738 research outputs found
Quantum Physics and Signal Processing in Rigged Hilbert Spaces by means of Special Functions, Lie Algebras and Fourier and Fourier-like Transforms
Quantum Mechanics and Signal Processing in the line R, are strictly related
to Fourier Transform and Weyl-Heisenberg algebra. We discuss here the addition
of a new discrete variable that measures the degree of the Hermite functions
and allows to obtain the projective algebra io(2). A Rigged Hilbert space is
found and a new discrete basis in R obtained. The operators {O[R]} defined on R
are shown to belong to the Universal Enveloping Algebra UEA[io(2)] allowing, in
this way, their algebraic discussion. Introducing in the half-line a
Fourier-like Transform, the procedure is extended to R^+ and can be easily
generalized to R^n and to spherical reference systems.Comment: 12 pages, Contribution to the 30th International Colloquium on Group
Theoretical Methods in Physics, July 14-18, 2014, Gent (Belgium
SU(2), Associated Laguerre Polynomials and Rigged Hilbert Spaces
We present a family of unitary irreducible representations of SU(2) realized
in the plane, in terms of the Laguerre polynomials. These functions are similar
to the spherical harmonics defined on the sphere. Relations with an space of
square integrable functions defined on the plane, , are analyzed. We
have also enlarged this study using rigged Hilbert spaces that allow to work
with iscrete and continuous bases like is the case here.Comment: 10 page
Intertwining Symmetry Algebras of Quantum Superintegrable Systems
We present an algebraic study of a kind of quantum systems belonging to a
family of superintegrable Hamiltonian systems in terms of shape-invariant
intertwinig operators, that span pairs of Lie algebras like or
. The eigenstates of the associated Hamiltonian
hierarchies belong to unitary representations of these algebras. It is shown
that these intertwining operators, related with separable coordinates for the
system, are very useful to determine eigenvalues and eigenfunctions of the
Hamiltonians in the hierarchy. An study of the corresponding superintegrable
classical systems is also included for the sake of completness
Symmetry Groups, Quantum Mechanics and Generalized Hermite Functions
This is a review paper on the generalization of Euclidean as well as pseudo-Euclidean groups of interest in quantum mechanics. The Weyl–Heisenberg groups, Hn, together with the Euclidean, En, and pseudo-Euclidean Ep,q, groups are two families of groups with a particular interest due to their applications in quantum physics. In the present manuscript, we show that, together, they give rise to a more general family of groups, Kp,q, that contain Hp,q and Ep,q as subgroups. It is noteworthy that properties such as self-similarity and invariance with respect to the orientation of the axes are properly included in the structure of Kp,q. We construct generalized Hermite functions on multidimensional spaces, which serve as orthogonal bases of Hilbert spaces supporting unitary irreducible representations of groups of the type Kp,q. By extending these Hilbert spaces, we obtain representations of Kp,q on rigged Hilbert spaces (Gelfand triplets). We study the transformation laws of these generalized Hermite functions under Fourier transform
Unitary-Preserving Holography in dS
This letter introduces a unitarity-preserving holographic correspondence
within a -dimensional de Sitter (dS) spacetime, distinctly challenging
the prevailing notion that the holographic framework of dS falls short in
maintaining unitarity. The proposed approach is rooted in the geometry of the
complex dS spacetime and leverages the inherent properties of the (global)
dS plane waves, as defined within their designated tube domains.Comment: 6 pages, no figur
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