Clock synchronization and quantum phase: a joint revisitation

Relativity and quantum mechanics each contain a quantity that no experiment can fix: the simultaneity of distant events and the phase of a quantum state. They are usually discussed in separate literatures. We revisit them together, starting from a structural fact: in a stationary frame a resynchronization of clocks, t → t + f(x), acts on a state of energy ω as the local phase exp(iωf), with the same structure as an electromagnetic gauge transformation, energy playing the role of charge. For a charged clock the two freedoms combine into a single connection. We then follow two routes. Keeping time as an external coordinate, we ask how the convention is distributed: in rotating interferometers with electrostatic potentials varying along the paths, only the total holonomy and the Sagnac delay are invariant, and the attribution of the phase to the rotation or to the field is conventional. Treating time as a relation between quantum systems, in the Page–Wootters framework, we ask why distant simultaneity is conventional: the relative reading of two clocks is observable only under a joint constraint, i.e. after an exchange of energy. The two routes converge on a single statement: between separated systems, only what is closed is physical, either geometrically, by a loop, or dynamically, by an exchange; offsets and splittings that refer to open separations are conventional. No new prediction is claimed; all statements are checked against the standard field equations.