the quantized Einstein's equation. An important point is that this rotational model in the universe can provide the maximum boundary in the universe, which agrees well with the concept of special relativity. Thus, even with the acceleration in the universe, the velocity must not dominate over the speed of light c. In fact, according to eq. (32), the acceleration decreases with radius r. Thus, our universe size is finite, which guarantees conservations of energy. Considering this rotational model in the universe, it is allowed that we claim that the true radial acceleration, which has been conventionally discussed, is not occurring in the universe. Note that the standard big bang model cannot explain the abovementioned facts (i.e., acceleration, finite size of the universe, and no contradiction for the special relativity concept.) Result section will conduct actual calculation using eq. (33).

Cosmic microwave background (CMB)
First, to the formula of the Prank emission , eq. (34), an angular frequency ω, eq. (30), is substituted, which was derived from the analytical solution of the QEG equation.
where T and kB denote the temperature and Boltzmann constant, respectively.
Thus when the exponential function in eq. (34) becomes e −1 , the following equation holds.
In this equation, the temperature T0 implies one of Prank emission. We claim that a photon of CMB is derived from the energy gap, which fluctuates in the energy level of the vacuum [2] and is related to e-neutrino self-energy. That is, considering that r in eq. (35) is a wavelength, this wavelength can be derived from the fluctuations in the energy level of the vacuum.  Figure 3 shows the iteration that photons are absorbed or emitted to or from the energy gap. This implies that CMB photons are created and absorbed everywhere in the universe and thus this paper claims that CMB photons are not the source of the birth in terms of big bang on our universe.
In Result section, the actual calculation will be conducted using e-neutrino self-energy.

Analytical derivation of the gravity wave
In the solution of the QEG equation, = 0 is a condition of the generation of gravity wave. Note that the index i is arbitrary. Thus, this implies that previous everywhere rotations ceases. That is, a macroscopic rotation ceased [10,11].
Moreover, depending on the variable z, variable angular frequency ω is varied. Considering the cylindrical coordinates, in this case, the center of the previous rotation is considered. That is, z=0 is assumed. This implies that the maximum is considered.
Moreover, for a secondary condition, the zero-point energy is converted to photon by the product of the fine-structure constant α because it is needed to convert the energy level to an energy gap.
When the uncertain relation is introduced, angular frequency-ω equation dependent on Δt is derived. Thus, In this equation, the distributed relationship regarding λc is employed.
Next, the strain hmax is considered.
This definition is translated to As mentioned in Section 2.2, this velocity vc is the critical speed of an electron, which is equal to that of an e-neutrino. Therefore this implies consideration of the quantized space-times in terms of gravity (magnetic) fields. Moreover, Considering these equations, Therefore the Chirp signal is where 00 is defined as a constant 1[s] because the angular frequency ω is variable dependent on time Δt.
In Result section, this Chirp signal up will be depicted.

The picture of the unified field in terms of an electromagnetic and a gravity field by rotations of quantized space-times
Again the solution of the QEG equation is where z=0 is assumed.
As mentioned, this equation implies that a quantized time-space's rotation. Concerning the gravity, furthermore, the general wave function is considered: Note that, as mentioned, a deferential and integral become merely a division and product in the quantized space-time [1].
Each of the above equations is substituted with the basic eq. (29), and thus the general electric field, magnetic field, and gravity field equations are derived as follows: 2 = ( 2 ) 2 − Considering the concept of quantized space-times, An electrostatic potential Φ is introduced.
Herein, the following relation is assumed.
where v implies an arbitrary and rotational velocity, not the speed of light c.
Considering the cyclotron angular frequency, Consequently, The first term is neglected, and ≡ is assumed. Then, the following potential energy for gravity is introduced.
where g is variable and δ is the Dirac function. Thus, where v ≡ denotes a relative velocity between two charged particles, because a vector r in the Dirac function implies relative distance between the two charged particles.
When relative velocity vR is zero, generally two charged particles experience strong attractive force with each other, as Lorentz force. For example, this attractive force creates a Cooper pair in hightemperature superconductors [12].
Therefore, when a relative velocity vR is assumed to be zero, it is approximated as In this equation, g should be considered as variable. Note that, as 2 has the meaning only when vR is not zero but large, the above conclusive equation does not have this factor.
Considering this volume integral, we take volume integral to the above process eq. (74). Note that, only in this case, sphere coordinates are considered, because r implies a relative distance.
Thus, finally, we obtain a Newtonian equation.
Considering the above, the basic solution becomes where | − | = ∫ | The deferential is revived and the number 1 is ignored.
At this time, the following Lorentz conservation is assumed: That is, At this time, the sign + is employed.
. From the similar process, the following simultaneous equations are formed.
Note that E and B are not magnitude, but arbitrary components of vectors regardless of any cooperates.
Similarly, from the division of , the differential is revived and the number 1 is ignored. In this case, the signis employed.
Thus, the above process becomes As mentioned, the ratio E/B is Thus, In view of vector analysis, this equation can be generalized to 3-demention.
This is how we derived the Maxwell's forth equation.
In Result section, we will summarize these processes and results.

The masses of 3 leptons
From our previous paper [2], the combination energy (i.e., Lorentz force) in terms of two embedded electrons in a quantized space-time, i.e., the magnetic field (also, gravity field) energy UB is estimated as | | = |−8.0 × 10 −10 | J.
As mentioned, this energy gives the rest energy of τ-particles. Considering this particle is a fermion, where Mτ is the mass of a τ-particle.
Comparison with the measurement [9] makes me find that the theoretical value has the same order as the measurement but a little bit large, which is in agreement with the measurement. This is because, in the theoretical value, the gravity interaction between two τ-particles includes due to their large masses. That is, strictly speaking, in eq. (118), a very small term regarding gravity interaction between two τ-particles should be added. As mentioned, we also claim that there are dark energies, but they have attractive interactions due to the gravity, not repulsive interactions.
Let us consider the case of μ-particles.
From our previous paper regarding superconductivity [13], the spin interaction V between up-and down-spin electrons is expressed as And e denotes the charge of an electron.
Considering that both μ-particle is a fermion and that from Fig. 1, the relative distance in the abovementioned equation (120-1) is the diameter of a quantized space-time 2 , the rest energy is derived as where Mμ denotes the mass of a μ-particle.
Compared with the measurement [9], the agreement is sufficient.
Note that a real electron appears as a result of lifetimes and collapses of τ-or μ-particle.
The significance of the above discussion is that this paper clarified the reason why leptons have 3generations, from the view of the basic configuration of quantized space-times (Fig.1). In previous paper [2], we calculated self-energy of 3-generation neutrinos. Thus combining with these results, now we obtained the comprehensive understanding of why leptons have 3-generations.

Acceleration-expansion universe
In the Theory section, we derived the following conclusion.
In this equation, r and Δt are listed in Table 1.
The latest measurement [14] is The agreement is sufficient.

Emission of CMB
Theory section derived the following unique angular frequency.
Thus, when the exponential function in eq. (34) becomes e −1 , the following equation was held.
In this equation, the temperature T0 implies one of Prank emission.
When r in eq. (35) is considered as a wavelength, the source of this wavelength λ comes from the fluctuation energy gap, which is related to an e-neutrino self-energy [2]. The e-neutrino self-energy is expressed as the following equation [2], 2 , = 0.025 = 4.0 × 10 −21 .
, implies the energy level for an e-neutrino. Thus, it is necessary to obtain a photon from this energy level. In this case, the product of the fine-structure constant α to this energy level makes it be photon energy gap ħω.
The derived wavelength λ is substituted to eq. (35), considering that r is equal to the wavelength λ.
This value agrees with the temperature of CMB [15] 3

.5 Depict of gravitation wave (Chirp signal)
The derived equation in Theory section is again, where 00 is defined as a constant 1[s] because the angular frequency ω is dependent on time Δt. Figure 5 shows the result of this analytical calculation of the gravity wave. As shown in this figure, this calculation agrees with the measurements [10]. Considering strain hmax implies a quantized spacetime , the gravity wave is the universal phenomenon. Moreover, hmax was derived by z=0 of the solution of the quantized Einstein's equation, and thus the gravity wave has an anisotropic property [16].
For vector potential, where n=1.
By the way, generally where dv denotes the volume difference.
dQ ≡ e. Thus, Therefore, as every elementary physics text mentions, the standard Coulomb potential forms: Note that variable r here is a relative distance between two charged particles because of the introduction of the Dirac function having position vector r.
Thus, following equation is satisfied.
Thus it is allowed to generalize this equation to 3-demention: Considering the satisfactions both of eq. (134) and (138), the Poisson equation for vector potential is automatically proved. That is, because the charge density and the continuity equations have been proved, the current density, i.e., the Poisson equation for vector potential as eq. (68) has also been proved.

Newtonian equation
In Theory section, we derived Thus for the embedded two electrons in Fig. 1, and in eq. (78), Then, the potential is resulted as This value is approximately equal to UB in eq. (117).
That is, using eq. (118), Thus, this energy is approximately the rest energy of the τ-particle [9], and thus also implies the energy of a quantized space-time in terms of magnetic or gravity field, which agrees with the measurement and which does not contradict the theory of this paper.
Note that the above Newtonian equation has a different shape from the standard Newtonian gravity equation, which is usually taught in high-schools. However, although the standard Newtonian gravity equation is applied in the scale of the solar system, it is unnatural to consider that it can be applied even in the quantum scale, because every physic equation generally has application scales. For example, the equation "ma = F" is well applied in macroscopic scales, but it cannot be applied in the scales less than an atomic one. Thus, the success of Schrodinger's equation in applying to the H atom comes not from the fact that the value of the standard Newtonian gravity equation is too small but from the fact that it is already considered inapplicable to the scale of an atom.

Derivation of the time-dependent Maxwell's equations
As mentioned many times, the source is the solution of the QEG equation: Using this equation, we in the Theory section converted divisions of the quantized space-times and into the standard differentials [1]. Considering the Lorentz conservation regarding the differentials, we derived the following equations: Therefore, we claim that, in a real sense, the above two equations are the same.

Summary of key points of this study
Only by introducing quantized space-times derived from the zero-point energy, electromagnetic and gravity fields including dark energy are analytically well explained, involving the quantization of the Einstein's gravity (QEG) equation. In this point, concepts are only the zero-point energy and the conservation of angular momentum of quantized space-times.

Analytical solution of the QEG equation
Analytical solutions of the QEG equation result in various significant results: To begin with, the significance of quantizing Einstein's gravity equation, which can be derived by the introductions of quantized time-spaces and the zero-point energy, enables us to obtain the analytical solution (not numerical solution), and this analytical solution describes every electromagnetic and gravity fields uniformly. According to our previous paper [1,3], the weak and strong interactions essentially equal to electromagnetic fields with consideration of the zero-point energy. Thus, this paper reinforced the results of our previous paper [1,3], which describes the unified field theory in terms of particle physics, according to that only the source of every field is the zero-point energy.
Moreover, the solutions of the QEG equation well describes the existing main problems in terms of the universe.

Acceleration a(r) resulting in acceleration-expansion universe
From the solution of the QEG equation, the specific form of acceleration ar was derived. The calculation agrees well with the measurements. The significance is that this paper first succeeded in describing acceleration-expansion universe without any additional concepts. Because acceleration ar depends on the distance r, this paper claims that a Hubble constant generally varies on distances. For example, although this paper calculated it for 1Mps, its value varies at 10 Mps.
Without introducing the dark energy, which exhibits the repulsive forces, only Einstein's gravity equation well explains acceleration-expansion in the universe. However, it does not imply that there is no dark energy. On the contrary, this paper proved the existence of the dark energy as well-known particles, which obey gravitational attractive forces. An important point is that the model of the universe with acceleration is not radial acceleration, but originated from a rotational model. Thus, this paper claims that our universe has a rotational model with a unique form of acceleration, while the standard big bang mode assumes the singularity and thus infinite energy. If the infinite energy was accepted, all our general physical equations and laws would be meaningless.

CMB emission
Analytical solution of the QEG equation also describes the emission of CMB. The result implies that we are not employing the standard big bang model. We derived the CMB emission and a unique angular frequency, which is a result of the solution of the QEG equation and which results in a rotational model of the universe. The significance is that the emissions and absorptions of CMB photons are occurring everywhere in our universe, and they are directly related to the e-neutrino selfenergy, which fluctuates in the energy level of the vacuum. Thus, CMB can be described without the standard big bang model, and thus this paper claims that the measured CMB does not have the meaning of initial time of the birth of the universe.

Unified field in terms of electromagnetic and gravity fields
The analytical solution of the QEG equation also describes the unified field in terms of electromagnetic and gravity fields. This solution implies rotations of quantized space-times both in terms of electric field and magnetic field (gravity field). The results lead to the Poisson equations regarding electrostatic potential, vector potential and gravity potential, respectively. These equations result in the Coulomb equation, Biot-Savart's law (the derivation of this law is described in every physics text), and Newtonian gravity equation, respectively.
Moreover, using the concepts of photons, the inductions both from electric field to magnetic field and from magnetic field to electric field are derived. Thus, the time-dependent Maxwell's equations are described.
In short, the existing Einstein's gravity equation already contained both the properties of electromagnetic and gravity fields. Thus, we claim that to obtain the unified filed theory, it is not needed to expand the existing Einstein's gravity equations like 5-dimention.
The most important point is that all equations from electromagnetic and gravity fields come from the conservation law of angular momentum in terms of quantized space-times. Moreover, as mentioned in our previous paper [1,3], the weak and strong interactions are, in a real sense, equal to electromagnetic fields, and thus all the microscopic fields and basic equations stem from the conservation law of angular momentum in terms of quantized space-times. That is, there is the only source of the zero-point energy to create all the fields.
Furthermore, the result of the analytical solution of the QEG equation leads to analytical derivation of gravity waves automatically. The significance is that, although thus far this gravity waves were obtained only by the numerical analysis of the existing Einstein's gravity equation, now we derived the same gravity wave from the pure analytical solution of the Einstein's gravity equation. The reason comes from the fact that we succeeded in the quantization of the Einstein's gravity equation.

3-Generation of leptons
Considering the basic configuration including quantized space-times both in terms of electric field and magnetic (gravity) field and the collapse of this configuration, we derived both rest energies of a τ-particle and a μ-particle, which basically agree with the measurements. Considering the real electron is result of the collapses of a τ-particle or of a μ-particle, we have now succeeded in giving the reason why leptons have 3-generations. The above implies that the concepts of quantized space-times in terms of electric, magnetic or gravity field with the zero-point energy can be proved by the measurements. In our previous paper regarding neutrinos [2], we described 3-generration of neutrino, i.e., the oscillation of neutrinos, under the assumption that the masses of 3 leptons are known in advance. However, we have now clarified all the masses of 3 leptons without an assumption, and the most important mystery of why elementary particles have 3-generations was uncovered.

Conclusion
This paper described that, by the introduction of quantized space-times derived from the zero-point With the combination of the results from our previous paper [1,3], we now could reinforce the unified field theory in terms of particle physics that only the concepts of the zero-point energy and quantized space-times describe all the fields (i.e., electromagnetic field, gravity field, weak interaction, strong interaction, leptons, neutrinos, quarks, protons and neutrons, and so on).
We select the zero-point energy (i.e., the basic configuration of quantized space-times) as the ultimate source, which describes almost all fields, including the masses of W and Z bosons, but there is a Higgs boson. This paper and our previous papers did not describe this particle. As a follow-up, it is necessary to achieve a consistent description, including this boson.

Theoretical Calculation of the Specific Charge e/m
The specific charge e/m has been measured very accurately, while there is few theoretical report to calculate it. Herein let us calculate the specific charge e/m theoretically without numerical calculations or fitting methods. This calculation will be able when the concept of a quantized spacetime is employed.
Generally cyclotron angular frequency ωc is Thus, we attempt to provide ωc and magnetic field B0, respectively.
First, in a static magnetic field, considering the analogy from a dc motor, a magnetic flux Φ is generally is expressed as where S and t denote the area and time, respectively.
The above implies a self-rotation-like spin. The initial voltage V0 comes from the energy gap of the vacuum, i.e., the zero-point energy; where 0 ≡ 1 rad is assumed, because generally a quantized space-time has no internal metrics.
That is, an arc and radius in a quantized space-time are not distinguished from each other.
As every physical text reports, the measurement value is Compared with each other, the agreement is sufficient.