Stephen Blaha, Ph.D. M PubHsMtw

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1 Quantum Big Bang Cosmology: Complex Space-time General Relativity, Quantum Coordinates,"Dodecahedral Universe, Inflation, and New Spin 0, 1 / 2,1 & 2 Tachyons & Imagyons Stephen Blaha, Ph.D. M PubHsMtw

2 CONTENTS WAYS OF READING THIS BOOK XV IF YOU HAVE AN INTEREST IN EARLY COSMOLOGY AT T = 0 xv... INTEREST IN DARK ENERGY AND DARK MATTER xv... INTEREST IN TWO-TIER QUANTUM FIELD THEORY : xv... INTEREST IN COMPLEX SPACE-TIME & GENERAL RELATIVITY xv... INTEREST IN ABSOLUTE SPACE-TIME xv... INTEREST IN QUANTUM COORDINATES xv... INTEREST IN MANY COMPLEX VARIABLES RESULTS xv... INTEREST IN THE DODECAHEDRAL SYMMETRY OF THE UNIVERSE xv... INTEREST IN INFLATION xv 1. INTRODUCTION CURRENT STANDARD COSMOLOGICAL MODEL 3 /./. / General Features Some Highlights of the History of the Universe Zero Seconds At lo' 12 Second At lo" 6 Second '.: At 1 Second At 3-4 Minutes At 300,000 Years PROBLEMS OF THE STANDARD COSMOLOGICAL MODEL ; The Extreme Simplicity of the Universe The Lackofa Theory of Quantum Gravity The Question of Inflation Dark Matter, Dark Energy, and other Dark Issues Why an Accelerating Universe? Before the Beginning of the Current Universe How Does the Universe End? Other Universes? " THE THEME OF THIS BOOK 6 2. INERTIAL REFERENCE FRAMES & ABSOLUTE SPACE-TIME INTRODUCTION 9 i

3 2.2 NEWTONIAN MECHANICS EMBODIES AN ABSOLUTE SPACE-TIME MACH'S PRINCIPLE EMBODIES AN ABSOLUTE SPACE-TIME GENERAL RELATIVITY EMBODIES AN ABSOLUTE SPACE-TIME THE CASE FOR ABSOLUTE SPACE-TIME EXPERIMENTAL DETERMINATION OF AN ABSOLUTE REFERENCE FRAME COMPLEX SPACE-TIME INTRODUCTION THE FUNDAMENTAL QUADRATIC DIFFERENTIAL FORM COMPLEX SPACE-TIME! COMPLEX COORDINATE TRANSFORMATIONS ANALYTICITY OF THE COMPLEX COORDINATES / Motion of a Particle The Complex Affine Connection Complex Covariant Derivatives The Complex Curvature Tensor The Interval Between Events in Complex Space-time Intervals with a Small Imaginary Part COVARIANT DERIVATIVES ALONG A LINE PARALLEL TRANSPORT AND LINE INTEGRALS Parallel Transport Line Integrals in Complex Space-time Line Integrals in the Complex Plane of One Coordinate Line Integrals in Several Coordinates in Complex Space-time Line Integral in Several Complex Coordinates around a Closed Curve - Generalized Cauchy Theorem Parallel Transport of a Vector around a Closed Curve in Several Variables Generalization of Cauchy's Theorem to Curved, Complex Space-time One Complex Variable Cauchy Theorem in Curved Complex Space-time Several Complex Variables Cauchy Theorem in Curved Complex Space-time Parallel Transport of a Vector Around a Closed Curve in the Complex Plane of a Coordinate Path Independence of Parallel Transported Vectors in Complex Space-time Path Independence of Line Integrals of Vectors in Complex Space-time COSMIC CONFINEMENT & LAYERS OF UNIVERSES IN COMPLEX SPACE-TIME Quark Confinement Many Universes Theories 41 ii

4 3.8.3 Cosmic Confinement of Energy to a Universe Mechanism for Cosmic Confinement Island Universes in Complex Space-time Quantum Tunneling between Universes? INTEGRATION IN COMPLEX SPACE-TIME & AN ANTI-HOLOMORPHIC EXTENSION OF SPACE-TIME EIGHT DIMENSIONAL REAL REPRESENTATION OF COMPLEX SPACE-TIME RELATION OF 8-DIMENSIONAL REAL SPACE-TIME JACOBIANS TO COMPLEX SPACE-TIME JACOBIANS COORDINATE TRANSFORMATIONS IN 4 COMPLEX DIMENSIONS DIMENSIONAL COMPLEX SPACE-TIME DIMENSIONAL COMPLEX COORDINATE TRANSFORMATIONS Matrix Representation of 4 Complex Dimensions Coordinate Transformations Matrix Representation of 8 Complex Dimensions Coordinate Transformations THE COMPLEX SPACE-TIME DETERMINANT G VOLUME INTEGRALS USING COMPLEX COORDINATES GAUSS'THEOREM FOR COMPLEX SPACE-TIME : Complex Space-time Gauss' Theorem EXAMPLES OF GAUSS' THEOREM IN COMPLEX SPACE-TIME : Four-dimensional, Flat, Complex Space-time Gauss' Theorem Example: a Hypercube Two-dimensional, Flat, Complex Space-time Gauss' Theorem Example: a Complex Sphere SOME OTHER USEFUL RELATIONS FOR COMPLEX VARIABLE INTEGRATIONS One Complex Variable Version of Green's Theorem., An Area Integral J INTERPRETATION OF CURVATURE IN 4-DIMENSIONAL COMPLEX SPACE-TIME Transformation to Eight-dimensional Real Space-time Calculation of the Equivalent Real Space-time Curvature tensor Spinor Formulation of 8-dimensional Space-time GENERAL RELATIVITY IN COMPLEX SPACE-TIME COMPLEX GENERAL RELATIVITY THE COMPLEX SPACE-TIME SCHWARZSCHILD SOLUTION Static Isotropic Solution The Schwarzschild Solution for Complex Space-time Equations of Motion of a Particle in Complex Schwarzschild Space-time 69 iii

5 E < 1 (bound orbit) 71 E = l 71 E > 1 (unbound orbit) Particle Motion as a Function of Time in Complex Schwarzschild Space-time : 73 E < 1 (bound orbit) 73 E=l 73 E> 1 (unbound orbit) 74 E < 1 (bound orbit) 74 E=l 74 E > 1 (unbound orbit) Real Space-time Solutions for the Radial Motion Quantum Mechanical "Sampling" of Complex Space-time Complex Space-time Solutions for the Radial Motion Real Radial Trajectory near the Schwarzschild Radius Complex Radial Trajectory near the Schwarzschild Radius EFFECTS OF MASSES LOCATED AT COMPLEX COORDINATES SPONTANEOUS "CREATION" OF MASS IN REAL SPACE-TIME FROM COMPLEX SPACE-TIME THE COMPLEX SPACE-TIME ROBERTSON-WALKER METRIC Symmetries of the Complex Robertson-Walker Metric MAPS OF COMPLEX SPACE-TIME TO REAL SPACE-TIME The Trivial Map Non-trivial Maps from Complex Space-time to Real Space-time PROPERTIES OF 4-DIMENSIONAL REAL SPACE-TIME EMBEDDED IN COMPLEX, OR 8-DIMENSIONAL REAL, SPACE-TIME BLACK HOLES AND THE LOSS OF INFORMATION PROBLEM QUANTUM COORDINATES PHYSICS "FLIP-FLOPS" A SPACE-TIME COORDINATES "FLIP-FLOP"? QUANTUM COORDINATES AND QUANTUM FIELD THEORY DIVERGENCES QUANTUM COORDINATES AND THEIR MEASUREMENT REQUIREMENTS FOR ACCEPTABLE QUANTUM COORDINATES QUANTUM COORDINATES & EXPERIMENTALLY VERIFIED ABSOLUTE SPACE-TIME RATIONALE FOR THE FORM OF QUANTUM COORDINATES A PRECIS OF TWO-TIER QUANTUM FIELD THEORY 99 iv

6 7.1 INTRODUCTION OUTLINE OF TWO-TIER SCALAR PARTICLE QUANTUM FIELD THEORY Scalar Particle Lagrangian Scalar Particle Quantization Coordinate Quantization Particle States and the ^Propagator The Feynman Propagator Large Distance Behavior of the Feynman Propagator Short Distance Behavior of Two-Tier Theories Perturbation Theory An Auxiliary Asymptotic Field Transformation Between O(y) and <fi(x(y)) Model Lagrangian with <p 4 Interaction In-states and Out-States (j> In-Field, <j> Out-Field, The S Matrix * LSZ Reduction for Scalar Fields The U Matrix Reduction of Time Ordered <p Products Nature of the Resulting Two-tier Scalar Quantum Field Theory TWO-TIER QUANTUM GRAVITY Lagrangian for Quantum Gravity and Unified Standard Model Quantum Gravity - Scalar Particle Model Lagrangian, Quantum Gravity with a Flat background Metric A Quantum Gravity-Scalar Particle Model Path Integral Finiteness of Quantum Gravity Unitarity of Quantum Gravity Scalar Particle Model The Mass Scale M c Planck Scale Physics Quantum Foam Measurement of the Quantum Gravity Field Measurement of Time Intervals Vacuum Fluctuations in the Gravitation Fields Two-Tier Gravitational Potential vs. Newton's Potential Black Holes.' 143

7 7.4 INVARIANCE OF TWO-TIER THEORIES UNDER SPECIAL RELATIV1STIC & GENERAL RELATIVISTIC TRANSFORMATIONS Two-tier Quantum Field Theories and Special Relativity Gauge Invariance of the Full Lagrangian Invariance of Two-tier QFT under Special Relativistic Transformations Demonstration that the Expectation Values of Y Field Products in the Lorentz Gauge Equal their Value in the Transverse Gauge Lorentz Invariants Matrix Elements Dirac Metric Operator CURVED SPACE-TIME GENERALIZATION OF TWO-TIER QUANTUM GRAVITY WHY ARE THE Y FIELD DYNAMICS INDEPENDENT OF THE GRAVITATIONAL FIELD? UNIFIED THEORY OF THE STANDARD MODEL AND QUANTUM GRAVITY Preliminary Unifications Unified Two-Tier Standard Model and Quantum Gravity Lagrangian "Low Energy" Behavior A Negative Degree of Divergence - A Finite Unified Theory Unitarity * GENERAL FORM OF TWO-TIER FREE PARTICLE PROPAGATORS OF ANY SPIN Scalar Particle Propagator Spin 'A Particle Propagator Massless Spin 1 Particle Propagator Spin 2 Particle Propagator A GENERALIZATION OF THE ROBERTSON-WALKER METRIC TO THE CASE OF A TWO-DIMENSIONAL MAXIMALLY SYMMETRIC SUBSPACE ); THE ROBERTSON-WALKER METRIC A GENERALIZATION OF THE ROBERSON-WALKER METRIC THE EINSTEIN EQUATIONS FOR THE GENERALIZED ROBERTSON-WALKER METRIC THE DIFFERENTIAL EQUATIONS FOR THE GENERALIZED SCALE FACTOR A(T, R), THE SOLUTION FOR THE GENERALIZED SCALE FACTOR A(T, R) THE SOLUTION EXPRESSED IN THE ORIGINAL RADIAL COORDINATE EQUIVALENCE OF THE GENERAL SOLUTION WITH THE ROBERTSON-WALKER SOLUTION HUBBLE'S LAW IN THE GENERALIZED ROBERTSON-WALKER MODEL AT THE BEGINNING OF TIME 179 INTRODUCTION 179 vi

8 9.1 THE SITUATION ATT = TWO-TIER QUANTUM MODEL FOR THE BEGINNING OF THE UNIVERSE Einstein Equations Near t YBlack-Body Coherent States Expectation Value of Y in Coherent States Blackbody Expectation Values of the Scale Factor A (t, X) and the Invariant Interval df Representation and Approximations for Y BB (t, $$ Some Representations of YBB Approximate Solution for Y BB THE TIME SUBFACTOR OF THE SCALE FACTOR A(T) NEART = A COMPLEX BLACKBODY TEMPERATURE NEAR T = THE NATURE OF THE UNIVERSE NEART = Behavior of the Complete Scale Factor B(t,j) Near t = Behavior of the Expectation Value of the Scale Factor A m (0,j) near t = THE SCALE FACTOR FROM T = 0 TO THE PRESENT INTRODUCTION A THE BEHAVIOR OF THE SCALE FACTOR A(T) AFTER THE BIG BANG PERIOD General Form ofa(t) Scale Factor Einstein Equation The Explosive Growth Epoch The Expanding Epoch TWO-TIER MODEL AT THE BEGINNING OF TIME The CMB Temperature The Generalized Robertson-Walker Scale Factor The Temperature of the Early Universe in the Generalized Robertson-Walker Metric Consistency of YBB Approximation with the Resulting Temperature near t = Plots of the Scale Factor from t = 0 to the Present THE INTERPRETATION OF THE COMPLEX SCALE FACTOR TIME EVOLUTION OF THE HUBBLE RATE THE BIG BANG EPOCH THE T = 0 BIG BANG SCENARIO ; THE RADIUS OF THE UNIVERSE IN THE BIG BANG EPOCH LocalizationofParticlesatt = THE VOLUME OF THE UNIVERSE AS A FUNCTION OF TIME Simple Example of a Volume Calculation in Complex Space 232 vii

9 The Volume of the Universe for 0<t< t e Alternate Physically Intuitive Calculation of the Volumeofthe Universe for 0 < t < t c The Calculation of the Volume of the Universe for t> t c ENERGY DENSITY OF THE UNIVERSE Impact of the Radial Dependence of the Energy Density Is THERE A BARRIER BETWEEN REAL AND IMAGINARY SPACE: ENERGY CONFINEMENT TO REAL SPACE? CMB TEMPERATURE SINCET = FORCES "REVERSED" IN THE RADIATION-DOMINATED PHASE IS AN INFLATIONARY PERIOD NEEDED? THE INFLATION CONCEPT The Guth Notion of "Initial Conditions " and Boltzmann 's Notion of Statistical Mechanics ARE THE "PROBLEMS" THAT INFLATION "SOLVES" REALLY PROBLEMS The Impact of the Uncertainty Principle on an Expanding Universe The Size of an Elementary Particle at the Beginning of Time." Compton Wavelength of the Univefse There is No Horizon Problem in the Neighborhood of t = No Higgs Bosons-No Magnetic Monopole Problem The Flatness Problem A PROPER VOLUME INFINITESIMAL INVARIANT AND A POSSIBLE DODECAHEDRAL UNIVERSE SUGGESTIONS OF A DOCDECAHEDRAL-SHAPED UNIVERSE The Dodecahedron is One of the Platonic Solids. J AN INVARIANT INFINITESIMAL VOLUME/AREA ELEMENT AN ICOSAHEDRAL SYMMETRY OF THE PROPER INFINITESIMAL VOLUME A Mapping of S } Terms to a Two Digit Representation Two-Digit Labeling of the Dodecahedron Finite Rotation (Icosahedral) Group of the Dodecahedron Effect of Icosahedral Rotations on the Invariant Volume Element IMPLICATIONS OF THE ICOSAHEDRAL GROUP INVARIANCE IN 3-DIMENSIONAL COMPLEX SPACE Spaces without Rotational Symmetry THE UNIVERSE AS A SPHERE IN 4-DIMENSIONAL SPACE IMPLICATIONS OF THE ICOSAHEDRAL GROUP INVARIANCE IN 3-DIMENSIONAL REAL SPACE 275 TM 14. TACHYONS, BRAIDYONS, AND IMAGYONS IN COMPLEX SPACE-TIME 276 viii

10 14.1 ON THE REALITY OF THE IMAGINARY THE TACHYON CONCEPT MAP FROM COMPLEX SPACE-TIME TO 8 REAL DIMENSIONS DIMENSIONAL (2 + 6) RELATIVITY Some Lorentz Subgroups of the Lorentz Group SO(4,4) SO(4,4) Generators and Algebra 26* SO(4,4) Lorentz Transformations Connectedness ofso(4,4) Observational Issues Associated with SO(3,1) Tachyons Observational Issues Associated with SO(4,4) Tachyons Braidyons Tachyons Imagyons DIMENSIONAL SCALAR FIELDS Braidyon Wave Packets in 4-Space-time Tachyon Wave Packets in 4-Space-time Massive Feinberg-type Tachyon Wave Packets in 4-dimensional Space-time Variable Mass Tachyon Wave Packets in 4-dimensional Imaginary Space-time Imagyon Wave Packets in 4-Space-time DIMENSIONAL FREE SCALAR QUANTUM FIELD THEORY Free Scalar Braidyon Quantum Field Theory Free Scalar Tachyon Quantum Field Theory Massive Feinberg-type Tachyon Quantum Field Theory in 4-dimensional Imaginary Space-time3O "Variable Mass" Scalar Tachyon Quantum Field Theory Free Scalar Imagyon Quantum Field Theory : Faster-than-Light Behavior of Imagyons -The Tachyonic Effect Interacting Scalar Imagyon Quantum Field Theories DlMENSIONAL SPIN Vz TACHYON/lMAGYON QUANTUM FIELD THEORY Three Types of Spin 'A TachySpinor Wave Packets Spin 'A Tachyon Wave Packets Spin 'A Imagyon Wave Packets Real Space-time Spin 'A Tachyon/Imagyons Free Spin 'A Tachyon/Imagyon Quantum Field Theory Imagyon QFT based on Conventional Quantum Field Theory Fermion States Spin '/ 2 Imagyon Propagator '. 332 ix

11 Interacting Spin 'A Imagyons Two-Tier Spin 'A Imagyon Quantum Field Theory SPIN 1 TACHYON/IMAGYON QUANTUM FIELD THEORY Interacting Spin 1 Imagyon Quantum Field Theories Imagyon QED & Electroweak Theory: Imagyon Vector Bosons SPIN 2 TACHYON/IMAGYON PARTICLES - MASSIVE GRAVITONS? Massive Graviton Two-Tier QFT GENERAL FORM OF TACHYON/IMAGYON PROPAGATORS IN REAL SPACE-TIME A NEW COMPLEX UNIVERSE WITH AN IMAGINARY PART AND TACHYON/IMAGYON PARTICLES PROBLEMS RESOLVED BY OUR TACHYON/IMAGYON QFT IMAGYONS & THE BIG BANG EPOCH IMAGINARY SPACE IN THE BIG BANG EPOCH SIGN OF THE ENERGY DENSITY FOR BRAIDYONS AND TACHYONS IN REAL AND IMAGINARY SPACE- TIME Real Space-time Energy Density Imaginary Space-time Energy Density Positivity of Energy Densities in Complex Space-time INTERPRETATION OF THE ENERGY DENSITY IN THE BIG BANG EPOCH CONCLUDING REMARKS 358 APPENDIX A. A FINITE ROTATIONAL INVARIANCE GROUP OF 3-DIMENSIONAL SURFACE AREA ELEMENTS 359 A.I THE 2-SURFACE ELEMENT IN 3-SPACE 359 A.2 THE CASE OF A 3-DIMENSIONAL SPACE WITH ONE COMPLEX AND ONE REAL DIMENSION 362 A.3 A MAPPING OF THE FINITE ROTATION GROUP TO A REGULAR FIGURE 364 APPENDIX B. SOME PHYSICAL CONSTANTS USED IN CALCULATING NUMERICAL EXPRESSIONS 366 SOME READINGS 368 INDEX 372

PRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in

$PRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in$ LONDON BEIJING HONG TSINGHUA Report and Review in Physics Vol2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China NEW JERSEY \Hp SINGAPORE World Scientific