unit time: $$E_{\rm Casimir}=\frac{\Gamma[V]}{\int_{\tau=\tau_-}^{\tau_+}\,d\tau}$$

The “Worldline Numerics” approach by Gies, Langfeld, and Moyaerts [@Gies:2003] is based on re-writing the logarithm in the effective action as an integral a la $\ln
(p/q) \rightarrow \int_0^{\infty} \frac{dT}{T}\left(\exp(-px)-\exp(-qx)\right)$ and re-expressing the operator trace as a Feynman path integral. This leads, for a real scalar field of mass $m$ interacting only with the external potential $V\!$, to an expression for the effective action that is numerically tractable via Monte Carlo methods. The key expression from [@Gies:2003], which we repeat here for the convenience of the readers, is: $$\label{eq:effective-action}
\Gamma_\Lambda[V] = - \frac 12 \frac 1{(4\pi)^{2}}\int_{1/\Lambda^2}^\infty
\frac{dT}{T^{3}} e^{-m^2T}\int d^4x
\left[\left\langle W_V[y;x,T]\right\rangle_y - 1\right]$$ where an UV cut-off regulator $\Lambda$ has been introduced. Here, the expectation value $\langle\cdot\rangle_y$ is the ensemble average over all closed loop (c.l.) gaussian random walks $y:[0;1]\mapsto\mathbb R, y(0)=y(1)$ of “Wilson loops” re-scaled to proper time $T$. Let the statistical weight of the loop $y$ be $$p[y]=\exp\left(-\int_{t=0}^{t=1}dt\,\dot y(t)^2/4\right)$$ then: $$\left\langle W_V[y;x,T]\right\rangle_y=
\frac{\int_{y\;\rm c.l.}\mathcal{D}y\,W_V[y;x,T]\,p[y]}
{\int_{y\;\rm c.l.}\mathcal{D}y\,p[y]}$$ where $W_V$ depends on the *path* $y$ and on position shift $x$ and propertime $T$: $$W_V[y;x,T]=\exp\left(-T\int_{t=0}^{t=1}dt\,V(x+\sqrt{T}y(t))\right).$$

From this expression for the effective action of a free scalar interacting with a potential, Casimir forces can be obtained by using 