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Space and the material world could be created
out of nothing but noise. That's the startling conclusion of a new theory
that attempts to explain the stuff of reality.
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 G -- del
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 G -- del 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 G -- del'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
G -- del'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 G -- del's work, and made a suggestive analogy. He called G -- del'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, G -- del had concocted
a statement that asserted that it was not itself provable. So G -- del'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 recognized 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 neighbors in the tree, second nearest
neighbors, 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. Tree roots: pseudo-objects
link up into random trees, which link into ever larger structures. The
hierarchy of neighbors is just like that of points in 3D space
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 Schr -- dinger'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 behavior
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 behavior 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 G -- del 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."
http://www.newscientist.com/features/features.jsp?id=ns22273
New Scientist magazine,
26 February 2000
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