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3083 lines
107 KiB
Markdown
3083 lines
107 KiB
Markdown
# Proven Equations of the Universe
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> Equations that have withstood every experimental probe. None are "exact" — each has a known domain of validity — but within those domains they are unfalsified to extraordinary precision.
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---
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## 1. Maxwell's Equations (Electromagnetism, 1861–1865)
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### 1.1 The Four Equations (Heaviside–Hertz Form)
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**Differential form (SI units):**
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```
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∇ · E = ρ / ε₀ Gauss's law (electric)
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∇ · B = 0 Gauss's law (magnetic) — no magnetic monopoles
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∇ × E = −∂B / ∂t Faraday-Lenz law of induction
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∇ × B = μ₀J + μ₀ε₀ ∂E/∂t Ampère's law with Maxwell's displacement current
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```
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**Integral form:**
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```
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∮_S E·dA = Q_enc / ε₀
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∮_S B·dA = 0
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∮_C E·dl = −d/dt ∫_S B·dA
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∮_C B·dl = μ₀ I_enc + μ₀ε₀ d/dt ∫_S E·dA
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```
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**Relativistic (covariant) form — tensor notation:**
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```
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∂_μ F^{μν} = μ₀ J^ν inhomogeneous equations
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∂_μ F̃^{μν} = 0 homogeneous (Bianchi identity)
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F_{μν} = ∂_μ A_ν − ∂_ν A_μ field strength tensor
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F̃^{μν} = (1/2) ε^{μνρσ} F_{ρσ} dual tensor
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F_{μν} = ⌈ 0 −E_x/c −E_y/c −E_z/c ⌉
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| E_x/c 0 −B_z B_y |
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| E_y/c B_z 0 −B_x |
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⌊ E_z/c −B_y B_x 0 ⌋
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```
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### 1.2 Scalar and Vector Potentials
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```
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B = ∇ × A magnetic field from vector potential
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E = −∇φ − ∂A/∂t electric field from scalar + vector potentials
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A^μ = (φ/c, A) 4-potential
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F_{μν} = ∂_μ A_ν − ∂_ν A_μ
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```
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**Gauge invariance:** The fields E, B are unchanged under:
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```
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A_μ → A_μ + ∂_μ Λ(x) arbitrary scalar function Λ
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φ → φ − ∂Λ/∂t
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A → A + ∇Λ
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```
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**Gauge choices:**
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```
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Coulomb gauge: ∇ · A = 0
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Lorenz gauge: ∂_μ A^μ = 0 (1/c² ∂φ/∂t + ∇·A = 0)
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Temporal gauge: φ = 0
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```
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### 1.3 Lagrangian Formulation
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```
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ℒ_EM = −¼ F_{μν} F^{μν} − J^μ A_μ
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S_EM = ∫ d⁴x ℒ_EM
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δS = 0 → ∂_μ F^{μν} = μ₀ J^ν (Euler-Lagrange)
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```
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Coupled to matter: replace `∂_μ → D_μ = ∂_μ + i e A_μ` (minimal coupling, QED).
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### 1.4 Wave Equation and Speed of Light
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From Maxwell's equations in vacuum (ρ = 0, J = 0):
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```
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∇² E − (1/c²) ∂²E/∂t² = 0
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∇² B − (1/c²) ∂²B/∂t² = 0
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c = 1 / √(μ₀ε₀) = 299,792,458 m/s (exact, defines meter)
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```
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Maxwell's displacement current term `μ₀ε₀ ∂E/∂t` was the theoretical prediction that electromagnetic waves exist and travel at `c`. Hertz confirmed it in 1887.
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### 1.5 Poynting's Theorem (Energy Conservation)
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```
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∂u/∂t + ∇ · S = −J · E
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u = (1/2) ε₀ E² + (1/2μ₀) B² energy density (J/m³)
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S = (1/μ₀) E × B Poynting vector (W/m², energy flux)
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S^μ = (u c, S) energy-momentum 4-vector
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∂_μ S^μ = −F^{μν} J_ν covariant form
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```
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**Electromagnetic momentum:**
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```
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p_EM = ∫ ε₀ (E × B) dV = ∫ S/c² dV
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```
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### 1.6 Polarization, Permittivity, Permeability in Media
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**Constitutive relations:**
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```
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D = ε₀ E + P = ε E (ε = ε_r ε₀, permittivity tensor)
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H = B/μ₀ − M = B/μ (μ = μ_r μ₀, permeability tensor)
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J = σ E (Ohm's law, conductivity)
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∇ · D = ρ_free
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∇ · B = 0
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∇ × E = −∂B/∂t
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∇ × H = J_free + ∂D/∂t
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```
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### 1.7 Radiation: Lienard-Wiechert Potentials
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Retarded potentials for a point charge `q` with trajectory `r_q(t)`:
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```
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φ(r, t) = (q / 4πε₀) [1 / (R − R·v/c)]_ret
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A(r, t) = (μ₀ q / 4π) [v / (R − R·v/c)]_ret
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R = r − r_q(t_ret)
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t_ret = t − |R|/c retarded time
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```
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**Lienard-Wiechert fields:**
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```
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E = (q/4πε₀) [ (R̂−v/c)(1−v²/c²) / γ²R²(1−R̂·v/c)³ + R̂×((R̂−v/c)×a) / c²R(1−R̂·v/c)³ ]_ret
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└── velocity field (Coulomb + SR correction) ──┘ └── acceleration field (radiation) ──┘
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B = (R̂/c) × E
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```
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**Larmor formula (non-relativistic radiation power):**
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```
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P = (q² a²) / (6πε₀ c³) (J/s = W)
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```
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**Liénard formula (relativistic generalization):**
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```
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P = (q² γ⁶ / 6πε₀ c³) [a² − (v × a)²/c²]
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```
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**Radiation reaction (Abraham-Lorentz force):**
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```
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F_rad = (q² / 6πε₀ c³) d³r/dt³ (pathological — pre-acceleration)
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```
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### 1.8 Electrodynamic Stress-Energy Tensor
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```
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Θ^{μν} = (1/μ₀) [F^μ_α F^{να} + (1/4) g^{μν} F_{αβ} F^{αβ}]
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Θ^{00} = u energy density
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Θ^{0i} = S^i / c momentum density (×c)
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Θ^{ij} = −ε₀ E_i E_j − (1/μ₀) B_i B_j + (1/2) δ_{ij} (ε₀E² + B²/μ₀) Maxwell stress tensor
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```
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**Radiation pressure on a perfect absorber:** `P = I/c` where `I = |S|`.
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### 1.9 Green's Function for the Wave Equation
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```
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(∇² − (1/c²) ∂²/∂t²) G(r,t; r',t') = −δ³(r−r') δ(t−t')
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G_ret(r,t; r',t') = δ(t − t' − |r−r'|/c) / (4π|r−r'|) retarded
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G_adv(r,t; r',t') = δ(t − t' + |r−r'|/c) / (4π|r−r'|) advanced
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```
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General solution for potentials with source `f(r,t)`:
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```
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ψ(r,t) = ∫ d³r' dt' G(r,r'; t,t') f(r', t')
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```
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### 1.10 Experimental Verification
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| Test | Precision | Status |
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|------|-----------|--------|
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| Coulomb's law (inverse square) | `1/r^{2±δ}` with δ < 10⁻¹⁶ | Confirmed |
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| Photon mass limit | m_γ < 10⁻¹⁸ eV/c² | Confirmed |
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| Magnetic monopole | None detected | Absence confirmed |
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| Displacement current | Hertz experiment, all radio | Confirmed |
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| c = 1/√(μ₀ε₀) | Measured to 10⁻⁹ | Confirmed |
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| Poynting vector | Energy balance in every antenna | Confirmed |
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| Lienard-Wiechert | Synchrotron radiation, undulators | Confirmed |
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| Abraham-Lorentz | Qualitative features in laser-plasma | Confirmed (limited precision) |
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**Domain:** Classical electromagnetism. Valid for field strengths ≪ Schwinger limit (`E_c = m²c³/eℏ ≈ 1.3×10¹⁸ V/m`). Unifies electricity, magnetism, and optics. **Every electronic device, radio, laser, and MRI machine is a continuous experimental verification.**
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---
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## 2. Einstein Field Equations (General Relativity, 1915)
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### 2.1 Fundamental Equation
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```
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G_{μν} + Λ g_{μν} = (8πG / c⁴) T_{μν}
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G_{μν} = R_{μν} − ½ R g_{μν} Einstein tensor
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R_{μν} = R^ρ_{μρν} Ricci tensor
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R = g^{μν} R_{μν} Ricci scalar (curvature scalar)
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```
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**Trace-reversed form:**
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```
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R_{μν} = (8πG/c⁴) [T_{μν} − ½ g_{μν} T] + Λ g_{μν}
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T = g^{μν} T_{μν}
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```
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### 2.2 Riemann Curvature Tensor
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```
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R^ρ_{σμν} = ∂_μ Γ^ρ_{νσ} − ∂_ν Γ^ρ_{μσ} + Γ^ρ_{μλ} Γ^λ_{νσ} − Γ^ρ_{νλ} Γ^λ_{μσ}
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Γ^ρ_{μν} = ½ g^{ρλ} (∂_μ g_{νλ} + ∂_ν g_{μλ} − ∂_λ g_{μν}) Christoffel symbols
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```
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**Symmetries of Riemann (in 4D, 20 independent components):**
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```
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R_{ρσμν} = −R_{σρμν} = −R_{ρσνμ} = R_{μνρσ} (antisymmetries)
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R_{ρσμν} + R_{ρμνσ} + R_{ρνσμ} = 0 (Bianchi identity, algebraic)
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∇_λ R_{ρσμν} + ∇_ν R_{ρσλμ} + ∇_μ R_{ρσνλ} = 0 (Bianchi identity, differential)
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```
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Contracting to Einstein tensor:
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```
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∇^μ G_{μν} = 0 → ∇^μ T_{μν} = 0 (conservation of stress-energy)
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```
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### 2.3 Geodesic Equation (Motion in Curved Spacetime)
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```
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d²x^μ/dτ² + Γ^μ_{αβ} dx^α/dτ · dx^β/dτ = 0
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τ = proper time: dτ² = −g_{μν} dx^μ dx^ν
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For a test mass (T^{μν} = 0 elsewhere):
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∇_u u = 0 u^μ = dx^μ/dτ is 4-velocity
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```
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**Equivalence principle:** In a freely falling frame, `g_{μν} → η_{μν}` and `Γ → 0` locally → SR physics.
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### 2.4 Einstein-Hilbert Action
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```
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S = (c⁴ / 16πG) ∫ d⁴x √(−g) (R − 2Λ) + S_matter
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g = det(g_{μν})
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δS/δg^{μν} = 0 → G_{μν} + Λ g_{μν} = (8πG/c⁴) T_{μν}
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T^{μν} = (2 / √(−g)) δS_matter / δg_{μν}
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```
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**Gibbons-Hawking-York boundary term:**
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```
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S_total = S_EH + S_GHY
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S_GHY = (c⁴ / 8πG) ∫_{∂M} d³y √(|h|) ε K
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K = extrinsic curvature of boundary
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ε = ±1 (timelike/spacelike boundary)
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h = induced metric on boundary
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```
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Necessary for a well-posed variational principle with fixed boundary metric.
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### 2.5 Palatini (First-Order) Formalism
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Treat `g_{μν}` and `Γ^ρ_{μν}` as independent variables:
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```
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S_Palatini = (c⁴/16πG) ∫ d⁴x √(−g) g^{μν} R_{μν}(Γ)
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δS/δΓ → Γ = Levi-Civita connection (metric compatibility)
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δS/δg → Einstein field equations
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```
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In vacuum GR, this is equivalent to the standard formulation. Extended to Einstein-Cartan theory with torsion.
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### 2.6 Linearized Gravity and Gravitational Waves
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**Perturbation expansion:** `g_{μν} = η_{μν} + h_{μν}` with `|h_{μν}| ≪ 1`.
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**Trace-reversed perturbation:**
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```
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h̄_{μν} = h_{μν} − ½ η_{μν} h h = η^{μν} h_{μν}
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```
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**Linearized Einstein equations (Lorenz gauge ∂^μ h̄_{μν} = 0):**
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```
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□ h̄_{μν} = −(16πG/c⁴) T_{μν}
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□ = η^{μν} ∂_μ ∂_ν = −(1/c²) ∂²/∂t² + ∇²
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```
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**Gravitational wave in TT gauge (transverse-traceless):**
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```
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h_{μν}^{TT} = [ 0 0 0 0 ] e^{i(kz − ωt)}
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[ 0 h_+ h_× 0 ]
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[ 0 h_× −h_+ 0 ]
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[ 0 0 0 0 ]
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h_+ = plus polarization (stretches/squeezes along x,y axes)
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h_× = cross polarization (stretches/squeezes at 45°)
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```
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### 2.7 Quadrupole Formula (Gravitational Radiation)
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Energy carried away by gravitational waves:
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```
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dE/dt = (G / 5c⁵) Σ_{i,j} ⟨d³Q_{ij}/dt³ · d³Q_{ij}/dt³⟩
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Q_{ij} = ∫ d³x ρ(x) (x_i x_j − (1/3) δ_{ij} r²) mass quadrupole moment
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```
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**Luminosity (full formula):**
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```
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L_GW = (G/5c⁵) ⟨Q̈_{ij} Q̈^{ij}⟩
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For a binary system (masses M₁, M₂, separation a):
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L_GW = (32/5) (G⁴/c⁵) (M₁² M₂² (M₁+M₂) / a⁵)
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```
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LIGO first detection (GW150914, 2015-09-14): two ~30 M_⊙ black holes merging at ~1.3 billion ly. Peak luminosity ~3.6×10⁴⁹ W — briefly outshining the entire observable universe.
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**Orbital decay (binary inspiral):**
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```
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dE_orb/dt = −L_GW → da/dt ∝ −1/a³ → "chirp" frequency increase
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f_GW = (c³ / G) [(5/256) (M_chirp/c²)^{-5/3} (t_coal − t)^{-3/8}]^{3/8}
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M_chirp = (M₁ M₂)^{3/5} / (M₁+M₂)^{1/5} chirp mass
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```
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### 2.8 Exact Solutions
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**Schwarzschild metric (1916) — static, spherical, vacuum (Λ=0):**
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```
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ds² = −(1 − r_s/r) c² dt² + dr²/(1 − r_s/r) + r² dΩ²
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r_s = 2GM / c² Schwarzschild radius
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dΩ² = dθ² + sin²θ dφ²
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Event horizon at r = r_s.
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Coordinate singularity at r = r_s (removable by Kruskal-Szekeres coordinates).
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Physical singularity at r = 0.
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```
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**Kerr metric (1963) — rotating, stationary, axisymmetric:**
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```
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ds² = −(1 − r_s r/Σ) c² dt² + (Σ/Δ) dr² + Σ dθ²
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+ (r² + a² + r_s r a² sin²θ/Σ) sin²θ dφ² − (2 r_s r a sin²θ/Σ) c dt dφ
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Σ = r² + a² cos²θ
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Δ = r² − r_s r + a²
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a = J / Mc spin parameter (m)
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J = angular momentum
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Event horizons at r_± = (r_s/2) ± √((r_s/2)² − a²)
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Ergosphere: region between r_+ and static limit where no observer can remain stationary.
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Penrose process extracts energy from ergosphere (up to ~29% of rest mass for extreme Kerr a→r_s/2).
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```
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**Kerr-Newman metric — charged rotating black hole:**
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```
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Same form as Kerr with Δ = r² − r_s r + a² + r_Q²
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r_Q² = G Q² / (4πε₀ c⁴)
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```
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**Reissner-Nordström metric — charged, non-rotating:**
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```
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ds² = −(1 − r_s/r + r_Q²/r²) c² dt² + dr²/(1 − r_s/r + r_Q²/r²) + r² dΩ²
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Two horizons for Q < M (in geometric units).
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Extremal black hole: r_s = 2r_Q → degenerate horizon.
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```
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**FLRW metric (Friedmann-Lemaître-Robertson-Walker) — homogeneous, isotropic cosmology:**
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```
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ds² = −c² dt² + a²(t) [ dr²/(1 − k r²) + r² dΩ² ]
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k = +1 (closed/spherical), 0 (flat/Euclidean), −1 (open/hyperbolic)
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a(t) = scale factor
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```
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**de Sitter space — vacuum with Λ > 0:**
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```
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ds² = −(1 − Λr²/3) c² dt² + dr²/(1 − Λr²/3) + r² dΩ²
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Static patch. Horizon at r = √(3/Λ).
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Exponential expansion: a(t) ∝ exp( H t ), H = c √(Λ/3).
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```
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### 2.9 ADM Formalism (3+1 Decomposition)
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Split spacetime into foliation of spacelike hypersurfaces Σ_t:
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```
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ds² = −N² c² dt² + γ_{ij} (dx^i + N^i c dt)(dx^j + N^j c dt)
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N = lapse function (rate of proper time vs coordinate time)
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N^i = shift vector (shift of spatial coordinates between slices)
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γ_{ij} = 3-metric on Σ_t
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```
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**Hamiltonian constraint:**
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```
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R(³) + K² − K_{ij} K^{ij} = 16πG/c⁴ · ρ
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K_{ij} = (1/2N)(∂_t γ_{ij} − D_i N_j − D_j N_i) extrinsic curvature
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R(³) = Ricci scalar of γ_{ij}
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ρ = energy density measured by Eulerian observer
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```
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**Momentum constraint:**
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```
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D_j (K^{ij} − γ^{ij} K) = 8πG/c⁴ · J^i
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```
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These are elliptic constraint equations solved on each slice. Evolution equations are hyperbolic.
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### 2.10 Post-Newtonian Approximation
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Expand for slow motion, weak field: `(v/c) ∼ ε`, `GM/rc² ∼ ε²`.
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```
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1PN order: corrections of order ε² to Newtonian
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2PN order: ε⁴, etc.
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Full equations of motion for binary systems known to 4PN order.
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```
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|
||
Essential for LIGO/Virgo template waveforms, pulsar timing (e.g., Hulse-Taylor binary PSR B1913+16 — orbital decay matches GR prediction to <0.2%).
|
||
|
||
### 2.11 Experimental Verification
|
||
|
||
| Test | Experiment | Precision | Status |
|
||
|------|-----------|-----------|--------|
|
||
| Perihelion precession (Mercury) | Optical astrometry | 43"/century, <0.1% | Confirmed |
|
||
| Light deflection (Eddington 1919) | Solar eclipse, VLBI | 0.01% today | Confirmed |
|
||
| Gravitational redshift | Pound-Rebka (1960), GPS, ACES | 10⁻⁵ (Pound), 10⁻⁶ (GP-A) | Confirmed |
|
||
| Shapiro time delay | Viking, Cassini | 10⁻⁵ | Confirmed |
|
||
| Frame-dragging (Lense-Thirring) | Gravity Probe B, LAGEOS | ~10% | Confirmed |
|
||
| Gravitational waves | LIGO/Virgo (2015+) | SNR > 20 in loud events | Confirmed |
|
||
| Black hole shadow | Event Horizon Telescope (2019) | 40 μas resolution | Confirmed |
|
||
| Equivalence principle | MICROSCOPE (2022) | 10⁻¹⁵ | Confirmed |
|
||
| Binary pulsar orbital decay | PSR B1913+16, PSR J0737-3039 | 0.2% | Confirmed |
|
||
| Strong-field tests | LIGO ringdown, EHT | Ongoing | Passed so far |
|
||
|
||
**Domain:** Classical gravity = spacetime curvature. Tested from ~10⁻⁴ m to ~10²⁶ m (cosmological). Breaks down at Planck scale (~10⁻³⁵ m) where quantum effects become non-negligible.
|
||
|
||
---
|
||
|
||
## 3. Schrödinger Equation (Non-relativistic Quantum Mechanics, 1926)
|
||
|
||
### 3.1 Time-Dependent and Time-Independent Forms
|
||
|
||
```
|
||
iℏ ∂/∂t |ψ⟩ = Ĥ |ψ⟩ time-dependent Schrödinger equation
|
||
|
||
Ĥ = −(ℏ²/2m) ∇² + V(r, t) Hamiltonian operator
|
||
|
||
Ĥ ψ_n(r) = E_n ψ_n(r) time-independent (stationary state)
|
||
|ψ(t)⟩ = e^{−iĤt/ℏ} |ψ(0)⟩ time evolution (unitary)
|
||
```
|
||
|
||
**Probability interpretation (Born rule):**
|
||
```
|
||
ρ(r, t) = |ψ(r, t)|² = ψ* ψ probability density
|
||
∫ d³r |ψ|² = 1 normalization (conserved)
|
||
```
|
||
|
||
**Probability current:**
|
||
```
|
||
j = (ℏ / 2mi) (ψ* ∇ψ − ψ ∇ψ*) probability flux
|
||
|
||
∂ρ/∂t + ∇ · j = 0 continuity equation
|
||
```
|
||
|
||
### 3.2 Canonical Commutation Relations
|
||
|
||
```
|
||
[x̂_i, p̂_j] = iℏ δ_{ij} fundamental quantization postulate
|
||
p̂ = −iℏ ∇ momentum operator in position rep.
|
||
[x̂_i, x̂_j] = [p̂_i, p̂_j] = 0
|
||
|
||
[x̂, p̂_x^n] = iℏ n p̂_x^{n-1}
|
||
[p̂, f(x̂)] = −iℏ df/dx
|
||
|
||
Δx Δp ≥ ℏ/2 Robertson-Schrödinger uncertainty
|
||
|
||
Generalized: ΔA ΔB ≥ (1/2) |⟨[Â, B̂]⟩| for any Hermitian operators
|
||
```
|
||
|
||
### 3.3 Harmonic Oscillator (Exact Solution)
|
||
|
||
```
|
||
Ĥ = p̂²/(2m) + (1/2) m ω² x̂²
|
||
|
||
E_n = ℏω (n + 1/2) n = 0, 1, 2, ...
|
||
|
||
Zero-point energy E₀ = ½ ℏω (measurable — Casimir effect, quantum optics)
|
||
```
|
||
|
||
**Ladder operators (Dirac method):**
|
||
```
|
||
â = √(mω/2ℏ) x̂ + i p̂ / √(2mℏω) annihilation
|
||
↠= √(mω/2ℏ) x̂ − i p̂ / √(2mℏω) creation
|
||
|
||
[â, â†] = 1
|
||
Ĥ = ℏω (↠â + 1/2) = ℏω (N̂ + 1/2)
|
||
|
||
N̂ |n⟩ = n |n⟩ number operator
|
||
↠|n⟩ = √(n+1) |n+1⟩
|
||
â |n⟩ = √n |n−1⟩
|
||
|n⟩ = (â†)^n / √(n!) |0⟩
|
||
```
|
||
|
||
**Wavefunctions:**
|
||
```
|
||
ψ_n(x) = (1 / √(2^n n!)) · (mω/πℏ)^{1/4} · H_n(√(mω/ℏ) x) · e^{−mωx²/2ℏ}
|
||
```
|
||
|
||
### 3.4 Hydrogen Atom (Exact Solution)
|
||
|
||
```
|
||
Ĥ = −(ℏ²/2μ) ∇² − e²/(4πε₀ r) μ = m_e m_p / (m_e+m_p) reduced mass
|
||
|
||
E_n = −(μ e⁴ / 32π² ε₀² ℏ²) · 1/n² = −R_y / n²
|
||
|
||
R_y = 13.605693122994 eV Rydberg energy (CODATA 2018)
|
||
|
||
Bohr radius: a₀ = 4πε₀ ℏ² / (μ e²) ≈ 5.29177210903×10⁻¹¹ m
|
||
```
|
||
|
||
**Quantum numbers:**
|
||
```
|
||
n = 1, 2, 3, ... principal
|
||
l = 0, 1, ..., n−1 orbital angular momentum
|
||
m_l = −l, ..., +l magnetic
|
||
m_s = ±½ spin
|
||
|
||
Degeneracy: 2n² per principal level (including spin).
|
||
```
|
||
|
||
**Spherical harmonics Y_l^m(θ,φ):**
|
||
```
|
||
ψ_{nlm}(r,θ,φ) = R_{nl}(r) Y_l^m(θ,φ)
|
||
|
||
R_{nl}(r) ∝ (2r/na₀)^l L_{n−l−1}^{2l+1}(2r/na₀) e^{−r/na₀}
|
||
```
|
||
|
||
### 3.5 Angular Momentum Algebra
|
||
|
||
```
|
||
L̂ = r̂ × p̂ orbital angular momentum operator
|
||
|
||
[L̂_i, L̂_j] = iℏ ε_{ijk} L̂_k
|
||
[L², L̂_i] = 0
|
||
|
||
L² |l,m⟩ = ℏ² l(l+1) |l,m⟩
|
||
L_z |l,m⟩ = ℏ m |l,m⟩
|
||
|
||
Spin-½ (S) = Pauli matrices:
|
||
σ_x = [0 1] σ_y = [0 −i] σ_z = [1 0]
|
||
[1 0] [i 0] [0 −1]
|
||
|
||
Ŝ_i = (ℏ/2) σ_i
|
||
|
||
[σ_i, σ_j] = 2i ε_{ijk} σ_k
|
||
{σ_i, σ_j} = 2 δ_{ij} I
|
||
|
||
Total angular momentum: Ĵ = L̂ + Ŝ
|
||
Addition: |l−s| ≤ j ≤ l+s
|
||
```
|
||
|
||
### 3.6 Density Matrix and Mixed States
|
||
|
||
```
|
||
ρ̂ = Σ_k p_k |ψ_k⟩⟨ψ_k| density operator (mixed state)
|
||
Tr[ρ̂] = 1
|
||
|
||
⟨Â⟩ = Tr[ρ̂ Â] expectation value
|
||
|
||
iℏ ∂ρ̂/∂t = [Ĥ, ρ̂] von Neumann (Liouville-von Neumann) equation
|
||
|
||
Pure state: ρ̂² = ρ̂, Tr[ρ̂²] = 1
|
||
Mixed state: Tr[ρ̂²] < 1
|
||
|
||
Reduced density matrix: ρ̂_A = Tr_B[ρ̂_AB] for subsystems
|
||
```
|
||
|
||
### 3.7 Ehrenfest Theorem (Quantum-Classical Bridge)
|
||
|
||
```
|
||
d/dt ⟨A⟩ = (1/iℏ) ⟨[Â, Ĥ]⟩ + ⟨∂Â/∂t⟩
|
||
|
||
For position and momentum:
|
||
d⟨x⟩/dt = ⟨p⟩/m
|
||
d⟨p⟩/dt = −⟨∇V(x̂)⟩ quantum Newton's 2nd law
|
||
|
||
Only equals classical if V varies slowly over ψ-packet width.
|
||
```
|
||
|
||
### 3.8 Time-Independent Perturbation Theory
|
||
|
||
**Non-degenerate — first order:**
|
||
```
|
||
Ĥ = Ĥ₀ + λ Ŵ
|
||
|
||
E_n^{(1)} = ⟨ψ_n^{(0)}|Ŵ|ψ_n^{(0)}⟩
|
||
|ψ_n^{(1)}⟩ = Σ_{k≠n} [⟨ψ_k^{(0)}|Ŵ|ψ_n^{(0)}⟩ / (E_n^{(0)}−E_k^{(0)})] |ψ_k^{(0)}⟩
|
||
```
|
||
|
||
**Second order energy:**
|
||
```
|
||
E_n^{(2)} = Σ_{k≠n} |⟨ψ_k^{(0)}|Ŵ|ψ_n^{(0)}⟩|² / (E_n^{(0)}−E_k^{(0)})
|
||
```
|
||
|
||
**Degenerate case:** Diagonalize Ŵ in degenerate subspace.
|
||
```
|
||
det[⟨ψ_{n,i}^{(0)}|Ŵ|ψ_{n,j}^{(0)}⟩ − E^{(1)} δ_{ij}] = 0
|
||
```
|
||
|
||
### 3.9 WKB Approximation (Semiclassical)
|
||
|
||
```
|
||
ψ(x) ∼ (C/√p(x)) exp(± i/ℏ ∫ p(x') dx')
|
||
|
||
p(x) = √(2m(E − V(x)))
|
||
|
||
Connection formula at turning point (x_t where p(x_t)=0):
|
||
ψ(x) matches exponentially decaying → oscillatory or vice versa.
|
||
|
||
Bohr-Sommerfeld quantization:
|
||
∮ p dx = 2πℏ (n + γ) n = 0,1,2,... γ = Maslov index
|
||
```
|
||
|
||
### 3.10 Scattering Theory
|
||
|
||
**Lippmann-Schwinger equation:**
|
||
```
|
||
|ψ^{(+)}⟩ = |φ⟩ + (E − Ĥ₀ + iε)^{-1} V̂ |ψ^{(+)}⟩
|
||
|
||
|φ⟩ = incident plane wave
|
||
```
|
||
|
||
**Scattering amplitude and differential cross-section:**
|
||
```
|
||
dσ/dΩ = |f(θ,φ)|²
|
||
|
||
Partial wave expansion (spherically symmetric potential):
|
||
f(θ) = (1/k) Σ_{l=0}^∞ (2l+1) e^{iδ_l} sin δ_l P_l(cos θ)
|
||
|
||
k = √(2mE)/ℏ
|
||
δ_l = phase shift
|
||
|
||
σ_total = (4π/k²) Σ_{l=0}^∞ (2l+1) sin² δ_l
|
||
```
|
||
|
||
**Born approximation (first-order):**
|
||
```
|
||
f(θ,φ) = −(2m/ℏ²) · (1/4π) ∫ d³r e^{−i q·r} V(r)
|
||
|
||
q = k_final − k_initial momentum transfer
|
||
```
|
||
|
||
### 3.11 Variational Principle
|
||
|
||
```
|
||
E_0 ≤ ⟨ψ_trial|Ĥ|ψ_trial⟩ / ⟨ψ_trial|ψ_trial⟩ for any trial function
|
||
|
||
δ[⟨ψ|Ĥ|ψ⟩ − E⟨ψ|ψ⟩] = 0 Euler-Lagrange → exact SE
|
||
```
|
||
|
||
Ritz method: expand `|ψ⟩ = Σ c_i |φ_i⟩` → generalized eigenvalue problem `H c = E S c`.
|
||
|
||
### 3.12 Quantum Tunneling
|
||
|
||
```
|
||
Transmission coefficient (WKB):
|
||
T ≈ exp( −2/ℏ ∫_{x₁}^{x₂} √(2m(V(x)−E)) dx ) for E < V_max
|
||
|
||
Gamow factor in α-decay:
|
||
T ∼ exp( −2π Z₁ Z₂ e² / (4πε₀ ℏ v) )
|
||
|
||
Explains Geiger-Nuttall law (α-decay half-life vs energy).
|
||
```
|
||
|
||
### 3.13 Experimental Verification
|
||
|
||
| Test | System | Precision | Status |
|
||
|------|--------|-----------|--------|
|
||
| Hydrogen spectrum | Balmer, Lyman, etc. | 10⁻¹⁰ (1S-2S transition) | Confirmed |
|
||
| Harmonic oscillator | Trapped ions, molecular vibrations | ~10⁻⁴ | Confirmed |
|
||
| Tunneling | STM, α-decay, tunnel diodes | Qualitative + quantitative | Confirmed |
|
||
| Scattering | Cross-section measurements | Percent level | Confirmed |
|
||
| Born rule | Double-slit, quantum eraser | Countless experiments | Confirmed |
|
||
| Superposition | SQUIDs, trapped ions, molecules | Decoherence timescale confirmed | Confirmed |
|
||
| Zero-point energy | Casimir effect | <1% | Confirmed |
|
||
| Entanglement | Bell-test violations > 40σ | > 40σ | Confirmed |
|
||
|
||
**Domain:** All non-relativistic quantum systems (v ≪ c, particle number conserved). Extends seamlessly to Schrödinger field theory (many-body QM) and, with second quantization, to non-relativistic QFT. **Not a single experimental counterexample.**
|
||
|
||
---
|
||
|
||
## 4. Dirac Equation (Relativistic Spin-½, 1928)
|
||
|
||
### 4.1 Fundamental Equation
|
||
|
||
```
|
||
(iℏ γ^μ ∂_μ − mc) ψ = 0
|
||
|
||
γ^μ matrices satisfy: {γ^μ, γ^ν} = γ^μ γ^ν + γ^ν γ^μ = 2 g^{μν} I₄
|
||
|
||
g^{μν} = diag(−1, +1, +1, +1) west-coast (mostly-minus) metric
|
||
g^{μν} = diag(+1, −1, −1, −1) east-coast / Bjorken-Drell
|
||
```
|
||
|
||
**Feynman slash notation:** `∂̸ = γ^μ ∂_μ`, `p̸ = γ^μ p_μ`, etc.
|
||
|
||
**Conjugate spinor:**
|
||
```
|
||
ψ̄ = ψ† γ⁰
|
||
```
|
||
|
||
**Lagrangian:**
|
||
```
|
||
ℒ_Dirac = ψ̄ (iℏ c ∂̸ − mc²) ψ
|
||
```
|
||
|
||
### 4.2 Gamma Matrix Representations
|
||
|
||
**Dirac (standard) representation:**
|
||
```
|
||
γ⁰ = [ I 0 ] γ^i = [ 0 σ_i ]
|
||
[ 0 −I ] [ −σ_i 0 ]
|
||
|
||
γ⁵ = i γ⁰ γ¹ γ² γ³ = [ 0 I ]
|
||
[ I 0 ]
|
||
```
|
||
|
||
**Weyl (chiral) representation:**
|
||
```
|
||
γ⁰ = [ 0 −I ] γ^i = [ 0 σ_i ]
|
||
[ −I 0 ] [ −σ_i 0 ]
|
||
|
||
γ⁵ = [ I 0 ]
|
||
[ 0 −I ] (diagonal — eigenstates are chirality eigenstates)
|
||
```
|
||
|
||
**Majorana representation:** All γ^μ purely imaginary → real solutions possible.
|
||
|
||
### 4.3 Plane Wave Solutions
|
||
|
||
**Positive-energy (particle) spinors:**
|
||
```
|
||
ψ^{(+)}(x) = u^{(s)}(p) e^{−i p·x/ℏ}
|
||
|
||
u^{(s)}(p) = √(E+mc²) [ φ^{(s)} ] E = +√(p²c² + m²c⁴)
|
||
[ σ·p̂ c / (E+mc²) φ^{(s)} ]
|
||
|
||
φ^{(1)} = [1] φ^{(2)} = [0] 2-spinor basis
|
||
[0] [1]
|
||
```
|
||
|
||
**Negative-energy (antiparticle) spinors:**
|
||
```
|
||
ψ^{(−)}(x) = v^{(s)}(p) e^{+i p·x/ℏ}
|
||
|
||
v^{(s)}(p) = √(E+mc²) [ σ·p̂ c / (E+mc²) η^{(s)} ]
|
||
[ η^{(s)} ]
|
||
|
||
where η^{(s)} = iσ² φ^{(s)*}
|
||
```
|
||
|
||
**Normalization:** `ū^{(r)} u^{(s)} = 2mc δ_{rs}`, `Σ_s u^{(s)} ū^{(s)} = p̸ + mc`.
|
||
|
||
### 4.4 Discrete Symmetries
|
||
|
||
**Parity (P):**
|
||
```
|
||
P ψ(t, r) P^{-1} = γ⁰ ψ(t, −r)
|
||
|
||
Spinor bilinear transformation: ψ̄ψ → +ψ̄ψ (scalar), ψ̄γ⁵ψ → −ψ̄γ⁵ψ (pseudoscalar)
|
||
```
|
||
|
||
**Charge conjugation (C):**
|
||
```
|
||
C ψ C^{-1} = i γ² ψ*
|
||
|
||
C = i γ² γ⁰ (in Dirac rep.)
|
||
C^{-1} γ^μ C = −(γ^μ)^T
|
||
|
||
Majorana condition: ψ = ψ^C ≡ C ψ̄^T (particle = own antiparticle)
|
||
```
|
||
|
||
**Time reversal (T):**
|
||
```
|
||
T ψ(t, r) T^{-1} = γ¹ γ³ ψ(−t, r) (antiunitary)
|
||
|
||
T i T^{-1} = −i
|
||
```
|
||
|
||
**CPT Theorem:** The combined CPT transformation is an exact symmetry of any local, Lorentz-invariant QFT. Violation of CPT has never been observed. Limits: mass difference `|m_K⁰ − m_K̄⁰|/m_K < 10⁻¹⁸`.
|
||
|
||
### 4.5 Bilinear Covariants
|
||
|
||
16 independent 4×4 matrices → 16 bilinear forms, classified by Lorentz transformation:
|
||
|
||
```
|
||
Scalar: ψ̄ ψ (1 component)
|
||
Pseudoscalar: ψ̄ γ⁵ ψ (1)
|
||
Vector: ψ̄ γ^μ ψ (4)
|
||
Axial-vector: ψ̄ γ^μ γ⁵ ψ (4)
|
||
Tensor: ψ̄ σ^{μν} ψ (6) σ^{μν} = (i/2)[γ^μ, γ^ν]
|
||
─────────────────────────────────────────
|
||
Total: 16 independent bilinears
|
||
```
|
||
|
||
**Gordon decomposition (current):**
|
||
```
|
||
ψ̄ γ^μ ψ = (i/2m) [ψ̄ ∂^μ ψ − (∂^μ ψ̄) ψ] + (1/m) ∂_ν (ψ̄ σ^{μν} ψ)
|
||
|
||
└── convection current ──┘ └── spin current ──┘
|
||
```
|
||
|
||
### 4.6 Non-Relativistic Reduction (Pauli Equation)
|
||
|
||
Expand in powers of `v/c`:
|
||
|
||
```
|
||
iℏ ∂ψ/∂t = [ (p − eA)²/2m + eφ − (eℏ/2m) σ·B − (p⁴/8m³c²) + ... ] ψ
|
||
|
||
Pauli spin term: −μ · B with μ = (eℏ/2m) σ = g_s (eℏ/4m) σ, g_s = 2
|
||
```
|
||
|
||
### 4.7 Relativistic Hydrogen Fine Structure
|
||
|
||
Iterating the reduction yields:
|
||
|
||
```
|
||
ΔE_{FS} = (R_y α²/n³) [ 1/(j+½) − 3/(4n) ]
|
||
|
||
Fine structure constant: α = e²/(4πε₀ ℏc) ≈ 1/137.035999084
|
||
|
||
Term Formula Origin
|
||
───── ─────── ──────
|
||
Relativistic −(α²R_y/n³) (n/(l+½)−3/4) kinetic energy expansion
|
||
Spin-orbit +(α²R_y/n³) [j(j+1)−l(l+1)−3/4] / [2l(l+½)(l+1)] Ŝ·L coupling
|
||
Darwin +(α²R_y/n³) δ_{l0} zitterbewegung smearing
|
||
```
|
||
|
||
**Lamb shift (2S_{1/2} − 2P_{1/2} in hydrogen):**
|
||
```
|
||
ΔE_Lamb ≈ 1057.8 MHz ≈ 4.37 μeV
|
||
|
||
From QED radiative corrections (vacuum polarization + electron self-energy).
|
||
Measured by Lamb & Retherford (1947) — confirmed QED as correct relativistic QFT.
|
||
```
|
||
|
||
### 4.8 Electron g-Factor and Anomalous Magnetic Moment
|
||
|
||
```
|
||
μ = g (eℏ/4m) σ/2
|
||
|
||
g_Dirac = 2 exactly, from Dirac equation
|
||
|
||
g_exp / 2 = 1.00115965218091(26) CODATA 2018
|
||
|
||
a_e = (g−2)/2 measured to 3×10⁻¹³
|
||
|
||
QED prediction:
|
||
a_e^{QED} = α/2π − 0.328478... (α/π)² + 1.181241... (α/π)³ − 1.912... (α/π)⁴ + ...
|
||
|
||
Agreement: 1 part in 10¹² — the most precisely tested prediction in physics.
|
||
```
|
||
|
||
### 4.9 Weyl Equation (Massless Fermions)
|
||
|
||
```
|
||
iℏ σ^μ ∂_μ ψ_L = 0 (left-handed Weyl spinor)
|
||
iℏ σ̄^μ ∂_μ ψ_R = 0 (right-handed)
|
||
|
||
σ^μ = (I, σ_i) σ̄^μ = (I, −σ_i)
|
||
|
||
Chirality = helicity for massless particles:
|
||
Left-handed (ψ_L): spin antiparallel to momentum
|
||
Right-handed (ψ_R): spin parallel to momentum
|
||
```
|
||
|
||
Neutrinos were long thought massless Weyl fermions. Neutrino oscillations → nonzero mass → beyond-minimal SM.
|
||
|
||
### 4.10 Klein Paradox
|
||
|
||
For a potential step `V > 2mc²`, the reflection coefficient `|R|² > 1` in single-particle Dirac theory. Resolution: QFT pair production — the potential creates electron-positron pairs. No violation of unitarity in QED.
|
||
|
||
### 4.11 Experimental Verification
|
||
|
||
| Test | Precision | Status |
|
||
|------|-----------|--------|
|
||
| Electron g−2 | 3×10⁻¹³ | Matches QED+EW+hadronic |
|
||
| Positron existence | Anderson 1932 | Confirmed |
|
||
| Fine structure in hydrogen | ~10⁻¹⁰ | Confirmed |
|
||
| Lamb shift | ~10⁻⁶ | Confirmed |
|
||
| Antiparticle properties | m_ē = m_e to < 10⁻¹² | Confirmed |
|
||
| CPT symmetry | Kaon mass difference < 10⁻¹⁸ | Confirmed |
|
||
| Zitterbewegung | Observable in trapped-ion simulations | Confirmed (simulated) |
|
||
|
||
**Domain:** Relativistic spin-½ particles (all quarks and leptons). The foundation of fermionic QFT.
|
||
|
||
---
|
||
|
||
## 5. Newton's Laws of Motion (1687)
|
||
|
||
### 5.1 The Three Laws
|
||
|
||
```
|
||
1st Law (Inertia): An object at rest stays at rest, and an object in motion stays in motion
|
||
with constant velocity, unless acted upon by a net external force.
|
||
|
||
2nd Law: F = dp/dt = d(mv)/dt (general form)
|
||
F = m a (constant mass)
|
||
|
||
3rd Law: F_{A→B} = −F_{B→A} (action = reaction, equal & opposite)
|
||
```
|
||
|
||
### 5.2 Relativistic Generalization
|
||
|
||
```
|
||
dp^μ/dτ = F^μ 4-force = proper time derivative of 4-momentum
|
||
|
||
p^μ = m u^μ = (γmc, γmv) 4-momentum
|
||
u^μ = dx^μ/dτ = (γc, γv) 4-velocity, dt/dτ = γ
|
||
|
||
F^μ = γ (F·v/c, F) relation between 3-force and 4-force
|
||
|
||
For constant mass in SR:
|
||
F = d(γmv)/dt = γ³ m a_∥ + γ m a_⊥ (transverse mass γm, longitudinal γ³m)
|
||
```
|
||
|
||
### 5.3 Lagrangian and Hamiltonian Mechanics (Generalized Newton)
|
||
|
||
**Principle of least action:**
|
||
```
|
||
S[q] = ∫_{t₁}^{t₂} L(q, q̇, t) dt
|
||
δS = 0 → Euler-Lagrange equations
|
||
|
||
d/dt (∂L/∂q̇_i) − ∂L/∂q_i = 0 for each generalized coordinate
|
||
```
|
||
|
||
**For a particle:** `L = T − V = ½ m q̇² − V(q)` → `m q̈ = −dV/dq = F`.
|
||
|
||
**D'Alembert's principle (virtual work):**
|
||
```
|
||
Σ_i (F_i − ṗ_i) · δr_i = 0 virtual displacements δr_i consistent with constraints
|
||
|
||
→ leads to Lagrange's equations for constrained systems.
|
||
```
|
||
|
||
**Hamilton's equations:**
|
||
```
|
||
H(q, p, t) = p_i q̇_i − L Legendre transform
|
||
p_i = ∂L/∂q̇_i canonical momentum
|
||
|
||
q̇_i = ∂H/∂p_i
|
||
ṗ_i = −∂H/∂q_i
|
||
|
||
dH/dt = ∂H/∂t (conserved if H has no explicit t-dependence)
|
||
```
|
||
|
||
**Poisson bracket formulation:**
|
||
```
|
||
{A, B}_PB = Σ_i (∂A/∂q_i · ∂B/∂p_i − ∂A/∂p_i · ∂B/∂q_i)
|
||
|
||
df/dt = {f, H}_PB + ∂f/∂t time evolution of any phase-space function
|
||
```
|
||
|
||
### 5.4 Rigid Body Dynamics (Euler's Equations)
|
||
|
||
```
|
||
I dω/dt + ω × (I ω) = τ Euler's equations for rigid body rotation
|
||
|
||
I = inertia tensor (3×3), τ = torque vector
|
||
|
||
In principal axes (I = diag(I₁, I₂, I₃)):
|
||
I₁ ω̇₁ − (I₂−I₃) ω₂ ω₃ = τ₁
|
||
I₂ ω̇₂ − (I₃−I₁) ω₃ ω₁ = τ₂
|
||
I₃ ω̇₃ − (I₁−I₂) ω₁ ω₂ = τ₃
|
||
```
|
||
|
||
**Angular momentum:** `L = I ω`, `dL/dt = τ`.
|
||
|
||
**Poinsot's theorem:** Torque-free motion — angular velocity vector precesses in body frame around the angular momentum vector.
|
||
|
||
### 5.5 Continuum Mechanics (Cauchy's Stress Principle)
|
||
|
||
```
|
||
ρ d²u/dt² = ∇ · σ + f_body (Cauchy momentum equation)
|
||
|
||
∂σ_{ij}/∂x_j + f_i = ρ ü_i (index form)
|
||
|
||
σ = stress tensor (Pa), u = displacement vector
|
||
```
|
||
|
||
Specialize to:
|
||
- **Elastic solids:** `σ = C : ε` (Hooke's law generalized — stiffness tensor)
|
||
- **Fluids:** `σ = −p I + μ(∇v + ∇v^T) + λ (∇·v) I` (Newtonian constitutive relation → Navier-Stokes)
|
||
- **Electrodynamics:** `σ_{ij}^{EM} = −ε₀E_iE_j − (1/μ₀)B_iB_j + ½δ_{ij}(ε₀E²+B²/μ₀)` (Maxwell stress)
|
||
|
||
### 5.6 Conservation Laws from Newton's Laws
|
||
|
||
```
|
||
Momentum conservation: dP/dt = F_ext (Σ forces = rate of change of total momentum)
|
||
Angular momentum: dL/dt = τ_ext (Σ torques = rate of change of angular momentum)
|
||
Center of mass: M R̈_cm = F_ext (center of mass moves like a point particle)
|
||
```
|
||
|
||
These are the low-velocity limits of the corresponding Noether symmetries.
|
||
|
||
### 5.7 Experimental Domain
|
||
|
||
- **Validity:** All macroscopic systems with v ≪ c and weak gravity (Φ/c² ≪ 1).
|
||
- **Transition:** Relativistic corrections needed at v/c ≳ 0.01 (GPS satellites at 14,000 km/h need both SR + GR corrections = ~38 μs/day).
|
||
- **Quantum limit:** Position-momentum uncertainty prevents simultaneous perfect determination of both — but expectation values obey Ehrenfest's theorem which exactly mirrors Newton's 2nd law.
|
||
|
||
**Falsification status:** Never falsified within domain. Relativity and QM did not falsify Newton — they revealed him as a low-energy limiting case.
|
||
|
||
---
|
||
|
||
## 6. Conservation of Energy (First Law of Thermodynamics)
|
||
|
||
### 6.1 The First Law
|
||
|
||
```
|
||
dU = δQ − δW internal energy change
|
||
|
||
In a closed system (no heat/work exchange):
|
||
ΔU = 0 E_total = constant
|
||
|
||
In differential form:
|
||
dU = T dS − p dV + Σ_i μ_i dN_i chemical potential μ_i for species i
|
||
```
|
||
|
||
### 6.2 Thermodynamic Potentials (Legendre Transforms)
|
||
|
||
```
|
||
Internal energy: U(S,V,N)
|
||
Enthalpy: H(S,p,N) = U + pV
|
||
Helmholtz free energy: F(T,V,N) = U − TS
|
||
Gibbs free energy: G(T,p,N) = U + pV − TS = H − TS
|
||
|
||
Differentials:
|
||
dH = T dS + V dp + Σ μ_i dN_i
|
||
dF = −S dT − p dV + Σ μ_i dN_i
|
||
dG = −S dT + V dp + Σ μ_i dN_i
|
||
```
|
||
|
||
### 6.3 Maxwell Relations
|
||
|
||
From equality of cross-derivatives (d²U = exact differential):
|
||
|
||
```
|
||
(∂T/∂V)_S = −(∂p/∂S)_V
|
||
(∂T/∂p)_S = +(∂V/∂S)_p
|
||
(∂S/∂V)_T = +(∂p/∂T)_V
|
||
(∂S/∂p)_T = −(∂V/∂T)_p
|
||
```
|
||
|
||
These relate seemingly unconnected quantities (e.g., how entropy changes with volume = how pressure changes with temperature). All experimentally confirmed.
|
||
|
||
### 6.4 Specific Heat Relations
|
||
|
||
```
|
||
C_V = T (∂S/∂T)_V = (∂U/∂T)_V
|
||
C_p = T (∂S/∂T)_p = (∂H/∂T)_p
|
||
|
||
C_p − C_V = −T (∂V/∂T)_p² / (∂V/∂p)_T = T V α² / κ_T
|
||
|
||
α = thermal expansion coefficient, κ_T = isothermal compressibility
|
||
```
|
||
|
||
**Equipartition theorem (classical):**
|
||
```
|
||
Each quadratic degree of freedom contributes ½ k_B T to energy.
|
||
C_V = (f/2) R per mole for f degrees of freedom.
|
||
```
|
||
|
||
### 6.5 Noether Derivation: Time Translation → Energy
|
||
|
||
```
|
||
S[φ] = ∫ d⁴x ℒ(φ, ∂_μ φ)
|
||
|
||
Under infinitesimal time translation: x^μ → x^μ + ε δ₀^μ
|
||
|
||
Noether current: J^μ = (∂ℒ/∂(∂_μ φ)) δφ − T^μ_ν ε^ν
|
||
|
||
where the canonical stress-energy tensor is:
|
||
T^μ_ν = (∂ℒ/∂(∂_μ φ)) ∂_ν φ − δ^μ_ν ℒ
|
||
|
||
E = ∫ d³x T⁰_₀ conserved charge = energy
|
||
```
|
||
|
||
### 6.6 Conservation in General Relativity
|
||
|
||
```
|
||
∇_μ T^{μν} = 0 covariant conservation
|
||
|
||
This does NOT imply a globally conserved energy in curved spacetime.
|
||
Energy is not globally defined in GR — only local conservation.
|
||
The "energy of the gravitational field" is not a tensor.
|
||
```
|
||
|
||
**Komar mass (stationary spacetimes):**
|
||
```
|
||
M_K = −(1/8πG) ∮_{S²_∞} ∇^μ ξ^ν dS_{μν} ξ^ν = timelike Killing vector
|
||
```
|
||
|
||
**ADM mass (asymptotically flat):**
|
||
```
|
||
M_ADM = (1/16πG) ∮_{S²_∞} (∂_j h_{ij} − ∂_i h_{jj}) n^i dA
|
||
```
|
||
|
||
### 6.7 Quantum Energy
|
||
|
||
```
|
||
Ĥ |E⟩ = E |E⟩ energy eigenvalue equation
|
||
|
||
⟨Ĥ⟩ = ⟨ψ|Ĥ|ψ⟩ expectation value — constant if Ĥ is time-independent
|
||
|
||
In QFT, the Hamiltonian is:
|
||
Ĥ = ∫ d³x : T^{00} :
|
||
|
||
Vacuum expectation value: ⟨0|T^{μν}|0⟩ = ρ_vac g^{μν}
|
||
ρ_vac ∝ Λ (cosmological constant problem: observed ρ_vac ~ 10⁻¹²⁰ × QFT prediction)
|
||
```
|
||
|
||
### 6.8 Zero-Point Energy and Casimir Effect
|
||
|
||
```
|
||
E₀ = ½ ℏω per mode
|
||
|
||
Casimir force between two parallel conducting plates (area A, separation d):
|
||
F = −(π² ℏc / 240 d⁴) A (attractive)
|
||
|
||
P_Casimir = F/A = 1.3×10⁻³ Pa at d=1μm measured to ~1%
|
||
```
|
||
|
||
### 6.9 Experimental Status
|
||
|
||
Energy conservation: **zero violations ever observed**. Apparent violations (beta decay spectrum → neutrino predicted by Pauli 1930, discovered 1956) were resolved by discovering new particles.
|
||
|
||
In GR, Wheeler's "geon" and "mass without mass" ideas do not violate energy conservation — they are just nonlocal in gravitational energy definition.
|
||
|
||
**Domain:** Universal — classical, quantum, relativistic, cosmological. Derived from time-translation symmetry of physical laws.
|
||
|
||
---
|
||
|
||
## 7. Second Law of Thermodynamics
|
||
|
||
### 7.1 The Second Law
|
||
|
||
```
|
||
dS_total ≥ 0 entropy of an isolated system never decreases
|
||
|
||
dS = δQ_rev / T Clausius definition of entropy change
|
||
|
||
For irreversible processes: dS > δQ_irr / T
|
||
```
|
||
|
||
### 7.2 Boltzmann Entropy
|
||
|
||
```
|
||
S = k_B ln Ω (Boltzmann, 1877)
|
||
|
||
Ω = number of microstates corresponding to given macrostate
|
||
k_B = 1.380649×10⁻²³ J/K (exact, defines kelvin since 2019)
|
||
```
|
||
|
||
### 7.3 Gibbs Entropy (Statistical Mechanics)
|
||
|
||
```
|
||
S = −k_B Σ_i p_i ln p_i classical discrete distribution
|
||
|
||
S = −k_B ∫ f(p,q) ln f(p,q) dΓ continuous phase space
|
||
|
||
S = −k_B Tr[ρ̂ ln ρ̂] quantum (von Neumann entropy)
|
||
|
||
Maximized by uniform distribution (microcanonical) / canonical (Boltzmann) / grand canonical.
|
||
```
|
||
|
||
### 7.4 Shannon Entropy (Information Theory, 1948)
|
||
|
||
```
|
||
H(X) = −Σ_i p(x_i) log₂ p(x_i) bits
|
||
|
||
Relationship: S = k_B ln 2 · H thermodynamic entropy = 0.957×10⁻²³ J/K per bit
|
||
```
|
||
|
||
### 7.5 Boltzmann H-Theorem (1872)
|
||
|
||
```
|
||
H(t) = ∫ d³p f(p,t) ln f(p,t)
|
||
|
||
dH/dt ≤ 0 H always decreases (or constant at equilibrium)
|
||
|
||
H = −S/k_B + constant → dS/dt ≥ 0
|
||
```
|
||
|
||
Proves that the Boltzmann equation implies the 2nd Law. The "arrow of time" emerges from molecular chaos (Stosszahlansatz).
|
||
|
||
### 7.6 Loschmidt's Paradox and Resolution
|
||
|
||
**Paradox:** Microscopic equations of motion are time-reversible. Where does irreversibility come from?
|
||
|
||
**Resolution:** The H-theorem relies on the Stosszahlansatz (molecular chaos assumption) — correlations are discarded after each collision. This is a coarse-graining. The apparent irreversibility emerges from:
|
||
- Low-entropy initial conditions (Past Hypothesis)
|
||
- Dynamical instability (Lyapunov exponents → rapid information scrambling)
|
||
- Coarse-graining (observables don't resolve micro-details)
|
||
|
||
### 7.7 Fluctuation Theorems (1990s–present)
|
||
|
||
**Crooks Fluctuation Theorem (1999):**
|
||
```
|
||
p(W) / p_rev(−W) = exp( (W − ΔF) / k_B T )
|
||
|
||
p(W) = probability of work W during forward process
|
||
ΔF = free energy difference between initial and final states
|
||
```
|
||
|
||
**Jarzynski Equality (1997):**
|
||
```
|
||
⟨exp(−W / k_B T)⟩ = exp(−ΔF / k_B T)
|
||
|
||
Averages over nonequilibrium trajectories recover equilibrium free energy differences.
|
||
```
|
||
|
||
These theorems **generalize** the 2nd Law — they describe fluctuations where `dS < 0` is probabilistically possible but exponentially unlikely for macroscopic systems.
|
||
|
||
**Experimental verification:** RNA pulling experiments, colloidal particle trapping, single-molecule force spectroscopy.
|
||
|
||
### 7.8 Landauer's Principle (1961)
|
||
|
||
```
|
||
Erasing 1 bit of information dissipates AT LEAST k_B T ln 2 joules of heat.
|
||
|
||
Physical basis: information is physical — logical irreversibility → thermodynamic irreversibility.
|
||
```
|
||
|
||
Verified experimentally (Bérut et al., Nature 2012).
|
||
|
||
Resolves Maxwell's demon: the demon must erase its memory to operate cyclically → this inevitably generates `≥ k_B T ln 2` per erased bit → 2nd Law holds.
|
||
|
||
### 7.9 Entropy in Physical Systems
|
||
|
||
**Mixing entropy (ideal gases):**
|
||
```
|
||
ΔS_mix = −k_B (N₁ ln x₁ + N₂ ln x₂) x_i = mole fraction
|
||
```
|
||
|
||
**Phase transitions:**
|
||
```
|
||
ΔS_vaporization = L_v / T_b L_v = latent heat
|
||
|
||
Trouton's rule: ΔS_vap ≈ 85 J/(mol·K) for many liquids at boiling point.
|
||
```
|
||
|
||
**Configurational entropy (polymers, glasses):**
|
||
```
|
||
S_conf = k_B ln Ω_conf e.g., number of chain conformations
|
||
```
|
||
|
||
**Residual entropy of ice:** `S(0) ≈ 3.4 J/(mol·K)` — Pauling's estimate for proton disorder. Confirmed experimentally.
|
||
|
||
### 7.10 Black Hole Entropy (Generalized Second Law)
|
||
|
||
```
|
||
S_BH = k_B A / 4ℓ_P² Bekenstein-Hawking (1972–74)
|
||
|
||
ℓ_P = √(ℏG/c³) ≈ 1.616255×10⁻³⁵ m Planck length
|
||
|
||
d/dt (S_BH + S_matter) ≥ 0 Generalized Second Law (GSL)
|
||
```
|
||
|
||
### 7.11 Heat Death and the Arrow of Time
|
||
|
||
The 2nd Law implies a future state of maximum entropy — "heat death":
|
||
- All free energy exhausted
|
||
- Uniform temperature everywhere
|
||
- No macroscopic work possible
|
||
- The universe reaches thermodynamic equilibrium
|
||
|
||
**Multiple arrows of time** all derive from the low-entropy initial condition:
|
||
- Thermodynamic arrow (entropy increase)
|
||
- Cosmological arrow (universe expansion)
|
||
- Psychological arrow (we remember the past, not the future)
|
||
- Causal arrow (causes precede effects)
|
||
|
||
### 7.12 Experimental Status
|
||
|
||
| Test | System | Status |
|
||
|------|--------|--------|
|
||
| Heat engines | Every engine since Newcomen (1712) | Efficiency ≤ Carnot — confirmed |
|
||
| Fluctuation theorems | Single-molecule biophysics | Confirmed |
|
||
| Landauer's principle | Micromagnetic bit manipulation | Confirmed |
|
||
| Maxwell's demon | Information engines (Toyabe et al. 2010) | Resolved |
|
||
| H-theorem | Molecular dynamics simulations | Confirmed |
|
||
| Entropy of black holes | Gravitational wave ringdown, analog gravity | Indirectly supported |
|
||
| Entropy increase | Every macroscopic process, every living organism | Universally observed |
|
||
|
||
**Domain:** Any system with many degrees of freedom. A statistical law — not absolute at the microscale — but overwhelmingly probable at macroscopic scales. **No macroscopic violation ever observed.**
|
||
|
||
---
|
||
|
||
## 8. Planck–Einstein Relation (Quantum of Action, 1900–1905)
|
||
|
||
### 8.1 The Fundamental Quantum Relations
|
||
|
||
```
|
||
E = hν = ℏω photon energy
|
||
p = h/λ = ℏk photon momentum
|
||
|
||
h = 6.62607015×10⁻³⁴ J·s (exact, defines kg since 2019)
|
||
ℏ = h/2π = 1.054571817×10⁻³⁴ J·s
|
||
```
|
||
|
||
**Compton wavelength:**
|
||
```
|
||
λ_C = h / mc electron: 2.4263102389×10⁻¹² m
|
||
```
|
||
|
||
### 8.2 Planck's Blackbody Radiation Law (1900)
|
||
|
||
**Spectral radiance (energy per unit time, area, solid angle, frequency):**
|
||
```
|
||
B_ν(ν, T) = (2hν³ / c²) · 1 / [exp(hν/k_B T) − 1] W·sr⁻¹·m⁻²·Hz⁻¹
|
||
|
||
B_λ(λ, T) = (2hc² / λ⁵) · 1 / [exp(hc/λk_B T) − 1] W·sr⁻¹·m⁻²·m⁻¹
|
||
```
|
||
|
||
**Derivation:** Quantize the electromagnetic field oscillators → energy per mode = hν/(e^{hν/kT}−1). Sum over all modes with density of states `g(ν)dν = (8πν²/c³) dν`.
|
||
|
||
**Limits:**
|
||
```
|
||
hν ≪ k_B T: B_ν → (2ν²/c²) k_B T Rayleigh-Jeans law (classical)
|
||
hν ≫ k_B T: B_ν → (2hν³/c²) e^{−hν/kT} Wien approximation
|
||
```
|
||
|
||
### 8.3 Consequences of Planck's Law
|
||
|
||
**Wien's Displacement Law (1893):**
|
||
```
|
||
λ_max T = 2.897771955...×10⁻³ m·K wavelength of peak emission
|
||
|
||
ν_max / T = 58.789... GHz/K frequency of peak
|
||
```
|
||
|
||
**Stefan-Boltzmann Law (1879–1884):**
|
||
```
|
||
j* = σ T⁴ total radiated power per unit area
|
||
|
||
σ = (2π⁵ k_B⁴) / (15 h³ c²) Stefan-Boltzmann constant
|
||
= 5.670374419×10⁻⁸ W·m⁻²·K⁻⁴ (CODATA 2018)
|
||
```
|
||
|
||
**Photon number density (blackbody):**
|
||
```
|
||
n_γ = (2 ζ(3) / π²) (k_B T / ℏc)³ ≈ 20.28 (T/1K)³ cm⁻³
|
||
```
|
||
|
||
**Energy density:**
|
||
```
|
||
u = a T⁴ a = 4σ/c = 7.5657×10⁻¹⁶ J·m⁻³·K⁻⁴
|
||
```
|
||
|
||
### 8.4 The Photoelectric Effect (Einstein, 1905)
|
||
|
||
```
|
||
K_max = hν − φ kinetic energy of ejected electron
|
||
|
||
φ = work function of metal (minimum energy to eject electron)
|
||
hν_0 = φ threshold frequency
|
||
|
||
K_max ≥ 0 → requires ν > ν_0 regardless of light intensity.
|
||
```
|
||
|
||
**Key predictions confirming photons, not classical waves:**
|
||
1. K_max depends only on ν, not intensity.
|
||
2. Threshold frequency ν_0 exists.
|
||
3. No time delay — emission is instantaneous (vs. minutes for classical energy accumulation).
|
||
4. Slope of K_max vs ν = h (Planck's constant — measured by Millikan 1916).
|
||
|
||
### 8.5 Compton Scattering (1923)
|
||
|
||
```
|
||
λ' − λ = (h / m_e c) (1 − cos θ) Compton shift
|
||
|
||
λ'_max = λ + 2h/m_e c full backscatter (θ=π)
|
||
|
||
Δλ_max ≈ 0.00486 nm independent of incident wavelength
|
||
```
|
||
|
||
**Derivation:** Photon + electron, relativistic energy-momentum conservation:
|
||
```
|
||
hν + m_e c² = hν' + γ m_e c²
|
||
hν/c = (hν'/c) cos θ + γ m_e v cos φ
|
||
0 = (hν'/c) sin θ − γ m_e v sin φ
|
||
```
|
||
|
||
Eliminating φ and v yields Δλ. Experimentally: detected recoil electron in coincidence with scattered photon (Bothe-Geiger 1925) — confirmed photon as particle.
|
||
|
||
### 8.6 de Broglie Wavelength (1923–1924)
|
||
|
||
```
|
||
λ = h / p = h / (γ m v) for any massive particle
|
||
|
||
Non-relativistic: λ = h / √(2mE)
|
||
Electron at 100 eV: λ ≈ 0.12 nm (atomic-scale diffraction)
|
||
```
|
||
|
||
**Davisson-Germer experiment (1927):** Electron diffraction from nickel crystal → interference pattern exactly matching de Broglie wavelength prediction. Confirmed wave nature of matter.
|
||
|
||
**Modern:** Neutron diffraction, He-atom scattering, Bose-Einstein condensate interference, molecule interferometry (up to >2000 atoms — C₆₀ buckyballs, tailored organic molecules >25,000 amu).
|
||
|
||
### 8.7 Electromagnetic Field Quantization (QED)
|
||
|
||
**Single-mode field quantization:**
|
||
```
|
||
Ê(r,t) = E₀ (â e^{i(k·r−ωt)} + ↠e^{−i(k·r−ωt)})
|
||
|
||
E₀ = √(ℏω / 2ε₀ V) field amplitude per photon
|
||
```
|
||
|
||
**Fock (number) states:**
|
||
```
|
||
↠|n⟩ = √(n+1) |n+1⟩ create photon
|
||
â |n⟩ = √n |n−1⟩ annihilate photon
|
||
N̂ |n⟩ = n |n⟩ N̂ = ↠â
|
||
|
||
⟨n|Ê|n⟩ = 0 zero mean field
|
||
⟨n|ʲ|n⟩ = E₀² (n + ½) nonzero variance = zero-point fluctuations
|
||
```
|
||
|
||
**Coherent states (laser light, Glauber 1963):**
|
||
```
|
||
|α⟩ = e^{−|α|²/2} Σ_{n=0}^∞ (α^n / √(n!)) |n⟩
|
||
|
||
â |α⟩ = α |α⟩ eigenvalue of annihilation operator
|
||
⟨n⟩ = |α|² = mean photon number
|
||
Δn = |α| = √⟨n⟩ → Poissonian photon statistics
|
||
```
|
||
|
||
**Thermal state:**
|
||
```
|
||
ρ̂_th = (1/Z) Σ_n e^{−β ℏω n} |n⟩⟨n|
|
||
⟨n⟩ = 1 / (e^{ℏω/kT} − 1) Bose-Einstein distribution
|
||
```
|
||
|
||
### 8.8 Photon Momentum and Radiation Pressure
|
||
|
||
```
|
||
p_γ = hν / c = E_γ / c
|
||
|
||
Radiation pressure on perfect absorber: P_rad = I / c
|
||
Radiation pressure on perfect reflector: P_rad = 2I / c (momentum reversal)
|
||
I = intensity (W/m²)
|
||
|
||
Solar radiation pressure at 1 AU: P_sun ≈ 4.6 μPa
|
||
Solar sail acceleration: a = 2η I / (c σ) η = efficiency, σ = areal density
|
||
```
|
||
|
||
**Photon recoil in atomic transitions:** `v_recoil = hν / (m c)` — critical for laser cooling, optical molasses, Bose-Einstein condensates.
|
||
|
||
### 8.9 Planck Units (Derived Quantities from ℏ, G, c)
|
||
|
||
```
|
||
Planck length: ℓ_P = √(ℏG/c³) ≈ 1.616255×10⁻³⁵ m
|
||
Planck time: t_P = √(ℏG/c⁵) ≈ 5.391247×10⁻⁴⁴ s
|
||
Planck mass: m_P = √(ℏc/G) ≈ 2.176434×10⁻⁸ kg (≈ 1.22×10¹⁹ GeV)
|
||
Planck energy: E_P = √(ℏc⁵/G) ≈ 1.9561×10⁹ J (≈ 1.22×10¹⁹ GeV)
|
||
Planck temperature: T_P = √(ℏc⁵/(G k_B²)) ≈ 1.416784×10³² K
|
||
```
|
||
|
||
### 8.10 Experimental Verification
|
||
|
||
| Test | Experiment | Precision | Status |
|
||
|------|-----------|-----------|--------|
|
||
| Blackbody spectrum | Any thermal radiation, CMB | 10⁻⁵ | Confirmed |
|
||
| Photoelectric effect | Millikan 1916, photoemission spectroscopy | Percent | Confirmed |
|
||
| Compton scattering | Compton 1923, γ-ray astronomy | <1% | Confirmed |
|
||
| de Broglie wavelength | Davisson-Germer, electron microscopy | Confirmed |
|
||
| Photon statistics | Hanbury Brown-Twiss, single-photon sources | Confirmed |
|
||
| Casimir effect (zero-point) | Lamoreaux 1997, MEMS experiments | ~1% | Confirmed |
|
||
| Photon recoil | Laser cooling — sub-μK temperatures | Confirmed |
|
||
| CMB blackbody | COBE/FIRAS (1990) | 50 ppm | Confirmed |
|
||
| Wave-particle duality | Double-slit with electrons, atoms, molecules | Confirmed |
|
||
|
||
**Domain:** All quantum systems. The fundamental granularity of energy and action. Underpins quantum mechanics, QED, and quantum optics.
|
||
|
||
---
|
||
|
||
## 9. The Standard Model Lagrangian
|
||
|
||
### 9.1 Complete Lagrangian (Before Symmetry Breaking)
|
||
|
||
```
|
||
ℒ_SM = ℒ_gauge + ℒ_fermion + ℒ_Higgs + ℒ_Yukawa + ℒ_gauge-fix + ℒ_ghost
|
||
|
||
Gauge group: SU(3)_c × SU(2)_L × U(1)_Y
|
||
```
|
||
|
||
### 9.2 Gauge Sector
|
||
|
||
```
|
||
ℒ_gauge = −¼ G_a^{μν} G^a_{μν} − ¼ W_i^{μν} W^i_{μν} − ¼ B^{μν} B_{μν}
|
||
|
||
G_a^{μν} = ∂^μ G_a^ν − ∂^ν G_a^μ + g_s f_{abc} G_b^μ G_c^ν SU(3) — 8 gluons
|
||
W_i^{μν} = ∂^μ W_i^ν − ∂^ν W_i^μ + g ε_{ijk} W_j^μ W_k^ν SU(2) — 3 W bosons
|
||
B^{μν} = ∂^μ B^ν − ∂^ν B^μ U(1) — B boson
|
||
|
||
g_s → strong coupling (α_s = g_s²/4π)
|
||
g → weak isospin coupling
|
||
g' → weak hypercharge coupling
|
||
```
|
||
|
||
### 9.3 Fermion Sector
|
||
|
||
```
|
||
ℒ_fermion = i Σ_f ψ̄_f D̸ ψ_f
|
||
|
||
Covariant derivative: D_μ = ∂_μ − i g_s G_μ^a T^a − i g W_μ^i τ^i/2 − i g' Y B_μ
|
||
|
||
T^a → SU(3) generators (λ^a/2 for triplets, 0 for singlets)
|
||
τ^i/2 → SU(2) generators (Pauli matrices/2 for doublets, 0 for singlets)
|
||
Y → hypercharge quantum number
|
||
```
|
||
|
||
**Fermion content (3 generations):**
|
||
|
||
```
|
||
SU(3)_c SU(2)_L U(1)_Y Q = T_3 + Y
|
||
──────── ─────── ───── ────────────
|
||
Q_Lᵢ: 3 2 +1/6 +2/3, −1/3 left-handed quark doublet (u_L, d_L)
|
||
u_Rᵢ: 3 1 +2/3 +2/3 right-handed up-type
|
||
d_Rᵢ: 3 1 −1/3 −1/3 right-handed down-type
|
||
L_Lᵢ: 1 2 −1/2 0, −1 left-handed lepton doublet (ν_L, e_L)
|
||
e_Rᵢ: 1 1 −1 −1 right-handed charged lepton
|
||
ν_Rᵢ: 1 1 0 0 right-handed neutrino (optional)
|
||
```
|
||
|
||
### 9.4 Higgs Sector (Electroweak Symmetry Breaking)
|
||
|
||
```
|
||
ℒ_Higgs = |D_μ Φ|² − V(Φ) D_μ = ∂_μ − i g W_μ^i τ^i/2 − i g' Y B_μ
|
||
|
||
Φ = [ φ⁺ ] Y_Φ = +1/2
|
||
[ φ⁰ ]
|
||
|
||
V(Φ) = −μ² |Φ|² + λ |Φ|⁴ μ² > 0, λ > 0
|
||
|
||
Minimum (vacuum expectation value):
|
||
⟨Φ⟩ = [ 0 ] |⟨Φ⟩|² = v²/2 = μ²/(2λ)
|
||
[ v/√2 ]
|
||
|
||
v ≈ 246.21971 GeV from Fermi constant G_F measured in muon decay.
|
||
G_F / (√2) = 1/(2 v²)
|
||
```
|
||
|
||
### 9.5 Mass Generation After Symmetry Breaking
|
||
|
||
**SU(2)_L × U(1)_Y → U(1)_EM**
|
||
|
||
```
|
||
Massive gauge bosons:
|
||
W^± = (W¹ ∓ i W²) / √2 M_W = g v / 2 ≈ 80.377 ± 0.012 GeV
|
||
Z⁰ = (g W³ − g' B) / √(g² + g'²) M_Z = v √(g²+g'²) / 2 ≈ 91.1876 ± 0.0021 GeV
|
||
|
||
Massless gauge boson:
|
||
A = (g' W³ + g B) / √(g² + g'²) M_γ = 0 (photon, unbroken U(1)_EM)
|
||
|
||
Weak mixing angle (Weinberg angle):
|
||
tan θ_W = g' / g
|
||
sin² θ_W = 0.23121 ± 0.00004 (on-shell scheme)
|
||
≈ 0.23141 (MS-bar, m_Z scale)
|
||
|
||
M_W = M_Z cos θ_W ρ = M_W²/(M_Z² cos² θ_W) = 1 at tree level
|
||
```
|
||
|
||
**Fermion masses (Yukawa couplings):**
|
||
```
|
||
ℒ_Yukawa = −Y_u^{ij} Q̄_Lⁱ Φ̃ u_Rʲ − Y_d^{ij} Q̄_Lⁱ Φ d_Rʲ − Y_e^{ij} L̄_Lⁱ Φ e_Rʲ + h.c.
|
||
|
||
Φ̃ = i τ² Φ* = [ φ⁰* ] transforms as Φ with Y = −1/2
|
||
[ −φ⁻ ]
|
||
|
||
After EWSB: m_f = Y_f · v / √2
|
||
|
||
CKM mixing (Cabibbo-Kobayashi-Maskawa):
|
||
The Yukawa matrices are not diagonal in the gauge basis → quark mass eigenstates mix.
|
||
CKM matrix V_{CKM} (3×3 unitary, 4 parameters):
|
||
|V_ud| = 0.97435 |V_us| = 0.22500 |V_ub| = 0.00369
|
||
|V_cd| = 0.22486 |V_cs| = 0.97349 |V_cb| = 0.04182
|
||
|V_td| = 0.00857 |V_ts| = 0.04110 |V_tb| = 0.999118
|
||
```
|
||
|
||
**PMNS mixing (neutrinos):** If neutrinos have Dirac mass, analogous 3×3 matrix with mixing angles θ₁₂ ≈ 33°, θ₂₃ ≈ 45°, θ₁₃ ≈ 8.5°.
|
||
|
||
**Higgs boson mass:**
|
||
```
|
||
M_H² = 2 λ v²
|
||
|
||
m_H = 125.25 ± 0.17 GeV (CMS+ATLAS combined)
|
||
λ ≈ 0.129 Higgs self-coupling
|
||
```
|
||
|
||
### 9.6 Faddeev-Popov Gauge Fixing and Ghosts
|
||
|
||
```
|
||
ℒ_gauge-fix = −(1/2ξ_G) (∂^μ G_μ^a)² − (1/2ξ_W) (∂^μ W_μ^i)² − (1/2ξ_B) (∂^μ B_μ)²
|
||
|
||
ξ_i → gauge parameters (ξ→0: Landau gauge, ξ→1: Feynman gauge, ξ→∞: unitary gauge)
|
||
|
||
ℒ_ghost = Σ_{G,W} [c̄^a ∂^μ D_μ^{ab} c^b]
|
||
|
||
Ghost fields c^a are anticommuting scalars (Fermi statistics, Bose kinematics).
|
||
Required for perturbative unitarity in non-abelian gauge theories.
|
||
```
|
||
|
||
### 9.7 Accidental Symmetries
|
||
|
||
The SM Lagrangian has global symmetries that are NOT imposed but follow from the gauge structure and renormalizability:
|
||
|
||
```
|
||
Baryon number (B): conserved at classical level.
|
||
Violated by non-perturbative effects (sphalerons) — ΔB = ΔL = 3 at T ≫ 100 GeV.
|
||
|
||
Lepton number (L): separately L_e, L_μ, L_τ conserved (no neutrino oscillations in minimal SM).
|
||
Violated by neutrino masses → charged lepton flavor violation possible but unobserved.
|
||
```
|
||
|
||
### 9.8 Renormalizability ('t Hooft & Veltman, 1971–72)
|
||
|
||
The SM with spontaneous symmetry breaking is renormalizable. All divergences can be absorbed into a finite set of counterterms:
|
||
|
||
```
|
||
Counterterm Lagrangian:
|
||
δℒ = δZ_gauge (kinetic terms) + δZ_fermion (kinetic terms) + δm (mass)
|
||
+ δλ (couplings) + δv (VEV)
|
||
|
||
Renormalization group equations (RGEs) determine running of all couplings.
|
||
```
|
||
|
||
### 9.9 Key Precision Tests
|
||
|
||
**Muon anomalous magnetic moment (g−2)_μ:**
|
||
```
|
||
a_μ^{EXP} = 116 592 061(41) × 10⁻¹¹ (Fermilab + BNL)
|
||
a_μ^{SM} = 116 591 810(43) × 10⁻¹¹ (2020 White Paper)
|
||
|
||
Tension: 251(59) × 10⁻¹¹ → ~4.2σ discrepancy. Possible new physics or underestimated hadronic corrections.
|
||
```
|
||
|
||
**Electroweak precision observables (LEP, SLC, Tevatron, LHC):**
|
||
```
|
||
M_W = 80.377 ± 0.012 GeV
|
||
M_Z = 91.1876 ± 0.0021 GeV
|
||
Γ_Z = 2.4952 ± 0.0023 GeV
|
||
σ_h⁰ = 41.480 ± 0.033 nb
|
||
R_l = 20.767 ± 0.025
|
||
A_FB^{0,b} = 0.0992 ± 0.0016
|
||
```
|
||
|
||
Global fit to all EWPO agrees with SM at <1σ across all observables.
|
||
|
||
**Higgs properties:**
|
||
```
|
||
σ(pp→H) = 1.02 ± 0.05 × SM (overall signal strength)
|
||
μ_γγ = 1.10 ± 0.08 × SM
|
||
μ_ZZ* = 1.01 ± 0.08 × SM
|
||
μ_WW* = 1.00 ± 0.08 × SM
|
||
μ_ττ = 0.91 ± 0.09 × SM
|
||
μ_bb̄ = 1.04 ± 0.14 × SM
|
||
```
|
||
|
||
All Higgs couplings consistent with SM predictions. CP properties: pure CP-even (0⁺⁺) favored; CP-odd/mixed disfavored at >3σ.
|
||
|
||
**Domain:** All known fundamental particles and the electromagnetic, weak, and strong forces (except gravity). The most precisely tested physical theory in history.
|
||
|
||
---
|
||
|
||
## 10. Yang–Mills Gauge Theory (1954)
|
||
|
||
### 10.1 Field Strength and Covariant Derivative
|
||
|
||
```
|
||
F_μν^a = ∂_μ A_ν^a − ∂_ν A_μ^a + g f^{abc} A_μ^b A_ν^c
|
||
|
||
D_μ = ∂_μ − i g A_μ^a T^a gauge-covariant derivative
|
||
|
||
[f^{abc}] = structure constants of Lie algebra
|
||
[T^a, T^b] = i f^{abc} T^c Lie algebra
|
||
```
|
||
|
||
**Yang-Mills Lagrangian:**
|
||
```
|
||
ℒ_YM = −¼ F_μν^a F^{a μν} gauge-invariant kinetic term
|
||
|
||
Gauge transformation:
|
||
A_μ → U A_μ U⁻¹ + (i/g) U ∂_μ U⁻¹ U = exp(i g α^a(x) T^a)
|
||
F_μν → U F_μν U⁻¹ transforms covariantly (adjoint)
|
||
```
|
||
|
||
### 10.2 SU(N) Structure Constants
|
||
|
||
```
|
||
For SU(2): f^{abc} = ε^{abc} Levi-Civita (1 generator)
|
||
T^a = τ^a/2 Pauli matrices
|
||
|
||
For SU(3): f^{abc}: 123 (1), 147 (1/2), 156 (−1/2), 246 (1/2),
|
||
257 (1/2), 345 (1/2), 367 (−1/2),
|
||
458 (√3/2), 678 (√3/2)
|
||
|
||
d^{abc}: 118 (1/√3), 146 (1/2), 157 (1/2), 228 (1/√3),
|
||
247 (−1/2), 256 (1/2), 338 (1/√3), 344 (1/2),
|
||
355 (1/2), 366 (−1/2), 377 (−1/2), 448 (−1/2√3),
|
||
558 (−1/2√3), 668 (−1/2√3), 778 (−1/2√3), 888 (−1/√3)
|
||
```
|
||
|
||
### 10.3 Self-Interactions
|
||
|
||
The `g f^{abc}` term in `F_μν^a` produces:
|
||
|
||
**Three-gluon vertex (momentum space):**
|
||
```
|
||
V_{μνρ}^{abc}(p,q,r) = −g f^{abc} [g_{μν}(p−q)_ρ + g_{νρ}(q−r)_μ + g_{ρμ}(r−p)_ν]
|
||
with p+q+r=0 (all momenta incoming)
|
||
```
|
||
|
||
**Four-gluon vertex:**
|
||
```
|
||
V_{μνρσ}^{abcd} = −i g² [ f^{abe} f^{cde} (g_{μρ}g_{νσ}−g_{μσ}g_{νρ})
|
||
+ f^{ace} f^{bde} (g_{μν}g_{ρσ}−g_{μσ}g_{νρ})
|
||
+ f^{ade} f^{bce} (g_{μν}g_{ρσ}−g_{μρ}g_{νσ}) ]
|
||
```
|
||
|
||
These are the source of asymptotic freedom (antiscreening) — unique to non-abelian theories.
|
||
|
||
### 10.4 Gauge Invariance of the YM Lagrangian
|
||
|
||
```
|
||
F_μν → U F_μν U⁻¹ → Tr(F_μν F^{μν}) = invariant
|
||
Tr(T^a T^b) = ½ δ^{ab} (normalization)
|
||
```
|
||
|
||
### 10.5 Classical Solutions — Instantons
|
||
|
||
Finite-action Euclidean solutions (Belavin, Polyakov, Schwartz, Tyupkin 1975):
|
||
|
||
```
|
||
A_μ(x) = (1/g) (x²/(x²+ρ²)) U⁻¹ ∂_μ U (BPST instanton, ρ=scale size)
|
||
|
||
Topological charge (winding number):
|
||
Q = (g²/32π²) ∫ d⁴x F_μν^a F̃^{a μν} ∈ ℤ
|
||
|
||
Action: S = 8π²|Q|/g²
|
||
|
||
Instanton transitions: ΔQ = ΔB = ΔL (in SM) — violates baryon number.
|
||
Strong CP problem from θ F\tilde{F} term in YM Lagrangian.
|
||
```
|
||
|
||
**Domain:** Non-abelian gauge invariance is the organizing principle behind QCD, the electroweak theory, and most beyond-SM proposals (GUTs, technicolor, etc.).
|
||
|
||
---
|
||
|
||
## 11. Noether's Theorem (1918)
|
||
|
||
### 11.1 Statement of the Theorem
|
||
|
||
```
|
||
Every continuous (differentiable) symmetry of the action S = ∫ L dt corresponds to a conserved current.
|
||
|
||
If δS = 0 under transformation φ → φ + ε Δφ (locally parametrized by ε^a(x)),
|
||
then there exist conserved currents J_a^μ satisfying:
|
||
|
||
∂_μ J_a^μ = 0 on-shell (when equations of motion are satisfied).
|
||
|
||
Conserved charge: Q_a = ∫ d³x J_a⁰
|
||
dQ_a/dt = 0
|
||
```
|
||
|
||
### 11.2 Derivation (Field Theory)
|
||
|
||
Consider an infinitesimal global symmetry transformation:
|
||
|
||
```
|
||
x^μ → x^μ + ε^a X_a^μ(x)
|
||
φ_i(x) → φ_i(x) + ε^a Ψ_{i,a}(x)
|
||
|
||
Noether current (first theorem):
|
||
J_a^μ = Σ_i [∂ℒ/∂(∂_μ φ_i)] (Ψ_{i,a} − ∂_ν φ_i X_a^ν) + ℒ X_a^μ
|
||
```
|
||
|
||
### 11.3 Symmetry-Conservation Dictionary
|
||
|
||
```
|
||
──────────────────────────────────────────────────────────
|
||
Symmetry Conserved Quantity Exact?
|
||
──────────────────────────────────────────────────────────
|
||
Time translation (t→t+ε) Energy (E) Yes
|
||
Spatial translation (x→x+ε) Momentum (p) Yes
|
||
Rotation (x→R·x) Angular momentum (L) Yes
|
||
U(1) gauge phase Electric charge (Q) Yes
|
||
SU(2) weak isospin Weak isospin current Broken (SSB)
|
||
SU(3) color Color charge Exact
|
||
SU(3)_L×SU(3)_R chiral (QCD) Axial/vector currents Approx. (SSB + anomaly)
|
||
Lorentz boost Center-of-mass motion Yes
|
||
Scale/dilatation Dilatation current Broken by anomaly (QCD)
|
||
Supersymmetry Supercurrent Broken (if realized)
|
||
Baryon number (accidental) Baryon number (B) Classical; violated nonpert.
|
||
Lepton number (accidental) Lepton number (L) Violated by ν mass
|
||
──────────────────────────────────────────────────────────
|
||
```
|
||
|
||
### 11.4 Noether's Second Theorem (Local/Gauge Symmetries)
|
||
|
||
For local gauge symmetries (ε^a(x) is an arbitrary function of x):
|
||
|
||
```
|
||
The second theorem gives identities (Bianchi identities in GR, Slavnov-Taylor in QFT)
|
||
relating the equations of motion — constraints on dynamics, not conserved charges.
|
||
```
|
||
|
||
This explains why gauge symmetries do not produce independent conserved charges in the same way.
|
||
|
||
### 11.5 Consequence: Why Conservation Laws Are Rock-Solid
|
||
|
||
Noether's theorem is a mathematical theorem given Lagrangian dynamics. It can only fail if:
|
||
1. The symmetry is NOT a symmetry of the Lagrangian.
|
||
2. The derivation of the Euler-Lagrange equations from the action fails.
|
||
3. The system is not Lagrangian (e.g., dissipative forces with no potential).
|
||
|
||
In all Lagrangian theories (all of fundamental physics), the symmetry-conservation link is absolute.
|
||
|
||
**Domain:** Every physical theory expressible in Lagrangian/Hamiltonian form — effectively all of fundamental physics. Not a testable hypothesis — a mathematical identity.
|
||
|
||
---
|
||
|
||
## 12. Friedmann Equations (Cosmology, 1922)
|
||
|
||
### 12.1 The Two Friedmann Equations
|
||
|
||
```
|
||
H² ≡ (ȧ/a)² = (8πG/3) ρ − kc²/a² + Λc²/3 First (expansion rate)
|
||
|
||
ä/a = −(4πG/3) (ρ + 3p/c²) + Λc²/3 Second (acceleration)
|
||
|
||
H = Hubble parameter
|
||
a(t) = scale factor
|
||
k = +1, 0, −1 (closed, flat, open)
|
||
```
|
||
|
||
**Fluid equation (conservation of stress-energy from Friedmann + 2nd):**
|
||
```
|
||
ρ̇ + 3H (ρ + p/c²) = 0
|
||
|
||
For matter (p=0): ρ_m ∝ a^{-3}
|
||
For radiation (p=ρc²/3): ρ_r ∝ a^{-4}
|
||
For dark energy (p=−ρc²): ρ_Λ = const.
|
||
```
|
||
|
||
### 12.2 Cosmological Parameters (ΛCDM — Planck 2018)
|
||
|
||
```
|
||
H₀ = 67.4 ± 0.5 km/s/Mpc Hubble constant
|
||
Ω_m = 0.315 ± 0.007 matter density parameter
|
||
Ω_Λ = 0.6847 ± 0.0073 dark energy density parameter
|
||
Ω_b = 0.0493 ± 0.0006 baryon density parameter
|
||
Ω_k = 0.001 ± 0.002 curvature (consistent with flat)
|
||
|
||
Ω_m + Ω_Λ + Ω_k = 1
|
||
|
||
Age of universe: t₀ = 13.797 ± 0.023 Gyr
|
||
```
|
||
|
||
**Redshift relation:**
|
||
```
|
||
a(t) = 1 / (1+z)
|
||
|
||
1 + z = λ_obs / λ_emit
|
||
```
|
||
|
||
### 12.3 Distance Measures
|
||
|
||
**Comoving distance:** `χ(z) = c ∫_0^z dz'/H(z')`
|
||
|
||
```
|
||
Luminosity distance: d_L = (1+z) χ
|
||
Angular diameter distance: d_A = χ / (1+z)
|
||
Distance modulus: μ = 5 log₁₀(d_L/10pc)
|
||
```
|
||
|
||
**BAO (Baryon Acoustic Oscillations):** Standard ruler at `r_d ≈ 147 Mpc` (comoving sound horizon at drag epoch). Measured in galaxy surveys (SDSS, DESI) — consistent with ΛCDM.
|
||
|
||
### 12.4 Thermal History
|
||
|
||
```
|
||
T(z) = T₀ (1+z) T₀ = 2.72548 ± 0.00057 K (CMB)
|
||
|
||
Key epochs:
|
||
z ~ 1100 (T~3000K): Recombination — CMB emitted, universe becomes neutral (~380,000 yr)
|
||
z ~ 3400 (T~0.9eV): Matter-radiation equality (~50,000 yr)
|
||
z ~ 10⁹ (T~1MeV): Big Bang Nucleosynthesis (~3 min → H, He, Li)
|
||
z ~ 10¹⁵ (T~100GeV): Electroweak phase transition (~10⁻¹¹ s)
|
||
```
|
||
|
||
### 12.5 BBN (Big Bang Nucleosynthesis)
|
||
|
||
Primordial abundances predicted:
|
||
|
||
```
|
||
Y_p = 0.24709 ± 0.00025 Helium-4 mass fraction
|
||
D/H = (2.527 ± 0.030) × 10⁻⁵ Deuterium
|
||
³He/H = (1.1 ± 0.2) × 10⁻⁵ Helium-3
|
||
⁷Li/H = (1.6 ± 0.3) × 10⁻¹⁰ Lithium
|
||
```
|
||
|
||
All except ⁷Li (2–3σ tension, possibly astrophysical or new physics) agree with ΛCDM+BBN predictions using η (baryon-to-photon ratio) from CMB.
|
||
|
||
### 12.6 The Cosmological Constant Problem
|
||
|
||
```
|
||
Observed: ρ_Λ ≈ (2.3 × 10⁻³ eV)⁴ ≈ 6 × 10⁻¹⁰ J/m³
|
||
QFT prediction (zero-point sum up to Planck scale): ρ_vac ~ (10¹⁸ GeV)⁴
|
||
|
||
ρ_obs / ρ_vac ~ 10⁻¹²⁰ worst prediction in physics
|
||
```
|
||
|
||
**Domain:** Homogeneous, isotropic cosmology on scales >~100 Mpc. FLRW metric. ΛCDM fits all cosmological datasets (CMB, BAO, SNe, LSS, cluster counts) at the ~1% level.
|
||
|
||
---
|
||
|
||
## 13. Klein–Gordon Equation (Relativistic spin-0, 1926)
|
||
|
||
### 13.1 Equation
|
||
|
||
```
|
||
(□ + m²c²/ℏ²) φ(x) = 0 where □ = ∂_μ ∂^μ = −(1/c²)∂²/∂t² + ∇²
|
||
|
||
Derived from relativistic energy-momentum: E² = p²c² + m²c⁴
|
||
Substituting E→iℏ∂/∂t, p→−iℏ∇ → (iℏ∂/∂t)² φ = [(−iℏ∇)²c² + m²c⁴] φ
|
||
```
|
||
|
||
### 13.2 Lagrangian and Conserved Current
|
||
|
||
```
|
||
ℒ_KG = ½ (∂_μ φ)(∂^μ φ) − ½ (m²c²/ℏ²) φ²
|
||
|
||
Noether current (U(1) symmetry φ→e^{iα}φ):
|
||
j^μ = i (φ* ∂^μ φ − φ ∂^μ φ*) for complex scalar field
|
||
∂_μ j^μ = 0 charge conservation
|
||
|
||
Energy-momentum tensor:
|
||
T^{μν} = (∂^μ φ)(∂^ν φ) − g^{μν} ℒ
|
||
```
|
||
|
||
### 13.3 Plane Wave Solutions
|
||
|
||
```
|
||
φ(x) = A e^{i(p·x − Et)/ℏ} with E = ±√(p²c² + m²c⁴)
|
||
|
||
Negative-energy solutions: reinterpreted as antiparticles in QFT.
|
||
```
|
||
|
||
### 13.4 Non-Relativistic Limit
|
||
|
||
```
|
||
φ = e^{−imc² t/ℏ} ψ factor out rest-energy oscillation
|
||
|
||
|∂²ψ/∂t²| ≪ mc²/ℏ |∂ψ/∂t| → KG → Schrödinger:
|
||
|
||
iℏ ∂ψ/∂t = −(ℏ²/2m) ∇² ψ
|
||
```
|
||
|
||
**Domain:** Relativistic scalar particles — pions, kaons, Higgs boson (before EWSB), axions (candidate), inflaton (candidate). Describes spin-0 particles. Used in QFT as field equation for spin-0 quantized fields. The Higgs field's dynamics before and after EWSB are governed by KG + self-interaction.
|
||
|
||
---
|
||
|
||
## 14. Heisenberg Uncertainty Principle (1927)
|
||
|
||
### 14.1 Standard Formulations
|
||
|
||
```
|
||
Δx · Δp ≥ ℏ/2 position-momentum
|
||
ΔE · Δt ≥ ℏ/2 energy-time (requires care — time is not an operator)
|
||
Δθ · ΔL ≥ ℏ/2 angle-angular momentum (cyclic variables)
|
||
ΔN · Δφ ≥ ½ photon number-phase
|
||
|
||
General Robertson-Schrödinger inequality:
|
||
ΔA · ΔB ≥ (1/2) |⟨[Â, B̂]⟩| for any two Hermitian operators
|
||
|
||
ΔA · ΔB ≥ (1/2) |⟨Â B̂ + B̂ Â⟩ − 2⟨Â⟩⟨B̂⟩|² (more robust — Schödinger)
|
||
```
|
||
|
||
### 14.2 Derivation (Cauchy-Schwarz)
|
||
|
||
```
|
||
Given Hermitian operators Â, B̂:
|
||
|⟨ψ| B̂|ψ⟩|² ≤ ⟨ψ|²|ψ⟩ ⟨ψ|B̂²|ψ⟩ (Cauchy-Schwarz)
|
||
|
||
Let Â' = Â − ⟨Â⟩, B̂' = B̂ − ⟨B̂⟩ → ΔA ΔB ≥ ½|⟨[Â,B̂]⟩|
|
||
```
|
||
|
||
### 14.3 Energy-Time "Uncertainty"
|
||
|
||
The energy-time relation is different — `t` is not a Hermitian operator in standard QM (Pauli's theorem). Several precise formulations:
|
||
|
||
**Mandelstam-Tamm (1945):**
|
||
```
|
||
ΔE · τ ≥ ℏ/2
|
||
|
||
τ = ΔA / |d⟨Â⟩/dt| lifetime of an observable A
|
||
ΔE = energy uncertainty of the state
|
||
```
|
||
|
||
**Decaying state:**
|
||
```
|
||
dP/dt = −Γ P exponential decay
|
||
P(t) = e^{−Γt} = e^{−t/τ}
|
||
|
||
Γ = ℏ/τ = energy width of unstable state
|
||
ΔE · τ ≈ ℏ
|
||
```
|
||
|
||
### 14.4 Physical Consequences
|
||
|
||
- **Zero-point energy:** Harmonic oscillator ground state has `E₀ = ½ ℏω` because `Δx Δp ≥ ℏ/2` prevents `x=0, p=0` simultaneously.
|
||
- **Quantum tunneling:** Uncertainty in energy allows short-lived borrowing → tunneling through barriers.
|
||
- **Linewidths:** `Γ = ℏ/τ` — short-lived states (hadronic resonances, τ~10⁻²³ s) have GeV-scale widths.
|
||
- **Limit on measurement precision:** Any measurement that determines one observable more precisely increases uncertainty in its conjugate.
|
||
|
||
### 14.5 Experimental Tests
|
||
|
||
| Test | Result |
|
||
|------|--------|
|
||
| Neutron interferometry | Δx Δp confirmed |
|
||
| Spontaneous emission linewidth | Γ = ℏ/τ confirmed |
|
||
| Squeezed states in quantum optics | Δx₁ < ℏ/2Δp₁ while Δx₂ > ℏ/2Δp₂ — below SQL |
|
||
| Weak measurement + postselection | Apparent violation is consistent with UP when measurement disturbance accounted for |
|
||
|
||
**Domain:** All quantum systems. A kinematical theorem following from operator non-commutation. Not a limitation of measurement technology — a fundamental property of quantum states.
|
||
|
||
---
|
||
|
||
## 15. Pauli Exclusion Principle + Spin-Statistics Theorem
|
||
|
||
### 15.1 Statement
|
||
|
||
```
|
||
Fermions (half-integer spin): total wavefunction antisymmetric under exchange
|
||
ψ(x₁, ..., x_i, ..., x_j, ..., x_N) = −ψ(x₁, ..., x_j, ..., x_i, ..., x_N)
|
||
|
||
Bosons (integer spin): total wavefunction symmetric under exchange
|
||
ψ(x₁, ..., x_i, ..., x_j, ..., x_N) = +ψ(x₁, ..., x_j, ..., x_i, ..., x_N)
|
||
|
||
Consequence for fermions (Pauli principle):
|
||
No two identical fermions can occupy the same quantum state simultaneously.
|
||
```
|
||
|
||
### 15.2 Spin-Statistics Theorem (Fierz 1939, Pauli 1940)
|
||
|
||
In **relativistic QFT**, the spin-statistics connection is a **theorem**, not an assumption:
|
||
|
||
```
|
||
Microcausality + Lorentz invariance + positive-definite energy + locality
|
||
⇒ half-integer spin → Fermi-Dirac statistics (anticommutators for field operators)
|
||
⇒ integer spin → Bose-Einstein statistics (commutators for field operators)
|
||
```
|
||
|
||
Proof relies on `(−1)^{2s}` factor from Lorentz transformation of fields. Violation of spin-statistics would violate causality.
|
||
|
||
### 15.3 Occupation Number Formalism
|
||
|
||
**Fermions (Fermi-Dirac statistics):**
|
||
```
|
||
n_i ∈ {0, 1} occupancy per single-particle state
|
||
|
||
⟨n_i⟩ = 1 / [e^{(E_i−μ)/k_B T} + 1] Fermi-Dirac distribution
|
||
```
|
||
|
||
**Bosons (Bose-Einstein statistics):**
|
||
```
|
||
n_i ∈ {0, 1, 2, ...} any integer occupancy
|
||
|
||
⟨n_i⟩ = 1 / [e^{(E_i−μ)/k_B T} − 1] Bose-Einstein distribution
|
||
```
|
||
|
||
### 15.4 Physical Consequences of the Pauli Principle
|
||
|
||
1. **Periodic table** — electron shells fill progressively; chemical properties from outermost shell.
|
||
2. **Stability of matter** (Dyson-Lenard theorem): fermionic electrons prevent collapse — without Pauli, all electrons would fall to 1s and matter would be ~10⁵ times smaller.
|
||
3. **Neutron star stability** — neutron degeneracy pressure supports stars against gravitational collapse up to ~2–3 M_⊙ (Tolman-Oppenheimer-Volkoff limit). Above this → black hole.
|
||
4. **White dwarf stability** — electron degeneracy pressure supports up to ~1.4 M_⊙ (Chandrasekhar limit).
|
||
5. **Fermi energy:** `E_F = (ℏ²/2m)(3π²n)^{2/3}` — conduction electrons occupy states up to E_F (several eV in metals).
|
||
6. **Nucleon shell model** — nuclear magic numbers from spin-orbit coupled shell filling.
|
||
|
||
### 15.5 Experimental Constraints on Pauli Violation
|
||
|
||
```
|
||
"VIP" experiment (Gran Sasso): searched for Pauli-forbidden X-ray transitions
|
||
Limit: probability of Pauli violation < 4.5×10⁻²⁹
|
||
|
||
Borexino: search for Pauli-forbidden nuclear transitions in ¹²C
|
||
β²/2 < 2.6×10⁻³⁷ (Pauli violation parameter)
|
||
```
|
||
|
||
**No violation ever detected.** The Pauli principle is one of the most stringently tested laws in physics.
|
||
|
||
**Domain:** All quantum identical particles. A theorem in relativistic QFT; experimentally unfalsified to extreme precision.
|
||
|
||
---
|
||
|
||
## 16. Feynman Path Integral (1948)
|
||
|
||
### 16.1 The Fundamental Formula
|
||
|
||
```
|
||
⟨x_f, t_f | x_i, t_i⟩ = ∫ D[x(t)] exp( i S[x] / ℏ )
|
||
|
||
S[x] = ∫_{t_i}^{t_f} dt L(x, ẋ, t) classical action
|
||
|
||
Path measure D[x(t)]:
|
||
∫ D[x(t)] ≡ lim_{N→∞} Π_{k=1}^{N-1} ∫ dx_k (m / 2πiℏΔt)^{N/2}
|
||
Δt = (t_f−t_i)/N
|
||
```
|
||
|
||
### 16.2 Equivalence to Schrödinger Equation
|
||
|
||
The path integral propagator:
|
||
```
|
||
K(x_f, t_f; x_i, t_i) ≡ ⟨x_f|e^{−iĤ(t_f−t_i)/ℏ}|x_i⟩
|
||
|
||
ψ(x_f, t_f) = ∫ dx_i K(x_f, t_f; x_i, t_i) ψ(x_i, t_i)
|
||
```
|
||
|
||
Infinitesimal time evolution → Schrödinger equation.
|
||
|
||
### 16.3 Classical Limit ℏ → 0
|
||
|
||
Stationary phase approximation:
|
||
```
|
||
δS = 0 → classical trajectory dominates path integral
|
||
|
||
Semiclassical expansion:
|
||
K ∼ Σ_{classical paths} A e^{iS_cl/ℏ}
|
||
|
||
A = √(det ∂²S/∂x_i ∂x_f) Van Vleck determinant
|
||
```
|
||
|
||
### 16.4 Euclidean (Imaginary Time) Path Integral
|
||
|
||
```
|
||
t → τ = i t Wick rotation
|
||
|
||
⟨x_f, τ_f | x_i, τ_i⟩ = ∫ D[x(τ)] exp( −S_E[x] / ℏ )
|
||
|
||
S_E[x] = ∫_{τ_i}^{τ_f} dτ [ (m/2)(dx/dτ)² + V(x) ]
|
||
|
||
Path integral becomes well-defined (Gaussian convergence) → statistical mechanics analogy.
|
||
```
|
||
|
||
### 16.5 QFT Path Integral
|
||
|
||
```
|
||
Z[J] = ∫ D[φ] exp( i ∫ d⁴x [ℒ(φ) + J φ] )
|
||
|
||
Generating functional for correlation functions:
|
||
⟨0|T{φ(x₁)...φ(x_n)}|0⟩ = (1/i^n) δ^n Z[J] / δJ(x₁)...δJ(x_n) |_{J=0}
|
||
|
||
Feynman diagrams emerge from perturbative expansion of exp(i∫ℒ_int).
|
||
```
|
||
|
||
### 16.6 Gaussian Integrals (Free Field)
|
||
|
||
```
|
||
∫ D[φ] exp( −½ ∫ d⁴x φ(x) K(x,y) φ(y) ) ∝ (det K)^{-1/2}
|
||
|
||
Propagator: ⟨φ(x) φ(y)⟩ = K^{-1}(x,y) = ∫ d⁴p e^{ip·(x−y)} / (p² − m² + iε)
|
||
```
|
||
|
||
**Domain:** Equivalent formulation of quantum mechanics and quantum field theory. Yields identical predictions to operator formalism. Foundation of lattice QFT, instanton calculus, and semiclassical methods.
|
||
|
||
---
|
||
|
||
## 17. Navier–Stokes Equations (Fluid Dynamics, 1822–1845)
|
||
|
||
### 17.1 The Equations
|
||
|
||
**Compressible, Newtonian fluid:**
|
||
|
||
```
|
||
ρ (∂v/∂t + v·∇v) = −∇p + μ ∇²v + (μ_v + μ/3) ∇(∇·v) + ρ g + f_ext
|
||
|
||
∂ρ/∂t + ∇ · (ρ v) = 0 continuity (mass conservation)
|
||
|
||
ρ = density
|
||
v = velocity field
|
||
p = pressure
|
||
μ = dynamic (shear) viscosity
|
||
μ_v = bulk (dilatational) viscosity
|
||
```
|
||
|
||
### 17.2 Incompressible Navier-Stokes (ρ = const)
|
||
|
||
```
|
||
∂v/∂t + (v·∇) v = −(1/ρ) ∇p + ν ∇²v + g + f_ext/ρ
|
||
|
||
∇ · v = 0
|
||
|
||
ν ≡ μ/ρ = kinematic viscosity
|
||
```
|
||
|
||
### 17.3 Dimensionless Form: Reynolds Number
|
||
|
||
Non-dimensionalize: `v* = v/U`, `p* = p/(ρU²)`, `t* = t U/L`, `x* = x/L`:
|
||
|
||
```
|
||
∂v*/∂t* + (v*·∇*) v* = −∇* p* + (1/Re) ∇*² v*
|
||
|
||
Re ≡ U L / ν Reynolds number
|
||
|
||
Re ≪ 1: laminar flow (viscosity dominates) — Stokes flow
|
||
Re ~ 10³–10⁵: transition to turbulence
|
||
Re ≫ 1: turbulent flow (inertia dominates)
|
||
```
|
||
|
||
**Physical examples:**
|
||
| Flow | Re | Regime |
|
||
|------|-----|--------|
|
||
| Swimming bacterium | 10⁻⁵ | Stokes |
|
||
| Blood in capillary | 10⁻³ | Stokes |
|
||
| Swimming fish | 10⁵ | Turbulent |
|
||
| Airplane wing | 10⁷ | Turbulent |
|
||
| Atmospheric weather | 10¹¹ | Fully turbulent |
|
||
|
||
### 17.4 Exact Solutions
|
||
|
||
**Poiseuille flow (pressure-driven pipe flow):**
|
||
```
|
||
v_z(r) = (G / 4μ) (R² − r²) G = −dp/dz
|
||
Q = π G R⁴ / 8μ volumetric flow rate
|
||
```
|
||
Hagen-Poiseuille law. Confirmed to incredible precision — used in viscometry.
|
||
|
||
**Couette flow (shear between moving plates):**
|
||
```
|
||
v_x(y) = U y / h shear rate γ̇ = U/h
|
||
τ = μ γ̇ shear stress
|
||
```
|
||
|
||
**Stokes flow (creeping flow past a sphere):**
|
||
```
|
||
F_drag = 6π μ R U Stokes drag law
|
||
C_D = 24 / Re drag coefficient for Re ≪ 1
|
||
```
|
||
|
||
### 17.5 Vorticity Formulation
|
||
|
||
```
|
||
ω ≡ ∇ × v vorticity vector
|
||
|
||
∂ω/∂t + v·∇ ω = ω·∇ v + ν ∇²ω vorticity transport
|
||
|
||
For incompressible 2D flow: ω·∇v = 0 → purely advection-diffusion.
|
||
```
|
||
|
||
**Helicity:** `H = ∫ v·ω d³x` — conserved in ideal fluid (ν→0).
|
||
|
||
### 17.6 Turbulence and the Kolmogorov Theory (1941)
|
||
|
||
**Energy cascade:** Energy injected at large scale L → cascades through inertial range → dissipated at Kolmogorov scale η.
|
||
|
||
```
|
||
Kolmogorov length scale: η = (ν³/ε)^{1/4}
|
||
Kolmogorov time scale: τ_η = (ν/ε)^{1/2}
|
||
Kolmogorov velocity: v_η = (ν ε)^{1/4}
|
||
|
||
ε = energy dissipation rate per unit mass
|
||
|
||
Re = (L/η)^{4/3}
|
||
```
|
||
|
||
**Kolmogorov energy spectrum (inertial range):**
|
||
```
|
||
E(k) = C_K ε^{2/3} k^{-5/3} C_K ≈ 1.5 (Kolmogorov constant)
|
||
|
||
Valid for: 1/L ≪ k ≪ 1/η
|
||
```
|
||
|
||
**Structure functions:**
|
||
```
|
||
⟨|v(x+r) − v(x)|^p⟩ ∝ r^{ζ_p} ζ_p = p/3 (K41)
|
||
= p/3 − τ_p/3 (intermittency corrections)
|
||
```
|
||
|
||
Observed in wind tunnels, oceanographic data, atmospheric measurements, and pipe flow over ~5 decades.
|
||
|
||
### 17.7 Bernoulli's Equation (Inviscid, Steady, Incompressible Along Streamline)
|
||
|
||
```
|
||
p + ½ ρ v² + ρ g z = constant along streamline
|
||
|
||
Inviscid, incompressible, steady flow.
|
||
Generalized: ½ v² + ∫ dp/ρ + Φ = constant (compressible, Φ = body force potential).
|
||
```
|
||
|
||
### 17.8 Continuum Hypothesis Validity
|
||
|
||
Knudsen number: `Kn = λ/L` where `λ` = mean free path, `L` = characteristic length.
|
||
```
|
||
Kn < 0.01: continuum (Navier-Stokes valid)
|
||
0.01 < Kn < 0.1: slip-flow regime
|
||
0.1 < Kn < 10: transition regime
|
||
Kn > 10: free molecular flow (Boltzmann/BGK needed)
|
||
```
|
||
|
||
Atmospheric mean free path at sea level: λ ≈ 68 nm.
|
||
|
||
### 17.9 The Millennium Prize Problem
|
||
|
||
Existence and smoothness of solutions to the 3D incompressible Navier-Stokes equations remain unproven. Despite this, the equations are used to ~10 decimal precision in engineering every day — a deep mathematical mystery.
|
||
|
||
**Domain:** Newtonian fluids (water, air at subsonic speeds, oils, most common liquids and gases). Underpins aerodynamics, hydrodynamics, meteorology, oceanography, hemodynamics, and industrial fluid processing.
|
||
|
||
---
|
||
|
||
## 18. Black Hole Thermodynamics (Bekenstein–Hawking, 1972–1974)
|
||
|
||
### 18.1 The Four Laws of Black Hole Mechanics (Bardeen-Carter-Hawking 1973)
|
||
|
||
```
|
||
0th Law: Surface gravity κ is constant over the event horizon of a stationary black hole.
|
||
1st Law: dM = (κ/8πG) dA + Ω_H dJ + Φ_H dQ
|
||
2nd Law: dA/dt ≥ 0 (Hawking area theorem, 1971)
|
||
3rd Law: κ cannot be reduced to zero by any finite process.
|
||
```
|
||
|
||
**Mapping to thermodynamics:**
|
||
```
|
||
E ↔ M c² energy ↔ mass
|
||
T ↔ κc²ℏ/(2πk_B) Hawking temperature
|
||
S ↔ k_B c³ A / (4Gℏ) Bekenstein-Hawking entropy
|
||
```
|
||
|
||
### 18.2 Bekenstein-Hawking Entropy and Hawking Temperature
|
||
|
||
```
|
||
S_BH = k_B A / 4ℓ_P² = k_B c³ A / (4Gℏ)
|
||
|
||
T_H = ℏc³ / (8πGMk_B) Schwarzschild BH
|
||
T_H = ℏc κ / (2πk_B) general stationary BH
|
||
|
||
A = 4π r_s² = 16π G² M² / c⁴ Schwarzschild horizon area
|
||
|
||
T_H(Schwarzschild) = 6.2×10⁻⁸ K × (M⊙/M) negligible for stellar BHs
|
||
|
||
A = 8π G²/c⁴ [M² + M√(M²−a²−Q²)] Kerr-Newman horizon area
|
||
```
|
||
|
||
### 18.3 Hawking Radiation (1974)
|
||
|
||
**Particle creation in curved spacetime:**
|
||
```
|
||
⟨N_{ωlm}⟩ = Γ_{ωlm} / [exp(2πω/κ) ∓ 1] Planckian spectrum
|
||
|
||
Γ_{ωlm} = greybody factor (absorption probability)
|
||
|
||
Lifetime for Schwarzschild BH:
|
||
τ_evap ∼ M³ / (3 α ℏ c⁴/G²) ≈ 10⁶⁷ yr × (M/M⊙)³
|
||
|
||
τ_evap ≈ 10⁻¹⁷ s for M = 10¹⁵ g (primordial BH, if they exist).
|
||
```
|
||
|
||
**Information paradox:** Hawking radiation appears thermal → loss of quantum information. Resolution debated: complementarity, firewalls, fuzzballs, ER=EPR, island formula, holography.
|
||
|
||
### 18.4 Generalized Second Law (GSL)
|
||
|
||
```
|
||
d/dt (S_BH + S_matter) ≥ 0
|
||
|
||
S_BH dominates for macroscopic black holes:
|
||
S_BH (M⊙ BH) ≈ 10⁷⁷ k_B vs S_CMB (observable universe) ≈ 10⁸⁹ k_B
|
||
```
|
||
|
||
GSL has passed all tests accessible with current technology (thought experiments, gravitational wave ringdown tests of area theorem at ~97% confidence for GW150914).
|
||
|
||
**Domain:** Semiclassical gravity on black hole horizons. Hawking temperature is too small for direct astrophysical detection. LIGO/Virgo ringdown constrains area increase. Analog gravity (sonic BHs in BECs, water waves) observes analogue Hawking radiation.
|
||
|
||
---
|
||
|
||
## 19. Weinberg–Salam Electroweak Unification (1967–1968)
|
||
|
||
### 19.1 The Gauge Structure
|
||
|
||
```
|
||
Gauge group: SU(2)_L × U(1)_Y
|
||
|
||
Spontaneous symmetry breaking: SU(2)_L × U(1)_Y → U(1)_EM
|
||
|
||
Gauge bosons before SSB:
|
||
W_μ^i (i=1,2,3): SU(2)_L gauge fields, coupling g
|
||
B_μ: U(1)_Y gauge field, coupling g'
|
||
|
||
Higgs field: Φ = [φ⁺, φ⁰]^T, Y=+1/2, SU(2) doublet
|
||
```
|
||
|
||
### 19.2 Covariant Derivative and Mass Generation
|
||
|
||
```
|
||
D_μ Φ = (∂_μ − i g W_μ^i τ^i/2 − i g' Y B_μ) Φ
|
||
|
||
After Φ acquires VEV ⟨Φ⟩ = (0, v/√2)^T:
|
||
|
||
|D_μ Φ|² → mass terms for W^±, Z⁰:
|
||
|
||
M_W = g v / 2 = 80.379 ± 0.012 GeV (Particle Data Group 2022)
|
||
M_Z = (v/2)√(g²+g'²) = 91.1876 ± 0.0021 GeV
|
||
|
||
Photon remains massless:
|
||
A_μ = (g' W_μ³ + g B_μ) / √(g²+g'²) M_γ = 0
|
||
|
||
Weak mixing angle:
|
||
cos θ_W = M_W / M_Z → sin² θ_W = 1 − M_W²/M_Z²
|
||
sin² θ_W = 0.23121 ± 0.00004 (on-shell scheme)
|
||
≈ 0.23141 (MS-bar at m_Z)
|
||
```
|
||
|
||
### 19.3 Weak Currents
|
||
|
||
**Charged current (W^±):**
|
||
```
|
||
J_CC^{+μ} = Σ_{gen} (ν̄_L γ^μ e_L + ū_L γ^μ d_L)
|
||
|
||
ℒ_CC = (g / 2√2) J_CC^{+μ} W_μ^+ + h.c.
|
||
|
||
Fermi constant (from muon decay):
|
||
G_F / √2 = g² / (8 M_W²) → G_F = 1.1663787 × 10⁻⁵ GeV⁻² (CODATA 2018)
|
||
|
||
v = 1 / √(√2 G_F) = 246.21971 GeV
|
||
```
|
||
|
||
**Neutral current (Z⁰):**
|
||
```
|
||
J_NC^μ = ψ̄ γ^μ (T³ − sin² θ_W Q) ψ
|
||
|
||
Vector coupling: g_V = T³ − 2 Q sin² θ_W
|
||
Axial coupling: g_A = T³
|
||
|
||
ℒ_NC = (g / 2 cos θ_W) J_NC^μ Z_μ
|
||
```
|
||
|
||
**Electromagnetic current:**
|
||
```
|
||
J_EM^μ = Q ψ̄ γ^μ ψ Q = T³ + Y (electric charge)
|
||
|
||
e = g sin θ_W = g' cos θ_W
|
||
α = e² / 4π ≈ 1/137.035999084
|
||
```
|
||
|
||
### 19.4 Key Predictions and Discoveries
|
||
|
||
```
|
||
1973: Neutral currents discovered at Gargamelle (CERN) — first confirmation of electroweak model.
|
||
|
||
1983: W⁺, W⁻, Z⁰ discovered at UA1/UA2 (CERN Spp̄S):
|
||
W bosons in p̄p → ℓ ±ν
|
||
Z boson in p̄p → ℓ⁺ ℓ⁻
|
||
Direct Nobel Prize to Rubbia & van der Meer (1984).
|
||
|
||
M_W prediction (before discovery): 80–83 GeV
|
||
M_W measured: 80.379 GeV
|
||
M_Z prediction (using sin² θ_W): ~90 GeV
|
||
M_Z measured: 91.1876 GeV
|
||
|
||
Number of light neutrino species from Z line shape at LEP:
|
||
N_ν = 2.9840 ± 0.0082 consistent with exactly 3.
|
||
```
|
||
|
||
### 19.5 Electroweak Precision Tests
|
||
|
||
LEP/SLD/Tevatron/LHC global fit (PDG 2022):
|
||
|
||
```
|
||
Observable Measured SM Prediction Pull (σ)
|
||
──────────────── ──────── ──────────── ────────
|
||
M_W (GeV) 80.379 ± 0.012 80.358 ± 0.006 +0.3
|
||
Γ_W (GeV) 2.085 ± 0.042 2.091 ± 0.001 −0.1
|
||
M_Z (GeV) 91.1876 ± 0.0021 91.1875 ± 0.0021 0.0
|
||
Γ_Z (GeV) 2.4952 ± 0.0023 2.4947 ± 0.0009 +0.2
|
||
σ_had⁰ (nb) 41.480 ± 0.033 41.478 ± 0.008 0.0
|
||
R_l 20.767 ± 0.025 20.744 ± 0.018 +0.8
|
||
A_FB^l 0.0171 ± 0.0010 0.01627 ± 0.00018 +0.9
|
||
A_l (SLD) 0.1513 ± 0.0021 0.1475 ± 0.0008 +1.8
|
||
sin² θ_W^eff 0.23153 ± 0.00016 0.23149 ± 0.00013 +0.2
|
||
```
|
||
|
||
Overall χ²/ndf ≈ 22/15 — excellent fit. The 1.8σ deviation in A_l (SLD) is the most notable tension.
|
||
|
||
### 19.6 Anomalous Triple Gauge Couplings (Beyond SM Test)
|
||
|
||
```
|
||
ℒ_WWV = i g_WWV [ g₁^V (W_μν^+ W^{−μ} − W_μν^− W^{+μ}) V^ν
|
||
+ κ_V W_μ^+ W_ν^− V^{μν}
|
||
+ (λ_V/M_W²) W^{−ν}_μ W^{+ρ}_ν V^μ_ρ ]
|
||
|
||
SM values at tree level: g₁^Z = g₁^γ = 1, κ_Z = κ_γ = 1, λ_Z = λ_γ = 0
|
||
|
||
LHC measurements: all consistent with SM within 1–2σ.
|
||
```
|
||
|
||
**Domain:** Unifies weak force (β-decay) with electromagnetism at ~100 GeV energy scale. The gauge structure SU(2)_L × U(1)_Y spontaneously broken to U(1)_EM by the Higgs mechanism. Confirmed to per-mille level at LEP/SLC/LHC.
|
||
|
||
---
|
||
|
||
## 20. Quantum Chromodynamics (QCD, 1973)
|
||
|
||
### 20.1 The Lagrangian
|
||
|
||
```
|
||
ℒ_QCD = Σ_{f=1}^{6} ψ̄_f (i D̸ − m_f) ψ_f − ¼ G_a^{μν} G^a_{μν} + ℒ_θ
|
||
|
||
D_μ = ∂_μ − i g_s A_μ^a T^a (covariant derivative, SU(3)_c)
|
||
T^a = λ^a / 2 (Gell-Mann matrices, 8 generators)
|
||
```
|
||
|
||
The sum runs over 6 quark flavors: up, down, strange, charm, bottom, top (masses from ~2 MeV to ~173 GeV).
|
||
|
||
### 20.2 Color Gauge Field Strength
|
||
|
||
```
|
||
G_a^{μν} = ∂^μ A_a^ν − ∂^ν A_a^μ + g_s f_{abc} A_b^μ A_c^ν
|
||
```
|
||
|
||
The structure constants `f_{abc}` of SU(3) encode gluon self-interaction — the **three-gluon** and **four-gluon vertices**. This is the source of all non-abelian behavior. No photon analogue exists in QED.
|
||
|
||
```
|
||
Three-gluon vertex: g_s f_{abc} [g^{μν}(k₁−k₂)^ρ + g^{νρ}(k₂−k₃)^μ + g^{ρμ}(k₃−k₁)^ν]
|
||
Four-gluon vertex: −i g_s² [f_{abe}f_{cde}(g^{μρ}g^{νσ}−g^{μσ}g^{νρ}) + permutations]
|
||
```
|
||
|
||
### 20.3 Feynman Rules (Perturbative QCD)
|
||
|
||
```
|
||
Quark propagator: i(γ^μ p_μ + m) / (p² − m² + iε)
|
||
Gluon propagator (Feynman): −i g_{μν} δ_{ab} / (k² + iε)
|
||
(in covariant gauge, needs ghost cancellation — Faddeev-Popov procedure)
|
||
Ghost propagator: i δ_{ab} / (k² + iε)
|
||
Quark-gluon vertex: −i g_s γ^μ T^a
|
||
Ghost-gluon vertex: g_s f_{abc} p^μ (p = outgoing ghost momentum)
|
||
```
|
||
|
||
BRST symmetry ensures unitarity of the gauge-fixed theory. Ghosts are unphysical but necessary for loop calculations — they cancel unphysical timelike/longitudinal gluon polarizations.
|
||
|
||
### 20.4 Running Coupling and the Beta Function
|
||
|
||
```
|
||
α_s(Q²) ≡ g_s²(Q²) / 4π
|
||
|
||
β(α_s) = ∂α_s / ∂ ln μ = −(b₀/2π) α_s² − (b₁/4π²) α_s³ − ...
|
||
|
||
b₀ = 11 − (2/3) n_f (one-loop coefficient)
|
||
b₁ = 102 − (38/3) n_f (two-loop coefficient)
|
||
```
|
||
|
||
For `n_f = 6` (all quark flavors active): `b₀ = 7`, so `β < 0` → **asymptotic freedom**.
|
||
|
||
```
|
||
α_s(Q²) ≈ 4π / [b₀ ln(Q²/Λ_QCD²)] (leading-order solution)
|
||
|
||
Λ_QCD ≈ 210 ± 14 MeV (MS-bar scheme)
|
||
```
|
||
|
||
| Scale | α_s value | Technique |
|
||
|-------|-----------|-----------|
|
||
| m_τ (1.78 GeV) | 0.33 ± 0.01 | τ decays |
|
||
| m_Z (91.2 GeV) | 0.1180 ± 0.0009 | global electroweak fit |
|
||
| LHC (1 TeV) | ~0.09 | jet cross-sections |
|
||
| LHC (10 TeV) | ~0.07 | extrapolation |
|
||
|
||
Confirmed: α_s decreases with energy over 4 orders of magnitude. The running is **logarithmic**, not a phase transition.
|
||
|
||
### 20.5 Color Confinement
|
||
|
||
No free colored particle has ever been observed. Conjectured mechanisms:
|
||
|
||
**Wilson loop area law (lattice QCD):**
|
||
|
||
```
|
||
⟨W(C)⟩ ∼ exp( −σ · Area(C) ) at large loop size
|
||
σ ≈ (440 MeV)² ≈ 1 GeV/fm string tension
|
||
```
|
||
|
||
This produces a linear potential at large distances:
|
||
|
||
```
|
||
V_QQ̄(r) ≈ σ r − (4/3) α_s / r + constant (Cornell potential)
|
||
```
|
||
|
||
The linear term means infinite energy to separate quarks to infinity → confinement. When the string stretches beyond ~1 fm, `V(r) > 2 m_q` and pair-creation (`q q̄` from vacuum) breaks the string — **hadronization**.
|
||
|
||
**Polyakov loop** (order parameter):
|
||
```
|
||
⟨L⟩ = 0 in confined phase (Z(3) center symmetry unbroken)
|
||
⟨L⟩ ≠ 0 in deconfined phase (Z(3) broken, T > T_c)
|
||
```
|
||
|
||
**Deconfinement transition temperature:**
|
||
```
|
||
T_c ≈ 155–165 MeV ≈ 1.8 × 10¹² K (from lattice QCD)
|
||
```
|
||
Cross-over at physical quark masses (not a sharp phase transition). The quark-gluon plasma (QGP) existed in the early universe for the first ~10 μs and is recreated in heavy-ion collisions at RHIC and LHC.
|
||
|
||
### 20.6 Chiral Symmetry and Its Breaking
|
||
|
||
In the limit `m_u, m_d → 0` (chiral limit), the QCD Lagrangian has an exact global symmetry:
|
||
|
||
```
|
||
SU(2)_L × SU(2)_R × U(1)_V × U(1)_A
|
||
```
|
||
|
||
- `U(1)_V` → **baryon number** (exact)
|
||
- `U(1)_A` → broken by **axial anomaly** (instanton effects, η' mass)
|
||
- `SU(2)_L × SU(2)_R` → **spontaneously broken** by quark condensate:
|
||
|
||
```
|
||
⟨ψ̄ ψ⟩ ≡ ⟨ū u⟩ = ⟨d̄ d⟩ ≈ −(250 MeV)³ ≠ 0
|
||
|
||
SU(2)_L × SU(2)_R → SU(2)_V (isospin)
|
||
```
|
||
|
||
Goldstone's theorem → 3 massless pseudoscalar bosons. Since `m_u, m_d` are small but non-zero, the pions have small masses:
|
||
|
||
```
|
||
m_π² = −(m_u + m_d) ⟨ψ̄ ψ⟩ / f_π² (Gell-Mann–Oakes–Renner relation)
|
||
|
||
f_π ≈ 92.2 MeV (pion decay constant, measured from π⁺ → μ⁺ ν_μ)
|
||
m_π⁰ = 134.977 MeV
|
||
m_π± = 139.570 MeV
|
||
```
|
||
|
||
Extending to SU(3) flavor (including strange quark):
|
||
|
||
```
|
||
SU(3)_L × SU(3)_R → SU(3)_V (octet of pseudoscalar mesons)
|
||
m_K² = −(m_s + m_{u,d}) ⟨ψ̄ ψ⟩ / f_K² / 2
|
||
```
|
||
|
||
The proton mass decomposition (from lattice QCD + phenomenological analysis):
|
||
|
||
```
|
||
M_proton ≈ 938.272 MeV
|
||
|
||
Trace anomaly (gluon field energy): ~90–95% (scale anomaly in QCD)
|
||
Quark kinetic energy + masses: ~5–10%
|
||
Quark masses (Higgs coupling): ~1–2% ≈ 9 MeV (σ_πN term)
|
||
```
|
||
|
||
**Only ~1% of your mass comes from the Higgs mechanism.** The rest is pure QCD binding energy.
|
||
|
||
### 20.7 Chiral Perturbation Theory (χPT)
|
||
|
||
Low-energy effective field theory of QCD (E ≪ 4πf_π ≈ 1.2 GeV):
|
||
|
||
```
|
||
ℒ_χPT = (f_π²/4) Tr[∂_μ U ∂^μ U†] + (f_π²/4) Tr[χ U† + U χ†] + ...
|
||
|
||
U = exp(i π^a λ^a / f_π) (nonlinear sigma model field)
|
||
χ = 2B₀ M (M = quark mass matrix, B₀ = −⟨ψ̄ ψ⟩/f_π²)
|
||
```
|
||
|
||
Expands in powers of `(p/Λ_χ)²` where `Λ_χ ≈ 4πf_π`. Matches to QCD order-by-order. Used for low-energy ππ scattering, pion-nucleon interactions, and lattice extrapolations.
|
||
|
||
### 20.8 U(1)_A Anomaly and the Strong CP Problem
|
||
|
||
The axial anomaly:
|
||
|
||
```
|
||
∂_μ J_5^μ = (g_s² N_f / 16π²) G_a^{μν} G̃_a_{μν} (Adler-Bell-Jackiw)
|
||
|
||
G̃_a^{μν} = (1/2) ε^{μνρσ} G_a^{ρσ} (dual field strength)
|
||
```
|
||
|
||
The θ-term allowed by gauge invariance:
|
||
|
||
```
|
||
ℒ_θ = θ (g_s² / 64π²) ε^{μνρσ} G_a^{μν} G_a^{ρσ} (CP-violating)
|
||
```
|
||
|
||
The neutron electric dipole moment constrains:
|
||
|
||
```
|
||
|θ̄| = |θ_QCD + Arg det M_q| < 10⁻¹⁰
|
||
|
||
d_n < 1.8 × 10⁻²⁶ e·cm (90% CL, experimental bound)
|
||
→ |θ̄| ≲ 10⁻¹⁰
|
||
```
|
||
|
||
This is the **strong CP problem**: why is θ̄ so small when it could be O(1)? Leading solution: Peccei-Quinn mechanism → **axion** (actively searched for by ADMX, CAST, etc.).
|
||
|
||
### 20.9 Hadron Spectrum
|
||
|
||
**Mesons (q q̄ bound states):**
|
||
|
||
```
|
||
Lightest pseudoscalar octet (J^P = 0⁻):
|
||
π⁰, π⁺, π⁻ (u, d only)
|
||
K⁺, K⁰, K̄⁰, K⁻ (u,d + s)
|
||
η (mixing: (uū+d d̄−2ss̄)/√6)
|
||
η′ (U(1)_A anomaly gives extra mass)
|
||
|
||
Vector meson nonet (J^P = 1⁻):
|
||
ρ⁰, ρ⁺, ρ⁻, ω, K*⁺, K*⁰, K̄*⁰, K*⁻, φ
|
||
|
||
Scalar mesons (J^P = 0⁺) and higher excitations extend to ~3 GeV
|
||
```
|
||
|
||
**Baryons (3-quark bound states, qqq):**
|
||
|
||
```
|
||
Nucleon octet (J^P = ½⁺): p, n, Λ, Σ⁺, Σ⁰, Σ⁻, Ξ⁰, Ξ⁻
|
||
Delta decuplet (J^P = ³⁄₂⁺): Δ⁺⁺, Δ⁺, Δ⁰, Δ⁻, Σ*⁺, Σ*⁰, Σ*⁻, Ξ*⁰, Ξ*⁻, Ω⁻
|
||
```
|
||
|
||
The Ω⁻ was predicted by the quark model (SU(3) flavor) and discovered in 1964 — one of QCD's early triumphs before QCD existed.
|
||
|
||
All masses up to ~2.5 GeV have been computed in lattice QCD with <1% error, including the nucleon mass.
|
||
|
||
### 20.10 Exotic Hadrons (Tetraquarks, Pentaquarks, Glueballs)
|
||
|
||
QCD permits color-singlet states beyond `q q̄` and `qqq`:
|
||
|
||
**Tetraquarks** (q q q̄ q̄): Z_c(3900), Z_c(4430), X(3872) — many confirmed at BESIII, LHCb, Belle. The X(3872) sits within 0.1 MeV of the D⁰ D̄*⁰ threshold.
|
||
|
||
**Pentaquarks** (q q q q q̄): P_c(4380), P_c(4450) → observed by LHCb in Λ_b → J/ψ p K decays (2015, updated 2019 with 3 narrow states).
|
||
|
||
**Glueballs** (gg, ggg — pure gauge excitations):
|
||
- Lightest predicted scalar glueball: `J^PC = 0⁺⁺`, m ≈ 1.5–1.7 GeV (lattice QCD)
|
||
- Candidates: f₀(1500), f₀(1710) — but mixing with ordinary mesons makes unambiguous identification difficult.
|
||
- Tensor glueball (`2⁺⁺`, m ≈ 2.4 GeV) — also predicted, not confirmed.
|
||
|
||
**Hybrid mesons** (q q̄ g): π₁(1600) with `J^PC = 1⁻⁺` (exotic quantum numbers impossible for `q q̄`). Evidence from COMPASS and GlueX experiments.
|
||
|
||
### 20.11 Deep Inelastic Scattering, Parton Distribution Functions, Factorization
|
||
|
||
**DIS kinematics** (e⁻ + p → e⁻ + X):
|
||
|
||
```
|
||
Q² = −q² (virtuality of exchanged photon)
|
||
x = Q² / (2 P·q) (Bjorken-x, momentum fraction of struck parton)
|
||
ν = P·q / M_p (energy transfer in target rest frame)
|
||
W² = M_p² + Q²(1/x − 1) (invariant mass of hadronic final state)
|
||
```
|
||
|
||
**Structure functions:**
|
||
|
||
```
|
||
d²σ / dx dQ² = (4πα² / x Q⁴) [ (1−y) F₂(x,Q²) + y² F₁(x,Q²) ]
|
||
|
||
F₁(x,Q²) = (1/2) Σ_q e_q² [q(x,Q²) + q̄(x,Q²)] (Callan-Gross relation for spin-½)
|
||
F₂(x,Q²) = x Σ_q e_q² [q(x,Q²) + q̄(x,Q²)]
|
||
```
|
||
|
||
Callan-Gross (`F₂ = 2x F₁`) confirmed at SLAC (1969) → quarks are spin-½.
|
||
|
||
**DGLAP evolution** (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi):
|
||
|
||
```
|
||
∂q(x,Q²)/∂ ln Q² = (α_s/2π) ∫_x¹ (dz/z) [P_{qq}(z) q(x/z,Q²) + P_{qg}(z) g(x/z,Q²)]
|
||
|
||
∂g(x,Q²)/∂ ln Q² = (α_s/2π) ∫_x¹ (dz/z) [P_{gq}(z) Σ q(x/z,Q²) + P_{gg}(z) g(x/z,Q²)]
|
||
```
|
||
|
||
Splitting functions at LO:
|
||
```
|
||
P_{qq}(z) = (4/3) (1+z²)/(1−z)_+ + 2 δ(1−z)
|
||
P_{qg}(z) = (1/2) [z² + (1−z)²]
|
||
P_{gq}(z) = (4/3) [1 + (1−z)²]/z
|
||
P_{gg}(z) = 6 [z/(1−z)_+ + (1−z)/z + z(1−z)] + (11/2 − n_f/3) δ(1−z)
|
||
```
|
||
|
||
These predict how PDFs scale with Q². Confirmed from HERA (≈1 GeV²) to LHC (≈10⁴ GeV²).
|
||
|
||
**Factorization theorem:**
|
||
```
|
||
dσ_{AB→X} = Σ_{a,b} ∫ dx_a dx_b f_a/A(x_a, μ_F) f_b/B(x_b, μ_F) · dσ̂_{ab→X}(μ_R, μ_F)
|
||
```
|
||
Short-distance (`dσ̂`, calculable in pQCD) and long-distance (PDFs, universal/non-perturbative but measurable) factorize at leading twist. Foundation of all LHC precision physics.
|
||
|
||
### 20.12 Jets, Event Shapes, and Infrared Safety
|
||
|
||
A **jet** is a collimated spray of hadrons from a fragmenting high-energy parton. Jet algorithms:
|
||
|
||
**Anti-k_T algorithm** (Cacciari-Salam-Soyez, 2008):
|
||
```
|
||
d_{ij} = min(p_{Ti}^{-2}, p_{Tj}^{-2}) · ΔR_{ij}² / R² (d_{iB} = p_{Ti}^{-2})
|
||
Merge smallest d_{ij}; if d_{iB} < d_{ij}, i becomes a jet.
|
||
```
|
||
|
||
Jet cross-sections measured at LHC agree with NNLO QCD predictions to ~5% over 8 orders of magnitude.
|
||
|
||
**Event shape variables** (e⁻e⁻ colliders):
|
||
```
|
||
Thrust: T = max_{n̂} (Σ_i |p_i·n̂|) / (Σ_i |p_i|) (T→1 for two back-to-back jets)
|
||
C-parameter: C = 3(λ₁λ₂ + λ₂λ₃ + λ₃λ₁) (linearized momentum tensor)
|
||
Broadening: B_T, B_W (transverse/w.r.t thrust axis)
|
||
```
|
||
|
||
N³LL resummation + NNLO fixed-order matches LEP data to sub-percent precision.
|
||
|
||
**Infrared and collinear safety:** Observables must be insensitive to soft gluons and collinear splittings. This ensures finite perturbative predictions. All standard jet/event variables are IRC-safe.
|
||
|
||
### 20.13 Quark-Gluon Plasma (QGP)
|
||
|
||
Above T ≈ 155 MeV, hadrons "melt" into a deconfined medium of quarks and gluons. Heavy-ion collisions (Au-Au at RHIC, Pb-Pb at LHC) produce droplets of QGP.
|
||
|
||
**Signatures:**
|
||
|
||
**Jet quenching** — high-pT partons lose energy traversing the medium:
|
||
```
|
||
ΔE ∼ C_R (α_s/4) q̂ L² (BDMPS energy loss, radiative)
|
||
q̂ ∼ 1–10 GeV²/fm (transport coefficient, extracted from data)
|
||
```
|
||
Manifested as dijet energy asymmetry and suppression of high-pT hadrons (R_AA < 1):
|
||
|
||
```
|
||
R_AA(p_T) = (dN_AA/dp_T) / [N_coll · (dN_pp/dp_T)]
|
||
```
|
||
R_AA ≈ 0.2–0.5 at RHIC/LHC central collisions — strong suppression.
|
||
|
||
**Elliptic flow (v₂):** pressure-driven anisotropy in non-central collisions. The QGP behaves as a near-perfect fluid (nearly inviscid):
|
||
|
||
```
|
||
η/s ≈ 1/4π ≈ 0.08 (shear viscosity / entropy density)
|
||
```
|
||
This is conjectured to be a lower bound from AdS/CFT (Kovtun-Son-Starinets). The QGP is the most perfect fluid known.
|
||
|
||
**Quarkonium suppression (Matsui-Satz, 1986):** Debye screening in QGP dissolves quarkonium states sequentially:
|
||
```
|
||
J/ψ dissolves at T ≈ 1.5 T_c (tightly bound, survives moderate QGP)
|
||
ψ' dissolves at T ≈ 1.1 T_c (loosely bound, "melts" early)
|
||
Υ(1S) survives to > 2 T_c (very tightly bound, bottomonia thermometers)
|
||
```
|
||
Observed as sequential suppression pattern at SPS, RHIC, LHC.
|
||
|
||
**Electromagnetic probes:** Real and virtual photons escape the QGP without further interaction → direct thermometer. Thermal photon v₂ and direct photon spectra at RHIC/LHC are consistent with hydrodynamics + QGP radiation.
|
||
|
||
### 20.14 QCD Phase Diagram
|
||
|
||
```
|
||
Temperature
|
||
↑
|
||
200 MeV ──── QUARK-GLUON PLASMA ──────
|
||
| \ \
|
||
155 MeV| \ CROSSOVER \ (1st order?)
|
||
| \ \
|
||
| ──── HADRONS ───────────────
|
||
| (confined, chiral broken)
|
||
|
|
||
|←────── μ_B ──────────────────────→
|
||
0 μ_B ~900 MeV μ_B
|
||
(LHC) (neutron stars)
|
||
```
|
||
|
||
- **Crossover** at μ_B ≈ 0 (confirmed by lattice QCD — no critical point at zero density).
|
||
- **Critical endpoint** predicted at μ_B ≈ 300–500 MeV, T ≈ 120–160 MeV (Beam Energy Scan at RHIC searching for it).
|
||
- **First-order phase transition** at large μ_B (cold, dense matter — neutron star interiors).
|
||
- **Color superconductivity**: at μ_B ≳ 400 MeV and low T, quarks form Cooper pairs → CFL (Color-Flavor-Locked) phase deep in neutron star cores.
|
||
|
||
### 20.15 Lattice QCD
|
||
|
||
Monte Carlo evaluation of the Euclidean path integral on a discrete spacetime grid:
|
||
|
||
```
|
||
⟨O⟩ = (1/Z) ∫ D[U] D[ψ] D[ψ̄] O[U, ψ, ψ̄] exp(−S_E[U, ψ, ψ̄])
|
||
|
||
S_E^g = β Σ_□ (1 − (1/3) Re Tr U_□) (Wilson gauge action, β = 6/g²)
|
||
S_E^q = ψ̄ D_W ψ (Wilson/Staggered/DWF/Overlap fermions)
|
||
```
|
||
|
||
**Spectrum results:** Nucleon mass, pion mass, kaon mass, Δ mass, Ω⁻ mass, and excited-state masses computed with <1% systematic error. Light hadrons agree with experiment.
|
||
|
||
**Hadronic contributions to g−2:**
|
||
```
|
||
a_μ^{HVP} = 692.8 ± 2.4 × 10⁻¹⁰ (lattice QCD, BMW collaboration 2020)
|
||
```
|
||
Tension with R-ratio dispersive method (~2σ). Crucial for interpreting the Muon g−2 experiment at Fermilab.
|
||
|
||
**Computational cost:** Scaling as `a^{-n} V L_t` with `a` the lattice spacing (continuum limit `a → 0`). Full QCD at physical pion mass requires petaflop-scale computing.
|
||
|
||
### 20.16 Soft-Collinear Effective Theory (SCET)
|
||
|
||
Effective field theory for QCD with highly boosted particles. Separates dynamics into distinct momentum regions:
|
||
|
||
```
|
||
SCET_I: p² ∼ (Qλ², Qλ², Q²) (e.g. B → ππ, endpoint regions)
|
||
SCET_II: p² ∼ (Qλ², Qλ, Q) (e.g. Drell-Yan at small q_T)
|
||
|
||
Collinear fields: ξ_n(x) (momentum along light-cone direction n)
|
||
Soft fields: q_s(x) (low-momentum modes)
|
||
```
|
||
|
||
SCET factorizes multi-scale processes and enables resummation of large Sudakov logarithms (e.g. `ln Q/m_b`, `ln τ Q`). Essential for precision B-physics at LHCb and Belle II.
|
||
|
||
### 20.17 Experimental Verification Summary
|
||
|
||
| Prediction | Test | Precision | Status |
|
||
|------------|------|-----------|--------|
|
||
| Asymptotic freedom | HERA, LHC DIS | α_s running over 4 decades | Confirmed |
|
||
| Jet cross-sections | LHC, Tevatron, LEP | 5% agreement with NNLO | Confirmed |
|
||
| DGLAP evolution | HERA → LHC | PDFs scale correctly across 10³ in Q² | Confirmed |
|
||
| Confinement | Absence of free quarks | Limit: σ < 10⁻²¹ cm² for free quarks | Confirmed |
|
||
| Chiral symmetry breaking | Pion mass, GOR relation | <1% | Confirmed |
|
||
| Hadron spectrum (light) | Lattice QCD | <1% for ground states | Confirmed |
|
||
| Hadron spectrum (excited) | Lattice + experiment | ~1–5% | Confirmed |
|
||
| Quark-gluon plasma | RHIC, LHC heavy-ion | v₂, R_AA, jet quenching | Confirmed |
|
||
| Tetraquarks / Pentaquarks | BESIII, LHCb, Belle | >5σ observations | Confirmed |
|
||
| Glueballs | Lattice predicts, exp. ambiguous | Candidates but no unambiguous ID | Active |
|
||
| Strong CP (θ ≪ 1) | nEDM bounds | θ̄ < 10⁻¹⁰ | Confirmed (problem remains) |
|
||
| Factorization | LHC, Tevatron | Global PDF fits consistent | Confirmed |
|
||
| Higgs production via ggF | LHC | ~10% agreement with NNLO QCD | Confirmed |
|
||
| α_s(m_Z) | Global average | 0.1180 ± 0.0009 | Confirmed |
|
||
|
||
**Domain:** The strong nuclear force. SU(3)_c non-abelian gauge theory. Tested from femtometer scales (nucleon structure) to LHC energies (~10 TeV). **Zero falsifications of any core prediction.**
|
||
|
||
---
|
||
|
||
## 21. Lorentz Invariance (Special Relativity, 1905)
|
||
|
||
```
|
||
ds² = η_μν dx^μ dx^ν = −c²dt² + dx² + dy² + dz²
|
||
Physical laws are identical in all inertial frames.
|
||
c is the same in all inertial frames.
|
||
```
|
||
|
||
**Domain:** Flat spacetime. Tested to extreme precision by Michelson-Morley, Kennedy-Thorndike, Hughes-Drever, modern optical-resonator experiments, and every particle accelerator ever built. Lorentz violation is constrained to < 10⁻¹⁷ in some parameters.
|
||
|
||
---
|
||
|
||
## 22. Einstein's Mass–Energy Equivalence (1905)
|
||
|
||
```
|
||
E² = (mc²)² + (pc)²
|
||
E = γmc² (massive particle)
|
||
E = pc (massless particle)
|
||
```
|
||
|
||
**Domain:** Every relativistic system. Tested in nuclear reactions (mass defect ≈ energy release), particle-antiparticle annihilation, and every synchrotron/cyclotron. E = mc² is the low-momentum limit.
|
||
|
||
---
|
||
|
||
## 23. Fermi's Golden Rule (Perturbation Theory)
|
||
|
||
```
|
||
Γ_{i→f} = (2π/ℏ) |⟨f|Ĥ'|i⟩|² ρ(E_f)
|
||
```
|
||
|
||
**Domain:** Weak perturbations in QM. Underpins calculation of decay rates, scattering cross-sections, transition probabilities. Used in every branch of quantum physics. Derived from time-dependent perturbation theory — exact in the limit of weak coupling and long times.
|
||
|
||
---
|
||
|
||
## 24. Boltzmann Transport Equation (1872)
|
||
|
||
```
|
||
∂f/∂t + v·∇_r f + (F/m)·∇_v f = (∂f/∂t)_coll
|
||
```
|
||
|
||
**Domain:** Non-equilibrium statistical mechanics. Underpins plasma physics, semiconductor transport, neutron diffusion, galactic dynamics. The H-theorem (entropy increase) is a direct consequence.
|
||
|
||
---
|
||
|
||
## 25. Quantization of Electric Charge / Dirac Quantization Condition
|
||
|
||
```
|
||
Q_e = −e (electron charge, exactly e)
|
||
Magnetic monopole charge g satisfies: e·g = 2πℏn (n ∈ ℤ)
|
||
```
|
||
|
||
**Domain:** All of electromagnetism. Charge is quantized in units of e/3 (quark confinement hides fractional charges). The Dirac condition shows that if *one* magnetic monopole exists anywhere, *electric charge must be quantized everywhere*. No monopole found yet, but the condition is a theorem.
|
||
|
||
---
|
||
|
||
## 26. Optical Theorem (Unitarity)
|
||
|
||
```
|
||
Im[ f(θ=0) ] = (k/4π) σ_total
|
||
```
|
||
|
||
**Domain:** All scattering processes. A consequence of unitarity (probability conservation). Tested in every scattering experiment. Forward scattering amplitude's imaginary part directly gives the total cross-section.
|
||
|
||
---
|
||
|
||
## 27. Einstein Coefficients (1917)
|
||
|
||
```
|
||
A_21 = spontaneous emission rate
|
||
B_12 = absorption coefficient
|
||
B_21 = stimulated emission coefficient
|
||
|
||
B_12/B_21 = g₂/g₁
|
||
A_21/B_21 = 8πhν³/c³
|
||
```
|
||
|
||
**Domain:** Atomic/molecular transitions. Derivation requires detailed balance and Planck's law. Underpins lasers, astrophysical spectroscopy, and atomic clocks. The ratio relations are exact consequences of thermodynamic equilibrium.
|
||
|
||
---
|
||
|
||
## 28. Equivalence Principle (Weak, Einstein Equivalence)
|
||
|
||
```
|
||
Inertial mass = Gravitational mass
|
||
(universality of free fall)
|
||
```
|
||
|
||
**Domain:** All gravitating bodies. Tested by Eötvös, Dicke, Braginsky, MICROSCOPE satellite to ~10⁻¹⁵. No violation. The foundation of GR.
|
||
|
||
---
|
||
|
||
## Summary Table
|
||
|
||
| # | Equation / Law | Domain | Last Falsified |
|
||
|---|---------------|--------|----------------|
|
||
| 1 | Maxwell | Classical E&M | Never |
|
||
| 2 | Einstein Field Equations | Gravity (GR) | Never |
|
||
| 3 | Schrödinger | Non-rel QM | Never |
|
||
| 4 | Dirac | Rel spin-½ | Never |
|
||
| 5 | Newton's 2nd (F=dp/dt) | Low-velocity mechanics | Never (relativistic correction, not falsified) |
|
||
| 6 | Energy Conservation | Universal | Never |
|
||
| 7 | 2nd Law of Thermo | Macroscopic systems | Never |
|
||
| 8 | E = hν | Quantum systems | Never |
|
||
| 9 | Standard Model Lagrangian | Particle physics | Never (neutrino masses are the only SM extension) |
|
||
| 10 | Yang-Mills | Gauge theory | Never |
|
||
| 11 | Noether's Theorem | All of physics | Mathematical theorem — can't be falsified |
|
||
| 12 | Friedmann | Cosmology | Never (ΛCDM fits all data) |
|
||
| 13 | Klein-Gordon | Rel spin-0 | Never |
|
||
| 14 | Heisenberg Uncertainty | Quantum systems | Never |
|
||
| 15 | Pauli Exclusion | Quantum statistics | Never |
|
||
| 16 | Path Integral | QM / QFT | Never |
|
||
| 17 | Navier-Stokes | Fluid dynamics | Never |
|
||
| 18 | Black Hole Thermo | Semiclassical gravity | Not directly falsifiable yet |
|
||
| 19 | Electroweak Unification | Particle physics | Never |
|
||
| 20 | QCD | Strong force | Never |
|
||
| 21 | Lorentz Invariance | Spacetime | Never |
|
||
| 22 | E² = (mc²)² + (pc)² | Relativity | Never |
|
||
| 23 | Fermi's Golden Rule | QM perturbation | Never |
|
||
| 24 | Boltzmann Transport | Non-equilibrium stat mech | Never |
|
||
| 25 | Charge Quantization | E&M | Never |
|
||
| 26 | Optical Theorem | Scattering | Never |
|
||
| 27 | Einstein Coefficients | Atomic transitions | Never |
|
||
| 28 | Equivalence Principle | Gravity | Never |
|
||
|
||
---
|
||
|
||
## What These Equations Do NOT Explain (Open Problems)
|
||
|
||
- **Quantum gravity** — GR and QM are mutually inconsistent at the Planck scale.
|
||
- **Dark matter** — evidence is overwhelming (rotation curves, CMB, lensing, bullet cluster), but no particle identification.
|
||
- **Dark energy** — Λ fits data, but the *value* is 10¹²⁰ smaller than QFT vacuum energy prediction.
|
||
- **Baryon asymmetry** — why does the universe contain matter, not equal matter/antimatter?
|
||
- **Neutrino masses** — require physics beyond the minimal SM (seesaw mechanism? Dirac? Majorana?).
|
||
- **Strong CP problem** — why is the QCD θ-angle < 10⁻¹⁰? (axion?)
|
||
- **Hierarchy problem** — why is the Higgs mass so light compared to the Planck scale?
|
||
- **Initial conditions** — what set the entropy and homogeneity of the early universe? (Inflation fits data but mechanism is speculative.)
|
||
- **Interpretation of QM** — the equations work; what they *mean* is debated (Copenhagen, Many-Worlds, de Broglie-Bohm, QBism).
|
||
- **Measurement problem** — why does "observation" collapse the wavefunction?
|
||
|
||
Every equation above continues to survive. The questions live in the gaps *between* them.
|