# Reality from maximizing overlap in the future-included real action theory

## TL;DR

This paper proves that in a future-included real action quantum theory, the normalized matrix element of a Hermitian operator becomes real when the future state maximizes the transition amplitude, suggesting a more natural quantum formalism.

## Contribution

It extends a previous theorem to the real action case, showing the normalized matrix element's reality under maximization, and discusses its implications for quantum theory formalism.

## Key findings

- Normalized matrix element becomes real when transition amplitude is maximized.
- Maximization principle does not strongly constrain future and past states in real action theory.
- Normalized matrix element may be more natural than expectation value in quantum formalism.

## Abstract

In the future-included real action theory whose path runs over not only past but also future, we demonstrate a theorem, which states that the normalized matrix element of a Hermitian operator $\hat{\cal O}$ defined in terms of the future state at the final time $T_B$ and the fixed past state at the initial time $T_A$ becomes real for the future state selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. This is a special version of our previously proposed theorem for the future-included complex action theory. We find that though the maximization principle leads to the reality of the normalized matrix element in the future-included real action theory, it does not specify the future and past states so much as in the case of the future-included complex action theory. In addition, we argue that the normalized matrix element seems to be more natural than the usual expectation value. Thus we speculate that the functional integral formalism of quantum theory could be most elegant in the future-included complex action theory.