Casimir piston for massless scalar fields in three dimensions

We study the Casimir piston for massless scalar fields obeying Dirichlet boundary conditions in a three-dimensional cavity with sides of arbitrary lengths $a$, $b$, and $c$ where $a$ is the plate separation. We obtain an exact expression for the Casimir force on the piston valid for any values of the three lengths. As in the electromagnetic case with perfect-conductor conditions, we find that the Casimir force is negative (attractive) regardless of the values of $a$, $b$, and $c$. Though cases exist where the interior contributes a positive (repulsive) Casimir force, the total Casimir force on the piston is negative when the exterior contribution is included. We also obtain an alternative expression for the Casimir force that is useful computationally when the plate separation $a$ is large.

Figure 1

Casimir piston in three dimensions.

Figure 2

Casimir force $F$ versus $a/c$ for the case $b/c=1$ where $a$ is the plate separation and $b$ and $c$ are the sides of the base. The force is in units of $1/c^2$. The force is large and negative at small values of $a/c$ and remains negative with its magnitude decreasing quickly to zero as $a/c$ increases. One obtains a similar plot for any value of $b/c$ (see the 3D plot Fig. 3).

Figure 3

3D plot of Casimir force $F$ versus $a/c$ and $b/c$. The force is in units of $1/c^2$. For a given $b/c$, the profile is the same as in the 2D plot: the force is large and negative at small values of $a/c$ and remains negative with its magnitude decreasing quickly to zero as $a/c$ increases. The value of $b/c$ shifts the magnitude of the force towards larger values as it increases.