Classical and Quantum solutions in Scalar field cosmology via the Eisenhart lift and linearization

TL;DR

This paper demonstrates how the Eisenhart lift can be used to recast scalar-field cosmology in a geodesic framework, enabling linearization of the classical dynamics for an exponential potential through conformally flat Eisenhart metrics. In the quantum regime, the extended system yields a Wheeler-DeWitt equation whose solutions reveal new wavefunction structures tied to symmetry breaking in the lifted space. The work connects geometric methods with quantum cosmology, providing explicit classical solutions and new quantum solutions arising from conformal symmetries, and discusses potential extensions to include matter content. Overall, the approach offers a principled route to analyze both classical and quantum aspects of scalar-field cosmology via higher-dimensional geometric embeddings.

Abstract

This study introduces a novel approach for solving the cosmological field equations within scalar field theory by employing the Eisenhart lift. The field equations are reformulated as a system of geodesic equations for the Eisenhart metric. In the case of an exponential potential, the Eisenhart metric is shown to be conformally flat. By applying basic geometric principles, a new set of dynamical variables is identified, allowing for the linearization of the field equations and the derivation of classical cosmological solutions. However, the quantization of the Eisenhart system reveals a distinct set of solutions for the wavefunction, particularly in the presence of symmetry breaking at the quantum level.