Exact Conservation Laws of the Gradient Expanded Kadano#--Baym Equations.pdf

Exact Conservation Laws of the Gradient Expanded Kadanoff–Baym Equations J. Knoll1, Yu. B. Ivanov1,2 and D. N. Voskresensky1,3 1Gesellschaft fu?r Schwerionenforschung mbH, Planckstr. 1, 64291 Darmstadt, Germany 2Kurchatov Institute, Kurchatov sq. 1, Moscow 123182, Russia 3Moscow Institute for Physics and Engineering, Kashirskoe sh. 31, Moscow 115409, Russia E-mail: J.Knoll@gsi.de, Y.Ivanov@gsi.de, D.Voskresensky@gsi.de It is shown that the Kadanoff–Baym equations at consistent first-order gradient approximation reveal exact rather than approximate conservation laws related to global symmetries of the system. The conserved currents and energy–momentum tensor coincide with corresponding Noether quantities in the local approximation. These exact conservations are valid, provided a Φ derivable approximation is used to describe the system, and possible memory effects in the collision term are also consistently evaluated up to first-order gradients. 1. INTRODUCTION Non-equilibrium Green function techniques, developed by Schwinger, Kadanoff, Baym and Keldysh [1–4], provide the appropriate concepts to study the space–time evolution of many-particle quantum systems. This formalism finds now applications in various fields, such as quantum chromo- dynamics [5,6], nuclear physics, in particular heavy ion collisions [7–23], as- trophysics [11,24,25], cosmology [26], spin systems [27], lasers [28], physics of plasma [29,30], physics of liquid 3He [31], critical phenomena, quenched random systems and disordered systems [32], normal metals and super- conductors [24,33,34], semiconductors [35], tunneling and secondary emis- sion [36], etc. For actual calculations certain approximation steps are necessary. In many cases perturbative approaches are insufficient, like for systems with strong couplings as treated in nuclear physics. In such cases, one has to resum certain sub-series of diagrams in order to obtain a reasonable approx- imation scheme. In contrast to perturbation theory, f