Gauge transformation
A change A → A + ∇λ, V → V − ∂λ/∂t in the potentials that leaves all physical fields E and B unchanged. The freedom that defines what 'gauge' means.
Definition
A gauge transformation is a redefinition of the electromagnetic potentials that leaves the physical fields untouched. Since B = ∇×A and the curl of any gradient is zero, you can replace A with A + ∇λ for any scalar function λ(r,t) without changing B. To keep E unchanged at the same time you also need to redefine V → V − ∂λ/∂t. Different choices of λ produce different "gauges" — different vector potentials that all describe the same physical electromagnetic field.
The freedom is convenience, but a powerful one. The Coulomb gauge fixes ∇·A = 0 and is the natural choice for magnetostatics — the integral A(r) = (μ₀/4π) ∫ J(r')/|r−r'| dV' satisfies ∇·A = 0 automatically when J is steady, and A becomes mathematically as well-behaved as the electric scalar potential. The Lorenz gauge ∇·A + (1/c²) ∂V/∂t = 0 is the natural choice for radiation problems, where it makes Maxwell's equations decouple into separate wave equations for V and each component of A. The Weyl gauge sets V = 0 entirely, which is what's commonly used in quantum field theory. None of these choices changes any measurable result; they only change how convenient the math looks.
The deep physical content is that gauge freedom is not a mathematical accident but a fundamental symmetry of the universe. In quantum mechanics, the wavefunction's phase is unobservable, and a gauge transformation A → A + ∇λ is exactly compensated by a phase rotation ψ → exp(iqλ/ℏ) ψ of the wavefunction. The requirement that physics be invariant under this kind of local phase change forces the existence of the electromagnetic field — gauge symmetry derives the photon. The same idea generalised to non-abelian gauge groups produces the strong nuclear force (SU(3) gauge symmetry) and the electroweak theory (SU(2)×U(1)). The whole architecture of the Standard Model is built from gauge transformations.
History
Hermann Weyl introduced the word "gauge" (German: Eichung, "calibration") in 1918 in a failed attempt to derive electromagnetism from a local rescaling symmetry of distances; the original idea didn't work, but the name stuck. In 1929 Weyl reformulated the symmetry as a local phase change of the quantum wavefunction — the modern electromagnetic gauge transformation. Yang and Mills generalised it to non-abelian gauge symmetries in 1954, and the resulting Yang–Mills theory became the framework of the Standard Model in the 1970s.