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Semi-Group Property of Propagators and the Gauge Invariance of the 1 st Kind

. SEMINÁŘ TEORETICKÉ HO ODD. FZ Ú SLOVANKA 14. ÚNORA 2006. 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. Introduction.

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Semi-Group Property of Propagators and the Gauge Invariance of the 1 st Kind

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  2. SEMINÁŘ TEORETICKÉHO ODD. FZÚ SLOVANKA 14. ÚNORA 2006 Semi-Group Property of Propagators and the Gauge Invariance of the 1st 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

  3. Introduction non-equilibrium propagators why "semigroup rule" at all? A little primer on SG rule and quasiparticles

  4. Non-equilibrium propagators Even out of equilibrium and in the presence of external fields, the definition is the same: The propagators do not contain the whole story, but they have some advantages: The universal equal-time boundary condition The Dyson equation involving only retarded quantities has a pronounced causal structure, with times ordered from right to the left. Consequence: no singular terms in the self-energy even for correlated initial conditions. Semi-Group Property …

  5. How I praised the propagators in Kiel (Favorable) properties of propagators • Propagators  quantum coherence and memory in QTE • Simpler than the particle correlation function • … propagation of one-particle excitations • …  generalized spectral density • … weakly dependent on the particle distribution • … Dyson eqs. with a strict causal structure • Propagators decisive for possible use of the quasi-particle picture • General properties of propagators modelling without detailed solution of the Dyson eqs. Semi-Group Property …

  6. How I praised the propagators in Kiel (Favorable) properties of propagators • Propagators  quantum coherence and memory in QTE • Simpler than the particle correlation function • … propagation of one-particle excitations • …  generalized spectral density • … weakly dependent on the particle distribution • … Dyson eqs. with a strict causal structure • Propagators decisive for possible use of the quasi-particle picture • General properties of propagators modelling without detailed solution of the Dyson eqs. previous slide Semi-Group Property …

  7. How I praised the propagators in Kiel (Favorable) properties of propagators • Propagators  quantum coherence and memory in QTE • Simpler than the particle correlation function • … propagation of one-particle excitations • …  generalized spectral density • … weakly dependent on the particle distribution • … Dyson eqs. with a strict causal structure • Propagators decisive for possible use of the quasi-particle picture • General properties of propagators modelling without detailed solution of the Dyson eqs. previous slide Semi-Group Property …

  8. How I praised the propagators in Kiel (Favorable) properties of propagators • Propagators  quantum coherence and memory in QTE • Simpler than the particle correlation function • … propagation of one-particle excitations • …  generalized spectral density • … weakly dependent on the particle distribution • … Dyson eqs. with a strict causal structure • Propagators decisive for possible use of the quasi-particle picture • General properties of propagators modelling without detailed solution of the Dyson eqs. previous slide SG rule Semi-Group Property …

  9. Semi-group rule for propagators This is a postulated property of the propagator, not an exact result. … rule means indeed a composition rule: from two bits of the propagator put together their summary effect Semi-group means the following: associative non-commutative grupoid with unity, but with no inversion element. The grupoid operation is a plain (operator) multiplication at a fixed time, no time integration is involved. This is very different from the NGF symbolic multiplication. Compare SGR with How could anybody even conceive of such a strange "rule"? Semi-Group Property …

  10. Motivation for the Semi-group rule Old idea: GKBA and the "semi-group property" of propagators are closely related…In fact,SG  GKBA Semi-group property is exact for free particles (even with U(t) ): It is closely related with the quasi-particle model  Primer Semi-Group Property …

  11. Primer: SG rule and quasi-particles quasi-particles, equilibrium free particles, equilibrium quasi-particles, external fields free particles, external fields Semi-Group Property …

  12. Primer: SG rule and quasi-particles quasi-particles, equilibrium free particles, equilibrium quasi-particles, external fields free particles, external fields ? Semi-Group Property …

  13. Primer: SG rule and quasi-particles quasi-particles, equilibrium free particles, equilibrium quasi-particles, external fields free particles, external fields ? Semi-Group Property …

  14. Primer: SG rule and quasi-particles quasi-particles, equilibrium free particles, equilibrium quasi-particles, external fields free particles, external fields natural extension: semi-group rule Semi-Group Property …

  15. Primer: SG rule and quasi-particles quasi-particles, equilibrium free particles, equilibrium quasi-particles, external fields free particles, external fields but NOT really correct natural extension: semi-group rule Semi-Group Property …

  16. Ways to correct the semi-group rule Semi-group rule (desired behavior): Quasi-particle composition rule (also just desired): If the quasi-particle picture is valid in equilibrium, this can be proved. Out of equilibrium, this rule itself serves to define the QP behavior… a double task. A vertex correction must appear for dressed propagators of the form: Our aim will be to derive this vertex correction, the Renormalized SG rule.

  17. Ways to correct the semi-group rule Semi-group rule (desired behavior): Quasi-particle composition rule (also just desired): If the quasi-particle picture is valid in equilibrium, this can be proved. Out of equilibrium, this rule itself serves to define the QP behavior… a double task. A vertex correction must appear for dressed propagators of the form: Our aim will be to derive this vertex correction, the Renormalized SG rule.

  18. Ways to correct the semi-group rule Semi-group rule (desired behavior): Quasi-particle composition rule (also just desired): If the quasi-particle picture is valid in equilibrium, this can be proved. Out of equilibrium, this rule itself serves to define the QP behavior… a double task. A vertex correction must appear for dressed propagators of the form: Our aim will be to derive this vertex correction, the Renormalized SG rule.

  19. Ways to correct the semi-group rule Semi-group rule (desired behavior): Quasi-particle composition rule (also just desired): If the quasi-particle picture is valid in equilibrium, this can be proved. Out of equilibrium, this rule itself serves to define the QP behavior… a double task. A vertex correction must appear for dressed propagators of the form: Our aim will be to derive this vertex correction, the Renormalized SG rule.

  20. Ways to correct the semi-group rule Semi-group rule (desired behavior): Quasi-particle composition rule (also just desired): If the quasi-particle picture is valid in equilibrium, this can be proved. Out of equilibrium, this rule itself serves to define the QP behavior… a double task. A vertex correction must appear for dressed propagators of the form: Our aim will be to derive this vertex correction, the Renormalized SG rule. Then try

  21. Renormalized semi-group rule direct derivation of the RSG rule meaning of the result Bogolyubov time hierarchy

  22. Deriving the RSG rule DYSON EQUATION

  23. Deriving the RSG rule DYSON EQUATION INTEGRATION AREA G R = 0

  24. Deriving the RSG rule DYSON EQUATION INTEGRATION AREA split the integration area at an intermediate time G R = 0

  25.   Deriving the RSG rule DYSON EQUATION split the integration area at an intermediate time   

  26. Deriving the RSG rule Each of the integrals has a different and specific role:   

  27. Deriving the RSG rule Each of the integrals has a different and specific role:    For the free GF, the SG rule is valid and can be applied … KEY STEP

  28. Deriving the RSG rule Each of the integrals has a different and specific role:   

  29. Deriving the RSG rule Each of the integrals has a different and specific role:   

  30. Deriving the RSG rule Each of the integrals has a different and specific role:   

  31. Deriving the RSG rule Each of the integrals has a different and specific role: unknown unknown  unperturbed GF   unperturbed GF

  32. Deriving the RSG rule Each of the integrals has a different and specific role:   

  33. Deriving the RSG rule Each of the integrals has a different and specific role:   

  34. Deriving the RSG rule Each of the integrals has a different and specific role:   

  35. Resulting RSG rule RENORMALIZED SEMI-GROUP RULE Semi-Group Property …

  36. Integration range for RSG rule RENORMALIZED SEMI-GROUP RULE VERTEX Semi-Group Property …

  37. Integration range for RSG rule RENORMALIZED SEMI-GROUP RULE off-diagonal integration range VERTEX Semi-Group Property …

  38. Discussion of the RSG rule • universal vertex, derived with almost no effort and no specific properties of the GF • off-diagonal vertex, linking in a smeared fashion propagation in the past and in the future • Similar to the Dyson Equation, but NO free GF • In fact, looks pretty much like a linear response • we would like to understand all these features Semi-Group Property …

  39. Off-diagonal vertexThe past and the future are in the RSG rule linked in a wedge-like area with the tip at the dividing time.If the quasi-particle formation time is short, the connection is nearly local in time Semi-Group Property …

  40. Once more from Kiel: Parallels G E N E R A L S C H E M E 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 Semi-Group Property …

  41. Bogolyubov time hierarchy Semi-Group Property …

  42. Bogolyubov time hierarchy X collision duration time transport relaxation time transport language Semi-Group Property …

  43. Bogolyubov time hierarchy X transport language propagator language this inequality …. a necessary condition for the quasi-particle picture Semi-Group Property …

  44. Integration range for RSG rule: QP formation time the "dominant" strip Semi-Group Property …

  45. Integration range for RSG rule: QP formation time the "dominant" strip Questions about Can the integral be neglected? Or does it become a -like vertex? Semi-Group Property …

  46. Integration range for RSG rule: QP formation time the "dominant" strip Questions about Can the integral be neglected? Or does it become a -like vertex? That is, is the limit of the RSG the plain SG or the QPM rule ??? Semi-Group Property …

  47. RSG rule as a linear responseThis is an alternative interpretation of the formula.In the next section, both interpretations will be linked.Here, we present a heuristic argument. Semi-Group Property …

  48. … as a linear response RSG Rule: Semi-Group Property …

  49. … as a linear response Rewrite RSG R using step functions: Semi-Group Property …

  50. … as a linear response Rewrite RSG R using step functions: Compare with linear response to a local field as known from the KB Book: Semi-Group Property …

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