# Formalism of a harmonic oscillator in the future-included complex action theory

## TL;DR

This paper explores a complex action harmonic oscillator model within a future-included framework, classifying its phases, constructing eigenstates and coherent states, and deriving an effective $Q$-Hermitian Hamiltonian using a maximization principle.

## Contribution

It introduces a novel complex harmonic oscillator model with a future-included formalism, classifies its phases, and formulates a $Q$-Hermitian effective Hamiltonian based on a maximization principle.

## Key findings

- Classification of the model into several phases based on parameters
- Construction of eigenstates and coherent states in a modified inner product
- Derivation of an effective $Q$-Hermitian Hamiltonian using the maximization principle

## Abstract

In a special representation of complex action theory that we call ``future-included'', we study a harmonic oscillator model defined with a non-normal Hamiltonian $\hat{H}$, in which a mass $m$ and an angular frequency $\omega$ are taken to be complex numbers. In order for the model to be sensible some restrictions on $m$ and $\omega$ are required. We draw a phase diagram in the plane of the arguments of $m$ and $\omega$, according to which the model is classified into several types. In addition, we formulate two pairs of annihilation and creation operators, two series of eigenstates of the Hamiltonians $\hat{H}$ and $\hat{H}^\dag$, and coherent states. They are normalized in a modified inner product $I_Q$, with respect to which the Hamiltonian $\hat{H}$ becomes normal. Furthermore, applying to the model the maximization principle that we previously proposed, we obtain an effective theory described by a Hamiltonian that is $Q$-Hermitian, i.e. Hermitian with respect to the modified inner product $I_Q$. The generic solution to the model is found to be the ``ground'' state. Finally we discuss what the solution implies.