A remark on gauge invariance in wavelet-based quantum field theory

TL;DR

This paper explores gauge invariance within wavelet-based quantum field theory, introducing a gauge transformation for scale-dependent fields and deriving related Ward-Takahashi identities to advance understanding of gauge regularization techniques.

Contribution

It develops a gauge theory framework using wavelet transforms, defining gauge transformations for scale-dependent fields, and deriving Ward-Takahashi identities in this context.

Findings

Defined gauge transformations for wavelet-based scale-dependent fields.

Derived Ward-Takahashi identities for the wavelet-based gauge theory.

Provided a foundation for regularization in gauge theories using wavelets.

Abstract

Wavelet transform has been attracting attention as a tool for regularization of gauge theories since the first paper of (Federbush, Progr. Theor. Phys. 94, 1135, 1995), where the integral representation of the fields by means of the wavelet transform was suggested: $$A_{\mu}(x) = \frac{1}{C_\psi} \int_{\R_+ \times\R^d} \frac{1}{a^d} g \left(\frac{x-b}{a} \right) A_{\mu a}(b) \frac{dad^db}{a},$$ with $A_{\mu a}(b)$ being understood as the fields $A_\mu$ measured at point $b\in \R^d$ with resolution $a\in\R_+$. In present paper we consider a wavelet-based theory of gauge fields, provide a counterpart of the gauge transform for the scale-dependent fields: $A_{\mu a}(x)\to A_{\mu a}(x)+\d_\mu f_a(x)$, and derive the Ward-Takahashi identities for them.