Quantum field theory 1: Basics in mathematics and physics

Description

Content: Cover -- Contents -- Part I. Introduction -- Prologue -- 1. Historical Introduction -- 1.1 The Revolution of Physics -- 1.2 Quantization in a Nutshell -- 1.3 The Role of Gottingen -- 1.4 The Gottingen Tragedy -- 1.5 Highlights in the Sciences -- 1.6 The Emergence of Physical Mathematics -- a New Dimension of Mathematics -- 1.7 The Seven Millennium Prize Problems of the Clay Mathematics Institute -- 2. Phenomenology of the Standard Model for Elementary Particles -- 2.1 The System of Units -- 2.2 Waves in Physics -- 2.3 Historical Background -- 2.4 The Standard Model in Particle Physics -- 2.5 Magic Formulas -- 2.6 Quantum Numbers of Elementary Particles -- 2.7 The Fundamental Role of Symmetry in Physics -- 2.8 Symmetry Breaking -- 2.9 The Structure of Interactions in Nature -- 3. The Challenge of Different Scales in Nature -- 3.1 The Trouble with Scale Changes -- 3.2 Wilson's Renormalization Group Theory in Physics -- 3.3 Stable and Unstable Manifolds -- 3.4 A Glance at Conformal Field Theories -- Part II. Basic Techniques in Mathematics -- 4. Analyticity -- 4.1 Power Series Expansion -- 4.2 Deformation Invariance of Integrals -- 4.3 Cauchy's Integral Formula -- 4.4 Cauchy's Residue Formula and Topological Charges -- 4.5 The Winding Number -- 4.6 Gauss' Fundamental Theorem of Algebra -- 4.7 Compactification of the Complex Plane -- 4.8 Analytic Continuation and the Local-Global Principle -- 4.9 Integrals and Riemann Surfaces -- 4.10 Domains of Holomorphy -- 4.11 A Glance at Analytic S-Matrix Theory -- 4.12 Important Applications -- 5. A Glance at Topology -- 5.1 Local and Global Properties of the Universe -- 5.2 Bolzano's Existence Principle -- 5.3 Elementary Geometric Notions -- 5.4 Manifolds and Diffeomorphisms -- 5.5 Topological Spaces, Homeomorphisms, and Deformations -- 5.6 Topological Quantum Numbers -- 5.7 Quantum States -- 5.8 Perspectives -- 6. Many-Particle Systems in Mathematics and Physics -- 6.1 Partition Function in Statistical Physics -- 6.2 Euler's Partition Function -- 6.3 Discrete Laplace Transformation -- 6.4 Integral Transformations -- 6.5 The Riemann Zeta Function -- 6.6 The Casimir Effect in Quantum Field Theory and the Epstein Zeta Function -- 6.7 Appendix: The Mellin Transformation and Other Useful Analytic Techniques by Don Zagier -- 7. Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory -- 7.1 Geometrization of Physics -- 7.2 Ariadne's Thread in Quantum Field Theory -- 7.3 Linear Spaces -- 7.4 Finite-Dimensional Hilbert Spaces -- 7.5 Groups -- 7.6 Lie Algebras -- 7.7 Lie's Logarithmic Trick for Matrix Groups -- 7.8 Lie Groups -- 7.9 Basic Notions in Quantum Physics -- 7.10 Fourier Series -- 7.11 Dirac Calculus in Finite-Dimensional Hilbert Spaces -- 7.12 The Trace of a Linear Operator -- 7.13 Banach Spaces -- 7.14 Probability and Hilbert's Spectral Family of an Observable -- 7.15 Transition Probabilities, S-Matrix, and Unitary Operators ...