Relationship between the usual formulation of a massless scalar field theory and its formulation in terms of a two-form potential

We show that the two possible formulations for a divergence- and curl-free vector field, namely, in terms of a scalar or a two-form potential, are equivalent, both as free quantum field theories even in the presence of a background gravitational field and in their coupling to the gravitational field. An apparent disparity in the extrema of the actions in the Euclidean formulations of these theories is resolved by showing that if boundary conditions that are common to the two formulations (which are the only boundary conditions for which the two theories can be compared) are imposed the only extrema of either action has a zero value for the vector field and thus for the stress-energy tensor.