Switching quantum reference frames in the N-body problem and the absence of global relational perspectives
Published 2018-09-13Version 1
Given the importance of quantum reference systems 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 quantum reference systems, which is valid in both fields. Here, we continue with such a unifying approach, begun in arxiv:1809.00556, whose key ingredients is a gravity-inspired symmetry principle, which 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 theory. By contrast, a perspective relative to a specific frame corresponds to a gauge choice and the associated reduced phase and Hilbert space. Quantum reference frame switches thereby amount to a symmetry transformation. In the quantum theory, they require a transformation that takes one from the Dirac to a reduced quantum theory and we show that it amounts to a trivialization of the constraints and a subsequent projection onto the classical gauge fixing conditions. We illustrate this method in the relational $N$-body problem with rotational and translational symmetry. This model is particularly interesting because it features the Gribov problem so that globally valid gauge fixing conditions are impossible which, in turn, implies also that globally valid relational frame perspectives are absent in both the classical and quantum theory. These challenges notwithstanding, we exhibit how one can systematically construct the quantum reference frame transformations for the three-body problem.