# 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? Sep 10, 2013 at 14:20
• This question (v1) seems too broad. Please narrow down the area of physics that the question is supposed to be about. Sep 15, 2013 at 21:46

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 $R_{\mu \nu \rho \sigma}=0$.