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Principles of Quantum Universe V.N. Pervushin, A.E. Pavlov Laboratory of Theoretical Physics Joint Institute for Nuclear Research, Dubna February 4, 2014 arXiv:1402.0354v1 [gr-qc] 3 Feb 2014 Contents 1 Introduction 15 1.1 Whatisthisbookabout?. 15 1.2 Program ........................... 20 1.3 Does the creation and evolution of the Universedependonanobserver? . 28 1.4 Contents........................... 56 2 Initial data and frames of reference 62 2.1 Unitsofmeasurement . 62 2.2 Nonrelativisticmechanicsofaparticle . 64 2.3 FoundationsofSpecialRelativity. 66 2.3.1 Actionofarelativisticparticle . 66 2.3.2 Dynamicsofarelativisticparticle . 68 2.3.3 Geometrodynamics of a relativistic particle . 72 2.3.4 Reduction of geometrodynamics to the Planck’s relativistic dynamics (1906) . 75 2.3.5 Quantum anomaly of geometric interval . 77 2.3.6 How does the invariant reduction differfromthechoiceofgauge? . 80 2 CONTENTS 3 2.4 Homogeneous approximation of GeneralRelativity . 82 2.4.1 Radiation-dominated cosmological model . 82 2.4.2 Arrow of conformal time as quantum anomaly . 86 2.5 Standardcosmologicalmodels . 87 2.6 Summary........................... 89 3 Principles of symmetry of physical theories 94 3.1 Irreducible representations of Lorentzgroup ........................ 94 3.2 Irreducible representations of Poincar´egroup. 97 3.3 Weylgroup..........................101 3.4 ConformalgroupC . .104 3.5 Conformal invariant theories ofgravitation ........................106 3.6 Affine group A(4) ......................117 3.7 Fundamental elements of base Minkowskian space M .................118 3.8 Summary...........................120 4 Nonlinearrealizationsofsymmetrygroups 123 4.1 DifferentialformsofCartan. 123 4.2 Structuralequations. .132 4.3 Exponentialparametrization . 135 4.4 Algebraic and dynamical principlesofsymmetry . .136 4 CONTENTS 4.5 Theory of gravitation as nonlinear realization of A(4) C ...................140 ⊗ 4.5.1 Derivation of action of General Relativity . 140 4.5.2 Differences between the standard General Relativ- ity and the nonlinear realization of A(4) C . 144 ⊗ 4.6 Summary...........................148 5 Hamiltonian formulation of the theory of gravity 151 5.1 Foliation4=3+1. .151 5.2 Hamiltonian formulation of GR intermsofCartanforms . .156 5.3 ProblemsofHamiltonianformulation . 162 5.4 Exact solution of the Hamiltonian constraint ..........................167 5.4.1 Statementoftheproblem . 167 5.4.2 Lagrangianformalism . .168 5.4.3 Hamiltonianformalism . 172 5.5 Summary...........................175 6 AmodelofanemptyUniverse 180 6.1 AnemptyUniverse . .180 6.2 The Supernovae data in ConformalCosmology. .184 6.3 Thehierarchyofcosmologicalscales . 194 6.4 Special Relativity – General Relativity correspondence . .197 6.5 The arrow of time as a consequence of the postulate of vacuum............................200 CONTENTS 5 6.6 CreationoftheUniverse . .204 6.7 Summary...........................209 7 Quantization of gravitons in terms of Cartan forms 215 7.1 Affinegravitons ... ... ... .. ... ... ... .. .215 7.2 Comparisonwithmetricgravitons . 220 7.3 Vacuumcreationofaffinegravitons . 224 7.4 Summary...........................230 8 Mathematical principles of description of the Universe 233 8.1 Theclassicaltheoryofgravitation . 233 8.2 Foundations of quantum theory ofgravity ..........................235 8.2.1 The irreducible unitary representation of the group A(4) C ...............235 ⊗ 8.2.2 Casimir’svacuum . .239 8.2.3 An approximation of a nearly empty Universe . 240 8.3 Summary...........................246 9 CreationofmatterintheUniverse 250 9.1 TheBigBangorthevacuumcreation? . 250 9.1.1 Statementoftheproblem . 250 9.1.2 Observational data on the CMB radiationorigin . .253 9.2 Vacuumcreationofscalarbosons . 256 9.3 Physicalstatesofmatter . .260 9.4 QUmodificationofS-matrixinQFT . 264 6 CONTENTS 9.5 Summary...........................271 10ReducedphasespaceQCD 278 10.1 Topologicalconfinement. 278 10.2 Quark–hadronduality. .283 10.3 ChiralsymmetrybreakinginQCD . 286 10.4Summary...........................296 11 QU modification of the Standard Model 301 11.1 SMLagrangian . .301 11.2 The condensate mechanism of Higgs boson mass . 304 11.3 EstimationoftheHiggsbosonmass . 311 11.4Summary...........................314 12 Electroweak vector bosons 318 12.1 Cosmological creation of electroweak vectorbosons ........................318 12.2 SourcesofCMBradiationanisotropy . 327 12.3 BaryonasymmetryoftheUniverse . 330 12.4Summary...........................334 13 Conformal cosmological perturbation theory 346 13.1 The equations of the theory of perturbations . ... ... ... .. ... ... ... .. .346 13.2 The solution of the equations for smallfluctuations . .348 13.3Summary...........................351 CONTENTS 7 14 Cosmological modification of Newtonian dynamics 353 14.1 Freemotion in conformallyflat metric . 353 14.2 The motion of a test particle inacentralfield ... ... ... .. ... ... ... .. .357 14.3 The Kepler problem in the Conformaltheory . .359 14.4 Capture ofaparticlebyacentralfield . 361 14.5 The problem of dark matter inSuperclusters . .363 14.6 The Kepler problem in the generalized Schwarzschildfield. .367 14.7 Quantum mechanics of a particle inConformalcosmology. .372 14.8Summary...........................374 15 Afterword 378 15.1 QuestionsofGenesis . .378 15.2 Generaldiscussionofresults . 381 15.2.1 Resultsofthework . .381 15.2.2 Discussion . .387 AReducedAbelianfieldtheory 390 A.1 ReducedQED ........................390 A.1.1 Actionandframeofreference. 390 A.1.2 Eliminationoftimecomponent . 392 A.1.3 Elimination of longitudinal component . 393 A.1.4 Staticinteraction . .393 8 CONTENTS A.1.5 Comparison of radiation variables with the Lorentz gaugeones ......................394 A.2 Reducedvectorbosontheory . 397 A.2.1 Lagrangianandreferenceframe. 397 A.2.2 Eliminationoftimecomponent . 398 A.2.3 Quantization. .400 A.2.4 Propagatorsandcondensates . 402 B Quantum field theory for bound states 408 B.1 Ladderapproximation. .408 B.2 Bethe–SalpeterEquations . 414 C Abel – Plana formula 422 D Functional Cartan forms 428 D.1 Dynamicalmodelwith high derivatives . 428 D.2 VariationalDeRhamcomplex . 433 E Dynamicsofthemixmastermodel 437 E.1 DynamicsoftheMisnermodel . 437 E.2 Kovalevskiexponents . .439 Preface to the first English edition The Russian edition of the present book was published in June 2013. It just happened that it was the time between two significant dates: in 2011 the Nobel Prize was awarded “for the discovery of the accelerated expansion of the Universe through observations of distant Supernovae” and in 2013 the Nobel Prize was awarded for “the theoretical discovery of a mechanism that contributes to our understanding of the origin of the mass of subatomic particles”. Both these formulations left the questions about the explanations of these phenomena in the framework of the fundamental principles open. Our book is devoted to attempts to explain the observed long dis- tances to the Supernovae and the small value of the Higgs particle mass by the principles of affine and conformal symmetries and the vacuum postulate. Both these phenomena are described by quantum gravity in the form of joint irreducible unitary representations of the affine and con- formal symmetry groups. These representations were used in our book to classify physical processes in the Universe, including its origin from the vacuum. The representations of the Poincar´egroup were used in the same way by Wigner to classify particles and their bound states. We are far from considering our understanding of the “distant Super- novae” and the “origin of the mass of subatomic particles” to be conclu- sive, but we do not abandon hope that the present revised and enlarged English edition encourages a deeper and worthier investigation of these open questions in the future. The authors express their appreciation and gratitude to the coauthors of the papers on which this book is based. The authors are grateful to I.V. Kronshtadtova and G.G. Sandukovskaya for proofreading the text of the book. The authors are grateful to Academician V.A. Matveev, Professors V.V. Voronov and M.G. Itkis for the support. V.N. Pervushin A.E. Pavlov December, 2013 Dubna Preface to the first Russian edition1 This monograph is based on papers published during last 25 years by the authors and lectures delivered by one of the authors (V.P.) at the universities of Graz (Austria), Berlin, Heidelberg, Rostock (Germany), New Delhi (India), Fairfield, the Argonne National Laboratory (USA), the physical faculty of Moscow State University and in the Joint Institute for Nuclear Research. The main goal of the authors is to bring readers into the interesting and intriguing problem of description of modern ex- perimental and observational data in the framework of ideas and methods elaborated until 1974 by the founders of the modern relativistic classical and quantum physics. The distinction of our approach from the stan- dard ones consists in using conformal symmetry: everywhere, from the horizon of the Universe to quarks, we use scale-invariant versions of mod- ern theories on the classical level with dimensionless coupling constants, breaking scale invariance only at the quantum level by normal ordering of products of field operators. The method of classification of novae data, obtained in the last fifteen years in cosmology and high–energy physics, essentially uses quantum theories and representations. From here the title of our book originated: “Principles of Quantum Universe”. Let us briefly present the content of the
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