Quantum Electrodynamics of Strong Fields: With an Introduction into Modern Relativistic Quantum Mechanics (Theoretical and Mathematical Physics)

The fundamental goal of physics is an understanding of the forces of nature in their simplest and most general terms. Yet there is much more involved than just a basic set of equations which eventually has to be solved when applied to specific problems. We have learned in recent years that the structure of the ground state of field theories (with which we are generally concerned) plays an equally funda- mental role as the equations of motion themselves. Heisenberg was probably the first to recognize that the ground state, the vacuum, could acquire certain prop- erties (quantum numbers) when he devised a theory of ferromagnetism. Since then, many more such examples are known in solid state physics, e. g. supercon- ductivity, superfluidity, in fact all problems concerned with phase transitions of many-body systems, which are often summarized under the name synergetics. Inspired by the experimental observation that also fundamental symmetries, such as parity or chiral symmetry, may be violated in nature, it has become wide- ly accepted that the same field theory may be based on different vacua. Practical- ly all these different field phases have the status of more or less hypothetical models, not (yet) directly accessible to experiments. There is one magnificent ex- ception and this is the change of the ground state (vacuum) of the electron-posi- tron field in superstrong electric fields.

1. Introduction.- 1.1 The Charged Vacuum.- 1.2 From Theory to Experimental Verification.- 1.2.1 Superheavy Quasimolecules.- 1.2.2 Nuclear Sticking.- 1.2.3 K-Shell Ionization.- 1.3 Theoretical Developments.- 1.4 Historical Annotations on the Vacuum.- 1.4.1 The Concept of Vacuum.- 1.4.2 The Vacuum in Strong Fields.- 1.5 The Vacuum in Modern Quantum Physics.- 1.5.1 Pion Condensation.- 1.5.2 Strong Gravitational Fields.- 1.5.3 Vacuum Structure of Strongly Interacting Fermions and Bosons.- Bibliographical Notes.- 2. The Wave Equation for Spin-1/2 Particles.- 2.1 The Dirac Equation.- 2.2 The Free Dirac Particle.- 2.3 Single-Particle Interpretation of Plane (Free) Dirac Waves.- 2.4 The Dirac Particle Coupled to Electromagnetic Fields — Non-Relativistic Limits and Spin of the Dirac Equation.- 2.5 Lorentz Covariance of the Dirac Equation.- 2.5.1 Formulation of Covariance (Form Invariance).- 2.5.2 Determining the ?(â) Operator for Infinitesimal Lorentz Transformations.- 2.5.3 The ?(â) Operator for Finite Lorentz Transformations.- 2.5.4 Finite, Proper Lorentz Transformations.- 2.5.5 The ? Operator for Finite Lorentz Transformations.- 2.5.6 The Four-Current Density.- 2.6 Spinor Under Space Inversion (Parity Transformation).- 2.7 Bilinear Covariants of Dirac Spinors.- 2.8 Gauge Invariant Coupling of Electromagnetic and Spinor Field.- Bibliographical Notes.- 3. Dirac Particles in External Potentials.- 3.1 A Dirac Particle in a One-Dimensional Square Well Potential.- 3.2 A Dirac Particle in a Scalar, One-Dimensional Square Well Potential.- 3.3 A Dirac Particle in a Spherical Well.- 3.4 Solutions of the Dirac Equation for a Coulomb and a Scalar 1/r Potential.- 3.4.1 Pure Scalar Potential.- 3.4.2 Pure Coulomb Potential.- 3.4.3 Coulomb and Scalar Potential of Equal Strength (?? = ??).- 3.5 Stationary Continuum Waves for a Dirac Particle in a Coulomb Potential.- Bibliographical Notes.- 4. The Hole Theory.- 4.1 The “Dirac Sea”.- 4.1.1 Historical Context.- 4.2 Charge Conjugation Symmetry.- 4.3 Charge Conjugation of States in External Potential.- 4.3.1 Historical Note.- 4.4 Parity and Time-Reversal Symmetry.- 4.4.1 Parity Invariance.- 4.4.2 Time-Reversal Symmetry.- Bibliographical Notes.- 5. The Klein Paradox.- 5.1 The Klein Paradox in the Single-Particle Interpretation of the Dirac Equation.- 5.2 Klein’s Paradox and Hole Theory.- Bibliographical Notes.- 6. Resonant States in Supercritical Fields.- 6.1 Resonances in the Negative Energy Continuum.- 6.2 One Bound State Diving into One Continuum.- 6.2.1 Filled K Shell.- 6.2.2 Empty K Shell.- 6.3 Two and More Bound States Imbedded in One Continuum.- 6.4 One Bound State Imbedded in Several Continua.- 6.5 Overcritical Continuum States.- 6.5.1 Continuum Solutions for Extended Nuclei.- 6.5.2 Comments on the Point Nucleus Problem for Z? > |?|.- 6.5.3 The Physical Phase Shifts.- 6.5.4 Resonances in the Lower Continuum for Z > Zcr.- 6.5.5 The Vacuum Charge Distribution.- 6.6 Some Useful Mathematical Relations.- 6.6.1 A Different Choice of Phases.- Bibliographical Notes.- 7. Quantum Electrodynamics of Weak Fields.- 7.1 The Non-Relativistic Propagator.- 7.2 The S Matrix.- 7.3 Propagator for Electrons and Positrons.- 7.4 Relativistic Scattering Theory.- Bibliographical Notes.- 8. The Classical Dirac Field Interacting with a Classical Electromagnetic Field — Formal Properties.- 8.1 Field Equations in Hamiltonian Form.- 8.2 Conservation Laws.- 8.3 Representation by Energy Eigenmodes.- 8.3.1 Time-Independent Potentials.- 8.3.2 Explicitly Time-Dependent Potentials.- 8.4 The Elementary Field Functions.- Bibliographical Notes.- 9. Second Quantization of the Dirac Field and Definition of the Vacuum.- 9.1 Canonical Quantization of the Dirac Field.- 9.2 Fock Space and the Vacuum State (I).- 9.3 Poincaré Invariance of the Quantum Theory.- 9.4 Gauge Invariance and Discrete Symmetries.- 9.5 The Vacuum State (II).- 9.6 The Feynman Propagator.- 9.7 Charge and Energy of the Vacuum (I).- 9.8 Charge and Energy of the Vacuum (II).- 9.9 Appendix: Feynman Propagator for Time-Dependent Fields.- Bibliographical Notes.- 10. Evolution of the Vacuum State in Supercritical Potentials.- 10.1 The In/Out Formalism.- 10.2 Evolution of the Vacuum State.- 10.3 Decay of a Supercritical K Vacancy — Projection Formalism.- 10.4 Decay of the Neutral Vacuum — Schrödinger Picture.- 10.5 The Vacuum in a Constant Electromagnetic Field.- 10.6 Quantum Electrodynamics in Strong Macroscopic Fields.- 10.7 Klein’s Paradox Revisited.- Bibliographical Notes.- 11. Superheavy Quasimolecules.- 11.1 Heavy-Ion Collisions: General Remarks.- 11.2 The Two-Centre Dirac Equation.- 11.3 The Critical Distance Rcr.- Bibliographical Notes.- 12. The Dynamics of Heavy-Ion Collisions.- 12.1 Rutherford Scattering.- 12.2 Expansion in the Quasi-Molecular Basis.- 12.3 Heavy-Ion Collisions: A Quantal Description.- 12.4 The Semiclassical Approximation.- 12.5 Collisions with Nuclear Interaction.- 12.6 Status of Numerical Calculations.- Bibliographical Notes.- 13. Experimental Test of Supercritical Fields in Heavy-Ion Collisions.- 13.1 Establishing Superheavy Quasimolecules.- 13.2 Positron Spectrometers.- 13.2.1 The “Orange”-Type ? Spectrometer.- 13.2.2 Solenoidal Transport Systems.- 13.3 Background Effects Creating Positrons.- 13.4 Positron Experiments I: Gross Features.- 13.5 Positron Experiments II: Deep Inelastic Collisions.- 13.6 Positron Experiments III: Narrow Structures in the Positron Spectrum.- 13.7 Giant Nuclear Systems and Spontaneous Positron Emission.- Bibliographical Notes.- 14. Vacuum Polarization.- 14.1 Vacuum-Current Density: Perturbative Expansion.- 14.2 Gauge Invariance and Vacuum Polarization.- 14.3 Charge Renormalization.- 14.4 Explicit Form of the Polarization Function.- 14.5 Vacuum Polarization Effects in Atoms.- Bibliographical Notes.- 15. Vacuum Polarization: Arbitrarily Strong External Potentials.- 15.1 Green’s Function for Arbitrarily Strong External Potentials.- 15.2 Vacuum Polarization Charge Density.- 15.3 Vacuum Polarization in External Fields of Arbitrary Strengths.- 15.4 Real and Virtual Vacuum Polarization.- Bibliographical Notes.- 16. Many-Body Effects in QED of Strong Fields.- 16.1 Self-Consistent Hartree-Fock Equations.- 16.2 Self-Energy Effects in Atoms.- 16.3 Self-Energy in Superheavy Atoms.- 16.4 Supercharged Vacuum.- 16.5 The Problem of a Supercritical Point Charge.- 16.5.1 Overcritical Single-Particle States.- 16.5.2 Screening Effects of the Vacuum Charge.- 16.5.3 Influence of Heavier Leptons.- 16.6 Klein’s Paradox Revisited.- Bibliographical Notes.- 17. Bosons Bound in Strong Potentials.- 17.1 The Klein-Gordon Field.- 17.2 Alternate Form of the Klein-Gordon Equation.- Bibliographical Notes.- 18. Subcritical External Potentials.- 18.1 Quantization of the Klein-Gordon Field with External Fields.- 18.2 (Quasi) Particle Representation of the Operators.- 18.3 The Fock Space and Diagonalization of the Hamiltonian.- 18.4 The Coulomb Problem for Spin-0 Particles.- Bibliographical Notes.- 19. Overcritical Potential for Bose Fields.- 19.1 The Critical Potentials.- 19.2 The True Ground State and Bose Condensation.- 19.3 Solutions of the Condensate Equations.- Bibliographical Notes.- 20. Strong Yang-Mills Fields.- 20.1 Quantum Chromodynamics.- 20.2 Gluon Condensates in Strong Colour Fields.- 20.3 The “Magnetic” Vacuum of QCD.- 20.4 Spontaneous Quark Pair Production and Fission of Quark Bags.- Bibliographical Notes.- 21. Strong Fields in General Relativity.- 21.1 Dirac Particles in a Gravitational Field.- 21.2 Limiting Charge of Black Holes.- 21.3 Uniform Acceleration and Rindler Space.- 21.4 Event Horizon and Thermal Particle Spectrum.- Bibliographical Notes.- References.