Worldline numerics for energy-momentum tensors in Casimir geometries

Abstract

We develop the worldline formalism for computations of composite operators such as the fluctuation induced energy-momentum tensor. As an example, we use a fluctuating real scalar field subject to Dirichlet boundary conditions. The resulting worldline representation can be evaluated by worldline Monte-Carlo methods in continuous spacetime. We benchmark this worldline numerical algorithm with the aid of analytically accessible single-plate and parallel-plate Casimir configurations, providing a detailed analysis of statistical and systematic errors. The method generalizes straightforwardly to arbitrary Casimir geometries and general background potentials.

Worldline formalism for a confined scalar field

“…In this context, worldline based numerical analysis has been used to determine the range of validity of the scheme known as proximity force approximation [35,36]. In [37,38] the method was tested by computing the positive-energy conditions in various Casimir settings. These numerical methods are based on a Monte Carlo generation of worldline ensembles which, apart from providing an intuitive picture of the nonlocal nature of quantum fluctuations, is comparatively cheap due to its probabilistic nature (see [39,40,41,42]); we consider our analytic expressions could be used to test numerical computations in spherical geometries.…”

Propagator from nonperturbative worldline dynamics

“…This v lines algorithm is a generalization of the efficient v loops algorithm introduced in [48] which generates corresponding closed worldlines. Open worldlines can also efficiently be generated by a variant of the d loop algorithm [18] that also works for open lines [58,59], however, the following v lines algorithm can be employed for an arbitrary number of points (for d lines or loops they always come in powers of 2).…”

“…where we have restricted ourselves to l < j < m and defined a shifted initial position α 0 := b l y 0 + 2ip/N . Consequently we are left with a pair of recursion relations analogous to (58) with c j taking the rôle of the b j , except for the fact that the initial condition is c l = 1. In other words, the initial condition corresponds in this case to a condition on the coefficients where the first potential insertion occurs.…”

Non-perturbative Quantum Propagators in Bounded Spaces

“…This is the case, for example, for many scattering problems and contact interactions [7][8][9] and varied models of lattice structure in condensed matter [10,11]. In the relativistic case, constrained path integrals arise in the context of the Casimir effect where one is interested in trajectories that touch the conducting plates [12][13][14][15], and for Dirichlet boundary conditions imposed in some spatial region [16][17][18][19][20][21][22][23][24][25] where the contributions from paths passing through this region should be removed. A fresh approach to calculating, either analytically, approximately or numerically, such path integrals will therefore find wide application.…”