Reality from maximizing overlap in the future-included theories

TL;DR

This paper reviews a theorem in future-included quantum theories with complex and real actions, showing that under certain conditions, the normalized matrix element becomes real and evolves with a $Q$-Hermitian Hamiltonian, suggesting a potentially elegant formulation of quantum mechanics.

Contribution

It introduces a theorem relating normalized matrix elements to $Q$-Hermitian operators in future-included theories, highlighting conditions for real-valued, time-evolving quantities.

Findings

Normalized matrix element becomes real under $Q$-Hermitian operators.

Time evolution is governed by a $Q$-Hermitian Hamiltonian.

Future-included complex action theory may be the most elegant quantum theory.

Abstract

In the future-included complex and real action theories whose paths run over not only the past but also the future, we briefly review the theorem on the normalized matrix element of an operator $\hat{\cal O}$, which is defined in terms of the future and past states with a proper inner product $I_Q$ that makes a given Hamiltonian normal. The theorem states that, provided that the operator $\hat{\cal O}$ is $Q$-Hermitian, i.e. Hermitian with regard to the proper inner product $I_Q$, the normalized matrix element becomes real and time-develops under a $Q$-Hermitian Hamiltonian for the past and future states selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. Discussing what the theorem implicates, we speculate that the future-included complex action theory would be the most elegant quantum theory.