AI 中文总结
研究量子N粒子态自由度问题，发现经典物理中可通过约束消除非物理自由度，量子力学中则不能，所有3N个规范自由度都是物理的，非关系变量导致广义不确定性关系，探讨了对关系模型及量子引力研究的影响。
AI 中文摘要
在牛顿时空中，经典N粒子系统的规范描述需要3N个自由度。但并非所有自由度都是物理的，由于净动量守恒，只有3(N - 1)个加速度是独立的。可通过三个约束消除非物理的质心变量，这类似于规范固定过程。在经典物理中，物理自由度数量等于独立关系自由度数量。而在量子力学中并非如此，经典净动量的算符类似物不能用于施加限制自由度的约束而不损失物理信息，所有3N个规范自由度都是物理的，尽管只有3(N - 1)个是关系的。非关系变量会导致广义不确定性关系，其非海森堡项定义了量子参考系在实空间和动量空间的伽利略不变展宽。还讨论了该结果对关系模型近期工作的影响以及对更广泛关系计划的影响，特别是其与量子引力研究的相关性。
英文摘要
In Newtonian spacetime, the canonical description of a classical $N$-particle system requires $3N$ degrees of freedom. Not all of these are physical, however, since the conservation of the net momentum implies that only $3(N-1)$ accelerations are independent. Hence, three constraints can be used to eliminate the unphysical centre-of-mass variables, at the level of the Lagrangian, leaving only the subset of observable displacements and momenta, which are relational. Imposing the constraints does not change the dynamics of these variables, at the classical level, and is analogous to a gauge-fixing procedure, which removes redundancy in the description of the system. In classical physics, therefore, the number of physical degrees of freedom equals the number of independent relational degrees of freedom. Here, we show that this is not the case in quantum mechanics. While an operator-analogue of the classical net momentum exists, it cannot be used to impose constraints that restrict the degrees of freedom in the theory, without a loss of physical information. This means that all $3N$ canonical degrees of freedom are physical, even though only $3(N-1)$ of them are relational. We explore the physical consequences of the non-relational variables and show that they give rise to generalised uncertainty relations (GURs), for the relational quantities that define the quantum reference frame (QRF). Hence, it is shown that the non-relational degrees of freedom refer to the frame itself and that the non-Heisenberg terms in the GURs define its Galilean-invariant spreads, in both real space and momentum space. The implications of this result for recent work on relational models, including the ``perspective neutral'' framework for QRFs, are discussed. Its implications for the wider relational program, and, in particular, the relevance of the latter to quantum gravity research, are also critically assessed
Comments30 pages of main text, 25 pages of appendices, and 8 pages of references. No tables, no figures. Updated Acknowledgements (v2)
