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Spacetime Quantum Reference Frames and superpositions of proper times
Quantum ( IF 6.4 ) Pub Date : 2021-07-22 , DOI: 10.22331/q-2021-07-22-508 Flaminia Giacomini 1
Quantum ( IF 6.4 ) Pub Date : 2021-07-22 , DOI: 10.22331/q-2021-07-22-508 Flaminia Giacomini 1
Affiliation
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning.
Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.
Concretely, we consider a system of $N$ relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the $N$ particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
中文翻译:
时空量子参考系和适当时间的叠加
在广义相对论中,时空的描述依赖于理想化的杆和钟,它们确定了一个参考系。在任何具体场景中,参考系都与物理系统相关联,最终本质上是量子的。因此,物理定律的相对论描述需要考虑这样的量子参考系(QRF),通过它可以赋予时空操作意义。
在这里,我们引入了时空量子参考系的概念,它与时空中的量子粒子相关联。这种表述的优点是平等对待空间和时间,并允许我们从另一个量子系统的角度描述一组量子系统的动力学演化,其中其余物理系统演化的参数重合以粒子的适当时间作为 QRF。至关重要的是,两个不同 QRF 中的适当时间与标准变换无关,但它们可能处于一个相对于另一个的量子叠加中。
具体来说,我们考虑一个弱引力场中的 $N$ 相对论量子粒子系统,并引入一个永恒的公式,其中 $N$ 粒子的全局状态看起来“冻结”,但动力学演化在关系方面恢复数量。粒子的位置和动量希尔伯特空间用于通过转换到粒子的局部坐标系来固定 QRF,使得度量在 QRF 的原点是局部惯性的。内部希尔伯特空间对应于时钟空间,它在粒子的局部坐标系中保持适当的时间。由于这种完全关系的构造,我们展示了剩余粒子如何从 QRF 的角度在关系变量中动态演化。当忽略外部自由度时,此处提出的构造包括用于非交互时钟的 Page-Wootters 机制。最后,我们发现在 QRF 中可以观察到引力红移的量子叠加和狭义相对论时间膨胀的量子叠加。
更新日期:2021-09-06
Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.
Concretely, we consider a system of $N$ relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the $N$ particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
中文翻译:
时空量子参考系和适当时间的叠加
在广义相对论中,时空的描述依赖于理想化的杆和钟,它们确定了一个参考系。在任何具体场景中,参考系都与物理系统相关联,最终本质上是量子的。因此,物理定律的相对论描述需要考虑这样的量子参考系(QRF),通过它可以赋予时空操作意义。
在这里,我们引入了时空量子参考系的概念,它与时空中的量子粒子相关联。这种表述的优点是平等对待空间和时间,并允许我们从另一个量子系统的角度描述一组量子系统的动力学演化,其中其余物理系统演化的参数重合以粒子的适当时间作为 QRF。至关重要的是,两个不同 QRF 中的适当时间与标准变换无关,但它们可能处于一个相对于另一个的量子叠加中。
具体来说,我们考虑一个弱引力场中的 $N$ 相对论量子粒子系统,并引入一个永恒的公式,其中 $N$ 粒子的全局状态看起来“冻结”,但动力学演化在关系方面恢复数量。粒子的位置和动量希尔伯特空间用于通过转换到粒子的局部坐标系来固定 QRF,使得度量在 QRF 的原点是局部惯性的。内部希尔伯特空间对应于时钟空间,它在粒子的局部坐标系中保持适当的时间。由于这种完全关系的构造,我们展示了剩余粒子如何从 QRF 的角度在关系变量中动态演化。当忽略外部自由度时,此处提出的构造包括用于非交互时钟的 Page-Wootters 机制。最后,我们发现在 QRF 中可以观察到引力红移的量子叠加和狭义相对论时间膨胀的量子叠加。