Advanced Quantum Field Theoy

Advanced Quantum Field Theoy - Advanced Quantum Field...

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Unformatted text preview: Advanced Quantum Field Theory Lent Term 2007 Prof Hugh Osborn Lecture notes (unofficial) L A T E Xtypeset by Steffen Gielen Trinity College, Cambridge Modified on: June 29, 2007 1 These notes should be expected to contain loads of mistakes due to the typesetters incapability to understand in detail all of the material that is covered herein. Contents 1 Path Integrals 3 1.1 Standard Approach to Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 3 1.2 Path Integral in One-Particle Quantum Mechanics . . . . . . . . . . . . . . . . . 4 1.2.1 Example: The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 A Few Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Gauss ian Integrals and Extensions Over Multi-Dimensional Coordinates . . . . . 12 1.4 Non- Gauss ian Integrals and Perturbation Expansions . . . . . . . . . . . . . . . 15 2 Functional Methods in Quantum Field Theory 19 2.1 Free Scalar Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Interacting Scalar Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Feynman Rules for Interacting Scalar Field Theory . . . . . . . . . . . . . 23 2.2.2 Connected and Disconnected Graphs . . . . . . . . . . . . . . . . . . . . . 26 2.2.3 One Particle Irreducible Graphs . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Fermi onic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 Gauss ian Integrals for Grassmann Variables . . . . . . . . . . . . . . . . 33 2.3.2 The Fermi onic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.3 Path Integral Formalism for Fermi onic Fields . . . . . . . . . . . . . . . 35 2.3.4 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 Methods of Calculation and Treatment of Divergences in Feynman Ampli- tudes 38 3.1 Relation to Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Calculation of Feynman Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 Wick Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.2 Apperance of Divergences in Feynman Integrals . . . . . . . . . . . . . . 42 3.3 Regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.1 Dimensional Regularisation for One-Loop Graphs . . . . . . . . . . . . . . 46 3.3.2 Dimensional Regularisation for Two-Loop Graphs . . . . . . . . . . . . . 51 3.4 The Renormalisation Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4.1 Bare Lagrang ians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4.2 The Callan-Symanzik Equation . . . . . . . . . . . . . . . . . . . . . . 56 3.4.3 Evolution of Coupling Constants . . . . . . . . . . . . . . . . . . . . . . . 59 3.5 Effective Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 Gauge Theories...
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Advanced Quantum Field Theoy - Advanced Quantum Field...

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