An introduction to symmetries and quantum field theory Jean-No¨elFuchs [email protected] Updated September 10, 2017 Contents 1 Introduction 5 1.1 Requirements.............................................5 1.2 Why quantum field theory?.....................................5 1.2.1 Quantum + relativity ) QFT...............................5 1.2.2 Many-body problem in condensed matter physics.....................6 1.2.3 What is a field?........................................7 1.3 Symmetries as a leitmotiv and guiding principle..........................7 1.4 Natural units, dimensional analysis and orders of magnitude in high-energy physics......8 1.5 Logic of the course: the menu....................................9 2 Space-time symmetries and geometrical objects 11 2.1 Rotation group............................................ 11 2.1.1 O(3) and SO(3) groups................................... 11 2.1.2 Representations of SO(3).................................. 14 2.2 Lorentz group............................................. 18 2.2.1 Metric of flat spacetime................................... 18 2.2.2 Interval between two events in spacetime......................... 19 2.2.3 Isometries of spacetime: the Lorentz group........................ 19 2.2.4 Lorentz algebra........................................ 21 2.2.5 Representations of the Lorentz group............................ 24 2.3 Poincar´egroup and field representations.............................. 26 2.3.1 Spacetime translations and field representations...................... 27 2.3.2 Lorentz transformation of a scalar field.......................... 27 2.3.3 Poincar´ealgebra....................................... 28 2.3.4 Lorentz transformation of a 4-vector field......................... 28 2.3.5 Lorentz transformation of a spinor field.......................... 28 2.3.6 Summary........................................... 29 2.3.7 Representations of the Poincar´egroup on single particle states............. 29 2.4 Conclusion.............................................. 29 3 Classical fields, symmetries and conservation laws 30 3.1 Lagrangian formalism........................................ 30 3.1.1 Action and Lagrangian.................................... 30 3.1.2 Equations of motion..................................... 31 3.1.3 Conjugate field and Hamiltonian.............................. 32 3.2 Symmetries and conservation laws................................. 32 3.2.1 Symmetries of the action.................................. 32 3.2.2 Proof of Noether's theorem................................. 33 3.2.3 Explicit form of the Noether current............................ 34 3.2.4 Examples........................................... 34 1 3.3 Scalar fields and the Klein-Gordon equation............................ 37 3.3.1 Real scalar field........................................ 37 3.3.2 Complex scalar field..................................... 39 3.4 Electromagnetic field and the Maxwell equation.......................... 40 3.4.1 Covariant form of the Maxwell equations......................... 40 3.4.2 Free electromagnetic field and gauge invariance...................... 41 3.4.3 Gauge choices......................................... 43 3.4.4 Energy-momentum tensor.................................. 44 3.4.5 Coupling to matter and electric charge conservation as a consequence of gauge invariance 44 3.4.6 Gauging a symmetry..................................... 45 3.4.7 Advanced geometrical description of the electromagnetic field.............. 47 3.5 Massless spinor fields and the Weyl equations........................... 48 3.5.1 Left Weyl field, helicity and the Weyl equation...................... 48 3.5.2 Right Weyl field....................................... 49 3.6 Spinor field and the Dirac equation................................. 50 3.6.1 Parity and the Dirac Lagrangian.............................. 50 3.6.2 The Dirac equation and γ matrices............................. 50 3.6.3 Chirality operator...................................... 51 3.6.4 The Clifford algebra and different representations of the γ matrices........... 52 3.6.5 General solution of the Dirac equation: mode expansion................. 54 3.6.6 Conserved currents and charges............................... 56 3.6.7 Coupling to the electromagnetic field............................ 57 3.6.8 The αj and β matrices and the Dirac equation as the square root of the KG equation 58 3.6.9 Conclusion on the Dirac equation.............................. 59 3.6.10 Bonuses: extra material................................... 61 3.7 Conclusion on classical fields.................................... 62 4 Canonical quantization of fields 64 4.1 Real scalar fields........................................... 64 4.1.1 Quantum field........................................ 64 4.1.2 Particle interpretation and Fock space........................... 65 4.1.3 Vacuum energy and normal ordering............................ 67 4.1.4 Fock space and Bose-Einstein statistics.......................... 68 4.1.5 Summary on canonical quantization............................ 68 4.2 Complex scalar field and anti-particles............................... 69 4.3 Dirac field............................................... 70 4.3.1 Anticommutators: spinor fields are weird......................... 70 4.3.2 Fock space, Fermi-Dirac statistics and the spin-statistics relation............ 73 4.3.3 U(1) charge and anti-particles................................ 73 4.3.4 Charge conjugation...................................... 73 4.3.5 Motivation for anticommutation.............................. 74 4.3.6 Spin.............................................. 75 4.3.7 Causality and locality.................................... 75 4.4 Propagators and force as exchange of particles.......................... 75 4.4.1 What is a propagator?.................................... 75 4.4.2 Force as an exchange of virtual particles.......................... 76 4.4.3 A first glance at Feynman diagrams............................ 77 2 5 Spontaneous symmetry breaking 78 5.1 Introduction: SSB and phase transitions.............................. 78 5.2 Goldstone mechanism: SSB of a global symmetry......................... 79 5.2.1 Classical groundstate..................................... 79 5.2.2 Excitations above the groundstate............................. 80 5.2.3 What is the number of Goldstone modes?......................... 81 5.2.4 SSB of a discrete symmetry and quantum tunneling................... 82 5.3 Higgs mechanism: SSB of a gauge symmetry........................... 83 5.3.1 Superconductivity and the Meissner effect......................... 85 3 Foreword These are lecture notes for the course \Symmetries and quantum field theory" given at the master 2 \concepts fondamentaux de la physique, parcours de physique quantique" in Paris. It is work in progress and comments are most welcome. There are 12 or 13 sessions consisting of a 1.5h lecture and a 1.5h exercise session (under the supervision of Giuliano Orso, [email protected]) from September 7 to December 21, 2017. Location: between towers 55 and 56, 1st floor, room 1-02, in Universit´ePierre et Marie Curie (Jussieu, Paris) Thursday afternoon from 14:00 to 15:30 (lecture), then a 30' break and from 16:00 to 17:30 (exercise session) Recommended books: • Textbooks: The three documents that I most used to prepare these lecture notes are the following. B. Delamotte, Un soup¸conde th´eoriedes groupes: groupe des rotations et groupe de Poincar´e[1] M. Maggiore, A modern introduction to quantum field theory [2] L.H. Ryder, Quantum field theory [3] • More advanced: A. Zee, Quantum field theory in a nutshell [4] • Less advanced (popular science reading): A. Zee, Fearful symmetry: the search for beauty in modern physics [5] 4 Chapter 1 Introduction V. Weisskopf: \There are no real one-particle systems in nature, not even few-particle systems. The existence of virtual pairs and of pair fluctuations show that the days of fixed particle number are over." These notes serve as an introduction to relativistic quantum fiel theory (but they often discuss the connection to condensed matter physics as well). They start by discussing the symmetries of flat spacetime (Lorentz and Poincar´egroups) and constructing the natural geometrical objects of spacetime, such as scalars, vectors and spinors. Then classical field theory is reviewed before quantizing the non-interacting fields using canonical quantization. The notes end with a short presentation of spontaneous symmetry breaking. 1.1 Requirements Quantum mechanics including representations of the rotation group and the notion of spin Special relativity in covariant notation Analytical mechanics (Lagrangian, action, Euler-Lagrange equations, etc.) Notions on group theory and representations Electromagnetism (wave propagation, gauge invariance) 1.2 Why quantum field theory? 1.2.1 Quantum + relativity ) QFT Wedding of quantum physics and relativity imposes the language of quantum field theory (QFT). Indeed, quantum mechanics is formulated to describe a fixed number of particles. But this is only possible non- relativistically. The uncertainty relation ∆E∆t & ~ permits the violation of the energy conservation law (by an amount ∆E) for a short time (∆t) and relativity theory permits the conversion of energy into matter (E = mc2 states that the rest energy is given by the mass). Therefore the number of particles (even massive) can not be fixed: (virtual) particles may appear or disappear out of nothing (vacuum). We therefore