Reality from maximizing overlap in the periodic complex action theory

Abstract We study the periodic complex action theory (CAT) by imposing a condition in future-included CAT where time integration is performed from past to future, and extend normalized matrix element of an operator $\hat{\mathcal {O}}$, which called weak value real theory, another expression $\langle \hat{\mathcal {O}} \rangle _{\mathrm{periodic}~\mathrm{time}}$. present two theorems stating that _{\mathrm{periodic}~\mathrm{time}}$ becomes for {O}}$ being Hermitian with regard modified inner product makes given non-normal Hamiltonian $\hat{H}$ normal. The first theorem holds period tp case number eigenstates having maximal imaginary part B eigenvalues just one, while second one stands selected such absolute transition amplitude maximized ≤ 0 |B| much smaller than distances between any parts $\hat{H}$. latter proven via number-theoretical argument suggests that, if our universe periodic, then even could be adjustment parameter determined Feynman path integral. This variant type maximization principle we previously proposed.