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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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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions .
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Error Analysis
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Nuclear Spin
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One Dimensional Wave Equation .
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Phase and Group Velocity
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Planck Radiation 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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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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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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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 2 Spinors - Part 1 ---------------- "No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the 'square root' of geometry and, just as understanding the square root of -1 took centuries, the same might be true of spinors." - Sir Michael Atiyah Spinors naturally describe spin 1/2 objects in physics. A 2π rotation does not result in the same quantum state. However, a 4π rotation does. This is due to a theorem in topology called ORIENTATION ENTANGLEMENT. This is illustrated below. After a 2π rotation, the spiral flips between clockwise and counterclockwise orientations. It returns to its original configuration after spinning a full 4π. Animation courtesy of Wikipedia Before we launch into the subject let us review some of the important matrices and identities we will need. The Dirac Matrices ------------------ The Dirac basis: - -   | 1 0 : 0 0 |   | 0 1 : 0 0 | - - γ0 = | .....:...... | = | I 0 |   | 0 0 : -1 0 | | 0 -I |   | 0 0 : 0 -1 | - - - - The Chiral (Weyl) basis: - -   | 0 0 : 1 0 |   | 0 0 : 0 1 | - - γ0 = | .....:..... | = | 0 I |   | 1 0 : 0 0 | | I 0 |   | 0 1 : 0 0 | - - - - The Dirac and Chiral bases: - -   | 0 0 : 0 1 |   | 0 0 : 1 0 | - - γ1 = | ......:...... | = | 0 σx |   | 0 -1 : 0 0 | | -σx 0 |   | -1 0 : 0 0 | - - - - - -   | 0 0 : 0 -i |   | 0 0 : i 0 | - - γ2 = | ......:...... | = | 0 σy |   | 0 i : 0 0 | | -σy 0 |   | -i 0 : 0 0 | - - - - - -   | 0 0 : 1 0 |   | 0 0 : 0 -1 | - - γ3 = | ......:...... | = | 0 σz |   | -1 0 : 0 0 | | -σz 0 |   | 0 1 : 0 0 | - - - - The σ matrices are the PAULI SPIN matrices. Useful identities: Both the Dirac and Chiral bases obey the following: 1. (γ0) = γ0 2. (γ0)-1 = γ0 3. γ0γμγ0 = (γμ) 4. (γi) = -γi 5. {γμν} = 2ημνI - The CLIFFORD ALGEBRA. 6. [γμν] = 2γμγν for μ ≠ ν, 0 otherwise.    = 2γμγν - 2ημν 7. All matrices are unitary (γμμ) = I) 8. Only γ0 is symmetric (γ0 = (γ0)T). 9. γμγν = -γνγμ 10. γ0 is hermitian but γi anti-hermitian (γi) = -γi. Multiplying γμ by i makes γi hermitian but then γ0 becomes anti-hermitian. 11. γ0γμγν = γμγνγ0 Note: Moving through an odd number of γ's changes the sign. Moving through an even number of γ leaves the sign unchanged. 12. γ0γμγν = -γνγμγ0 SO(3) Review ------------ The following three basic rotation matrices rotate vectors by an angle θ about the x, y, or z-axis, in 3D. - -   | 1 0 0 | R1(θ) = | 0 cos(θ) -sin(θ) |   | 0 sin(θ) cos(θ) | - - - -   | cos(θ) 0 sin(θ) | R2(θ) = | 0 0 0 |   | -sin(θ) 0 cos(θ) | - - - -   | cos(θ) -sin(θ) 0 | R3(θ) = | sin(θ) cos(θ) 0 |   | 0 0 1 | - - With generators: - -   | 0 0 0 | G1(θ) = | 0 0 i |   | 0 -i 0 | - - - -   | 0 0 -i | G2(θ) = | 0 0 0 |   | i 0 0 | - - - -   | 0 i 0 | G3(θ) = | -i 0 0 |   | 0 0 0 | - - And Lie algebra: [Gi,Gj] = iεijkGk SU(2) and SO(3) --------------- Are there other G's that we can write down that satisfy the Lie algebra? The answer is yes. These are the Pauli matrices of SU(2). The commutation relationship (Lie algebra) is: [σij] = 2iεijkσk We can write this equivalently as: (1/4)[σij] = iεijkσk/2 For SO(3) rotational transformation on 3D vectors are represented by: ΛR = exp(iθG) If we write: Gi = σi/2 We can perform a rotation as follows: R = exp(iεijkσk/2) If we let εijk = θ, this can be expanded as a Taylor series as follows: R(θ) = 1 + iσθ/2 - (σθ/2)2/2 - i(σθ/2)3/6 + (σθ/2)4/24 ... = 1 - (σθ/2)2/2 + (σθ/2)4/24 + i{σθ/2 - (σθ/2)3/6} Now σieven = 1 and σiodd = σi Therefore, R(θ) = 1 - (θ/2)2/2 + (θ/2)4/24 + iσ{θ/2 - (θ/2)3/6} ... = Icos(θ/2) + iσsin(θ/2) - - = Icos(θ/2) + i| 0 1 |sin(θ/2) for σ1 | 1 0 | - - - - = Icos(θ/2) + i| 0 -i |sin(θ/2) for σ2 | i 0 | - - - - = Icos(θ/2) + i| 1 0 |sin(θ/2) for σ3 | 0 -1 | - - If we pick σ1 we get: - - R(θ) = exp(iσ1θ/2) = | cos(θ/2) isin(θ/2) |   | isin(θ/2) cos(θ/2) | - - with detR(θ) = +1 and RR = I Now, unlike SO(3) transformations, which are real 3 x 3 matrices acting on 3 vectors, we have 2 x 2 matrices that are complex. In this case it doesn't make sense for such matrices to act on vectors with real coordinates (x, y and z). So what do they act on? It is at this point we introduce a new 2 component object called a SPINOR, χ, that transforms as χ' = exp(θ.σ/2)χ. Spinors are abstract mathematical constructs that do not have coordinates like a vector and should not be regarded as being physically rotatable in spacetime like a vector. However, as we will see in the Spinors - Part 2 notes, spinors can be combined to form scalars and vectors that do transform in the 'traditional' way. In this sense, the spinor representation is more fundamental than the vector representation with coordinates x,y and z. Compare to a SO(3) rotation about the x axis. - - | 1 0 0 | = | 0 cos(θ) -sin(θ) | | 0 sin(θ) cos(θ) | - - Now when θ is small (i.e. near the identity) we get: - - SU(2): Λ = | 1 0 | | 0 1 | - - - - SO(3): Λ = | 1 0 0 | | 0 1 0 | | 0 0 1 | - - Therefore, SU(2) ~ SO(3) near the identity meaning that they have the same Lie algebra. Mathematically we say that the groups are 'locally isomorphic', meaning that as long as we consider only small rotations, we can’t detect any difference between the two. However, when θ = 2π we get: - - SU(2): Λ = | -1 0 | | 0 -1 | - - - - SO(3): Λ = | 1 0 0 | | 0 1 0 | | 0 0 1 | - - Of course, at 4π both agree! So this transformation has the effect of a 3D rotation in space. However, a rotation by 2π only rotates the object by 180°. This is clearly different from how a vector in spacetime would behave. SO(3,1) Review -------------- The following three basic rotation matrices rotate vectors by an angle θ about the t, x, y, or z-axis, in 4D.   - -   | 0 0 0 0 | J1 = | 0 0 0 0 |   | 0 0 0 -i |   | 0 0 i 0 |   - - Rotation about y:   - -   | 0 0 0 0 | J2 = | 0 0 0 i |   | 0 0 0 0 |   | 0 -i 0 0 |   - - Rotation about z:   - -   | 0 0 0 0 | J3 = | 0 0 -i 0 |   | 0 i 0 0 |   | 0 0 0 0 |   - - And Lie algebra: [Ji,Jj] = iεijkJk [ji,Kj] = iεijkKk [Ki,Kj] = -iεijkJk The algebra can be recast more simply be redefining: J+i = (1/2)(Ji + iKi) J-i = (1/2)(Ji - iKi) The commutation relations are: [J+i,J+j] = iεijkJ+k [J-i,J-j] = iεijkJ-k [J+i,J-j] = 0 From these calculations it is clear that J+ and J- form their own groups which follow the same commutation rules as SO(3). Thus, we can write: SO(3,1) = SO(3) x SO(3) But from before we found that SO(3) ~ SU(2). Therefore, SO(3,1) ~ SU(2) x SU(2) Note: Technically, we should write so(3,1) etc. instead of SO(3,1) etc. since we are talking about the Lie algebra of the groups. Since a rotation of 0 to 2π covers all of SO(3) but only one half of SU(2), SU(2) is referred to as the DOUBLE COVER of SO(3,1). SU(2) has 2 dimensions and, therefore, SO(3,1) has 2 x 2 = 4 dimensions as we would expect. So what this means is that a Hilbert space that is Lorentz invariant can be expressed as a pair of SU(2) representations as follows: Spin 0: (0,0) = Scalar Spin 1/2: (1/2,0) = Left handed spinor Spin 1/2: (0,1/2) = Right handed spinor So we can have 2 spin 1/2 representations. One transforms under left but not right and vice versa. These are the WEYL SPINORS that we discuss in the next section. Spinors in 4 Dimensions ----------------------- To get to get to 4D we replace the Pauli matrices with the Gamma matrices. The commutator becomes: Sρσ = (1/4)[γρσ] Then it can be shown that the commutatator [Sμν,Sρσ] satisfies the Lorentz algebra. [Sρσ,Sτν] = ηστSρν - ηρτSσν + ηρνSστ - ησνSρτ The Chiral Basis ---------------- There are many possible versions of the γ matrices that satisfy the Clifford algebra. However, it turns out there is one unique irreducible representation of the Clifford algebra known as the Chiral or Weyl representation. - - In the Chiral basis γ0 = | 0 I |   | I 0 | - - Rotations are given by:    - - - - - - - - Sij = (1/4)| 0 σi || 0 σj | - | 0 σj || 0 σi |    | -σi 0 || -σj 0 | | -σj 0 || -σi 0 |    - - - - - - - - - - = (1/4)| [σji] 0 | | 0 [σji] | - - - -    = -(i/2)εijk| σk 0 |       ^ | 0 σk |       | - -       | The 1/2 will be a critical factor in the behaviour of rotations. And, for boosts: S01 = (1/2)γ0γi Thus, - - - - S0i = (1/2)| 0 1 || 0 σj |    | 1 0 || -σj 0 | - - - -    -   -    = (1/2)| -σi 0 |    | 0 σi |    -   - The chiral basis has the advantage that the generators are in block diagonal form and hence the represenation is irreducible (see the note on Basic Representation Theory). We can see that this would not be the case in the Dirac representation where, - - γ0 = | I 0 |   | 0 -I | - - And boosts are be given by: - - - - S0i = (1/2)| 1 0 || 0 σj |    | 0 -1 || -σj 0 | - - - - - -    = (1/2)| 0 σi |    | σi 0 | - - Therefore, the reducibility is not manifest. The fact that the Weyl representation is irreducible is essentially the motivation for using this representation for our calculations. Infinitesimal Generators for Rotations -------------------------------------- We had from before (after renaming indeces): - - Sij = -(i/2)εijk| σk 0 |       | 0 σk | - - A Lorentz transformation for a VECTOR is given by: Λ = exp((1/2)ΩijMij) (Mij is complex) Where Ωij 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. Note: Both Ωij and Mij are antisymmetric. Therefore, in index notation: ΩijMij = ΩijMij + ΩjiMji Since Ωij = -Ωji and Mij = -Mji the second term is additive. Hence the need for the factor of (1/2). We can create the equivalent for our SPINOR as: S[Λ] = exp((1/2)ΩijSij) If we let, Ωij = -Ωji = -εijkθk (accounting for the factor of 2) - - and Sij = (i/2)| σk 0 |    | 0 σk | - - The rotation transformation for the spinor, S[Λ] becomes: S[Λ] = exp(-iθSij) Now, Sij is a diagonal matrix. Therefore, - - S[Λ] = | exp(-(i/2)θσk) 0 | | 0 exp(-(i/2)θσk) | - - - - = | cos(θ/2) - isin(θ/2) 0 | | 0 cos(θ/2) - isin(θ/2) | - - We can also derive this in a different way: exp(-iθS/2) = 1 - iθS/2 - (θS)2/2.2! + i(θS)3/2.3! + (θS)4/2.4!... etc. = [1 + (θS)2/2.2! + (θS)4/2.4! ...] - i[θS/2 + (θS)3/2.3! ...] Now (S)even = I and (S)odd = S. Thus, = [1 + (θ)2/4 + (θ)4/48 ...] - iS[θ/2 + (θ)3/12 ...] = Icos(θ/2) - iSijsin(θ/2) - - = | [cos(θ/2) - isin(θ/2)] 0 | | 0 [cos(θ/2) + isin(θ/2)] | - - Spinor Fields ------------- Just as the Lorentz transform operates on 4-vectors, we need an object for these matrices to act on. This object is the DIRAC SPINOR field, ψα(x). It has 4 complex components labelled by α = 1, 2, 3, 4. Under a Lorentz transformation this field transforms as: ψα(x) -> S[Λ]αβψβ(x)(Λ-1x) Where S[Λ] = exp((1/2)ΩijSij) And, Λ = exp((1/2)ΩijMij) We can form two 2D representations by writing: - - ψDIRAC = m| ψL |       | ψR | - - ψL and ψR each have 2 components and are called the left handed and right handed WEYL SPINORS. In the Weyl basis, the particle is regarded as being massless with the spin quantized parallel or anti-parallel to the direction of motion (see the note on Helicity and Chirality). So the transformation is: - - - - | [cos(θ/2) - isin(θ/2)] 0 || ψL | | 0 [cos(θ/2) + isin(θ/2)] || ψR | - - - - Therefore, the left and right components behave the same under rotations. For a rotation of 2π about the z-axis this becomes: - - - - - - | -1 0 || ψL | = | -ψL | | 0 -1 || ψR | | -ψR | - - - - - - Which means: S(θ + 2π) = -S(θ) as we found before. Infinitesimal Generators for Boosts ----------------------------------- Again, from before, we had: S01 = (1/2)γ0γi - - In the Chiral basis γ0 = | 0 I |   | I 0 | - - Thus, - - - - S0i = (1/2)| 0 1 || 0 σj |    | 1 0 || -σj 0 | - - - -    -   -    = (1/2)| -σi 0 |    | 0 σi |    -   - We follow the same procedure from before except that Ω0i = -Ωi0 = ξ. We get: - - S[Λ] = exp(ξS0i) = | exp(ξσ/2) 0 |    | 0 exp(-ξσ/2) | - - Note: We don't have to worry about the factor of (1/2) here because we are only summing over one index. So the transformation is: - - - - | exp(ξσ/2) 0 || ψL | | 0 exp(-ξσ/2) || ψR | - - - - Therefore, the left and right components behave oppositely under boosts. We can summarize these transformations in exponential form in the following way. ψL -> exp(-iθ.σ/2 + ξ.σ/2)ψL ψR -> exp(-iθ.σ/2 - ξ.σ/2)ψR