Basis for the Generalization of Physics to a Different Number of Dimensions

Question

I am reading this really interesting book by Zwiebach called "A First Course in String Theory". Therein, he generalizes the laws of electrodynamics to the cases where dimensions are not 3+1. It's an intriguing idea but the way he generalizes seems like an absolute guess with no sound basis. In particular, he generalizes the behavior of electric fields to the case of 2 spatial and 1 temporal dimensions by maintaining $\vec{\nabla}. \vec{E} = \rho$. But I struggle to understand why. I could have maintained that $|\vec{E}|$ falls off as the square of the inverse of the distance from the source. Essentially, there is no way to differentiate between the Coulomb's law and the Gauss's law in the standard 3+1 dimensions--so how can I prefer one over the other in the other cases? To me, it seems like it becomes purely a matter of one's taste as to which mathematical form seems more generic or deep--based on that one guesses which form would extend its validity in the cases with the number of dimensions different than that in which the experiments have been performed. But, on the other hand, I think there should be a rather sensible reason behind treating the laws in the worlds with a different number of dimensions this way--considering how seriously physicists talk about these things. So, I suppose I should be missing something. What is it?

Answer 1

Great question. First of all, you're absolutely right that until we find a universe with a different number of dimensions in the lab, there's no single "right" way to generalize the laws of physics to different numbers of dimensions - we need to be guided by physical intuition or philosophical preference.

But there are solid theoretical reasons for choosing to generalize E&M to different numbers of dimensions by choosing to hold Maxwell's equations "fixed" across dimensions, rather than, say, Coulomb's law, the Biot-Savart law, and the Lorentz force law. For one thing, it's hard to fit magnetism into other numbers of dimensions while keeping it as a vector field - the defining equations of 3D magnetism, the Lorentz force law and the Biot-Savart law, both involve cross products of vectors, and cross products can only be formulated in three dimensions (and also seven, but that's a weird technicality and the 7D cross product isn't as mathematically nice as the 3D one).

For another thing, a key theoretical feature of 3D E&M is that it is Lorentz-invariant and therefore compatible with special relativity, so we'd like to keep that true in other numbers of dimensions. And the relativistically covariant form of E&M much more directly reduces to Maxwell's equations in a given Lorentz frame than to Coulomb's law.

For a third thing, 3D E&M possess a gauge symmetry and can be formulated in terms of the magnetic vector potential (these turn out to be very closely related statements). If we want to keep this true in other numbers of dimensions, then we need to use Maxwell's equations rather than Coulomb's law.

These reasons are all variations on the basic idea that if we transplanted Coulomb's law into other numbers of dimensions, then a whole bunch of really nice mathematical structure that the 3D version possesses would immediately fall apart.

Answer 2

Here is one line of reasoning: E&M is supposed to be a fundamental theory. Having an action principle may facilitate developing a consistent quantum theory. The structure of the Maxwell Lagrangian density $$ {\cal L}~=~-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + j^{\mu}A_{\mu}$$ does not depend on the spacetime dimension $n$. It possesses Lorentz symmetry. It possesses gauge symmetry (if $d_{\mu}j^{\mu}=0$). Its Euler-Lagrange equations imply a Gauss law, and hence in the electrostatic limit, a Coulomb force law that falls off as $1/r^{n-2}$. For more details, see also my related Phys.SE answers here & here.