Everything around you – from tables and trees to distant stars and the great diversity of animal and plant life – is built from a small set of elementary particles. According to established scientific theories, these particles fall into two basic and deeply distinct categories: bosons and fermions.

Bosons are sociable. They happily pile into the same quantum state, that is, the same combination of quantum properties such as energy level, like photons do when they form a laser. Fermions, by contrast, are the introverts of the particle world. They flat out refuse to share a quantum state with one another. This reclusive behaviour is what forces electrons to arrange themselves in layered atomic shells, ultimately giving rise to the structure of the periodic table and the rich chemistry it enables.

At least, that’s what we assumed. In recent years, evidence has been accumulating for a third class of particles called ‘anyons’. Their name, coined by the Nobel laureate Frank Wilczek, gestures playfully at their refusal to fit into the standard binary of bosons and fermions – for anyons, anything goes. If confirmed, anyons wouldn’t just add a new member to the particle zoo. They would constitute an entirely novel category – a new genus – that rewrites the rules for how particles move, interact, and combine. And those strange rules might one day engender new technologies.

Although none of the elementary particles that physicists have detected are anyons, it is possible to engineer environments that give rise to them and potentially harness their power. We now think that some anyons wind around one another, weaving paths that store information in a way that’s unusually hard to disturb. That makes them promising candidates for building quantum computers – machines that could revolutionise fields like drug discovery, materials science, and cryptography. Unlike today’s quantum systems that are easily disturbed, anyon-based designs may offer built-in protection and show real promise as building blocks for tomorrow’s computers.

Philosophically, however, there’s a wrinkle in the story. The theoretical foundations make it clear that anyons are possible only in two dimensions, yet we inhabit a three-dimensional world. That makes them seem, in a sense, like fictions. When scientists seek to explore the behaviours of complicated systems, they use what philosophers call ‘idealisations’, which can reveal underlying patterns by stripping away messy real-world details. But these idealisations may also mislead. If a scientific prediction depends entirely on simplification – if it vanishes the moment we take the idealisation away – that’s a warning sign that something has gone wrong in our analysis.

So, if anyons are possible only through two-dimensional idealisations, what kind of reality do they actually possess? Are they fundamental constituents of nature, emergent patterns, or something in between? Answering these questions means venturing into the quantum world, beyond the familiar classes of particles, climbing among the loops and holes of topology, detouring into the strange physics of two-dimensional flatland – and embracing the idea that apparently idealised fictions can reveal deeper truths.

Bosons and fermions differ from one another in various ways. But if we want to understand anyons, the characteristic that interests us goes by the name of ‘quantum statistics’, which concerns the rules of engagement that dictate how particles behave when grouped together and distributed in single-particle states. Experimentally speaking, only two kinds of quantum statistics have been found so far – one for fermions and one for bosons. And each is defined by what happens when two identical particles swap places.

To unpack what this means, let’s first consider a contrasting case. In classical physics, the state of the system is just the set of numbers for quantities like position and momentum that lets you predict how an object like a baseball will move next. Imagine a bucket of baseballs, then. You could label them Ball One, Ball Two, and so on. If Ball One is at the top of the bucket and Ball Two is at the bottom, that’s one distinct arrangement. If you swap them, you have created a new, physically different arrangement. In classical physics, these two different situations correspond to two different states, and you can distinguish between them by observing how Ball Two and Ball One exchanged locations. In fact, you can ‘tag’ every classical particle – every baseball in our example – and follow its motion along a path.

So far so good. But in quantum mechanics the story is different.

When you have a collection of particles, quantum mechanics doesn’t describe them one by one. It assigns a quantum state to the entire system, which can take the form of a field-like entity known as a ‘wavefunction’ that expresses the probabilities associated with various observable processes. It encodes, for instance, all possible energies, momenta, positions and, importantly, how the particles populate the available single-particle states. You might think about single-particle states as describing one particle’s possibilities, and the general state as describing all particles and their correlations.

What is it about quantum theory that entails two basic categories of particles?

Take a system of identical particles like electrons, where the state is represented by a wavefunction. What is the relationship between the state of the system and another new state where we swapped the locations of particles One and Two? Is it like the baseballs in our bucket?

Here’s the rub. In quantum mechanics, unlike in classical physics, there is no experiment that can be conducted, and no observation that we can make, that will allow us to distinguish between these two systems. We can’t ‘tag’ an electron and follow it along a path. Although quantum mechanics can represent these two situations differently, exchanging identical quantum particles can’t change anything that you measure experimentally or observe, so our theoretical description of states should respect this symmetry. This idea is known as ‘permutation invariance’.

Moreover, if permutation invariance implied that there are only two allowed behaviours of the quantum state when you swap identical particles, then there are only two kinds of quantum statistics – with only two corresponding classes of particles.

So, given a system of identical particles, what is it about quantum theory that entails two basic categories of particles?

The standard story is as follows. Represent the state with a wavefunction and examine how it changes when we swap two particles as with the baseballs. Permutation invariance implies that this system’s quantum state can be affected in only two ways. Either the swap leaves the wavefunction unchanged, giving us bosons, or it reverses its sign (from plus to minus or vice versa), producing fermions. Mathematically, this difference is tracked by a built-in multiplier – the plus or minus sign in our case – that records how the state responds to a particle swap. If this multiplier could take on different intermediate values, it would mean various particle classes, but the claim is that this marker can take only two forms. When you swap two identical bosons, the wavefunction appears exactly the same. It’s like tossing two identical baseballs into the air and catching them in opposite hands – nobody........

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