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…