Lagrangian/Hamiltonian field theory over spacetime: action principle for fields, Euler–Lagrange field equations, stress–energy tensor, Noether currents, continuum symmetries. Classical substrate beneath QFT and GR.
Classical Field Theory
Action principle (field theory)
Dynamics of a field φ(x) stationarise S[φ] = ∫ d⁴x ℒ(φ, ∂_μ φ). Equivalent to the Euler–Lagrange field equations via δS = 0.
Euler–Lagrange field equations
∂_μ (∂ℒ/∂(∂_μ φ)) − ∂ℒ/∂φ = 0: EOM derived from stationarising the action over field configurations.
Classical stress–energy tensor
Canonical T^{μν} = (∂ℒ/∂(∂_μ φ)) ∂^ν φ − η^{μν} ℒ; conserved by translation invariance (Noether). Belinfante–Rosenfeld improvement makes…
Noether currents (field theory)
Each continuous symmetry φ → φ + ε δφ of the action yields a conserved current j^μ with ∂_μ j^μ = 0 and conserved charge Q = ∫ j⁰ d³x.
Classical scalar field
Real (or complex) field φ(x) with Lagrangian ℒ = ½ (∂_μ φ)(∂^μ φ) − V(φ). Protypical classical field; Lorentz-invariant action under η.
Continuum mechanics
Classical field theory of continuous matter: Cauchy stress tensor, strain tensor, conservation of mass/momentum/energy; specialises to…
Elastic wave equation
Linear PDE governing small-amplitude displacement u in an isotropic elastic solid: ρ ∂²_t u = (λ + μ) ∇(∇·u) + μ Δu. Predicts longitudinal…
Electromagnetic Lagrangian
ℒ_EM = −(1/4μ₀) F^{μν} F_{μν} − A^μ J_μ (in natural units). Euler–Lagrange equations reproduce sourced Maxwell's equations.
Complex scalar field
Field φ = (φ_1 + i φ_2)/√2 with ℒ = (∂_μ φ*)(∂^μ φ) − m²|φ|². U(1) global symmetry yields a conserved Noether charge; spontaneous breaking…
Massive vector field (Proca)
Classical field theory of a massive spin-1 field A^μ with ℒ = −(1/4) F^{μν} F_{μν} + ½ m² A^μ A_μ. Breaks U(1) gauge invariance explicitly.
Canonical field quantisation
Prescription [φ(x,t), π(y,t)] = iℏ δ³(x−y), [π(x,t), π(y,t)] = [φ(x,t), φ(y,t)] = 0. Upgrades classical fields to operator-valued…
Ghost fields (BRST formalism)
Anticommuting scalar Grassmann fields (Faddeev–Popov ghosts, c, c̄) introduced to make gauge-fixed path integrals unitary. BRST charge…
Noether theorems (1st + 2nd)
Global symmetries → conserved currents; local (gauge) symmetries → constraints among EoM. Foundation of conservation laws in field theory.
EM dual tensor & Bianchi
F̃^µν = (1/2)ε^µναβ F_αβ; source-free Maxwell = ∂_µ F̃^µν = 0 (Bianchi identity). Electric-magnetic duality for source-free fields.
Stress-energy tensor from translations
T^μν = ∂ℒ/∂(∂_μφ) ∂^ν φ - η^μν ℒ; symmetric Belinfante form; source of gravity; conservation ∂_μ T^μν = 0 from Poincaré invariance.
Klein–Gordon scalar field
(□+m²)φ = 0 from ℒ = ½(∂φ)² - ½m²φ²; plane-wave solutions with E² = p²+m²; basis for scalar QFT.
Dirac spinor field (classical)
(iγ^μ∂_μ - m)ψ = 0; bilinear currents ψ̄γ^μψ; chiral decomposition ψ = ψ_L + ψ_R; anticommuting in classical Grassmann treatment.
Sine-Gordon model
ℒ = ½(∂φ)² + (m²/β²)(cos βφ - 1); kink solitons; equivalent to massive Thirring model via bosonization; integrable in 1+1D.
φ⁴ scalar theory
ℒ = ½(∂φ)² - ½m²φ² - (λ/4!)φ⁴; spontaneous symmetry breaking for m²<0; kink soliton between ±v vacua in 1+1D.
Nonlinear σ-model
Fields taking values on manifold G/H with kinetic term Σg_ij(φ)∂φ^i∂φ^j; describes Goldstone bosons; chiral perturbation theory for pions.
Topological solitons via π_n(G/H)
Homotopy classes of asymptotic configurations classify solitons: domain walls π_0, vortices π_1, monopoles π_2, instantons π_3.
Derrick's theorem
Scaling argument: no stable static localized scalar solitons in d≥2 spatial dimensions; evaded by gauge fields, topology, time-dependence.
Hamiltonian density and conjugate momenta
π = ∂ℒ/∂(∂_t φ); ℋ = πφ̇ - ℒ; Poisson brackets {φ(x),π(y)} = δ(x-y); canonical quantization basis.
Second quantization (classical view)
Expand field in modes φ = Σ(a_k e^(-ikx) + a_k* e^(ikx))/√(2ω_k); a_k → operators upon quantization; occupation-number basis.
Gauge covariant derivative
D_μ = ∂_μ - igA_μ^a T^a acts covariantly: (D_μψ) → U (D_μψ) under local gauge transformation; minimal coupling prescription.
Canonical vs symmetric stress-energy
Canonical T^μν from Noether may be asymmetric for spin ≠ 0; Belinfante–Rosenfeld improvement makes symmetric; both couple to gravity…
Heavy-particle effective Lagrangian
Integrate out heavy field H: L_eff = L_light + (coupling)² O(light)/M_H² + ...; Fermi 4-fermion theory as EFT of W exchange.
Kaluza–Klein reduction
5D gravity on S¹ gives 4D gravity + Maxwell + scalar; mode expansion φ(x,y) = Σφ_n(x)e^(iny/R) with masses n/R; origin of extra-dimensional…