Quaternionic Analysis, Representation Theory and Physics

TL;DR

This paper develops quaternionic analysis through representation theory of conformal groups, linking it to quantum mechanics and field theory, and extends classical formulas to Minkowski space with new physical interpretations.

Contribution

It introduces a quaternionic version of classical analysis formulas, connects quaternionic analysis to physics, and explores unitarity and representation theory in this context.

Findings

Extension of Cauchy-Fueter and Poisson formulas to Minkowski space

Connection between quaternionic analysis and quantum field theory structures

Proposal of a quaternionic analogue of the second order pole Cauchy formula

Abstract

We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of quaternions. Representation theory also suggests a quaternionic version of the Cauchy formula for the second order pole. Remarkably, the derivative appearing in the complex case is replaced by the Maxwell equations in the quaternionic counterpart. We also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory, such as the spectrum of the hydrogen atom, polarization of vacuum, one-loop Feynman integrals. We also make some further conjectures. The main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis, representation theory and four-dimensional physics.