Scalar field dynamics in a BTZ background with generic boundary conditions

The revisit the dynamics of a massive scalar field in a BTZ background taking into account the lack of global hyperbolicity of the spacetime. We approach this issue using the strategy of Ishibashi and Wald which finds a unique smooth solution as the causal evolution of initial data, each possible evolution corresponding to a positive self-adjoint extension of certain operator in a Hilbert space on the initial surface. Moreover, solutions obtained this way are the most general ones satisfying a few physically-sensible requirements. This procedure is intimately related to the choice of boundary conditions and the existence of bound states. We find that the scalar field dynamics in the (effective) mass window $-3/4\leq m_e^2\ell^2<0$ can be well-defined within a one-parametric family of distinct boundary conditions ($-3/4$ being the conformally-coupled case), while for $m_e^2\ell^2\geq 0$ the boundary condition is unique (only one self-adjoint extension is possible). It is argued that there is no sensible evolution possible for $m_e^2\ell^2<-1$, and also shown that in the range $m_e^2\ell^2 \in [-1,-3/4)$ there is a U$(1)$ family of allowed boundary conditions, however, the positivity of the self-adjoint extensions is only motivated but not proven. We focus mainly in describing the dynamics of such evolutions given the initial data and all possible boundary conditions, and in particular we show the energy is always positive and conserved.