ÆTHEREAL MULTIVERSE Selected Problems of Lorentz Symmetry Violation, Quantum Cosmology, and Quantum Gravity ŁUKASZ ANDRZEJ GLINKA International Institute for Applicable Mathematics & Information Sciences Hyderabad (India) & Udine (Italy) arXiv:1102.5002v2 [physics.gen-ph] 8 Mar 2011 Draft of Book Contents Preface iv Prologue vi I Lorentz Symmetry Violation 1 1 Deformed Special Relativity 2 A Thelineardeformation . 3 A1 TheDiracequationandthenewalgebra . 3 A2 AnotherIdentifications . 8 B TheSnyder–SidharthHamiltonian . 18 B1 TheCaseofFermions. 20 B2 TheCaseofBosons .................. 25 C TheModifiedComptonEffect . 31 C1 The Relativistic Approach . 32 C2 The Relativistic Limit. The Lensing Hypothesis. 36 C3 BoundsontheModified ComptonEquation . 39 D TheDispersionalGeneralization . 43 2 The Neutrinos: Masses & Chiral Condensate 49 A OutlookonNoncommutativeGeometry . 49 B Massiveneutrinos....................... 57 C TheCompton–PlanckScale . 65 D The Global Effective Chiral Condensate . 67 E Conclusion ........................... 72 3 The Neutrinos: Energy Renormalization & Integrability 74 A Introduction .......................... 74 B Energyrenormalization. 76 C The Integrability Problem for the Dirac equations . 87 D The Integrability Problem for the massive Weyl equations 92 i ŁUKASZ ANDRZEJ GLINKA: ÆTHEREAL MULTIVERSE D1 TheDiracbasis .................... 94 D2 TheWeylbasis..................... 95 D3 Thespace-timeevolution . 96 D4 Probability density. Normalization . 99 E The Ultra-Relativistic Massive Neutrinos . 102 II Quantum General Relativity 107 4 The Quantum Cosmology 108 A Introduction .......................... 108 B TheClassicalUniverse . 110 C QuantizationofHamiltonianConstraint . 128 D The Multiverse Thermodynamics . 134 E TheEarlyLightMultiverse . 137 F Summary............................ 150 5 The Inflationary Multiverse 152 A TheInflationaryCosmology . 152 B ThePowerLawInflaton . 157 C TheHiggs–HubbleInflaton . 158 D TheChaoticSlow–RollInflation. 161 E ThePhononicHubbleInflaton . 166 F TheInflatonConstant . 172 6 Review of Quantum General Relativity 174 A 3+1SplittingofGeneralRelativity . 174 B Geometrodynamics: Classical and Quantum . 180 C TheWheelerSuperspace . 191 D TheProblemsofGeometrodynamics . 206 E OtherApproaches ....................... 210 7 Global One-Dimensionality Conjecture 213 A Introduction .......................... 213 B The Γ-Scalar-Flat Space-times . 215 C TheAnsatzforWaveFunctionals . 226 D Field Quantization in Static Fock Space . 236 E SeveralImplications . 240 E1 TheGlobal1DWaveFunction . 240 ii ŁUKASZ ANDRZEJ GLINKA: ÆTHEREAL MULTIVERSE E2 The Unitary Three-Manifolds . 245 E3 TheFourierAnalysis . 247 E4 QuantumCorrelations . 253 8 The Invariant Global Dimension 255 A TheInvariantGlobalQuantumGravity . 255 B TheOne-DimensionalDiracEquation . 257 C The Cauchy-Like Wave Functionals . 259 D ProblemI:InvertedTransformation . 267 E Problem II: The Hilbert Space and Superposition . 268 F ProblemIII:TheProblemofTime. 278 9 Examples of Invariant 1D Wave Functions 279 A TheMinkowskiSpace-time. 279 B TheKasnerSpace-time . 281 C TheSchwarzschildSpace-time. 282 D The(Anti-)DeSitterSpace-time . 283 E The (Anti-) De Sitter–Schwarzschild Space-time . 286 F TheKerrSpace-Time ..................... 288 G TheKerr–NewmanSpace-time . 289 H The Reissner–Nordström Space-time . 299 I TheGödelSpace-time. .. .. .. .. .. .. .. .. .. 301 J The Einstein–Rosen Gravitational Waves . 305 K The Taub–Newman–Unti–Tamburino Space-time . 306 10 The Functional Objective Geometry 308 A EffectiveScalarCurvature . 308 B TheNewton–CoulombPotential . 314 C Boundary Conditions for The Wave Functionals . 326 11 Ab Initio Thermodynamics of Space Quanta Æther 334 A EntropyI:TheAnalyticApproach. 335 B EntropyII:TheAlgebraicApproach . 345 Epilogue 352 Bibliography 354 Index 406 iii Preface A thought is an idea in transit. Pythagoras Æthereal Multiverse is my personal research attempt to demonstrate the productive realization of the fusion of two fundamental concepts of Antiquity, i.e. Æther and Multiverse which lay the foundations of Aristotelian and Epicurean–Islamic systems, respectively. It is my deep conviction that this combination enables the fruitful description of physical Reality, which allows to understand constructively the an- thropic everything, i.e. all what can be observed and detected by a man and all devices produced by humankind. From my standpoint the constructive proposal for the fusion involves the essential theoretical symbols of the physical Reality: Quantum Mechanics, Quantum Field Theory, General Relativity, and Thermodynamics. On the one hand this monograph collects advanced developments of certain elementary knowledge of theoretical and mathematical physics. On the other hand, however, the presented deductions are performed step-by-step and often include detailed calculations. In this manner this book is available for the readers interested in development and applications of the fundamental knowledge. Intentionally the content is divided onto two independent lines which have arose in my research work of the years 2006-2010. Part I, Lorentz Symmetry Violation, con- tains 3 chapters and is devoted to discussion of several applications of the noncommutative geometry of the Snyder quantized space-time. Strictly speaking I shall focus in some detail on the basic consequences of what I call the Snyder–Sidharth Hamiltonian constraint, i.e. the modification of Special Relativity arising from the noncommutative ge- ometry. Particularly the Compton effect and the massive neutrinos model are discussed. Part II, Quantum General Relativity, contains 8 chapters and is much more inhomogeneous. This part presents my ap- proach to quantum cosmology and quantum gravity, and my point of iv ŁUKASZ ANDRZEJ GLINKA: ÆTHEREAL MULTIVERSE view on inflationary cosmology. I present the version of quantum ge- ometrodynamics strictly based on the Wheeler–DeWitt equation, which leads to constructive and consistent deductions. Particularly two ap- proaches to entropy calculation, which lead to distinguished formula- tions of thermodynamics of space quanta Æther, are discussed. The greatest motivation to this book is the 2010’ book of an experi- enced mathematical physicist Robert W. Carroll [1] who expressed his opinion about certain part of my research results (. ) we sketch some work of L. Glinka et al on thermodynamics and quantum gravity. This involves bosonic strings and quantum f ield the- ory (QFT) and is speculative (but very interesting). (. ) we deal with some fundamental articles by L. Glinka for which some general theory is also motivated by theoretical material involving second quantization and Bogoliubov transformations (. ) we suggest printing out the latest versions and working from them - we can only give a sketch here and remark that some of the work has a visionary nature which is valuable in itself. This book collects all my non-coauthored research results rigorously revisited and enriched by necessary updates. Numerous typos and technical mistakes are improved. The necessity of the improvement follows from my deep conviction that the only mathematical truth re- sults in the constructive theoretical and mathematical physics. This book develops also my philosophical standpoint on theoretical physics involving the philosophical interpretation needed for the new physics. My grateful acknowledgements belong to numerous senior scien- tists and scholars who helped me kindly in the various aspects related to this book. Professor Robert W. Carroll granted me by a number of valuable discussions and comments, and mailed a hard copy of his 2010’ book. Professor Sir Harold W. Kroto gave matter-of-fact discus- sions, and included my views into his GEOSET programme. Essential comments and discussions due to Professor Burra G. Sidharth signifi- cantly helped in edition of the primary manuscript of this book. Discus- sions with Dr. Andrej B. Arbuzov and Professor Alessandro De Angelis benefitted during my research work were helpful. Comments from Pro- fessor Wojciech H. Zurek were also valuable and constructive. I dedicate this book to all Friends of mine. Comments and discus- sion are welcome and invited to [email protected]. Łukasz Andrzej Glinka v Prologue It is a dogma of the Roman Church that the existence of God can be proved by natural reason. Now this dogma would make it impossible for me to be a Roman Catholic. If I thought of God as another being like myself, outside myself, only inf initely more powerful, then I would regard it as my duty to defy him. Ludwig Wittgenstein The U-turn to Antiquity In the second decade of the 21th century technological progress defines evolution of humankind. Present civilizations are based on techno- cratic ideas which are often overly formal. Such a state of things re- jects freedom of thinking and focuses on creation of new barriers, called standards and norms. The obstacles follow from promotion of feudal submissiveness toward various traditions. This fight with rationalism results in inhibition of mental growth of individuals, and leads to evi- dent intellectual poverty. The irrationalism ruins efforts of rationalists, establishes and stabilizes illusions and delusions, and effectively re- sults in the permanent damage of individual mentality. Such a regime has been produced the emergence of social multiple disabilities, and above all a regular de-evolution of humankind. On the other hand there is detectable the vast U-turn to the ideas of the most fundamental epoch in the history of