Positronium is the positron-electron bound state. The structure of positronium is governed almost completely by QED; strong and weak corrections are small compared to electrodynamic effects. Positromium is also accessible to high precision experiments. Thus positronium is an ideal laboratory for studying QED and the bound state formalism in quantum field theory. The main quantities of physical interest in positronium are energy levels and decay rates. Several energy level splittings of n=1 and n=2 states have been measured with uncertainties on the order of a few MHz. Present theoretical evaluations of these splittings agree with experiment, with terms of order m(alpha)6 = 18.7 MHz still uncalculated. The n=1 spin-singlet and spin triplet decay rates have also been measured. The spin-triplet rate is several standard deviations away from the present theoretical value. There is an evident need to extend the theoretical calculations of energy levels and decay rates in positronium by another power of alpha. It is proposed to calculate some of these corrections and to make progress on the calculation of others. In particular it is planned to complete the calculation of n=1 and n=2 positronium energy levels through order m(alpha)6. Comparison of experiment and theory at this level will provide a detailed test of the bound state formalism and of QED.
