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In this work, we show that, for the $i\phi^{3}$ scalar field theory, theirexists a contradiction between the assumption that the field is real and thefact that the quantized as well as the classical fields have to satisfy theKlein-Gordon equation. In solving the Klein-Gordon equation for the theoryunder investigation, we realized that the field is a pure imaginary solitarywave which spoils out the non-Hermiticity of the theory. Thus, instead of beingnon-Hermitian, the $i\phi^{3}$ scalar field theory is a kind of aHermitian-Lee-Wick theory which suffers from the existence of the famous ghoststates and instability problems. We applied a Canonical transformation toobtain a Non-Hermitian and non-$\mathcal{PT}$-symmetric representation whichleads to the invalidity of the previous trials in the literature to cure theghost states problem. Moreover, the solitonic solution is a non-topological onewhich is a very strange result to appear for a one component field theory. Toaccount for this strange result, we conjecture that the $i\phi^{3}$ scalarfield theory has an equivalent Hermitian and non-Lee wick theory that have aconserved Noether current.

On the $i\phi^{3}$ $\mathcal{PT}$-symmetric Scalar Field Theory