Planck scale, Dirac delta function and ultraviolet divergence

TL;DR

This paper introduces a novel self-consistent method using a redefined delta function to address ultraviolet divergences in quantum field theory, aligning with the Planck length's minimal scale.

Contribution

The authors propose a new delta function ${oldsymbol{ riangle}}_P$ that regularizes divergences while respecting the Planck length, leading to convergent propagators and modified quantum relations.

Findings

New convergent Feynman propagators for Klein-Gordon, Dirac, and Maxwell fields.

Elimination of ultraviolet divergence in quantum field calculations.

Consistency with the Planck length scale in quantum theory.

Abstract

The Planck length is the minimum length which physical law do not fail. The Dirac delta function was created to deal with continuous range issue, and it is zero except for one point. Thus contradict the Planck length. Renormalization method is the usual way to deal with divergence difficulties. The authors proposed a new way to solve the problem of ultraviolet divergence and this method is self-consistent with the Planck length. For this purpose a redefine function ${\delta}_P$ in position representation was introduced to handle with the canonical quantization. The function ${\delta}_P$ tends to the Dirac delta function if the Planck length goes to zero. By logical deduction the authors obtains new commutation/anticommutation relations and new Feynman propagators which are convergent. This article will deduce the new Feynman propagators for the Klein-Gordon field, the Dirac field and the Maxwell field. Through the new Feynman propagators, we can eliminate ultraviolet divergence.