Physics

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Algebra, Geometry, and Quantisation

An ongoing collection of notes exploring algebraic and geometric aspects of quantisation.

• Classical Dynamics and Geometry
  • The Action Principle
  • Canonical Structures
  • Motivating Symplectic Geometry
  • More Geometry
    • Manifolds, Vectors, and 1-Forms
    • Higher-Degree Forms and Exterior Derivatives
    • Symplectic Geometry, Revisited
    • Contractions, Lie Derivatives, and Integration
  • Hamiltonian Symmetries and Moment Maps
    • Generators and Conserved Charges
    • Group Actions and Infinitesimal Symmetries
    • Moment Maps
    • Dual Algebras
    • Noether’s Theorem, Revisited
• Groups, Algebras, and Representations
• The Heisenberg Algebra
• Schur-Weyl Duality
  • Partitions, Young Diagrams, and Tableaux
  • Representations of the Permutation Group
  • Tensor Spaces and Commuting Actions
• The Oscillator Formalism
  • Two Dimensions
  • Higher Dimensions
  • Completeness?
  • Three-Dimensional Oscillators, Revisited
  • The Theta Correspondence
  • Dual Reductive Pairs and Applications
  • Supergroups
• The Hall Effect
  • The Landau Problem
  • Integer Hall Effect
  • Fractional Hall Effect
• Bosonisation
• Geometric Quantisation
  • Topological Effects
  • Coherent States
  • Kähler Manifolds and Quotient Spaces
• Kirillov’s Orbit Method
• Localisation Theorems
• Fermions, Clifford Algebras, and Spinors

Other Notes

A collection of shorter or independent notes that do not yet belong to a larger project.

• The Noether Current for Electric-Magnetic Duality