Formalism of a harmonic oscillator in the future-included complex action theory
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Publication:5163110
DOI10.1093/PTEP/PTZ047zbMath1479.81020arXiv1902.01424OpenAlexW2963286881MaRDI QIDQ5163110
Publication date: 8 November 2021
Published in: Unnamed Author (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/1902.01424
General topics in linear spectral theory for PDEs (35P05) Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) Coherent states (81R30) Operator algebra methods applied to problems in quantum theory (81R15) Nonselfadjoint operator theory in quantum theory including creation and destruction operators (81Q12)
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