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We consider the continuation of free and interacting scalar field theory tonon-integer spacetime dimension d. We find that the correlation functions inthese theories are necessarily incompatible with unitarity (or with reflectionpositivity in Euclidean signature). In particular, the theories containnegative norm states unless d is a positive integer. These negative norm statescan be obtained via the OPE from simple positive norm operators, and aretherefore an integral part of the theory. At the Wilson-Fisher fixed point thenon-unitarity leads to the existence of complex anomalous dimensions. Wedemonstrate that they appear already at leading order in the epsilon expansion.

Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions