$3D$ scalar model as a $4D$ perfect conductor limit: Dimensional reduction and variational boundary conditions

Under dimensional reduction, a system in $D$ spacetime dimensions will not necessarily yield its $D-1$-dimensional analog version. Among other things, this result will depend on the boundary conditions and the dimension $D$ of the system. We investigate this question for scalar and Abelian gauge fields under boundary conditions that obey the symmetries of the action. We apply our findings to the Casimir piston, an ideal system for detecting boundary effects. Our investigation is not limited to extra dimensions and we show that the original piston scenario proposed in 2004, a toy model involving a scalar field in $3D$ ($2+1$) dimensions, can be obtained via dimensional reduction from a more realistic $4D$ electromagnetic (EM) system. We show that for perfect conductor conditions, a $D$-dimensional EM field reduces to a $D-1$ scalar field and not its lower-dimensional version. For Dirichlet boundary conditions, no theory is recovered under dimensional reduction and the Casimir pressure goes to zero in any dimension. This “zero Dirichlet” result is useful for understanding the EM case. We then identify two special systems where the lower-dimensional version is recovered in any dimension: systems with perfect magnetic conductor (PMC) and Neumann boundary conditions. We show that these two boundary conditions can be obtained from a variational procedure in which the action vanishes outside the bounded region. The fields are free to vary on the surface and have zero modes, which survive after dimensional reduction.

Figure 1

piston geometry: (a) Piston geometry in two spatial dimensions. The length $s$ is taken to be infinite, $a$ is the plate separation and $b$ is the length of the second side. The $3D$ Dirichlet piston is a scalar field obeying Dirichlet boundary conditions on all sides. (b) Piston geometry in three spatial dimensions. The plate separation is $a$ and $s$ is taken to be infinite. The $4D$ PEC piston is an EM field obeying perfectly conducting conditions on all walls. In the limit as one of the sides ($b$ or $c$) goes to zero, we recover the $3D$ Dirichlet piston.