The discrete time oscillator

The harmonic oscillator is perhaps the most important system under observation (SUO) in mathematical physics: its mathematics underpins the physics of quantum optics and relativistic quantum field theory (QFT), from free-particle fields to full-blown string theory. Remarkably, even the Coulombic interaction responsible for the structure of atoms in three dimensions can be modelled in terms of two coupled harmonic oscillators, each moving in a plane (Cornish, 1984). The continuous time (CT) harmonic oscillator has the great merit of being completely solvable from every theoretical direction in classic mechanics (CM) or quantum mechanics (QM).

The DT harmonic oscillator is also completely solvable classically and quantum mechanically. However, the range of dynamical behaviour is greater in DT than in CT, there being found three possible modes of behaviour in the former theory in contrast to the one mode in the latter. None of the three DT modes is strictly periodic except under very special circumstances when parameters and initial conditions take on special values. One of the three DT modes, the elliptic mode, is bounded, whilst the other two modes, the parabolic and hyperbolic modes, are unbounded. In DT QFT, the onset of the hyperbolic regime introduces a natural cutoff for particle momentum, which may be of potential significance as a natural regularization mechanism. This is discussed in later chapters.