Regularization, renormalization, and dimensional analysis: Dimensional regularization meets freshman E&M

Abstract

We illustrate the dimensional regularization technique using a simple example from electrostatics. This example illustrates the virtues of dimensional regularization without the complications of a full quantum field theory calculation. We contrast the dimensional regularization approach with the cutoff regularization approach, and demonstrate that dimensional regularization preserves translational symmetry. We then introduce minimal subtraction and modified minimal subtraction schemes to renormalize the result. Finally, we consider dimensional transmutation as encountered in the case of compact “extra dimensions.”