Wavelet field decomposition and UV `opaqueness'

TL;DR

This paper introduces a wavelet-based formalism to incorporate UV 'opaqueness' in field theories, effectively localizing interactions within a fundamental length scale and ensuring UV regulation and unitarity.

Contribution

It develops a wavelet decomposition approach that models UV 'opaqueness', leading to a new way of regulating and localizing interactions in quantum field theories.

Findings

Fields are expandable only in scaling parts of wavelet analysis.

Interactions are delocalized within opaque regions that decay rapidly.

Effective Feynman rules resemble string field theory, ensuring UV regulation and unitarity.

Abstract

A large body of work over several decades indicates that, in the presence of gravitational interactions, there is loss of localization resolution within a fundamental ( $\sim$ Planck) length scale $\ell$. We develop a general formalism based on wavelet decomposition of fields that takes this UV `opaqueness' into account in a natural and mathematically well-defined manner. This is done by requiring fields in a local Lagrangian to be expandable in only the scaling parts of a (complete or, in a more general version, partial) wavelet Multi-Resolution Analysis. This delocalizes the interactions, now mediated through the opaque regions, inside which they are rapidly decaying. The opaque regions themselves are capable of discrete excitations of $\sim 1/\ell$ spacing. The resulting effective Feynman rules, which give UV regulated and (perturbatively) unitary physical amplitudes, resemble those of string field theory.