A Monte Carlo approach to the worldline formalism in curved space

Abstract

We present a numerical method to evaluate worldline (WL) path integrals defined on a curved Euclidean space, sampled with Monte Carlo (MC) techniques. In particular, we adopt an algorithm known as YLOOPS with a slight modification due to the introduction of a quadratic term which has the function of stabilizing and speeding up the convergence. Our method, as the perturbative counterparts, treats the non-trivial measure and deviation of the kinetic term from flat, as interaction terms. Moreover, the numerical discretization adopted in the present WLMC is realized with respect to the proper time of the associated bosonic point-particle, hence such procedure may be seen as an analogue of the time-slicing (TS) discretization already introduced to construct quantum path integrals in curved space. As a result, a TS counter-term is taken into account during the computation. The method is tested against existing analytic calculations of the heat kernel for a free bosonic point-particle in a D-dimensional maximally symmetric space.

Non-perturbative Quantum Propagators in Bounded Spaces

“…This information can be useful for analytic and numerical estimations of the kernel for systems whose path integral cannot be computed in close form, since it can indicate which trajectories, or regions of space, will be dominant in its determination. In fact the hit function was already sampled in works based on worldline numerics [4] and the extension considered here could shed new light on the undersampling problem encountered in [35][36][37]. Looking further ahead, the methods we describe here are well-adapted to the problem of estimating the Casimir energy for arbitrary surface geometries, even for partially conducting plates.…”

Dimensional regularization for the particle transition amplitude in curved space

“…On the one hand, TS is the most straightforward, as it is rooted directly to the first-principled formulation of the path integral, which originates from the multiple slicing of the (operatorial) particle transition amplitude. As such, it is also the natural one to be used in non-perturbative numeric (Monte Carlo) approaches [42]. Moreover, in perturbative computations it requires no integration by parts (i.b.p.)…”

New Techniques for Worldline Integration

“…Although most applications of the formalism are based on an analytical calculation of the path integral, efficient algorithms have also been developed for the direct numerical computation of worldline path integrals [47,60]. Applications include scalar bound states [70], Casimir energies with Dirichlet boundary conditions [61], Schwinger pair-creation rates [59] and the curved-space heat kernel [37].…”