UV AND IR behaviour for QFT and LCQFT with fields as operator valued distributions: Epstein and Glaser revisited

Following Epstein-Glaser's work we show how a QFT formulation based on operator valued distributions (OPVD) with adequate test functions treats original singularities of propagators on the diagonal in a mathematically rigourous way. Thereby UV and/or IR divergences are avoided at any stage, only a finite renormalization finally occurs at a point related to the arbitrary scale present in the test functions. Some well known UV cases are exemplified. The power of the IR treatment is shown for the free massive scalar field theory developed in the (conventionally hopeless) mass perturbation expansion. It is argued that the approach should prove most useful for non-pertubative methods where the usual determination of counterterms is elusive.