Gauge invariance is the basis of the modern theory of electroweak and strong interactions (the so-called standard model). A number of authors have discussed the ideas and history of quantum guage theories, beginning with the 1920s, but the roots of gauge invariance go back to the year 1820 when electromagnetism was discovered and the first electrodynamic theory was proposed. We describe the 19th century developments that led to the discovery that different forms of the vector potential (differing by the gradient of a scalar function) are physically equivalent, if accompanied by a change in the scalar potential: $\mathbf{A}\to {\mathbf{A}}^{\prime }=\mathbf{A}+\nabla \chi ,\hspace{0.5em}\Phi \to {\Phi }^{\prime }=\Phi -\partial \chi /c\partial t.\mathrm{}$ L. V. Lorenz proposed the condition ${\partial }_{\mu }{A}^{\mu }=0\mathrm{}$ in the mid-1860s, but this constraint is generally misattributed to the better known H. A. Lorentz. In the work in 1926 on the relativistic wave equation for a charged spinless particle in an electromagnetic field by Schrödinger, Klein, and Fock, it was Fock who discovered the invariance of the equation under the above changes in A and Φ if the wave function was transformed according to $\psi \to {\psi }^{\prime }=\psi \exp (ie\chi /ħc).\mathrm{}$ 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 of non-Abelian gauge theories started in 1954 with the paper by Yang and Mills on isospin gauge invariance.

DOI: http://dx.doi.org/10.1103/RevModPhys.73.663

• Published 14 September 2001