A demonstration of the necessity of regularization in order to avoid inconsistent results in quantum field theory

TL;DR

This paper highlights the importance of regularization in quantum field theory to prevent mathematical inconsistencies, demonstrating that improper handling can lead to incorrect results.

Contribution

It identifies a specific inconsistency in a previous derivation and shows how proper regularization resolves this issue.

Findings

The original derivation contains a mathematical inconsistency.

Regularization resolves the inconsistency in the derivation.

Proper mathematical treatment is essential for consistent results in quantum field theory.

Abstract

We will examine a particular mathematical derivation in a paper by P. Falkensteiner and H. Grosse (F&G) [1]. In [1] a quantity "delta(A)" is defined. This quantity is generated when the normal ordered generalized charge operator undergoes a unitary transformation. Using standard mathematical techniques F&G convert "delta(A)" from its original form to another form which is suppose to be equivalent. It will be shown here that the two forms are not equivalent and that there is a mathematical inconsistency in their derivation. We will examine the source of this inconsistency and show that it can be resolved by proper regularization of the mathematical expressions.