Dirac Equation in Riemannian Manifolds: An Explicit Analytical Solution in AdS(1, 2) Spacetime

We present and discuss the covariant spin-connection Dirac equation in riemannian manifolds. As a particular case we present the Anti-de Sitter n-spacetime, [AdS(n)], which is, together with the de Sitter and Minkowski, the most maximallysymmetric space with positive cosmological constant $\Lambda$. Some asymptotic properties are also discussed. We parameterize AdS(1,2) manifold with cartesian coordinates associated to tangent spaces and introduce the conformal time. After constructing a suitable 2-dimensional Dirac gamma matrices representation, we can separate the Dirac equation and solve it, obtaining a hypergeometric equation. We obtain the Dirac spinor in AdS(1,2) spacetime explicitly, as $\lambda_x$-periodic function of space and a hypergeometric function of time and the radius of AdS(1,2) spacetime.