Introduction to the Classical Theory of Particles and Fields

485 Pages·2007·3.03 MB·English

by  Boris Kosyakov

Introduction totheClassicalTheoryofParticlesandFields
Boris Kosyakov
Introduction
to the Classical Theory
of Particles and Fields
ABC
Dr.BorisPavlovichKosyakov
RussianFederalNuclearCenter
InstituteofTheoreticaland
MathematicalPhysics
MirAvenue37
Sarov
607190NizhniiNovgorodRegion
Russia
E-mail:[email protected]
LibraryofCongressControlNumber:2006932179
ISBN-10 3-540-40933-5SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-40933-5SpringerBerlinHeidelbergNewYork
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare
liableforprosecutionundertheGermanCopyrightLaw.
SpringerisapartofSpringerScience+BusinessMedia
springer.com
(cid:1)c Springer-VerlagBerlinHeidelberg2007
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
Typesetting:bytheauthorandtechbooksusingaSpringerLATEXmacropackage
Coverdesign:eStudioCalamarS.L.,F.Steinen-Broo,Pau/Girona,Spain
Printedonacid-freepaper SPIN:11801504 56/techbooks 543210
Preface
This book is intended as an introduction to gauge field theory for advanced
undergraduate and graduate students in high energy physics. The discussion
is restricted to the classical (nonquantum) theory. Furthermore, general rela-
tivity is outside the limits of this book.
Myinitialplanwastoreviewtheself-interactionprobleminclassicalgauge
theories with particular reference to the electrodynamics of point electrons
and Yang–Mills interactions of point quarks. The first impetus to summarize
currentaffairsinthisproblemcametomefromthelateProfessorAsimBarut
ataconferenceonmathematicalphysicsinMinsk,Belarus,duringthesummer
of 1994. He pointed out that developing a unified approach to self-interaction
intheclassicalcontextmighthelptoilluminatethefarmoreinvolvedquantum
versionofthisproblem.Theideaofwritingareviewofthiskindcameupagain
in my discussions with Professor Rudolf Haag during a physics conference at
the Garda Lake, in the autumn of 1998. He advised me to extend the initial
project to cover all attendant issues so that the review would meet the needs
of senior students.
Self-interactionisarealchallenge.Traditionally,studentsbecomeawareof
thisprobleminthecourseofquantumfieldtheory.Theyencounternumerous
divergencesoftheSmatrix,andrecognizethemasastimulusforunderstand-
ing the procedure of renormalization. But even after expending the time and
effort to master this procedure one may still not understand the physics of
self-interaction. For example, it is difficult to elicit from textbooks whether
the orders of divergence are characteristic of the interaction or are an arti-
fact of the perturbative method used for calculations. On the other hand, the
structure of self-interaction is explicit in solvable models. We will see that
manyclassicalgaugetheoryproblemscanbecompletely orpartlyintegrated.
In contrast, quantum field theory has defied solvability, with the exception of
two- and three-dimensional models.
The self-interacting electron was of great concern to fundamental physics
duringthe20thcentury.However,classicalaspectsofthisproblemaregradu-
allyfadingfromthecollective consciousnessoftheoreticalphysics.Textbooks
VI Preface
which cover this topic in sufficient detail are rare. Among them, the best
known is the 1965 Rohrlich’s volume. This excellent review represents the
state of the art in the mid-1960s. Since then many penetrating insights into
this subject have been gained, in particular exact retarded solutions of the
Yang–Mills–Wong theory. It is therefore timely to elaborate a unified view of
the classical self-interaction problem. The present work is a contribution to
this task.
Thebookis,ratherarbitrarily,dividedintotwoparts.Thefirstpart,which
involves Chaps. 1–5 and 7, is a coherent survey of special relativity and field
theory, notably the Maxwell–Lorentz and Yang–Mills–Wong theories. In ad-
dition, Mathematical Appendices cover the topics that are usually beyond
the standard knowledge of advanced undergraduates: Cartan’s differential
forms, Lie groups and Lie algebras, γ-matrices and Dirac spinors, the con-
formal group, Grassmannian variables, and distributions. These appendices
are meant as pragmatic reviews for a quick introduction to the subject, so
that the reader will hopefully be able to read the main text without resorting
to other sources.
The second part of this book, stretching over Chaps. 6 and 8–10, focuses
on the self-interaction problem. The conceptual basis of this study is not
entirely conventional. The discussion relies heavily on three key notions: the
rearrangement of the initial degrees of freedom resulting in the occurrence of
dressed particles, and spontaneous symmetry deformation.
We now give an outline of this book.
Chapter 1 discusses special relativity. Following the famous approach of
Minkowski, we treat it as merely the geometry of four-dimensional pseu-
doeuclideanspacetime.Section1.1offersthephysicalmotivationofthispoint
of view, and introduces Minkowski space. Mathematical aspects of special
relativity are then detailed in Sects. 1.2 through 1.6.
Chapter 2 covers the relativistic mechanics of point particles. Newton’s
secondlawisembeddedinthefour-dimensionalgeometryofMinkowskispace
to yield the dynamical law of relativistic particles. We define the electromag-
netic field through the Lorentz force law, and the Yang–Mills field through
the Wong force law. Electromagnetic field configurations are classified ac-
cording to their algebraic properties. We develop a regular method of solving
the equation of motion for a charged particle driven by a constant and uni-
form electromagnetic field. Section 2.5 reviews the Lagrangian formalism of
relativistic mechanical systems. Reparametrization invariance is studied in
Sect. 2.6. It is shown that a consistent dynamics is possible not only for mas-
sive, but also for massless particles. Section 2.7 explores the behavior of free
spinning particles. Since the rigorous two-particle problem in electrodynam-
ics is a formidable task, we pose a more tractable approximate problem, the
so-called relativistic Kepler problem. We then analyze a binary system com-
posed of a heavy magnetic monopole and a light charged particle. Collisions
and decays of relativistic particles are briefly discussed in the final section.
Preface VII
Chapter 3 gives a derivation of the equation of motion for the electro-
magnetic field, Maxwell’s equations. We show that some of the structure of
Maxwell’s equations is dictated by the geometrical features of our universe,
in particular the fact that there are three space dimensions. The residual in-
formation translates into four assumptions: locality, linearity, the extended
action-reaction principle, and lack of magnetic monopoles.
Chapter4coverssolutionstoMaxwell’sequations.Itbeginsbyconsidering
static electric and constant magnetic fields. Some general properties of solu-
tionstoMaxwell’sequationsaresummarizedinSect.4.2.Freeelectromagnetic
fieldsarethenexaminedinSect.4.3.WeusetheGreen’sfunctiontechniqueto
solve the inhomogeneous wave equation in Sect. 4.4. The Li´enard–Wiechert
field appears as the retarded solution to Maxwell’s equations with a point
source moving along an arbitrary timelike smooth world line. A method of
solving Maxwell’s equations without resort to Green’s functions is studied in
Sect. 4.7. This method will prove useful later in solving the Yang–Mills equa-
tions.WeshowthattheretardedelectromagneticfieldF generatedbyasingle
arbitrarily moving charge is invariant under local SL(2,R) transformations.
This is the same as saying there is a frame in which the Li´enard–Wiechert
field F appears as a pure Coulomb field at each observation point. The chap-
ter concludes with a discussion of the electromagnetic field due to a magnetic
monopole.
Chapter 5 covers the Lagrangian formalism of general field theories, with
emphasisonsystemsofchargedparticlesinteractingwiththeelectromagnetic
field.Muchattentionisgiventosymmetriesandtheirassociatedconservation
laws in electrodynamics. These symmetries are of utmost importance in the
theoryoffundamentalinteractions.Thereadermaywishtofamiliarizehimself
or herself with these concepts early in the study of field theory; the Maxwell–
Lorentztheoryseemstobeagoodtestingground.Anoverviewofstringsand
branes completes this chapter. This material may be useful in its own right,
and as an application of the calculus of variations to systems that combine
mechanical and field-theoretic features.
Chapter 6 treats self-interaction in electrodynamics. We begin with the
Goldstone and Higgs models to illustrate the mechanism of rearrangement
whereby the original degrees of freedom appearing in the Lagrangian are re-
arranged to give new, stable modes. We then introduce the basic concept
of radiation, and derive energy-momentum balance showing that mechanical
and electromagnetic degrees of freedom are rearranged into dressed particles
andradiation.TheLorentz–Diracequationgoverningadressedparticleisdis-
cussed in Sect. 6.4. Two alternative ways of deriving this equation are given
in Sect. 6.5.
The essentials of classical gauge theories are examined in Chap. 7. Sec-
tion 7.1 introduces the Yang–Mills–Wong theory of point particles interact-
ing with gauge fields, in close analogy with the Maxwell–Lorentz theory. We
briefly review a Lagrangian framework for the standard model describing the
VIII Preface
threefundamentalforcesmediatedbygaugefields:electromagnetic,weak,and
strong. Section 7.3 outlines gauge field dynamics on spacetime lattices.
Exact solutions to the Yang–Mills equations are the theme of Chap. 8.
It seems impossible to cover all known solutions. Many of them are omit-
ted, partly because these solutions are of doubtful value in accounting for
thesubnuclearrealmandpartlybecausetheyarecoveredelsewhere.Theem-
phasis is on exact retarded solutions to the Yang–Mills equations with the
source composed of several colored point particles (quarks) moving along ar-
bitrary timelike world lines. The existence of two classes of exact solutions
distinguished by symmetry groups is interpreted as a feature of the Yang–
Mills–Wong theory pertinent to the description of two phases of subnuclear
matter.
Chapter 9 deals with selected issues concerning self-interaction in gauge
theories. The initial degrees of freedom in the Yang–Mills–Wong theory are
shown to rearrange to give dressed quarks and Yang–Mills radiation. We ad-
dressthequestionofwhethertherenormalizationprocedureusedfortreating
the self-interaction problem is self-consistent. A plausible explanation for the
paradoxes of self-interaction in the Maxwell–Lorentz theory is suggested in
Sect. 9.3.
To comprehend electrodynamics as a whole, one should view this the-
ory from different perspectives in a wider context. For this purpose Chap.
10 generalizes the principles underlying mechanics and electrodynamics. The
discussion begins with a conceivable extension of Newtonian particles (gov-
erned by the second order equation of motion) to systems whose equations
of motion contain higher derivatives, so-called ‘rigid’ particles. Most if not
all of such systems exhibit unstable behavior when coupled to a continuum
force field. Electrodynamics in various dimensions is another line of general-
izations. Two specific examples, D +1 = 2 and D +1 = 6, are examined
in some detail. If Maxwell’s equations are preserved, then a consistent de-
scription for D+1 = 6 is attained through the use of rigid particle dynam-
ics with acceleration-dependent Lagrangians. With these observations, we re-
vise Ehrenfest’s famous question: ‘In what way does it become manifest from
the fundamental laws of physics that space has three dimensions?’ Nonlinear
versions of electrodynamics, such as the Born–Infeld theory, are analyzed in
Sect. 10.4. We modify the Maxwell–Lorentz theory by introducing a nonlocal
formfactorintheinteractionterm.Thefinalsectionoutlinesthedirect-action
approachinwhichtheinteractionsofparticlesaresuchthattheysimulatethe
electromagnetic field between them.
With rare exceptions, each section has problems to be solved. Some prob-
lemsexploreequationsthatappearinthemaintextwithoutderivation,while
other problems introduce additional ideas or techniques. The problems are
an integral part of the book. Many of them are essential for the subsequent
discussion. The reader is invited to read every problem and look for its so-
lution. When running into difficulties with a particular problem, the reader
may consult the answer or hint. References to problems are made by writing
Preface IX
thenumberofthesectioninfrontofthenumberoftheproblem,forexample,
Problem 10.4.2 for the second problem of Sect. 10.4.
Each chapter ends with Notes where some remarks and references for fur-
ther reading can be found. The reader should be warned that these Notes do
notpretendtoprovideacompleteguidetothehistoryofthesubject.These-
lection of the literature sources is a matter of the author’s personal taste and
abilities.Preferenceisgiventothemostfrequentlycitedbooks,representative
reviews, classical original articles, and papers that are useful in some sense.
Referencesarelistedbythenameoftheauthor(s)andtheyearofpublication.
I am indebted to many people with whom I have discussed the is-
sues addressed in this book. I am especially thankful to Professors Irina
Aref’eva, Vladislav Bagrov, Asim Barut, Iosif Buchbinder, Gariˇı Efimov,
Dmitriˇı Gal’tsov, Iosif Khriplovich, Vladimir Nesterenko, Lev Okun, Valeriˇı
Rubakov, and Georgiˇı Savvidi for their illuminating remarks. E-mail corre-
spondences with Professors Terry Goldman, Matej Pavˇsiˇc, Martin Rivas, and
Fritz Rohrlich were of great benefit to this project.
I thank Professor Rudolf Haag for his kind encouragement, interesting
comments, and letter of support to the International Science and Technology
Center (ISTC) with his recommendation to allot funds for writing this book.
The financial support from ISTC, under the project # 1560, during the
time the first version of this book was written, is gratefully acknowledged.
Finally,IamgratefulfortheassistanceofProfessorRichardWoodard.He
was my foreign collaborator in the ISTC project. Although the role of foreign
collaborators in project implementation is sometimes a formality, Richard
voluntarily shouldered proofreading of this rather long manuscript. He read
carefully all parts of the text, made numerous corrections of my wording,
and pointed to several mathematical mistakes. His comments, criticism, and
suggestions concerning the most difficult physical issues are of inestimable
help. In spite of his best efforts, errors minor and major, and obscurities are
solely my responsibility.
July, 2006 Boris Kosyakov
Sarov
Contents
1 Geometry of Minkowski Space............................. 1
1.1 Spacetime .............................................. 1
1.2 Affine and Metric Structures.............................. 10
1.3 Vectors, Tensors, and n-Forms ............................ 22
1.4 Lines and Surfaces....................................... 32
1.5 Poincar´e Invariance...................................... 38
1.6 World Lines ............................................ 43
Notes....................................................... 48
2 Relativistic Mechanics ..................................... 51
2.1 Dynamical Law for Relativistic Particles.................... 52
2.2 The Minkowski Force .................................... 58
2.3 Invariants of the Electromagnetic Field..................... 65
2.4 Motion of a Charged Particle in Constant
and Uniform Electromagnetic Fields ....................... 69
2.5 The Principle of Least Action. Symmetries
and Conservation Laws................................... 75
2.6 Reparametrization Invariance ............................. 90
2.7 Spinning Particle ........................................ 98
2.8 Relativistic Kepler Problem...............................104
2.9 A Charged Particle Driven by a Magnetic Monopole .........110
2.10 Collisions and Decays ....................................113
Notes.......................................................118
3 Electromagnetic Field .....................................123
3.1 Geometric Contents of Maxwell’s Equations.................124
3.2 Physical Contents of Maxwell’s Equations ..................127
3.3 Other Forms of Maxwell’s Equations.......................135
Notes.......................................................139
XII Contents
4 Solutions to Maxwell’s Equations ..........................141
4.1 Statics .................................................141
4.2 Solutions to Maxwell’s Equations: Some General Observations.152
4.3 Free Electromagnetic Field ...............................157
4.4 The Retarded Green’s Function ...........................167
4.5 Covariant Retarded Variables .............................174
4.6 Electromagnetic Field Generated by a Single Charge
Moving Along an Arbitrary Timelike World Line ............179
4.7 Another Way of Looking at Retarded Solutions .............183
4.8 Field Due to a Magnetic Monopole ........................187
Notes.......................................................191
5 Lagrangian Formalism in Electrodynamics .................195
5.1 Action Principle. Symmetries and Conservation Laws ........195
5.2 Poincar´e Invariance......................................206
5.3 Conformal Invariance ....................................216
5.4 Duality Invariance.......................................225
5.5 Gauge Invariance........................................228
5.6 Strings and Branes ......................................235
Notes.......................................................245
6 Self-Interaction in Electrodynamics........................249
6.1 Rearrangement of Degrees of Freedom......................249
6.2 Radiation ..............................................258
6.3 Energy-Momentum Balance...............................265
6.4 The Lorentz–Dirac Equation..............................274
6.5 Alternative Methods of Deriving the Equation of Motion
for a Dressed Charged Particle ............................278
Notes.......................................................283
7 Lagrangian Formalism for Gauge Theories .................285
7.1 The Yang–Mills–Wong Theory ............................285
7.2 The Standard Model.....................................294
7.3 Lattice Formulation of Gauge Theories .....................298
Notes.......................................................305
8 Solutions to the Yang–Mills Equations.....................307
8.1 The Yang–Mills Field Generated by a Single Quark..........309
8.2 Ansatz .................................................317
8.3 The Yang–Mills Field Generated by Two Quarks ............320
8.4 The Yang–Mills Field Generated by N Quarks ..............326
8.5 Stability................................................331
8.6 Vortices and Monopoles ..................................334
8.7 Two Phases of the Subnuclear Realm ......................343
Notes.......................................................348