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

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.

• The paper demonstrates that only Hamiltonians quadratic in momenta can be geometrized using the Eisenhart–Duval lift, establishing a no-go theorem for non-natural Hamiltonians in quantum cosmologies.
• It shows that polymer quantization in Loop Quantum Cosmology leads to non-polynomial momentum dependence, which fundamentally obstructs the ED geometric formalism.
• The work highlights a structural limitation in applying classical geometric methods to quantum-corrected cosmological models, suggesting the need for alternative frameworks such as Finsler geometry.

Structural Limitations of Eisenhart–Duval Lifts in Quantum Cosmology

Overview and Context

The paper "A No-Go Theorem for Quantum Cosmologies with Non-natural Hamiltonians" (2605.13921) rigorously addresses the kinematical obstruction to geometrizing cosmological dynamics beyond classical natural Hamiltonian systems. The Eisenhart–Duval (ED) lift is a geometric formalism, originally developed to encode the trajectories of classical mechanical systems, specifically those with Hamiltonians quadratic in canonical momenta, as null geodesics in extended Lorentzian spacetimes. This powerful geometric interpretation facilitates analysis of symmetries, integrability, and hidden structures by connecting dynamics to the properties of the underlying metric. However, the paper demonstrates that quantum cosmological models—particularly those arising from polymer quantization, as in Loop Quantum Cosmology (LQC)—yield effective Hamiltonians with non-polynomial momentum dependence, which invalidates the ED construction.

Eisenhart–Duval Construction and Natural Hamiltonians

The ED lift operates on classical Hamiltonian systems defined by a "natural" structure—namely, a kinetic term quadratic in the momenta and a potential term dependent on configuration variables. The procedure utilizes Brinkmann and Bargmann spacetimes, featuring a covariantly constant null vector field. The geodesic flow in the lifted spacetime projects onto the Hamiltonian flow in the original phase space.

For the ED lift to faithfully encode the dynamics, the essential requirement is:

$H(q, p) = \frac{1}{2} g^{ij}(q) p_i p_j + V(q)$

The geodesic Hamiltonian in the extended spacetime is invariably quadratic in all canonical momenta. Projection preserves the polynomial degree; if the original Hamiltonian is not quadratic, no Lorentzian metric can reproduce its flow via the null geodesic structure. This algebraic restriction is structural, not merely technical, and universal across dimensionalities of minisuperspace reductions.

Non-natural Hamiltonians in Quantum Cosmology

Quantum cosmological models, especially those arising from LQC via polymer modifications, result in Hamiltonians of the form:

$$H_{\mathrm{LQC}}(q, p) = A(q) \frac{\sin^2(\lambda p)}{\lambda^2} + V(q)$$

Such Hamiltonians have highly non-polynomial dependence on canonical momenta $p$, invalidating the natural form required by the ED lift. The paper asserts a no-go theorem: any single-degree-of-freedom, deparametrized minisuperspace Hamiltonian that is not quadratic in the canonical momentum cannot admit an ED lift. This applies structurally and cannot be remedied by local or perturbative expansions; the incompatibility is global and inherent.

Implications for Cosmological Models and Bounce Scenarios

Effective quantum cosmological models—those intending to encode singularity resolution via quantum corrections, such as non-singular bounce cosmologies—often violate the quadratic canonical structure in their Hamiltonians. The paper stresses that prior use of the ED lift for classical cosmological dynamics (e.g., FLRW models) is not extensible to polymerized systems in LQC. Consequently, the geometric arsenal afforded by Bargmann structures (interpretation of trajectories as null geodesics, conservation laws via Killing vectors, integrability via curvature properties) is unavailable for quantum-corrected cosmologies with non-natural Hamiltonians.

The broader implication is categorical: quantum-corrected dynamics that evade canonical quadratic form cannot be geometrized via the ED paradigm, delineating a structural boundary for the utility of metric-based geometric frameworks in cosmological modeling. This limits the interpretative power of geometric methods, including the connection to Schrödinger symmetry or hidden integrability.

Consideration of Finslerian Extensions and EMP Rewriting

The potential mitigation of the obstruction via Finsler geometry is acknowledged. Finsler frameworks accommodate non-quadratic dependence on momenta, but no universal Finsler analogue for the ED lift is recognized. Transcending Lorentzian geometry to Finslerian settings would necessarily depart from the structure of the Bargmann spacetime and would demand novel mathematical developments beyond current literature.

Similarly, reformulating quantum-corrected Friedmann equations as generalized Ermakov–Milne–Pinney (EMP) equations does not circumvent the theorem. While differential equations may allow for EMP-type rewriting, the ED lift is fundamentally tied to Hamiltonian structure, not to the autonomy or specific form of second-order ODEs.

Theoretical and Practical Impact

The absence of an ED lift for quantum-corrected cosmological Hamiltonians possesses practical consequences: geometric analysis of dynamical stability, symmetry, and integrability through lifted metric spaces is obstructed in LQC and analogous quantum cosmologies. The paper provides a rigorous demarcation of the scope of geometrization in gravitational and cosmological theory, highlighting the necessity for new geometric formalisms—or alternative interpretative tools—when confronting non-natural Hamiltonian systems.

Theoretically, this result underscores the structural rigidity of metric-based geometrization schemes, guiding future research toward either Finslerian generalizations or other non-metric structures capable of accommodating quantum-corrected dynamics. There is also potential for further mathematical investigation into the possibility of Finsler or Hamilton geometry-based lifts.

Conclusion

This paper defines a structural no-go theorem: Only cosmological dynamics governed by a Hamiltonian quadratic in canonical momenta can be geometrized via the Eisenhart–Duval construction. Quantum-corrected models in Loop Quantum Cosmology, with non-natural Hamiltonians, fall outside the scope of this geometric framework. The obstruction is algebraic and geometric, global rather than local, and marks a categorical limitation on the application of metric spacetime lifts to quantum cosmological models. Future advances may require the development of Finslerian or other generalized geometric structures to restore the interpretative tools otherwise provided by the ED lift (2605.13921).