Dirac field in AdS2 and representations of SL̃(2,R)

Abstract

We analyze a massive spinor field satisfying the Dirac equation in the universal covering space of two-dimensional anti-de Sitter space. In order to obtain well-defined dynamics for the classical field despite the lack of global-hyperbolicity of the spacetime, we impose a suitable set of boundary conditions that render the spatial component of the Dirac operator self-adjoint. Then, we find which of the solution spaces obtained by imposing the self-adjoint boundary conditions are invariant under the action of the isometry group of the spacetime manifold, namely, the universal covering group of SL(2,R). The invariant solution spaces are then identified with unitary irreducible representations of this group using the classification given by Pukánszky [Math. Ann. 156, 96–143 (1964)]. We determine which of these correspond to invariant positive- or negative-frequency subspaces and, thus, result in a vacuum state invariant under the isometry group after canonical quantization. Additionally, we examine the invariant theories obtained from the self-adjoint boundary conditions, which result in a non-invariant vacuum state, identifying the unitary representation this state belongs to.