Edited sentence: Although this cartoon ignores some technical details, it retains the physical phenomena that are important here.If both clocks are sped up by the same amount, the phase relationship between them is unchanged, and the results of experiments are the same.In summary, gauge transformations attain their full importance in the contect of quantum mechanics, where they apply to both electromagnetic waves and to electron waves.Not only that, but it is not even necessary to change the speed of each clock by a fixed amount.Note that in these experiments, the only quantity that affects the result is the difference in phase between the two parts of the electron wave.Suppose we imagine the two parts of the electron wave as tiny clocks, each with a single hand that sweeps around in a circle, keeping track of its phase.This is another example of a gauge transformation: it is local, and it does not change the results of experiments.This would have no effect on the result of the experiment, since the final observation of the location of the electron occurs at a single place and time, so that the phase shift in each electron's "clock" would be the same, and the two effects would cancel out.We could change the angle of the hand on each clock by a varying amount f, where f could depend on both the position in space and on time