Every clock is a physical system and thereby ultimately quantum. A naturally arising question is thus how to describe time evolution relative to quantum clocks and, specifically, how the dynamics relative to different quantum clocks are related. This is a particularly pressing issue in view of the multiple choice facet of the problem of time in quantum gravity, which posits that there is no distinguished choice of internal clock in generic general relativistic systems and that different choices lead to inequivalent quantum theories. Exploiting a recent unifying approach to switching quantum reference systems (arXiv:1809.00556, arXiv:1809:05093), we exhibit a systematic method for switching between different clock choices in the quantum theory. We illustrate it by means of the parametrized particle, which, like gravity, features a Hamiltonian constraint. We explicitly switch between the quantum evolution relative to the non-relativistic time variable and that relative to the particle's position, which requires carefully regularizing the zero-modes in the so-called time-of-arrival observable. While this toy model is simple, our approach is general and, in particular, directly amenable to quantum cosmology. It proceeds by systematically linking the reduced quantum theories relative to different clock choices via the clock-choice-neutral Dirac quantized theory, in analogy to coordinate changes on a manifold. This method overcomes the multiple choice problem here, showing that it is actually a multiple choice feature of the complete relational quantum theory, taken as the conjunction of Dirac and reduced quantized theories. Precisely this conjunction permits to consistently switch between different temporal reference systems, which is a prerequisite for a quantum notion of general covariance.

中文翻译：

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如何在关系量子时钟之间切换

每个时钟都是一个物理系统，因此最终是量子的。因此，一个自然产生的问题是如何描述相对于量子时钟的时间演化，特别是相对于不同量子时钟的动力学是如何相关的。鉴于量子引力中时间问题的多项选择方面，这是一个特别紧迫的问题，它假定在一般广义相对论系统中没有明显的内部时钟选择，不同的选择导致不等价的量子理论。利用最近切换量子参考系统的统一方法 (arXiv:1809.00556, arXiv:1809:05093)，我们展示了一种在量子理论中不同时钟选择之间切换的系统方法。我们通过参数化粒子来说明它，它像重力一样，具有哈密顿约束。我们明确地在相对于非相对论时间变量的量子演化和相对于粒子位置的量子演化之间切换，这需要在所谓的到达时间可观测值中仔细调整零模式。虽然这个玩具模型很简单，但我们的方法是通用的，特别是直接适用于量子宇宙学。它通过时钟选择中性狄拉克量子化理论系统地将相对于不同时钟选择的简化量子理论联系起来，类似于协调流形上的变化。这种方法克服了这里的多选问题，表明它实际上是完全关系量子理论的多选特征，是狄拉克和约简量子化理论的结合。