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Properties from discretization



From the time symmetry one can conclude that any solution must either diverge or loop through a repeated sequence that includes the initial conditions. The restriction to looping or divergence follows from the discreteness (there are a finite number of states) and causality (each new state is completely determined by the 2 (or N depending on the differencing scheme) previous states. The loop must include the initial state because of time symmetry. At any time one can reverse the sequence of the last 2 (or N) states and the entire history will be repeated in reverse. Thus any loop must include the initial conditions.

The time required for a given system to repeat an exact sequence of states based on the number of possibilities easily makes astronomical numbers appear minute. However if there are only a small number of stable structures and the loops do not need to be exact but only produce states close to a stable attractor then we can get a form of structural conservation law.

For large field values this model can approximate the corresponding differential equation to an arbitrarily high precision. As the intensity decreases with an initial perturbation spreading out in space a limit will be reached when this is no longer possible. Thus something like field quantization exists. Eventually the disturbance will break up into separate structures that move apart from each other. Each of these structures must have enough total energy to maintain structural stability. This may require that they individually continue to approximate the differential equation to high accuracy. Such a process is consistent with quantum mechanics in predicting field quantization. It differs from quantum mechanics in limiting the spatial dispersion of the wave function of a single photon. It suggests that the wave function we use in our calculations models both this physical wave function and our ignorance of the exact location of this physical wave function.

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