Reality from maximizing overlap in the periodic complex action theory

Abstract

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