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Randomness???



Random Reality 26 February 2000.
Comments???

IF YOU COULD LIFT A CORNER of the veil that shrouds reality, what would you
see beneath? Nothing but randomness, say two Australian physicists.
According to Reginald Cahill and Christopher Klinger of Flinders University
in Adelaide, space and time and all the objects around us are no more than
the froth on a deep sea of randomness.
Perhaps we shouldn't be surprised that randomness is a part of the Universe.
After all, physicists tell us that empty space is a swirling chaos of
virtual particles. And randomness comes into play in quantum theory--when a
particle such as an electron is observed, its properties are randomly
selected from a set of alternatives predicted by the equations.
But Cahill and Klinger believe that this hints at a much deeper randomness.
"Far from being merely associated with quantum measurements, this randomness
is at the very heart of reality," says Cahill. If they are right, they have
created the most fundamental of all physical theories, and its implications
are staggering. "Randomness generates everything," says Cahill. "It even
creates the sensation of the 'present', which is so conspicuously absent
from today's physics."
Their evidence comes from a surprising quarter--pure mathematics. In 1930,
the Austrian-born logician Kurt Gödel stunned the mathematical world with
the publication of his incompleteness theorem. It applied to formal
systems--sets of assumptions and the statements that can be deduced from
those assumptions by the rules of logic. For example, the Greeks developed
their geometry using a few axioms, such as the idea that there is only one
straight line through any pair of points. It seemed that a clever enough
mathematician could prove any theorem true or false by reasoning from
axioms.
But Gödel proved that, for most sets of axioms, there are true theorems that
cannot be deduced. In other words, most mathematical truths can never be
proved.
This bombshell could easily have sent shock waves far beyond mathematics.
Physics, after all, is couched in the language of maths, so Gödel's theorem
might seem to imply that it is impossible to write down a complete
mathematical description of the Universe from which all physical truths can
be deduced. Physicists have largely ignored Gödel's result, however. "The
main reason was that the result was so abstract it did not appear to connect
directly with physics," says Cahill.
But then, in the 1980s, Gregory Chaitin of IBM's Thomas J. Watson Research
Center in Yorktown Heights, New York, extended Gödel's work, and made a
suggestive analogy. He called Gödel's unprovable truths random truths. What
does that mean? Mathematicians define a random number as one that is
incompressible. In other words, it cannot be generated by an algorithm--a
set of instructions or rules such as a computer program--that is shorter
than the number. Chaitin defined random truths as ones that cannot be
derived from the axioms of a given formal system. A random truth has no
explanation, it just is.
Chaitin showed that a vast ocean of such truths surrounds the island of
provable theorems. Any one of them might be stumbled on by accident--an
equation might be accidentally discovered to have some property that cannot
be derived from the axioms--but none of them can be proved. The chilling
conclusion, wrote Chaitin in New Scientist, is that randomness is at the
very heart of pure mathematics (24 March 1990, p 44).
To prove his theorem, Gödel had concocted a statement that asserted that it
was not itself provable. So Gödel's and Chaitin's results apply to any
formal system that is powerful enough to make statements about itself.
"This is where physics comes in," says Cahill. "The Universe is rich enough
to be self-referencing--for instance, I'm aware of myself." This suggests
that most of the everyday truths of physical reality, like most mathematical
truths, have no explanation. According to Cahill and Klinger, that must be
because reality is based on randomness. They believe randomness is more
fundamental than physical objects.
At the core of conventional physics is the idea that there are
"objects"--things that are real, even if they don't interact with other
things. Before writing down equations to describe how electrons, magnetic
fields, space and so on work, physicists start by assuming that such things
exist. It would be far more satisfying to do away with this layer of
assumption.
This was recognised in the 17th century by the German mathematician
Gottfried Leibniz. Leibniz believed that reality was built from things he
called monads, which owed their existence solely to their relations with
each other. This picture languished in the backwaters of science because it
was hugely difficult to turn into a recipe for calculating things, unlike
Newton's mechanics.
But Cahill and Klinger have found a way to do it. Like Leibniz's monads,
their "pseudo-objects" have no intrinsic existence--they are defined only by
how strongly they connect with each other, and ultimately they disappear
from the model. They are mere scaffolding.
The recipe is simple: take some pseudo-objects, add a little randomness and
let the whole mix evolve inside a computer. With pseudo-objects numbered 1,
2, 3, and so on, you can define some numbers to represent the strength of
the connection between each pair of pseudo-objects: B12 is the strength of
the connection between 1 and 2; B13 the connection between 1 and 3; and so
on. They form a two-dimensional grid of numbers--a matrix.
The physicists start by filling their matrix with numbers that are very
close to zero. Then they run it repeatedly through a matrix equation which
adds random noise and a second, non-linear term involving the inverse of the
original matrix. The randomness means that most truths or predictions of
this model have no cause--the physical version of Chaitin's mathematical
result. This matrix equation is largely the child of educated guesswork, but
there are good precedents for that. In 1932, for example, Paul Dirac guessed
at a matrix equation for how electrons behave, and ended up predicting the
existence of antimatter.
When the matrix goes through the wringer again and again, most of the
elements remain close to zero, but some numbers suddenly become large.
"Structures start forming," says Cahill. This is no coincidence, as they
chose the second term in the equation because they knew it would lead to
something like this. After all, there is structure in the Universe that has
to be explained.
The structures can be seen by marking dots on a piece of paper to represent
the pseudo-objects 1, 2, 3, and so on. It doesn't matter how they are
arranged. If B23 is large, draw a line between 2 and 3; if B19 is large,
draw one between 1 and 9. What results are "trees" of strong connections,
and a lot of much weaker links. And as you keep running the equation,
smaller trees start to connect to others. The network grows.
The trees branch randomly, but Cahill and Klinger have found that they have
a remarkable property. If you take one pseudo-object and count its nearest
neighbours in the tree, second nearest neighbours, and so on, the numbers go
up in proportion to the square of the number of steps away (click on
thumbnail graphic below). This is exactly what you would get for points
arranged uniformly throughout three-dimensional space. So something like our
space assembles itself out of complete randomness. "It's downright creepy,"
says Cahill. Cahill and Klinger call the trees "gebits", because they act
like bits of geometry.


They haven't proved that this tangle of connections is like 3D space in
every respect, but as they look closer at their model, other similarities
with our Universe appear. The connections between pseudo-objects decay, but
they are created faster than they decay. Eventually, the number of gebits
increases exponentially. So space, in Cahill and Klinger's model, expands
and accelerates--just as it does in our Universe, according to observations
of the recession of distant supernovae. In other words, Cahill and Klinger
think their model might explain the mysterious cosmic repulsion that is
speeding up the Universe's expansion.
And this expanding space isn't empty. Topological defects turn up in the
forest of connections--pairs of gebits that are far apart by most routes,
but have other shorter links. They are like snags in the fabric of space.
Cahill and Klinger believe that these defects are the stuff we are made of,
as described by the wave functions of quantum theory, because they have a
special property shared by quantum entities: nonlocality. In quantum theory,
the properties of two particles can be correlated, or "entangled", even when
they are so far apart that no signal can pass between them. "This ghostly
long-range connectivity is apparently outside of space," says Cahill. But in
Cahill and Klinger's model of reality, there are some connections that act
like wormholes to connect far-flung topological defects.
Even the mysterious phenomenon of quantum measurement can be seen in the
model. In observing a quantum system any detector ought to become entangled
with the system in a joint quantum state. We would see weird quantum
superpositions like Schrödinger's alive-and-dead cat. But we don't.
How does the quantum state "collapse" to a simple classical one? In Cahill
and Klinger's model, the nonlocal entanglements disappear after many
iterations of the matrix equation. That is, ordinary 3D space reasserts
itself after some time, and the ghostly connection between measuring device
and system is severed.
This model could also explain our individual experience of a present moment.
According to Einstein's theory of relativity, all of space-time is laid out
like a four-dimensional map, with no special "present" picked out for us to
feel. "Einstein thought an explanation of the present was beyond theoretical
physics," says Cahill. But in the gebit picture, the future is not
predetermined. You never know what it will bring, because it is dependent on
randomness. "The present is therefore real and distinct from an imagined
future and a recorded past," says Cahill.
Sand castles
But why can't we detect this random dance of the pseudo-objects? "Somehow,
in the process of generating reality, the pseudo-objects must become hidden
from view," says Cahill. To simulate this, the two physicists exploited a
phenomenon called self-organised criticality.
Self-organised criticality occurs in a wide range of systems such as growing
sand piles. Quite spontaneously, these systems reach a critical state. If
you drop sand grains one by one onto a sand pile, for instance, they build
up and up into a cone until avalanches start to happen. The slope of the
side of the cone settles down to a critical value, at which it undergoes
small avalanches and big avalanches and all avalanches at all scales in
between. This behaviour is independent of the size and shape of the sand
grains, and in general it is impossible to deduce anything about the
building blocks of a self-organised critical system from its behaviour. In
other words, the scale and timing of avalanches doesn't depend on the size
or shape of the sand grains.
"This is exactly what we need," says Cahill. "If our system self-organises
to a state of criticality, we can construct reality from pseudo-objects and
simultaneously hide them from view." The dimensionality of space doesn't
depend on the properties of the pseudo-objects and their connections. All we
can measure is what emerges, and even though gebits are continually being
created and destroyed, what emerges is smooth 3D space. Creating reality in
this way is like pulling yourself up by your bootstraps, throwing away the
bootstraps and still managing to stay suspended in mid-air.
This overcomes a problem with the conventional picture of reality. Even if
we discover the laws of physics, we are still left with the question: where
do they come from? And where do the laws that explain where they come from
come from? Unless there is a level of laws that explain themselves, or turn
out to be the only mathematically consistent set--as Steven Weinberg of the
University of Texas at Austin believes--we are left with an infinite
regression. "But it ceases to be a problem if self-organised criticality
hides the lowest layer of reality," says Cahill. "The start-up
pseudo-objects can be viewed as nothing more than a bundle of weakly linked
pseudo-objects, and so on ad infinitum. But no experiment will be able to
probe this structure, so we have covered our tracks completely."
Other physicists are impressed by Cahill and Klinger's claims. "I have never
heard of anyone working on such a fundamental level as this," says Roy
Frieden of the University of Arizona in Tucson. "I agree with the basic
premise that 'everything' is ultimately random, but am still sceptical of
the details." He would like to see more emerge from the model before
committing himself. "It would be much more convincing if Cahill and Klinger
could show something physical--that is, some physical law--emerging from
this," says Frieden. "For example, if this is to be a model of space, I
would expect something like Einstein's field equation for local space
curvatures emerging. Now that would be something."
"It sounds rather far-out," says John Baez of the University of California
at Riverside. "I would be amazed--though pleased--if they could actually do
what you say they claim to."
"I've seen several physics papers like this that try to get space-time or
even the laws of physics to emerge from random structures at a lower level,"
says Chaitin. "They're interesting efforts, and show how deeply ingrained
the statistical point of view is in physics, but they are difficult,
path-breaking and highly tentative efforts far removed from the mainstream
of contemporary physics."
What next? Cahill and Klinger hope to find that everything--matter and the
laws of physics--emerges spontaneously from the interlinking of gebits. Then
we would know for sure that reality is based on randomness. It's a
remarkable ambition, but they have already come a long way. They have
created a picture of reality without objects and shown that it can emerge
solely out of the connections of pseudo-objects. They have shown that space
can arise out of randomness. And, what's more, a kind of space that allows
both ordinary geometry and the non-locality of quantum phenomena--two
aspects of reality which, until now, have appeared incompatible.
Perhaps what is most impressive, though, is that Cahill and Klinger are the
first to create a picture of reality that takes into account the fundamental
limitations of logic discovered by Gödel and Chaitin. In the words of
Cahill: "It is the logic of the limitations of logic that is ultimately
responsible for generating this new physics, which appears to be predicting
something very much like our reality."

Jim Green
mailto:JMGreen@sisna.com
http://users.sisna.com/jmgreen