PROGRESS IN NONEQUILIBRIUM GREENS FUNCTIONS III Kiel August

PROGRESS IN NON-EQUILIBRIUM GREEN'S FUNCTIONS III Kiel August 22 – 25, 2005 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 III Kiel August 22 – 25, 2005 Between Green's Functions and Transport Equations: Reconstruction Theorems and Role of Initial Conditions the B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha

Topical Problems in Statistical Physics TU Chemnitz, November 30, 2005 Between Green's Functions and Transport Equations: Correlated Initial Condition for Restart Process A. Kalvová, Acad. Sci. of CR, Praha B. B. Velický, Charles University and Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha

Topical Problems in Statistical Physics TU 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. B. Velický, Charles University and Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha

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

(Non-linear) quantum transport non-equilibrium problem Many-body system many-body Hamiltonian Initial state many-body density matrix External disturbance additive operator TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 6

(Non-linear) quantum transport non-equilibrium problem Many-body system many-body Hamiltonian Initial state many-body density matrix External disturbance additive operator Response one-particle density matrix TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 7

(Non-linear) quantum transport non-equilibrium problem Many-body system many-body Hamiltonian Initial state many-body density matrix External disturbance additive operator Response one-particle density matrix Quantum Transport Equation a closed equation for generalized collision term TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 8

(Non-linear) quantum transport non-equilibrium problem Many-body system many-body Hamiltonian Initial state many-body density matrix External disturbance additive operator Response one-particle density matrix Quantum Transport Equation a closed equation for interaction term TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 9

(Non-linear) quantum transport non-equilibrium problem Many-body system many-body Hamiltonian Initial state many-body density matrix External disturbance additive operator Response one-particle density matrix Quantum Transport Equation a closed equation for interaction term QUESTIONS q existence, construction of q incorporation of the initial condition TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 10

This talk: orthodox study of quantum transport using NGF TWO PATHS DIRECT use a NGF solver INDIRECT use NGF to construct a Quantum Transport Equation TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 11

Lecture on NGF This talk: orthodox study of quantum transport using NGF TWO PATHS DIRECT use a NGF solver INDIRECT use NGF to construct a Quantum Transport Equation TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 12

Lecture on NGF…continuation This talk: orthodox study of quantum transport using NGF TWO PATHS DIRECT use a NGF solver TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 13

Lecture on NGF…continuation This talk: orthodox study of quantum transport using NGF TWO PATHS DIRECT use a NGF solver Real time NGF choices TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 14

This talk: orthodox study of quantum transport using NGF TWO PATHS DIRECT use a NGF solver INDIRECT use NGF to construct a Quantum Transport Equation 15

Standard approach based on GKBA Real time NGF our choice GKBE TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 16

Standard approach based on GKBA Real time NGF our choice GKBE Specific physical approximation -- self-consistent form TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 17

Standard approach based on GKBA Real time NGF our choice GKBE Specific physical approximation -- self-consistent form Elimination of by an Ansatz widely used: KBA (for steady transport), GKBA (transients, optics) TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 18

Standard approach based on GKBA Real time NGF our choice GKBE Specific physical approximation -- self-consistent form Elimination of by an Ansatz GKBA Lipavsky, Spicka, Velicky, Haug + Frankfurt team, TU Chemnitz Nov 30, 2005 Vinogradov, Rostock school, Between GF and Transport Equations … Horing Jauho, … 19

Standard approach based on GKBA Real time NGF our choice GKBE Specific physical approximation -- self-consistent form Elimination of by an Ansatz GKBA Resulting Quantum Transport Equation TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 20

Standard approach based on GKBA Real time NGF our choice GKBE Specific physical approximation -- self-consistent form Elimination of by an Ansatz GKBA Resulting Quantum Transport Equation TU Chemnitz Nov 30, 2005 Famous examples: • Levinson eq. (hot electrons) • Optical quantum Bloch eq. Between GF and Transport Equations … (optical transients) 21

Act I reconstruction TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 22

Exact formulation -- Reconstruction Problem GENERAL QUESTION: conditions under which a many-body interacting system can be described in terms of its single-time single-particle characteristics TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 23

Exact formulation -- Reconstruction Problem GENERAL QUESTION: conditions under which a many-body interacting system can be described in terms of its single-time single-particle characteristics Reminiscences: BBGKY, Hohenberg-Kohn Theorem TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 24

Exact formulation -- Reconstruction Problem GENERAL QUESTION: conditions under which a many-body interacting system can be described in terms of its single-time single-particle characteristics Reminiscences: BBGKY, Hohenberg-Kohn Theorem Here: time evolution of the system TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 25

Exact formulation -- Reconstruction Problem New look on the NGF procedure: Eliminate by an Ansatz GKBA … in fact: express time diagonal , a double-time correlation function, by its Any Ansatz is but an approximate solution… ¿Does an answer exist, exact at least in principle? TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 26

Reconstruction Problem – Historical Overview INVERSION SCHEMES TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 27

Reconstruction Problem – Historical Overview INVERSION SCHEMES TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 28

Parallels GENERAL SCHEME LABEL Bogolyubov Postulate/Conjecture: typical systems are controlled by a hierarchy of times separating the initial, kinetic, and hydrodynamic stages. A closed transport equation holds for TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 29

Parallels GENERAL SCHEME LABEL Bogolyubov Postulate/Conjecture: typical systems are controlled by a hierarchy of times separating the initial, kinetic, and hydrodynamic stages. A closed transport equation holds for TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 30

Parallels GENERAL SCHEME LABEL TDDFT 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 theorem. This is an existence theorem, systematic implementation based on the use of the closed time path generating functional. TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 31

Parallels GENERAL SCHEME LABEL TDDFT 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 theorem. This is an existence theorem, systematic implementation based on the use of the closed time path generating functional. TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 32

Parallels GENERAL SCHEME LABEL Schwinger 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 TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 33

Parallels GENERAL SCHEME LABEL Schwinger 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 TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 34

Parallels: Lessons for the Reconstruction Problem GENERAL SCHEME LABEL NGF Reconstruction Theorem q „Bogolyubov“: importance of the time hierarchy REQUIREMENT Characteristic times should emerge in a constructive manner during the reconstruction procedure. q „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. q „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. 35

Reconstruction Problem – Summary INVERSION SCHEMES TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 36

Reconstruction Problem – Summary INVERSION SCHEMES TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 37

Reconstruction theorem : Reconstruction equations Keldysh IC: simple initial state permits to concentrate on the other issues DYSON EQUATIONS Two well known “reconstruction equations” easily follow: RECONSTRUCTION EQUATIONS LSV, Vinogradov … application! TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 38

Reconstruction theorem : Reconstruction equations Keldysh IC: simple initial state permits to concentrate on the other issues DYSON EQUATIONS Two well known “reconstruction equations” easily follow: RECONSTRUCTION EQUATIONS q Source terms … the Ansatz q For t=t' … tautology TU Chemnitz Nov 30, 2005 … input Between GF and Transport Equations … 39

Reconstruction theorem: Coupled equations GKB EQ. DYSON EQ. TU Chemnitz Nov 30, 2005 RECONSTRUCTION EQ. Between GF and Transport Equations … 40

Reconstruction theorem: operational description NGF RECONSTRUCTION THEOREM determination of the full NGF restructured as a DUAL PROCESS quantum transport equation TU Chemnitz Nov 30, 2005 reconstruction equations Dyson eq. Between GF and Transport Equations … 41

Reconstruction theorem: formal statement NGF RECONSTRUCTION THEOREM determination of the full NGF restructured as a DUAL PROCESS quantum transport equation reconstruction equations Dyson eq. "THEOREM" The one-particle density matrix and the full NGF of a process are in a bijective relationship, TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 42

Act II reconstruction and initial conditions NGF view TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 43

General initial state For an arbitrary initial state Problem of determination of at start from the NGF 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 TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 44

General initial state For an arbitrary initial state Problem of determination of at start from the NGF G extensively studied Fujita Hall Danielewicz … Wagner Morozov&Röpke … Morawetz Klimontovich Kremp … Bonitz&Semkat … Take over the relevant result for : The self-energy depends on the initial state (initial correlations) has singular components TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 45

General initial state: Structure of TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 46

General initial state: Structure of Danielewicz notation TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 47

General initial state: Structure of Danielewicz notation TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 48

General initial state: A try at the reconstruction GKB EQ. DYSON EQ. RECONSTRUCTION EQ.

General initial state: A try at the reconstruction GKB EQ. DYSON EQ. To progress further, narrow down the selection of the initial states RECONSTRUCTION EQ.

Initial state for restart process To progress further, narrow down the selection of the initial states Special situation: Process, whose initial state coincides with intermediate state of a host process (running) Aim: to establish relationship between NGF of the host and restart process TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 51

Restart at an intermediate time Let the initial time be , the initial state . In the host NGF the Heisenberg operators are TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 52

Restart at an intermediate time We may choose any later time as the new initial time. For times the resulting restart GF should be consistent. Indeed, with we have TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 53

Restart at an intermediate time We may choose any later time as the new initial time. For times the resulting GF should be consistent. Indeed, with we have tes a t ls ia t i n f i t 0 o ily ying m fa var e l r o o h f w TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 54

Restart at an intermediate time NGF is invariant with respect to the initial time, the self-energies must be related in a specific way for Important difference … causal structure of the Dyson equation … develops singular parts at as a condensed information about the past TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 55

Restart at an intermediate time NGF is invariant with respect to the initial time, the self-energies must be related in a specific way for Important difference … causal structure of the Dyson equation … develops singular parts at as a condensed information about the past TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 56

Restart at an intermediate time NGF is invariant with respect to the initial time, the self-energies must be related in a specific way for Important difference … causal structure of the Dyson equation … develops singular parts at as a condensed information about the past TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 57

Restart at an intermediate time NGF is invariant with respect to the initial time, the self-energies must be related in a specific way for Important difference … causal structure of the Dyson equation … develops singular parts at as a condensed information about the past 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. TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 58

Intermezzo Time-partitioning TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 59

Time-partitioning: general method Special position of the (instant-restart) time t 0 -Separates the whole time domain into the past and the future - past - future notion … in reconstruction equation TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 60

Time-partitioning: general method Special position of the (instant-restart) time t 0 -Separates the whole time domain into the past and the future - past future notion … in reconstruction equation RECONSTRUCTION EQ. TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 61

Time-partitioning: general method Special position of the (instant-restart) time t 0 -Separates the whole time domain into the past and the future - past future notion … in reconstruction equation RECONSTRUCTION EQ. TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 62

Time-partitioning: general method Special position of the (instant-restart) time t 0 -Separates the whole time domain into the past and the future -past future notion … in reconstruction equation RECONSTRUCTION EQ. past TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 63

Time-partitioning: general method Special position of the (instant-restart) time t 0 -Separates the whole time domain into the past and the future -past future notion … in reconstruction equation RECONSTRUCTION EQ. future TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 64

Time-partitioning: general method Special position of the (instant-restart) time t 0 -Separates the whole time domain into the past and the future - past - future notion … in reconstruction equation for G < TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 65

Time-partitioning: general method Special position of the (instant-restart) time t 0 -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 TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 66

Time-partitioning: general method Special position of the (instant-restart) time t 0 -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 67

Time-partitioning: general method Special position of the (instant-restart) time t 0 -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 TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 68

Time-partitioning: general method Special position of the (instant-restart) time t 0 -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 TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 69

Partitioning in time: formal tools Past and Future with respect to the initial (restart) time TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 70

Partitioning in time: formal tools Past and Future with respect to the initial (restart) time Projection operators TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 71

Partitioning in time: formal tools Past and Future with respect to the initial (restart) time Projection operators Double time quantity X …four quadrants of the two-time plane TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 72

Partitioning in time: for propagators 1. Dyson eq. 2. Retarded quantity only for 3. Diagonal blocks of TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 73

Partitioning in time: for propagators …continuation 4. Off-diagonal blocks of -free propagator corresponds to a unitary evolution multiplicative composition law semigroup rule … time local operator TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 74

Partitioning in time: for propagators …continuation 4. Off-diagonal blocks of -free propagator corresponds to a unitary evolution multiplicative composition law semigroup rule … time local operator TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 75

Partitioning in time: for propagators …continuation 4. Off-diagonal blocks of -free propagator corresponds to a unitary evolution multiplicative composition law semigroup rule … time local operator TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 76

Partitioning in time: for propagators …continuation 4. Off-diagonal blocks of -free propagator corresponds to a unitary evolution multiplicative composition law semigroup rule … time local operator TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 77

Partitioning in time: for propagators …continuation 4. Off-diagonal blocks of -free propagator corresponds to a unitary evolution multiplicative composition law semigroup rule time-local factorization TU Chemnitz Nov 30, 2005 vertex correction: universal form (gauge invariance) … time local linkoperator past-future non-local in time Between GF and Transport Equations … 78

Partitioning in time: for propagators …continuation 4. Off-diagonal blocks of -free propagator corresponds to a unitary evolution multiplicative composition law semigroup rule time-local factorization TU Chemnitz Nov 30, 2005 Corr ecte sem d igro up r ule vertex correction: universal form (gauge invariance) … time local linkoperator past-future non-local in time Between GF and Transport Equations … 79

Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy 2. Propagators … split into four blocks … by partitioning expressions …(diagonal) past blocks only TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 80

Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy 2. Propagators TU Chemnitz Nov 30, 2005 … split into four blocks … by partitioning expressions Between GF and Transport Equations … 81

Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy 2. Propagators … split into four blocks … by partitioning expressions …diagonals of GF’s TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 82

Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy 2. Propagators … split into four blocks … by partitioning expressions …off-diagonals of selfenergies TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 83

Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy 2. Propagators TU Chemnitz Nov 30, 2005 … split into four blocks … by partitioning expressions Between GF and Transport Equations … 84

Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy 2. Propagators TU Chemnitz Nov 30, 2005 … split into four blocks … by partitioning expressions Between GF and Transport Equations … 85

Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy 2. Propagators … split into four blocks … by partitioning expressions ls na o iag F f. G ’s o d … TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 86

Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy 2. Propagators TU Chemnitz Nov 30, 2005 … split into four blocks … by partitioning expressions Between GF and Transport Equations … ls na o iag ergy d en off … self of 87

Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy 2. Propagators TU Chemnitz Nov 30, 2005 … split into four blocks !!! onal n o pti diag e c Ex uture f reu t u F … by partitioning expressions Between GF and Transport Equations … ls na o iag ergy d en off … self of 88

Partitioning in time: restart corr. function HOST PROCESS RESTART PROCESS TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 89

Partitioning in time: restart corr. function HOST PROCESS RESTART PROCESS TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 90

Partitioning in time: restart corr. function HOST PROCESS RESTART PROCESS future memory of the past folded down into the future by partitioning TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 91

Partitioning in time: restart corr. function HOST PROCESS RESTART PROCESS future memory of the past folded down into the future by partitioning TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 92

Partitioning in time: initial condition Singular time variable fixed at restart time TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 93

Partitioning in time: initial condition TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 94

Partitioning in time: initial condition TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 95

Partitioning in time: initial condition TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 96

Partitioning in time: initial condition TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 97

Partitioning in time: initial condition TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 98

Partitioning in time: initial condition … omited initial condition, Keldysh limit TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 99

Partitioning in time: initial condition … with uncorrelated initial condition, TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 100

Partitioning in time: initial condition … with uncorrelated initial condition, TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 101

Partitioning in time: initial condition TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 102

Restart correlation function: initial conditions continuous time variable t > t 0 TU Chemnitz Nov 30, 2005 singular time variable fixed at t = t 0 Between GF and Transport Equations … 103

Restart correlation function: initial conditions uncorrelated initial condition. . . KELDYSH singular time variable fixed at t = t 0 TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 104

Restart correlation function: initial conditions correlated initial condition. . . DANIELEWICZ TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 105

Restart correlation function: initial conditions host continuous self-energy. . . KELDYSH initial correlations correction MOROZOV &RÖPKE TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 106

Act III applications: restarted switch-on processes pump and probe signals. . TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 107

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 t 0 - . . TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 109

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 t 0 - . . TU Chemnitz Nov 30, 2005 Between GF and Transport Equations … 110

THE END