Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070
Date: Fri, 17 Dec 1993 04:05:32 GMT
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
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
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
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
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.