Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces

TL;DR

This paper develops a unified trigonometric framework for complex Hermitian symmetric spaces, encompassing various known spaces and their quantum state space applications, through a universal group equation.

Contribution

It introduces a single basic trigonometric group equation that captures the trigonometry of the entire family of complex Hermitian symmetric spaces, highlighting universality and self-duality.

Findings

Unified trigonometric equations for complex Hermitian spaces

Special cases recover known results for CP^N and CH^N

Applications to quantum state space trigonometry

Abstract

This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex Hermitian' type and rank-one. The complex Hermitian elliptic CP^N and hyperbolic CH^N spaces, their analogues with indefinite Hermitian metric and some non-compact symmetric spaces associated to SL(N+1,R) are the generic members in this family. The method encapsulates trigonometry for this whole family of spaces into a single "basic trigonometric group equation", and has 'universality' and '(self)-duality' as its distinctive traits. All previously known results on the trigonometry of CP^N and CH^N follow as particular cases of our general equations. The physical Quantum Space of States of any quantum system belongs, as the complex Hermitian space member, to this parametrised family; hence its trigonometry appears as a rather particular case of the equations we obtain.