Spectral properties of Wick power series for a free field with an indefinite metric

The properties of infinite series in Wick powers of a free field whose two-point correlation function has singular infrared behavior and does not satisfy the positivity condition are investigated. If these series are defined on an appropriate functional domain, then the fields they converge to satisfy all conditions of the pseudo-Wightman formalism. For series convergent only on analytic test functions in the momentum representation, the spectral condition is formulated using the previously introduced notion of the carrier cone of an analytic functional. A suitable generalization of the Paley–Wiener–Schwartz theorem is used to prove that this condition is satisfied.