The branch‐cut cosmology: A topological canonical quantum‐mechanics approach

Abstract

In this contribution, we sketch a branch‐cut quantum formulation of the Wheeler‐DeWitt equation analytically continued to the complex plane. As a starting point, we base our approach on the Hořava‐Lifshitz formulation of gravity, which employs higher spatial‐derivative terms of the spacetime curvature for renormalization reasons. Following standard procedures, the quantization of the Lagrangian density is achieved by raising the Hamiltonian, the dynamical variable , which represents the the branch‐cut complex scale factor, and the conjugate momentum pln to the category of operators. We arrive at a Schrödinger‐type equation with a non‐linear potential. Solutions are then obtained and discussed for different potential parameterizations. The results reinforce the conception of a quantum leap between the contraction and expansion phases of the branch‐cut universe, in good agreement with the Bekenstein criterion.

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:…”

“…where the variable change u(t) ≡ ln −1 [𝛽(t)], with du ≡ dln −1 [𝛽(t)], was introduced. K = K i𝑗 g i𝑗 represents in expression (3) the trace of the extrinsic curvature tensor K i𝑗 (Bodmann et al 2023a;Bodmann et al 2023b;Hess et al 2023):…”

“…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.…”

The branch‐cut quantum gravity with a self‐coupling inflation scalar field: Dynamical equations

“…The recombination era would have occurred approximately 380 million years after cosmic inflation when the Universe reached a temperature of approximately T∼0.26 eV. Branch cut gravity (BCG) corresponds to an extension of the ontological domain of General Relativity to the complex plane (Bodmann et al 2022;Bodmann et al 2023a;Bodmann et al 2023b;Einstein 1916Einstein , 1917Hess et al 2023;Zen Vasconcellos et al 2019;Zen Vasconcellos et al 2021a, 2021bZen Vasconcellos et al 2023), having been developed as a theoretical alternative proposal to the inflation model (Guth 1981(Guth , 2004. BCG is based on the mathematical technique of augmentation and the notion of closure and existential completeness (Manders 1989), which have proven to be extremely useful in both quantum mechanics (Aharonov & Bohm 1959;Dirac 1937;Wu et al 2021) and in pseudocomplex general relativity (pc-GR) Hess (2017); Hess & Boller (2020); Hess & Greiner (2009), with direct physical and cosmological manifestations.…”