Wavelet-based quantum field theory

Abstract:
We construct a Euclidean field theory for the fields φ_Δx(x) that depend on both position and resolution. The Feynman diagrams in such theory become finite under the assumption that there should be no scales in internal lines smaller than the minimal of scales of external lines. The construction is performed using the continuous wavelet transform with the basic wavelet being understood as an apparatus function of an abstract measuring device. The transition from the newly constructed theory to a standard Euclidean field theory is achieved by integration over the scale arguments and shows that the source of divergences in standard theory is the integration over all scales which is unfeasible for it would require infinitely high energies.