  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.
