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Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces

Ramon Ortega·Mariano Santander
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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.

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Ramon Ortega, Mariano Santander. 2001-12-14. Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces. https://doi.org/10.1088/0305-4470%2F35%2F37%2F303

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