Vacuum Fluctuations and Boundary Conditions in a Global Monopole

We study the vacuum fluctuations of a massless scalar field $\hat{\Psi}$ on the background of a global monopole. Due to the nontrivial topology of the global monopole spacetime, characterized by a solid deficit angle parametrized by $\eta^2$, we expect that $\left<\hat{\Psi}^2\right>_{\text{ren}}$ and $\left<\hat{T}_{\mu\nu}\right>_{\text{ren}}$ are nonzero and proportional to $\eta^2$, so that they annul in the Minkowski limit $\eta\to0$. However, due to the naked singularity at the monopole core, the evolution of the scalar field is not unique. In fact, they are in one to one correspondence with the boundary conditions which turn into self-adjoint the spatial part of the wave operator. We show that only Dirichlet boundary condition corresponds to our expectations and gives zero contribution to the vacuum fluctuations in Minkowski limit. All other boundary conditions give nonzero contributions in this limit due to the nontrivial interaction between the field and the singularity.