Ultraviolet divergences: Effective field theory (EFT)

Only local relativistic quantum field theories (QFT) are considered: the action that appears in the field integral is the integral of a classical Lagrangian density, function of fields and their derivatives (taken at the same point). Physical quantities can be calculated as power series in the various interactions. As a consequence of locality, infinities appear in perturbative calculations, due to short-distance singularities, or after Fourier transformation, to integrals diverging at large momenta: one speaks of ultraviolet (UV) divergences. These divergences are peculiar to local QFT: in contrast to classical mechanics or non-relativistic quantum mechanics (QM) with a finite number of particles, a straightforward construction of a QFT of point-like objects with contact interactions is impossible. A local QFT, in a straightforward formulation, is an incomplete theory. It is an effective theory, which eventually (perhaps at the Planck's scale?), to be embedded in some non-local theory, which renders the full theory finite, but where the non-local effects affect only short-distance properties (an operation sometimes called UV completion). The impossibility to define a QFT without an explicit reference to an external short scale is an indication of a non-decoupling between short- and long-distance physics. The forms of divergences are investigated to all orders in perturbation theory using power counting arguments.