Heribert Weigert

# Course Outline

### 400nW Relativistic Quantum Mechanics: Course Outline (2011)

• Contents
• Introduction
• Notation
• Matrix Lie groups
• Definitions
• Parametrizing the Lie algebras, dimensions
• Orthogonal and unitary Lie-groups and algebras, a summary
• Representations, the fundamental and adjoint representations
• Representations of a Lie group
• Representations of a Lie algebra
• Examples of representations
• Exercises on group theory (warm up problems)
• From Galilei to Einstein
• Galilei invariance in classical mechanics
• On to special relativity
• The Lorentz and Poincaré groups
• Exercises on Lorentz transformations
• Physical invariants of interest
• The Klein-Gordon field - elements of field theory
• Relativistic quantum field theory and second quantization: why and what for
• Classical field theory
• Exercises on classical field theory
• The Klein-Gordon field in Hamiltonian quantization: Schrödinger picture and particle interpretation
• The Klein-Gordon field in space-time
• Heisenberg picture and time evolution
• Microscopic causality
• Complex fields and the starting point to a particle antiparticle interpretation
• Exercises on the quantized, free Klein-Gordon theory
• The interacting Klein-Gordon field: $\phi^4$-theory
• The S-matrix and time ordering
• Exercises on propagators and interacting theories
• Path-integrals for $\phi^4$-theory
• The idea of generating functionals
• The free theory
• The interacting theory
• First steps in perturbation theory
• Feynman rules in momentum space
• Scattering amplitudes and cross sections
• The Dirac equation and fermion fields
• SL(2,C) as the universal cover of SO(1,3)
• Constructing the Dirac equation
• Dirac Lagrangian, bilinear invariants and chiral symmetries
• Free particle solutions of the Dirac equation
• Quantization by anticommutators: fermions
• The Dirac propagator
• Classical gauge theory
• Abelian and nonabelian local gauge invariance
• The classical Higgs mechanism
• Remarks on polarization states in massive and massless QED
• The Higgs-Kibble model
• Solutions to (selected) exercises