Relativistic Quantum Fields W + ¯b s¯ t¯ t¯ g s s¯ Lecture notes Based on the course given by Riccardo Rattazzi First redaction, Johnny Espin, 2012 Second redaction, Riccardo Rattazzi, 2016 Contents I Foundations and Matter5 1 Introduction 6 1.1 What is quantum field theory?......................................6 1.2 Why quantum field theory?........................................6 1.3 Units of measure..............................................8 1.4 Overview of particle physics........................................9 2 Classical Field Theory 13 2.1 Lagrangian mechanics........................................... 13 2.2 From mechanics to field theory...................................... 15 2.3 Least action principle in field theory................................... 16 2.4 Hamiltonian formalism........................................... 18 3 Symmetries 19 3.1 Group theory................................................ 19 3.1.1 Groups............................................... 19 3.1.2 Schur’s lemma........................................... 23 3.1.3 Lie algebras............................................. 25 3.2 Lorentz and Poincaré groups....................................... 27 3.2.1 Construction............................................ 27 3.2.2 Raising and lowering........................................ 30 3.2.3 Lie algebra............................................. 31 3.2.4 Representations........................................... 32 3.3 Noether’s theorem............................................. 35 3.3.1 Statement.............................................. 35 3.3.2 The stress-energy tensor...................................... 39 1 4 Scalar fields 43 4.1 The Klein-Gordon field........................................... 43 4.2 Quantisation of the Klein-Gordon field.................................. 44 4.2.1 Continuum limit.......................................... 45 4.2.2 Hilbert space............................................ 48 4.3 Properties of the Klein-Gordon field................................... 49 4.3.1 Relativistic normalisation of states and measure........................ 49 4.3.2 States localised in space...................................... 51 4.3.3 Time evolution........................................... 51 4.4 The charged scalar field.......................................... 54 4.4.1 Construction............................................ 54 4.4.2 Internal current and charge.................................... 54 4.4.3 Quantisation............................................ 55 5 Spinor fields 57 5.1 Spinor representations of the Lorentz group............................... 57 5.2 Miscellaneous about spinors........................................ 58 5.2.1 Parity................................................ 58 ↑ 5.2.2 Relation between SL(2, C) and L+ ................................ 61 5.3 Covariant wave equations......................................... 64 5.3.1 Weyl spinors............................................ 64 5.3.2 Majorana spinors.......................................... 66 5.3.3 Dirac spinors............................................ 67 5.3.4 Lagrangians............................................. 68 5.4 Path towards quantisation......................................... 70 5.4.1 Classical solutions......................................... 70 5.4.2 Chirality and helicity....................................... 71 5.5 Quantisation of the Dirac field...................................... 73 5.5.1 How one should not quantise the Dirac field........................... 73 5.5.2 The right quantisation method.................................. 75 5.5.3 Hilbert space............................................ 76 5.6 Properties of the Dirac field........................................ 77 5.6.1 Time evolution........................................... 77 2 5.6.2 Internal charge........................................... 77 6 Causality in quantum field theory 78 II Forces and Interactions 79 7 Vector Fields 80 7.1 Classical Electrodynamics......................................... 80 7.2 Exercise................................................... 82 7.3 A fresh look at Maxwell equations: counting degrees of freedom.................... 83 7.4 A first step towards quantization..................................... 85 7.5 Quantization à la Gupta-Bleuler..................................... 85 7.6 Quantization of massive vector fields................................... 89 7.6.1 Massive Vector Fields....................................... 89 7.6.2 Canonical quantization...................................... 91 7.6.3 Heisenberg Picture Field...................................... 94 7.6.4 Polarization vectors........................................ 94 7.7 Polarisations of the vector field...................................... 96 7.7.1 One-particle states......................................... 97 7.7.2 Operators and fields transformations............................... 99 7.7.3 Polarisations............................................ 101 8 Discrete symmetries 103 8.1 Introduction................................................. 103 8.2 Parity.................................................... 104 8.2.1 Foreword.............................................. 104 8.2.2 Spin 0................................................ 105 8.2.3 Spin 1/2............................................... 107 8.2.4 Spin 1................................................ 109 8.3 Time reversal................................................ 110 8.3.1 Foreword.............................................. 110 8.3.2 Spin 0................................................ 111 8.3.3 Spin 1/2............................................... 112 8.3.4 Spin 1................................................ 113 3 8.4 Charge conjugation............................................. 113 8.4.1 Foreword.............................................. 113 8.4.2 Spin 0................................................ 114 8.4.3 Spin 1/2............................................... 114 8.4.4 Spin 1................................................ 115 9 Interacting Fields 116 9.1 Asymptotic States............................................. 116 9.2 The S-Matrix................................................ 119 9.2.1 Lippmann-Schwinger Equation.................................. 119 9.2.2 Symmetries of the S-Matrix.................................... 122 9.3 Phenomenology............................................... 124 9.3.1 Cross Sections........................................... 124 9.3.2 Decay Rates............................................ 124 9.4 Amplitudes in Perturbation Theory.................................... 126 9.4.1 Perturbative Expansion...................................... 126 9.4.2 Dyson Series Representation of the S-Matrix.......................... 127 9.4.3 Wick’s Theorem.......................................... 131 9.4.4 Feynman Rules and Feynman Diagrams............................. 134 A Proof of Wick’s Theorem 143 4 Part I Foundations and Matter 5 Chapter 1 Introduction Quantum field theory (QFT) is the theoretical “apparatus” that is needed to describe how nature works at its most fundamental level, that is to say at the shortest distances we have explored. In such domain physics is described by elementary particles and by their interactions, and QFT beautifully accounts for that. It also turns out that the notion of Fundamentality is here paired with the notion of Simplicity. As we shall see towards the end of this course, after having digested the necessary set of mathematical and physical concepts (≡formalism), the laws of particle physics can be written in a few lines with absolute precision and greatest empirical adequacy. In a sense particle physics is orthogonal to the sciences that deal with complexity. 1.1 What is quantum field theory? Technically, quantum field theory is the application of quantum mechanics to dynamical systems of fields, very much like basic quantum mechanics concerns the quantisation of dynamical systems of particles. Therefore, while quantum mechanics deals with mechanical systems with a finite number of degrees of freedom, quantum field theory describes the quantisation of systems with infinitely many degrees of freedom. Specifically this course is devoted to relativistic QFT. Relativistic QFTs explain the existence of particles and describe their mutual interactions. The fact that nature at its most basic level consists of particles can thus be viewed merely as a consequence of relativistic QFT. The domains of application of the latter in modern physics are quite broad: from the study of collisions among elementary particles in high energy accelerators, to early Universe cosmology. For instance, the primordial density fluctuations that later gave rise to structures like galaxies, the origin of dark matter or black-hole radiation are all described by relativistic QFT. Nevertheless there are also applications of quantum field theory to non-relativistic systems, in particular in condensed-matter physics: superfluidity, superconductivity, quantum Hall effect, ... 1.2 Why quantum field theory? We just outlined what QFT describes, but why do we need a field theory to describe particles? Is this a necessity? It basically is: relativistic QFT is the only way to reconcile ordinary quantum mechanics with special relativity. We can understand this necessity with several qualitative arguments: • Special relativity postulates the existence of a limiting speed. This implies that instantaneous