The exponential of the two-dimensional massless scalar field as an infrared Jaffe field

The definition of the Wick exponential of the massless scalar field in two dimensions as an operator-valued distribution is discussed in the Krein space realization of the field. It is shown that the exponential Wick power series converges strongly on a suitably dense domain, provided that it is smeared with test functions which satisfy some growth conditions in configuration space. Such conditions are similar to the high energy bounds previously introduced by Jaffe for the definition of strictly localizable fields. In this case it is found that the Wick exponential is strictly localizable in momentum space.