This book describes Relativistic Quantum Mechanics, starting from the free field equations for spin-zero particles and for spin-one-half particles, leading to the Klein–Gordon equation and Dirac equations. Interactions of these particles with the electromagnetic field through minimal coupling are introduced as well as other interactions between particles. It includes the calculation of the fundamental processes of Quantum Electrodynamics by means of Feynman's propagator theory, which allows for a proper treatment of diverse scattering and particle creation processes. In addition to this, a number of special topics are discussed, such as spontaneous symmetry breaking, the global and local cases, the Higgs mechanism, axion–photon interactions using techniques borrowed from scalar QED, pair creation in a strong external electric field, the two-dimensional representation of the Klein–Gordon propagator, bound states in the Greens functions approach, and the Breit equation for bound states. Also, the photon–electron interactions are treated in the context of a symmetric treatment within electrons and photons for eg. Compton scattering, pair creation and pair annihilation. Finally, non-abelian gauge theories, the Glashow–Weinberg–Salam model, some electroweak processes, and Feynman diagrams are also discussed.
Author(s): Eduardo Guendelman, David Owen
Edition: 1
Publisher: World Scientific
Year: 2022
Language: English
Pages: 212
Tags: Relativistic Quantum Mechanics, Poincare Transformation, Klein-Gordon Equation, Dirac Equation, Photon-Elektron Interaction, Feynman Diagrams, Gauge Theories, Weak Interactions
Contents
Preface
Chapter 1. Preliminaries
Chapter 2. The Relativistic Principle
2.1 Poincaré Transformation
2.1.1 Tensors with an arbitrary of up and down indices
2.2 Relativistic Dynamics
2.2.1 Principle of least action
2.2.2 Energy and momentum
2.2.3 External electromagnetic field
Chapter 3. Klein–Gordon Equation and Its Physical Applications
3.1 Free Particles and Relativistic Schrödinger Equation
3.2 Difficulties with the Relativistic Schrödinger Equation in a Strong External Potential
3.3 The Klein–Gordon Equation, Boundary Conditions and Propagator
3.4 Complex Fields and Klein–Gordon Equation
3.4.1 Klein–Gordon field under a Lorentz and Poincare Transformation
3.5 Scalar Particles in the Presence of an External Electromagnetic Field
3.5.1 Klein paradox
3.5.2 Perturbation expansion
3.5.3 Special topic: Pair creation in a strong uniform electric field
3.5.4 Scattering in an external field
3.5.5 Coulomb scattering of a charged scalar particle
3.6 The Lagrangian Approach and Conserved Currents
3.6.1 Introducing the electromagnetic interaction with charged scalar fields in the Lagrangian approach
3.7 Introduction to Axions in a Magnetic Field
3.7.1 Action and equations of motion
3.7.2 The continuous axion–photon duality symmetry and the scalar QED analogy
3.7.3 The particle antiparticle representation of axions and photons and their splitting in an external magnetic field
3.7.4 Conclusions
3.8 Special Topic: Two-Dimensional Propagator for a Scalar Particle
3.8.1 Hydrogen-like pionic atoms
3.9 Spontaneous Symmetry Breaking
3.9.1 Spontaneous breaking of a global symmetry
3.9.2 Spontaneous symmetry breaking of a local symmetry
Chapter 4. The Dirac Equation
4.1 Projection Operators
4.1.1 Orthonormal plane waves
4.1.2 Charge conjugation symmetry
4.1.3 Dirac’s hole theory
4.2 Dirac Propagator
4.2.1 Solution of the Dirac equation from solutions of the Klein–Gordon equation
4.2.2 Dirac propagator from the Klein–Gordon propagator
4.3 CPT Symmetry
4.4 Scattering of Dirac Particles
4.4.1 Scattering of a Dirac particle by an external field
4.4.2 Coulomb scattering of an electron
4.4.3 Trace theorems of gamma matrices and spin averaged over Coulomb cross-section
4.4.4 Scattering of Dirac particles by their mutual electromagnetic interaction
4.4.5 The annihilation of electron–positron into muon and antimuon
4.5 A Dirac Particle Bound in an External Field: Applications to Atomic Physics
4.5.1 Magnetic dipole moment of the electron
4.5.2 First relativistic corrections and atomic physics
4.6 Gauge Transformation and Form of the Propagator
4.6.1 Gauge transformation of the photon propagator
4.7 Special Topic: Bound States in the Green’s Function Theory Approach
4.8 Special Topic: Breit Equation
Chapter 5. Photon–Electron Interactions
5.1 Free Photon and Electron Solutions and Their Propagators
5.2 S-matrix
5.2.1 Compton scattering: “The Klein–Nishina formula”
5.2.2 Annihilation of an electron–positron into two photons
5.2.3 Electron–positron pair creation from two photons
5.2.4 Paradoxes concerning e+e− production from photons in the Universe
5.3 Massive Vector Bosons
5.4 Electron–Electron Scattering
Chapter 6. Feynman Diagrams
6.1 Feynman Graphs for Tree Diagrams
Chapter 7. Non-abelian Gauge Theories
7.1 Non-abelian Gauge Transformations
7.1.1 Non-abelian global transformations
7.1.2 Quarks
7.1.3 Non-abelian local transformations
7.1.4 Non-abelian Higgs phenomena
Chapter 8. Weak Interactions
8.1 Extending the Glashow–Weinberg–Salam Model to the Three Families of Quarks and Leptons
8.1.1 Neutrino masses and oscillations
8.2 Strong Interactions
8.2.1 The color degree of freedom
Bibliography
