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Generalized Heisenberg-Weyl Groups and Hermite Functions

by Enrico Celeghini, Manuel Gadella and Mariano A del Olmo

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Submission summary

Authors (as registered SciPost users): Mariano A. del Olmo
Submission information
Preprint Link: scipost_202212_00018v1  (pdf)
Date accepted: 2023-08-11
Date submitted: 2022-12-06 13:10
Submitted by: del Olmo, Mariano A.
Submitted to: SciPost Physics Proceedings
Proceedings issue: 34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022)
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approach: Theoretical


A generalisation of Euclidean and pseudo-Euclidean groups is presented, where the Weyl-Heisenberg groups, well known in quantum mechanics, are involved. A new family of groups is obtained including all the above-mentioned groups as subgroups. Symmetries, like self-similarity and invariance with respect to the orientation of the axes, are properly included in the structure of this new family of groups. Generalized Hermite functions on multidimensional spaces, which serve as orthogonal bases of Hilbert spaces supporting unitary irreducible representations of these new groups, are introduced.

Published as SciPost Phys. Proc. 14, 023 (2023)

Submission & Refereeing History

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Submission scipost_202212_00018v1 on 6 December 2022

Reports on this Submission

Anonymous Report 1 on 2022-12-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202212_00018v1, delivered 2022-12-30, doi: 10.21468/SciPost.Report.6408


The paper presents a unified approach of (symmetry) group generalizing the (pseudo-)Euclidean groups and based on the structural properties of the Heisenberg group. The construction is useful for physical applications, combining symmetry and invariance properties, provide an additional link with the theory of orthogonal functions.


As the procedure seems to be quite algorithmic, a brief comment on the n-dimensional case is missed, albeit the cases presented (1D and 2D) are quite explicit.


The paper meets the requirements for publication in the ICGTMP-34 proceedings. The paper is concisely written. Merely at the conclusions, the expression "origine groups" should better be replaced by "give rise to new groups".

Requested changes

Besides the above-mentioned correction, on line 13 of the Conclusions, replace Gelfand by Gel'fand.

  • validity: good
  • significance: good
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: good

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