Nonequilibrium quantum field theory and the lattice. Jürgen Berges Darmstadt University of Technology

Transcription

1 Nonequilibrium quantum field theory and the lattice Jürgen Berges Darmstadt University of Technology

2 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

3 I. Motivation

4 Heavy ion collisions Relativistic Heavy Ion Collider (BNL) Facility for Antiproton and Ion Research (GSI) Large Hadron Collider (CERN)

5 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;...

6 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;...

7 Fast thermalization? New properties (sqgp)? Shuryak, Zahed, Phys. Rev. C70 (2004) ;... Fast thermalization from kinetic theory? Xu, Greiner, Phys. Rev. C 71 (2005) ;... Prethermalization? Different quantities effectively thermalize... on different time scales: Early equation of state Hydrodynamics Berges, Borsanyi, Wetterich, Phys. Rev. Lett. 93 (2004) Plasma instabilities: Mrowczynski, Phys. Lett. B 314 (1993) 118 Arnold, Moore, Yaffe, Phys. Rev. Lett. 94 (2005) ; Rebhan, Romatschke, Strickland, Phys. Rev. Lett. (2005) ; Romatschke, Venugopalan, Phys. Rev. Lett. 96 (2006) ;...

8 Early Universe End of Inflation reheating CMB far-from-equilibrium `initial state `entropy production thermal spectrum with fluctuations time

9 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) Arrizabalaga, Smit, Tranberg, JHEP 0410 (2004) 017 Parametric resonance reheating: explosive particle production quasistationary evolution

10 Quasistationary evolution leads to extremely slow thermal equilibration non-thermal fixed points Prethermalization Berges, Borsanyi, Wetterich, Phys. Rev. Lett. 93 (2004) Podolsky, Felder, Kofman, Peloso, Phys. Rev. D 73 (2006) ; SU(2) SU(2) quark-meson model (2PI 1/N F to NLO): t pt t damp t eq Prerequisite for hydrodynamics! t pt Approximatively thermal equation of state after t pt t relax t eq!

11 Ultra-cold quantum gases Tunable BEC self-interaction! Strong coupling (Feshbach resonance) Attract Repel B-field Measure BEC size, shape: B(t) faster than atom motion: OD 1 a = 70 a 0 BEC remnant 0 OD a 0 In trap focussed burst atoms OD a 0 Cornish et al. Phys. Rev. Lett. 85 (2000) µm

12 Ultracold atomic gas dynamics of 23 Na in 1D Gasenzer, Berges, Schmidt, Seco, PRA 72 (2005) Method: 2PI 1/N expansion Berges, NPA 699 (2002) 847 t

13 II. Nonequilibrium quantum fields

14 Standard QFT techniques fail out of equilibrium `Secularity uniform approximations in time require infinite pert. orders `Universality 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...

15 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!

16 Time evolution equations spectral function h[φ,φ]i statistical propagator h{φ,φ}i Nonequilibrium: Equilibrium/Vacuum: (fluct.-diss. relation)

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18 Nonequilibrium instability: (parametric resonance) Nonperturbative!

19 III. Quantum fields on a lattice Real time: non-positive definite probability measure!

20 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 (t 5 ) to compute quantum averages!

21 discretization to second order in conjugate momenta have Gaussian distribution; randomly refresh after every single step Langevin dynamics, Parisi, Wu 81; with white noise,

22 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!

23 Simulating nonequilibrium quantum fields Berges, Stamatescu, Phys. Rev. Lett. 95 (2005) Scalar λφ 4 -theory: λ= 0 λ 0 ta t -1 classical starting configuration (t 5 = 0), Langevin updating takes into account quantum corrections ta t -1

24 Convergence: same initial (t = 0) conditions null starting configuration (t 5 = 0) ta t -1 apparently good convergence properties run-away trajectories much suppressed by smaller step-size Langevin time

25 Precision tests Anharmonic quantum oscillator: real-time thermal equilibrium comparison with solution of Schrödinger equation <ϕ(0)ϕ(t)> stochastic Schrödinger: (real contour) (complex contour) weak coupling stochastic Schrödinger t <ϕ(0)ϕ(t)> strong coupling short real-time contour: good agreement of stochastic quantization and `exact results t Berges, Borsanyi, Sexty, Stamatescu, in preparation

26 Fixed points of the Langevin flow Stationary solutions at late t 5 fulfill: similarly for,,... infinite set of Dyson-Schwinger equations for n-point functions!

27 0 thermal fixed point Dyson-Schwinger equation: t final =1 t= Langevin time LHS (0,0) RHS (0,0) LHS (0,t) RHS (0,t) LHS (t,t) RHS (t,t) LHS RHS fulfilled by both thermal as well as non-unitary fixed point (symmetrized) Re G(t,t) t final =1 t final =2-0.1 LHS (0,0) RHS (0,0) t final =2 LHS (0,t) RHS (0,t) t=0.375 LHS (t,t) RHS (t,t) Langevin time Im G(t,t) contour point index non-unitary fixed point

28 Nonabelian gauge theory Real-time lattice action: (plaquette) with anisotropic couplings,, Langevin dynamics:, (not g µν for Minkowski theory!)

29 SU(2) gauge theory on a contour: thermal fixed point only approximate (intermediate Langevin times)! spatial plaquette average Euclidean contour tilt tan(α)=2.2 tan(α)=1.1 tan(α)= Langevin time 5 4 ϑ crossover 3 2 τ + 1 τ + =0.125 τ + =0.25 τ + =2, symmetric Contour tilt: tan(α)

30 Dyson-Schwinger equation for plaquette: (, ) 2 2(N 1) N LHS µ = i N Σ +γ βµγ { µ γ RHS µ γ γ γ Schwinger-Dyson equations thermal crossover LHS RHS non-unitary 1 N µ µ } Langevin time

31 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

32 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)

33 Limitations of kinetic theory Based on Berges, Borsányi, Phys. Rev. D74 (2006) gradient expansion in, memory loss (t 0, s 0 (-, ) with X 0 finite) (quasiparticle picture) Lowest-order gradient expansion: Imaginary part real part of self-energy

34 NLO gradient expansion: with and Poisson brackets

35 Quantitative example weak-coupling g 2 φ 4 -model, 2PI three-loop occupation number p transverse p longitudinal characteristic anisotropy measure: (isotropy F 0)

36 : valid kinetic description t damp LO/NLO results only quantitative after t damp (memory loss) not suitable for studying fast thermalization (t t damp )

37 : : valid kinetic description valid kinetic description t damp t damp NLO gradient corrections insignificant for F (cf. isotropization) NLO gradient corrections significant for F (cf. thermalization) NLO results quantitative for t & t damp