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