13 research outputs found
Coherent states and geodesics: cut locus and conjugate locus
The intimate relationship between coherent states and geodesics is pointed
out. For homogenous manifolds on which the exponential from the Lie algebra to
the Lie group equals the geodesic exponential, and in particular for symmetric
spaces, it is proved that the cut locus of the point is equal to the set of
coherent vectors orthogonal to . A simple method to calculate the
conjugate locus in Hermitian symmetric spaces with significance in the coherent
state approach is presented. The results are illustrated on the complex
Grassmann manifold.Comment: 19 pages, enlarged version, 14 pages, Latex + some macros from Revtex
+ some AMS font
Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces
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.Comment: 46 pages, LaTe
On metric-connection compatibility and the signature change of space-time
We discuss and investigate the problem of existence of metric-compatible
linear connections for a given space-time metric which is, generally, assumed
to be semi-pseudo-Riemannian. We prove that under sufficiently general
conditions such connections exist iff the rank and signature of the metric are
constant. On this base we analyze possible changes of the space-time signature.Comment: 18 standard LaTeX 2e pages. The packages AMS-LaTeX and amsfonts are
require