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Mathematical Principles of Theoretical Physics

(Rick Simeone) #1



  • 1 General Introduction

  • 1.1 Challenges of Physics and Guiding Principle.

  • 1.2 Law of Gravity, Dark Matter and Dark Energy.

  • 1.3 First Principles of Four Fundamental Interactions

  • 1.4 Symmetry and Symmetry-Breaking

  • 1.5 Unified Field Theory Based On PID and PRI

  • 1.6 Theory of Strong Interactions.

  • 1.7 Theory of Weak Interactions

  • 1.8 New Theory of Black Holes

  • 1.9 The Universe

  • 1.10 Supernovae Explosion and AGN Jets.

  • 1.11 Multi-Particle Systems and Unification.

  • 1.12 Weakton Model of Elementary Particles

  • 2 Fundamental Principles of Physics

  • 2.1 Essence of Physics

  • 2.1.1 General guiding principles

  • 2.1.2 Phenomenological methods

  • 2.1.3 Fundamental principles in physics.

  • 2.1.4 Symmetry.

  • 2.1.5 Invariance and tensors

  • 2.1.6 Geometric interaction mechanism

  • 2.1.7 Principle of symmetry-breaking

  • 2.2 Lorentz Invariance

  • 2.2.1 Lorentz transformation.

  • 2.2.2 Minkowski space and Lorentz tensors.

  • 2.2.3 Relativistic invariants.

  • 2.2.4 Relativistic mechanics

  • 2.2.5 Lorentz invariance of electromagnetism.

  • 2.2.6 Relativistic quantum mechanics

  • 2.2.7 Dirac spinors

  • 2.3 Einstein’s Theory of General Relativity

  • 2.3.1 Principle of general relativity.

  • 2.3.2 Principle of equivalence

  • 2.3.3 General tensors and covariant derivatives viii CONTENTS

  • 2.3.4 Einstein-Hilbert action

  • 2.3.5 Einstein gravitational field equations.

  • 2.4 Gauge Invariance

  • 2.4.1 U( 1 )gauge invariance of electromagnetism

  • 2.4.2 Generator representations ofSU(N)

  • 2.4.3 Yang-Mills action ofSU(N)gauge fields

  • 2.4.4 Principle of gauge invariance.

  • 2.5 Principle of Lagrangian Dynamics (PLD)

  • 2.5.1 Introduction.

  • 2.5.2 Elastic waves.

  • 2.5.3 Classical electrodynamics

  • 2.5.4 Lagrangian actions in quantum mechanics.

  • 2.5.5 Symmetries and conservation laws.

  • 2.6 Principle of Hamiltonian Dynamics (PHD).

  • 2.6.1 Hamiltonian systems in classical mechanics.

  • 2.6.2 Dynamics of conservative systems.

  • 2.6.3 PHD for Maxwell electromagnetic fields

  • 2.6.4 Quantum Hamiltonian systems.

  • 3 Mathematical Foundations

  • 3.1 Basic Concepts

  • 3.1.1 Riemannian manifolds

  • 3.1.2 Physical fields and vector bundles

  • 3.1.3 Linear transformations on vector bundles

  • 3.1.4 Connections and covariant derivatives.

  • 3.2 Analysis on Riemannian Manifolds.

  • 3.2.1 Sobolev spaces of tensor fields.

  • 3.2.2 Sobolev embedding theorem.

  • 3.2.3 Differential operators.

  • 3.2.4 Gauss formula

  • 3.2.5 Partial Differential Equations on Riemannian manifolds

  • 3.3 Orthogonal Decomposition for Tensor Fields

  • 3.3.1 Introduction.

  • 3.3.2 Orthogonal decomposition theorems.

  • 3.3.3 Uniqueness of orthogonal decompositions.

  • 3.3.4 Orthogonal decomposition on manifolds with boundary.

  • 3.4 Variations with divA-Free Constraints

  • 3.4.1 Classical variational principle

  • 3.4.2 Derivative operators of the Yang-Mills functionals

  • 3.4.3 Derivative operator of the Einstein-Hilbert functional.

  • 3.4.4 Variational principle with divA-free constraint.

  • 3.4.5 Scalar potential theorem

  • 3.5 SU(N)Representation Invariance.

  • 3.5.1 SU(N)gauge representation

  • 3.5.2 Manifold structure ofSU(N).

  • 3.5.3 SU(N)tensors CONTENTS ix

  • 3.5.4 Intrinsic Riemannian metric onSU(N).

  • 3.5.5 Representation invariance of gauge theory.

  • 3.6 Spectral Theory of Differential Operators

  • 3.6.1 Physical background.

  • 3.6.2 Classical spectral theory

  • 3.6.3 Negative eigenvalues of elliptic operators

  • 3.6.4 Estimates for number of negative eigenvalues.

  • 3.6.5 Spectrum of Weyl operators

  • 4 Unified Field Theory

  • 4.1 Principles of Unified Field Theory

  • 4.1.1 Four interactions and their interaction mechanism.

  • 4.1.2 General introduction to unified field theory

  • 4.1.3 Geometry of unified fields

  • 4.1.4 Gauge symmetry-breaking.

  • 4.1.5 PID and PRI

  • 4.2 Physical Supports to PID

  • 4.2.1 Dark matter and dark energy.

  • 4.2.2 Non well-posedness of Einstein field equations

  • 4.2.3 Higgs mechanism and mass generation

  • 4.2.4 Ginzburg-Landau superconductivity.

  • 4.3 Unified Field Model Based on PID and PRI

  • 4.3.1 Unified field equations based on PID.

  • 4.3.2 Coupling parameters and physical dimensions.

  • 4.3.3 Standard form of unified field equations.

  • 4.3.4 Potentials of the weak and strong forces.

  • 4.3.5 Gauge-fixing problem

  • 4.4 Duality and decoupling of Interaction Fields.

  • 4.4.1 Duality

  • 4.4.2 Gravitational field equations derived by PID.

  • 4.4.3 Modified QED model.

  • 4.4.4 Strong interaction field equations

  • 4.4.5 Weak interaction field equations.

  • 4.5 Strong Interaction Potentials

  • 4.5.1 Strong interaction potential of elementary particles

  • 4.5.2 Layered formulas of strong interaction potentials

  • 4.5.3 Quark confinement.

  • 4.5.4 Asymptotic freedom

  • 4.5.5 Modified Yukawa potential.

  • 4.5.6 Physical conclusions for nucleon force.

  • 4.5.7 Short-range nature of strong interaction

  • 4.6 Weak Interaction Theory

  • 4.6.1 Dual equations of weak interaction potentials.

  • 4.6.2 Layered formulas of weak forces.

  • 4.6.3 Physical conclusions for weak forces

  • 4.6.4 PID mechanism of spontaneous symmetry breaking. x CONTENTS

  • 4.6.5 Introduction to the classical electroweak theory.

  • 4.6.6 Problems in WS theory.

  • 5 Elementary Particles

  • 5.1 Basic Knowledge of Particle Physics.

  • 5.1.1 Classification of particles.

  • 5.1.2 Quantum numbers

  • 5.1.3 Particle transitions

  • 5.1.4 Conservation laws

  • 5.1.5 Basic data of particles

  • 5.2 Quark Model

  • 5.2.1 Eightfold way.

  • 5.2.2 Irreducible representations ofSU(N)

  • 5.2.3 Physical explanation of irreducible representations

  • 5.2.4 Computations for irreducible representations

  • 5.2.5 Sakata model of hadrons.

  • 5.2.6 Gell-Mann-Zweig’s quark model.

  • 5.3 Weakton Model of Elementary Particles

  • 5.3.1 Decay means the interior structure.

  • 5.3.2 Theoretical foundations for the weakton model

  • 5.3.3 Weaktons and their quantum numbers.

  • 5.3.4 Weakton constituents and duality of mediators

  • 5.3.5 Weakton confinement and mass generation

  • 5.3.6 Quantum rules for weaktons

  • 5.4 Mechanisms of Subatomic Decays and Electron Radiations.

  • 5.4.1 Weakton exchanges.

  • 5.4.2 Conservation laws

  • 5.4.3 Decay types.

  • 5.4.4 Decays and scatterings

  • 5.4.5 Electron structure.

  • 5.4.6 Mechanism of bremsstrahlung.

  • 5.5 Structure of Mediator Clouds Around Subatomic Particles

  • 5.5.1 Color quantum number.

  • 5.5.2 Gluons

  • 5.5.3 Color algebra.

  • 5.5.4 w∗-color algebra

  • 5.5.5 Mediator clouds of subatomic particles

  • 6 Quantum Physics

  • 6.1 Introduction.

  • 6.2 Foundations of Quantum Physics.

  • 6.2.1 Basic postulates.

  • 6.2.2 Quantum dynamic equations.

  • 6.2.3 Heisenberg uncertainty relation and Pauli exclusionprinciple.

  • 6.2.4 Angular momentum rule

  • 6.3 Solar Neutrino Problem. CONTENTS xi

  • 6.3.1 Discrepancy of the solar neutrinos.

  • 6.3.2 Neutrino oscillations

  • 6.3.3 Mixing matrix and neutrino masses

  • 6.3.4 MSW effect.

  • 6.3.5 Massless neutrino oscillation model

  • 6.3.6 Neutrino non-oscillation mechanism.

  • 6.4 Energy Levels of Subatomic Particles

  • 6.4.1 Preliminaries

  • 6.4.2 Spectral equations of bound states.

  • 6.4.3 Charged leptons and quarks

  • 6.4.4 Baryons and mesons

  • 6.4.5 Energy spectrum of mediators

  • 6.4.6 Discreteness of energy spectrum.

  • 6.5 Field Theory of Multi-Particle Systems

  • 6.5.1 Introduction.

  • 6.5.2 Basic postulates forN-body quantum physics.

  • 6.5.3 Field equations of multi-particle systems

  • 6.5.4 Unified field model coupling matter fields.

  • 6.5.5 Atomic spectrum.

  • 7 Astrophysics and Cosmology

  • 7.1 Astrophysical Fluid Dynamics

  • 7.1.1 Fluid dynamic equations on Riemannian manifolds.

  • 7.1.2 Schwarzschild and Tolman-Oppenheimer-Volkoff (TOV) metrics.

  • 7.1.3 Differential operators in spherical coordinates.

  • 7.1.4 Momentum representation

  • 7.1.5 Astrophysical Fluid Dynamics Equations

  • 7.2 Stars.

  • 7.2.1 Basic knowledge

  • 7.2.2 Main driving force for stellar dynamics

  • 7.2.3 Stellar interior circulation

  • 7.2.4 Stellar atmospheric circulations

  • 7.2.5 Dynamics of stars with variable radii

  • 7.2.6 Mechanism of supernova explosion

  • 7.3 Black Holes.

  • 7.3.1 Geometric realization of black holes.

  • 7.3.2 Blackhole theorem

  • 7.3.3 Criticalδ-factor

  • 7.3.4 Origin of stars and galaxies.

  • 7.4 Galaxies.

  • 7.4.1 Introduction.

  • 7.4.2 Galaxy dynamics.

  • 7.4.3 Spiral galaxies

  • 7.4.4 Active galactic nuclei (AGN) and jets

  • 7.5 The Universe

  • 7.5.1 Classical theory of the Universe xii CONTENTS

  • 7.5.2 Globular universe with boundary.

  • 7.5.3 Spherical Universe without boundary

  • 7.5.4 New cosmology

  • 7.6 Theory of Dark Matter and Dark energy

  • 7.6.1 Dark energy and dark matter phenomena

  • 7.6.2 PID cosmological model and dark energy.

  • 7.6.3 PID gravitational interaction formula

  • 7.6.4 Asymptotic repulsion of gravity

  • 7.6.5 Simplified gravitational formula

  • 7.6.6 Nature of dark matter and dark energy.

  • Index

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