the position dependency of the potential $V(x)$ to model the geometry, and calculating energy changes associated with changes to the geometry.

While the applicability of this model for the calculation of real Casimir forces is questionable (even for perfect conductors) as the physics of photons is quite different from that of a scalar field, the remarkable conceptual simplicity of the above expressions certainly warrants a deeper investigation of its properties and potential utility, for it might actually allow a (yet undiscovered) generalization to the photon case. For the electromagnetic case, one would naturally want to start with investigations of perfect conductor surfaces, and the (obvious) scalar pendant of this is a potential $V(x)$ that suppresses all quantum fluctuations *inside* the given bodies. It is not difficult to see that one may alternatively restrict the potential to have non-zero values only close to the surfaces of bodies, taking $$V(x) = \lambda \int_\Sigma d^2\sigma \delta^3(x-x_\sigma)$$ and considering the limit $\lambda\to\infty$. Then, $W_V[y;x,T]$ reduces to: $$\exp\left[-T\int_0^1 dt V(x+\sqrt T y(t))\right]
  =
  \begin{cases}
    1 & \text{Loop pierces a surface}\\
    0 & \text{Loop does not pierce a surface.}
  \end{cases}$$ Substituting $s=\sqrt{T}$ to eliminate the square root, and using translation invariance of the integral to ensure 