Table Of ContentMathematical Principles of Theoretical Physics
TianMa
ShouhongWang
March 30,2015
2
To our families
ii
Preface
The objectivesofthis bookare to deriveexperimentallyverifiablelawsofNature basedon
a few fundamentalmathematicalprinciples, and to providenew insights and solutionsto a
number of challenging problems of theoretical physics. This book focuses mainly on the
symbioticinterplaybetweentheoreticalphysicsandadvancedmathematics.
The greatsuccess and experimentalverificationof both Albert Einstein’s theory of rel-
ativity and quantum mechanics have placed them as cornerstonesof modern physics. The
fundamentalprinciplesofbothrelativityandquantummechanicsarethestartingpointofthe
studyundertakeninthisbook.
JamesClerkMaxwell’sdiscoveryoftheMaxwellequationsmarksthebeginningofthe
field theory and the gauge theory. The quantum electrodynamics(QED) is beautifully de-
scribedbytheU(1)abeliangaugetheory. Thenon-abelianSU(N)gaugetheorywasorigi-
natedfromtheearly workof (Weyl, 1919;Klein, 1938;YangandMills, 1954). Physically,
gauge invariance refers to the conservation of certain quantum property of the interacting
physicalsystem. Such quantumpropertyof N particles cannotbe distinguished for the in-
teraction. Consequently, the energy contribution of these N particles associated with the
interactionisinvariantunderthegeneralSU(N)phase(gauge)transformations.
Thesuccessandexperimentalsupportofthegaugetheoryindescribingtheelectromag-
netism, thestrongandtheweakinteractionsclearlydemonstratethattheprincipleofgauge
invarianceisindeedaprincipleofNature.
Wehave,atourdisposal,theprincipleofgeneralrelativity,theprincipleofLorentzinvari-
ance(specialtheoryofrelativity),andtheprincipleofgaugeinvariance.Thesearesymmetry
principles. Wecanshowthatthesesymmetryprinciples,togetherwiththesimplicityoflaws
ofNature,dictatetheactionsofthefourfundamentalinteractionsofNature: thegravity,the
electromagnetism,thestrongandtheweakinteractions.
Modern theoretical physics also faces great challenges and mysteries. For example, in
astrophysicsandcosmology,themostimportantchallengesandmysteriesinclude1)theun-
explaineddarkmatteranddarkenergyphenomena,2)theexistenceandpropertiesofblack
holes, 3)the structureandoriginof ourUniverse,and 4)the mechanismof supernovaeex-
plosionandactivegalacticnucleusjets. Inparticlesphysics,thenatureofHiggsfields,quark
confinementandtheunificationoffourfundamentalinteractionsareamongthegrandchal-
lenges.
Thebreakthroughofourworkpresentedinthisbookandinasequenceofpaperscomes
from our recent discovery of three new fundamental principles: the principle of interac-
tiondynamics(PID),theprincipleofrepresentationinvariance(PRI)(MaandWang,2014e,
iii
iv
2015a,2014h),andtheprincipleofsymmetry-breaking(PSB)(MaandWang,2014a).
Basically,PIDtakesthevariationoftheLagrangianactionundertheenergy-momentum
conservationconstraints. Forgravity,PIDisthedirectconsequenceofthepresenceofdark
energyanddarkmatter. Fortheweakinteraction,PIDistherequirementofthepresenceof
theHiggsfield. Forthestronginteraction,wedemonstratedthatPIDistheconsequenceof
thequarkconfinementphenomena.
PRI requires that the gauge theory be independentof the choices of the representation
generators. These representation generators play the same role as coordinates, and in this
sense,PRIisacoordinate-freeinvariance/covariance,reminiscentoftheEinsteinprincipleof
generalrelativity.Inotherwords,PRIispurelyalogicrequirementforthegaugetheory.
PSB offers an entirely different route of unification from the Einstein unification route
which uses large symmetry group. The three sets of symmetries — the generalrelativistic
invariance,theLorentzandgaugeinvariances,aswellas theGalileo invariance—are mu-
tually independentand dictate in partthe physicallaws in differentlevels of Nature. For a
systemcouplingdifferentlevelsofphysicallaws,partofthesesymmetriesmustbebroken.
Thesethreenewprincipleshaveprofoundphysicalconsequences,and,inparticular,pro-
videanewrouteofunificationforthefourinteractions:
1) thegeneralrelativityandthegaugesymmetriesdictatetheLagrangian;
2) the couplingof the fourinteractionsis achievedthrough PID and PRI in
theunifiedfieldequations,whichobeythePGRandPRI,butbreaksponta-
neouslythegaugesymmetry;
3) theunifiedfieldmodelcanbeeasilydecoupledtostudyindividualinterac-
tion,whentheotherinteractionsarenegligible;and
4) theunifiedfieldmodelcouplingthematterfieldsusingPSB.
Themainpartofthisbookistoestablishsuchafieldtheory,andtoprovideexplanations
and solutionsto a numberof challengingproblemsand mysteries such as those mentioned
above.
Anotherpartofthebookisonbuildingaphenomenologicalweaktonmodelofelementary
particles, based on the new field theory that we established. This model explains all the
subatomicdecaysandscattering,andgivesrisetonewinsightsonthestructureofsubatomic
particles.
Ournewfieldtheoryisbasedsolelyonafewfundamentalprinciples. Itagreeswithall
therelatedexperimentsandobservations,andsolvesmanychallengingproblemsinmodern
physics. Inthissense,itshouldreflectthetruthoftheNature!
Inretrospect,mostattemptsinmodernphysicsregardingtothechallengesandmysteries
mentioned earlier focus on modifying, on an ad hoc basis, the basic Lagrangian actions.
Even with careful tuning, these artificial modifications often lead to unsolvable difficulties
andconfusionsonmodel-buildingandontheunderstandingoftherelatedphenomena. The
PID approach we discovered is the first principle approach, required by the phenomenaas
thedarkmatter,darkenergy,quarkconfinement,andtheHiggsfield. Itleadstodualfields,
whicharenotachievablebyothermeans.Hencewebelievethatthisistheveryreasonbehind
theconfusionsandchallengesthatmodernphysicshasfacedformanydecades.
Chapter 1, General Introduction, is written for everyone. It synthesizes the main ideas
and results obtained in this book. Chapter 2 developsthe general view and basic physical
v
backgroundsneededforlaterchapters. Chapter3providestheneededmathematicalfounda-
tionsforPID,PRIandthegeometryfortheunifiedfields. Morephysicalorientedreaderscan
jumpdirectlytolaterchapters,Chapters4-7,forthemainphysicaldevelopments.
The research presented in this book was supported in part by grants from the US NSF
and the Chinese NSF. The authors are most grateful for their advice, encouragement, and
consistentsupportfromLouisNirenberg,RogerTemam,andWenyuanChenthroughoutour
scientific career. Our warm thanks to Jie Shen, Mickael Chekroun, Xianling Fan, Kevin
Zumbrun,WenMastersandRezaMalek-Madanifortheirsupport,insightfuldiscussionsand
appreciationforhiswork.
Lastbutnotleast,wewouldliketoexpressourgratitudetoourwives,LiandPing,and
ourchildren,Jiao,MelindaandWayne,fortheirloveandappreciation. Inparticular,weare
mostgratefulforLiandPing’sunflinchingsupportandunderstandingduringthelongcourse
ofouracademiccareer.
March30,2015
Chengdu,China TianMa
Bloomington,IN ShouhongWang
vi
Contents
1 GeneralIntroduction 1
1.1 ChallengesofPhysicsandGuidingPrinciple. . . . . . . . . . . . . . . . . . 1
1.2 LawofGravity,DarkMatterandDarkEnergy . . . . . . . . . . . . . . . . . 3
1.3 FirstPrinciplesofFourFundamentalInteractions . . . . . . . . . . . . . . . 6
1.4 SymmetryandSymmetry-Breaking . . . . . . . . . . . . . . . . . . . . . . 11
1.5 UnifiedFieldTheoryBasedOnPIDandPRI . . . . . . . . . . . . . . . . . 13
1.6 TheoryofStrongInteractions . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 TheoryofWeakInteractions . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8 NewTheoryofBlackHoles . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.9 TheUniverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.10 SupernovaeExplosionandAGNJets . . . . . . . . . . . . . . . . . . . . . . 25
1.11 Multi-ParticleSystemsandUnification . . . . . . . . . . . . . . . . . . . . . 26
1.12 WeaktonModelofElementaryParticles . . . . . . . . . . . . . . . . . . . . 28
2 FundamentalPrinciplesofPhysics 33
2.1 EssenceofPhysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.1 Generalguidingprinciples . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.2 Phenomenologicalmethods . . . . . . . . . . . . . . . . . . . . . . 35
2.1.3 Fundamentalprinciplesinphysics . . . . . . . . . . . . . . . . . . . 36
2.1.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.5 Invarianceandtensors . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1.6 Geometricinteractionmechanism . . . . . . . . . . . . . . . . . . . 42
2.1.7 Principleofsymmetry-breaking . . . . . . . . . . . . . . . . . . . . 44
2.2 LorentzInvariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.1 Lorentztransformation . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.2 MinkowskispaceandLorentztensors . . . . . . . . . . . . . . . . . 47
2.2.3 Relativisticinvariants . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2.4 Relativisticmechanics . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2.5 Lorentzinvarianceofelectromagnetism . . . . . . . . . . . . . . . . 54
2.2.6 Relativisticquantummechanics . . . . . . . . . . . . . . . . . . . . 56
2.2.7 Diracspinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3 Einstein’sTheoryofGeneralRelativity . . . . . . . . . . . . . . . . . . . . 60
2.3.1 Principleofgeneralrelativity. . . . . . . . . . . . . . . . . . . . . . 60
2.3.2 Principleofequivalence . . . . . . . . . . . . . . . . . . . . . . . . 61
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viii CONTENTS
2.3.3 Generaltensorsandcovariantderivatives . . . . . . . . . . . . . . . 63
2.3.4 Einstein-Hilbertaction . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.3.5 Einsteingravitationalfieldequations. . . . . . . . . . . . . . . . . . 68
2.4 GaugeInvariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.4.1 U(1)gaugeinvarianceofelectromagnetism . . . . . . . . . . . . . . 70
2.4.2 GeneratorrepresentationsofSU(N) . . . . . . . . . . . . . . . . . . 72
2.4.3 Yang-MillsactionofSU(N)gaugefields . . . . . . . . . . . . . . . 74
2.4.4 Principleofgaugeinvariance. . . . . . . . . . . . . . . . . . . . . . 78
2.5 PrincipleofLagrangianDynamics(PLD) . . . . . . . . . . . . . . . . . . . 79
2.5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.5.2 Elasticwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.5.3 Classicalelectrodynamics . . . . . . . . . . . . . . . . . . . . . . . 82
2.5.4 Lagrangianactionsinquantummechanics . . . . . . . . . . . . . . . 86
2.5.5 Symmetriesandconservationlaws . . . . . . . . . . . . . . . . . . . 89
2.6 PrincipleofHamiltonianDynamics(PHD). . . . . . . . . . . . . . . . . . . 93
2.6.1 Hamiltoniansystemsinclassicalmechanics . . . . . . . . . . . . . . 93
2.6.2 Dynamicsofconservativesystems . . . . . . . . . . . . . . . . . . . 96
2.6.3 PHDforMaxwellelectromagneticfields . . . . . . . . . . . . . . . 100
2.6.4 QuantumHamiltoniansystems . . . . . . . . . . . . . . . . . . . . . 101
3 MathematicalFoundations 107
3.1 BasicConcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.1.1 Riemannianmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.1.2 Physicalfieldsandvectorbundles . . . . . . . . . . . . . . . . . . . 113
3.1.3 Lineartransformationsonvectorbundles . . . . . . . . . . . . . . . 116
3.1.4 Connectionsandcovariantderivatives . . . . . . . . . . . . . . . . . 119
3.2 AnalysisonRiemannianManifolds. . . . . . . . . . . . . . . . . . . . . . . 123
3.2.1 Sobolevspacesoftensorfields . . . . . . . . . . . . . . . . . . . . . 123
3.2.2 Sobolevembeddingtheorem . . . . . . . . . . . . . . . . . . . . . . 126
3.2.3 Differentialoperators . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.2.4 Gaussformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.2.5 PartialDifferentialEquationsonRiemannianmanifolds . . . . . . . 133
3.3 OrthogonalDecompositionforTensorFields . . . . . . . . . . . . . . . . . 135
3.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.3.2 Orthogonaldecompositiontheorems . . . . . . . . . . . . . . . . . . 136
3.3.3 Uniquenessoforthogonaldecompositions . . . . . . . . . . . . . . . 140
3.3.4 Orthogonaldecompositiononmanifoldswithboundary. . . . . . . . 143
3.4 Variationswithdiv -FreeConstraints . . . . . . . . . . . . . . . . . . . . . 144
A
3.4.1 Classicalvariationalprinciple . . . . . . . . . . . . . . . . . . . . . 144
3.4.2 DerivativeoperatorsoftheYang-Millsfunctionals . . . . . . . . . . 146
3.4.3 DerivativeoperatoroftheEinstein-Hilbertfunctional . . . . . . . . . 147
3.4.4 Variationalprinciplewithdiv -freeconstraint . . . . . . . . . . . . . 150
A
3.4.5 Scalarpotentialtheorem . . . . . . . . . . . . . . . . . . . . . . . . 154
3.5 SU(N)RepresentationInvariance. . . . . . . . . . . . . . . . . . . . . . . . 156
3.5.1 SU(N)gaugerepresentation . . . . . . . . . . . . . . . . . . . . . . 156
3.5.2 ManifoldstructureofSU(N) . . . . . . . . . . . . . . . . . . . . . . 157
