Continuous Wavelet Transform in Quantum Field Theory

TL;DR

This paper explores the use of continuous wavelet transforms to compute Green functions in various quantum field theories, introducing a scale-dependent approach that regularizes Green functions and incorporates causality conditions.

Contribution

It extends previous work by applying the continuous wavelet transform to multiple quantum field theories, transforming local fields into scale-dependent fields to improve Green function calculations.

Findings

Green functions become finite with the wavelet-based regularization.

The method introduces causality conditions related to the scale parameter.

Applicable to scalar, QED, and QCD theories.

Abstract

We describe the application of the continuous wavelet transform to calculation of the Green functions in quantum field theory: scalar $\phi^4$ theory, quantum electrodynamics, quantum chromodynamics. The method of continuous wavelet transform in quantum field theory presented in M.Altaisky Phys. Rev. D81(2010)125003 for the scalar $\phi^4$ theory, consists in substitution of the local fields $\phi(x)$ by those dependent on both the position $x$ and the resolution $a$. The substitution of the action $S[\phi(x)]$ by the action $S[\phi_a(x)]$ makes the local theory into nonlocal one, and implies the causality conditions related to the scale $a$, the region causality J.D. Christensen and L. Crane, J.Math. Phys 46 (2005) 122502. These conditions make the Green functions $G(x_1,a_1,..., x_n,a_n)=<\phi_{a_1}(x_1)...\phi_{a_n}(x_n)> $ finite for any given set of regions by means of an effective cutoff scale $A=\min (a_1,...,a_n)$.