Gauss's law, the manifestations of gauge fields, and their impact on local observables

Detlev Buchholz, Fabio Ciolli, Giuseppe Ruzzi, Ezio Vasselli
December 21, 2022
Within the framework of the universal algebra of the electromagnetic field, the impact of globally neutral configurations of external charges on the field is analyzed. External charges are not affected by the field, but they induce localized automorphisms of the universal algebra. Gauss's law implies that these automorphisms cannot be implemented by unitary operators involving only the electromagnetic field, they are outer automorphisms. The missing degrees of freedom can be incorporated in an enlargement of the universal algebra, which can concretely be represented by exponential functions of gauge fields and an abelian algebra describing the external charges. In this manner, gauge fields manifest themselves in the framework of gauge invariant observables. The action of the automorphisms on the vacuum state gives rise to representations of the electromagnetic field with vanishing global charge, which are locally disjoint from the vacuum representation. This feature disappears in the enlarged universal algebra of the electromagnetic field. The energy content of the states is well defined in both cases and bounded from below. The passage from these globally neutral states to charged states and the determination of their energy content are also being discussed.