Retarded functions, dispersion relations, and Cutkosky rules at zero and finite temperature

Some properties at zero and finite temperature in the real-time formalism of a sum of graphs, previously shown in some examples to be equivalent to the retarded product, are discussed. For the general 2-, 3-, and 4-point functions we demonstrate that expressing this sum in the form of the definition of the retarded product leads to identities that can be interpreted as dispersion relations in energies. By deriving a set of Cutkosky rules, we show that the weights involved in these relations arise in finding the imaginary part of this sum, and thus have an interpretation in terms of reaction rates. We contrast this interpretation of the imaginary part of this sum with that of the time-ordered product. We also indicate how dispersionlike relations can be derived for the time-ordered product.