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There are many reasons to suspect that at the most fundamental level nature is discrete. These go way beyond the philosophical arguments I have already described for the nonexistence of infinite or continuous structures. Perhaps the most compelling is finiteness in . The information content of any finite region containing a finite amount of energy is finite. It would be an extraordinary extravagance to use continuous structures to embed finite information. Everything we know about nature leads to the conclusion that she abhors such extravagance.
If one assumes nature at the most fundamental level is both discrete and simple one is led to consider models in which space and time are described by discrete points connected in a topological structure such as a grid. Although we visualize this as a structure in physical space it is important to understand that the topological arrangement of points does not exist in space but is the basis for space. Spatial relations are determined by the topological connection of points and not vice versa. Further the metric or distance function that we observe is not just a function of these topological connections. If relativity is even approximately true the distance we observe between two points must be a function of this topological structure and the physical state defined on the topological structure.
If that physical state is discrete it can be described be a set of integers or a single integer at each point. The laws of physics would then be reducible to mathematical rules for describing a future state in terms of earlier states. If these rules are to be simple they must be local. That is the state of a point at some time is determined by the state at a small number of previous time steps and at a small number of previous locations that are in the immediate neighborhood of this point. Models of this type are discrete field models . They are the discrete analogue of the continuous field models that Einstein spent most of his life trying to reduce physics to.
If one suspects that physics can be reduced to a discrete field model then one is led almost inevitably to a set of rules for describing how that field evolves. The wave equation is universal in physics describing both the evolution of the electromagnetic field and quantum mechanical wave function for particles with no rest mass. Discretized rules for describing state evolution that approximates the classical wave function are known as discretized finite difference approximations to the wave equation. The remarkable thing about discretized finite difference approximations to the wave equation is that they may account for all of physics including all the properties of all the fundamental particles. A model that one can easily right down on a half a sheet of paper may be sufficient to explain all of physics.
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