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 $\varphi$. We follow a functional approach, whereby nonlocal terms in the action for $\varphi$, concentrated on the mirrors’ loci, appear after functional integration of the microscopic degrees of freedom. This action for $\varphi$, 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 nonvanishing 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 $\varphi$.

Figure 1

Singularities of $f(p^0)$ [Eq. (38)] in the complex $p^0$ plane. Simple poles are depicted as filled dots, while the branch cuts are represented by dashed lines. We have introduced the notation $u^{\pm }=u{p}^1\pm \mathrm{\Omega }$.

Figure 2

Imaginary part of the effective action for thin mirrors, as a function of $u$, with $\mathrm{\Omega }a=0.01$. $A$ is the global factor $A=\frac{g^{4}T\mathrm{\Sigma }(\mathrm{\Omega }a{)}^3{\pi }^2}{4a^3{\mathrm{\Omega }}^6}$. The imaginary part of the effective action, and hence the dissipative effects, are strongly suppressed for small values of the velocity between the plates.

Figure 3

Imaginary part of the effective action for half-spaces, as a function of $u$, with $\mathrm{\Omega }a=0.01$. $A$ is the global factor $A=\frac{g^{4}T\mathrm{\Sigma }(\mathrm{\Omega }a{)}^5{\pi }^2}{16a^3{\mathrm{\Omega }}^8}$. The imaginary part of the effective action, and hence the dissipative effects, are strongly suppressed for small values of the velocity between the plates.

Figure 4

Modulus of the dissipative force, as a function of the relative velocity of the plates, for $\mathrm{\Omega }a=0.01$. The global factor is $A=\frac{g^4(\mathrm{\Omega }a{)}^5}{4{\mathrm{\Omega }}^6\pi a^4}$. The frictional force is practically negligible for small values of the velocity between the plates.

Figure 5

Modulus of the dissipative force for half-spaces, as a function of the relative velocity of the plates, for $\mathrm{\Omega }a=0.01$. The global factor is $A=\frac{g^4(\mathrm{\Omega }a{)}^5}{16{\mathrm{\Omega }}^8\pi a^4}$. The frictional force is practically negligible for small values of the velocity between the plates.