Eisenhart lifts and symmetries of time-dependent systems

TL;DR

This paper explores the symmetries and quantization of time-dependent dissipative systems using Eisenhart lifts to Bargmann spacetimes, revealing new insights into their geometric structure and quantum behavior.

Contribution

It introduces a geometric framework for analyzing time-dependent systems via Eisenhart lifts, connecting symmetries, quantization, and cosmological models.

Findings

Eisenhart lift provides a geometric perspective on dissipative systems.

Reparametrization corresponds to conformal rescaling of the Bargmann metric.

Quantum mechanics emerges naturally from curved spacetime techniques.

Abstract

Certain dissipative systems, such as Caldirola and Kannai's damped simple harmonic oscillator, may be modelled by time-dependent Lagrangian and hence time dependent Hamiltonian systems with $n$ degrees of freedom. In this paper we treat these systems, their projective and conformal symmetries as well as their quantisation from the point of view of the Eisenhart lift to a Bargmann spacetime in $n+2$ dimensions, equipped with its covariantly constant null Killing vector field. Reparametrization of the time variable corresponds to conformal rescalings of the Bargmann metric. We show how the Arnold map lifts to Bargmann spacetime. We contrast the greater generality of the Caldirola-Kannai approach with that of Arnold and Bateman. At the level of quantum mechanics, we are able to show how the relevant Schr\"odinger equation emerges naturally using the techniques of quantum field theory in curved spacetimes, since a covariantly constant null Killing vector field gives rise to well defined one particle Hilbert space. Time-dependent Lagrangians arise naturally also in cosmology and give rise to the phenomenon of Hubble friction. We provide an account of this for Friedmann-Lemaitre and Bianchi cosmologies and how it fits in with our previous discussion in the non-relativistic limit.