A computational TOE (theory of everything) requires a universe with
discrete, as opposed to continuous, space, time and state. Any
discrete model for the universe must approximate the continuous
wave equation to extremely high accuracy. The wave equation plays a
central role in physical theory. It is the solution to
Maxwell's equations and the Klein Gordon (or
Relativistic Schrödinger) Equation for a particle with zero
rest mass. The interaction of waves as fields are a primary means
of information transfer in the physical universe. Thus
understanding the properties of discretized finite difference
approximations to the wave equation is likely to play an important
role in developing a digital TOE. In addition the equations are
simple enough to be considered as a possible definition of a
computational TOE.
No discrete equation can model the continuous wave equation
exactly because the amplitude of a continuous wave decreases to an
arbitrarily small value as it propagates. As a result
discretization must introduce nonlinearities at something like the
Planck time and distance scales. Thus detailed predictions about
fundamental particles are difficult to make. However there are
emergent properties that mimic aspects of quantum mechanics. One
important result from these models is a possible explanation for
apparent experimental violations of Bell's
inequality. This explanation arises naturally from solving the
problem of reconciling quantum randomness with absolute
conservation laws.