Rainbow metric from quantum gravity

TL;DR

This paper proposes a general mechanism where quantum cosmology leads to an energy-dependent 'rainbow' metric, causing particles of different energies to experience different spacetime geometries, without relying on a specific quantum gravity theory.

Contribution

It introduces a universal approach to derive energy-dependent metrics from quantum space-time models based on quantum field theory, independent of specific quantum gravity frameworks.

Findings

Quantum space-time can be replaced by a classical, energy-dependent metric.

Different energy quanta propagate on different metrics, akin to a refractive medium.

The approach is based solely on the wave-function of quantum space-time, not specific theories.

Abstract

In this letter, we describe a general mechanism for emergence of a rainbow metric from a quantum cosmological model. This idea is based on QFT on a quantum space-time. Under general assumptions, we discover that the quantum space-time on which the field propagates can be replaced by a classical space-time, whose metric depends explicitly on the energy of the field: as shown by an analysis of dispersion relations, quanta of different energy propagate on different metrics, similar to photons in a refractive material (hence the name "rainbow" used in the literature). In deriving this result, we do not consider any specific theory of quantum gravity: the qualitative behavior of high-energy particles on quantum space-time relies only on the assumption that the quantum space-time is described by a wave-function $\Psi_o$ in a Hilbert space $\mathcal{H}_G$.