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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

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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 III Kiel August 22 – 25, 2005

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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, Praha V. Špička, Acad. Sci. of CR, Praha Topical Problems in Statistical Physics TU Chemnitz, November 30, 2005

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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, Praha V. Špička, Acad. Sci. of CR, Praha Topical Problems in Statistical Physics TU Chemnitz, November 30, 2005

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

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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

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Between GF and Transport Equations … 7 TU Chemnitz Nov 30, 2005 (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

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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 generalized collision term Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix

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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 Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix interaction term

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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 Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix QUESTIONS existence, construction of incorporation of the initial condition interaction term

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

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

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

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

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

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

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

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Between GF and Transport Equations … 18 TU Chemnitz Nov 30, 2005 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)

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Between GF and Transport Equations … 19 TU Chemnitz Nov 30, 2005 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, Vinogradov, Horing Haug + Frankfurt team, Rostock school, Jauho, …

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Between GF and Transport Equations … 20 TU Chemnitz Nov 30, 2005 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

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Between GF and Transport Equations … 21 TU Chemnitz Nov 30, 2005 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 Famous examples: Levinson eq. (hot electrons) Optical quantum Bloch eq. (optical transients)

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

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

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Between GF and Transport Equations … 24 TU Chemnitz Nov 30, 2005 Exact formulation -- Reconstruction Problem G ENERAL Q UESTION : 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

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Between GF and Transport Equations … 25 TU Chemnitz Nov 30, 2005 Exact formulation -- Reconstruction Problem G ENERAL Q UESTION : 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

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

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

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

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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 LABEL Bogolyubov

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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 LABEL Bogolyubov

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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. N OTES : 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 LABEL TDDFT

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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. N OTES : 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 LABEL TDDFT

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Between GF and Transport Equations … 33 TU Chemnitz Nov 30, 2005 Closed Time Contour Generating Functional (Schwinger): Used to invert the relation E XAMPLES OF U SE : 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 LABEL Schwinger

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Between GF and Transport Equations … 34 TU Chemnitz Nov 30, 2005 Closed Time Contour Generating Functional (Schwinger): Used to invert the relation E XAMPLES OF U SE : 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 LABEL Schwinger

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35 TU Chemnitz Nov 30, 2005 „Bogolyubov“: importance of the time hierarchy R EQUIREMENT Characteristic times should emerge in a constructive manner during the reconstruction procedure. „TDDFT“ : analogue of the Runge - Gross Theorem R EQUIREMENT Consider a general non-local disturbance U in order to obtain the full 1-DM as its dual. „Schwinger“: explicit reconstruction procedure R EQUIREMENT 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 LABEL NGF Reconstruction Theorem

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

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

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

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

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

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

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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 THEOREM determination of the full NGF restructured as a DUAL PROCESS quantum transport equation reconstruction equations Dyson eq. Reconstruction theorem: formal statement

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

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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

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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 Morawetz

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

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

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

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

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

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Between GF and Transport Equations … 51 TU Chemnitz Nov 30, 2005 Initial state for restart process 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 To progress further, narrow down the selection of the initial states Special situation :

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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

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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 Restart at an intermediate time

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Between GF and Transport Equations … 54 TU Chemnitz Nov 30, 2005 We may choose any later time as the new initial time. For times the resulting GF should be consistent. Indeed, with we have Restart at an intermediate time whole family of initial states for varying t 0

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Between GF and Transport Equations … 55 TU Chemnitz Nov 30, 2005 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

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Between GF and Transport Equations … 56 TU Chemnitz Nov 30, 2005 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 … causal structure of the Dyson equation … develops singular parts at as a condensed information about the past

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Between GF and Transport Equations … 57 TU Chemnitz Nov 30, 2005 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 … causal structure of the Dyson equation … develops singular parts at as a condensed information about the past

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Between GF and Transport Equations … 58 TU Chemnitz Nov 30, 2005 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 … causal structure of the Dyson equation … develops singular parts at as a condensed information about the past

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

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Between GF and Transport Equations … 60 TU Chemnitz Nov 30, 2005 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

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Between GF and Transport Equations … 61 TU Chemnitz Nov 30, 2005 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.

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Between GF and Transport Equations … 62 TU Chemnitz Nov 30, 2005 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.

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Between GF and Transport Equations … 63 TU Chemnitz Nov 30, 2005 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

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Between GF and Transport Equations … 64 TU Chemnitz Nov 30, 2005 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

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Between GF and Transport Equations … 65 TU Chemnitz Nov 30, 2005 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 <

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Between GF and Transport Equations … 66 TU Chemnitz Nov 30, 2005 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 G R

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67 TU Chemnitz Nov 30, 2005 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 G R CORR. SEMIGR. RULE

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Between GF and Transport Equations … 68 TU Chemnitz Nov 30, 2005 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 G R

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Between GF and Transport Equations … 69 TU Chemnitz Nov 30, 2005 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 G R - past - future notion … in restart NGF unified description— time-partitioning formalism

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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

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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 Projection operators

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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 Projection operators Double time quantity X …four quadrants of the two-time plane

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

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Between GF and Transport Equations … 74 TU Chemnitz Nov 30, 2005 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

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Between GF and Transport Equations … 75 TU Chemnitz Nov 30, 2005 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

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Between GF and Transport Equations … 76 TU Chemnitz Nov 30, 2005 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

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Between GF and Transport Equations … 77 TU Chemnitz Nov 30, 2005 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

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Between GF and Transport Equations … 78 TU Chemnitz Nov 30, 2005 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 time-local factorization vertex correction: universal form (gauge invariance) link past-future non-local in time

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Between GF and Transport Equations … 79 TU Chemnitz Nov 30, 2005 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 time-local factorization vertex correction: universal form (gauge invariance) link past-future non-local in time Corrected semigroup rule

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

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

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

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

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

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

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

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Between GF and Transport Equations … 87 TU Chemnitz Nov 30, 2005 Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy … split into four blocks 2. Propagators … by partitioning expressions …off-diagonals of selfenergy

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Between GF and Transport Equations … 88 TU Chemnitz Nov 30, 2005 Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy … split into four blocks 2. Propagators … by partitioning expressions …off-diagonals of selfenergy Exception!!! Future-future diagonal

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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NEXT TIME

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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 t 0 - ....

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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 t 0 - ....

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THE END

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