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We revisit the previously unsolved problems of ensuring Lorentz invarianceand non-perturbative unitarity in Lee-Wick theories. We base our discussion onan ultraviolet completion of QED by Lee-Wick ghost fields, which is argued tobe asymptotically safe. We argue that as long as the state space is based upona suitable choice of distributions of a type invented by Gel'fand and Shilov,the Lee-Wick ghosts can be eliminated while preserving Lorentz invariance toproduce a unitary theory. The method for eliminating ghosts is in principlenon-perturbatively well-defined, in contrast with some previous proposals. Wealso point out a second, independent mechanism for producing a unitary theory,based on a covariant constraint on the maximum four-momentum, which would implyan amusing connection, based on naturalness, between the coupling constant andthe hierarchy of scales in the theory. We further emphasize that the resultingtheory is causal, and point out some analogies between between the behaviour ofLee-Wick ghost degrees of freedom and black holes.

Unitarity, Lorentz invariance and causality in Lee-Wick theories: An asymptotically safe completion of QED