The Dirac Fermion of a Monopole Pair (MP) Model

1. Introduction

At the fundamental level of matter, particles are described by wave-particle duality, charges and their spin property. These properties are revealed from light interactions and are pursued by the application of relativistic quantum field theory (QFT) [1,2]. The theory of special relativity defines lightspeed, c to be constant in a vacuum and the rest mass, m of particles, $m=E/c^2$ with E equal to energy. By definition, the particle-like property of light waves is massless photons possessing spin 1 of neutral charge. Any differences to the spin, charge and mass-energy equivalence provide the inherent properties of the particles at the fundamental level and this is termed causality [3,4]. Based on QFT, particles are considered to be fields permeating space at less than lightspeed. There is a level of indetermination towards unveiling their charge and spin property, while the wave-particle characteristic is depended on the instrumental set-up [5,6]. The definition counteracts the deterministic viewpoint of non-relativistic Schrödinger’s electron field, $\psi$, which is extremely useful in describing precisely the probability of future events of an electron in orbit of the atom [7]. First, it does not account for the spin property of the particles. Second, $\psi$ is classically applied to physical waves such as for the water waves. Thus, it is difficult to imagine wavy form of particles freely permeating space without interactions and this somehow collapses to a point at observation [8].

At the atomic state, the energy is radiated in discrete energy forms in infinitesimal steps of Planck’s radiation, ±h. The interpretation is consistent with observations except for the resistive nature of proton decay [9]. With this set-back posed by the nucleus, the preferred quest is to make non-relativistic equations become relativistic due to the shared properties of both matter and light at the fundamental level as mentioned above.

Beginning with Klein-Gordon equation [10], the energy and momentum operators of Schrödinger equation,

$$\widehat{E}=i\hslash \frac{\partial }{\partial t}, \widehat{p}=-i\hslash \mathbf{𝛻},$$

(1)

are adapted in the expression,

$$\left({\hslash }^2\frac{{\partial }^2}{\partial t^2}-c^2{\hslash }^2{\mathbf{𝛻}}^2+m^2{c}^4\right)\psi \left(t,\overline{x}\right)=0.$$

(2)

Equation 2 incorporates special relativity, $E^2=p^2{c}^2+m^2{c}^4$ for mass-energy equivalence, $\mathbf{𝛻}$ is the del operator in 3D space, ℏ is reduced Planck constant and $i$ is an imaginary number, $i=\sqrt{-1}$. Only one component is considered in Equation 2 and it does not take into account the negative energy contribution from antimatter. In contrast, the Hamiltonian operator, $\widehat{H}$ of Dirac equation [11] for a free particle is,

$$\widehat{H}\psi =\left(-i\mathbf{𝛻}.\mathit{a}\mathit{}+m\beta \right)\psi .$$

(3)

The $\psi$ has four-components of fields, $i$ with vectors of momentum,$\nabla$ and gamma matrices, α, β represent Pauli matrices and unitarity. The concept is akin to, e⁺ e^– ⟶ 2𝛾, where the electron annihilates with its antimatter to produce two gamma rays. Antimatter existence is observed in Stern-Gerlach experiment and positron from cosmic rays. While the relativistic rest mass is easy to grasp, how fermions acquire mass other than Higgs field remains yet to be solved at a satisfactory level [12]. But perhaps, the most intriguing dilemma is offered by the magnetic spin ±1/2 of the electron and how this translates to a Dirac fermion of four-component spinor field. Such a case remains a very complex topic, whose intuitiveness in terms of a proper physical entity remains lacking and it is often described by Dirac process of either Dirac belt trick [13] or Balinese cup trick [14]. At the moment, this cannot be easily resolved either by experiments or QFT application the conventional way based on empirical data acquisition from experiments.

2. Limitations of quantum field theory

Yukawa definition for the relativistic range of interaction, R = ℏ/mc adapts the uncertainty principle [15]. This is incorporated into the underlying non-abelian Yang-Mills theory of the Standard Model (SM), where relativistic mass, m = 0 sustains unitarity and gauge invariance [16]. Such definition accounts very well for the propagation of electromagnetic field in space, quark confinement and the property of asymptotic freedom but at the expense of infinite, R [17]. Similarly, m ≠ 0 draws divergent terms to the Fourier Transform integral, $\int d^{4}k$ with k equal to 4^th dimensional variable [18]. Renormalization is assumed by particle self-interaction with exchanges of virtual photons such as for the Dirac fermion of four-component spinor field as shown by Feynman path integral diagrams [1,2]. The SM lagrangian adapts such corrective measures for the infinite high-order terms, where the transition from m = 0 to m > 0 is attributed to dynamic chiral symmetry breaking like the Higgs mechanism to render particles’ masses. The theory has seen tremendous success, whereby it ably accounts for all observations made so far into the electroweak force interactions. However, the resistive nature of the Higgs boson from protons-protons collisions to decay into observable supersymmetry partners [9,19] of fermions and bosons to fulfil its scalar field, Φ of quartic self-interacting terms [20] offers the hierarchy problem [21]. The non-zero mass of the vacuum state provided by the Higgs boson does not match with its predicted value when adapting free parameters like W^± and Z bosons with fermions and other bosons constrained by observations. The difference is of several orders of magnitude and this brings into question the completeness of the SM to effectively describe matter at the fundamental level. Furthermore, with QFT there is no distinction between field and particle in Hilbert space analogous to mass-energy equivalence. Thus, physicists are fairly constrained to the ethos, ‘shut up and calculate’ when dealing with the enormity of empirical data acquired in experiments so far. Without any way forward, a million-dollar quest is offered for someone who can properly address the underlying Yang-Mills theory of the SM as one of the seven millennium problems in mathematics [22].

So a revisitation of Copenhagen interpretation of quantum mechanics [23] considers an electron to be a charged elementary particle. It is in superposition states of spin ±1/2 for electron-positron pairing with the outcome depended on observation as illustrated by the Schrödinger cat narrative [24]. Its momentum(p) and position(x) are non-commutable towards the limit of Planck scale defined by the uncertainty principle, Δx.Δp ≥ $ћ/2$. Its evolution with time is provided by Schrödinger $\psi$ without accounting for the actual state of the spin property [7]. How such basic information gets translated to relativistic Dirac fermion by preserving its Hamiltonian is limited by QFT such as the SM described above. In this study, an unorthodox approach is considered, where the transformation of a physical object such as an electron to a Dirac fermion of a complex spinor field is conceptualized in Hilbert space of 4D space-time. Gravity is considered localized in a multiverse of the proposed MP model at a hierarchy of scales, while the electromagnetic field permeates space. Supplemented by several postulates, the model is shown to be in general agreement to both relativistic and non-relativistic aspects of the hydrogen atom based on existing knowledge in physics. Such findings, if considered could consolidate properly the SM and possibly pave future research paths to probe physics beyond.

3. The conceptualization process

3.1. Euclidean and Minkowski space-times

In Figure 1a, an alternative perspective on how the electron of the hydrogen of a Bohr model can be converted to a Dirac fermion of four-component spinor in non-abelian Euclidean space-time is offered. Its spin ±1/2 in superposition states of Minkowski space-time is demonstrated in Figure 1b. Based on these illustrations, a number of postulates are drawn from the first principle of space-time.

1)

The electron’s orbit of time reversal is defined by h of sinusoidal wavy form and is linked to Bohr orbits (BOs) into n-dimensions of energy levels. Its transformation to an elliptic form is embedded within a MP field of clockwise precession and this insinuates spherical inertia frame with unitarity, λ sustained. Excitation from external light interaction assumes, $E=nhv$, whereas the probability of locating the electron is defined by De Broglie relationship, λ = h/p. From Euclidean space-time, the cyclic BO (e.g., positions 1 and 5) in Figure 1a is transformed to angular momentum in Minkowski space-time (Figure 1b). At 360° rotation from positions 0 to 3, maximum twists from the torque of the BOs provide a complex spinor field. Thus, the electron with spin up flips to spin down at position 4 to generate a positron. The unfolding process for another 360° rotation from positions 5 to 8 restores the electron to its original state. This allows the electron to undergo 720° rotation within a classical spherical rotation of 360° at lightspeed analogous to the Dirac belt trick. Such a scenario offers both local realism and entanglement of electron-positron pairing in violation of lightspeed. In multielectron atoms, multiple MP fields are assumed for the electrons distribution.

Figure 1. The MP model [25]. (a) In flat space, a spinning electron’s (green dot) orbit is sinusoidal (green curve) of time reversal. It is embedded within an elliptic MP field (grey area) of a magnetic field, B. Its clockwise precession (black arrows) generates a circular electric field, E of inertia frame defined by λ, and this produces Euclidean 4D space-time. With precession, the electron shifts from positions 0 to 4 at 360° rotation to generate a positron at maximum twists of the BOs. The unfolding process flips the electron to shift in its positions from 5 to 8 for another 360° rotation to assume its original state. In this way, the electron undergoes 720° rotation to assume a dipole moment (±) within a classical spherical rotation of 360° at lightspeed analogous to Dirac belt trick. (b) The BOs defined by the pairings of the numbered positions such as 1,5 and 3,7 translate to angular momentum (purple dotted lines) of spin ±1/2 of a pair of light-cones (navy colored) in Minkowski space-time. This caters for the twisting and unfolding, while the BOs in degeneracy are projected towards singularity at the center.

Figure 1. The MP model [25]. (a) In flat space, a spinning electron’s (green dot) orbit is sinusoidal (green curve) of time reversal. It is embedded within an elliptic MP field (grey area) of a magnetic field, B. Its clockwise precession (black arrows) generates a circular electric field, E of inertia frame defined by λ, and this produces Euclidean 4D space-time. With precession, the electron shifts from positions 0 to 4 at 360° rotation to generate a positron at maximum twists of the BOs. The unfolding process flips the electron to shift in its positions from 5 to 8 for another 360° rotation to assume its original state. In this way, the electron undergoes 720° rotation to assume a dipole moment (±) within a classical spherical rotation of 360° at lightspeed analogous to Dirac belt trick. (b) The BOs defined by the pairings of the numbered positions such as 1,5 and 3,7 translate to angular momentum (purple dotted lines) of spin ±1/2 of a pair of light-cones (navy colored) in Minkowski space-time. This caters for the twisting and unfolding, while the BOs in degeneracy are projected towards singularity at the center.

2)

The electron offers chiral symmetry, where its orthogonal projection, $\frac{1}{2}\left(1\pm i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3\right)$ at positions, 0 to 3 is reduced to spin, $\frac{1}{2}\left(1\pm i{\gamma }^1{\gamma }^3\right)$ for the pair of light-cones. Its Hermitian is P(0→8) = ${\int }_{\tau }{\psi }^{*}\widehat{H}\psi d\tau ,$with $\tau$ equal to time. The MP model of unitarity gauge obeys the “natural units”, ℏ = c = 1 with the electron’s orbit is accorded to Euler’s formula, $e^{i\pi }$ + 1 = 0 for continual precession (e.g., Figure 2). Inversion of symmetry at the center violates parity to generate polarization states, ±1 as the dipole moment of MP field. The electron’s position, $iħ$ within a sphere (e.g., Figure 3) incorporates the uncertainty principle, ΔE.Δt ≥ ħ/2 or Δx.Δp ≥ ħ/2 of time invariance. In this way, Schrödinger $\psi$ of an electron cloud model is transformed to a Dirac fermion of a complex four-component spinor field.

3)

The Dirac four-component spinor, $\psi =\left(\genfrac{}{}{0pt}{}{{\psi }_0}{\begin{array}{c}{\psi }_1\\ \genfrac{}{}{0pt}{}{{\psi }_2}{{\psi }_3}\end{array}}\right)$ assumes its own antimatter at positions, 0, 1, 2 and 3 in Hilbert space when combined with positions 4 to 8 (Figure 1b). The enclosed area defined by the electron shift from positions 0 to 3 is of Euclidean space. Where positions 1 and 3 converge at either position 0 or 2 offers geodesic motion of non-Euclidean space on the surface of the sphere and this somewhat resembles the equivalence principle. By reduction measures such as wave function collapse, only two positions ${\psi }_1$ and ${\psi }_3$ generate the spin, ±1/2 property of a pair of light-cones. The outcome of the spin is determined by Born’s probabilistic interpretation, ${\left|\psi \right|}^2$, for the wave function collapse scenario (e.g., Figure 3), where the past or future paths of the electron from positions 0 to infinity are not accounted for at observations.

3.2. Gravity and multiverse

4)

How gravity becomes applicable within the MP model is demonstrated in Figure 2. The electron path of sinusoidal wave is of time reversal. The overriding clockwise precession of the MP field induces spherical inertia frames at lightspeed. The generated outline forms the event horizon of elasticity comparable to a black hole. Any external observations are assumed at lightspeed to overcome the spinning inertia frame of reference. Einsteinian gravity then becomes applicable, where matter curves space-time and space-time tells matter how to move [26]. The metric tensor in Minkowksi space-time of angular momenta in Hilbert space is assumed towards singularity at the center (Figure 1b). Any external light interaction with the electron-positron pair at 720° rotation or 360° spherical rotation can somehow translate to gravitational wave types by conservation of angular momenta with the nucleus confined to a black hole scenario (Figure 2). Spherical polarization from a complementary pair of photons over large distance would generate entanglement. Each clockwise precession stage is defined by Ω of a von Neumann entropy state. The microcanonical ensemble for the entropy, S = k In Ω offers exponential rise with continuous precession and this is approximated to the k value, where the space-time metric tensor along the BOs are also included.

5)

In a multiverse of the models at a hierarchy of energy scales, the procession ensues in the following manner, nucleus => atom => planet => star => galaxy. Though there is clear distinction to matter between the scales such as life on Earth and gluons at the nucleus, it is possible that the space-time structural frame offered by the MP model is about the same. So the relationship of the electron to an atom is comparable to satellite to planet, planet to star and possibly star to galaxy. In this way, electromagnetic field permeates space, whereas observation of the MP model’s structural frame appears to be an emergent property of matter from external light interactions.

6)

At the solar scale, only a segment of the spherical curvature of the model is observed by redshift such as the rotation of Earth during lunar eclipse. For Mercury, its unusual perihelion precession deviates about 43 seconds of arc per century from Newtonian theory and this is well accounted by general relativity [27]. However, the explicit solution is more complex and restricted to a planet, while relativity does not explain how the effect of space-time curvature on the inertia frame by precession reconciles with singularity. If, however, the MP model is considered, each planet’s sinusoidal orbit of time reversal against overall clockwise precession is attained at a n-dimension of the sun at the solar scale [25]. Precession is applicable to all the planets, where distinction can be made wherever possible unlike the microscale. For distal objects, considerable redshifts are expected for light waves propagating in space. Such a possibility could explain rotation of galaxies, gravitational lensing, black hole and so forth when these are not ably catered for by general relativity. So with gravity confined to matter in space, the electromagnetic field permeates a multiverse of the models at a hierarchy of scales.

The above postulates with respect to the transformation of the electron to Dirac fermion within the MP model of 4D space-time are pertinent to the tenets of physics. The model is shown to link conventional interpretation of quantum mechanics to gravity, which is generally difficult to attain the conventional way. Based on these interpretations, both relativistic and non-relativistic aspects of the atomic state are explored from an alternative perspective in accordance with existing knowledge in physics.

4. Theoretical outcomes and interpretations

The dynamics of the MP model of 4D space-time physically incorporate well the transformation of the electron to Dirac fermion with both relativistic and non-relativistic features sustained (see Section 3). Some of the possible outcomes are demonstrated in the following order; spinor and helicity, basics of Schrödinger $\psi$ and wave function collapse scenario. Towards the end, the implications to the SM and spin-orbit coupling splitting process are explored.

4.1. Chirality of the MP model

The vector gauge invariance of the Dirac fermion fields, $\psi$ exhibit chiral symmetry by the relationships,

$${\psi }_L\to e^{i\theta L}{\psi }_L$$

(4)

or

$${\psi }_R\to e^{i\theta L}{\psi }_R.$$

(5)

The exponential factor, iθ provides both the position, i and angular momenta, θ, for the unitary rotations of right-handedness (R) or positive helicity and left-handedness (L) or negative helicity. Both helicities are projections operators acting on the spinors such as,

$$P_L=\frac{1}{2} \left(1-{\gamma }_5\right)$$

(6)

and

$$P_R=\frac{1}{2} \left(1-{\gamma }_5\right),$$

(7)

where ${\gamma }_5$ is Dirac matrices of eigenstates for the spinor field at fixed momentum. The usual properties of the projection operators are: L + R = 1; RL = LR = 0; L² = L and R² = R. Without getting into the intricate details, somehow Equations 4 to 7 are accommodated within the MP model (Figure 1a,b). For example, the electron’s orbit at 360° rotation from positions 0 to 3 within a clockwise precession of the MP field insinuates a positron of positive helicity at position 4 (Figure 3a). Maximum twists from the torque of the BOs provide a complex spinor field equivalent to ${\gamma }_5$, where spin up flips over to spin down in anticlockwise direction of time reversal to precession (Figure 3b). The unfolding process at 720° rotation restores the electron of negative helicity with spin up at position 8 or 0. The twisting and unfolding process is attained within a classical rotation of 360° at lightspeed and this compares well with Dirac belt trick. Hence, the isospin of electron allows for transformation of charge conjugation, parity and time reversal simply known as CPT symmetry. Its orbit occupies a hemisphere and this equates to spin 1/2, while its negativity is attributed to clockwise precession (Figure 1a and Figure 3a). Two successive rotations of the electron in violation of lightspeed is identified by $iћ$ and the vector axial current allows for polarization of ±1 of the model depicting chirality. How these descriptions are applicable to matter in a multiverse at a hierarchy of scales (Postulate 5) is beyond the scope of this paper.

4.2. The basis of Schrödinger wave function

The wave-particle duality of the electron is described by the Bohr model of the hydrogen atom and is applicable to Figure 4. By considering the electron to be a charged particle, its orbit of

time reversal due to Einsteinian gravity (Figure 2) is expected to curve the clockwise precession of the MP field to induce spherical inertia frame of reference (e.g., Figure 2). Its evolution with time adheres to the generic non-linear Schrödinger equation,

$$i\hslash \frac{\partial \psi }{\partial t}\left(x,t\right)=\widehat{H}\psi \left(x, t\right).$$

(8)

Equation 8 is of first order in space-time and ℏ describes the sinusoidal form of the orbit (Postulate 1). The Hamiltonian for the electron-positron pair (Equation 3) is demonstrated in Figure 1a,b and Figure 3a. In 3D space, the non-relativistic Schrödinger’s electron field,$\psi$ is defined by the spherical polar coordinates, Ω, Φ, θ with respect to the Cartesian coordinates, x, y, z (Figure 4). The angular component is assigned to the BOs and is demarcated by θ. A diagonal pair of BOs encompasses a pair of light-cones into Minkowski space-time is applicable to Pauli matrices (Figure 1b). The magnetic momentum, Φ is related to the BO and is of a cylindrical boundary within the MP field. The precession of the model then assumes the integrals [29],

$${\int }_0^{\infty }{\int }_0^{\pi }{\int }_0^{2\pi }f\left(r\right)r^{2}sin\theta drd\theta d\varphi ={\int }_0^{\infty }f\left(r\right){4\pi r}^{2}dr,$$

(9)

where the polar coordinates for the axes are, $x=rsin\theta cos\varphi$, $y=rsin\theta sin\varphi$ and $z=rcos\theta$ (Figure 4). In high-order Poincaré sphere, the electron vector path of a spinor loop (Figure 1b) is assumed by Berry framework state [30,31,32], $\psi \left(\mathit{r}\right)$ of a circuit (C) at polarization with respect to clockwise precession of the MP field. Each precession stage provides additional phase to the dynamics of the model. With radiation given off, Equation 4 is alternatively interpreted as [30],

$$\gamma \left(C\right)=-{\int }_0^{\infty }{\int }_C^{\infty }d\mathit{S}.\mathit{V}\left(\mathit{r}\right),$$

(10)

where S is the surface area with$d\mathit{S}={\rho }^{2}sin\theta d\rho d\theta d\varphi \widehat{\rho }$ and ρ is the density of the BOs within the MP field defined by Ω (Figure 2). By substituting the vector potential, $\mathit{V}\left(\mathit{r}\right)=\frac{\mathit{l}-(\mathit{m}+2\mathit{\sigma })}{{4\mathit{\rho }}^2}\widehat{\rho }$ for the electron path (Figure 2), the resulting Berry phase is given by,

$$\gamma \left(C\right)=\frac{l-(m+2\sigma )}{4}\Omega ,$$

(11)

where $l$ is a function of $\theta$, r, and by precession it provides integer order of high-order Poincaré sphere [31], ${\mathcal{P}}_l$. m is a function of $\varphi$ aligned to the BO and is assumed in perpendicular direction to the electron’s position (Figure 4). σ is defined by the twisting and unfolding of degenerate states of BOs to 720° rotation of Dirac belt trick to generate the polarization states, ±1 (Figure 1a). By geometric comparison, this somehow mimics helical cholesteric ordering by left- or right-handedness [31], $\chi =\pm 1$ observed in chiral liquid crystals. The particle’s unique optical state, ${\psi }_l$ of ${\mathcal{P}}_l$ is described by,

$${\psi }_l\left(a,b\right)={ae}^{-il\phi }{c}_{+}+{be}^{+il\phi }{c}_{+},$$

(12)

where the spinor, $\phi$ represents the influx of the magnetic field at the BOs in degeneracy in

Minkowski space-time (Figure 1b). The electron path at 720° rotation (Figure 1a) has two degrees of freedom for the generation of electron ($a)$ and the positron $\left(b\right)$. The clockwise precession of the electron, $iћ$ obeys the Euler’s form, ${\psi }^{i\theta }=cos\theta +i sin\theta$ (Postulate 2). These explanations relate to the dynamics of the MP model and they can be further explored into depth for both spin photonics and complex Schrödinger equation as a function of time.

4.3. Wave function collapse scenario

The electron of the MP model of a hydrogen atom provides a local gauge group (Figure 5). Its elliptical orbit aligned on a straight path in Minkowki space-time is of circular polarized plane

[32] of a magnetic field, B and is orthogonal to a cyclic electric field, E. The transition at positions 0 to 8 (Figure 1a,b) transforms the electron to a global gauge symmetry at observation. In this way, the electron assumes both local realism and entanglement. Multiple slits are envisioned for the electron path (Figure 3). The translation of its hemisphere to multi-order Feynman diagrams (Figure 4a,b) and this accommodates the orthogonal duality of E and B. The model of a gauge group obeys the “natural units”, ℏ = c =1 (Postulate 2), where other physical constants like the fine-structure constant, α, vacuum permittivity, ${\epsilon }_0$, reduced Planck

Figure 6. The general consequences of the electron-positron pair. (a) Orthogonal relationship of E and B for eight-order Feynman diagrams of the electron-self interaction within a hemisphere [33] is somewhat comparable to Figure 3 representation in Minkowski space-time. For example, the arrowed horizontal line represents the actual electron path, which acquires electron-positron pair. Wavy lines are virtual photons with the circles assigned to assumed precession stages of the MP field in Hilbert space. The applicability of the physical constants like, ${\epsilon }_0$ and c are demonstrated, while ℏ and $e^2$ are provided in Figure 3. (b) The processes described are transferrable to a generic Feynman diagram inclusive of time reversal.

Figure 6. The general consequences of the electron-positron pair. (a) Orthogonal relationship of E and B for eight-order Feynman diagrams of the electron-self interaction within a hemisphere [33] is somewhat comparable to Figure 3 representation in Minkowski space-time. For example, the arrowed horizontal line represents the actual electron path, which acquires electron-positron pair. Wavy lines are virtual photons with the circles assigned to assumed precession stages of the MP field in Hilbert space. The applicability of the physical constants like, ${\epsilon }_0$ and c are demonstrated, while ℏ and $e^2$ are provided in Figure 3. (b) The processes described are transferrable to a generic Feynman diagram inclusive of time reversal.

constant, ℏ and lightspeed, c become applicable. The electron charge, $e^2=4\pi {\epsilon }_0\hslash c\alpha$ and the reciprocal of the mystical dimensionless value [34,35], $\alpha$ as 1/137 can be accounted for by the model. Light ray, $\gamma$ of circular polarized plane (e.g., Equation 5) provides the four-component spinor field, $i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ of Dirac fermion ($i$) (e.g., Figure 3). ${\gamma }^0$is time in asymmetry at position 0 of Minkowski space-time and the rest are exponentials of Dirac matrices in repetitive process for the electron in orbit. Its reduction to ${\gamma }^{\mathrm{1,3}}$ relates to the antisymmetric product of one-particle solutions [36] in the general form,

$$\psi \left(x_1,x_3\right)={\phi }_1\left(x_1\right){\phi }_3\left(x_3\right)-{\phi }_1\left(x_3\right){\phi }_3\left(x_1\right).$$

(13a)

$$\widehat{P}\psi \left(x_1,x_3\right)={\phi }_1\left(x_3\right){\phi }_3\left(x_1\right)-{\phi }_1\left(x_1\right){\phi }_3\left(x_3\right),$$

$$=-\psi \left(x_1,x_3\right).$$

(13b)

$\widehat{p}$ is the probability operator for the Hermitian of the spin 1/2 property. $\phi$ is provided in Equation 10 and it represents both u- and v-type particles respectively in either inward or outward direction (e.g., Figure 3a and Figure 4). The outcome of $\pm \psi \left(1/2\right)$ is deterministic for the wave function collapse (Equation 13b). These interpretations are somehow consistent with experiments violating Bell’s inequality tests [37,38] supposing that multielectron are applicable to multiple MP fields for many-body system (Postulate 1) and their interactions with photons pairs can somehow correlate at a distance.

5. Implications to the basics of Standard Model

5.1. Electromagnetism and electron spin-orbit splitting

Transformation of U(1) gauge symmetry of infinite number of degrees of freedom for the electromagnetic field is of time invariance accommodates both radial and angular wave distributions (Figure 7a–c(i),(ii). The generation of E from 180° classical rotation orthogonal to B at positions 0 to 4 or 4 to 8 of reverse spin in repetitive mode at the event

horizon of the gauge field somewhat mimes a black hole radiation (Postulates 2 and 4). The variation in the curvature of the event horizon of inertia frame is enlarged in a multiverse at a hierarchy of scales (Postulate 6) so the electromagnetic field of linear time can contract or expand in accordance with classical Maxwell’s equation of the general form,

$$\mathbf{𝛻} . \mathit{E}\left(\psi \right)=-\frac{\partial \mathit{B}}{\partial t}\left(\psi \right)=m_j\hslash \left(\psi \right),$$

(14)

where $\mathbf{𝛻}$ is the divergence of the dipole moment of the MP field during precession and $m_j$ is the spin momentum of the electron. The shift in the electron’s position about the BO of angular momentum (Postulate 1) is offered by Faraday’s relationship,

$$\mathbf{𝛻} . \mathit{B}\left(\psi \right)=U_{0}J+U_0{\epsilon }_0 \frac{\partial \mathit{E}}{\partial t}\left(\psi \right),$$

(15a)

and

$$\mathbf{𝛻} . \mathit{E}\left(\psi \right)=\frac{\rho }{{\epsilon }_0}\left(\psi \right).$$

(15b)

The density, ρ of the vacuum state within the model is constrained to spherical configurations of precession, Ω (Figure 4). Ω x Ω* is Hermitian of discrete space-time defined by ℏ (Equation 6).

5.2. Nuclear spin and the Standard Model lagrangian

The spin quantum number, Ι for the nucleus is either an integer or a half-integer. For the hydrogen atom, its nuclear spin 1/2 in respond to the applied external magnetic field is shown in Figure 8 with respect to Figure 7c(i),(ii). The magnitude of its angular momentum is quantized,

${\rm I}={\left\{{\rm I}\left({\rm I}+1\right)\right\}}^{1/2}\hslash$ with the component of the angular moment, $m_{{\rm I}}\hslash$. The orientations of the spin and hence, its magnetic moment is 2Ι + 1 relative to an axis, where the hydrogen has two orientations. These interpretations are applicable to the electron spin with further details inclusive of lamb shift offered in Appendix A. In a similar manner, spin-orbit coupling triplet state of Zeeman effect can be explored within the confinement of the proposed model. The explanations briefly demonstrate the concept of the multiverse of the models at a hierarchy of scale, from the nucleus to beyond the solar system (Postulate 5). On this basis, it is plausible to apply the model to the SM. The in depth mathematical framework of the SM is quite complex with its compact form shown in Figure 9.

The electromagnetic field strenght tensor, F is related to the transformation of the vector axial current (Figure 3a) to linear time at observation (Figure 7a–c(i),(ii)), while the index, α for Pauli matrices is applicable to angular momenta into Minkowski space-time, 𝜇, 𝜈 (Figure 1b). The electron’s isospin of 720° rotation within a classical rotation of 360° at positions 0 to 8 during precession (Figure 3a) offers the Dirac operator, and is subject to correction imposed by hermitian conjugate (h.c). The matter fields of fermions acquire mass by Yukawa coupling to the Higgs field, Φ during electroweak symmetry breaking. The flavor from mixing of the matter fields is sourced from a pool of three leptons, six quarks, three Cabibbo-Kobayashi-Maskawa angles and a CP-violating phase [42]. These properties in a simplified version are intuitively applicable to the electron helical states (Figure 3a,b), where quantization of the sinusoidal wave of unidirecitonal is linked to the BOs defined by Φ and these offer both quadratic and quartic couplings by precession comparable to the Higgs field (Figure 4). Such assumption implies unitarity for the conservation of angular momenta at interactions, where removal of the electron generates a particle-hole symmetry for the vacuum expectation value like the Higgs boson. The continuity of particle-hole (electron) in orbit at high energy would then mimic Nambu-Goldstone bosons [43] as massive version of the photons. In this way, the MP model is applicable to the gauge symmetry, ${U\left(1\right)}_Y{ \mathrm{x} SU\left(2\right)}_L{\mathrm{x} SU\left(3\right)}_C$ of the SM. The broken symmetries of the electroweak force is given by both abelian hypercharge, Y of the electromagnetic field and non-abelian left-handed, L fermions of a complex 2-vector of chiral flavor symmetry (e.g., Figure 3a). Only the color, C change of charges for the strong nuclear force of non-abelian remains unbroken.

By the explanations offered above, it is plausible to speculate that the electroweak nuclear decay like beta decay, $n^{°}\to p+W^{-}\to uud+e^{-}+\overline{v}$, reinforces the models for both the quark and electron in a multiverse at a hierarchy of scales (Postulate 5). If three generations of both the leptons and quarks are massive versions of the electron and the quark in an onion-like structure of MP models, then W^± and Z° bosons mediate their interactions comparable to the photons (e.g., Figure 3a and Figure 7a,b). In this case, the transition between the models in the atom

can somehow relate to either neutrinos at 720° rotation or antineutrinos for 360° rotation in Hilbert space. A possible quark model strictly related to the precession stages of the electron (Figure 3a) is shown in Figure 10. The discussions on how it is applicable to mesons and baryons are briefly summarized on the Wikipedia [44], beginning with the quark field of the types,

$$q_L\to e^{i\theta \left(x\right)}{q}_L$$

(16a)

and

$$q_R\to e^{i\theta \left(x\right)}{q}_R.$$

(16b)

Equations 16a and b are relatable to Equations 4 and 5 in a multiverse of the models. By substituting $q=\left[\begin{array}{c}u\\ d\end{array}\right]$ and $\overline{q}=\left[\begin{array}{c}\overline{u}\\ \overline{d}\end{array}\right]$ with respect to Figure 10, the baryogensis lagrangian of the strong force then becomes,

$$\mathcal{L}={\overline{q}}_{L}i{Dq}_L+{\overline{q}}_{R}i{Dq}_R+{\mathcal{L}}_{gluons,}$$

(17)

where D is the Dirac operator for both matter and antimatter. The 2 x 2 Dirac and Pauli matrices of diagonal relationships is assigned to the BOs in Minkowski space-time. The explicit form of Equation 17 by incorporating both the left- and right-handed spinors of u- and v- types particles in Hilbert space (Figure 5) is of the form,

$$\mathcal{L}={\overline{u}}_{L}i{Du}_L+{\overline{u}}_{R}i{Du}_R+{{\overline{d}}_{L}i{Dd}_L+{\overline{d}}_{R}i{Dd}_R+\mathcal{L}}_{gluons}.$$

(18)

The reduction of Equation 18 to depict the Dirac process is,

$$\mathcal{L}=\overline{u}iDu+\overline{d}iDd+{\mathcal{L}}_{gluons}.$$

(19)

Based on Figure 10, a plethora of particle types can be expected during proton-proton collisions, where the three generations of quarks and leptons are probably massive versions of the electron and quark in an onion-like structure of the model towards singularity.

6. Conclusion

The dynamics of the MP model of 4D space-time considered in this study for the hydrogen atom is somehow able to transform the electron to Dirac fermion of a complex four-component spinor field. Gravity is considered localized in a multiverse of the models at a hierarchy of scales, while electromagnetic field permeate space. Such interpretations though speculative to some extent, the model allows for exceptionally oversimplified intuitions to very complex themes covered by both relativistic and non-relativistic QFTs from the atom to the universe at large. As Erwin Schrödinger, one of the founders of modern physics ably puts it this way [45], “The task is not so much to see what no-one has yet seen, but to think what nobody has yet thought, about that which everybody sees.” These presentations to an extent adapts such a concept from an alternative perspective and this is also comparable to the quest of unifying both quantum mechanics and general relativity. That is trying to integrate what we already know into a proper holistic standpoint before venturing into the unknowns to probe physics beyond the SM and the more complex reality of the physical world.