A quantum theory of spacetime in spinor formalism and the physical reality of cross-ratio representation: the equation of density parameters of dark energy, matter, and ordinary matter is derived: Ω _M ² = 4 Ω _b Ω _Λ

ABSTRACT

By theorizing the physical reality through the deformation of an arbitrary cross-ratio, we leverage Galois differential theory to describe the dynamics of isomonodromic integratable system. We found a new description of curvature of spacetime by the equivalency of isomonodromic integratable system and Penrose’s spinor formalism of general relativity. Using such description, we hypothetically quantize the curvature of spacetime (gravity) and apply to the problem of the evolution of the universe. The Friedmann equation is recovered and compared so that the mathematical relationship among dark energy, matter (dark matter + ordinary matter), and ordinary matter, ${\Omega }_M{2}^{}\simeq 4{\Omega }_b{\Omega }_{\Lambda }$, is derived; the actual observed results are compared to this equation (calculated Ω_M = 0.33 vs. observed Ω_M = 0.31); the model might explain the origin of dark energy and dark matter of the evolution of the universe.

Main article text

INTRODUCTION

We looked for the simplest mathematical object to identify the underlying reality of nature, and we found it to be cross-ratio. By defining cross-ratio over Riemann sphere, infinity is just another usual point; similarly, there shouldn’t be any point in the universe more special than others. However, the variety of nature must be realized as a condition for such an underlying object.

In “Cross-ratio arbitrariness and the constraint to the parameter space of projective space basis” section, the article explains how potential physical varieties come from different representations of the same thing – cross-ratio deformation. So, the cross-ratio consists of both simplicity and variety. A successful example is like Einstein’s masterpiece – general theory of relativity. Although Einstein’s field equation is simple, many interesting solutions emerged.

In “Cassidy’s work on isomonodromic system” section, we introduce Galois differential theory and related Cassidy’s work. It is a mathematical machinery to manifest the deformation of cross-ratio. Cassidy’s work consists of introducing a 2 by 2 matrix differential equation and related isomonodromic integratable system, so it can describe the deformation. By such machinery, we formulate an alternative theory of the dynamics of curvature of spacetime to recover the spinor general relativity equivalent counterpart (for which a brief introduction is given in “Brief overview of spinor formulation of general relativity” section). By such connection, we hypothetically claim the origin of spacetime is from the isomonodromic integratable system, and spacetime is more fundamentally described by the curvature rather than metric or coordinated mathematical framework, that is, spinor formulation of general relativity might be more fundamental than classical general relativity; a similar argument was postulated by Penrose (1960).

In “As an application to the problem of modeling the universe evolution” section, we apply the calculation of the deformation of the isomonodromic integratable system with certain simplified conditions so a solution is found. The solution is used to recover Friedmann equation and related density parameters such that observed and calculated results are compared. This proposes an explanation of the origin of dark matter and dark energy without new kind of matter or energy, as they are new kind of gravitational field of spacetime’s curvature.

BRIEF OVERVIEW OF SPINOR FORMULATION OF GENERAL RELATIVITY

Penrose’s spinor approach to general relativity

Spinor formulaism of general relativity (i.e., Spinor GR) (Penrose, 1960) adopted a coordinate-free approach. The correspondence between tensors and spinors is obtained by the use of a hybrid spinor Σ_Ν^CD′ (2 × 2 Hermitian matrix per CD index, denoted as Σ_Ν in short) (Einstein’s summation notation is used in this article):

$${\Sigma }_{{\rm M}}{A^{}}_{C^{\prime }}{\Sigma }_{{\rm N}}{{\text{BC}}^{\prime }}^{}+{\Sigma }_{{\rm N}}{A^{}}_{C^{\prime }}{\Sigma }_{{\rm M}}{{\text{BC}}^{\prime }}^{}=g_{{\rm M}{\rm N}}{\epsilon }^{\text{AB}},\mathrm{for}A,B,C^{\prime }=0,1;{\rm M},{\rm N}=0,1,2,3,$$  (1)

where ${\epsilon }^{\text{BC}}$ represents 2 × 2 skew-symmetric “metric” spinor (components’ values are 0 or ± 1). To raise or lower spinor index, either one of ${\epsilon }^{\text{BC}}\text{, }{\epsilon }_{\text{BC}}$, ${\epsilon }^{B^{\prime }{C}^{\prime }}$, and ${\epsilon }_{B^{\prime }{C}^{\prime }}$ is used:

$${\Xi }^A={\epsilon }^{\text{AB}}{\Xi }_B,{\Xi }_B={\Xi }^A{\epsilon }_{\text{AB}}$$  (2)

where Ξ denotes spinor. Primed indexes refer to the complex conjugate spin space. Please note that

$${\epsilon }^{\text{BD}}{\epsilon }_{\text{DF}}=-{\Delta }^B{F}_{},{\Delta }^B{F}_{}=0(\mathrm{if }B\ne F),\mathrm{or }1(\mathrm{if}B=F).$$  (3)

$${\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}\stackrel{\mathrm{def}}{=}{g}^{{\rm M}{\rm N}}{\Sigma }_{{\rm N}}{{\text{CD}}^{\prime }}^{}{\epsilon }_{\mathrm{CA}}{\epsilon }_{D^{\prime }{B}^{\prime }}$$  (4)

is Hermitian because g^ΜΝ (g is the metric) is real.

Spinor GR requires the covariant derivatives of ${\Sigma }_{{\rm N}}{{\text{CD}}^{\prime }}^{}$ , g_ΜΝ, and $({\epsilon }_{\mathrm{CA}}{\epsilon }_{D^{\prime }{B}^{\prime }})$ to be zero.

The spinor equivalent of tensor G_ΜΝ is stated as

$$G_{{\text{AC}}^{\prime }{\text{BD}}^{\prime }}={\Sigma }^{{\rm M}}{{\text{AC}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{BD}}^{\prime }}_{}{G}_{{\rm M}{\rm N}},G_{{\rm M}{\rm N}}={\Sigma }_{{\rm M}}{{\text{AC}}^{\prime }}^{}{\Sigma }_{{\rm N}}{{\text{BD}}^{\prime }}^{}{G}_{{\text{AC}}^{\prime }{\text{BD}}^{\prime }}$$  (5)

Curvature tensors, symmetry and Einstein tensors in spinor general relativity

By $R_{{\mathrm{AE}}^{\prime }{\text{BF}}^{\prime }{\mathrm{CG}}^{\prime }{\mathrm{DH}}^{\prime }}$ (spinor version of Riemann-Christoffel tensor R_ΜΝΡΣ), the curvature in spinor GR is defined by curvature spinors - Χ_ABCD and ${\Phi }_{{\mathrm{ABG}}^{\prime }{H}^{\prime }}$ (Penrose, 1960):

$$R_{{\text{AE}}^{\prime }{\text{BF}}^{\prime }{\mathrm{CG}}^{\prime }{\mathrm{DH}}^{\prime }}=\frac{1}{2}\left({{\rm X}}_{\text{ABCD}}{\epsilon }_{E^{\prime }{F}^{\prime }}{\epsilon }_{G^{\prime }{H}^{\prime }}+{\epsilon }_{\text{CD}}{\Phi }_{{\text{ABG}}^{\prime }{H}^{\prime }}{\epsilon }_{E^{\prime }{F}^{\prime }}+{\epsilon }_{\text{AB}}{\Phi }^{\star }{E^{\prime }{F}^{\prime }\text{CD}}_{}{\epsilon }_{G^{\prime }{H}^{\prime }}+{\epsilon }_{\text{AB}}{\epsilon }_{\text{CD}}{{\rm X}}^{\star }{E^{\prime }{F}^{\prime }{G}^{\prime }{H}^{\prime }}_{}\right)$$

($\star$ indicates complex conjugate), which satisfy the following equations (Penrose, 1960):

$$\begin{array}{l}\left({\nabla }_{{\text{AH}}^{\prime }}{\nabla }_B{H^{\prime }}^{}+{\nabla }_{{\text{BH}}^{\prime }}{\nabla }_A{H^{\prime }}^{}\right){\Xi }_F={{\rm X}}_{\text{ABFE}}{\Xi }^E\\ \left({\nabla }_{{\mathrm{AD}}^{\prime }}{\nabla }^A{C^{\prime }}_{}+{\nabla }_{{\text{AC}}^{\prime }}{\nabla }^A{D^{\prime }}_{}\right){\Xi }_F={\Phi }_{\text{FE}{D}^{\prime }{C}^{\prime }}{\Xi }^E\end{array}$$  (6)

where ${\nabla }_{A{G}^{\prime }}$ and ${\nabla }_B{G^{\prime }}^{}$ are spinor covariant derivatives and raised spinor derivatives, respectively. They can be modified to

$$\begin{array}{c}{\epsilon }^{\text{GF}}\left({\nabla }_{A{H}^{\prime }}{\nabla }_B{H^{\prime }}^{}+{\nabla }_{B{H}^{\prime }}{\nabla }_A{H^{\prime }}^{}\right){\Xi }_F={\epsilon }^{\text{GF}}{{\rm X}}_{\text{ABFE}}{\Xi }^E\to \left({\nabla }_{A{H}^{\prime }}{\nabla }_B{H^{\prime }}^{}+{\nabla }_{B{H}^{\prime }}{\nabla }_A{H^{\prime }}^{}\right){\Xi }^G={{\rm X}}_{\text{AB}}{G^{}}_E{\Xi }^E\\ {\epsilon }^{\text{GF}}\left({\nabla }_{A{D}^{\prime }}{\nabla }^A{C^{\prime }}_{}+{\nabla }_{A{C}^{\prime }}{\nabla }^A{D^{\prime }}_{}\right){\Xi }_F={\epsilon }^{\text{GF}}{\Phi }_{\text{FE}{D}^{\prime }{C}^{\prime }}{\Xi }^E={\epsilon }^{\text{GF}}\left({\Phi }^{\star }{D^{\prime }{C}^{\prime }\text{FE}}_{}\right){\Xi }^E\to \left({\nabla }_{A{D}^{\prime }}{\nabla }^A{C^{\prime }}_{}+{\nabla }_{A{C}^{\prime }}{\nabla }^A{D^{\prime }}_{}\right){\Xi }^G={\Phi }^{\star }{{D^{\prime }{C}^{\prime }}_{}}^G{E}_{}{\Xi }^E\end{array}$$  (7)

for which we use the symmetry property of Χ_AB(FE) and ${\Phi }_{D^{\prime }{C}^{\prime }(\text{FE})}$, ${\Phi }_{D^{\prime }{C}^{\prime }\text{FE}}={\Phi }^{\star }{\text{FE}{D}^{\prime }{C}^{\prime }}_{}$ (Penrose,1960).

By Bianchi identity of Riemann curvature,

$${\partial }_{{\rm T}}{R}_{{\rm M}{\rm N}{\rm P}\Sigma }+{\partial }_{{\rm P}}{R}_{{\rm M}{\rm N}\Sigma {\rm T}}+{\partial }_{\Sigma }{R}_{{\rm M}{\rm N}{\rm T}{\rm P}}=0$$

it implies the relationship of curvature spinors, which is called Bianchi identity of spinor GR (Penrose, 1960):

$${\nabla }^B{D^{\prime }}_{}{{\rm X}}_{\text{FEAB}}={\nabla }_A{C^{\prime }}^{}{\Phi }_{{\text{FED}}^{\prime }{C}^{\prime }}$$  (8)

One can further modify it by the symmetry of (Penrose, 1960)

$${{\rm X}}_{\text{FEAB}}={{\rm X}}_{\text{ABFE}},{\Phi }_{\text{FE}{D}^{\prime }{C}^{\prime }}={\Phi }^{\star }{D^{\prime }{C}^{\prime }\text{FE}}_{}$$

to

$$\begin{array}{c}\to {\nabla }^B{D^{\prime }}_{}{{\rm X}}_{\text{ABFE}}={\nabla }_A{C^{\prime }}^{}{\Phi }^{\star }{D^{\prime }{C}^{\prime }\text{FE}}_{}\\ \to {\nabla }^B{D^{\prime }}_{}{{\rm X}}_{\text{ABFE}}{\epsilon }^{\text{GF}}={\nabla }_A{C^{\prime }}^{}{\Phi }^{\star }{D^{\prime }{C}^{\prime }\text{FE}}_{}{\epsilon }^{\text{GF}}\\ \to {\nabla }^B{D^{\prime }}_{}{{\rm X}}_{\text{AB}}{G^{}}_E={\nabla }_A{C^{\prime }}^{}{\Phi }^{\star }{{D^{\prime }{C}^{\prime }}_{}}^G{E}_{}\end{array}$$

Note that we can express Bianchi identity as

$${\nabla }^B{D^{\prime }}_{}{{\rm X}}_{\text{AB}}{G^{}}_E={\nabla }_A{C^{\prime }}^{}{\Phi }^{\star }{{D^{\prime }{C}^{\prime }}_{}}^G{E}_{}$$  (9)

where ${{\rm X}}_{\text{AB}}{G^{}}_E$ and ${\Phi }^{\star }{{D^{\prime }{C}^{\prime }}_{}}^G{E}_{}$ are 2 × 2 traceless matrices if considering G and E as the matrix indexes because of the symmetry property of Χ_AB(FE) and ${\Phi }^{\star }{D^{\prime }{C}^{\prime }(\text{FE})}_{}$ (Penrose, 1960) (trivially proved by contracting ${{\rm X}}_{\text{ABFE}}{\epsilon }^{\text{GF}}$ and ${\Phi }^{\star }{D^{\prime }{C}^{\prime }\text{FE}}_{}{\epsilon }^{\text{GF}}$). The author claims this form (or in matrix equivalent), and Equation (7) is more fundamental to the original version of Equations (6) and (8).

By the symmetry of Riemann curvature R_ΜΝΡΣ, the curvature spinor Χ_FEAB needs to obey (Penrose, 1960)

$$\Lambda \stackrel{\mathrm{def}}{=}\frac{1}{2}{{\rm X}}_{\text{AB}}{\text{AB}}^{}\mathrm{ is real}.$$  (10)

Penrose (1960) mentioned that the geometry of a Riemannian space (with signature +,−,−,−) can be described by spinor terms with the role of curvature tensor being taken over spinors Χ_FEAB and ${\Phi }_{\text{FE}{D}^{\prime }{C}^{\prime }}$ satisfying their symmetry property, Bianchi identity (Equation 8), and Equation (6).

Finally, Ricci spinor is equivalent to Ricci tensor (Penrose, 1960):

$$R_{\text{A}{\text{C}}^{\prime }\text{B}{\text{D}}^{\prime }}=\Lambda {\epsilon }_{\text{AB}}{\epsilon }_{\text{CD}}-{\Phi }_{\text{AB}{\text{C}}^{\prime }{\text{D}}^{\prime }}$$  (11)

Ricci scalar is (Penrose, 1960)

$$R=4\Lambda$$  (12)

Einstein spinor is equivalent to Einstein tensor (Penrose, 1960):

$$G_{\text{A}{C}^{\prime }\text{B}{\text{D}}^{\prime }}=-\Lambda {\epsilon }_{\text{AB}}{\epsilon }_{\text{CD}}-{\Phi }_{\text{AB}{\text{C}}^{\prime }{D}^{\prime }}$$  (13)

APPLICATION OF GALOIS DIFFERENTIAL THEORY TO SPINOR FORMULATION OF GENERAL RELATIVITY

Cassidy’s work on isomonodromic system

For a commuting set of derivations $\frac{\partial }{\partial z_i}$ (i=0,...,m), there exists an integratable (parameterized second-order linear differential) system [Cassidy & Singer, 2005: Prop6.3(3)]

$$\frac{\partial }{\partial z_i}Y=B_{z_i}Y$$  (14)

with integratability condition: $[B_{z_i},B_{z_j}]=\frac{\partial }{\partial z_i}{B}_{z_j}-\frac{\partial }{\partial z_j}{B}_{z_i}(i,j=0,\mathrm{...},m)$, where Y is in GL₂(f(z_i)), $B_{z_0}={\mathrm{sl}}_2\left(f\left(z_i\right)\right)$, $B_{z_k}={\mathrm{gl}}_2\left(f\left(z_i\right)\right)$ (k=1,…,m), f(z_i) is a set of functions which $\frac{\partial }{\partial z_i}$ acts on. The variable set {z_k}, for k=1,…,m, is called parameters-set. This system is called isomonodromic system. Furthermore, the related Gal-group is a proper Zariski dense subgroup of SL₂(c^z0), where Gal-group is the parameterized Picard–Vessiot group (PPV-group) and c^z0 is the differential constant of $\frac{\partial }{\partial z_0}$ (but function of parameters-set) or called $\frac{\partial }{\partial z_0}$-invariant. Y is called the fundamental solution matrix.

The PPV-group is a ∂-automorphism group such that Gal ∈ PPV-group if Gal:Y → Y and Gal ∂ = ∂ Gal $\forall$∂. Therefore, Gal(Y) is another fundamental solution matrix to the system.

Let us define the Lie algebra corresponding to one-parameter subgroup of Gal by G through exponential map:

$$\mathrm{Gal}=\mathrm{Exp}[{\rm K}G],\mathrm{here}{\rm K}\mathit{\text{ is a real parameter}}$$

By expanding the exponential map with parameter Κ → 0 and commuting property of Gal ∂ (i.e., Gal ∂ = ∂ Gal), the commuting property of Lie algebra G and ∂ exists:

$$G\partial =\partial G,\forall \partial$$  (15)

It is obvious that G ∈ sl₂(c^z0).

There is a property of integratability condition that change of variables from parameter-set z_i to ${z_{{\rm M}}}^{\prime }$ preserves this property (by change of variables) if there exists smooth maps $\frac{\partial z_i}{\partial {z_{{\rm M}}}^{\prime }}$ for $z_i\left({z_{{\rm M}}}^{\prime }\right)$. The transformed new integratable system (together with $\frac{\partial }{\partial z_0}Y=B_{z_0}Y$) is

$$\frac{\partial }{\partial {z_{{\rm M}}}^{\prime }}Y={B^{\prime }}_{z_{{\rm M}}}Y,\mathrm{where}{B^{\prime }}_{z_{{\rm M}}}={\displaystyle {\sum }_i\frac{\partial z_i}{\partial {z^{\prime }}_{{\rm M}}}}{B}_{z_i}\mathrm{for}{\rm M}=1,\mathrm{...},n;i=1,\mathrm{...},m$$  (16)

note that n is not necessarily equal to m.

Cross-ratio arbitrariness and the constraint to the parameter space of projective space basis

A cross-ratio (https://en.wikipedia.org/wiki/Cross-ratio) can be defined by a function of four points on Riemann sphere:

$$\mathrm{Cross-}\mathrm{Ratio}\equiv \frac{\left(z_0-z_2\right)\left(z_1-z_3\right)}{\left(z_1-z_2\right)\left(z_0-z_3\right)}$$

If we allow any arbitrary cross-ratio fixed value to be assigned, those four points are then constrained (only three points are free to move). If we further assume any arbitrary reference fixed point to be assigned, say for z₀, then those other points are also constrained by four real parameters (x₀, x₁, x₂, x₃), that is,

$$\mathrm{Cross}-\mathrm{Ratio}=\frac{\left(z_0-z_2\left(x_0,x_1,x_2,x_3\right)\right)\left(z_1\left(x_0,x_1,x_2,x_3\right)-z_3\left(x_0,x_1,x_2,x_3\right)\right)}{\left(z_1\left(x_0,x_1,x_2,x_3\right)-z_2\left(x_0,x_1,x_2,x_3\right)\right)\left(z_0-z_3\left(x_0,x_1,x_2,x_3\right)\right)}$$

So, the arbitrariness of description of a cross-ratio allows four real parameters. Physically, we mean the principle of cross-ratio arbitrariness is natural because no particular point over Riemann sphere is more special and no reference point is more special than others.

By Yoshida (Chapters I and IV), the realization of cross-ratio has direct relationship with hypergeometric function (three regular singular points):

$$z(1-z)\frac{d^{2}y}{{\mathrm{dz}}^2}+(c-(a+b+1)y)\frac{\mathrm{dy}}{\mathrm{dz}}-aby=0,\mathit{\text{where a,b,c are complex numbers}}$$

Since hypergeometric function is a solution of related hypergeometric differential equation, so if we deform the hypergeometric differential equation, we also deform the related hypergeometric function and its underlying cross-ratio realization.

Referring to Hypergeometric function (https://en.wikipedia.org/wiki/Hypergeometric_function), any second-order differential equation with three regular singular points can be converted to hypergeometric differential equation by change of variables, so we can describe the hypergeometric differential equation by second-order differential equation. Leveraging Cassidy’s work in previous section, we can deform the hypergeometric differential equation by the deformation of second-order differential equation. Physically, we mean we can deform the realization of cross-ratio by deformation of related isomonodromy integratable system by a deformation space of four real parameters:

$${\partial }_{{\rm M}}Y=\frac{\partial }{\partial x_{{\rm M}}}Y=\left({\partial }_{{\rm M}}\mathrm{Id}\right)Y=B_{{\rm M}}Y\mathrm{or }{\partial }_{{\rm M}}{\Delta }^E{F}_{}{Y}^{\mathrm{FG}}={\mathbb{D}}_{{\rm M}}{E^{}}_F{Y}^{\mathrm{FG}}=B_{{\rm M}}{E^{}}_F{Y}^{\mathrm{FG}}$$  (17)

the later is in spinor form, where $B_{{\rm M}}\left(z_0,x_{{\rm N}}\right)={\displaystyle {\sum }_i\frac{\partial z_i\left(x_{{\rm M}}\right)}{\partial x_{{\rm M}}}}{B}_{z_i}\left(z_0,z_j\right)$ for Μ, Ν = 0,…,3;i,j = 1,…,3; x_Ν ∈ Reals.

Id is 2 × 2 identity matrix; ${\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F\equiv {\partial }_{{\rm M}}{\Delta }^E{F}_{}$; note that B_Μ(z₀, x_Ν) and B_zi(z₀,z_j) are matrices gl₂, and ${\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F,B_{{\rm M}}{E^{}}_F$ are hybrid spinors like Σ^Μ_AB′ in spinor GR (Robert, Chapter 13).

Operators of integratability condition and quantized Bianchi equation for curvature spinors

Based on the isomonodromy integratable system formulated in previous section, we look for a set of the Lie algebras associated with Gal that can produce the integratability. In order to do so, some fundamental operators, such as derivation operator and integratability operator, are introduced. They are related to mathematical objects in spinor GR so that Bianchi identity is recovered by an operator equation. Finally, the equivalency of spinor GR and this theory is summarized.

Let us define spinor (derivation) operator:

$${\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }}{E^{}}_F\equiv {\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F$$  (18)

and ${\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F$ is the differential hybrid spinor operator from previous section (so it is associated with an integratable system), not covariant derivative.

We demand ${\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}$ (denoted as ${\Sigma }^{{\rm M}}$ in short) is the differential constant of ${\widehat{\mathbb{D}}}_{{\rm N}}$ in the sense that ${\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F-invariant|{\widehat{\mathbb{D}}}_{{\rm M}}{{\text{E}}^{}}_{\text{F}}\left({\Sigma }^{{\rm N}}{{\text{AB}}^{\prime }}_{}\text{X}\right)\text{=}{\Sigma }^{{\rm N}}{{\text{AB}}^{\prime }}_{}\left({\widehat{\mathbb{D}}}_{{\rm M}}{{\text{E}}^{}}_{\text{F}}\text{X}\right)$ for any spinor X, so it is consistent with spinor GR requirement (covariant derivative of ${\Sigma }^{{\rm N}}{{\text{AB}}^{\prime }}_{}$ is zero) except $\nabla \to \widehat{\mathbb{D}}$. ${\Sigma }^{{\rm N}}{{\text{AB}}^{\prime }}_{}$ defined here satisfies the same relationship in Equation (4).

Note that commutative ∂_Μ implies commutative ${\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }}{E^{}}_F:{\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }}{E^{}}_F{\widehat{\mathbb{D}}}_{{\text{CD}}^{\prime }}{F^{}}_G={\widehat{\mathbb{D}}}_{{\text{CD}}^{\prime }}{E^{}}_F{\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }}{F^{}}_G$.

By Equations (16) and (18) as well as ${\Sigma }^{{\rm M}}{\text{AB}}_{}$ being ${\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F-\text{invariant}$, there is a commuting set of spinor operator for an integratable system:

$${\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }}{E^{}}_F{Y}^{\text{FG}}=\mathbb{H}{ }_{{\text{AB}}^{\prime }}{E^{}}_F{Y}^{\text{FG}}\leftrightarrow {\widehat{\mathbb{D}}}^{\star }{{\text{BA}\prime }_{}}^{E^{\prime }}{F^{\prime }}_{}{Y}^{\star }{F^{\prime }{G}^{\prime }}^{}={\mathbb{H}}^{\star }{{{\text{BA}}^{\prime }}_{}}^{E^{\prime }}{F^{\prime }}_{}{Y}^{\star }{F^{\prime }{G}^{\prime }}^{},$$  (19)

where

$$\mathbb{H}{ }_{{\text{AB}}^{\prime }}{E^{}}_F\equiv {\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{B}_{{\rm M}}{E^{}}_F$$

(↔ indicates complex conjugate, so prime and unprime indices are interchanged), and note that the ordering between prime and unprime indices are irrelevant (Robert, Chapter 13).

The integratability condition is used to facilitate a definition on a set of spinor operator.

${\widehat{J}}_{\text{AB}\prime \text{CD}\prime }{E}^{}{}_F$ is called integratability operator (or, in short, int operator), acting on Y |

$${\widehat{J}}_{{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}_{}}{}^E{F}_{}{Y}^{\text{FG}}=\left(\mathbb{H}{ }_{{\text{AB}}^{\prime }}{E^{}}_G\mathbb{H}{ }_{{\text{CD}}^{\prime }}{G^{}}_F-{\mathbb{H}}_{{\text{CD}}^{\prime }}{E^{}}_G\mathbb{H}{ }_{{\text{AB}}^{\prime }}{G^{}}_F\right)Y^{\text{FG}}=\mathbb{H}{ }_{[{\text{AB}}^{\prime }}{E^{}}_G\mathbb{H}{ }_{{\text{CD}}^{\prime }]}{G^{}}_F{Y}^{\text{FG}},$$

and

$${\widehat{J}}^{\star }{{{\text{BA}}^{\prime }{\text{DC}}^{\prime }}_{}}^{E^{\prime }}{}_{F^{\prime }}{Y}^{\star }{F^{\prime }{G}^{\prime }}^{}={\mathbb{H}}^{\star }{}_{{\text{[BA}}^{\prime }}{{E^{\prime }}^{}}_{{G^{\prime }}_{}{\text{H}}^{\star }{\text{DC}}^{\prime }]}{G^{\prime }}^{}{}_{F^{\prime }}{Y}^{\star {\text{F}}^{\prime }{G}^{\prime }}$$

[AB’ and CD’] are anti-commutating indices notation as usual in spinor GR. So, by integratability condition (14), this implies

$$\begin{array}{c}{\widehat{J}}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{}^E{F}_{}{Y}^{\text{FG}}=\left({\widehat{\mathbb{D}}}_{[{\text{AB}}^{\prime }}{E^{}}_G\left({\mathbb{H}}_{{\text{CD}}^{\prime }]}{G^{}}_F\right)\right)Y^{\text{FG}},\\ {\widehat{J}}^{\star }{{\text{BA}}^{\prime }{\text{DC}}^{\prime }}_{}{}^{E^{\prime }}{F^{\prime }}_{}{Y}^{\star F^{\prime }{G}^{\prime }}=\left({\widehat{\mathbb{D}}}^{\star }{{[{\text{BA}}^{\prime }}_{}}^{E^{\prime }}{G^{\prime }}_{}\left({\mathbb{H}}^{\star }{{{\text{DC}}^{\prime }]}_{}}^{G^{\prime }}{F^{\prime }}_{}\right)\right)Y^{\star {\text{F}}^{\prime }{G}^{\prime }}{}_{}\end{array}$$  (20)

There are a total of four sets of spinor operators from the contraction of $\widehat{J}$ and ${\widehat{J}}^{\star }$, which have a close relationship with int operators and curvature spinors:

$$\begin{array}{c}{\widehat{\mathcal{J}}}_{\text{AC}}{E^{}}_F\equiv {\widehat{J}}_{{{\text{AB}}^{\prime }C}_{}}{B^{\prime }E}^{}{}_F={\epsilon }^{B^{\prime }{D}^{\prime }}{\widehat{J}}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F,\\ {\widehat{\mathcal{J}}}^{\star }{{A^{\prime }{C}^{\prime }}_{}}^{E^{\prime }}{F^{\prime }}_{}={\widehat{J}}^{\star }{}_{{\text{BA}}^{\prime }}{B^{}}_{C^{\prime }}{{E^{\prime }}^{}}_{F^{\prime }}={\epsilon }^{\text{BD}}\widehat{J}{}^{\star }{{{\text{BA}}^{\prime }{\text{DC}}^{\prime }}_{}}^{E^{\prime }}{F^{\prime }}_{},\\ {\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{E^{}}_F\equiv {\widehat{J}}_{{\text{AB}}^{\prime }}{}^A{{D^{\prime }}_{}}^E{F}_{}={\epsilon }^{\text{AC}}{\widehat{J}}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F,\\ {\widehat{\mathcal{M}}}^{\star }{{\text{BD}}_{}}^{E^{\prime }}{F^{\prime }}_{}={\widehat{J}}^{\star }{}_{{\text{BA}}^{\prime }D}{A^{\prime }{E}^{\prime }}^{}{}_{F^{\prime }}={\epsilon }^{A^{\prime }{C}^{\prime }}{\widehat{J}}^{\star }{}_{{\text{BA}}^{\prime }{\text{DC}}^{\prime }}{{E^{\prime }}^{}}_{F^{\prime }},\end{array}$$  (20.b)

Specifically defining a bar-spinor: ${\overline{\mathcal{J}}}_{\text{BD}}{{E^{\prime }}^{}}_{F^{\prime }}\equiv \widehat{J}{}^{\star }{{{\text{BA}}^{\prime }D}_{}}^{A^{\prime }{E}^{\prime }}{F^{\prime }}_{}={\widehat{\mathcal{M}}}^{\star }{{\text{BD}}_{}}^{E^{\prime }}{F^{\prime }}_{}$, one can see the ${\overline{\mathcal{J}}}_{\text{BD}}{{E^{\prime }}^{}}_{F^{\prime }}$ has symmetric contraction over prime indices of ${\widehat{\mathcal{J}}}_{\text{AC}}{E^{}}_F$ except over complex space: that is, $Y^{{\star }^{F^{\prime }{H}^{\prime }}}$.

Note that ${\widehat{\mathcal{J}}}_{(\text{AC})}{E^{}}_F$ and ${\widehat{\mathcal{M}}}_{(B^{\prime }{D}^{\prime })}{E^{}}_F$ have symmetric property over their first two indices because

$${\widehat{\mathcal{J}}}_{\text{AC}}{E^{}}_F={\epsilon }^{B^{\prime }{D}^{\prime }}\widehat{J}{}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E^{}}_F={\epsilon }^{D^{\prime }{B}^{\prime }}\widehat{J}{}_{{\text{CD}}^{\prime }{\text{AB}}^{\prime }}{E^{}}_F={\widehat{\mathcal{J}}}_{\text{CA}}{E^{}}_F$$

,

where we use the property of anti-symmetry of ${\epsilon }^{D\prime B\prime }$ and ${\widehat{J}}_{{\text{ABCD}}^{\prime }}{E}^{}{}_F=-{\widehat{J}}_{{\text{CD}}^{\prime }{\text{AB}}^{\prime }}{E}^{}{}_F$.

Let us look for an integratable system with the Lie algebra of PPV-group generated by spinor operators ${\widehat{\mathcal{J}}}_{\text{AC}}{E^{}}_F,{\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{E^{}}_F$ on Y^FG:

$${\widehat{\mathcal{J}}}_{\text{AC}}{E^{}}_F{Y}^{\text{FG}}={\mathcal{J}}_{\text{AC}}{E^{}}_F{Y}^{\text{FG}},$$

where ${\mathcal{J}}_{\text{AC}}{E^{}}_F$ is a spinor without derivation;

$${\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{E^{}}_F{Y}^{\text{FG}}={\mathcal{M}}_{B^{\prime }{D}^{\prime }}{E^{}}_F{Y}^{\text{FG}}$$

where ${\mathcal{M}}_{B^{\prime }{D}^{\prime }}{E^{}}_F$ is a spinor without derivation;

such that ${\mathcal{J}}_{\text{AC}}{E^{}}_F$ and ${\mathcal{M}}_{B^{\prime }{D}^{\prime }}{E^{}}_F$ are matrices (if considering E and F as the matrix indices) of Gal Lie algebra G. By requirements in Equation (15), but with association to ${\widehat{\mathbb{D}}}_{C,D^{\prime }}{F^{}}_G$ instead of ∂, commuting property of $\text{G}(\widehat{\mathcal{J}},\widehat{\mathcal{M}})$ against $\widehat{\mathbb{D}}$ reads:

$$\left({\widehat{\mathcal{J}}}_{\text{AB}}{E^{}}_F{\widehat{\mathbb{D}}}_{{\text{CD}}^{\prime }}{F^{}}_G\right)Y^{\text{GH}}=\left({\widehat{\mathbb{D}}}_{{\text{CD}}^{\prime }}{E^{}}_F{\widehat{\mathcal{J}}}_{\text{AB}}{F^{}}_G\right)Y^{\text{GH}},\left({\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{E^{}}_F{\widehat{\mathbb{D}}}_{{\text{AC}}^{\prime }}{F^{}}_G\right)Y^{\text{GH}}=\left({\widehat{\mathbb{D}}}_{{\text{AC}}^{\prime }}{E^{}}_F{\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{F^{}}_G\right)Y^{\text{GH}}$$

Let’s define an fundamental equation as:

$${\epsilon }^{\text{BC}}\left({\widehat{\mathcal{J}}}_{\text{AB}}{E^{}}_F{\widehat{\mathbb{D}}}_{{\text{CD}}^{\prime }}{F^{}}_G\right)Y^{\text{GH}}={\epsilon }^{B^{\prime }{C}^{\prime }}\left({\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{E^{}}_F{\widehat{\mathbb{D}}}_{{\text{AC}}^{\prime }}{F^{}}_G\right)Y^{\text{GH}}$$  (21)

or equivalently:

$$\left({\widehat{\mathcal{J}}}_{\text{AB}}{E^{}}_F{\widehat{\mathbb{D}}}^B{{D^{\prime }}_{}}^F{G}_{}\right)Y^{\text{GH}}={\left({\overline{\mathcal{J}}}_{\text{DB}}{{E^{\prime }}^{}}_{F^{\prime }}{\widehat{\mathbb{D}}}^{\star }{B^{}}_{A^{\prime }}{{F^{\prime }}^{}}_{G^{\prime }}\right)}^{\star }{Y}^{\text{GH}}$$

after applying the commuting property of $(\widehat{\mathcal{J}},\widehat{\mathcal{M}})$ against $\widehat{\mathbb{D}}$ above and using symmetry of ${\widehat{\mathcal{M}}}_{(B^{\prime }{D}^{\prime })}{F^{}}_G$:

$$\left({\widehat{\mathbb{D}}}^B{{D^{\prime }}_{}}^E{F}_{}{\widehat{\mathcal{J}}}_{\text{AB}}{F^{}}_G\right)Y^{\text{GH}}=\left({\widehat{\mathbb{D}}}_A{{C^{\prime }E}^{}}_F{\widehat{\mathcal{M}}}_{D^{\prime }{C}^{\prime }}{F^{}}_G\right)Y^{\text{GH}}$$

This is the quantized version (i.e., operator-wise equivalent) of Bianchi identity which describes the differentiation on curvature spinors Φ, Χ (compared to Equation (9) $${\nabla }^B{D^{\prime }}_{}{{\rm X}}_{\text{AB}}{G^{}}_E={\nabla }_A{C^{\prime }}^{}{\Phi }^{\star }{{D^{\prime }{C}^{\prime }}_{}}^G{E}_{}$$)), while the spinor covariant derivative is replaced by spinor derivation operator $\widehat{\mathbb{D}}$, and Χ, ${\Phi }^{\star }$ are replaced by operators $\widehat{\mathcal{J}}$ and $\widehat{\mathcal{M}}$, respectively. The quantization here refers to changing the classical equation of physical observable (spinor derivative of curvature in spinor GR) to operator equation (e.g., operator ${\widehat{\mathbb{D}}}^B{{D^{\prime }}_{}}^E{F}_{}{\widehat{\mathcal{J}}}_{\text{AB}}{F^{}}_G$ acting on Y^GH), like Schrodinger’s equation that KE + PE = E →$\left(\frac{{\widehat{p}}^2}{2m}+V\right)\Psi =E\Psi$.

As a short summary, we look for an integratable system, where G is generated by the int operator $\widehat{J}$, and G is a solution to the fundamental (equation 21). This system leads to quantized Bianchi identity.

Let us define a set of spinor operators ${\widehat{\Delta }}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F$ such that

$$\left({\widehat{\Delta }}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F-{\widehat{\Delta }}_{{\text{CD}}^{\prime }{\text{AB}}^{\prime }}{E}^{}{}_F\right)Y^{\text{FH}}=\left({\widehat{\mathbb{D}}}_{[{\text{AB}}^{\prime }}{E^{}}_G\left({\text{H}}_{{\text{CD}}^{\prime }]}{G^{}}_F\right)\right)Y^{\text{FH}}$$  (22)

By contracting over B’ and D’, we get

$${\epsilon }^{B^{\prime }{D}^{\prime }}\left({\widehat{\Delta }}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F-{\widehat{\Delta }}_{{\text{CD}}^{\prime }{\text{AB}}^{\prime }}{E}^{}{}_F\right)Y^{\text{FH}}=\left({\widehat{\Delta }}_{{\text{AB}}^{\prime }C}{B^{\prime }E}^{}{}_F+{\widehat{\Delta }}_{{\text{CD}}^{\prime }A}{D^{\prime }E}^{}{}_F\right)Y^{\text{FH}}={\epsilon }^{B^{\prime }{D}^{\prime }}\left({\widehat{\mathbb{D}}}_{[{\text{AB}}^{\prime }}{E^{}}_G\left(\text{H}{*}_{{\text{CD}}^{\prime }]}{G^{}}_F\right)\right)Y^{\text{FH}}.$$

Using the integratability condition property (Equation 20) for $\widehat{J}$ operator on the right-hand side, we get

$$\left({\widehat{\Delta }}_{{\text{AB}}^{\prime }C}{B^{\prime }E}^{}{}_F+{\widehat{\Delta }}_{{\text{CD}}^{\prime }A}{D^{\prime }E}^{}{}_F\right)Y^{\text{FH}}={\epsilon }^{B^{\prime }{D}^{\prime }}\left({\widehat{J}}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F\right)Y^{\text{FH}}$$

The right-hand side is just ${\widehat{\mathcal{J}}}_{\text{AC}}{E^{}}_F$ (Equation 20b), and the equation becomes

$$\left({\widehat{\Delta }}_{{\text{AB}}^{\prime }C}{B^{\prime }E}^{}{}_F+{\widehat{\Delta }}_{{\text{CD}}^{\prime }A}{D^{\prime }E}^{}{}_F\right)Y^{\text{FH}}={\widehat{\mathcal{J}}}_{\text{AC}}{E^{}}_F{Y}^{\text{FH}}$$  (23)

Let us do the same for (Equation 22), but contracting over A and C:

$${\epsilon }^{\text{AC}}\left({\widehat{\Delta }}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F-{\widehat{\Delta }}_{{\text{CD}}^{\prime }{\text{AB}}^{\prime }}{E}^{}{}_F\right)Y^{\text{FH}}={\epsilon }^{\text{AC}}\left({\widehat{\mathbb{D}}}_{[{\text{AB}}^{\prime }}{E^{}}_G\left({\text{H}}_{{\text{CD}}^{\prime }]}{G^{}}_F\right)\right)Y^{\text{FH}}$$

After the same operations, we get

$$\left(\widehat{\Delta }{}_{{\text{AB}}^{\prime }}{A^{}}_{D^{\prime }}{E^{}}_F+\widehat{\Delta }{}_{{\text{AD}}^{\prime }}{A^{}}_{B^{\prime }}{E^{}}_F\right)Y^{\text{FH}}={\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{E^{}}_F{Y}^{\text{FH}}$$  (24)

By decomposing Y^FH as matrix into Ξ^F and Η^H (both are column vectors of dimension 2) into Equations (23) and (24): Y → Ξ ⊕ Η, we get: $\left(\widehat{\Delta }{}_{{\text{AB}}^{\prime }C}{{B^{\prime }E}^{}}_F+\widehat{\Delta }{}_{{\text{CD}}^{\prime }A}{{D^{\prime }E}^{}}_F\right){\Xi }^F={\mathcal{J}}_{\text{AC}}{E^{}}_F{\Xi }^F,\left(\widehat{\Delta }{}_{{\text{AB}}^{\prime }}{A^{}}_{D^{\prime }}{E^{}}_F+\widehat{\Delta }{}_{{\text{AD}}^{\prime }}{A^{}}_{B^{\prime }}{E^{}}_F\right){\Xi }^F={\mathcal{M}}_{B^{\prime }{D}^{\prime }}{E^{}}_F{\Xi }^F$ (and same for Η). They are equivalent to the equations of curvature spinors in Equation (7) if

$${\widehat{\Delta }}_{{\text{AB}}^{\prime }C}{B^{\prime }E}^{}{}_F,\widehat{\Delta }{}_{{\text{AB}}^{\prime }}{A^{}}_{D^{\prime }}{E^{}}_F\to {\nabla }_{A,B^{\prime }}{\nabla }_C{B^{\prime }}^{},{\nabla }_{A,B^{\prime }}{\nabla }^A{D^{\prime }}_{}$$  (25)

and

$${\mathcal{J}}_{\text{AC}}{E^{}}_F={{\rm X}}_{\text{AC}}{E^{}}_F,{\mathcal{M}}_{B^{\prime }{D}^{\prime }}{E^{}}_F={\Phi }^{\star }{{B^{\prime }{D}^{\prime }}_{}}^E{F}_{}(\text{not the operators})$$  (26)

In conclusion, spinor GR requires a well-defined spinor covariant derivative ∇∇. ∇∇ gives raise to the relationship as Equation (7), so that curvature spinors Χ and Φ are well established, which obey Bianchi identity (Equation 9). These conditions are said to be equivalent to “generate” spacetime manifold of {+,–,–,–} (Penrose, 1960) (except certain torsionness symmetric properties). In this article, the author claims that the spinor covariant derivative ∇∇ is corresponding to the operator $\widehat{\Delta }$; curvature spinors Φ and Χ are generated by $\widehat{\mathcal{J}}$ and $\widehat{\mathcal{M}}$ operators on Y; Bianchi identity is satisfied by the operators $\widehat{\mathcal{J}}$, $\widehat{\mathcal{M}}$ in such quantized form.

To illustrate the correspondences between spinor GR and this part of the theory (i.e., isomonodromic system):

$$\text{○Spinor covariant dervatives: }\nabla \nabla \leftrightarrow {\widehat{\Delta }}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F$$  (27)

$$\text{○Curvatures}:{\Phi }^{\star },{\rm X}\leftrightarrow \widehat{\mathcal{J}},\widehat{\mathcal{M}}$$  (28)

$$\begin{array}{l}\text{○Bianchi identity }(\text{dynamics of curvatures}):{\nabla }^B{D^{\prime }}_{}{{\rm X}}_{\text{AB}}{G^{}}_E={\nabla }_A{C^{\prime }}^{}{\Phi }^{\star }{{D^{\prime }{C}^{\prime }}_{}}^G{E}_{}\leftrightarrow \left({\widehat{\mathbb{D}}}^B{{D^{\prime }}_{}}^E{F}_{}{\widehat{\mathcal{J}}}_{\text{AB}}{F^{}}_G\right)Y^{\text{GH}}\\ =\left({\widehat{\mathbb{D}}}_A{{C^{\prime }E}^{}}_F{\widehat{\mathcal{M}}}_{D^{\prime }{C}^{\prime }}{F^{}}_G\right)Y^{\text{GH}}\end{array}$$  (29)

Note that $\widehat{\mathcal{J}}$ and $\widehat{\mathcal{M}}$ are just linear combinations of int operators, which must exist in an integratable system, while Y is a special solution to quantized Bianchi equation, which is a result of commuting relationship of $\widehat{\mathcal{J}}$, $\widehat{\mathcal{M}}$ and $\widehat{\mathbb{D}}$. The consequence is that, we can only claim that the equivalence of spinor GR to this theory is up to particular set of solution Y and its associated values of $\widehat{\mathcal{J}}$ and $\widehat{\mathcal{M}}$, that is, spinor GR is equivalent to this part of the theory in such quantized sense.

The symmetry of curvature spinor

Certain symmetries of curvature spinors are described by Penrose in previous sections, and those symmetries are consequence of classical general relativity. We show that certain symmetries are naturally arisen by the operators introduced in previous section, while some others are torsion related and needed to be met additionally. However, those additional conditions are not considered as mandatary because the key objective is to describe that spacetime’s curvature can be manifested by the deformation of isomonodromy integrable system, not to explain the naturalness of torsion.

From Equation (26) and the traceless property of matrices generated from int operators (Equation 20), it is concluded that ${{\rm X}}_{\text{AB}}{G^{}}_E$ and ${\Phi }^{\star }{{D^{\prime }{C}^{\prime }}_{}}^G{E}_{}$ are traceless matrices (if considering G and E as the matrix indices), and

$$\text{int operators}\widehat{\mathcal{J}}\to \text{symmetry of \hspace{0.17em}}{\rm X},\varphi :{{\rm X}}_{\text{AB}(\text{FE})}\text{and }{\Phi }^{\star }{D^{\prime }{C}^{\prime }(\text{FE})}_{}$$  (30)

Because of the symmetry properties of ${\widehat{\mathcal{J}}}_{(\text{AC})}{E^{}}_F$ and ${\widehat{\mathcal{M}}}_{(B^{\prime }{D}^{\prime })}{E^{}}_F$, it implies trivially

$${{\rm X}}_{(\text{AB})\text{FE}}\text{\hspace{0.17em} and \hspace{0.17em}}{\Phi }^{\star }{(D^{\prime }{C}^{\prime })\text{FE}}_{}$$  (31)

Besides these two symmetries, spinor GR also requires symmetry:

$${{\rm X}}_{\text{ABFE}}={{\rm X}}_{\text{FEAB}}\text{\hspace{0.17em} and \hspace{0.17em}}{\Phi }^{\star }{D^{\prime }{C}^{\prime }\text{FE}}_{}={\Phi }_{{\text{FED}}^{\prime }{C}^{\prime }}$$

,

and the Λ reality (Equation 10):

$$\Lambda =\frac{{{\rm X}}_{\text{AB}}{\text{AB}}^{}}{2}=\frac{1}{2}{{\rm X}}_{\text{ABCD }}{\epsilon }^{\text{AC}}{\epsilon }^{\text{BD}}=\frac{1}{2}{\epsilon }^{\text{BD}}{{\rm X}}_{\text{AB}}{A^{}}_D=\frac{1}{2}{\epsilon }^{\text{BD}}{\mathcal{J}}_{\text{AB}}{A^{}}_D$$  (32)

By Gomez-Lobo and Martin-Garcia (2012), GR symmetry (Equation 31) and Λ reality are consequences of the torsion-free property of original GR theory. However, torsion free is known that it is not a mandatory requirement; for example, Einstein–Carton theory removes such constraints.

In conclusion, to recover GR symmetry Χ_ABFE=Χ_FEAB and ${\Phi }^{\star }{D^{\prime }{C}^{\prime }\text{FE}}_{}={\Phi }_{{\text{FED}}^{\prime }{C}^{\prime }}$ and Λ reality, there are conditions for spinors generated by operators $\widehat{\mathcal{J}}$ and $\widehat{\mathcal{M}}$ to meet. However, the discussion of sufficient and necessary conditions for $\widehat{\mathcal{J}}$ and $\widehat{\mathcal{M}}$ to meet these symmetries is not covered yet in this paper.

Translation to ordinary Ricci, Einstein tensors, and Bianchi equation

The purpose of this section is to show the explicit links between related mathematical objects spinor GR (spinors of Ricci and Einstein tensors) and the isomonodromy integratable counterparts. Because those are key components of classical general relativity, showing explicit relationships helps visualize the equivalency and claim the origin of related tensors.

Recall the spinor equivalences of Ricci tensor (Equation 11), Ricci scalar (Equation 12), and Einstein tensor (Equation 13): $R_{{\text{AC}}^{\prime }{\text{BD}}^{\prime }}=\Lambda {\epsilon }_{\text{AB}}{\epsilon }_{C^{\prime }{D}^{\prime }}-{\Phi }_{{\text{ABC}}^{\prime }{D}^{\prime }}$, R = 4 Λ, and $G_{{\text{AC}}^{\prime }{\text{BD}}^{\prime }}=-\Lambda {\epsilon }_{\text{AB}}{\epsilon }_{C^{\prime }{D}^{\prime }}-{\Phi }_{{\text{ABC}}^{\prime }{D}^{\prime }}$. The following are $\mathcal{J}$ and $\mathcal{M}$ equivalences:

$$R_{{\text{AC}}^{\prime }{\text{BD}}^{\prime }}=\left(\frac{1}{2}{\epsilon }^{\text{FH}}{\mathcal{J}}_{\text{GF}}{{\text{G}}^{}}_{\text{H}}\right){\epsilon }_{\text{AB }}{\epsilon }_{{\text{C}}^{\prime }{\text{D}}^{\prime }}-{\left({\mathcal{M}}_{{\text{A}}^{\prime }{\text{B}}^{\prime }}{{\text{E}}^{}}_{\text{D }}{\epsilon }_{\text{EC}}\right)}^{\star }$$  (33)

$$R=2{\epsilon }^{\text{BD}}{\mathcal{J}}_{\text{AB}}{A^{}}_D$$  (34)

$$G_{{\text{AC}}^{\prime }{\text{BD}}^{\prime }}=-\left(\frac{1}{2}{\epsilon }^{\text{FH}}{\mathcal{J}}_{\text{GF}}{{\text{G}}^{}}_{\text{H}}\right){\epsilon }_{\text{AB }}{\epsilon }_{C^{\prime }{D}^{\prime }}-{\left({\mathcal{M}}_{{\text{A}}^{\prime }{\text{B}}^{\prime }}{{\text{E}}^{}}_{\text{D }}{\epsilon }_{\text{EC}}\right)}^{\star }$$  (35)

Follow the ordinary way of spinor GR to translate from spinor to tensor (Equation 5), Einstein tensor is equivalent to

$$G_{{\rm M}{\rm N}}={\Sigma }_{{\rm M}}{{\text{AC}}^{\prime }}^{}{\Sigma }_{{\rm N}}{{\text{BD}}^{\prime }}^{}\left(-\left(\frac{1}{2}{\epsilon }^{\text{FH}}{\mathcal{J}}_{\text{GF}}{{\text{G}}^{}}_{\text{H}}\right){\epsilon }_{\text{AB }}{\epsilon }_{C^{\prime }{D}^{\prime }}-{\left({\mathcal{M}}_{A^{\prime }{B}^{\prime }}{E^{}}_D{\epsilon }_{\text{EC}}\right)}^{\star }\right).$$  (36)

If we use the property of ${\mathcal{M}}_{A^{\prime }{B}^{\prime }}{E^{}}_D{\epsilon }_{\text{EC}}={\mathcal{M}}_{B^{\prime }{A}^{\prime }}{E^{}}_D{\epsilon }_{\text{EC}}$, and ${\mathcal{M}}_{A^{\prime }{B}^{\prime }}{E^{}}_D{\epsilon }_{\text{EC}}={\mathcal{M}}_{A^{\prime }{B}^{\prime }}{E^{}}_C{\epsilon }_{\text{ED}}$, for which the symmetries are proved in the previous section, then

$$\begin{array}{c}{G}_{{\rm M}{\rm N}}={\Sigma }_{{\rm M}}{{\text{AC}}^{\prime }}^{}{\Sigma }_{{\rm N}}{{\text{BD}}^{\prime }}^{}\left(-\left(\frac{1}{2}{\epsilon }^{\text{FH}}{\mathcal{J}}_{\text{GF}}{{\text{G}}^{}}_{\text{H}}\right){\epsilon }_{\text{AB }}{\epsilon }_{{\text{C}}^{\prime }{\text{D}}^{\prime }}-{\left({\mathcal{M}}_{{\text{B}}^{\prime }{\text{A}}^{\prime }}{{\text{E}}^{}}_{\text{C }}{\epsilon }_{\text{ED}}\right)}^{\star }\right)=\\ {\Sigma }_{{\rm N}}{{\text{BD}}^{\prime }}^{}{\Sigma }_{{\rm M}}{{\text{AC}}^{\prime }}^{}\left(-\left(\frac{1}{2}{\epsilon }^{\text{FH}}{\mathcal{J}}_{\text{GF}}{{\text{G}}^{}}_{\text{H}}\right){\epsilon }_{\text{AB }}{\epsilon }_{{\text{C}}^{\prime }{\text{D}}^{\prime }}-{\left({\mathcal{M}}_{{\text{B}}^{\prime }{\text{A}}^{\prime }}{{\text{E}}^{}}_{\text{C }}{\epsilon }_{\text{ED}}\right)}^{\star }\right)=G_{{\rm N}{\rm M}}\end{array}$$

(where, in the last right-hand side, we rename the dummy indices). So, it is a clear symmetric for G_(ΜΝ) as expected.

Bianchi identity (Equation 9) in spinor GR describes the dynamics of curvature. Unlike Einstein field equation, which the dynamic of curvature is described by solving (up to) second-order differential equation of metric tensor, spinor GR handles the same problem by first-order differential equation – Bianchi identity (Penrose, 1960). However, spinor GR is coordinate-free yet a classical theory; there is no metric differential analog. Nevertheless, quantized Bianchi equation do provide quantized analog to spinor GR as

$${\nabla }^B{D^{\prime }}_{}{{\rm X}}_{\text{AB}}{G^{}}_E={\nabla }_A{C^{\prime }}^{}{\Phi }^{\star }{{D^{\prime }{C}^{\prime }}_{}}^G{E}_{}\leftrightarrow \left({\widehat{\mathbb{D}}}^B{{D^{\prime }}_{}}^E{F}_{}{\widehat{\mathcal{J}}}_{\text{AB}}{F^{}}_G\right)Y^{\text{GH}}=\left({\widehat{\mathbb{D}}}_A{{C^{\prime }E}^{}}_F{\widehat{\mathcal{M}}}_{D^{\prime }{C}^{\prime }}{F^{}}_G\right)Y^{\text{GH}}.$$

Once we solve the operators $\widehat{\Delta }$, $\widehat{\mathbb{D}}$, $\widehat{\mathcal{J}}$, $\widehat{\mathcal{M}}$, and fundamental solution matrix Y, the dynamics of curvature spinors Χ and Φ can be found by equations Equation (26). The ordinary curvature components such as Ricci tensors and Ricci scalar as well as Einstein tensors can be found.

From last three chapters of this section, the overall approach of spacetime dynamics is done by first identifying related classical curvature spinors and the derivatives equations (i.e., Bianchi identity), then manifesting the spacetime by a deeper structure from isomonodromy system; finally, illustrating related tensors such as Ricci and Einstein tensors equivalence by associated spinors by associated objects from isomonodromy system. Physically, the approach claims a more fundamental reality from isomonodromy system over classical spacetime just like Penrose mentioned that spinor GR may be more deep-rooted than tensors (Penrose, 1960).

Einstein condition and the connection to quantum operators

It is well known that Einstein’s theory (theory of general relativity) does not assign a definite stress–energy tensor to the gravitational field.

When solving Einstein field equation, we guess the form of metric and energy–momentum–stress tensor T_ΑΒ, because the physical interpretation to define T_ΑΒ is not yet known without knowing the metric first. In this article, the author uses the same approach to define T_ΑΒ but claims that the curvature is caused by $\widehat{\Delta }$ on Y, which constitutes energy–momentum–stress distribution. As required, the energy–momentum conservation is guaranteed in “quantized sense” by quantized Bianchi equation (just as in classical GR, Bianchi identity guarantees conservation). However, it is expected that there must exist a condition that local energy/mass and momentum density do contribute proportional to G_ΑΒ as Einstein field equation: $G_{{\rm A}{\rm B}}=-8\Pi \mathcal{G}{T}_{{\rm A}{\rm B}}$ ($\mathcal{G}$ is gravitational constant).

As G_ΑΒ is related to ${\Sigma }_{{\rm A}}{{\text{EG}}^{\prime }}^{}{\Sigma }_{{\rm B}}{{\text{FH}}^{\prime }}^{}$ multiplying linear combination of Λ (by constraction of $\mathcal{J}$) and enties of ${\mathcal{M}}^{\star }$ (Equation 36), let’s examinate how to find $\mathcal{J}$ and $\mathcal{M}$. To find $\mathcal{J}$ and $\mathcal{M}$, which are contractions of int operators $\widehat{J}$ on Y (Equation 20), Equation (22) explains that we can get them by ${\widehat{\Delta }}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F$. Considering the form of the operator $\widehat{\Delta }$:

$${\widehat{\Delta }}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F={\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }}{E^{}}_G{\widehat{\mathbb{D}}}_{{\text{CD}}^{\prime }}{G^{}}_F+{\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{CD}}^{\prime }}_{}({\Gamma }_{{\rm M}{\rm N}}{E^{}}_G{\widehat{\mathbb{D}}}_{{\rm N}}{G^{}}_F),$$  (37)

where ${\Gamma }_{{\rm M}{\rm N}}{E^{}}_G$ is the unknown hybrid-spinor. This implies that int operator $\widehat{\text{J}}$ can be found by

$${\widehat{J}}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_F{Y}^{\text{FH}}=\left({\widehat{\Delta }}_{[{\text{AB}}^{\prime }{\text{CD}}^{\prime }]}{E}^{}{}_F\right)Y^{\text{FH}}={\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{CD}}^{\prime }}_{}\left({\Gamma }_{{\rm M}{\rm N}}{E^{}}_G{\widehat{\mathbb{D}}}_{{\rm N}}{G^{}}_F-{\Gamma }_{{\rm N}{\rm M}}{E^{}}_G{\widehat{\mathbb{D}}}_{{\rm M}}{G^{}}_F\right)Y^{\text{FH}}$$  (38)

for which we use the fact that Μ and Ν are dummy indexes, and ${\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_G$ and ${\widehat{\mathbb{D}}}_{{\rm N}}{G^{}}_F$ commute; this implies $\mathcal{J}$ and $\mathcal{M}$ are related to terms of $\left({\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{CD}}^{\prime }}_{}\left({\Gamma }_{{\rm M}{\rm N}}{{\text{E}}^{}}_{\text{G}}{\widehat{\mathbb{D}}}_{{\rm N}}{{\text{G}}^{}}_{\text{F}}-{\Gamma }_{{\rm N}{\rm M}}{{\text{E}}^{}}_{\text{G}}{\widehat{\mathbb{D}}}_{{\rm M}}{{\text{G}}^{}}_{\text{F}}\right)Y^{\text{FH}}\right){\left({\text{Y}}^{-1}\right)}_{\text{HK}}$, where (Y⁻¹)_HK is the inverse of Y^FH. The introduction of $\widehat{\Delta }$ (Equation 37) is to change from dealing with ${\mathbb{H}}_{{\text{AB}}^{\prime }}{E^{}}_G$ to${\Gamma }_{{\rm M}{\rm N}}{E^{}}_G$, while this form $\widehat{\Delta }$ seems not unique.

Let $B_{{\rm N}}{E^{}}_F$ is invertible (if considering E and F are considered as the matrix indices), that is, in GL, we may naturally define the hybrid-spinor ${\Gamma }_{{\rm M}{\rm N}}{E^{}}_G$ by

$${\Gamma }_{{\rm M}{\rm N}}{E^{}}_G=\left({\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F\left(B_{{\rm N}}{F^{}}_H\right)\right){\left(B_{{\rm N}}{-1}^{}\right)}^H{G}_{}\text{.}$$

So it implies

$$\begin{array}{c}{\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{CD}}^{\prime }}_{}\left({\Gamma }_{{\rm M}{\rm N}}{E^{}}_G{\widehat{\mathbb{D}}}_{{\rm N}}{G^{}}_F-{\Gamma }_{{\rm N}{\rm M}}{E^{}}_G{\widehat{\mathbb{D}}}_{{\rm M}}{G^{}}_F\right)Y^{\text{FH}}=\\ {\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{CD}}^{\prime }}_{}\left({\Gamma }_{{\rm M}{\rm N}}{E^{}}_G{B}_{{\rm N}}{G^{}}_F-{\Gamma }_{{\rm N}{\rm M}}{E^{}}_G{B}_{{\rm M}}{G^{}}_F\right)Y^{\text{FH}}={\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{CD}}^{\prime }}_{}\left({\widehat{\mathbb{D}}}_{[{\rm M}}{E^{}}_G\left(B_{{\rm N}]}{G^{}}_F\right)\right)Y^{\text{FH}}\\ =\left({\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{\widehat{\mathbb{D}}}_{[{\rm M}}{E^{}}_G\left({\Sigma }^{{\rm N}}{{\text{CD}}^{\prime }}_{}{B}_{{\rm N}]}{G^{}}_F\right)\right)Y^{\text{FH}}=({\widehat{\mathbb{D}}}_{[{\text{AB}}^{\prime }}{E^{}}_G({\mathbb{H}}_{{\text{CD}}^{\prime }]}{G^{}}_F))Y^{\text{FH}}\end{array}$$

where we use core equation (Equation 17) and condition, that is, ${\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}$is $\widehat{\mathbb{D}}$-invariant. So, we get the int operator working as desired.

For scalar case of ${\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F$, we currently consider (${\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F\equiv {\partial }_{{\rm M}}{\Delta }^E{F}_{}$ as previously stated):

$${\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F{Y}^{\text{FH}}={\widehat{\mathbb{D}}}_{{\rm M}}{Y}^{\text{EH}}=\frac{{\widehat{p}}_{{\rm M}}}{i\hslash }{Y}^{\text{FH}}$$  (39)

where ${\widehat{p}}_{{\rm M}}$ is the usual quantum energy, momentum operator, while, for simplicity, p_i (i=1,2,3) is negative to usual momentum; we can plug it into the following expression and get the relationship:

$$\begin{array}{c}{\Sigma }_{{\rm A}}{{\text{EG}}^{\prime }}^{}{\Sigma }_{{\rm B}}{{\text{FH}}^{\prime }}^{}\frac{{\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{CD}}^{\prime }}_{}}{\mathit{i}\hslash }\left({\Gamma }_{{\rm M}{\rm N}}{E^{}}_G{\widehat{p}}_{{\rm N}}-{\Gamma }_{{\rm N}{\rm M}}{E^{}}_G{\widehat{p}}_{{\rm M}}\right)=\\ {\Sigma }_{{\rm A}}{{\text{EG}}^{\prime }}^{}{\Sigma }_{{\rm B}}{{\text{FH}}^{\prime }}^{}\frac{\left({\Sigma }^{[{\rm M}}{{\text{AB}}^{\prime }}_{}{\Sigma }^{{\rm N}]}{{\text{CD}}^{\prime }}_{}\right)}{i\hslash }{\Gamma }_{{\rm M}{\rm N}}{E^{}}_G{\widehat{p}}_{{\rm N}}~\text{entry of }{G}_{{\rm A}{\rm B}}~\text{entry of}-8\Pi \mathcal{G}{T}_{{\rm A}{\rm B}}\end{array}$$

note that we choose to ignore the complex conjudge sign $\star$ of $\mathcal{M}$ and detail of the indexes per entity as we just want to illustrate the possible condition to match T_ΑΒ (as operator-sense) of Einstein field equation.

It motivates the definition of a hybrid-spinor density to match the unit of Einstein field equation:

$${\Gamma }_{{\rm M}{\rm N}}{E^{}}_G\equiv ih\mathcal{G}[\text{Density}]$$  (40)

such that $\widehat{\Delta }$ with ${\Gamma }_{{\rm M}{\rm N}}{E^{}}_G$ density generates the curvature by the source of local energy–momentum from related ordinary quantum mechanics operators (iħ∂_Μ). The detail of ${\Sigma }_{{\rm M}}{{\text{AB}}^{\prime }}^{},{\Sigma }^{{\rm N}}{{\text{CD}}^{\prime }}_{}$ and ${\Gamma }_{{\rm M}{\rm N}}{E^{}}_G$ results in the freedom that one can assume the structure of T_ΑΒ as we normally solve problems in classical general relativity (demonstrated in later sections). Nevertheless, one must solve the operators and required equations in section 3.3, in order to find ${\Gamma }_{{\rm M}{\rm N}}{E^{}}_G$ density for a complete solution.

The generalization to gauge operators for spacetime

The gauge theory adopted in this theory is, as usual, to allow certain transformation of mathematical objects including the solution (Y here) such that same set of equations are invariant. So, if we found those gauge transformation, a new dynamic description of the curvature of spacetime exists, which we call it generalization to the gauge operators (or called fields).

The core theory of this article naturally suggests the spinor operation over 2 × 2 matrix (the base mathematical field) instead of scalar because of core equation (Equation 17). So in this section (ONLY), the author illustrates the equations in spinor over 2 × 2 matrix, that is, core equation reads from ${\mathbb{D}}_{{\rm M}}{E^{}}_F{Y}^{\text{FG}}=B_{{\rm M}}{E^{}}_F{Y}^{\text{FG}}$ to ${\mathbb{D}}_{{\rm M}}Y=B_{{\rm M}}Y$, so it is easy to see the natural simplicity.

Gauge operators (or fields) $\stackrel{def}{=}$ operators used to define the transformation of following objects as well as Y simultaneously while keeping following equations invariant:

 Objects to transform by gauge fields: ${\widehat{\mathbb{D}}}_{{\rm M}}$, ${\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}$, ${\mathbb{H}}_{{\text{AB}}^{\prime }}$, B_Μ, ${\widehat{\mathcal{J}}}_{\text{AB}}$

 Equations to be invariant:

 commutativity of $\widehat{\mathbb{D}}$:  $[{\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }},{\widehat{\mathbb{D}}}_{{\text{CD}}^{\prime }}]\text{Y}=0$

 $\widehat{\mathbb{D}}$ invariant:  ${\widehat{\mathbb{D}}}_{{\rm N}}{\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}\text{X}={\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}{\widehat{\mathbb{D}}}_{{\rm N}}X$ (${\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}$ denotes as generalized${\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}$, so it is 2 × 2 matrix in general $\forall$ indices of A,B’, Μ)

 Equations of system (Equation 19):  ${\widehat{\mathbb{D}}}_{\text{AB}}Y={\mathbb{H}}_{\text{AB}}$Y, ${\widehat{\mathbb{D}}}_{{\rm M}}Y=B_{{\rm M}}$Y

 Integratability: [B_Μ,B_Ν] = ${\widehat{\mathbb{D}}}_{{\rm M}}{B}_{{\rm N}}$ –${\widehat{\mathbb{D}}}_{{\rm N}}{B}_{{\rm M}}$

 Γ operator (if any): ${\Gamma }_{{\rm M}{\rm N}}=\left({\widehat{\mathbb{D}}}_{{\rm M}}{B}_{{\rm N}}\right)B_{{\rm N}}{-1}^{}$

 commutativity of $\widehat{\mathcal{J}}$ and $\widehat{\mathcal{M}}$:   $[{\widehat{\mathcal{J}}}_{\text{AB}}{\widehat{\mathbb{D}}}_{{\text{CD}}^{\prime }}]\text{Y}=0$, $[{\widehat{\mathcal{M}}}_{A^{\prime }{B}^{\prime }}{\widehat{\mathbb{D}}}_{{\text{CD}}^{\prime }}]\text{Y}=0$

 Fundamental equation (Equation 21): ${\epsilon }^{\text{BC}}\left({\widehat{\mathcal{J}}}_{\text{AB}}{\widehat{\mathbb{D}}}_{{\text{CD}}^{\prime }}\right)Y={\epsilon }^{B^{\prime }{C}^{\prime }}\left({\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{\widehat{\mathbb{D}}}_{{\text{AC}}^{\prime }}\right)Y$ or

 $\left({\widehat{\mathcal{J}}}_{\text{AB}}{\widehat{\mathbb{D}}}^B{D^{\prime }}_{}\right)Y={\left({\overline{\mathcal{J}}}_{\text{DB}}{\widehat{\mathbb{D}}}^{\star }{B^{}}_{A^{\prime }}\right)}^{\star }Y$(original form)

One can easily observe that only certain transformation is allowed because the equations above must be simultaneously satisfied. Physically, it means only certain gauge field related to transformation of curvature of spacetime is allowed. In the next section, we will exercise this generalization in certain way to apply to evolution model of the universe.

A key symmetric structure of fundamental equation in second form $\left({\widehat{\mathcal{J}}}_{\text{AB}}{\widehat{\mathbb{D}}}^B{D^{\prime }}_{}\right)Y={\left({\overline{\mathcal{J}}}_{\text{DB}}{\widehat{\mathbb{D}}}^{\star }{B^{}}_{A^{\prime }}\right)}^{\star }Y$ is obvious that the author defines this in the original version. As it has Hermitian conjugate structure if considering the indices of A’D and some $\overline{\mathcal{J}}$ and $\widehat{\mathbb{D}}$.

AS AN APPLICATION TO THE PROBLEM OF MODELING THE UNIVERSE EVOLUTION

In this section, some forms of operators and solution Y are deployed such that a simple solution of equation (21) is obtained. An equation explaining the evolution model of the universe is derived; usual Friedmann equation is compared as well as density parameters of dark matter, ordinary matter, and dark energy is proposedly originated; furthermore, the prediction of density parameter of ordinary matter is compared to the observation of Planck data (Carroll & Ostlie, 2007; Density data: http://hyperphysics.phy-astr.gsu.edu/hbase/astro/denpar.html). At last, certain equations of unsolved fields are discussed for the consistency of the theory.

Basic cosmological model

There are some assumptions on the form of Y^GH, ${\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F$, and ${\Gamma }_{{\rm M}{\rm N}}{E^{}}_F$ as well as the new gauge field – hybrid spinor ${\Lambda }_{{\rm M}}{E^{}}_F$ to the application of a cosmological model we interest:

$$Y^{\text{GH}}\to Y^{\text{GH}}(t)=\left(\begin{array}{cc}{\mathbb{e}}^{{\rm A}(t)}& 0\\ 0& {\mathbb{e}}^{{\rm A}(t)}\end{array}\right),({\rm A}(t)\text{is a scalar function of }t)$$

$${\widehat{\mathbb{D}}}_{{\rm N}}{E^{}}_F\to {\partial }_{{\rm N}}{\Delta }^E{F}_{}+a{\Lambda }_{{\rm N}}{E^{}}_F,{\Lambda }_0{E^{}}_F\text{is }t-\text{independent},$$

parameter space are t = x₀, x = {x₁, x₂, x₃}; ${\Lambda }_{{\rm N}}{E^{}}_F(t,x)$ are 2 × 2 commutating matrices (if considering E and F as the matrix indices) such that [Λ_Μ, Λ_Ν] = 0, and a(t) is a real scalar field for t-dependent;

$${\Gamma }_{{\rm M}{\rm N}}{E^{}}_F\to a(t)\text{aX}(x){\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{E^{}}_F,$$  (41)

where aX(x) is the real scalar field for x-dependent only, ${\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{{\text{E}}^{}}_{\text{F}}$ is a constant spinor such that

$$\begin{array}{c}{\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{{\text{E}}^{}}_{\text{F}}=0\text{ for }{\rm N}\ne 0,\\ {\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{{\text{E}}^{}}_{\text{F}}=\stackrel{c}{\Gamma }{\Delta }^{\text{E}}{\text{F}}_{}\text{is a real scalar constant if}{\rm N}=0.\end{array}$$

Finally, we assume $B_{{\rm M}}{E^{}}_F$ is an invertible matrix (if considering E and F as the matrix indices).

Let’s examinate ${\widehat{J}}_{{\text{EF}}^{\prime }{\text{GH}}^{\prime }}{I}^{}{}_K{Y}^{\text{KL}}$ from Equation (38):

$$\begin{array}{c}{\widehat{\mathrm{J}}}_{{\text{EF}}^{\prime }{\text{GH}}^{\prime }}{\mathrm{I}}^{}{}_{\mathrm{K}}{Y}^{\text{KL}}={\Sigma }^{{\rm M}}{{\text{EF}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{GH}}^{\prime }}_{}\left({\Gamma }_{{\rm M}{\rm N}}{{\mathrm{I}}^{}}_G{\widehat{\mathbb{D}}}_{{\rm N}}{{\mathrm{G}}^{}}_K-{\Gamma }_{{\rm N}{\rm M}}{{\mathrm{I}}^{}}_G{\widehat{\mathbb{D}}}_{{\rm M}}{{\mathrm{G}}^{}}_K\right){\mathrm{Y}}^{\text{KL}}\\ =(a\text{aX}\left({\Sigma }^{{\rm M}}{{\text{EF}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{GH}}^{\prime }}_{}\right)(a\left({\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{I^{}}_G{\Lambda }_{{\rm N}}{G^{}}_K-{\stackrel{c}{\Gamma }}_{{\rm N}{\rm M}}{I^{}}_G{\Lambda }_{{\rm M}}{G^{}}_K\right)+{\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{I^{}}_K{\partial }_{{\rm N}}-{\stackrel{c}{\Gamma }}_{{\rm N}{\rm M}}{I^{}}_K{\partial }_{{\rm M}})Y^{\text{KL}}\end{array}$$

where we use the commutativity property of $\widehat{\mathbb{D}}$, form of $\widehat{\mathbb{D}}$, form of ${\Gamma }_{{\rm M}{\rm N}}{E^{}}_F$ (Equation 41), and ${\Sigma }^{{\rm M}}{{\text{EF}}^{\prime }}_{}$ is $\widehat{\mathbb{D}}$-invariant. We further apply the form of ${\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{E^{}}_F$ (Equation 41) that ${\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{E^{}}_F=0\text{for }{\rm N}\ne 0$ and Y^KL is t-dependent only:

${\widehat{J}}_{{\text{EF}}^{\prime }{\text{GH}}^{\prime }}{I}^{}{}_K{Y}^{\text{KL}}=a^2\text{aX}(x)\stackrel{c}{\Gamma }\left(\left(\underset{i=1}{\stackrel{3}{\Sigma }}{\Sigma }^i{{\text{EF}}^{\prime }}_{}\right){\Sigma }^0{{\text{GH}}^{\prime }}_{}-{\Sigma }^0{{\text{EF}}^{\prime }}_{}\left(\underset{i=1}{\stackrel{3}{\Sigma }}{\Sigma }^i{{\text{GH}}^{\prime }}_{}\right)\right){\Lambda }_0{I^{}}_K(x)Y^{\text{KL}}(t).$

Let us study the overall t-scaling (i.e., t-dependent) effect of the equation by introducing length scaling factor l from a:

$$\begin{array}{c}a\equiv l^{-3}[\text{Density \hspace{0.17em} scaling \hspace{0.17em} factor}]\text{\hspace{0.17em} to \hspace{0.17em}}[\text{length scaling \hspace{0.17em} factor}].\\ {\widehat{J}}_{{\text{EF}}^{\prime }{\text{GH}}^{\prime }}{I}^{}{}_K{Y}^{\text{KL}}=\stackrel{c}{\Gamma }{\Sigma }_{{\text{EF}}^{\prime }{\text{GH}}^{\prime }}\frac{aX(x){\Lambda }_0{I^{}}_K(x)}{l{(t)}^6}{Y}^{\text{KL}}(t),\end{array}$$  (42)

where a new spinor from combination of ${\Sigma }^{{\rm M}}{{\text{AB}}^{\prime }}_{}:{\Sigma }_{{\text{EF}}^{\prime }{\text{GH}}^{\prime }}\equiv \left(\underset{i=1}{\stackrel{3}{\Sigma }}{\Sigma }^i{{\text{EF}}^{\prime }}_{}\right){\Sigma }^0{{\text{GH}}^{\prime }}_{}-{\Sigma }^0{{\text{EF}}^{\prime }}_{}\left(\underset{i=1}{\stackrel{3}{\Sigma }}{\Sigma }^i{{\text{GH}}^{\prime }}_{}\right).$

Then we get $\mathcal{J}$ and $\mathcal{M}$ by $\widehat{\mathcal{J}},\widehat{\mathcal{M}}$ and the inverse of Y^GH:

$$\begin{array}{c}{\mathcal{J}}_{\text{AC}}{E^{}}_F=\left({\widehat{\mathcal{J}}}_{\text{AC}}{E^{}}_G{Y}^{\text{GH}}\right){\left(Y^{-1}\right)}_{\text{HF}}=\\ \left({\epsilon }^{B^{\prime }{D}^{\prime }}{\widehat{J}}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_G{Y}^{\text{GH}}\right){\left(Y^{-1}\right)}_{\text{HF}}=\left({\epsilon }^{B^{\prime }{D}^{\prime }}\stackrel{c}{\Gamma }{\Sigma }_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}\frac{\text{aX}(x){\Lambda }_0{E^{}}_G(x)}{l{(t)}^6}{Y}^{\text{GH}}\right){\left(Y^{-1}\right)}_{\text{HF}}={\Sigma }_{\text{AC}}\left(\frac{\stackrel{c}{\Gamma }\text{aX}(x){\Lambda }_0{E^{}}_F(x)}{l{(t)}^6}\right),\\ {\mathcal{M}}_{B^{\prime }{D}^{\prime }}{E^{}}_F=\left({\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{E^{}}_G{Y}^{\text{GH}}\right){\left(Y^{-1}\right)}_{\text{HF}}=\\ \left({\epsilon }^{\text{AC}}{\widehat{J}}_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}{E}^{}{}_G{Y}^{\text{GH}}\right){\left(Y^{-1}\right)}_{\text{HF}}=\left({\epsilon }^{\text{AC}}\stackrel{c}{\Gamma }{\Sigma }_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}\frac{\text{aX}(x){\Lambda }_0{E^{}}_G(x)}{l{(t)}^6}{Y}^{\text{GH}}\right){\left(Y^{-1}\right)}_{\text{HF}}={\Sigma }_{B^{\prime }{D}^{\prime }}\left(\frac{\stackrel{c}{\Gamma }\text{aX}(x){\Lambda }_0{E^{}}_F(x)}{l{(t)}^6}\right),\end{array}$$

where ${\Sigma }_{\text{AC}}\equiv {\epsilon }^{B^{\prime }{D}^{\prime }}\stackrel{c}{\Gamma }{\Sigma }_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }},{\Sigma }_{B^{\prime }{D}^{\prime }}\equiv {\epsilon }^{\text{AC}}\stackrel{c}{\Gamma }{\Sigma }_{{\text{AB}}^{\prime }{\text{CD}}^{\prime }}$.

Let’s examinate the Einstein spinor $G_{{\text{AC}}^{\prime }{\text{BD}}^{\prime }}$ (Equation 35) based on these assumptions and the calculation above. $G_{{\text{AC}}^{\prime }{\text{BD}}^{\prime }}$ is contributed by Λ and ${\mathcal{M}}^{\star }{{{\text{ABC}}^{\prime }}_{}}^{E^{\prime }}$; Λ is contributed by ${\mathcal{J}}_{\text{ABC}}{B}^{}$ Equation 32, and Einstein tensor is derived by $G_{{\rm M}{\rm N}}={\Sigma }_{{\rm M}}{{\text{EF}}^{\prime }}^{}{\Sigma }_{{\rm N}}{{\text{GH}}^{\prime }}^{}{G}_{{\text{EF}}^{\prime }{\text{GH}}^{\prime }}$. The t-dependency of ${\mathcal{J}}_{\text{AC}}{E^{}}_F$ and ${\mathcal{M}}_{B^{\prime }{D}^{\prime }}{E^{}}_F$ is clearly proportional to l⁻⁶. Refering to the Equation (1 and 4), Σ^Μ and Σ_Ν balance each others (because of the metric and its inverse) so we can neglect the overall t-scaling effect of ${\Sigma }_{{\rm M}}{{\text{EF}}^{\prime }}^{}{\Sigma }_{{\rm N}}{{\text{FG}}^{\prime }}^{}\times {\Sigma }_{\text{AC}}$, or ${\Sigma }_{{\rm M}}{{\text{EF}}^{\prime }}^{}{\Sigma }_{{\rm N}}{{\text{FG}}^{\prime }}^{}\times {\Sigma }_{B^{\prime }{D}^{\prime }}$ from G_ΜΝ; it makes sense, as in spinor approach of GR, the curvature dynamics is reflected by Χ, Φ curvature spinors, not${\Sigma }_{{\rm M}}{A,C^{\prime }}^{}$,${\stackrel{{\rm N}}{\Sigma }}_{E,F^{\prime }}$. One can also choose ${\Sigma }^0{{\text{GH}}^{\prime }}_{}$ and ${\Sigma }^i{{\text{EF}}^{\prime }}_{}$ to be the near constants to make computation easy; however, one must check the consistency of the choice with the core equations (refer to section: the generalization to gauge operators for spacetime).

In conclusion, G_ΜΝ is contributed by the term

$$\frac{\stackrel{c}{\Gamma }\text{aX}(x){\Lambda }_0{E^{}}_F(x)}{l{(t)}^6},$$

and by the following definition (to be consistent with Einstein equation):

$$\mathit{\text{Ordinary matter density tensor}}\stackrel{\mathit{\text{def}}}{=}\mathit{\text{ordinary stress-energy denisity tensor}}\stackrel{\mathit{\text{def}}}{=}-\frac{G_{{\rm M}{\rm N}}}{8\Pi \mathcal{G}}$$

the t-dependency of ordinary matter density in this case is proportional to $\frac{1}{l{(t)}^6}$. Note that, the ordinary matter density defined here is to match Einstein equation for this model only.

Calculation of scaling factor and comparison to Friedmann equation

Solving Equation (21, the fundamental equation) is the core task to get the dynamics of the cosmological model, or more generally the physical structure of spacetime. The first step is to calculate the term, ${\widehat{\mathcal{J}}}_{\text{CD}}{E^{}}_F{\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }}{F^{}}_G$, which is left side of the fundamental equation. After using the form of $\widehat{\mathbb{D}}$, ${\Gamma }_{{\rm M}{\rm N}}{E^{}}_F$ (41) and $\widehat{\mathbb{D}}$-invariant, it implies:

$$\begin{array}{l}{\widehat{\mathcal{J}}}_{\text{CD}}{E^{}}_F{\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }}{F^{}}_G{Y}^{\text{GH}}={\epsilon }^{F^{\prime }{H}^{\prime }}\left({\Sigma }^{\Gamma }{{\text{AB}}^{\prime }}_{}{\Sigma }^{{\rm M}}{{\text{CF}}^{\prime }}_{}{\Sigma }^{{\rm N}}{{\text{DH}}^{\prime }}_{}\right)\times \\ a\text{aX}(a^2{\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{E^{}}_F{\Lambda }_{\Gamma }{F^{}}_I{\Lambda }_{{\rm N}}{I^{}}_G-a^2{\stackrel{c}{\Gamma }}_{{\rm N}{\rm M}}{E^{}}_F{\Lambda }_{{\rm M}}{F^{}}_I{\Lambda }_{\Gamma }{I^{}}_G\\ +a{\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{E^{}}_F{\Lambda }_{\Gamma }{F^{}}_G{\partial }_{{\rm N}}+a{\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{E^{}}_F{\Lambda }_{{\rm N}}{F^{}}_G{\partial }_{\Gamma }+a{\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{E^{}}_F{\partial }_{{\rm N}}\left({\Lambda }_{\Gamma }{F^{}}_G\right)\\ -a{\stackrel{c}{\Gamma }}_{{\rm N}{\rm M}}{E^{}}_F{\Lambda }_{\Gamma }{F^{}}_G{\partial }_{{\rm M}}-a{\stackrel{c}{\Gamma }}_{{\rm N}{\rm M}}{E^{}}_F{\Lambda }_{{\rm M}}{F^{}}_G{\partial }_{\Gamma }-a{\stackrel{c}{\Gamma }}_{{\rm N}{\rm M}}{E^{}}_F{\partial }_{{\rm M}}\left({\Lambda }_{\Gamma }{F^{}}_G\right)+{\partial }_{{\rm N}}(a){\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{E^{}}_F{\Lambda }_{\Gamma }{F^{}}_G\\ -{\partial }_{{\rm M}}(a){\stackrel{c}{\Gamma }}_{{\rm N}{\rm M}}{E^{}}_F{\Lambda }_{\Gamma }{F^{}}_G+{\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{E^{}}_G{\partial }_{\Gamma ,{\rm N}}-{\stackrel{c}{\Gamma }}_{{\rm N}{\rm M}}{E^{}}_G{\partial }_{\Gamma ,{\rm M}})Y^{\text{GH}}\end{array}.$$

Applying ${\stackrel{c}{\Gamma }}_{{\rm M}{\rm N}}{E^{}}_F$ for Ν ≠ 0 (condition 41), Y^GH is t-dependent only, commuting property of ${\Lambda }_{{\rm M}}{E^{}}_F$ (if considering E and F as the matrix indices), and the following equations from later discussion (Equation 45):

$$\begin{array}{c}{a}^{\prime }{\Lambda }_i{E^{}}_F=a{\partial }_i\left({\Lambda }_0{E^{}}_F\right)-a{\partial }_0\left({\Lambda }_i{E^{}}_F\right)\\ a{\Lambda }_i{E^{}}_G{\Lambda }_0{G^{}}_F=-{\partial }_i\left({\Lambda }_0{E^{}}_F\right)\end{array}$$

for i= 1,2,3 (a′ is t-derivative of a), we get:

$$\begin{array}{l}{\widehat{\mathcal{J}}}_{\text{CD}}{E^{}}_F{\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }}{F^{}}_G{Y}^{\text{GH}}=\text{aaX}\stackrel{c}{\Gamma }{\epsilon }^{F^{\prime }{H}^{\prime }}\left({\Sigma }_{{\text{CF}}^{\prime }{\text{DH}}^{\prime }}\right)(a^2{\Sigma }^0{{\text{AB}}^{\prime }}_{}{\Lambda }_0{E^{}}_F{\Lambda }_0{F^{}}_G{Y}^{\text{GH}}+a^{\prime }{\Sigma }^0{{\text{AB}}^{\prime }}_{}{\Lambda }_0{E^{}}_G{Y}^{\text{GH}}+\\ 2a{\Sigma }^0{{\text{AB}}^{\prime }}_{}{\Lambda }_0{E^{}}_G{(Y^{\text{GH}})}^{\prime }+a{\Sigma }^i{{\text{AB}}^{\prime }}_{}{\Lambda }_i{E^{}}_G{\left(Y^{\text{GH}}\right)}^{\prime }+a{\Sigma }^0{{\text{AB}}^{\prime }}_{}{\partial }_0({\Lambda }_0{E^{}}_G)Y^{\text{GH}}+{\Sigma }^0{{\text{AB}}^{\prime }}_{}{\left(Y^{\text{EH}}\right)}^{\prime \prime })\end{array},$$

where Y′,Y′ denote first and second t-derivatives of Y.

By right side of fundamental equation:

$$\left({\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{E^{}}_F{\widehat{\mathbb{D}}}_C{{D^{\prime }F}^{}}_G\right)Y^{\text{GH}}={\epsilon }^{D^{\prime }{A}^{\prime }}\left({\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{E^{}}_F{\widehat{\mathbb{D}}}_{C{A}^{\prime }}{F^{}}_G\right)Y^{\text{GH}}.$$

Combing both sides, we get ${\epsilon }^{\text{DA}}$ $\left({\widehat{\mathcal{J}}}_{\text{CD}}{E^{}}_F{\widehat{\mathbb{D}}}_{{\text{AB}}^{\prime }}{F^{}}_G\right)Y^{\text{GH}}=$ ${\epsilon }^{A^{\prime }{D}^{\prime }}\left({\widehat{\mathcal{M}}}_{B^{\prime }{D}^{\prime }}{E^{}}_F{\widehat{\mathbb{D}}}_{C,A^{\prime }}{F^{}}_G\right)Y^{\text{GH}}$ as

$${\epsilon }^{\text{DA}}a\text{aX}\stackrel{c}{\Gamma }{\epsilon }^{F^{\prime }{H}^{\prime }}\left({\Sigma }_{{\text{CF}}^{\prime }{\text{DH}}^{\prime }}\right)(a^2{\Sigma }^0{{\text{AB}}^{\prime }}_{}{\Lambda }_0{E^{}}_F{\Lambda }_0{F^{}}_G{Y}^{\text{GH}}+a^{\prime }{\Sigma }^0{{\text{AB}}^{\prime }}_{}{\Lambda }_0{E^{}}_G{Y}^{\text{GH}}+2a{\Sigma }^0{{\text{AB}}^{\prime }}_{}{{\Lambda }_0{E^{}}_G(Y^{\text{GH}})}^{\prime }+a{\Sigma }^i{{\text{AB}}^{\prime }}_{}{\Lambda }_i{E^{}}_G{\left(Y^{\text{GH}}\right)}^{\prime }+a{\Sigma }^0{{\text{AB}}^{\prime }}_{}{\partial }_0({\Lambda }_0{E^{}}_G)Y^{\text{GH}}+{\Sigma }^0{{\text{AB}}^{\prime }}_{}{\left(Y^{\text{EH}}\right)}^{\prime \prime })={\epsilon }^{D^{\prime }{A}^{\prime }}a\text{aX}\stackrel{c}{\Gamma }{\epsilon }^{\text{FH}}\left({\Sigma }_{{\text{FB}}^{\prime }{\text{HD}}^{\prime }}\right)(a^2{\Sigma }^0{{\text{CA}}^{\prime }}_{}{\Lambda }_0{E^{}}_F{\Lambda }_0{F^{}}_G{Y}^{\text{GH}}+a^{\prime }{\Sigma }^0{{\text{CA}}^{\prime }}_{}{\Lambda }_0{E^{}}_G{Y}^{\text{GH}}+2a{\Sigma }^0{{\text{CA}}^{\prime }}_{}{{\Lambda }_0{E^{}}_G(Y^{\text{GH}})}^{\prime }+a{\Sigma }^i{{\text{CA}}^{\prime }}_{}{\Lambda }_i{E^{}}_G{\left(Y^{\text{GH}}\right)}^{\prime }+a{\Sigma }^0{{\text{CA}}^{\prime }}_{}{\partial }_0({\Lambda }_0{E^{}}_G)Y^{\text{GH}}+{\Sigma }^0{{\text{CA}}^{\prime }}_{}{\left(Y^{\text{EH}}\right)}^{\prime \prime })$$

Using the form of $Y^{\text{GH}}=\left(\begin{array}{cc}{e}^{{\rm A}(t)}& 0\\ 0& e^{{\rm A}(t)}\end{array}\right),$

differentiating it results in

$${\left(Y^{\text{GH}}\right)}^{\prime }\to Y^{\text{GH}}\varphi ,{\left(Y^{\text{GH}}\right)}^{\prime \prime }\to Y^{\text{GH}}\left({\varphi }^2+{\varphi }^{\prime }\right),$$

where Α′ ≡ ϕ. After substituting these results to the equation above, all terms are matrices operating on Y^GH only (if considering G,H as the matrix indices), so we can take it out as Y^GH is in GL. The equation becomes

$$\begin{array}{l}{\epsilon }^{\text{DA}}a\text{aX}\stackrel{c}{\Gamma }{\epsilon }^{F^{\prime }{H}^{\prime }}\left({\Sigma }_{{\text{CF}}^{\prime }{\text{DH}}^{\prime }}\right)({\Sigma }^0{{\text{AB}}^{\prime }}_{}(a^2{\Lambda }_0{E^{}}_F{\Lambda }_0{F^{}}_G+{\Lambda }_0{E^{}}_G\left(a^{\prime }+2a\varphi \right)\\ +a{\partial }_0\left({\Lambda }_0{E^{}}_G\right)+\left({\varphi }^2+{\varphi }^{\prime }\right){\Delta }^E{G}_{})+{\Sigma }^i{{\text{AB}}^{\prime }}_{}a\left(\varphi {\Lambda }_i{E^{}}_G\right))\\ ={\epsilon }^{D^{\prime }{A}^{\prime }}a\text{aX}\stackrel{c}{\Gamma }{\epsilon }^{\text{FH}}\left({\Sigma }_{{\text{FB}}^{\prime }{\text{HD}}^{\prime }}\right)({\Sigma }^0{{\text{CA}}^{\prime }}_{}(a^2{\Lambda }_0{E^{}}_F{\Lambda }_0{F^{}}_G+{\Lambda }_0{E^{}}_G\left(a^{\prime }+2a\varphi \right)\\ +a{\partial }_0\left({\Lambda }_0{E^{}}_G\right)+\left({\varphi }^2+{\varphi }^{\prime }\right){\Delta }^E{G}_{})+{\Sigma }^i{{\text{CA}}^{\prime }}_{}a\left(\varphi {\Lambda }_i{E^{}}_G\right)).\end{array}$$

Finally, we further assume the following conditions in order to look for a simple solution:

$$\left\{\varphi {\Lambda }_i{E^{}}_G\to 0\right\}({\Lambda }_i{E^{}}_G\text{coupling with}\varphi \text{contributing very little to the overall dynamics}),$$

and note that ${\Lambda }_0{F^{}}_G$ is t-independent. Then, there exists a solution:

$$0=a^2{\Lambda }_0{E^{}}_F{\Lambda }_0{F^{}}_G+\left(a^{\prime }+2a\varphi \right){\Lambda }_0{E^{}}_G+\left({\varphi }^2+{\varphi }^{\prime }\right){\Delta }^E{G}_{}.$$

Let’s pick an abelian matrix structure for ${\Lambda }_0{F^{}}_G$ (if considering F and G as the matrix indices)below:

$${\Lambda }_0{F^{}}_G=\left(\begin{array}{cc}s(x)& y(x)\\ y(x)& s(x)\end{array}\right),$$

and we have the set of differential equations for the solution:

$$\begin{array}{c}0=2ay(as+\varphi )+y{a}^{\prime }\\ 0=a^2{s}^2+a^2{y}^2+2as\varphi +{\varphi }^2+s{a}^{\prime }+{\varphi }^{\prime },\end{array}$$

and we pick a solution that a ≠ 0:

$$a^{\prime }=-2a(as+\varphi ),{\varphi }^{\prime }=a^2\left(s^2-y^2\right)-{\varphi }^2$$

To get an equation to compare with usual Friedmann equation, let’s converting the solution to length scaling factor l and getting a scalar $\mathcal{H}\equiv \frac{l^{\prime }}{l}$:

$$l^{\prime }=\frac{2\left(l^3\varphi +s\right)}{3l^2},\mathcal{H}=\frac{2\left(l^3\varphi +s\right)}{3l^3}$$

The equation to be compared with Friedmann equation is

$${\left(\frac{\mathcal{H}(l,\varphi )}{{\mathcal{H}}_0}\right)}^2=\frac{4s^2}{9l^6{\mathcal{H}}_0^2}+\frac{8s\varphi }{9l^3{\mathcal{H}}_0^2}+\frac{4{\varphi }^2}{9{\mathcal{H}}_0^2},$$  (43)

while Friedmann equation is (spatial curvature k is assumed to be zero, Λ_cos is usual cosmological constant, Ω_n are the usual density parameters for different types of matter):

$${\left(\frac{\mathcal{H}(l)}{{\mathcal{H}}_0}\right)}^2={\displaystyle \sum _n{\Omega }_n}{l}^{-n}=\frac{8\Pi \mathcal{G}}{3{\mathcal{H}}_0{2}^{}}\left({\displaystyle \sum _n{{\rm P}}_n}\left(l^{-n}\right)\right)+\frac{{\Lambda }_{\cos }}{3{\mathcal{H}}_0{2}^{}}$$

Densities of dark energy, matter, and ordinary matter (Ω_Λ, Ω_M, Ω_b)

As usual in Λ CDM model, we identify the zeroth and negative power terms of scaling factor (l⁰, l⁻³ and l⁻⁶), in equation ${\left(\frac{\mathcal{H}(l,\varphi )}{{\mathcal{H}}_0}\right)}^2$ (Equation 43), so there are three types of matter to be responsible for the evolution of the scaling factor, and their density parameters are namely Ω_Λ, Ω_M, and Ω_b, respectively. The Ω_b is proposedly to be responsible for ordinary matter density as the t-dependency of ordinary matter density is proportional to l⁻⁶ in preceding discussion:

$${\Omega }_b=\frac{4{\mathbb{s}}^2}{9{\mathcal{H}}_0^2},{\Omega }_{\Lambda }=\frac{4{\varphi }^2}{9{\mathcal{H}}_0^2},{\Omega }_M=\frac{8\mathbb{s}\varphi }{9{\mathcal{H}}_0^2}.$$

${\Omega }_M{2}^{}\simeq 4{\Omega }_b{\Omega }_{\Lambda }$ (approximation − equal sign because of the approximation of the model)

To compare observation results from observation data (Carroll & Ostlie, 2007; Density data: http://hyperphysics.phy-astr.gsu.edu/hbase/astro/denpar.html), Ω_Λ and Ω_b data are used:

$${\Omega }_{\Lambda }\to 0.69,{\Omega }_b\to 0.04;$$

the calculated and observed Ω_M are

$$\text{Calculated \hspace{0.17em}}{\Omega }_M=0.33\text{ \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} vs \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} Observed \hspace{0.17em}}{\Omega }_M=0.31(\text{upper \hspace{0.17em} bound \hspace{0.17em} for \hspace{0.17em} best \hspace{0.17em} fit})$$

The observation results of Ω_b, Ω_Λ, Ω_M motivate the speculation for this model to claim the relationship of densities; it is similar that Maxwell speculated that electromagnetic wave is the light itself because the speed of electromagnetic wave from his equations was found accidentally the same as light speed.

The author claims that, more accurate relationship among those sources of evolution of the universe should be manageable after certain assumption is removed such as time-independent of Λ₀. Moreover, complete proof of calculation of Ω_b requires particle theory, which is also not covered in this article.

Spinor gauge field-${\Lambda }_{{\rm M}}{E^{}}_F$ dynamics

In preceding discussion of calculation of the solution for fundamental equation (Equation 21), a set of PDE related to ${\Lambda }_{{\rm M}}{E^{}}_F$ is used. In fact, they are the results of core constraints of the theory.

Let us look at the commutativity of $\widehat{\mathbb{D}}$ after appying gauge field${\Lambda }_{{\rm M}}{E^{}}_F$:

$${\widehat{\mathbb{D}}}_{A,B^{\prime }}{E^{}}_F{\widehat{\mathbb{D}}}_{C,D^{\prime }}{F^{}}_G{Y}^{\text{GH}}={\widehat{\mathbb{D}}}_{C,D^{\prime }}{E^{}}_F{\widehat{\mathbb{D}}}_{A,B^{\prime }}{F^{}}_G{Y}^{\text{GH}}$$

,

there exist a solution

$$\left[{\partial }_{{\rm N}}{\Delta }^E{F}_{}+a{\Lambda }_{{\rm N}}{E^{}}_F,{\partial }_{{\rm M}}+a{\Lambda }_{{\rm M}}{E^{}}_F\right]Y^{\text{FH}}=0$$

so$\left({\partial }_{[{\rm N}}\left(a{\Lambda }_{{\rm M}}{E^{}}_F\right)Y^{\text{FH}}+a^2{\Lambda }_{{\rm N}}{E^{}}_F{\Lambda }_{{\rm M}]}{F^{}}_G{Y}^{\text{GH}}\right)=0$;

by ${\Lambda }_{{\rm M}}{E^{}}_F$ is abelian (if considering E and F as the matrix indices), it implies

$${\partial }_{[{\rm N}}\left(a{\Lambda }_{{\rm M}]}{E^{}}_G\right)Y^{\text{GH}}=0.$$  (44)

Then, let us look at integratability of the system: $B_{[{\rm M}}{{\text{E}}^{}}_{\text{F}}{B}_{{\rm N}]}{{\text{F}}^{}}_{\text{G}}={\widehat{\mathbb{D}}}_{[\text{Μ}}{{\text{E}}^{}}_{\text{F}}{\text{B}}_{{\rm N}]}{{\text{F}}^{}}_{\text{G}}$. As ${\Gamma }_{{\rm M}{\rm N}}{{\text{E}}^{}}_{\text{F}}=0$ for Ν ≠ 0 , and it provides ${\widehat{\mathbb{D}}}_{{\rm M}}{{\text{E}}^{}}_{\text{F}}{B}_{{\rm N}}{{\text{F}}^{}}_{\text{G}}$ by Equation (38) ${\Gamma }_{{\rm M}{\rm N}}{E^{}}_G=\left({\widehat{\mathbb{D}}}_{{\rm M}}{E^{}}_F\left(B_{{\rm N}}{F^{}}_H\right)\right){\left(B_{{\rm N}}{-1}^{}\right)}^H{G}_{}$, so we get

$${\widehat{\mathbb{D}}}_{{\rm M}}{{\text{E}}^{}}_{\text{F}}{B}_i{{\text{F}}^{}}_{\text{G}}=0,{\text{B}}_{[\text{i}}{{\text{E}}^{}}_{\text{F}}{\text{B}}_{\text{j}]}{{\text{F}}^{}}_{\text{G}}=0,{\text{B}}_{[0}{{\text{E}}^{}}_{\text{F}}{\text{B}}_{i]}{{\text{F}}^{}}_{\text{G}}=-a\text{aX}\stackrel{c}{\Gamma }{B}_0{{\text{E}}^{}}_{\text{G}}(\text{for \hspace{0.17em}}i,j=1,2,3;{\rm M}=0,1,2,3)\text{.}$$

By the core equation ${\widehat{\mathbb{D}}}_i{{\text{E}}^{}}_{\text{F}}{Y}^{\text{FG}}=B_{\text{i}}{{\text{E}}^{}}_{\text{F}}{\text{Y}}^{\text{FG}}$, $B_{\text{i}}{{\text{E}}^{}}_{\text{F}}$ is solved by$\left({\partial }_{\text{i}}{\Delta }^{\text{E}}{\text{F}}_{}+\text{a}{\Lambda }_{\text{i}}{{\text{E}}^{}}_{\text{F}}\right)\text{Y}{(\text{t})}^{\text{FG}}=B_{\text{i}}{{\text{E}}^{}}_{\text{F}}\text{Y}{(\text{t})}^{\text{FG}}$; $\to$ exist a solution

$$B_i{E^{}}_F=a{\Lambda }_i{E^{}}_F,$$

it is consistent with $B_{[i}{{\text{E}}^{}}_{\text{F}}{\text{B}}_{j]}{{\text{F}}^{}}_{\text{G}}=0$ and ${\Lambda }_i{E^{}}_F$ abelian property (if considering E and F as the matrix indices). ${\widehat{\mathbb{D}}}_{{\rm M}}{{\text{E}}^{}}_{\text{F}}{B}_{\text{i}}{{\text{F}}^{}}_{\text{G}}=0,$ equation leads to:

$$\begin{array}{c}\text{a }{\Lambda }_{\text{j}}{{\text{E}}^{}}_{\text{F}}{\Lambda }_{\text{i}}{{\text{F}}^{}}_{\text{G}}+{\partial }_{\text{j}}\left({\Lambda }_{\text{i}}{{\text{E}}^{}}_{\text{G}}\right)==0\\ {\text{a}}^2{\Lambda }_0{{\text{E}}^{}}_{\text{F}}{\Lambda }_{\text{i}}{{\text{F}}^{}}_{\text{G}}+{\text{a}}^{\prime }{\Lambda }_{\text{i}}{{\text{E}}^{}}_{\text{G}}+\text{a}{\partial }_0\left({\Lambda }_{\text{i}}{{\text{E}}^{}}_{\text{G}}\right)==0& \text{.}\end{array}$$

Combining the above equation and Equation (44), the dynamics of gauge field ${\Lambda }_{{\rm M}}{E^{}}_F$ is got:

$$\left(\begin{array}{c}a{\partial }_j\left({\Lambda }_i{E^{}}_F\right)=a{\partial }_i\left({\Lambda }_j{E^{}}_F\right)\\ a{\partial }_0\left({\Lambda }_i{E^{}}_F\right)=-a^{\prime }{\Lambda }_i{E^{}}_F+a{\partial }_i\left({\Lambda }_0{E^{}}_F\right)\\ {\partial }_i\left({\Lambda }_j{E^{}}_G\right)=-a{\Lambda }_j{E^{}}_F{\Lambda }_i{F^{}}_G\\ {\partial }_i\left({\Lambda }_0{E^{}}_G\right)=-a{\Lambda }_i{E^{}}_F{\Lambda }_0{F^{}}_G\end{array}\right)$$  (45)

for i,j=1,2,3(a′ is t-derivative of a).

Please note that the calculation of cosmological model in this section does not include solving gauge field-${\Lambda }_{{\rm M}}{E^{}}_F$; however, the theory requires that the dynamics of ${\Lambda }_{{\rm M}}{E^{}}_F$ must exist in order to be consistent.

DISCUSSION

This article provides another approach of describing the physical nature by simplicity of cross-ratio. However, the complete correspondence between cross-ratio deformation and isomonodromic integratable system used in this article is not yet covered. Potentially, additional property beyond our spacetime might emerge.

The description of quantum mechanical wave function is also not discussed, and this is what this theory should extend to. With such description and spacetime description in this article, a more complete picture of matter and spacetime should be reviewed.

Finally, the author proposes that the physical reality is the general deformation of same cross-ratio.

References

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2. Cassidy P, Singer M. 2005. Galois theory of parameterized differential equations and linear differential algebraic groups. arXiv:math/0502396arxiv.org

3. Gomez-Lobo A. G.-P, Martin-Garcia J. M. 2012. Spinors: A Mathematica package for doing spinor calculus in general relativity. Computer Physics Communications. Vol. 183:2214–2225

4. Penrose R. 1960. Chapters 1–3 of “A spinor approach to general relativity”. Annals of Physics. Vol. 10:171–201

5. Robert M. W. General Relativity. Chicago, IL: University of Chicago Press. 1984

6. Yoshida M. Hyper geometric functions, my love: Modular interpretations of configuration spaces. Springer Fachmedien Wiesbader (1997). Germany:

Competing interests

The authors declare no competing interests.

Publishing notes

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