Transients in Quantum Transport: B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of.

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Transients in Quantum Transport: B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha SEMINÁŘ TEORETICKÉHO ODD. FZÚ SLOVANKA 14. ÚNORA 2006

Transients in Quantum Transport: I. Semi-Group Property of Propagators and the Gauge Invariance of the 1 st Kind B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha SEMINÁŘ TEORETICKÉHO ODD. FZÚ SLOVANKA 14. ÚNORA 2006

Transients in Quantum Transport: II. Correlated Initial Condition for Restart Process by Time Partitioning A. Kalvová, Acad. Sci. of CR, Praha B. Velický, Charles University and Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha SEMINÁŘ TEORETICKÉHO ODD. FZÚ SLOVANKA 21. ÚNORA 2006

Transients in Quantum Transport: II. Correlated Initial Condition for Restart Process by Time Partitioning Progress in Non-Equilibrium Green’s Function III, Kiel Aug 22, 2005 Topical Problems in Statistical Physics, TU Chemnitz, Nov 30, 2005 SEMINÁŘ TEORETICKÉHO ODD. FZÚ SLOVANKA 21. ÚNORA 2006

Transients in Quantum Transport II... 5 Teor. Odd. FZÚ 21.II.2006 Prologue

Transients in Quantum Transport II... 6 Teor. Odd. FZÚ 21.II.2006 (Non-linear) quantum transport non-equilibrium problem many-body Hamiltonian many-body density matrix additive operator Many-body system Initial state External disturbance

Transients in Quantum Transport II... 7 Teor. Odd. FZÚ 21.II.2006 (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

Transients in Quantum Transport II... 8 Teor. Odd. FZÚ 21.II.2006 (Non-linear) quantum transport non-equilibrium problem Quantum Transport Equation a closed equation for generalized collision term Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix

Transients in Quantum Transport II... 9 Teor. Odd. FZÚ 21.II.2006 (Non-linear) quantum transport non-equilibrium problem Quantum Transport Equation a closed equation for 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 many-particle initial condition interaction term

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 This talk: orthodox study of quantum transport using NGF TWO PATHS DIRECT use a NGF solver Lecture on NGF… continuation

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Lecture on NGF… continuation Real time NGF choices This talk: orthodox study of quantum transport using NGF TWO PATHS DIRECT use a NGF solver

14 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Standard approach based on GKBA  Real time NGF our choice

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Standard approach based on GKBA  Real time NGF our choice D YSON E QUATIONS Keldysh IC: simple initial state permits to concentrate on the other issues

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Standard approach based on GKBA  Real time NGF our choice  GKBE

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Standard approach based on GKBA  Real time NGF our choice  GKBE  Specific physical approximation -- self-consistent form

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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)

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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)

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Act I reconstruction

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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?

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 INVERSION SCHEMES Reconstruction Problem – Historical Overview

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 INVERSION SCHEMES Reconstruction Problem – Historical Overview

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Postulate/Conjecture: typical systems are controlled by a hierarchy of times separating the initial, kinetic, and hydrodynamic stages. A closed transport equation holds for Parallels G E N E R A L S C H E M E LABEL Bogolyubov

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Postulate/Conjecture: typical systems are controlled by a hierarchy of times separating the initial, kinetic, and hydrodynamic stages. A closed transport equation holds for Parallels G E N E R A L S C H E M E LABEL Bogolyubov

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Runge – Gross Theorem: Let be local. Then, for a fixed initial state, the functional relation is bijective and can be inverted. 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Runge – Gross Theorem: Let be local. Then, for a fixed initial state, the functional relation is bijective and can be inverted. 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

35 Teor. Odd. FZÚ 21.II.2006  „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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 INVERSION SCHEMES Reconstruction Problem – Summary

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 INVERSION SCHEMES Reconstruction Problem – Summary

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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!

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Reconstruction theorem: Coupled equations DYSON EQ. GKB EQ. RECONSTRUCTION EQ.

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Reconstruction theorem: operational description NGF RECONSTRUCTION THEOREM determination of the full NGF restructured as a DUAL PROCESS quantum transport equation  reconstruction equations Dyson eq.

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 "THEOREM" The one-particle density matrix and the full NGF of a process are in a bijective relationship, NGF RECONSTRUCTION THEOREM determination of the full NGF restructured as a DUAL PROCESS quantum transport equation  reconstruction equations Dyson eq. Reconstruction theorem: formal statement

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Act II reconstruction and initial conditions NGF view

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 For an arbitrary initial state at start from the NGF Problem of determination of G extensively studied Fujita  Hall  Danielewicz  …  Wagner  Morozov&Röpke … Klimontovich  Kremp  …  Bonitz&Semkat, Morawetz … Take over the relevant result for : The self-energy depends on the initial state (initial correlations) has singular components General initial state

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 General initial state: Structure of Structure of

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Structure of General initial state: Structure of singular time variable fixed at t = t 0 continuous time variable t > t 0

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Danielewicz notation Structure of General initial state: Structure of singular time variable fixed at t = t 0 continuous time variable t > t 0

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Danielewicz notation Structure of General initial state: Structure of singular time variable fixed at t = t 0 continuous time variable t > t 0

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

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

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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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 :

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Let the initial time be, the initial state. In the host NGF the Heisenberg operators are Restart at an intermediate time

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 We may choose any later time as the new initial time. For times the resulting restart GF should be consistent. Indeed, with we have Restart at an intermediate time

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Act III. Time-partitioning

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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.

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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.

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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 <

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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 renormalized semigroup rule G R

67 Teor. Odd. FZÚ 21.II.2006 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 renormalized semigroup rule G R RENORM. SEMIGROUP RULE

68 Teor. Odd. FZÚ 21.II.2006 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 < RENORM. SEMIGROUP RULE last time - past - future notion … in renormalized semigroup rule G R

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 renormalized semigroup rule G R RENORM. SEMIGROUP RULE

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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 renormalized semigroup rule G R

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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 restart NGF unified description— time-partitioning formalism - past - future notion … in renormalized semigroup rule G R

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Partitioning in time: formal tools Past and Future with respect to the initial (restart) time

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Partitioning in time: formal tools Past and Future with respect to the initial (restart) time Projection operators

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Partitioning in time: for propagators 1. Dyson eq. 2. Retarded quantity only for 3. Diagonal blocks of

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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 renormalized semi-group rule

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy … split into four blocks 2. Propagators … by partitioning expressions

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy … split into four blocks 2. Propagators … by partitioning expressions

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Partitioning in time: for corr. function Question: to find four blocks of 1. Selfenergy … split into four blocks 2. Propagators … by partitioning expressions

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 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

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 restart Partitioning in time: restart corr. function HOST PROCESS RESTART PROCESS

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 restart Partitioning in time: restart corr. function HOST PROCESS RESTART PROCESS

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 restart Partitioning in time: restart corr. function HOST PROCESS RESTART PROCESS future memory of the past folded down into the future by partitioning

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 restart Partitioning in time: restart corr. function HOST PROCESS RESTART PROCESS future memory of the past folded down into the future by partitioning

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 initial condition Partitioning in time: initial condition Singular time variable fixed at restart time

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 initial condition Partitioning in time: initial condition

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 initial condition Partitioning in time: initial condition

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 initial condition Partitioning in time: initial condition

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 initial condition Partitioning in time: initial condition

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 initial condition Partitioning in time: initial condition

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 initial condition Partitioning in time: initial condition … omitted initial condition, Keldysh limit

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 initial condition Partitioning in time: initial condition … with uncorrelated initial condition,

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 initial condition Partitioning in time: initial condition … with uncorrelated initial condition,

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 initial condition Partitioning in time: initial condition

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Act IV applications: restarted switch-on processes pump and probe signals....

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Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Conclusions time partitioning as a novel general technique for treating problems, which involve past and future with respect to a selected time semi-group property as a basic property of NGF dynamics full self-energy for a restart process including all singular terms expressed in terms of the host process GF and self-energies result consistent with the previous work (Danielewicz etc.) explicit expressions for host switch-on states (from KB -- Danielewicz trajectory to Keldysh with t 0  - ....

Transients in Quantum Transport II Teor. Odd. FZÚ 21.II.2006 Conclusions time partitioning as a novel general technique for treating problems, which involve past and future with respect to a selected time semi-group property as a basic property of NGF dynamics full self-energy for a restart process including all singular terms expressed in terms of the host process GF and self-energies result consistent with the previous work (Danielewicz etc.) explicit expressions for host switch-on states (from KB -- Danielewicz trajectory to Keldysh with t 0  - ....

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