Trans-Planckian Physics and Inflation: An Introduction to Renormalizable and Background-Free Quantum Gravity

Preface
Contents
1 What Quantum Gravity Should Reveal
2 Renormalizable and Background-Free Quantum Gravity
	2.1 Quantum Gravity Action and Effective Action
		2.1.1 Diffeomorphism Invariance Demands Conformal Anomaly
		2.1.2 Quantization and Effective Action
	2.2 Reality of Fields and Unitarity
	2.3 Physical Constants and Spacetime Phase Transition
	2.4 Several Comments on Renormalization
		2.4.1 Historical Background of Weak-Field Approximation
		2.4.2 Renormalization in Different Perspectives
		2.4.3 QED and Landau Singularity
	2.5 Appendix: More About Effective Action
		2.5.1 QCD Effective Action in Curved Space
		2.5.2 Quantum Gravity Effective Action
3 Conformal Invariance as Background Freedom
	3.1 BRST Conformal Invariance
	3.2 Background Freedom and Unitarity
		3.2.1 On Historical Background
4 Physical Meaning of Hamiltonian/Momentum Constraints
	4.1 About Gravitational Equations of Motion
	4.2 What Diffeomorphism Invariance Shows
		4.2.1 No Absolute Time
		4.2.2 No Zero-Point Energy
5 Renormalization Using Dimensional Regularization
	5.1 How to Determine Quantum Gravity Action
	5.2 Quantum Gravity Action in DD Dimensions
		5.2.1 Renormalization Group Equations and Beta Functions
		5.2.2 Kinetic Terms, Interactions, and Counterterms
		5.2.3 Gauge Fixing and Propagators
	5.3 Finiteness of Energy-Momentum Tensors
	5.4 Main Results of Loop Calculations
		5.4.1 Non-renormalization Theorem Z_\phi =1Zφ=1
		5.4.2 Beta Function Results
	5.5 Cosmological Term and Einstein-Hilbert Term
	5.6 Effective Potentials and Physical Constants
		5.6.1 Derivation of Physical Cosmological Constant
	5.7 Appendix: Hathrell's Renormalization Group Equations
		5.7.1 Normal Products and Renormalization Group Equations
		5.7.2 Determination of Gravitational Counterterms
		5.7.3 Forms of Conformal Anomalies
6 BRST Conformal Algebra and Physical States
	6.1 Canonical Quantization on \mathbb{R} \times S^3mathbbR timesS3
	6.2 Generators of Conformal Algebra
		6.2.1 BRST Formulation and Physical State Conditions
	6.3 Construction of Physical States
		6.3.1 Primary States
		6.3.2 Physical States in Quantum Gravity
	6.4 Appendix: Two-Dimensional Quantum Gravity on \mathbb{R} \times S^1mathbbR timesS1
		6.4.1 Liouville Action, Virasoro Algebra, and Physical States
		6.4.2 BRST Formulation
7 Quantum Gravity Inflation
	7.1 Stable Inflationary Solution
	7.2 Big Bang as Spacetime Phase Transition
		7.2.1 After Phase Transition
	7.3 Entropy of The Universe
8 Localized Massive Excitations of Quantum Gravity
	8.1 Equations Describing Inside of Excitations
	8.2 Static Spherical Excitations
9 What the Cosmological Constant Problem Is
	9.1 Discovery of Cosmological Constant
	9.2 The Cosmological Constant Problem and Its Answer
10 Amplitude Reduction of Spacetime Fluctuations and Primordial Spectra
	10.1 Nonlinear Evolution Equations for Scalar Fluctuations
	10.2 How to Handle Nonlinear Terms
	10.3 Simplification of Nonlinear Terms and Numerical Evaluation
	10.4 Appendix: How to Solve Simplified Evolution Equation
11 Topology and Quantum Gravity
	11.1 Differential Geometry and Fermions
	11.2 \mathbb{CP}^2mathbbCP2 Gravitational Instanton
	11.3 Fermions on \mathbb{CP}^2mathbbCP2 and Spin Structures
	11.4 Generalized Spin Structures
	11.5 Particle Models with Generalized Spin Structures
	11.6  Appendix: Addenda to Differential Geometry
		11.6.1  Tools for Instantons
		11.6.2  Eguchi-Hanson Metric
		11.6.3  Spin 1/2 Dirac Index (Atiyah-Singer Index)
12 Simplicial Quantum Gravity
	12.1 Dynamical Triangulation Method
		12.1.1 Two-Dimensional Simplicial Quantum Gravity
		12.1.2 Four-Dimensional Simplicial Quantum Gravity
	12.2 Grand-Canonical Monte-Carlo Quantization
		12.2.1 Numerical Calculation Results
	12.3 On Three-Dimensional Models
Appendix  Finale
Appendix A Useful Gravitational Formulas
Appendix B About Definitions of Fermions in Curved Space
Appendix C Energy-Momentum Tensor for Each Sector
Appendix D Fundamental Constants
Appendix  References
Index