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

Automated summary

This paper reviews the generalization of Euclidean and pseudo-Euclidean groups in quantum mechanics. The study finds that these groups give rise to a more general family of groups, with Euclidean and pseudo-Euclidean groups as subgroups. The paper also constructs generalized Hermite functions on multidimensional spaces and investigates their transformation laws under Fourier transform.