Multidisciplinary Approach to Quantum Field Theory, Volume 1: An introduction

198 Pages·2022·7.936 MB·English

by  Michael Ogilvie

A Multidisciplinary Approach
to Quantum Field Theory,
Volume 1
An introduction
Online at: https://doi.org/10.1088/978-0-7503-3227-9

A Multidisciplinary Approach
to Quantum Field Theory,
Volume 1
An introduction
Michael Ogilvie
Department of Physics, Washington University, St. Louis, MO, USA
IOP Publishing, Bristol, UK
ªIOPPublishingLtd2022
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ISBN 978-0-7503-3227-9(ebook)
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ISBN 978-0-7503-3228-6(myPrint)
ISBN 978-0-7503-3226-2(mobi)
DOI 10.1088/978-0-7503-3227-9
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To Judy, for everything

Contents
Preface xi
Acknowledgement xiv
Author biography xv
1 Introduction to quantum field theory 1-1
1.1 Natural units 1-1
1.2 The simple harmonic oscillator in classical mechanics 1-2
1.3 The harmonic oscillator in quantum mechanics 1-3
1.4 Photons 1-6
1.5 Paths to quantum field theory 1-9
Reference 1-10
2 Quantum mechanics and path integrals 2-1
2.1 Classical mechanics and fields 2-1
2.1.1 The Lagrangian formalism 2-2
2.1.2 Functional differentiation 2-3
2.1.3 Symmetry in classical mechanics 2-5
2.1.4 The Hamiltonian formalism 2-7
2.2 Quantum mechanics 2-8
2.2.1 Time evolution in the Schrödinger picture 2-9
2.2.2 The propagator for a free nonrelativistic particle* 2-9
2.2.3 The Heisenberg representation 2-11
2.2.4 Interactions 2-12
2.3 The Feynman path integral for one degree of freedom 2-13
2.3.1 Defining the path integral 2-13
2.3.2 Matrix elements and time ordering 2-16
2.3.3 Generating functions 2-19
2.3.4 The simple harmonic oscillator 2-20
2.3.5 Wick’s theorem 2-23
2.3.6 Perturbation theory and Feynman diagrams 2-24
Problems 2-30
Bibliography 2-32
vii
AMultidisciplinaryApproachtoQuantumFieldTheory,Volume1
3 Classical fields 3-1
3.1 Wave equations in classical mechanics and quantum mechanics 3-1
3.2 Special relativity 3-2
3.2.1 Geometry of spacetime 3-4
3.2.2 Lorentz transformations of fields 3-5
3.3 The Lagrangian formalism for fields 3-6
3.3.1 The Klein–Gordon equation 3-7
3.3.2 Maxwell’s equations 3-9
3.3.3 The Schrödinger equation 3-10
3.4 Continuous symmetries in classical field theory 3-11
3.4.1 Example: translation in space and time 3-13
3.5 The Hamiltonian formalism 3-15
3.6 Causality 3-15
Problems 3-19
4 Free quantum fields 4-1
4.1 The Feynman path integral for field theories 4-1
4.2 Free scalar fields 4-3
4.3 Another approach to the functional integral 4-5
4.4 Interpretation of Z[0] for free fields 4-5
4.5 Vacuum energy examples 4-6
4.5.1 Casimir effect 4-7
4.5.2 Energy of field interacting with a static source 4-8
4.6 Fock space 4-9
4.7 Relativistic invariance and Fock space 4-13
4.8 Free quantum fields in Fock space 4-15
4.9 The canonical commutation relations and causality 4-16
4.10 Equivalence to the functional integral formalism 4-18
4.11 Continuous symmetries in quantum field theories 4-18
Problems 4-21
Further reading 4-21
5 Interacting quantum fields 5-1
5.1 Perturbation theory and Feynman diagrams 5-2
5.2 Feynman diagrams in position space 5-4
5.3 Feynman diagrams in momentum space 5-6
viii
AMultidisciplinaryApproachtoQuantumFieldTheory,Volume1
5.4 Scattering theory 5-8
5.5 A toy model of nucleons and pions 5-10
5.5.1 The NN and N¯N¯ scattering amplitude 5-11
5.5.2 The NN¯ scattering amplitude 5-13
5.5.3 Mandelstam variables and crossing symmetry 5-13
5.5.4 Four more processes: Nϕ → Nϕ, N¯ϕ → N¯ϕ, NN¯ → ϕϕ 5-16
and ϕϕ → NN¯
5.6 The CPT theorem 5-17
5.7 Cross-sections and decay rates 5-18
5.7.1 Decay rates 5-20
5.7.2 Cross-sections 5-22
Problems 5-24
Reference 5-25
6 Renormalization 6-1
6.1 Mass renormalization 6-2
6.2 Coupling constant renormalization 6-6
6.3 Field renormalization 6-13
6.4 Renormalization: a systematic process 6-14
6.5 Renormalizability 6-16
6.6 Matrix elements and the LSZ reduction formula 6-19
Problems 6-20
Bibliography 6-21
7 Symmetries and symmetry breaking 7-1
7.1 Internal symmetries 7-1
7.1.1 Introduction to spontaneous symmetry breaking 7-3
7.2 Spontaneous symmetry breaking and perturbation theory 7-7
7.3 Broken continuous symmetries and Goldstone bosons 7-9
7.3.1 Examples of Goldstone bosons 7-10
7.4 Renormalization of models with spontaneous symmetry breaking 7-12
Problems 7-14
8 Fermions 8-1
8.1 Introduction to the Dirac equation 8-1
8.2 Representations of the Lorentz group 8-3
ix