The Dirac system on the Anti-de Sitter Universe

Details The Dirac system on the Anti-de Sitter Universe The Dirac system on the Anti-de Sitter Universe

2007-06-09

arxiv.org/abs/0706.1315v3

Commun.Math.Phys.283:127-167,2008

Physics

Mathematical Physics

33 pages

Scientific paper

10.1007/s00220-008-0564-8

We investigate the global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, {\it a priori}, well-posed. Nevertheless we can prove that there exists unitary dynamics, but its uniqueness crucially depends on the ratio beween the mass $M$ of the field and the cosmological constant $\Lambda>0$ : it appears a critical value, $\Lambda/12$, which plays a role similar to the Breitenlohner-Freedman bound for the scalar fields. When $M^2\geq \Lambda/12$ there exists a unique unitary dynamics. In opposite, for the light fermions satisfying $M^2<\Lambda/12$, 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 prove a result of equipartition of the energy.

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