Talking about different physical theories in different dimensions is one of physicists favorite pastimes. Thus, for instance, we often move the theory to the infinite dimensional case or to the case of dimension slightly greater or less than some integer dimension. Often we work with low dimensional theories because of constraints–for example we do this when we talk about an electron gas trapped in a two surface or a one dimensional line or even a zero dimensional dot. But sometimes we work in different dimensions to see if we can gain insight into the theory in a dimension where we can’t seem to make much progress. Such is the case for a quantum theory of gravity. Our success in understanding quantum gravity in two spatial dimensions plus one time dimension (2+1) has been far better than our success in understanding quantum gravity in three spatial dimensions plus one time dimension (3+1). Why?
Well there is an easy way to see why quantum gravity in 2+1 dimensions is quite a different beast than quantum gravity in 3+1 dimensions. Let’s look at the 3+1 dimensional case first. At a fixed time, the spatial metric has six degrees of freedom (it’s a real tensor and it’s a symmetric tensor and it’s in dimension three.) But the laws of general relativity are invariant under general coordinate changes. This means that there are four gauge degrees of freedom which correspond to the choice of four spacetime coordinates. Thus there are two physical degrees of freedom in 3+1 dimensional gravity.
But what happens in 2+1 dimensional gravity? At a fixed time, the spatial metric now has three degrees of freedom (it’s a real tensor and it’s a symmetric tensor and it’s in dimension two.) But now there are three gauge degrees of freedom. Thus in 2+1 dimensional gravity there are no physical degrees of freedom!
In fact what happens in 2+1 is that the curvature tensor vanishes! Now recall that if we parallel transport a vector around an area where the curvature tensor vanishes, then the vector doesn’t change. Thus if we work in a spacetime where there all loops encose areas, then parallel transport will be path independent and there will be a global notion of parallelism. Quite a boring theory right? Well yes, if you only consider simply connected spacetimes (i.e. spacetimes where all loops are continuously contractible to a point.) But if you consider spacetimes which have noncontractible loops (think of a torus and the circles which form circumferences of this torus) then the parallel transport around one of these noncontractible loops doesn’t enclose an area. Now the geometry of this flat spacetime is characterized by the results of parallel transport around noncontratible loops (holonomies.) So right away we see that gravity in 2+1 dimensions will be an interesting theory when we allow topologically nontrivial spacetimes. In fact, when we construct the solutions and quantize gravity in 2+1 dimensions we are led to a topological quantum field theory! Actually things get quite interesting in 2+1 dimensions when we try to quantize the theory. In fact there are many different approaches to this quantizations, and, strangely, not all of these are consistent (this is why you have to pay attention when all these mathematical physicists go on and on about all these different methods to quantize classical theories!)
Well enough quantum gravity for today. Just remember, theories of physics are never as complicated as most theoretical physicists would like you to believe.
O brave new quantum world!