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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 Mechanics

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Blackbody Radiation .

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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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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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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Poiseuille's Law
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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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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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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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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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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
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Vacuum Energy .

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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Basis One-forms .
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Einstein's Field Equations
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Geodesics
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Gravitational Waves
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Hyperbolic Motion and Rindler Coordinates .
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Quantum Gravity
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Ricci Decomposition .
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Ricci Flow .
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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 Dirac Equation in Curved Spacetime .
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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 Light Cone .
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The Metric Tensor .
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The Principle of Least Action in Relativity .
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Vierbein (Frame) Fields

Group Theory

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Basic Group Theory .
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Basic Representation Theory .
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Building Groups From Other Groups .
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Sets, Groups, Modules, Rings and Vector Spaces
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Symmetric Groups .
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The Integers Modulo n Under + and x .

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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Binomial Theorem (Pascal's Triangle)
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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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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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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 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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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Nuclear Physics

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Radioactive Decay

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity, Chirality and Weyl Spinors .
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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 .

Probability and Statistics

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Box and Whisker Plots
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Buffon's Needle .
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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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Density Operators and Mixed States .
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Entangled States .
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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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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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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions .
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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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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 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 (Faraday) Tensor .
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Energy and Momentum in Special Relativity, E = mc2 .
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Invariance of the Velocity of Light .
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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 Continuity Equation .
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The Lorentz Group .

Statistical Mechanics

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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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Bardeen–Cooper–Schrieffer Theory
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BCS Theory
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Cooper Pairs
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Introduction to Superconductivity .
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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
Last modified: April 25, 2022 ✓

Lorentz Transform ----------------- The Lorentz transformation converts between two different observers' measurements of space and time, where one observer is moving a CONSTANT velocity with respect to the other. Coordinate frames that are moving at a constant velocity with respect to one another are called INERTIAL reference frames. Special Relativity does not apply to the situation where the frames are accelerating with respect to each other. This is the subject of General Relativity. In classical physics, the only conversion believed necessary was x' = x - vt, describing how the origin of one observer's coordinate system slides through space with respect to the other's, at speed v and along the x-axis of each frame. According to SR this is only a good approximation at speeds small compared to the speed of light. At higher speeds the result is not just an offsetting of the x coordinates but times, lengths and masses are also effected. Consider 2 cartesian coordinate systems (t,x,y,z) and (t',x',y',z') such that the x and x' axes are coincident and the other 2 axes are parallel. Let (t,x,y,z) be a fixed reference frame and let (t',x',y',z') be a reference frame moving relative to the fixed frame. The reference frames coincide at t = t' = 0. For a frame boosted along the x- axis). Time Dilation ------------- The Lorentz transformation for time is: t - vx/c2 t' = ------------ √(1 - v2/c2) = γ(t - vx/c2) Consider a time interval: t1' - t2' = γ{t1 - vx1/c2 - t2 + vx2/c2} If the time measurements in the moving frame are made at the same location, the expression reduces to: t12' = γt12 Example: Consider 2 people - Bob and Alice. Bob's frame is stationary and Alice's frame is moving with v = 0.5c, therefore γ = 1.154. Alice measures 1s in her frame (t' = 1) but when Bob looks at her clock he sees. t = t'/γ = 1/1.154 = 0.866s Likewise, if Bob measures 1s in his frame (t = 1) then Alice will observe Bob's time to be: t' = γt = 1.154s Therefore, a clock in an inertial reference frame moving relative to an observer in a stationary reference frame appears to run more slowly. Length Contraction ------------------ The Lorentz Transformation for spacial coordinates is: x' = ct' = γ(ct - vx/c) = γ(x - vt) Consider a length measurement: x1' - x2' = γ{x1 - vt1 - x2 + vt2} If the length measurements in the moving frame are made at the same time, the expression reduces to: x12' = γx12 Example: Again, consider Bob and Alice under the same conditions. Therefore, x = x'/γ = 0.866m x' = γx = 1.154m Therefore, a meter stick in an inertial reference frame moving relative to an observer in a stationary reference frame appears to be shorter. Proper Time and Proper Length ----------------------------- The proper time, τ, and proper length, L0, is defined as the time and length that is always measured in the observer’s own frame regardless of whether that frame is moving or not. This is because an observer is always at rest in his own reference frame and cannot be moving relative to himself. Thus a clock or meter stick moving along a world line will show the proper time to the observer carrying the clock or meter stick. Velocity Addition ----------------- Consider the inverse transformations: x = γ(x' + vt') t = γ(t' + vx'/c2) w = x/t = γ(x' + vt')/γ(t' + vx'/c2) = (x' + vt')/(t' + vx'/c2) ÷ RHS by t'. Let u = x'/t': w = (u + v)/(1 + vu) For small v and/or u this becomes the standard Newtonian formula for relative velocity. The Lorentz Transformation in 4-vector Form ------------------------------------------- The Lorentz Transform can also be written as a matrix (see the note on the Lorentz Group for more details). For a boost of a contravariant vector in the x'-direction we get: - - - - | γ -βγ 0 0 || ct | | -βγ γ 0 0 || x | where β = v/c | 0 0 1 0 || y | | 0 0 0 1 || z | - - - - - - | γct - βγx | = | -βγct + γx | | y | | z | - - Likewise, for a boost of a covariant vector in the x'-direction we get: - - - - | γ βγ 0 0 ||-ct | | βγ γ 0 0 || x | | 0 0 1 0 || y | | 0 0 0 1 || z | - - - - - - | γβx - γct | = | γx - βγct | | y | | z | - - Note: The boost matrix for a covariant vector is the inverse of the boost for a contravariant vector. (see notes on the Lorentz Group). Therefore, the product of the 2 resulting vectors is equal to: - - - - | (γct - βγx) (-βγct + γx) y z|| γβx - γct | - - | γx - βγct | | y | | z | - - = (γct - βγx)(γβx - γct) + (-βγct + γx)(γx - βγct) + y2 + z2 = γ2(ct - βx)(βx - ct) + γ2(-βct + x)(x - βct) + y2 + z2 = γ2[(ct - βx)(βx - ct) + (-βct + x)(x - βct)] + y2 + z2 = γ2[(-c2t2 - β2x2 + 2ctβx) + (-2βctx + x2 + β2c2t2)] + y2 + z2 = γ2[-c2t2 - β2x2 + x2 + β2c2t2] + y2 + z2 = γ2[-c2t2(1 - β2) + x2(1 - β2] + y2 + z2 = γ2(1 - β2)[-c2t2 + x2] + y2 + z2 = -c2t2 + x2 + y2 + z2 Note: the above matrix will be different for a boost in either the y or z direction. Energy and Momentum ------------------- Energy and momentum can be combined into a single 4-vector, (E/c,px,py,pz). Therefore, - - - - | γ -βγ 0 0 || E/c  | | -βγ γ 0 0 || px | | 0 0 1 0 || py | | 0 0 0 1 || pz | - - - - - - | γE/c - βγpx | = | -βγE/c + γpx | | py | | pz | - - Mass (Energy) Contraction ------------------------- E'/c = γE/c - βγpx = γ(E/c - vpx/c) ∴ E' = γ(E - vpx) ∴ m'c2 = γ(mc2 - vpx) ∴ m' = γ(m - vpx/c2) For a particle at rest px = 0. Therefore, m = γm0 Note: There is another derivation of this equation that relies on analyzing the elastic collision of 2 masses in the moving frame. This derivation uses the formula for relativistic velocity addition derived above in conjunction with the conservation of linear momentum law. -----> v t' | | u u | o -> <- o Before | 1 2 | <- oo -> After | -u u | -------------- x' u1 = (u + v)/(1 + uv/c2) u2 = (-u + v)/(1 - uv/c2) m1u1 + m2u2 = (m1 + m2)v After a considerable amount of math one arrives at: m1/m2 = √(1 - u22/c2)/√(1 - u12/c2) If u2 = 0 then m2 = m0 and we get: m1 = m0/√(1 - u12/c2) If we then let u1 = we get: m = m0/√(1 - v2/c2) Momentum Contraction -------------------- px' = -βγE/c + γpx = γ(px - Ev/c2) Rapidity and Boosts ------------------- The Lorentz transformations for boosts can also be derived in a way that resembles circular rotations in 3d space using the hyperbolic functions. For the moment consider rotations in circular space: A rotation of x and y by θ around the z axis is given by: x' = xcosθ + ysinθ y' = -xsinθ + ycosθ z' = z t' = t The length of the vector, R, has to be the same in both frames. Therefore, x2 + y2 = x'2 + y'2 The rotation can also be written as the matrix. - - | cosθ sinθ 0 0| R = |-sinθ cosθ 0 0| | 0 0 1 0| | 0 0 0 1| - - Boosts ------ Consider a boost along the x coordinate. t t' | | / x = vt | | / | |/ o A | +----- x' | / | / | / | / ------------------- x The origin of the moving frame is moving on the line x = vt. The point A in the prime frame is at (x',t'). Therefore: x' = x - vt t = t' But this doesn't work for a light ray x = ±ct because x' = ct - vt = (c - v)t and x' = -ct - vt = -(c + v)t violates the idea that the laws of physics are the same in all inertial reference frames. To compensate for the fact that the speed of light in free space has to be the same value in all inertial reference frames. We start with, x2 - c2t2 = x'2 - c2t'2 This is the equation of a hyperbola. Instead, of using circular geometry using sines and cosines we now need to switch to the hyperbolic functions sinh and cosh. The transformations that satisfy x2 - c2t2 = x'2 - c2t'2 are: x' = xcoshζ - ctsinhζ t' = -(x/c)sinhζ + tcoshζ y' = y z' = z In the prime frame, x' = x - vt, we can write: x - vt = xcoshζ - ctsinhζ ∴ x - xcoshζ = -ctsinhζ + vt ∴ x(1 - coshζ) = t(v - csinhζ) ∴ v(1 - coshζ) = (v - csinhζ) ∴ vcoshζ = csinhζ ∴ v = ctanhζ or tanhζ = v/c tanh2ζ = v2/c2 Now, tanh2ζ = 1 - sech2ζ ∴ sech2 = 1 - v2/c2 ∴ 1/cosh2ζ = 1 - v2/c2 ∴ coshζ = 1/√(1 - v2/c2) Now, sinhζ = tanhζcoshζ = (v/c)/√(1 - v2/c2) Substituting into the original coordinate transformation formulas gives: x' = x/√(1 - v2/c2) - (v/c)ct/√(1 - v2/c2) = (x - vt)/√(1 - v2/c2) ζ is the RAPIDITY defined as tanh-1β where β = v/c. ζ is the hyperbolic angle. It is analagous to but does not have the dimensions of velocity. In summary, the connections between γ, β and ζ are: β = v/c = tanhζ γ = 1/√(1 - v2/c2/) = coshζ βγ = (v/c)/√(1 - v2/c2) = sinhζ Rotational Transform -------------------- In addition to boosts we can also have rotations about the 3 different spatial axes. In this case the Lorentz transformation matrix is: - - | 1 0 0 0 | | 0 x x x | where the x's represent a 3D rotation matrix. | 0 x x x | | 0 x x x | - - In general there can be combination of boosts and rotations. Active versus Passive Transformations ------------------------------------- An active transformation is a transformation which actually changes the physical position of a point. A passive transformation is a change in the coordinate system in which the object is described. Lorentz Transformation of Fields -------------------------------- For a basic scalar field: φ(x) -> φ'(x) = φ(Λ-1x) The inverse appears in the argument because we want to express φ'(x) in terms of the 'untransformed' field, φ. In other words, the transformed field evaluated at the transformed point gives the same value as the original field evaluated at the point before it was transformed. For example consider a rotation, R, of a field about the z axis. z | | | | (1,0,0) --------o---- x / A / B o (0,1,0) / y The point A goes to B. So from B's perspective, A looks like R-1(0,1,0) = (1,0,0). The definition of a Lorentz invariant theory is that if φ(x) solves the equations of motion then φ(Λ-1x) also solves the equations of motion. The derivative of φ(Λ-1x) can be found as follows: Let yν = (Λ-1)νμxμ We can use the Chain rule to write: (∂'φ(xμ)/∂xμ) = (∂yν)/∂xμ)(∂φ(yν)/∂yν) Now, ∂yν/∂xμ = ∂(Λ-1x)νμ/∂xμ = (Λ-1)νμ So we end up with: (∂φ'(xμ)/∂xμ) = (Λ-1)νμ(∂φ(yν)/∂yν)    ≡ (Λ-1)νμνφ(y) Therefore, the derivative transforms like a vector which is not surprising since derivative of a scalar field is a vector. We can find the second derivative as follows: (∂φ(x))2 = ∂μφ(x)∂νφ(x)ημν   = (Λ-1)ρμρφ(y)(Λ-1)σνσφ(y)ημν   = ∂ρφ(y)∂σφ(y)ηρσ since ΛρμΛσνημν = ηρσ   = (∂φ(y))2 We now show that the Klein-Gordon, Dirac, and Weyl equations are Lorentz invariance. Klein-Gordon: (∂2 + m2)φ = 0 (∂2 + m2)φ'(x) -> ((Λ-1)νμν-1)νμν + m2)φ(Λ-1x) = (gνσνσ + m2)φ(Λ-1x) = (∂2 + m2)φ(Λ-1x) = 0 Dirac: (iγμμ - m)ψ = 0 In the case of the Dirac field the spinor also carries an orientation that can be rotated or boosted in spacetime. Therefore, ψα = S[Λ]αβψβ-1x) (iγμμ - m)ψ(x) -> (iγμ-1)νμν - m)S[Λ]ψ(Λ-1x) Multiply from the right by S[Λ]S[Λ]-1 = I to get: = S[Λ]S[Λ]-1(iγμ-1)νμν - m)S[Λ]ψ(Λ-1x) = S[Λ](iS[Λ]-1γμS[Λ](Λ-1)νμν - m)S[Λ]ψ(Λ-1x) = S[Λ](i(Λ)μσγσ-1)νμν - m)S[Λ]ψ(Λ-1x) = S[Λ]{iγνν - m)ψ(Λ-1x) = 0 Weyl: iσμμψL = 0 iσμμψL -> (iσμ-1)νμν)S[Λ]ψL-1x) Using the same process as above we get: = S[Λ]{iσννL-1x) = 0