107 KiB
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:
- K_max depends only on ν, not intensity.
- Threshold frequency ν_0 exists.
- No time delay — emission is instantaneous (vs. minutes for classical energy accumulation).
- 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:
- The symmetry is NOT a symmetry of the Lagrangian.
- The derivation of the Euler-Lagrange equations from the action fails.
- 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 ℏωbecauseΔx Δp ≥ ℏ/2preventsx=0, p=0simultaneously. - 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
- Periodic table — electron shells fill progressively; chemical properties from outermost shell.
- 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.
- Neutron star stability — neutron degeneracy pressure supports stars against gravitational collapse up to ~2–3 M_⊙ (Tolman-Oppenheimer-Volkoff limit). Above this → black hole.
- White dwarf stability — electron degeneracy pressure supports up to ~1.4 M_⊙ (Chandrasekhar limit).
- Fermi energy:
E_F = (ℏ²/2m)(3π²n)^{2/3}— conduction electrons occupy states up to E_F (several eV in metals). - 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.