A Wheeler–DeWitt Quantum Approach to the Branch-Cut Gravitation with Ordering Parameters

Abstract

In this contribution to the Festschrift for Prof. Remo Ruffini, we investigate a formulation of quantum gravity using the Hořava–Lifshitz theory of gravity, which is General Relativity augmented by counter-terms to render the theory regularized. We are then led to the Wheeler–DeWitt (WDW) equation combined with the classical concepts of the branch-cut gravitation, which contemplates as a new scenario for the origin of the Universe, a smooth transition region between the contraction and expansion phases. Through the introduction of an energy-dependent effective potential, which describes the space-time curvature associated with the embedding geometry and its coupling with the cosmological constant and matter fields, solutions of the WDW equation for the wave function of the Universe are obtained. The Lagrangian density is quantized through the standard procedure of raising the Hamiltonian, the helix-like complex scale factor of branched gravitation as well as the corresponding conjugate momentum to the category of quantum operators. Ambiguities in the ordering of the quantum operators are overcome with the introduction of a set of ordering factors α, whose values are restricted, to make contact with similar approaches, to the integers α=[0,1,2], allowing this way a broader class of solutions for the wave function of the Universe. In addition to a branched universe filled with underlying background vacuum energy, primordial matter and radiation, in order to connect with standard model calculations, we additionally supplement this formulation with baryon matter, dark matter and quintessence contributions. Finally, the boundary conditions for the wave function of the Universe are imposed by assuming the Bekenstein criterion. Our results indicate the consistency of a topological quantum leap, or alternatively a quantum tunneling, for the transition region of the early Universe in contrast to the classic branched cosmology view of a smooth transition.

The branch‐cut quantum gravity with a self‐coupling inflation scalar field: The wave function of the Universe

“…The Hořava-Lifshitz action,  HL , is given by (Abreu et al 2019;Bertolami & Zarro 2011;Bodmann et al 2023a;Bodmann et al 2023b;Cordero et al 2019;García-Compeán & Mata-Pacheco 2022;Hess et al 2023;Hořava 2009;Vieira et al 2020):…”

“…In the BCG formulation, the action  HL depends on the scalar curvature of the branched Universe, , and on its derivatives, in different orders (Abreu et al 2019;Bertolami & Zarro 2011;Bodmann et al 2023a;Bodmann et al 2023b;Cordero et al 2019;García-Compeán & Mata-Pacheco 2022;Hess et al 2023;Hořava 2009;Vieira et al 2020). In expression (3), g i denote running coupling constants, M P is the Planck mass, ∇ i represents covariant derivatives, and the branching Ricci components of the three-dimensional metrics may be determined by imposing a maximum symmetric surface foliation (Hess et al 2023) which gives:…”

“…As a fundamental result, quantum density perturbations lead to the eternal inflation as a result of a repulsive form of gravity, driving the acceleration of cosmic expansion. Branch cut gravity (BCG) in turn, a theoretical alternative to the inflation model, based on the mathematical augmentation technique of closure and existential completeness (Manders 1989), represents an analytically continuous extension of general relativity (Einstein 1916(Einstein , 1917 to the complex plane (Bodmann et al 2022;Bodmann et al 2023a;Bodmann et al 2023b;Hess et al 2023;Zen Vasconcellos et al 2019;Zen Vasconcellos et al 2021a;Zen Vasconcellos et al 2021b;Zen Vasconcellos et al 2023). Such mathematical procedures have proven extremely useful both in quantum mechanics (Dirac 1937), with direct physical manifestations (Aharonov & Bohm 1959;Wu et al 2021) and in pseudocomplex general relativity (pc-GR) (Hess 2017;Hess et al 2016;Hess & Boller 2020;Hess & Greiner 2009), allowing to identify a suppression mechanism of the primordial gravitational singularity and to the prediction of the existence of dark energy outside and inside cosmic mass distributions.…”

“…The branch-cut formulation corresponds to the complexification of the Friedman-Lemaître-Robertson-Walker (FLRW) metric (Friedman 1922;Lemaître 1927;Robertson 1935;Walker 1937), resulting in a sum of field equations associated to continuously distributed single-poles with infinitesimal residues, arranged along a line in the complex plane (for details, see ref. Bodmann et al 2022;Bodmann et al 2023a;Bodmann et al 2023b;Hess et al 2023;Zen Vasconcellos et al 2019Zen Vasconcellos et al 2021a;Zen Vasconcellos et al 2021b). Through a Riemann integration process, this complexification gives rise to a new scale factor, denoted as ln −1 (𝛽(t)), which characterizes a topological foliated spacetime structure.…”

“…In this work, based on a recently developed commutative formulation that combines the branch-cut cosmology, the Wheeler-DeWitt equation, and the Hořava-Lifshitz quantum gravity (Bodmann et al 2023a;Bodmann et al 2023b), considering a mini-superspace framework, we study the implications of an inflaton-type scalar field and the corresponding potential in the acceleration of the Universe.…”