Mathematics

Reference notes on the mathematical structures used throughout the physics sections. These pages are companion material — physics docs link in for definitions and structure theorems, and the math pages stay self-contained and general.

Contents

1. Group Theory: Definitions and Axioms — group axioms, Lie groups and Lie algebras, representations, and the universal cover; the foundation behind the Lorentz, Poincaré, and internal symmetry groups ($U(1)$, $SU(2)$, $SU(3)$) used in QFT/preliminaries.md.
2. Clifford Algebras — the abstract definition and structure theory of $\mathrm{Cl}(V,Q)$, the gamma-matrix and Dirac-spinor algebra underlying relativistic-fermion fields; supplies the algebra used by QED/historical.md.
3. Topology, Manifolds, and Smooth Structure — point-set topology, smooth manifolds, tangent spaces, and smooth maps; the prerequisites that physics texts typically take for granted.

Reading order

The dependency arrows run

$$\text{topology-manifolds}⟶\text{group-theory}⟶\text{clifford-algebra},$$

so newcomers should read in that order. Readers who only need a specific definition can jump in anywhere — each page is cross-linked to its prerequisites.