Open system dynamics in interacting quantum field theories

TL;DR

This work develops and analyzes a Redfield master-equation approach to open quantum field theories, treating a system scalar field $\phi$ coupled to an environment field $\chi$ on Minkowski space via bilinear ($\lambda\phi\chi$) and nonlinear ($\lambda\phi\chi^{2}$) interactions. By deriving the Redfield equation, the authors formulate coupled differential equations for equal-time two-point correlators and explore Markovian, RWA, and Dyson-series limits to compare non-Markovian dynamics with standard perturbation theory. They find that bilinear coupling exhibits persistent memory and Markovian approximations fail, while nonlinear coupling yields a rapidly decaying environment correlator and a late-time Redfield solution that aligns with the Markovian limit; renormalization is essential in the nonlinear case, introducing a scale $\mu_{\mathrm{r}}$ and Lamb-shift corrections. Overall, the Redfield framework provides a perturbative resummation that can outperform naive second-order Dyson results and offers a controlled path to incorporating non-Markovian effects in open quantum field theories.

Abstract

A quantum system that interacts with an environment generally undergoes nonunitary evolution described by a non-Markovian or Markovian master equation. In this paper, we construct the non-Markovian Redfield master equation for a quantum scalar field that interacts with a second field through a bilinear or nonlinear interaction on a Minkowski background. We use the resulting master equation to set up coupled differential equations that can be solved to obtain the equal-time two-point function of the system field. We show how the equations simplify under various approximations including the Markovian limit and argue that the Redfield equation-based solution provides a perturbative resummation to the standard second-order Dyson series result. For the bilinear interaction, we explicitly show that the Redfield solution is closer to the exact solution compared to the perturbation theory-based one. Further, the environment correlation function is oscillatory and nondecaying in this case, making the Markovian master equation a poor approximation. For the nonlinear interaction, on the other hand, the environment correlation function is sharply peaked and the Redfield solution matches that obtained using a Markovian master equation in the late-time limit.

Figures (6)

• Figure 1: The environment correlation as a function of elapsed time $t_{1}$ in a $\lambda \phi \chi$ interacting theory, computed for $M = 3 m$ and $k = m$, and normalized to unity at $t_{1} = 0$.
• Figure 2: The time-averaged absolute error in the two-point function calculated using the Redfield equation and perturbation theory as a function of $M/m$ in a $\lambda \phi \chi$ interacting theory, for $\lambda = M m /2$ and $k = 0$, and averaged over the time interval $\sqrt{2 \lambda} t \in [0,10]$.
• Figure 3: The relative error in the two-point function calculated using the Redfield equation and perturbation theory as a function of time in a $\lambda \phi \chi$ interacting theory, for three relatively large values of $\lambda$, $k = m$, and $M = 3 m$. Expectedly, the error is suppressed for smaller values of $\lambda$.
• Figure 4: The two-point function calculated using the Redfield equation, perturbation theory, the Redfield equation in the Markovian limit, and the Redfield equation under the RWA as a function of time in a $\lambda \phi \chi$ interacting theory, for $\lambda = m^{2}/2$, $k = m$, and $M = 3 m$, and normalized by the free theory solution, $\mathcal{G}_{k,0} = (2 \omega_{k})^{-1}$.
• Figure 5: The environment correlation \ref{['eq:PhiChi2EnvCorrIntegrated']} as a function of elapsed time $t_{1}$ in a $\lambda \phi \chi^{2}$ interacting theory, computed for $\epsilon = 1/10$ and $k = m$, and normalized to unity at $t_{1} = 0$.