Reality and hermiticity from maximizing overlap in the future-included complex action theory

TL;DR

This paper explores a complex action theory with future-included states, demonstrating that maximizing transition amplitude leads to real, Hermitian evolution and providing a method to construct a consistent $Q$-Hermitian framework.

Contribution

It introduces a theorem linking maximized transition amplitudes to reality and Hermiticity in a future-included complex action theory, and proposes a way to formulate $Q$-Hermitian operators.

Findings

Normalized matrix elements become real under $Q$-Hermitian operators.

Time evolution is governed by a $Q$-Hermitian Hamiltonian when transition amplitude is maximized.

A procedure to construct $Q$-Hermitian Hamiltonians and conserved currents is provided.

Abstract

In the complex action theory whose path runs over not only past but also future we study a normalized matrix element of an operator $\hat{\cal O}$ defined in terms of the future state at the latest time $T_B$ and the past state at the earliest time $T_A$ with a proper inner product that makes normal a given Hamiltonian that is non-normal at first. We present a theorem that states that, provided that the operator $\hat{\cal O}$ is $Q$-Hermitian, i.e., Hermitian with regard to the proper inner product, 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. Furthermore, we give a possible procedure to formulate the $Q$-Hermitian Hamiltonian in terms of $Q$-Hermitian coordinate and momentum operators, and construct a conserved probability current density.