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

Abstract

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 [1], whose key ingredients are an operational language and 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 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 a general mechanical particle model, namely the relational N -body problem in three-dimensional space with rotational and translational symmetry. This model is particularly interesting because it features the generic 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. We will show that the constraint surface is topologically non-trivial and foliated by three-, five-and six-dimensional gauge orbits, 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 systematically construct the quantum reference frame transformations for the three-body problem.

Transformation of Spin in Quantum Reference Frames

“…displacement of the translation, velocity of the boost) is replaced by the quantum operator acting on the QRF, so that the transformation generalizes the usual coordinate transformations to a "superposition of coordinate transformations". Some of these QRF transformations have subsequently been rederived starting from a gravity inspired symmetry principle in a perspective neutral model [23] and extended to three-dimensional problems with translational and rotational invariance [26].…”

Quantum superposition of spacetimes obeys Einstein's Equivalence Principle

“…[19] and further developed in Refs. [20][21][22][23][24][25][26][27][28], allows us to reconcile these fundamental principles. As a consequence, the spatial superposition of massive bodies is not to be considered as giving rise to inconsistencies between General Relativity and Quantum Theory.…”

“…Notice that, in this view, the position of a system in a specific choice of coordinates is not an abstract labelling of a manifold, but is defined as a distance between two physical systems, one of which serves as the origin of the coordinate system. We have introduced the notion of a Quantum Reference Frame (QRF) [18][19][20][21][22][23][24][25][26][27][28], i.e., a reference frame associated to a quantum system, whose state can be in a superposition or entangled with other physical systems. The key feature of the formalism that we exploit here is the generalisation of the standard diffeomorphism to a quantum superposition of classical diffeomorphisms.…”

Spacetime Quantum Reference Frames and superpositions of proper times

“…Refs. [26][27][28] adopt a symmetry principle to describe a Galilean system in a fully relational way, and Refs. [29,30] extend the formalism to the special relativistic regime, applying it to concrete problems in Relativistic Quantum Information.…”