Would-Be Local Fields

General quantum field theory is formulated for the case when the Wightman distributions can grow in momentum space as the exponential of a covariant polynomial. Appropriate spaces of test functions are introduced, and it is shown that the vacuumexpectation values can be written in terms of various associated tempered distributions, which enjoy some of the properties of ordinary Wightman distributions; in particular, they can be represented as boundary values of functions holomorphic in the usual extended tubes. Notions of locality for the tempered distributions can be introduced, which are sufficient to imply the PCT theorem and theorems on the connection between spin and statistic for the non-tempered fields. It is shown how a Haag-Ruelle theory of asymptotic states and fields may be set up. A possible line of generalisation is illustrated by the special example of fields of the type χ (□) A (x), where A is a tempered field, and χ an entire analytic function of finite exponential order.