Detailed Course Outline

Table of Contents

  1. Overview of Math
  2. Overview of Classical mechanics
  3. Overview of the Theory of Relativity
  4. Overview of Relativistic Electromagnetic Theory in Covariant Form
  5. Overview of Lie Algebras & Groups
  6. The Heisenberg group – Foundations of quantum theory
  7. The Harmonic Oscillator group 
  8. The Rotation group O3 = SU2 
  9. The Lorentz group – particle theory
  10. The Poincare group – particle theory
  11. XPM group – relativistic position operators
  12. Internal Symmetry – SUn 
  13. TCP & discrete symmetry groups
  14. The General Linear & Affine Group
  15. The DeSitter Group
  16. The Markov Group
  17. Foundations of Lie Algebras and Lie Groups
  18. Course Summary and Conclusions
  19. Applications of the Markov group to Fibonacci numbers
  20. Applications of the Markov group to Logic, Numbers, & Information
  21. Network Theory
  22. Applications of Information theory to Quantum Theory

Personal Notes on Details Within Each Section

  1. Overview of Math
    1. The Integers
    2. Rational Numbers & Fields
    3. Polynomials
    4. Real Numbers
    5. Complex Numbers
    6. Groups
    7. Vectors and Vector Spaces
    8. Algebra of Matrices
    9. Linear Groups
    10. Determinants and Canonical Forms
    11. Boolean Algebras & Lattices
    12. Transfinite Arithmetic
    13. Rings & Ideals
    14. Algebraic Number Fields
    15. Galois Theory
  2. Overview of Classical mechanics
    1. Introduction to classical mechanics
      1. Newtons Equations
        1. The philosophy of the representation of a small component of matter at a given point in space time, given forces, that one can then understand the whole.
        2. Newtons second law with the concept of force keep us from having to compute an infinite number of derivatives
        3. Solution to harmonic  oscillator:  Hook’s law and friction
      2. The Four fundamental forces – force arises from the exchange of particles
        1. Newtons law of gravitation
          1. Solution to inverse square potential – Kepler orbits
        2. Lorentz Force – Electromagnetism
          1.  Action at a distance
          2. Coulomb force
      3. Lagrangian formulation - & how to deal with constraints with generalized coordinates.
      4. Principle of Least Action – a beautiful mathematical expression of classical mechanics
      5. Hamiltonian formulation – a expression of classical mechanics that lays a foundation for quantum theory.
      6. Poisson Bracket formulation – an extension of the Hamiltonian theory in terms of bracket expressions that predate Lie algebra theory.
    2. Electromagnetism
      1. The second entity in addition to the matter in pre 1900 physics 
      2. Lorentz force,
      3. Maxwell’s equations – integral form –applications of each
      4. Maxwell’s equations – differential form
  3. Overview of the Theory of Relativity
    1. Derivation of plane waves from Maxwell equations & velocity of light = a constant
    2. The violation of Newtonian mechanics via addition of velocities
    3. Michaelson Morley experiment
    4. Assumptions of relativity – the relativity of  inertial frames, constant c, linear transformation
    5. Derivation of Lorentz transformation
    6. Formulation of 4 vectors, tensors, etc
    7. Lorentz contraction & time dilation
    8. Proper time, invariant mass & the connection to energy and momentum
    9. Classification of particles by sign of energy, mass, velocity.
    10. General relativity
  4. Overview of Relativistic Electromagnetic Theory in Covariant Form
    1. Electromagnetic field tensor, four potential & four current
    2. Covariant form of Maxwell’s equations
    3. Relativistic force equation
    4. Magnetic monopoles
    5. Derivation of standard form of Maxwell’s equations
  5. Overview of Lie Algebras & Groups
    1. Foundations of Lie Groups and Algebras
      1. Definition of Lie group and associated algebra
        1. Structure constants
        2. Jacobi Identity
      2. Foundational definitions, properties, & classifications
    2. Representation Theory – Dirac notation
      1. Cartan subalgebra – maximum Abelian subalgebra
      2. Raising & lowering operators
      3. Casmir operators
      4. Weight and root diagrams
    3. Overview of semisimple Lie algebras
    4. Lie Algebras which are not semisimple
  6. The Heisenberg group – Foundations of quantum theory
    1. The Heisenberg Lie algebra
    2. The foundations of quantum mechanics
    3. The uncertainty principle – information
    4. The Fourier transform as a directional cosine
    5. The equations of Schrödinger, Klein Gordon & Dirac
  7. The Harmonic Oscillator group
    1. Creation & Destruction Operators
  8. The Rotation group O3 = SU2
    1. Angular momentum
    2. Spherical harmonics
  9. The Lorentz group – particle theory
  10. The Poincare group – particle theory
  11. XPM group – relativistic position operators
  12. Internal Symmetry – SUn
  13. TCP & discrete symmetry groups
  14. The General Linear & Affine Group
  15. The DeSitter Group
  16. The Markov Group
    1. Discussion of Markov processes
    2. Derivation of the Markov Lie group
    3. Properties of the Markov group
  17. Foundations of Lie Algebras and Lie Groups
  18. Course Summary and Conclusions
  19. Applications  of the Markov group to Fibonacci numbers
    1. Set up of the problem
    2. Differential equation for GL(n,R)
    3. Fibonacci functions
    4. Similar classes of functions
    5. Quantization to get Fibonacci sequence
    6. Concept of information
  20. Applications of the Markov group to Logic, Numbers, & Information
    1. Entropy, information, & uncertainty
      1. Numerical error & error propagation
      2. Uncertain logic elements (probabilities rather than 1/0
      3. Uncertainty principle
      4. Meaning of information below the bit level
    2. Definition of a bittor in logic
    3. Multiplication – formation of new product
    4. Linear combination defined for closure with bittor coefficients
    5. Definition of information on bittors
    6. Definition of a bittor number
      1. Uncorrelated outer product
      2. Error is flat square distribution
      3. Representation as a binary number (infinite digression of bittors)
      4. Error in any position allowed
    7. Addition of bittor numbers
    8. Multiplication of bittor numbers
    9. Java program to add and multiply
    10. Laws of information
  21. Network Theory
    1. Introduction to types of networks
    2. The connectivity matrix for undirected graphs
    3. Connectivity as a Markov Lie algebra
      1. Eigenvalues & eigenvectors
      2. Interpretation as a dynamical system – information conserved
    4. Generalizations
      1. Non-uniform bandwidths but symmetric
      2. Asymmetric graphs and directed graphs
      3. Growth and decay (sources & sinks) at nodes
    5. Topological classifications
      1. Self connectivity vectors & matrix
      2. Mutual connectivity matrices – eigenvectors & eigenvalues
      3. Proposed classification for topologies
      4. Research area – prove this is isomorphic to topologies
      5. Research area – use Mathematica to enumerate and count lower orders
    6. Applications
      1. Research area – study applications to electrical grids & internet
  22. Applications of Information theory to Quantum Theory
    1. Generalization of the definition of Information to integral of P^2
    2. Proposed conservation law of Information for Heisenberg algebra
    3. Proposed conservation of Information for Rotation group & spin
    4. Research problem – study conservation law for Lie groups
    5. Research problem – what symmetry transformation does I generate?
    6. Research Problem – can this conservation be used in applications and to help to solve problems.