[Paper Review] Origin of a classical space in guantum cosmologies

A. A. Kirillov, G. V. Serebryakov|ArXiv.org|Dec 27, 2000

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper investigates how vector fields influence the emergence of classical spacetime in quantum cosmologies, particularly in multidimensional gravity. Using a generalized Bianchi-I model with vector fields, it shows that classical space arises when the horizon size matches the smallest characteristic scale—either inhomogeneity or vector field-induced scale—while vector fields eliminate the initial compactification phase seen in vacuum models, stabilizing early expansion in all dimensions.

ABSTRACT

The influence of vector fields on the origin of a classical space in quantum cosmologies and on the possible compactification process in multidimensional gravity is investigated. It is shown that all general features of the transition between classical and quantum regimes of the evolution can be obtained within the simplest Bianchi-I model for arbitrary number of dimensions. It is shown that the classical space appears when the horizon size reaches the smallest of characteristic scales (the characteristic scale of inhomogeneity or a scale associated with vector fields). In multidimensional case the presence of vector fields completely removes the initial stage of the compactification process which takes place in the case of vacuum models.

Motivation & Objective

• To understand the origin of classical spacetime in quantum cosmologies, especially in the presence of vector fields.
• To examine how vector fields affect the transition from quantum to classical evolution in multidimensional gravity.
• To investigate whether vector fields can eliminate the initial compactification stage observed in vacuum models.
• To analyze the role of characteristic scales—inhomogeneity and vector field-induced scales—in determining the onset of classical spacetime.
• To establish a connection between quantum fluctuations and the formation of a stable, quasi-isotropic background geometry.

Proposed method

• Formulates a Hamiltonian action for a vector field coupled to gravity in n-dimensional spacetime using Planck units.
• Applies a generalized Kasner-like parametrization to decompose metric and momentum variables into anisotropic components.
• Neglects spatial derivatives near the singularity (BKL-like approximation), reducing the system to a homogeneous Bianchi-I model.
• Uses the Schrödinger equation to describe quantum evolution, with solutions expressed as superpositions of eigenstates ϕJ.
• Derives the probabilistic distribution P(y, τ) = |Ψ(y, τ)|2 to analyze quantum fluctuations and background metric expectation values.
• Analyzes moments of scale functions ⟨aM_i⟩ and their dependence on the anisotropy parameter Q, particularly in the limit g → 0.

Experimental results

Research questions

• RQ1 At what scale does classical spacetime emerge in quantum cosmologies with vector fields?
• RQ2 How do vector fields alter the initial evolution of multidimensional universes compared to vacuum models?
• RQ3 Does the presence of vector fields eliminate the initial compactification phase observed in vacuum quantum cosmologies?
• RQ4 What determines the stability of the background geometry during the quantum-to-classical transition?
• RQ5 How does the structure of the configuration space (billiard geometry) change with vector fields in n > 3 dimensions?

Key findings

• Classical spacetime emerges when the horizon size reaches the smallest characteristic scale—either inhomogeneity scale or vector field-induced scale (analogous to Jeans length).
• In the presence of vector fields, the initial compactification stage—observed in vacuum models—disappears, leading to monotonic expansion in all spatial directions.
• The anisotropy parameters Q remain non-negative (0 ≤ Q ≤ 1), preventing negative anisotropy and ensuring stable, quasi-isotropic evolution.
• Quantum fluctuations diverge as g → 0, with ⟨δ²⟩ ∼ (ln 1/g*)^{f(n)+1}, indicating instability of the average geometry in the quantum regime.
• The transition to a stable classical background occurs at t ∼ (L_n)^{J} k_n^{-1} (g ∼ 1), when anisotropy functions become small perturbations.
• In classical theory, the same background formation mechanism applies, with minor corrections due to the replacement f(n) → f(n) − 2 in fluctuation estimates.