Lie's Gauge / Gauge Theory
Overview
Framework for fundamental forces via local symmetry. The Lagrangian is invariant under gauge transformations; this constrains the form of interactions. Unifies electromagnetism (U(1)), weak (SU(2)), strong (SU(3)).
Era: 1920s–
Proposed at: Princeton, USA
Formal structure
Domain: Gauge group G, principal bundle P → M, connection A
L = -¼ Tr(F_μν F^μν) + ψ̄(iD̸ - m)ψ (Yang–Mills + matter)
Constraints: F = dA + A ∧ A; D_μ ψ = ∂_μ ψ + A_μ ψ (covariant derivative)
Objective: δS = 0 ⇒ Yang–Mills equations, Dirac equation