Dirac sea for bosons. II - Study of the naive vacuum theory for the toy model world prior to filling the negative energy sea

We use an idea presented in a previous paper "Dirac Sea for Bosons. I"[Prog. Theor. Phys. 113 (2005), 603] in which, for the first time, we perform a naive second quantization with respect to both negative and positive energy for the Klein-Gordon equation analogous to that with respect to the unfilled Dirac sea for fermions, in order to study as a toy model this naive second quantization theory. The model we treat is not physically realistic, because it employs a Fock space whose norm square possesses indefinite signature, but it nevertheless has interesting possibilities: Although the naive (quantization) theory exhibits a spontaneous breakdown of the usual CPT-symmetry, we show that it possesses a different type of symmetry formulated by a CPT-like theorem. We prove this theorem and present the properties of this "CPT-like" symmetry possessed by our theory.