The problem of understanding the role of large gauge transformations in thermal field theories has recently inspired a number of studies of a one dimensional field theory. Such work has led to the conclusion that gauge invariance is restored only when the entire perturbation expansion can be summed. A careful reexamination of that model is shown, h0owever, to lead to vastly different conclusions when the constraint implied by the field equations is explicitly taken into account. In particular it is found that none of the relevant propagators has any temperature dependence and that the effective action is essentially trivial. A generalization of the model to include bosons as well as fermions is also solved with qualitatively identical results being obtained.