arXiv subjects
Publications and source records attributed to Stefan Berceanu.
A realization of coherent state Lie algebras by first-order differential operators with holomorphic polynomial coefficients on Kähler coherent state orbits is presented. Explicit formulas involving the Bernoulli numbers and the structure constants for the semisimple Lie groups are proved.
For simple Lie groups, the only homogeneous manifolds $G/K$, where $K$ is maximal compact subgroup,for which the phase of the scalar product of two coherent state vectors is twice the symplectic area of a geodesic triangle are the hermitian symmetric spaces. An explicit calculation of the multiplicative factor on the complex Grassmann manifold and its noncompact dual is presented.It is shown that the multiplicative factor is identical with the two-cocycle considered by A. Guichardet and D. Wigner for simple Lie groups.
For arbitrary quantizable compact Kaehler manifolds, relations between the geometry given by the coherent states based on the manifold and the algebraic (projective) geometry realised via the coherent state mapping into projective space, are studied. Polar divisors, formulas relating the scalar products of coherent vectors on the manifold with the corresponding scalar products on projective space (Cauchy formulas), two-point, three-point and more generally cyclic m-point functions are discussed. The three-point function is related to the shape invariant of geodesic triangles in projective space.
The coherent states are viewed as a powerful tool in differential geometry. It is shown that some objects in differential geometry can be expressed using quantities which appear in the construction of the coherent states. The following subjects are discussed via the coherent states: the geodesics, the conjugate locus and the cut locus; the divisors; the Calabi's diastasis and its domain of definition; the Euler-Poincaré characteristic of the manifold, the number of Borel-Morse cells, Kodaira embeding theorem....
For homogeneous simply connected Hodge manifolds it is proved that the set of coherent vectors orthogonal to a given one is the divisor responsible for the homogeneous holomorphic line bundle of the coherent vectors. In particular, for naturally reductive spaces, the divisor is the cut locus.
Different topics on the differential geometry of the complex Grassmann manifold are surveyed in relation to the coherent states. A calculation of the tangent conjugate locus and conjugate locus in the complex Grassmann manifold is presented. The proofs use the Jordan's stationary angles. Also various formulas for the distance on the complex Grassmann manifold are furnished.
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 $0$ is equal to the set of coherent vectors orthogonal to $\vert 0>$. 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.
A geometric characterization of transition amplitudes between coherent states, or equivalently, of the hermitian scalar product of holomorphic cross sections in the associated D - M tilda - module, in terms of the embedding of the cohe- rent state manifold M-tilda into a projective Hilbert space is proposed. Cohe- rent state manifolds endowed with a homogeneous kaehler structure are conside- red. Using the coherent state approach, an effective method to find the cut loci on symmetric manifolds and generalized symmetric manifolds M-tilda is proposed. The CW - complex structure of coherent state manifolds of flag type is discussed. Recent results of Anandan and Aharonov are commented vis-a-vis of last century constructions in projective geometry. Calculations with signi- ficance in coherent state approch furnish explicit proofs of the results announ- ced by Y. C. Wong on conjugate locus in complex Grassmann manifold.