Formulation of Abelian Gauge Theories without Regulators

In this, the first of three papers, we present the essential features of a treatment of the U( 1) -Higgs model based upon a regulator free, momentum space subtraction scheme. The principle new results which follow from this approach are: (1) the fields of the theory satisfy their classical equations of motion, (2) the source of the vector meson field, f(x), is a finite, conserved current, (3) the Higgs theory passes “smoothly” over to a Goldstone boson theory when the vector meson coupling constant (e) is set equal to zero, (4) the conserved current is gauge invariant and can be used as an interpolating field for the stable one particle states of the theory, (5) one can define a generalized Higgs model wherein only part of the vector meson mass comes from spontaneous breakdown; this theory has the features of the usual Higgs model, is ghost free but it keeps its Goldstone boson. This paper is devoted to stating precisely what can be proven and to establishing the relationship of our treatment to classical field theory ideas on the one hand and other quantum field theoretic treatments on the other. Further details concerning the techniques adopted and proofs that various limiting procedures are well-defined are given in parts II and III.