Quantum radiation force on a moving mirror with Dirichlet and Neumann boundary conditions at vacuum, finite temperature and coherent states

We consider a real massless scalar field in a two-dimensional spacetime, satisfying Dirichlet or Neumann boundary condition at the instantaneous position of a moving boundary. For a relativistic law of motion, we show that Dirichlet and Neumann boundary conditions yield the same radiation force on a moving mirror when the initial field state is invariant under time translations. We obtain the exact formulas for the energy density of the field and the radiation force on the boundary for vacuum, thermal and coherent state. In the nonrelativistic limit, our results coincide with those found in the literature.