Phase - crowlogic/arb4j GitHub Wiki
In the context of quantum mechanics, a unimodular complex multiplier is frequently referred to as a Phase (or a PhaseAngle), and it is regarded in the quantum formalism as something that is not directly observable. The subtle change that converts Weyl’s ingenious but extraordinary idea into a key ingredient of modern physics is to replace Weyl’s real positive scale factor – or Gauge – into the ComplexPhase of quantum mechanics.
For these historical reasons, the term gauge has stuck, even though it might well have been more appropriate to have referred to Weyl’s theory, as modified in this way, as a PhaseTheory, and a GaugeConnection as a PhaseConnection.
RogerPenrose thinks it might well confuse more people than it would help if we were to change the terminology in this way now but I think it is well worth it to go ahead and change the terminology.