N体问题中的量子参考系切换与全局关系性视角的缺失
Switching quantum reference frames in the N-body problem and the absence of global relational perspectives
AI总结：
本研究延续统一方法，基于对称性原理构建视角中立结构以实现量子参考系切换，在存在Gribov问题的关系性N体模型中完成3体QRF变换构建，揭示狄拉克与约化量子化的不等价性。
摘要：
鉴于量子参考系（QRFs）对量子物理和引力物理的重要性，有必要开发一种系统方法，用于在相对于不同QRF选择的物理描述之间进行切换，该方法在两个领域中均有效。本文延续了arxiv:1809.00556中提出的这种统一方法，其核心要素是一项对称性原理，该原理要求物理具有关系性。得益于规范相关的冗余性，这催生了一种视角中立结构，该结构同时包含所有参考系选择，且可通过它一致地切换不同参考系视角。用约束系统的语言表述，该视角中立结构在经典层面是约束曲面，在狄拉克量子化理论中是规范不变希尔伯特空间。相比之下，相对于特定参考系的视角对应于一种规范选择以及相关的约化相空间和约化希尔伯特空间。因此，QRF的变换相当于一种规范变换。我们证明它们采取“量子坐标变换”的形式。我们在一个通用力学模型中对此进行了说明，即具有旋转和平移对称性的三维空间关系性N体问题。该模型尤为有趣，因为它存在Gribov问题，因此不存在全局有效的规范固定条件，也就不存在全局的关系性参考系视角。约束曲面在拓扑上是非平凡的，由3维、5维和6维的规范轨道叶化，其中低维轨道是一组零测度集。N体问题也不允许存在全局有效的正则共轭狄拉克可观测量对。尽管存在这些挑战，我们仍展示了如何为3体问题构建QRF变换。我们的构建还通过其与QRF视角的相互作用，为狄拉克量子化与约化量子化的普遍不等价性提供了新的见解。
英文摘要：
Given the importance of quantum reference frames (QRFs) to both quantum and gravitational physics, it is pertinent to develop a systematic method for switching between the descriptions of physics relative to different choices of QRFs, which is valid in both fields. Here we continue with such a unifying approach, begun in arxiv:1809.00556, whose key ingredient is a symmetry principle, which enforces physics to be relational. Thanks to gauge related redundancies, this leads to a perspective-neutral structure which contains all frame choices at once and via which frame perspectives can be consistently switched. Formulated in the language of constrained systems, the perspective-neutral structure is the constraint surface classically and the gauge invariant Hilbert space in the Dirac quantized theory. By contrast, a perspective relative to a specific frame corresponds to a gauge choice and the associated reduced phase and Hilbert space. QRF changes thus amount to a gauge transformation. We show that they take the form of `quantum coordinate changes'. We illustrate this in a general mechanical model, namely the relational $N$-body problem in 3D space with rotational and translational symmetry. This model is especially interesting because it features the Gribov problem so that globally valid gauge fixing conditions, and hence relational frame perspectives, are absent. The constraint surface is topologically non-trivial and foliated by 3-, 5- and 6-dimensional gauge orbits, where the lower dimensional orbits are a set of measure zero. The $N$-body problem also does not admit globally valid canonically conjugate pairs of Dirac observables. These challenges notwithstanding, we exhibit how one can construct the QRF transformations for the 3-body problem. Our construction also sheds new light on the generic inequivalence of Dirac and reduced quantization through its interplay with QRF perspectives.
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