Switching quantum reference frames in the N-body problem and the absence of global relational perspectives

Given the importance of quantum reference systems to both quantum and grav-itational physics, it is pertinent to develop a systematic method for switching be-tween the descriptions of physics relative to different choices of quantum reference systems. Here, we expand on a unifying approach, begun in [1], which blends the operational language of [2] with a gravity-inspired symmetry principle. The latter enforces physics to be relational and leads, thanks to gauge related redundancies, 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 turns out to be the constraint surface classically and the gauge invariant Hilbert space in the Dirac quantized the-ory. By contrast, a perspective relative to a specific frame corresponds to a gauge choice and involves a symmetry reduction procedure to an associated reduced phase and Hilbert space. Quantum reference frame switches thereby amount to a particular gauge transformation. Here, we show that they take the form of 'quantum coordinate changes'. We illustrate this method in a general mechanical particle model, namely the relational N-body problem in three-dimensional space with rotational and translational sym-metry. This model is particularly interesting because it features an analog of the generic Gribov problem so that globally valid gauge fixing conditions, and thereby relational frame perspectives, are absent. We will show that the constraint surface is topologically non-trivial and foliated by three-, five-and six-dimensional gauge or-bits, where the lower dimensional orbits are a set of measure zero. In consequence, the N-body problem also does not admit globally valid canonically conjugate pairs of Dirac observables. These challenges notwithstanding, we exhibit how one can con-struct the quantum reference frame transformations for the three-body problem. Our construction also sheds new light on the generic inequivalence of Dirac and reduced quantization through its interplay with quantum frame perspectives.