Introduction to Nonequilibrium Quantum Field Theory

TL;DR

This work develops and applies the nonequilibrium quantum field theory framework based on nPI (notably 2PI) generating functionals to describe far-from-equilibrium dynamics from first principles. By systematically resumming classes of diagrams via loop and $1/N$ expansions, it provides a conserving, non-secular approach that captures early-time amplification (e.g., parametric resonance), scattering, memory effects, and the approach toward thermalization, including prethermalization. The formalism extends to fermions and nonabelian gauge theories, establishing an equivalence hierarchy among $n$PI actions and connecting to kinetic theory, while illustrating with scalar $O(N)$ and chiral quark-meson models. It demonstrates that beyond mean-field, scattering and off-shell processes drive rapid equilibration and reveal universal late-time behavior governed by energy densities, with numerical implementations feasible on lattices and clusters. Overall, the framework offers a principled, nonperturbative path from nonequilibrium evolution to thermalization applicable to high-energy, cosmological, and many-body contexts.

Abstract

There has been substantial progress in recent years in the quantitative understanding of the nonequilibrium time evolution of quantum fields. Important topical applications, in particular in high energy particle physics and cosmology, involve dynamics of quantum fields far away from the ground state or thermal equilibrium. In these cases, standard approaches based on small deviations from equilibrium, or on a sufficient homogeneity in time underlying kinetic descriptions, are not applicable. A particular challenge is to connect the far-from-equilibrium dynamics at early times with the approach to thermal equilibrium at late times. Understanding the ``link'' between the early- and the late-time behavior of quantum fields is crucial for a wide range of phenomena. For the first time questions such as the explosive particle production at the end of the inflationary universe, including the subsequent process of thermalization, can be addressed in quantum field theory from first principles. The progress in this field is based on efficient functional integral techniques, so-called n-particle irreducible effective actions, for which powerful nonperturbative approximation schemes are available. Here we give an introduction to these techniques and show how they can be applied in practice. Though we focus on particle physics and cosmology applications, we emphasize that these techniques can be equally applied to other nonequilibrium phenomena in complex many body systems.

Figures (29)

• Figure 1: Topologically distinct diagrams in the 2PI loop expansion up to five-loop order for $\phi = 0$. The suppressed prefactors are given in Eq. (\ref{['eq:2PIGfiveloop']}).
• Figure 2: Graphical representation of the $\phi$--dependent contributions for $\Gamma_2 \equiv 0$. The crosses denote field insertions $\sim \phi_a\phi_a$ for the left figure, which contributes at leading order, and $\sim \phi_a\phi_b$ for the right figure contributing at next-to-leading order.
• Figure 3: LEFT: Comparison of the LO and NLO time dependence of the equal-time correlation modes $F(t,t;p)$ for the "tsunami" initial condition (\ref{['eq:initialtsF']})--(\ref{['eqtsunamidist']}). The importance of scattering included in the NLO approximation is apparent: the non-thermal LO fixed points become unstable and the "tsunami" decays, approaching thermal equilibrium at late times. RIGHT: Effective particle number distribution for a "tsunami" in the presence of a thermal background. The solid line shows the initial distribution which for low and for high momenta follows a Bose-Einstein distribution, i.e. ${\rm Log}[1+1/n_p(0)] \simeq \epsilon_p(0) / T_0$. At late times the non-thermal distribution equilibrates and approaches a straight line with inverse slope $T_{\rm eq} \, >\, T_0$.
• Figure 4: LEFT: Shown is the time-dependent mass term $M^2(t)$ in the LO approximation for three different couplings following a "quench". All quantities are given in units of appropriate powers of the initial-time mass $M(0)$. RIGHT: Time dependence of the equal-time zero-mode $F(t,t;p=0)$ after a "quench" (see text for details). The inset shows the mass term $M^2(t)$, which includes a sum over all modes. The dotted lines represent the Hartree approximation (LO$^+$), while the solid lines give the NLO results. The coupling is $\lambda/6N = 0.17 \, M^2(0)$ for $N=4$.
• Figure 5: LEFT: Shown is the evolution of the unequal-time correlation $F(t,0;p=0)$ after a "quench". Unequal-time correlation functions approach zero in the NLO approximation and correlations with early times are effectively suppressed ($\lambda/6N = (5/6N\simeq 0.083) \, M^2(0)$ for $N=10$). In contrast, there is no decay of correlations with earlier times for the LO approximation. RIGHT: The logarithmic plot of $|\rho(t,0;p=0)|$ and $|F(t,0;p=0)|$ as a function of time $t$ shows an oscillation envelope which quickly approaches a straight line. At NLO the correlation modes therefore approach an exponentially damped behavior. (All in units of $M(0)$.)