Initial Singularity of the Little Bang Kenji Fukushima (RIKEN BNL Research Center) Ref: K.Fukushima, F.Gelis, L.McLerran: hep-ph/0610416

Big Bang Fluctuations Inflation Reheating (Instability) (Thermalization)

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

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

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

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

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

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

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

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??

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

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 ?

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

Boundary Conditions Light-Cone Singularity Source Singularity Ex.

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

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

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.

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?