Properties of non-hermitian quantum field theories

In this paper I discuss quantum systems whose Hamiltonians are non-Hermitian but whose energy levels are all real and positive. Such theories are required to be symmetric under CPT, but not symmetric under P and T separately. Recently, quantum mechanical systems having such properties have been investigated in detail. In this paper I extend the results to quantum field theories. Among the systems that I discuss are -phi(4) and iphi(3) theories. These theories all have unexpected and remarkable properties. I discuss the Green's functions for these theories and present new results regarding bound states, renormalization, and nonperturbative calculations.