量子参考系的广群方法
A groupoidal approach to quantum reference frames
AI总结：
针对弯曲时空缺乏全局对称群导致常规群基量子参考系不适用的问题，提出广群基量子参考框架，构建了相对化映射与关系可观测量，拓展了弯曲时空关系量子场论并为关系量子杨-米尔斯理论奠基。
AI中文摘要：
我们建立了弯曲时空中基于广群的关系量子场论（RQFT）的运动学基础与算子代数基础。实际上，常规的基于群的量子参考系（QRF）形式体系无法直接适用于一般的弯曲洛伦兹背景，因为这类时空通常不存在全局对称群，或全局对称群规模过小。我们构建了适用于连续广群的QRF概念，由此得到了广群相对化映射与关系可观测量，并证明局域化极限可恢复普通的非关系描述。我们构造了典范的锐广群QRF，它对应于基于$L^2(G)$的理想群QRF的广群版本；进一步证明对于局部紧群，广群QRF构造可退化为标准的操作型QRF形式体系。作用广群QRF仅对特定类别的正算子值测度（POVMs）场构成 torsor QRF，且torsor相对化映射仅适用于系统可观测量的常算子场。\n我们回顾了闵可夫斯基时空中RQFT的基础，证明了进一步关联RQFT与怀特曼（Wightman）QFT的新结果：协变POVM相对于拟不变的σ有限正博雷尔测度$μ$具有$μ$连续性，因此关系量子场可理解为逐点定义的核关于QRF统计量的涂抹。我们发展了弯曲时空中的RQFT，指出庞加莱群的正确替代物是时空的庞加莱广群；还说明了该框架如何通过阿蒂亚（Atiyah）广群进一步拓展到内部规范对称性与关系规范协变量子场论，为构建关系量子杨-米尔斯场论迈出了第一步。
英文摘要：
We develop the kinematical and operator-algebraic foundations of a groupoid-based relational quantum field theory (RQFT) on curved spacetimes. Indeed, the usual group-based quantum reference frame (QRF) formalism is not directly suited to generic curved Lorentzian backgrounds as global symmetry groups are typically absent or too small. We formulate a notion of QRF for a continuous groupoid. This yields a groupoid relativization map and relational observables. We show that a localization limit recovers the ordinary non-relational description. We construct canonical sharp groupoid QRFs, which form the groupoidal counterpart of the ideal group QRFs based on $L^2(G)$. We further prove that the groupoid QRF construction reduces to the standard operational QRF formalism for locally compact groups. The action groupoid QRFs are torsor QRFs only for specific classes of fields of positive operator-valued measures (POVMs), and the torsor relativization map only applies to constant operator fields of system observables. We review the foundations of RQFT in Minkowski spacetime. We prove new results that further link RQFT to Wightman QFT. We show that covariant POVMs are $μ$-continuous with respect to quasi-invariant $σ$-finite positive Borel measures $μ$. Thus, relational quantum fields can be understood as the smearing of pointwise-defined kernels with respect to the QRF's statistics. We develop RQFT in curved spacetime, where we argue that the correct replacement for the Poincaré group is the Poincaré groupoid of the spacetime. We also indicate how the framework extends further to internal gauge symmetry and relational gauge-covariant quantum field theory via Atiyah groupoids, providing a first step towards formulating a relational quantum Yang-Mills field theory.
