Фізика квантової теорії поля

⚛️ Квантові поля

Field quantization

ϕ(x) → operator ϕ̂(x). Creation a†, annihilation a operators. Commutation [a, a†] = 1. Fock space.

Vacuum fluctuations

<0|ϕ²|0> ≠ 0. Casimir effect: force between plates. Zero-point energy.

Particle-antiparticle

ϕ = ϕ+ + ϕ-: positive, negative frequency. a: annihilation, b†: antiparticle creation.

Scattering

S-matrix: |out⟩ = S|in⟩. Transition amplitudes. Cross sections, decay rates.

⚡ QED

Lagrangian

L = -¼F_μνF^μν + ψ̄(iγ^μD_μ - m)ψ. U(1) gauge. D_μ = ∂_μ - ieA_μ. Covariant derivative.

Feynman rules

Photon propagator: -iημν/(p²+iε). Fermion: i/(p̸-m+iε). Vertex: -ieγ^μ.

Loop corrections

Self-energy, vertex, vacuum polarization. g-2: muon anomalous magnetic moment. Precision tests.

Renormalization

Electric charge, mass renormalization. Running α(Q²). Landau pole.

🔬 QCD

SU(3) gauge

Quarks: 3 colors (R,G,B). Gluons: 8 generators. Yang-Mills Lagrangian. Color confinement.

Asymptotic freedom

Running αs(Q²): αs → 0 для Q² → ∞. β < 0. Perturbative QCD на high energies.

Confinement

Quarks confined in hadrons. String tension σ ≈ 0.9 GeV/fm. Lattice QCD.

Lattice

Non-perturbative: discrete spacetime. Wilson loops, hadron masses. Compute-intensive.

🔬 Renormalization

Divergences

UV divergences: loop integrals ~ ∫ d⁴k/k⁴. Regularization: cutoff Λ, dimensional.

Counterterms

Renormalization: mass, charge, wavefunction. δm, δe, Z factors. Finite physical quantities.

Running couplings

α(Q²), αs(Q²). β-function: dα/d(ln Q) = β(α). Asymptotic freedom: β < 0.

Effective field theory

Low-energy effective theories. Heavy particles decouple. Operator expansion.

📊 Feynman diagrams

Perturbation theory

Amplitudes: iM = Σ diagrams. Tree-level, loops. g = coupling, perturbative expansion.

Vertices

3-point, 4-point vertices. QED: eeγ. QCD: qqg, ggg. λϕ⁴: 4-point.

Propagators

Particle: i/(p²-m²+iε). Photon: -iημν/(p²+iε). Fermion: i(p̸+m)/(p²-m²+iε).

Calculations

Momentum integration. Feynman parameters. Passarino-Veltman. Computer algebra.

🔬 Gauge theories

U(1) gauge

QED: local gauge invariance. A_μ → A_μ + ∂_μΛ. Phase rotation ψ → e^(ieΛ)ψ.

Non-Abelian

SU(3) QCD, SU(2) weak. Generators: T_a. Covariant derivative. Gluon self-interaction.

Higgs mechanism

Spontaneous gauge symmetry breaking. Mass generation: W±, Z⁰. H boson.

Standard Model

SU(3)×SU(2)×U(1). Quarks, leptons. Yukawa couplings. CP violation.

📊 Графіки та діаграми

Running couplings

α(Q²) evolution:

Asymptotic freedom: αs decreases @ high Q. Confinement @ low Q.

Feynman diagram

e⁻e⁺ → μ⁻μ⁺:

s-channel: annihilation. t-channel: scattering. Bhabha scattering.

α(Q²) = α(μ²) / (1 + β₀/(2π)·α(μ²)·ln(Q²/μ²)) QED: α ≈ 1/137 @ me QCD: αs(Q²) → 0 @ Q → ∞ ΛQCD ~ 200 MeV

🧪 Практичні приклади

Приклад 1: QED precision

Muon g-2: a_μ = (g-2)/2. QED calculation, experiment agree ~10⁻⁶. Loop corrections.

Приклад 2: Lattice QCD

Hadron masses: p, n, π, K. Computation-intensive. Agreement with experiment.

Приклад 3: α(Q²) running

α(Q² = 10³ GeV²) ≈ 1/128 vs 1/137 @ me. Renormalization. Landau pole.

Приклад 4: e⁺e⁻ → hadrons

R = σ(e⁺e⁻→hadrons)/σ(μ⁺μ⁻). QCD running αs. Jet production.

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Квантова теорія поля: квантування полів

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