How fractals offer a new way to see the quantum realm | Explained Premium

Fractal geometry is used to study quantum systems in non-integer dimensions. Fractal patterns are seen in nature, from fingerprints to lightning. Fractals are used in data compression, antenna design, and more. In the quantum realm, they are a potential source of information that are otherwise hard to access because of the limiting rules of quantum physics. This much has been elucidated in materials science research in which scientists have found fractal designs emerging entirely as a result of quantum effects.

Quantum physics is too weird for many people to understand, and part of the weirdness is due to some of its counter-intuitive features. For example, many quantum phenomena are bound by Heisenberg’s uncertainty principle, so it is impossible to know them with great certainty. According to this principle, we can’t obtain the information about a particle’s position, say, until we actively check for it.

This is unlike that of, say, a football that has been kicked: we can calculate its position based on the information that we get from Newton’s laws. In other words, gaining information about a particle means collapsing its wavefunction. The wavefunction is a mathematical object that contains information about the particle, and ‘collapsing’ it means forcibly modifying it in a way that yields that information.

Before we obtain the information about a particle’s location, however, it can be said to be in more than one place, and possibly in ‘contact’ with other particles even if they are physically quite far away.

Uncertainty is an inherent feature of all systems. It is independent of the precision or the accuracy with which the system is measured. It just is there, as an implicit element of the system’s existence. And it has forced physicists to find a practical approach to study quantum systems in ways that can circumvent the limitations it imposes.

One of the ways in which physicists have responded is via the so-called non-integer small dimensions, a.k.a. fractal dimensions. The dimensionality of a quantum system is an important thing to bear in mind when physicists study its properties. For instance, electrons in a one-dimensional system form a Luttinger liquid (not a liquid per se but a model that describes the electrons’ liquid-like behaviour); in a two-dimensional system, the particles exhibit the Hall effect (the conductor develops a side-to-side voltage in the presence of a top-to-bottom electric field and a perpendicular magnetic field).

The question obviously arises: How would a quantum system behave in non-integer or fractal dimensions?

Physicists use the fractal geometry approach to study quantum systems in dimensions like 1.55 or 1.58, or in fact anything between one and two dimensions.