Historical roots of gauge invariance, J.D. Jackson and L.B. Okun (2001)
Abstract. Gauge invariance is the basis of the modern theory of the electroweak and strong interactions (the so called Standard Model). The roots of gauge invariance go back to the year 1820 when electromagnetism was discovered and the first electrodynamic theory was proposed. Subsequent developments led to the discovery that different forms of vector potential result in the same observable forces. The partial arbitrariness of the vector potential A brought fourth various restrictions on it. div A was proposed by J.C. Maxwell; 4-div A =0 was proposed L.V. Lorenz in the middle off 1860's. In most of the modern texts the later condition is attributed to H.A. Lorentz, who half a century later was one of the key figures in the final formulation of classical electrodynamics. In 1926 a relativistic quantum-mechanical equation for charged spinless particles was formulated by E. Schrödinger, O.Klein, and V. Fock. The latter discovered that this equations is invariant with respect to multiplication of the wave function by a phase factor (...) with the accompanying additions to the scalar potential (...) and the vector potential (...). In 1929 H. Weyl proclaimed this invariance as a general principle and called it Eichinvarianz in German and gauge invariance in English. The present era on non-abelian gauge theories started in 1954 with the paper by C.N. Yang and R.L. Mills.