1. Initial Singularity of the Little Bang
Kenji Fukushima
(RIKEN BNL Research Center)
Ref: K.Fukushima, F.Gelis, L.McLerran: hep-ph/

2. Big Bang Fluctuations Inflation Reheating (Instability)
(Thermalization)

3. Little Bang Space-Time Evolution of the Little Bang
Instability
 thermalization
Quantum Fluctuations
after collision
Quantum Fluctuations before collision
Fluctuations (seeds)  Instability  Thermalization
Fluctuations (seeds)  Instability  Thermalization

4. Fluctuations and Instability
Time Evolution of Fluctuations under Instability
Stable
c.f. Ordering process in continuous transition
Potential
Wavefunction
Instability
Classical evolution is a good approximation unless the instability is weak (i.e. potential is flat).
Singularity
depending on the problem

5. Formula (after singularity)
Quantum Mechanics
Similar results by S.Jeon (05)
tF is some time after singularity
WKB (semi-classical) approximation
xFc is a solution of the classical equations of motion at time tF
Wigner function characterizes the spectrum of quantum fluctuations
Typically

6. Derivation (1/3) Expansion around the classical path
assuming strong instability (WKB approximation)

7. Derivation (2/3) c.f. Schwinger-Keldysh formalism
Only the surface terms remain due to the equations of motion.
without approximation

8. Derivation (3/3) Definitions Variable Changes  Formula is derived
(neglecting the associated Jacobian)
 Formula is derived

9. Solvable Example Inverted Harmonic Oscillator Initial Wavefunction
with
Wigner Function
Classical Path

10. Application to the Little Bang
Formula
Initial background fields
Classical Equations of Motion
Right-moving Nucleus
Left-moving Nucleus
These equations have been solved numerically by Romatschke-Venugopalan.
Instability w.r.t. h-dependent fields  Weibel Instability??

11. Weibel Instability Seed of Instability Instability
electron motion under Bz
resulting current density e Bz
electron Bz e
Fluctuations in Bz Bz Bz
Initial Bz is amplified by current density.
Seed of Instability
specified by Wavefunction or Wigner function
Instability

12. Fluctuations before Collision
Zero-Point Oscillation of Empty Steady State from Infinite Past
Time-independent Schroedinger Equation
Ground-state Wavefunction at t~0- (where Ah=0)
Gauss Law
Cut-Off ?

13. How to pass over the singularity?
Classical Part
Kovner-McLerran-Weigert (’95)
Empty
source singularity
from
or the Gauss law

14. Boundary Conditions
Light-Cone Singularity
Source Singularity
Ex.

15. How to pass over the singularity?
Fluctuation Part
Integration over the Schroedinger Equation from t=0- to z=0+
Hsing is the singular part of Hamiltonian containing d(t)
Boundary Condition
Initial singularity shifts the fields by the classical background

16. Fluctuation Spectrum
Wigner Function characterizing initial fluctuations
Metric ~t2
Gauss law

17. Procedure
Generate the initial conditions with fluctuations A+a, E+e satisfying
Solve the classical equations of motion with this initial condition.
Take the average over the distribution of fluctuations a and e.

18. Obstacle…?
Large fluctuations for almost homogeneous components. They should be suppressed by non-linearity.
Large fluctuations for furiously inhomogeneous components.
What results from our initial conditions for the Glasma instability?