Physics
Physics is the first section to make the cross-referenced structure visible. Every claim here points back to a specific mathematical result — Lagrangians to variational calculus, field theory to representation theory, statistical mechanics to measure theory.
Units (§09–13)
09.01.01— Kinematics — position, velocity, acceleration shipped09.01.02— Newton's laws of motion shipped09.01.03— Conservation laws — energy, momentum, angular momentum shipped09.01.04— Two-body central-force problem, Kepler orbits, and Rutherford scattering shipped09.01.05— Newton's Law of Gravitation: Gravitational Field, Potential, and the Shell Theorem shipped09.01.06— Friction, Normal Forces, and Constraints: Holonomic and Non-holonomic Systems shipped09.01.07— Non-inertial Frames: Centrifugal Force, Coriolis Force, and Foucault's Pendulum shipped09.01.08— Variable-Mass Systems: The Rocket Equation (Tsiolkovsky) and Momentum Transport shipped09.02.01— The action principle and variational calculus shipped09.02.02— Euler-Lagrange equations shipped09.02.03— Constrained Lagrangian Systems: Lagrange Multipliers and D'Alembert's Principle shipped09.02.04— Small Oscillations: Normal Modes and the Eigenvalue Problem for Coupled Oscillators shipped09.02.05— Lagrangian Perturbation Theory: Secular Terms and the Method of Multiple Scales shipped09.02.06— Field Lagrangians and the Continuum Limit: Elastic Media and the Wave Equation as a Field Theory shipped09.03.01— Noether's theorem — symmetries and conservation laws shipped09.03.02— Cyclic Coordinates, Ignorable Degrees of Freedom, and Routh's Procedure shipped09.03.03— Quantum free particle as a representation of shipped09.03.04— Discrete Symmetries in Mechanics: Parity, Time-Reversal, and Conservation Laws shipped09.03.05— Galilean Symmetry Group and the Ten Integrals of Motion shipped09.04.01— Legendre transform — from Lagrangian to Hamiltonian shipped09.04.02— Hamilton's equations shipped09.04.03— Poisson Brackets: Structure, the Jacobi Identity, and Bracket Algebra of Conserved Quantities shipped09.04.04— Liouville's Theorem and the Incompressibility of Phase-Space Flow shipped09.04.05— Symplectic Structure, the Symplectic Form, and Darboux's Theorem shipped09.04.06— Hamiltonian Perturbation Theory: Averaging and Adiabatic Invariants shipped09.04.07— Complex structures and quantization; squeezed states shipped09.04.08— Classical Spin: The SO(3) Poisson Bracket and Euler's Equations for Rigid Body Rotation shipped09.04.09— The Hamilton-Jacobi Equation as the Eikonal Limit: WKB and Geometric Optics shipped09.05.01— Canonical transformations shipped09.05.02— Hamilton-Jacobi equation shipped09.05.03— Generating Functions for Canonical Transformations: Four Types (F1, F2, F3, F4) shipped09.05.04— Point Transformations and Their Canonical Lifts shipped09.05.05— The Symplectic Group Sp(2n, R) and Its Role in Linear Canonical Transformations shipped09.06.01— Action-angle variables shipped09.06.02— The Liouville-Arnold Theorem: Integrability and the Existence of Action-Angle Variables shipped09.06.03— Completely Integrable Systems: The Toda Lattice and the Kepler Problem Revisited shipped09.06.04— Lax Pairs and the Inverse Scattering Method: Solitons in Classical Mechanics shipped09.07.01— Continuum Mechanics and Field Theory shipped09.07.02— Elastic Waves in Solids: Longitudinal and Transverse Modes, the Cauchy Stress Tensor shipped09.07.03— Ideal Fluid Mechanics: Euler's Equations, Bernoulli's Equation, and Vorticity shipped09.07.04— Viscous Flow: The Navier-Stokes Equations, Reynolds Number, and Laminar-Turbulent Transition shipped09.08.01— KAM theorem and chaos shipped09.08.02— Phase-Space Structure of Chaos: Poincare Sections, Homoclinic Tangles, and Horseshoe Maps shipped09.08.03— Lyapunov Exponents: Definition, Computation, and the Chaotic Double Pendulum shipped09.08.04— Period-doubling route to chaos: the logistic map and Feigenbaum universality shipped09.08.05— Strange attractors: the Lorenz system, fractal dimension, and sensitive dependence shipped10.01.01— Coulomb's law and Gauss's law shipped10.01.02— Laplace equation and boundary value problems shipped10.01.03— Conductors, capacitance, and electrostatic energy shipped10.01.04— Dielectrics, polarization P, and the electric displacement D shipped10.01.05— Multipole expansion of the electrostatic potential shipped10.02.01— Biot-Savart law and Ampere's law shipped10.02.02— Magnetic vector potential, gauge freedom, and the Coulomb gauge shipped10.02.03— Magnetization, H-field, and magnetic materials: diamagnets, paramagnets, ferromagnets shipped10.02.04— Boundary conditions at interfaces: normal and tangential components of E, D, B, H shipped10.02.05— Magnetic scalar potential and the demagnetization factor shipped10.03.01— Faraday's law and electromagnetic induction shipped10.03.02— Faraday's law in integral and differential form: mutual and self-inductance, energy in inductors shipped10.03.03— Energy and momentum in the electromagnetic field: Poynting vector, Maxwell stress tensor, conservation laws shipped10.03.04— Displacement current, the continuity equation, and the complete Maxwell equations in matter shipped10.03.05— Energy and momentum in the electromagnetic field: the Poynting vector and radiation pressure shipped10.04.01— Maxwell's equations in differential form shipped10.04.02— EM waves and the wave equation shipped10.04.03— Plane waves in matter: dispersion relations, skin depth, and the complex refractive index shipped10.04.04— Waveguides and transmission lines: TE, TM, TEM modes and the cutoff frequency shipped10.04.05— Cavities and resonant modes: the quality factor Q and normal-mode expansion shipped10.05.01— Special relativity — postulates and Lorentz transformations shipped10.05.02— Relativistic kinematics and dynamics shipped10.05.03— Four-velocity, four-momentum, and the relativistic energy-momentum identity shipped10.06.01— Covariant electrodynamics — Faraday tensor shipped10.06.02— The four-current, charge-current density, and covariant continuity shipped10.06.03— The electromagnetic stress-energy tensor and covariant energy-momentum conservation shipped10.06.04— Transformation of E and B fields under Lorentz boosts shipped10.07.01— Radiation from accelerating charges — Larmor formula shipped10.07.02— Electric dipole radiation: angular distribution, polarization, and total radiated power shipped10.07.03— Magnetic dipole and electric quadrupole radiation: multipole expansion of radiation fields shipped10.07.04— Radiation reaction: the Abraham-Lorentz force and the self-energy problem shipped10.07.05— Synchrotron radiation: relativistic circular motion and the 1/gamma^4 power formula shipped10.07.06— Radiation — antennas, scattering, and radiation reaction shipped10.08.01— Green's function method for the scalar and vector potentials: retarded, advanced, and Feynman propagators shipped10.08.02— Jefimenko's Equations: Exact Retarded Solutions for E and B Given Arbitrary Sources shipped10.08.03— Diffraction: Kirchhoff Integral, Fraunhofer and Fresnel Regimes shipped10.08.04— Geometrical Optics as the Short-Wavelength Limit: Eikonal Equation and Ray Tracing shipped10.08.05— Macroscopic Maxwell Equations: Linear Response, Kramers-Kronig Relations, and Causality shipped11.01.01— First and second laws of thermodynamics shipped11.01.02— Thermodynamic potentials and Legendre transforms shipped11.01.03— Third law of thermodynamics: Nernst heat theorem, absolute entropy, and zero-point entropy shipped11.01.04— Thermodynamic stability: convexity, Le Chatelier's principle, and the spinodal curve shipped11.01.05— Equations of state: virial expansion, van der Waals gas, and the law of corresponding states shipped11.01.06— Thermodynamic cycles: Carnot efficiency, Otto and Diesel cycles, and entropy production shipped11.02.01— Maxwell-Boltzmann distribution from kinetic theory shipped11.02.02— Transport phenomena: diffusion, viscosity, thermal conductivity, and the mean free path shipped11.02.03— The Boltzmann transport equation: H-theorem, the approach to equilibrium, and the BGK approximation shipped11.02.04— Fluctuations and Noise: The Fluctuation-Dissipation Theorem and Nyquist's Formula shipped11.03.01— Microcanonical ensemble shipped11.03.02— Grand canonical ensemble: chemical potential, fugacity, and variable particle number shipped11.03.03— Equivalence of Ensembles in the Thermodynamic Limit: Legendre Structure and Equivalence Conditions shipped11.03.04— Isothermal-Isobaric (NPT) Ensemble: Gibbs Free Energy as the Natural Potential shipped11.04.01— Canonical ensemble and partition function shipped11.04.02— Souriau Gibbs state on a symplectic -space shipped11.04.03— Factorization of the partition function: independent subsystems and the grand potential shipped11.04.04— Classical ideal gas partition function: Sackur-Tetrode entropy and the Gibbs paradox shipped11.04.05— Harmonic Oscillator in a Thermal Bath: Heat Capacity and the Einstein Solid shipped11.05.01— Bose-Einstein distribution shipped11.05.02— Fermi-Dirac distribution and electron gas shipped11.05.03— Blackbody radiation: Planck distribution, Stefan-Boltzmann law, Wien displacement law shipped11.05.04— Bose-Einstein condensation and the critical temperature shipped11.05.05— Fermi gas: heat capacity, electron specific heat, and Pauli paramagnetism shipped11.05.06— Photon gas, phonon gas, and the Debye model of solids shipped11.06.01— Ising model and phase transitions shipped11.06.02— Mean-field theory of phase transitions: the van der Waals and Curie-Weiss models shipped11.06.03— Landau Theory: Order Parameter, Symmetry Breaking, and the Free Energy Expansion shipped11.06.04— Fluctuations Beyond Mean Field: The Ginzburg Criterion and Ginzburg-Landau Theory shipped11.06.05— Scaling Hypothesis: Widom Scaling, Kadanoff Block Spins, and Data Collapse shipped11.06.06— Transfer Matrix Method: Exact Solution of the 1D Ising Model and Onsager's Legacy shipped11.07.01— Critical phenomena and renormalization group shipped11.07.02— The renormalization group — Wilson's framework shipped11.07.03— Block-Spin Renormalization: Wilson's Real-Space RG, Fixed Points, and Relevant Operators shipped11.07.04— Kosterlitz-Thouless Transition: Topological Phase Transitions in Two Dimensions shipped11.07.05— The Epsilon Expansion, RG Flows Near Four Dimensions, Critical Exponents, and Conformal Field Theory shipped12.01.01— Wave-particle duality and the double-slit shipped12.01.02— Stern-Gerlach and spin-1/2 shipped12.01.03— WKB approximation: tunneling, Bohr-Sommerfeld quantization, and connection formulas shipped12.01.04— Probability current, continuity equation, and the flow interpretation of the wavefunction shipped12.01.05— Spreading of the free Gaussian wave packet: group velocity and dispersion shipped12.02.01— Hilbert-space formalism of quantum mechanics shipped12.02.02— Operators, observables, and Hermiticity shipped12.02.03— Density matrix and pure / mixed states shipped12.02.04— Measurement postulates: Born rule, wavefunction collapse, and the projection postulate shipped12.03.01— Schrödinger and Heisenberg pictures shipped12.03.02— Time-dependent perturbation theory: Fermi's golden rule and transition rates shipped12.03.03— The interaction picture, Dyson series, and time-ordered exponentials shipped12.04.01— Particle in a box shipped12.04.02— Quantum harmonic oscillator shipped12.04.03— The finite square well: bound states, tunneling, and resonant scattering shipped12.04.04— The delta-function potential: exactly solvable bound state and scattering shipped12.05.01— Angular momentum operators and SU(2) representations shipped12.05.02— Spherical harmonics and Legendre polynomials shipped12.05.03— Addition of angular momenta and Clebsch-Gordan coefficients shipped12.05.04— Free Klein-Gordon scalar quantum field shipped12.05.05— Free Dirac spin-1/2 quantum field shipped12.05.06— Free Maxwell / massive vector fields; photon and Proca shipped12.05.07— Molecular vibrations and spectroscopic selection rules via symmetry shipped12.06.01— Hydrogen atom bound states shipped12.06.02— The isotropic harmonic oscillator in three dimensions: Cartesian and spherical solutions shipped12.06.03— The Runge-Lenz vector, SO(4) symmetry, and the accidental degeneracy of hydrogen shipped12.06.04— Crossing symmetry; CPT theorem at the -matrix level shipped12.07.01— Time-independent perturbation theory shipped12.07.02— Time-dependent perturbation theory and Fermi's golden rule shipped12.07.03— Variational method (Rayleigh-Ritz) in quantum mechanics shipped12.07.04— WKB approximation and Bohr-Sommerfeld quantisation shipped12.07.05— Stark and Zeeman effects in LL3 framing shipped12.07.07— Adiabatic theorem and Berry phase preview shipped12.07.08— Berry phase and the geometric phase shipped12.08.01— Scattering Theory shipped12.08.02— Born approximation and the Lippmann-Schwinger equation shipped12.08.03— Partial-wave expansion and phase shifts shipped12.08.04— Inelastic collisions and the distorted-wave Born approximation shipped12.09.01— Identical Particles and Many-Body Quantum Mechanics shipped12.09.02— Exchange interaction and the helium atom shipped12.09.03— Hartree-Fock self-consistent field method shipped12.09.04— Multi-electron atomic structure and LS coupling shipped12.09.05— Diatomic molecule and the Born-Oppenheimer approximation shipped12.10.01— Path integral formulation of quantum mechanics shipped12.10.02— Euclidean path integrals, instantons, and tunneling between vacua shipped12.10.03— The semiclassical approximation: stationary phase, van Vleck determinant, and Gutzwiller trace formula shipped12.11.01— Dirac equation and relativistic spin shipped12.11.02— Klein-Gordon equation in external EM field: Coulomb and uniform-magnetic cases shipped12.11.03— Dirac equation in a Coulomb field shipped12.11.04— Klein paradox shipped12.11.05— Furry's theorem and charge-conjugation symmetry of QED shipped12.12.01— Canonical Quantum Field Theory shipped12.12.02— Coulomb gauge vs Lorenz gauge in QED shipped12.12.03— Compton scattering and the Klein-Nishina formula shipped12.12.04— Møller scattering (electron-electron) shipped12.12.05— Bhabha scattering (electron-positron) shipped12.12.06— Bethe-Heitler bremsstrahlung and pair production shipped12.13.01— Bosonic Fock space and second quantisation shipped12.13.02— Fermionic Fock space, Pauli exclusion, and anticommutators shipped12.13.03— Cluster decomposition and the connected S-matrix shipped12.14.01— CCR algebra, Weyl algebra, and quasi-free states shipped12.14.02— The Heisenberg group, the Schrödinger representation, and Stone–von Neumann as quantization shipped12.15.01— Time-reversal symmetry and Kramers' degeneracy shipped12.15.02— Parity, discrete-symmetry groups, and the Wigner-Eckart theorem shipped12.15.03— CP violation and the CKM matrix shipped12.16.01— Electron self-energy and mass renormalization at one loop shipped12.16.02— One-loop QED vertex function and the anomalous magnetic moment shipped12.16.03— Vacuum polarization at one loop and the Uehling potential shipped12.16.04— Lamb shift from one-loop QED shipped12.16.05— Infrared divergences and the Bloch-Nordsieck cancellation in QED shipped12.16.06— Power counting, the superficial degree of divergence, and renormalizability classification shipped12.17.01— Density matrix, pure states, and mixed states shipped12.17.02— Entanglement, Schmidt decomposition, and entanglement entropy shipped12.17.03— Bell inequalities, CHSH inequality, and the Tsirelson bound shipped12.17.07— Quantum teleportation and superdense coding shipped12.18.01— The Higgs mechanism: spontaneously broken gauge symmetry shipped12.18.02— The Goldstone theorem and effective Goldstone Lagrangians shipped12.18.03— Asymptotic freedom and the running gauge coupling shipped12.18.04— Theta-vacua, the vacuum angle, and the strong-CP problem shipped12.18.05— The chiral (Adler-Bell-Jackiw) anomaly from the triangle diagram shipped12.18.06— Operator product expansion and short-distance behaviour shipped12.18.13— Vortices (Nielsen-Olesen / Abrikosov flux tubes) shipped12.18.16— Lattice gauge theory and confinement (QFT pointer) shipped12.19.01— The supersymmetry algebra: Coleman-Mandula, Haag-Lopuszanski-Sohnius, and the graded extension of Poincare shipped12.19.02— Superspace, superfields, and Berezin integration shipped12.19.03— Supermultiplets and the Wess-Zumino model: Bose-Fermi degeneracy, the superpotential, and the auxiliary F-field shipped12.19.04— Super-Yang-Mills and the non-renormalization theorem: the vector superfield, Wess-Zumino gauge, , and supergraphs shipped12.19.05— Spontaneous SUSY breaking: the Goldstino theorem, O'Raifeartaigh and Fayet-Iliopoulos, the supertrace sum rule, and the field-theory Witten index shipped12.19.06— Supersymmetric QCD and Seiberg duality: the moduli space of vacua, holomorphy, and N=1 electric-magnetic duality shipped12.20.01— Condensed-matter physics — band structure, phonons, and emergent order shipped12.20.02— Superconductivity — Cooper pairs, BCS theory, and macroscopic quantum order shipped12.20.03— Magnetism, Mott physics, and topological phases of matter shipped12.21.01— The Standard Model — electroweak unification and QCD shipped12.21.02— Electroweak theory and the Higgs mechanism shipped13.01.01— The equivalence principle shipped13.01.02— The equivalence principle — weak, strong, Einstein, and experimental tests shipped13.02.01— Tensors on smooth manifolds shipped13.02.02— Geodesics and parallel transport shipped13.02.03— Cartan tetrad and spin-connection formulation of general relativity shipped13.03.01— Riemann curvature tensor shipped13.03.02— Ricci and scalar curvature, and the Einstein tensor shipped13.03.03— The ADM 3+1 formalism and the Hamiltonian constraint shipped13.04.01— Einstein field equations shipped13.04.02— Einstein-Hilbert action and variational derivation of the Einstein equations shipped13.04.03— Palatini first-order variational formulation of general relativity shipped13.04.04— Stress-energy tensor as functional derivative of the matter action shipped13.05.01— Schwarzschild solution shipped13.05.02— Orbits in Schwarzschild geometry shipped13.05.03— Solar-system tests of general relativity: perihelion precession, light bending, Shapiro time delay, gravitational redshift, frame-dragging shipped13.05.04— Kerr black hole, ergosphere, and the Penrose process shipped13.06.01— Black Holes shipped13.06.03— Black hole thermodynamics: the four laws, Bekenstein-Hawking entropy, and the area theorem shipped13.06.04— Hawking radiation: Bogoliubov derivation, thermal spectrum, and black-hole evaporation shipped13.07.01— Linearized GR and gravitational waves shipped13.07.02— Null infinity, the BMS group, and the Bondi-Sachs mass-loss formula shipped13.07.03— Interferometric gravitational-wave detectors: LIGO, Virgo, KAGRA shipped13.08.01— FLRW cosmology and Friedmann equations shipped13.08.02— Cosmology — FLRW, inflation, nucleosynthesis, CMB, and structure shipped13.08.03— Cosmological inflation: slow-roll scalar fields and the origin of structure shipped13.08.04— Gravitational lensing: strong, weak, and microlensing — Einstein rings, dark matter maps, and exoplanet detection shipped13.09.01— Globally hyperbolic Lorentzian manifolds shipped13.09.02— Klein-Gordon equation on a globally hyperbolic spacetime shipped13.09.03— Hadamard states via the wave-front-set criterion shipped13.09.04— Existence of Hadamard states via the FNW deformation argument shipped13.09.05— Hadamard states by pseudo-differential calculus (Gérard-Wrochna) shipped13.09.06— Wick polynomials in curved spacetime via Hadamard parametrix subtraction shipped13.09.07— Time-ordered products and Hollands-Wald renormalisation on curved spacetimes shipped13.09.08— Bunch-Davies state on de Sitter spacetime shipped13.09.09— Unruh effect via the Bisognano-Wichmann theorem shipped13.09.10— Hartle-Hawking and Unruh states on Schwarzschild shipped13.09.11— Quantum energy inequalities (Fewster) shipped13.09.12— The Peierls bracket and the covariant phase space of an interacting field theory shipped
Five-branch structure
§09 classical mechanics · §10 electromagnetism & special relativity · §11 statistical mechanics (physics flavor) · §12 quantum mechanics & QFT · §13 general relativity & cosmology. Plus existing §03 math-physics border units (Clifford, spin, Dirac, Yang-Mills, Atiyah-Singer) and §08 math-stat-mech (Onsager, RG, lattice gauge). See the physics plan for branch-by-branch detail.