First-order quantum perturbation theory and Colombeau generalized functions

TL;DR

This paper reformulates first-order quantum perturbation theory involving Dirac delta functions within Colombeau algebra, providing a rigorous mathematical foundation for operations like squaring distributions, and confirming physical results align with experimental data.

Contribution

It introduces a Colombeau algebra framework to rigorously define products of distributions in quantum perturbation theory, resolving longstanding mathematical ambiguities.

Findings

Dirac delta functions are shown to belong to a subset with well-defined squares.

The Colombeau algebra provides a consistent interpretation of distribution products.

Results agree with experimental data and standard quantum mechanics.

Abstract

The electromagnetic scattering of a spin-0 charged particle off a fixed center is calculated in first-order quantum perturbation theory. This implies evaluating the square of a `Dirac delta-function,' an operation that is not defined in Schwartz distribution theory, and which in elementary text-books is dealt with according to `Fermi's golden rule.' In this paper these conventional calculations are carefully reviewed, and their crucial parts reformulated in a Colombeau algebra -- in which the product of distributions is mathematically well defined. The conclusions are: (1) The Dirac delta-function insuring energy conservation in first order perturbation theory belongs to a particular subset of representatives of the Schwartz distribution defined by the Dirac measure. These particular representatives have a well-defined square, and lead to a physically meaningful result in agreement with the data. (2) A truly consistent mathematical interpretation of these representatives is provided by their redefinition as Colombeau generalized functions. This implies that their square, and therefore the quantum mechanical rule leading from amplitudes to probabilities, is rigorously defined.