# HET seminar: Keiichi Nagao

Speaker:  Keiichi Nagao

Title: Formalism of harmonic oscillator in the future-included complex action theory

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, which is described by a
Hamiltonian that is $Q$-Hermitian, i.e., Hermitian with respect to the modified inner product $I_Q$.
The solution to the model is found to be the vacuum state.
Finally we discuss what the solution implies.
This work is based on the work with Holger Bech Nielsen (arXiv:1902.01424 [quant-ph]).