Abstract
For quantum field theories that do not satisfy the Wightman positivity condition, a Hilbert space structure condition is proposed, which guarantees a Krein structure for the space of states associated to the Wightman functions. The analogous problem for the Schwinger function is also discussed, as well as the conditions that ensure the analytic continuation of a Krein structure in the Euclidean case and vice versa. The Gupta–Bleuler formulation of free quantum electrodynamics is discussed as an example.
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