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Lecture Notes on Quantum Field Theory

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Lecture Notes on Quantum Field Theory Draft for Internal Circulations: v1: Spring Semester, 2012, v2: Spring Semester, 2013 v3: Spring Semester, 2014, v4: Spring Semester, 2015 Lecture Notes on Quantum Field Theory Yong Zhang1 School of Physics and Technology, Wuhan University, China (Spring 2015) Abstract These lectures notes are written for both third-year undergraduate students and first-year graduate students in the School of Physics and Technology, University Wuhan, China. They are mainly based on lecture notes of Sidney Coleman from Harvard and lecture notes of Michael Luke from Toronto and lecture notes of David Tong from Cambridge and the standard textbook of Peskin & Schroeder, so I do not claim any originality. These notes certainly have all kinds of typos or errors, so they will be updated from time to time. I do take the full responsibility for all kinds of typos or errors (besides errors in English writing), and please let me know of them. The third version of these notes are typeset by a team including: Kun Zhang2, Graduate student (Id: 2013202020002), Participant in the Spring semester, 2012 and 2014 Yu-Jiang Bi3, Graduate student (Id: 2013202020004), Participant in the Spring semester, 2012 and 2014 The fourth version of these notes are typeset by Kun Zhang, Graduate student (Id: 2013202020002), Participant in the Spring semester, 2012 and 2014-2015 1yong [email protected] 2kun [email protected] [email protected] 1 Draft for Internal Circulations: v1: Spring Semester, 2012, v2: Spring Semester, 2013 v3: Spring Semester, 2014, v4: Spring Semester, 2015 Acknowledgements I thank all participants in class including advanced undergraduate students, first-year graduate students and French students for their patience and persis- tence and for their various enlightening questions. I especially thank students who are willing to devote their precious time to the typewriting of these lecture notes in Latex. Main References to Lecture Notes * [Luke] Michael Luke (Toronto): online lecture notes on QFT (Fall, 2012); the link to Luke's homepage * [Tong] David Tong (Cambridge): online lecture notes on QFT (October 2012); the link to Tong's homepage * [Coleman] Sidney Coleman (Harvard): online lecture notes; the link to Coleman's lecture notes. the link to Coleman's teaching videos; * [PS] Michael E. Peskin and Daniel V. Schroeder: An Introduction to Quantum Field Theory, 1995 Westview Press; * [MS] Franz Mandl and Graham Shaw: Quantum Field Theory (Second Edition), 2010 John Wiley & Sons, Ltd. * [Zhou] Bang-Rong Zhou (Chinese Academy of Sciences): Quantum Field Theory (in Chinese), 2007 Higher Education Press; Main References to Homeworks * Luke's Problem Sets #1-6 or Tong's Problem Sets #1-4. Research Projects * See Yong Zhang's English and Chinese homepages. 2 To our parents and our teachers! To be the best researcher is to be the best person first of all: Respect and listen to our parents and our teachers always! 3 Quantum Field Theory (I) focuses on Feynman diagrams and basic concepts. The aim of this course is to study the simulation of both quantum field theory and quantum gravity on quantum computer. 4 Contents I Quantum Field Theory (I): Basic Concepts and Feynman Diagrams 11 1 Overview of Quantum Field Theory 12 1.1 Definition of quantum field theory . 12 1.2 Introduction to particle physics and Feynman diagrams . 13 1.2.1 Feynman diagrams in the pseudo-nucleon meson interaction . 13 1.2.2 Feynman diagrams in the Yukawa interaction . 18 1.2.3 Feynman diagrams in quantum electrodynamics . 20 1.3 The canonical quantization procedure . 22 1.4 Research projects in this course . 22 2 From Classical Mechanics to Quantum Mechanics 24 2.1 Classical particle mechanics . 24 2.1.1 Lagrangian formulation of classical particle mechanics . 24 2.1.2 Hamiltonian formulation of classical particle mechanics . 25 2.1.3 Noether's theorem: symmetries and conservation laws . 25 2.1.4 Exactly solved problems . 27 2.1.5 Perturbation theory . 28 2.1.6 Scattering theory . 28 2.1.7 Why talk about classical particle mechanics in detail . 29 2.2 Advanced quantum mechanics . 29 2.2.1 The canonical quantization procedure . 29 2.2.2 Fundamental principles of quantum mechanics . 30 2.2.3 The Schr¨oedinger,Heisenberg and Dirac picture . 31 2.2.4 Non-relativistic quantum many-body mechanics . 33 2.2.5 Compatible observable . 34 2.2.6 The Heisenberg uncertainty principle . 34 2.2.7 Symmetries and conservation laws . 35 2.2.8 Exactly solved problems in quantum mechanics . 35 2.2.9 Simple harmonic oscillator . 36 2.2.10 Time-independent perturbation theory . 37 2.2.11 Time-dependent perturbation theory . 37 2.2.12 Angular momentum and spin . 39 2.2.13 Identical particles . 39 2.2.14 Scattering theory in quantum mechanics . 40 5 3 Classical Field Theory 41 3.1 Special relativity . 41 3.1.1 Lorentz transformation . 41 3.1.2 Four-vector calculation . 42 3.1.3 Causality . 43 3.1.4 Mass-energy relation . 44 3.1.5 Lorentz group . 45 3.1.6 Classification of particles . 46 3.2 Classical field theory . 46 3.2.1 Concepts of fields . 46 3.2.2 Electromagnetic fields . 47 3.2.3 Lagrangian, Hamiltonian and the action principle . 48 3.2.4 A simple string theory . 49 3.2.5 Real scalar field theory . 50 3.2.6 Complex scalar field theory . 51 3.2.7 Multi-component scalar field theory . 52 3.2.8 Electrodynamics . 53 3.2.9 General relativity . 54 4 Symmetries and Conservation Laws 56 4.1 Symmetries play a crucial role in field theories . 56 4.2 Noether's theorem in field theory . 56 4.3 Space-time symmetries and conservation laws . 58 4.3.1 Space-time translation invariance and energy-momentum tensor . 58 4.3.2 Lorentz transformation invariance and angular-momentum tensor . 60 4.4 Internal symmetries and conservation laws . 64 4.4.1 Definitions . 64 4.4.2 Noether's theorem . 65 4.4.3 SO(2) invariant real scalar field theory . 65 4.4.4 U(1) invariant complex scalar field theory . 67 4.4.5 Non-Abelian internal symmetries . 68 4.5 Discrete symmetries . 69 4.5.1 Parity . 69 4.5.2 Time reversal . 70 4.5.3 Charge conjugation . 70 5 Constructing Quantum Scalar Field Theory 71 5.1 Quantum mechanics and special relativity . 71 5.1.1 Natural units . 71 5.1.2 Particle number unfixed at high energy (Special relativity) . 72 5.1.3 No position operator at short distance (Quantum mechanics) . 73 5.1.4 Micro-Causality and algebra of local observables . 74 5.1.5 A naive relativistic single particle quantum mechanics . 75 5.2 Comparisons of quantum mechanics with quantum field theory . 76 5.3 Definition of Fock space . 76 5.4 Rotation invariant Fock space . 77 5.4.1 Fock space . 77 5.4.2 Occupation number representation (ONR) . 79 5.4.3 Hints from simple harmonic oscillator (SHO) . 80 6 5.4.4 The operator formalism of Fock space in a cubic box . 80 5.4.5 Drop the box normalization and take the continuum limit . 81 5.5 Lorentz invariant Fock space . 81 5.5.1 Lorentz group . 81 5.5.2 Definition of Lorentz invariant normalized states . 82 5.5.3 Lorentz invariant normalized state . 82 5.6 Constructing quantum scalar field theory . 84 5.6.1 Notation . 84 5.6.2 Constructing quantum scalar field . 85 5.7 Canonical quantization of classical field theory . 87 5.7.1 Classical field theory . 87 5.7.2 Canonical quantization . 89 5.7.3 Vacuum energy, Casimir force and cosmological constant problem . 91 5.7.4 Normal ordered product . 93 5.8 Symmetries and conservation laws . 94 5.8.1 SO(2) invariant quantum real scalar field theory . 94 5.8.2 U(1) invariant quantum complex scalar field theory . 95 5.8.3 Discrete symmetries . 96 5.8.3.1 Parity . 96 5.8.3.2 Time reversal . 96 5.8.3.3 Charge conjugation . 97 5.8.3.4 The CPT theorem . 98 5.9 Micro-causality (Locality) . 98 5.9.1 Real scalar field theory . 98 5.9.2 Complex scalar field theory . 101 5.10 Remarks on canonical quantization . 101 6 Feynman Propagator, Wick's Theorem and Dyson's Formula 103 6.1 Retarded Green function . 103 6.1.1 Real scalar field theory with a classical source . 103 6.1.2 The retarded Green function . 104 6.1.3 Analytic formulation of D (x y)................... 105 R − 6.1.4 Canonical quantization of a real scalar field theory with a classical source . 107 6.2 Advanced Green function . 108 6.3 The Feynman Propagator D (x y)...................... 109 F − 6.3.1 Definition of Feynman propagator with time-ordered product . 109 6.3.2 Definition of Feynman propagator with contractions . 110 6.3.3 Definition with the Green function . 111 6.3.4 Definition with contour integrals . 111 6.3.5 Definition of Feynman propagator with i" 0+ prescription . 113 ! 6.4 Wick's theorem . 113 6.4.1 The Time-ordered product . 113 6.4.2 Example for Wick's theorem n = 3 and n =4............. 114 6.4.3 The Wick theorem . 115 6.4.4 Remarks on Wick's theorem . ..
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