Geometric phase and modulus relations for SU(n) matrix elements in the defining representation

TL;DR

This paper derives relations between the modulus and phase of SU(n) matrix elements in the fundamental representation, with illustrations for SU(2) and discussions on phase singularities, complementing previous results on ray space amplitudes.

Contribution

It introduces new relations between modulus and phase for SU(n) amplitudes in the fundamental representation, expanding understanding of their geometric phase properties.

Findings

Relations between modulus and phase for SU(n) amplitudes derived

Illustrations provided for the SU(2) case

Discussion on phase singularities and superoscillatory behavior

Abstract

A set of relations between the modulus and phase is derived for amplitudes of the form $\mels{\hatu(x)}$ where $\hat{U}(x) \in SU(n)$ in the fundamental representation and $x$ denotes the coordinates on the group manifold. An illustration is given for the case $n=2$ as well as a brief discussion of phase singularities and superoscillatory phase behavior for such amplitudes. The present results complement results obtained previously \cite{PMrel1} for amplitudes valued on the ray space ${\cal R} = {\mathbb C}P^n$. The connection between the two is discussed.