Reality from maximizing overlap in the periodic complex action theory

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 $⟨\hat{\mathcal{O}}{⟩}_{\mathrm{periodic}\mathrm{time}}$. We present two theorems stating that $⟨\hat{\mathcal{O}}{⟩}_{\mathrm{periodic}\mathrm{time}}$ 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 $t_p$ 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 $t_p$ selected such that the absolute value of the transition amplitude is maximized in a case where $B\le 0$ 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.

arXiv commentLatex 14 pages, typos corrected, presentation improved, the final version to appear in Prog.Theor.Exp.Phys

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Cited by in corpus (1)

• Automatic hermiticity for mixed states