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Quantum Field Theory Notes QFT Lectures Notes Jared Kaplan Department of Physics and Astronomy, Johns Hopkins University Abstract Various QFT lecture notes to be supplemented by other course materials. Contents 1 Introduction 2 1.1 What are we studying? . .2 1.2 Useful References and Texts . .3 2 Fall Semester 4 2.1 Philosophy . .4 2.2 Review of Lagrangians and Harmonic Oscillators . .8 2.3 Balls and Springs . 11 2.4 Special Relativity and Anti-particles . 17 2.5 Canonical Quantization and Noether's Theorem . 24 2.6 Free Scalar Quantum Field Theory with Special Relativity . 27 2.7 Dimensional Analysis, or Which Interactions Are Important? . 33 2.8 Interactions in Classical Field Theory with a View Towards QFT . 36 2.9 Overview of Scattering and Perturbation Theory . 39 2.10 Relating the S-Matrix to Cross Sections and Decay Rates . 41 2.11 Old Fashioned Perturbation Theory . 46 2.12 LSZ Reduction Formula { S-Matrix from Correlation Functions . 51 2.13 Feynman Rules as a Generalization of Classical Physics . 56 2.14 The Hamiltonian Formalism for Perturbation Theory . 59 2.15 Feynman Rules from the Hamiltonian Formalism . 64 2.16 Particles with Spin 1 . 72 2.17 Covariant Derivatives and Scalar QED . 78 2.18 Scattering and Ward Identities in Scalar QED . 83 2.19 Spinors and QED . 87 2.20 Quantization and Feynman Rules for QED . 95 2.21 Overview of `Renormalization' . 100 2.22 Casimir Effect . 102 2.23 The Exact 2-pt Correlator { Kallen-Lehmann Representation . 106 2.24 Basic Loop Effects { the 2-pt Correlator . 109 2.25 General Loop Effects, Renormalization, and Interpretation . 116 2.26 Large Logarithms and Renormalization Flows . 121 2.27 QM Example of Wilsonian Renormalization . 128 2.28 Wilsonian Renormalization Flows . 130 2.29 Path Integrals . 134 2.30 Generating Function(al)s and Feynman Rules . 140 2.31 Path Integrals and Statistical Physics . 146 2.32 Soft Limits and Weinberg's Theorem . 148 1 3 Spring Semester 151 3.1 Effective Actions, Renormalization, and Symmetry . 151 3.2 Schwinger-Dyson from the PI and Ward Identities . 153 3.3 Discrete Symmetries and Spinors . 157 3.4 More on Spin and Statistics . 164 3.5 QED Vacuum Polarization and Anomalous Magnetic Moment . 169 3.6 Renormalization and QED . 176 3.7 IR Divergences and Long Wavelength Physics . 182 3.8 Implications of Unitarity . 191 3.9 Interlude on Lie Groups and Lie Algebras . 197 3.10 Overview of Lie Algebra Classification and Representations . 201 3.11 Spontaneously Broken Global Symmetries and Goldstone Bosons . 207 3.12 Non-Abelian Gauge Theories . 216 3.13 Quantization of Yang-Mills Theories . 222 3.14 Renormalization in YM and Asymptotic Freedom . 225 3.15 Higgs Mechanism . 227 3.16 Lattice Gauge Theory, QCD, and Confinement . 234 3.17 Parton Model and Deep Inelastic Scattering . 238 3.18 DIS, CFTs, and the OPE . 245 3.19 Anomalies { A Summary . 249 3.20 Anomalies as Almost Local Effects . 251 3.21 A Lecture on Cosmological Perturbation Theory . 254 1 Introduction 1.1 What are we studying? Almost everything. • The Standard Model of particle physics, which accounts for all observed phenomena on length scales larger than ∼ 10−18 meters, is a quantum field theory. In traditional QFT courses one mostly learns how to compute the probabilities for various scattering processes in the standard model. • General Relativity can also be viewed as a quantum field theory, or an ‘effective quantum field theory' describing the long-distance behavior of a massless spin 2 particle, the graviton, as it interacts with the particles from the rest of the standard model. (The division between GR and the SM is linguistic/historical, and not very sensible coneptually.) • The perturbations in the energy density in the early universe, which seeded structure formation, seem to have arisen from quantum fluctuations of a quantum field that we refer to as `the inflaton’. 2 • The behavior of systems near phase transitions, especially the universality of such systems. For example, systems of spins behave in the same way as water near its critical point { both are well-described by a simple quantum field theory, which we will study. • The universal long-distance theory of metals, as Fermi liquids of weakly interacting electrons. This accounts for the universal properties of metals as shiny materials that are good at conducting electricity and heat, and that superconduct at low temperatures. • Fluids and superfluids. Strangely enough, the latter are easier to understand than the former. • Sound waves in materials, such as metals and crystals, as we will discuss. • The quantum hall effect can be described using a very simple `topological' quantum field theory, which is like electrodynamics, but much simpler. What aren't we studying? Basically, all that's left are some systems from condensed matter physics that have a finite size, or a detailed lattice structure that's crucially important for the physics. Or systems full of dirt (e.g. systems exhibiting localization). Systems with very little symmetry usually aren't well described by QFT. 1.2 Useful References and Texts Here's a list of useful references with a brief summary of what they are good for. • Official Course Textbook: Quantum Field Theory and the Standard Model, by Matt Schwartz • Weinberg's Quantum Theory of Fields { A profound, instant classic, which you should eventually absorb as completely as possible. It's not used as a primary textbook for a first course in QFT because of its sophistication and its various ideosyncrasies. • Zee's QFT in a Nutshell { This book provides a wealth of the conceptual information about QFT and its applications; it can't be used as a text because it's not very systematic, and it doesn't teach you how to calculate anything. • Howard Georgi's Lie Algebras and Particle Physics { One major stumbling block for students learning QFT is that an understanding of Lie Groups and their Representation Theory is absolutely essential, and yet this sujbect is almost never taught. This book is the most useful way to learn Lie Theory... I've read mathematics books on the subject, but they are far less useful because they rarely compute anything. • Banks Modern Quantum Field Theory { This is an interesting alternative text, which basically forces readers to teach themselves QFT by working through a number of exercises. If you're (re-)learning QFT on your own, or want a different, more self-directed perspective, this would probably be great... doing all the exercises would insure that you really understand the subject. 3 • Srednicki's Quantum Field Theory { This seems to be a well-liked standard text based on the path integral. If you want to see everything developed from that perspective, this is probably the reference for you. • Peskin & Schroeder's Introduction to QFT { This has been a standard text for teaching for a long time, because it immediately involves students in relevant particle physics calculations... it's a great reference for computing Feynman diagrams and for some Standard Model subjects. • Weinberg's Quantum Mechanics { this is a good reference for background on Lagrangian and Hamiltonian mechanics, as applied to Quantum Mechanics and Canonical Quantization (see chapter 9). It also has a lot of background on scattering. Here are a few more advanced texts that are a natural place to go after this course, or for independent reading this year: • Sidney Coleman's Aspects of Symmetry { This is a classic text with many important and fascinating advanced topics in QFT. • Polyakov's Gauge Fields and Strings { This is basically Polyakov's notebook... deep and fascinating, touching on condensed matter and particle physics. • Wess and Bagger Supersymmetry { Classic, bare-bones text on supersymmetry. • Slava Rychkov's CFT Lectures notes and my lectures notes on AdS/CFT from fall 2013. • Donoghue, Golowich, & Holstein's Dynamics of the Standard Model { Nice book about all sorts of standard model phenomenology and processes, especially the description of low-energy QCD phenomena. • Whatever you find interesting and engaging from Inspire searches, hearsay, etc. 2 Fall Semester 2.1 Philosophy Why is Quantum Field Theory the way it is? Does it follow inevitably from a small set of more fundamental principles? These questions have been answered by two very different, profoundly compatible philosophies, which I will refer to as Wilsonian and Weinbergian. Together with the Historical approach to QFT, we have three philosophies that motivate the introduction and quantization of spacetime fields. Now a bit about each philosophy. The Historical philosophy arose from the discovery of quantum mechanics, and the subsequent realization that theories should be `quantized'. This means that for any given classical theory of physics, we should look for a quantum mechanical theory that reduces to the classical theory in a classical limit. Since Maxwell's theory of the electromagnetic field was known long before the 4 discovery of QM, physicists in the early twentieth century had the immediate task of `quantizing' it. There is a general procedure for quantizing a classical theory, called `Canonical Quantization' (Canonical just means standard; you can read about these ideas in chapters 2 and 6 of Shankar's Quantum Mechanics textbook). So one can motivate QFT by attempting to quantize classical field theories, such as the theory of the electromagnetic field. We will start with a simpler theory in this course, but the motivation remains. The Wilsonian philosophy is based on the idea of zooming out. Two different physical systems that look quite different at short distances may behave the same way at long distances, because most of the short distance details become irrelevant. In particular, we can think of our theories as an expansion in `short=L, where `short is some microscopic distance scale and L is the length scale relevant to our experiment.
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