We propose a model to identify the quantum/classical boundary. The model introduces a spontaneous collapse of state superposition: $\frac{d}{dt} \rho_{ij} =-\frac{i}{\hbar}[H,\rho]_{ij}-\rho_{ij}/\tau_{ij}$. Different from other collapse models, the collapsing scale $\tau_{ij}$ here does not contain a universal parameter, but is specified by the two states $| i\rangle $ and $ | j\rangle$: If each state is {\em in principle} repeatedly readable (typically by a QND measurement), then $\tau_{ij}$ is the {\em potentially} needed measuring time to discriminate the two states, and the collapse occurs spontaneously {\em without} any actual monitoring. Otherwise, $\tau_{ij}=\infty$, which means no collapse and everlasting superposition. This happens if one state is not repeatedly readable, or if the two states cannot possibly be discriminated in a particular circumstance (for example in the Rabi oscillation). Detailed analysis shows that for a "trapped Schr{\"o}dinger's cat", the superposition of $|{\rm here} \rangle$ and $| {\rm there} \rangle $ is forbidden if $E D \gg 4\pi \hbar c$, and allowed if $E D \le 4\pi \hbar c$, where $D$ is the trap separation and $ E$ is the energy gap, which can be estimated with $ M v^2$. The model also constrains a "free Schr{\"o}dinger's cat" to display double-slit interference if $p\theta D\ge 8\hbar$, where $p= Mv$, $\theta $ is the angle spanned by the two trajectories, and $D$ is the slit separation. In contrast, this model sets no limit on the coherent length of massless photon, thus the arm of a Michelson interferometer can be arbitrarily long. The spontaneous collapse which we propose can occur for an isolated system, and parallels the decoherence induced by interaction with environment.

中文翻译：

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重复可读状态、自发坍缩和量子/经典边界

我们提出了一个模型来识别量子/经典边界。该模型引入了状态叠加的自发崩溃：$\frac{d}{dt} \rho_{ij} =-\frac{i}{\hbar}[H,\rho]_{ij}-\rho_{ij }/\tau_{ij}$。与其他折叠模型不同，这里的折叠尺度$\tau_{ij}$不包含通用参数，而是由两个状态$|指定。我\rangle $和$ | j\rangle$：如果每个状态{\em 原则上} 可重复读取（通常通过 QND 测量），则 $\tau_{ij}$ 是 {\em 可能} 需要的测量时间来区分这两种状态，并且崩溃自发发生 {\em 没有} 任何实际监控。否则，$\tau_{ij}=\infty$，表示没有塌陷和永久叠加。如果一个状态不能重复读取，就会发生这种情况，或者如果在特定情况下（例如在拉比振荡中）不可能区分这两种状态。详细分析表明，对于“被困的 Schr{\”odinger 的猫”，如果 $ED \gg 4，则 $|{\rm here} \rangle$ 和 $| {\rm there} \rangle $ 的叠加是被禁止的\pi \hbar c$，如果 $ED \le 4\pi \hbar c$ 则允许，其中 $D$ 是陷阱分离，$ E$ 是能隙，可以用 $ M v^2$ 估计. 如果$p\theta D\ge 8\hbar$，该模型还约束“自由Schr{\”odinger's cat”显示双缝干涉，其中$p= Mv$，$\theta$是角度由两条轨迹跨越，$D$ 是狭缝间隔。相比之下，该模型对无质量光子的相干长度没有限制，因此迈克尔逊干涉仪的臂可以任意长。