A question of discrete space-time, part 4

Dear members,

Part 1 proposed certain postulates for a discrete space-time model with
probabilistic motion.

Part 2 derived certain relativistic emergent properties from this
model.

Part 3 derived certain quantum mechanical emergent properties from this
model.

Unfortunately, none of these emergent properties of a discrete
space-time model are sufficiently different from continuum physics to
allow for a concrete test that will distinguish between the two.

We also have the issue that this discrete space-time model has no
explicit wave functions. If a unit particle is to have any wave-like
properties, they will need to be emergent properties derived from the
model's basic postulates. In a sense, this requires some sort of
resolution to the dilemma of the wave-particle duality.

The search continues...

If apparent continuum motion were really a series of discrete
probabilistic motions between 0% and 100% within a discrete space-time
lattice, then there should be an inherent dispersion in the actual path
of any object.

To illustrate the inherent dispersion in binomial probability, consider
the example of a fair coin. Since it is fair, it should land heads 50%
of the time, but if you were to flip it 100 times, it would not be
likely to come up heads precisely every other time. In the short run,
it will tend to have random runs of heads and tails, while in the long
run it will tend to land heads pretty close to 50% of the time. This
means that in the short run, the number of heads would be expected to
get ahead of the average part of the time, while at other times, the
number of heads would be expected to lag behind the average.

This inherent dispersion seems trivial when dealing with a fair coin,
but consider the probabilistic motion of a unit particle with a 50%
probability of motion at each moment of time, within a discrete
space-time lattice. Since the unit particle has a 50% probability of
motion, then over the span of several trillion moments, it should have
a world-line slope very close to -2. In the short run, however, the
world-line will tend to fluctuate, with greater or lesser dispersion
around the central "average" slope of -2.

As a visual aid, I will actually flip a coin 19 times and record the
resulting "world-line".

1 2 3 4 5 6 7 8 Position
01 |_o_|___|___|___|___|___|___|___|___|___|___|___|___|___|___|___|
02 |___|_x_|___|___|___|___|___|___|___|___|___|___|___|___|___|___|
03 |___|_x_|___|___|___|___|___|___|___|___|___|___|___|___|___|___|
04 |___|___|_x_|___|___|___|___|___|___|___|___|___|___|___|___|___|
05 |___|___|___|_x_|___|___|___|___|___|___|___|___|___|___|___|___|
06 |___|___|___|___|_x_|___|___|___|___|___|___|___|___|___|___|___|
07 |___|___|___|___|___|_x_|___|___|___|___|___|___|___|___|___|___|
08 |___|___|___|___|___|_x_|___|___|___|___|___|___|___|___|___|___|
09 |___|___|___|___|___|_x_|___|___|___|___|___|___|___|___|___|___|
10 |___|___|___|___|___|_x_|___|___|___|___|___|___|___|___|___|___|
11 |___|___|___|___|___|_x_|___|___|___|___|___|___|___|___|___|___|
12 |___|___|___|___|___|_x_|___|___|___|___|___|___|___|___|___|___|
13 |___|___|___|___|___|_x_|___|___|___|___|___|___|___|___|___|___|
14 |___|___|___|___|___|___|_x_|___|___|___|___|___|___|___|___|___|
15 |___|___|___|___|___|___|___|_x_|___|___|___|___|___|___|___|___|
16 |___|___|___|___|___|___|___|_x_|___|___|___|___|___|___|___|___|
17 |___|___|___|___|___|___|___|___|_x_|___|___|___|___|___|___|___|
18 |___|___|___|___|___|___|___|___|_x_|___|___|___|___|___|___|___|
19 |___|___|___|___|___|___|___|___|_x_|___|___|___|___|___|___|___|
20 |___|___|___|___|___|___|___|___|_x_|___|___|___|___|___|___|___|
Time

What we get is a random wavy line, rather than a straight line, due to
the probability and dispersion involved. Even with two or three
spatial dimensions, and regardless of whatever geometry we may choose
to associate with such a 2-D or 3-D lattice, this kind of a random walk
using probabilistic motion will generally result in a random wavy line,
as motion along one spatial dimension gets ahead, or lags behind
another spatial dimension. The resulting world line would not be
expected to be perfectly straight because of the probability and
dispersion involved.

At this point, I would like to make a distinction between the type of
randomness that might be expected from a stand-alone unit particle vs.
the type of randomness that might be expected from a world-line
considered as an entity in its own right.

If you remember in part 1 of this thread, it was postulated that: (#4)
The world-line of a particle is more important than the stand-alone
particle itself. At the time, this was less a reflection of the
seemingly arbitrary notion that a unit particle within a discrete
space-time lattice should not move more than one space per time, but
more to the notion that the world-line of a unit particle is an entity
/ structure in its own right, and therefore it should logically not
have any gaps.

In this context, a stand-alone particle might be expected to have a
truly random, jittery, Brownian motion type of dispersion, not affected
by anything that had gone on in the prior moments. A world-line
structure, on the other hand, might be expected to have a slightly more
smoothed sort of dispersion, where any given moment might be affected
by prior moments. This world-line structure might not be totally
random as in Brownian motion, nor totally straight as with infinitely
low dispersion, but may be somewhere in between.

If I had to choose between a stand-alone particle having a jittery sort
of Brownian motion type of random dispersion vs. a particle's
world-line structure with random, but smoothed dispersion, I would put
my money on the latter.

This is just a guess, of course, so we will record it (along with the
previous three guesses) as a postulate in order to keep moving forward,
and update the fourth postulate from this:

Postulate #4 (temporary) - The world-line of a particle is more
important than the stand-alone particle itself.

to this:

Postulate #4 (final) - The world-line structure of a particle is not
perfectly straight, but a random wavy line.

The average fluctuation length, calculated from trillions of such
fluctuations for a given particle, could depend on the properties of
the particle forming the world-line structure. Since higher
dispersions are related to increased entropy, and increase entropy is
related to lower energy states, then a unit particle with a lower
energy state might have a higher dispersion, and conversely, a unit
particle with a higher energy state might have a lower dispersion.
This would predict that the path of a high-energy particle should be
fairly direct, with a low amount of dispersion about its mean path,
i.e., small fluctuations resulting in a nearly straight path. The path
of a low-energy particle should be less direct, with a higher amount of
dispersion about its mean path, i.e., larger fluctuations resulting in
a more meandering path. Only a particle with infinitely low dispersion
would consistently travel in a perfectly straight line.

This fifth emergent property of discrete space-time, wherein particles
follow random wavy paths that depend on the energy of the particle,
might correspond with the wave theory of light in conventional
continuum physics. Conversely, if a photon were a particle that
traveled in a wavy path, giving it wave-like properties, that may help
explain the wave-particle duality of light. It may also help to
explain how light waves can pass through a vacuum, where there is no
material substance to carry the "wave".

None of the four previously proposed emergent properties of a discrete
space-time model were sufficiently different from continuum physics to
allow for a concrete test that will distinguish between the two, but
this fifth emergent property, where particles are predicted to follow
random wavy paths, instead of straight lines, seems to be substantially
different from current continuum physics.

Dear Igor, if you are reading - I believe that this prediction is
original, and could be the basis of a concrete test to distinguish
between discrete physics and continuum physics. Or we may have just
run into a brick wall.

Regards,
Ed Hanna

.

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