Continuous wavelet transform in quantum field theory

Abstract

We describe the application of the continuous wavelet transform to calculation of the Green functions in quantum field theory: scalar ${\ensuremath{\phi}}^{4}$ theory, quantum electrodynamics, and quantum chromodynamics. The method of continuous wavelet transform in quantum field theory, presented by Altaisky [Phys. Rev. D 81, 125003 (2010)] for the scalar ${\ensuremath{\phi}}^{4}$ theory, consists in substitution of the local fields $\ensuremath{\phi}(x)$ by those dependent on both the position $x$ and the resolution $a$. The substitution of the action $S[\ensuremath{\phi}(x)]$ by the action $S[{\ensuremath{\phi}}_{a}(x)]$ makes the local theory into a nonlocal one and implies the causality conditions related to the scale $a$, the region causality [J. D. Christensen and L. Crane, J. Math. Phys. (N.Y.) 46, 122502 (2005)]. These conditions make the Green functions $G({x}_{1},{a}_{1},\dots{},{x}_{n},{a}_{n})=⟨{\ensuremath{\phi}}_{{a}_{1}}({x}_{1})\dots{}{\ensuremath{\phi}}_{{a}_{n}}({x}_{n})⟩$ finite for any given set of regions by means of an effective cutoff scale $A=\mathrm{min}({a}_{1},\dots{},{a}_{n})$.