Manifestly Retarded Formalism for Out-of-Equilibrium Thermal Field Theories I. Dadić Ruđer Bošković Institute, Zagreb

1. Manifestly RetardedFormalism for Out-of-Equilibrium Thermal Field Theories I. Dadić Ruđer Bošković Institute, Zagreb Talk given at “2nd Croatian – Hungarian Meeting”, 30. August -3. September, 2007., Rab, Croatia

2. R/A formalism in early 1990ies P.Aurenche and T.B. Becherawy, P. Aurenche, E. Petitgirard and T. del Rio Gaztelurrutia, M.A. van Eijck and Ch. G. van Weert, T. Ewans,F. Guerin • mainly for equilibrium • propagators R/A, vertices include density distributions • not directly connected to present work Infinite (Keldysh) time path - pinching T. Altherr and D. Seibert, P. F. Bedaque, M. leBellac and H. Mabilat, A. Niegawa, ID, A. Jakovac,… Finite time path - no adiabatic switching of interaction • Boyanovsky with collaborators

3. 1 2 Green functions

4. R, A, K Basis Rename index “2” into “-1” Then in (1,-1) basis

5. Projection to R/A basis Two point function Finite time path Projected functions

6. Fourier and Wigner transforms

9. Particles number Number of particles at the time X0 - inclusive - correspond to M2 - at t   corresponds to “Fermi´s golden rule”

10.       Absence of “source” vertices Only even power of contributes  at least one DR !!!!  no contribution from “source” vertices  internal vertex cannot have local maximal time  no closed diagrams

11. Turn all propagators to retarded; ignore all the other properties t xo yo xo  yo All lines retarded ● R sink normal source (absent) Graphical representation

12. arbitrary graph sink graphs with outgoing line vanish at equal time

13. equal time contributing graph sink sink 6 point Greens function external lines in pairs

14. =+1 (x0) (y0) x0n ● =-1 +1 DR (x,y) +1 Integrate time variables of internal vertices

15. implies Remaining exponentials connected to external vertices, x0je Equal time procedure x0jet

16. j j+1 j-1 P0 integrals   0 small finite number 2   !!!!

17. 0 Energy nonconservation at sink sink i Conservation of nonconservation Equal time limit x0let

18. P0 P0´ Regularisation - remains only Energy nonconserving term - works only at equal time! • no pinching two denominators have different energy variables vanishes !!!

19. Large X0 = t limit D. Bojanovski at all., Phys. Rev.D 60 (1999) 065003 -in relativistic theories subleading terms may have infinite coefficients -naive recipe for “adiabatic switching”: keep only leading terms

20. Conclusion - out of equilibrium in R/A - pinching  sink vertices energy not conserved at sink - regulation at equal time limit  elimination of energy conserving term of sink - large t limit enabled - retarded propagator regulated when it is subgraph of multipoint equal time Greens function

21. References I. Dadic, Proceedings of Dubrovnik, 2004. I. Dadic, Proceedings of XI Chris Engelbercht SummerSchool in TF., Ed. J. Cleymans & H. B. Geyer, FGSchook,edited by Springer, 1998. I. Dadic, Proceedings of TFT 98, e-print, xxx. Lanl.gov/html/hep-ph/9811467/schedule/html I. Dadic, Phys. Rev. D 59 (1999), 125012-1-14 I. Dadic, Phys. Rev. D 63 (2001), 024011 I. Dadic, Erratum Phys. Rev. D66(2002), 049903 I. Dadic, Nucl. Phys. A702 (2002) 356C I. Dadic, hep-ph/0103025 D. Kuic, I. Dadic, in preparation

22.               Examples of diagrams g2order Photon production from QGP pairing of nonsymmetric diagrams g4 order

23.      g6order

24.             Sinks in Swinger-Dyson equation K,R

25.         Double sinks  

26.                Sinks in retarded propagator none Sinks in 4-point Green functions - dypletion production 2-particle scattering

27.        Bhabha scattering  