We have formulated a theory of massive electrodynamics which admits gauge invariance of the second kind. In so doing, a massless scalar field is needed. However, if the conserved-current condition ∂_μj^μ=0 is satisfied, this scalar field has no dynamical consequences. This theory then reduces to the conventional theory. When the conserved-current condition is not satisfied, a generalized current must still be conserved in order to have consistency of the theory. In an addendum, a brief discussion of the generalized Stueckelberg formulation is given.
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York‐Peng Yao (1965) studied this question.