Edited sentence: ("Abelian" means that it has the commutative property, and a group (mathematics) is a mathematical structure such as addition or multiplication., one in which the calculated predictions of all physically measurable quantities are finite.Another way of putting this is that a phase angle of, say, 5° is considered to be completely equivalent to an angle of 365°.Except for the "wrap-around" property, the algebraic properties of this mathematical structure are exactly the same as those of the ordinary real numbers, and in particular, they obey the commutative law, θ+φ=φ+θ.If a gauge transformation θ is applied to the electron waves, for example, then one must also apply a corresponding transformation to the potentials that describe the electromagnetic waves.In mathematical terminology, this is a certain Abelian group, called U(1).The description of the electons in the subsection above as little clocks is in effect a statement of the mathematical rules according to which the phases of electrons are to be added and subtracted: they are to be treated as ordinary numbers, except that in the case where the result of the calculation falls outside the range of 0≤θ<360°, we force it to "wrap around" into the allowed range, which covers a circle.The gauge symmetry is also intimately related to conservation of charge, and is required in order to make quantum electrodynamics a renormalizable theory, i.These two gauge symmetries are in fact intimately related.This is an empirically testable statement about the interference patterns formed by electron waves, and experiments verify that it is true