Redshift Academy

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Astronomy

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Astronomical Distance Units .
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Celestial Coordinates .
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Celestial Navigation .
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Location of North and South Celestial Poles .

Chemistry

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Avogadro's Number
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table .

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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Center of Mass Frame
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Comparison Between Gravitation and Electrostatics
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Compton Effect .
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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions .
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Electric Fields
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Error Analysis
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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity .
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases .
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Kirchoff's Laws
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Laplace's and Poisson's Equations
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Lorentz Force Law
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Maxwell's Equations
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Moments and Torque
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Nuclear Spin
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One Dimensional Wave Equation .
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Pascal's Principle
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Phase and Group Velocity
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Planck Radiation Law .
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Poiseuille's Law
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Radioactive Decay
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Refractive Index
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Rotational Dynamics
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Simple Harmonic Motion
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Specific Heat, Latent Heat and Calorimetry
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Stefan-Boltzmann Law
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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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Wien's Displacement Law
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Young's Modulus

Climate Change

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Keeling Curve .

Cosmology

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Baryogenesis
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Cosmic Background Radiation and Decoupling
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CPT Symmetries
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Dark Matter
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Friedmann-Robertson-Walker Equations
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Geometries of the Universe
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes .
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Olbers' Paradox
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Penrose Diagrams
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Planck Units
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Stephen Hawking's Last Paper .
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Stephen Hawking's PhD Thesis .
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The Big Bang Model

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets
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Capital Structure
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Dividend Discount Formula
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

Game Theory

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The Truel .

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Geodesics
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Gravitational Time Dilation
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Gravitational Waves
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One-forms
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Quantum Gravity
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Relativistic, Cosmological and Gravitational Redshift
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Ricci Decomposition
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Ricci Flow
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Stress-Energy Tensor
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Stress-Energy-Momentum Tensor
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Tensors
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The Area Metric
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The Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Metric Tensor
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Vierbein (Frame) Fields
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World Lines Refresher

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory .
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Euler-Lagrange Equation
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Hamiltonian Formulation .
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem

Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Arithmetic and Geometric Sequences and Series
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Asymptotes
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Augmented Matrices and Cramer's Rule
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Basic Group Theory
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Basic Representation Theory
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Binomial Theorem (Pascal's Triangle)
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Building Groups From Other Groups
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Completing the Square
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Complex Numbers
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Composite Functions
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Conformal Transformations .
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Conjugate Pair Theorem
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Contravariant and Covariant Components of a Vector
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Derivatives of Inverse Functions
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Double Angle Formulas
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Eigenvectors and Eigenvalues
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Euler Formula for Polyhedrons
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Factoring of a3 +/- b3
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Fourier Series and Transforms .
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Fractals
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Gauss's Divergence Theorem
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Grassmann and Clifford Algebras .
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Heron's Formula
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Index Notation (Tensors and Matrices)
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Inequalities
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Integration By Parts
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Introduction to Conformal Field Theory .
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Inverse of a Function
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Law of Sines and Cosines
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Line Integrals, ∮
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Logarithms and Logarithmic Equations
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Matrices and Determinants
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Matrix Exponential
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Mean Value and Rolle's Theorem
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Modulus Equations
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Orthogonal Curvilinear Coordinates .
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Parabolas, Ellipses and Hyperbolas
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Piecewise Functions
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Polar Coordinates
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Polynomial Division
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Quaternions 1
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Quaternions 2
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Regular Polygons
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Related Rates
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Sets, Groups, Modules, Rings and Vector Spaces
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Similar Matrices and Diagonalization .
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Spherical Trigonometry
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Stirling's Approximation
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Sum and Differences of Squares and Cubes
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Symbolic Logic
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Symmetric Groups
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Tangent and Normal Line
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Taylor and Maclaurin Series .
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The Essential Mathematics of Lie Groups
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The Integers Modulo n Under + and x
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The Limit Definition of the Exponential Function
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Tic-Tac-Toe Factoring
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Trapezoidal Rule
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Unit Vectors
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Vector Calculus
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity and Chirality
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Klein-Gordon and Dirac Equations
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Regularization and Renormalization
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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors .
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The Vacuum Catastrophe

Probability and Statistics

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Box and Whisker Plots
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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Chi Squared Goodness of Fit
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Conditional Probability
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Confidence Intervals
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Data Types
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Expected Value
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Monte Carlo Methods
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Non Parametric Tests
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One-Way ANOVA
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Pearson Correlation
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Permutations and Combinations
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Pooled Variance and Standard Error
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Probability Distributions
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Probability Rules
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Sample Size Determination
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Sampling Distributions
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption

Quantum Computing

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The Qubit .

Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory

Quantum Mechanics

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Basic Relationships
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Bell's Theorem
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Bohr Atom
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Clebsch-Gordan Coefficients .
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Commutators
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Dyson Series
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Electron Orbital Angular Momentum and Spin
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Entangled States
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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions
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Pauli Exclusion Principle
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Pauli Spin Matrices
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Photoelectric Effect
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Position and Momentum States
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Probability Current
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Schrodinger Equation for Hydrogen Atom
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Schrodinger Wave Equation
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Schrodinger Wave Equation (continued)
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Observer Effect
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The Quantum Harmonic Oscillator .
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids .
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Fermi-Dirac Statistics .
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Intrinsic and Extrinsic Semiconductors
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors .
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Electromagnetic 4 - Potential
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Energy and Momentum, E = mc2
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Lorentz Invariance
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Lorentz Transform
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Lorentz Transformation of the EM Field
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Newton versus Einstein
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Spinors - Part 1 .
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Spinors - Part 2 .
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The Lorentz Group
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Velocity Addition

Statistical Mechanics

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Black Body Radiation
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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas

String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification

Superconductivity

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BCS Theory
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Introduction to Superconductors
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1 .
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3 .
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
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Formulas
Last modified: January 26, 2018

Regularization and Renormalization ---------------------------------- Regularization and Renormalization are a collection of enormously important and mathematically difficult techniques that are employed in QFT to solve the problem of divergences in Feynman loop integrals. Because the mathematics is lengthy and difficult, we will skip a lot of the more complicated calculations in favor of illustrating the concepts. We start by saying that regularization and renormalization are 2 different things. Simply put, Regularization makes a divergent expression (quantity) finite. Renormalization discards this expression (makes it zero). Regularization essentially assumes the existence of 'new' unknown physics at unobserved scales (small size or large energy levels) and enables current theories to give accurate predictions as an "effective theory" within its intended scale of use. We will concentrating on 2 of the most popular techiques in use today - DIMENSIONAL REGULARIZATION and the MINIMAL SUBTRACTION RENORMALIZATION scheme. Dimensional Regularization -------------------------- In general, Feynman loop integrals have the form: I = (1/2π)4∫ dk4p/(2π)4 1/((k2 - m2))n These integrals are either logarithmically or quadratically divergent. To evaluate these types of integrals and get them into the form required for dimensional regularization requires the use of 3 techniques - FEYNMAN PARAMETERIZATION: Feynman noted that, 1 1 δ(x1 + ... + xn - 1) 1/(A1 ... An) = (n - 1)!∫dx1 ... ∫dxn -------------------- 0 0 [x1A1 + ... + xnAn]n For n = 2 we get: 1 1 δ(x + y - 1) 1/(AB) = ∫dx ∫dy ------------- 0 0 [xA + yB]2 If we let y = 1 - x => δ(x + 1 - x - 1) = δ(0) = 1 And the integral becomes: 1 1 1 1/(AB) = ∫dx ∫dy ----------------- 0 0 [xA + (1 - x)B]2 - WICK ROTATION: After introducing Feynman parameters and completing the square, one is often left with an integral over a loop 4-momentum in Minkowski space. For example: I = ∫d4k/(2π)4 i/(k2 - m2)n = ∫d3k ∫dk0 1/(k02 - k2 - m2)n However, this has poles at k0 = ±√(k2 + m2) To rectify this the term iε is added to displace these poles away from the integration path. k0 plane: Im(k0) | | x | iε | -----------+---------- Re(k0) | iε | x | | x = poles of the Feynman propagator. So the integral now becomes the integral of a contour in the complex plane: I = ∫d3k∫dk01/(k02 - k2 - m2 + iε)n If we rotate Re(k0) counter clockwise by 90° we do not encounter any problem with crossing the poles. If we do this we end up with: (kE)2 = (k0)2 + (k1)2 + (k2)2 + (k3)2. This is the momentum in Euclidean space. Now the iε term no longer plays a role and we can set ε = 0. This rotation is refered to as a Wick rotation. Once Wick rotated, the integrals can be evaluated in standard ways. ∫d4k/(2π)4 1/(k2 - m2 + iε)2 => -i∫d4k/(2π)4 1/(kE2 - m2)2 The Wick rotation is just a trick to do integrals. It has no physical meaning. - TENSOR INTEGRALS: Often integrals have terms such as kμkν in the numerator: Iμν = d4k/(2π)4 kμkν/(k2 - m2)n    = gμμ∫d4k/(2π)4 k2/(k2 - m2)n If there is just one k in the numerator we get: ∫d4k/(2π)4 kp/(k2 - m2)n = 0 This is because we are integrating over all k and the integrand is antisymmetric under k -> -k. The only terms we need to keep are those with even powers of k in the numerator. Dimensional regularization assumes that the spacetime dimension is not 4 but is analytically continued to 4 - ε. As an example, for the case where n = 2, we write: I = lim με∫ d4 - εk/(2π)4 - ε 1/(k2 - Δ)2 ε -> 0 Where μ is an arbitrary parameter with the dimension of energy that forces the result to be dimensionless. The integral is now convergent and has the solution: (-1)2i Γ(2-d/2) 1 I = με --------------- ----- (4π)d/2Γ(2) Δ2-d/2 Where Γ() is the Gamma function and 1/Δ2-d/2 can be expressed using the series representation:  ∞ x-n = Σnm(-ln(x))  m=0    = 1 - nlog(x) ... Which leads to: 1/Δ2-d/2 = 1 - (2 - d/2)lnΔ If we now replace d with 4 - ε we get: iΓ(ε/2) 1 I = με/2 ---------- -- (4π)2-ε/2Γ(2) Δε/2 iΓ(ε/2) = ---------- (μ2/Δ)ε/2 (4π)2-ε/2Γ(2) Expansion of Γ around ε = 0 (using Weierstrass's definition of Γ) gives: Γ(ε/2) = 2/ε - γE + O(ε) + ... Where γE is the Euler-Mascheroni constant (~ 0.5772). Noting that Γ(2) = 1 we get: (i/16π2)(2/ε - γE)(1 - εlnΔ) = (i/16π2)(2/ε - ln(μ2/Δ) - γE - εγEln(μ2/Δ)) = (i/16π2)(2/ε - ln(μ2/Δ) - γE - O(ε)) The final result becomes: I = (i/16π2)[2/ε - γE + ln(4π) + ln(μ2/Δ)] = (i/16π2)[2/ε + ln{4πexp(-γE2/Δ}]   _ = (i/16π2)[2/ε + ln{μ2/Δ}] ... 1. _ Where μ2 = 4πexp(-γE2 Other useful integrals are: (-1)1i Γ(1-d/2) 1 ∫d4k/(2π)4 1/(k2 - Δ) = -------------- ----- (4π)d/2Γ(1) Δ1-d/2 Γ(ε/2-1) = Γ(ε/2)/(ε/2 - 1) = -2/ε + γE - 1 1/Δ1-d/2 = 1/Δε/2-1 = 1/Δε/21/Δ-1 = (1 - (ε/2)ln(μ2/Δ))Δ -(i/16π2)(-2/ε + γE - 1)(1 - (ε/2)ln(μ2/Δ))Δ = (i/16π2)Δ(2/ε - ln(μ2/Δ) - γE + 1 + O(ε)) Therefore, I = (iΔ/16π2)[2/ε - ln(μ2/Δ) + ln(4π) - γE + 1] ... 2. (-1)2-1i Γ(1-d/2) d 1 ∫d4k/(2π)4 k2/(k2 - Δ)2 = ---------------- --- ------ (4π)d/2Γ(2) 2 Δ1-d/2 Γ(ε/2-1) = -2/ε + γE - 1 1/Δ1-d/2 = 1/Δε/2-1 = 1/Δε/21/Δ-1 = (1 - (ε/2)ln(μ2/Δ))Δ -(i/16π2)(-2/ε + γE - 1)(2 - ε/2)(1 - (ε/2)ln(μ2/Δ))Δ = (i/16π2)2Δ(2/ε - ln(μ2/Δ) - γE + 1 + O(ε)) Therefore, I = ((iΔ/8π2)[2/ε - ln(μ2/Δ) + ln(4π) - γE + 1] ... 3. (-1)2-1i Γ(1-d/2) gμν 1 ∫d4k/(2π)4 kμkν/(k2 - Δ)2 = ---------------- -- ------ (4π)d/2Γ(2) 2 Δ1-d/2 Γ(ε/2-1) = -2/ε + γE - 1 1/Δ1-d/2 = 1/Δε/2-1 = 1/Δε/21/Δ-1 = (1 - (ε/2)ln(μ2/Δ))Δ -(igμν/16π2)(-2/ε + γE - 1)(2 - ε)(1 - εln(μ2/Δ))Δ = (i/16π2)2gμνΔ(2/ε - ln(μ2/Δ) - γE + 1 + O(ε)) Therefore, I = ((igμνΔ/8π2)[2/ε - ln(μ2/Δ) + ln(4π) - γE + 1] ... 4. The general forms of above solutions is: I = (-1)ni(1/4π)d/2(1 or d/2 or gμν)f(Γ,Δ) The (-1)ni term represents th NOTE: The integrals are specified in Minkowski space. The solutions involving the Γ functions have already taken into account the Wick rotation via the (-1)ni term where n is the power of the denominator, (k2 - Δ)n. The main advantage of dimensional regularization is that it preserves gauge invariance. Now, there is still a divergence in this expression as ε -> 0. How do we get rid of it?. There are 2 renormalization schemes that can be applied. The Minimal Subtraction (MS) scheme or, more commonly, the Modified Minimal Subtraction (MS) scheme. Both methods introduce a COUNTERTERM, δ, into the Lagrangian that removes the 2/ε or the 2/ε - γE + ln(4π) terms respectively. With dimensional regularization the δ's take the form: L k δ = ΣckΣakii k=1 i=1 Where L is the number of loops, c is the coupling term and ak is a numerical coefficient. For 1 loop: δ = (g2/16π2)[a1/ε] Therefore, in our simple case: δ = g2/16π2ε for MS    __ δ = (g2/16π2)(2/ε - γE + ln(4π)) for MS Photon and Electron Self Energy ------------------------------- To illustrate these techniques, we will focus on the simpler theory of QED. In particlar, will consider the electron and photon self energy to 1 loop only in our calculations. Consider the QED Lagrangian: _ L = ψ0(iγμDμ - m00 - (1/4)(FμνFμν)0 _ _ _ = iψ0γμ∂ψ0 + ψ0γμe0A0ψ0 - m0ψ0ψ0 - (1/4)(FμνFμν)0 The Feynman rules are: Vertex (A): -ieγμ Fermionic propagator (B): (iγμp + m)/(p2 - m2 + iε) Photon propagator (C): -igμν/p2 Electron Self Energy, Σ(p) -------------------------- αμk + m) γβ(-igμν) Σ(p) = (-ie)2∫d4k/(2π)4 ---------------------------- (k2 - m2 + iε)((p - k)2 + iε) Applying Feynman parameters to our integral leads to: 1 (2γμk - 4m) Σ(p) = e2∫d4k/(2π)4 ∫dx --------------------------------- 0 [(k2 - m2)(1 - x) + (p - k)2x + iε]2 For the denominator, D, we get: D = k2 - m2 - k2x + m2x + p2x + k2x - 2kpx = k2 - m2 + m2x + p2x - 2kpx = k2 - m2 + m2x + p2x2 - p2x2 + p2x - 2kpx = (k - px)2 - m2 + m2x - p2x2 + p2x = (k - px)2 - Δ Where Δ = (1 - x)(m2 - p2x) Shifting k -> k + px results in: 1 (2γμk + 2γμpx - 4m) Σ(p) = e2∫dx ∫d4k/(2π)4 ----------------- 0 [k2 - Δ + iε]2 Now, from before we said ∫dk k/(...) = 0. Therefure, 1 1 Σ(p) = e2∫dx (2γμpx - 4m) ∫d4k/(2π)4 ------------- 0 [k2 - Δ + iε]2 We can now use equation 1. from above to get: 1 _ Σ(p) = (iα/4π)∫dx (2γμpx - 4m)(2/ε + ln(μ2/Δ)) 0 Where α = e2/4π __ After applying MS we get: 1 ΣR(p) = (iα/4π)∫dx (2γμpx - 4mR)(ln(μ2/Δ)) 0 1 = (iα/4π)∫dx (2γμpx - 4mR)(ln(μ2/(1 - x)(mR2 - p2x)) 0 Photon Self Energy, Π(p) ------------------------ For fermionic loops the rule is to multiply the propagators and take the trace of the matrix product. The probability amplitude for 1 loop is: Tr{γαμp - γμk + m)γβνk + m)} Π(p) = -(-ie)2∫d4p/(2π)4 --------------------------------- ((p - k)2 - m2 + iε)(p2 - m2 + iε) Note: There is a -ve in front of the e2 term because it represents a fermionic loop. The sign difference between fermionic and bosonic loops can be seen in the following diagram. Bosons Fermions ---------- ---------- A B AB A B AB - - -- - - -- + + + + - - - - + - + - Therefore, bosonic loops make a positive contribution to the vacuum energy while fermionic loops make a negative contribution. The diagram also illustrates the SPIN-STATISTICS THEOREM. If A and B swap positions there is no change in sign for the bosons but there is a change for the fermions. Thus, φ(x1)φ(x2) = φ(x2)φ(x1) While ψ(x1)ψ(x2) = -ψ(x2)ψ(x1) For the numerator, N, using various γ matrix identites yields: N = pμ(pν - kν) + pν(pμ - kμ) - gμν(p.(p - k) - m2) = 2pμpν - pμkν - pνkμ - gμν(p.(p - k) - m2) The integral becomes: N Π(p) = -4e2∫d4p/(2π)4 -------------------------------- ((p - k)2 - m2 + iε)(p2 - m2 + iε) We can now apply Feynman parameters to get: 1 N Π(p) = -4e2∫d4p/(2π)4∫dx ------------------------------------ 0 [x((p - k)2 - m2) + (1 - x)(p2 - m2)]2 For the denominator, D, we get: D = [p2x + k2x - 2pkx - m2x + p2 - m2 - p2x + m2x]2 = [k2x - 2pkx + p2 - m2]2 = [k2x - 2pkx + k2x2 - k2x2 + p2 - m2]2 = [(p - kx)2 - k2x2 + k2x - m2]2 = [(p - kx)2 - k2x(x - 1) - m2]2 = [(p - kx)2 - Δ]2 Where Δ = m2 - k2x(1 - x) We can now make the shift p -> p + kx to get for the numerator, N: For the 2pμpν term we get: 2pμpν = 2(pμ + xkμ)(pν + xkν)    = 2(pμpν + pμxkν + pνxkμ + x2kμkν) For the pμkν and pνkμ terms we get: pμkν -> (pμ + kμx)kν = pμkν + kμkνx pνkμ -> (pν + kνx)kμ = pνkμ + kνkμx For the gμν(p.(p - k) - m2) term we get: gμν(p + kx)(p + kx - k) - m2) = gμν(p2 + kpx - kp + pxk + k2x2 - k2x - m2) Putting it all together for the numerator we get: N = 2(pμpν + pμxkν + pνxkμ + x2kμkν) - (pμkν + kμkνx) - (pνkμ + kνkμx) - (gμν(p2 + kpx - kp + pxk + k2x2 - k2x - m2)) Neglecting terms linear in p leads to: N = 2pμpν + 2x2kμkν - kμkνx - kνkμx - (gμν(p2 + k2x2 - k2x - m2)) = 2pμpν + 2x2kμkν - 2kμkνx - (gμν(p2 + k2x2 - k2x - m2)) = 2pμpν + 2kμkνx(x - 1) - gμν(p2 + k2x(x - 1) - m2) = 2pμpν - 2kμkνx(1 - x) - gμν(p2 - k2x(1 - x) - m2) Now, in d dimensions gμνgμν = d. Therefore, dpμkν = gμνgμνpμkν = p2gμν ∴ pμpν = (1/d)p2gμν So we can write: N = (2/d)p2gμν - kμkν2x(1 - x) - gμν(p2 - k2x(1 - x) - m2) = (-1 + 2/d)p2gμν - kμkν2x(1 - x) - gμν(-k2x(1 - x) - m2) Lets take this term by term. For the 1st term we get: ∫d4p/(2π)4(-1 + 2/d)p2gμν/(p2 - Δ)2 = (-1 + 2/d)gμν∫d4p/(2π)4 p2/(p2 - Δ)2 = -igμν(d/2)(-1 + 2/d)(d/2)Γ(1-d/2)(1/Δ)1-d/2/(4π)d/2 = -igμν(1 - d/2)Γ(1-d/2)(1/Δ)1-d/2/(4π)d/2 Now Γ(2-d/2) = (1-d/2)Γ(1-d/2) and Δ1-d/2 = Δ2-d/2Δ. Therefore, ∫d4p/(2π)4(-1 + 2/d)p2gμν/(p2 - Δ)2 = -iΓ(2-d/2)(1/Δ)2-d/2/(4π)d/2gμνΔ For the 2nd term we get: kμkν∫d4p/(2π)4 1/(p2 - Δ)2 = ikμkνΓ(2-d/2)(1/Δ)2-d/2/(4π)d/2 For the 3rd term we get: gμν(-k2x(1 - x) - m2)∫d4p/(2π)4 1/(p2 - Δ)2 = igμν(-k2x(1 - x) - m2)Γ(2-d/2)(1/Δ)2-d/2/(4π)d/2 Therefore, Π(k) = iΓ(2-d/2)(1/Δ)2-d/2/(4π)d/2[-gμνΔ - kμkν - gμν(-k2x(1 - x) - m2) = i[-gμν(m2 - k2x(1 - x)) - 2x(1 - x)kμkν - gμν(-k2x(1 - x) - m2)] = i[-gμν(-2k2x(1 - x)) - 2x(1 - x)kμkν] = i[gμνk2 - kμkν]∫dx x(1 - x)(-8e2/(4π)d/2)∫dx x(1 - x)Γ(2-d/2)(1/Δ)2-d/2 We can now write this as: Π(k) = (gμνk2 - kμkν)iπ(k2) Where, π(k2) = -8e2/(4π)d/2∫dx x(1 - x)Γ(2-d/2)(1/Δ)2-d/2   = -8e2/(4π)d/2∫dx x(1 - x)(2/ε - ln(μ2/Δ) + ln(4π) - γE - O(ε)) Π(k) is a rank 2 tensor formed from the 4-momentum of the photon and gμν. The only way to build such a tensor is to make a linear combination of gμν and kμkν. It, therefore, has the form: Πμν(k) = Agμν + Bkμkν We now need to introduce the WARD IDENTITY into the discussion. Ward Identity ------------- Let Π(k) = εμ(k)Πμ(k) be the amplitude for some QED process involving an external photon with momentum, k, and εμ(k) is the polarization vector of the photon. The Ward identity is: kμΠμ(k) = 0 It tells us that the photon's polarization parallel to its direction of propagation doesn't contribute to the scattering amplitude. If we now apply the Ward identity to Πμν we get: kμΠμν = Akμgμν + Bk2kν     = Akν + Bk2pν = 0     = kν(A + Bk2) = 0 For kν ≠ 0 we get A = -Bk2 Substituting back into our original equation gives: Πμν(k) = -Bk2gμν + Bkμkν    = (-k2gμν + kμkν)B    = (-k2gμν + kμkν)π(k2) Where, 1 π(k2) = -8e2/(4π)d/2∫dx x(x - 1)(2/ε - ln(μ2/Δ) + ln(4π) - γE - O(ε)) 0 So the contribution to the self energy by the photon itself is 0. Zero energy implies 0 mass which is required by gauge invariance. Therefore, the Ward identity is closely related to gauge invariance. With the Ward identity, the self energy of the photon is logarithmically divergent. Without it, the self energy would be quadratically divergent. We will see this in the next section on Yukawa theory where we calculate the 1 loop fermionic correction to the massive Higgs boson. Yukawa Theory ------------- For the Yukawa 1 loop correction our integral is similar to the photon self energy integral except that the coupling constant becomes g, the photon is replaced by the Higgs boson and trace of the numerator becomes simpler. Tr[γαμp - γμk + m)γβνk + m)] = p.(p - k) + m2   = 4(p2 - pk + m2) The integral becomes: (p2 - pk + m2) Π(p) = -4g2∫d4p/(2π)4 -------------------------------- ((p - k)2 - m2 + iε)(p2 - m2 + iε) We can now apply Feynman parameters to get: 1 (p2 - pk + m2) Π(p) = -4g2∫d4p/(2π)4∫dx ------------------------------------ 0 [x((p - k)2 - m2) + (1 - x)(p2 - m2)]2 For the denominator calculation is identical to the photon self energy case: D = [(p - kx)2 - Δ]2 Where Δ = m2 - k2x(1 - x) Again, we can now make the shift p -> p + kx to get for the numerator, N: N = (p + kx)2 - k(p + kx) + m2 = p2 + 2kpx + k2x2 - pk - k2x + m2 = p2 - k2x(1 - x) + 2kpx - pk + m2 And for the denominator, D: D = [p2 - Δ]2 Therefore, the integral becomes: 1 p2 - k2x(1 - x) + 2kpx - pk + m2 Π(k) = -4g2∫dx ∫d4p/(2π)4 -------------------------------- 0 [p2 - Δ]2 As before we can omit the terms linear in p and rearrange to get: 1 p2 + m2 - k2x(1 - x) Π(k) = -4g2∫dx ∫d4p/(2π)4 -------------------- 0 [p2 - Δ]2 1 p2 Π(k) = -4g2[∫dx ∫d4p/(2π)4 -------- 0 [p2 - Δ]2 1 1 - ∫dx (m2 - k2x(1 - x)) ∫d4p/(2π)4 ---------] 0 [p2 - Δ]2 Π(k) = (-4g2)(-i/(4π)d/2[(d/2)Γ(1-d/2)(1/Δ)1-d/2 + ΔΓ(2-d/2)(1/Δ)2-d/2] Now Γ(2-d/2)(1/Δ)(2-d/2) = (1 - d/2)Γ(1-d/2)(1/Δ)(1-d/2) Therefore, Π(k) = -4g2/(4π)d/2[(-i(d/2)Γ(1-d/2)(1/Δ)(1-d/2) - (-i)(1 - d/2)(d/2)Γ(1-d/2)(1/Δ)(1-d/2)] = -4g2/(4π)d/2[-iΓ(1-d/2)(d/2) - (-i)(1 - d/2)(Γ(1-d/2))(1/Δ)(1-d/2)] = -4g2/(4π)d/2[-iΓ(1-d/2)((d/2) - (1 - d/2))(1/Δ)(1-d/2)] = -4g2/(4π)d/2[-i[(d - 1)Γ(1-d/2)(1/Δ)1-d/2] = -4g2/(4π)d/2[-i(3 - ε)Γ(1-d/2)(1/Δ)1-d/2] = -4g2(-i/(4π)d/2)[(3 - ε)Δ(2/ε - ln(μ2/Δ) + ln(4π) - γE + 1) = -4g2(-i)(1/(4π)d/2)[(3 - ε)Δ + (3 - ε)Δ(2/ε - ln(μ2/Δ) + ln(4π) - γE)] = -4g2(-i)(1/(4π)d/2)[3Δ + 3Δ(2/ε - ln(μ2/Δ) + ln(4π) - γE) - Δ] = -4g2(-i)(1/(4π)d/2)[2Δ + 3Δ(2/ε - ln(μ2/Δ) + ln(4π) - γE)] = -4g2(-i)[(1/8π2)(k2/6 + m2) + (3/16π2)∫dx (m2 - k2x(1 - x)(2/ε - ln(μ2/Δ) + ln(4π) - γE)] Check: Using equations 2. and 3. from above we get: Π(k) = -4g2[-i/8π2Δ(2/ε - ln(μ2/Δ) + ln(4π) - γE + 1) - i/16π2(2/ε - ln(μ2/Δ) + ln(4π) - γE)] = -4g2(-i)[1/8π2Δ + 1/8π2Δ(2/ε - ln(μ2/Δ) - ln(4π) - γE) + 1/16π2(2/ε - ln(μ2/Δ) + ln(4π) - γE)] = -4g2(-i)[1/8π2Δ + 3/16π2(2/ε - ln(μ2/Δ) + ln(4π) - γE)] After MS we get: Π(k) = -4g2(-i)[1/8π2(k2/6 + m2) - 3/16π2(ln(μ2/Δ))] This is quadratically divergent. This is significant in the context of the GAUGE HIERARCHY PROBLEM disussed in the notes on SUPERSYMMETRY.