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Transients in Quantum Transport:
B. Velický, Charles University and Acad. Sci. of CR, Praha
A. Kalvová, Acad. Sci. of CR, PrahaV. Špička, Acad. Sci. of CR, Praha
SEMINÁŘ TEORETICKÉHO ODD. FZÚSLOVANKA 14. ÚNORA 2006
Transients in Quantum Transport:I. Semi-Group Property of Propagators
and the Gauge Invariance of the 1st Kind
B. Velický, Charles University and Acad. Sci. of CR, Praha
A. Kalvová, Acad. Sci. of CR, PrahaV. Špička, Acad. Sci. of CR, Praha
SEMINÁŘ TEORETICKÉHO ODD. FZÚSLOVANKA 14. ÚNORA 2006
Transients in Quantum Transport: II. Correlated Initial Condition for Restart Process
by Time Partitioning
A. Kalvová, Acad. Sci. of CR, Praha
B. Velický, Charles University and Acad. Sci. of CR, PrahaV. Špička, Acad. Sci. of CR, Praha
SEMINÁŘ TEORETICKÉHO ODD. FZÚSLOVANKA 21. ÚNORA 2006
Transients in Quantum Transport: II. Correlated Initial Condition for Restart Process
by Time Partitioning
Progress in Non-Equilibrium Green’s Function III, Kiel Aug 22, 2005Topical Problems in Statistical Physics, TU Chemnitz, Nov 30, 2005
SEMINÁŘ TEORETICKÉHO ODD. FZÚSLOVANKA 21. ÚNORA 2006
Transients in Quantum Transport II ... 5 Teor. Odd. FZÚ 21.II.2006
Prologue
Transients in Quantum Transport II ... 6 Teor. Odd. FZÚ 21.II.2006
(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 P0 ( ) for U t t t ( )tU
Transients in Quantum Transport II ... 7 Teor. Odd. FZÚ 21.II.2006
0( ) for t t t
(Non-linear) quantum transport non-equilibrium problem Many-body system
Initial state
External disturbance
Response
many-body Hamiltonian
many-body density matrix
additive operator
one-particle density matrix
H
0 0 0at ( )t t t P P0 ( ) for U t t t ( )tU
Transients in Quantum Transport II ... 8 Teor. Odd. FZÚ 21.II.2006
(Non-linear) quantum transport non-equilibrium problem
Quantum Transport Equation a closed equation for ( )t
drift [ ( ); ]tt
generalized collision term
Many-body system
Initial state
External disturbance
Response
many-body Hamiltonian
many-body density matrix
additive operator
one-particle density matrix
H
0 0 0at ( )t t t P P
0( ) for t t t 0 ( ) for U t t t ( )tU
Transients in Quantum Transport II ... 9 Teor. Odd. FZÚ 21.II.2006
(Non-linear) quantum transport non-equilibrium problem
Quantum Transport Equation a closed equation for ( )t
drift [ ( ); ]tt
Many-body system
Initial state
External disturbance
Response
many-body Hamiltonian
many-body density matrix
additive operator
one-particle density matrix
H
0 0 0at ( )t t t P P
0( ) for t t t
QUESTIONS existence, construction of incorporation of the many-particle
initial condition
0P
interaction term
0 ( ) for U t t t ( )tU
Transients in Quantum Transport II ... 10 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 11 Teor. Odd. FZÚ 21.II.2006
This talk: orthodox study of quantum transport using NGF
TWO PATHS
DIRECT
INDIRECT
†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
Lecture on NGF
Transients in Quantum Transport II ... 12 Teor. Odd. FZÚ 21.II.2006
This talk: orthodox study of quantum transport using 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
Transients in Quantum Transport II ... 13 Teor. Odd. FZÚ 21.II.2006
Lecture on NGF…continuation
Real time NGF choices Kadanoff and BaymKeldysh
,, ,, Langreth and Wilk, ins
R A
R A
G GG G G GG G G
This talk: orthodox study of quantum transport using NGF
TWO PATHS
DIRECT †0(1,1') Tr( (1) (1')) (1) i (1,1)G G CTP
use a NGF solver
14
TWO PATHS
DIRECT
INDIRECT
†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
This talk: orthodox study of quantum transport using NGF
Transients in Quantum Transport II ... 15 Teor. Odd. FZÚ 21.II.2006
Standard approach based on GKBA Real time NGF our choice Langreth and , Wilkins,R AG G G
Transients in Quantum Transport II ... 16 Teor. Odd. FZÚ 21.II.2006
Standard approach based on GKBA Real time NGF our choice Langreth and , Wilkins,R AG G G
DYSON EQUATIONS
1 1 1 10 0( ) ( ) ( ) ( )R R R R A A A AG G G G G G G
Keldysh IC: simple initial state permits to concentrate on the other issues
<G ( , ) d d ( , ) ( , ) ( , )t t
R At t t t G t t t t G t t
Transients in Quantum Transport II ... 17 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 18 Teor. Odd. FZÚ 21.II.2006
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
Specific physical approximation -- self-consistent form R
A
G GG
GR
A
[ ] G
Transients in Quantum Transport II ... 19 Teor. Odd. FZÚ 21.II.2006
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
Specific physical approximation -- self-consistent form R
A
G GG
GR
A
[ ] G
Elimination of by an Ansatz
widely used: KBA (for steady transport), GKBA (transients, optics)
G
Transients in Quantum Transport II ... 20 Teor. Odd. FZÚ 21.II.2006
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
drift [ ( ); | , ]R At G Gt
Specific physical approximation -- self-consistent form R
A
G GG
GR
A
[ ] G
Elimination of by an Ansatz
GKBA
G
( , ') ( , ') ( ') ( ) ( , ')R AG t t G t t t t G t t
Resulting Quantum Transport Equation
Transients in Quantum Transport II ... 21 Teor. Odd. FZÚ 21.II.2006
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
drift [ ( ); | , ]R At G Gt
Specific physical approximation -- self-consistent form R
A
G GG
GR
A
[ ] G
Elimination of by an Ansatz
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)
Transients in Quantum Transport II ... 22 Teor. Odd. FZÚ 21.II.2006
Act I
reconstruction
Transients in Quantum Transport II ... 23 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 24 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 25 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 26 Teor. Odd. FZÚ 21.II.2006
Exact formulation -- Reconstruction Problem
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?
Transients in Quantum Transport II ... 27 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 28 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 29 Teor. Odd. FZÚ 21.II.2006
Postulate/Conjecture:typical systems are controlled by a hierarchy of times
separating the initial, kinetic, and hydrodynamic stages.A closed transport equation
holds for
ParallelsG E N E R A L S C H E M E
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
LABELBogolyubov
drift [ ( ); ]tt
C H
0 C .t t
Transients in Quantum Transport II ... 30 Teor. Odd. FZÚ 21.II.2006
Postulate/Conjecture:typical systems are controlled by a hierarchy of times
separating the initial, kinetic, and hydrodynamic stages.A closed transport equation
holds for
ParallelsG E N E R A L S C H E M E
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
LABELBogolyubov
drift [ ( ); ]tt
C H
0 C .t t
Transients in Quantum Transport II ... 31 Teor. Odd. FZÚ 21.II.2006
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.
ParallelsG E N E R A L S C H E M E
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
LABELTDDFT
C
U t 0[ ]n U
0 ,t t
Transients in Quantum Transport II ... 32 Teor. Odd. FZÚ 21.II.2006
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.
ParallelsG E N E R A L S C H E M E
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
LABELTDDFT
C
U t 0
0 ,t t
( )n t
[ ]n U
Transients in Quantum Transport II ... 33 Teor. Odd. FZÚ 21.II.2006
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
ParallelsG E N E R A L S C H E M E
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
LABELSchwinger
0 0i d ( ( ) ) i d ( ( ) )i ( , )
0e Tr e e
( ) ( )
t t
t tU X U XW U U
tU U U U U U
W WU t U t
T TH H
P
X
[ ]n U
Transients in Quantum Transport II ... 34 Teor. Odd. FZÚ 21.II.2006
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
ParallelsG E N E R A L S C H E M E
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
LABELSchwinger
0 0i d ( ( ) ) i d ( ( ) )i ( , )
0e Tr e e
( ) ( )
t t
t tU X U XW U U
tU U U U U U
W WU t U t
T TH H
P
X
( )n t
[ ]n U
35 Teor. Odd. FZÚ 21.II.2006
„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
G E N E R A L S C H E M EREDUCTION additive ( )
0 0
0
at ( ) ( ) ( ) Tr ( )
( ) for
t
t
t tt t X t X t
U t t t
H U XP XP
LABELNGF
ReconstructionTheorem
C , ,
Transients in Quantum Transport II ... 36 Teor. Odd. FZÚ 21.II.2006
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 tBOGOLYUBOV
SCHWINGER GENERATING FUNCTIONAL
TIME-DEPENDENT DENSITY FUNCTIONAL
RUNGE - GROSS THEOREM
Reconstruction Problem – Summary
Transients in Quantum Transport II ... 37 Teor. Odd. FZÚ 21.II.2006
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 – Summary
G
Transients in Quantum Transport II ... 38 Teor. Odd. FZÚ 21.II.2006
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 tt t t
R 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!
Transients in Quantum Transport II ... 39 Teor. Odd. FZÚ 21.II.2006
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 tt t t
R 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
Transients in Quantum Transport II ... 40 Teor. Odd. FZÚ 21.II.2006
Reconstruction theorem: Coupled equations
DYSON EQ.R AG G G
GKB EQ.
equal times
drift
A R R A
tG 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
tt t
R 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
Transients in Quantum Transport II ... 41 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 42 Teor. Odd. FZÚ 21.II.2006
"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
Transients in Quantum Transport II ... 43 Teor. Odd. FZÚ 21.II.2006
Act II
reconstructionand initial conditions
NGF view
Transients in Quantum Transport II ... 44 Teor. Odd. FZÚ 21.II.2006
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 , Morawetz …
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 tG G G
0[ | ]U
P
Transients in Quantum Transport II ... 45 Teor. Odd. FZÚ 21.II.2006
General initial state: Structure of
0 0 0
0 0 0 0
( , ') i ( ) ( ) ( ' )( , ') ( , ) ( ' ) ( , ') ( , ') ( )
( , ') ( , ')
t t t t t t tt t t t t t t t t t t t
t t t t
Structure of
Transients in Quantum Transport II ... 46 Teor. Odd. FZÚ 21.II.2006
0 0 0
0 0 0 0
( , ') i ( ) ( ) ( ' )( , ') ( , ) ( ' ) ( , ') ( , ') ( )
( , ') ( , ')
t t t t t t tt t t t t t t t t t t t
t t t t
Structure of
General initial state: Structure of
singular time variable fixed at
t = t0
continuous time variable
t > t0
Transients in Quantum Transport II ... 47 Teor. Odd. FZÚ 21.II.2006
Danielewicz notation
0 0 0
0 0 0 0
( , ') i ( ) ( ) ( ' )( , ') ( , ) ( ' ) ( , ') ( , ') ( )
( , ') ( , ')
t t t t t t tt t t t t t t t t t t t
t t t t
Structure of
General initial state: Structure of
singular time variable fixed at
t = t0
continuous time variable
t > t0
Transients in Quantum Transport II ... 48 Teor. Odd. FZÚ 21.II.2006
Danielewicz notation
0 0 0
0 0 0 0
( , ') i ( ) ( ) ( ' )( , ') ( , ) ( ' ) ( , ') ( , ') ( )
( , ') ( , ')
t t t t t t tt t t t t t t t t t t t
t t t t
Structure of
0t0t
t
't
General initial state: Structure of
singular time variable fixed at
t = t0
continuous time variable
t > t0
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.
DANIELEWICZ CORRECTION
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
Transients in Quantum Transport II ... 52 Teor. Odd. FZÚ 21.II.2006
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:
Transients in Quantum Transport II ... 53 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 54 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 55 Teor. Odd. FZÚ 21.II.2006
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 initial states
for varying t 0
Transients in Quantum Transport II ... 56 Teor. Odd. FZÚ 21.II.2006
Restart at an intermediate time
† †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
0t0t t
Transients in Quantum Transport II ... 57 Teor. Odd. FZÚ 21.II.2006
0(1,1') (1,1') (1,1')t tG G G
NGF is invariant with respect to the initial time,
the self-energies must be related in a specific way for
Important difference
Restart at an intermediate time
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
0t0t t
Transients in Quantum Transport II ... 58 Teor. Odd. FZÚ 21.II.2006
0(1,1') (1,1') (1,1')t tG G G
NGF is invariant with respect to the initial time,
the self-energies must be related in a specific way for
Important difference
Restart at an intermediate time
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
0t0t t
Transients in Quantum Transport II ... 59 Teor. Odd. FZÚ 21.II.2006
Act III.
Time-partitioning
Transients in Quantum Transport II ... 60 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 61 Teor. Odd. FZÚ 21.II.2006
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 equationRECONSTRUCTION EQ.
1 2 1 1 2 2
1 2 1 1'
2
'
'
2
'
( , ) ( , ) ( )
d d ( , ) ( , )
' '
( , ')
d d ( , ) ( ,
'
) ( , )
'
'
t
tt
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
Transients in Quantum Transport II ... 62 Teor. Odd. FZÚ 21.II.2006
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 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
Transients in Quantum Transport II ... 63 Teor. Odd. FZÚ 21.II.2006
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 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
Transients in Quantum Transport II ... 64 Teor. Odd. FZÚ 21.II.2006
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 equationRECONSTRUCTION EQ.
'
'
2 2 2
2
1 1 1'
21 1 1 2'
( , ) ( , ) ( )
d )d ( , ) ( , ')
d ) ( , ')
'
( ,
d ( ,
' '
) ( ,
't
R
tt
R
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
Transients in Quantum Transport II ... 65 Teor. Odd. FZÚ 21.II.2006
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 for G<
Transients in Quantum Transport II ... 66 Teor. Odd. FZÚ 21.II.2006
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 for G<
- past - future notion … in renormalized semigroup rule GR
67 Teor. Odd. FZÚ 21.II.2006
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 for G<
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 renormalized semigroup rule GR
RENORM. SEMIGROUP RULE
68 Teor. Odd. FZÚ 21.II.2006
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 for G<
RENORM. SEMIGROUP 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
last time
- past - future notion … in renormalized semigroup rule GR
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 for G<
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
t
't
't
t
tt
t
t
- past - future notion … in renormalized semigroup rule GR
RENORM. SEMIGROUP RULE
Transients in Quantum Transport II ... 70 Teor. Odd. FZÚ 21.II.2006
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 for G<
- past - future notion … in renormalized semigroup rule GR
Transients in Quantum Transport II ... 71 Teor. Odd. FZÚ 21.II.2006
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 for G<
- past - future notion … in restart NGF
unified description—time-partitioning formalism
- past - future notion … in renormalized semigroup rule GR
Transients in Quantum Transport II ... 72 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 73 Teor. Odd. FZÚ 21.II.2006
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
0 0
et
( ) ( ) ( ') ( ) ( ') ( ) ( ')( , ') ( , ') ( , ')
t t t t t t I t t t t I t t t t It t t t t t
P F 1P F 1Projection operators
Transients in Quantum Transport II ... 74 Teor. Odd. FZÚ 21.II.2006
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
0 0
et
( ) ( ) ( ') ( ) ( ') ( ) ( ')( , ') ( , ') ( , ')
t t t t t t I t t t t I t t t t It t t t t t
P F 1P 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
Transients in Quantum Transport II ... 75 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: for propagators1. Dyson eq.
0 0R R R R RG G G G
2. Retarded quantity R ( , ') 0X t t only for 't t
0RX P F3. 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 PF F F F F F F F
Transients in Quantum Transport II ... 76 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 77 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: for propagators …continuation
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
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
Transients in Quantum Transport II ... 78 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: for propagators …continuation
0 0R R R R RG G G G
R RR R RR GG GL GG RF P F P F FPF PFP 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
Transients in Quantum Transport II ... 79 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: for propagators …continuation
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
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
Transients in Quantum Transport II ... 80 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: for propagators …continuation
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
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
time-local factorization
vertex correction: universal form
(gauge invariance) link past-future
non-local in timewidth 2 Q
Transients in Quantum Transport II ... 81 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: for propagators …continuation
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
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
time-local factorization
vertex correction: universal form
(gauge invariance) link past-future
non-local in timewidth 2 Q
renormalized semi-group rule
Transients in Quantum Transport II ... 82 Teor. Odd. FZÚ 21.II.2006
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
Transients in Quantum Transport II ... 83 Teor. Odd. FZÚ 21.II.2006
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 expressionsR 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
Transients in Quantum Transport II ... 84 Teor. Odd. FZÚ 21.II.2006
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 expressionsR AG G G P P P P P P
( )R A AA AG GLGG G FP P PF PP FF F…diagonals of GF’s
Transients in Quantum Transport II ... 85 Teor. Odd. FZÚ 21.II.2006
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 expressionsR AG G G P P P P P P
( )R A AAAG G LG G G FPP F P P FF P F…off-diagonals of selfenergies
Transients in Quantum Transport II ... 86 Teor. Odd. FZÚ 21.II.2006
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 expressionsR 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 R R RG G G G L G F P F F P P F F P P
Transients in Quantum Transport II ... 87 Teor. Odd. FZÚ 21.II.2006
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 expressionsR 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 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
Transients in Quantum Transport II ... 88 Teor. Odd. FZÚ 21.II.2006
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 expressionsR 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 R R RG G G G L G F P F F P P F F P P( )
( )
( ) ( )
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 FF F F FF F F FF F
F F
P P
P P FP P
F
…diagonals of GF’s
Transients in Quantum Transport II ... 89 Teor. Odd. FZÚ 21.II.2006
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 expressionsR 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 R R RG G G G L G F P F F P P F F P P( )
( )
( ) ( )
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 FF FF F
P PP
P
PF PF F F
…off-diagonals of
selfenergy
Transients in Quantum Transport II ... 90 Teor. Odd. FZÚ 21.II.2006
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 expressionsR 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 R R RG G G G L G F P F F P P F F P P( )
( )
( ) ( )
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 FF FF F
P PP
P
PF PF F F
…off-diagonals of
selfenergy
Exception!!!
Future-future diagonal
Transients in Quantum Transport II ... 91 Teor. Odd. FZÚ 21.II.2006
( )
( )
( ) ( )
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 FF FF F
P PP
P
PF PF F F
Partitioning in time: restartrestart corr. function 0t
G
R AG G G HOST PROCESS
RESTART PROCESS0 0R A
t tG G G F F F F
Transients in Quantum Transport II ... 92 Teor. Odd. FZÚ 21.II.2006
( )
( )
( ) ( )
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 FF FF F
P PP
P
PF PF F F
Partitioning in time: restartrestart corr. function 0t
G
R AG G G HOST PROCESS
RESTART PROCESS0 0R A
t tG G G F F F F
0t
Transients in Quantum Transport II ... 93 Teor. Odd. FZÚ 21.II.2006
( )
( )
( ) ( )
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 FF FF F
P PP
P
PF PF F F
Partitioning in time: restartrestart corr. function 0t
G
R AG G G HOST PROCESS
RESTART PROCESS0 0R A
t tG G G F F F F
0t
future
memory of the past folded
down into the future by
partitioning
Transients in Quantum Transport II ... 94 Teor. Odd. FZÚ 21.II.2006
( )
( )
( ) ( )
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 FF FF F
P PP
P
PF PF F F
Partitioning in time: restartrestart corr. function 0t
G
R AG G G HOST PROCESS
RESTART PROCESS0 0R A
t tG G G F F F F
0initial conditionst F F
future
memory of the past folded
down into the future by
partitioning
Transients in Quantum Transport II ... 95 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: initial conditioninitial condition 0tG
0initial conditionst F F
0 0[ ]t t
00 0 0( , ') i ( ) ( ) ( ' )
tt t t t t t t
Singular time variable fixed at restart time 0t t
Transients in Quantum Transport II ... 96 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: initial conditioninitial condition 0tG
0initial conditionst F F
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 FR RG F P P F R RL G P P F
Transients in Quantum Transport II ... 97 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: initial conditioninitial condition 0tG
0initial conditionst F F
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 FR RG F P P F R RL G P P F
Transients in Quantum Transport II ... 98 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: initial conditioninitial condition 0tG
0initial conditionst F F
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 FR RG F P P F R RL G P P F
Transients in Quantum Transport II ... 99 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: initial conditioninitial condition 0tG
0initial conditionst F F
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 FR RG F P P F R RL G P P F
Transients in Quantum Transport II ... 100 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: initial conditioninitial condition 0tG
0initial conditionst F F
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 FR RG F P P F R RL G P P F
Transients in Quantum Transport II ... 101 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: initial conditioninitial condition 0tG
0initial conditionst F F
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 FR RG F P P F R RL G P P F
… omitted initial condition, 0
[ ]t
0t t Keldysh limit
Transients in Quantum Transport II ... 102 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: initial conditioninitial condition 0tG
0initial conditionst F F
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 FR RG F P P F R RL G P P F
… with uncorrelated initial condition,
Transients in Quantum Transport II ... 103 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: initial conditioninitial condition 0tG
0initial conditionst F F
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 FR RG F P P F R RL G P P 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
Transients in Quantum Transport II ... 104 Teor. Odd. FZÚ 21.II.2006
Partitioning in time: initial conditioninitial condition 0tG
0initial conditionst F F
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 FR RG F P P F R RL G P P F
Transients in Quantum Transport II ... 105 Teor. Odd. FZÚ 21.II.2006
Act IVapplications:
restarted switch-on processespump and probe signals
....
NEXT TIME
Transients in Quantum Transport II ... 107 Teor. Odd. FZÚ 21.II.2006
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 -
....
Transients in Quantum Transport II ... 108 Teor. Odd. FZÚ 21.II.2006
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