Renormalization, wavelets, and the Dirichlet-Shannon kernels

Abstract

In constructive quantum field theory (CQFT) it is customary to first regularize the theory at finite UV and IR cutoff. Then one first removes the UV cutoff using renormalization techniques applied to families of CQFTs labeled by finite UV resolutions and then takes the thermodynamic limit. Alternatively, one may try to work directly without IR cutoff. More recently, wavelets have been proposed to define the renormalization flow of CQFTs which is natural as they come accompanied with a multiresolution analysis. However, wavelets so far have been mostly studied in the noncompact case. Practically useful wavelets that display compact support and some degree of smoothness can be constructed on the real line using Fourier space techniques but explicit formulas as functions of position are rarely available. Compactly supported wavelets can be periodized by summing over period translates keeping orthogonality properties but still yielding to rather complicated expressions which generically lose their smoothness and position locality properties. It transpires that a direct approach to wavelets in the compact case is desirable. In this contribution we show that the Dirichlet-Shannon kernels serve as a natural scaling function to define generalized orthonormal wavelet bases on tori or copies of real lines, respectively. These generalized wavelets are smooth, are simple explicitly computable functions, display quasilocal properties close to the Haar wavelet, and have compact momentum support. Accordingly they have a built-in cutoff in both position and momentum, making them very useful for renormalization applications.