Nonlocal form of quantum off-shell kinetic equation
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NUCLEI Theory
Nonlocal Form of Quantum Off-Shell Kinetic Equation* Yu. B. Ivanov** and D. N. Voskresensky*** Gesellschaft fur ¨ Schwerionenforschung mbH, Darmstadt, Germany Received December 1, 2008
Abstract—A new nonlocal form of the off-shell kinetic equation is derived. While being equivalent to the Kadanoff–Baym and Botermans–Malfliet formulations in the range of formal applicability, it has certain advantages beyond this range. It possesses more accurate conservation laws for Noether quantities than those in the Botermans–Malfliet formulation. At the same time the nonlocal form, similarly to the Botermans–Malfliet one, allows application of the test-particle method for its numerical solution, which makes it practical for simulations of heavy-ion collisions. The physical meaning of the time–space nonlocality is clarified. PACS numbers: 24.10.Cn, 05.60.-k, 05.70.Ln DOI: 10.1134/S1063778809070096
1. INTRODUCTION The work of Belyaev and Budker [1], who demonstrated the Lorentz invariance of the relativistic distribution function and derived relativistic Fokker– Planck kinetic equation, stands in the line of achievements of the kinetic theory. This work entered many textbooks and found numerous applications in various fields. Presently the relativistic transport concepts are a conventional tool to analyze the dynamics of dense and highly excited matter produced in relativistic heavy-ion collisions. A great progress was also achieved in microscopic foundation of the kinetic theory. The appropriate frame for description of nonequilibrium processes within the real-time formalism of quantum-field theory was developed by Schwinger, Kadanoff, Baym, and Keldysh [2–4]. The formalism allows extensions of the quantum kinetic picture beyond conventional approximations (like the quasiparticle one). This is caused by the quest for dynamical treatment of broad resonances as well as stable particles which acquire a considerable mass width because of collisional broadening. The above mentioned applications request for development of approximate self-consistent schemes possessing conservation laws being at least approximately satisfied [5–11]. Based on these schemes, numerical transport methods for treatment of the offshell dynamics have been developed [12–14].
Two slightly different forms of the Kadanoff–Baym equations expanded up to first-order space–time gradients are now used: the proper Kadanoff–Baym (KB) form, as it follows right after the gradient expansion of exact KB equations [3], and the Botermans– Malfliet (BM) one, as it follows after a modification of a Poisson-bracket term in the KB equation [15]. Both the KB and BM forms coincide in the first-order gradient approximation but differ in higher orders. Both forms have their advantages and disadvantages [10]. The KB form possesses exact conservation laws for the Noether current and the energy–momentum [9, 10], however, it does not allow the efficient testparticle method to be applied to its numeric solution. The BM form is very suitable for the test-particle method [12–14] but only approxi
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