A No-Go Theorem for Quantum Cosmologies with Non-natural Hamiltonians

TL;DR

This paper proves a no-go theorem showing that non-natural, non-quadratic Hamiltonians in quantum cosmology cannot be geometrized using the Eisenhart-Duval lift, highlighting a fundamental limitation in this approach.

Contribution

The authors establish a structural no-go theorem demonstrating the impossibility of geometrizing non-quadratic Hamiltonian dynamics in quantum cosmology via Eisenhart-Duval lifts.

Findings

Non-quadratic Hamiltonians cannot be embedded into null geodesics of higher-dimensional spacetime.

Loop Quantum Cosmology models with polymer modifications fall outside the ED lift framework.

The result reveals a fundamental limitation of metric geometrization in quantum cosmological models.

Abstract

The Eisenhart-Duval lift (ED) geometrizes classical dynamics by embedding their trajectories into null geodesics of a higher-dimensional Lorentzian spacetime. However, such a construction requires a natural Hamiltonian, that is, quadratic in the canonical momenta. As a consequence, mini-superspace cosmological models governed by non-natural Hamiltonians cannot admit an ED lift. Effective models in Loop Quantum Cosmology provide a concrete example: polymer-modified Hamiltonians become non-polynomial in the momenta and therefore fall outside the metric framework of the ED lift. We thus establish a kinematical no-go theorem: non-quadratic cosmological dynamics cannot be geometrized via ED constructions. Quantum-corrected bounce models therefore illustrate a structural limitation of metric geometrization within the ED framework.