# Proven Equations of the Universe > 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. --- ## 1. Maxwell's Equations (Electromagnetism, 1861–1865) ### 1.1 The Four Equations (Heaviside–Hertz Form) **Differential form (SI units):** ``` ∇ · E = ρ / ε₀ Gauss's law (electric) ∇ · B = 0 Gauss's law (magnetic) — no magnetic monopoles ∇ × E = −∂B / ∂t Faraday-Lenz law of induction ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t Ampère's law with Maxwell's displacement current ``` **Integral form:** ``` ∮_S E·dA = Q_enc / ε₀ ∮_S B·dA = 0 ∮_C E·dl = −d/dt ∫_S B·dA ∮_C B·dl = μ₀ I_enc + μ₀ε₀ d/dt ∫_S E·dA ``` **Relativistic (covariant) form — tensor notation:** ``` ∂_μ F^{μν} = μ₀ J^ν inhomogeneous equations ∂_μ F̃^{μν} = 0 homogeneous (Bianchi identity) F_{μν} = ∂_μ A_ν − ∂_ν A_μ field strength tensor F̃^{μν} = (1/2) ε^{μνρσ} F_{ρσ} dual tensor F_{μν} = ⌈ 0 −E_x/c −E_y/c −E_z/c ⌉ | E_x/c 0 −B_z B_y | | E_y/c B_z 0 −B_x | ⌊ E_z/c −B_y B_x 0 ⌋ ``` ### 1.2 Scalar and Vector Potentials ``` B = ∇ × A magnetic field from vector potential E = −∇φ − ∂A/∂t electric field from scalar + vector potentials A^μ = (φ/c, A) 4-potential F_{μν} = ∂_μ A_ν − ∂_ν A_μ ``` **Gauge invariance:** The fields E, B are unchanged under: ``` A_μ → A_μ + ∂_μ Λ(x) arbitrary scalar function Λ φ → φ − ∂Λ/∂t A → A + ∇Λ ``` **Gauge choices:** ``` Coulomb gauge: ∇ · A = 0 Lorenz gauge: ∂_μ A^μ = 0 (1/c² ∂φ/∂t + ∇·A = 0) Temporal gauge: φ = 0 ``` ### 1.3 Lagrangian Formulation ``` ℒ_EM = −¼ F_{μν} F^{μν} − J^μ A_μ S_EM = ∫ d⁴x ℒ_EM δS = 0 → ∂_μ F^{μν} = μ₀ J^ν (Euler-Lagrange) ``` Coupled to matter: replace `∂_μ → D_μ = ∂_μ + i e A_μ` (minimal coupling, QED). ### 1.4 Wave Equation and Speed of Light From Maxwell's equations in vacuum (ρ = 0, J = 0): ``` ∇² E − (1/c²) ∂²E/∂t² = 0 ∇² B − (1/c²) ∂²B/∂t² = 0 c = 1 / √(μ₀ε₀) = 299,792,458 m/s (exact, defines meter) ``` 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. ### 1.5 Poynting's Theorem (Energy Conservation) ``` ∂u/∂t + ∇ · S = −J · E u = (1/2) ε₀ E² + (1/2μ₀) B² energy density (J/m³) S = (1/μ₀) E × B Poynting vector (W/m², energy flux) S^μ = (u c, S) energy-momentum 4-vector ∂_μ S^μ = −F^{μν} J_ν covariant form ``` **Electromagnetic momentum:** ``` p_EM = ∫ ε₀ (E × B) dV = ∫ S/c² dV ``` ### 1.6 Polarization, Permittivity, Permeability in Media **Constitutive relations:** ``` D = ε₀ E + P = ε E (ε = ε_r ε₀, permittivity tensor) H = B/μ₀ − M = B/μ (μ = μ_r μ₀, permeability tensor) J = σ E (Ohm's law, conductivity) ∇ · D = ρ_free ∇ · B = 0 ∇ × E = −∂B/∂t ∇ × H = J_free + ∂D/∂t ``` ### 1.7 Radiation: Lienard-Wiechert Potentials Retarded potentials for a point charge `q` with trajectory `r_q(t)`: ``` φ(r, t) = (q / 4πε₀) [1 / (R − R·v/c)]_ret A(r, t) = (μ₀ q / 4π) [v / (R − R·v/c)]_ret R = r − r_q(t_ret) t_ret = t − |R|/c retarded time ``` **Lienard-Wiechert fields:** ``` 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 └── velocity field (Coulomb + SR correction) ──┘ └── acceleration field (radiation) ──┘ B = (R̂/c) × E ``` **Larmor formula (non-relativistic radiation power):** ``` P = (q² a²) / (6πε₀ c³) (J/s = W) ``` **Liénard formula (relativistic generalization):** ``` P = (q² γ⁶ / 6πε₀ c³) [a² − (v × a)²/c²] ``` **Radiation reaction (Abraham-Lorentz force):** ``` F_rad = (q² / 6πε₀ c³) d³r/dt³ (pathological — pre-acceleration) ``` ### 1.8 Electrodynamic Stress-Energy Tensor ``` Θ^{μν} = (1/μ₀) [F^μ_α F^{να} + (1/4) g^{μν} F_{αβ} F^{αβ}] Θ^{00} = u energy density Θ^{0i} = S^i / c momentum density (×c) Θ^{ij} = −ε₀ E_i E_j − (1/μ₀) B_i B_j + (1/2) δ_{ij} (ε₀E² + B²/μ₀) Maxwell stress tensor ``` **Radiation pressure on a perfect absorber:** `P = I/c` where `I = |S|`. ### 1.9 Green's Function for the Wave Equation ``` (∇² − (1/c²) ∂²/∂t²) G(r,t; r',t') = −δ³(r−r') δ(t−t') G_ret(r,t; r',t') = δ(t − t' − |r−r'|/c) / (4π|r−r'|) retarded G_adv(r,t; r',t') = δ(t − t' + |r−r'|/c) / (4π|r−r'|) advanced ``` General solution for potentials with source `f(r,t)`: ``` ψ(r,t) = ∫ d³r' dt' G(r,r'; t,t') f(r', t') ``` ### 1.10 Experimental Verification | Test | Precision | Status | |------|-----------|--------| | Coulomb's law (inverse square) | `1/r^{2±δ}` with δ < 10⁻¹⁶ | Confirmed | | Photon mass limit | m_γ < 10⁻¹⁸ eV/c² | Confirmed | | Magnetic monopole | None detected | Absence confirmed | | Displacement current | Hertz experiment, all radio | Confirmed | | c = 1/√(μ₀ε₀) | Measured to 10⁻⁹ | Confirmed | | Poynting vector | Energy balance in every antenna | Confirmed | | Lienard-Wiechert | Synchrotron radiation, undulators | Confirmed | | Abraham-Lorentz | Qualitative features in laser-plasma | Confirmed (limited precision) | **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.** --- ## 2. Einstein Field Equations (General Relativity, 1915) ### 2.1 Fundamental Equation ``` G_{μν} + Λ g_{μν} = (8πG / c⁴) T_{μν} G_{μν} = R_{μν} − ½ R g_{μν} Einstein tensor R_{μν} = R^ρ_{μρν} Ricci tensor R = g^{μν} R_{μν} Ricci scalar (curvature scalar) ``` **Trace-reversed form:** ``` R_{μν} = (8πG/c⁴) [T_{μν} − ½ g_{μν} T] + Λ g_{μν} T = g^{μν} T_{μν} ``` ### 2.2 Riemann Curvature Tensor ``` R^ρ_{σμν} = ∂_μ Γ^ρ_{νσ} − ∂_ν Γ^ρ_{μσ} + Γ^ρ_{μλ} Γ^λ_{νσ} − Γ^ρ_{νλ} Γ^λ_{μσ} Γ^ρ_{μν} = ½ g^{ρλ} (∂_μ g_{νλ} + ∂_ν g_{μλ} − ∂_λ g_{μν}) Christoffel symbols ``` **Symmetries of Riemann (in 4D, 20 independent components):** ``` R_{ρσμν} = −R_{σρμν} = −R_{ρσνμ} = R_{μνρσ} (antisymmetries) R_{ρσμν} + R_{ρμνσ} + R_{ρνσμ} = 0 (Bianchi identity, algebraic) ∇_λ R_{ρσμν} + ∇_ν R_{ρσλμ} + ∇_μ R_{ρσνλ} = 0 (Bianchi identity, differential) ``` Contracting to Einstein tensor: ``` ∇^μ G_{μν} = 0 → ∇^μ T_{μν} = 0 (conservation of stress-energy) ``` ### 2.3 Geodesic Equation (Motion in Curved Spacetime) ``` d²x^μ/dτ² + Γ^μ_{αβ} dx^α/dτ · dx^β/dτ = 0 τ = proper time: dτ² = −g_{μν} dx^μ dx^ν For a test mass (T^{μν} = 0 elsewhere): ∇_u u = 0 u^μ = dx^μ/dτ is 4-velocity ``` **Equivalence principle:** In a freely falling frame, `g_{μν} → η_{μν}` and `Γ → 0` locally → SR physics. ### 2.4 Einstein-Hilbert Action ``` S = (c⁴ / 16πG) ∫ d⁴x √(−g) (R − 2Λ) + S_matter g = det(g_{μν}) δS/δg^{μν} = 0 → G_{μν} + Λ g_{μν} = (8πG/c⁴) T_{μν} T^{μν} = (2 / √(−g)) δS_matter / δg_{μν} ``` **Gibbons-Hawking-York boundary term:** ``` S_total = S_EH + S_GHY S_GHY = (c⁴ / 8πG) ∫_{∂M} d³y √(|h|) ε K K = extrinsic curvature of boundary ε = ±1 (timelike/spacelike boundary) h = induced metric on boundary ``` Necessary for a well-posed variational principle with fixed boundary metric. ### 2.5 Palatini (First-Order) Formalism Treat `g_{μν}` and `Γ^ρ_{μν}` as independent variables: ``` S_Palatini = (c⁴/16πG) ∫ d⁴x √(−g) g^{μν} R_{μν}(Γ) δS/δΓ → Γ = Levi-Civita connection (metric compatibility) δS/δg → Einstein field equations ``` In vacuum GR, this is equivalent to the standard formulation. Extended to Einstein-Cartan theory with torsion. ### 2.6 Linearized Gravity and Gravitational Waves **Perturbation expansion:** `g_{μν} = η_{μν} + h_{μν}` with `|h_{μν}| ≪ 1`. **Trace-reversed perturbation:** ``` h̄_{μν} = h_{μν} − ½ η_{μν} h h = η^{μν} h_{μν} ``` **Linearized Einstein equations (Lorenz gauge ∂^μ h̄_{μν} = 0):** ``` □ h̄_{μν} = −(16πG/c⁴) T_{μν} □ = η^{μν} ∂_μ ∂_ν = −(1/c²) ∂²/∂t² + ∇² ``` **Gravitational wave in TT gauge (transverse-traceless):** ``` h_{μν}^{TT} = [ 0 0 0 0 ] e^{i(kz − ωt)} [ 0 h_+ h_× 0 ] [ 0 h_× −h_+ 0 ] [ 0 0 0 0 ] h_+ = plus polarization (stretches/squeezes along x,y axes) h_× = cross polarization (stretches/squeezes at 45°) ``` ### 2.7 Quadrupole Formula (Gravitational Radiation) Energy carried away by gravitational waves: ``` dE/dt = (G / 5c⁵) Σ_{i,j} ⟨d³Q_{ij}/dt³ · d³Q_{ij}/dt³⟩ Q_{ij} = ∫ d³x ρ(x) (x_i x_j − (1/3) δ_{ij} r²) mass quadrupole moment ``` **Luminosity (full formula):** ``` L_GW = (G/5c⁵) ⟨Q̈_{ij} Q̈^{ij}⟩ For a binary system (masses M₁, M₂, separation a): L_GW = (32/5) (G⁴/c⁵) (M₁² M₂² (M₁+M₂) / a⁵) ``` 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. **Orbital decay (binary inspiral):** ``` dE_orb/dt = −L_GW → da/dt ∝ −1/a³ → "chirp" frequency increase f_GW = (c³ / G) [(5/256) (M_chirp/c²)^{-5/3} (t_coal − t)^{-3/8}]^{3/8} M_chirp = (M₁ M₂)^{3/5} / (M₁+M₂)^{1/5} chirp mass ``` ### 2.8 Exact Solutions **Schwarzschild metric (1916) — static, spherical, vacuum (Λ=0):** ``` ds² = −(1 − r_s/r) c² dt² + dr²/(1 − r_s/r) + r² dΩ² r_s = 2GM / c² Schwarzschild radius dΩ² = dθ² + sin²θ dφ² Event horizon at r = r_s. Coordinate singularity at r = r_s (removable by Kruskal-Szekeres coordinates). Physical singularity at r = 0. ``` **Kerr metric (1963) — rotating, stationary, axisymmetric:** ``` ds² = −(1 − r_s r/Σ) c² dt² + (Σ/Δ) dr² + Σ dθ² + (r² + a² + r_s r a² sin²θ/Σ) sin²θ dφ² − (2 r_s r a sin²θ/Σ) c dt dφ Σ = r² + a² cos²θ Δ = r² − r_s r + a² a = J / Mc spin parameter (m) J = angular momentum Event horizons at r_± = (r_s/2) ± √((r_s/2)² − a²) Ergosphere: region between r_+ and static limit where no observer can remain stationary. Penrose process extracts energy from ergosphere (up to ~29% of rest mass for extreme Kerr a→r_s/2). ``` **Kerr-Newman metric — charged rotating black hole:** ``` Same form as Kerr with Δ = r² − r_s r + a² + r_Q² r_Q² = G Q² / (4πε₀ c⁴) ``` **Reissner-Nordström metric — charged, non-rotating:** ``` ds² = −(1 − r_s/r + r_Q²/r²) c² dt² + dr²/(1 − r_s/r + r_Q²/r²) + r² dΩ² Two horizons for Q < M (in geometric units). Extremal black hole: r_s = 2r_Q → degenerate horizon. ``` **FLRW metric (Friedmann-Lemaître-Robertson-Walker) — homogeneous, isotropic cosmology:** ``` ds² = −c² dt² + a²(t) [ dr²/(1 − k r²) + r² dΩ² ] k = +1 (closed/spherical), 0 (flat/Euclidean), −1 (open/hyperbolic) a(t) = scale factor ``` **de Sitter space — vacuum with Λ > 0:** ``` ds² = −(1 − Λr²/3) c² dt² + dr²/(1 − Λr²/3) + r² dΩ² Static patch. Horizon at r = √(3/Λ). Exponential expansion: a(t) ∝ exp( H t ), H = c √(Λ/3). ``` ### 2.9 ADM Formalism (3+1 Decomposition) Split spacetime into foliation of spacelike hypersurfaces Σ_t: ``` ds² = −N² c² dt² + γ_{ij} (dx^i + N^i c dt)(dx^j + N^j c dt) N = lapse function (rate of proper time vs coordinate time) N^i = shift vector (shift of spatial coordinates between slices) γ_{ij} = 3-metric on Σ_t ``` **Hamiltonian constraint:** ``` R(³) + K² − K_{ij} K^{ij} = 16πG/c⁴ · ρ K_{ij} = (1/2N)(∂_t γ_{ij} − D_i N_j − D_j N_i) extrinsic curvature R(³) = Ricci scalar of γ_{ij} ρ = energy density measured by Eulerian observer ``` **Momentum constraint:** ``` D_j (K^{ij} − γ^{ij} K) = 8πG/c⁴ · J^i ``` These are elliptic constraint equations solved on each slice. Evolution equations are hyperbolic. ### 2.10 Post-Newtonian Approximation Expand for slow motion, weak field: `(v/c) ∼ ε`, `GM/rc² ∼ ε²`. ``` 1PN order: corrections of order ε² to Newtonian 2PN order: ε⁴, etc. Full equations of motion for binary systems known to 4PN order. ``` 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) |⟨ 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⟩, 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.