Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces

Details Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces

2001-12-14

arxiv.org/abs/math-ph/0112030v1

J.Phys.A35:7877-7918,2002

Physics

Mathematical Physics

46 pages, LaTeX

Scientific paper

10.1088/0305-4470/35/37/303

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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