Between Green's Functions and Transport Equations

B. Velický, Charles University and Acad. Sci. of CR, Praha

A. Kalvová, Acad. Sci. of CR, Praha

V. Špička, Acad. Sci. of CR, Praha

PROGRESS IN NON-EQUILIBRIUM GREEN'S FUNCTIONS IIIKiel August 22 – 25, 2005

Between Green's Functions and Transport

Equations: Reconstruction Theorems and the Role of Initial

Conditions

B. Velický, Charles University and Acad. Sci. of CR, Praha

V. Špička, Acad. Sci. of CR, Praha

PROGRESS IN NON-EQUILIBRIUM GREEN'S FUNCTIONS IIIKiel August 22 – 25, 2005

Between Green's Functions and Transport Equations:

Correlated Initial Condition for Restart Process

B. Velický, Charles University and Acad. Sci. of CR, PrahaV. Špička, Acad. Sci. of CR, Praha

Topical Problems in Statistical PhysicsTU Chemnitz, November 30, 2005

Between Green's Functions and Transport Equations:

Correlated Initial Condition for Restart Process

Time Partitioning for NGF

B. Velický, Charles University and Acad. Sci. of CR, PrahaV. Špička, Acad. Sci. of CR, Praha

Topical Problems in Statistical PhysicsTU Chemnitz, November 30, 2005

Between GF and Transport Equations … 5 TU Chemnitz Nov 30, 2005

Prologue

(Non-linear) quantum transport non-equilibrium problem

many-body Hamiltonian

many-body density matrix

additive operator

Many-body system

Initial state

External disturbance

0 0 0at ( )t t t P P

0 ( ) for U t t t ( )tU

0( ) for t t t

Many-body system

Initial state

Response

additive operator

one-particle density matrix

0 0 0at ( )t t t P P

0 ( ) for U t t t ( )tU

Quantum Transport Equation a closed equation for ( )t

drift [ ( ); ]tt

generalized collision term

Many-body system

Initial state

Response

additive operator

0 0 0at ( )t t t P P

0( ) for t t t 0 ( ) for U t t t ( )tU

drift [ ( ); ]tt

Many-body system

Initial state

Response

additive operator

0 0 0at ( )t t t P P

0( ) for t t t

interaction term

0 ( ) for U t t t ( )tU

drift [ ( ); ]tt

Many-body system

Initial state

Response

additive operator

0 0 0at ( )t t t P P

0( ) for t t t

QUESTIONS existence, construction of incorporation of the initial

condition

interaction term

0 ( ) for U t t t ( )tU

This talk: orthodox study of quantum transport using NGF

TWO PATHS

INDIRECT

DIRECT †0(1,1') Tr( (1) (1')) (1) i (1,1)G G CTP

use a NGF solver

use NGF to construct a Quantum Transport Equation

TWO PATHS

DIRECT

INDIRECT

†0(1,1') Tr( (1) (1')) (1) i (1,1)G G CTP

use a NGF solver

Lecture on NGF

TWO PATHS

DIRECT †0(1,1') Tr( (1) (1')) (1) i (1,1)G G CTP

use a NGF solver

Lecture on NGF…continuation

Real time NGF choices Kadanoff and Baym

Keldysh

, Langreth and Wilk, ins

G G G G

TWO PATHS

DIRECT †0(1,1') Tr( (1) (1')) (1) i (1,1)G G CTP

use a NGF solver

TWO PATHS

DIRECT

INDIRECT

†0(1,1') Tr( (1) (1')) (1) i (1,1)G G CTP

use a NGF solver

Standard approach based on GKBA

Real time NGF our choice Langreth and , Wilkins,R AG G G

GKBEequal times

drift A R R AG G G Gt

GKBEequal times

Specific physical approximation -- self-consistent form R

GKBEequal times

Elimination of by an Ansatz

widely used: KBA (for steady transport), GKBA (transients, optics)

GKBEequal times

( , ') ( , ') ( ')'

( ) ( , ')'

t t t tG t t G t t t t G t t

Lipavsky, Spicka, Velicky, Vinogradov, Horing

Haug + Frankfurt team, Rostock school, Jauho, …

GKBEequal times

drift [ ( ); | , ]R At G Gt

( , ') ( , ') ( ') ( ) ( , ')R AG t t G t t t t G t t

Resulting Quantum Transport Equation

GKBEequal times

drift [ ( ); | , ]R At G Gt

( , ') ( , ') ( ') ( ) ( , ')R AG t t G t t t t G t t

Resulting Quantum Transport EquationFamous examples:•Levinson eq. (hot electrons)•Optical quantum Bloch eq. (optical transients)

reconstruction

Exact formulation -- Reconstruction Problem

GENERAL QUESTION: conditions under which a many-body interacting systemcan be described in terms of its single-time single-particle characteristics

Reminiscences: BBGKY, Hohenberg-Kohn Theorem

Here: time evolution of the system

Eliminate by an Ansatz

GKBA ( , ') ( , ') ( ') ( ) ( , ')R AG t t G t t t t G t t

… in fact: express , a double-time correlation function, by its time diagonal

( , ')G t t

i ( ) ( , .)t G t t

New look on the NGF procedure:

Any Ansatz is but an approximate solution…

¿Does an answer exist, exact at least in principle?

INVERSION SCHEMES

REDUCTION additive ( )0 0

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

( , )n x tBOGOLYUBOV

SCHWINGER GENERATING FUNCTIONAL

TIME-DEPENDENT DENSITY FUNCTIONAL

RUNGE - GROSS THEOREM

Reconstruction Problem – Historical Overview

INVERSION SCHEMES

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

Reconstruction Problem – Historical Overview

Postulate/Conjecture:typical systems are controlled by a hierarchy of times

separating the initial, kinetic, and hydrodynamic stages.A closed transport equation

holds for

Parallels

G E N E R A L S C H E M E

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

Bogolyubov

drift [ ( ); ]tt

0 C .t t

Postulate/Conjecture:typical systems are controlled by a hierarchy of times

separating the initial, kinetic, and hydrodynamic stages.A closed transport equation

holds for

Parallels

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

Bogolyubov

drift [ ( ); ]tt

0 C .t t

Runge – Gross Theorem:Let be local. Then, for a fixed initial state , the functional relation is bijective and can be inverted.

NOTES: U must be sufficiently smooth. no enters the theorem. This is an existence theorem, systematic implementation based on the use of the closed time path generating functional.

Parallels

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

U t 0[ ]n U

0 ,t t

Runge – Gross Theorem:Let be local. Then, for a fixed initial state , the functional relation is bijective and can be inverted.

NOTES: U must be sufficiently smooth. no enters the theorem. This is an existence theorem, systematic implementation based on the use of the closed time path generating functional.

Parallels

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

0 ,t t

( )n t

[ ]n U

Closed Time Contour Generating Functional (Schwinger):

Used to invert the relation EXAMPLES OF USE:Fukuda et al. … Inversion technique based on Legendre transformation Quantum kinetic eq.Leuwen et al. … TDDFT context

Parallels

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

Schwinger0 0

i d ( ( ) ) i d ( ( ) )i ( , )0e Tr e e

( ) ( )

t tU X U XW U U

tU U U U U U

U t U t

T TH H

[ ]n U

Closed Time Contour Generating Functional (Schwinger):

Used to invert the relation EXAMPLES OF USE:Fukuda et al. … Inversion technique based on Legendre transformation Quantum kinetic eq.Leuwen et al. … TDDFT context

Parallels

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

Schwinger0 0

i d ( ( ) ) i d ( ( ) )i ( , )0e Tr e e

( ) ( )

t tU X U XW U U

tU U U U U U

U t U t

T TH H

( )n t

[ ]n U

35 TU Chemnitz Nov 30, 2005

„Bogolyubov“: importance of the time hierarchy

REQUIREMENT Characteristic times should emerge in a constructive manner during the reconstruction procedure.

„TDDFT“ : analogue of the Runge - Gross Theorem

REQUIREMENT Consider a general non-local disturbance U in order to obtain the full 1-DM as its dual.

„Schwinger“: explicit reconstruction procedure

REQUIREMENT A general operational method for the reconstruction (rather than inversion in the narrow sense). Its success in a particular case becomes the proof of the Reconstruction theorem at the same time.

Parallels: Lessons for the Reconstruction Problem

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

NGFReconstruction

Theorem

INVERSION SCHEMES

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

( )n tBOGOLYUBOV

Reconstruction Problem – Summary

INVERSION SCHEMES

at ( ) ( ) ( ) Tr ( )

( ) for

t tt t X t X t

U t t t

H U XP XP

Reconstruction Problem – Summary

Reconstruction theorem :Reconstruction equationsKeldysh IC: simple initial state permits to concentrate on the other issues

DYSON EQUATIONS

1 1 1 10 0( ) ( ) ( ) ( )R R R R A A A AG G G G G G G

Two well known “reconstruction equations” easily follow:

RECONSTRUCTION EQUATIONS

1 2 1 1 2 2 2 1 1 1 2 2'

1 2 1 1 2 2 2 1 1 1 2'

( , ')

( , ') ( ') ( ) ( , ')

d d ( , ) ( , ) ( , ') d d ( , ) ( , ) ( , ')

d d ( , ) ( , ) ( , ') d d ( , ) ( , ) (

' 'R A

t t t tR A R A

t t tR R A A

G t t t t G t t

t t G t t t t G t t t t G t t t t G t t

t t G t t t t G t t t t G t t t

2 , ')t

LSV, Vinogradov … application!

DYSON EQUATIONS

1 1 1 10 0( ) ( ) ( ) ( )R R R R A A A AG G G G G G G

Two well known “reconstruction equations” easily follow:

RECONSTRUCTION EQUATIONS

1 2 1 1 2 2 2 1 1 1 2 2'

1 2 1 1 2 2 2 1 1 1 2'

( , ')

( , ') ( ') ( ) ( , ')

d d ( , ) ( , ) ( , ') d d ( , ) ( , ) ( , ')

d d ( , ) ( , ) ( , ') d d ( , ) ( , ) (

' 'R A

t t t tR A R A

t t tR R A A

G t t t t G t t

t t G t t t t G t t t t G t t t t G t t

t t G t t t t G t t t t G t t t

2 , ')t

Source terms … the Ansatz

For t=t' … tautology … input

Reconstruction theorem :Reconstruction equationsKeldysh IC: simple initial state permits to concentrate on the other issues

Reconstruction theorem: Coupled equations

DYSON EQ.R AG G G

GKB EQ.

equal times

A R R A

G G G G

RECONSTRUCTION EQ.

1 2 1 1 2 2'

( , ') ( , ') ( ')

d d ( , ) ( , ) ( , ')

d d ( , ) ( , ) ( ,

t tR A

t tR R

G t t G t t t

t t G t t t t G t t

t t G t t t t G

Reconstruction theorem: operational description

NGF RECONSTRUCTION THEOREMdetermination of the full NGF restructured as a

DUAL PROCESS

quantum transport equation

reconstruction equations

Dyson eq.

,R AG G

"THEOREM" The one-particle density matrix and the full NGF of a process are in a bijective relationship,

NGF RECONSTRUCTION THEOREMdetermination of the full NGF restructured as a

DUAL PROCESS

quantum transport equation

reconstruction equations

Dyson eq.

,R AG G

Reconstruction theorem: formal statement

Act II

reconstructionand initial conditions

NGF view

For an arbitrary initial state at start from the NGF

Problem of determination of G extensively studied

Fujita Hall Danielewicz … Wagner Morozov&Röpke …

Klimontovich Kremp … Bonitz&Semkat …

Take over the relevant result for :

The self-energy

depends on the initial state (initial correlations)

has singular components

General initial state

†0(1,1') iTr( (1) (1'))G CTP

0 0 t tP

for Keldysh limit

for an arbitrary t

G G G t

0[ | ]U

For an arbitrary initial state at start from the NGF

Problem of determination of G extensively studied

Fujita Hall Danielewicz … Wagner Morozov&Röpke …

Klimontovich Kremp … Bonitz&Semkat …

Take over the relevant result for :

The self-energy

depends on the initial state (initial correlations)

has singular components

General initial state

†0(1,1') iTr( (1) (1'))G CTP

0 0 t tP

for Keldysh limit

for an arbitrary t

G G G t

0[ | ]U

Morawetz

General initial state: Structure of

0 0 0 0

( , ') i ( ) ( ) ( ' )

( , ') ( , ) ( ' ) ( , ') ( , ') ( )

( , ') ( , ')

t t t t t t t

t t t t t t t t t t t t

t t t t

Structure of

Danielewicz notation

0 0 0 0

( , ') i ( ) ( ) ( ' )

( , ') ( , ) ( ' ) ( , ') ( , ') ( )

( , ') ( , ')

t t t t t t t

t t t t

Structure of

Danielewicz notation

0 0 0 0

( , ') i ( ) ( ) ( ' )

( , ') ( , ) ( ' ) ( , ') ( , ') ( )

( , ') ( , ')

t t t t t t t

t t t t

Structure of

General initial state: A try at the reconstruction

DYSON EQ.R AG G G

GKB EQ.

equal times

drift A R R A

A R R A A R

G G G Gt

G G G G G G

1 2 1 1 2 2'

0( , ') ( , ') ( ')

d d ( , ) ( , ) ( , ')

d d ( , ) ( , )

( , ')

t tR A

t tR R

G t t G t t t

t t G t t t t G t t

t t G t t t

RECONSTRUCTION EQ.

General initial state: A try at the reconstruction

DYSON EQ.R AG G G

GKB EQ.

equal times

drift A R R A

A R R A A R

G G G Gt

G G G G G G

1 2 1 1 2 2'

0( , ') ( , ') ( ')

d d ( , ) ( , ) ( , ')

d d ( , ) ( , )

( , ')

t tR A

t tR R

G t t G t t t

t t G t t t t G t t

t t G t t t

RECONSTRUCTION EQ.

To progress further,

narrow down the selection of the initial states

Initial state for restart process

Process, whose initial state coincides withintermediate state of a host process (running)

Aim: to establish relationship between NGF of the host and restart process

To progress further, narrow down the selection of the initial states

Special situation:

Let the initial time be , the initial state . In the host NGF

the Heisenberg operators are

Restart at an intermediate time

†(1) ( , ) ( ) ( , ), (1')t t x t t K K

†0(1,1') Tr( (1) (1'))G CTP

i ( , ') ( ( )) ( , '), ( , )t t t t t t t t K H U K K 1

We may choose any later time as the new initial time.For times the resulting restart GF should be consistent. Indeed, with

we have† †

0 0 0 0(1,1') Tr( (1) (1')) Tr( ( ) (1| ) (1' | ))G t t t C CT TP P

Restart at an intermediate time0t t

0 0, 't t t t

0 0 0 0

†0 0 0 0

( ) ( , ) ( , ),

(1| ) ( , ) ( ) ( , ), (1' | )

t t t t t

t t t x t t t

K KP PK K

We may choose any later time as the new initial time.For times the resulting GF should be consistent. Indeed, with

we have† †

0 0 0 0(1,1') Tr( (1) (1')) Tr( ( ) (1| ) (1' | ))G t t t C CT TP P

Restart at an intermediate time0t t

0 0, 't t t t

0 0 0 0

†0 0 0 0

( ) ( , ) ( , ),

(1| ) ( , ) ( ) ( , ), (1' | )

t t t t t

t t t x t t t

K KP PK K

whole family of initia

l states

for varying t 0

† †0 0 0 0(1,1') Tr( (1) (1')) Tr( ( ) (1| ) (1' | ))G t t t C CT TP P

NGF is invariant with respect to the initial time,

the self-energies must be related in a specific way for

Important difference

0 0, 't t t t

, ,( , ') ( , ')

( , ') ( , ')

R A R At t t t

t t t t

… causal structure of the Dyson equation

… develops singular parts at as a condensed information about the past

0(1,1') (1,1') (1,1')t tG G G

0 0, 't t t t

, ,( , ') ( , ')

( , ') ( , ')

R A R At t t t

t t t t

0(1,1') (1,1') (1,1')t tG G G

0 0, 't t t t

, ,( , ') ( , ')

( , ') ( , ')

R At t t t

t t t t

0(1,1') (1,1') (1,1')t tG G G

Objective and subjective components of the initial correlations

The zone of initial correlations of wanders with our choice of the initial time; if we do not know about the past, it looks to us like real IC.

0 0, 't t t t

, ,( , ') ( , ')

( , ') ( , ')

R A R At t t t

t t t t

Intermezzo

Time-partitioning

Time-partitioning: general method

Special position of the (instant-restart) time t0

-Separates the whole time domain into the past and the future

- past - future notion … in reconstruction equation

- past future notion … in reconstruction equationRECONSTRUCTION EQ.

1 2 1 1 2 2

1 2 1 1'

( , ) ( , ) ( )

d d ( , ) ( , )

( , ')

d d ( , ) ( ,

) ( , )

G t G t

t t G t t t t G t t

t G t t t t G t t

- past future notion … in reconstruction equationRECONSTRUCTION EQ.

2 2 21

d ) ( , ')

( , ) ( , ) ( )

d ( , ) ( ,

d (d ), ) (

t t G t t

G t G t

t G t t t

t G t t

-past future notion … in reconstruction equationRECONSTRUCTION EQ.

2 2 21

d ) ( , ')

( , ) ( , ) ( )

d ( , ) ( ,

d (d ), ) (

t t G t t

G t G t

t G t t t

t G t t

-past future notion … in reconstruction equationRECONSTRUCTION EQ.

1 1 1'

21 1 1 2'

( , ) ( , ) ( )

d )d ( , ) ( , ')

d ) ( , ')

G t G t

t G t t t

t t G t t

t t t G t

future

- past - future notion … in reconstruction equation for G<

- past - future notion … in corrected semigroup rule GR

67 TU Chemnitz Nov 30, 2005

CORR. SEMIGR. RULE

1 2 1 1 2

( , ') i ( , ) ( , ')

d d ( , ) ( , ) (

G t t G t G t

t t G t t t t

'' 't t t

- past - future notion … in restart NGF

unified description—

time-partitioning formalism

Partitioning in time: formal tools

Past and Future with respect to the initial (restart) time 0t0tt

t0 0( ) ( ) ( ) ( )t t t t t t t t

pas futur

( ) ( ) ( ') ( ) ( ') ( ) ( ')

( , ') ( , ') ( , ')

t t t t t t I t t t t I t t t t I

t t t t t t

P F 1Projection operators

t0 0( ) ( ) ( ) ( )t t t t t t t t

pas futur

( ) ( ) ( ') ( ) ( ') ( ) ( ')

( , ') ( , ') ( , ')

t t t t t t I t t t t I t t t t I

t t t t t t

P F 1Projection operators

Double time quantity X X= X X X X P P P F F P F F

…four quadrants of the two-time plane

Partitioning in time: for propagators

1. Dyson eq.0 0

R R R R RG G G G

2. Retarded quantity R ( , ') 0X t t only for 't t

0RX P F

3. Diagonal blocks of RG

R R R R R

G G G G

P P P P P P P P

F F F F F F F F

Partitioning in time: for propagators …continuation

0 0R R R R RG G G G

0 0 ( ) ( )R R R R RG G G G F P F P F F P F P P

4. Off-diagonal blocks of RG-free propagator corresponds to a unitary evolution multiplicative composition law semigroup rule

0 0 0 0 0 0( , ') i ( , ) ( , ') 'R R RG t t G t t G t t t t t

0 0 0R R R RG G L G F P F F P P

0 0( , ') i ( ) ( ')RL t t t t t t I … time local operator

0 0R R R R RG G G G

R R R R R RG G L G G G RF P F F P P F F F P P P

0 0R R R R RG G G G

R RR R RR GG GL GG RF P F P F FPF PFP P

1 2 1 20 1 2( , ) ( , )( , ')( , ') i d '( ,d , ( ))t

RGG t t G tG t t t t tt t tt G tt

0 0R R R R RG G G G

1 2 1 20 1 2( , ) ( , )( , ')( , ') i d '( ,d , ( ))t

0 0R R R R RG G G G

time-local factorization

vertex correction: universal form

(gauge invariance) link past-future

non-local in timewidth 2 Q

1 2 1 20 1 2( , ) ( , )( , ')( , ') i d '( ,d , ( ))t

0 0R R R R RG G G G

time-local factorization

vertex correction: universal form

(gauge invariance) link past-future

non-local in timewidth 2 Q

Corrected semigroup rule

Partitioning in time: for corr. function G

R AG G G Question: to find four blocks of G

1. Selfenergy … split into four blocks

2. Propagators ,R AG G … by partitioning expressions

R AG G G P P P P P P …(diagonal) past blocks only

R AG G G P P P P P P

( )R A A A AG G G G L G P F P P F F P P F F

( )R A AA AG GLGG G FP P PF PP FF F

…diagonals of GF’s

( )R A AAAG G LG G G FPP F P P FF P F

…off-diagonals of selfenergies

( )R A R R RG G G G L G F P F F P P F F P P

( )R A R R RG G G G L G F P F F P P F F P P( )

( ) ( )

R A A A A

R R R R A

R R R A A A

G G L G

G L G G

G L G L G

F F F F F F

F F P P F F

( ) ( )

F F F F

…diagonals of GF’s

( ) ( )

R R A A

G G GL

F F F F

PF PF F F

…off-diagonals of

selfenergy

( ) ( )

R R A A

G G GL

F F F F

PF PF F F

…off-diagonals of

selfenergy

Exception!!!

Future-future diagonal

( ) ( )

R R A A

G G GL

F F F F

PF PF F F

Partitioning in time: restartrestart corr. function 0t

R AG G G HOST PROCESS

RESTART PROCESS0 0

R At tG G G F F F F

( ) ( )

R R A A

G G GL

F F F F

PF PF F F

RESTART PROCESS0 0

R At tG G G F F F F

( ) ( )

R R A A

G G GL

F F F F

PF PF F F

RESTART PROCESS0 0

R At tG G G F F F F

future

memory of the past folded

down into the future by

partitioning

( ) ( )

R R A A

G G GL

F F F F

PF PF F F

RESTART PROCESS0 0

R At tG G G F F F F

0initial conditionst F F

future

memory of the past folded

down into the future by

partitioning

Partitioning in time: initial conditioninitial condition 0t

0 0[ ]t t

00 0 0( , ') i ( ) ( ) ( ' )

tt t t t t t t

Singular time variable fixed at restart time 0t t

0 0[ ]t t

0t F F

R AL G L P PR AG F P P F R AL G P P FR AG L F P P

A AG L F P PA AG F P P F

R RG F P P FR RL G P P F

0 0[ ]t t

0t F F

0 0[ ]t t

0t F F

0 0[ ]t t

0t F F

0 0[ ]t t

0t F F

0 0[ ]t t

0t F F

… omited initial condition, 0

0t t Keldysh limit

0 0[ ]t t

0t F F

… with uncorrelated initial condition,

0 0[ ]t t

0t F F

… with uncorrelated initial condition,

d d ( , ) ( , ) ( , )t t

t t t t G t t t t

0 0 0i ( ) ( ) ( )t t t t t

0 0[ ]t t

0t F F

RestartRestart correlation function: initial conditions

continuous time variable

t > t0

singular time variable fixed at

t = t0

singular time variable fixed at

t = t0

00 0 0( , ') i ( ) ( ) ( ' )

tt t t t t t t

uncorrelated initial condition ... KELDYSH

<o o 0 0( , ') ( , ) ( ' )

tt t t t t t

correlated initial condition ... DANIELEWICZ

<o 0 o 0( , ') ( ) ( , )

tt t t t t t

<( , ') ( , ')t

t t t t

host continuous self-energy ... KELDYSHinitial correlations correction MOROZOV &RÖPKE

Act III

applications:restarted switch-on processes

pump and probe signals....

NEXT TIME

Conclusions• time partitioning as a novel general technique for

treating problems, which involve past and future with respect to a selected time

• semi-group property as a basic property of NGF dynamics

• full self-energy for a restart process including all singular terms expressed in terms of the host process GF and self-energies

• result consistent with the previous work (Danielewicz etc.)

• explicit expressions for host switch-on states (from KB -- Danielewicz trajectory to Keldysh with t0 -

Conclusions• time partitioning as a novel general technique for

treating problems, which involve past and future with respect to a selected time

• semi-group property as a basic property of NGF dynamics

• full self-energy for a restart process including all singular terms expressed in terms of the host process GF and self-energies

• result consistent with the previous work (Danielewicz etc.)

• explicit expressions for host switch-on states (from KB -- Danielewicz trajectory to Keldysh with t0 -

THE END

Between Green's Functions and Transport Equations

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