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arXiv:hep-ph/0205271

Thermalization of an Interacting Quantum Field in the CTP-2PI Next-to-leading-order Large N Scheme

Esteban Calzetta, B. L. Hu

Published May 23, 2002• 30 citations
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Abstract

In this paper we use an O(N)-invariant scalar field of unbroken symmetry to investigate whether an interacting quantum field at the next-to-leading order Large $N$ approximation may show signs of thermalization. We develop the closed time-path (CTP) two-particle irreducible (2PI) effective action in powers of 1/N, retaining up to next to leading order (O(1)) terms, and write down the corresponding (truncated) Schwinger-Dyson equations for its two point function. We show that in this approximation, the only translation invariant solutions to the Schwinger - Dyson equations are thermal. This provides a useful temperature concept without invoking a heat bath. When combined with the familiar Kadanoff-Baym approach to quantum kinetic theory our result shows that at this order of approximation thermalization can occur, at least if initial conditions are smooth enough that a derivative expansion is valid. Our analytic result provides support for similar claims in recent literature based on numerical evidence.

DOI
10.48550/arxiv.hep-ph/0205271

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In this paper we use an O(N)-invariant scalar field of unbroken symmetry to investigate whether an interacting quantum field at the next-to-leading order Large NNN approximation may show signs of thermalization. We develop the closed time-path (CTP) two-particle irreducible (2PI) effective action in powers of 1/N, retaining up to next to leading order (O(1)) terms, and write down the corresponding (truncated) Schwinger-Dyson equations for its two point function. We show that in this approximation, the only translation invariant solutions to the Schwinger - Dyson equations are thermal. This provides a useful temperature concept without invoking a heat bath. When combined with the familiar Kadanoff-Baym approach to quantum kinetic theory our result shows that at this order of approximation thermalization can occur, at least if initial conditions are smooth enough that a derivative expansion is valid. Our analytic result provides support for similar claims in recent literature based on numerical evidence.
We present a theoretical study of thermalization of quantum fields with Kadanoff-Baym (KB) equation. First we introduce a field theoretical technique known as 2 Particle Irreducible (2PI) effective action and review derivation of equation of motion for mean fields and fluctuations, then derive Kadanoff-Baym equation in scalar and gauge theories. In order to analyze thermalization processes we introduce relativistic entropy current based on the first order gradient expansion of Kadanoff-Baym equation. Then we show that in taking into account next-to-leading order (NLO) self-energy of the coupling expansion in φ4 theory (in symmetric phase <φ^> = 0), it is possible to prove the H-theorem within the 1st order gradient expansion. We also show that it is possible to prove the H-theorem for KB equation with NLO self-energy of 1/N expansion in O(N) theory (in symmetric phase). Furthermore we suggest a possibility for H-theorem to be satisfied in KB equation with leading-order (LO) self-energy of the coupling expansion in non-Abelian gauge theory. In d+1 dimensions (d＞2) LO particle number changing processes, such as 0-to-3, 1-to-2 and 2-to-1, which are prohibited in the normal Boltzmann approach with on-shell particles, may contribute to entropy production in KB approach although we must deal with gauge dependence and infrared divergence to complete the proof. Next we perform numerical simulations in spatially homogeneous configurations to investigate thermalization properties of the system in 1+1 and 2+1 dimensions for φ4 theory with NLO self-energy of the coupling expansion and 1+1 dimensions for O(N) theory with NLO self-energy of 1/N expansion. We also estimate the time evolution of the kinetic entropy in the case of scalar φ4 and O(N) theory. Then we find that at later times X0≫1/m, where m represents mass of particles, the kinetic entropy increases monotonically and approaches the equilibrium value, although the limited time interval at the earlier times X0～1/m invalidates the use of it. No thermalization occurs in the simulations with the normal Boltzmann equation with 2-to-2 collision term in 1+1 dimensions, while it occurs in the simulations with KB equation in 1+1 dimensions. It is due to two types of offshell effects: memory effects and spectral functions with the decay width.
We rewrite the Martin-Siggia-Rose (MSR) formalism for the statistical dynamics of classical fields in a covariant second order form appropriate for the statistical dynamics of relativistic field theory. This second order formalism is related to a rotation of Schwinger's closed time path (CTP) formalism for quantum dynamics, with the main difference being that certain vertices are absent in the classical theory. These vertices are higher order in an ℏ\hbarℏ expansion. The structure of the second order formulation of the Schwinger Dyson (S-D) equations is identical to that of the rotated CTP formalism apart from initial conditions on the Green's functions and the absence of these vertices. We then discuss self-consistent truncation schemes based on keeping certain graphs in the two-particle irreducible effective action made up of bare vertices and exact Green's functions.
We present kinetic equations that describe the evolution of O(N)-symmetric real scalar quantum fields out of thermal equilibrium in a systematic nonperturbative approximation scheme. This description starts from the 1/N-expansion of the 2PI effective action to next-to-leading order, which includes scattering and memory effects. From this starting point one is lead to evolution equations for the propagator, which are nonlocal in time. Numerical solutions showed that the propagator depends only very slightly on the center coordinates already after moderate times, and that correlations between earlier and later times are suppressed exponentially, which causes an effective memory loss. Exploiting these two observations, we combine a first order gradient expansion with a Wigner transformation to derive our kinetic equations, which are local in time, from the nonlocal evolution equations. In contrast to standard descriptions based on loop expansions, our kinetic equations remain valid even for nonperturbatively large fluctuations. Additionally, employing a quasi-particle approximation, we eventually arrive at a generalized Boltzmann equation.
This is the first of a series of papers which describe the functional-integral approach to the study of the statistical and kinetic properties of nonequilibrium quantum fields in flat and curved spacetimes. In this paper we treat a system of self-interacting bosons described by \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ scalar fields in flat space. We adopt the closed-time-path (CTP or ``in-in'') functional formalism and use a two-particle irreducible (2PI) representation for the effective action. These formalisms allow for a full account of the dynamics of quantum fields, and put the correlation functions on an equal footing with the mean fields. By assuming a thermal distribution we recover the real-time finite-temperature theory as a special case. By requiring the CTP effective action to be stationary with respect to variations of the correlation functions we obtain an infinite set of coupled equations which is the quantum-field-theoretical generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Truncation of this series leads to dissipative characteristics in the subsystem. In this context we discuss the nature of dissipation in interacting quantum fields.To one-loop order in a perturbative expansion of the CTP effective action, the 2PI formalism yields results equivalent to the leading 1/N expansion for an O(N)-symmetric scalar field. To higher-loop order we introduce a two-time approximation to separate the quantum-field effects of radiative correction and renormalization from the statistical-kinetic effects of collisions and relaxation. In the weak-coupling quasiuniform limit, the system of nonequilibrium quantum fields can subscribe to a kinetic theory description wherein the propagators are represented in terms of relativistic Wigner distribution functions. From a two-loop calculation we derive the Boltzmann equation for the distribution function and the gap equation for the effective mass of the quasiparticles. One can define an entropy function for the quantum gas of quasiparticles which satisfies the H theorem. We also calculate the limits to the validity of the binary collision approximation from a three-loop analysis. The theoretical framework established here can be generalized to nonconstant background fields and for curved spacetimes.
I review the use of the 2PI effective action in nonequilibrium quantum field theory. The approach enables one to find approximation schemes which circumvent long-standing problems of non-thermal or secular (unbounded) late-time evolutions encountered in standard loop or 1/N expansions of the 1PI effective action. It is shown that late-time thermalization can be described from a numerical solution of the three-loop 2PI effective action for a scalar ϕ4ϕ^4ϕ4--theory in 1+1 dimensions (with Jurgen Cox, hep-ph/0006160). Quantitative results far from equilibrium beyond the weak coupling expansion can be obtained from the 1/N expansion of the 2PI effective action at next-to-leading order (NLO), calculated for a scalar O(N) symmetric quantum field theory (hep-ph/0105311). Extending recent calculations in classical field theory by Aarts et al. (hep-ph/0007357) and by Blagoev et al. (hep-ph/0106195) to N>1N>1N>1 we show that the NLO approximation converges to exact (MC) results already for moderate values of NNN (with Gert Aarts, hep-ph/0107129). I comment on characteristic time scales in scalar quantum field theory and the applicability of classical field theory for sufficiently high initial occupation numbers.
We consider the nonequilibrium evolution of an O(N)-symmetric scalar quantum field theory using a systematic two-particle irreducible 1/N-expansion to next-to-leading order, which includes scattering and memory effects. The corresponding ``full Kadanoff-Baym equations'' are solved numerically without further approximations. This allows one to obtain a controlled nonperturbative description of far-from-equilibrium dynamics and the late-time approach to quantum thermal equilibrium. Employing in addition a first-order gradient expansion for the Wigner transformed correlators we derive kinetic equations. In contrast to standard descriptions based on loop expansions, our equations remain valid for nonperturbatively large fluctuations. As an application, we discuss the fluctuation dominated regime following parametric resonance in quantum field theory.