The Haar measure of the $p$-adic rotation group $\textrm{SO}(3)_p$ via nautical angles
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914526249615360 |
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| author | Guglielmi, Lorenzo Mancini, Stefano Parisi, Vincenzo Svampa, Ilaria |
| author_facet | Guglielmi, Lorenzo Mancini, Stefano Parisi, Vincenzo Svampa, Ilaria |
| contents | We study the explicit construction of the Haar measure on the compact $p$-adic rotation group $\textrm{SO}(3)_p$ by nautical (Cardano) parametrization. Exploiting its topological group isomorphism with $\mathbb{H}_p^\times/\mathbb{Q}_p^\times$ of $p$-adic quaternions modulo scalars, we derive the corresponding change of variables formulas and compute the associated Jacobian in the $p$-adic setting, which we combine with the known Haar measure on the multiplicative group of $p$-adic quaternions $\mathbb{H}_p^\times$. This yields an explicit formula for the normalized Haar measure on $\textrm{SO}(3)_p$ in nautical coordinates, with a factorized density in the three angles. Our construction provides a concrete tool suited for applications of non-Archimedean models where an explicit angular description of invariant integration is required. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01385 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Haar measure of the $p$-adic rotation group $\textrm{SO}(3)_p$ via nautical angles Guglielmi, Lorenzo Mancini, Stefano Parisi, Vincenzo Svampa, Ilaria Mathematical Physics We study the explicit construction of the Haar measure on the compact $p$-adic rotation group $\textrm{SO}(3)_p$ by nautical (Cardano) parametrization. Exploiting its topological group isomorphism with $\mathbb{H}_p^\times/\mathbb{Q}_p^\times$ of $p$-adic quaternions modulo scalars, we derive the corresponding change of variables formulas and compute the associated Jacobian in the $p$-adic setting, which we combine with the known Haar measure on the multiplicative group of $p$-adic quaternions $\mathbb{H}_p^\times$. This yields an explicit formula for the normalized Haar measure on $\textrm{SO}(3)_p$ in nautical coordinates, with a factorized density in the three angles. Our construction provides a concrete tool suited for applications of non-Archimedean models where an explicit angular description of invariant integration is required. |
| title | The Haar measure of the $p$-adic rotation group $\textrm{SO}(3)_p$ via nautical angles |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2605.01385 |