Every quantum system (whether an atom, a crystal, or the entire Universe) has its own set of allowed energies, like the steps of a ladder. The hotter the system, the more readily particles climb to higher rungs. Physicists have figured out how to compare two such “ladders” from different theories: they introduced the Boltzmann–Wasserstein distance — a number that shows how much the energy spectra of two worlds differ.
To compute this distance, one must find a special bridge between the two sets of energies — a wormhole in the doubled space. Think of two sheets of paper with ladders drawn on them: a wormhole is a way to overlay the sheets so that the steps of one ladder align with those of the other in the most economical manner. It turns out the best way is simply to order the energies in ascending order and pair them up, as if we were sorting two decks of cards by rank. In the language of gravity, such a wormhole looks like two black holes connected by a throat, through which one can pass from one theory to the other.
When two theories differ only by a small perturbation (say, a weak interaction is added) the distance reduces to the time average of the thermal jitter of that perturbation. If the perturbation averages to zero (for instance, it changes sign under reflection), then a more subtle characteristic comes to the fore — the four-point function, something like the correlation of jitters at different times. This resembles measuring not just the loudness of noise but its “roughness.”
This approach unexpectedly links seemingly distant fields: optimal transport theory (how to move cargo with minimal cost), black hole physics, and quantum mechanics. It provides a new tool for quantitatively comparing different quantum theories and may shed light on how gravity emerges from quantum effects.
The Wasserstein distance, which underpins this work, was originally devised for very down-to-earth problems: how to optimally shovel a pile of sand into a hole of the same shape but elsewhere. In physics, it has become a measure of similarity between quantum states.
In Greg Egan's novel Schild's Ladder, different universes with different laws of physics are connected via wormholes, the new metric could measure how dissimilar they are.