A question of discrete space-time.

John Baez Oct 22 1999, 3:00 am
Subject: Re: chronons?

<snip>

> There's certainly no simple reason why there *must* be a minimum
> unit of time, because there are mathematically consistent models
> where is not. However, none of these takes gravity and quantum
> mechanics into account, so one might speculate that for some reason
> unknown to us now, a full theory of physics must have some kind of
> temporal discreteness to it. Not many respectable people are studying
> such models these days - but one of them is 't Hooft.

Dear Dr. Baez (and others),

I've been studying discrete space-time, and would like to pose a rather
basic question, leading to progressively more elaborate questions.

It should be possible to start with the basic postulate that space and
time come in discrete units that cannot be divided (postulate #1), and
see if there are any natural consequences that can be predicted from
first principles, especially any concrete testable predictions.

If we take the simplest case - one dimension of space and one dimension
of time (Feynman's checkerboard), in order to easily discuss basic
principles without getting into issues of geometry, we could use the
following visual aid to represent a portion of discrete space-time with
64 "unit cells". If a "unit particle" originates at position-1 time-1
(represented here by the letter "o"), and it did not move position, it
would still move in time. The string of x's at position-1 time-2,
position-1 time 3, and ending up at position-1 time-8, would represent
the unit particle's world-line over the course of 8 "moments":

1 2 3 4 5 6 7 8 Position
1 |_o_|___|___|___|___|___|___|___|
2 |_x_|___|___|___|___|___|___|___|
3 |_x_|___|___|___|___|___|___|___|
4 |_x_|___|___|___|___|___|___|___|
5 |_x_|___|___|___|___|___|___|___|
6 |_x_|___|___|___|___|___|___|___|
7 |_x_|___|___|___|___|___|___|___|
8 |_x_|___|___|___|___|___|___|___|
Time

If the unit particle originates at position-1 time-1 and it moves one
position every moment of time, it would be in constant motion, and end
up at position-8 time-8 with a world-line slope of -1:

1 2 3 4 5 6 7 8 Position
1 |_o_|___|___|___|___|___|___|___|
2 |___|_x_|___|___|___|___|___|___|
3 |___|___|_x_|___|___|___|___|___|
4 |___|___|___|_x_|___|___|___|___|
5 |___|___|___|___|_x_|___|___|___|
6 |___|___|___|___|___|_x_|___|___|
7 |___|___|___|___|___|___|_x_|___|
8 |___|___|___|___|___|___|___|_x_|
Time

I've tried to keep the setup simple, and I've left some stuff out, but
here is my question:

A unit particle in constant motion within a discrete space-time lattice
would have a maximum speed limit of one position per moment, and this
should be a testable prediction -- a piece of evidence that space-time
may be discrete?

Of course, physicists already know that there is a maximum speed that
cannot be exceeded (the speed of light), so this experiment has already
been done, so to speak, and we know the results.

If space-time were a continuum, however, the gridline constraints of
the visual aid above would be removed, and there would be no reason, a
priori, that you couldn't take a ruler and draw a world line with a
slope of -1/2 or -1/3 or as arbitrarily flat as you wanted to,
representing any arbitrarily high speed.

In other words, it seems to me that using continuum physics, there is
no reason for there to be a maximum speed limit within the universe,
but that using discrete physics, there is a perfectly logical reason
for there to be one.

Does this make sense, or am I being overly simplistic?

PS. I have read (and enjoyed) Baez's crackpot index, and am trying like
the dickens to (mostly) avoid the 35 items listed.

Regards,
Ed Hanna

.

Loading