The mathematical apparatus shared across physics: Hilbert spaces for QM, Lie groups for symmetry, distributions and Green's functions for PDEs, tensor analysis for relativity, special functions and integral transforms as the universal…
mathematical-physics
Hilbert space (physics use)
Complete inner-product vector space (typically L²) used as the state space of a quantum system. Pure states are unit rays; observables are…
Distribution theory (Schwartz)
Generalised functions: continuous linear functionals on smooth compactly-supported test functions. Makes objects like the Dirac delta,…
Lie groups in physics
Continuous symmetry groups with smooth manifold structure — SU(2) for spin, SU(3) for colour, SO(3,1) for spacetime Lorentz…
Lie algebra (physics)
Tangent space at the identity of a Lie group, equipped with a bracket [X,Y] = XY − YX. Generators of infinitesimal symmetry…
Green's function
Impulse-response of a linear differential operator: L_x G(x,x') = δ(x-x'). Convolution with the source then solves Lu = f. Foundational…
Variational calculus
Search for extremals of a functional S[q] = ∫ L dt; stationarity δS=0 yields the Euler–Lagrange equations. Underlies Lagrangian/Hamiltonian…
Tensor analysis on manifolds
Calculus of multilinear maps on differentiable manifolds — covariant derivatives, connection coefficients, curvature tensors. Mathematical…
Spinors
Two-component complex objects transforming under the double cover SL(2,C) of the proper Lorentz group. Required to describe…
Sturm–Liouville theory (physics use)
Spectral theory of self-adjoint second-order ODEs: eigenfunctions are orthogonal with respect to a weight, eigenvalues are real and…
Special functions of mathematical physics
Bessel, Legendre, Hermite, Laguerre, hypergeometric, elliptic — the canonical solutions to PDEs separated in standard coordinate systems,…
Integral transforms in physics
Linear maps between function spaces by integration against a kernel — Fourier, Laplace, Mellin, Hankel, wavelet. Diagonalise translation,…
Differential forms
Antisymmetric covariant tensor fields, with exterior derivative d satisfying d²=0. Cleans up Maxwell's equations to dF=0, d⋆F=⋆J; gives…
Haag-Kastler axioms (1964)
R Haag-D Kastler 1964 algebraic QFT axioms; nets-of-algebras; modern Haag 1992 'Local Quantum Physics' standard reference.
BCFW recursion (2005)
Britto-Cachazo-Feng-Witten 2005 on-shell-recursion; modern amplituhedron Arkani-Hamed-Trnka 2014; revolutionized perturbative QFT.
QFT integrability
Yangian symmetry + Y-system + thermodynamic-Bethe-ansatz; modern N=4 SYM all-loop predictions Beisert 2010.
Yang-Mills mass gap
Clay 2000 Millennium Problem; non-perturbative existence of mass-gap in 4D Yang-Mills; lattice-QCD numerical evidence; analytic proof open.
Conformal bootstrap (2008+)
Rattazzi-Rychkov-Tonni-Vichi 2008 numerical; Kos-Poland-Simmons-Duffin 2014 Ising 3D; modern lightcone + analytic bootstrap.
VOA (Borcherds 1986)
R Borcherds 1986 vertex-operator-algebras (Fields 1998); monstrous-moonshine; modern modular-tensor-category basis 2D-CFT chiral-algebras.
Hilbert space (vN 1932)
J von Neumann 1932 'Mathematische Grundlagen der Quantenmechanik'; foundational text for QM Hilbert-space formulation.
Noether (1918)
E Noether 1918 conservation-laws ↔ continuous-symmetries; modern foundational text in quantum-field-theory + GR + lattice-gauge.
Dirac delta (1930)
P Dirac 1930 (Nobel 1933) δ-function distribution; modern Schwartz 1950 distribution-theory + tempered + Sobolev spaces.
Hawking-Penrose (1970)
S Hawking-R Penrose 1970 (Penrose Nobel 2020) singularity-theorems; modern modern positive-mass-Schoen-Yau + Penrose process.
Haag-Kastler axioms (1964)
R Haag-D Kastler 1964 algebraic-QFT; modern modular-Tomita-Takesaki + von-Neumann-algebras + entropy + KMS-state classification.
Yang-Mills (1954)
C N Yang-R Mills 1954 non-abelian gauge-theory; modern foundational of Standard-Model + Clay-millennium-mass-gap problem.