1. Hawking Radiation and Vacuum Polarization Sang Pyo Kim ( 金相杓 ) Kunsan Nat’l Univ. & YITP, Kyoto Univ. 竹原理論物理硏究會, June 06-08, 2011 [AS & NCU & NTHU & Kinki seminars]

2. Outline Introduction Spin Statistics Inversion in QED Vacuum Polarization and Hawking Radiation Conclusion

3. Introduction Lectures at YITP May 24-25, 2011

4. QED Vacuum Polarization Scalar QED: Weisskopf/Schwinger effective action per volume and per time in a constant E-field Spinor QED: Heisenberg-Euler/Schwinger effective action per volume and per time in a constant E-field

5. QED Vacuum Persistence Spinor QED: Schwinger pair production in a constant E-field Scalar QED: Schwinger pair production

6. Nonperturbative Aspect of QED & sQG [SPK, JHEP11(2007)048] Davies-Unruh Effect & Pair Production Schwinger Mechanism QEDQCD Hawking Radiation Black Holes de Sitter/ Expanding Universe

7. One-Loop Effective Actions The in-out formalism via the Schwinger variational principle [Schwinger, PNAS(‘51); DeWitt, Phys. Rep. (‘75), The Global Approach to Quantum Field Theory (‘03)] The vacuum persistence (twice of the imaginary part ) and the mean number of produced pairs

8. Bogoliubov Transformation & In-Out Formalism The Bogoliubov transformation between the in-state and the out-state, equivalent to the S-matrix, Commutation relations from quantization rule (CTP): Particle (pair) production

9. Out-Vacuum from In-Vacuum For bosons, the out-vacuum is the multi-particle states of but unitary inequivalent to the in-vacuum: The out-vacuum for fermions:

10. Out-Vacuum from S-Matrix The out-vacuum in terms of the S-matrix (evolution operator) The diagrammatic representation for pair production

11. Effective Actions at T=0 & T Zero-temperature effective action for scalar and spinor [SKP, Lee, Yoon, PRD 78, 105013 (‘08); 82, 025016 (‘10)] finite-temperature effective action for scalar and spinor [SKP, Lee, Yoon, PRD 82, 025016 (‘10)]

12. Spin Statistics Inversion in QED

13. Spin Statistics in QFT The spin statistics theorem -Bosons: Bose-Einstein distribution and commutator -Fermions: Fermi-Dirac distribution and anticommutator The vacuum persistence for boson production takes the form of spinor QED [Stephens, AP193 (’89)] The vacuum persistence for fermion production takes the form of scalar QED

14. Spin-Statistics Inversion The vacuum polarization of scalar QED can be written as a spectral function times the Fermi-Dirac distribution and that for spinor QED as a spectral function times the Bose-Einstein distribution [Muller, Greiner, Rafelski, PLA 63 (‘77)]

15. Spin-Statistics Inversion The vacuum persistence in terms of the transverse energy [Hwang, SPK, PRD 80, 065004(‘09)] The vacuum persistence in terms of the instanton action or the worldline instanton

16. Vacuum Polarization and Hawking Radiation SPK & W-Y. Pauchy Hwang, “Vacuum Polarization and Persistence on the Black Hole Horizon,” [arXiv:1103.5264]

17. Vacuum Persistence for BH Hawking radiation of bosons and fermions in a charged rotating black hole Vacuum persistence for zero amplification (R J = 0) 2Im(W) as ln(Z), the logarithm of the partition function, plus the vacuum energy implies that 2Im(W)/V is the pressure of boson and fermion gas [Ritus, (‘84), (‘98)]

18. Effective Action for Black Hole Schwarzschild black hole in d-dimensions Bogoliubov coefficients for a massless boson [DeWitt, Phys.Rep.19 (‘75)] Effective action

19. Vacuum Polarization for BH Hawking radiation in a Schwarzschild black hole is effectively two-dimensional one, in the (t-r) spacetime Reduced effective action for bosons and fermions Vacuum polarization as a thermal effective action Renormalized vacuum polarization for massive emission

20. Vacuum Persistence and Gravitational Anomalies Vacuum persistence = decay rate of vacuum due to Hawking radiation or Schwinger mechanism Trace anomalies = Hawking radiation and Schwinger mechanism [Christensen, Fulling, PRD 15 (’77); Dittrich, Sieber, JPA 21 (‘88)] Vacuum persistence for bosons and fermions equals to the total flux of Hawking radiation from gravitational anomalies [Robinson, Wilczek, PRL 95 (‘05); Iso, Umetsu, Wilczek, PRL 96 (‘06)].

21. Duality between BH and QED

22. Conclusion The nonperturbative effective actions for QED and gravity in the in-/out formalism al a Schwinger variational principle. The effective action for pair production are characterized by two prominent aspects: the vacuum polarization (real part) and persistence (imaginary part) The effective action for Schwarzschild black hole, modulo number of states, is dual (equivalent) to QED effective action in constant E. Duality of gravity coupling and gauge coupling? (Davies- Unruh temperature)