Applications of the worldline Monte Carlo formalism in quantum mechanics

Abstract

In recent years efficient algorithms have been developed for the numerical computation of relativistic single-particle path integrals in quantum field theory. Here, we adapt this "worldline Monte Carlo" approach to the standard problem of the numerical approximation of the non-relativistic path integral, resulting in a formalism whose characteristic feature is the fast, non-recursive generation of an ensemble of trajectories that is independent of the potential, and thus universally applicable. The numerical implementation discretises the trajectories with respect to their time parametrisation but maintains a continuous spatial domain. In the case of singular potentials, the discretised action gets adapted to the singularity through a "smoothing" procedure. We show for a variety of examples (the harmonic oscillator in various dimensions, the modified Pöschl-Teller potential, delta-function potentials, the Coulomb and Yukawa potentials) that the method allows one to obtain fast and reliable estimates for the Euclidean propagator and use them in a certain time window suitable for extracting the ground state energy. As an aside, we apply it for studying the classical limit where nearly classical trajectories are expected to dominate in the path Email addresses: jedwards@ifm.umich.mx (James P. Edwards), gerberu@itp.unibe.ch (Urs Gerber), schubert@ifm.umich.mx (Christian Schubert), mtrejo@ifm.umich.mx (Maria Anabel Trejo), ttsiftsi@matmor.unam.mx (Thomai Tsiftsi), axel@ifm.umich.mx (Axel Weber)integral. We expect the advances made here to be useful also in the relativistic case.

Worldline formalism for a confined scalar field

“…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

“…For this, we generalize a technique introduced in [18] to the computation of the propagator. Similar methods have also been applied to numerical worldline computations of Schwinger pair production [53] and recently to high-accuracy results of quantum mechanical potential problems [54,55]. In the present case, the use of a suitable analytic fit function for the PDF also allows to extrapolate the nonperturbative results to parameter regimes where a direct simulation is computationally expensive.…”

A Monte Carlo approach to the worldline formalism in curved space

“…Thus, instead of using the full (kinetic + potential) action as a weight to select the points on the worldline, only the kinetic part is used. As explained in the recent work by Edwards et al [31], the reason for such a choice is universality, which then allows the application of the method to a variety of different cases, regardless of the specific potential involved. Moreover, therein they propose an efficient algorithm, called YLOOPS, to generate closed worldlines around a point, 1 which is an improved version of the alghoritms VLOOPS and DLOOPS originally designed in [26] and [28] respectively.…”

“…Secondly, each worldline has to be discretized: this is usually done with respect to the affine parameter, taking a number N of points per worldline. As pointed out in [31], here discretization is not realized on the points x(τ ) of the manifold, 3 but directly on the…”

“…affine parameter τ . To fulfill the above two requirements, what is usually done in WLMC approaches [24,[26][27][28]31] is to compute worldline averages like .3) respectively. In this way, the exponential e −S KIN characterizing each worldline can be used as a weight function to build the worldlines used for the calculation.…”