This chapter is devoted to examining the physical and speculative roots of the notion of gauge fields, reviewing early attempts at applying this attractive notion to various physical processes, and explaining the reasons why these heroic attempts failed.

Gauge invariance

The idea of gauge invariance, as we mentioned in section 5.3, originated in 1918, from Weyl's attempt to unify gravity and electromagnetism, based on a geometrical approach in four-dimensional spacetime (1918a, b). Weyl's idea was this. In addition to the requirement of GTR that coordinate systems have only to be defined locally, the standard of length, or scale, should also only be defined locally. So it is necessary to set up a separate unit of length at every spacetime point. Weyl called such a system of unit-standards a gauge system. In Weyl's view, a gauge system is as necessary for describing physical events as a coordinate system. Since physical events are independent of our choice of descriptive framework, Weyl maintained that gauge invariance, just like general covariance, must be satisfied by any physical theory. However, Weyl's original idea of scale invariance was abandoned soon after its proposal, since its physical implications appeared to contradict experiments. For example, as Einstein pointed out, this concept meant that spectral lines with definite frequencies could not exist.

Despite the initial failure, Weyl's idea of a local gauge symmetry survived, and acquired new meaning with the emergence of quantum mechanics (QM). As is well known, when classical electromagnetism is formulated in Hamiltonian form, the momentum Pμ is replaced by the canonical momentum (Pμ-eAμ/C).