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

Physicists use the fractal geometry approach to study quantum systems in dimensions like 1.26 or 1.58

November 17, 2023 10:30 am | Updated 10:30 am IST

A fractal Mandelbrot set.

A fractal Mandelbrot set. | Photo Credit: Emeritus22 (CC BY-SA 4.0)

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.

What are fractal dimensions?

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.

Fractality is ubiquitous in nature, if also sometimes hidden from plain sight. A shape is fractal if it exhibits self-similarity, i.e. if parts of it at a smaller scale resemble parts at a larger scale. Such shapes can be easily produced by repeatedly modifying its edges using simple rules. Consider the Koch snowflake – a shape that begins as an equilateral triangle, and in each subsequent step, every side becomes the base for a new triangle. After many steps, a fractal snowflake appears.

The first four steps to building a Koch snowflake.

The first four steps to building a Koch snowflake. | Photo Credit: Wxs (CC BY-SA 3.0)

The higher the ‘value’ of a fractal’s dimension, the greater is its ability to fill space as its shape evolves. For example, the Koch snowflake has a fractal dimension of around 1.26.

What do fractals look like?

On the macroscopic scale, fractals can be seen as irregular, complex patterns at all scales and in all views, near or far. Some of the more remarkable examples of such patterns include the design of human fingerprints, the stumps of trees, in the shells of snails, the system of human veins, the network of rivers as seen from high up, the splitting of veins in a plant leaf, the edges of a snowflake, a bolt of lightning going branching off in different directions, the shapes of clouds, the mixing of liquids of different viscosity, the way tumours grow in the body, and so on.

There are fractals in the quantum realm as well. In a study published in 2019, for example, researchers from Switzerland and the U.S. used X-rays to study the magnetic properties of a compound called neodymium nickel oxide. They erased its magnetic order (the parts of its internal order imposed by magnetic fields) and then restored it. To their surprise, they found that parts of the material’s insides where the magnetisation was in the same direction – called magnetic domains – had a fractal arrangement. They also found that the domains reappeared in almost the same positions they were in before they were erased, as if the material had a memory. All these effects were due to the material’s quantum physical properties.

Another example of fractal behaviour at the microscopic scale is available in graphene – a single-atom thick sheet of carbon atoms linked to each other. In this setting, the surface density pattern of electrons has an almost fractal distribution.

What are the applications of fractality?

Historically, the first attempt to apply fractal analysis in physics was for Brownian motion – the rapid, random, zigzagging motion of small particles suspended in a liquid medium, like pollen in water. As such, the value of fractals is that they describe a new kind of order in systems that we may have otherwise overlooked. They pave the way to potential new insights from otherwise familiar shapes like lines, planes, and points, in the unfamiliar milieu of a space with non-integer dimensions.

Researchers have also used the concept of fractality in data compression, such as to reduce the size of an image when storing it, and to design more compact antennae without compromising their performance. Some have also used fractality to study patterns in galaxies and planets and, in cell biology, to make sense of some bacteria cultures. Fractal geometry has also found applications in chromatography and ion-exchange processes, among others.

Fractals are rooted in geometry but – like the fractal growth of branches on trees – they have far-reaching implications, more so as they interact with different natural processes in a variety of settings. There are self-similar structures around us that become increasingly complex with time. You just need to slow down and look closer, and you might just glean some information that brings some quantum mystery into focus.

Qudsia Gani is an assistant professor in the Department of Physics, Government College for Women, Srinagar.

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