Heisenberg Fields which Vanish on Domains of Momentum Space

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Abstract
If a local Heisenberg field vanishes, or where appropriate has an infinite zero, on one of the momentum space domains A, p2 = —a2; B, 0 ≤ p2 < m2, and p = 0; or C, p2 > M2, then the field is a generalized free field. Counter examples show that this conclusion cannot be drawn if the field vanishes on the momentum space domains D, 0 ≤ M12 < p2 < M22, p ≠ 0; or E, p = 0. It follows that if two fields in the same Borchers class are equal on one of the domains A, B, or C, then the fields differ at most by a generalized free field in their Borchers class.

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