Energy levels of QED in a Euclidean formulation

Hydrogen-like energy levels of scalar and spinor QED are calculated using a Euclidean functional approach. The matter field is integrated over. Stationary points of the resulting effective action already yield a hydrogen like level structure for the energy. There is an interesting difference between the scalar and the spinor case. Whereas for spinors the conventional results are reproduced, the calculation for scalars yields a fine structure which is opposite in sign to the conventional one and has no critical singularity at Zα=1/2. The crucial structural difference between the two cases is that, for scalars, minima for the gauge invariant energy are not extrema of the action, even for time independent fields.