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String Theory

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Extra Dimensions

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Introduction to String Theory

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Strings in Curved Spacetime

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Introduction to Superconductors

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Superconductivity (Lectures 1 - 10)

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Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Introduction to Supersymmetry

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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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Last modified: February 7, 2019


Index Notation (Tensors and Matrices)
-------------------------------------

Problems With Matrices
----------------------

1.  They can be big and difficult to manipulate.

2.  They do not commute so the order of multiplication 
    is important.

3.  Some things cannot be represented by matrices or 
    matrix algebra.

Consider the 2D rotation matrix:

 -  -      -   -     -          - -   -   
| x¹ | -> | x^1' | = |  cosθ sinθ || x¹ |
| x² |    | x^2' |   | -sinθ cosθ || x² |
 -  -      -   -     -          - -   -  

                     -                -
                  = |  cosθx¹ + sinθx² |
                    | -sinθx¹ + cosθx² |
                     -                -

Rewrite the rotation matrix as:

 -         -
| Λ^1'₁  Λ^1'₂ |
| Λ^2'₁  Λ^2'₂ |
 -         -

xⁱ -> x^i' = ΣΛ^i'_jx^j  
          ^j

       = Λ^i'_jx^j using Einstein's notation.

       = Λ^i'₁x¹ + Λ^i'₂x²

So we get:

x^1'  = cosθx¹ + sinθx²

and,

x^2'  = -sinθx¹ + cosθx² 

In index notation the order doesn't matter.  Thus,

x^jΛ^i'_j ≡ Λ^i'_jx^j

However, this not true for matrices.

 -   -     -  -  -          -
| x^1' | = | x¹ ||  cosθ sinθ |
| x^2' |   | x² || -sinθ cosθ |
 -   -     -  -  -          -

which is an illegal operation.

Converting Index to Matrix Notation
-----------------------------------

To get the matrix form we can rearrange the terms
but we need to ensure that like indeces are adjacent 
to each other before contraction.  This is because
the product of 2 matrices is only defined if the
number of columns in the first matrix is equal to
the number of rows in the second matrix, i.e.

         (m x n)(n x m) = m x k

Therefore,  A^m_nBⁿ_m will correspond to a legal matrix 
operation while Bⁿ_mA^m_n will not.

Example:

M^μν = A^μ_σA^ν_ρM^σρ:

    = A^ν_ρA^μ_σM^σρ:

    = A^ν_ρB^μρ where B^μρ = A^μ_σM^σρ

    = A^ν_ρ(B^ρμ)^T 

    = (AB^T)^νμ

    = ((AB^T)^νμ)^T = (BA^T)^μν

But B = AM.  Therefore,

M' = AMA^T