Non-perturbative quantum fields

The physics of the early universe

Quantum field theory has developed in the second half of the last century into a successful description of the fundamental interactions, culminating in the Standard Model of the strong, electromagnetic and weak interactions between quarks and leptons. A striking area where quantum field theory has become gradually more important is the physics of the early universe. According to our current understanding, the observable part of universe 'started' in a phase that requires quantum gravity for its description, then it went through an early rapid expansion, called inflation, which was followed by production of particles that thermalized into a very hot plasma. Subsequently the universe cooled down and went through a number of phase transitions, including the electroweak and QCD transitions. Much later nucleosynthesis took place, the newly formed nuclei combined with electrons into atoms, while freeing photons which are now being observed in the cosmic microwave background radiation, and gravitational collapse led to the formation galaxies and stars. A stream of accurate new data is expected to arrive in the near future from various space observatories of the large scale structure, which may be interpreted within the framework of inflation, and the effects of early phase transitions.

Baryogenesis

A key area of investigation is furthermore baryogenesis, the origin of the surplus of matter over antimatter in the universe. This may have happened shortly after inflation, through leptogenesis in heavy neutrino decay, or later near the time of the electroweak transition. The physics of baryogenesis is very rich and necessitates a detailed understanding of the interplay of lepton and baryon non-conservation, violation of matter-antimatter symmetry (CP violation), and out-of-equilibrium dynamics in field theory, This has led to a lot of theoretical effort, for example on determining the properties of the electroweak transition and on the rate of violation of baryon number in the Standard Model. Such understanding is also vital for the QCD transition which is being studied experimentally with heavy ion colliders.

Phases and phase transitions

The study of phases and phase transitions in field theory is crucial for understanding the important events that took place in our early universe. Our understanding of finite-temperature field theory from first principles has increased substantially, in spite of the breakdown of perturbative methods. At this point the work of the Van Weert and collaborators should be mentioned [6].

On the question, what phases may exist in complicated gauge field theories and what their low energy excitations are, a lot is known already, allowing for effective model building. The role of topological excitations (defects), their dynamics and formation during phase transitions is being studied under the Fundamental Interactions program by Bais and Smit. These applications of field theory all correspond to non-equilibrium situations, involving non-linear phenomena, which are notoriously hard to deal with by analytical means. Van Weert has been involved with similar applications to the dynamics of phase transitions in condensed matter systems, such as He, and he is also an well-known expert in the relativistic domain of quantum fields at high temperature and kinetic theory.

Computer simulation

A powerful tool to help us in tackling some of these problems is computer simulation. Smit has pioneered lattice formulations of quantum fields, which is crucial for numerical work, and has with his collaborators been involved now for some fifteen years in numerical simulations of QCD, the electroweak sector of the Standard Model, fluctuating geometries laying the ground work for quantum gravity, and typical field theoretic models used for testing ideas. The emphasis has at first been on fundamental aspects of the lattice discretisations, such as the species doubling phenomenon of fermions, which are crucial for numerical simulations of the Standard Model. In quantum geometry, using dynamical triangulations, some important results have been obtained on the road to a non-perturbative quantum formulation of Einstein's theory of gravity [10,11,12].

The extension of these methods to non-equilibrium situations, which is needed for applications such as the generation of the observed baryon asymmetry, consequences of cosmological phase transitions, and the transition of the quark-gluon plasma into hadrons in heavy ion collisions, is a great challenge. Monte Carlo methods, which are so powerful for systems in equilibrium, cannot be used anymore. Various approximation schemes are being developed to cope with this difficult situation. For several applications the classical approximation can be quite good and it has been used in numerical simulations of the Higgs sector of the Standard Model [9] and simplified models [7]. Its relation to the quantum world has been carefully examined in [8]. More recently we have made progress in the much more difficult problem of non-perturbative real-time dynamics in the quantum domain, using an inhomogeneous 'Hartree ensemble approximation' [5], aiming at applications to cosmology and heavy-ion collisions [3,4]. Improvements on Hartree approximations can be formulated within so-called -derivable approximations [2]. Incorporating CP violation in studies of baryogenesis is another challenge that is currently being undertaken [1].

Other topics

The heavy-ion collisions constitute currently a very active field of research, driven by experimental results at Brookhaven with the RHIC machine and at CERN in the past with the SPS and in the near future with the LHC, with groups involved at NIKHEF and Utrecht (ALICE). Phenomenological aspects of finite temperature QCD related to the heavy-ion collision experiments are studied by Koch (who holds a parttime appointment at the UvA) and coworkers.

Finally, a related topic which is being pursued, is the early phase of neutron-star formation in supernova's. Neutron stars and stellar black holes originate from the gravitational collapse of massive stars, of which a supernova is the observational signal. The neutrino detections of SN1987A have opened a window to look right into the core of the event and reveal conditions such as prevailed within the first second after the Big Bang. The early phase of neutron star formation is neutrino-driven, and indeed forms a rare and spectacular instance of a macroscopic manifestation of the weak interaction! Van den Horn is actively involved in this interesting front of interdisciplinary physics through our participation in CHEAF (Center for High Energy Astrophysics). The question of neutrino masses and oscillations, with the spectacular results recently obtained in the Super Kamiokande experiment, is a subject of interest on which Gaemers' research is focussed (we should mention that Gaemers contribution to the program fundamental interactions has been limited because after his directorship (2 terms) of the NIKHEF he became the dean of the WINS-Department (Mathematics, Informatics, Physics and Astronomy)).

The research described above has been funded by the FOM/NWO through its program Fundamental Interactions, through the 'beleidsruimte' and some European networks. A regular series of meetings at the NIKHEF and the UvA on our own work and topics of relevance was held in 2000 and 2001, which brought a number of interested researchers from the region together. The UvA group is also taking part in the ESF network COSLAB, which pursues the many conceptual parallels between the physics of the early universe and that of certain condensed matter systems.

Future plans

In the coming period Smit and collaborators intend to continue his research into quantum fields out-of-equilibrium, and to concentrate on applications to the currently very active fields of observational cosmology and heavy-ion collision experiments as described above.

Bibliography

1

A. Tranberg, Baryogenesis from a quench in 1+1 and 3+1 dimensions, seminar, NorFa Network Meeting on 'Theoretical Particle Physics and Cosmology', Copenhagen, 25-26 January 2002.

2

A. Arrizabalaga and J. Smit, Gauge-fixing dependence of Phi-derivable approximations, arXiv:hep-ph/0207044.

3

M. Salle, Kinks in the Hartree ensemble approximation, seminar at Helsinki University, Helsinki, Finland, 21 February 2002.

4

M. Salle, J. Smit, J.C. Vink, Initial conditions for simulated 'tachyonic preheating' and the Hartree ensemble approximation, To be published in the proceedings of COSMO-01, Rovaniemi, Finland.

5

M. Salle, J. Smit, The Hartree ensemble approximation revisited: the symmetric phase, arXive:hep-ph/0208139.
M. Salle, J. Smit, J.C. Vink, Staying Thermal with Hartree Ensemble Approximations, Nucl. Phys. B625 (2002) 495.
M. Salle, J. Smit, J.C. Vink, Thermalization in a Hartree ensemble approximation to quantum field dynamics, Phys. Rev. D64 (2001) 025016.
M. Salle, J. Smit, J.C. Vink, Scalar field dynamics: classical, quantum and in between, Proceedings of the meeting Strong and Electroweak Matter 2000, C.P. Korthals Altes ed. (World Scientific, Singapore 2001) 321-326.
M. Salle, J. Smit, J.C. Vink, Finite temperature simulations from quantum field dynamics?, Proceedings of the XVIIIth International Symposium on Lattice Field Theory, Nucl. Phys. (Proc. Suppl.) 94 (2001) 431-434.
M. Salle, J. Smit, J.C. Vink, Damping and the Hartree ensemble approximation, Proceedings of the XVIIIth International Symposium on Lattice Field Theory, Nucl. Phys. (Proc. Suppl.) 94 (2001) 427-430.
M. Salle, J. Smit, J.C. Vink, Twin Peaks, Proceedings of the meeting Strong and Electroweak Matter 2000, C.P. Korthals Altes ed. (World Scientific, Singapore 2001) 327-331.

6

A. Arrizabalaga, B. J. Nauta, and Ch. G. van Weert, Transversality of logarithmic divergences in the classical SU() self-energy, Phys. Lett. B491 (2000) 214-218.
B.J. Nauta, Dynamics of Gauge Fields at High Temperature, promotor: prof.dr. Ch.G. van Weert, Universiteit van Amsterdam, Amsterdam, November 8, 2000, UvA, 128 pages, proefschrift.
B. J. Nauta and Ch. G. van Weert, Effective classical theory for real-time SU() gauge theories at high temperature, Nucl. Phys. B571 (2000) 151-163.
B. J. Nauta, Counterterms for linear divergences in classical real-time gauge theories at high temperatures, Nucl.Phys. B575 (2000) 383-396.
B. J. Nauta, Effect of CP-violation on the sphaleron rate, Phys. Lett. B478 (2000) 275-279.
B.J. Nauta and Ch.G. van Weert, Time Dimensional Reduction in Thermal Field Theory, Proceedings of the 5th International Workshop on Thermal Field Theories and Their Applications, U. Heinz (ed.), hep-ph/9811469, 1998.

7

G.A.P.T. Aarts, Dynamics in Equilibrium and Nonequilibrium Quantum Field Theory, PhD thesis, 4 October 1999, Universiteit Utrecht, advisor J. Smit.
G. Aarts and J. Smit, Particle Production and Effective Thermalization in Inhomogeneous Mean Field Theory, Phys.Rev. D61 (2000) 025002.
G. Aarts and J. Smit, Real time dynamics with fermions on a lattice, Nucl. Phys. B555 (1999) pp. 355-394.

8

G. Aarts and J. Smit, Classical Approximation for Time Dependent Quantum Field Theory: Diagrammatic Analysis for Hot Scalar Fields, Nucl. Phys. B511 (1998) pp. 451-478.
G. Aarts and J. Smit, Finiteness of hot Classical Scalar Field Theory and the Plasmon Damping Rate, Phys.Lett. B393 (1997) 395-402.

9

W.H. Tang, Numerical Calculation of the Baryon Number Violation Rate in Hot Electroweak Theory, PhD thesis, 1 September 1998, Universiteit van Amsterdam, advisor J. Smit.
W.H. Tang and J. Smit, High-temperature behavior of the Chern-Simons diffusion rate in the 1+1 dimensional abelian Higgs model, Nucl. Phys. B540 (1999) pp. 437-456.
W.H. Tang and J. Smit, Numerical Study of Plasmon Properties in the SU(2) Higgs Model, Nucl. Phys. B510 (1998) pp. 401-420.

10

S. Bilke and G. Thorleifsson, Simulating four-dimensional simplicial gravity using degenerate triangulations, Phys. Rev. D59 (1999) 124008.
S. Bilke, Z. Burda, A. Krzywicki, B. Petersson, K. Petrov, J. Tabaczek and G. Thorleifsson, The strong-coupling expansion in simplicial quantum gravity, Nucl. Phys. Proc. Suppl. 73 (1999) 798-800.
S. Bilke, Z. Burda, A. Krzywicki, B. Petersson, G. Thorleifsson, 4-D Simplicial Quantum Gravity interacting with Gauge Matter Fields, Phys. Lett. B418 (1998) pp. 266-272.
S. Bilke, Z. Burda, A. Krzywicki, B. Petersson, G. Thorleifsson, 4-D Simplicial Quantum Gravity: Matter Fields and the Corresponding Effective Action, Phys. Lett. B432 (1998) pp. 279-286.

11

P. Bialas, Correlations in Fluctuating Geometries, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 739-742.
P. Bialas, Z. Burda, B. Petersson, J. Tabaczek Appearance of mother universe and singular vertices in random geometries, Nucl. Phys. B495 (1997) 463-476.
P. Bialas, Long range correlations in branched polymers, Phys. Lett. B373 (1996) 289-295.
P. Bialas, Z. Burda, A. Krzywicki, B. Peterson, Focusing on the fixed point of 4d simplicial gravity, Nucl. Phys. B472 (1996) 293-308.
P. Bialas, Z. Burda, Phase transition in fluctuating branched geometry, Phys. Lett. B384 (1996) 75-80.

12

Bas V. de Bakker, 'Simplicial quantum gravity', PhD thesis, Amsterdam, 13 September 1995, advisor J. Smit.

Bas V. de Bakker, Jan Smit, Gravitational binding in 4D dynamical triangulation, Nucl.Phys. B484 (1997) 476-494.