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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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Kinematics
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Kinetic Theory of Gases .
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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: November 18, 2021 ✓

Spinors - Part 1 Spinors - Part 2 ---------------- In Spinors - Part 1 we identified the following: 1. The generators, S>ρσ, are defined as: Sρσ = (1/4)[γρσ]    = (1/2)γργσ - (1/2)ηρσ 2. The Lorentz transformation for a spinor, S[Λ] S[Λ] = exp((1/2)ΩρσSρσ) 3. The Lorentz transformation for a vector, Λ Λ = exp((1/2)ΩρσMρσ) 4. The Sμν operates operate on a spinor field, ψ. Under a Lorentz transformation these fields transform as: ψα = S[Λ]αβψβ 5. (Mρσ)μν = ηρμδσν - ησμδρν 6. {γμν} = 2ημν ... CLIFFORD ALGEBRA. 7. (γ0) = γ0 8. (γ0)-1 = γ0 9. γ0γμγ0 = (γμ) 10. (γi) = -γi In addition, we will need: 11. [Sμνμ] = γμηνρ - γνηρμ Proof: [Sμνμ] = (1/2)[γμγνρ]     = (1/2)γμγνγρ - (1/2)γργμγν     = (1/2)γμνρ} - (1/2)γμγργν     - (1/2){γρνν + (1/2)γμγργν Use 6. to get:     = γμηνρ - γνηρμ 12. [Sμν,Sρσ] = ηνρSμσ - ημρSνσ + ημσSνρ - ηνσSμρ ... LORENTZ ALGEBRA Proof: [Sμν,Sρσ] = (1/2)[Sμνργσ]      = (1/2)[Sμνρσ + (1/2)γρ[Sμνσ] Use 7. to get:      = γμγσηνρ - γνγσηρμ + γργμηνσ - γργνησμ Rearrange 1. γμγσ = 2Sμσ + ημσ and substitute to get: [Sμν,Sρσ] = ηνρSμσ - ημρSνσ + ημσSνρ - ηνσSμρ The Dirac Adjoint ----------------- Now that we have a field, ψ, we need to construct a Lorentz scalar and a Lorentz invariant equation of motion. If we try and construct a scalar ψψ we get:. ψψ -> S[Λ]ψψS[Λ] The problem with this is that S[Λ]S[Λ] ≠ 1 and is certainly not real. To see this consider: S[Λ] = exp((1/2)ΩρσSρσ) S[Λ] = exp((1/2)Ωρσ(Sρσ)) So to be unitary we require (Sρσ) = -Sρσ Unfortunately, this is not possible because γ0 is hermitian whereas the γi's are anti-hermitian. Therefore, there is no way to pick γμ such that all are anti-hermitian. However, there is a way around this. Consider: γ0S[Λ]γ0 = γ0[exp(iΩμνSμν)]γ0     = γ0exp((-i/2)Ωμν(Sμν)γ0     = γ0(1 - (i/2)Ωμν(Sμν)0 Multiplying from the left and the right gives: γ0S[Λ]γ0 = 1 - (i/2)Ωμνγ0(Sμν)γ0     = exp(-(i/2)Ωμνγ0(Sμν)γ0) Now, (Sμν) = (1/4)[(γμ),(γν)]     = (1/4)((γμ)ν) - (γν)μ))     = (1/4)((γ0γμγ0)(γ0γνγ0) - (γ0γνγ0)(γ0γμγ0))     = (1/4)((γ0γμγ0)(γ0γνγ0) + (γ0γμγ0)(γ0γνγ0))     = (1/2)(γ0γμγνγ0)     = γ0(1/2)(γμγν0 Therefore, (Sμν) = γ0Sμνγ0 Returning to where we left off and using this result we get: γ0S[Λ]γ0 = exp(-(i/2)ΩμνSμν)     = S[Λ]-1 Multiplying both sides from the left and the right by γ0 gives: S[Λ] = γ0S[Λ]-1γ0 With this in mind, we now define the DIRAC ADJOINT: _ ψ = ψγ0 Under a boost we get: _ ψψ -> ψS[Λ]γ0S[Λ]ψ = ψγ0S[Λ]-1γ0γ0S[Λ]ψ = ψγ0S[Λ]-1S[Λ]ψ = ψγ0ψ This looks like: - - | ψ1 | ψ = | ψ2 | | ψ3 | | ψ4 | - - - - ψ = (ψ*)T = | ψ1* ψ2* ψ3* ψ4* | - - _ - - ψ = ψγ0 = | ψ1* ψ2*3*4* | - - _ ψψ = ψ1ψ1* + ψ2ψ2* - ψ3ψ3* - ψ4ψ4* Again, this is completely different to the vector case, xμxμ. -     - - - |x0 x1 x2 x3 || x0 | = t2 - x2 - y2 - z2 -     - | x1 |      | x2 |      | x3 |      - - Bilinears --------- In the previous section we formed a scalar from 2 spinors. Are they other objects we can form by combining 2 spinors? The answer is yes. Spinors can be combined to form scalars, vectors, tensors and more. The products of two spinors are called BILINEARS. Scalar ------ To form a scalar (spin 0) from 2 spinors one writes: _ ψψ = ψγ0ψ Under a Lorentz transformation, _ ψψ -> ψ-1x)S[Λ]γ0S[Λ]ψ(Λ-1x) = ψ-1x)γ0ψ(Λ-1x) _ = ψ(Λ-1x)ψ(Λ-1x) _ So ψψ does indeed transform as a scalar. Vector ------ To form a vector (spin 1) from 2 spinors one writes: _ ψγμψ Under a Lorentz transformation, _ ψγμψ -> ψS[Λ]-1γμS[Λ]ψ For this to be true we need S[Λ]-1γμS[Λ] = Λμνγν ... 13. Where, Λ = exp((1/2)ΩρσMρσ) ~ 1 + (1/2)ΩρσMρσ ... and, S[Λ] = exp((1/2)ΩρσSρσ) ~ 1 + (1/2)ΩρσSρσ ... We work infinitesimally with Ωρσ = ερσ. Therefore, (1 - (1/2)ερσSρσμ(1 + (1/2)ερσSρσ) = (1 + (1/2)ερσMρσν (1 - (1/2)ερσSρσ)(γμ + (1/2)ερσSρσγμ) = (1 + (1/2)ερσMρσν γμ + (1/2)ερσSρσγμ - (1/2)ερσSρσγμ + O(ερσ2) = γμ + (1/2)ερσMρσγν (Sρσγμ - Sρσγμ) = Mρσγν [Sρσμ] = (Mρσ)μνγν Using 11. on the LHS and 5. on the RHS gives: γμηνρ - γνηρμ = (ηρμδσν - ησμδρνν γμηνρ - γνηρμ = ηρμγσ - ησμγρ _ So we have proven 13. and ψγμψ does indeed transform as a vector. Tensor ------ To form a tensor (spin 2) from two spinors one writes: _ ψSμνψ Under a Lorentz transformation, _ ψSμνψ -> ψS[Λ]-1SμνS[Λ]ψ _ = ψS[Λ]-1((1/2)[γμν])S[Λ]ψ _ = (1/2)ψS[Λ]-1μγν - γνγμ)S[Λ]ψ _ = ψ(1/2)(S[Λ]-1γμS[Λ]S[Λ]-1γνS[Λ] - S[Λ]-1γνS[Λ]S[Λ]-1γμS[Λ])ψ Now from 13. S[Λ]-1γμS[Λ] = Λμνγν ψSμνψ -> ψ(1/2)(ΛμαγαΛνβγβ - ΛνβγβΛμαγα)ψ _ = ψ(1/2)ΛμαΛνβαβ]ψ _ = ΛμαΛνβψSαβψ _ So ψSμνψ does indeed transform as a tensor. Generalization -------------- Consider the expression: _ ψΓAψ We have already found expressions for scalars, vectors and tensors. Can we find ΓA's that create other objects that transform correctly under a Lorentz transformation. Again, the answer is yes. We can write ΓA in terms of the following combinations of 16 γ matrices. ΓA = 1      : scalar (1 component) γμ     : vector (4 components) Sμν = (i/2)[γμν] : Tensor (6 components) γμγ5    : pseudo-vector (4 components) γ5     : pseudo-scalar (4 component) Spinor Indeces -------------- Consider the Weyl spinors mentioned above and define: - - ψL = | ξα |   | 0  | - - - - ψR = | 0  |   | _  |   | χα'| - - - - ψ = ψL + ψR = | ξα |    | _  |    | χα'| - - This is the VAN DER WAERDEN NOTATION. α = left-handed spinor index. α' = right handed spinor index. _ does not mean conjugate - it is purely notation. We can further define: _ ξα' = (ξα) and, _ χα = (χα') -_ - ψ = | ξα' |   | χα  | - - _ Now, ψ = γ0ψ - - - - Where γ0 = | 0 I2 | = | 0 1 | in the Weyl representation.   | I2 0 | | 1 0 | - - - - _ - - - _ - ψ = | 0 1 || ξα' | | 1 0 || χα  | - - - - - _ - = | χα ξα' | - - Spinor indices are raised and lowered using the antisymmetric symbol, ε, that has the properties: ε12 = -ε21 = ε21 = -ε12 = 1 ε11 = ε22 = ε11 = ε22 = 0 εabεbc = δac = 1 εabεbc = δac = 1 εa'b'εb'c' = δa'c' = 1 εa'b'εb'c' = δa'c' = 1 In matrix form: - - εαβ = εα'β' = | 0 1 | = iσ2        | -1 0 | - - - - εαβ = εα'β' = | 0 -1 | = -iσ2        | 1 0 | - - We can use these operators as follows: _ _ _ _ ξα = εαβξβ ξα = εαβξβ χα' = εα'β'χβ' χα' = εα'β'χβ' Therefore, εαβξβ = -ξα - - - - - - | 0 1 || ξα | = | 0  | | -1 0 || 0  | | -ξα | - - - - - - εαβξβ = ξα - - - - - - | 0 -1 || ξα | = | 0  | | 1 0 || 0  | | ξα | - - - - - - Note: εαβ = εα'β' etc. εα'β and εαβ' are not allowed. Therefore, the Dirac Lagrangion excluding the mass term is: - _ - - - - - _ _ _ L = i| χα ξα' || 0 σμ || ∂μξα  | = iχασμμχα' + ξα'σμμξα -     - | _  || _   |     | σμ 0 || ∂μχα' | - - - - ^ | γμ in the Weyl representation. Spinor Transformations ---------------------- Let us now reconstruct the spinor transformation laws that incorporate indeces. - - - - In the Weyl basis γμ = | 0 σk | -> | 0 σμ |   | -σk 0 | | _  |    - - | σμ 0 | - - Where, σμ = (1,σi) is substituted for σk _ σμ = (1,-σi) is substituted for -σk Which for rotations leads to: - - - - Sμν = (1/4)| 0 σμ || 0 σν |    | _   || _   |    | σμ 0 || σν 0 | - - - - - _ _ - = (1/4)| σμσν - σνσμ 0 | |      _ _ | | 0 σμσν - σνσμ | - - The objects that obey the Lorentz algebra and generate the desired rotations are given by: _ _ (Sμν)αβ = (1/4)(σμσν - σνσμ)αβ _ _ _ (Sμν)α'β' = (1/4)(σμσν - σνσμ)α'β' So these correspond to: ξα -> (exp((1/2)ΩμνSμν))αβξβ and, _ _ χα' -> (exp((1/2)ΩμνSμν))α'β'χβ' Which is the same as the rotational transformation that we derived in Spinors - Part 1. Parity ------ Parity is the operation of changing coordinates as: x0 -> x0 xi -> -xi Under parity, the left and right-handed spinors are exchanged. Under parity, rotations don’t change sign but boosts flip sign. Therefore, PψL,R(t,x) = ψR,L(t,-x)     = γ0ψR,L(t,-x) if ψ(t,x) satisfies the Dirac equation, then the parity transformed spinor γ0ψ(t,-x) also satisfies the Dirac equation, meaning: (iγ0t + iγii - m)γ0ψ(t,-x) = γ0(iγ0t - iγii - m)γ0ψ(t,-x) Where the extra minus sign from passing γ0 through γi is compensated by the derivative acting on -x instead of +x. Thus, in the Weyl basis we have: - - - - - _ - ψP = | 0 1 || ξα  | = | χα' |   | 1 0 || _   | | ξα  |   - - | χα' | - - - - Charge Conjugation ------------------ Charge conjugation is the operation of changing every particle into its antiparticle. It is a discrete symmetry and represents the 'C' in the CPT (Charge, Parity, Time Reversal) symmetry. Charge conjugation is accomplished using the charge conjugation operator, C: _ ψC = CψT Where C is defined as the matrix: - - - - C = | iσ2 0 | = | εαβ 0  | | 0 -iσ2 | | 0 εα'β'| - - - - C satisfies CT = C = C-1 = -C and has the property C-1ψC = ψC So for our spinor, ψ, we get: _ - - ψT = | χα  |   | _   |   | ξα' | - - - - - - - - | εαβ 0  || χα  | = | εαβχα   | | 0 εα'β'|| _   | |       _ | -     - | ξα' | | εα'β'ξα'| - - - - - - = | χβ | | _  | | ξβ'| - - - - - - | ξα  | | χβ | | _   | <- h.c. -> | _  | | χα' | | ξβ'| - - - - Therefore, the Hermitian conjugate of any left handed Weyl spinor is a right handed Weyl spinor and vice- versa. Scalars, Vectors and Tensors ---------------------------- Scalars ------- We can form a scalar out of 2 Weyl spinors as follows: _ _ ξχ ≡ ξα'χα' _ = ξα'εα'β'χβ' _ = -χβ'εα'β'ξα' (- because χ and ξ anticommute) _ = χβ'εβ'α'ξα' _ = χβ'ξβ' _ = χξ Similarly, ξχ ≡ ξαχα = ξαεαβχβ = -χβεαβξα (- because χ and ξ anticommute) = χβεβαξα = χβξβ = χξ Note the positive sign in accordance with the spin statistics theorem. Vectors ------- Rewrite: σμ = (σμ)αα' _ _ σμ = (σμ)αα' Such that, _ σμαα' = εαβεα'β'σμββ' We can now form a vector out of 2 Weyl spinors as follows: ξσμχ ≡ ξασμαα'χα' _ = -χα'σμαα'ξα _ = -εα'β'χβ'σμαα'εαβξβ _ = -εα'β'χβ'σμαα'εαβξβ _ = -χβ'β'α'εβαμαα'ξβ _ _ = -χβ'σμβ'βξβμαα' = (1,σi) and σμαα' = (1,-σi)) __ = -χσμξ Note the negative sign in accordance with the spin statistics theorem.