A Generalization of Fermat's Principle for Classical and Quantum Systems

TL;DR

This paper extends Fermat's principle to complex dynamical systems in classical and quantum spaces, proposing a generalized principle based on metric distances and invariance of velocity, with specific validated cases.

Contribution

It introduces a generalized Fermat's principle applicable to high-dimensional dynamical spaces, supported by proofs for quantum and classical harmonic oscillator systems.

Findings

Fermat's principle holds in quantum and classical harmonic oscillator systems.

An exception occurs in a charged particle's configuration space in a magnetic field.

The principle is valid in a rotating frame for charged particles in magnetic fields.

Abstract

The analogy between dynamics and optics had a great influence on the development of the foundations of classical and quantum mechanics. We take this analogy one step further and investigate the validity of Fermat's principle in many-dimensional spaces describing dynamical systems (i.e., the quantum Hilbert space and the classical phase and configuration space). We propose that if the notion of a metric distance is well defined in that space and the velocity of the representative point of the system is an invariant of motion, then a generalized version of Fermat's principle will hold. We substantiate this conjecture for time-independent quantum systems and for a classical system consisting of coupled harmonic oscillators. An exception to this principle is the configuration space of a charged particle in a constant magnetic field; in this case the principle is valid in a frame rotating by half the Larmor frequency, not the stationary lab frame.