Nonequilibrium quantum field theory
and the lattice
Jürgen Berges
Darmstadt University of Technology
Content
I. Motivation
fast thermalization in heavy ion collisions
early universe instabilities and prethermalization
strongly coupled quantum gases
II. Nonequilibrium dynamics
two-particle irreducible expansions
limitations of (semi-)classical descriptions
III. Real-time quantum fields on a lattice
real-time stochastic quantization
nonabelian gauge theory
I. Motivation
Heavy ion collisions
Relativistic Heavy Ion Collider (BNL)
Facility for Antiproton and
Ion Research (GSI)
Large Hadron Collider (CERN)
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; ...
Far-from-equilibrium dynamics
Heavy-ion collisions (BNL,CERN,GSI) explore strong interaction matter
starting from a transient nonequilibrium state
• Thermalization ?
Properties of the equilibrium
phase diagram of QCD ?
Braun-Munzinger, Redlich, Stachel,
QGP3 (2004) 491; ...
• Theoretical justification of early
local thermal equilibrium?
Hydrodynamics after .1 fm/c?
Kolb, Heinz, QGP3 (2004) 634; ...
Fast thermalization?
F New properties (sQGP)?
Shuryak, Zahed, Phys. Rev. C70 (2004) 021901; ...
F Fast thermalization from kinetic theory?
Xu, Greiner, Phys. Rev. C 71 (2005) 064901; ...
F Prethermalization? Different quantities effectively thermalize
on different time scales: Early equation of state → Hydrodynamics
Berges, Borsanyi, Wetterich, Phys. Rev. Lett. 93 (2004) 142002
Plasma instabilities:
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; ...
...
Early Universe
End of Inflation
reheating
far-from-equilibrium
`initial´ state
`entropy´
production
time
CMB
thermal spectrum
with fluctuations
Reheating
• Explosive particle production from nonequilibrium instabilities
CLASSICAL: Traschen, Brandenberger, PRD 42 (1990) 2491;
Kofman, Linde, Starobinsky, PRL 73 (1994) 3195;
Khlebnikov, Tkachev, PRL 77 (1996) 219; ...
Vergleiche: Parametrische Resonanz in der klassischen Mechanik
QUANTUM: Berges, Serreau, PRL 91 (2003) 111601
Arrizabalaga, Smit, Tranberg, JHEP 0410 (2004) 017
Parametric resonance reheating:
explosive particle
production
quasistationary
evolution
• Quasistationary evolution leads to extremely slow thermal equilibration
→ non-thermal fixed points
• Prethermalization
Berges, Borsanyi, Wetterich, Phys. Rev. Lett. 93 (2004) 142002
Podolsky, Felder, Kofman, Peloso, Phys. Rev. D 73 (2006) 023501;…
SU(2)×SU(2) ‘quark-meson‘ model (2PI 1/NF to NLO):
tpt
tdamp
Prerequisite for hydrodynamics!
teq
tpt
F Approximatively thermal equation of state after tpt ¿ trelax ¿ teq!
Ultra-cold quantum gases
• Tunable BEC self-interaction!
Strong coupling (Feshbach resonance)
Attract
Repel
B-field
OD
⇒ Measure BEC size, shape:
1
a = 70 a0
⇒ B(t) faster than atom motion:
BEC remnant
OD
0
1
0
550 a0
OD
In trap focussed burst atoms
1
0
3000 a0
Cornish et al. Phys. Rev. Lett. 85 (2000) 1795
480 µm
Ultracold atomic gas dynamics of 23Na in 1D
Gasenzer, Berges, Schmidt,
Seco, PRA 72 (2005) 063604
Method: 2PI 1/N expansion
Berges, NPA 699 (2002) 847
t
II. Nonequilibrium quantum fields
Standard QFT techniques fail out of equilibrium
`Secularity´
`Universality´
• uniform approximations in time
require infinite pert. orders
• nonlinear dynamics necessary
for late-time thermalization
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 ...
Luttinger, Ward ´60; Baym ´62; Cornwall, Jackiw, Tomboulis ´74
E.g. scalar N-component field theory to NLO in 2PI 1/N-expansion:
Berges ´02 ; Aarts, Ahrensmeier, Baier, Berges, Serreau ´02
includes
NLO 1PI !
Time evolution equations
spectral function ∼ h[Φ,Φ]i
statistical propagator ∼ h{Φ,Φ}i
Nonequilibrium:
Equilibrium/Vacuum:
(fluct.-diss. relation)
Nonequilibrium instability:
(parametric resonance)
p
n
o
N
e!
v
i
t
ba
r
u
t
er
III. Quantum fields on a lattice
Real time:
non-positive definite
probability measure!
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!
:
• discretization to second order in
• conjugate momenta have Gaussian distribution; randomly refresh
after every single step → Langevin dynamics
,
Parisi, Wu ’81; …
with white noise
,
Real-time stochastic quantization
Klauder ’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!
Simulating nonequilibrium quantum fields
Berges, Stamatescu, Phys. Rev. Lett. 95 (2005) 202003
Scalar λφ4-theory:
λ=0
t at-1
λ≠0
classical starting configuration (t5 = 0), Langevin
updating takes into account quantum corrections
t at-1
Convergence:
• same initial (t = 0)
conditions
’null’ starting configuration (t5 = 0)
t at-1
⇒ apparently good
convergence properties
⇒ ‘run-away’ trajectories
much suppressed by
smaller step-size
Langevin time
Precision tests
0.25
Anharmonic quantum oscillator:
stochastic
Schrödinger: (real contour)
(complex contour)
0.2
<ϕ(0)ϕ(t)>
• real-time thermal equilibrium
• comparison with solution
of Schrödinger equation
0.15
weak coupling
0.1
0.05
0.4
0
stochastic
Schrödinger
0.35
0
0.2
0.3
0.4
0.5
t
0.3
<ϕ(0)ϕ(t)>
0.1
0.25
• short real-time contour:
0.2
strong coupling
0.15
⇒ good agreement of stochastic
quantization and `exact´ results
0.1
0.05
0
−0.05
0
0.5
1
1.5
2
t
2.5
3
3.5
4
Berges, Borsanyi, Sexty, Stamatescu,
in preparation
Fixed points of the Langevin flow
Stationary solutions at late t5 fulfill:
⇒
similarly for
,
,…
⇒
...
⇒
⇒
infinite set of Dyson-Schwinger equations for n-point functions!
0
thermal fixed point
-0.02
Dyson-Schwinger equation:
-0.04
-0.06
LHS (0,0)
RHS (0,0)
LHS (0,t)
RHS (0,t)
LHS (t,t)
RHS (t,t)
tfinal=1
-0.08
t=0.375
-0.1
0
1
2
3
4
5
6
LHS
7
8
Langevin time
0.05
RHS
• fulfilled by both thermal as well as
non-unitary fixed point (symmetrized)
0.5
Re G(t,t)
0
Im G(t,t)
-0.05
-0.1
LHS (0,0)
RHS (0,0)
LHS (0,t)
RHS (0,t)
LHS (t,t)
RHS (t,t)
tfinal=2
-0.15
t=0.375
-0.2
0
1
2
3
4
5
Langevin time
6
tfinal=1
tfinal=2
0.4
0.3
0.2
0.2
0.1
0
-0.1
-0.2
0
5
10
15
20
contour point index
25
non-unitary fixed point
7
8
30
Nonabelian gauge theory
Real-time lattice action:
with anisotropic couplings
(plaquette)
,
,
Langevin dynamics:
,
(not ∼ gµν for Minkowski theory!)
1
SU(2) gauge theory
on a contour:
Euclidean
contour tilt tan(α)=2.2
tan(α)=1.1
tan(α)=0.6
spatial plaquette average
0.9
• thermal fixed point
only approximate
(intermediate Langevin
times) !
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
1
6
5
ϑcrossover
4
τ+
3
2
τ+=0.125
τ+=0.25
τ+=2 , symmetric
1
0
0.1
1
Contour tilt: tan(α)
10
2
3
Langevin time
4
5
Dyson-Schwinger equation for plaquette:
(
,
RHS
LHS
2(N 2− 1)
N
)
µ
Σ
= i
βµγ
N +
−γ
{
µ
γ
γ
−
µ
γ
Schwinger-Dyson equations
9
− 1
N
LHS
RHS
8
thermal
7
6
crossover
5
4
non-unitary
3
2
0
0.5
1
1.5
2
2.5
Langevin time
3
3.5
4
µ
γ
−
µ
}
Conclusions
• Loop-, or 1/N-expansions of 2PI effective action suitable
to resolve secularity and universality
⇒ Far-from-equilibrium dynamics & thermalization in QFT
⇒ Limited range of validity of kinetic approaches
• 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-time
Dyson-Schwinger equations, however, solutions not unique
⇒ Short-time evolution of scalar fields
⇒ Thermal fixed point unstable for SU(2) gauge theory
Nonequilibrium Dynamics in
Particle Physics and Cosmology
Kavli Institute for Theoretical Physics, Santa Barbara
Jan. 14 to March 28, 2008
Organizers:
J. Berges (Darmstadt), L. Kofman (CITA), L. Yaffe (U. of Washington)
Limitations of kinetic theory
Berges, Borsányi, Phys. Rev. D74 (2006) 045022
Based on
• gradient expansion in
,
• memory loss (t0 → ∞, s0 ∈ (-∞,∞) with X0 finite)
• (quasiparticle picture)
Lowest-order gradient expansion:
Imaginary part real part
of self-energy
NLO gradient expansion:
with
and Poisson brackets
Quantitative example
occupation
number
ptransverse
• weak-coupling g2φ4-model, 2PI three-loop
plongitudinal
⇒
• characteristic anisotropy measure:
(isotropy → ∆F ≡ 0)
:
valid kinetic description
tdamp
• LO/NLO results only quantitative after tdamp (memory loss)
→ not suitable for studying fast thermalization (t ¿ tdamp )
:
:
valid kinetic description
tdamp
valid kinetic description
tdamp
• NLO gradient corrections insignificant for ∆F (cf. isotropization)
• NLO gradient corrections significant for F (cf. thermalization)
• NLO results quantitative for t & tdamp