A gauge theory is a field theory in which the fields and potentials are described by a symmetry group (the gauge group). A gauge theory arises when a particular symmetry is imposed on a theory even when the parameter labeling the symmetry is allowed to vary over spacetime.
The earliest physical theory to have a gauge symmetry was classical electrodynamics – the theory of electricity and magnetism originated by James Clerk Maxwell. It turns out that electrodynamic theory is invariant under redefinition of the electrostatic potential, which corresponds eventually to the law of conservation of electric charge. The mathematics of gauge theories was not fully developed, however, until the early 20th century by the German mathematician Hermann Weyl. In quantum theory, for example, the phase of the wave functions describing a system can be changed by an arbitrary amount without altering the physical content or structure of the theory, provided that all the wave functions are changed in the same way, everywhere in space. To gauge this symmetry the phase change is allowed to vary over spacetime. In order to maintain this as a symmetry of the theory it turns out that the electromagnetic force must be introduced. Thus quantum electrodynamics (QED) is an example of a gauge theory. The different wave functions can also be allowed to have different phase changes and this corresponds to their electric charge.
In a gauge theory there is a group of transformations of the field variables (gauge transformations) that leaves the basic physics of the quantum field unchanged. This condition, called gauge invariance, gives the theory a certain symmetry, which governs its equations. In short, the structure of the group of gauge transformations in a particular gauge theory entails general restrictions on the way in which the field described by that theory can interact with other fields and elementary particles. The classical theory of the electromagnetic field, proposed by Maxwell in 1864, is the prototype of gauge theories, though the concept of gauge transformation was not fully developed until the early 20th century by Weyl. In Maxwell's theory the basic field variables are the strengths of the electric and magnetic fields, which may be described in terms of auxiliary variables (e.g., the scalar and vector potentials). The gauge transformations in this theory consist of certain alterations in the values of those potentials that do not result in a change of the electric and magnetic fields. This gauge invariance is preserved in the modern theory of electromagnetism called quantum electrodynamics, or QED. Modern work on gauge theories began with the attempt of the American physicists Chen Ning Yang and Robert L. Mills (1954) to formulate a gauge theory of the strong interaction. The group of gauge transformations in this theory dealt with the isospin of strongly interacting particles. In the late 1960s Steven Weinberg, Sheldon Glashow, and Abdus Salam developed a gauge theory that treats electromagnetic and weak interactions in a unified manner. This theory, now commonly called the electroweak theory, has had notable success and is widely accepted. During the mid-1970s much work was done toward developing quantum chromodynamics (QCD), a gauge theory of the interactions between quarks (see quark). For various theoretical reasons, the concept of gauge invariance seems fundamental, and many physicists believe that the final unification of the fundamental interactions (i.e., gravitational, electromagnetic, strong, and weak) will be achieved by a gauge theory. The Standard Model is also built out of a gauge theory.