Dirac Fermion of a Monopole Pair (MP) Model of 4D Space-Time and Its Wider Implications

A. Non-Relativistic Theory to Relativistic Field Theory

B. Relevance of Quantum Space-Time in Dirac Theory

A. Unveiling of Dirac Belt Trick and CPT Symmetry

field mimicking Dirac’s string of a small magnet (Figure 1a). With clockwise precession, the torque

Figure 1. The MP model [14]. (a) In flat Euclidean space-time, arrow of time is aligned with the principal axis or z-axis of the MP as time axis in asymmetry of a clock face. A spinning electron (green dot) in orbit of sinusoidal form (green curve) assumes time reversal and it is normalized to an elliptical MP field (black area) mimicking Dirac’s string of a small magnet. Clockwise precession of the MP field (black arrows) against time reversal generates an inertia frame, λ. The electron at position 0 is assigned center of mass and is subjected to Newton’s first law of motion, $F=ma$. The shift in the electron’s position from positions 0 to 4 at 360° rotation offers maximum twist and the electron is converted to a positron. The unfolding process from positions 5 to 8 for another 360° rotation restores the electron to its original state at position 0 analogous to DBT and this is associated with zero-point energy (ZPE). A dipole moment (±) of an electric field, E is induced at the total of 720° rotation between two interchangeable hemispheres of the MP so both CPT symmetry and Pauli exclusion principle are sustained. (b) Twisting and unfolding by electron-positron transition generates hyperbolic solenoid of spin angular momentum, ±1/2 (navy colored pair of light-cones) into Minkowski space-time. This levitates into n-dimensions (purple dotted diagonal lines) between the hemispheres and are linked to smooth manifolds of BO perpendicular to the light-cones. The undisturbed helical solenoid of a magnetic field, B by unitarity is referenced to the hypersurface of the MP field. (c) The precession stages, $\Omega$, at spherical lightspeed is polarized by electron-positron transition with qubits, 0 and ±1 assumed at positions, 0, 4 and 8 at the vertices of the MP field. The polar coordinates (r, θ, Φ) with respect to the electron’s position in space are applicable to Schrödinger wave function. (d) Hyperbolic surface of the light-cone of holonomic is contained within hypersurface of the MP field towards position 0 of ZPE at N-1. Balancing out of charges at homomorphic positions 1, 3 and 5, 7 is isomorphic to compact topological torus of BO. The BO is defined by $\varphi$ (white loops) and dissected orthonormal by $\theta$ (yellow circles) from hypersurface of the MP field by clockwise precession. The reach of singularity towards the nucleus is constrained by the irreducible spinor field (insert centered image). The embedded terms and equations are described in the text.

Figure 1. The MP model [14]. (a) In flat Euclidean space-time, arrow of time is aligned with the principal axis or z-axis of the MP as time axis in asymmetry of a clock face. A spinning electron (green dot) in orbit of sinusoidal form (green curve) assumes time reversal and it is normalized to an elliptical MP field (black area) mimicking Dirac’s string of a small magnet. Clockwise precession of the MP field (black arrows) against time reversal generates an inertia frame, λ. The electron at position 0 is assigned center of mass and is subjected to Newton’s first law of motion, $F=ma$. The shift in the electron’s position from positions 0 to 4 at 360° rotation offers maximum twist and the electron is converted to a positron. The unfolding process from positions 5 to 8 for another 360° rotation restores the electron to its original state at position 0 analogous to DBT and this is associated with zero-point energy (ZPE). A dipole moment (±) of an electric field, E is induced at the total of 720° rotation between two interchangeable hemispheres of the MP so both CPT symmetry and Pauli exclusion principle are sustained. (b) Twisting and unfolding by electron-positron transition generates hyperbolic solenoid of spin angular momentum, ±1/2 (navy colored pair of light-cones) into Minkowski space-time. This levitates into n-dimensions (purple dotted diagonal lines) between the hemispheres and are linked to smooth manifolds of BO perpendicular to the light-cones. The undisturbed helical solenoid of a magnetic field, B by unitarity is referenced to the hypersurface of the MP field. (c) The precession stages, $\Omega$, at spherical lightspeed is polarized by electron-positron transition with qubits, 0 and ±1 assumed at positions, 0, 4 and 8 at the vertices of the MP field. The polar coordinates (r, θ, Φ) with respect to the electron’s position in space are applicable to Schrödinger wave function. (d) Hyperbolic surface of the light-cone of holonomic is contained within hypersurface of the MP field towards position 0 of ZPE at N-1. Balancing out of charges at homomorphic positions 1, 3 and 5, 7 is isomorphic to compact topological torus of BO. The BO is defined by $\varphi$ (white loops) and dissected orthonormal by $\theta$ (yellow circles) from hypersurface of the MP field by clockwise precession. The reach of singularity towards the nucleus is constrained by the irreducible spinor field (insert centered image). The embedded terms and equations are described in the text.

B. Dynamics of Space-Time Geometry in an Atom

• The MP model is limited to light-matter coupling at observation with the matter property attributed to the electron’s position of a point-particle. The vertices of the elliptical MP field by clockwise precession at positions, 0, 4 and 8 at a point-boundary offer moduli spaces of Hermitian forms by the electron-positron transition. The positions respectively accommodate polarization states, 0, 1 and hypercharge –1 for the two hemispheres of ±1/2 with respect to the electron shift in positions 0 → 8 in accordance with Pauli exclusion principle. Unitarity by Euler’s formula, $e^{i\pi }$ + 1 = 0 in real space is sustained.
• Polarization of other moduli spaces of vertices devoid of the electron by clockwise precession mimics creation and annihilation of virtual particles in a vacuum. Real particle emergence with spin angular momentum is assumed at positions 1, 3 or its conjugates positions, 5 and 7 (Figure 1b) with the electron popping in and out of existence by the process of DBT. The Hamiltonian, P(0→8) = ${\int }_{\tau }{\psi }^{*}\widehat{H}\psi d\tau$ is sustained for the electron shift in position and its transition to a positron with the time taken equal to $\tau$.
• The electron at the vertex of the MP field at position 0 mimics monopole instanton (MI). It offers the base point for transmission of vortex electron through a helical solenoid which intersects topological torus of BO at n-dimensions by levitation between two hemispheres of the MP field. The base point presents COM reference frame from the electron-positron transition with time reversal orbit of the electron balanced out by clockwise precession in accordance with Newton’s second law of motion. The COM is about Planck length ~ 1.62 x 10^-35 meters) is aligned to the principal axis or z-axis of the MP field as nuclear isospin (about 1.1 angstroms for the hydrogen atom) and it depicts arrow of time in asymmetry. Infinitesimal radiation of Planck constant, $\hslash$ is assumed from twisting and unfolding process of the electron-positron transition by DBT at COM. It can also cater for ZPE of the hydrogen atom at ultraviolet range in a potential well of a harmonic oscillator, $E_v={\displaystyle \frac{1}{2}} \hslash \omega$ at a value of -13.60 electronvolts.
• In flat Euclidean space-time of inertia frame (Figure 1a), any deviation from z-axis of the MP field by precession is trivial, $\delta \left(z-z^{\text{'}}\right)=⟨z|z\text{'}⟩$ (Figure 1d) and time remains absolute from a background of clock face with distances between any two points being positive and COM is assigned to point-boundary at position 0 as mentioned above. The electron position in space within the model is defined by the Cartesian coordinates, x, y, z. At positions 1 and 3 or 5 and 7, electron-positron transition of superposition states obey Born’s rule,${\left|\psi \right|}^2$ and acquires spin angular momentum to mimic Weyl spinor of Minkowski space-time (Figure 1b). Hyperbolic surface of the light-cone of holonomic is contained within hypersurface of the MP field of Euclidean space towards position 0 of ZPE at N-1. Balancing out of charges at homomorphic positions 1, 3 and 5, 7 is isomorphic to compact topological torus of BO. By clockwise precession, time is relative, i.e., $z-z^{\text{'}}\equiv ct-c{t}^{\text{'}}={tan}^{-1}v/c=\alpha$ with $\alpha =\theta$ (Figure 1c). By Lorentz transformation in 1D for accelerated reference frame, the volume or dipole moment of the MP field remains constant so, $t^{\text{'}}=\gamma (t-vx/c^2)$ and $x^{\text{'}}=\gamma (x-vt/c^2)$ with Lorentz factor, $\gamma =1/\left(\sqrt{1-v^2/c^2}\right)$ and $y^{\text{'}}=y$ and $z^{\text{'}}=z$. Its invariance, $t=\gamma (t\text{'}-vx\text{'}/c^2)$ and $x=\gamma (x\text{'}-vt\text{'}/c^2)$ sustains the unitarity of the model. Observation by light-MP model coupling is confined to the electron and its motion and this can relate to Lorentz transformation with any outgoing radiation being of linear time. In this way, 4D Minkowski space-time (Figure 1b) is applicable to multidimension of Euclidean space-time (Figure 1a) with COM by Newtonian gravity assigned to position 0 as mentioned above. Cancellation of charges for the conjugate positions, 1, 3 and 5, 7 transforms the hyperbolic surface of light-cone to compact topological torus (Figure 1c) and these levitates between the hemispheres of the MP field into extra dimensions.
• Singularity towards the nucleus is constrained by the dipole moment of the MP field of an irreducible spinor generated by the electron-positron transition at positions 0 to 3 in repetition. Stochastic behavior from the emergence of envelop solitons at positons 2 and 6 insinuates a chaotic system. Combined with the quantum critical region of the light-cones at positions 1 and 3, the quest towards quantum critical point at the baseline of the hemisphere remains elusive by quantum chaos of butterfly effect for the hydrogen atom (e.g., Figure 1a). Such intuition is important in the pursuit of quantum critical point in condensed matter physics for an array of atoms in one-dimensional line.
• The electron is a point-particle at constant motion, mv and this offers wave-particle duality for the overall spherical reference frame, $\lambda =h/mv$ in accordance with de Broglie relationship. In this case, ±h is assigned to the electron-positron transition of a sinusoidal orbital. The electron is defined by the wave function, $\psi$ and its orbit of time reversal by time derivative is accorded to Schrödinger equation, $i\hslash {\displaystyle \frac{\partial }{\partial t}}\left(\psi \left(\overrightarrow{r}\right)\theta \left(t\right)\right)$. It is a fundamental equation with the basis vector, $\overrightarrow{r}$ and $\theta$ assumed into Minkowski space-time (Figure 1c) against z-axis as time axis and x-axis equal to distance.
• The electron’s spin-charge of superposition states is linked to BO of topological torus defined by $\varphi$ and this is projected in the x-y plane (Figure 1c). Its inner product, ${⟨\psi |\varphi ⟩}^{*}=⟨\psi |\varphi ⟩$ sustains unitarity for the electron of weak isospin into n-dimensions of topological torus by disturbance (Figure 1d). The electron’s position can be split into both radial and angular wave functions, $\psi \left(r,\theta ,\varphi \right)=R_{n,l}\left(r\right)Y_l^{m_l}(\theta ,\varphi )$. The radial part,$R_{n,l}$ is attributed to the principal quantum number, n and angular momentum, l of a light-cone with respect to r (Figure 1c). The angular part, $Y_l^{m_l}$ of degenerate states, ${\pm m}_l$ is assigned to the BO defined by both $\theta$ and $\varphi$ (Figure 1d).
• The BO of a constant structure, ɑ is orthogonal to z-axis by linearization (Figure 1d). Its tangential link to electron-positron pair of anticommutation is, $⟨a_j|a_k⟩=\int dx{\psi }_{a_j}^{*}\left(x\right){\psi }_{a_k}\left(x\right)={\delta }_{jk}$ for continuous derivation of MP field due to clockwise precession. Linear translation of n-dimensions or n-levels by Fourier transform is by the generalized state, $E=nhv$ and this is standardized to the z-axis. By continuity, the sum of expansion coefficients, $C_n$, from the electron’s position offers the expectant value, $\left|\psi \right.⟩=\sum _n{C}_n/a_n⟩$ with its probability given as, ${⟨a_n|\psi ⟩}^2$.

A. Non-Relativistic Wave Function

B. Wave Function Collapse

C. Quantized Hamiltonian

A. Dirac Theory

The helical property of the electron within the MP model towards the generation of Dirac fermion by the process of DBT is expounded in here. The elliptical MP field of Dirac’s string is warped and unwarped by twisting and unfolding process (Figure 5a and Figure 5b). As a result, the electron-positron transition is restricted to a pair of interchangeable hemispheres (Figure 5c and Figure 5d). At spherical lightspeed, these combine to generate an irreducible spinor (Figure 5e) and this is applicable to a spherical geometry (Figure 5f) incorporating Euler’s formula for hyperbolic and parabolic complex number with respect to the electron’s position about BO in orbit. Components of Pauli matrices, $i, 0, -1, 1$are distinct from those of Dirac matrices, ${\psi }_{0\to 3}$. The latter is depended on the shift in the electron’s position, 0 → 3 for the transition to positron by DBT. In the former, the matrices are [6] traceless, Hermitian and of unitarity with determinant –1 and satisfy the relations $\sum _j{\widehat{e}}_j{\widehat{e}}_j^T=\left[e_i{e}_j\right]=c_{ij}^k{e}_k$ with $c_{ij}^k$ is expansion coefficients at angular frequency, ${\omega }^2=k/m$ for the electron of a body-mass in continuous oscillation mode. The unit matrices, ${\widehat{e}}_k$ is applied to the electron-positron pair, $i$ and $j$ is the angular momentum along BO (Figure 5f) at positions 1 and 3. Anti-commutation and its square for the electron-positron transition offers the identity matrix, ${\sigma }_0\equiv I$ and is relevant to Clifford multiplication. These provide orthonormal basis like, ${\widehat{e}}_i,{\widehat{e}}_j$ for the vector space of Clifford algebra as described in Equations (5 to 7). In this case, breaking of CPT symmetry in discrete form for charge-parity emanates from the chirality of the electron in a hemisphere versus the devoid hemisphere. Time reversal symmetry is sustained for electron-positron transition at spherical lightspeed. These properties are revealed by light-matter coupling and are incorporated into Dirac equation of the extended form, coupling, this can be described by the famous Dirac equation,

Figure 5. Exposition of DBT within the MP model. (a) Light-matter coupling is reduced to the electron’s position of an irreducible spinor field of close loop generated within a hemisphere at spherical lightspeed. The electron’s (green dot) shift in position of time reversal is balanced out by clockwise precession of the MP model (see also Figure 1a). The position of the particle on a straight path (colored lines) is referenced to the MP field of elliptical shape (centered image). (b) Maximum twist is attained at position 4 as detectable energy and the unfolding process by another 360° rotation for a total of 720° rotation restores the electron at position 8 or 0 akin to DBT. (c) Precession normalizes the loop to generate an electron of positive helicity or right-handedness. Spin up vector at the point-boundary correlates with the direction of precession. (d) At position 4, the electron flips to a positron of negative helicity or left-handedness. The spin down vector is in opposite direction to the direction of precession to begin the unfolding process. Transposition is attained at 720° rotation, $\sum _j{\widehat{e}}_j{\widehat{e}}_j^T=\left[e_i{e}_j\right]=c_{ij}^k{e}_k$ as described in the text. (e) Irreducible spinor field at spherical lightspeed. Cancelation of charges at conjugate positions, 1, 3 and 5, 7 from on-shell momentum offers close loops of BOs into 3D of discrete form to stabilize the electron to only generate either spin up or spin down states respectively at position 0 and 4. The point of singularity at the center is evaded by shift in the electron orbit. The slight tilt at position 4 compared to position 0 is attributed to energy loss from the electron-positron transition in the form, $E=hv=g\beta B$. (f) Polarization of the model either horizontally or vertically with respect to the electron-positron pair, ±i at position 0 by 720° rotation generates qubits 0, 1 and hypercharge of –1 respectively at positions 0, 8 and 4 (see also Figure 4a). These are relevant to classical computing and accessibility to quantum computing for the nucleons are restricted by the chaotic system of envelop solitons at positions 2 and 6. Image (e) adapted from ref. [23].

Figure 5. Exposition of DBT within the MP model. (a) Light-matter coupling is reduced to the electron’s position of an irreducible spinor field of close loop generated within a hemisphere at spherical lightspeed. The electron’s (green dot) shift in position of time reversal is balanced out by clockwise precession of the MP model (see also Figure 1a). The position of the particle on a straight path (colored lines) is referenced to the MP field of elliptical shape (centered image). (b) Maximum twist is attained at position 4 as detectable energy and the unfolding process by another 360° rotation for a total of 720° rotation restores the electron at position 8 or 0 akin to DBT. (c) Precession normalizes the loop to generate an electron of positive helicity or right-handedness. Spin up vector at the point-boundary correlates with the direction of precession. (d) At position 4, the electron flips to a positron of negative helicity or left-handedness. The spin down vector is in opposite direction to the direction of precession to begin the unfolding process. Transposition is attained at 720° rotation, $\sum _j{\widehat{e}}_j{\widehat{e}}_j^T=\left[e_i{e}_j\right]=c_{ij}^k{e}_k$ as described in the text. (e) Irreducible spinor field at spherical lightspeed. Cancelation of charges at conjugate positions, 1, 3 and 5, 7 from on-shell momentum offers close loops of BOs into 3D of discrete form to stabilize the electron to only generate either spin up or spin down states respectively at position 0 and 4. The point of singularity at the center is evaded by shift in the electron orbit. The slight tilt at position 4 compared to position 0 is attributed to energy loss from the electron-positron transition in the form, $E=hv=g\beta B$. (f) Polarization of the model either horizontally or vertically with respect to the electron-positron pair, ±i at position 0 by 720° rotation generates qubits 0, 1 and hypercharge of –1 respectively at positions 0, 8 and 4 (see also Figure 4a). These are relevant to classical computing and accessibility to quantum computing for the nucleons are restricted by the chaotic system of envelop solitons at positions 2 and 6. Image (e) adapted from ref. [23].

$$\left(i{\gamma }^0{\displaystyle \frac{\partial }{\partial t}}+cA{\displaystyle \frac{\partial }{\partial x}}+cB{\displaystyle \frac{\partial }{\partial y}}+cC{\displaystyle \frac{\partial }{\partial z}}-{\displaystyle \frac{{mc}^2}{\hslash }}\right)\psi \left(t,\overrightarrow{x}\right),$$

(21)

where c acts on the coefficients A, B and C and transforms them to ${\gamma }^1{, \gamma }^2$and ${\gamma }^3$. The exponentials of $\gamma$ are denoted i for off-diagonal Pauli matrix for the light-cone of irreducible spinor and ${\gamma }^0$ to 0, 1 polarization states at position 0 of COM reference frame. These are demonstrated in Equation (7). Hypercharge, –1 is expected at position 4 along z-axis as nuclear isospin and ${\sigma }^i$ can relate to oscillations from on-shell momentum of BOs of topological torus into n-dimensions (Figure 2a) with anticommutation relationship, $e^{+}\left(\psi \right)\ne e^{-}\left(\overline{\psi }\right)$ of chiral symmetry. The associated vector gauge invariance for the electron-positron transition in continuum form exhibits the following relationships,

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

(22a)

B. Weyl Spinor and Majorana Fermions

C. Lorentz Transformation

A. Space-Time Fabric of an Elliptical Orbit

B. Internal Structure by Lie Group Representation

and

$$g+\left(-g\right)=i,$$

(36b)

where $i$ is spin matrix for both spin ±1/2. The form, $g_1+g_3\ne g_5+g_7$ due to electron-positron (±$g$) transition and radiation loss, $E=nhv$ tangential to the manifolds of the spherical model. For linear transformation along z-axis in 1D space, the hyperbolic surface of the light-cones are compacted to isomorphic BOs with respect to the particle position, $\left|\psi \right.⟩=\sum _n{C}_n/a_n⟩$ (e.g., Figure 1d and Figure 7). By intermittent precession, the inner product of r is a scalar and relates to the boundary of the MP field in the form,

$${\overrightarrow{r}}_1 {. \overrightarrow{r}}_2=\left|{\overrightarrow{r}}_1 \right|\left|{ \overrightarrow{r}}_2\right|cos\theta$$

(37)

where rotation of both vectors preserve the lengths and relative angles (e.g., Figure 1c). By assigning rotation matrix, R to Equation (37), its transposition becomes,

$${\left(R{r}_1\right)}^T\left(R{r}_2\right)=r_1^T{r}_{2}I,$$

(38)

where the identity matrix, $I=R^T\times R$ by reduction and compaction of vector bundles in homogenous spaces spaces (Figure 7). The rotation by $\overrightarrow{r}$ is a four 3-fold axis in 3D and its matrices

Figure 7. The basis of vectors to multivectors, matrices, tensors and Fourier transform for the electron position (P) in space (see also Figure 1c). Time axis is assumed along z-axis as the principal axis of the MP field and distances projected in the x-y directions. Fourier transform of tensor matrices along z-axis is by commutation, $pZ$ to $Z$. The electron of a resultant vector, $\overrightarrow{r}$ of four 3-fold rotational axes (shaded orange plane) in 3D of a cube can relate to the vector space, $\overrightarrow{J}=\overrightarrow{L}+\overrightarrow{S}$ offered in Figure 3a and Figure 3b. These are ascribed to 4-gradient Dirac operator, $\mathsf{\nabla }$ for vectors to multivector of n-dimensions by the cube translation along z-axis. Stress-energy tensor of BO mimicking rotation into forward time, $\tau =\overrightarrow{r}\times F$ by the emergence of spin up is balanced out by spin down of time reversal, $L=\overrightarrow{r}\times p$. The spin rotation matrices applicable to Clifford algebra are shown to the left. The half-integer spins of SU(2) group provide double cover (bivector) with shift in both $\theta$ and $\varphi$ for BOs of topological torus (Figure 1d) between two interchangeable hemispheres of the MP field. These are relevant to the Lie group ladder operators, $G\left(g\right)$ and $\widehat{G}\left(\widehat{g}\right)$ into n-dimension. Some key features of the dimensional cube are expounded to the right.

Figure 7. The basis of vectors to multivectors, matrices, tensors and Fourier transform for the electron position (P) in space (see also Figure 1c). Time axis is assumed along z-axis as the principal axis of the MP field and distances projected in the x-y directions. Fourier transform of tensor matrices along z-axis is by commutation, $pZ$ to $Z$. The electron of a resultant vector, $\overrightarrow{r}$ of four 3-fold rotational axes (shaded orange plane) in 3D of a cube can relate to the vector space, $\overrightarrow{J}=\overrightarrow{L}+\overrightarrow{S}$ offered in Figure 3a and Figure 3b. These are ascribed to 4-gradient Dirac operator, $\mathsf{\nabla }$ for vectors to multivector of n-dimensions by the cube translation along z-axis. Stress-energy tensor of BO mimicking rotation into forward time, $\tau =\overrightarrow{r}\times F$ by the emergence of spin up is balanced out by spin down of time reversal, $L=\overrightarrow{r}\times p$. The spin rotation matrices applicable to Clifford algebra are shown to the left. The half-integer spins of SU(2) group provide double cover (bivector) with shift in both $\theta$ and $\varphi$ for BOs of topological torus (Figure 1d) between two interchangeable hemispheres of the MP field. These are relevant to the Lie group ladder operators, $G\left(g\right)$ and $\widehat{G}\left(\widehat{g}\right)$ into n-dimension. Some key features of the dimensional cube are expounded to the right.

C. Visualization of 2D Manifolds into 4D Space-Time

D. Space-Time Curvature

Based on the descriptions of the geometry of MP model of 4D space-time offered so far in the preceding sections, an alternative interpretation of Einstein field equation described in ref. [14] is reproduced in Figure 8 for the planetary model at a higher hierarchy of scale. The curvature of space-time of elliptical orbits undergoing clockwise precession does not require framework of space-time fabric. Time reversal orbit of Earth is balanced out by clockwise precession of its

Figure 8. The components of Einstein field equation are applied to the MP model of 4D space-time [14]. At the solar scale, the planet, $\psi$ shifts in its position at point X of position 0 as COM of ZPE towards position 1. As a consequence, it warps space-time akin to the DBT (see also Figure 1a) and the emergence of angular momentum,$\overrightarrow{J}$ at position 1 and 3 of superposition states mimics stress-energy tensor, $T_{\mu ,\nu }$. Note, because of the irreducible spinor presented by the MP field mimicking Dirac’s string, space-time does not curve towards singularity. Instead, the Hamiltonian of quantized states (green outline) of moduli space undergo the process of DBT to generate Riemann surface, $R_{\mu ,\nu }$.

Figure 8. The components of Einstein field equation are applied to the MP model of 4D space-time [14]. At the solar scale, the planet, $\psi$ shifts in its position at point X of position 0 as COM of ZPE towards position 1. As a consequence, it warps space-time akin to the DBT (see also Figure 1a) and the emergence of angular momentum,$\overrightarrow{J}$ at position 1 and 3 of superposition states mimics stress-energy tensor, $T_{\mu ,\nu }$. Note, because of the irreducible spinor presented by the MP field mimicking Dirac’s string, space-time does not curve towards singularity. Instead, the Hamiltonian of quantized states (green outline) of moduli space undergo the process of DBT to generate Riemann surface, $R_{\mu ,\nu }$.