On Orthogonal Projections of Symplectic Balls

Nuno C. Dias; Maurice A. de Gosson; João N. Prata

doi:10.5802/crmath.542

Article de recherche - Géométrie et Topologie, Physique mathématique

On Orthogonal Projections of Symplectic Balls
[Sur les projections orthogonales de boules symplectiques]

Nuno C. Dias ^1,² ; Maurice A. de Gosson ³ ; João N. Prata ^1,²

¹ Grupo de Física Matemática, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
² Escola Superior Náutica Infante D. Henrique. Av. Eng. Bonneville Franco, 2770-058 Paço d’Arcos, Portugal
³ University of Vienna, Faculty of Mathematics (NuHAG), Vienna, Austria

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 217-227.

Résumé
Abstract

Nous étudions les projections orthogonales de boules symplectiques dans $ℝ^{2 n}$ sur des sous-espaces complexes. En particulier, nous montrons que ces projections sont elles-mêmes des boules symplectiques sous une certaine hypothèse de complexité. Notre résultat principal est une amélioration d’un résultat récent et très intéressant d’Abbondandolo et Matveyev, qui étend la version linéaire du théorème de non-plongement de Gromov. Nous utilisons une approche conceptuellement plus simple où le complément de Schur d’une matrice joue un rôle central. Une application aux traces partielles de matrices densité est donnée.

We study the orthogonal projections of symplectic balls in $ℝ^{2 n}$ on complex subspaces. In particular we show that these projections are themselves symplectic balls under a certain complexity assumption. Our main result is a refinement of a recent very interesting result of Abbondandolo and Matveyev extending the linear version of Gromov’s non-squeezing theorem. We use a conceptually simpler approach where the Schur complement of a matrix plays a central role. An application to the partial traces of density matrices is given.

Reçu le : 2023-03-09
Accepté le : 2023-07-20
Publié le : 2024-05-02

DOI : 10.5802/crmath.542

Keywords: Symplectic ball, orthogonal projection, Gromov’s non-squeezing theorem
Mots-clés : boule symplectique, projection orthogonale, théorème de non-plongement de Gromov

Affiliations des auteurs :

Nuno C. Dias ^{1, 2} ; Maurice A. de Gosson ³ ; João N. Prata ^{1, 2}

¹ Grupo de Física Matemática, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
² Escola Superior Náutica Infante D. Henrique. Av. Eng. Bonneville Franco, 2770-058 Paço d’Arcos, Portugal
³ University of Vienna, Faculty of Mathematics (NuHAG), Vienna, Austria

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Nuno C. Dias and Maurice A. de Gosson and Jo\~ao N. Prata},
     title = {On {Orthogonal} {Projections} of {Symplectic} {Balls}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {217--227},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.542},
     language = {en},
}

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Nuno C. Dias; Maurice A. de Gosson; João N. Prata. On Orthogonal Projections of Symplectic Balls. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 217-227. doi : 10.5802/crmath.542. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.542/

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