Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915691585601536 |
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| author | Beauvillain, Mathieu Oblak, Blagoje Petropoulos, Marios |
| author_facet | Beauvillain, Mathieu Oblak, Blagoje Petropoulos, Marios |
| contents | Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19840 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms Beauvillain, Mathieu Oblak, Blagoje Petropoulos, Marios Quantum Physics High Energy Physics - Theory Mathematical Physics Classical Analysis and ODEs Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds. |
| title | Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms |
| topic | Quantum Physics High Energy Physics - Theory Mathematical Physics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2512.19840 |