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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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.

9. 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

10. 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

11. 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

12. 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

13. 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

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