Quantum Field Theory

167 Pages·2017·1.71 MB·English

by  Professor Harald Fritzsch

QUANTUM
FIELD THEORY
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Q UANTUM
FI ELD TH EORY
Harald Fritzsch
Ludwig Maximilian University of Munich, Germany
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Fritzsch, Harald, 1943– author.
Title: Quantum field theory / Harald Fritzsch (Ludwig Maximilian University of Munich, Germany).
Description: Singapore ; Hackensack, NJ : World Scientific, [2017] |
Includes bibliographical references and index.
Identifiers: LCCN 2016049881| ISBN 9789813141728 (hardcover ; alk. paper) |
ISBN 9813141727 (hardcover ; alk. paper)
Subjects: LCSH: Quantum field theory.
Classification: LCC QC174.45 .F767 2017 | DDC 530.14/3--dc23
LC record available at https://lccn.loc.gov/2016049881
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Translation from the German language edition Quantenfeldtheorie — Wie man beschreibt, was die Welt
im Innersten zusammenhält by Harald Fritzsch. © Springer Verlag, Berlin Heidelberg, 2015. Springer-
Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media. All Rights Reserved.
Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Desk Editor: Ng Kah Fee
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Email: [email protected]
Printed in Singapore
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Contents
Chapter 1. Introduction 1
Chapter 2. Mechanics 9
Chapter 3. Classical Fields 19
Chapter 4. Quantum Theory 27
Chapter 5. Group Theory 37
Chapter 6. Free Scalar Fields 43
Chapter 7. Free Spinor Fields 53
Chapter 8. Free Vector Fields 63
Chapter 9. Perturbation Theory 67
Chapter 10. Quantum Electrodynamics 75
Chapter 11. Symmetry 89
Chapter 12. Gauge Theories 101
v
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vi Quantum Field Theory
Chapter 13. Quantum Chromodynamics 105
Chapter 14. Spontaneous Symmetry Breaking 121
Chapter 15. Electroweak Interactions 127
Chapter 16. Grand Unification 145
Appendix 157
Bibliography 159
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Chapter 1
Introduction
From Classical Physics to Modern Physics
The classical mechanics is valid for systems where all velocities are
much less than the velocity of light. If the velocities are close to
the velocity of light, the classical mechanics must be replaced by the
relativistic mechanics, introduced by Albert Einstein in 1905.
In the relativity theory the speed of light is a fundamental
constant:
c = 299792458m/sec.
This implies that the flow of time depends on the reference system.
Energyandmassarerelatedtoeachother—massis“frozen”energy.
For a mass at rest the corresponding energy is given by the equation
of Einstein: E = mc2.
The Lorentz transformations are transformations between the
space-time coordinates of one system of reference and another
one. Space and time are unified to the four-dimensional space-
time. The Lorentz transformations can be regarded as rotations in
the four-dimensional space-time.
Electrodynamics is another important classical field of physics,
which describes the behavior of electric charges and of electromag-
netic fields. The theory of classical electrodynamics was introduced
by James Clerk Maxwell in 1864. The basic equations of electrody-
namics are the Maxwell equations.
1
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2 Quantum Field Theory
Quantum Mechanics
Besides relativistic mechanics, quantum mechanics is another
important branch of modern physics. Quantum physics started in
1900, when Max Planck wrote a paper on the quantization of the
energy in electromagnetic processes. He introduced the new funda-
mental constant h.
In 1905 Albert Einstein proposed that the energy of light is
quantized. Light is a collection of particles, which are called photons.
The energy of a photon is given by the product of Planck’s constant
h and the frequency of the light.
In 1924 Louis de Broglie published a theory of matter waves.
A particle, e.g. an electron, is at the same time also a wave, just
like the case of a photon. The wave length is given by the ratio of
Planck’s constant and themomentum of theelectron. Thuselectrons
andphotonsaresimilar,butphotonsmoveatthespeedoflight,while
electrons can have any speed smaller than the speed of light.
Werner Heisenberg discovered in 1927 that in quantum physics
observablescannotbepreciselydetermined;theyhaveafundamental
uncertainty, which is related to Planck’s constant h. For example,
the product of the uncertainty of the position of a particle and
the uncertainty of the momentum cannot be smaller than Planck’s
constant:
∆x·∆p ∼ h.
In 1926 Erwin Schro¨dinger introduced the wave mechanics. He
interpreted the matter waves as the wave functions of particles.
The time evolution of these wave functions is described by a
differential equation, the Schr¨odinger equation. He showed that his
wave mechanics is equivalent to the matrix mechanics. Following
this school of thought, Max Born interpreted the square of the wave
function as the probability density.
In 1914 James Chadwick discovered that energy and momentum
were not conserved in the beta decay of atomic nuclei. For many
years this phenomenon was not understood. In 1930 Wolfgang
Pauli suggested that in the beta decay not only an electron was
emitted, but also a neutral particle, which could not be observed.
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Introduction 3
The energy and the momentum of this particle, later called the
neutrino, would be the observed missing energy and momentum.
Pauli assumed that neutrinos could never be observed directly. But
in 1956 they were discovered by Clyde Cowan and Frederick Reines
while investigating the neutrino emission of a big reactor in South
Carolina.
Quantum Electrodynamics
In1929WernerHeisenbergandWolfgangPauliwentonestepfurther
fromquantumphysicsandquantizedtheelectromagneticfield,which
started the development of quantum electrodynamics (QED). In this
theory the electromagnetic forces are generated by the exchange of
photons.
The Schro¨dinger equation cannot be used to describe relativis-
tic effects, since it contains the second derivatives of the three-
dimensional space, butonly the firstderivative of time. A relativistic
field equation was introduced by Paul Dirac in 1928. In the Dirac
equation only the first derivatives with respect to space and time
appear. Dirac noticed that his equation was only consistent if the
electron has an antiparticle, later called the positron. The positron
was discovered in 1932 in the cosmic rays.
In quantum electrodynamics there is a serious problem. An elec-
tron can emit virtual photons, but can also absorb them afterwards.
If one calculates this process, one obtains a divergent result — the
charge and also the mass of an electron are infinite. This problem
was solved by Julian Schwinger, Richard Feynman and Freeman
Dyson.
Anelectronwithouttheelectromagnetic interactionhasa“naked
mass” and a “naked charge.” If the electromagnetic interaction is
introduced, the contributions to the mass and to the charge are
infinite. If one assumes that the naked mass and the naked charge
are also infinite, butwith a negative sign, the sum of the naked mass
plus the corrections should be equal to the observed mass of the
electron. The same can be done for the charge. Then all infinities
have disappeared, due to the renormalization of the mass and the
charge.