Is there a smallest length?

Probing a smaller distance requires more energy, and enough energy in a small enough region makes a black hole. Beyond a point, trying to look closer makes the region you are looking at larger.

That argument does not depend on any particular theory of quantum gravity, which is why the Planck length is taken seriously as a boundary even though nothing is known about what lies at it.

If space were grainy, light of different energies might travel at very slightly different speeds. Over a few metres this is undetectable; over billions of light years, from a gamma-ray burst, it accumulates.

Observations of distant bursts have found photons of very different energies arriving within seconds of each other after travelling for billions of years, which constrains the simplest versions of energy-dependent speed at or beyond the Planck scale. Not everyone reads the bursts that way: one group finds a regularity in the Fermi delays that it takes as a variation at a few hundredths of the Planck scale, while the LHAASO observatory’s analysis of the brightest burst yet recorded puts the bound for the linear case at about eight times the Planck energy.

That is a real experimental constraint on models of quantum gravity, obtained without any accelerator. It pushes the simplest energy-dependent models past the scale they were built around. That is not the same as closing them out, because a single burst cannot separate a delay picked up on the way from one the source imposed.

In 2009 a gamma-ray burst whose light had been travelling for about seven billion years produced photons across a range of energies, and a space telescope recorded them.

A photon of 31 billion electronvolts arrived within a second of the lower-energy ones. Over that distance, that puts a limit on a linear energy-dependent variation in the speed of light at or beyond the Planck scale itself.

A single burst, about two seconds long, constraining a class of models of the structure of space at a scale no accelerator can reach.

Some approaches describe space as built from discrete elements with a smallest area and volume. Others keep space continuous and change what distance means at short scales. Others treat spacetime as emerging from something more fundamental.

These are not variations on one picture. They disagree about whether the question, how short can a length be, is even meaningful.

No candidate theory of quantum gravity has an experimental result in its favour. The observations described above constrain possibilities without selecting one.

This is one of the largest open problems in fundamental physics, and it has been open, without an experimental foothold that discriminates between the candidates, for the better part of a century.