包含复杂action理论的未来最大化重叠的实在性和厄米性
Reality and hermiticity from maximizing overlap in the future-included complex action theory
摘要：
在复作用量理论中，其路径不仅经过过去，而且经过未来，我们研究了算子$\hat{\cal O}$的归一化矩阵元，该算子由最近时刻的未来状态$T_B$和最早时刻的过去状态$T_A$定义，并具有一个适当的内积，使得一个给定的哈密顿量是正规的，而该哈密顿量起初是非正规的.我们提出了一个定理，该定理指出，如果算子$\hat{\cal O}$是$Q$ -厄米特算子，即，关于适当的内积的厄米算符，归一化矩阵元素变为实数，并且在用于选择的过去和未来状态的$Q$ -厄米算符哈密尔顿算子下时间发展，使得从过去状态到未来状态的跃迁幅度的绝对值最大化。此外，我们给出了一种用$Q$ -Hermite坐标和动量算符表示$Q$-Hermite哈密顿量的可能方法，并构造了一个守恒的几率流密度.
英文摘要：
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.
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