A non group theoretic proof of completeness of arbitrary coherent states D(α) | f > Abstract A new proof for the completeness of the coherent states D(α) | f > for the Heisenberg Weyl group and the groups SU (2) and SU (1, 1) is presented. Generalizations of these results and their consequences are disussed.

We show that the Schrödinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extended Weyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians.

In this paper a new quasi-triangular Hopf algebra as the quantum double of the Heisenberg-Weyl algebra is presented.Its universal R-matrix is built and the corresponding representation theory are studied with the explict construction for the representations of this quantum double. 1Permanet address:Physics Dpartment,Northeast Normal University, Changchun 130024,P.R.China 2Permanet address:Theor...

The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the c...

In this paper we extend the Balian–Low theorem, which is a version of the uncertainty principle for Gabor (Weyl–Heisenberg) systems, to functions of several variables. In particular, we first prove the Balian–Low theorem for arbitrary quadratic forms. Then we generalize further and prove the Balian– Low theorem for differential operators associated with a symplectic basis for the symplectic for...

This paper is concerned with generalizations and specific applications of the coorbit space theory based on group representations modulo quotients that has been developed quite recently. We show that the general theory applied to the affine Weyl–Heisenberg group gives rise to families of smoothness spaces that can be identified with α-modulation spaces.

The unitary representations of the Canonical group that is defined to be the semidirect product of the unitary group with the Weyl-Heisenberg group, !!1, 3" ! "!1, 3""s #!1, 3" are studied. The Canonical group is equivalently written as the semi-direct product of the special unitary group with the Oscillator group !!1, 3" ! $"!1, 3""s %&!1, 3" . The non-abelian Weyl-Heisenberg group represents ...

We study singly-generated wavelet systems on R that are naturally associated with rank-one wavelet systems on the Heisenberg group N . We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset I of the dual of N , we give an explicit construction for Parseval frame wavelets that are associated with I . We say that g ∈ L(I×R) is Ga...

It is known that if $$d^2$$ vectors in a d-dimensional Hilbert space H form symmetric, informationally complete, positive operator-valued measure (SIC-POVM), then the tensor squares of these an equiangular tight frame symmetric subspace $$H\otimes H$$ . We prove that, for any SIC-POVM Weyl–Heisenberg group covariant type (WH-type), this can be obtained by projecting WH-type basis onto subspace....

This paper aims to explore the inherent connection between Heisenberg groups, quantum Fourier transform (QFT) and (quasi-probability) distribution functions. Distribution functions for continuous and finite quantum systems are examined from three perspectives and all of them lead to Weyl–Gabor–Heisenberg groups. The QFT appears as the intertwining operator of two equivalent representations aris...