Quantum Field Theory: Standard Model, Feynman Diagrams & Renormalization

1. Second Quantization & Fields

Quantum field theory (QFT) combines quantum mechanics with special relativity. Fields are operator-valued distributions: φ̂(x) creates/annihilates particles at spacetime point x. Canonical quantization: impose [φ̂(x,t), π̂(y,t)] = iℏδ³(x-y). Klein-Gordon equation: (□ + m²)φ = 0 for scalar fields. Dirac equation: (iγᵘ∂_μ - m)ψ = 0 for spin-½ fermions. Fock space: |n₁, n₂, ...⟩ states with variable particle number. Creation/annihilation operators: [â_k, â†_k'] = δ_{kk'}. Spin-statistics theorem: bosons (integer spin) → commutation, fermions (half-integer) → anticommutation. Noether's theorem: continuous symmetry → conserved current.

2. Gauge Symmetries & Interactions

Gauge principle: promoting global symmetry to local requires introducing gauge fields. QED: U(1) gauge symmetry → photon field A_μ. Covariant derivative: D_μ = ∂_μ + ieA_μ. Lagrangian: L = ψ̄(iγᵘD_μ - m)ψ - ¼F_μνF^μν. Electroweak: SU(2)_L × U(1)_Y → Higgs mechanism breaks to U(1)_EM. W±, Z⁰ bosons acquire mass: M_W = 80.4 GeV, M_Z = 91.2 GeV. QCD: SU(3)_C gauge symmetry → 8 gluons (carry color charge). Asymptotic freedom (Gross, Politzer, Wilczek, Nobel 2004): α_s decreases at high energy. Confinement: quarks cannot exist as free particles — only hadrons (baryons: qqq, mesons: qq̄).

3. Feynman Diagrams & Perturbation Theory

Feynman diagrams: graphical representation of terms in perturbative expansion. Vertices: interaction points (coupling constants). Propagators: internal lines represent virtual particles. External lines: incoming/outgoing real particles. QED vertex: electron-photon coupling (strength α = e²/4π ≈ 1/137). Rules: assign momenta, apply vertex factors, integrate over loop momenta. Tree-level: lowest order, no loops. One-loop: first quantum correction. Cross sections: |M|² integrated over phase space via Fermi's golden rule. Example: e⁺e⁻ → μ⁺μ⁻ at tree level: σ = 4πα²/(3s). Higher orders: increasingly accurate but computationally intensive — state-of-art: 5-loop QED, 4-loop QCD.

4. Renormalization

Loop integrals diverge: ultraviolet (UV) divergences from high-momentum modes. Regularization: dimensional regularization (d = 4-ε), cutoff Λ. Renormalization: absorb divergences into redefined (physical) parameters: mass, charge, field strength. Running coupling constants: α_em(M_Z) = 1/128 (vs. 1/137 at low energy). QCD: α_s(M_Z) = 0.118, increases at low energy → confinement. Beta function: β(g) = μ∂g/∂μ. Asymptotic freedom: β < 0 for non-Abelian gauge theories. Wilson's renormalization group: integrating out high-energy modes systematically. Effective field theory: valid below cutoff scale, non-renormalizable terms suppressed by powers of Λ. QED: most precisely tested theory — g-2 of electron agrees to 12 significant figures.

5. Standard Model & Beyond

Standard Model: SU(3)_C × SU(2)_L × U(1)_Y gauge theory. Matter: 3 generations of quarks (u,d,c,s,t,b) and leptons (e,μ,τ + neutrinos). Force carriers: γ, W±, Z⁰, 8 gluons. Higgs boson (125.1 GeV, discovered 2012 at LHC). 19 free parameters. Successes: predicts everything measured in particle physics. Limitations: doesn't include gravity, dark matter, dark energy, neutrino masses (need extension), matter-antimatter asymmetry, strong CP problem. Beyond SM: supersymmetry (SUSY), extra dimensions, grand unified theories (SU(5), SO(10)), string theory, loop quantum gravity. No BSM physics found at LHC (as of 2026) up to ~14 TeV.

Try it live

Everything above runs in your browser — open SPH Fluid and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.