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arXiv · 2605.13921

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

Christine C. Dantas
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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. A few qualitative comments on the possibility of St\"ackel-lifting a single-degree-of-freedom non-natural Hamiltonian in quantum cosmology is also presented.

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Christine C. Dantas. 2026-05-13. A No-Go Theorem for Quantum Cosmologies with Non-natural Hamiltonians. https://doi.org/10.1142/s0218271826410142

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