Canonical and path-integral quantisation of fields; particle content, propagators, Feynman diagrams, renormalisation, spontaneous symmetry breaking, effective-field-theory methodology. Operational substrate of the Standard Model.
Quantum Field Theory
Second quantisation
Reformulation of many-body QM in terms of creation/annihilation operators a†(k), a(k) acting on Fock space. Fermions/bosons differentiated…
Fock space
Direct sum of n-particle Hilbert spaces ⊕_n 𝓗^{⊗n}_S (Sym for bosons, Antisym for fermions). State space of a non-interacting QFT.
Feynman propagator
Time-ordered two-point function Δ_F(x−y) = ⟨0|T{φ(x) φ(y)}|0⟩, encoding free-field propagation with Feynman's iε prescription.
Feynman diagram
Graphical shorthand for terms in the perturbative S-matrix expansion; vertices = interactions, internal lines = propagators, external legs…
Renormalisation
Systematic redefinition of bare parameters (mass, coupling, wavefunction) to absorb UV divergences and yield finite predictions at a chosen…
Spontaneous symmetry breaking
Symmetry of the Lagrangian not shared by the vacuum: ⟨0|φ|0⟩ ≠ 0. Produces Nambu–Goldstone bosons (continuous global) or gauge-boson…
Goldstone theorem
Spontaneous breaking of a continuous global symmetry produces one massless spin-0 particle per broken generator. Bypassed by the Higgs…
Wick's theorem
Decomposition of time-ordered products of free fields into normal-ordered products plus sums over all possible pairwise contractions…
LSZ reduction formula
Expresses S-matrix elements in terms of on-shell-amputated n-point correlation functions: S = ∏ (−i)(∂² + m²)/√Z applied to ⟨0|T{φ…φ}|0⟩.
Quantum electrodynamics (QED)
Renormalisable gauge theory of electrons and photons. Predictions (g−2 of electron) match experiment to 12 decimal places — most…
Quantum chromodynamics (QCD)
Non-Abelian SU(3) gauge theory of quarks and gluons. Asymptotic freedom at high energies; confinement at low energies (colourless hadrons).
Asymptotic freedom
In non-Abelian gauge theories with few enough flavours (QCD: N_f < 16.5), the running coupling g(μ) → 0 as μ → ∞. Gross–Wilczek–Politzer…
Effective field theory
Systematic low-energy Lagrangian including all operators consistent with symmetries, suppressed by powers of Λ (cutoff). Chiral…
Wightman axioms
Axiomatic formulation of QFT: fields as operator-valued tempered distributions satisfying Poincaré covariance, spectral condition, local…
Anomalies in QFT
Classical symmetries broken by quantization. ABJ axial anomaly (π⁰→γγ), gauge anomalies cancel in SM, 't Hooft anomaly matching,…
Instantons
Euclidean solutions with finite action mediating tunneling between degenerate vacua. BPST instanton in SU(2); θ-vacuum, strong-CP problem,…
LSZ reduction formula
Connects S-matrix elements to time-ordered vacuum expectations: amputate external legs and put on-shell; basis of perturbative QFT…
Källén–Lehmann spectral representation
G(p²) = ∫₀^∞ dμ² ρ(μ²)/(p²-μ²+iε); positivity of spectral density ρ(μ²) ≥ 0 constrains field content, ghost-freedom.
Ward–Takahashi identity
k_μ Γ^μ(p,p+k) = S⁻¹(p+k) - S⁻¹(p); QED gauge invariance ensures Z_1 = Z_2, photon remains massless to all orders.
Slavnov–Taylor identities
Non-abelian generalization of Ward identities; consequence of BRST symmetry; ensures gauge invariance of renormalized Yang–Mills.
BRST symmetry & quantization
Nilpotent Grassmann-odd symmetry Q² = 0 after Faddeev-Popov gauge fixing; physical states = Q-cohomology; ensures unitarity of gauge…
Faddeev–Popov ghosts
Functional determinant det(δG/δω) exponentiated as Grassmann integral → anti-commuting scalar ghosts; cancel unphysical gauge-boson…
Dimensional regularization
Analytically continue loop integrals to d = 4-2ε; preserves gauge/Lorentz symmetry; poles in ε absorbed into counterterms; MS̄ scheme.
Callan–Symanzik equation
(μ∂_μ + β∂_g + γ_ϕ)Γ = 0; renormalization-group flow of couplings and anomalous dimensions; controls high-energy behavior.
Wilson exact RG
Integrate out shells of high-momentum modes to get flow of infinitely many couplings; fixed points classify universality; Polchinski…
Trace (conformal) anomaly
Classically conformal theory: ⟨T^μ_μ⟩ = β(g)/2g G² + γ_m m ψ̄ψ + ...; nonzero via loops; relates to running couplings.
Adler–Bell–Jackiw anomaly
Axial current has anomalous divergence ∂_μ j^μ_5 = (e²/16π²) F F̃; explains π⁰ → γγ rate (1/α²); cancels in SM between quarks & leptons.
Spin-statistics theorem
Integer-spin fields commute at spacelike separation (bosons); half-integer anticommute (fermions); follows from locality + positivity.
Haag's theorem
Interaction picture doesn't exist in QFT: free and interacting fields cannot be unitarily equivalent. Motivates axiomatic (Wightman)…
Reeh–Schlieder theorem
Vacuum is cyclic for local field algebra of any open region; extreme entanglement of QFT vacuum; implications for information in curved…
Unruh effect
Uniformly accelerating observer sees Minkowski vacuum as thermal at T = ℏa/(2πck_B); Bogoliubov transformation of Rindler modes.
Schwinger pair production
Γ/V = (e²E²/4π³)Σe^(-nπm²/eE)/n² in strong constant E-field; nonperturbative; critical field E_c = m²c³/eℏ ≈ 1.3×10¹⁸ V/m.
Cutkosky cutting rules
Discontinuities of amplitude = sum over cuts through intermediate on-shell states; generalizes optical theorem; unitarity at each order.
Conformal bootstrap
Crossing symmetry + unitarity of 4-pt functions constrain CFT data; precision determination of 3D Ising critical exponents to 10⁻⁵.
Spectral positivity & Osterwalder–Schrader
Reflection positivity on Euclidean side ↔ unitarity on Lorentzian; basis for rigorous construction (constructive QFT, lattice).
Atiyah–Segal topological quantum field theory
Quantum-field-theory application of L0 symmetric monoidal functor and monoidal category. Atiyah's 1988 axioms (and Segal's conformal…
Connes–Kreimer Hopf-algebraic renormalization
Quantum-field-theory application of L0 adjunction and monad. Connes & Kreimer 1998–2000: the combinatorics of BPHZ renormalization is…