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Astronomy

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Celestial Coordinates
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Celestial Navigation
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Distance Units
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Location of North and South Celestial Poles

Chemistry

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Avogadro's Number
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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Center of Mass Frame
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Comparison Between Gravitation and Electrostatics
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Compton Effect
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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions
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Electric Fields
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Error Analysis
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Gauss's Law of Universal Gravity
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Gravity - Force and Acceleration
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Kinematics
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Kinetic Theory of Gases
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Maxwell's Equations
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Moments and Torque
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One Dimensional Wave Equation
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Phase and Group Velocity
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Planck Radiation Law
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Radioactive Decay
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Refractive Index
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Rotational Dynamics
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Simple Harmonic Motion
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Specific Heat, Latent Heat and Calorimetry
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Stefan-Boltzmann Law
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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect
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Wien's Displacement Law
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Young's Modulus

Climate Change

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Keeling Curve

Cosmology

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Penrose Diagrams
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Baryogenesis
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Cosmic Background Radiation and Decoupling
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CPT Symmetries
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Dark Matter
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Geometries of the Universe
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes
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Planck Units
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Stephen Hawking's Last Paper
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Stephen Hawking's PhD Thesis
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The Big Bang Model

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets
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Capital Structure
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Dividend Discount Formula
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Curvature and Parallel Transport
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Geodesics
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Gravitational Time Dilation
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Quantum Gravity
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Relativistic, Cosmological and Gravitational Redshift
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Tensors
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The Area Metric
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The Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Metric Tensor
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Vierbein (Frame) Fields
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World Lines Refresher

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory
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Euler-Lagrange Equation
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem

Macroeconomics

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Mathematics

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Amplitude, Period and Phase
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Basic Group Theory
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Binomial Theorem (Pascal's Triangle)
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Building Groups From Other Groups
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Complex Numbers
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Contravariant and Covariant Components of a Vector
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Grassmann and Clifford Algebras
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Integration By Parts
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Introduction to Conformal Field Theory
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Inverse of a Function
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Law of Sines and Cosines
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The Essential Mathematics of Lie Groups
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The Integers Modulo n Under + and x
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The Limit Definition of the Exponential Function
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Tic-Tac-Toe Factoring
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Trapezoidal Rule
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Unit Vectors
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Vector Calculus
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity and Chirality
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Klein-Gordon and Dirac Equations
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Regularization and Renormalization
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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors
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The Vacuum Catastrophe

Probability and Statistics

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Box and Whisker Plots
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Factor Analysis
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Programming and Computer Science

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Hashing
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How this site works ...
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Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory

Quantum Mechanics

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Basic Relationships
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Bell's Theorem
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Bohr Atom
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Clebsch-Gordan Coefficients
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Commutators
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Dyson Series
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Electron Orbital Angular Momentum and Spin
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Entangled States
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Heisenberg Uncertainty Principle
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Photoelectric Effect
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Position and Momentum States
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Probability Current
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Schrodinger Equation for Hydrogen Atom
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Schrodinger Wave Equation (continued)
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Observer Effect
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The Qubit
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids
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Fermi-Dirac Statistics
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Intrinsic and Extrinsic Semiconductors
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors
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Electromagnetic 4 - Potential
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Energy and Momentum, E = mc2
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Lorentz Invariance
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Lorentz Transform
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Lorentz Transformation of the EM Field
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Newton versus Einstein
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Spinors - Part 1
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Spinors - Part 2
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The Lorentz Group
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Velocity Addition

Statistical Mechanics

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Black Body Radiation
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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas

String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification

Superconductivity

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BCS Theory
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Introduction to Superconductors
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

test

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test

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
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Formulas
Last modified: January 28, 2020

Introduction to Conformal Field Theory -------------------------------------- Conformal Group --------------- The conformal group of a space is the group of transformations from the space to itself that preserve angles. Associativity: (Ω12).Ω3 = Ω1.(Ω2).Ω31 + Ω2) + Ω3 = Ω1 + (Ω2) + Ω3 Inverse: Ω1-1 = I Ω1 + Ω-1 = 0 Closure: Ω12 = Ω3 Ω1 + Ω2 = Ω3 All conformal groups are Lie groups. Rescaling versus Coordinate Transformations ------------------------------------------- Coordinates Transformations --------------------------- The equation: gμν(x) = (∂x'ρ/∂xμ)(∂x'σ/∂xν)gρσ(x') represents an active coordinate transformation of the metric from frame x' -> frame x. This means that the components of the metric change under the transformation. Weyl Transformation ------------------- The equation: gμν(x) -> Ω(x)gμν Where Ω(x) is real-valued positive scalar function defined on the manifold, represents a scaling of the metric that changes the proper distances at each point by a factor and the factor may depend on the place – but not on the direction of the line whose proper distance we measure (because Ω is a scalar). In other words, the Weyl transformation takes us to a coordinate system where the metric has the same form as the one we started with, but the points have all been moved around and pushed closer together or further apart depending on the scale factor (aka conformal factor). To summarize, it is a transformation which acts on the metric tensor itself and not on the coordinates. Conformal Transformation ------------------------ A conformal transformation can now be defined as a coordinate transformation which acts on the metric as a Weyl transformation. In other words, it is a coordinate transformation that produces the same result as scaling the metric by Ω(x). Therefore, (∂x'ρ/∂xμ)(∂x'σ/∂xν)gρσ(x') = Ω(x)gμν(x) ^ ^ | | Coordinate transformation Weyl transformation Metrics obeying this equation are said to be conformally equivalent. Consider the infinitesimal transformation x'μ = xμ + εμ(x) Expanding the LHS using ∂xμ/∂xν = δμν: ∂x'ρ/∂xμ = ∂x'ρ/∂xμ + ∂ερ/∂xμ = δρμ + ∂ερ/∂xμ Likewise, ∂x'σ/∂xν = δσν + ∂εσ/∂xν Therefore, gρσρμ + ∂ερ/∂xμ)(δσν + ∂εσ/∂xν) = Ωgμν ∴ gρσρμδσν + δρμ(∂εσ/∂xν) + (∂ερ/∂xμσν + O(ε2)) = Ωgμν ∴ gμν + gρσδρμ(∂εσ/∂xν) + (∂ερ/∂xμ)gρσδσν) = Ωgμν ∴ gμν + (∂εμ/∂xν) + (∂εν/∂xμ) = Ωgμν Therefore, gμν + Δgμν = Ωgμν ... 1a. Where, Δgμν = (∂εμ/∂xν) + (∂εν/∂xμ) ... 1b. Now, (∂εμ/∂xν) + (∂εν/∂xμ) must be proportional to gμν Therefore, (∂εμ/∂xν) + (∂εν/∂xμ) = kgμν ... 2. Now taking the trace of both sides gives: gμν((∂εμ/∂xν) + (∂εν/∂xμ)) = gμνkgμν ∴ gμν((∂εμ/∂xν) + (∂εν/∂xμ)) = kδμμ ∴ gμν(∂νεμ + ∂μεν) = kδμμ ∴ ∂μεμ + ∂νεν = kδμμ ∴ 2∂μμε = kd Now ∂μμε = ∂0ε0 + ∂1ε1 + ... + ∂dεd ≡ ∂.ε Therefore, 2(∂.ε) = kd or, k = 2(∂.ε)/d Therefore 2. becomes, ∂νεμ + ∂μεν = (2∂.ε)/d)gμν = Δgμν ... 3. 2-dimensions ------------ The case where d = 2 is of special interest. If we let gμν = δμν we can write 3. as: (∂μεν + ∂νεμ) = (∂.ε)δμν μ = ν gives: ∂1ε1 = ∂2ε2 and μ ≠ ν gives: ∂1ε2 = -∂2ε1 In other words, εμ satisfies the Cauchy-Riemann equations. Combining 3. with 1a. and 1b. gives: gμν + gμν(2/d)∂.ε = Ωgμν ∴ gμν(1 + (2/d)∂.ε) = Ωgμν Or, by comparing: Ω = 1 + (2/d)∂.ε Note that: ds2 -> ds2 + (∂μεν + ∂νεμ)dxμdxν We now need one more equation. We start with ∂μεν + ∂νεμ = (2/d)(∂.ε)gμν and take the derivative of both sides w.r.t ∂ν. ∂ν(∂μεν + ∂νεμ) = ∂ν((2/d)(∂.ε)gμν) ∴ ∂μνεν + ∂ννεμ = (2/d)∂μ(∂.ε) ∴ ∂μ(∂.ε) + ∂ννεμ = (2/d)∂μ(∂.ε) Now take the derivative of both sides of this w.r.t. ∂ν. ∂ν(∂μ(∂.ε) + ∂ννεμ) = (2/d)∂νμ(∂.ε) ∴ ∂μν(∂.ε) + ∂νννεμ = (2/d)∂νμ(∂.ε) ∴ ∂μν(∂.ε) + □∂νεμ = (2/d)∂νμ(∂.ε) ... 4. Where □ is the d'Alembert operator = ∂νν. Now, ∂νεμ = (1/2)(∂νεμ + ∂μεν) = (1/d)(∂.ε)gμν Using this, 4. becomes: ∂μν(∂.ε) + □(1/d)(∂.ε)gμν = (2/d)∂νμ(∂.ε) Which simplifies to: ∂μν(∂.ε)d + □(∂.ε)gμν = 2∂νμ(∂.ε) Rearranging: gμν(□ + (d - 2)∂μν)(∂.ε) = 0 Conformal Manifolds ------------------- The Weyl Tensor --------------- The Riemann tensor, Rabcd, keeps track of how much a vector that is parallel transported around a small parallelogram deviates from the its initial direction after returning to its original position. All the information about the curvature of a Riemann manifold is contained in the Riemann tensor. The Ricci tensor represents the amount by which the volume of an object differs from its volume in Euclidean space in the presence of matter. The Ricci tensor Rbd is obtained by contracting in the first and third indeces of the Riemann tensor. The energy/matter source of Ricci curvature is the stress-energy tensor, Tμν. The Weyl tensor is the part of the curvature that exists if there is no matter present. It is basically the Riemann tensor with all of its contractions removed. It differs from the Ricci tensor in that it does not convey information on how the volume of the object changes, but rather only how the shape of the body is distorted by tidal forces. Therefore, (i) If the Ricci tensor is zero but Weyl tensor is not, the object will be deformed by tidal forces without changing its volume. (ii) If the Weyl tensor is zero but the Ricci tensor is not, the object will only change its volume but not its shape. The Weyl tensor measures the deviation of a Riemannian manifold from conformal flatness. If it vanishes, the manifold is locally conformally equivalent to a flat manifold. The fancy way of saying this is that a manifold is conformally flat if each point has a neighborhood, U, that can be mapped to flat space by a conformal transformation, gab -> Ω2ηab ---- ^ | conformally flat form So, ds2 = Ω2δabdxadxb This does not mean that the manifold itself is necessarily flat. The Weyl tensor has the special property that it is invariant under conformal changes. That is: C'abcd = Ω2Cabcd For this reason the Weyl tensor is also called the conformal tensor. The Weyl tensor can be shown to be 0 for 2 and 3 dimensions. Therefore, any 2D or 3D Riemannian manifold is conformally flat. For example, the 2-sphere can be stereographically projected onto a 2D plane as shown below. Therefore, the 2-sphere is locally conformally flat but is not itself flat for obvious reasons.