Anti De Sitter Deformation of Quaternionic Analysis and the Second Order Pole

TL;DR

This paper extends quaternionic analysis by introducing a conformally invariant anti de Sitter deformation, providing new quaternionic Cauchy formulas related to quantum field theory infinities.

Contribution

It develops a one-parameter deformation of quaternionic analysis preserving conformal invariance and extends classical formulas to this new setting.

Findings

Deformation preserves conformal invariance.

Quaternionic Cauchy formulas are extended to the deformed setting.

Connections to quantum field theory infinities are established.

Abstract

This is a continuation of a series of papers [FL1, FL2, FL3], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we continue to study the quaternionic analogues of Cauchy's formula for the second order pole. These quaternionic analogues are closely related to regularization of infinities of vacuum polarization diagrams in four-dimensional quantum field theory. In order to add some flexibility, especially when dealing with Cauchy's formula for the second order pole, we introduce a one-parameter deformation of quaternionic analysis. This deformation of quaternions preserves conformal invariance and has a geometric realization as anti de Sitter space sitting inside the five-dimensional Euclidean space. We show that many results of quaternionic analysis - including the Cauchy-Fueter formula - admit a simple and canonical deformation. We conclude this paper with a deformation of the quaternionic analogues of Cauchy's formula for the second order pole.