Geometrical phases on hermitian symmetric spaces

2008

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.

arXiv (Cornell University), 1999

On certain manifolds, the phase which appears in the scalar product of two coherent state vectors is twice the symplectic area of the geodesic triangle determined by the corresponding points on the manifold and the origin of the system of coordinates. This result is proved for compact Hermitian symmetric spaces using the generalization via coherent states of the shape invariant for geodesic triangles and re-obtained on the complex Grassmannian by brute-force calculation.

arXiv preprint math/9903190, 1999

Abstract: On certain manifolds, the phase which appears in the scalar product of two coherent state vectors is twice the symplectic area of the geodesic triangle determined by the corresponding points on the manifold and the origin of the system of coordinates. This ...

Geometriae Dedicata, 1997

This paper contains a proof of the following property of compact irreducible Hermitian symmetric spaces. If H=G/K where G is a compact simply connected simple Lie group, T a maximal torus of G and F(T,H)=|E 1,...,E m is the fixed point set of T on H, then for each pair E i , E j there is a 2-dimentional sphere N ij ⊂ H such that E i and E j are antipodal points of N ij.

Journal of Symplectic Geometry, 2015

Inspired by the work of G. Lu [34] on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer-Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also compute the Gromov width and the Hofer-Zehnder capacity for Cartan domains and their products.

Pramana, 1997

We study the behaviour of the geometric phase under isometries of the ray space. This leads to a better understanding of a theorem first proved by Wigner: isometries of the ray space can always be realised as projections of unitary or anti-unitary transformations on the Hilbert space. We suggest that the construction involved in Wigner's proof is best viewed as an use of the Pancharatnam connection to "lift" a ray space isometry to the Hilbert space.

Journal of Geometry and Physics, 2007

The geometry of Grassmann manifolds Gr K (H), of orthogonal projection manifolds P K (H) and of Stiefel bundles St(K , H) is reviewed for infinite dimensional Hilbert spaces K and H. Given a loop of projections, we study Hamiltonians whose evolution generates a geometric phase, i.e. the holonomy of the loop. The simple case of geodesic loops is considered and the consistence of the geodesic holonomy group is discussed. This group agrees with the entire U (K) if H is finite dimensional or if dim(K) ≤ dim(K ⊥). In the remaining case we show that the holonomy group is contained in the unitary Fredholm group U ∞ (K) and that the geodesic holonomy group is dense in U ∞ (K).

Transformation Groups, 2007

We characterize irreducible Hermitian symmetric spaces which are not of tube type both in terms of the topology of the space of triples of pairwise transverse points in the Shilov boundary, and of two invariants which we introduce, the Hermitian triple product and its complexification.

Journal of Geometry and Physics, 1993

We show that an inner symmetric space with a compatible Hermitian structure is necessarily Hermitian symmetric, and the Hermitian structure must be invariant. This last result was known for some of the spaces of classical type and conjectured to be true for all compact Hermitian symmetric spaces in [2].

1995

The almost Hermitian symmetric structures include several important geometries,e.g. the conformal, projective, quaternionicor almost Grassmannianones. The conformal case is known best and several e cient techniques have been worked out in the last 90 years. The present note provides links of the development presented in CSS1, CSS2, CSS3, CSS4] to several other approaches and it suggests extensions of some techniques to all geometries in question.