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

**Authors:** Christine C. Dantas  
**arXiv:** [2605.13921](https://arxiv.org/abs/2605.13921)  
**Date:** 2026-05-13  
**Summary page:** https://www.summarizepaper.com/fr/arxiv-id/2605.13921v1/

## 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.