Cambridge University Press 978-1-108-49399-4 — Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information Introduction to Quantum Field Theory Quantum field theory provides a theoretical framework for understanding fields and the particles associated with them, and is the basis of particle physics and condensed matter research. This graduate-level textbook provides a comprehensive introduction to quantum field theory, giving equal emphasis to operator and path-integral formalisms. It covers modern research such as helicity spinors, BCFW construction, and generalized unitarity cuts, as well as treating advanced topics including BRST quantization, loop equations, and finite-temperature field theory. Various quantum fields are described, including scalar and fermionic fields, abelian vector fields and quantum electrodynamics (QED), and finally non-abelian vector fields and quantum chromodynamics (QCD). Applications to scattering cross-sections in QED and QCD are also described. Each chapter ends with exercises and an important concepts section, allowing students to identify the key aspects of the chapter and test their understanding. Hora¸tiu Nastase˘ is a Researcher at the Institute for Theoretical Physics at the State University of São Paulo, Brazil. To date, his career has spanned four continents. As an undergraduate he studied at the University of Bucharest and Copenhagen University. He later completed his Ph.D. at the State University of New York, Stony Brook, before moving to the Insti- tute for Advanced Study, Princeton University, New Jersey, where his collaboration with David Berenstein and Juan Maldacena defined the pp-wave correspondence. He has also held research and teaching positions at Brown University, Rhode Island and the Tokyo Institute of Technology. He has published three other books with Cambridge University Press: Introduction to the AdS/CFT Correspondence (2015), String Theory Methods for Condensed Matter Physics (2017), and Classical Field Theory (2019). © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-49399-4 — Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-49399-4 — Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information Introduction to Quantum Field Theory HORA¸TIU NASTASE˘ State University of São Paulo, Brazil With material from unpublished notes by Jan Ambjorn and Jens Lyng Petersen © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-49399-4 — Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108493994 DOI: 10.1017/9781108624992 © Cambridge University Press 2020 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2020 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Nastase, Horatiu, 1972– author. Title: Introduction to quantum field theory / Horatiu Nastase (Universidade Estadual Paulista, Sao Paulo). Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2020. | Includes bibliographical references and index. Identifiers: LCCN 2019006491| ISBN 9781108493994 (alk. paper) | ISBN 1108493998 (alk. paper) Subjects: LCSH: Quantum field theory. Classification: LCC QC174.45 .N353 2020 | DDC 530.14/3–dc23 LC record available at https://lccn.loc.gov/2019006491 ISBN 978-1-108-49399-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-49399-4 — Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information To the memory of my mother, who inspired me to become a physicist © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-49399-4 — Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-49399-4 — Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information Contents Preface page xxv Acknowledgments xxvi Introduction xxvii I Quantum Fields, General Formalism, and Tree Processes 1 1 Review of Classical Field Theory: Lagrangians, Lorentz Group and its Representations, Noether Theorem 3 1.1 What is and Why Do We Need Quantum Field Theory? 3 1.2 Classical Mechanics 6 1.3 Classical Field Theory 6 1.4 Noether Theorem 8 1.5 Fields and Lorentz Representations 9 Further Reading 11 Exercises 12 2 Quantum Mechanics: Harmonic Oscillator and Quantum Mechanics in Terms of Path Integrals 13 2.1 The Harmonic Oscillator and its Canonical Quantization 13 2.2 The Feynman Path Integral in Quantum Mechanics in Phase Space 15 2.3 Gaussian Integration 18 2.4 Path Integral in Configuration Space 19 2.5 Correlation Functions 19 Further Reading 21 Exercises 22 3 Canonical Quantization of Scalar Fields 23 3.1 Quantizing Scalar Fields: Kinematics 23 3.2 Quantizing Scalar Fields: Dynamics and Time Evolution 25 3.3 Discretization 27 3.4 Fock Space and Normal Ordering for Bosons 28 3.4.1 Fock Space 28 3.4.2 Normal Ordering 29 3.4.3 Bose–Einstein Statistics 30 Further Reading 30 Exercises 31 vii © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-49399-4 — Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information viii Contents 4 Propagators for Free Scalar Fields 32 4.1 Relativistic Invariant Canonical Quantization 32 4.2 Canonical Quantization of the Complex Scalar Field 33 4.3 Two-Point Functions and Propagators 35 4.4 Propagators: Retarded and Feynman 37 4.4.1 Klein–Gordon Propagators 37 4.4.2 Retarded Propagator 38 4.4.3 Feynman Propagator 38 Further Reading 39 Exercises 40 5 Interaction Picture and Wick Theorem for λφ4 in Operator Formalism 41 5.1 Quantum Mechanics Pictures 41 5.1.1 Schrödinger Picture (Usual) 42 5.1.2 Heisenberg Picture 42 5.1.3 Dirac (Interaction Picture) 42 5.2 Physical Scattering Set-up and Interaction Picture 43 5.2.1 λφ4 Theory 43 5.3 Evolution Operator and the Feynman Theorem 44 5.4 Wick’s Theorem 47 Further Reading 49 Exercises 49 6 Feynman Rules for λφ4 from the Operator Formalism 50 6.1 Diagrammatic Representation of Free Four-Point Function 50 6.2 Interacting Four-Point Function: First-Order Result and its Diagrammatic Representation 51 6.3 Other Contractions and Diagrams 52 6.4 x-Space Feynman Rules for λφ4 53 6.5 p-Space Feynman Rules and Vacuum Bubbles 54 6.5.1 Canceling of the Vacuum Bubbles in Numerator vs. Denominator in Feynman Theorem 55 Further Reading 57 Exercises 57 7 The Driven (Forced) Harmonic Oscillator 59 7.1 Set-up 59 7.2 Sloppy Treatment 60 7.3 Correct Treatment: Harmonic Phase Space 64 Further Reading 66 Exercises 67 8 Euclidean Formulation and Finite-Temperature Field Theory 68 8.1 Phase-Space and Configuration-Space Path Integrals and Boundary Conditions 68 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-49399-4 — Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information ix Contents 8.2 Wick Rotation to Euclidean Time and Connection with Statistical Mechanics Partition Function 70 8.3 Quantum-Mechanical Statistical Partition Function and Correlation Functions 72 8.4 Example: Driven Harmonic Oscillator 73 Further Reading 75 Exercises 75 9 The Feynman Path Integral for a Scalar Field 77 9.1 Euclidean Formulation 77 9.2 Perturbation Theory 78 9.3 Dyson Formula for Perturbation Theory 79 9.4 Solution of the Free Field Theory 80 9.5 Wick’s Theorem 81 Further Reading 82 Exercises 82 10 Wick’s Theorem for Path Integrals and Feynman Rules Part I 83 10.1 Examples 83 10.2 Wick’s Theorem: Second Form 86 10.3 Feynman Rules in x Space 87 Further Reading 89 Exercises 89 11 Feynman Rules in x Space and p Space 91 11.1 Proof of the Feynman Rules 91 11.2 Statistical Weight Factor (Symmetry Factor) 92 11.3 Feynman Rules in p Space 93 11.4 Most General Bosonic Field Theory 95 Further Reading 97 Exercises 97 12 Quantization of the Dirac Field and Fermionic Path Integral 98 12.1 The Dirac Equation 98 12.2 Weyl Spinors 100 12.3 Solutions of the Free Dirac Equation 101 12.4 Quantization of the Dirac Field 103 12.5 The Fermionic