# Is space-time a lattice?

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