Inconsistency of Minkowski higher-derivative theories

We show that Minkowski higher-derivative quantum field theories are generically inconsistent, because they generate nonlocal, non-hermitian ultraviolet divergences, which cannot be removed by means of standard renormalization procedures. By Minkowski theories we mean theories that are defined directly in Minkowski spacetime. The problems occur when the propagators have complex poles, so that the correlation functions cannot be obtained as the analytic continuations of their Euclidean versions. The usual power counting rules fail and are replaced by much weaker ones. Self-energies generate complex divergences proportional to inverse powers of D'Alembertians. Three-point functions give more involved nonlocal divergences, which couple to infrared effects. We illustrate the violations of the locality and hermiticity of counterterms in scalar models and higher-derivative gravity.