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Nonequilibrium Quantum-Field Dynamics and Off-Shell Transport

S. Juchem

Published Jun 12, 2015• 2 citations
OpenAlex

Abstract

We solve the Kadanoff-Baym equations for nonequilibrium initial configurations of the φ 4-theory in 2+1 dimensions and compare to explicit solutions of generalized transport equations for the same theory. The latter transport equations are derived from the Kadanoff-Baym equation in a first order gradient expansion in phase space and explicitly retain the off-shell dynamics as inherent in the timedependent spectral functions. The solutions of these equations compare very well with the exact solutions of the full Kadanoff-Baym equations with respect to the occupation numbers of the individual modes, the spectral evolution as well as the chemical equilibration process. Furthermore, the proper equilibrium off-shell distribution is reached for large times contrary to the quasiparticle Boltzmann limit. We additionally present a direct comparison of the solution of the generalized transport equations in the Kadanoff-Baym and Botermans-Malfliet form; both solutions are found to agree very well with each other. PACS: 06.60.+w; 05.70.Ln; 11.10.Wx; 24.10.-i; 25.75.-q Keywords: Many-body theory; finite temperature field theory; out-of-equilibrium quantum field theory; nonequilibrium statistical physics; relativistic heavy-ion collisions Part of the PhD thesis of S. Juchem

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We solve the Kadanoff-Baym equations for nonequilibrium initial configurations of the φ 4-theory in 2+1 dimensions and compare to explicit solutions of generalized transport equations for the same theory. The latter transport equations are derived from the Kadanoff-Baym equation in a first order gradient expansion in phase space and explicitly retain the off-shell dynamics as inherent in the timedependent spectral functions. The solutions of these equations compare very well with the exact solutions of the full Kadanoff-Baym equations with respect to the occupation numbers of the individual modes, the spectral evolution as well as the chemical equilibration process. Furthermore, the proper equilibrium off-shell distribution is reached for large times contrary to the quasiparticle Boltzmann limit. We additionally present a direct comparison of the solution of the generalized transport equations in the Kadanoff-Baym and Botermans-Malfliet form; both solutions are found to agree very well with each other. PACS: 06.60.+w; 05.70.Ln; 11.10.Wx; 24.10.-i; 25.75.-q Keywords: Many-body theory; finite temperature field theory; out-of-equilibrium quantum field theory; nonequilibrium statistical physics; relativistic heavy-ion collisions Part of the PhD thesis of S. Juchem
The quantum time evolution of ${\ensuremath{\varphi}}^{4}$-field theory for a spatially homogeneous system in 2+1 space-time dimensions is investigated numerically for out-of-equilibrium initial conditions on the basis of the Kadanoff-Baym equations including the tadpole and sunset self-energies. Whereas the tadpole self-energy yields a dynamical mass, the sunset self-energy is responsible for dissipation and an equilibration of the system. In particular we address the dynamics of the spectral (``off-shell'') distributions of the excited quantum modes and the different phases in the approach to equilibrium described by Kubo-Martin-Schwinger relations for thermal equilibrium states. The investigation explicitly demonstrates that the only translation invariant solutions representing the stationary fixed points of the coupled equation of motions are those of full thermal equilibrium. They agree with those extracted from the time integration of the Kadanoff-Baym equations for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}.$ Furthermore, a detailed comparison of the full quantum dynamics to more approximate and simple schemes such as that of a standard kinetic (on-shell) Boltzmann equation is performed. Our analysis shows that the consistent inclusion of the dynamical spectral function has a significant impact on relaxation phenomena. The different time scales that are involved in the dynamical quantum evolution towards a complete thermalized state are discussed in detail. We find that far off-shell $1\ensuremath{\leftrightarrow}3$ processes are responsible for chemical equilibration, which is missed in the Boltzmann limit. Finally, we briefly address the case of (bare) massless fields. For sufficiently large couplings $\ensuremath{\lambda}$ we observe the onset of Bose condensation, where our scheme within symmetric ${\ensuremath{\varphi}}^{4}$ theory breaks down.
Boltzmann equations are often used to study the thermal evolution of particle reaction networks. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after a relativistic heavy ion collision. However, Boltzmann equations are only a classical approximation of the quantum thermalization process which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the full Kadanoff-Baym equations. Therefore, we present in this thesis a detailed comparison of Boltzmann and Kadanoff-Baym equations in the framework of relativistic quantum field theories in 3+1 space-time dimensions. In a first step, for simplicity we consider a real scalar Phi^4 quantum field theory and in a second step we generalize our results to a chirally invariant Yukawa-type quantum field theory including fermions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Apart from quantitative discrepancies, on a qualitative level the universality respected by Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations. Furthermore, Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. This separation of time scales is absent for Boltzmann equations.
Boltzmann equations are often used to describe the nonequilibrium time-evolution of many-body systems in particle physics. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after a relativistic heavy ion collision. However, Boltzmann equations are only a classical approximation of the quantum thermalization process, which is described by so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the complete Kadanoff-Baym equations. Therefore, we present in this article a detailed comparison of the Boltzmann and the Kadanoff-Baym equations in the framework of a chirally invariant Yukawa-type quantum field theory including fermions and scalars. The obtained numerical results reveal significant differences between both types of equations. Apart from quantitative differences, on a qualitative level the late-time universality respected by the Kadanoff-Baym equations is severely restricted in the case of the Boltzmann equations. Furthermore, the Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. In contrast to this standard Boltzmann equations cannot describe the process of quantum-chemical equilibration, and consequently also cannot feature the above separation of time scales.
(Classical) Boltzmann equations suffer from several shortcomings as compared to their quantum mechanical generalizations, the so-called Kadanoff-Baym equations. Nevertheless, in practical calculations approximations to Boltzmann equations are widely used. A prominent example is the computation of the baryon asymmetry of the universe. In this work, we present a detailed comparison between the Kadanoff-Baym and Boltzmann equations in the framework of a scalar phi 4 quantum field theory. We solve the Kadanoff-Baym and the Boltzmann equations numerically in 3 + 1 dimensions and compare their predictions on the evolution of systems out of thermal equilibrium for various initial conditions. The obtained numerical solutions reveal significant discrepancies between both types of equations. Most notably, the universality respected by Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations.
Boltzmann equations are often used to study the thermal evolution of particle reaction networks. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after relativistic heavy ion collisions. However, Boltzmann equations are only a classical approximation of the quantum thermalization process which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the full Kadanoff-Baym equations. Therefore, we present in this paper a detailed comparison between the Kadanoff-Baym and Boltzmann equations in the framework of a scalar ${\ensuremath{\Phi}}^{4}$ quantum field theory in 3+13+13+1 space-time dimensions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Apart from quantitative discrepancies, on a qualitative level the universality respected by the Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations. Furthermore, the Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. This separation of time scales is absent for the Boltzmann equation.
The nonequilibrium dynamics of quantum fields is an initial-value problem, which can be described by Kadanoff-Baym equations. Typically, and, in particular, when numerical solutions are demanded, these Kadanoff-Baym equations are restricted to Gaussian initial states. However, physical initial states are non-Gaussian correlated initial states. In particular, renormalizability requires the initial state to feature nnn-point correlations that asymptotically agree with the vacuum correlations at short distances. In order to identify physical nonequilibrium initial states, it is therefore a precondition to describe the vacuum correlations of the interacting theory within the nonequilibrium framework. In this paper, Kadanoff-Baym equations for non-Gaussian correlated initial states describing vacuum and thermal equilibrium are derived from the 2PI effective action. A diagrammatic method for the explicit construction of vacuum and thermal initial correlations from the 2PI effective action is provided. We present numerical solutions of Kadanoff-Baym equations for a real scalar ${\ensuremath{\Phi}}^{4}$ quantum field theory, which take the thermal initial four-point correlation as the leading non-Gaussian correction into account. We find that this minimal non-Gaussian initial condition yields an approximation to the complete equilibrium initial state that is quantitatively and qualitatively significantly improved as compared to Gaussian initial states.