The Science of Discworld IV

theories. And that brings us to the search for a unified field theory, or Theory of Everything, to which Einstein devoted many years of thought – without success. Somehow, relativity and quantum mechanics have to bemodified to produce a consistent theory that agrees with each of them in the relevant domain.
    Today’s front runner is string theory, which replaces point particles by tiny multidimensional shapes, as we discussed in
The Science of Discworld III
. Some versions of string theory require space to be nine-dimensional, so spacetime has to be ten-dimensional. The extra six dimensions of space are thought either to be curled up so tightly that we don’t notice them, or inaccessible to humans – in the same way that A. Square could not travel out of Flatland unaided, but needed a shove from the Sphere to push him into the third dimension. The formulations of string theory currently in vogue also introduce new ‘super-symmetry’ principles, predicting the existence of a host of new ‘sparticles’ to match the known particles. So an electron is paired with a selectron, and so on. So far, however, this prediction has not been confirmed. The LHC has looked for sparticles, but so far it has found precisely none of them.
    One of the latest unification attempts, refreshingly different from most that have gone before, takes us neatly back to Flatland. The idea, common in mathematics and often fruitful, is to take inspiration from a cut-down toy problem. If unifying relativity and quantum mechanics is too hard using three-dimensional space, why not simplify the problems by looking at the non-physical but mathematically informative case of two-dimensional space? Plus one of time, naturally. It’s clear enough where to start. In order to unify two theories, you have to have two theories to unify. So what would gravity look like in Flatland, and what would quantum mechanics look like in Flatland? We hasten to add that Flatland, here, need not be A. Square’s Euclidean plane. Any two-dimensional space, any surface, would do. Indeed, other topologies are essential to get anything interesting.
    It’s straightforward to write down sensible analogues of Einstein’s field equations when space is a surface. It’s very close to what Gauss did when he started the whole thing off, and his ant would have no trouble in devising the right equations, since they’re all about curvature. Thereare obvious analogies to follow; all you do is replace the number three by the number two at key points. In Roundworld, the Polish physicist Andrzej Staruszkiewicz wrote such equations down in 1963.
    It turns out that gravity in two dimensions differs significantly from gravity in three. In three dimensions relativity predicts the existence of gravitational waves, which propagate at the speed of light. But there are no gravitational waves in two dimensions. In three dimensions, relativity predicts that any mass bends space into a rounded bump, so that anything that passes nearby will follow a curved path, as if it were attracted by Newtonian gravity. And an object that was at rest will fall into the gravitational well of the mass concerned. In two dimensions, however, gravity bends space into a cone. Moving bodies are deflected, but bodies at rest simply remain at rest. In three dimensions massive bodies collapse under their own gravitation to form black holes. In two dimensions, this is impossible.
    These differences are things we can live with, but in three dimensions, gravitational waves are a useful way to link relativity to quantum theory. The absence of gravitational waves in two dimensions causes headaches, because it means there is nothing to quantise – nothing to use as a starting point for a quantum-mechanical formulation. Gravity should correspond to hypothetical particles called gravitons, and in quantum theory particles have ghostly companions – waves. No waves, no gravitons. But in 1989 Edward Witten, one of the architects of string theory, ran into other quantum problems involving fields that do not propagate waves. Two-dimensional gravity is like that, and it opened his eyes to a missing ingredient.
    Topology.
    Even when gravity can’t travel as a wave, it can have a huge effect on the shape of space. Witten’s experience with topological quantum field theories, where just this ingredient arises, suggested a way forward. The humble torus, in many ways the simplest non-trivial topological space, plays a key role.