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arXiv:2105.02880

On the numerical evaluation of real-time path integrals: Double exponential integration and the Maslov correction

Rosenfelder, R.

Published May 6, 2021• 1 citation
arXivHTMLPDF

Abstract

Ooura's double exponential integration formula for Fourier transforms is applied to the oscillatory integrals occuring in the path-integral description of real-time Quantum Mechanics. Due to an inherent, implicit regularization multi-dimensional Gauss-Fresnel integrals are obtained numerically with high precision but modest number of function calls. In addition, the Maslov correction for the harmonic oscillator is evaluated numerically with an increasing number of time slices in the path integral thereby clearly demonstrating that the real-time propagator acquires an additional phase $ - π/2 $ each time the particle passes through a focal point. However, in the vicinity of these singularities an overall small damping factor is required. Prospects of evaluating scattering amplitudes of finite-range potentials by direct numerical evaluation of a real-time path integral are discussed.

DOI
10.48550/arxiv.2105.02880

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Ooura's double exponential integration formula for Fourier transforms is applied to the oscillatory integrals occuring in the path-integral description of real-time Quantum Mechanics. Due to an inherent, implicit regularization multi-dimensional Gauss-Fresnel integrals are obtained numerically with high precision but modest number of function calls. In addition, the Maslov correction for the harmonic oscillator is evaluated numerically with an increasing number of time slices in the path integral thereby clearly demonstrating that the real-time propagator acquires an additional phase −π/2- π/2−π/2 each time the particle passes through a focal point. However, in the vicinity of these singularities an overall small damping factor is required. Prospects of evaluating scattering amplitudes of finite-range potentials by direct numerical evaluation of a real-time path integral are discussed.
The Feynman path integral has revolutionized modern approaches to quantum physics. Although the path integral formalism has proven very successful and spawned several approximation schemes, the direct evaluation of real-time path integrals is still extremely expensive and numerically delicate due to its high-dimensional and oscillatory nature. We propose an efficient method for the numerical evaluation of the real-time world-line path integral for theories where the potential is dominated by a quadratic at infinity. This is done by rewriting the high-dimensional oscillatory integral in terms of a series of low-dimensional oscillatory integrals, that we efficiently evaluate with Picard-Lefschetz theory or approximate with the eikonal approximation. Subsequently, these integrals are stitched together with a series of fast Fourier transformations to recover the lattice regularized Feynman path integral. Our method directly applies to problems in quantum mechanics, the word-line quantization of quantum field theory, and quantum gravity.
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Recent progress in numerical methods for evaluating the real-time path integral in dissipative harmonic environments is reviewed. Quasi-adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi-adiabatic propagator path integral is evaluated by efficient system-specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunneling dynamics in the condensed phase are described.
Abstract. In this paper, we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the L 2 transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each carries at most a finite number of singularities and discontinuities. 1. Introduction. In his paper(see [22] footnote 13), Feynman observed that by using wave functions, ill-defined oscillatory integrals can be given rigorous meaning. We will use Feynman’s observation and use wave functions to provide a convergence factor in the derivation of a real time propagator that takes the form of an L 2 transition probability amplitude and the derivation of a real time, time sliced Feynman
The direct transformation from the polygonal to the Fourier series representation of Feynman path integrals, via a change of integration variables, is effected explicitly for cases where the Lagrangian is of the form L(x,ẋ,t)= 1/2 ẋ2−V(x,t). This transformation involves a ‘‘functional Jacobian’’ stemming solely from the velocity term in the Lagrangian; this is because N-segment polygonal paths of the type contributing significantly to the integral, and their N-term Fourier series approximants, coalesce together as N→∞, but not their derivatives. We also consider integrals over paths with fixed means T−1∫T0 dt x(t) =const. The usefulness of the Fourier representation is illustrated with the harmonic oscillator case V(x)= 1/2 ω2x2, in both the free and fixed means situations; in particular, the Fourier evaluation of the path integrals trivially determines the large time phases (Maslov indices), and the ranges of ω2 values for which the integrals are finite or infinite in the imaginary time cases.
Path integrals are a powerful method for calculating real time, finite temperature, and ground state properties of quantum systems. By exploiting some remarkable properties of the symmetric Trotter formula and the discrete Fourier transform, we arrive at a high-accuracy method for removing ``time slice'' errors in Trotter-approximated propagators. We provide an explicit demonstration of the method applied to the two-body density matrix of 4He^{4}\mathrm{He}4He. Our method is simultaneously fast, high precision, and computationally simple and can be applied to a wide variety of quantum propagators.