Lectures on Matrix Field Theory

358 Pages·2017·1.784 MB·English

by  Badis Ydri

Lecture Notes in Physics 929
Badis Ydri
Lectures on
Matrix Field
Theory
Lecture Notes in Physics
Volume 929
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Badis Ydri
Lectures on Matrix Field
Theory
123
BadisYdri
InstituteofPhysics
BMAnnabaUniversity
Annaba,Algeria
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-319-46002-4 ISBN978-3-319-46003-1 (eBook)
DOI10.1007/978-3-319-46003-1
LibraryofCongressControlNumber:2016959278
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To myfatherforhiscontinuoussupport
throughouthislife...
SaadYdri
1943–2015
Alsoto my...
Nour
Preface
The subject of matrix field theory involves matrix models, noncommutative
geometry,fuzzyphysicsandnoncommutativefieldtheory,andtheirinterplay.
These lecture notes contain a systematic construction of matrix models of
quantum field theories with noncommutative and fuzzy geometries. Emphasis is
placed on the matrix formulation of noncommutative and fuzzy spaces and on
thenonperturbativetreatmentofthecorrespondingfieldtheories.Inparticular,the
phase structure of noncommutative phi-four theory is treated in great detail, and
an introduction to noncommutative gauge theory is given. The text has evolved
partly from my own personalnotes on the subject and partly from lectures given,
intermittently, to my doctoral students during the past few years. Thus, the list
of topics, while not necessarily representing the exact state of the art, reflects the
researchinterestsoftheauthorandtheeducationalgoalsofAnnabaUniversity.
The references included are not meant to be comprehensiveor exhaustive, but
theywillprovideasolidbibliographyandareliableguidetobackgroundreading.
Small parts of these lectures have already appeared in various preprintson the
arXiv.ReferencetothisSpringerpublicationismadethere.
The bookisprimarilywrittenasa self-studyguideforpostgraduatestudents—
with the aim of pedagogically introducing them to key analytical and numerical
toolsaswellasusefulphysicalmodelsinapplications.Lastbutnotleast,Idedicate
thisworktomyfatherSaadYdri,1943–2015,forhiscontinuoussupport.
Annaba,Algeria BadisYdri
July2016
vii
Contents
1 IntroductoryRemarks ...................................................... 1
1.1 Noncommutativity,FuzzinessandMatrices........................... 1
1.2 NoncommutativityinQuantumMechanics ........................... 3
1.3 MatrixYang-MillsTheories............................................ 5
1.4 NoncommutativeScalarFieldTheory................................. 8
References..................................................................... 12
2 TheNoncommutativeMoyal-WeylSpacesRd ............................ 19
(cid:2)
2.1 HeisenbergAlgebraandCoherentStates.............................. 19
2.1.1 RepresentationsoftheHeisenberg-WeylAlgebra............ 19
2.1.2 CoherentStates................................................. 21
2.1.3 Symbols......................................................... 25
2.1.4 WeylSymbolandStarProducts............................... 30
2.2 NoncommutativityfromaStrongMagneticField .................... 36
2.3 NoncommutativeMoyal-WeylSpacetimes............................ 41
2.3.1 Algebra,WeylMap,DerivationandIntegral/Trace........... 41
2.3.2 StarProductandScalarAction................................ 46
2.3.3 TheLangmann-Szabo-ZaremboModels...................... 47
2.3.4 DualityTransformationsandMatrixRegularization ......... 52
2.3.5 TheGrosse-WulkenhaarModel ............................... 55
2.3.6 ASphereBasis ................................................. 58
2.4 OtherSpaces............................................................ 59
2.4.1 TheNoncommutative/FuzzyTorus............................ 59
2.4.2 TheFuzzyDiscofLizzi-Vitale-Zampini...................... 68
References..................................................................... 70
3 TheFuzzySphere............................................................ 73
3.1 QuantizationofS2...................................................... 73
3.1.1 TheAlgebraC1.S2/andtheCoadjointOrbit
SU.2/=U.1/ .................................................... 73
3.1.2 TheSymplecticFormdcos(cid:2) ^d(cid:3) ........................... 75
3.1.3 QuantizationoftheSymplecticFormonS2 .................. 77
ix
x Contents
3.2 CoherentStatesandStarProductonFuzzyS2 ....................... 80
N
3.3 TheFlatteningLimitofR2............................................. 88
(cid:2)
3.3.1 FuzzyStereographicProjection ............................... 88
3.3.2 CoherentStatesandPlanarLimit.............................. 91
3.3.3 TechnicalDigression........................................... 94
3.4 TheFuzzySphere:ASummary........................................ 96
3.5 FuzzyFieldsandActions .............................................. 103
3.5.1 ScalarActiononS2 ............................................ 103
N
3.5.2 ExtensiontoS2 (cid:2)S2 .......................................... 103
N N
2
3.6 IntroducingFuzzyCP ................................................. 106
3.7 FuzzyFermions......................................................... 108
3.7.1 ContinuumDiracOperators ................................... 109
3.7.2 FuzzyDiracOperators......................................... 112
References..................................................................... 117
4 QuantumNoncommutativePhi-Four...................................... 119
4.1 TheUV-IRMixing ..................................................... 119
4.2 TheStripePhase........................................................ 123
4.2.1 TheDisorderedPhase.......................................... 123
4.2.2 TheOrderedPhase............................................. 129
4.2.3 ThePhaseStructure:ALifshitzTriplePoint ................. 132
4.2.4 Stripesin2Dimension......................................... 137
4.3 TheSelf-DualNoncommutativePhi-Four............................. 149
4.3.1 IntegrabilityandExactSolution............................... 149
4.3.2 NonperturbativeUV-IRMixing ............................... 161
4.4 NoncommutativePhi-FourontheFuzzySphere...................... 163
4.4.1 ActionandLimits .............................................. 163
4.4.2 TheEffectiveActionandThe2-PointFunction.............. 164
4.4.3 The4-PointFunctionandNormalOrdering.................. 167
4.4.4 ThePhaseStructureandEffectivePotential.................. 169
4.4.5 FuzzyS2(cid:2)S2andPlanarLimit:FirstLook.................. 172
4.4.6 MoreScalarActionsonMoyal-WeylPlaneand
FuzzySphere................................................... 176
4.5 MonteCarloSimulations............................................... 185
4.5.1 FuzzySphere:AlgorithmsandPhaseDiagram............... 185
4.5.2 FuzzyTorus:DispersionRelations............................ 190
4.6 InitiationtotheWilsonRenormalizationGroup...................... 191
4.6.1 TheWilson-FisherFixedPointinNCˆ4..................... 191
4.6.2 TheNoncommutativeO.N/Wilson-FisherFixedPoint...... 203
4.6.3 TheMatrixFixedPoint ........................................ 203
References..................................................................... 204
5 TheMultitraceApproach................................................... 207
5.1 PhaseStructureofFuzzyandNoncommutativeˆ4 .................. 207
5.2 NoncommutativePhi-FourRevisited.................................. 209
Contents xi
5.2.1 TheMoyal-WeylPlaneR2 ................................... 209
(cid:2);(cid:4)
5.2.2 TheFuzzySphereS2 ........................................ 211
N;(cid:4)
5.3 MultitraceApproachontheFuzzySphere............................ 213
5.4 TheRealQuarticMultitraceMatrixModelonR2 andS2 ....... 221
(cid:2);(cid:4) N;(cid:4)
5.4.1 Setup............................................................ 221
5.4.2 TheEffectiveMatrixAction................................... 226
5.5 MatrixModelSolution................................................. 236
5.5.1 Scaling.......................................................... 236
5.5.2 SaddlePointEquation.......................................... 239
5.6 TheRealQuarticMatrixModel ....................................... 242
5.6.1 FreeTheory..................................................... 243
5.6.2 TheSymmetricOne-Cut(Disordered)Phase................. 243
5.6.3 TheTwo-Cut(Non-uniform-Ordered)Phase ................. 246
5.6.4 TheAsymmetricOne-Cut(Uniform-Ordered)Phase ........ 249
5.7 MulticutSolutionsoftheMultitraceMatrixModel .................. 250
5.7.1 TheOne-CutPhase............................................. 250
5.7.2 TheTwo-CutPhase ............................................ 254
5.7.3 TheTriplePoint................................................ 255
5.8 ThePlanarTheory...................................................... 260
5.8.1 TheWignerSemicircleLaw................................... 260
5.8.2 IntroducingFuzzyProjectiveSpaces.......................... 264
5.8.3 FuzzyToriRevisited ........................................... 266
5.9 TheNon-perturbativeEffectivePotentialApproach.................. 269
References..................................................................... 273
6 NoncommutativeGaugeTheory ........................................... 277
6.1 GaugeTheoryonMoyal-WeylSpaces ................................ 277
6.2 RenormalizedPerturbationTheory.................................... 279
6.2.1 TheEffectiveActionandFeynmanRules .................... 279
6.2.2 VacuumPolarization........................................... 283
6.2.3 TheUV-IRMixingandTheBetaFunction................... 286
6.3 QuantumStability ...................................................... 289
6.3.1 EffectivePotential.............................................. 289
6.3.2 ImpactofSupersymmetry ..................................... 291
6.4 Initiationto NoncommutativeGaugeTheoryonthe
FuzzySphere............................................................ 294
6.5 GaugeTheoryonTheNoncommutativeTorus........................ 299
6.5.1 TheNoncommutativeTorusTd Revisited..................... 299
(cid:2)
6.5.2 U.N/GaugeTheoryonTd .................................... 301
(cid:2)
6.5.3 TheWeyl-’tHooftSolution ................................... 302
6.5.4 SL.d;Z/Symmetry............................................. 305
6.5.5 MoritaEquivalence............................................. 307
References..................................................................... 312