A generalization of Fermat's principle for classical and quantum systems

作者：Elsayed Tarek A^*

来源：Physics Letters, Section A: General, Atomic and Solid State Physics , 2014, 378(44): 3205-3209.

DOI：10.1016/j.physleta.2014.09.032

• Fermat's principle
• Variational principles in classical and quantum dynamics
• Geometry of physical systems
• Time

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.

• 出版日期2014-9-12