We are used to the idea of three dimensions of space, plus one of time. That's not the same as saying there are only four dimensions. We perceive three dimensions of space directly, and are aware of moving along a fourth dimension of time. But what if there are other dimensions, completely beyond our perception? Mathematicians and theoretical physicists dream up models of the world using extra dimensions all the time, but why?
In science, it's considered bad form to just make stuff up for no good reason. But there are serious theoretical reasons for tinkering with the notion of extra dimensions. The world is certainly very different and strange at the sub-atomic scale, and our everyday assumptions of what is normal are a poor guide. In quantum mechanics, where scientists seek to comprehend the physics beneath the physical world, all kinds of weird things can be going on.
We keep talking about "particles", but this word doesn't adequately sum up the type of matter that particle physicists deal with. In physics, particles aren't usually tiny bits of stuff. When you start talking about fundamental particles like quarks that have a volume of zero, or virtual particles that have no volume and pop in and out of existence just like that, it is stretching the everyday meaning of the word "particle" a bit far. Thinking about particles as points sooner or later leads the equations up a blind alley. Understanding what is happening at the smallest scale of matter needs a new vocabulary, new maths, and very possibly new dimensions.
This is where string theory comes in. In string theory fundamental particles aren't treated as zero-dimensional points. Instead they are one-dimensional vibrating strings or loops. The maths is hair-raising, and the direct evidence non-existent, but it does provide a way out of the current theoretical cul-de-sac. It even provides a route to unifying gravity with the other three fundamental forces - a problem which has baffled the best brains for decades. The problem is, you need to invoke extra dimensions to make the equations work in string-theory and its variants: 10 spacetime dimensions to be precise. Or 11 (M-theory). Or maybe 26. In any case, loads more dimensions than 4.
So where are they then? One idea is that they are right under our noses, but compacted to the quantum scale so that they are imperceptible. "Hang on a minute", you might think,"How can you ever prove the existence of something that, by definition, is impossible to perceive?" It's a fair point, and there are scientists who criticize string theory for its weak predictive power and testability. Leaving that to one side, how can you conceptualize extra dimensions?
Imagine an ant crawling along a length of string, sometimes along the top, sometimes along the bottom. From a distance, you'd say the ant was moving in one dimension - along the string. From the ant's viewpoint, it's quite happily moving in three dimensions. You see the general idea. On the scale of quarks and leptons there could be all sorts of extra dimensions.
Revealing extra dimensions means using small enough "ants", in other words particles whose wavelengths are short enough to "notice" the extra dimensions. That means very high energies, probably even higher than the LHC achieves. But it is possible that the LHC experiments could see some phenomena that indirectly support the existence of these extra dimensions, which would give string theory a major boost.