Why, when and how to simulate the out-of-equilibrium quantum dynamics of gauge and fermion fields Anders Tranberg Niels Bohr International Academy Discovery Center Niels Bohr Institute. Numerical Cosmology 2012, 17.-20. July 2012, CTC, DAMTP, Cambridge, UK Why? Gauge fields and fermions... Why? (pre-4.7.2012) Elementary physics is described in terms of quantum fields. All known elementary particles are either gauge particles or fermions. LHC@CERN Why? (post-4.7.2012) Elementary physics is described in terms of quantum fields. All but one of the known elementary particles are either gauge particles or fermions. And the single scalar interacts with all the rest (except gluons). Higgs-Englert-Brout-Kibble-Guralnik-Hagen-Anderson-Nambu Fields in Cosmology ● Inflation -> scalar field dynamics ( ) ● Primordial curvature perturbations -> quantum fluctuations of light scalar field ( ) ● Sometimes: = Cosmological phase transitions ● Nucleosynthesis (QCD) ● Confinement (QCD) ● Chiral symmetry breaking (QCD) ● Electroweak symmetry breaking ● Baryogenesis (EW?) ● Magnetic field creation (EW?) ● Gravitational waves (EW?) ● (P)Reheating after inflation (EW?) ● MSSM, GUT, Monopoles, Cosmic strings, Oscillons, ... Equilibrium Out of equilibrium Non-perturbative initial-value problem: -> Numerics! When? Quantum dynamics, as opposed to... Classical, Classical and Classical Classical field theory: No h-bar, no quantum fluctuations -> classical equations of motion. Applies to: – – – – Inflaton, curvaton, sort-of. Galactic magnetic fields (macroscopic fields). Gravity (no alternative, currently). Cosmic strings (?) Classical limit... Classical, Classical and Classical ● Classical field = quantum averaged “mean” field -> Equation of motion from minimizing quantum effective action. Exact! But effective action is usually truncated in a diagram expansion. Applies to: – – Inflaton, curvaton dynamics. Preheating (2PI). Quantum effective action... Classical, Classical and Classical ● Free quantum dynamics of bosonic field: Operator Complex number Classical eom, average over statistical ensemble N_e. Also for interacting, non-linear evolution! Classical approximation... Example: Tachyonic preheating ● Hybrid inflation: ● “Waterfall field” unstable: Modes allow “classical approx.” Tachyonic preheating Example: Tachyonic preheating Arrizabalaga, Smit, AT (2004) Skullerud, Smit, AT (2003) Fermions ● Pauli principle -> ● Fermions always quantum! ● But fermion eom also always linear (~Dirac equation)! ● Fermion bilinear correlator: Time-independent operators -> Initial condition (vacuum) How? To simulate quantum fermions out of equilibrium... Expensive fermions ● Gauge-scalar-fermion theory: – Treat gauge and scalar fields classically – Do fermions quantum ● One “field” per lattice site! Scales as N^(2 x 3)...in parallel. ● And then average over gauge-scalar ensemble... Aarts, Smit: 1998-9 Cheap(er) fermions ● complex numbers Only need fermion bilinears. NotRandom, individualGaussian mode functions . ● Introduce two fields in x-space (M, F): ● Both obey Dirac equation. And bilinears: ● Classical bosons. 2 x Nq fermion fields in parallel. Ensemble averaged bilinears into bosonic eoms. Is 2 x Nq < N x N x N? Borsanyi, Hindmarsh: 2009 Simple numerical implementation ● Lattice fermions -> doublers. Wilson term -> ok! ● Bosonic fields on all processors. Updated everywhere. ● Fermion realizations distributed Nq/np per processor. ● ● Essentially a few MPI_Allreduce per timestep to gather fermion bilinears for the bosonic updates. Lattice 32^3 x size(double) x 2 x 2 x 4 x 4 x Nq/np (+bosons) -> Nq/np ~ 50/GB ● Alternative: Distribute volume, all modes local? Lots of nearestneighbour points to pass(!) Relevant for very large volumes? Example: Electroweak Baryogenesis ● Baryogenesis = creation of the matter/antimatter asymmetry. ● Breaking of C, CP, P, B and departure from equilibrium. -> Electroweak sector of Standard Model! ● Electroweak B-breaking through quantum anomaly: Example: Electroweak Baryogenesis 3+1D, SU(2)-Higgs + u, d, e, ( Σαφφ ι ν, ΑΤ: ϑΗΕΠ 02(2012)102) How? To simulate quantum bosonic fields, too... Quantum effective action ● 2PI effective action: Feynman diagrams: perturbative expansion. Cannot describe topological defects, confinement. Does give quantum thermalization! Simple numerical implementation ● ● ● Put fields on a spatial lattice. Go to momentum space. Distribute over second time-coordinate (typically 400-2000 timesteps). At each time-step, compute self-energies by Fourier transform. Sum over time-steps stored locally. ● MPI_sum between procs, then update on proc -> t'. ● Lattice 32^3 -> symmetries N = 969. N x size(double) x Nt^2 x NG, NG = number of correlator components needed, ~ 10. -> ~100 GB. ● Runtime ~ N x Nt^3. ● Scalars and fermions ok. Gauge fields difficult (gauge invariance, ...) Example: Scalar with expansion AT: 2008 Conclusions (but not quite done yet...) ● ● ● ● Why? The Standard Model is a theory of fermions and gauge fields...and one scalar. Unlikely that all new physics is only scalars. Fields are quantum. The Universe expands -> out of equilibrium. When? When all else fails: gauge fields at small occupation numbers, or close to equilibrium; fermions. Otherwise use classical approx. How? Discretize on a lattice. Either classical approx. for bosons and cheap fermions; or ultimately 2PI/quantum effective action for everybody. Standard model: 24 fermions, 4 scalars, 1 U(1), 1 SU(2), 1 SU(3) = (384 x Nq + 40) x size(double) x N x N x N ● N = 64 -> 800 Mbyte x Nq. Nq = 10000? -> 8Tbyte. (210 Tbyte) ● Buy a large computer...or ask a friend who has one... Other issues ● ● ● Renormalization of quantum correlators. Expanding Universe: Redshifting of modes. Comoving lattice? Physical lattice? Inflation?! Self-consistent Friedmann equation: Renormalization of energymomentum tensor. Semi-classical gravity? Quantum gravity? ● Non-linear, inhomogeneous metrics? ● Single-user codes. Resources for optimization? ● Optimization tool box/rules of thumb/check-list? ● Access to flexible supercomputing resources? COSMOSIX@DAMTP ?! Thank you! Numerical Cosmology 2012, 17.-20. July 2012, CTC, DAMTP, Cambridge, UK