A Fractal Quantum World?

By Mark Buchanan
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.

This is 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 quantum puzzles.

The Invariant Set Hypothesis

T.N. Palmer
arXiv:0812.1148v3

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.

The elements 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.

Combining 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.