Thanks to David Broadhurst for stressing the following point during my lecture at a summer school in Les Houches.

Whilst introducing gauge fields from noncommutative spin manifolds (aka spectral triples) I first explained how the Dirac operator can be seen as a metric on a (possibly noncommutative) space described via Connes’ distance formula. Then the action of a unitary in the algebra of coordinates was given as a gauge transformation on the Dirac operator, generating a pure gauge field.

What David noticed was that this is in compelling agreement with Weyl’s old idea of gauge invariance. Indeed, the term *Eichinvarianz *was preceded by *MaÎ²stabinvarianz* in the original work (see Yang’s review below). This is precisely the notion captured by noncommutative geometry: a gauge transformation actually acts on the metric (the Dirac operator) but leaves the distance function invariant.