Singularities in quantum field theory

The short-range behaviour of certain Feynman integrals reveals mathematical properties which are not those of either functions or distributions - they contain terms which are more singular than distributions and possess inherent ambiguities. Two classes of singularities exist: To the first one belong all those singularities which have a physical meaning in the sense that in a convergent (regluarized) quantum field theory they contribute to observable quantities, frequently as renormalization constants. Most of the singularities of the second, the spurious type, violate the symmetries of the Lagrangian. It is demonstrated that they are associated with certain mathematical difficulties of unregularized theories. Much of the analysis deals with the isolation of singularities of this type and with the study of the properties of the singular products of distribution. It is argued that the four-dimensional integration leading to the S-matrix in the perturbation expansion must be carried out over an open domain which leaves out the contributions from singularities of the contact type, that is terms proportional to delta4(x-y). (author)