Skip to main content

Research Repository

Advanced Search

Real-time quantum dynamics, path integrals and the method of thimbles

Mou, Zong-Gang; Saffin, Paul M.; Tranberg, Anders; Woodward, Simon


Zong-Gang Mou

Anders Tranberg

Professor of Synthetic Organic Chemistry


Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy's theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.


Mou, Z., Saffin, P. M., Tranberg, A., & Woodward, S. (2019). Real-time quantum dynamics, path integrals and the method of thimbles. Journal of High Energy Physics, 2019(6),

Journal Article Type Article
Acceptance Date Jun 10, 2019
Online Publication Date Jun 19, 2019
Publication Date 2019-06
Deposit Date Jul 8, 2019
Publicly Available Date Jul 11, 2019
Journal Journal of High Energy Physics
Electronic ISSN 1029-8479
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 2019
Issue 6
Article Number 94
Keywords Lattice quantum field theory, Nonperturbative effects, Stochastic processes
Public URL
Publisher URL
Additional Information Received: 11 March 2019; Accepted: 10 June 2019; First Online: 19 June 2019


You might also like

Downloadable Citations