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We study the solutions to the Dirac equation for the massive spinor field inthe universal covering space of two-dimensional anti-de Sitter space. Forcertain values of the mass parameter, we impose a suitable set of boundaryconditions which make the spatial component of the Dirac operator self-adjoint.Then, we use the transformation properties of the spinor field under theisometry group of the theory, namely, the universal covering group of$\mathrm{SL}(2,\mathbb{R})$, in order to determine which self-adjoint boundaryconditions are invariant under this group. We identify the correspondingsolution spaces with unitary irreducible representations of this group usingthe classification given by Pukanzki, and determine which of these correspondto invariant positive- and negative-frequency subspaces and, hence, in a vacuumstate invariant under the isometry group. Finally, we examine the cases wherethe self-adjoint boundary condition leads to an invariant theory withnon-invariant vacuum state and determine the unitary representation to whichthe vacuum state belongs.

Dirac field in $\mathrm{AdS}_2$ and representations of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$