Symmetry Groups, Quantum Mechanics and Generalized Hermite Functions

Enrico Celeghini, Manuel Gadella and Mariano A. del Olmo
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Enrico Celeghini: Dipartimento di Fisica, Università di Firenze and INFN-Sezione di Firenze, 150019 Sesto Fiorentino, FI, Italy
Manuel Gadella: Departamento de Física Teórica, Atómica y Optica and IMUVA, Universidad de Valladolid, 47011 Valladolid, Spain
Mariano A. del Olmo: Departamento de Física Teórica, Atómica y Optica and IMUVA, Universidad de Valladolid, 47011 Valladolid, Spain

Mathematics, 2022, vol. 10, issue 9, 1-21

Abstract: 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, H n , together with the Euclidean, E n , and pseudo-Euclidean E p , 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, K p , q , that contain H p , q and E p , 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 K p , 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 K p , q . By extending these Hilbert spaces, we obtain representations of K p , q on rigged Hilbert spaces (Gelfand triplets). We study the transformation laws of these generalized Hermite functions under Fourier transform.

Keywords: Euclidean and pesudo-Euclidean symmetry groups; generalized Hermite functions; rigged Hilbert spaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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