Differential Geometry on SU(3) with Applications to Three State Systems

TL;DR

This paper computes invariant vector fields, forms, and Haar measure on SU(3) using Euler coordinates, then applies these to analyze pure states and geometric phases in three-state quantum systems.

Contribution

It introduces a detailed Euler angle parameterization of SU(3) and applies it to quantum state geometry and phase analysis.

Findings

Derived explicit expressions for invariant vector fields and forms on SU(3)

Calculated the Haar measure in Euler coordinates

Applied geometric framework to three-state quantum systems

Abstract

The left and right invariant vector fields are calculated in an ``Euler angle'' type parameterization for the group manifold of SU(3), referred to here as Euler coordinates. The corresponding left and right invariant one-forms are then calculated. This enables the calculation of the invariant volume element or Haar measure. These are then used to describe the density matrix of a pure state and geometric phases for three state systems.