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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: February 2, 2018

Quantum Flavordynamics - SU(3) ------------------------------ SU(3) is a 3 x 3 matrix operating on ψi, ψj and ψk quark field operators. This representation is called the '3' and is referred to as the FUNDAMENTAL representation. Likewise, we can identify complex conjugate matrices operating on the anti quarks field operators, ψi*, ψj* and ψk*. This   _ representation is called the '3' (≡ 3*) and is referred to as the CONJUGATE _ representation of SU(3). Thus, ψi,j,k ≡ ψi,j,k*. The i, j and k notation can represent u, d and s quarks or R, G and B colored quarks - the math is the same. Mesons ------ We can construct a state composed of 2 quarks by taking the tensor product: 3 ⊗ 3 This has 9 states. If you work this out you get the curious result that,    _ 3 ⊗ 3 = 3 ⊕ 6 _ The 3 represents the antisymmetric states, the 6 represents the symmetric states. Note: The analogy for spin 1/2 particles (i.e. electrons) would be 2 ⊗ 2 = 1 ⊕ 3 where the 1 is antisymmetric and the 3 are the symmetric states. Using the same approach we we can also construct a state consisting of a quark and an anti quark.    _ 3 ⊗ 3 = 1 ⊕ 8 1 is an antisymmetric SINGLET that is completely invariant (transforms into itself) under SU(3) rotation. The 8 respresents the symmetric states that transform in the same way as the group generators. The 1 is referred to as the TRIVIAL representation and the 8 is referred to as the ADJOINT representation. Baryons ------- Continuing from before, we can also construct a 3 quark state as follows: 3 ⊗ 3 ⊗ 3 _ = (3 ⊕ 6) ⊗ 3 _ = (3 ⊗ 3) ⊕ (6 ⊗ 3) = 1 ⊕ 8 ⊕ 8 ⊕ 10 Again, 1 is an antisymmetric singlet that is completely invariant under SU(3) rotation. The 8s are 'mixed' symmetry states and the 10 represents the symmetric states that transform in the same way as the group generators. The lower energy 8 is the BARYON OCTET containing the protons and neutrons. The 10 corresponds to the BARYON DECUPLET. This can be summarized in terms of the Gell-Mann's EIGHTFOLD WAY. The Eightfold Way ------------------ The Eightfold Way is a theory organizing baryons and mesons into octets based on combinations of u, d and s quarks. The overall wavefunctions are constructed as follows: |ψ> = |ψFlavor> ⊗ |ψSpin> ⊗ |ψSpace> The flavor, spin and space parts are symmetric under interchange of any 2 quarks. The total momentum, J, is given by: J = L + S Physicists are generally only interested in studying the ground states of particles where L = 0. A single quark can have Sz = ± 1/2 2 quarks can have their spins aligned as Sz = -1, 0, 1 3 quarks can have their spins aligned as Sz = -3/2, -1/2, 1/2, 3/2 The Pseudoscalar Meson Octet (1 ⊕ 8) ------------------------------------- _ Pseudoscalar mesons have the qq spins antialigned. Therefore, J = 0. _ _ _ |ηo> = (1/√6)(|uu> + |dd> - 2|ss>) _ _ _ |η'> = (1/√3)(|uu> + |dd> + |ss>) _ _ |πo> = (1/√2)(|uu> - |dd>) The Vector Meson Octet (1 ⊕ 8) ------------------------------- _ Vector mesons have 0 the qq spins aligned. Therefore, J = 1. _ _ |ρo> = (1/√2)(|uu> - |dd>) _ |φo> = |ss>) _ _ |ωo> = (1/√2)(|uu> + |dd>) The Baryon Octet (1 ⊕ 8 ⊕ 8 ⊕ 10) ---------------------------------- For the baryon octet we choose J = 1/2. The principles of the Eightfold Way also applied to the isospin 3/2 baryons, forming a decuplet. The Baryon Decuplet (1 ⊕ 8 ⊕ 8 ⊕ 10) ------------------------------------- For the baryon decuplet we choose J = 3/2. The charges (green) are computed using the Gell-Mann - Nishijimi formula: Q = I3 + (1/2)(B + S) where B is the baryon number: B = (1/3)(nq - nq) = (# of quarks - # of antiquarks) and S is the strangeness quantum number. The masses (blue) follow the GELL-MANN - OKUBO mass formula that describes an approximate relation between meson and baryon masses. Baryons: (N + Ξ)/2 = (3Λ + Σ)/4 (N + Ξ)/2 ~ 1129 MeV (3Λ + Σ)/4 ~ 1135 MeV Mesons: _ [((K- + K0)/2)2 + ((K+ + K0)/2)2]/2 = (3η2 + π2)/4 _ [((K- + K0)/2)2 + ((K+ + K0)/2)2]/2 ~ 246 x 103 MeV (3η2 + π2)/4 ~ 230 x 103 MeV SU(3) FLAVOR transformations move around an 8/10 dimensional vector space which correspond to the 8/10 particles in the baryon octet/decuplet. All particles in the octet would have equal mass if we were to only consider SU(3). However, SU(3) flavor symmetry contains SU(2) isospin as a subgroup. We can see this by looking at the Gell-Mann matrices which include the Pauli spin matrices as indicated in orange. - -   | 0 1 0 | λ1 = | 1 0 0 |   | 0 0 0 | - - - -   | 0 -i 0 | λ2 = | i 0 0 |   | 0 0 0 | - - - -   | 1 0 0 | λ3 = | 0 -1 0 |   | 0 0 0 | - - - -   | 0 0 1 | λ4 = | 0 0 0 |   | 1 0 0 | - - - -   | 0 0 -i | λ5 = | 0 0 0 |   | i 0 0 | - - - -   | 0 0 0 | λ6 = | 0 0 1 |   | 0 1 0 | - - - -   | 0 0 0 | λ7 = | 0 0 -i |   | 0 i 0 | - - - -   | 1 0 0 | λ8 = 1/√3 | 0 1 0 |   | 0 0 -2 | - - The SU(3) flavor symmetry is not exact because the quarks have different rest masses and different electroweak interactions. Thus, interchanging quarks results in changes to the overall mass*. * The masses for the different particle species is not just the cumulative mass of their quarks. The vast majority of the mass is due to the kinetic energy of the quarks and to the energy of the gluon fields that bind the quarks together. Let look in detail at some examples of the isospin (flavor) multiplets that make up the above octets and decuplets. Isospin Multiplets ------------------ Recall the allowed spin/isospin states are as follows: | | | + +3/2 | | + +1 | | | | | | + +1/2 | + +1/2 | | | | + 0 | + 0 | | | | | | + -1/2 | + -1/2 | | | | | | | | | | + -1 + -3/2 and, nstates = (2l + 1) where l is the spin/isospin +---+------+------+------+------+ | | Spin | I3 | Q | J | +---+------+------+------+------+ | u | 1/2 | 1/2 | 2/3 | 1/2 | +---+------+------+------+------+ | d | 1/2 | -1/2 | -1/3 | 1/2 | +---+------+------+------+------+ | s | 1/2 | 0 | -1/3 | 1/2 | +---+------+------+------+------+ | _ | | | | | | u | 1/2 | -1/2 | -2/3 | 1/2 | +---+------+------+------+------+ | _ | | | | | | d | 1/2 | 1/2 | 1/3 | 1/2 | +---+------+------+------+------+ | _ | | | | | | s | 1/2 | 0 | 1/3 | 1/2 | +---+------+------+------+------+ The idea of isospin multiplets is best illustrated using the following example. Consider isospin 1. There are 3 states +1, -1 and 0. These states can be constructed in the following way: 1. |uu> with I3 = 1 2. |dd> with I3 = -1 3. |ud + du> with I3 = 0 and I1, I2 ≠ 0 4. |ud - du> with I3 = I1 = I2 = 0 1, 2 and 3 form a multiplet. 4 is a singlet state. In the multiplet, the states transform into each other under a SU(2) transformation. In the singlet case, the state just transforms into itself. Physically isospin multiplets are families of hadrons with approximately equal masses. All particles within a multiplet, have the same spin angular momentum, J, but differ in their electric charges. An isospin projection (denoted by I3) is associated with each charged state. Δ Particles ----------- Δ particles consist of 3 quarks therefore there are 8 possible combinations. They form an isospin, I, 3/2 multiplet. Their spins are aligned to create J = ±3/2. According to the rule, n = (2l + 1), they are allowed to have 4 states. These are: Δ++ has I3 = 3/2, J = 3/2 and Q = 2 Δ+ has I3 = 1/2, J = 3/2 and Q = 1 Δ0 has I3 = -1/2, J = -3/2 and Q = 0 Δ- has I3 = -3/2, J = -3/2 and Q = -1 The Δs have similar mass to each other but they are much heavier than either the proton or the neutron. Δs decay. Consider Δ++. It decays in the following manner: Pions are very short lived ~ 10-23s Historically, the discovery of the Δ3/2 was very important because it led to the question "how do 3 fermions with spin 1/2 occupy the same state without violating the Pauli Exclusion principle?". This also applies to protons and neutrons which are comprised of 2 identical quarks. The answer lies in the emergence of 'COLOR CHARGE' and QCD. The color charge is assigned as follows: +---------+----+---+----+----+----+---+ | | L  | R || L  | R || L  | R | +---------+----+---+----+----+----+---+ | Red | u  | d || c  | s || t  | b | +---------+----+---+----+----+----+---+ | Blue | u  | d || c  | s || t  | b | +---------+----+---+----+----+----+---+ | Green | u  | d || c  | s || t  | b | +---------+----+---+----+----+----+---+ +---------+----+---+----+----+----+---+ | Leptons | νe | e || νμ | μ | ντ || τ | +---------+----+---+----+----+----+---+ Weak interaction symmetries (SU(2)) act horizontally and never vertically. All particles in a row are transformed at the same time. Transitions from R column to L column emits a W- gauge boson. Transitions from L column to R column emits a W+ gauge boson. Strong interaction symmetries (SU(3)) act vertically and never horizontally. All particles in a column are transformed at the same time. This can be summarized as follows: The fact that SU(3) and SU(2) and U(1) don't mix is expressed as: SU(3)⊗SU(2)⊗U(1) Note: Leptons are part of SU(2) and do not participate in the color symmetry. The net color in hadrons must vanish. Since color is an unobserved property, all hadrons must be colorless objects. Therefore, protons and neutrons must consist of red, blue and green quarks in superpositions that result in no color. Protons and Neutrons -------------------- Protons and neutrons consist of 3 quarks therefore there are 8 possible combinations. They form an isospin 1/2 multiplet. Their spins are aligned to create J = ±1/2. According to the rule, nstates = (2l + 1), they are allowed to have 2 states. These are: p has I3 = 1/2, J = 1/2 and Q = 1 n has I3 = -1/2, J = -1/2 and Q = 0 NOTE: Technically speaking, we should use the full description to describe the various combinations of quarks that make up the various particles. For brevity, however, it is conventional to use a simpler shorthand notation. Under this notation a protons would be written as 'uud' and a neutron as 'ddu' with the above constructions implied. Pions ----- Pions are combination of quark anti-quark pairs. They form an isospin 1 multiplet. Their spins are aligned to create J = 0. According to the rule, nstates = (2l + 1), they are allowed to have 3 states. These are: π+ has I3 = 1, J = 0 and Q = 1 π0 has I3 = 0, J = 0 and Q = 0 π- has I3 = -1, J = 0 and Q = -1 Quantum Chromodynamics - SU(3) ------------------------------ Historically, the idea of color as a quantum number came about to overcome the violation of the Pauli Exclusion principle that was apparent with combining identical quarks into particles like the proton and neutron. As mentioned previously, color states also be described by the 3 and 3 representations (i, j, k ≡ R, G, B). In this case, the singlets are referred to as COLOR SINGLETS. We can construct the overall wavefunction for a particle as follows: |ψ> = |ψFlavor> ⊗ |ψSpin> ⊗ |ψSpace> ⊗ |ψColor> All observable particles in nature are color singlets. Furthermore, the overall wavefunction must be antisymmetric under the exchange of any 2 quarks. The flavor, space and spin parts are symmetric under interchange of any pair of labels. This implies that the color part must be asymmetric and colorless. In the case of mesons we can construct a colorless antisymmetric state as follows:   _ _ _ ψColorM = (1/√3){RR + GG + BB} In the case of baryons we can construct a colorless antisymmetric state via a Slater determinant as follows.       | R1 G1 B1 | ψColorB = (1/√6)| R2 G2 B2 |       | R3 G3 B3 |    = (1/√6){R1G2B3 - R1B2G3 + G1B3R2 - G1R3B2 + B1R2G3 - B1G2R3} The antisymmetric colour combination is the only combination of 3 quark states that has no net color. Gluons ------ Force between quarks: The exchange of a gluon between 2 quarks. This represents the force between quarks. Force between gluons: Exchange of gluon between 2 gluons. This represent a forces between gluons. THUS, GLUONS ARE DIFFERENT FROM PHOTONS IN THAT THEY CAN INTERACT WITH EACH OTHER. Gluons BEHAVE like quark anti-quark pairs with respect to their color symmetry. They are NOT mesons but we can use the same representations. Thus we can write:    _ 3 ⊗ 3 = 1 ⊕ 8 There are many ways of presenting the 8. One commonly used list is: _ _ _ _ _ _ (1/√2)(RB + BR) (i/√2)(RB - BR) (1/√2)(RR - GG) _ _ _ _ (1/√2)(BG + GB) (i/√2)(BG - GB) _ _ _ _ _ _ _ (1/√2)(GR + RG) (i/√2)(GR - RG) (1/√6)(RR + GG - 2BB) _ _ _ (1/√3)(RR + GG + BB) which is colorless. These are the combined states that are constructed according to the principle of superposition. If one were somehow able to make a direct measurement of the color of a gluon in one of these states, there would be a 50% chance of it having a particular color charge. We can construct LADDER OPERATORS that map these eigenstates into the gluon octet. Ladder operators were introduced in the context of angular momentum and rotation so it is also appropriate to use them in the context of rotations associated with SU(3). T± = T ± iT' Thus, for example: _ _ _ _ _ (1/√2)(RB + BR) + i(i/√2)(RB - BR) => BR and, _ _ _ _ _ (1/√2)(RB + BR) - i(i/√2)(RB - BR) => BR Not surprisingly, the gluon colour wavefunctions have a similar form to the mesons. The singlet does not appear in nature and we can disregard it for now. We are left with thr 8 gluons that constitute the strong force. We can combine 2 gluons as follows: 8 ⊗ 8 = 1 ⊕ 63 The color singlet state of the 2 gluon combinations corresponds to a GLUEBALL. The Nature of the Strong Interaction ------------------------------------ Gluons carry a color charge. The color force resulting from this charge does not follow an inverse square force like the electromagnetic force. In fact it behaves in the opposite way and increases linearly with distance. This behavior is responsible for QUARK CONFINEMENT. As a quark-antiquark pair separates, the chromoelectric flux lines associated with the gluon field between them attract each other to form flux tubes rather like stretched chewing gum. As the distance increases, more and more energy is required to pull them apart. At some point, it becomes energetically favorable for the tube to 'snap' rather than extend further, and a new quark–antiquark pair spontaneously appears. In so doing, energy is conserved because the energy of the color-force field is converted into the mass of the new quarks, and the color-force field can 'relax' back to an unstretched state. In reality, in high energy particle accelerators, the tube actually snaps in multiple places forming a spray of hadrionic particles referred to as a JET. This process of jet formation is referred to a HADRONIZATION. Quark confinement is the reason individual quarks are never observed in nature. The color force appears to exert little force at short distances so that the quarks are like free particles within the confining boundary of the color force and only experience the strong confining force when they begin to get too far apart. The behavior is referred to as term ASYMPTOTIC FREEDOM. Gauge symmetry prohibits gluons to have mass and as far as we know the SU(3) gauge symmetry is not broken (see the Higgs mechanism). Massless particles are mediators for long range forces such as electromagnetism. However, the range of the strong interaction is limited by confinement. Like photons, gluons are spin 1 particles with 0 conventional charge. Nuclear Force -------------- The nuclear force is the force that hold the nucleons together. In 1935 (long before QCD) Hideki Yukawa modelled the nuclear force in terms of the YUKAWA POTENTIAL given by: V(r) = -g2exp(-kmr)/r where g and k are constants, m is the mass and r is the radial distance. The nuclear force is maximally attractive between nucleons at distances of about 1 fermi (10-15 metres) between their centers, but exponentially decreases at distances beyond this. At about 2.5 fermi the force becomes negligible. At distances less than 0.7 fermi, the nuclear force becomes repulsive. The repulsive component is responsible for the size of the nucleus. The nuclear force is stronger than the Coulomb force between protons at distance of less than 1.7 fermi. Nucleons have a radius of about 0.8 fermi. Based on the range of the force, Yukawa calculated that the virtual particles associated with the nuclear force should have a mass in the neighborhood of 100 MeV. This led to the eventual discovery of the pion. With the development of QCD, the nuclear force is considered to be a residual effect of the more fundamental strong force. As previously stated quarks and gluons are confined within nucleons. However, some combinations of quarks and gluons can 'leak' away from nucleons, in the form of pions. As a result, the nuclear force is sometimes called the residual strong force. Pions are different from gluons in that gluons are massless and are confined within the nucleons. In contrast, pions have a mass of about 140 MeV and are not confined. The 'mechanism' of the nuclear force is illustrated in the following graphic (courtesy of Wikipedia) The Origin of Mass in QCD ------------------------- When we talk about combinations of quarks in the hadrons we are referring to VALENCE quarks. In reality the picture is much more complicated. Virtual quark-antiquark pairs known as SEA quarks form when gluons split. This process also occurs in reverse so that the annihilation of 2 sea quarks produces a gluon. The result is a constant flux of gluon splits and creations inside the hadron. This flux is referred to as the quark/gluon condensate and characterizes the QCD VACUUM state. The VACUUM EXPECTATION LEVEL of the QCD vacuum is given by: _ VEV = <ψiψi> where ψi is the quark field operator summed over i (flavor). _ _ _ = <uu + dd + ss> It is possible to imagine a valence quark as a CURRENT quark that is 'dressed' by their interactions with this condensate. Valence quarks are often referred to as CONSTITUENT quarks with the following approximate masses: Constituent Mass ---------------------- Up quark 336 MeV Down quark 340 MeV Strange quark 486 MeV Charm quark 1550 MeV Bottom quark 4730 MeV Top quark 177000 MeV The bulk of the mass of the hadrons comes from these high energy interactions with the QCD vacuum. The current quarks themselves are very light and get their mass via the HIGGS MECHANISM. The process of aquiring mass is related to CHIRAL SYMMETRY breaking. Chiral Symmetry Breaking ------------------------ We have said before that the strong force behaves differently to other forces in that it becomes weaker as the distance decreases. This is equivalent to saying that the coupling to the strong field, αS, gets smaller as the distance becomes smaller. This is referred to as RUNNING coupling. The running coupling introduces a dimensional parameter, ΛQCD, which sets the scale at which the coupling constant becomes large and the physics becomes highly non linear and NON- PERTURBATIVE requiring extremely intense numerical computations such as LATTICE QCD. Lattice QCD is similar to lattice QED in that fields representing quarks are defined by lattice cells, while the gluon fields are defined by the propagators connecting neighboring cells. This approximation approaches continuum QCD as the spacing between lattice cells is reduced to zero. Below ΛQCD the physics becomes PERTURBATIVE and perturbation methods can be used to solve complex problems. In fact, ΛQCD simply sets the scale for strong interaction physics. Experimentally it is determined that ΛQCD ~ 200 MeV. With this it is possible to introduce the notion of light and heavy quarks. Briefly speaking, light quarks are ones having masses much smaller than ΛQCD, and heavy quarks having masses much larger than ΛQCD. Therefore, under this regime u, d and s quarks qualify as light quarks whereas the c, b and t quarks may be regarded as heavy. To understand the physics of the u and d quarks, it is convenient to consider a theoretical limit in which their masses are exactly zero. The quarks can be right-handed or left-handed. The chirality can be selected using 5th Dirac matrix: γ5 = γ0γ1γ2γ3 ψLf = (1/2)(1 - γ5f ψRf = (1/2)(1 + γ5f where f labels the flavor (u or d). The total quark field is simply a linear combination of the two, ψ = ψL + ψR Without mass there is no coupling between the L and R quarks and they live in 2 separate worlds (see the Dirac equation). Because the u and d quarks are both massless, they may be regarded as two independent states of the same object forming a two component spinor in 'isospin space' in the same way that the proton and neutron can be regarded as 2 states of the same particle if charge is ignored. - - - - | u'L,R | = UL,R| uL,R | | d'L,R |     | dL,R | - - - - UL,R are 2 x 2 unitary matrices: U(2)L⊗U(2)R The quark part of the QCD Lagrangian is: _ L = ψ(iγμDμ - m)ψ when m = 0 this reduces to: _ L = ψiγμDμψ It is easy to see that the Lagrangian is invariant under under these transformations. U(2)L⊗U(2)R can be decomposed into: SU(2)L⊗SU(2)R⊗U(1)V⊗U(1)A Where V and A represent the VECTOR and AXIAL symmetries. Vector means that L and R transform in the same way. Axial means L and R transform in the opposite way. Therefore, U(1)V acts as: ψL -> exp(iθ)ψL and ψR -> exp(iθ)ψR and U(1)A acts as: ψL -> exp(iθ)ψL and ψR -> exp(-iθ)ψR Applying Noether's theorem to the Lagrangian shows that the conserved quantity for U(1) is electric charge. Charge is related to the baryon number through the Gell-Mann-Nishijima formula (Q = I3 + (1/2)(B + S)) and therefore the baryon number can be interpreted as the conserved quantity. It turns out that this is true for U(1)V. However, U(1)A does not correspond to a conserved quantity because the symmetry is explicitly violated due to a quantum anomaly. This will not be explained here since it is not crucial to the ensuing discussion. If we just focus on the two SU(2) symmetries we can write: SU(2)L⊗SU(2)R This corresponds to 6 generators: 3 for left, TLa and 3 for right, TRa. These generators can be combined as follows: TLa + TRa corresponding to SU(2)V and TLa - TRa corresponding to SU(2)A In addition there are 6 conserved Noether currents that can be derived from the Lagrangian:    _ JμL,R,a = ψL,Rγμ(Ta/2)ψL,R with ∂μJμR,L = 0 These can be also be combined as follows to form the vector and axial currents:    _ Vμa = JμL,a + JμR,a = ψγμ(Ta/2)ψ and    _ Aμa = JμL,a - JμR,a = ψγ5γμ(Ta/2)ψ With corresponding charges: QVa = ∫d3(Ta/2)ψ and QAa = ∫d3γ5(Ta/2)ψ that are, likewise, generators of SU(2)L⊗SU(2)R In QCD, it is the vacuum that SPONTANEOUSLY breaks the chiral symmetry. In particular, it is the axial generators that get broken. Thus, we can write: SU(2)L⊗SU(2)R⊗U(1)V⊗U(1)A -> SU(2)V⊗U(1)V Where SU(2)V is the diagonal vector subgroup describing the isospin. Isospin is conserved in the strong interaction but not in the weak interaction because the strong interaction is agnostic to charge differences. Also, the implication from the Gell-Mann - Nishijimi equation is that if charge is conserved and baryon number is conserved, then I3 must also be conserved. Spontaneous chiral symmetry breaking by itself results in 3 massless NAMBU GOLDSTONE BOSONS (NGBs) - one for each generator of the SU(2)A symmetry that is broken. SU(2)V and U(1)V are unbroken. Likewise, we can write the following for SU(3) flavor symmetry involving u, d and s quarks if we assume that the s quark is very similar to the u and d quarks. The s quark is heavier than the u and d quarks but if we set the scale correctly (see renormalization) we can treat them in the same way. U(3)L⊗U(3)R decomposes to: SU(3)L⊗SU(3)R⊗U(1)V⊗U(1)A Again, if we just focus on the two SU(3) symmetries we can write: SU(3)L⊗SU(3)R In this case there are 16 generators. Again, the 8 axial generators get spontaneosly broken by the QCD vacuum to produce 8 NGBs. As before, SU(3)V and U(1)V are unbroken. Thus, SU(3)L⊗SU(3)R⊗U(1)V⊗U(1)A -> SU(3)V⊗U(1)V If the symmetry is EXPLICITLY broken as well as spontaneously broken, the NGBs are not massless and are termed PSEUDO NAMBU GOLDSTONE BOSONS (PNGBs). It is the quark masses that explicitly break the symmetry. The observed pion masses are obtained using CHIRAL PERTURBATION THEORY. This leads to the GELL-MANN - OAKES - RENNER relationship:   _ _ mπ2 = -(1/2fπ2)(mu + md)<uu + dd> + O(mu,d2) \ \ (explicit) (spontaneous) Where fπ is the experimentally determined PION DECAY CONSTANT (~ 93 MeV). Similarly, for the remaining members of the pseudoscalar octet: mK2 ∝ (ms + mu,d) mη2 ∝(mu + md + 4ms) Vector Mesons -------------- The quark color charges lead to a spin-spin interaction between quarks and antiquarks called the CHROMOMAGNETIC INTERACTION. The chromomagnetic energy depends on the relative spin orientations of the quark and the antiquarks. This is analagous to the electromagnetic interaction that results in the spliiting of energy levels. The equations are very similar. However, the spin-spin interaction is much much stronger than the electromagnetic case due to the much larger coupling constant for the strong force.