Correlations and Equilibration in Relativistic Quantum Systems

Abstract

In this article we study the time evolution of an interacting field theoretical system, i.e. φ 4 -field theory in 2+1 space-time dimensions, on the basis of the Kadanoff-Baym equations for a spatially homogeneous system including the self-consistent tadpole and sunset self-energies. We find that equilibration is achieved only by inclusion of the sunset self-energy. Simultaneously, the time evolution of the scalar particle spectral function is studied for various initial states. We also compare associated solutions of the corresponding Boltzmann equation to the full Kadanoff-Baym theory. This comparison shows that a consistent inclusion of the spectral function has a significant impact on the equilibration rates only if the width of the spectral function becomes larger than 1/3 of the particle mass. Furthermore, based on these findings, the conventional transport of particles in the on-shell quasiparticle limit is extended to particles of finite life time by means of a dynamical spectral function A(X, p, M 2 ). The off-shell propagation is implemented in the Hadron-String-Dynamics (HSD) transport code and applied to the dynamics of nucleus-nucleus collisions. ⋆ supported by GSI Darmstadt

Effect of memory on relaxation in a scalar field theory

“…5, δN (p = 0, t)/δN (p = 0, 0) is plotted as a function of t • T in the moderately large coupling regime, g(2πT ) = 0.69, 0.91 and 1.02. In this regime, relaxation for the non-Markovian case is faster than that for the Markovian case, which is consistent with the results from the KB equations in 2+1 dimensions [19].…”

“…The spectral function used in this paper is the quasiparticle one and cannot be applicable to the system where the relaxation time is shorter than 1/ω p . In order to study the effect of the finite width in the spectral function, it would be necessary to use the phenomenological treatment such as an extended quasiparticle picture [32,39], or to solve directly KB equations for two-point Green's functions [13][14][15][16]18,19]. Twotime Green's functions in KB equations involve the information about the spectral function having the finite width as well as the one-particle distribution function.…”

“…This is an extensive example of a relativistic quantum field theory which allows us to challenge theoretical calculations without encountering troubles of gauge invariance. Within the three-loop order for 2PI effective action, KB equation for G < on Schwinger-Keldysh contour [12,28,29] takes a following form in the spatially homogeneous case [13,19,30],…”

“…It is instructive to notice that, by letting the finite initial time t 0 → −∞ corresponding to the limit of complete collisions and using the identity sin(xt)/x −−−→ t→∞ πδ(x), Eq. (2.10) reduces to the standard on-shell Boltzmann equation [19,30]: .11) In this equation, the time integration in the original KB equation reduces to δ function which represents the energy conservation in each binary collisions. Therefore, the quantum effect involved in the retardation is considered to be related to the energy broadening.…”

“…The quantum Kadanoff-Baym (KB) equations describe the time evolution of the two-time Green's functions which involve the information about both the dynamical spectral function and the one-particle distribution function in the correlated quantum plasmas [12], and are designed to study time-dependent nonequilibrium phenomena [13][14][15][16][17][18][19][20][21][22][23][24][25][26]. In general, it is difficult to solve the KB equations directly due to two-time correlation.…”