Classical Field Theory

474 Pages·2018·7.21 MB·English

by  Florian Scheck

Graduate Texts in Physics
Florian Scheck
Classical
Field Theory
On Electrodynamics, Non-Abelian
Gauge Theories and Gravitation
Second Edition
Graduate Texts in Physics
Series editors
Kurt H. Becker, Polytechnic School of Engineering, Brooklyn, USA
Jean-Marc Di Meglio, Université Paris Diderot, Paris, France
Sadri Hassani, Illinois State University, Normal, USA
Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan
Richard Needs, University of Cambridge, Cambridge, UK
William T. Rhodes, Georgia Institute of Technology, Boca Raton, USA
Susan Scott, Australian National University, Acton, Australia
H. Eugene Stanley, Boston University, Boston, USA
Martin Stutzmann, TU München, Garching, Germany
Andreas Wipf, Friedrich-Schiller-Universität Jena, Jena, Germany
GraduateTextsinPhysicspublishescorelearning/teachingmaterialforgraduate-and
advanced-levelundergraduatecoursesontopicsofcurrentandemergingfieldswithin
physics, both pure and applied. These textbooks serve students at the MS- or
PhD-levelandtheirinstructorsascomprehensivesourcesofprinciples,definitions,
derivations,experimentsandapplications(asrelevant)fortheirmasteryandteaching,
respectively.Internationalinscopeandrelevance,thetextbookscorrespondtocourse
syllabisufficientlytoserveasrequiredreading.Theirdidacticstyle,comprehensive-
nessandcoverageoffundamentalmaterialalsomakethemsuitableasintroductions
orreferencesforscientistsentering,orrequiringtimelyknowledgeof,aresearchfield.
More information about this series at http://www.springer.com/series/8431
Florian Scheck
Classical Field Theory
On Electrodynamics, Non-Abelian
Gauge Theories and Gravitation
Second Edition
123
Florian Scheck
Mainz
Germany
ISSN 1868-4513 ISSN 1868-4521 (electronic)
Graduate Textsin Physics
ISBN978-3-662-55577-4 ISBN978-3-662-55579-8 (eBook)
https://doi.org/10.1007/978-3-662-55579-8
LibraryofCongressControlNumber:2017950040
1stedition:©Springer-VerlagBerlinHeidelberg2012
2ndedition:©Springer-VerlagGmbHGermany2018
Therightsbelongto“©Kandelaki”.Thedesignerboughttheprintingpermission.
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringer-VerlagGmbHGermany
Theregisteredcompanyaddressis:HeidelbergerPlatz3,14197Berlin,Germany
To Dörte
Preface
Traditionally one begins a course or a textbook on electrodynamics with an
extensivediscussionofelectrostatics,ofmagnetostatics,andofstationary currents,
before turning to the full time-dependent Maxwell theory in local form. In this
book, I choose a somewhat different approach: Starting from Maxwell’s equations
in integral form, that is to say, from the phenomenological and experimentally
verifiedbasis ofelectrodynamics, thelocal equations are formulatedand discussed
with their general time and space dependence right from the start. Static or sta-
tionary situations appear as special cases for which Maxwell’s equations split into
two more or less independent groups and thus are decoupled to a certain extent.
GreatimportanceisattachedtothesymmetriesoftheMaxwellequationsand,in
particular,theircovariancewithrespecttoLorentztransformations.Anothercentral
issueisthetreatmentofelectrodynamicsintheframeworkofclassicalfieldtheoryby
meansofaLagrangedensityandHamilton’sprinciple.Generalprinciplesthatwere
developedformechanics,appearinadeeperandmoregeneralapplicationthatcan
serveasamodelandprototypeforanyclassicalfieldtheory.Thefactthatthefields
of Maxwell theory, in general, depend on space and time makes it necessary to
enlargetheframeworkoftraditionaltensoranalysisinR3toexteriorcalculusonR4.
The venerable vector and tensor analysis that was designed for three-dimensional
Euclideanspaces,doesnotsufficeandmustbegeneralizedtohigherdimensionsand
to Minkowski signature. While the exterior product is the generalization of the
vectorproductinR3,Cartan’sexteriorderivativeisthenaturalgeneralizationofthe
curlinR3 and,bythesametoken,encompassesthefamiliaroperationsofgradient
and divergence.
Among the many applications of Maxwell theory I chose some characteristic
and,Ifelt,nowadaysparticularlyrelevantexamplessuchasanextensivediscussion
ofpolarizationofelectromagneticwaves,thedescriptionofGaussianbeams(these
are analytic solutions of the Helmholtz equation in paraxial approximation), and
optics of meta-materials with negative index of refraction. Regarding other,
more traditional applications I refer to the well-known, excellent textbooks by
J.D. Jackson, by L.D. Landau and E.M. Lifshits, and others.
vii
viii Preface
As a novel feature I take up in the fifth chapter, a further direction of great
importance for present-day physics: The construction of non-Abelian gauge theo-
ries.TheseYang–Millstheoriesastheyarecalled1,areessential andindispensable
for our present understanding of the fundamental interactions of nature. Although
these theories which are at the basis of the so-called standardmodel of elementary
particle physics,lead usfarinto quantized fieldtheory,theirconstruction andtheir
essentialfeaturesareofaclassicalnature,atleastaslongasoneconsidersonlythe
radiationfields,theanaloguesoftheMaxwellfields,andclassicalscalarfields,but
leaves out fermionic matter particles. Non-Abelian gauge theories are constructed
followingtheexampleofMaxwelltheory.Theybearsomesimilaritiestothelatter,
but exhibit also significant differences from it. Even the phenomenon of sponta-
neous symmetry breaking that preserves us from the appearance of too many
masslessfields,inessence,isaclassicalmechanism.Inviewofthegreatimpactof
gaugetheoriesonourunderstandingofthefundamentalinteractions,it wouldbea
loss not to do this step which builds on Maxwell theory in a most natural manner.
Chapter 6 gives an extensive phenomenological and geometric introduction to
general relativity and, hence, rounds off the description of all fundamental inter-
actions in the framework of classical field theory. Here too, I use consistently a
moderngeometriclanguagewhich—aftersomeinvestmentindifferentialgeometry
—allows for a transparent formulation of Einstein’s equations which is better
focused to its essentials than the older tensor analysis formulated in components
only. Thepresentsecond edition wasenlargedandrevisedinseveralrespects. The
most important additions were made in Chap. 6 to which I added the derivation
oftheSchwarzschildsolutionintheframeworkofCartan’sstructureequations,the
Kruskal continuation of this solution to the interior of the Schwarzschild radius, a
discussion of black holes and of rotating solutions and, finally, a description of
gravitational waves.
Much of the material included in this book was tried out in numerous lectures
that I gave at Johannes Gutenberg University over the years. I am grateful to the
studentswhohavefollowedthesecourses,fortheirquestionsandcomments,andto
theteachingassistantswhotookgoodcareofexerciseclasses,fortheirstimulating
questions and critical comments.
IowespecialthankstoImmanuelBlochforthediscussionswehadonGaussian
beamsandthefascinatingtopicofmeta-materialswithnegativeindexofrefraction,
and for his encouragement to include these modern applications. Special thanks
also Mario Paschke for stimulating discussions on original ideas and for hints at
some older butrelevantreferences, as well asto Andrès Reyes-Lega for numerous
discussions over the years and for careful reading of major parts ofthe book. I am
alsoverygratefultoMaximilianBeckerwhoreadverycarefullyseveralpartsofthe
book, notably the new sections in Chap. 6, and made numerous suggestions for
their presentation.
1FirstideaswerepublishedbyOskarKlein,Z.Physik37(1926)895.ItisreportedthatWolfgang
Paulidevelopedthemindependentlybutdidnotpublishthem.
Preface ix
The cooperation with Springer-Verlag in Heidelberg and with the production
team of LE-TeX in Leipzig was excellent and efficient. The encouragement, the
editorial care and the friendship offered by Dr. Thorsten Schneider from
Springer-Verlag is gratefully acknowledged.
Mainz, Germany Florian Scheck
December 2017
Contents
1 Maxwell’s Equations. .... ..... .... .... .... .... .... ..... .... 1
1.1 Introduction ... .... ..... .... .... .... .... .... ..... .... 1
1.2 Gradient, Curl, and Divergence . .... .... .... .... ..... .... 2
1.3 The Integral Theorems for the Case of R3 . .... .... ..... .... 7
1.4 Maxwell’s Equations in Integral Form .... .... .... ..... .... 11
1.4.1 The Law of Induction.. .... .... .... .... ..... .... 11
1.4.2 Gauss’ Law. ..... .... .... .... .... .... ..... .... 13
1.4.3 The Law of Biot and Savart . .... .... .... ..... .... 15
1.4.4 The Lorentz Force. .... .... .... .... .... ..... .... 16
1.4.5 The Continuity Equation.... .... .... .... ..... .... 17
1.5 Maxwell’s Equations in Local Form.. .... .... .... ..... .... 21
1.5.1 Induction Law and Gauss’ Law... .... .... ..... .... 21
1.5.2 Local Form of the Law of Biot and Savart.. ..... .... 22
1.5.3 Local Equations in all Systems of Units .... ..... .... 24
1.5.4 The Question of Physical Units... .... .... ..... .... 24
1.5.5 The Equations of Electromagnetism in the
SI-System .. ..... .... .... .... .... .... ..... .... 27
1.5.6 The Gaussian System of Units ... .... .... ..... .... 29
1.6 Scalar Potentials and Vector Potentials.... .... .... ..... .... 34
1.6.1 A few Formulae from Vector Analysis . .... ..... .... 34
1.6.2 Construction of a Vector Field from its Source
and its Curl. ..... .... .... .... .... .... ..... .... 40
1.6.3 Scalar Potentials and Vector Potentials . .... ..... .... 41
1.7 Phenomenology of the Maxwell Equations. .... .... ..... .... 46
1.7.1 The Fundamental Equations and their Interpretation .... 46
1.7.2 Relation between Displacement Field and Electric
Field .. .... ..... .... .... .... .... .... ..... .... 49
1.7.3 Relation between Induction and Magnetic Fields .. .... 52
1.8 Static Electric States. ..... .... .... .... .... .... ..... .... 55
1.8.1 Poisson and Laplace Equations... .... .... ..... .... 55
1.8.2 Surface charges, Dipoles, and Dipole Layers. ..... .... 62
xi