Efficient evaluation of real-time path integrals

TL;DR

The paper introduces an efficient numerical method for evaluating real-time path integrals in quantum physics, reducing computational complexity by decomposing high-dimensional oscillatory integrals into manageable low-dimensional ones.

Contribution

It presents a novel approach combining Picard-Lefschetz theory and Fourier transforms to efficiently evaluate real-time path integrals for quadratic-dominant potentials.

Findings

Significantly reduces computational cost of path integral evaluation.

Applicable to quantum mechanics, quantum field theory, and quantum gravity.

Demonstrates improved numerical stability over traditional methods.

Abstract

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.