A functional approach to quantum friction: effective action and dissipative force

Abstract

We study the Casimir friction due to the relative, uniform, lateral motion of two parallel semitransparent mirrors coupled to a vacuum real scalar field, φ. We follow a functional approach, whereby nonlocal terms in the action for φ, concentrated on the mirrors’ locii, appear after functional integration of the microscopic degrees of freedom. This action for φ, 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 φ.