Abstract
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.
Highlights
In [2, 3], it was shown that the real-time path integral can be computed through a Generalized Thimble Method, based on complexifying the field variables
Given that we are studying a quantum system, there will be an ensemble of initial positions and velocities described by an initial density matrix, but we will see that we are able to separate the path integral into a two-step sampling procedure; for each member of the initial condition ensemble, we may compute a well-defined contribution to the path integral using the Generalized Thimble Method, and subsequently average over the initial condition ensemble in a straightforward way
Real-time quantum dynamics is well-defined in terms of the Schwinger-Keldysh formalism, and the classical-statistical approximation often does very well in some cases
Summary
With real variables φi, and I is a function of all φi. As in the Feynman path integral, the exponent could be purely imaginary, so that the integrand is oscillatory with a constant amplitude. We can improve the convergence of the integral through complexifying φi and, because of Cauchy’s theorem, we can deform the real integration cycle into the complex. Plane and still obtain the same result for the integral. In the following we shall use φi to denote the real field, and φi shall denote the complexified field. The initial integration manifold is Rn, parametrized by φi. This integration cycle is deformed to a surface in Cn with n real dimensions, parametrized by φi
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