Introduction

The term ‘gauge’ refers in its most general everyday connotation to a system of measuring physical quantities, for example by comparing a physical magnitude with a standard or ‘unit’. Changing the gauge would then refer to changing the standard. The original idea of a gauge as introduced by Weyl in his (1918) in an attempt to provide a geometrical interpretation of the electromagnetic field was to consider the possibility of changing the standard of ‘length’ in a four-dimensional generalization of Riemannian geometry in an arbitrary local manner, so that the invariants of the new geometry were specified not just by general coordinate transformations but also by symmetry under conformal rescaling of the metric. The result was, in general, a non-integrability or path dependence of the notion of length which could be identified with the presence of an electromagnetic field. In relativistic terms this meant that, unacceptably, the frequencies of spectral lines would depend on the path of an atom through an electromagnetic field, as was pointed out by Einstein.

With the development of wave mechanics the notion of gauge invariance was revived by Weyl himself (1929) following earlier suggestions by Fock and by London, so as to apply to the non-integrability of the phase of the Schrödinger wave function, effectively replacing a scale transformation eα(x) by a phase transformation eiα(x).