# Key difference between 3D and 4D solutions

In many papers in theoretical physics especially in the more advanced theories, I notice that physicists solve problems in 3 dimensions (2 spatial+ 1 temporal).

In some specific papers (I can't seem to recall exactly which ones), some models solved in three dimensions gave finite solutions, whereas 4D (3+1) were either hard to solve or gave divergent results.

I don't seem to understand why adding a 3rd spatial dimension should cause problems if 2 spatial dimensions gave no problems. Intuitively, it seems straightforward matter to extrapolate 4D solutions from 3D. Then what is the essential difference between the 2?

-
Are you talking about quantum/statistical field theories? –  Michael Brown Sep 10 '13 at 14:20
Yes and some topics in GTR and quantum gravity. –  dj_mummy Sep 10 '13 at 15:02
This question (v1) seems too broad. Please narrow down the area of physics that the question is supposed to be about. –  Qmechanic Sep 15 '13 at 21:46

I don't know which examples you are talking about but there are a couple reasons to study toy models in 2+1 dimensions. The quickest answer would be that when doing QFT, having one less spatial dimension helps with convergence problems of loop integrals.

The difference adding an extra dimension causes in QFT is probably more pronounced in going from 1+1 to 2+1 dimensions. In 1+1 dimensions, we have only boosts, no rotations, so the structure of the symmetries is much simpler. In addition, in 1+1 dimension there is only forward and backward scattering, which evades an assumption of the Coleman-Mandula theorem (analyticity of S-Matrix) and essentially allows for exactly solvable S-matrices. You lose all of this in moving to 2+1 dimensions.

Probably one of the more important examples of a 2+1 toy model would be 3D gravity. It turns out that gravity in 3 dimensions is very different from gravity in 4 dimensions. Essentially, in 3 dimensions the Riemann curvature tensor is completely determined by the Ricci tensor, so that $R_{\mu \nu}=0 \implies R_{\mu \nu \rho \sigma}=0$. The Einstein equation says $R_{\mu \nu}\propto(T_{\mu \nu} -\frac{1}{2}Tg_{\mu \nu})$ so that vacuum solutions in 3D satisfy not only $R_{\mu \nu}=0$ but also $R_{\mu \nu \rho \sigma}=0$.

This means that 3D pure gravity does not have local propagating degrees of freedom. This is a dramatic simplification that allows you to get much more mathematical control over the theory and lets you test ideas about quantization. Going to 4D gives you local gravitational degrees of freedom and makes the theory much more intractable.

-
I know that 1+1->2+1 is a big deal in QFT. Which is why my question was only about 2+1->3+1. Are you saying that 2+1->3+1 requires only trivial changes to theory in QFT? And that 2+1->3+1 is only significant in GTR and quantum gravity? –  dj_mummy Sep 16 '13 at 7:03
No there are other examples, I think gravity is just one of the more important ones. For instance, the Chern Simons term AdA+2/3A^3 is a 3 form and so in 3 dimensions you may study this as a topological field theory or add the term to non-topological field theories and for existence give the photon a mass. I think there are also differences when studying topological solutions in gauge theories. Because of the difference in space-time dimension, you end up look at a different homotopy group of the gauge group when looking for special solutions. –  Dan Sep 17 '13 at 0:19