A Generalization of Fermat's Principle for Classical and Quantum Systems
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.
Cite
@article{arxiv.1302.7185,
title = {A Generalization of Fermat's Principle for Classical and Quantum Systems},
author = {Tarek A. Elsayed},
journal= {arXiv preprint arXiv:1302.7185},
year = {2014}
}
Comments
Revised edition; accepted for publication in Physics Letters A