Casimir dynamics: Interactions of surfaces with codimension $>1$ due to quantum fluctuations

We study the Casimir force between defects (branes) of codimension larger than 1 due to quantum fluctuations of a scalar field $\varphi$ living in the bulk. We show that the Casimir force is attractive and that it diverges as the distance between the branes approaches a critical value $L_c$. Below this critical distance $L_c$ the vacuum state $\varphi =0$ of the theory is unstable, due to the birth of a tachyon, and the field condenses.

Figure 1

The interaction energy, in arbitrary units, for three delta functions on the line as a function of the position $x$ of one of them and $m=0$. $\alpha =-1$ for all three deltas, one delta is held fixed at $x=0$, another at $x=5$.

Figure 2

Interaction energy $\mathcal{E}$ (the continuous line is $\mathrm{Re}\mathcal{E}$ and the dashed line is $-\mathrm{Im}\mathcal{E}$) in units of $1/L_c$, for two delta functions as a function of their distance $L$. (a) The ${\mathbb{R}}^3$ case with $m=0$. (b) The ${\mathbb{R}}^2$ case with $m/M=2$.