From paul@mtnmath.UUCP Sat Dec 18 15:21:08 1993 Newsgroups: sci.physics.research From: paul@mtnmath.UUCP (Paul Budnik uunet!mtnmath!paul)

Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070 Date: Fri, 17 Dec 1993 04:05:32 GMT Lines: 103 It may seem unlikely that space-time is a lattice It contradicts fundamental assumptions in physics. 1. Special relativity can only be approximately true. 2. There must be absolute preferred directions. 3. There is an absolute frame of reference in effect an `ether'. One might think the experimental evidence against this possibility is overwhelming. This is obviously false. We can use a lattice to approximate a continuous model with arbitrarily accuracy by choosing the grid of the lattice sufficiently small. If space-time is a lattice its dimensions are likely to be on the order of the Plank time and distance scales. Thus we are talking about a grid much finer than can be used in a practical numerical simulation. Such a grid could approximate a continuous model to an extraordinarily high accuracy far beyond our current ability to detect. On the other hand it is possible that we have detected effects from our motion relative to an absolute frame of reference but have misinterpreted the evidence. The symmetry violations observed in weak interaction could be an example of this. For a model to be fully discrete one must require that any functions defined on the lattice be discrete. For example one should limit the function to integer values. Reasons for considering this class of models include the following. 1. There exist absolute time and distance *scales* in nature. 2. Quantization of observables is a natural result in such a model. 3. Current physical models suggest that the information content of any finite space-time region is finite. This suggest that continuous models are not needed to account for physical reality. 4. There are a number of properties of discrete space-time models that may account for some of the stranger effects observed in nature. The first three are obvious but the fourth requires some explanation. The natural starting point for a discrete model is the wave equation. This single equation models both the classical electromagnetic field and the Klein Godon equation for the photon. A fully discrete model would use a finite difference approximation to the wave equation, i. e. one that has been modified to map integers to integers by some mechanism for truncation or rounding. Models of this type have some interresting properties. 1. Any initial disturbance that is large enough and smooth enough will initially be an accurate approximation to a solution of the continuous differential equation. However the disturbance will not be able to diffuse indefinitely eventually it will break up into minimal stable dynamic structures. These structure may diverge from each other but will not individually diffuse over larger regions. 2. The nonlinearity can introduce chaotic like behavior. Over long periods of time even a confined (non diffusing system) will accumulate roundoff error in a seemingly random way. 3. In spite of the random chaotic behavior of solutions they will obey absolute conservation laws if the model is time reversible. This is because any initial disturbance must eventually diverge or loop through the same sequence of states. The time period for a repeated sequence will be astronomical if we consider all possible sequences. However there are likely to be a small number of stable solutions like attractors in chaos theory that most initial disturbances eventually converge to. The looping will be confined to a comparatively small number of sequences that are part of these `attractors'. 4. There are likely to be chaotic like transformations between these attractors that resemble the nonlinear changes that occur in nature (quantum collapse) that are not modeled by any existing theory. This is of course speculative. To develop this physical theory will require a new branch of mathematics that investigates the properties of discrete solutions to differential equations as mathematical entities in their own right i. e. not just as approximations to a continuous equation. Existing continuous models are unable to deal with some aspects of physical. reality. The nonlinear changes that appear to occur to the wave function in nature (quantum collapse) are not describable by any existing theory. They are relegated to interpretations. The EPR paradoxes derive from the combination of absolute conservation laws and statistical laws of observation. Discrete models suggest a resolution of these paradoxes may be in a deterministic model in which the information that enforces the conservation laws is stored in holographic fashion throughout the wave function and not `attached' to a particle at a particular position. Perhaps it is considerations like these that led Einstein to conclude that such models may be needed. I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures. In that case *nothing* remains of my entire castle in the air gravitation theory included, [and of] the rest of modern physics. -- Einstein in a 1954 letter to Besso, quoted from: "Subtle is the Lord", Abraham Pais, page 467. Paul Budnik