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Unique properties of discrete models

The finite difference equation can only approximate the continuous differential equation for a limited number of time steps. Eventually it will behave quite differently. In the continuous case the amplitude of the wave spreads out at ever greater distances decreasing in amplitude to arbitrarily small levels. In the discrete case there is a limit beyond which this cannot happen.

Discrete systems either diverge or go through a repeating sequence of states. A system that does not diverge has only a finite number of states and therefore must at some point loop through the same sequence. However even small discrete models have an enormous number of states. One hundred integers each with a range of 100 allows for $100^{100}$ possible states. The time before one must repeat a state in even a very small system can easily exceed the age of the universe.

For a finite difference equation to be a candidate for a physical model it must form stable dynamic structures that go through a repeated sequence of similar states. Such structures could lead to a more complete theory of the fundamental particles of physics. They would exhibit chaotic like behavior.7.9The truncation function $T$ defined in Table 7.1 is nonlinear. This could induce chaotic like effects. It is plausible that some fully discrete approximations to the wave equation would lead to a variety of dynamically stable structures. These are structures that repeat a similar but not necessarily identical sequence of states. They would be relatively stable in that small perturbations would not significantly affect their average behavior. An initial burst of energy would break up into such structures. These structures could transform into one another under appropriate conditions and with constraints on what transformations were allowed. The chaotic like randomness of their behavior would be fully deterministic but knowing the exact integer value at every point in space would in most circumstances be impossible. It is only these exact values that would support fully deterministic predictions.

The model in Table 7.1 has exact time symmetry. That imposes a strong conservation law that puts limits on possible transformations It implies that all transformations are reversible. Swap the values at two successive time steps and the previous sequence of events will reoccur in reverse order.

This is all speculative but such structures could provide an explanation for wave particle duality and support a physical wave function collapse. There are many difficulties with this possibility. One aspect of quantum mechanics, quantum entanglement (discussed in Section 7.11), contradicts any model of this class. We discuss the experiments to test this in Section 8.6.

The great difficulty with these class of models is their enormous complexity. The basic rules are simple but any attempt to model even the smallest of fundamental particles would require an enormous simulation. Of necessity mathematicians and physicists work with mathematical models they can solve. Nature is under no such constraint. We further discuss the possible behavior of discrete models in Section 8.7.

One objection to this class of models is the crude truncation toward zero. No doubt if such models are the way nature works there is something more elegant involved. Edward Fredkin has pursed a potentially more elegant approach in using Cellular Automaton as models[54]. The problem with cellular automaton is simulating the wave equation that can grow to very large amplitudes. There are solutions but they may not be elegant either.


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