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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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Gauss's Law of Universal Gravity .
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Gravity - Force and Acceleration
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Kinetic Theory of Gases .
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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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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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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: May 6, 2020

Relativistic Quantum Field Theory --------------------------------- Canonical Representation ------------------------ One of the key insights of quantum field theory is to do away with classical mathematical forms of the wave function by listing how many particles (quanta) are in each momentum state, k. These occupation numbers are the analog of the Slater determinant in QM. This referred to as the CANONICAL REPRESENTATION of QFT. We can specify the state of a system as: |n(p1) n(p2) n(p2) .... n(p)> where, n(pi) = number of quanta with momentum pi These states are referred to as FOCK States. Examples: |0> = |0,0,0,0,0> a1|0> = |1,0,0,0,0> a2a1|0> = |0,1,0,0,0> |0> = |0,0,0,0,0> a1a1|0> = |2,0,0,0,0> Consider the periodic loop. Momentum (and frequency) are quantized where p = 2πn/L. The waves moving on the loop are equivalent to a collection of harmonic oscillators - one harmonic oscillator for each value of n. For each harmonic there is a number of quanta n(pn) represented by the occupation number. Field Operators --------------- In QM, the general solution to the wave equation is a superposition of plane waves (Fourier expansion of basis vectors): ψ(x) = ΣpApexp(ipx) where Ap is an amplitude term. and, ψ*(x) = ΣpAp*exp(-ipx) By analogy, in QFT we can write: Ψ(x) = Σpapexp(-ipx) where ap is the creation operator and Ψ(x) = Σpapexp(ipx) where ap is the annihilation operator The field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is a quantum-mechanical amplitude for finding a particle in some position. However, they are closely related. We use capitals to denotes the field operators. Ψ(x)/Ψ(x) create/destroy a particle at a particular point in space. ap/ap create/annihilate particles with certain momenta. Field operators act by applying the Fourier transform to the creation and annihilation operators. Examples: Pre-existing particle, n5 = 1 |0 0 0 0 1 0 0> Create a new particle in momentum state p = 3 at x. Thus, n3 -> 1, n5 = 1 Σpa3exp(-i3x)|0 0 0 0 1 0 0> => e-i3x|0 0 1 0 1 0 0> Create another new particle in momentum state p = 5 at the same x. Thus, n5 -> 2 Σpa5exp(-i5x)|0 0 1 0 1 0 0> => √2e-i5x|0 0 1 0 2 0 0> The factor of √2 comes from a|n> = √(n+1)|n+1> Creation and Annihilation Operators on Bra Vectors -------------------------------------------------- The rule for the creation and annihilation operators when they operate on bra vectors is the opposite of the case for ket vectors. Thus, the creation operator acts on a bra as an annihilation operator and vice versa. <n|a => <n-1|√n and <n|a => <n+1|√n+1 What is the meaning of ∫Ψ(x)Ψ(x)dx ? ------------------------------------- Expand in terms of basis states. = ∫Σpape-ipxΣqaqeiqxdx = ∫ΣpapΣqaqei(q-p)xdx = ∫ΣpapΣqaqδpqdx = ∫Σpapap - the occupation number is apap = the number of particles at x. Therefore, Ψ(x)Ψ(x) is the density of particles at x. Time Dependent Schrodinger Equation ----------------------------------- Differentiate, Ψ w.r.t. t and x: ∂Ψ/∂t = Σp(-iω)apexp(i(px - ωt)) ∂Ψ/∂x = Σp(ip)apexp(i(px - ωt)) ∂2Ψ/∂x2 = Σp(ip)2apexp(i(px - ωt)) So we can write: ∂Ψ/∂t/∂2Ψ/∂x2 = Σp(-iω)apexp(i(px - ωt))/Σp(ip)2apexp(i(px - ωt))   = (-iω)/(ip)2 Which leads to:   (ip)2∂Ψ/∂t = (-iω)∂2Ψ/∂x2   p2∂Ψ/∂t = (iω)∂2Ψ/∂x2 Now E = hω = p2/2m Therefore p2 = 2mω/h (2mω/h)∂Ψ/∂t/∂ = (iω)∂2Ψ/∂x2 ∂Ψ/∂t =(ih/2m)∂2Ψ/∂x2 Note: In both cases we could have equally integrated over position/time to demonstrate conservation of momentum and conservation of energy respectively. Heisenberg Uncertainty Principle in QFT --------------------------------------- Analagous to the relationship in QM, the creation and annihilation for position and momentum are Fourier transforms of each other. Thus: Ψ(x) = Σpapexp(-ipx) and Ψ(x) = Σpapexp(ipx) and ap = ΣxΨ(x)exp(ipx) and ak = ΣxΨ(x)exp(-ipx) This shows that to create a state of definite position, it is necessary to sum over many momentum states. Conversely, to create a state of definite momentum it is necessary to sum over many position states. Free Field Quantization ----------------------- It is well known that the standard equations in field theory for scalar and fermionic fields are respectively the Klein-Gordon and Dirac equations. Massive Spin 0 Real Scalar Fields --------------------------------- Consider the Klein-Gordon equation (h = c = 1): ∂2φ/∂t2 - ∇2φ + m2φ = 0 This can be derived using the relativistic relationship (h = c = 1): E2 - p2 - m2 = 0 <=> -ημνpμpν + m2 = 0 where ημν = (-, +, +, +) and pμ -> i∂/∂xμ (p0 -> i∂/∂t) In QM solutions to the K-G equation are of the form φ = exp(i(kx - ωt)) with the constraint that: ω = √(p2 + m2) {from E = hω = √(p2c2 + m2c4)} Canonical Commutation Relations ------------------------------- Let's look at the commutator, [x,p] = ihδij. In QM this is referred to as the CANONICAL COMMUTATION RELATION and defines the relationship between canonical conjugate quantities (quantities which are related through the Fourier transform). The quantity p is often written as π and is referred to as the momentum conjugate to x. By analogy with this we can write a similar relationship for the commutators for quantum fields: [φ(x),π(y)] = ihδ(3)(x - y) [φ(x),φ(y)] = [π(x),π(y)] = 0 The Lagrangian for the K-G equation is (h = c = 1): L = (1/2)∂μφ∂μφ - (1/2)m2φ2 The quantity, π, for the fields is obtained using the Euler-Lagrange equations generally written as: ∂μ(∂L/∂(∂μφ)) - ∂L/∂φ = 0 The momenta canonically conjugate to the coordinates of the field (or simply the canonical momentum) is defined as: . π = ∂L/∂φ For the K-G equation we get: . . π = ∂L/∂φ = φ Therefore: . [φ(x),π(y)] ≡ [φ(x),φ(y)] = ihδ3(x - y) The quantization of the harmonic oscillator can now be applied to the free scalar field by writing φ as a linear sum (Fourier expansion) of creation and annihilation operators. Using the expessions for the creation and annihilation operators for the harmonic oscillator, we can write: ap = √(ω/2)(φ + iπ/√(2ω) or (1/√(2ω)(ωφ + iπ) and ap = √(ω/2)(φ - iπ/√(2ω) or (1/√(2ω)(ωφ - iπ) Therefore, φ = (1/√(2ω))(apexp(i(px - ωt) + apexp(-i(px - ωt)) and . π = φ = -i√(ω/2)(apexp(i(px - ωt) - apexp(-i(px - ωt)) Thus, the field operator equations are: φ(x) = ∫(d3p/(2π)3) (1/√(2ω))[apexp(ipx) + apexp(-ipx)] and . π(x) = φ(x) = ∫(d3p/(2π)3) (-i√(ω/2))[apexp(ipx) - apexp(-ipx)] The commutator [ap,aq] is a little tricky to derive. It can be shown to be: [ap,aq] = (2π)3δ(3)(p - q) These creation and annihilation operators represent a more abstract form of the operators that we derived for the Quantum Harmonic oscillator. We can summarize the commutators for the real scalar field as follows: [ap,aq] = (2π)3δ(3)(p - q) [φ(x),π(y)] = ihδ(3)(x - y) [ap,aq] = [ap,aq] = 0 [φ(x),φ(y)] = [π(x),π(y)] = 0 The operators φ(x) and π(y) depend on space and not time. This is the SCHRODINGER PICTURE. In the HEISENBERG PICTURE the operators change with time. However, the operators in the two pictures agree at a fixed time, say, t = 0. The commutation relations become EQUAL TIME commutation relations in the Heisenberg picture. [φ(x,t),π(y,t)] = ihδ(3)(x - y) [φ(x,t),φ(y,t)] = [π(x),π(y)] = 0 We can show this as follows: ap(t) = exp(iHt)ap(0)exp(-iHt) Now, from before we had: H = ωapap Therefore, i[H,ap(t)] = -iωap(t) [H,ap(t)] = -ωap(t) Similarly, [H,ap(t)] = ωap(t) Plugging these into the Heisenberg equation, i[H,ap(t)] = ∂ap(t)/∂t, yields: ∂ap(t)/∂t = -ωap(0) with solution, ap(t) = ap(0)exp(-iωt) and, ∂ap(t)/∂t = ωap(t) with solution, ap(t) = ap(0)exp(iωt) Thus, the field operator equations in the Heisenberg picture are: φ(x,t) = ∫(d3p/(2π)3) (1/√(2ω))[exp(-iωt)apexp(ipx) + exp(iωt)apexp(-ipx)] = ∫(d3p/(2π)3) (1/√(2ω))[apexp(ip.x) + apexp(-ip.x)] Where p is now a 4-vector instead of a 3-vector. p.x = px - ωt It is easy to that this satisfies the K-G equation. . Now, we said from before that π = φ. Therefore, . φ(x,t) = ∫(d3p/(2π)3) (1/√(2ω))[-iωapexp(ip.x) + iωapexp(-ip.x)] = ∫(d3p/(2π)3) (1/√(2ω))(-iω)[apexp(ip.x) - apexp(-ip.x)] = ∫(d3p/(2π)3) (-i√(ω/2))[apexp(ip.x) - apexp(-ip.x)] = π Which is what we had before, so everything checks out! Non Relativistic Limit ---------------------- In the non-relativistic limit the K-G equation yields the Schrodinger equation. We can see this by replacing E = √(p2 + m2) with the non-relativistic relationship E = p2/2m. We get: ihtφ - (1/2m)∇2φ = 0 The Lagrangian for the SE is: L = φ(iht - (h2/2m)∇2)φ Where, . π = ∂L/∂φ = iφ Therefore: [φ(x),π(y)] ≡ [φ(x),iψ(y)] Now, [φ(x),iψ(y)] = φ(x)iφ(y) - iφ(y)φ(x)   = i(φ(x)φ(y) - φ(y)φ(x))   = i[φ(x),φ(y)] Therefore, by comparison: [φ(x),φ(y)] = hδ(x - y) Relativistic Normalization -------------------------- The formulation involving 4-vectors is clearly Lorentz invariant. But what about 3-vectors? The Dirac Delta function in 3 dimensions is defined as: δ(3)(p - q) = (1/2π)3∫ d3x exp(i(p - q).x) Likewise in spacetime it is defined as: δ(4)(p - q) = (1/2π)4∫ d4x exp(i(p - q).x) δ(4) is Lorentz invariant but δ(3) is not. How do we make it so? Consider: δ(4)(p - q) = δ(Ep - Eq(3)(p - q) Now consider; δ(Ep2 - Eq2) Using the following property of the delta function: δ(x2 - α2) = (1/|2α|)[δ(x + α) + δ(x - α)] We can write: δ(Ep2 - Eq2) = (1/|2Eq|)[δ(Ep + Eq) + δ(Ep - Eq)] If we disregard the negative energies we get: δ(Ep - Eq) = 2Eqδ(Ep2 - Eq2) Therefore, δ(4)(p - q) = 2Eqδ(Ep2 - Eq2(3)(p - q) Now δ(Ep2 - Eq2) is Lorentz invariant so the implication is that the invariant form of δ(3)(p - q) is 2Eqδ(3)(p - q) Now by definition ∫d3p δ(3)(p - q) = 1. Therefore, to compensate for the Lorentz invariant for we need to divide by 2Eq. Thus, ∫(d3p/2Eq) 2Eqδ(3)(p - q) = 1 Therefore, the relativistically normalized momentum states are given by: <p|q> = (2π)32ωδ(3)(p - q) |p> = √(2ω)|p> = √(2ω)ap|0> <q| = <q|√(2ω) = <0|√(2ω)ap Note: Some texts also define relativistically normalized creation operators as: ap = a(p)/√(2ω). This results in the 1/2ω factor seen in these texts. We will not assume this throughout this document. The Scalar Propagator --------------------- We can use the canonical approach to write the propagator as: G(x,y) = <0|annihilate particle at x, create particle at y|0> Thus, y -> x The propagator can now be written as: G(x,y) = <0|φ(x)φ(y)|0> By plugging in the equations for φ(x), φ(y) we end up with: <0|φ(x)φ(y)|0> = ∫(d3pd3q/(2π)6) (1/√(4ωpωq))<0|ap,aa|0>exp(-ipx + iqy) Which, after some math, yields: <0|φ(x)φ(y)|0> = ∫(d3p/(2π)3) (1/2ω)exp(-ip(x - y)) Until now, we integrated only over 3-momentum, with p0 fixed by the 'on-shell' relationship, E2 = ω2 = p02 = p2 + m2. To accomodate 'off-shell' virtual particles we need to integrate over the 4-momentum. This also makes the propagator manifestly Lorentz invariant. Therefore, we now integrate over 4-dimensional Minkowski spacetime. Thus, G(x,y) = ∫(d4p/(2π)4) exp(-ip(x-y))/(p2 - m2) However, as it stands this integral is ill-defined because the integrand has 2 poles at p0 = ±√(p2 + m2). The final expression for the propagator will depend on the choice of contour. A contour going under the left pole and over the right pole gives the FEYNMAN PROPAGATOR. Thus, the contour looks like: Instead of specifying the contour, it is standard to write the Feynman propagator as: G(x,y) = ∫(d4p/(2π)4) exp(-ip(x-y))/(p2 - m2 + iε) This way of writing the propagator is called the "iε prescription". The iε term has the effect of shifting the poles slightly off the real axis, so the integral is equivalent to the contour along the real p0 axis. The Feynman propagator satisfies: (∂2φ/∂t2 - ∇2φ + m2)G(x,y) = -iδ(4)(x - y) So G(x,y) is a Green's function for the Klein-Gordon operator. Solutions to G(x,y) over the chosen contour yield: GF(x,y) = limε->0∫(d4p/(2π)4) exp(-ip(x-y)/(p2 - m2 + iε)   = -(1/4π)δ(s) + (m/8π√s)H1(2)(m√s) for s ≥ 0   = -(im/4π2√(-s)K1(m√(-s)) for s < 0 Where s := (x0 - y0) - (x - y), H1(2)(m√s) is a Hankel function and K1(m√(-s)) is a modified Bessel function. The Fourier transform of the position space propagators can be thought of as propagators in momentum space. This is simpler than the position space form. GF(p) = -i/(p2 - m2 + iε) Massive Spin 0 Complex Scalar Fields ------------------------------------ Complex fields are required to accomodate charge. φ carries a charge of +1. φ carries a charge of -1 We can write φ in terms of real and imaginary parts. φ = (1/√2)(φR + iφI) The K-G Lagrangian becomes (h = c = 1): L = ∂μφ∂μφ - m2φφ = (1/2)∂μφRμφR + (1/2)∂μφLμφL - (1/2)m2φR2 - (1/2)m2φL2 Which is 2 uncoupled real scalar fields. . . . . π = ∂L/∂φ = φ and π = ∂L/∂φ = φ Therefore: . [φ(x),π(y)] ≡ [φ(x),φ(y)] The equation of motion associated to the above Lagrangian is the very same Klein-Gordon equation. However, now complex solutions of φ(x) are allowed. In order to describe boson particles with electric charge we need to introduce a new pair of creation and annihilation operators, b and b which create and annihilate antiparticles. Therefore, bp creates a particle of momentum p. cp creates an antiparticle of momentum p. The field operator equations are: φ(x) = ∫(d3p/(2π)3) (1/√(2ω))[bpexp(ip.x) + cpexp(-ip.x)] φ(x) = ∫(d3p/(2π)3) (1/√(2ω))[bpexp(-ip.x) + cpexp(ip.x)] . π(x) = φ(x) = ∫(d3p/(2π)3) (i(√(ω/2))[bpexp(-ip.x) - cpexp(ip.x)]   . π(x) = φ(x) = ∫(d3p/(2π)3) (-i(√(ω/2))[bpexp(ip.x) - cpexp(-ip.x)] We can summarize the same-time commutators for the complex field as follows: [bp,bq] = [cp,cq] = (2π)3δ(3)(p - q) [φ(x),π(y)] = [φ(x),π(y)] = ihδ(x - y) All other combinations of bp, bp, cp, cp = 0 All other combinations of φ(x), φ(x), π(y), π(y) = 0 Conserved Quantities -------------------- It is easy to see that the Lagrangian is invariant under the transformation (φ -> φexp(iθ) and φ* -> φ*exp(-iθ)). This implies there is a symmetry and a conserved quantity. Now e = 1 + iε ... Taylor series Thus, φ -> φ + iεφ -> φ + εfφ where fφ = iφ φ* -> φ* - iεφ* -> φ* - εfφ* where fφ* = -iφ* Look for a conserved quantity using Noether's Theorem. . . πφ = ∂L/∂φ = φ* . . πφ* = ∂L/∂φ* = φ Conserved quantity = ∫[πφfφ + πφ*fφ*]d3x . . = i∫[φ*φ - φφ*]d3x = i∫[πφ - π*φ*]d3x The quantity in [] is the charge density of the field, ρ. Another way to write this is: . ρ = Im(φφ*) ∴ Q = ∫ρd3x We can also write a 4-vector for current, jμ, as: jμ = Im(φ*μφ) ∂μjμ = Im(∂μ*μφ)) ∂μjμ = Im(∂μφ*μφ + φ*μμφ) The first term is real so the imaginary part is 0. The second term can be written as: -φ*m2φ since ∂μμφ + m2φ = 0 Again this is real. We can therefore conclude that: ∂μjμ = 0 This is the CONTINUITY EQUATION. Substituting the equations for the field operators yields: Q = ∫d3p/(2π)3 (cpcp - bpbp) The Complex Scalar Propagator ----------------------------- The real scalar propagator describes the creation of a particle at y and its annihilation at x. The question is how how to deal with antiparticles*. Feynman decided that the propagator must be made up of 2 parts - particles are represented by the case x0 > y0 and antiparticles are represented by x0 < y0. In order to include the latter it is necessary to introduce the WICK TIME 0RDERING symbol, T. T is defined as: Tφ(x)φ(y) = φ(x)φ(y) for x0 > y0 = φ(y)φ(x) for x0 < y0 The propagator can now be written as: G(x,y) = <0|Tφ(x)φ(y)|0> = <0|Θ(x0 - y0)φ(x)φ(y) + Θ(y0 - x0)φ(y)φ(x)|0> * Although we need complex fields to represent antiparticles, this interpretation is also there for a real scalar field because the particle is its own antiparticle. For a complex field: G(x,y) = <0|Tφ(x)φ(y)|0> Where, Tφ(x)φ(y) = φ(x)φ(y) for x0 > y0   = φ(y)φ(x) for x0 < y0 Therefore, G(x,y) = <0|Θ(x0 - y0)φ(x)φ(y) + Θ(y0 - x0(y)φ(x)|0> The propagator of the complex scalar field can be calculated in the same fashion as for the non-complex case. This can be seen from: <φ(x)φ(y)> = (1/2)[<φR(x)φR(y) - iφR(x)φL(y) + φL(x)φR(y) + φL(x)φL(y)>]   = (1/2)[<φR(x)φR(y) + φL(x)φL(y)>]   = <φ(x)φ(y)> The resulting Feynman propagator in 4-dimensional Minkowski spacetime is: G(x,y) = -∫(d4p/(2π)4) exp(-ip(x-y))/(p2 - m2 + iε) The interpretation now is that the amplitude for the particle to propagate from y -> x cancels the amplitude for the antiparticle to travel from x -> y. In fact, this interpretation is also there for the real scalar field because, as stated, the particle is its own antiparticle. As before, the propagator is equal to the Green's function of the non-complex case. As stated previously, the Fourier transform of the position space propagators can be thought of as propagators in momentum space. GF(p) = -i/(p2 - m2 + iε) Massive Fermionic Fields ------------------------ The Dirac equation is: (iγμμ - m)ψ = 0 The simplest solutions to the DE are plane waves u(p)exp(-ikx) and v(p)exp(ikx) where u(p) and v(p) are Dirac spinors. The Dirac Lagrangian is (h = c = 1): _ L = ψ(iγμμ - m)ψ _ L = ψ(iγ0∂ψ/∂t + iγ0γj0∂ψ/∂x - mψ) _ ψ = γ0ψ. The ψ by itself does not result in a Lorentz invariant Lagrangian and Lorentz invariant Dirac equation after the Euler- Lagrange equations have been applied to minimize the action. Although, the Dirac equation is first order in derivatives, it is the magic of the γ matrices that makes the Lagrangian Lorentz invariant! _ ψ(x) creates a Fermion. ψ(x) annihilates a Fermion. The canonical momenta are: . _ π = ∂L/∂ψ = iψγ0 = iγ0ψγ0 = iψ0)2 = 1) . _ π = ∂L/∂ψ = iψγ0 = iγ0ψγ0 = 0 Therefore: {ψ(x),π(y)} ≡ {ψ(x),iψ(y)} Now, {ψ(x),iψ(y)} = ψ(x)iψ(y) + iψ(y)ψ(x)   = i(ψ(x)ψ(y) + ψ(y)ψ(x))   = i{ψ(x),ψ(y)} The anticommutator obeys the same rule as the commutator in that: {ψ(x),π(y)} = ihδ(x - y) Therefore, by comparison: {ψ(x),ψ(y)} = hδ(x - y) The field operator equations are: ψ(x) = ∫(d3p/(2π)3) (1/√(2ω)Σs[bpsus(p)exp(ip.x) + cps†vs(p)exp(-ip.x)] ψ(x) = ∫(d3p/(2π)3) (1/√(2ω)Σs[bps†us†(p)exp(-ip.x) + cpsvs†(p)exp(ip.x)] π(x) = iψ = ∫(d3p/(2π)3) (i/√(2ω)Σs[bps†us†(p)exp(-ip.x) + cpsvs†(p)exp(ip.x)] π(x) = 0 bps creates a fermion of momentum p and spin s. cps creates an antifermion of momentum p and spin s. We can summarize the anti-commutators for the fermionic field as follows: {bpr,bqs†} = {cpr,cqs†} = (2π)3δrsδ(3)(p - q) {ψ(x)α,ψ(y)β} = hδαβδ(x - y) All other combinations of bp, bp, cp, cp = 0 All other combinations of ψ(x) and ψ(x) = 0 Where r and s represents the spin and α and β are the spinor indeces (u1, u2, v1, v2). Conserved Quantities -------------------- It is easy to see that the Lagrangian is invariant under the _ _ transformation (φ -> exp(iθ)φ and φ -> φexp(-iθ)). This implies there is a symmetry and a conserved quantity. We can derived the conserved quantity in 2 ways. Start with the Dirac equation: (iγμμ + m)ψ = 0 ... 1. Which expands to be: (iγ00ψ + iγiiψ + mψ) = 0 The Hermitian conjugate is: (i∂0ψγ0† + i∂iψγi† - mψ†) = 0 Now, γ0† = γ0 and γi† = -γi Therefore, (i∂0ψγ0 + i∂iψ(-γi) - mψ†) = 0 Multiply by γ0 from the right and make use of the defining relationship: γ0γi + γiγ0 = 2η0iI = 0 ∴ γ0γi = -γiγ0 To get:   _    _  _ (i∂0ψγ0 + ∂iψiγi - mψ) = 0 Or, _  ψ(i∂μγμ - m) = 0 ... 2. _ Where ψ = γ0ψ This is the ADJOINT DIRAC EQUATION. _ Multiplying equation 1. by ψ from the left and equation 2 by ψ from the right and adding gives: _   _ ψ(iγμμ + m)ψ + ψ(i∂μγμ - m)ψ = 0 _   _ ψ(γμμψ) + (ψ∂μγμ)ψ = 0 Now, the 2nd term on the LHS is interpreted as being right to left so this is equivalent to: _     _   _ ψγμμψ + (∂μψ)γμψ = ∂μ(ψγμψ) = ∂μ(jVμ) = 0 Again, this is the CONTINUITY EQUATION where jVμ is the 4-VECTOR PROBABILITY CURRENT. The PROBABILITY DENSITY is defined as:    _ ρ = jV0 = ψγ0ψ Therefore we can write, jVμ = (ρ,jVi) The other way to get at the conserved quantity is similar to the method already encountered with the complex scalar field. We can write: _ L = ψ(iγμμ + m)ψ _   _ = iψγμμψ + mψψ The canonical momentum is: . _ π = ∂L/∂ψ = iψγ0 _ _ π = ∂L/∂(∂0ψ) = 0 The transformations are: ψ -> ψ - iεψ -> ψ - εf where f = -iψ _ _ _ _ _ _ ψ -> ψ + iεψ -> ψ + εf where f = iψ Look for a conserved quantity using Noether's Theorem. __ Conserved quantity = ∫[πf + πf]d3x _ = ∫[iψγ0(-iψ)]d3x _ = ∫[ψγ0ψ]d3x Again, the quantity in [] is the charge density of the field, ρ. ∴ Q = ∫ρd3x Substituting the equations for the field operators yields: ψ(x) = ∫ d3p Σs[cpscps - dps†dp] Axial Vector Current -------------------- When m = 0, the Dirac Lagrangian also has an extra internal symmetry which rotates left and right-handed fermions in opposite directions. We can go through a similar procedure to the above to get: ψ -> exp(iαγ5)ψ _ _ ψ -> ψexp(iαγ5)    _ jAμ = ψγ5γμψ The Fermionic Propagator ------------------------ For a Dirac field: _ S(x,y) = <0|Tψ(x)ψ(y)|0> Where, _ Tψ(x)ψ(y) = ψ(x)ψ(y) for x0 > y0 _ = -ψ(y)ψ(x) for x0 < y0 The - sign is required for Lorentz invariance. Therefore,   _ _ S(x,y) = <0|Θ(x0 - y0)ψ(x)ψ(y) + Θ(y0 - x0)ψ(y)ψ(x)|0> The propagator of the Dirac field can be calculated in the same fashion as before. SF(x,y) = -∫(d4p/(2π)4) γμpμexp(-ip(x-y))/(p2 - m2 + iε) Where γμpμ = γ0p0 + γjpj This satisfies: (iγμμ + m)SF(x,y) = iδ(4)(x - y) So SF(x,y) is a Green's function for the Dirac operator. As stated previously, the Fourier transform of the position space propagator can be thought of as propagators in momentum space. SF(p) = (γμpμ + m)/(p2 - m2 + iε) Massless Spin 1 Vector Fields (The Electromagnetic Field) --------------------------------------------------------- There is no equivalent equation to the Klein-Gordon, Dirac or Schrodinger equations that describes the wavefunction of the photon. This is because photons always behave relativistically and can be freely emitted and absorbed. Instead, a single photon is described quantum mechanically by Maxwell's equations, where the solutions are taken to be complex. The electromagnetic 4 potential is defined as: Aμ = (φ/c, -A) Where φ is the electric scalar potential and A is the magnetic vector potential. The B and E fields in terms of φ and A are: B = ∇ x A E = -∇φ - ∂A/∂t We can also define an asymmetric tensor, Fμν such that, Fμν = ∂μAν - ∂νAμ - - | 0 Ex/c Ey/c Ez/c | |-Ex/c 0 -Bz By | = Fμν |-Ey/c Bz 0 -Bx | |-Ez/c -By Bx 0 | - - - - | 0 -Ex/c -Ey/c -Ez/c | | Ex/c 0 -Bz By | = Fμν | Ey/c Bz 0 -Bx | | Ez/c -By Bx 0 | - - From these we get Maxwell's Equations in the absence of charge sources: ∇.B = 0 ∂B/∂t = -∇ x E ∇.E = 0 ∂E/∂t = -∇ x B The Lagrangian for Maxwell's equations is: L = (-1/4)FμνFμν - JμAμ In the absence of charge sources is this becomes: L = (-1/4)FμνFμν ≡ (-1/2)(B2 - E2) = (-1/2)(∂μAνμAν - ∂νAμμAν) Substituting this into the Euler-Lagrange equations for a field gives: ∂μ(∂L/∂(∂μAν) = ∂L/∂Aν     = 0 So the E-L equation becomes: ∂μ(∂μAν - ∂νAμ) = 0 ∂μμAν - ∂μνAμ = 0 ∂μFμν = 0 Now, the electromagnetic field is a vector field that can be decomposed into longitudinal and transverse components. Longitudinal components do not have a curl and are parallel to the direction of motion. Transverse components do not have divergence and are perpendicular to the direction of motion. Thus, ∇.FT = 0 and ∇ x FL = 0 This is referred to as the HELMHOLTZ DECOMPOSITION. Since ∇.E = ∇.B = 0 in the absence of sources, both E and B are transverse. Gauge Fixing ------------ Different representative configurations of a physical state are called different gauges. Picking a gauge is rather like picking coordinates that are adapted to a particular problem. Moreover, different gauges often reveal slightly different aspects of a problem. Since the physical fields E and B are not modified by a gauge transformation, we may say that the potentials φ and A contain both a physical part (because the physical fields can be computed from them) and a nonphysical part (which changes when we do a gauge transformation). We will look at 2 gauges: the Coulomb gauge and the Lorentz gauge. Choosing the Coulomb gauge makes A into a transverse field that correctly restricts the photon to 2 degrees of freedom corresponding to the 2 polarization states. However the field commutation relations are a little more involved as is the propagator. The Coulomb gauge is also not Lorentz invariant although the final result is. Choosing the Lorentz gauge results in 4 degrees of freedom that need to be dealt with, but the field commutators follow the familiar form and the propagator is simpler. The Lorentz gauge is also manifestly Lorentz invariant from the outset. Coulomb Gauge ------------- E = -∇φ - ∂A/∂t. Therefore, ∇.E = -∇2φ - (∂(∇.A)/∂t) = 0 In the Coulomb gauge ∇.A = 0 so the equation of motion becomes: ∇2φ = 0 Let us pause for a moment to look at this. A is a 4-vector with the time component set to 0 (in other words a 3-vector). This allows the remaining part of the potential to satisfy the condition ∇.A = 0 (≡ ∂iAi = 0). We can also use ∂μμAν - ∂μνAμ = 0 to derive the equations of motion. The second term is ∂ν(∇.A) = 0 leaving the first term ∂μμAν ≡ ∇2A = 0. The momentum conjugate to Aμ is: . π0 = ∂L/∂A0 = 0 . πi = ∂L/∂Ai = -F0i = Ei The field operator equations are similar to the real scalar field: A(x) = ∫(d3p/(2π)3) (1/√(2ω))Σrξr[aprexp(ip.x) + apr†exp(-ip.x)] and πi = E(x) = ∫(d3p/(2π)3) (1/√(2ω))(-iω)Σrξr[aprexp(ip.x) - apr†exp(-ip.x)]   = ∫(d3p/(2π)3) (-i√(ω/2))Σrξr[aprexp(ip.x) - apr†exp(-ip.x)] Where ξr are the 2 POLARIZATION VECTORS. These polarization vectors obey the rules ξ12 = ξ1.p = ξ2.p = 0 and ξ12 = ξ22 = 1 and thereby satisfy the transversality condition imposed by the gauge. r = 1 to 2. We can summarize the commutators for the electromagnetic field as follows: [apr,aqs†] = (2π)3δrsδ(3)(p - q) [apr,aqs] = [apr†,aqs†] = 0 Which is the same as the usual commutation rules for creation and anihilation operators. However, the field commutation rules are a little more complicated. [Ai(x),Ej(y)] = ihδij(x - y)    = ihij - pipj/p2(3)(x - y)    = ihij - ∂ij/∇2(3)(x - y) since p = -i∂/∂x [Ai(x),Aj(y)] = [Ei(x),Ej(y)] = 0 Where δij(x - y) is the TRANSVERSE DELTA FUNCTION. The TDF acts like a δ function on fields with 0 divergence and projects an arbitrary vector field onto its transverse part. The projection is most easily constructed in Fourier space. FT(x) = ∫(d3p/(2π)3) (δij - pipj/p2) FL(p) exp(ipx) The term in () also satisfies the following completeness relationship. Σξirξjr = (δij - pipj/p2) Projection Operators and Completeness ------------------------------------- Recall from the discussion of projection operators in Quantum Mechanics (Bell's Theorem). I = Σn|i><i| = ΣnPi where Pi is the projection onto i. Also recall from QM the completeness relationship. ψ(x) = Σnanψn(x) where an = ∫dx' ψn*(x')ψ(x') Therefore, ψ(x) = Σn(∫dx' ψn*(x')ψ(x'))ψn(x) = ∫dx' (Σnn*(x')ψn(x))ψ(x') This is an equation of the form: ψ(x) = ∫dx' G(x,x')ψ(x') where G is Green's function. Consider the eigenvalue equation D|ψn> = λnn> where D is a linear differential operator. We can write: DG(x,x') = λnG(x,x') Where G(x,x') is a Green's Function such that: DG(x,x') ≡ λnG(x,x') = δ(x - x') Now λn is just a scalar so we can write λnψ(x) = λn∫dx' δ(x - x') ψ(x') By comparison we see that: δ(x - x') = Σnn*(x')ψn(x)) Returning to the completeness relationship for the polarization vectors we see: Σξirξjr ≡ Σ|ξir><ξjr| = |ξi1><ξj1| + |ξi2><ξj2| This is a projection onto the sub space spanned by the polarisation vectors. This sub space is orthogonal to the momentum. The Lorentz Gauge ----------------- We now take time into consideration and replace 3-vectors with 4-vectors associated with Minkowski spacetime.The Lorentz gauge is manifestly Lorentz invariant and is defined as: ∇.A + ∂φ/∂t = 0 Which can be written in space-time as: ∂μAμ = 0 The equation of motion is found from: ∂μμAν - ∂μνAμ = 0 Which, after imposing the gauge becomes: ∂μμAν = 0 These equations of motion can also be derived from the following Lagrangian: L = (-1/4)FμνFμν - (1/2)(∂μAμ)2 The momentum conjugate to Aμ is: . π0 = ∂L/∂A0 = -∂μAμ . πi = ∂L/∂Ai = -F0i = Ei The field operator equations are once again: Aμ(x) = ∫(d3p/(2π)3) (1/√(2ω))Σrξr[aprexp(ip.x) + apr†exp(-ip.x)] and Eμ(x) = ∫(d3p/(2π)3) (-i√(ω/2))Σrξr[aprexp(ip.x) - apr†exp(-ip.x)] But this time since there is no restriction regarding transverse and longitudinal components, ξr now represents 4 polarization 4-vectors versus 2 3-vector polarization vectors in the Coulomb gauge. These can be characterized as 2 tranverse, 1 longitudinal and 1 time-like. Therefore, r = 0 to 3. We can summarize the commutators in this gauge as follows: [apμ,aqν†] = -(2π)3gμνδ(3)(p - q) [apμ,aqν] = [apμ†,aqν†] = 0 gμν is necessary to achieve Lorentz invariance. The -ve sign is problematic because it implies negative probabilities in the Hilbert space we are working in. The minus is entirely due to the time component (note π0 = -∂μAμ. The challenge is to 'fix' this problem and also deal with the 2 'extra' degrees of freedom that we have. The solution is to impose the gauge condition ∂μAμ = 0. However, if we impose the gauge condition at the Lagrangian level we end up where we started in the Coulomb gauge. The answer is to impose this as a constraint on the physical states associated with the tranvese photon. This is done by splitting Aμ(x) into 2 separate operators. Aμ(x) = Aμ+(x) + Aμ-(x) Therefore, ∂μAμ(x) = ∂μAμ+(x) + ∂μAμ-(x) Where, ∂μAμ+(x) = ∫(d3p/(2π)3) (1/√(2ω))(ip)Σrξr[aprexp(ip.x)] represents the +ve frequency (annhilation). and ∂μAμ-(x) = ∫(d3p/(2π)3) (1/√(2ω))(-ip)Σrξr[apr†exp(-ip.x)] represents the -ve frequency (creation) We then impose the Lorentz condition on the physical states through the eigenvalue equations: ∂μAμ+|Ψ> = 0|Ψ and <Ψ|∂μAμ- = <Ψ|0> This ensures the condition <Ψ|∂μAμ|Ψ> = 0 so that the expectation value of ∂μAμ is 0. This is referred to as the GUPTA-BLEULER condition. Now, we want get rid of the 0 and 3 components since we know these are 'unphysical' states. Let us write: ∂μAμ+|Ψ> = ∫(d3p/(2π)3) (1/√(2ω))(ip)Σrξr[aprexp(ip.x)]|Ψ> = 0 We can write p.ξ1 = p.ξ2 = 0 for the transverse components and p.ξ0 = -p.ξ3 for the time and longitudinal components. Thus, we end up with ∫(d3p/(2π)3) (1/√(2ω))(ip)Σrξr[(ap3 - ap0)exp(ip.x)]|Ψ> = 0 Where r is now 0 and 3. We immediately conclude that: (ap3 - ap0)|Ψ> = 0 which leads to the crucial identity: <Ψ|ap3†ap3|Ψ> = <Ψ|ap0†ap0|Ψ> So only the transverse modes survive and contribute to the dynamics. Therefore, it is consistent to implement the Lorenz gauge condition and indeed decouples the unwanted degrees of freedom from the physical Hilbert space. With this condition applied the commutation relations are restored to their familiar form: [apμ,aqν†] = (2π)3gμνδ(3)(p - q) [apμ,aqν] = [apμ†,aqν†] = 0 [Aμ(x),Eν(y)] = ihgμνδ(3)(x - y) [Aμ(x),Aν(y)] = [Eμ(x),Eν(y)] = 0 The Photon Propagator --------------------- In the Coulomb gauge the propagator is: G(x,y) = <0|TAμ(x)Aν(y)|0> = ∫(d4p/(2π)4) i/(p2 + iε)(δij - pipj/p2)exp(-ip.(x-y)) In the Lorentz gauge the propagator is: G(x,y) = <0|TAμ(x)Aν(y)|0> = ∫(d4p/(2π)4) i/(p2 + iε)exp(-ip.(x-y)) As stated previously, the Fourier transform of the position space propagators can be thought of as propagators in momentum space. Thus, in the Lorentz gauge we get: G(p) = -igμν/(p2 + iε) Massive Spin 1 Vector Fields ---------------------------- In the discussion of Quantum Field theory we have talked about real and complex scalar fields, fermionic fields and the electromagnetic field. The scalar fields represent massive spin 0 particles, the fermionic field represents massive half-integer spin particles and the electromagnetic field represents massless spin 1 particles. The question is "how do we represent massive spin 1 particles?". The W± and Z0 gauge bosons are examples of particles (also called "vector bosons") that fall into this category. It seems logical that somehow we should try and incorporate mass into the spin 1 electromagnetic vector field. However, simply adding a mass term to Lagrangian as follows spoils the invariance. L = (-1/4)FμνFμν - (1/2)m2AμAμ This is discussed here However, there is a way to include an extra field such that the resulting Lagrangian invariant and includes a massive vector field. Consider the Lagrangian L = (-1/4)FμνFμν - (1/2)(∂μφ + mAμ)(∂μφ + mAμ) Where φ is a real scalar field. This is the STUECKELBERG Lagrangian. It is an extension to the Standard Model that provides a mechanism for mass generation in an Abelian gauge theory while preserving gauge invariance. In essence it provides a mass to the physical photon. It is a special case of the Higgs mechanism where the Higgs excitations are so large they can be ignored. The procedure for quantizing the field is the same as we have previously encountered. The math is similar and will not be repeated here. As a fotnote it is worth noting that while the Stueckelberg 'trick' works for the Abelian case, the Higgs mechanism involving spontaneous symmetry breaking remains the only presently known way to give masses to non-Abelian vector fields. Massive Vector Field Propagator -------------------------------- Like the electromagnetic field, the propagator for a massive vector field is dependant on the gauge chosen. In the Feynman gauge this is G(p) = gμν/(p2 - m2 + iε) The Field Hamiltonians ---------------------- The Legendre transformation yields: . H = πφ - L . π = ∂L/∂φ H = ∫d3x H' Real Scalar ----------- L = (1/2)∂μφ∂μφ - (1/2)m2φ2 = (1/2)∂2φ/∂t2 + (1/2)∇2φ - (1/2)m2φ2 . . π = ∂L/∂φ = φ . H' = πφ - L . . = φ2 - (1/2)φ2 - (1/2)(∂φ/∂x)2 - (1/2)m2φ2 . = (1/2)φ2 - (1/2)(∂φ/∂x)2 - (1/2)m2φ2 Substituting the expression for φ and π and noting that ω2 = p2 + m2 φ(x) = ∫(d3p/(2π)3√(2ω)) (1/√(2ω))[apexp(-ipx) + apexp(ipx)] and π(x) = (-i/2)∫(d3p/(2π)3 [apexp(-ipx) - apexp(ipx)] Leads to: Hrenorm' = ∫d3p/(2π)3 ωp apap The order of the creation operators is important to achieve a vacuum expectation value of 0. The creation and annihilation must satisfy <0|ap = 0 and|ap|0> = 0. This requires that when taking the product of quantum fields, or equivalently their creation and annihilation operators, all creation operators are placed to the left of all annihilation operators in the product. This is referred to as NORMAL ORDERING. The normal ordered string of operators is written as: :apapapap: = apapapap Note also that the ∫ -> ∞ unless a cutoff in the momentum is imposed. This is discussed in the section on Feynman diagrams. Complex Scalar Field -------------------- Using a similar procedure to the above we get: Hrenorm' = (1/2)∫d3p/(2π)3 ωp (bpbp + cpcp) Schrodinger ----------- L = ψ(i∂/∂t + ∇2/2m)ψ   . = iψψ + ψ2ψ/2m . π = ∂L/∂ψ = iψ . H' = πψ - L   .  . = iψψ - iψψ + ψ2ψ/2m = ψ(-∇2/2m)ψ This is just the expectation value of the operator. <H> = <φ|H|φ> = ∫d3x ψ[p2/2m]ψ Now, H|ψm> = Emm>   = ωmm> (h = 1) E = ∫dx ψ ω ψ = ∫d3x ω Σmamψm Σnanψn = ∫d3x ω Σmnamanψmψn = ∫d3x ω Σmnamanδmn = ∫d3x ω Σpapap = ∫d3x d3/(2π)3 ωp apap = ∫d3x H' Fermionic --------- _ L = Ψ(iγμμ - m)Ψ _ = Ψ(iγ0∂Ψ/∂t + iγ0γi0∂Ψ/∂x - mΨ) _ .   _ _ = Ψiγ0Ψ + Ψiγi∇Ψ - ΨmΨ . _ π = ∂L/∂Ψ = iγ0Ψ . H' = πΨ - L   _. _  . _    _ = iγ0ΨΨ - Ψiγ0Ψ - ΨiγiiΨ + ΨmΨ _    _ = -ΨiγiiΨ + ΨmΨ _ = Ψ(-iγii + m)Ψ This is just the expectation value of the operator (-iγii + m) Consider: (iγii - m)Ψ = 0 Rewrite as: (iγ00 + iγii - m)u(p)exp(-ipx) = 0 Or, since p0 = i∂/∂t and pi = i∂/∂x, (γ0p0 + γipi - m)u(p)exp(-ipx) = 0 Rearranging, γ0p0u(p)exp(ip.x) = (-γipi + m)u(p)exp(ip.x) and -γ0p0v(p)exp(-ip.x) = (γipi + m)v(p)exp(-ip.x) Now, Ψ = ∫(d3p/(2π)3) (1/√(2ω))Σs[bpsus(p)exp(ip.x) + cps†vs(p)exp(-ip.x)] Which leads to: (-iγii + m)Ψ = ∫(d3p/(2π)3) (1/√(2ω)) Σs[bps(-iγii + m)us(p)exp(ip.x) + cps†(iγii + m)vs(p)exp(-ip.x)] Which leads to: (-iγii + m)Ψ = ∫(d3p/(2π)3) (1/√(2ω)) Σs[bps0p0)us(p)exp(ip.x) - cps†0p0)vs(p)exp(-ip.x)] (-iγii + m)Ψ = ∫(d3p/(2π)3) (1/√(2ω)) (γ0p0) Σs[bpsus(p)exp(ip.x) - cps†vs(p)exp(-ip.x)] Now from Special Relativity p0 is equivalent to the energy E (= ω). Thus, (-iγii + m)Ψ = ∫(d3p/(2π)3) (√(ω/2)) γ0 Σs[bpsus(p)exp(ip.x) - cps†vs(p)exp(-ip.x)] Now, Ψ(x) = ∫(d3p/(2π)3) (1/√(2ω))Σs[bps†us†(p)exp(-ip.x) + cpsvs†(p)exp(ip.x)] Which leads to: Ψ(-iγii + m)Ψ = ∫(d3pd3q/(2π)6) (√(ω/2))(1/√(2ω)) γ0r[bprur(p)exp(ip.x) - cpr†vr(p)exp(-ip.x)] Σs[cqs†us†(q)exp(-iq.x) + dqsvs†(q)exp(iq.x)]) Ψ(-iγii + m)Ψ = ∫(d3pd3q/(2π)6) (1/2) γ0r[bprur(p)exp(ip.x) - cpr†vr(p)exp(-ip.x)] Σs[cqs†us†(q)exp(-iq.x) + dqsvs†(q)exp(iq.x)]) _ Now Ψ = γ0Ψ and (γ0)2 = 1 Therefore, _ Ψ(-iγii + m)Ψ = ∫(d3pd3q/(2π)6) (1/2) (Σr[bprur(p)exp(ip.x) - cpr†vr(p)exp(-ip.x)] Σs[cqs†us†(q)exp(-iq.x) + dqsvs†(q)exp(iq.x)]) After manipulation and making use of the inner product relationship between u and v we end up with: H' = ∫(d3/(2π)3) ω(bps†bps + cps†cps) Electromagnetic --------------- From the electromagnetic tensor we get: L = (-1/4)FμνFμν = (-1/2)(B2 - E2) In the Coulomb gauge the canonical momenta are: π0 = ∂L/∂A0 = 0 .  . . . πi = ∂L/∂Ai = Ai = Ei (same as real field with φ -> A) H' = πAi - L . H' = (Ai)2 - L = E2 + (1/2)(B2 - E2) = (1/2)E2 + (1/2)B2 We can also get this another way: . H' = (1/2)A2 + (1/2)(∂A/∂x)2 + (1/2)m2A2 Now m = 0 so ω2 = p2 . H' = (1/2)A2 + (1/2)(∂A/∂x)2 Now, A = ∫(d3p/(2π)3) (1/√(2ω))Σrξr[aprexp(ip.x) + apr†exp(-ip.x)] Therefore, ∂A/∂x =∫(d3p/(2π)3) (i/√(2ω))(ip) Σrξr[aprexp(ip.x) - apr†exp(-ip.x)] Now p = -i∇ ∴ ip = ∇ and B = ∇ x A. Therefore, B = ip x A and we can also write: B(x) = ∫(d3p/(2π)3) (i/√(2ω))Σr{∇ x ξr[aprexp(ip.x) - apr†exp(-ip.x)]} Therefore, (∂A/∂x)2 ≡ B2 So we again get: H' = (1/2)E2 + (1/2)B2 After substituting in the expressions for E and B we get: H' = ∫(d3p/(2π)3) ω Σrapr†apr Where r = 1 -> 2 Note: In the Lorentz gauge we would get: H' = ∫(d3p/(2π)3) ω Σrapr†apr Where r = 0 -> 3 However, the 0 and 3 components cancel leaving the Coulomb result.