arXiv 1902.014242019-02quant-phhep-th

含未来复action理论中谐振子的形式

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

Keiichi Nagao, Holger Bech Nielsen

备注：: Latex 42 pages, 3 figures, typos corrected, presentation improved, the final version to appear in Prog.Theor.Exp.Phys

摘要：

在复作用量理论的一个特殊表示中，我们称之为“包含未来的”，我们研究了一个定义为非正规哈密顿量$\hat{H}$的谐振子模型，其中质量$m$和角频率$ω$取为复数.为了使模型合理，需要对$m$和$ω$进行一些限制。我们在$m$和$ω$的自变量平面上绘制了相图，根据相图将模型分为几种类型。此外，我们还提出了两对湮灭算符和产生算符，两系列哈密顿量$\hat{H}$和$\hat{H}^†$的本征态，以及相干态。它们在修改的内积$I_Q$中被归一化，相对于该修改的内积，哈密顿量$\hat{H}$变得正常。此外，应用我们以前提出的最大化原理，我们得到了一个有效的理论描述的哈密尔顿是$Q$ -厄米特，即厄米特关于修改的内积$I_Q$。该模型的一般解被认为是“基态”。最后，我们讨论解决方案意味着什么。

英文摘要：

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 $ω$ are taken to be complex numbers. In order for the model to be sensible some restrictions on $m$ and $ω$ are required. We draw a phase diagram in the plane of the arguments of $m$ and $ω$, 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 $\hat{H}^†$, 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.

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