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Astronomy

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Astronomical Distance Units .
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Celestial Coordinates .
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Celestial Navigation .
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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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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity .
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases .
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Kirchoff's Laws
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Laplace's and Poisson's Equations
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Lorentz Force Law
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Maxwell's Equations
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Moments and Torque
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Nuclear Spin
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One Dimensional Wave Equation .
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Pascal's Principle
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Phase and Group Velocity
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Planck Radiation Law .
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Poiseuille's 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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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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Friedmann-Robertson-Walker Equations
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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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Olbers' Paradox
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Penrose Diagrams
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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

Game Theory

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The Truel .

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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Gravitational Waves
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One-forms
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Quantum Gravity
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Relativistic, Cosmological and Gravitational Redshift
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Ricci Decomposition
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Ricci Flow
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Stress-Energy Tensor
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Stress-Energy-Momentum Tensor
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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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Hamiltonian Formulation .
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem

Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Arithmetic and Geometric Sequences and Series
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Asymptotes
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Augmented Matrices and Cramer's Rule
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Basic Group Theory
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Basic Representation Theory
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Binomial Theorem (Pascal's Triangle)
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Building Groups From Other Groups
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Completing the Square
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Complex Numbers
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Composite Functions
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Conformal Transformations .
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Conjugate Pair Theorem
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Contravariant and Covariant Components of a Vector
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Derivatives of Inverse Functions
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Double Angle Formulas
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Eigenvectors and Eigenvalues
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Euler Formula for Polyhedrons
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Factoring of a3 +/- b3
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Fourier Series and Transforms .
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Fractals
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Gauss's Divergence Theorem
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Grassmann and Clifford Algebras .
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Heron's Formula
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Index Notation (Tensors and Matrices)
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Inequalities
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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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Line Integrals, ∮
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Logarithms and Logarithmic Equations
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Matrices and Determinants
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Matrix Exponential
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Mean Value and Rolle's Theorem
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Modulus Equations
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Orthogonal Curvilinear Coordinates .
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Parabolas, Ellipses and Hyperbolas
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Piecewise Functions
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Polar Coordinates
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Polynomial Division
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Quaternions 1
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Quaternions 2
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Regular Polygons
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Related Rates
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Sets, Groups, Modules, Rings and Vector Spaces
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Similar Matrices and Diagonalization .
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Spherical Trigonometry
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Stirling's Approximation
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Sum and Differences of Squares and Cubes
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Symbolic Logic
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Symmetric Groups
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Tangent and Normal Line
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Taylor and Maclaurin Series .
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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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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Chi Squared Goodness of Fit
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Conditional Probability
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Confidence Intervals
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Data Types
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Expected Value
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Monte Carlo Methods
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Non Parametric Tests
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One-Way ANOVA
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Pearson Correlation
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Permutations and Combinations
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Pooled Variance and Standard Error
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Probability Distributions
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Probability Rules
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Sample Size Determination
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Sampling Distributions
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption

Quantum Computing

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The Qubit .

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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Ladder Operators .
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Multi Electron Wavefunctions
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Pauli Exclusion Principle
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Pauli Spin Matrices
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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
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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 Quantum Harmonic Oscillator .
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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

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

The Essential Mathematics of Lie Groups --------------------------------------- Abstract Lie Groups ------------------- Lie groups lie at the intersection of two fundamental fields of mathematics: group theory and differential geometry. They are finite dimensional DIFFERENTIABLE MANIFOLDS, G, together with a group structure on G such that the group law, * (+ or .), and inversion are smooth maps: μ: G x G -> G (g1,g2) -> g1*g2 inverse: G -> G g -> g-1 The axioms regarding the identity and associativity also apply. Lie Algebra ----------- Differentiable manifolds have tangent spaces. As such a Lie group G has a tangent space, g. At first this is just a vector space but the group structures on the manifold give rise to an additional structure of the tangent space. In a vector space, we cannot, in general, multiply vectors, but in the case of a tangent space to a Lie group we can. Specifically, the tangent space is equipped with a multiplication defined by the Lie bracket, i.e. x.y -> [x,y]. This additional structure of the tangent space satisfies: Closure: [a,b] = c (= 0 for abelian case) Jacobi Identity: [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 All together this additional structure is called a LIE ALGEBRA. More formally we can say a Lie algebra over a Field, F, is a vector space over F endowed with an operation [ , ]. In this context a field is a set with additive and multiplicative operations together with an additive and multiplicative inverse (for non elements). A field as described. should not be confused with a vector field. Lie algebras are easier to calculate and locally are a good approximation to the object they represent. It is worth reiterating that the Lie algebra is distinct from the group itself. Group operations (. or +) pertain to G. In a group, you multiply (or add) 2 elements together to get another element of the group. In the corresponding algebra, which lives in the tangent space, you take 2 elements of the algebra, and commute them to get another element of the algebra. The commutator is not defined on G itself. In the additive case, G is abelian and [,] = 0 since a + b = b + a. Lie groups are CONTINUOUS groups. As a result, multiplication tables don't work. Instead, it is the commutator in the algebra that plays a role similar to the multiplication law for the group. Example: Consider the group of real numbers under multiplication. The real line of positive numbers is a manifold of dimension 1. Equivalently it is a vector space over the field R of real numbers (that is, over itself) of dimension 1 (Note: A Vector Space is any set of elements that can be added together and multiplied by a scalar. The set of real numbers satisfy these axioms and therefore forms a vector space.) G = (ℝ\{0},.) r1 = 2.03 and r2 = 3.06 Identity: 1 Multiplication: 2.03 x 3.06 = 5.09 ∈ G Associativity: 2.03 x 3.06 = 3.06 x 2.03 Inverse: 2.03(1/2.03) = 3.06(1/3.06) = 1 It is easy to see that the commutator equals 0. [2.03,3.06] = 2.03 x 3.06 - 3.06 x 2.03 = 0 Also, [1.01,[2.03,3.06]] + [2.03,[3.06,1.01]] + [3.06,[1.01,2.03]] = 0 So G = (ℝ\{0},.) is an Abelian Lie group. We will now look at Lie algebras in more detail. We start by recognizing that manifolds are also associated with vector fields. Manifolds and Vector Fields --------------------------- Prerequisite: 'The Essential Mathematics of General Relativity'. A vector field on a manifold can be defined as a function that takes a point, P, in a manifold, M, and gives back a vector, v, in TP in M. Thus, v: M -> TPM such that vP ∈ TPM for all P ∈ M. Since a vector field is an assignment of a tangent space to each point in M, we can write the vector fields, X and Y, at P in terms of tangent vectors: X = xii and Y = yii For example, in 2D we might have the vector field X(x,y): - - - - X(x,y) = | xy || ∂x | ≡ xy∂x + (x + y)∂y | x + y || ∂y | - - - - At (2,1) and (2,2) we get the vectors | 2 | and | 3 | - - - - (2,3) o (4,4) o | 4 | which are ^ and ^ respectively. | 4 | | / - - (2,1) o (2,2) o Dual Space ---------- dxji = δji Where dxj are the dual basis vectors. Therefore, dxjX = dxjxii = xidxji = xiδji = xj When evaluated on a vector field, X, the dual basis vectors return the components of X. The Directional Derivative on Vector Fields ------------------------------------------- Tangent vector ≡ directional derivative, v = d/dλ = (dxi/dλ)(∂/∂xi) = xii Where xi = dxi/dλ are the components and ∂i = ∂/∂xi are the bases. Let f be a smooth differentiable scalar function on the manifold, C(M) then, Xf = xiif Represents the change in f along a curve in the direction of the tangent vector, v. This produces a real number at the point P. SO THE VECTOR FIELD, X, CAN BE INTERPRETED AS A DERIVATIVE OPERATOR. X also obeys the product rule (fg)' = f'g + fg'. Therefore, X(fg) = (Xf)g + f(Xg) Example. - - Suppose we have the tangent vector | 1 | = V and | 2 | - - we want to find the value of f = x2 + y3 at the point (1,0). Xf = 1∂xf + 2∂yf = (1)2x + (2)3y2 At (1,0) we get: Xf = 2 So the DD of f in the direction of the tangent vector, V, at the point (1,0) is 2. A concrete example of this would be the following. Suppose X is a wind velocity field across the North American continent and f is a temperature described by a scalar function defined over the same area. An airborne temperature probe caught in the wind field would measure how the temperature changes as it drifts across the continent. Compare the DD with the divergence and curl: divF = ∂F1/∂x + ∂F2/∂y = y + 1 curlF = (∂F2/∂x - ∂F1/∂y)k = (1 - x)k The Lie Derivative on Vector Fields ----------------------------------- Let X be a collection of all fields on M (i.e. X ∈ Γ(M)). Let f be a function on M (i.e. f ∈ C(M)). The Lie derivative of f, LXf, is given by: LXf = xiif So the Lie derivative for a scalar is the directional derivative discussed above. The Lie derivative of a vector can be be found as follows: γ(λ) is called an INTEGRAL CURVE of the vector field. Integral curves for an electric field or magnetic field are known as field lines, and integral curves for the velocity field of a fluid are known as streamlines. Consider: xi' = xi + dxi = xi + xidλ since xi = dxi/dλ A vector, vi, transforms as: vi'(x') = (∂xi'/∂xj)vj(x) = [(∂xi'/∂xj) + (∂xi'/∂xj)dλ]vj(x) = [δij + (∂jxi)dλ]vj(x) = vi(x) + (∂jxi)vi(x)dλ We now use the expand vi(x + dx) as a Taylor series f(x + dx) ~ f(x) + f'(x)dx vi(x + dx) ~ vi(x) + dxjjvi(x) But, dxi = xidλ. Therefore, vi(x + dx) ~ vi(x) + xjjvi(x)dλ LXvi = lim (vi(x + dx) - vi(x))/dλ dλ->0 = lim [[vi(x) + xjjvi(x)dλ] dλ->0 - [vi(x) + vi(x)(∂jxi)dλ]]/dλ Therefore, LXvi = xjjvi(x) - vj(x)(∂jxi) The Killing Vector ------------------ A vector field, K, is a Killing field if: LXg = 0 Where g is the metric. This means that when you move the metric g a small amount along K, g doesn't change. A typical use of the Killing Field is to express a symmetry in General Relativity (in which the geometry of spacetime as distorted by gravitational fields is viewed as a 4-dimensional Riemannian manifold). In a static configuration, in which nothing changes with time, the time vector will be a Killing vector, and thus the Killing field will point in the direction of the forward motion in time (i.e. g does mot change with time). Covariant vs Lie Derivative --------------------------- The covariant derivative allows vectors in different tangent spaces to be compared and relies on the manifold being equipped with a connection. If you have a vector X at one point, the connection tells you what are the parallel vectors at nearby points. In contrast, the Lie derivative evaluates the change of one vector field along the flow of another vector field. The Lie Bracket on 2 Vector Fields ---------------------------------- Consider 2 vector fields and the aforementioned scalar function, f: LX(LYf) = xii(yjjf) = xiiyjjf + xiyj2ijf And, LY(LXf) = yi(∂i(xjjf)) = yiixjjf + yixj2ijf The pesky term in each case is the ∂2ijf term because coordinate transformations never contain second derivatives. However, if we subtract the 2 results we get: [LX(LYf) - LY(LXf)] = {xi(∂iyj) - yi(∂ixj)}∂jf ... 1. The {} term is a new vector field with components made from the 2 old vector fields (i.e. Z = Zjj). We can write this is as: [LX(LYf) - LY(LXf)] = {(xii)(yjj) - (yii)(xjj)}f ... 2. Knowing that X = xii and Y = yii enables us to write: LX(LYf) - LY(LXf) = [X,Y]f ≡ X(Yf) - Y(Xf) We can also write 1. in the form: [X,Y] = [X,Y]jj where [ ]j ≡ { ... } This is the LIE BRACKET. Because it is a vector field it takes a point, P, in M and gives back a tangent vector, X. Indeed the RHS has the structure of a tangent vector. The Lie bracket is often written as: [X,Y] = JYX - JXY ... 3. where JY and JX are the JACOBEAN MATRICES of Y and X respectively. NOTE THAT THE LIE BRACKET OF 2 VECTOR FIELDS IS THE SAME AS THE LIE DERIVATIVE OF 2 VECTOR FIELDS. i.e. LXY = [X,Y] ----------------------------------------------------- Aside: A slightly simpler explanation of the above is: XY = Xμμ(Yνν) Using the product rule: = XμYν(∂2μν) + Xμ(∂μYν)∂ν There is no particular interpretation for the second derivative, and as we saw above, it does not transform nicely. But suppose we take the commutator: [X,Y] = XY - YX = Xμ(∂μYν)∂ν - Yμ(∂μXν)∂ν The second derivatives cancel leaving the directional derivative. [X,Y] = (Xμ(∂μYν) - Yμ(∂μXν))∂ν = [X,Y]νν Where [X,Y]ν = (Xμ(∂μYν) - Yμ(∂μXν)) are the components of a new vector field made from the 2 old vector fields. ----------------------------------------------------- Example: Consider the following 2 vector fields (tangent vectors) on the 2-sphere. The 2 sphere is a 2D surface that forms the boundary of a sphere in 3 dimensions. - - - - - - - - | y || ∂x | | 0 || ∂x | X = | -x || ∂y | Y = | z || ∂y | | 0 || ∂z | | -y || ∂z | - - - - - - - - Or, X = x11 + x22 + x33 Y = y11 + y22 + y33 Therefore, X = y∂1 - x∂2 and Y = z∂2 - y∂3 Using equation 2. from above: ΣΣ[xii,yjj] ij = [y∂x,z∂y] + [y∂x,-y∂z] + [-x∂y,z∂y] + [-x∂y,-y∂z] = y∂x(z)∂y - z∂y(y)∂x = -z∂x + y∂x(-y)∂z - (-y)∂z(y)∂x = 0 + (-x)∂y(z)∂y - z∂y(-x)∂y = 0 + (-x)∂y(-y)∂z - (-y)∂z(-x)∂y = x∂z = -z∂x + x∂z - - - - | -z || ∂x | = | 0 || ∂y | | x || ∂z | - - - - Using equation 3. from above: JX: - - | ∂x(y) ∂y(y) ∂z(y) | | ∂x(-x) ∂y(-x) ∂z(-x) | | ∂x(0) ∂y(0) ∂z(0) | - - JXY: - - - - - - | 0 1 0 || 0 | | z | | -1 0 0 || z | = | 0 | | 0 0 0 || -y | | 0 | - - - - - - Which is NOT a tangent vector. JY: - - | ∂x(0) ∂y(0) ∂z(0) | | ∂x(z) ∂y(z) ∂z(z) | | ∂x(-y) ∂y(-y) ∂z(-y) | - - JYX: - - - - - - | 0 0 0 || y | | 0 | | 0 0 1 || -x | = | 0 | | 0 -1 0 || 0 | | x | - - - - - - Which is also NOT a tangent vector. However, JYX - JXY gives: - - - - - - | 0 | | z | | -z | | 0 | - | 0 | = | 0 | | x | | 0 | | x | - - - - - - Which IS a tangent vector. So the 2 approaches give the same result. We can envisage X and Y as 2 velocity fields. Suppose we now move along X for a brief time t, then along Y for another brief interval t. Next we switch back to X, but with a minus sign for time t, and then to Y with a minus sign for time t. We have tried to retrace our path but we will fail to return to our exact starting point. Cross Product ------------- If we consider the special case of vector fields on the manifold R3, the set of all smooth vector fields forms an infinite-dimensional vector space. In this instance the Lie bracket relates a 3-dimensional subspace of vector fields on R3 to the cross product. Note: This is a different case from the previous example where 2 specific vector fields, X and Y, on the specific manifold S2 illustrates the general property that the Lie bracket of any two vector fields on a manifold is itself a vector field on the manifold. In that instance the resulting vector has nothing to do with the cross product of two vectors. Example: - - - - - - - - - - - - | 0 || ∂x | | z || ∂x | | -y || ∂x | X = | -z || ∂y | Y = | 0 || ∂y | Z = | x || ∂y | | y || ∂z | | -x || ∂z | | 0 || ∂z | - - - - - - - - - - - - JX: - - | ∂x(0) ∂y(0) ∂z(0) | | ∂x(-z) ∂y(-z) ∂z(-z) | | ∂x(y) ∂y(y) ∂z(y) | - - JXY: - - - - - - | 0 0 0 || z | | 0 | | 0 0 -1 || 0 | = | x | | 0 1 0 || -x | | 0 | - - - - - - JY: - - | ∂x(z) ∂y(z) ∂z(z) | | ∂x(0) ∂y(0) ∂z(0) | | ∂x(-x) ∂y(-x) ∂z(-x) | - - JYX: - - - - - - | 0 0 1 || 0 | | y | | 0 0 0 || -z | = | 0 | | -1 0 0 || y | | 0 | - - - - - - JYX - JXY gives: - - - - - - | y | | 0 | | y | | 0 | - | x | = | -x | | 0 | | 0 | | 0 | - - - - - - The cross product is defined as: (ux,uy,uz) ^ (vx,vy,vz) = (uyvz - uzvy,uzvx - uxvz,uxvy - uyvx) = |u||v|sinθ - - - - - - | 0,-z,y | ^ | z,0,-x | = | zx,zy,y2 | - - - - - - Now, x ^ y = z y ^ x = -z y ^ z = x z ^ y = -x z ^ x = y x ^ z = -y x2 = y2 = z2 = 0 Therefore, - - - - - - | 0,-z,y | ^ | z,0,-x | = | y,-x,0 | - - - - - - Left Invariant Vector Field --------------------------- Any Lie group, G, acts on itself by left multiplication. (recall that the only group operation is .). If h ∈ G is fixed, and g ∈ G, we denote this action by: Lg: h -> gh or, Lg(h) = gh This is interpreted as the translation of a point h by g. We will show the justification for this interpretation later when we discuss one-parameter subgroups. A vector field, X, is a function that takes a point, h, in G (which in this case is also a manifold) and gives back a vector, vh, in ThG. Consider a vector field, X, at h, (X)h We can do 2 things: 1. We can evaluate X at another point g. Therefore, X(Lgh) is (X)h evaluated at gh, (X)gh. 2. We can take the tangent vector of X at h, xh and apply the pushforward, Lg*, to xg to get the vector at gh. Therefore, Lg*xh = xgh Let's look at this in more detail. From the discussion 'The Essential Mathematics of General Relativity' we had that the pushforward of a tangent vector from a manifold M to a manifold N is given by: (φ*v)αα = (φ*)αμvμα Where (φ*)αμ is the Jacobean. If M and N are the same manifold then: (φ*)vαα = vμμ We now replace φ by L (for left), vαα with xa and multiply by g to get: Lg*xa = xga Therefore, Lg*, is a linear map from the tangent space at h to the tangent space at the point gh. Now we can define: Lg*(xh) := (Lg*(X))gh ^ ^ | | vector in vector ThG field at gh ≡ gdLgX (see later) This is true because a vector field assigns a tangent vector to every point on the manifold and vice-versa. Therefore, we can write 2 as: Lg*Xh = Xgh ... 3. If 1. and 3. yield the same result we say that X is LEFT INVARIANT VECTOR FIELD, i.e. Lg*Xh = X(Lg(h)) = (X)gh Therefore, the vector field at gh is the SAME as the vector field h. A LIVF satisfies the commutation diagram:   vh Lg* vgh   ThG -----------> TghG   ^ ^   | | Xh | | Xgh   | |   | |   G -------------> G   h Lg gh If we now let h = the element at the identity we get: Lg*XI = X(Lg(I)) = XgI Therefore, X at g is the same as the same as X at the identity. Note: We picked a tangent vector at the identity and then translated that vector to another point. However, we could have equally well picked an element that is not at the identity but working with the vector at the identity is much easier. Summarizing: The Lie algebra of a Lie group is defined to be the left invariant vector fields on the group. Integral Curves (aka Flow) of a Vector Field -------------------------------------------- An integral curve for the vector field, X, with an initial condition, P0, is a curve such that: dγ(t)/dt = Xγ(t) and, γ(0) = P0 What this says is that an integral curve, γ(t), is a curve with initial condition P0 such that for every point, P = γ(t), on the curve, the tangent vector, vγ(t), is equal to the value of the vector field, X, at P. That is, the vector field gives the value of the tangent to the path at every point. One-Parameter Subgroups ----------------------- Consider a left invariant vector field, XL. Let us denote the integral curve of XL as γL(t). By definition, XL assigns a tangent vector to each point on γL(t). γL(t) is referred to as a ONE PARAMETER SUBGROUP where an element of the subgroup is γL(t) for each t. dγL(t)/dt = vγL(t) But we know for a LIVF that, LγL(t)*v = vγL(t) Therefore, we can say, dγL(t)/dt ≡ LγL(t)* There are 2 ways we can move along the curve, translation by s and pushforward by γL(s). Translation by s gives γL(s + t). Therefore, dγL(s + t)/dt = vγL(s + t) Pushforward by γL(s) gives: LγL(s)*γL(t) = γL(s)vγL(t) = vγL(s)γL(t) Clearly, these 2 ways have to be equal. Therefore, γL(s + t) = γL(s)γL(t) THIS CAN ONLY BE SATISFIED IF γ(t) IS AN EXPONENTIAL FUNCTION. Note that: . γ(0) = 1 γ(t) at the identity. . γ(-t) = (γ(t))-1 This now also justifies the previous interpretation of Lg(a) = ga as a translation, i.e. γL(g)γL(a) ≡ γL(a + g) ----------------------------------------------------- Digression: What we have described is called a GROUP HOMOMORPHISM. A homomorphism is a way to compare 2 groups. Consider 2 groups, (G,*) and (H,◆), a group homomorphism from G to H is a function φ: G -> H such that for all x and v ∈ in G it holds that: φ(x*y) = φ(x)◆φ(y) Where * and ◆ are the group operations. Example: G = ℝ under + H = ℝ' under . φ: G -> H φ(x + y) = φ(x).φ(y) φ = exponential function. So φ is a homomorphism. A one-parameter subgroup of a Lie group is a group homomorphism i.e. a smooth map: γ: ℝ -> G where (ℝ,+) and (G,.) For which: γ(s + t) = γ(s)γ(t) ----------------------------------------------------- Recovering the Lie Group from its Lie Algebra --------------------------------------------- The tangent vector to the integral curve, γ(t), at the identity is given by: v = dγ(t)/dt|t=0 Where we have simplied the notation such that vγL ≡ v. Which has the solution: γ(t) = exp(tv) This is the ONE-PARAMETER SUBGROUP from before which can be written as a group element, X. So we can recover the Lie algebra from the Lie group by taking the differential of the group element evaluated at t = 0. One can also recover the Lie algebra by simply taking the logarithm of the group element. To show that these operations are equivalent, we need to prove that dX/dt|t=0 = lnX. d(exp(tv))/dt|t=0 = ln(exp(tv)) xexp(tv)|t=0 = v Therefore, v = v. The exponential and the logarithm give a 1-to-1 correspondence, continuous in both directions, between a neighborhood of 1 in any Lie group and a neighborhood of 0 in its Lie algebra. These operations are illustrated in the following diagram. We can linearize the one-parameter Lie subgroup γ(t) = exp(tx) by considering infinitessimal changes of the form: γ(t) ~ I + tx + O((tx)2) Which is just the Taylor series expansion of the exponential function. Matrix Lie Groups ----------------- Matrix Lie groups are a special class of abstract Lie groups which are also smooth differentiable manifolds. For a matrix Lie group the vector fields associated with manifolds are represented by matrices. The most interesting Lie groups turn out to be matrix groups. For matrices, one can regard the smooth manifold as the General Linear group GL(n,ℝ) which is the set of n x n invertible matrices with det ≠ 0. Indeed, sometimes this is written as M(n,ℝ). Having det ≠ 0 means that the matrices have an inverse and satisfy the other axioms of Lie groups. Hence, GL(n,ℝ) is a Lie group. Matrix Exponential ------------------ To move from the matrix Lie algebra to the Lie group we use the MATRIX EXPONENTIAL. The power series expansion of this is: M = exp(tm) = Σ(tm)k/k! X ∈ G, m ∈ g, k=0 Where M and m are now matrices. One can also move from the Lie group back to the Lie algebra by differentiating the matrix exponential: m = dM/dt|t=0 M ∈ G, m ∈ g, Alternatively, we can use the MATRIX LOGARITHM which is the inverse map of the matrix exponential (i.e. log = exp-1). The power series expansion is: m = ln(1 + M) = Σ(-1)k+1Mk/k M ∈ G, x ∈ g, k=1 This is a difficult calculation for most practical purposes. Fortunately, if M can be diagonalized to a matrix MD, then: M = PMDP-1 lnM = P(lnMD)P-1 Where the diagonal elements are ln(d11), ln(d22) etc. and all other entries are 0. For matrices that cannot be diagonalized one needs to find its JORDAN DECOMPOSITION and calculate the logarithm of the JORDAN BLOCKS. Jacobi Formula ------------- Jacobi's formula states: det(exp(x)) = exp(Tr(x)) Where x is a real or complex matrix. The Lie Bracket of Matrix Lie Groups ------------------------------------ Since for a matrix Lie groups the vector fields are represented by matrices, the Lie bracket corresponds to the usual commutator for a matrix group. We can demonstrate this by using the BAKER-CAMPBELL-HAUSDORFF formula: exp(z) = exp(x)exp(y) = exp(x + y + (1/2)[x,y] + ... ) Let W = [X,Y] Therefore: exp(tw) = [exp(tx),exp(ty)] = exp(tx)exp(ty) - exp(ty)exp(tx) = exp(tx + ty + (1/2)[tx,ty] + ... ) - exp(tx + ty + (1/2)[ty,tx] + ... ) = exp(tx + ty + (1/2)[tx,ty] + ... ) - exp(tx + ty - (1/2)[tx,ty] + ... ) From here we can either take the log of both sides to get (setting t = 1): w = (x + y + (1/2)[x,y] + ... ) - (x + y - (1/2)[x,y] + ... ) = [x,y] or, d{exp(tw))/dt|t=0 = [x,y] Examples -------- G = (SO(2),.): - - - - AB = | cosα -sinα || cosβ -sinβ | | sinα cosα || sinβ cosβ | - - - - - - = | cosαcosβ - sinαsinβ -cosαsinβ - sinαcosβ | | sinαcosβ + cosαsinβ -sinαsinβ + cosαcosβ | - - - - = | cos(α + β) -sin(α + β) | ∈ G | sin(α + β) cos(α + β) | - - With generator, Xg = dA(α)/dα|α = 0 - - = | 0 -1 | | 1 0 | - - [Xg,Xg] = 0 Therefore (SO(2),.) is an Abelian Lie group. G = (SO(3),.): - -   | 1 0 0 | Rx = | 0 cosθ -sinθ |   | 0 sinθ cosθ | - - - -   | cosθ 0 sinθ | Ry = | 0 1 0 |   | -sinθ 0 cosθ | - - - -   | cosθ -sinθ 0 | Rz = | sinθ cosθ 0 |   | 0 0 1 | - - With generators: - -   | 0 0 0 | Xx = | 0 0 -1 |   | 0 1 0 | - - - -   | 0 0 1 | Xy = | 0 0 0 |   | -1 0 0 | - - - -   | 0 -1 0 | Xz = | 1 0 0 |   | 0 0 0 | - - Rx.Ry is another rotation (∴ ∈ G) but it is not equal to Rz. Whereas [Xx,Xy] = Xz G = (SU(3),.): An SU(3) group element is obtained by exponentiating the 3 × 3 traceless Hermitian Gell-Mann matrices. Unfortunately the resulting expression is rather complicated. However, for λ1 - λ7 we can use the following formula: exp(itθλi/2) = I + iλisin(tθ/2) + λi2(cos(tθ/2) - 1) i.e. exp(itλ1/2) - - - -   | 1 0 0 | | 0 1 0 | exp(itλ1/2) = | 0 1 0 | + i| 1 0 0 |sin(tθ/2)   | 0 0 1 | | 0 0 0 | - - - - - - | 1 0 0 | + | 0 1 0 |(cos(tθ/2) - 1) | 0 0 0 | - - - - | cos(tθ/2) isin(tθ/2) 0 | = | isin(tθ/2) cos(tθ/2) 0 | = X | 0 0 1 | - - Note that det(exp(iλ1/2)) = exp(Tr(λ1)) = 1 (Jacobi's identity) and XX = I The eigenvalues/eigenvectors of this matrix are: - - - - - - | 0 | | -1 | | 1 | 1| 0 | exp(-itθ/2)| 1 | exp(itθ/2)| 1 | | 1 | | 0 | | 0 | - - - - - - The eigenvectors are linearly independent and therefore we can diagonalize X. - - - - | 0 -1 1 |    | 0 0 1 | P = | 0 1 1 | P-1 = | -1/2 1/2 0 | | 1 0 0 |    | 1/2 1/2 0 | - - - - After diagonalization (XD = P-1XP): - - | 1 0 0 | | 0 cos(tθ/2) - isin(tθ/2) 0 | | 0 0 cos(tθ/2) + isin(tθ/2) | - - - - | 1 0 0 | ≡ | 0 exp(-itθ/2) 0 | | 0 0 exp(itθ/2) | - - Take the logarithm: - - | 0 0 0 | | 0 -itθ/2 0 | = γ | 0 0 itθ/2 | - - Apply the similarity transform: λ1 = PγP-1 - - - - - - | 0 -1 1 || 0 0 0 || 0 0 1 | = | 0 1 1 || 0 -itθ/2 0 || -1/2 1/2 0 | | 1 0 0 || 0 0 itθ/2 || 1/2 1/2 0 | - - - - - - - - | 0 itθ/2 0 | = | itθ/2 0 0 | | 0 0 0 | - - = itθλ1/2 We get the same result if we find the derivative at t = 0. Thus, - -     | -(θ/2)sin(tθ/2) i(θ/2)cos(tθ/2) 0 | dX/dt|t=0 = | i(θ/2)cos(tθ/2) -(θ/2)sin(tθ/2) 0 |     | 0 0 1 |t=0 - - - - | 0 1 0 | = (iθ/2)| 1 0 0 | | 0 0 0 | - - LIVF: Lg*XI = gdX/dt|t=0 = gd(exp(tλ1))/dt|t=0 = gλ1 = (λ1)g where g ∈ G, v ∈ TgG - - So the LIVF is: | 0 1 0 | | 1 0 0 | | 0 0 0 | - - Finally, lets show that algebraic operations can only be performed in the tangent space and not on the manifold itself. In other words, elements have to be 'referred' back to the tangent space for alebraic operations. For example, [λ12] = iλ3 (f123 = 1) - -    | costθ + isintθ 0 0 | and X3 = exp(itθλ3) = | 0 costθ - isintθ 0 |    | 0 0 1 | - - - -      | iθ 0 0 | dX3/dt|t=0 = | 0 -iθ 0 | = iλ3      | 0 0 0 | - - If we try and perform the same operation on the manifold after exponentiation we get: - -    | -2isin2tθ 0 0 | [exp(itθλ1),exp(itθλ2)] = | 0 2isin2tθ 0 |    | 0 0 1 | - - With dX3/dt|t=0 = 0 Which clearly doesn't work! For SU(3), the group elements represent rotations of complex 3 component (u, d, s) vectors in an 8 dimensional space. Note that if we had performed the same calculation using Rz from SO(3) we would get the following LIVF. - - - - - - | 0 -1 0 || x | | -y | | 1 0 0 || y | = | x | | 0 0 0 || z | | 0 | - - - - - - Which is: XL = -y∂x + x∂y For SO(3), the group elements represent rotations of real 3 component (x, y, z) vectors in a 3 dimensional space. Group Actions ------------- Let G be a group and S be a set. Let g ∈ G and s ∈ S. One can construct a left action of g on s denoted as g.s as follows: g.s = gs Likewise, one can construct a right action as follows: s.g = g.s One can construct a left action from a right action by composing with the inverse operation of the group. Therefore, g.s = gs ≡ sg-1 We can think of the multiplication as the act of moving points around inside of S. For example, in D4, the symmetry group of the square, the elements move the set of vertices to achieve rotations and reflections. (see see D4 here) We will focus on left actions. A left group action has to satisfy the following 3 axioms. Let s ∈ S and g,h ∈ G then: 1. g.s ∈ S 2. e.s = s 3. g.(h.s) = (gh).s For example, from D4 it is easy to see that axiom 3. is satisfied i.e. d.[r.s] = [dr].s where s = {1,2,3,4}. G Acting on Itself ------------------ To make G act on itself we let S = G. For example, in D4 the element d, performs a reflection about the vertical axis and the element, r, performs a 90° rotation. The left action d.r = dr combines these 2 operations to achieve a reflection about the right diagonal. All possible operations can be seen in the Cayley table. For example, from the Cayley table for D4 it is easy to see that d.[r.r] = [dr].r. In addition, to the left action, a group also acts on itself is by conjugation. Therefore, g.x = gxg-1 x, g ∈ G Proof of axioms 2 and 3 for conjugation: 2. e.x = ese-1 3. g1.(g2.x) = g1.(g2xg2-1) since g2.x = g2xg2-1 = g1.p where p = g2xg2-1 = g1pg1-1 since g1.p = g1pg1-1 = g1(g2xg2-1)g1-1 = (g1g2)x(g2-1g1-1) = (g1g2)x(g1g2)-1 = (g1g2).x Q.E.D The conjugacy equation also describes matrix similarity. However, conjugacy in this context is more restrictive than similarity in that g, g-1 and x have to be in G. The Adjoint Representation -------------------------- We start with the conjugacy class equation and change the notation. We use upper case letters for elements of the group, G, and lower case letters for elements of the Lie algebra. Therefore, Z = YXY-1 X, Y, Z ∈ G The differential of the conjugation action, evaluated at the identity, is called the adjoint action. Therefore, AdYX = dZ/dt|t=0 X is obtained by exponentiating the generator x. Therefore, we can write: AdYX = d(YXY-1)/dt|t=0 = d(Yexp(tx)Y-1)/dt|t=0 = Yxexp(tx)Y-1|t=0 = YxY-1 Where Y, X ∈ G and x ∈ the Lie algebra. Therefore, the adjoint representation is a representation where the group elements in G act on a basis which is comprised of the group generators in the tangent space at the identity and not the column vectors that appears in the fundamental representation. In other words, in the adjoint representation, the Lie algebra itself forms the basis upon which G acts. Derivative of Ad = ad --------------------- We can pass from a representation of a Lie group G, to a representation of its Lie algebra by taking the derivative at the identity. adYx = d(AdYX)/dt|t=0 = d(YxY-1)/dt|t=0 = d(exp(ty)xexp(-ty))/dt|t=0 = yexp(ty)xexp(-ty) + exp(ty)x(-y)exp(-ty)|t=0 = yx - xy = [y,x] The ad operator is the adjoint action of the group generators on themselves. As one would expect, AdY and adY are related through the exponential map: AdY = exp(adY) Diagramatically, we can summarize these relationships in a commutative diagram. ad g --------> gl(g) : : : exp : exp v v G --------> GL(g) Ad Non-Abelian Gauge Theory ------------------------ Lie groups and Lie algebras play an important role in the various dynamical theories which govern the behaviour of particles - the gauge theories. We now present the relationship between Lie algebras and gauge theories. We have seen that the elements of a group are related to the Lie algebra via the exponential map, i.e. U = exp(tTa) We have used U in place of the group element, X, as a reminder that the matrices are unitary. We now want to look at the situation where t is not a constant but becomes a function of the spacetime coordinates, xμ. This means that the above formula becomes: U(xμ) = exp(t(xμ)Ta) Now, the group elements can act on a field, φ, where φ denotes an array of different fields written as a column vector (e.g. quark fields). The fields transform among themselves, just as the components of a three-dimensional vector transform among themselves under the group of rotations. In other words the fields form a basis and the action of the group on this basis constitutes a group representation. These fields transform as (dropping the μ and (x) for brevity): φ -> φ' = Uφ ∂μφ -> (∂μφ)' = U∂μφ + (∂μU)φ However, the second term is a problem. What we want is a derivative such that: Dμφ -> (Dμφ)' = UDμφ Try: Dμφ = ∂μφ - Wμφ Rearranging, Wμφ = ∂μφ - Dμφ Consider the transformation: Wμφ -> (Wμφ)' = (∂μφ)' - (Dμφ)' = U∂μφ + (∂μU)φ - UDμφ = {U∂μ - UDμ + (∂μU)}φ = {U(∂μ - Dμ) + (∂μU)}φ = {UWμ + (∂μU)}φ But φ = U-1φ'. Therefore, (Wμφ)' = {UWμU-1 + (∂μU)U(x)-1}φ' So Wμ transforms as: Wμ -> Wμ' = UWμU-1 + (∂μU)U-1 Dμφ = ∂μφ - igWμφ is called the GAUGE COVARIANT DERIVATIVE. Wμ is a matrix of the type generated by an infinitesimal gauge transformation and carries information regarding the group from one spacetime point to another. Wμ can be decomposed as: Wμ = WμaTa Where Wμa are the GAUGE FIELDS that act as coefficients to the generators, Ta. When these gauge fields are quantized, their quanta are called GAUGE BOSONS. The gauge covariant derivative differs from the 'ordinary' covariant derivative ∇μvρ = ∂μvρ + Γρμνvν in that the latter applies to transformations that behave properly under a change of basis. Now lets look at 2 successive transformations. Wμ -> Wμ' = U1WμU1-1 + (∂μU1)U1-1 Wμ -> Wμ'' = U2{U1WμU1-1 + (∂μU1)U1-1}U2-1 + (∂μU2)U2-1 = (U2U1)Wμ(U2U1)-1 + (∂μU2U1)(U2U1)-1 = U3WμU3-1 + (∂μU3)U3-1 Therefore, performing 2 gauge transformations is also a gauge transformation. In addition, the transformation with U = 1 corresponds to the identity transformation Wμ -> Wμ' = 1Wμ1-1 + (∂μ1)1-1 = Wμ And a transformation with U followed by a transformation with U-1 yields the identity. Wμ -> Wμ' = 1.1-1Wμ(1.1-1)-1 + (∂μ1.1-1)(1.1-1)-1 = 1Wμ This fits the definition of a group whose elements are the gauge transformations. This is called the GAUGE GROUP. Specifically, the first term in the transformation, Wμ -> Wμ' = U1WμU1-1 + (∂μU1)U1-1 is the conjugacy class equation that gives the adjoint representation of a Lie group. In the the adjoint representation U(1) has 1 one dimensional generator corresponding to the photon. SU(2) has 3 three dimensional generators that correspond to the W+/- and Z0 bosons. SU(3) has 8 eight dimensional generators corresponding to the gluons. The adjoint representation contrasts with the defining (fundamental) representations that describe the Fermionic particles (e-1, u, d and s quarks). For the second term U(x) displaced over an infinitesimally small distance dx will differ from U(x) by an infinitesimal gauge transformation. Therefore, we can use a Taylor series expansion to get U(x + dx) ~ U(x) + U'(x)dx + .... Thus, U(x + dx) = U(x) + ε(x)TaU(x)dx + ... Therefore, by comparison, ∂μU(x) = ε(x)TaU(x) So, ∂μU(x)U(x)-1 = ε(x)Ta As such it is also part of the Lie algebra. Field Strength, Gμν ------------------- We define: Gμν := [Dμ,Dν] = ∂μWν - ∂νWμ - [Wμ,Wν] Or, setting Gμν = GμνaTa, we get: GμνaTa = ∂μWνaTa - ∂νWμaTa - WμaWνb[Ta,Tb]     = ∂μWνaTa - ∂νWμaTa - WμaWνbfabcTc In the abelian case the commutator = 0 so Gμν = = ∂μWν - ∂νWμ