Nrqed in Bound States: Applying Renormalization to an Effective Field Theory

This is a pedagogical introduction to NRQED, a nonrelativistic approximation to quantum electrodynamics. It can be made to reproduce QED to any degree of precision when applied to nonrelativistic systems and is particularly powerful when applied to bound states. We will begin by explaining what makes bound states so special, and so difficult to treat using QED. We will then develop the most näıve nonrelativistic approximation of QED which will be shown to fail when applied to a simple example (hyperfine splitting in positronium). We will see what was missing to make the theory well-defined, and the resulting theory will be, essentially, NRQED. We will conclude by discussing what makes NRQED so much more powerful than QED in nonrelativistic bound states, and what the approach we followed can teach us about “new physics” beyond QED.