A Fractal Quantum World?
New Scientist, March 30, 2009
Edited by Andy Ross
AQuantum theory just seems too weird to believe. Tim Palmer argues that we
need some ideas from the science of fractals. With the mathematics of
fractals, the puzzles of quantum theory may be easier to understand.
Palmer says that gravity is the only fundamental physical process that can
destroy information. A star contains an enormous amount of information, but
if the star collapses under its own gravity to form a black hole, almost all
of that information vanishes.
As a system loses information, the
number of states you need to describe it diminishes. Finally the system
reaches a point where no more states can be lost. This subset of states is
known as an invariant set. Once a state lies in this subset, it stays in it.
Because black holes destroy information, Palmer suggests that the
universe has an invariant set. Complex systems are affected by chaos, and
the invariant set of a chaotic system is a fractal. Fractal invariant sets
have unusual geometric properties.
The invariant set of the universe
may have a fractal structure. If the universe is trapped in this subset of
all possible states, it might help to explain why the universe at the
quantum level seems so bizarre.
Quantum theory seems to say that
particles do not have any properties before they are measured. Instead,
quantum systems have properties only in the context of the particular
experiments performed on them.
In 1967, Simon Kochen and Ernst
Specker published a theorem. Say you choose to measure different properties
of a quantum system, such as the position or velocity of a quantum particle.
Each time you do so, you will find that your measurements agree with the
predictions of quantum theory. Kochen and Specker showed that it is
impossible to conceive a hypothesis that can make the same successful
predictions as quantum theory if the particles have pre-existing properties,
as would be the case in classical physics.
Many physicists conclude
that either you have to abandon the existence of any kind of objective
reality, and say instead that objects have no properties until they are
measured, or you have to accept that distant parts of the universe share a
spooky connection that allows them to share information even when they are
too far apart for light signals.
Palmer suggests a third possibility:
that the kinds of experiments considered by Kochen and Specker are simply
impossible to get answers from and hence irrelevant. If the invariant set
contains all the physically realistic states of the universe, any state that
isn't part of the invariant set cannot physically exist.
where the fractal nature of the invariant set matters. If a hypothetical
universe does not lie on the fractal, then that universe is not in the
invariant set and so it cannot physically exist.
Due to the spare and
wispy nature of fractals, even subtle changes in the hypothetical universes
could cause them to fall outside the invariant set. In this way, Palmer's
hypothesis may help to make some sense of quantum contextuality.
Palmer believes that quantum theory makes only statistical predictions
because it is blind to the intricate fractal structure of the invariant set.
Palmer is hoping his idea can account for quantum uncertainty and other
The Invariant Set Hypothesis
The Invariant Set Hypothesis proposes that states of physical reality belong
to, and are governed by, a non-computable fractal subset I of state space,
invariant under the action of some subordinate deterministic causal dynamics
D. The Invariant Set Hypothesis is motivated by key results in nonlinear
dynamical systems theory, and black hole thermodynamics.
of a reformulation of quantum theory are developed using two key properties
of I: sparseness and self-similarity. Sparseness is used to relate
counterfactual states to points not on I thus providing a basis for
understanding the essential contextuality of quantum physics. Self
similarity is used to relate the quantum state to oscillating coarse-grain
probability mixtures based on fractal partitions of I, thus providing the
basis for understanding the notion of quantum coherence.
these, an analysis is given of the standard mysteries of quantum theory:
superposition, nonlocality, measurement, emergence of classicality, the
ontology of uncertainty and so on. It is proposed that gravity plays a key
role in generating the fractal geometry of I.
Since quantum theory
does not itself recognize the existence of such a state-space geometry, the
results here suggest that attempts to formulate unified theories of physics
within a quantum theoretic framework are misguided; rather, a successful
quantum theory of gravity should unify the causal non-Euclidean geometry of
space time with the atemporal fractal geometry of state space.