Introduction to Quantum Field Theory and Matter under Extreme Conditions Prof. Dr. David Blaschke Institute for Theoretical Physics, University of Wroclaw, Poland Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, Russia Summer Semester 2007 Abstract The series of lectures gives an introduction to the modern formulation of quan- tum field theories using Feynman path integrals. The formalism is developed first for the vacuum case and is then generalized to the conditions of finite tem- peratures, densities and strong fields with special emphasis on phase transitions, processes of particle creation in Quantum Field Theories of strong, electro-weak and gravitational interactions. Applications of the formalism are considered in the physics of condensed matter, plasma physics, ultrarelativistic heavy-ion colli- sions, high-intensity optical and X-ray lasers and the physics of compact objects such as neutron stars and black holes. Contents 1 Quantum Fields at Zero Temperature 1 1.1 Introduction . 1 1.2 Minkowski Space Conventions . 3 1.2.1 Four Vectors . 3 1.2.2 Dirac Matrices . 4 1.3 Dirac Equation . 6 1.3.1 Free Particle Solutions . 6 1.4 Green Functions . 9 1.4.1 Free-Fermion Propagator . 10 1.4.2 Green Function for the Interacting Theory . 13 1.4.3 Exercises . 14 1.5 Path Integral in Quantum Mechanics . 14 1.6 Functional Integral in Quantum Field Theory . 16 1.6.1 Scalar Field . 17 1.6.2 Lagrangian Formulation of Quantum Field Theory . 19 1.6.3 Quantum Field Theory for a Free Scalar Field . 20 1.6.4 Scalar Field with Self-Interactions . 21 1.6.5 Exercises . 22 1.7 Functional Integral for Fermions . 23 1.7.1 Finitely Many Degrees of Freedom . 23 1.7.2 Fermionic Quantum Field . 26 1.7.3 Generating Functional for Free Dirac Fields . 28 1.7.4 Exercises . 31 1.8 Functional Integral for Gauge Field Theories . 31 1.8.1 Faddeev-Popov Determinant and Ghosts . 34 1.8.2 Exercises . 42 1.9 Dyson-Schwinger Equations . 42 1.9.1 Photon Vacuum Polarization . 42 1.9.2 Fermion Self Energy . 47 1.9.3 Exercises . 49 1.10 Perturbation Theory . 50 1.10.1 Quark Self Energy . 50 1 1.10.2 Dimensional Regularization . 53 1.10.3 Regularized Quark Self Energy . 57 1.10.4 Exercises . 59 1.11 Renormalized Quark Self Energy . 60 1.11.1 Renormalized Lagrangian . 60 1.11.2 Renormalization Schemes . 63 1.11.3 Renormalized Gap Equation . 66 1.11.4 Exercises . 68 1.12 Dynamical Chiral Symmetry Breaking . 68 1.12.1 Euclidean Metric . 68 1.12.2 Chiral Symmetry . 73 1.12.3 Mass Where There Was None . 75 References . 82 2 Quantum Fields at Finite Temperature and Density 84 2.1 Ensembles and Partition Function . 84 2.1.1 Partition function in Quantum Statistics and Quantum Field Theory . 85 2.1.2 Equivalence of Path Integral and Statistical Operator rep- resentation for the Partition function . 86 2.1.3 Bosonic Fields . 90 2.1.4 Fermionic Fields . 93 2.1.5 Gauge Fields . 95 2.2 Interacting Fermion Systems: Hubbard-Stratonovich Trick . 98 2.2.1 Walecka Model . 98 2.2.2 Nambu{Jona-Lasinio (NJL) Model . 98 2.2.3 Mesonic correlations at finite temperature . 106 2.2.4 Matsubara frequency sums . 107 3 Particle Production by Strong Fields 112 3.1 Introduction . 112 3.2 Dynamics of pair creation . 113 3.2.1 Creation of fermion pairs . 113 3.2.2 Creation of boson pairs . 118 3.3 Discussion of the source term . 119 3.3.1 Properties of the source term . 119 3.3.2 Numerical results . 121 3.4 Summary . 124 4 Problem sets - statistical QFT 127 4.1 Partition function - Introduction . 127 4.2 Partition function - Fermi systems . 128 4.3 Partition function - Quark matter . 129 5 Projects 130 5.1 Symmetry breaking. Goldstone-Theorem. Higgs-Effect . 130 5.1.1 Spontaneous symmetry breaking: Complex scalar field . 130 5.1.2 Electroweak symmetry breaking: Higgs mechanism . 131 Chapter 1 Quantum Fields at Zero Temperature 1.1 Introduction This is the first part of a series of lectures whose aim is to provide the tools for the completion of a realistic calculation in quantum field theory (QFT) as it is relevant to Hadron Physics. Hadron Physics lies at the interface between nuclear and particle (high en- ergy) physics. Its focus is an elucidation of the role played by quarks and gluons in the structure of, and interactions between, hadrons. This was once parti- cle physics but that has since moved to higher energy in search of a plausible grand unified theory and extensions of the so-called Standard Model. The only high-energy physicists still focusing on hadron physics are those performing the numerical experiments necessary in the application of lattice gauge theory, and those pushing at the boundaries of applicability of perturbative QCD or trying to find new kinematic regimes of validity. There are two types of hadron: baryons and mesons: the proton and neu- tron are baryons; and the pion and kaon are mesons. Historically the names distinguished the particle classes by their mass but it is now known that there are structural differences: hadrons are bound states, and mesons and baryons are composed differently. Hadron physics is charged with the responsibility of providing a detailed understanding of the differences. To appreciate the difficulties inherent in this task it is only necessary to re- member that even the study of two-electron atoms is a computational challenge. This is in spite of the fact that one can employ the Schr¨odinger equation for this problem and, since it is not really necessary to quantize the electromagnetic field, the underlying theory has few complications. The theory underlying hadron physics is quantum chromodynamics (QCD), and its properties are such that a simple understanding and simple calculations 1 are possible only for a very small class of problems. Even on the domain for which a perturbative application of the theory is appropriate, the final (observable) states in any experiment are always hadrons, and not quarks or gluons, so that complications arise in the comparison of theory with experiment. The premier experimental facility for exploring the physics of hadrons is the Thomas Jefferson National Accelerator Facility (TJNAF), in Newport News, Vir- ginia. Important experiments are also performed at the Fermilab National Ac- celerator Facility (FermiLab), in Batavia, Illinois, and at the Deutsches Elek- tronensynchrotron (DESY) in Hamburg. These facilities use high-energy probes and/or large momentum-transfer processes to explore the transition from the nonperturbative to the perturbative domain in QCD. On the basis of the introduction to nonperturbative methods in QFT in the vacuum, we will develop in the second part of the lecture series the tools for a generalization to the situation many-particle systems of hadrons at finite tem- peratures and densities in thermodynamical equilibrium within the Matsubara formalism. The third part in the series of lectures is devoted to applications for the QCD phase transition from hadronic matter to a quark-gluon plasma (QGP) in relativistic heavy-ion collisions and in the interior of compact stars. Data are provided from a completed program of experiments at the CERN-SPS and presently running programs at RHIC Brookhaven. The future of this direction will soon open a new domain of energy densities (temperatures) at CERN-LHC (2007) and at the future GSI facility FAIR, where construction shall be completed in 2015. The CBM experiment will then allow insights into the QCD phase transition at relatively low temperatures and high baryon densities, a situation which bears already similarities with the interior of compact stars, formed in supernova explosions and observed as pulsars in isolation or in binary systems. Modern astrophysical data have an unprecendented level of accuracy allowing for new stringent constraints on the behavior of the hadronic equation of state at high densities. The final part of lectures enters the domain of nonequilibrium QFT and will focus on a particular problem which, however, plays a central role: particle pro- duction in strong time-dependent external fields. The Schwinger mechanism for pair production as a strict result of quantum electrodynamics (QED) is still not experimentally verified. We are in the fortunate situation that developments of modern laser facilities in the X-ray energy domain with intensities soon reaching the Schwinger limit for electron- positron pair creation from vacuum will allow new insights and experimental tests of approaches to nonperturbative QFT in the strong field situation. These insights will allow generalizations for the other field theories such as the Standard Model and QCD, where still the puzzles of initial conditions in heavy-ion collisions and the origin of matter in the Universe remain to be solved. 1.2 Minkowski Space Conventions In the first part of this lecture series I will use the Minkowski metrics. Later I will employ a Euclidean metric because that is most useful and appropriate for nonperturbative calculations. 1.2.1 Four Vectors Normal spacetime coordinates are denoted by a contravariant four-vector: xµ := (x0; x1; x2; x3) (t; x; y; z): (1.1) ≡ Throughout: c = 1 = h¯, and the conversion between length and energy is just: 1fm = 1=(0:197327 GeV) = 5:06773 GeV−1 : (1.2) The covariant four-vector is obtained by changing the sign of the spatial compo- nents of the contravariant vector: x := (x ; x ; x ; x ) (t; x; y; z) = g xν ; (1.3) µ 0 1 2 3 ≡ − − − µν where the metric tensor is 1 0 0 0 2 0 1 0 0 3 gµν = − : (1.4) 0 0 1 0 6 7 6 − 7 6 0 0 0 1 7 4 − 5 µ ν ν The contracted product of two four-vectors is (a; b) := gµνa b = aν b : i.e., a contracted product of a covariant and a contravariant four-vector.