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
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:
Correlated Initial Condition for Restart Process
A. Kalvová, Acad. Sci. of CR, Praha
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
A. Kalvová, Acad. Sci. of CR, Praha
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
Between GF and Transport Equations … 6 TU Chemnitz Nov 30, 2005
(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
Between GF and Transport Equations … 7 TU Chemnitz Nov 30, 2005
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 P
0 ( ) for U t t t ( )tU
Between GF and Transport Equations … 8 TU Chemnitz Nov 30, 2005
(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
Between GF and Transport Equations … 9 TU Chemnitz Nov 30, 2005
(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
interaction term
0 ( ) for U t t t ( )tU
Between GF and Transport Equations … 10 TU Chemnitz Nov 30, 2005
(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 initial
condition
0P
interaction term
0 ( ) for U t t t ( )tU
Between GF and Transport Equations … 11 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 12 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 13 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 14 TU Chemnitz Nov 30, 2005
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
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
15
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
Between GF and Transport Equations … 16 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 17 TU Chemnitz Nov 30, 2005
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 G
G
GR
A
[ ] G
Between GF and Transport Equations … 18 TU Chemnitz Nov 30, 2005
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 G
G
GR
A
[ ] G
Elimination of by an Ansatz
widely used: KBA (for steady transport), GKBA (transients, optics)
G
Between GF and Transport Equations … 19 TU Chemnitz Nov 30, 2005
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 G
G
GR
A
[ ] G
Elimination of by an Ansatz
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, …
Between GF and Transport Equations … 20 TU Chemnitz Nov 30, 2005
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 G
G
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
Between GF and Transport Equations … 21 TU Chemnitz Nov 30, 2005
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 G
G
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)
Between GF and Transport Equations … 22 TU Chemnitz Nov 30, 2005
Act I
reconstruction
Between GF and Transport Equations … 23 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 24 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 25 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 26 TU Chemnitz Nov 30, 2005
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?
Between GF and Transport Equations … 27 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 28 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 29 TU Chemnitz Nov 30, 2005
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
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
LABEL
Bogolyubov
drift [ ( ); ]tt
C H
0 C .t t
Between GF and Transport Equations … 30 TU Chemnitz Nov 30, 2005
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
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
LABEL
Bogolyubov
drift [ ( ); ]tt
C H
0 C .t t
Between GF and Transport Equations … 31 TU Chemnitz Nov 30, 2005
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
G 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
LABEL
TDDFT
C
U t 0[ ]n U
0 ,t t
Between GF and Transport Equations … 32 TU Chemnitz Nov 30, 2005
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
G 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
LABEL
TDDFT
C
U t 0
0 ,t t
( )n t
[ ]n U
Between GF and Transport Equations … 33 TU Chemnitz Nov 30, 2005
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
G 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
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
Between GF and Transport Equations … 34 TU Chemnitz Nov 30, 2005
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
G 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
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
G 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
LABEL
NGFReconstruction
Theorem
C , ,
Between GF and Transport Equations … 36 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 37 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 38 TU Chemnitz Nov 30, 2005
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!
Between GF and Transport Equations … 39 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 40 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 41 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 42 TU Chemnitz Nov 30, 2005
"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
Between GF and Transport Equations … 43 TU Chemnitz Nov 30, 2005
Act II
reconstructionand initial conditions
NGF view
Between GF and Transport Equations … 44 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 45 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 46 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 47 TU Chemnitz Nov 30, 2005
Danielewicz notation
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
General initial state: Structure of
Between GF and Transport Equations … 48 TU Chemnitz Nov 30, 2005
Danielewicz notation
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
0t
0t
t
't
General initial state: Structure of
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
Between GF and Transport Equations … 51 TU Chemnitz Nov 30, 2005
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:
Between GF and Transport Equations … 52 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 53 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 54 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 55 TU Chemnitz Nov 30, 2005
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
0t
0t t
Between GF and Transport Equations … 56 TU Chemnitz Nov 30, 2005
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
0t
0t t
Between GF and Transport Equations … 57 TU Chemnitz Nov 30, 2005
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
… causal structure of the Dyson equation
… develops singular parts at as a condensed information about the past
0t
0t t
0
0
, ,( , ') ( , ')
( , ') ( , ')
t t
t
R A
t
R At t t t
t t t t
Between GF and Transport Equations … 58 TU Chemnitz Nov 30, 2005
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
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.
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
0t
0t t
0t
Between GF and Transport Equations … 59 TU Chemnitz Nov 30, 2005
Intermezzo
Time-partitioning
Between GF and Transport Equations … 60 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 61 TU Chemnitz Nov 30, 2005
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
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
Between GF and Transport Equations … 62 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 63 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 64 TU Chemnitz Nov 30, 2005
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
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
Between GF and Transport Equations … 65 TU Chemnitz Nov 30, 2005
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<
Between GF and Transport Equations … 66 TU Chemnitz Nov 30, 2005
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 corrected semigroup rule GR
67 TU Chemnitz Nov 30, 2005
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 corrected semigroup rule GR
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
Between GF and Transport Equations … 68 TU Chemnitz Nov 30, 2005
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 corrected semigroup rule GR
Between GF and Transport Equations … 69 TU Chemnitz Nov 30, 2005
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 corrected semigroup rule GR
- past - future notion … in restart NGF
unified description—
time-partitioning formalism
Between GF and Transport Equations … 70 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 71 TU Chemnitz Nov 30, 2005
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 I
t t t t t t
P F 1
P F 1Projection operators
Between GF and Transport Equations … 72 TU Chemnitz Nov 30, 2005
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 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
Between GF and Transport Equations … 73 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 74 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 75 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 76 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 77 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 78 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 79 TU Chemnitz Nov 30, 2005
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
Corrected semigroup rule
Between GF and Transport Equations … 80 TU Chemnitz Nov 30, 2005
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
Between GF and Transport Equations … 81 TU Chemnitz Nov 30, 2005
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
( )R A A A AG G G G L G P F P P F F P P F F
Between GF and Transport Equations … 82 TU Chemnitz Nov 30, 2005
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
( )R A AA AG GLGG G FP P PF PP FF F
…diagonals of GF’s
Between GF and Transport Equations … 83 TU Chemnitz Nov 30, 2005
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
( )R A AAAG G LG G G FPP F P P FF P F
…off-diagonals of selfenergies
Between GF and Transport Equations … 84 TU Chemnitz Nov 30, 2005
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
( )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
Between GF and Transport Equations … 85 TU Chemnitz Nov 30, 2005
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
( )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
Between GF and Transport Equations … 86 TU Chemnitz Nov 30, 2005
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
( )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 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
Between GF and Transport Equations … 87 TU Chemnitz Nov 30, 2005
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
( )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 F
F F
F F
P P
P
P
PF PF F F
…off-diagonals of
selfenergy
Between GF and Transport Equations … 88 TU Chemnitz Nov 30, 2005
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
( )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 F
F F
F F
P P
P
P
PF PF F F
…off-diagonals of
selfenergy
Exception!!!
Future-future diagonal
Between GF and Transport Equations … 89 TU Chemnitz Nov 30, 2005
( )
( )
( ) ( )
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
Between GF and Transport Equations … 90 TU Chemnitz Nov 30, 2005
( )
( )
( ) ( )
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
0t
Between GF and Transport Equations … 91 TU Chemnitz Nov 30, 2005
( )
( )
( ) ( )
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
0t
future
memory of the past folded
down into the future by
partitioning
Between GF and Transport Equations … 92 TU Chemnitz Nov 30, 2005
( )
( )
( ) ( )
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
0initial conditionst F F
future
memory of the past folded
down into the future by
partitioning
Between GF and Transport Equations … 93 TU Chemnitz Nov 30, 2005
Partitioning in time: initial conditioninitial condition 0t
G
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
Between GF and Transport Equations … 94 TU Chemnitz Nov 30, 2005
Partitioning in time: initial conditioninitial condition 0t
G
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 F
R RG F P P FR RL G P P F
Between GF and Transport Equations … 95 TU Chemnitz Nov 30, 2005
Partitioning in time: initial conditioninitial condition 0t
G
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 F
R RG F P P FR RL G P P F
Between GF and Transport Equations … 96 TU Chemnitz Nov 30, 2005
Partitioning in time: initial conditioninitial condition 0t
G
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 F
R RG F P P FR RL G P P F
Between GF and Transport Equations … 97 TU Chemnitz Nov 30, 2005
Partitioning in time: initial conditioninitial condition 0t
G
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 F
R RG F P P FR RL G P P F
Between GF and Transport Equations … 98 TU Chemnitz Nov 30, 2005
Partitioning in time: initial conditioninitial condition 0t
G
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 F
R RG F P P FR RL G P P F
Between GF and Transport Equations … 99 TU Chemnitz Nov 30, 2005
Partitioning in time: initial conditioninitial condition 0t
G
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 F
R RG F P P FR RL G P P F
… omited initial condition, 0
[ ]t
0t t Keldysh limit
Between GF and Transport Equations … 100 TU Chemnitz Nov 30, 2005
Partitioning in time: initial conditioninitial condition 0t
G
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 F
R RG F P P FR RL G P P F
… with uncorrelated initial condition,
Between GF and Transport Equations … 101 TU Chemnitz Nov 30, 2005
Partitioning in time: initial conditioninitial condition 0t
G
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 F
R RG F P P FR 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
Between GF and Transport Equations … 102 TU Chemnitz Nov 30, 2005
Partitioning in time: initial conditioninitial condition 0t
G
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 F
R RG F P P FR RL G P P F
Between GF and Transport Equations … 103 TU Chemnitz Nov 30, 2005
RestartRestart correlation function: initial conditions
continuous time variable
t > t0
singular time variable fixed at
t = t0
Between GF and Transport Equations … 104 TU Chemnitz Nov 30, 2005
RestartRestart correlation function: initial conditions
singular time variable fixed at
t = t0
00 0 0( , ') i ( ) ( ) ( ' )
tt t t t t t t
uncorrelated initial condition ... KELDYSH
Between GF and Transport Equations … 105 TU Chemnitz Nov 30, 2005
RestartRestart correlation function: initial conditions
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
Between GF and Transport Equations … 106 TU Chemnitz Nov 30, 2005
RestartRestart correlation function: initial conditions
0
<( , ') ( , ')t
t t t t
host continuous self-energy ... KELDYSHinitial correlations correction MOROZOV &RÖPKE
Between GF and Transport Equations … 107 TU Chemnitz Nov 30, 2005
Act III
applications:restarted switch-on processes
pump and probe signals....
NEXT TIME
Between GF and Transport Equations … 109 TU Chemnitz Nov 30, 2005
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 -
....
Between GF and Transport Equations … 110 TU Chemnitz Nov 30, 2005
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