Quantum Structure of dS Space In-Out Formalism Perspective on dS Radiation

1. QuantumStructure of dS SpaceIn-Out Formalism Perspective on dS Radiation Sang Pyo Kim Kunsan Nat’l Univ. & Nat’l Taiwan Univ. 1stLeCosPA Symposium February 6-9, 2012 National Taiwan University

2. Caveat In this talk, neither Holographic Entropy nor Hilbert Space of dS.

3. Outline • Introduction • de Sitter (Gibbons-Hawking) radiation • Quantum dynamics for a scalar field • In-out formalism perspective for dS radiation • Conclusion

4. Introduction

5. de Sitter Space Still Interesting? • The present and future universe dominated by Dark Energy will be described by an asymptotically dS space. • The universe with the cosmological constant  is a pure dS, has a cosmological horizon and has the Gibbons-Hawking’sdS radiation crossing the horizon. • The maximal symmetry of dS makes QFT more tractable than other spacetimes except for Minkoswki one. BUT we still do NOT comprehend the vacuum structure of dS space. The Bunch-Davies vacuum challenged by Polyakov [Polyakov, NPB834(2010); NPB797(2008)]..

6. Legacyof Dirac, Heisenberg and Schwinger on Vacuum • What is the (quantum) vacuum? (J. Rafelski, W-T. Ni) • What is a particle and its quantum state? (B. Unruh) • The Dirac vacuum or sea: filled with pairs of particles and antiparticles with all possible quantum numbers. • Heisenberg and Euler (‘36): spinor effective action, the instability of the Dirac sea against the emission of pairs. • Schwinger (‘51): vacuum polarization and Schwinger mechanism for emission of charged pairs in electric fields. • The 4th facility of ELI (Extreme Light Infrastructure, EU) to be completed by 2017 is highly likely to detect electron-positron pair production (a clean evidence of Dirac sea).

7. Quest for Vacuum and Pair Production[SPK, JHEP11(‘07)]

8. Connecting Schwinger Mechanism & Unruh Effect & Hawking Radiation • An intuitive way to understand particle (pair) production [Frolov & Novikov, Black Hole Physics (‘98)] • Schwinger pair production: • Unruh effect: • Hawking radiation: • dS (Gibbons-Hawking) radiation?

9. Gauge/Gravity Relation • The correspondence of pair production between the acceleration in Schwinger mechanism and the Hubble constant (curvature) in de Sitter space [SPK, JHEP11 (‘07)] • The relation between the gauge scattering amplitudes and the graviton scattering amplitudes [Bern, Living Rev. Rel. 5 (‘02)]

10. One-Loop Gauge/Gravity Relation • Massive scalar in a self-dual EM field in 4d-D = massive spinor in 2d-D AdS at one-loop [Basar, Dunne, JPA 43 (‘10)] • The Heisenberg-Euler/Schwinger effective actions in QED are better understood in the weak and strong field regime.

11. Gibbons-Hawking’sdS Radiation

12. Classical de Sitter Space • The planar/ global coordinates of dS space • The static coordinates The horizon is observer-dependent: The past and future null infinity (I at top and bottom) and the past and future horizons (diagonal lines) of an observer at the south (north) pole. [Spradlin et al, hep-th/0110007]

13. dS Space as a Black Hole? • Gibbons and Hawking [PRD15(‘77)]: a dS space has a surface gravity  = H at the cosmological horizon r = 1/H, the temperature T = H/(2) and the entropy S = A/4. • Jacobson [PRL75(‘95)]: the black hole thermodynamics seen by all local Rindler observers is equivalent to the Einstein equation [equilibrium condition]. • Cai and Kim [JHEP02(‘05)]: the black hole thermodynamics at the apparent horizon of an expanding FRW universe is equivalent to the Friedmann equation [nonequilibrium condition]. • An open question is, What is the origin of the cosmological entropy?

14. dS Radiation • The the Bunch-Davies vacuum for a massive scalar in the planar and the global coordinates • The in-vacuum and the out vacuum solutions

15. dS Radiation • The Bogoliubov transformation between the in-vacuum and the out-vacuum solutions • The Bogoliubov coefficients and pair production in the planar coordinates and the global coordinates

16. dS Radiation as Tunneling • The in-out formalism (t = ) predicts particle production only in even dimensions [Mottola, PRD 31 (‘85); Bousso et al, PRD 65 (‘02)]. • A massive scalar in a dSd+1 space:

17. dS Radiation as Tunneling • The Hamilton-Jacobi action in a complex time

18. Stokes Phenomenon • Two pairs of turning points • Hamilton-Jacobi action [Fig. adopted from Dumlu & Dunne, PRL 104 (2010)]

19. dS Radiation as Tunneling • Use the phase-integral approximation and find the mean number of produced particles [SPK, JHEP09(‘10)]. • The Stokes phenomenon explains the destructive interference between two Stokes’s lines in odd dimensions and the constructive interference in even dimensions [solitonic character].

20. Quantum Dynamics in dS Space TFD Workshop [hep-th/9511082]

21. Cauchy Problem in dS Space • The dS space in the planar coordinates and the global coordinates • The dS space is a globally hyperbolic spacetime, each constant-time hypersurface is spacelike and defines a Cauchy surface (t), and, therefore, the Cauchy problem is well-defined: the evolution or wave propagation

22. Quantum Dynamics in dS • The evolution operator on Cauchy surfaces • A complex scalar field (equivalent to two real scalars)

23. Quantum Dynamics in dS • The Schrodinger equation on the Cauchy surfaces • The Gaussian wave packets on the Cauchy surface (t)

24. Gaussian Wave Packets • The Gaussian wave packets for the Bunch-Davies vacuum in the planar and the global coordinates • The in-vacuum and the out vacuum solutions

25. The In-Out Formalism

26. Vacuum Persistence Amplitude • The Schwinger variational principle for the vacuum persistence amplitude and the effective action density [Schwinger (‘51), DeWitt (‘75)] • The in-out formalism is equivalent to the Feynman integral (Feynman propagator representation)

27. 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

28. 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 (Pauli blocking):

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

30. Vacuum Polarization at T=0 & T • Zero-temperature effective action for scalar and spinor from the gamma-function regularization [SKP, Lee, Yoon, PRD 78 (‘08); 82 (‘10); SPK, 84 (‘11)] • finite-temperature effective action for scalar and spinor [SKP, Lee, Yoon, PRD 82 (‘10)]

31. The In-Out Formalism for dS Radiation

32. Vacuum Persistence Amplitude • The vacuum persistence amplitude • The vacuum persistence for momentak and –k and another Bogoliubov coefficients

33. Kinematic Pair Production • The Hamiltonian for a free massive scalar has SU(1,1) algebra and the in-vacuum particle number operator • The quadratic variances • The number of in-vacuum particles carried by the wave packet is finite for the global coordinates but increases in the planar coordinates

34. Wave Packet for Out-Vacuum • The Gaussian wave packet for the out-vacuum solution • The vacuum persistence amplitude for GH radiation

35. Backreaction • The semiclassical Einstein equation • The back-reaction including the dS radiation • The number of in-vacuum particles or Gibbons-Hawking radiation carried by the wave packet has already been included in the back-reaction. Not many new physical issues from dS radiation for cosmology.

36. Conclusion • The Gibbons-Hawking’sdS radiation is more or less analogous to the Unruh effect: • The cosmological horizon is observer-dependent. • The dS radiation can be measured by a detector prescribed by the out-vacuum solution. • The in-out formalism is a good framework for QFT involving pair production and can be unified with quantum dynamics in a dS space. • The particle production from quantum dynamics has a pathology in the planar coordinates (increases as the increasing volume), though NOT in the global coordinates.