Principles of Discrete Time Mechanics: I. Particle Systems

TL;DR

This paper develops foundational principles for discrete time classical and quantum mechanics applied to particle systems, exploring variational principles, invariants, and quantum formulations including stroboscopic quantum mechanics and the quantised oscillator.

Contribution

It introduces a comprehensive framework for discrete time mechanics, integrating classical variational principles with quantum concepts like system amplitude and stroboscopic quantum theory.

Findings

Classical discrete time mechanics derived from Lagrangians and invariants.

Quantum discrete time mechanics constructed via system amplitude and stroboscopic methods.

Application to quantised inhomogeneous oscillator demonstrating the framework's utility.

Abstract

We discuss the principles to be used in the construction of discrete time classical and quantum mechanics as applied to point particle systems. In the classical theory this includes the concept of virtual path and the construction of system functions from classical Lagrangians, Cadzow's variational principle applied to the action sum, Maeda-Noether and Logan invariants of the motion, elliptic and hyperbolic harmonic oscillator behaviour, gauge invariant electrodynamics and charge conservation, and the Grassmannian oscillator. First quantised discrete time mechanics is discussed via the concept of system amplitude, which permits the construction of all quantities of interest such as commutators and scattering amplitudes. We discuss stroboscopic quantum mechanics, or the construction of discrete time quantum theory from continuous time quantum theory and show how this works in detail for the free Newtonian particle. We conclude with an application of the Schwinger action principle to the important case of the quantised discrete time inhomogeneous oscillator.