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A digital physics fantasy

There is no digital theory of physics. All efforts in this direction are in a primitive state. Developing a digital theory that makes macroscopic predictions is likely to be far more difficult than developing quantum mechanics was. For any digital theory that approximates the continuum will be impossible to simulate at a macroscopic scale with existing or foreseeable technology (see Section 7.1). Quantum mechanics was created by experimenters and theoreticians feeding each other. A more complete digital theory may require a trio of experimenters, theoreticians and engineers. The engineers will design the computers made possible by a deeper understanding of physics and thus create the simulation tools to further expand that understanding.

Although there is nothing close to a complete theory it is possible to construct an intuitive fantasy that illustrates how such a theory might account for the existing experimental record. At best the description that follows will turn out to be vaguely right in some respects. No doubt in others it will be precisely wrong. Still this description may be useful in giving physicists a sense of what might be possible. Quantum mechanics is so successful and so strange it has become hard to see how an alternative more complete theory might be possible. The hope is that the following description can convince at least a few that there could be an alternative.

Start with a discretized wave equation as described in Section 7.4. This is the relativistic Schrödinger equation for a single particle with zero rest mass. This is a very simple model that can be easily described on a half sheet of paper. I speculated in Section 8.7 about how this same model might lead to the relativistic Schrödinger equation for a single particle with rest mass.

Discrete systems can generate dynamically stable structures. These structures can transform into each other when perturbed something like a chaotic system moving between attractors. These structures are the particles of quantum mechanics. Reversibility without divergence produces a sort of structural conservation law. Reversibility can exist without absolute conservation at each time step. Thus individual observations can be pseudo random. But reversibility imposes global conservation laws. These laws are enforced in a combinatorially complex way that can only be understood with detailed knowledge of how the discrete states behave. From a macroscopic view the conservation laws are enforced with no visible mechanism to enforce them.

In classical physics higher dimensional models are required for deterministic systems in which we have only limited knowledge and can only model statistically. I think the same thing is true of quantum mechanics. The big difference with quantum mechanics are things like violations of Bell's inequality and quantum computing which are inconsistent with a simple local model in physical space.

Superposition seems to exist physically. Perhaps this can be explained by the chaotic like structural transformation of particles that can start any place. Keep in mind that these must be huge structures relative to the discretization of physical space if they approximate continuous structures to high accuracy. The same particle can start to transform at two or more different places simultaneously. Ultimately only one transformation can complete but for a while there is a physical superposition of states. Quantum collapse is the physical process of these structural transformations. These transformations have focal points in physical space and state space and the location of these focal points are the values observed experimentally. The uncertainty principle constrains how tightly focused these transformations can be in a given experiment. Ultimately spontaneous quantum collapse puts a limit on quantum computing. For problems that are complex enough you will only get a linear speed up. Bell's inequality is not violated at least not relative to the speed of causality in the discrete model. But the conservation laws are enforced by this extraordinarily complex combinatorial process that has reversibility at its core. One cannot understand how the correlations observed in existing experiments occur without a complete understanding of the discrete model.

There are two aspects of a fundamental theory that are generally considered to be independent. These are state evolution and initial conditions. Discrete models offer the possibility that a simple model could fully account for both. In a continuous solution to the wave equation an initial disturbance spreads our over an arbitrarily large volume becoming arbitrarily small in amplitude. A discrete model cannot do this. There is a lower limit to how small the amplitude can become. There are two possibilities as to how a fully discrete model that approximates the wave equation can behave in the long run. It can break up into independent components that separate from each other but do not themselves further disperse. This behavior could explain the quantization of electromagnetic energy. The other thing that may happen is that the wave front fills all of space with a residue of small but nonzero values.

These are not mutually exclusive alternatives. The diffusing wave function can do both. It can break up into components that have structural integrity and do not further diffuse and it can it also fill all of space with a small residue of nonzero values. The latter behavior can only happen in virgin space. A region of space can only become permeated with these residual values once.

This suggests the possibility of a divergent model that creates energy but only on the surface of an expanding sphere. It is possible that a very simply transformation of state model combined with a very simple initial state could account for all aspects of the universe. This would be the ultimate in simple explanations.

It is clearly possible to start with some simple discrete model that supports a Universal Turing Machine and program it so that eventually it will fill all of space with simulations that will grow in arbitrary complexity because for example they simulate every possible program. There is a simplest set of rules and initial conditions that will grow into arbitrary complexity. Perhaps it is precisely this simplest possible model that is the basis for all of physics.

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