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Comptes Rendus Mathématique
Volume 345, n° 8
pages 435-440 (octobre 2007)
Doi : 10.1016/j.crma.2007.09.011
Received : 16 May 2007 ;  accepted : 12 September 2007
The Dirac equation on the Anti-de-Sitter Universe
Lʼéquation de Dirac sur lʼunivers Anti-de Sitter
 

Alain Bachelot
Université de Bordeaux, Institut de mathématiques, 33405 Talence cedex, France 

Abstract

We investigate global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, a priori, well-posed. Nevertheless, this is the case when the mass of the field is large compared to the cosmological constant. In opposite, for the light fermions, we construct several asymptotic conditions at infinity, such that the problem becomes well-posed. In all the cases, the spectrum of the Hamiltonian is discrete. We also get a result of equipartition of the energy. To cite this article: A. Bachelot, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

The full text of this article is available in PDF format.
Résumé

Nous cherchons des solutions globales de lʼéquation de Dirac dans lʼunivers Anti-de Sitter. Comme cet espace nʼest pas globalement hyperbolique, le problème de Cauchy nʼest pas, a priori, bien posé. Nous montrons que cʼest toutefois le cas quand la masse du champ est grande par rapport à la constante cosmologique. En revanche, pour les faibles masses, nous construisons diverses conditions asymptotiques à lʼinfini, rendant le problème bien posé. Dans tous les cas, lʼhamiltonien a un spectre discret. On établit également un résultat dʼéquipartition de lʼénergie. Pour citer cet article : A. Bachelot, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

The full text of this article is available in PDF format.


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