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