Casimir forces in a T-operator approach

We explore the scattering approach to Casimir forces. Our main tool is the description of Casimir energy in terms of transition operators. The approach is valid for scalar fields as well as for electromagnetic fields. We provide several equivalent derivations of the formula presented by Kenneth and Klich [Phys. Rev. Lett. 97, 160401 (2006)]. We study the convergence properties of the formula and how to utilize it together with scattering data to compute the force. Next, we discuss the form of the formula in special cases such as the simplified form obtained when a single object is placed next to a mirror. We illustrate the approach by describing the force between scatterers in one dimension and three dimensions, where we obtain the interaction energy between two spherical bodies at all distances. We also consider the cases of scalar Casimir effect between spherical bodies with different radii as well as different dielectric functions.

Figure 1

Bodies $A$ and $B$.

Figure 2

Coordinate system used for the partial wave approach.

Figure 3

(Color online) The calculated Casimir energy of the (a) two Dirichlet spheres of radius $R$ at distance $a$ between their centers. [(b) and (c)] A Dirichlet sphere of radius $R$ whose center is at a distance $a/2$ from a Dirichlet/Neumann mirror. The graphs show $E/E_0$ as a function of ${\left(a/R-1\right)}^{-1}$, where $E_0$ is the large distance asymptotic expression of it. Specifically, (a) $E_0^S=-\frac{R^2}{4\pi {\left(a-2R\right)}^{2}a}$ and [(b) and (c)] $E_0^{D,N}=\mp \frac{R}{2\pi {\left(a-2R\right)}^2}$. At short distances, $E/E_0$ approach the PFA prediction, (a) $\frac{{\pi }^4}{360}\sim 0.27$, (b) $\frac{{\pi }^4}{180}\sim 0.54$, and (c) $\frac{7{\pi }^4}{1440}\sim 0.47$. The black curve shows the calculated exact result for $l_0\to \infty$. We extrapolated it to $a=2R$ and $a=\infty$ using the known asymptotics. The colored graphs show how the computed energy increases as a result of including partial waves of $l\le l_0$ where $l_0=0$ (red), $l_0=1$ (sky blue), $l_0=2$ (green), and $l_0=10$ (blue).

Figure 4

(Color online) The calculated Casimir energy of two (scalar) Dielectric spheres of radii $R_2=2R_1$ at centers distance $a=4R_1$ depicted as a function of their dielectric constant ${ϵ}_1={ϵ}_2$. The energy $E$ was normalized by the Dirichlet spheres result $E_0$ so that at $ϵ\to \infty$, we obtain $E/E_0=1$.

Figure 5

(Color online) EM Casimir energy of two conducting spheres of radius $R$ at distance $a$ between their centers. The graphs show $E/E_0$ as a function of ${\left(a/R-1\right)}^{-1}$, where $E_0$ is the large distance asymptotic expression of it. $E_0^S=-\frac{143R^6}{16\pi {\left(a-2R\right)}^2{a}^5}$. The black curve shows the calculated exact result for $j_0\to \infty$. We extrapolated it to $a=2R$ and $a=\infty$ using the known asymptotics. The colored graphs show the result of including partial waves of $j\le j_0$ where $j_0=1$ (red), $j_0=2$ (sky blue), $j_0=4$ (green), and $j_0=10$ (blue).

Figure 6

(Color online) Including partial waves of $j\le j_0$ results in error behaving roughly as $e^{-c{j}_0}$. The graph shows the constant $c$ as a function of the separation distance $d$. The blue dots correspond to our results for two conducting spheres (where $d=a-2R$) and the red dots to conducting sphere $+$ conducting plate (where $d=a/2-R$). At small distances, both cases give $c\sim 1.7d/R$.