Methods of bosonic and fermionic path integrals representations: continuum random geometry in quantum field theory

351 Pages·2009·5.34 MB·english

by  Luiz C L Botelho

M B F
ETHODS OF OSONIC AND ERMIONIC
P I R :
ATH NTEGRALS EPRESENTATIONS
C R G
ONTINUUM ANDOM EOMETRY IN
Q F T
UANTUM IELD HEORY
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M B F
ETHODS OF OSONIC AND ERMIONIC
P I R :
ATH NTEGRALS EPRESENTATIONS
C R G
ONTINUUM ANDOM EOMETRY IN
Q F T
UANTUM IELD HEORY
LUIZ C.L. BOTELHO
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LibraryofCongressCataloging-in-PublicationData
Botelho,LuizC.L.
Methodsofbosonicandfermionicpathintegralsrepresentations: continuumrandomgeometryin
quantumfieldtheory/LuizC.L.Botelho.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-60741-908-2(E-Book)
1. Pathintegrals.2. Integralrepresentations.I.Title.
QC174.17.P27B6792007
530.14’3–dc22
2007042443
PublishedbyNovaSciencePublishers,Inc.>NewYork
For all those who have been struggling, resisting and finally
prevailing in “Gulags” (even “Academic” ones.)
To Nelma, Rafael and Gabriel – always at my side.

Contents
AboutThisMonograph xi
1 Loop Space Path Integrals Representations for Euclidean Quantum Fields
PathIntegralsandtheCovariantPathIntegral 1
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. TheBosonicLoopSpaceFormulationoftheO(N)-ScalarFieldTheory . . 1
1.3. AFermionicLoopSpaceforQCD . . . . . . . . . . . . . . . . . . . . . . 4
1.4. Invariant Path Integration and the Covariant Functional Measure for Ein-
steinGravitation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.2. Invariant Integration . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.3. AQuantumPathMeasureforEinsteinTheory . . . . . . . . . . . . 8
2 PathIntegralsEvaluationsinBosonicRandomLoopGeometry-
AbelianWilsonLoops 35
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2. AbelianWilsonLoopInteraction atFiniteTemperature . . . . . . . . . . . 35
2.3. The Static Confining Potential for Q.C.D. in the Mandelstam Model
through PathIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 The Triviality – Quantum Decoherence of Quantum Chromodynamics SU(∞)
inthePresenceofanExternalStrongWhite-NoiseEletromagnetic Field 59
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2. TheTriviality–Quantum Decoherence Analysis . . . . . . . . . . . . . . 60
3.3. RandomSurfaceDynamicalFactorintheAnalytical Regularization Scheme 64
3.4. TheNon-relativistic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5. TheStaticConfiningPotentialinaTensorAxionModel . . . . . . . . . . . 69
3.6. The Confining Potential on the Axion-String Model in the Axion Higher-
EnergyRegion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.7. Aλφ4 StringFieldTheoryasaDynamicsofSelfAvoidingRandomSurfaces 75
4 TheConfiningBehaviourandAsymptoticFreedomfor QCD(SU(∞))-ACon-
stantGaugeFieldPathIntegralAnalysis 87
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2. TheModelandItsConfiningBehavior . . . . . . . . . . . . . . . . . . . . 88
vii
viii Contents
4.3. ThePath-IntegralTrivialityArgumentfortheThirringModelatSU(∞) . . 94
4.4. TheLoopSpaceArgumentfortheThirringModel
Triviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Triviality - Quantum Decoherence of Fermionic Quantum Chromodynamics
SU(N ) in the Presence of an External Strong U(∞) Flavored Constant noise
c
Field 105
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2. The Triviality - Quantum Decoherence Analysis for Quantum Chromody-
namics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 Fermions on the Lattice by Means of Mandelstam-Wilson Phase Factors: A
BosonicLatticePath-IntegralFramework 115
6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2. TheFramework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7 A Connection between Fermionic Strings and Quantum Gravity States – A
LoopSpaceApproach 121
7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2. TheLoopSpaceApproachforQuantumGravity . . . . . . . . . . . . . . . 122
7.3. TheWheeler-DeWittGeometrodynamical Propagator . . . . . . . . . . . 127
7.4. Aλφ4 Geometrodynamical FieldTheoryforQuantumGravity . . . . . . . 130
8 AFermionicLoopWaveEquationforQuantumChromodynamicsatN =+∞137
c
8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.2. TheFermionicLoopWaveEquation . . . . . . . . . . . . . . . . . . . . . 137
9 StringWaveEquationsinPolyakov’sPathIntegralFramework 141
9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.2. TheWaveEquationinCovariantParticleDynamics . . . . . . . . . . . . . 141
9.3. TheWaveEquationintheCovariantBosonicStringDynamics . . . . . . . 143
9.4. AStringSolutionfortheQCD[SU(∞)]BosonicContour
AverageEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.5. TheNeveu-SchwarzStringWaveEquation . . . . . . . . . . . . . . . . . . 150
10 ARandomSurfaceMembraneWaveEquationforBosonicQ.C.D.(SU(∞)) 163
10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.2. TheRandomSurfaceWaveFunctional . . . . . . . . . . . . . . . . . . . . 163
10.3. AConnection withQ.C.D(SU(∞)) . . . . . . . . . . . . . . . . . . . . . . 166
11 CovariantFunctionalDiffusionEquationforPolyakov’sBosonicString 173
11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
11.2. TheCovariantEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
11.3. TheWheeler-DeWittEquationasaFunctional DiffusionEquation . . . . 177
Contents ix
12 CovariantPathIntegralforNambu-GotoStringTheory 183
12.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
12.2. TheNambu-GotoFullPathIntegral . . . . . . . . . . . . . . . . . . . . . 183
13 Topological Fermionic String Representation for Chern-Simons Non-Abelian
GaugeTheories 191
13.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
13.2. TheFermionicStringRepresentation . . . . . . . . . . . . . . . . . . . . . 191
14 FermionicStringRepresentation fortheThree-DimensionalIsingModel 195
14.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
14.2. TheProposedStringTheory . . . . . . . . . . . . . . . . . . . . . . . . . 196
15 APolyakovFermionicString
asaQuantumStateofEinsteinTheoryofGravitation 201
15.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
15.2. TheQuantumGravityString . . . . . . . . . . . . . . . . . . . . . . . . . 201
16 AScatteringAmplitudeintheQuantumGeometryofFermionicStrings 209
16.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
16.2. TheScatteringAmplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
17 Path-IntegralBosonization
fortheThirringModelonaRiemannSurface 217
17.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
17.2. ThePath-IntegralBosonization onaRiemannSurface . . . . . . . . . . . . 217
18 APath-IntegralApproachforBosonicEffectiveTheories
forFermionFieldsinFour
andThreeDimensions 225
18.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
18.2. TheBosonicHigh-Energy EffectiveTheory . . . . . . . . . . . . . . . . . 225
18.3. TheBosonicLow-EnergyEffectiveTheory . . . . . . . . . . . . . . . . . 228
18.4. Polyakov’s Fermi-BoseTransmutation in3DAbelian-Thirring Model . . . 231
18.5. EffectiveFour-Dimensional BosonicActions–Some
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
18.6. TheTrivialityoftheAbelian-Thirring QuantumFieldModel . . . . . . . . 239
19 Domains ofBosonic FunctionalIntegrals andSomeApplicationsto theMath-
ematicalPhysicsofPathIntegralsandStringTheory 245
19.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
19.2. TheEuclideanSchwingerGenerating Functional
asaFunctional FourierTransform . . . . . . . . . . . . . . . . . . . . . . 246
19.3. TheSupportofFunctionalMeasures-TheMinlos
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
x Contents
19.4. SomeRigorousQuantumFieldPathIntegral
intheAnalyticalRegularization Scheme . . . . . . . . . . . . . . . . . . . 256
19.5. Remarks on the Theory of Integration of Functionals on Distributional
SpacesandHilbert-Banach Spaces . . . . . . . . . . . . . . . . . . . . . . 261
20 Non-linearDiffusioninRD
andinHilbertSpaces,aPathIntegralStudy 279
20.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
20.2. TheNon-linearDiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
20.3. TheLinearDiffusionintheSpaceL2(Ω) . . . . . . . . . . . . . . . . . . . 285
21 BasicsIntegralsRepresentationsinMathematicalAnalysisofEuclideanFunc-
tionalIntegrals 295
22 SupplementaryAppendixes 307
Index 331