Times in Quantum Kinetic Equations Evgeni Kolomeitsev Matej Bel University, Banska Bystrica

Boltzmann kinetic equation distribution of particles in the phase space binary collisions! drift term Between collisions “particles” move along characteristic determined by an external force Fext collision term: Assumptions: “Stosszahlansatz” (chaos ansatz) -- valid for times larger than a collision time; -- sufficiently long mean free path, but not too long. local in time and space! Conservation: Entropy increase entropy for BKE is

Standard solution methods: Chapman-Enskog (expansion around equlibrium) hydrodynamical limit (kinetic coefficients); Grad’s method of moments…. Modifications of BKE: Vlasov equation: plasma Landau collision integral for Coulomb interaction (divergent) Balescu-Lenard (1960) and Silin-Rukhadze(1961) finite collision integral medium polarization Derivation of kinetic equations: Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy Bogoliubov’s principle of weakening of correlations Quantum kinetic equation: Pauli blocking, derivation of QKE

Important assumption behind the above KE: fixed energy-momentum relation In heavy-ion collisions many assumption behind the BKE are not justified At low energies Elab < 250 AMeV only NN collisions At high energies many new particles are produced Resonance dynamics How to write kinetic equations for resonaces? Spectral functions for pions and nucleons in medium

Non-equilibrium Green’s functions vacuum or equilibrium medium in interaction picture: transition from the ground state to the ground state under action of evolution operator Wick theorem Only one type of Green’s functions! Diagram technique

The non-equilibrium theory is formulated on closed real-time contour non- equilibrium medium “observables” The non-equilibrium theory is formulated on closed real-time contour orders the operators according to a time parameter running along the time contour Schwinger, Keldysh Heisenberg picture interaction picture For any two-point function F, the contour values on the different branches define a 2x2-matrix function

4 Green’s functions in non-equilibrium inverse time ordering Wigner densities (no time ordering operations) Green functions are not independent ! Retarded Green’s function a “would-be” spectral density

free Green’s function G0 local source current Full Green’s function This equation is still exact and accounts for the full set of initial correlations contained in the density matrix. Feynman diagrammatic representation of the processes is not yet possible at this level. This description level should be still time reversible.

coarse-graining over very short times In order to proceed further one suggests that the typical interaction time tint for the change of the correlation functions is significantly shorter than the typical time, which determines the system evolution. Then, describing the system at times t-t0 << tint, one can neglect the short range correlations, which are supposed to die out beyond tint in line with the principle of weakening of initial and all short-range ~tint correlations. coarse-graining over very short times After dropping higher order correlations one can apply the standard Wick decomposition. Dyson equation is recovered c1= one-particle irreducible part

Transport scheme. Wigner transformation For any two-point function Difference of the Dyson equation and the corresponding adjoint equation Half-sum of the Dyson equation and the corresponding adjoint equation give the so-called “mass” equation

Gradient expansion Poisson brackets Physical notations We introduce quantities which are real and, in the quasi-homogeneous limit, positive, have a straightforward physical interpretation, much like for the Boltzmann equation. Wigner functions 4-phase-space distribution functions

Kinetic equation in the 1st gradient approximation Drift operator Collision term Gain term (production rate): Loss term (absorption rate): The “mass” equation gives a solution for the retarded Green’s function mass function: width

Three forms of the kinetic equation Equilibrium relation Three forms of the kinetic equation Kadanoff-Baym equation Botermans-Malfliet equation non-local form [Ivanov, Voskresensky] shifted collision term

Time shifts resonance decay time delay time can be positive (delay) and negative (advance) for Wigner resonance