The mode expansions and are written in terms of the plane-wave solutions of the associated wave equations. Also the Hamiltonians turned out to be just the sum of individual oscillator Hamiltonians for each mode frequency. As remarked earlier, the trouble with all these ‘real-life’ cases is that they involve significant complications due to spin; the corresponding fields then have several components, with attendant complexity in the solutions of the associated free-particle wave equations. Theories with dimensionless coupling constants, such as QED, are generally renormalizable, though invariably so. Theories whose coupling constants have positive mass dimension, as in the ABC model, are ‘super-renormalizable’, meaning that they have fewer basic divergences than ordinary renormalizable theories. In quantum mechanics, if operators representing physical observables commute with each other, then measurements of either observable can be performed without interfering with each other; the observables are said to be ‘compatible’.