1. 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

2. 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

3. 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

4. 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

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

17. 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

18. 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)

19. 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, …

20. 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

21. 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)

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

23. 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

24. 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

25. 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

26. 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?

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

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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

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

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

38. 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!

39. 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

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

41. 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.

42. 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

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

44. 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

45. 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

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

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

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

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

50. 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

51. 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 :

52. 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

53. 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

54. 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

55. 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

56. 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

57. 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

58. 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

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

60. 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

61. 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.

62. 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.

63. 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

64. 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

65. 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 <

66. 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

67. 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

68. 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

69. 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

70. 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

71. 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

72. 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

73. 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

74. 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

75. 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

76. 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

77. 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

78. 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

79. 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

80. 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

81. 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

82. 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

83. 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

84. 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

85. 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

86. 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

87. 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

88. 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

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

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

91. 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

92. 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

93. 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

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

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

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

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

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

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

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

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

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

103. 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

104. 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

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

106. 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

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

108. NEXT TIME

109. 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  - ....

110. 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  - ....

111. THE END