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Discretizing the wave equation

The simplest model for a local deterministic physical theory is a field function i.e. a function defined at each space time coordinate whose evolution is determined by the previous field values in the immediate neighborhood. The starting point for any theory like this must be the classical wave equation 

for that equation is universal in physics describing both electromagnetic effects and the relativistic quantum wave function (Klein Gordon equation  ) for the photon.

By `discretized' I mean an equation that is modified to map integers to integers. A modification is required because there is no finite difference approximation to the wave equation that can do this. The universality of the wave function requires that any discrete model for physics approximates this continuous model to extraordinary accuracy. Discretizing the finite difference equation adds a rich combinatorial structure that has a number of properties that suggest quantum mechanical effects. Perhaps the most obvious is that an initial disturbance cannot spread out or diffuse indefinitely as it does with the continuous equation. It must either diverge and fill all of space with increasing energy or break up into independent structures that will continue to move apart, i.e., become quantized.

We describe how to approximate the wave equation with a discretized finite difference equation. Let P be defined at each point in a 4 dimensional grid. To simplify the expression for we will adopt the following conventions. Subscripts will be written relative to and will be dropped if they are the same as this point. Thus is at the same position in the previous time step. is at the same time step and z coordinate and one position less on both the x and y axes.

The wave equation is approximated by the difference equation:

The difference equation discretizes space and time but not the function defined on this discrete manifold. The simplest approach to discretizing the function values is to constrain them to be integers. This requires either that be an integer or that some rounding scheme be employed that forces the product involving to be an integer. The former is not possible since it does not allow for solutions that approximate the differential equation.

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