Quantum Mechanics from General Relativity and the Quantum Friedmann Equation

TL;DR

This paper establishes that the nonlinear first Friedmann equation can be recast as a linear differential equation, with redshift interpreted as the first-order WKB expansion of a quantum cosmological equation. It introduces a quantum scale factor that regularizes singularities and governs quantum corrections to cosmic dynamics, and shows that the radiation era yields a dual description via Seiberg–Witten theory, linking cosmology to resurgence and complex metrics. The framework leverages Schwarzian derivatives and connects to quantum stationary Hamilton–Jacobi theory, suggesting a unified GR–QM perspective with potential applications to multi-fluid cosmologies and quantum gravity formalisms such as Wheeler–DeWitt. The results also illuminate potential ties to uniformization theory (Γ(2)) and to broader contexts like JT gravity and SYK models, highlighting deep geometric structures underlying cosmological evolution. Overall, the work broadens the conceptual bridge between gravity and quantum mechanics in cosmology, with concrete linear, quantum, and dual descriptions of cosmic dynamics.

Abstract

We demonstrate that the recently introduced linear equation, reformulating the first Friedmann equation, is the first-order WKB expansion of a quantum cosmological equation. This result shows a deeper underlying connection between General Relativity and Quantum Mechanics, pointing towards a unified framework. Solutions of this equation are built in terms of a scale factor encapsulating quantum effects on a free-falling particle. The quantum scale factor reshapes cosmic dynamics, resolving singularities at its vanishing points in several cases of interest. As an explicit example, we consider the radiation-dominated era and show that the quantum equation is dual to the one in Seiberg-Witten formulation, recently applied to black holes, and incorporates resurgence phenomena and complex metrics, as developed by Kontsevich, Segal, and Witten. This links to the invariance of time parametrization under $Γ(2)$ transformations of the dual wave function.

Figures (1)

• Figure 1: $a_k$ versus $a_{qk}$ for the radiation solution.