The string theory inspired worldline numerics approach to determine the Casimir effect is developed in detail in [5], and the critical aspects of the analysis technique are briefly summarized here for convenience. With the objective of evaluating the Casimir interaction energy \(E_{\text {Casimir}}\) (e.g. normalized) arising from the coupling of a real scalar quantum field \(\phi \) of finite mass m with a background potential V(x) that represents the Casimir geometry, the key equation from Section 2.1 of [5] is the effective action shown in Eq. (1).
$$\begin{aligned} \varGamma [V]= & {} -\tfrac{1}{2(4\pi )^{2}} \int _{1/\Lambda ^2}^\infty \tfrac{dT}{T^{3}} \, e^{-m^2T}\int d^4 x \nonumber \\&\times \left[ \Big \langle \,W_V[y(t);x]\, \Big \rangle _y -1\right] \end{aligned}$$
(1)
The expectation value in Eq. (1) is the average of the loop ensemble over all closed loops with Gaussian walks:
$$\begin{aligned} \Big \langle W_V[y(t);x]\Big \rangle _y = \frac{ \int \nolimits _{y(0)=y(1)} {\mathscr {D}}y\,\, W_V[y(t);x] \,e^{-\int _0^1 dt\, \dot{y}^2/4}}{\int \nolimits _{y(0)=y(1)} {\mathscr {D}}y\, \,e^{-\int _0^1 dt\, \dot{y}^2/4}}, \end{aligned}$$
(2)
where the following “Wilson loop” identity has been introduced with y representing the (unit) loop pathFootnote 4, x representing the position shift of the unit loop in model space, and T denoting the proper time and serves to scale the unit loops:
$$\begin{aligned} W_V[y(t);x]=\exp \left[ -T \int _0^1 dt\, V(x+\sqrt{T} y(t))\right] . \end{aligned}$$
(3)
Equipped with this information, one can calculate the (unrenormalized) Casimir energy as \({\mathscr {E}}=\varGamma /\int dx_0\) where the integral represents the “volume” in the time direction. When considering the Casimir force, the portion of the Casimir energy that has a dependency on the relative positions of the bounding geometries can be obtained by subtracting the energies of the single objects from the total Casimir energy.
$$\begin{aligned} {E_{\text {Casimir}}{:}{=}{\mathscr {E}}_{\text {V}_{1}+\text {V}_{2}+\cdots }-{\mathscr {E}}_{\text {V}_{1}}-{\mathscr {E}}_{\text {V}_{2}}-\cdots } \end{aligned}$$
(4)
The Casimir force can be obtained by taking the negative spatial derivative of this interaction energy, and further, this process has removed any UV divergences. In the Dirichlet limit \(\lambda \rightarrow \infty \) and for a massless scalar field with Dirichlet boundaries in D = 3+1, the worldline representation of the Casimir interaction energy boils down toFootnote 5:
$$\begin{aligned} E_{\text {Casimir}}=-\tfrac{1}{2(4\pi )^2}\int _0^{\infty } \tfrac{dT}{T^3} \int d^3 x_{CM} \langle \Theta _{\varSigma }[\mathbf{x }(\tau )] \rangle _x \end{aligned}$$
(5)
The worldline functional \(\Theta _{\varSigma }[\mathbf{x }(\tau )] = 0\) if the re-scaled unit loop does not intersect any Casimir geometry and \(\Theta _{\varSigma }[\mathbf{x }(\tau )]=1-n\) if the re-scaled loop intersects \(n\ge 1\) Casimir geometry. The numerical evaluation process requires two discretization steps. The first is the discretization of the path integral into an ensemble of \(n_{\text {L}}\) random paths \(\mathbf{x }_{\ell }(\tau )\), \(\ell = 1,\ldots ,n_{\text {L}}\) with each path forming a closed spacetime loop. The second is the discretization of the proper time interval \(\tau \in [0,T]\) into N steps such that an individual closed loop consists of N points per loop: \(\mathbf{x }_{\ell k } {:}{=}\mathbf{x }_{\ell }( k \cdot T/N), k =1,\ldots N\). Transporting and rescaling the ensemble of unit loops to a point \(\mathbf{x }_{\text {CM}}\) in the model takes the following form: \(\mathbf{x }_{\ell k }= \mathbf{x }_{\text {CM}}+\sqrt{T}\mathbf{y }_{\ell k }\). Applying these two discretizations to the Casimir interaction energy in Eq. (5) yields the following form:
$$\begin{aligned} E_{\text {Casimir}}=-\tfrac{1}{2(4\pi )^2}\int _0^{\infty } \tfrac{dT}{T^3} \int d^3 x_{\text {CM}} \frac{1}{n_{\text {L}}} \sum _{\ell =1}^{n_{\text {L}}}\Theta _{\varSigma }[ \mathbf{x }_{\text {CM}}+\sqrt{T}\mathbf{y }_{\ell }]. \end{aligned}$$
(6)
As the worldline numeric approach for the Casimir phenomenon is based on (massless) scalar fields, the technique can currently only assess idealized behaviour for bounding geometry and cannot assess any frequency dependence of materials. Additionally, the approach developed to date in the literature does not account for the impacts of temperature. However, it is still a very capable and appealing technique in that it can provide quick and fairly accurate assessments for very complicated geometries where analytic techniques are not practical.
Generating unit loops, computational approach, and implementation validation
The developers of the worldline numerics for the Casimir phenomenon explored numerous ways [5] to generate ensembles of unit loops with Gaussian distribution ranging from a heat bath kernel to random walks, and finally landing on a technique denoted as the “v-loop” algorithm. The curious reader is encouraged to review the referenced manuscript for a thorough discussion of the benefits and shortfalls of the different techniques explored. The “v-loop” technique was selected as it can computationally generate an ensemble of \(n_{\text {L}}\) each having \(\text {N}\) points per loop without having to perform multiple iterations on each loop to realize a closed random walk/worldline with the required statistical characteristics. Figure 4 shows several examples of unit-loops generated by the v-loop methodology ranging from a 100 point unit loop to a 5000 point unit loop. A summary of the computational procedure steps are provided here to facilitate the reader’s understanding of the “v-loop” approach:
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1.
generate \(N-1\) numbers \(w_i\) (\(i=1,\ldots ,N-1\)) with a Gaussian distribution \(e^{-w_i^2}\) (e.g. using Box–Müller method);
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2.
calculate \(N-1\) numbers \(\bar{v}_i\) by normalizing \(w_i\):
$$\begin{aligned} \bar{v}_1= & {} \sqrt{ \frac{2}{N}}\, w_1, \nonumber \\ \bar{v}_i= & {} \frac{2}{\sqrt{N}} \sqrt{ \frac{N+1-i}{N+2-i}}\quad w_i, \, \, i=2,\ldots ,N-1; \end{aligned}$$
(7)
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3.
calculate \(v_i\) for \(i=2,\ldots ,N-1\) with the following:
$$\begin{aligned} v_i=\bar{v}_i -\frac{1}{N+2-i}\, v_{i-1,1}, \nonumber \\ \text {where}\, v_{i-1,1} =\sum _{j=2}^{i-1} v_j; \end{aligned}$$
(8)
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4.
a unit loop \(\mathbf{y }\) can now be created by using:
$$\begin{aligned} y_1= & {} \frac{1}{N} \left( \bar{v}_1 -\sum _{i=2}^{N-1} \left( N-i+ \frac{1}{2} \right) v_i \right) , \nonumber \\ y_i= & {} y_{i-1} +v_i, \, \, i=2,\ldots , N-1, \nonumber \\ y_N= & {} -\sum _{i=1}^{N-1} y_i; \end{aligned}$$
(9)
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5.
this procedure is repeated \(n_{\text {L}}\) times to create the unit loop ensemble \(\mathbf{y }_{\ell }\) with \(\ell = 1,\ldots ,n_{\text {L}}\).
The benefit of this numeric worldline approach is that it can be used to address any type of geometry while other approaches such as Proximity-Force Approximation (PFA) are not as flexible. Additionally, the approach has no dependency on the choice of model grid spacing or grid choice. The answer for a single point of interest in space does not have any interdependency on any other model grid points and may be calculated in total isolation if that is all that is needed. Figure 5Footnote 6 provides a pictoral representation of the analysis process for a parallel plate Casimir cavity. As indicated in the figure, once the loop ensemble has been generated the computational process to calculate the Casimir interaction energy follows the below enumerated steps:
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1.
The loop ensemble is moved to each model grid point of interest and scaled using proper time until 2+ bodies in the model are pierced;
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2.
the scale at which an individual loop pierces 2+ bodies defines the integral limits for the Casimir interaction energy integral;
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3.
the energy at the geometric point of interest in the model is increased based on wavelength (loop scale) and loop weight factor;
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4.
this scaling process is repeated for each loop in the ensemble at a geometric point of interest in the model;
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5.
the above steps are repeated for each geometric point of interest in the model.
Validation of our implementation of the numeric worldline approach was done on a plate-plate case and a corresponding plate-sphere case and was compared to documented results in the literature [5]. For the reader’s awareness, the work documented in [5] conducts extensive analysis to compare analytic results to the numeric results produced by the worldline technique for the simple plate-plate scenario and plate-sphere scenario. The referenced study explored the impact of number of points per unit loop, number of unit loops in an ensemble, separation distance of geometries, coupling, and mass. It is not the intention of this paper to duplicate the viability of the overall worldline numerics approach as this has already been done in the literature as noted, rather the intention of this paper is to apply this very powerful and flexible technique to fairly complicated geometries where only numerical methods can effectively be used. In our validation effort, we confirmed that our model predict the correct Casimir force for a given plate-plate or sphere place scenario, and subsequently compared their Casimir interaction energy density results from their numeric worldline algorithm to our interaction energy density results from our numeric worldline algorithm. The subsequent more complicated geometries we consider forthwith as part of this work do not have trivial analytic solutions which is why the numeric worldline technique is employed.
A plot of the results from our implementation is provided in Fig. 6 for the two cases and the plot also includes a plate-blade case. The geometry of all three cases is such that the closest point of separation between all three cases is identical allowing for comparison of the results to evaluate the effects of curvature. The plots reflect the energy density as measured along a line normal to the plate-plate geometry and these geometric points of interest are the same for the plate-sphere and plate-blade cases. The colors of the Casimir energy density plots correspond to the colors of the toy geometry also overlayed on the plot facilitating comparison of the results and to clearly see the impact of curvature. Comparing our results to those in literature indicates that our algorithms are functioning properly.