When physicists say “3+1″they are not talking about the number “4”, but usually they are talking about spacetime: three spatial dimensions and one temporal dimension. One question which plagues theoretical physicists’ poor little minds is the question of why we see a macroscopic world of 3+1 dimensions. Mostly this is because physicists believe that at small enough length or time scales (large enough energies) the geometry of spacetime itself can exist in nontrivial states of connectivity. Thus we think of spacetimes at small enough scale as existing in all sorts of strange configurations (in some poor little region the spacetime may look like a 10 dimensional manifold, for instance.) “Spacetime foam” is what we call this strange state of affairs. How do we get from this spacetime foam up to where our experiments live and we seem to see a four dimensional universe?
Concerning this problem, I just today read the paper “Emergence of a 4D World from Causal Quantum Gravity,” by J. Ambjorn, J. Jurkiewicz, and R. Loll which was published in Physical Review Letters, (Volume 93, page 131301, 2004.) This paper attempts the following. Construct spacetime by glueing together a bunch of little four dimensional simplical spacetimes. Like I said earlier, if we glue a bunch of these four dimensional simplical spacetimes together, we get something which is not necessarily four dimensional. Now when we do this glueing we should insist on maintain causality (i.e. no closed time like curves and such.) So we can construct these crazy spacetimes, but what do they mean. Well now we associate with each of these spacetimes an amplitude. So there is some notion of an action S for the given simpical spacetime we have created and we assign to this an amplitude, Exp[iS]. Now what one would love to do is to sample over all of these crazy spacetimes and hence calculate the propogators for different such spacetimes. But this is hard. This is hard because of the fact that we have to sample over this crazy oscillating Exp[iS]. But sometimes it is not so hard. Sometimes it is possible to perform a “Wick” rotation and change Exp[iS] into Exp[-S]. This means the problem of calculating the total amplitude looks like adding up a bunch of different spacetimes with weights Exp[-S]: this looks just like classical statistical mechanics! What the authors of the above paper do is they insist that it is possible to perform such a rotation. They then perform Monte Carlo simulations of the resulting statistical mechanical system. And what do they find? They argue that what they find is that the resulting spacetime is indeed dominated by a spacetime of dimension “3+1!”
So starting out from something which had only a totally local sense of dimension (the original building blocks are “3+1”) you glue them together in pseudo-arbitrary (preserve causality, able to Wick rotate) ways (this is what is called “background independence”) and yet, you find, at the end of the day, that you have effectively a global “3+1” spacetime! Amazing, no?