From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: sci.physics
Subject: Re: Quantized Length
Summary: Einstein also thought that length may be quantized.
Date: 7 Sep 92 16:56:59 GMT
Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070

In article <1992Sep7.034433.19616@galois.mit.edu>, jbaez@riesz.mit.edu (John C. Baez) writes:
> ...
> The most naive way to obtain a smallest length scale, which goes back
> at least to the medieval Arabs (who loved geometric designs) is to
> assume that space is a lattice, that is, something like a
> checkerboard, on which particles can only move in discrete steps. 
> This idea is currently popular among fans of cellular automata, which
> are computers (or programs) that simulate worlds in which space is
> like this and time also occurs in discrete steps.  The problem with
> such theories, apart from the fact that there's no evidence for them,
> is that they have less symmetry than our world appears to have - we
> don't SEE any preferred "grid" in space, so it is rather upsetting to
> most physicists to assume there is one.  Computer scientists, however,
> who play with discrete stuff all day and get brainwashed :-), are
> often fond of this approach.

It may be only computer scientists who are fond of this approach, but
Einstein, at the end of his life also felt that something like this 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.

It is not that surprising that one would not see evidence for preferred
directions in space. When one does finite difference approximations to
differential equations one can get agreement with differential equations
to an arbitrarily high degree of accuracy by using a fine enough grid.
A grid with resolution on the order of the plank length is extremely fine!
Besides, are you so certain we have not seen such evidence. Is it not
possible the the asymmetry associated with weak interactions has
been misinterpreted and may be the first evidence of the absolute
space-time grid?

I think the approach that looks most promising is also the most simple
minded. It is not, strictly speaking, based on cellular automata.
Take a finite difference approximation to the wave equation and discretize
the field values, by rounding all results to integer values. The
fascinating thing about this simple minded idea is that so much of
quantum mechanics falls out directly at least at an intuitive level.
Photons must be quantized because an initial disturbance cannot spread
out indefinitely. There is a minimum intensity level at which wave like
behavior can be maintained.  There will be solutions to the difference
equation that are not solutions to the differential equation and some
of these will involve standing waves that are potentially models for
particles. When such structures interact they will transform into one
another in chaotic like nonlinear process that is a potential model for
quantum collapse. 

It is a huge gap to go from such intuitive possibilities to a physical
theory. Simulating three dimensional grids at the resolution of the
Plank distant is out of the question for any process that would have
macroscopic effects. Thus I don't expect many physicists to take this
approach seriously unless they are forced to consider radical new
possibilities. If tests of Bell's inequality begin to provide
evidence that quantum mechanics is false, as I think will happen
in the next few years, this may provide both the necessary motivation
and the experimental results that might make it possible to develop a
theory along these lines.

Paul Budnik