Many-body contributions to Green’s functions and Casimir energies

The multiple scattering formalism is used to extract irreducible $N$-body parts of Green’s functions and Casimir energies describing the interaction of $N$ objects that are not necessarily mutually disjoint. The irreducible $N$-body scattering matrix is expressed in terms of single-body transition matrices. The irreducible $N$-body Casimir energy is the trace of the corresponding irreducible $N$-body part of the Green’s function. This formalism requires the solution of a set of linear integral equations. The irreducible three-body Green’s function and the corresponding Casimir energy of a massless scalar field interacting with potentials are obtained and evaluated for three parallel semitransparent plates. When Dirichlet boundary conditions are imposed on a plate the Green’s function and Casimir energy decouple into contributions from two disjoint regions. We also consider weakly interacting triangular and parabolic wedges placed atop a Dirichlet plate. The irreducible three-body Casimir energy of a triangular and parabolic wedge is minimal when the shorter side of the wedge is perpendicular to the Dirichlet plate. The irreducible three-body contribution to the vacuum energy is finite and positive in all the cases studied.

Figure 1

Three parallel plates. Plates $i$ and $k$ are separated by distances $a_{ij}$ and $a_{jk}$ from the center plate $j$.

Figure 2

The distances $d_{12}$ and ${\overline{d}}_{12}$. The effective distance ${\overline{d}}_{12}$ is the shortest distance between the two points for a path that reflects off the Dirichlet plate at $z=a_3$. It also is the shortest distance between $(y_1,z_1)$ and a mirror image of the point $(y_2,z_2)$ with respect to the $z=a_3$ line.

Figure 3

Weakly interacting triangular wedge on a Dirichlet plate. The objects are of infinite extent in the $x$-direction. The weakly interacting sides of the wedge (in red) have finite length.

Figure 4

Casimir Landscape: $\mathcal{E}(\alpha ,\beta )$ as a function of the opening angles $\alpha$ and $\beta$ for a weakly interacting triangular wedge on a Dirichlet plate. The valley connecting the $\alpha >\beta$ region with the $\alpha <\beta$ region is an artifact caused by limited numerical accuracy. The valley should be replaced by a very thin and infinitely high wall describing the sharp change in energy when all surfaces overlap. On the right, the shapes of the triangles are matched to the respective regions of the $\alpha$-$\beta$ plane.

Figure 5

$\mathcal{E}(\tilde{a},\tilde{b})$ as a function of $\tilde{a}$ for fixed area, $A=h^2$. The irreducible three-body Casimir energy is minimal when the shorter side of the wedge is perpendicular to the Dirichlet plate ($\tilde{a}=0$ or $\tilde{b}=0$). The maximum in the intermediate region corresponds to the unstable equilibrium of an isosceles triangle. The dashed curves are the approximation $\mathcal{E}(\tilde{a},\tilde{b})\sim |\tilde{a}\tilde{b}|$ obtained by replacing the integrals in Eq. (103) with unity. The dotted curves are reflections about the $\mathcal{E}=0$ line.

Figure 6

Irreducible three-body Casimir energy of a semiweak triangular waveguide as a function of the cross-sectional area and perimeter. This energy vanishes and is minimal along the line $\tilde{p}=1+2\tilde{s}+\sqrt{1+4{\tilde{s}}^2}$.

Figure 7

Weakly interacting parabolic wedge on a Dirichlet plate.

Figure 8

Irreducible three-body Casimir energy for a parabolic waveguide of given cross-sectional area. See Fig. 5 for description.