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: ΩM2 = 4 Ωb ΩΛ

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, XM ’ 4Xb XK, is derived; the actual observed results are compared to this equation (calculated ΩM = 0.33 vs. observed ΩM = 0.31); the model might explain the origin of dark energy and dark matter of the evolution of the universe. 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 iso‐ monodromic 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 SOR-PHYS