We define the gravitational Spinor (GS) as a fully symmetric fourth-order spinor field operator ${\hat{\psi }}_{ABCD}$, which is independent of any composite form (e.g., constructed from Weyl spinors) and directly associated with the quantum Weyl tensor:

${\hat{C}}_{\mu \nu \rho \sigma }=\kappa {\hat{\psi }}_{ABCD}\left({\sigma }_{\mu \rho }^{AB}{\sigma }_{\nu \sigma }^{CD}-{\sigma }_{\mu \sigma }^{AB}{\sigma }_{\nu \rho }^{CD}\right)$ (1)

Here, ${\sigma }_{\mu }^{A{B}^{\prime }}$ is a spinor basis (Vierbein notation), ensuring Lorentz invariance [3] . This assumption stems from spinor geometry, which decomposes the Weyl tensor (describing free gravitational radiation) into a completely symmetric spinor ${\hat{\psi }}^{ABCD}$ [13] . GS independence means that it is treated as a fundamental field, rather than a derivative, allowing direct quantization [14] .

The above formula (1) assumes a purely left-handed GS (ignoring the $\bar{\psi }$ term). This is a simplification of mathematical physics, similar to the left-handed neutrinos in the Standard Model (ignoring the right-handed neutrinos), which simplifies the calculation. However, in spinor geometry and physics, both left and right rotations can also be considered. but if we consider it purely from a geometric perspective, it is $\kappa =1$ [3] . They are not the same equation, but can be transformed into the same equation under the action of GGE, becoming different perspectives on the same relationship: the spinor form is the geometric foundation, while the classical form is a version realized through physical processes (such as GGE).

So the meaning of $\kappa$ is the dimensionality matching: The Weyl tensor ${\hat{C}}_{\mu \nu \rho \sigma }$ has dimension ${\left[L\right]}^{-2}$, the spinor ${\hat{\psi }}_{ABCD}$ has dimension ${\left[L\right]}^{-1}$ (spin-2 field), ${\sigma }_{\mu }^{A{A}^{\prime }}$ is dimensionless, and $\kappa ~\sqrt{8\pi G}$ (dimension $\left[L\right]$) is the dimension of the equilibrium equation. In the GGE mapping, $\kappa$ does not need to be numerically adjusted because the GGE rotation ( $g_{UV}$) remains scale-invariant [5] .

The spin-2 property of GS stems from its spinor structure: in the spinor formalism, each pointless index $A$ contributes a helicity of +1/2 [15] . The total helicity of the four pointless indices is +2. For massless fields, helicity is equivalent to spin:

$\text{Helicity}=4\times \left(+1/2\right)=+2\Rightarrow \text{Spin}=2$

The massless property is derived from the equation of motion ${\partial }^{A{A}^{\prime }}{\hat{\psi }}_{ABCD}=0$: Applying ${\partial }_{A{A}^{\prime }}$, we obtain $\square {\hat{\psi }}_{ABCD}=0$ (the massless Klein-Gordon equation), which ensures light-speed propagation and long-range gravity, consistent with gravitons [16] .

The GS field ${\hat{\psi }}_{ABCD}$ transforms to an irreducible representation under the $SO\left(1,3\right)$ Lorentz group: completely symmetric ( ${\hat{\psi }}_{\left(ABCD\right)}={\hat{\psi }}_{ABCD}$, all indices are arranged identically) and traceless ( ${ϵ}^{AB}{\hat{\psi }}_{ABCD}=0$, where ${ϵ}^{AB}$ is the antisymmetric spinor metric). This ensures 10 degrees of freedom (4-dimensional spin-2 field), corresponding to the symmetry of the Weyl tensor [3] . The transformation rule is:

${{\hat{\psi }}^{\prime }}_{ABCD}={\Lambda }_A^{A^{\prime }}{\Lambda }_B^{B^{\prime }}{\Lambda }_C^{C^{\prime }}{\Lambda }_D^{D^{\prime }}{\hat{\psi }}_{A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }}$ (2)

where $\Lambda$ is the Lorentz spinor representation, ${\Lambda }_A^{A^{\prime }}$ is the transformation matrix of $SL\left(2,ℂ\right)$, which represents the effect of Lorentz transformation in spinor space [17] .

Table 1 shows the comparison between GS as a unique spinor representation of spin-2 field and traditional fields (such as photon vector form and graviton tensor form) [18] . Among them:

$P^{ABCD,EFGH}=\frac{1}{24}\left({ϵ}^{A(E}{ϵ}^{F)B}{ϵ}^{C(G}{ϵ}^{H)D}+\text{permutations}\right)$ [13] .

We now consider Canonical Quantization, Commutation Relation of ${\hat{\psi }}_{ABCD}$, Vacuum State and Excitations (Plane Waves as Gravitons):

a) Canonical quantization

Imagine that canonical quantization elevates the classical field ${\psi }_{ABCD}\left(x\right)$ to a quantum operator ${\hat{\psi }}_{ABCD}\left(x\right)$, defines its conjugate momenta, and imposes commutation relations to satisfy the principles of quantum mechanics. Since ${\hat{\psi }}_{ABCD}$ is a completely symmetric massless spin-2 field, satisfying the equation of motion ${\partial }^{A{A}^{\prime }}{\hat{\psi }}_{ABCD}=0$, we can refer to the quantization methods of massless high-spin fields (such as the $U\left(1\right)$ gauge field of the electromagnetic field or the graviton of linearized gravity). Here the spinor index $A$ (pointless) is associated with the positive helicity (left-handed component) of the particle, and $A^{\prime }$ (with a point) is associated with the negative helicity (right-handed component), ${\partial }^{A^{\prime }A}={\bar{\partial }}_{A^{\prime }A}$ (conjugate form); in the Lagrangian, ${\partial }_{A{A}^{\prime }}$ and ${\partial }^{A^{\prime }A}$ are complementary: ${\partial }_{A{A}^{\prime }}$ acts on pointless indices, and ${\partial }^{A^{\prime }A}$ acts on point indices, ensuring that the Lagrangian is a scalar (Lorentz invariant) [21] .

Specifically, we first consider the transition from Einstein-Hilbert action to linearized gravity:

The Einstein-Hilbert action of classical general relativity is:

$S_{EH}=\frac{1}{16\pi G}{\displaystyle \int {\text{d}}^{4}x\sqrt{-g}R}$ (3)

where $R$ is the Ricci scalar and $g=det\left(g_{\mu \nu }\right)$. In the weak field approximation (linearized gravity), the metric decomposes into $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$, where ${\eta }_{\mu \nu }$ is the Minkowski metric and $h_{\mu \nu }$ is a small perturbation. After linearization, the action simplifies to the quadratic form:

$S_{lin}={\displaystyle \int {\text{d}}^{4}x}\left[\frac{1}{4}{\partial }_{\lambda }{h}_{\mu \nu }{\partial }^{\lambda }{h}^{\mu \nu }-\frac{1}{2}{\partial }_{\mu }{h}^{\mu \nu }{\partial }_{\nu }h-\frac{1}{4}{\partial }_{\mu }h{\partial }^{\mu }h+\frac{1}{2}{\partial }_{\mu }{h}^{\mu \nu }{\partial }_{\lambda }{h}_{\nu }^{\lambda }\right]$ (4)

where $h=h_{\mu }^{\mu }$. In the transverse-traceless (TT) gauge, ${\partial }_{\mu }{h}^{\mu \nu }=0$, $h=0$, and the Lagrangian further simplifies to:

$L_{lin}=\frac{1}{4}{\partial }_{\lambda }{h}_{\mu \nu }{\partial }^{\lambda }{h}^{\mu \nu }$ (5)

This is the free-field Lagrangian of the massless spin-2 field, describing the graviton, satisfying the wave equation $\square h_{\mu \nu }=0$ [22] .

Next, we consider the rewriting of the spinor form:

$C_{\mu \nu \rho \sigma }=\kappa {\psi }_{ABCD}\left({\sigma }_{\mu \rho }^{AB}{\sigma }_{\nu \sigma }^{CD}-{\sigma }_{\mu \sigma }^{AB}{\sigma }_{\nu \rho }^{CD}\right)$

As mentioned earlier, ${\psi }_{ABCD}$ is a spin-2 field.

$h_{\mu \nu }~{ϵ}_{\mu \nu }^{CD}{\psi }_{ABCD}$ (6)

where ${ϵ}_{\mu \nu }^{CD}$ is the spinor-tensor mapping factor, involving the Pauli matrix ${\sigma }_{\mu }^{A{A}^{\prime }}$. To derive the Lagrangian, we assume that ${\psi }_{ABCD}$ is a fundamental field whose dynamics is governed by a Lagrangian of the Klein-Gordon form:

$L={\partial }_{A{A}^{\prime }}{\psi }^{ABCD}{\partial }^{A^{\prime }A}{\psi }_{ABCD}$ (7)

Why this form? This is because

Let us consider the derivation from the linearized Einstein-Hilbert action to the spinor form:

${\partial }^{A{A}^{\prime }}{\psi }_{ABCD}=0$ (8)

This is equivalent to $\square h_{\mu \nu }=0$ (contraction by the spinor index).

So we can assume that the Lagrangian (7):

$ℒ={\partial }_{A{A}^{\prime }}{\psi }^{ABCD}{\partial }^{A^{\prime }A}{\psi }_{ABCD}$

From the Euler-Lagrange equation (variation of ${\psi }_{ABCD}$):

${\partial }_{A{A}^{\prime }}\frac{\partial L}{\partial \left({\partial }_{A{A}^{\prime }}{\psi }^{ABCD}\right)}+{\partial }^{A^{\prime }A}\frac{\partial L}{\partial \left({\partial }^{A^{\prime }A}{\psi }_{ABCD}\right)}=0$ (9)

we can get:

${\partial }_{A{A}^{\prime }}{\partial }^{A^{\prime }A}{\psi }_{ABCD}+{\partial }^{A^{\prime }A}{\partial }_{A{A}^{\prime }}{\psi }^{ABCD}=0$ (10)

By the complementary effect of ${\partial }_{A{A}^{\prime }}$ and ${\partial }^{A^{\prime }A}$, Equation (10) becomes:

$2{\partial }_{A{A}^{\prime }}{\partial }^{A^{\prime }A}{\psi }_{ABCD}=0\Rightarrow {\partial }_{A{A}^{\prime }}{\partial }^{A^{\prime }A}{\psi }_{ABCD}=\square {\psi }_{ABCD}=0$ (11)

The massless wave equation $\square {\psi }_{ABCD}=0$ implies that the mass $m=0$ (for a massless object, $\left(\square +m^2\right)\psi =0$); physically, formula (8) ${\partial }^{A{A}^{\prime }}{\psi }_{ABCD}=0$ ensures the transverse degree of freedom (similar to ${\partial }^{\mu }{A}_{\mu }=0$ for photon), which is the origin of the above equation of motion; and $\square {\psi }_{ABCD}=0$ ensures propagation at the speed of light.

Let’s continue with the steps of canonical quantization:

1) Fields and conjugate momentum: ${\hat{\psi }}_{ABCD}\left(x\right)$ to be a fundamental field, its classical Lagrangian can be rewritten in spinor form from the linearized Einstein-Hilbert action. A natural candidate is:

$S={\displaystyle \int {\text{d}}^{4}x{\partial }_{A{A}^{\prime }}{\psi }^{ABCD}{\partial }^{A^{\prime }A}{\psi }_{ABCD}}$ (12)

where ${\psi }^{ABCD}={\psi }_{ABCD}$ is completely symmetric, ensuring spin-2 and massless properties. In canonical quantization, conjugate momentum is the derivative of the Lagrangian with respect to the time derivative, which can be simplified by considering $\delta S$: ${\partial }_{A{A}^{\prime }}\to {\partial }_{00}$ as the time part:

${\hat{\pi }}^{ABCD}\left(x\right)=\frac{\delta S}{\delta {\partial }_0{\hat{\psi }}_{ABCD}}\propto {\partial }^{A^{\prime }A}{\hat{\psi }}^{ABCD}$ (13)

Since ${\hat{\psi }}_{ABCD}$ has four spinor indices (each corresponding to helicity +1/2, total helicity +2), we need to ensure its symmetry and tracelessness to match the spin-2 field.

2) Gauge fixing: Massless spin-2 fields have gauge degrees of freedom (e.g., transverse traceless gauges in gravitons). Similarly, ${\hat{\psi }}_{ABCD}$ requires gauge fixing to eliminate redundant degrees of freedom, e.g., by constraining ${\partial }^{A{A}^{\prime }}{\hat{\psi }}_{ABCD}=0$ or by additional conditions (e.g., orthogonality of the spinor field).

3) Canonical commutative relation: To ensure consistency, the canonical

commutative relation $\left\{,\right\}\to \frac{1}{i\hslash }\left[,\right]$ is imposed:

$\left[{\hat{\psi }}_{ABCD}\left(x\right),{\hat{\pi }}^{EFGH}\left(y\right)\right]=i\hslash {\delta }_{ABCD}^{EFGH}{\delta }^{\left(3\right)}\left(x-y\right)$ (14)

where ${\delta }_{ABCD}^{EFGH}$ is a fully symmetric Kronecker tensor,

${\delta }_{ABCD}^{EFGH}=\frac{1}{24}{\displaystyle {\sum }_{perms}{\delta }_A^E{\delta }_B^F{\delta }_C^G{\delta }_D^H}$, reflecting the symmetry of ${\hat{\psi }}_{ABCD}$. Other commutators (such as $\left[\hat{\psi },\hat{\psi }\right]=0$, $\left[\hat{\pi },\hat{\pi }\right]=0$) remain in standard form.

(4) Construction details:

${\hat{\psi }}_{ABCD}\left(x\right)={\displaystyle \int \frac{{\text{d}}^{3}k}{{\left(2\pi \right)}^3}}\frac{1}{\sqrt{2{\omega }_k}}{{\displaystyle \sum }}_{hel=\pm 2}\left[{\hat{a}}_{k,hel}{ϵ}_{ABCD}^{hel}\left(k\right){\text{e}}^{-ik\cdot x}+{\hat{a}}_{k,hel}^{†}{ϵ}_{ABCD}^{hel*}\left(k\right){\text{e}}^{ik\cdot x}\right]$ (15)

where ${ϵ}_{ABCD}^{hel}$ is the polar spinor of spin-2, satisfying the symmetry and transverse conditions.

b) Vacuum state and excitation

c. Derivation of exchange relations:

$\left[{\hat{\psi }}_{ABCD}\left(x\right),{\hat{\psi }}^{EFGH}\left(y\right)\right]=0$, $\left[{\hat{\pi }}_{ABCD}\left(x\right),{\hat{\pi }}^{EFGH}\left(y\right)\right]=0$ (16)

and the canonical commutator:

$\left[{\hat{\psi }}_{ABCD}\left(x\right),{\hat{\pi }}^{EFGH}\left(y\right)\right]=i\hslash P_{ABCD}^{EFGH}{\delta }^{\left(3\right)}\left(x-y\right)$ (17)

where $P_{ABCD}^{EFGH}$ is the projection operator, ensuring complete symmetry and tracelessness. The specific form of the projection operator must be derived from the symmetries of ${\hat{\psi }}_{ABCD}$ (complete symmetry across all four indices) and the equations of motion, potentially involving a combination of spinor indices (e.g., Pauli-Lubanski vector analysis).

a) Two-point correlation function

${\Delta }^{ABCD,EFGH}\left(x-y\right)=\langle 0|T\left\{{\hat{\psi }}_{ABCD}\left(x\right),{\hat{\psi }}_{EFGH}\left(y\right)\right\}|0\rangle$ (18)

where $T$ represents the time-order operator. This describes the quantum amplitude of the GS as it propagates from 𝑦 to 𝑥, reflecting its dynamics as a massless spin-2 field.

(1) Equation of Motion: From ${\partial }^{A{A}^{\prime }}{\hat{\psi }}_{ABCD}=0$, we know that ${\hat{\psi }}_{ABCD}$ satisfies the massless wave equation, with the plane wave solution ${\hat{\psi }}_{ABCD}\left(k\right){\text{e}}^{-ik\cdot x}$ in momentum space, where $k^2=0$ and ${\omega }_k=\left|k\right|$. Its Green’s function is similar to that of the electromagnetic field or graviton propagator.

(2) Fourier transform: Convert a two-point function to momentum space:

${\Delta }^{ABCD,EFGH}\left(x-y\right)={\displaystyle \int \frac{{\text{d}}^{4}q}{{\left(2\pi \right)}^4}{\text{e}}^{-iq\cdot \left(x-y\right)}{\Delta }^{ABCD,EFGH}\left(q\right)}$ (19)

(3) Momentum space form:

${\Delta }^{ABCD,EFGH}\left(q\right)\propto \frac{P^{ABCD,EFGH}}{q^2+iϵ}$ (20)

where $P^{ABCD,EFGH}$ is the projection operator onto the completely symmetric, traceless spin-2 component, $q^2=0$ (massless condition), and $iϵ$ ensures causality.

(4) Projection operator $P^{ABCD,EFGH}$:

The projection operator $P^{ABCD,EFGH}$ ensures the complete symmetry and tracelessness of the GS field ${\hat{\psi }}_{ABCD}$ ( ${ϵ}_{AB}{\psi }^{ABCD}=0$), defined as:

$P^{ABCD,EFGH}=\frac{1}{24}\left({ϵ}^{A(E}{ϵ}^{F)B}{ϵ}^{C(G}{ϵ}^{H)D}+\text{permutations}\right)$ (21)

where, “permutations” refers to fully symmetric permutations of the indices { $A,B,C,D$} and { $E,F,G,H$}, totaling 24 entries. This ensures that $P^{ABCD,EFGH}$ projects onto a fully symmetric fourth-order spinor space (spin 2), and 1/24 is a normalization factor.

For the massless propagator ${\Delta }^{ABCD,EFGH}\left(q\right)=P^{ABCD,EFGH}/\left(q^2+iϵ\right)$. In the Feynman diagram, the vertex factor is $V_{vertex}\kappa \left(p^{\mu }{p}^{\nu }\right){\sigma }_{\mu }^{A{A}^{\prime }}{\sigma }_{\nu }^{B{B}^{\prime }}{P}_{ABCD}$, where $\kappa$ is the gravitational coupling, but in the context of EM-to-gravity coupling (GGE mechanism), it matches the $\kappa$ in the Weyl-EM relation (Equation (3)), while the P-constrained spinor index, meets the symmetry requirements of high-spin field theory [20] .

(5) Physical meaning:

In the low energy limit, through the spinor-tensor mapping ${\hat{h}}_{\mu \nu }~{ϵ}_{\mu \nu }^{CD}{\hat{\psi }}_{ABCD}$, the propagator degenerates into a graviton propagator of the form:

$\langle 0|T\left\{{\hat{h}}_{\mu \nu }\left(x\right),{\hat{h}}_{\rho \sigma }\left(y\right)\right\}|0\rangle \propto \frac{{\eta }_{\mu \rho }{\eta }_{\nu \sigma }+{\eta }_{\mu \sigma }{\eta }_{\nu \rho }-{\eta }_{\mu \nu }{\eta }_{\rho \sigma }}{q^2+iϵ}$ (22)

At high energies or nonlinear states, the propagator may require non-perturbative corrections to describe solitons or condensed states.

b) The connection between communicators and exchange relations

$\left[{\hat{\psi }}_{ABCD}\left(x\right),{\hat{\pi }}^{EFGH}\left(y\right)\right]=i\hslash P_{ABCD}^{EFGH}{\delta }^{\left(3\right)}\left(x-y\right)$ (23)

determines the form of the vacuum expectation value. The propagator ${\Delta }^{ABCD,EFGH}\left(x-y\right)$ is a Green function that satisfies:

${\partial }^{A{A}^{\prime }}{\Delta }_{ABCD,EFGH}\left(x-y\right)=0$ (24)

and is calculated using time order and commutative relations. The $1/q^2$ term in momentum space arises from wave propagation in the massless field, and the projection operator $P$ ensures the physical degree of freedom.

The Lagrangian of the GS field ${\hat{\psi }}_{ABCD}\left(x\right)$ Equation (7) is

$ℒ={\partial }_{A{A}^{\prime }}{\psi }^{ABCD}{\partial }^{A^{\prime }A}{\psi }_{ABCD}$

Using variational calculus we can generate Equation (11):

$\square {\psi }_{ABCD}={\partial }_{A{A}^{\prime }}{\partial }^{A^{\prime }A}{\psi }_{ABCD}=0$

Analogy to the Klein-Gordon field:

Spinor Generalization:

Propagator Analogy:

${\Delta }^{ABCD,EFGH}\left(q\right)\propto \frac{P^{ABCD,EFGH}}{q^2+iϵ}$

Where $P^{ABCD,EFGH}$ is a projection operator onto the completely symmetric, traceless spin-2 subspace. The introduction of $P$ is similar to the projection of the graviton propagator ( ${\eta }_{\mu \rho }{\eta }_{\nu \sigma }+{\eta }_{\mu \sigma }{\eta }_{\nu \rho }-{\eta }_{\mu \nu }{\eta }_{\rho \sigma }$), but using spinor language.

In short, the simplicity of the Klein-Gordon field (spin-0, single degree of freedom) is extended to spin-2 through spinors, introducing complex symmetries (4 index, complete symmetry, traceless) to describe gravitons or solitons. The spinor form (Penrose spinor) decomposes the tensor field (such as $h_{\mu \nu }$) into ${\psi }_{ABCD}$, through formula (1):

${\hat{C}}_{\mu \nu \rho \sigma }=\kappa {\hat{\psi }}_{ABCD}\left({\sigma }_{\mu \rho }^{AB}{\sigma }_{\nu \sigma }^{CD}-{\sigma }_{\mu \sigma }^{AB}{\sigma }_{\nu \rho }^{CD}\right)$

Elegantly encodes the Weyl tensor, connecting classical gravity and quantum fields. Its physical implication is that GS, as a spin-2 field, retains the simple dynamics of the Klein-Gordon field ( $\square \psi =0$), but introduces gravitational curvature (Weyl tensor) through the spinor index, bridging quantum mechanics and general relativity. Furthermore, through the GGE extension, GS can be excited from electromagnetic/weak/strong fields (such as two optical solitons generating one gravitational soliton), suggesting the potential for unified interactions [23] .

As mentioned above, the core of the GS framework is free-field dynamics (the Klein-Gordon analogy), but to describe gravitational solitons (localized solutions in the presence of highly nonlinear backgrounds), a nonlinear self-interaction term is introduced. This modifies the propagator (from a free-field $1/q^2$ form to include self-energy corrections) and allows for soliton solutions (stable, non-diffusing wave packets). The following is a theoretical derivation based on the Dyson-Schwinger equation and soliton theory.

Theoretical derivation steps:

(1) Introducing nonlinear terms:

$L_{free}={\partial }_{AA\text{'}}{\psi }^{ABCD}{\partial }^{A^{\prime }A}{\psi }_{ABCD}$ (25)

This yields $\square {\psi }_{ABCD}=0$.

$L_{int}=-\lambda {\left({\psi }_{ABCD}{\psi }^{ABCD}\right)}^2$ (26)

where $\lambda >0$ is the coupling constant (dimensional analysis: $\left[\lambda \right]=\left[L^0\right]$, size: $\lambda ~{\kappa }^2{M}_{Pl}^2~8\pi$, if related to GGE). This simulates nonlinearities in gravity (such as higher-order terms in the Einstein-Hilbert interaction.

$L=L_{free}+L_{int}$ (27)

(2) Modify the equation of motion:

$\frac{\partial L}{\partial {\psi }_{ABCD}}-{\partial }_{A{A}^{\prime }}\frac{\partial L}{\partial \left({\partial }_{A{A}^{\prime }}{\psi }_{ABCD}\right)}-{\partial }^{A^{\prime }A}\frac{\partial L}{\partial \left({\partial }^{A^{\prime }A}{\psi }_{ABCD}\right)}=0$ (28)

$L_{int}$ contribution $\frac{\partial L_{int}}{\partial {\psi }_{ABCD}}=-4\lambda {\psi }^{ABCD}\left({\psi }_{EFGH}{\psi }^{EFGH}\right)$ (index contraction).

$\square {\psi }_{ABCD}-4\lambda {\psi }^{ABCD}\left({\psi }_{EFGH}{\psi }^{EFGH}\right)=0$ (29)

The nonlinear terms introduce self-coupling, similar to nonlinear wave equations (such as the KdV or nonlinear Schrödinger equations). Other nonlinear forms are also allowed (such as ${\left({\psi }_{ABCD}{\psi }^{ABCD}\right)}^n$, EM coupling terms), but they must satisfy Lorentz invariance, dimensionality consistency, symmetry, and renormalization. The constraints are: scalar properties, dimension [ $L^{-4}$], spinor symmetry, and renormalization. $n=2$ is preferred.

(3) Modify the propagator:

${\Delta }^{-1}\left(q\right)={\Delta }_0^{-1}\left(q\right)-\Sigma \left(q\right)$ (30)

where ${\Delta }_0^{-1}\left(q\right)\propto P/q^2$ is the free propagator and $\Sigma \left(q\right)$ is the self-energy (the sum of the one-particle-irreducible diagram).

First-order perturbation: $\Sigma \left(q\right)\propto \lambda {\displaystyle \int \frac{{\text{d}}^{4}p}{{\left(2\pi \right)}^4}{\Delta }_0\left(p\right){\Delta }_0\left(q-p\right)}$ (bubble diagram

i.e., refers to the one-loop self-energy contribution involving two internal propagators), also considering Dyson equation (34), the free propagator

${\Delta }_0^{ABCD,EFGH}\left(q\right)=\frac{P^{ABCD,EFGH}}{q^2+iϵ}$, and the self-energy ${\Sigma }^{ABCD,EFGH}\left(q\right)=\Sigma \left(q\right)P^{ABCD,EFGH}$, we can get:

${\Delta }^{ABCD,EFGH}\left(q\right)=\frac{P^{ABCD,EFGH}}{q^2+iϵ+\Sigma \left(q\right)}$ (31)

where $\Sigma \left(q\right)$ introduces mass correction or attenuation (IR/UV behavior).

(4) Forming a soliton solution:

Substituting into the above equation $\square \psi -\lambda {\psi }^3=0$, we get:

$\left(k^2-{\omega }^2\right)f^{″}\left(\xi \right)-\lambda f{\left(\xi \right)}^3=0.$

Integrate to get:

$f^{\prime }{\left(\xi \right)}^2=\frac{\lambda }{2\left(k^2-{\omega }^2\right)}f{\left(\xi \right)}^4+C$

According to the boundary conditions ( $f\to 0$ at $\infty$), $C=0$, so the solution is:

$f\left(\xi \right)=\sqrt{\frac{2\left(k^2-{\omega }^2\right)}{\lambda }}\mathrm{sech}\left(\sqrt{\frac{\lambda }{2\left(k^2-{\omega }^2\right)}}\xi \right)$ (32)

This matches the background gravitational soliton ${\mathrm{sech}}^2\left(ku\right)$ [10] , nonlinear $\lambda$ balances dispersion, forming stable solitons.

Based on the background formula [9] , the following is a detailed derivation, including SymPy simulation. The process simulates the GGE transformation, mapping two electromagnetic solitons (optical solitons) to the gravitational gauge potential to form solitons. The derivation steps are as follows:

(1) Initial optical soliton:

Each laser emits an optical soliton, and the polarization is expressed as:

${\omega }_U={\mathrm{sech}}^2\left(ku\right)\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)T_{EM},k=0.1{\text{m}}^{-1}$ (33)

where $T_{EM}$ is the generator of the Lie algebra u(1)， $u=k_{\mu }{x}^{\mu }$, $k_{\mu }=\left(1,0,0,1\right)$, $e_{\mu }=\left(0,1,-1,0\right)$ (polarization vector).

(2) Polarization rotation:

Time dependence of $g_{UV}\left(t\right)$:

$g_{UV}=\left(\begin{array}{cc}\cos \left(\theta \left(t\right)\right)& -\sin \left(\theta \left(t\right)\right)\\ \sin \left(\theta \left(t\right)\right)& \cos \left(\theta \left(t\right)\right)\end{array}\right)$ (34)

$\frac{\text{d}{g}_{UV}}{\text{d}t}=\frac{\text{d}\theta \left(t\right)}{\text{d}t}\left(\begin{array}{cc}-\sin \left(\theta \left(t\right)\right)& -\cos \left(\theta \left(t\right)\right)\\ \cos \left(\theta \left(t\right)\right)& -\sin \left(\theta \left(t\right)\right)\end{array}\right)$ (35)

(3) GGE Transformation:

${\omega }_V\left(t\right)=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}\left(\frac{\text{d}{g}_{UV}}{\text{d}t}\right)$ (36)

$g_{UV}^{-1}{\omega }_U{g}_{UV}={\mathrm{sech}}^2\left(kt\right)\left(\begin{array}{cc}\cos \left(2\theta \right)& -\sin \left(2\theta \right)\\ -\sin \left(2\theta \right)& -\cos \left(2\theta \right)\end{array}\right)$ (37)

$g_{UV}^{-1}d{g}_{UV}=\frac{\text{d}\theta }{\text{d}t}\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ (38)

(4) Matching gravitational solitons:

${\mathrm{sech}}^2\left(ku\right)\left(\begin{array}{cc}A& B\\ B& -A\end{array}\right)$, $A=8/9$, $B=\sqrt{17}/9$ (39)

(5) SymPy Simulation results:

${\omega }_V={\mathrm{sech}}^2\left(ku\right)\left(\begin{array}{cc}\cos 2\theta & -\sin 2\theta \\ -\sin 2\theta & -\cos 2\theta \end{array}\right)J_{12}$ (40)

where $J_{12}$ is the generator of the Lie algebra so(1, 3), it can be proved that $g_{UV}$ above gives: $g_{UV}^{-1}{T}_{EM}{g}_{UV}=J_{12}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)=T_{EM}$.

In vacuum $R_{\mu \nu }=0$, the Riemann tensor $R_{\mu \nu \rho \sigma }=C_{\mu \nu \rho \sigma }$, the GS field ${\hat{\psi }}_{ABCD}$ satisfies the massless wave equation $\square {\hat{\psi }}_{ABCD}=0$ (or with the nonlinear term $\square {\hat{\psi }}_{ABCD}-4\lambda {\hat{\psi }}^{ABCD}\left({\hat{\psi }}_{EFGH}{\hat{\psi }}^{EFGH}\right)=0$), through the spinor-tensor mapping Equation (1):

${\hat{C}}_{\mu \nu \rho \sigma }=\kappa {\hat{\psi }}_{ABCD}\left({\sigma }_{\mu \rho }^{AB}{\sigma }_{\nu \sigma }^{CD}-{\sigma }_{\mu \sigma }^{AB}{\sigma }_{\nu \rho }^{CD}\right)$

The quantized Weyl tensor ${\hat{C}}_{\mu \nu \rho \sigma }$ here describes the vacuum gravitational fluctuations (gravitons or solitons).

$G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G{T}_{\mu \nu }$ (41)

where $G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }$. Quantization requires dealing with the quantum version of the curvature tensor $R_{\mu \nu \rho \sigma }$ and coupling it with the quantum matter field ${\hat{T}}_{\mu \nu }$.

$R_{\mu \nu \rho \sigma }=C_{\mu \nu \rho \sigma }+\frac{1}{2}\left(g_{\mu \rho }{R}_{\nu \sigma }-g_{\mu \sigma }{R}_{\nu \rho }-g_{\nu \rho }{R}_{\mu \sigma }+g_{\nu \sigma }{R}_{\mu \rho }\right)-\frac{1}{6}R\left(g_{\mu \rho }{g}_{\nu \sigma }-g_{\mu \sigma }{g}_{\nu \rho }\right)$ (42)

The above formula can be quantized as:

${\hat{R}}_{\mu \nu \rho \sigma }={\hat{C}}_{\mu \nu \rho \sigma }+\hat{R}icci+\hat{R}\text{term}$,

where GS ${\hat{\psi }}_{ABCD}$ is responsible for ${\hat{C}}_{\mu \nu \rho \sigma }$, and material coupling is responsible for $\hat{R}icci$.

In spinor geometry, Einstein’s equations can be expressed as equivalent spinor representations:

${\Phi }_{AB{A}^{\prime }{B}^{\prime }}=8\pi G{\Theta }_{AB{A}^{\prime }{B}^{\prime }}$ (43)

Where ${\Phi }_{AB{A}^{\prime }{B}^{\prime }}$ is the Ricci spinor ( $R_{\mu \nu }={\Phi }_{AB{A}^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$), ${\Theta }_{AB{A}^{\prime }{B}^{\prime }}$ is the energy-momentum spinor ( $T_{\mu \nu }={\Theta }_{AB{A}^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$). The Weyl spinor ${\psi }_{ABCD}$ remains independent (degrees of freedom). In vacuum ${\Phi }_{AB{A}^{\prime }{B}^{\prime }}=0$, the Equation (43) degenerates to ${\partial }^{A{A}^{\prime }}{\psi }_{ABCD}=0$.

${\hat{R}}_{\mu \nu \rho \sigma }={\hat{C}}_{\mu \nu \rho \sigma }+{\hat{\Phi }}_{AB{A}^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }\left({\sigma }_{C{C}^{\prime }}^{\rho }{\sigma }_{D{D}^{\prime }}^{\sigma }+{\sigma }_{C{C}^{\prime }}^{\sigma }{\sigma }_{D{D}^{\prime }}^{\rho }\right)+\Lambda {ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}{g}_{\mu [\rho }{g}_{\sigma ]\nu }+\cdots$, where ${\hat{C}}_{\mu \nu \rho \sigma }=\kappa {\hat{\psi }}_{ABCD}\left({\sigma }_{\mu \rho }^{AB}{\sigma }_{\nu \sigma }^{CD}-{\sigma }_{\mu \sigma }^{AB}{\sigma }_{\nu \rho }^{CD}\right)$(Equation (1)).

${\hat{T}}_{\mu \nu }={\partial }_{\mu }\hat{\varphi }{\partial }_{\nu }\hat{\varphi }+\frac{1}{2}{g}_{\mu \nu }\left({\partial }_{\lambda }\hat{\varphi }{\partial }^{\lambda }\hat{\varphi }-m^2{\hat{\varphi }}^2\right)+\frac{i}{2}\left[\widehat{\stackrel{¯}{\chi }}{\gamma }_{(\mu }{\nabla }_{\nu )}\hat{\chi }-\left({\nabla }_{(\mu }\widehat{\stackrel{¯}{\chi }}\right){\gamma }_{\nu )}\hat{\chi }\right]+{\hat{F}}_{\mu \rho }{\hat{F}}_{\nu }^{\rho }-\frac{1}{4}{g}_{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }+\cdots$

${\hat{\Phi }}_{AB{A}^{\prime }{B}^{\prime }}=8\pi G{\hat{\Theta }}_{AB{A}^{\prime }{B}^{\prime }}$ (44)

where ( ${\hat{\Phi }}_{AB{A}^{\prime }{B}^{\prime }}={\partial }_{C{C}^{\prime }}{\psi }^{ABCD}{ϵ}_{A^{\prime }{C}^{\prime }}+\cdots$), i.e., decomposition from curvature (42), and ${\hat{\Theta }}_{AB{A}^{\prime }{B}^{\prime }}={\hat{T}}_{\mu \nu }{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$.

Step 1: Spinor decomposition of the curvature tensor:

$\begin{array}{c}{R}_{\mu \nu \rho \sigma }={\psi }_{ABCD}{ϵ}_{A^{\prime }{B}^{\prime }}{ϵ}_{C^{\prime }{D}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }{\sigma }_{C{C}^{\prime }}^{\rho }{\sigma }_{D{D}^{\prime }}^{\sigma }+{\bar{\psi }}_{A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }}{ϵ}_{AB}{ϵ}_{CD}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }{\sigma }_{C{C}^{\prime }}^{\rho }{\sigma }_{D{D}^{\prime }}^{\sigma }\\ +{\Phi }_{AB{A}^{\prime }{B}^{\prime }}{ϵ}_{CD}{ϵ}_{C^{\prime }{D}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }\left({\sigma }_{C{C}^{\prime }}^{\rho }{\sigma }_{D{D}^{\prime }}^{\sigma }+{\sigma }_{C{C}^{\prime }}^{\sigma }{\sigma }_{D{D}^{\prime }}^{\rho }\right)+\Lambda {ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}{g}_{\mu [\rho }\text{}{g}_{\sigma ]\nu }\end{array}$ (45)

Where ${\psi }_{ABCD}$ is the Weyl spinor, fully symmetric, with 10 degrees of freedom (spin-2, free from gravitational radiation, GS main field). ${\Phi }_{AB{A}^{\prime }{B}^{\prime }}$ is the Ricci spinor, symmetric ( ${\Phi }_{AB{A}^{\prime }{B}^{\prime }}={\Phi }_{\left(AB\right)\left(A^{\prime }{B}^{\prime }\right)}$), with 9 degrees of freedom, derived from matter. $\Lambda =R/24$ is scalar curvature ( $R$ is the Ricci scalar). ${\sigma }_{A{A}^{\prime }}^{\mu }$ is the Vierbein (spinor basis), connecting tensors and spinors: $g_{\mu \nu }={\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }{ϵ}^{AB}{ϵ}^{A^{\prime }{B}^{\prime }}$.

$R_{\mu \nu }=R_{\mu \rho \nu }^{\rho }=T{r}_{CD}\left({\psi }_{ABCD}{ϵ}_{A^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }{\sigma }_{C{C}^{\prime }}^{\rho }{\sigma }_{D{D}^{\prime }}^{\sigma }+\text{Other items}\right)$

where the Weyl term ${\psi }_{ABCD}$ contracts to zero due to perfect symmetry ${\psi }_{\left(ABCD\right)}$:

${\psi }_{ABCD}{ϵ}^{CD}=0$ (46)

${\Phi }_{AB{A}^{\prime }{B}^{\prime }}{ϵ}_{CD}{ϵ}_{C^{\prime }{D}^{\prime }}{\sigma }_{C{C}^{\prime }}^{\rho }{\sigma }_{\rho }^{D{D}^{\prime }}={\Phi }_{AB{A}^{\prime }{B}^{\prime }}{g}_{\rho }^{\rho }=4{\Phi }_{AB{A}^{\prime }{B}^{\prime }}$ (47)

$\Lambda {ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}{ϵ}_{CD}{ϵ}_{C^{\prime }{D}^{\prime }}{\sigma }_{C{C}^{\prime }}^{\rho }{\sigma }_{\rho }^{D{D}^{\prime }}=\Lambda {ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}\cdot 4=\frac{R}{24}\cdot 4{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}=\frac{R}{6}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}$ (48)

$R_{\mu \nu }={\Phi }_{AB{A}^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }+\frac{R}{6}{g}_{\mu \nu }$ (49)

where $g_{\mu \nu }={ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$. Substituting into the Einstein tensor $G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }$, we get its spinor form:

$G_{\mu \nu }=\left({\Phi }_{AB{A}^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }+\frac{R}{6}{g}_{\mu \nu }\right)-\frac{1}{2}R{g}_{\mu \nu }={\Phi }_{AB{A}^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }+\frac{R}{3}{g}_{\mu \nu }$ (50)

Step 2: The spinor form of the energy-momentum tensor:

$T_{\mu \nu }={\Theta }_{AB{A}^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$ (51)

where ${\Theta }_{AB{A}^{\prime }{B}^{\prime }}$ is a symmetric spinor (9 degrees of freedom) derived from matter fields (e.g., scalar, fermionic, electromagnetic). For example:

Step 3: Spinor form Einstein equation:

${\Phi }_{AB{A}^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }-\frac{R}{3}{g}_{\mu \nu }=8\pi G{\Theta }_{AB{A}^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$ (52)

Noting that $g_{\mu \nu }={ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$, the scalar term can be written as:

$-\frac{R}{3}{g}_{\mu \nu }=-\frac{R}{3}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$

Therefore, equation (53) becomes:

$\left({\Phi }_{AB{A}^{\prime }{B}^{\prime }}-\frac{R}{3}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}\right){\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }=8\pi G{\Theta }_{AB{A}^{\prime }{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$ (53)

Since ${\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$ are linearly independent bases, the coefficients of Equation (53) must be equal:

${\Phi }_{AB{A}^{\prime }{B}^{\prime }}-\frac{R}{3}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}=8\pi G{\Theta }_{AB{A}^{\prime }{B}^{\prime }}$ (54)

To eliminate the scalar term $-\frac{R}{3}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}$, we use the trace constraint of the Einstein equation. Contracting $G_{\mu \nu }=8\pi G{T}_{\mu \nu }$ with $g_{\mu \nu }$, we get:

$G_{\mu }^{\mu }=R-\frac{1}{2}R\cdot 4=-R=8\pi G{T}_{\mu }^{\mu }$

Then calculate $T_{\mu }^{\mu }=T$:

$T=g^{\mu \nu }{T}_{\mu \nu }={\Theta }_{AB{A}^{\prime }{B}^{\prime }}{g}^{\mu \nu }{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }={\Theta }_{AB{A}^{\prime }{B}^{\prime }}{ϵ}^{AB}{ϵ}^{A^{\prime }{B}^{\prime }}{\sigma }_{\mu }^{A{A}^{\prime }}{\sigma }_{\nu }^{B{B}^{\prime }}{\sigma }_{A{A}^{\prime }}^{\mu }{\sigma }_{B{B}^{\prime }}^{\nu }$

$T={\Theta }_{AB{A}^{\prime }{B}^{\prime }}{ϵ}^{AB}{ϵ}^{A^{\prime }{B}^{\prime }}=\Theta$

where $\Theta ={\Theta }_{AB{A}^{\prime }{B}^{\prime }}{ϵ}^{AB}{ϵ}^{A^{\prime }{B}^{\prime }}$.

Therefore we have:

$-R=8\pi G\Theta \Rightarrow R=-8\pi G\Theta$

Substituting back to $-\frac{R}{3}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}$:

$-\frac{R}{3}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}=-\frac{-8\pi G\Theta }{3}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}=\frac{8\pi G\Theta }{3}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}$

Equation (54) becomes:

${\Phi }_{AB{A}^{\prime }{B}^{\prime }}+\frac{8\pi G\Theta }{3}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}=8\pi G{\Theta }_{AB{A}^{\prime }{B}^{\prime }}$ (55)

To derive the simplified spinor form of the Einstein equation, we decompose ${\Phi }_{AB{A}^{\prime }{B}^{\prime }}$ and ${\Theta }_{AB{A}^{\prime }{B}^{\prime }}$ into their trace-free and trace parts. The Ricci spinor ${\Phi }_{AB{A}^{\prime }{B}^{\prime }}$ is symmetric and trace-free in the spinor indices, satisfying ${\Phi }_{AB{A}^{\prime }{B}^{\prime }}{ϵ}^{AB}{ϵ}^{A^{\prime }{B}^{\prime }}=0$, as derived from the trace-free part of the Ricci tensor (Penrose & Rindler [3] , Section 4.6). Similarly, ${\Theta }_{AB{A}^{\prime }{B}^{\prime }}$ can be decomposed as:

${\Theta }_{AB{A}^{\prime }{B}^{\prime }}={\Theta }_{\left(AB{A}^{\prime }{B}^{\prime }\right)}+\frac{\Theta }{4}{ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}$

where ${\Theta }_{\left(AB{A}^{\prime }{B}^{\prime }\right)}$ is the trace-free part, and $\Theta ={\Theta }_{AB{A}^{\prime }{B}^{\prime }}{ϵ}^{AB}{ϵ}^{A^{\prime }{B}^{\prime }}$ is the trace. Substituting this decomposition into Equation (55):