We study the four most significant high precision observables of QED ---the anomalous electron and muon magnetic moments, the hydrogen Lamb shift and muonium hyperfine splitting--- in the context of SU(2) x U(1) gauge-invariant effective Lagrangians. The agreement between the theoretical predictions for these observables and the experimental data places bounds on the lowest dimension operators of the effective Lagrangians. We also place bounds on such effective operators using other experimental data. Comparison of the tow types of bounds allows us to discuss the potential of each one of the four high precision observables in the search for physics beyond the Standard Model. We find that the anomalous electron and muon magnetic moments are sensitive to new physics while the hydrogen Lamb shift and muonium hyperfine splitting are not.

12 pages, LaTeX, minor changes included, to appear in Zeit. fur Physics