Lorentz-covariant classical wave equations (Klein–Gordon, Dirac, Proca, Maxwell in tensor form), spinor/tensor/vector field representations of the Poincaré group, causality and microcausality.
Relativistic Field Theory
Klein–Gordon equation
Lorentz-covariant wave equation for a relativistic scalar field: (□ + m²c²/ℏ²) φ = 0. First attempt at a relativistic QM; positive and…
Dirac equation
First-order Lorentz-covariant wave equation for a spin-½ fermion: (iγ^μ ∂_μ − mc/ℏ) ψ = 0. Predicts antimatter and combines QM with…
Gamma matrices (Dirac algebra)
Four 4×4 complex matrices γ^μ satisfying the Clifford algebra {γ^μ, γ^ν} = 2 η^{μν} I; realise the spinor representation of the Lorentz…
Proca equation
Relativistic wave equation for a massive spin-1 field: ∂_μ F^{μν} + m² A^ν = 0. Describes massive photons (hypothetical) and W/Z bosons…
Poincaré group
Semidirect product of Lorentz group SO(1,3) with spacetime translations. Classifies particles by mass and spin (Wigner's classification)…
Wigner's classification of particles
Irreducible unitary representations of the Poincaré group are labelled by (m² ≥ 0, spin/helicity). Provides the Lorentz-covariant meaning…
Spinor field
Section of the spin bundle; transforms under the (½,0) ⊕ (0,½) representation of Spin(1,3). Dirac spinor = chiral-pair (ψ_L, ψ_R).
Microcausality
Commutator of local fields at space-like-separated points vanishes: [φ(x), φ(y)] = 0 for (x−y)² < 0. Codifies relativistic causality at…
Chirality and parity
Projectors P_L = ½(1 − γ⁵), P_R = ½(1 + γ⁵) split a Dirac spinor into left/right-handed components. Parity maps L ↔ R; weak interaction…
CPT theorem
Any local Lorentz-invariant QFT with a Hermitian Hamiltonian is invariant under the combined operation of charge-conjugation C, parity P,…
Majorana fermions
Fermions equal to their own antiparticle (ψ=ψc). Neutrinos might be Majorana (0νββ decay probes). Majorana zero modes in topological…
Supersymmetry
Symmetry exchanging bosons ↔ fermions with fermionic generator Qα. MSSM doubles SM particles; solves hierarchy problem if sparticles at…
CFT & AdS/CFT duality
Conformal field theories in d dim dual to quantum gravity in (d+1)-AdS. Maldacena 1997. Strong/weak coupling; black-hole entropy,…
Wigner little-group classification
Massive: SO(3) rotations → spin s with 2s+1 polarizations; massless: ISO(2) → helicity ±h; continuous spin representations excluded…
Weyl, Majorana, Dirac representations
Two-component Weyl ψ_L, ψ_R; Majorana = self-conjugate ψ = ψ^c; Dirac = charged 4-component; choices matter for neutrino nature.
Rarita–Schwinger equation (spin-3/2)
(iγ^μ∂_μ - m)ψ^ν = 0 with γ_νψ^ν = 0, ∂_νψ^ν = 0; describes gravitino in SUGRA; has Velo–Zwanziger pathologies when coupled to EM.
Bargmann–Wigner equations
Higher-spin wave equations via multiple Dirac-type factors on symmetric tensor-spinor; manifestly Lorentz covariant for arbitrary s.
Poincaré algebra
[P_μ, P_ν] = 0, [M_μν, P_ρ] = i(η_νρ P_μ - η_μρ P_ν), [M,M]~M; Casimirs P² = m², W² = -m²s(s+1).
Weinberg low-energy theorems
Soft photons factor: A(k→0) = A_0 Σ e_i ε·p_i/p_i·k; soft gravitons universally couple to energy-momentum regardless of spin.
Weinberg–Witten theorem
No massless particles of spin > 1/2 can carry covariantly conserved current; restricts emergent gauge bosons & gravitons.
2D conformal anomaly (Polyakov)
Trace anomaly T^μ_μ = -(c/12) R in 2D CFTs; basis of Liouville theory and string world-sheet consistency conditions.
Virasoro algebra
[L_m, L_n] = (m-n)L_{m+n} + (c/12)(m³-m)δ_{m+n,0}; conformal symmetry of 2D CFT with central charge c.
SUSY algebra (N-extended)
{Q_α, Q̄_β̇} = 2σ^μ_αβ̇ P_μ + central charges; fermion-boson symmetry; protects scalar masses, forbids EW hierarchy tuning.
Superspace and superfields
Extend spacetime by Grassmann coords θ, θ̄; superfield Φ(x,θ,θ̄) packages SUSY multiplet; chiral D̄Φ=0.
Supergravity (N=1 4D)
Gauge local SUSY → graviton g_μν + gravitino ψ_μ^α; Q̄Q = P locally → GR; cosmological constant tied to SUSY breaking.
Vasiliev higher-spin theory
Interacting massless fields of all spins in AdS; overcomes no-go results via non-locality; dual to vector models via HS/vector-model…
Twistor formalism
Points in Minkowski ↔ lines in twistor space ℂℙ³; massless fields as cohomology; MHV amplitudes localize on lines (Witten 2003).
BCFW recursion & on-shell methods
Tree amplitudes from on-shell-shifted factorizations ∑(Â_L · 1/P² · Â_R); avoids Feynman diagram explosion for multi-particle.