Quantum Field Theory I Lecture Notes ETH Zurich, 2019 HS Prof. N. Beisert © 2012{2020 Niklas Beisert. This document as well as its parts is protected by copyright. This work is licensed under the Creative Commons License \Attribution-NonCommercial-ShareAlike 4.0 International" (CC BY-NC-SA 4.0). To view a copy of this license, visit: https: //creativecommons.org/licenses/by-nc-sa/4.0/. The current version of this work can be found at: http://people.phys.ethz.ch/~nbeisert/lectures/. Contents Contents 3 Overview 5 0.1 Prerequisites . .7 0.2 Contents . .7 0.3 Disclaimer . .8 0.4 References . .9 0.5 Acknowledgements . .9 1 Classical and Quantum Mechanics 1.1 1.1 Classical Mechanics . 1.1 1.2 Hamiltonian Formulation . 1.2 1.3 Quantum Mechanics . 1.3 1.4 Quantum Mechanics and Relativity . 1.6 1.5 Conventions . 1.9 2 Classical Free Scalar Field 2.1 2.1 Spring Lattice . 2.1 2.2 Continuum Limit . 2.4 2.3 Relativistic Covariance . 2.6 2.4 Hamiltonian Field Theory . 2.8 3 Scalar Field Quantisation 3.1 3.1 Quantisation . 3.1 3.2 Fock Space . 3.4 3.3 Complex Scalar Field . 3.8 3.4 Correlators . 3.9 3.5 Sources . 3.14 4 Symmetries 4.1 4.1 Internal Symmetries . 4.1 4.2 Spacetime Symmetries . 4.5 4.3 Poincar´eSymmetry . 4.11 4.4 Poincar´eRepresentations . 4.13 4.5 Discrete Symmetries . 4.17 5 Free Spinor Field 5.1 5.1 Dirac Equation and Clifford Algebra . 5.1 5.2 Poincar´eSymmetry . 5.4 5.3 Discrete Symmetries . 5.7 5.4 Spin Statistics . 5.10 3 5.5 Graßmann Numbers . 5.13 5.6 Quantisation . 5.15 5.7 Complex and Real Fields . 5.19 5.8 Massless Field and Chiral Symmetry . 5.21 6 Free Vector Field 6.1 6.1 Classical Electrodynamics . 6.1 6.2 Gauge Fixing . 6.3 6.3 Particle States . 6.6 6.4 Casimir Energy . 6.12 6.5 Massive Vector Field . 6.16 7 Interactions 7.1 7.1 Interacting Lagrangians . 7.1 7.2 Interacting Field Operators . 7.6 7.3 Perturbation Theory . 7.9 8 Correlation Functions 8.1 8.1 Interacting Time-Ordered Correlators . 8.1 8.2 Time-Ordered Products . 8.2 8.3 Some Examples . 8.6 8.4 Feynman Rules . 8.13 8.5 Feynman Rules for QED . 8.17 9 Particle Scattering 9.1 9.1 Scattering Basics . 9.1 9.2 Cross Sections and Matrix Elements . 9.3 9.3 Electron Scattering . 9.4 9.4 Pair Production . 9.13 9.5 Loop Contributions . 9.16 10 Scattering Matrix 10.1 10.1 Asymptotic States . 10.1 10.2 S-Matrix . 10.5 10.3 Time-Ordered Correlators . 10.7 10.4 S-Matrix Reconstruction . 10.11 10.5 Unitarity . 10.15 11 Loop Corrections 11.1 11.1 Self Energy . 11.1 11.2 Loop Integral . 11.4 11.3 Regularisation and Renormalisation . 11.7 11.4 Counterterms . 11.10 11.5 Vertex Renormalisation . 11.14 Schedule of Lectures 10 4 Quantum Field Theory I Chapter 0 ETH Zurich, 2019 HS Prof. N. Beisert 12. 03. 2020 0 Overview Quantum field theory is the quantum theory of fields just like quantum mechanics describes individual quantum particles. Here, a the term “field” refers to one of the following: • A field of a classical field theory, such as electromagnetism. • A wave function of a particle in quantum mechanics. This is why QFT is sometimes called \second quantisation". • A smooth approximation to some property in a solid, e.g. the displacement of atoms in a lattice. • Some function of space and time describing some physics. Usually, excitations of the quantum field will be described by \particles". In QFT the number of these particles is not conserved, they are created and annihilated when they interact. It is precisely what we observe in elementary particle physics, hence QFT has become the mathematical framework for this discipline. This lecture series gives an introduction to the basics of quantum field theory. It describes how to quantise the basic types of fields, how to handle their quantum operators and how to treat (sufficiently weak) interactions. We will focus on relativistic models although most methods can in principle be applied to non-relativistic condensed matter systems as well. Furthermore, we discuss symmetries, infinities and running couplings. The goal of the course is a derivation of particle scattering processes in basic QFT models. This course focuses on canonical quantisation along the lines of ordinary quantum mechanics. The continuation of this lecture course, QFT II, introduces an alternative quantisation framework: the path integral.1 It is applied towards formulating the standard model of particle physics by means of non-abelian gauge theory and spontaneous symmetry breaking. What Else is QFT? There are many points of view. After attending this course, you may claim QFT is all about another 1000 ways to treat free particles and harmonic oscillators. True, these are some of the few systems we can solve exactly in theoretical physics; almost everything else requires approximation. After all, this is a physics course, not mathematics! If you look more carefully you will find that QFT is a very rich subject, you can learn about many aspects of physics, some of which have attained a mythological status: 1The path integral is much more convenient to use than canonical quantisation discussed here. However, some important basic concepts are not as obvious as in canonical quantisation, e.g. the notion of particles, scattering and, importantly, unitarity. 5 • anti-particles, anti-matter, • vacuum energy, • tachyons, • ghosts, • infinities, • mathematical (in)consistency. Infinities. How to deal with infinities? There is a famous quote due to Dirac about QED (1975): \I must say that I am very dissatisfied with the situation, because this so-called `good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small { not neglecting it just because it is infinitely great and you do not want it!" This is almost true, but QFT is neither neglecting infinities nor in an arbitrary way. Infinities are one reason why QFT is claimed to be mathematically ill-defined or even inconsistent. Yet QFT is a well-defined and consistent calculational framework to predict certain particle observables to extremely high precision. Many points of view; one is that it is our own fault: QFT is somewhat idealised; it assumes infinitely extended fields (IR) with infinite spatial resolution (UV);2 there is no wonder that the theory produces infinities. Still, it is better to stick to idealised assumptions and live with infinities than use some enormous discrete systems (actual solid state systems). There is also a physics reason why these infinities can be absorbed somehow: Our observable everyday physics should neither depend on how large the universe actually is (IR) nor on how smooth or coarse-grained space is (UV). We can in fact use infinities to learn about allowable particle interactions. This leads to curious effects: running coupling and quantum anomalies. More later, towards the end of the semester. Uniqueness. A related issue is uniqueness of the formulation. Alike QM, QFT does not have a unique or universal formulation. For instance, many meaningful things in QM/QFT are actually equivalence classes of objects. It is often more convenient or tempting to work with specific representatives of these classes. However, one has to bear in mind that only the equivalence class is meaningful, hence there are many ways to describe the same physical object. The usage of equivalence classes goes further, it is not just classes of objects. Often we have to consider classes of models rather than specific models. This is something we have to accept, something that QFT forces upon us. 2The UV and the IR are the two main sources for infinities. 6 We will notice that QFT does what it wants, not necessarily what we want. For example, we cannot expect to get what we want using bare input parameters. Different formulations of the same model naively may give different results. We must learn to adjust the input to the desired output, then we shall find agreement. We just have to make sure that there is more output than input; otherwise QFT would be a nice but meaningless exercise because of the absence of predictions. Another nice feature is that we can hide infinities in these ambiguities in a self-consistent way. Enough of Talk. Just some words of warning: We must give up some views on physics you have become used to, only then you can understand something new. For example, a classical view of the world makes understanding quantum mechanics harder. Nevertheless, one can derive classical physics as an approximation of quantum physics, once one understands the latter sufficiently well. Let us start with something concrete, we will discuss the tricky issues when they arise. Important Concepts. Some important concepts of QFT that will guide our way: • unitarity { probabilistic framework. • locality { interactions are strictly local. • causality { special relativity. • symmetries { exciting algebra and geometry. • analyticity { complex analysis. 0.1 Prerequisites Prerequisites for this course are the core courses in theoretical physics of the bachelor syllabus: • Classical mechanics (brief review in first lecture) • Quantum mechanics (brief review in first lecture) • Electrodynamics (as a simple classical field theory) • Mathematical methods in physics (HO, Fourier transforms, . ) 0.2 Contents 1. Classical and Quantum Mechanics (105 min) 2. Classical Free Scalar Field (110 min) 3.