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Nonequilibrium quantum field theoryand the lattice
Jürgen BergesDarmstadt University of Technology
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Content
I. Motivation fast thermalization in heavy ion collisionsearly universe instabilities and prethermalizationstrongly coupled quantum gases
II. Nonequilibrium dynamicstwo-particle irreducible expansionslimitations of (semi-)classical descriptions
III. Real-time quantum fields on a latticereal-time stochastic quantizationnonabelian gauge theory
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I. Motivation
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Facility for Antiproton and Ion Research (GSI)
Relativistic Heavy Ion Collider (BNL)
Large Hadron Collider (CERN)
Heavy ion collisions
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Phasediagramm
(schematic)
QCD critical point in the universality class of the Ising model!Berges, Rajagopal; Halasz et al.; Stephanov et al. `99 ... ; Lattice-QCD: Fodor, Katz `02; ...
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Far-from-equilibrium dynamics
Heavy-ion collisions (BNL,CERN,GSI) explore strong interaction matter starting from a transient nonequilibrium state
• Thermalization ?Properties of the equilibriumphase diagram of QCD ?Braun-Munzinger, Redlich, Stachel,
QGP3 (2004) 491; ...
• Theoretical justification of earlylocal thermal equilibrium? Hydrodynamics after .1 fm/c?
Kolb, Heinz, QGP3 (2004) 634; ...
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Fast thermalization?
Xu, Greiner, Phys. Rev. C 71 (2005) 064901; ...
Shuryak, Zahed, Phys. Rev. C70 (2004) 021901; ...
Mrowczynski, Phys. Lett. B 314 (1993) 118 Arnold, Moore, Yaffe, Phys. Rev. Lett. 94 (2005) 072302;Rebhan, Romatschke, Strickland, Phys. Rev. Lett. (2005) 102303;Romatschke, Venugopalan, Phys. Rev. Lett. 96 (2006) 062302; ...
Berges, Borsanyi, Wetterich, Phys. Rev. Lett. 93 (2004) 142002
New properties (sQGP)?
Plasma instabilities:
Prethermalization? Different quantities effectively thermalizeon different time scales: Early equation of state → Hydrodynamics
Fast thermalization from kinetic theory?
...
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Early Universe
End of Inflation reheating CMB
`entropy´ production
thermal spectrumwith fluctuations
far-from-equilibrium`initial´ state
time
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Reheating• Explosive particle production from nonequilibrium instabilities
Vergleiche: Parametrische Resonanz in der klassischen Mechanik
CLASSICAL: Traschen, Brandenberger, PRD 42 (1990) 2491; Kofman, Linde, Starobinsky, PRL 73 (1994) 3195;Khlebnikov, Tkachev, PRL 77 (1996) 219; ...
QUANTUM: Berges, Serreau, PRL 91 (2003) 111601 Arrizabalaga, Smit, Tranberg, JHEP 0410 (2004) 017
quasistationaryevolution
explosive particleproduction
Parametric resonance reheating:
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• Quasistationary evolution leads to extremely slow thermal equilibration→ non-thermal fixed points
• Prethermalization Berges, Borsanyi, Wetterich, Phys. Rev. Lett. 93 (2004) 142002Podolsky, Felder, Kofman, Peloso, Phys. Rev. D 73 (2006) 023501;…
SU(2)×SU(2) ‘quark-meson‘ model (2PI 1/NF to NLO):
Prerequisite for hydrodynamics!
tdamp teqtpt
tpt
Approximatively thermal equation of state after tpt ¿ trelax ¿ teq!
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Ultra-cold quantum gases
B-field
Attract Repel• Tunable BEC self-interaction!Strong coupling (Feshbach resonance)
⇒ Measure BEC size, shape: ⇒ B(t) faster than atom motion:
OD
0
1
OD
0
1
OD
0
1
550 a0
3000 a0
a = 70 a0
In trap focussed burst atoms
BEC remnant
480 µmCornish et al. Phys. Rev. Lett. 85 (2000) 1795
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Ultracold atomic gas dynamics of 23Na in 1D
Gasenzer, Berges, Schmidt, Seco, PRA 72 (2005) 063604
t
Method: 2PI 1/N expansionBerges, NPA 699 (2002) 847
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II. Nonequilibrium quantum fields
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Standard QFT techniques fail out of equilibrium
`Secularity´ `Universality´
• nonlinear dynamics necessaryfor late-time thermalization
• uniform approximations in timerequire infinite pert. orders
2-particle irreducible generating functionals
⇒ systematic 2PI loop-, coupling- or 1/N-expansions available
⇒ far-from-equilibrium dynamics as well as late-time thermalization in QFT
Berges, Cox ´01; Aarts, Berges ´01; Berges ´02; Cooper, Dawson, Mihaila ´03; Berges, Serreau ´03; Berges, Borsányi, Serreau ´03; Cassing, Greiner, Juchem ´03; Arrizabalaga, Smit, Tranberg ´04 ...
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Luttinger, Ward ´60; Baym ´62; Cornwall, Jackiw, Tomboulis ´74
E.g. scalar N-component field theory to NLO in 2PI 1/N-expansion:
includesNLO 1PI !
Berges ´02 ; Aarts, Ahrensmeier, Baier, Berges, Serreau ´02
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Time evolution equations
statistical propagator ∼ h{Φ,Φ}i
spectral function ∼ h[Φ,Φ]i
Nonequilibrium:
Equilibrium/Vacuum: (fluct.-diss. relation)
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Nonequilibrium instability:(parametric resonance)
Nonperturbative!
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III. Quantum fields on a lattice
Real time:
non-positive definite probability measure!
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Euclidean stochastic quantization• Classical Hamiltonian in (d+1)-dimensional space-time
• Expectation values for quantum theory with action :
,
• Replace canonical ensemble averages by micro-canonical:
Classical dynamics in ‘fifth‘-time (t5) to compute quantum averages!
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• discretization to second order in
• conjugate momenta have Gaussian distribution; randomly refreshafter every single step → Langevin dynamics
,
Parisi, Wu ’81; …
,with white noise
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Real-time stochastic quantizationKlauder ’83; Parisi ’83; Hüffel, Rumpf ’84; Okano, Schülke, Zheng ’91 …
Replace embedded d-dimensional Euclidean by Minkowskian action:
with d‘Alembertian
for Euclidean stochastic quantization⇒
for real-time stochastic quantization⇒
Langevin dynamics:
i.e. , in general complex!
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Simulating nonequilibrium quantum fieldsBerges, Stamatescu, Phys. Rev. Lett. 95 (2005) 202003
t at-1
t at-1
Scalar λφ4-theory:
classical starting configuration (t5 = 0), Langevinupdating takes into account quantum corrections
λ = 0
λ ≠ 0
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Convergence:
t at-1
Langevin time
apparently good convergence properties
• same initial (t = 0) conditions
’null’ starting configuration (t5 = 0)
‘run-away’ trajectoriesmuch suppressed by smaller step-size
⇒
⇒
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Precision tests
Berges, Borsanyi, Sexty, Stamatescu, in preparation
Anharmonic quantum oscillator:
• real-time thermal equilibrium
• comparison with solutionof Schrödinger equation weak coupling
strong coupling
0
0.05
0.1
0.15
0.2
0.25
0 0.1 0.2 0.3 0.4 0.5
<ϕ(
0)ϕ(
t)>
t
stochastic Schrödinger: (real contour)
(complex contour)
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.5 1 1.5 2 2.5 3 3.5 4
<ϕ(
0)ϕ(
t)>
t
stochasticSchrödinger
• short real-time contour:
⇒ good agreement of stochasticquantization and `exact´ results
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Fixed points of the Langevin flowStationary solutions at late t5 fulfill:
⇒
, , …similarly for
⇒
⇒
⇒
...
infinite set of Dyson-Schwinger equations for n-point functions!
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-0.2
-0.15
-0.1
-0.05
0
0.05
0 1 2 3 4 5 6 7 8
Langevin time
t=0.375
tfinal=2
LHS (0,0)RHS (0,0)LHS (0,t)RHS (0,t)LHS (t,t)RHS (t,t)
-0.1
-0.08
-0.06
-0.04
-0.02
0
0 1 2 3 4 5 6 7 8
Langevin time
t=0.375
tfinal=1
LHS (0,0)RHS (0,0)LHS (0,t)RHS (0,t)LHS (t,t)RHS (t,t)
thermal fixed point
LHS RHS
non-unitary fixed point
Dyson-Schwinger equation:
• fulfilled by both thermal as well as non-unitary fixed point (symmetrized)
-0.2-0.1
0 0.1 0.2
0 5 10 15 20 25 30
Im G
(t,t)
contour point index
0.2
0.3
0.4
0.5
Re
G(t
,t)
tfinal=1tfinal=2
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Nonabelian gauge theoryReal-time lattice action: (plaquette)
with anisotropic couplings
Langevin dynamics:
, ,
,
(not ∼ gµν for Minkowski theory!)
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0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5
spat
ial p
laqu
ette
ave
rage
Langevin time
Euclideancontour tilt tan(α)=2.2
tan(α)=1.1tan(α)=0.6
0
1
2
3
4
5
6
0.1 1 10
ϑ cro
ssov
er
Contour tilt: tan(α)
τ+=0.125τ+=0.25
τ+=2 , symmetric
τ+
SU(2) gauge theoryon a contour:
• thermal fixed point only approximate(intermediate Langevintimes) !
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,( )Dyson-Schwinger equation for plaquette:
µ
N2(N − 1)2
βµγ iN
1N
γµ
γµ
γ
µ
γ
µ
γ+−
}
{=
−
Σ −
−
LHS RHS
2
3
4
5
6
7
8
9
0 0.5 1 1.5 2 2.5 3 3.5 4
Sch
win
ger-
Dys
on e
quat
ions
Langevin time
LHSRHS
thermal
crossover
non-unitary
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Conclusions• Loop-, or 1/N-expansions of 2PI effective action suitable to resolve secularity and universality
⇒ Limited range of validity of kinetic approaches⇒ Far-from-equilibrium dynamics & thermalization in QFT
• 2PI 1/N-expansion provides quantitative description of nonperturbative dynamics as instabilities or critical phenomena ⇒ 2PI 1/N for SU(N) gauge theories?
• Nonperturbative lattice simulations of real-time quantum fields:
⇒ Stochastic quantization solves hierarchy of real-timeDyson-Schwinger equations, however, solutions not unique
⇒ Short-time evolution of scalar fields⇒ Thermal fixed point unstable for SU(2) gauge theory
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Nonequilibrium Dynamics in Particle Physics and Cosmology
Jan. 14 to March 28, 2008Kavli Institute for Theoretical Physics, Santa Barbara
Organizers: J. Berges (Darmstadt), L. Kofman (CITA), L. Yaffe (U. of Washington)
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Limitations of kinetic theory
• gradient expansion in
• memory loss (t0 →∞, s0 ∈ (-∞,∞) with X0 finite)
• (quasiparticle picture)
Based on Berges, Borsányi, Phys. Rev. D74 (2006) 045022
,
Lowest-order gradient expansion:
Imaginary part real partof self-energy
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NLO gradient expansion:
withand Poisson brackets
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Quantitative example• weak-coupling g2φ4-model, 2PI three-loop
occupationnumber
⇒
p tra
nsve
rse
plongitudinal
• characteristic anisotropy measure: (isotropy → ∆F ≡ 0)
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:
tdamp
valid kinetic description
• LO/NLO results only quantitative after tdamp (memory loss)→ not suitable for studying fast thermalization (t ¿ tdamp )
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tdamp
valid kinetic description
tdamp
valid kinetic description
: :
• NLO gradient corrections insignificant for ∆F (cf. isotropization)
• NLO gradient corrections significant for F (cf. thermalization)
• NLO results quantitative for t & tdamp