Scalar Quantum Field Theory with a Complex Cubic Interaction

In this Letter it is shown that an $i{\phi }^3$ quantum field theory is a physically acceptable model because the spectrum is positive and the theory is unitary. The demonstration rests on the perturbative construction of a linear operator $\mathcal{C}$, which is needed to define the Hilbert space inner product. The $\mathcal{C}$ operator is a new, time-independent observable in $\mathcal{P}\mathcal{T}$-symmetric quantum field theory.