between green's functions and transport equations

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

H

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

H

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


H

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


H

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


H

0 0 0at ( )t t t P P

0( ) for t t t

QUESTIONS existence, construction of incorporation of the initial

condition

0P

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


Lecture on NGF…continuation

Real time NGF choices Kadanoff and Baym

Keldysh

,

, ,

, Langreth and Wilk, ins

R A

R A

G G

G G G G

G G G


TWO PATHS

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

use a NGF solver

15

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

A

G G

G

GR

A

[ ] G




GKBEequal times



A

G G

G

GR

A

[ ] G

Elimination of by an Ansatz

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

G




GKBEequal times



A

G G

G

GR

A

[ ] G


GKBA

G

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

( ) ( , ')'

R A

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


A

G G

G

GR

A

[ ] G


GKBA

G

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

Resulting Quantum Transport Equation




GKBEequal times


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


A

G G

G

GR

A

[ ] G


GKBA

G

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


Act I

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




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

G

… 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

0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

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


0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

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


0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

t tt t X t X t

U t t t

H U XP XP

LABEL

Bogolyubov

drift [ ( ); ]tt

C H

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



0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

t tt t X t X t

U t t t

H U XP XP

LABEL

Bogolyubov

drift [ ( ); ]tt

C H

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



0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

t tt t X t X t

U t t t

H U XP XP

LABEL

TDDFT

C

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



0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

t tt t X t X t

U t t t

H U XP XP

LABEL

TDDFT

C

U t 0

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



0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

t tt t X t X t

U t t t

H U XP XP

LABEL

Schwinger0 0

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

( ) ( )

t t

t tU X U XW U U

tU U U U U U

W W

U t U t

T TH H

P

X

[ ]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



0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

t tt t X t X t

U t t t

H U XP XP

LABEL

Schwinger0 0

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

( ) ( )

t t

t tU X U XW U U

tU U U U U U

W W

U t U t

T TH H

P

X

( )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



0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

t tt t X t X t

U t t t

H U XP XP

LABEL

NGFReconstruction

Theorem

C , ,


INVERSION SCHEMES


0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

t tt t X t X t

U t t t

H U XP XP

( )n tBOGOLYUBOV




Reconstruction Problem – Summary


INVERSION SCHEMES


0

at ( ) ( ) ( ) Tr ( )

( ) for

t

t

t tt t X t X t

U t t t

H U XP XP





Reconstruction Problem – Summary

G


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

t t tR R A A

t

G 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 t G t t t

t t t

t

t

G

'

2 , ')t

t

t 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

t t tR R A A

t

G 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 t G t t t

t t t

t

t

G

'

2 , ')t

t

t 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

drift

A R R A

t

G G G G

RECONSTRUCTION EQ.

'

1 2 1 1 2 2'

'

1 2 1 1 2 2'

( , ') ( , ') ( ')

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

d d ( , ) ( , ) ( ,

'

')

R

t tR A

t

t tR R

t

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

t

t t

t


Reconstruction theorem: operational description

NGF RECONSTRUCTION THEOREMdetermination of the full NGF restructured as a

DUAL PROCESS

quantum transport equation

reconstruction equations

Dyson eq.

G

,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.

G

,R AG G

R

A

G G

G

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

G

0

0

for Keldysh limit

for an arbitrary t

R A

R A

G G G t

G G G

0[ | ]U

P


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

G

0

0

for Keldysh limit

for an arbitrary t

R A

R A

G G G t

G G G

0[ | ]U

P

Morawetz


General initial state: Structure of

0 0 0

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 0 0 0

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

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

( , ') ( , ')

t t t t t t t


t t t t

Structure of



Danielewicz notation

0 0 0

0 0 0 0

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

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

( , ') ( , ')

t t t t t t t


t t t t

Structure of

0t

0t

t

't


General initial state: A try at the reconstruction

DYSON EQ.R AG G G

GKB EQ.

equal times

equal times

drift A R R A

A R R A A R

G G G Gt

G G G G G G

0

0

'

1 2 1 1 2 2'

'

1 2 1 1 2 2'

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

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

d d ( , ) ( , )

'

( , ')

t

t

R

t tR A

t

t tR R

t

G t t G t t t

t t G t t t t G t t

t t G t t t

t t

G t t

t

t

RECONSTRUCTION EQ.

General initial state: A try at the reconstruction

DYSON EQ.R AG G G

GKB EQ.

equal times

equal times

drift A R R A

A R R A A R

G G G Gt

G G G G G G

0

0

'

1 2 1 1 2 2'

'

1 2 1 1 2 2'

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

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

d d ( , ) ( , )

'

( , ')

t

t

R

t tR A

t

t tR R

t

G t t G t t t

t t G t t t t G t t

t t G t t t

t t

G t 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

0P

†(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

t t

t

t

'tt


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

t

t

'tt

t

t

'tt

0t

0t


t

t

'tt

0t

0t

t

t

'tt

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

0

0

, ,( , ') ( , ')

( , ') ( , ')

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

0t

0t t


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





0 0, 't t t t

0

0

, ,( , ') ( , ')

( , ') ( , ')

t

t

R A R At t t t

t t t t



0t

0t t


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





0 0, 't t t t



0t

0t t

0

0

, ,( , ') ( , ')

( , ') ( , ')

t t

t

R A

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

0

0

, ,( , ') ( , ')

( , ') ( , ')

t

t

R A R At t t t

t t t t



0t

0t t

0t


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'

2

'

'

2

'

( , ) ( , ) ( )

d d ( , ) ( , )

' '

( , ')

d d ( , ) ( ,

'

) ( , )

'

'

t

t

t

R

tR A

tR R

t

G t G t

t t G t t t t G t t

t

tt

t G t t t t G t t

tt t





- past future notion … in reconstruction equationRECONSTRUCTION EQ.

1 1 1

1

'

2 2 2

'

2 2 21

'

'1

d ) ( , ')

( , ) ( , ) ( )

d ( , ) ( ,

d (d ), ) (

'

'

'

( , )

' '

,

t

t

tA

t

R

tR

tR R

t t G t t

G t G t

t G t t t

t G t t

t

t t

t

G t

tt

t t

t





-past future notion … in reconstruction equationRECONSTRUCTION EQ.

1 1 1

1

'

2 2 2

'

2 2 21

'

'1

d ) ( , ')

( , ) ( , ) ( )

d ( , ) ( ,

d (d ), ) (

'

'

'

( , )

' '

,

t

t

tA

t

R

tR

tR R

t t G t t

G t G t

t G t t t

t G t t

t

t t

t

G t

tt

t t

t

past





-past future notion … in reconstruction equationRECONSTRUCTION EQ.

'

'

2 2 2

2

1 1 1'

21 1 1 2'

( , ) ( , ) ( )

d )d ( , ) ( , ')

d ) ( , ')

'

( ,

d ( ,

' '

) ( ,

't

R

t

tR

t

tR

R

A

t

G t G t

t G t t t

t G t

t t G t t

t

t t

t t t G t

t t

t

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

''

''2

'

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

d d ( , ) ( , ) (

'

)

'

'

''

,t

R R R

tR R

t

t

R

G t t G t G t

t t G t t t t

t t

G 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




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

pas futur

0 0

et

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

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

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 1

P F 1Projection operators




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

pas futur

0 0

et

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

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

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 1

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

0 0

0 0

R R R R R

R R R R R

G G G G

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







0

0

0'

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

R Rt

R R

t

Rt

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





0 0R R R R RG G G G



0

0

0'

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

R Rt

R R

t

Rt






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



0

0

0'

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

R Rt

R R

t

Rt






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

R A A A A

R R R R A

R R R A A A

G G G

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 P P F F

F F P P F F









( )

( ) ( )

A A

R R

R R

R A

R A

R A

R A

A

R

AA

G

G

G

L

L

L L

G G

G G

G G

G G G

F F F F

F F F F

F F F F

F F

F F

P P

P P F

P P

F

…diagonals of GF’s









( )

( ) ( )

A A

R

R A

R A A

R

R R A A

R R A

R A

G G G

G G GL

L

L

G G

G GL

G

G

F F F F

F F

F F

F F

F F

F F

P P

P

P

PF PF F F

…off-diagonals of

selfenergy









( )

( ) ( )

A A

R

R A

R A A

R

R R A A

R R A

R A

G G G

G G GL

L

L

G G

G GL

G

G

F F F F

F F

F F

F F

F F

F F

P P

P

P

PF PF F F

…off-diagonals of

selfenergy

Exception!!!

Future-future diagonal


( )

( )

( ) ( )

A A

R

R A

R A A

R

R R A A

R R A

R A

G G G

G G GL

L

L

G G

G GL

G

G

F F F F

F F

F F

F F

F F

F F

P P

P

P

PF PF F F

Partitioning in time: restartrestart corr. function 0t

G

R AG G G HOST PROCESS

RESTART PROCESS0 0

R At tG G G F F F F


( )

( )

( ) ( )

A A

R

R A

R A A

R

R R A A

R R A

R A

G G G

G G GL

L

L

G G

G GL

G

G

F F F F

F F

F F

F F

F F

F F

P P

P

P

PF PF F F


G


RESTART PROCESS0 0

R At tG G G F F F F

0t


( )

( )

( ) ( )

A A

R

R A

R A A

R

R R A A

R R A

R A

G G G

G G GL

L

L

G G

G GL

G

G

F F F F

F F

F F

F F

F F

F F

P P

P

P

PF PF F F


G


RESTART PROCESS0 0

R At tG G G F F F F

0t

future

memory of the past folded

down into the future by

partitioning


( )

( )

( ) ( )

A A

R

R A

R A A

R

R R A A

R R A

R A

G G G

G G GL

L

L

G G

G GL

G

G

F F F F

F F

F F

F F

F F

F F

P P

P

P

PF PF F F


G


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

G


0 0[ ]t t

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

tt t t t t t t

Singular time variable fixed at restart time 0t t



G


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



G


0 0[ ]t t

0t F F






G


0 0[ ]t t

0t F F




… omited initial condition, 0

[ ]t

0t t Keldysh limit



G


0 0[ ]t t

0t F F




… with uncorrelated initial condition,



G


0 0[ ]t t

0t F F




… with uncorrelated initial condition,

0 0

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

R A

t t

t t t t G t t t t

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



G


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



0

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

tt t t t t t

correlated initial condition ... DANIELEWICZ

0

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

tt t t t t t



0

<( , ') ( , ')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 -

....

THE END

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