Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists

1 Daß ich erkenne, was die Welt im Innersten zusammenh¨ alt. Faust Concepts without intuition are empty, intuition without concepts is blind. Immanuel Kant (1724–1804) The greatest mathematicians like Archimedes, Newton, and Gauss have always been able to combine theory and applications into one. Felix Klein (1849–1925) The present comprehensive introduction to the mathematical and physical aspects of quantum ?eld theory consists of the following six volumes: Volume I: Basics in Mathematics and Physics Volume II: Quantum Electrodynamics Volume III: Gauge Theory Volume IV: Quantum Mathematics Volume V: The Physics of the Standard Model Volume VI: Quantum Gravity and String Theory. Since ancient times, both physicists and mathematicians have tried to und- stand the forces acting in nature. Nowadays we know that there exist four fundamental forces in nature: • Newton’s gravitational force, • Maxwell’s electromagnetic force, • the strong force between elementary particles, and • the weak force between elementary particles (e.g., the force responsible for the radioactive decay of atoms). In the 20th century, physicists established two basic models, namely, • the Standard Model in cosmology based on Einstein’s theory of general relativity, and • the Standard Model in elementary particle physics based on gauge theory. 1 So that I may perceive whatever holds the world together in its inmost folds.