Transcript A Vlasov Equation for Quantized Meson Field

Vlasov Equation for
Chiral Phase Transition
M. Matsuo and T. Matsui
Univ. of Tokyo ,Komaba
T
Quark-Gluon Plasma
Chiral Phase Transition
Hadron
CSC
μ
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Chiral Phase Transition in heavy ion collision:
Two Lorentz contracted nuclei are approaching
toward each other at the velocity of light.
After two nuclei pass each other,,,
Vacuum between two nuclei are excited and filled
with a quark-gluon plasma.
Vacuum chiral condensate has melted away in this
region.
As the system expands, the quark-gluon plasma will
hadronize and the chiral symmetry will be broken
spontaneously again.
We have growing chiral condensate and particle
excitations
Eventually, condensate will be repaired and
particles will fly apart with a frozen momentum
distribution.
≫We like to formulate a quantum transport theory
to describe the final stages.
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Our Physical Picture:
Particle excitation by quantizing the fields
In the past:
described by
Classical fields (coherent state)
(ref. Asakawa, Minakata, Muller)
Our formalism:
put incoherent particle excitations
by quantizing the fields
time-evolution
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Outline of the rest of this talk
I. Derivation of Coupled Equations
II. Uniform Equilibrium
III. Dispersion Relations: solutions in linearized approx.
around uniform equilibrium
IV. Open problems: How the system time-evolve?
I. derivation of coupled equations
to describe evolution of non-equilibrium systems
Classical field equation
for chiral condensate
II. Apply to
time-independent
equilibrium states
Coupled Eqn.
Quantum kinetic equation
for particle excitations
III. dispersion relations:
solutions in linearized
approx. around uniform
equilibrium
IV. time-evolution
>> Sorry! Now Working!
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Our Formalism
Heisenberg Equation of Motion for quantum fields
*Separate the fields into Condensate / Non-cond. part
*Statistical average with Gaussian density matrix
*odd power => 0
*4th-power decoupled
into the product of 2nd-powers
Equation of Motion
for the mean field
Classical field equation
for condensate
Equation of Motion
for fluctuation
in terms of the Wigner functions
Quantum kinetic equation
for non-condensate
(particle excitations)
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A simple model: phi^4 model
*Model:phi^4 model for quantized real scalar field
*Hamiltonian:
*Heisenberg eq. of scalar field:
Gaussian statistical average
Classical mean field equation
(“Non-linear Klein-Gordon eq.)
★This equation includes the effects of quantum fluctuation
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Wigner function …～Quantum Kinetic Equations～
*Define creation/annihilation operator:
(μ: physical particle mass)
*Construct Wigner function (quantum version of
number density distribution in phase space):
*Equation of Motion for F contains other “Wigner functions”
★ For a static uniform system,
G,Gbar can be eliminated by the Bogoliubov tr.
(corresponds to redefinition of particle mass) .
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Quantum Kinetic Equation for <a+a>
*Equation for f(p,r,t) in long wavelength limit (quantum Vlasov eq.)
l.h.s: Landau kinetic equation
Quasi-particle energy
Mean Field potential
Fluctuation of meson self-energy
which is not included in the particle mass
Relativistic drift term including the
effect of local change of particle mass
Vlasov term due to continuous acceleration
generated by the gradient of mean field
potential U
r.h.s: sink/source terms due to the local fluctuation of
“particle mass” can not be eliminated for a nonuniform system.
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Kinetic Equation for the Wigner function g
*Equation for the Wigner function g=<aa>
* no drift/Vlasov term for g
* purely quantum mechanical origin
* looks more like an equation of a simple ocsillator with frequency 2ε
Rapid oscillation between particle and “anti-particle”
r.h.s: Matter distribution f(p,r,t)
disturbs the oscillation as “external perturbation “
★ If the system is static and uniform, Up=0
 F and G are decoupled by Bogoliubov tr.
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Extension to O(N) model
Extend one component model to multi component model with
continuous symmetry (Chiral symmetry SU(2)L × SU(2)R ～ O(4) )
(i＝1～N)
N classical field equations
(non-linear Klein-Gordon eq.)
Highly non-linear eas.
*Define creation/annihilation ops.
& construct N×N Wigner functions:
N×N kinetic eqs
(Quantum Vlasov equations)
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Equilibrium States
Time-independent solution of Coupled equation for O(2)
(assuming only one component of the meson field φc0 has nonvanishing expectation value in equilibrium ):
gap equations
Difficulties
I.
1st order phase transition
- Always confronted with this problem
when using mean field approximation
II. Goldstone theorem is apparently violated.
(μ1≠0 ＆ μ2≠0)
-
0
Tc
T
We will show later missing Goldstone mode can be
found in the collective excitations of the system.

Dispersion relations
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Linearized Eqs. and Collective modes
Coupled non-linear eqs for
condensates & particle excitations
*linearization with respect
to small deviations
from equilibrium solutions
(assuming only one component of the condensates
φc0 has non-vanishing expectation value in equilibrium )
Coupled linear eqs for these fluctuations:
N decoupled sets of fluctuations
Dispersion relations:
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[N=2] Dispersion relations
Dispersion relation (σ-like mode)
in the direction of condensate
No collective branch
=> Meson excitation
1
1
1
2
1
1
1
2
w
w=k
Meson excitation
k
Dispersion relation (π-like mode)
in the direction perpendicular to condensate
w
Massless collective mode
=> Nambu-Goldstone boson
2 1
1 2
1
2
1
2
w=k
Goldstone theorem
is recovered!
k
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SUMMARY
* We have derived a coupled set of equations for quantized self-interacting real
scalar field (O(N) linear sigma model) containing equations for classical mean
field and Vlasov equations for particle excitations.
* We have studied dispersion relation of excitations and found
σ-like mode with mass
and π-like massless modes corresponding to the Nambu-Goldstone bosons.
Open problems:
* We have to solve the equations with more realistic initial condition for final
stages of evolution of nucleus-nucleus collision.
* Non-hydrodynamic collective flow may be generated by acceleration by mean field
gradient .
* This formalism gives a consistent framework for studying these problems.
Thank you very much indeed
for your kind attention!!
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Bose-Einstein dist.
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