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

TL;DR

This work develops a relational, gauge-theoretic framework for switching quantum reference frames in a 3D N-body system with translational and rotational invariance, highlighting a perspective-neutral (Dirac) quantum structure that contains all frame descriptions. Quantum frame changes are implemented as quantum coordinate maps that traverse through the perspective-neutral space, revealing that Dirac and reduced quantizations are generally inequivalent and that Dirac quantization should be privileged for quantum frame covariance. The approach demonstrates unitary, frame-dependent transformations for regular states and shows that entanglement is not frame-invariant, providing a concrete operational handle on how quantum information depends on the chosen relational description. The results extend prior one-dimensional QRF work to a full 3D setting and illuminate the interplay between gauge structure, Gribov-type obstructions, and the globalization of relational observables in quantum gravity and foundations.

Abstract

Given the importance of quantum reference frames (QRFs) 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 QRFs, which is valid in both fields. Here we continue with such a unifying approach, begun in arxiv:1809.00556, whose key ingredient is a symmetry principle, which enforces physics to be relational. Thanks to gauge related redundancies, this leads 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 is 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. QRF changes thus amount to a gauge transformation. We show that they take the form of `quantum coordinate changes'. We illustrate this in a general mechanical model, namely the relational $N$-body problem in 3D space with rotational and translational symmetry. This model is especially interesting because it features the Gribov problem so that globally valid gauge fixing conditions, and hence relational frame perspectives, are absent. The constraint surface is topologically non-trivial and foliated by 3-, 5- and 6-dimensional gauge orbits, where the lower dimensional orbits are a set of measure zero. The $N$-body problem also does not admit globally valid canonically conjugate pairs of Dirac observables. These challenges notwithstanding, we exhibit how one can construct the QRF transformations for the 3-body problem. Our construction also sheds new light on the generic inequivalence of Dirac and reduced quantization through its interplay with QRF perspectives.

Figures (3)

• Figure 1: (a) Illustration of the three-body problem and gauge invariant configuration degrees of freedom. Our three configuration Dirac observables from the (almost everywhere) conjugate pairs in (\ref{['conjDirac3d']}) are functions of these: $\rho_{BA},\rho_{CA}$ are simply the logs of the relative distances $r_{AB}=|\vec{q}_A-\vec{q}_B|,r_{AC}=|\vec{q}_A-\vec{q}_C|$ between $A$ and $B,C$, respectively, while $u$ is (minus) the cotangent of the relative angle between $B$ and $C$ as seen from $A$. (b) Illustration of a generic situation in the six-body problem as a triangulation. In order to fully localize particle $F$ relative to particles $A$ to $E$, three relative distances between $F$ and the rest are sufficient and not all relative distances are independent.
• Figure 2: Illustration of the gauge-fixing procedure (\ref{['constraintsChi']}--\ref{['constraintDiscrete12']}) for the three-body problem. We choose $A$ as the origin from which to measure all distances, $B$ to define the $z$-axis and $C$ to define the $x$-$z$-plane. This procedure works in the same way for $N>3$ particles by choosing a three-body subsystem.
• Figure 3: Diagram of the two quantization methods of sec. \ref{['quantum3D']} for the three-body problem and their relation (see also Table \ref{['tablenotations']} for a summary of the notations used). Each column represents a step from the Dirac to the reduced quantum theory, as explained in the main text. Vertical arrows on the left side correspond to constraint imposition. The horizontal arrows between Hilbert spaces are all isometries and correspond to the quantum reduction steps. They are a sequence of: (1) trivialization of the translation generators, (2) conditioning on the classical conditions $\vec{\chi}=0$ that fix the translational gauge freedom, (3) trivialization of the rotation generators, and (4) conditioning on the classical conditions $\vec{\phi}=0$ fixing the rotational gauge freedom. The two squared diagrams among Hilbert spaces on the lower left are commutative. For better visualization, we have summarized the relevant phase and Hilbert spaces appearing in this diagram in a table. The last reduction map $\bra{\vec{\phi}=0}$ is not globally defined owing to the global gauge fixing issues. It does, however, act as an isometry on its image.