49 research outputs found
Characterization of SU(1,1) coherent states in terms of affine group wavelets
The Perelomov coherent states of SU(1,1) are labeled by elements of the
quotient of SU(1,1) by the compact subgroup. Taking advantage of the fact that
this quotient is isomorphic to the affine group of the real line, we are able
to parameterize the coherent states by elements of that group or equivalently
by points in the half-plane. Such a formulation permits to find new properties
of the SU(1,1) coherent states and to relate them to affine wavelets.Comment: 11 pages, latex, to be published in J. Phys. A : Math. Ge
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
Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry
A new method to obtain trigonometry for the real spaces of constant curvature
and metric of any (even degenerate) signature is presented. The method
encapsulates trigonometry for all these spaces into a single basic
trigonometric group equation. This brings to its logical end the idea of an
absolute trigonometry, and provides equations which hold true for the nine
two-dimensional spaces of constant curvature and any signature. This family of
spaces includes both relativistic and non-relativistic homogeneous spacetimes;
therefore a complete discussion of trigonometry in the six de Sitter,
minkowskian, Newton--Hooke and galilean spacetimes follow as particular
instances of the general approach. Any equation previously known for the three
classical riemannian spaces also has a version for the remaining six
spacetimes; in most cases these equations are new. Distinctive traits of the
method are universality and self-duality: every equation is meaningful for the
nine spaces at once, and displays explicitly invariance under a duality
transformation relating the nine spaces. The derivation of the single basic
trigonometric equation at group level, its translation to a set of equations
(cosine, sine and dual cosine laws) and the natural apparition of angular and
lateral excesses, area and coarea are explicitly discussed in detail. The
exposition also aims to introduce the main ideas of this direct group
theoretical way to trigonometry, and may well provide a path to systematically
study trigonometry for any homogeneous symmetric space.Comment: 51 pages, LaTe
Subnormal operators regarded as generalized observables and compound-system-type normal extension related to su(1,1)
In this paper, subnormal operators, not necessarily bounded, are discussed as
generalized observables. In order to describe not only the information about
the probability distribution of the output data of their measurement but also a
framework of their implementations, we introduce a new concept
compound-system-type normal extension, and we derive the compound-system-type
normal extension of a subnormal operator, which is defined from an irreducible
unitary representation of the algebra su(1,1). The squeezed states are
characterized as the eigenvectors of an operator from this viewpoint, and the
squeezed states in multi-particle systems are shown to be the eigenvectors of
the adjoints of these subnormal operators under a representation. The affine
coherent states are discussed in the same context, as well.Comment: LaTeX with iopart.cls, iopart12.clo, iopams.sty, The previous version
has some mistake