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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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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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Gravity - Force and Acceleration
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Kinematics
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Kinetic Theory of Gases .
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Maxwell's Equations
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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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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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Lecture Notes on International Financial Management
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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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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Tensors
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The Area Metric
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The Equivalence Principal
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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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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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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: August 20, 2020

The Lorentz Group ----------------- The Lorentz group is the SO(3,1) symmetry group of Special Relativity. It is the extension of the rotation group SO(3) that mixes spatial directions to the group that mixes both space and time. The Lorentz group is a Lie group. x.x = xTηx - - - - = | t x y z || -t | - - | x | | y | | z | - - = xμημνxν = xμxν = t2 - x2 - y2 - z2 The Lorentz transform can be thought as a 4 x 4 matrix boosting or rotating the coordinates in inertial frame, S, to inertial frame, S'. Thus, xμ' = Λμνxν Lorentz transformation leave the inner product invariant. Thus, x'.x' = x.x The set of all matrices obeying ΛTηΛ = η form the LORENTZ GROUP SO(3,1). Proof: Consider the transformation of a vector, x: x -> Λx Define the DUAL representation which transforms as follows: ~    ~ x -> (Λ-1)Tx Therefore, ~    ~ xTx -> ((Λ-1)Tx))TΛx Using the property that (AB)T = BTAT we get: ~ -> xTΛ-1Λx ~ ~ Check xTx -> xTΛ-1Λx: - - - - - - | a | ~ | e | | γ -βγ 0 0 | x = | b | and x = | f | and Λ = | -βγ γ 0 0 | | c | | g | | 0 0 1 0 | | d | | h | | 0 0 0 1 | - - - - - - Now,       - -       | γ βγ 0 0 | Λ-1 = (1/(γ2 - β2γ2))| βγ γ 0 0 |       | 0 0 1 0 |       | 0 0 0 1 |       - - Now 1/(γ2 - β2γ2) = 1 Therefore, expanding both sides gives: - -  - - - - - - | e |T| γ βγ 0 0 || γ -βγ 0 0 || a | ae + bf + cg + dh = | f | | βγ γ 0 0 || -βγ γ 0 0 || b | | g | | 0 0 1 0 || 0 0 1 0 || c | | h | | 0 0 0 1 || 0 0 0 1 || d | - -  - - - - - - - - - - = | e f g h || a | - - | b | | c | | d | - - = LHS Q.E.D. ~ ~   ~ Note: If x is complex then (x)T)* ≡ x Define, ~ x = ηx where η is the metric. Therefore, ~ xTx = (ηx)Tx Again, using the property that (AB)T = BTAT we get: ~ xTx = xTηTx   = xTηx since η is always symmetric. Now consider the transformation: xTηx -> (Λx)TηΛx Again, using the property that (AB)T = BTAT we get: xTηx -> xTΛTηΛx   -> xTΛx i.f.f. ΛTηΛ = η Because elements with determinants ≠ +1 do not form a subgroup of SO(3,1), the determinant of Λ is required to equal +1. In 3D Euclidean space η is the 3 x 3 identity matrix (≡ δij). In 4D flat spacetime η is the 4 x 4 Minkowski metric. In 4D curved spacetime the situation is a little more complicated. Lorentz transformations only refer to transformations in the context of flat spacetime. In curved spacetime the assumption is that the spacetime can be considered to be locally flat. In other words, the metric around any given point is arbitrarily close to the Minkowski metric within a small area around that point. Essentially, there is a 4-vector in the tangent space at each spacetime point through which the particle's world line passes. Check ΛTηΛ = η: - - - - - - | γ -βγ 0 0 || -1 0 0 0 || γ -βγ 0 0 | | -βγ γ 0 0 || 0 1 0 0 || -βγ γ 0 0 | | 0 0 1 0 || 0 0 1 0 || 0 0 1 0 | | 0 0 0 1 || 0 0 0 1 || 0 0 0 1 | - - - - - - detΛ = γ2 - β2γ2 = 1 - - - - | γ -βγ 0 0 || -γ βγ 0 0 | | -βγ γ 0 0 || -βγ γ 0 0 | | 0 0 1 0 || 0 0 1 0 | | 0 0 0 1 || 0 0 0 1 | - - - - - - | -(γ2 - β2γ2) 0 0 0 | | 0 (γ2 - β2γ2) 0 0 | | 0 0    1 0 | | 0 0    0 1 | - - Now γ2 - β2γ2 = 1. Therefore, this reduces to: - - | -1 0 0 0 | | 0 1 0 0 | Q.E.D. | 0 0 1 0 | | 0 0 0 1 | - - Tensor Form ----------- We have seen that: xμ -> (x')μ = Λμνxν Where Λμν is the matrix - - | Λ00 Λ01 Λ02 Λ03 | | Λ10 Λ11 Λ12 Λ13 | | Λ20 Λ21 Λ22 Λ23 | | Λ30 Λ31 Λ32 Λ33 | - - and xμ and xν are 4-vectors. For a tensor with 2 indeces this becomes: Tμν -> (T')μν = ΛμσΛνρTσρ (We can think of this as (A')μ(B')ν = ΛμσAσΛνρBρ) Thus, for example: ησρ = ημνΛμσΛνρ Therefore, each index gets its own transformation. Consider: Tμν = ΛμσΛνρTσρ Λμσ and Λνρ are transformation matrices. The tensor components then transform as: Tμν = ΣΣΛμσΛνρTσρ σρ Where σ amd ρ run from 0 to 3 for 4D spacetime.    = Λμ0Λν0T00 + Λμ0Λν1T01 + Λμ1Λν0T10 + Λμ1Λν1T11 ... ΛμσΛνρTσρ This is referred to as the INDEX FORM. We can also write this in MATRIX FORM as: T' = ΛTΛT Proof: To get the matrix form we can rearrange the terms but we need to ensure that like indeces are adjacent to each other before contraction. Therefore, Tμν = ΛμσΛνρTσρ ≡ ΛμσTσρΛνρ ≡ TσρΛμσΛνρ Tμν = ΛμσΛνρTσρ    = ΛνρΛμσTσρ    = Λνρ(ΛT)μρ    = ΛνρBμρ where B = ΛT    = (ΛBT)νμ    = (BΛT)μν    = (ΛTΛT)μν Therefore, T' = ΛTΛT Inverse Notation ---------------- ΛTηΛ = η Multiply from left by η-1 to get: η-1ΛTηΛ = η-1η Therefore, η-1ΛTηΛ = I Multiply from the right by Λ-1 to get: η-1ΛTηΛΛ-1 = IΛ-1 Therefore, η-1ΛTη = Λ-1 Lets look at the LHS and add back in the indeces. Noting that (ημν)-1 = ημν and that matrix multiplication requires getting repeated indeces adjacent to each other. ηανμν)Tημβ -> ηανΛνμημβ = Λαβ This works for the Lorentz transformation because η has the property η = ηT = η-1. Therefore, in index notation (Λ-1)μν ≡ Λνμ Note that this is different to (ΛT)μν ≡ Λνμ - -    | γ -βγ 0 0 | Λμν = | -βγ γ 0 0 |    | 0 0 1 0 |    | 0 0 0 1 | - - - -    | γ βγ 0 0 | Λμσ = | βγ γ 0 0 |    | 0 0 1 0 |    | 0 0 0 1 | - - - -      | 1 0 0 0 | ΛμσΛμν = | 0 1 0 0 | = δσν      | 0 0 1 0 |      | 0 0 0 1 | - - If we define a contravariant 4-vector as (-ct,x,y,z) then the vector transforms as: Aμ = ΛμνAν A boost in the x direction is: - - - - - - | γ -βγ 0 0 || -ct | | -γ(ct + βx) | | -βγ γ 0 0 || x | = | γ(x + βct) | | 0 0 1 0 || y | | y | | 0 0 0 1 || z | | z | - - - - - - If we define a covariant 4-vector as (ct,x,y,z) then the vector transforms as: Aν = ΛνμAμ A boost in the x direction is: - - - - - - | γ βγ 0 0 || ct | | γ(ct + βx) | | βγ γ 0 0 || x | = | γ(x + βct) | | 0 0 1 0 || y | | y | | 0 0 0 1 || z | | z | - - - - - - The product of - - - - | -γ(ct + βx) γ(x + βct) y z || γ(ct + βx) | - - | γ(x + βct) | | y | | z | - - is: -(γ(ct + βx))2 + (γ(x + βct))2 + y2 + z2 γ2[-t2 - β2x2 - 2βxt + x2 + β2t2 + 2βxt] + y2 + x2 γ2[-t2(1 - β2) + x2(1 - β2)] + y2 + x2 γ2(1 - β2)[-t2 + x2] + y2 + x2 -t2 + x2 + y2 + x2 as expected (the invariant interval). Infinitesimal Group Generators ------------------------------ We now look at the generators used to create the actual transformations, Λ, that form the group. Consider: ημνΛμσΛνρ = ησρ The infinitesimal Lorentz transformation can be constructed as: Λμσ = δμσ + ωμσ and Λνρ = δνρ + ωνρ where ω is a small quantity and δνσ is the Kronecker delta in tensor form. Note: In spacetime δab or δab don't make sense. Since ηabδcc = ηac, δcc acts as the identity matrix. ημνμσ + ωμσ)(δνρ + ωνρ) = ημνμσδνρ + δμσωνρ + ωμσδνρ + O((ω)2)   = ημνμσδνρ + δμσωνρ + ωμσδνρ)   = ημνμσδνρ + δμσωνρ + δνρωμσ)   = ημνδμσδνρ + ημνδμσωνρ + ημνδνρωμσ Now δμσ = ημρηρσ and δνρ = ηνσησρ therefore ημνημρηρσηνσησρ = ησρ ημνμσ + ωμσ)(δνρ + ωνρ) = ησρ + ωσρ + ωρσ For this to be true ωσρ + ωρσ = 0. Therefore, ωσρ = -ωρσ. Thus, the matrices of the infinitesimal generators of the Lorentz transformations are antisymmetric. We can write a basis of these six 4 × 4 antisymmetric matrices corresponding to the 3 rotations and 3 boosts as: (Mρσ)μν = i(ηρμησν - ησμηρν) Where ρ and σ indicate which generator and μ and ν are the matrix row and column. Thus, Mσρ = -Mρσ and the 6 matrices are: M01 = -M10, M02 = -M20, M03 = -M30, M11 = -M11, M12 = -M21, M13 = -M31 In accordance with: 00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33 For example, (M01)01 = i(η00η11 - η10η01) = -i (M01)10 = i(η01η10 - η11η00) = i If we continue we get the following:      - -      | 0 -i 0 0 | (M01)μν = | i 0 0 0 |      | 0 0 0 0 |      | 0 0 0 0 |      - - If we use these matrices for anything practical (for example, if we want to multiply them together, or act on some field) we will typically need to lower one index. We can do this using the Minkowskic metric as follows: ηβν(Mρσ)μβ = i(ηρμησβηβν - ησμηρβηβν) (Mρσ)μν = i(ηρμδσν - ησμδρν) Therefore, for example:      - - - -      | 0 i 0 0 | | 1 0 0 0 | (M01)μν = | i 0 0 0 | after multiplying by η = | 0 -1 0 0 |      | 0 0 0 0 | | 0 0 -1 0 |      | 0 0 0 0 | | 0 0 0 -1 |      - - - - Or,      - -      | 0 1 0 0 | (M01)μν = i| 1 0 0 0 |      | 0 0 0 0 |      | 0 0 0 0 |      - - Mρσ are referred to as the GENERATORS of the Lorentz transformation. We are now in a position to write ωσρ from before as: ωμν = (1/2)Ωρσ(Mρσ)μν Where Ωρσ consists of 6 numbers corresponding to the 6 generators. They tell us what kind of Lorentz transformation we are performing (i.e., rotate by θ = π/7 about the z-direction and boost at speed v = 0.2c in the x direction. Ωρσ is also antisymmetric in the indeces. It is customary to define the rotation and boost generators using index notation as follows: Ji = (1/2)εijkMjk Ki = M0i The factor of (1/2) is needed to avoid double counting as we sum over both j and k. This is not an issue for Ki because the sum is only over one index. Thus, for J1 we get: J1 = (1/2)[ε123M23 + ε132M32] Since ε123 = -ε132 and M23 = -M32, the second term is additive. We can write: Rotation about x:   - -   | 0 0 0 0 | J1 = i| 0 0 0 0 | ≡ (M23)μν   | 0 0 0 -1 |   | 0 0 1 0 |   - - Rotation about y:   - -   | 0 0 0 0 | J2 = i| 0 0 0 1 | ≡ (M13)μν   | 0 0 0 0 |   | 0 -1 0 0 |   - - Rotation about z:   - -   | 0 0 0 0 | J3 = i| 0 0 -1 0 | ≡ (M12)μν   | 0 1 0 0 |   | 0 0 0 0 |   - - Boost along x:   - -   | 0 1 0 0 | K1 = i| 1 0 0 0 | ≡ (M01)μν   | 0 0 0 0 |   | 0 0 0 0 |   - - Boost along y:   - -   | 0 0 1 0 | K2 = i| 0 0 0 0 | ≡ (M02)μν   | 1 0 0 0 |   | 0 0 0 0 |   - - Boost along z:   - -   | 0 0 0 1 | K3 = i| 0 0 0 0 | ≡ (M03)μν   | 0 0 0 0 |   | 1 0 0 0 |   - - Lorentz generators can be added together, or multiplied by real numbers, to get more Lorentz generators. For example, - - | 0  Bx By Bz | B.K + θ.J = | Bx 0 z θy | | By θz 0 x | | Bzy θx 0  | - - Differential Operators ---------------------- Consider a boost along x. - - | γ -βγ 0 0 | Λ = | -βγ γ 0 0 | where β = v/c | 0 0 1 0 | | 0 0 0 1 | - - Λ = 1 + Bxf'(Λx) + ... = I + Bx∂Λx/∂Bx + ... The derivative of the matrix is the matrix of the entries differentiated with respect to the same variable. Thus, - -   | 0 -γ 0 0 | ∂Λ/∂Bx = | -γ 0 0 0 |   | 0 0 0 0 |   | 0 0 0 0 | - - - -   | 0 -1 0 0 |   = γ| -1 0 0 0 |   | 0 0 0 0 |   | 0 0 0 0 | - - = -γKx Therefore, we can write: Bx(γ) = I - iγβKx Therefore, by comparison: ∂B/∂β = -γKx When β is small (= δ), γ ~ 1 and we get: Bx(γ) = I - δKx Differential operators can also be used to represent infinitesimal generators. x' = xcosθ - ysinθ and y' = xsinθ + ycosθ For small θ's this becomes: x' = x - ydθ and y' = xdθ + y An arbitrary differentiable function F(x,y) then transforms as: F(x',y') = F(x - ydθ,xdθ + y) Using dF = (∂F/∂x)dθ + (∂F/∂y)dθ we get: F(x',y') = F(x,y) + x(∂F/∂y)dθ - y(∂F/∂x)dθ = F(x,y) + {x(∂F/∂y) - y(∂F/∂x)}dθ Therefore, we can associate infinitesimal rotations with the operator: R = x(∂F/∂y) - y(∂F/∂x) In general we can write: Mρσ = i(xρσ - xσρ) Or, in terms of J and K: Jx ≡ i(y∂z - z∂y) Jy ≡ i(z∂x - x∂z) Jz ≡ i(x∂y - y∂x) -Kx ≡ i(x∂t + t∂x) -Ky ≡ i(y∂t + t∂y) -Kz ≡ i(z∂t + t∂z) Consider a rotation about the z-axix. As noted before, the differential of a matrix is the differential of the indivividual elements. Thus, for a rotation about the z-axis: - - - - | 1 0 0 0 | | 0 0 0 0 | | 0 cosθ -sinθ 0 | -> | 0 -sinθ -cosθ 0 | | 0 sinθ cosθ 0 | | 0 cosθ -sinθ 0 | | 0 0 0 1 | | 0 0 0 0 | - - - - Therefore, - - - - - - | 0 0 0 0 || t | | 0 | | 0 -sinθ -cosθ 0 || x | = | -xsinθ - ycosθ | | 0 cosθ -sinθ 0 || y | | xcosθ - ysinθ | | 0 0 0 0 || z | | 0 | - - - - - - The RHS is equivalent to: i∂x(xcosθ - ysinθ) = icosθ and i∂y(xcosθ - ysinθ) = -isinθ Therefore, i(x∂y - y∂x) = -ixsinθ - iycosθ i∂x(xsinθ + ycosθ) = isinθ and i∂y(xsinθ + ycosθ) = icosθ Therefore, i(x∂y - y∂x) = ixcosθ - iysinθ Setting θ = 0 yields: - - | 0 0 0 0 | i| 0 0 -1 0 | = Jz | 0 1 0 0 | | 0 0 0 0 | - - Similarly, for a boost in the x direction: - - - - | coshζ -sinhζ 0 0 | | sinhζ -coshζ 0 0 | | -sinhζ coshζ 0 0 | -> | -coshζ sinhζ 0 0 | | 0 0 1 0 | | 0 0 0 0 | | 0 0 0 1 | | 0 0 0 0 | - - - - Therefore, - - - - - - | sinhζ -coshζ 0 0 || t | | tsinhζ - xcoshζ | | -coshζ sinhζ 0 0 || x | = | -tcoshζ + xsinhζ | | 0 0 0 0 || y | | 0 | | 0 0 0 0 || z | | 0 | - - - - - - i∂t(tcoshζ - xsinhζ) = icoshζ and i∂x(tcoshζ - xsinhζ) = -isinhζ Therefore, i(t∂x + x∂t) = -itsinhζ + ixcoshζ i∂t(-tsinhζ + xcoshζ) = -isinhζ and i∂x(-tsinhζ + xcoshζ) = icoshζ Therefore, i(t∂x + x∂t) = itcoshζ - ixsinhζ Setting ζ = 0 yields: - - | 0 1 0 0 | i| 1 0 0 0 | = Kx | 0 0 0 0 | | 0 0 0 0 | - - These are the classical generators of angular momentum generalized to include time. They illustrate how to pass between matrix and vector representations of elements of the Lie algebra. We showed before that (Mρσ)μν = i(ηρμησν - ησμηρν). If we multiply both sides by ημν we get: Mρσ = i(ηρσ - ησρ) Now, also from before, Mρσ = i(xρσ - xσρ) Therefore, we can conclude that: ηρσ ≡ xρσ and ησρ ≡ xσρ The Lorentz Algebra ------------------- It can be shown that the generators obey the LORENTZ ALGEBRA relations: [Mρσ,Mαβ] = i(ησαMρβ - ηραMσβ + ηρβMσα - ησβMρα) ... A. Proof: [Mρσ,Mαβ] = i2{(xρσ - xσρ)(xαβ - xβα) - (xαβ - xβα)(xρσ - xσρ)}      = -{(xρσxαβ - xρσxβα - xσρxαβ + xσρxβα) - (xαβxρσ - xαβxσρ - xβαxρσ + xβαxσρ)}      = -{(xρησαβ - xρησβα - xσηραβ + xσηρβα) - (xαηβρσ - xαηβσρ - xβηαρσ + xβηασρ)}      = -{xρησαβ - xρησβα - xσηραβ + xσηρβα - xαηβρσ + xαηβσρ + xβηαρσ - xβηασρ}      = -{ησα(xρβ - xβρ) - ηρα(xσβ - xβσ) + ηρβ(xσα - xασ) - ησβ(xρα - xαρ)} Using the fact that () = M/i = -iM this becomes: [Mρσ,Mαβ] = i{ησαMρβ - ηραMσβ + ηρβMσα - ησβMρα} We could also have derived this using i(ηρμδσν - ησμδρν) as: [Mρσ,Mαβ]μν = (Mρσ)μλ(Mαβ)λν - (Mαβ)μλ(Mρσ)λν        = i2{(ηρμδσλ - ησμδρλ)(ηαλδβν - ηβλδαν) - (ηαμδβλ - ηβμδαλ)(ηρλδσν - ησλδρν)}        = -{(ηρμδσληαλδβν - ηρμδσληβλδαν - ησμδρληαλδβν + ησμδρληβλδαν) - (ηαμδβληρλδσν - ηαμδβλησλδρν - ηβμδαληρλδσν + ηβμδαλησλδρν)} Contracting the middle δ and η gives: [Mρσ,Mαβ]μν = -{(ηρμησαδβν - ηρμησβδαν - ησμηραδβν + ησμηρβδαν) - (ηαμηβρδσν - ηαμηβσδρν - ηβμηαρδσν + ηβμηασδρν)} Rearranging and factoring gives: [Mρσ,Mαβ]μν = -{ησαρμδβν - ηβμδρν) - ησβρμδαν - ηαμδρν) - ηρασμδβν - ηβμδσν) + ηρβσμδαν - ηαμδσν)} Again, using the fact that () = M/i = -iM this becomes: [Mρσ,Mαβ]μν = i{ησα(Mρβ)μν - ησβ(Mρα)μν - ηρα(Mσβ)μν + ηρβ(Mσα)μν} The LIE ALGEBRA for the Lorentz group SO(3,1) can be obtained using A. as: (Mρσ)μν = i(ηρμησν - ησμηρν) [Ji,Jj] = iεijkJk [Ji,Kj] = iεijkKk [Ki,Kj] = -iεijkJk Note: Boosts alone cannot be a subgroup of SO(1,3) since combinations involving boosts result in rotations! Generation of a Lorentz Transform --------------------------------- Now that we have described the generators of the Lorentz group we need to explore how these generators actually produce a specific finite transformation, Λ We will go through the exercise for a boost in the x direction and a rotation around the x axis. The boost generator for the x direction, K1, is: - -   | 0 1 0 0 | K1 = i| 1 0 0 0 |   | 0 0 0 0 |   | 0 0 0 0 | - - A boost along x is constructed as n consecutive boosts of δ. Thus, ζ = nδ Λ = limn->∞(I + (iζ/n)K1)n = exp(iζK1) from the lemma limn->∞(1 + A/n)n = exp(A) But exp(iζK1) also can be written as the Taylor series: Λ = exp(iζK1) = 1 + iζK1 + (iζK1)2/2! - (iζK1)3/3! + (iζK1)4/4!   - (iζB1)5/5! + (iζB1)6/6! Now, let, - -   | 0 -1 0 0 | iK1 = | -1 0 0 0 | = B1   | 0 0 0 0 |   | 0 0 0 0 | - - So, - -    | 1 0 0 0 | B12 = | 0 1 0 0 |    | 0 0 0 0 |    | 0 0 0 0 | - - It is easy to show: B1 = B13 = B15 ... B12 = B14 = B16 ... Therefore, Λ = exp(ζB1) = 1 + (ζ + ζ3/3! + ζ5/5! ...)B1   + (ζ2/2! + ζ4/4! + ζ6/6! ...)B12   = 1 - B12 + (ζ + ζ3/3! + ζ5/5! ...)B1   + (1 + ζ2/2! + ζ4/4! + ζ6/6! ...)B12   = 1 - B12 + B1sinh(ζ) + B12cosh(ζ) - - - - - -    | 1 0 0 0 | | 1 0 0 0 | | 0 0 0 0 | I - B12 = | 0 1 0 0 | - | 0 1 0 0 | = | 0 0 0 0 |    | 0 0 1 0 | | 0 0 0 0 | | 0 0 1 0 |    | 0 0 0 1 | | 0 0 0 0 | | 0 0 0 1 | - - - - - - - -   | cosh(ζ) -sinh(ζ) 0 0 | Λ = exp(ζB1) = | -sinh(ζ) cosh(ζ) 0 0 |   | 0 0 1 0 |   | 0 0 0 1 | - - - -   | γ -βγ 0 0 |   = | -βγ γ 0 0 |   | 0 0 1 0 |   | 0 0 0 1 | - - The rotation generator around the x axis, J1, is:   - -   | 0 0 0 0 | J1 = i| 0 0 0 0 |   | 0 0 0 -1 |   | 0 0 1 0 |   - - A rotation by a finite angle, θ, is constructed as n consecutive rotations of θ/n. Thus, Λ = limn->∞(I + (iθ/n)J1)n = exp(iθJ1) from the lemma limn->∞(1 + A/n)n = exp(A) Again this can be expanded as a Taylor series: Λ = exp(iθJ1) = 1 + iθJ1 + (iθJ1)2/2! + (iθJ1)3/3! + (iθJ1)4/4!   + (iθJ1)5/5! + (iθJ1)6/6! Now, let, - -   | 0 0 0 0 | iJ1 = | 0 0 0 0 | = R1   | 0 0 0 1 |   | 0 0 -1 0 | - - So, - -    | 0 0 0 0 | R12 = | 0 0 0 0 |    | 0 0 -1 0 |    | 0 0 0 -1 | - - It is easy to show: R1 = -R13 = R15 ... R12 = -R14 = R16 ... Therefore, Λ = exp(θR1) = 1 + θR1 + θ2R12/2! - θ3R1/3! - θ4R12/4!   + θ5R1/5! + θ6R12/6!   = 1 + (θ - θ3/3! + θ5/5! ...)R1   - (-θ2/2! + θ4/4! - θ6/6! ...)R12   = 1 + R12 + (θ - θ3/3! + θ5/5! ...)R1   - (1 - θ2/2! + θ4/4! - θ6/6! ...)R12   = 1 - R12 + R1sin(θ) - R12cos(θ) - - - - - -    | 1 0 0 0 | | 0 0 0 0 | | 1 0 0 0 | I + R12 = | 0 1 0 0 | + | 0 0 0 0 | = | 0 1 0 0 |    | 0 0 1 0 | | 0 0 -1 0 | | 0 0 0 0 |    | 0 0 0 1 | | 0 0 0 -1 | | 0 0 0 0 | - - - - - - - -   | 1 0 0 0 | Λ = exp(θR1) = | 0 1 0 0 |   | 0 0 cos(θ) sin(θ) |   | 0 0 -sin(θ) cos(θ) | - - Summary ------- In summary, a finite Lorentz transformation can be written as: Λ = exp(iθ.J + iζ.K)