Functional approach to quantum friction: effective action and dissipative force

We study the Casimir friction due to the relative, uniform, lateral motion of two parallel semitransparent mirrors coupled to a vacuum real scalar field, $\phi$. We follow a functional approach, whereby nonlocal terms in the action for $\phi$, concentrated on the mirrors' locii, appear after functional integration of the microscopic degrees of freedom. This action for $\phi$, which incorporates the relevant properties of the mirrors, is then used as the starting point for two complementary evaluations: Firstly, we calculate the { in-out} effective action for the system, which develops an imaginary part, hence a non-vanishing probability for the decay (because of friction) of the initial vacuum state. Secondly, we evaluate another observable: the vacuum expectation value of the frictional force, using the { in-in} or Closed Time Path formalism. Explicit results are presented for zero-width mirrors and half-spaces, in a model where the microscopic degrees of freedom at the mirrors are a set of identical quantum harmonic oscillators, linearly coupled to $\phi$