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Astronomy

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Celestial Coordinates
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Chemistry

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Avogadro's Number
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Balancing Chemical Equations
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Classical Physics

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Archimedes 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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Rotational Dynamics
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Specific Heat, Latent Heat and Calorimetry
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The Gas Laws
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The Zeeman Effect
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Wien's Displacement Law
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Climate Change

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Cosmology

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Penrose Diagrams
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Inflation Theory
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Finance and Accounting

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Repurchase versus Dividend Analysis

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Einstein's Field Equations
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Tensors
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World Lines Refresher

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Macroeconomics

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Mathematics

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Amplitude, Period and Phase
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Building Groups From Other Groups
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Complex Numbers
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Integration By Parts
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The Integers Modulo n Under + and x
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Tic-Tac-Toe Factoring
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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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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors
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Probability and Statistics

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Factor Analysis
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Programming and Computer Science

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Hashing
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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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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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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 Qubit
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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

test

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test

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

Bosonic String Theory --------------------- Open Strings ------------ From classical physics: E = p2/2m + B where B is the rest energy (internal or binding energy). From SR for p << c E = √(p2c2 + m2c4) ... 1. = √(m2c4(1 + p2c2/m2c4)) = mc2 + p2c4m/2m2c4 from √(1 + ε) = 1 + ε/2 (binomial) = mc2 + p2/2m Light Cone Frame ---------------- To simplify matters, we want to be able to treat strings non-relativistically. To accomplish this we introduce the LIGHT CONE FRAME. The LCF is a 'trick' that wallows a relativistic system to be treated non-relativistically. Consider a large boost along the z axis such that pz -> c. Set c = 1. From 1. E = √(pz2 + px2 + py2 + m2) = pz√(1 + (px2 + py2 + m2)/pz2)) = pz(1 + (px2 + py2 + m2)/2pz2) = pz + p'2/2pz + m2/2pz pz is a conserved quantity (constant). It is the energy of the center of mass motion of the entire boosted string. It has no relevance to the internal energy of the string in the x, y coordinates therefore we can disregard it. Therefore, the energy of the string is: E = p'2/2pz + m2/2pz ... 2. For high pz, the energy is very small. This is consistent with E = ih∂/∂t which says that the internal motions of the system of the system of particles are changing slowly due to time dilation. The first term depends on x and y and is non-relativistic because the motion in the xy plane is not in the direction of the boost. It is equivalent to p2/2m with pz taking on the role of m. The second term does not depend on x and y and is equivalent to the binding or internal energy of the system. Essentially, using the LCF slows the oscillations of the string in the x, y plane. The LCF is also called the INFINITE MOMENTUM FRAME. Energy Stored in a String ------------------------- FROM THIS POINT ON WE WILL ONLY CONSIDER THE X COORDS. THERE IS A PARALLEL SET OF CALCULATIONS FOR THE Y COORDS. In the x dimension: . E = Σmxi2/2 + (k/2)(Δxi)2 Where Δxi = xi - xi+1 We make the following substitutions: N = π/Δσ k = N/π2 = 1/πΔσ m = μΔσ = N/π The energy associated with a point mass is: E = ΣμΔσxi2/2 + (Δσ/π)(Δxi)2 = Σ(N/π)Δσxi2/2 + (Δσ/2π)(Δxi)2 = (1/2π)ΔσΣN[xi2 + (Δxi)2] In the continuous limit, the energy associated with the entire string is: π E = (1/2π)∫(∂x/∂τ)2 + (∂x/∂σ)20 The Lagrangian is therefore: π L = (1/2π)∫(∂x/∂τ)2 - (∂x/∂σ)20 Now, in the light cone frame (see 2.) we equated the internal energy with m2. Therefore, π E = (1/2π)∫(∂x/∂τ)2 + (∂x/∂σ)2 dσ = m2 0 Boundary Conditions ------------------- Now we have to consider what is happening at the string endpoints. There are two possible boundary conditions. Either the ends are fixed so they cannot move. Or the ends are not fixed and are moving around in the spacetime background. Fixed ends are called Dirichlet boundary conditions. Free ends are called Neumann boundary conditions. We will see later that fixed ends are attached to BRANES. Dirichlet: x(0) = 0, x(π) = 0 Dirichlet conditions result in the reflection of waves at the endpoints with inversion. Neumann: ∂x(0)/∂σ = 0, ∂x(π)/∂σ = 0 Neumann conditions result in the reflection of waves at the endpoints without inversion. Now look at the boundary condition in more detail. | N .... N-1 | N is the endpoint. Force on N is F is: F = (k/2)Δx = (N/2π2)Δx = (N/2π2)(∂x/∂σ)Δσ = (N/2π2)(∂x/∂σ)π/N = (1/2π)(∂x/∂σ) From Newton's laws μ∂2x/∂τ2 = (1/N)∂2x/∂τ2. Therefore, (1/N)∂2x/∂τ2 = (1/2π)(∂x/∂σ) Or, ∂2x/∂τ2 = (N/2π)(∂x/∂σ) This implies that the acceleration will head towards infinity as N gets larger and larger. Therefore, the correct boundary conditions for the open string are the Neumann boundary conditions. Fourier Expansions ------------------ Strings can oscillate in any of a number of discrete modes. We want to know whether the different modes reproduce the characteristics of the familiar particles of the standard model. Fourier expansion for Neumann: x(σ,τ) = Σxn(τ)cos(nσ) n=0 The xn's are the Fourier coefficients. They depend on the proper time, τ and, as we shall see, represent the harmonic oscillator modes. Note: ∂sin(nσ)/∂σ = cos(nσ) = 0 at endpoints. π ∫cos(nσ)cos(mσ)dσ = 0 if m ≠ n 0 = π/2 if m = n = δmnπ/2 = π if n = m = 0 Consider the classical string energy again. π E = (1/2π)∫(∂x/∂τ)2 + (∂x/∂σ)20 . . (∂x/∂τ)2 = ΣnΣmxncos(nσ)xmcos(mσ) [Σ is 0 to ∞]   . . = (1/2π)ΣnΣmxnxm∫cos(nσ)cos(mσ) dσ [∫ is 0 to π] . . = x02/2 + (1/2π)Σnxn2(π/2) . . = x02/2 + (1/4)Σnxn2 (∂x/∂σ)2 = ΣnΣmnxnsin(nσ)mxmsin(mσ)   = (1/2π)ΣnΣmnmxnxm∫sin(nσ)sin(mσ) dσ   = (1/2π)Σnn2xn2∫sin(nσ)sin(mσ) dσ   = (1/2π)Σnn2xn2(π/2)   = (1/4)Σnn2xn2 Therefore the total energy is: . . E = x02/2 + (1/4)Σnxn2 + (1/4)Σnn2xn2 The first term is the KE of the centre of mass of the string. The second term is the KE of the points oscillating in the transverse direction. The third term is the PE due to tension between points on the string. We can equate the string oscillation mode,n, with ω in the following manner: E = hω = x02/2 + (1/4)Σnxn2 + (1/4)Σnn2xn2 With h = 1 and neglecting the first term we see: ω = (1/4)Σnxn2 + (1/4)Σnn2xn2 String Quantization ------------------- The second and third terms is the equation for the harmonic oscillator with frequency n (≡ ωn). These represent the internal energy of string identified with m2. We can write the Lagrangian as: . L = (1/4)Σnxn2 - (1/4)Σnn2xn2 The Hamiltonian is found from: H = p - L Where, . . p = ∂L/∂x = x/2 Therefore, H = p2 + n2xn2/4 = (nxn/2 + ipn)(nxn/2 - ipn) The commutator is: [(nxn/2 + ipn),(nxn/2 - ipn)] = n Therefore, [(1/√n)(nxn/2 + ipn),(1/√n)(nxn/2 - ipn)] = 1 These correspond to the creation and annihilation operators that obey the commutator: [a-,a+] = 1 Now, (nxn/2 + ipn) + (nxn/2 - ipn) = (√n)xn Therefore, xn = (an- + an+)/√n We can write: x(σ,τ) = Σnxn(τ)cos(nσ) = Σn((an- + an+)/√n)cos(nσ) If string theory is to be a theory of quantum gravity, then the average size of a string should be somewhere in the vicinity of the length scale of quantum gravity, called the Planck length, which is about 10-33 cm. This is referred to as the characteristic length scale. Particle Spin Angular Momentum ------------------------------- # of spin states for massive particles = 2J + 1. Spin Massive Massless ---- ------- -------- 0 1 1 1 3 2 2 5 2 Consider a spin 1 particle. For the massive case it is always possible to bring the particle to rest (catch up with it), rotate it and boost it in a direction perpendicular to the direction of motion. The spin can therefore be parallel, anti-parallel or perpendicular to the original direction of motion (i.e. a zero component along the direction of motion). For the massless case it is not possible to bring the particle to rest and do the same thing. Thus, the spin of a massless spin 1 particle is constrained to be parallel or anti-parallel to the direction of motion. The particle either has right-handed spin or left-handed spin. For the photon this manifests itself as the CIRCULAR POLARIZATION of light. The photon can also be LINEARLY POLARIZED. In both cases the polarization is always perpendicular to the direction of motion (transverse). For linear polarization, the polarization state can be written as: |PL> = |Px> + |Py> This would be a photon linearly polarized at some angle to the x and y axes for a photon propagating in the z direction. For circular polarization, the polarization state can be written as: |PRH> = |Px> + i|Py> |PLH> = |Px> + i|Py> Note that these are similar to the ladder operators associated with the spin angular momentum. Discrete Energy Spectrum of an Open String ------------------------------------------ From before we have said that energy, E, corresponds to m2. By analogy with the ground state energy of the harmonic oscillator ((1/2)hω), we can assign a ground state mass, m0, to the unexcited string. a and b refer to the x and y oscillators respectively. |0> is the ground state not the vacuum. a-n|0> = 0 b-n|0> = 0 a+1|0> = m02 + 1 unit of energy b+1|0> = m02 + 1 unit of energy (a+1 + b+1)|0> = m02 + 1 unit of energy a and b are identified with coordinates x and y in the following way: xn = (an- + an+)/√n yn = (bn- + bn+)/√n Therefore they have vector-like properties. In fact, they have similar properties to the polarization states of the photon. For linear polarization we can write, for example. a+1|0> + b+1|0> and for circular polarization. a+1|0> ± ib+1|0> This corresponds to spin angular momenta around the z axis of +1 and -1. The fact that there are only spin states of +1 and -1 and no 0 states implies that these objects could be PHOTONS. If this is true, however, then they must be massless implying that m02 = -1. Unfortunately, this doesn't work because the ground state energy would then be -1. A particle with -m02 is called a TACHYON - a hypothetical particle that always moves faster than light. This can be seen as follows: E = hω = √(p2c2 + m2c4) Now, the group velocity, v, is defined as ∂ω/∂k (≡ ∂E/∂p) With c = 1 we get: ∂E/∂p = p/√(p2 + m2) For positive m2, v is < c. For negative m2, v is > c. To see how the dilemna involving Tachyons can be solved we need to introduce the idea of increasing the number of spacial dimensions in the theory. This is discussed in a separate note.