Fermions From a Bosonic Medium? An Open Problem

The problem, in one paragraph

In our three-dimensional world, every particle belongs to one of two families. Bosons — photons, for example — are happy to pile into the same state; a laser beam is billions of photons doing exactly that. Fermions — electrons, protons, neutrons — refuse: no two can occupy the same state. That refusal is why atoms have shells, and a large part of why matter takes up room.

This framework pictures matter as patterns of fields living on a discrete substrate (see Occupied or Empty). The trouble, noted at the end of entry 01, is that the simplest version of that picture — small oscillations of a simple field — produces bosons. A framework that cannot produce electrons cannot produce chemistry. So the question is sharp: can a medium whose basic ripples like to share still give rise to particles that refuse?

1 · Why spin is the real question

Which family a particle belongs to is tied to its spin. Particles with integer spin (0, 1, 2, …) are bosons; particles with half-integer spin (½, 3/2, …) are fermions. In relativistic quantum theory this link is a theorem, not a coincidence — the spin–statistics theorem (Fierz, 1939; Pauli, 1940).

So, in a relativistic theory, “can it make fermions?” becomes “can it make spin-½?”. And spin-½ has a strange signature, met in entry 01: turn a spin-½ object through a full 360° and its wave changes sign; only after 720° is it truly back to where it began. In a relativistic theory, anything that produces fermions has to produce that 720° behaviour.

2 · Route one: twisted configurations of a field

In 1962 Tony Skyrme proposed something that sounded backwards. Take a field whose ordinary ripples are bosons — in his case, the field of particles called pions. Now look not at its small ripples but at a twisted configuration: a region where the field is wound up in a way that cannot be smoothed out without tearing it. Such a configuration is stable for topological reasons, the way a knot in a rope stays a knot. It behaves like a particle in its own right.

Skyrme suggested these twisted configurations — now called skyrmions — could describe protons and neutrons. Later work showed that such an object, made entirely from a boson field, can carry spin-½ and behave as a fermion (Finkelstein and Rubinstein, 1968; Witten, 1983). The twist makes the 720° behaviour possible; whether the object actually is a fermion is settled by an extra ingredient (in nuclear physics, the odd number of quark “colours”).

How seriously to take this:

• The mathematics is established. It is not in doubt that a twisted configuration of a boson field can be a fermion when the field has the right structure.
• The physics is an approximation. The Skyrme model is an effective description of nuclear physics. It reproduces properties of the proton and neutron to within roughly 30% (Adkins, Nappi and Witten, 1983) — useful, but not the fundamental theory.
• Twisted field configurations are real objects. Magnetic skyrmions — tiny stable whirls in the magnetisation of certain materials — were detected by neutron scattering (Mühlbauer and colleagues, 2009) and imaged directly a year later (Yu and colleagues, 2010). They are not fermions, but they show that topologically stable twists are not just mathematics.

3 · Route two: fermions that emerge on a lattice

The second route comes from the physics of materials, where particles often emerge as collective patterns of many simpler parts.

• In one dimension it is exact. A chain of tiny magnets (spins) can be rewritten, exactly, as a chain of fermions (Jordan and Wigner, 1928). Nothing is approximated: the same system, described two ways. (The fermions are non-local in terms of the original spins, and in one dimension the difference between the two families is partly a matter of description.)
• In two and three dimensions it is a design principle. There are lattice models built entirely from ingredients that are not themselves fermions, whose collective excitations are fermions — for example the lattice spin models of Levin and Wen (2003, 2005) and Kitaev’s honeycomb model (2006). In such models the emergent fermions come together with an emergent force field of a particular kind (a gauge field).

These results show that “the ingredients are bosonic” does not settle “the particles are bosonic”; the structure of the whole can decide it. But each of these models relies on specific local rules on a fixed lattice or network in which every site has a small, fixed set of neighbours, with carefully chosen rules. They do not arise from just any arrangement.

4 · Checking both routes against this framework

This framework’s current working choice for the substrate is a random scattering of discrete elements through spacetime, with a fixed average density — about one element per Planck-sized volume of spacetime — and no regular pattern. How do the two routes fare?

One premise is not ruled out. The spin–statistics theorem assumes that the laws look the same to every observer moving at constant speed (Lorentz invariance). A regular grid picks out a preferred frame; a random scattering of points through spacetime does not — no individual scattering selects a frame (Bombelli, Henson and Sorkin, 2009) — a statement about the substrate’s structure, not yet about any dynamics on it. The framework chose a random substrate mainly for a different reason: small violations of this symmetry at the smallest scales tend to leak into everyday physics, where they are excluded (Collins and colleagues, 2004). Either way, this removes one obstacle to the theorem. It does not show that the theorem’s other premises — local fields and microcausality — hold here; on such a substrate they would first have to emerge, and even defining fermion fields on a substrate of this kind is still largely open (Surya, 2019).

Route one runs into discreteness. Topological protection — “a knot cannot be undone without tearing” — is strictly true only for a smooth, continuous field. On a discrete substrate a twist can, in principle, unwind through the discreteness itself. It is protected only approximately, and only if the twisted configuration is much larger than the discreteness scale set by the substrate’s density. Whether that protection is good enough, and what field on this substrate would carry the right kind of twist, has not been worked out. Note also that the requirement itself — a winding number and a non-contractible rotation loop — can only be written down once a continuous field has emerged from the substrate. At that level it is the ordinary soliton argument and says nothing about this substrate in particular; a substrate-level definition of “twist” does not yet exist.

Route two runs into the same feature that helps the premise. The lattice models that produce fermions need each site to have a few fixed neighbours. A random substrate that avoids a preferred frame does have a frame-independent notion of nearest neighbour — a direct causal link — but every element has unboundedly many of them, spread along its light cone, rather than the small, fixed set the lattice models need. The property that keeps the theorem in play makes this route harder. Whether any rule can work on such a substrate is not known.

And any route must respect what is measured. Experiments find no internal structure in the electron down to scales far smaller than an atomic nucleus. A twisted-configuration electron would have to be smaller than that — possible in principle, since the substrate’s discreteness scale lies vastly below it. It would also have to be extraordinarily light for its size: naively, a configuration smaller than about 10⁻¹⁸ m carries an energy of order hundreds of GeV, while the electron’s mass corresponds to 0.511 MeV. Explaining that gap would be part of the problem.

Where this leaves the idea

Honest status. The useful outcome of this entry is a sharper problem, not a solution. Before, the framework simply assumed that its matter obeys the exclusion principle. Now the requirement can be written down: the substrate must carry a field whose stable patterns either have the right topological twist, robust against the substrate’s discreteness, or emerge collectively with fermionic behaviour — and the result must be as small and as light as a real electron. Until one of those is shown, the framework cannot claim to produce electrons — and that limitation applies to every page on this site that builds on it.

Last updated: October 4, 2026