Von Neumann's Quantization of General Relativity

TL;DR

This paper applies von Neumann's quantization procedure to General Relativity at the Planck epoch, deriving a quantum framework that incorporates the vacuum state as a quantum ensemble and explores implications for cosmological observations.

Contribution

It introduces a novel quantization approach for General Relativity using von Neumann's method, including initial data at the Planck epoch and vacuum state degeneracy.

Findings

Derivation of the quantum energy constraint for gravidynamics

Identification of vacuum degeneracy related to the Casimir effect

Discussion of gravidynamics' applicability to modern cosmological data

Abstract

Von Neumann's procedure is applied for quantization of General Relativity. We quantize the initial data of dynamical variables at the Planck epoch, where the Hubble parameter coincides with the Planck mass. These initial data are defined via the Fock simplex in the tangent Minkowskian space-time and the Dirac conformal interval. The Einstein cosmological principle is applied for the average of the spatial metric determinant logarithm over the spatial volume of the visible Universe. We derive the splitting of the general coordinate transformations into the diffeomorphisms (as the object of the second N\"other theorem) and the initial data transformations (as objects of the first N\"other theorem). Following von Neumann, we suppose that the vacuum state is a quantum ensemble. The vacuum state is degenerated with respect to quantum numbers of non-vacuum states with the distribution function that yields the Casimir effect in gravidynamics in analogy to the one in electrodynamics. The generation functional of the perturbation theory in gravidynamics is given as a solution of the quantum energy constraint. We discuss the region of applicability of gravidynamics and its possible predictions for explanation of the modern observational and experimental data.