Quantum Field Theory of Classically Unstable Hamiltonian Dynamics

TL;DR

This paper develops a quantum field theoretical framework for classically unstable Hamiltonian systems by linking geodesic deviation to a parametric harmonic oscillator, enabling analysis of fluctuations and thermodynamic implications.

Contribution

It introduces a novel quantum field theory approach to analyze classical instability using geodesic deviation and oscillator quantization.

Findings

Geodesic deviation equations are unitarily equivalent to parametric harmonic oscillators.

Quantization of these oscillators models quantum fluctuations in unstable systems.

Unstable dynamics are associated with calculable quantum fluctuations and potential thermodynamic effects.

Abstract

We study a class of dynamical systems for which the motions can be described in terms of geodesics on a manifold (ordinary potential models can be cast into this form by means of a conformal map). It is rigorously proven that the geodesic deviation equation of Jacobi, constructed with a second covariant derivative, is unitarily equivalent to that of a parametric harmonic oscillator, and we study the second quatization of this oscillator. The excitations of the Fock space modes correspond to the emission and absorption of quanta into the dynamical medium, thus associating unstable behavior of the dynamical system with calculable fluctuations in an ensamble with possible thermodynamic consequences.