Dynamics in anti-de Sitter spacetimes and representations of SL(2,R)

We present the analysis of the dynamics for a scalar field in the universal covering space of $N$--dimensional anti-de Sitter spacetime, $AdS_N$ ($N\geq 2$), and for a spinor field satisfying the Dirac equation in the universal covering space of two-dimensional anti-de Sitter spacetime, $AdS_2$. We apply a prescription for dynamics in static, non-globally hyperbolic spacetimes based on the theory of self-adjoint extensions of operators on Hilbert spaces. This prescription results in a family of field theories with a well-defined initial value problem despite the lack of global-hyperbolicity of the spacetime manifold. Then, we impose the invariance of the associated solution spaces under the infinitesimal action of the isometry group of $AdS_N$ ($\widetilde{\mathrm{SL}}(2,\mathbb{R})$ for $N=2$ and $\widetilde{\mathrm{SO}}(2,N-1)$ for $N\geq 3$) to determine which among the family of theories obtained by the prescription for dynamics can be used to construct a quantum field theory with a stationary vacuum state.