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arXiv:2502.21298 · Modern Physics Letters A

The quantum Newton’s bucket: Active and passive rotations in quantum theory

Augusto Facundes, Kayman Jhosef Carvalho Gonçalves, Giorgio Torrieri

Published Dec 2, 2025• 1 citation
arXivHTMLPDF

Abstract

Motivated both by classical physics problems associated with “Newton’s bucket” and recent developments related to QCD in rotating frames of reference relevant to heavy ion collisions, we discuss the difference between “active” and “passive” rotations in quantum systems. We examine some relevant potentials and give general symmetry arguments to give criteria where such rotations give the same results. We close with a discussion of how this can be translated to problems of current interest in quantum field theory and quantum gravity.

DOI
10.1142/s0217732325400012

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Motivated both by classical physics problems associated with “Newton’s bucket” and recent developments related to QCD in rotating frames of reference relevant to heavy ion collisions, we discuss the difference between “active” and “passive” rotations in quantum systems. We examine some relevant potentials and give general symmetry arguments to give criteria where such rotations give the same results. We close with a discussion of how this can be translated to problems of current interest in quantum field theory and quantum gravity.
The topic of quantum reference frames (QRFs) has attracted a great deal of attention in the recent literature. Potentially, the correct description of such frames is important for both the technological applications of quantum mechanics and for its foundations, including the search for a future theory of quantum gravity. In this letter, we point out potential inconsistencies in the mainstream approach to this subject and propose an alternative definition that avoids these problems. Crucially, we reject the notion that transformations between QRFs can be represented by unitary operators and explain the clear physical reasons for this. An experimental protocol, capable of empirically distinguishing between competing definitions of the term, is also proposed. The implications of the new model, for uncertainty relations, spacetime symmetries, gauge symmetries, the quantisation of gravity, and other foundational issues are discussed, and possible directions for future work in this field are considered.
Abstract Quantum reference frame as in the example of observing a quantum particle from another has been a topic of much recent interest. Quantum spatial translations, quantum rotations, and quantum Lorentz boosts in the sense have been studied to some extent. The article aims at using a consistent formulation of all that to give a full picture of what would be the symmetry of Special Quantum Relativity as special relativity as seen from quantum particles. We first look at a direct formulation of the mathematical quantum analog of any classical one-parameter Lie group of such transformations suggested by our Heisenberg picture thinking of quantum mechanics. A quantum rotation and a quantum Lorentz boost are presented. Analysis of the compositions of two such transformations shows unnatural results with various problems, especially with their commutators. From a physical point of view, we argue for a modified formulation. A quantum reference frame transformation, as the description of firstly the spacetime position of an object, idealized as a particle, as seen from another has to involve a full set of four-coordinate quantum quantities instead of one in a single coordinate only. Unlike the classical case, we illustrate that the analog of a change by the zero classical coordinate values has nontrivial effects in the transformation. A consistent full picture of the symmetry is then obtained. Results that could be interpreted essentially as the quantum analogs of the translation of a single coordinate observable, including angular ones for a rotation or a boost, can be retrieved from that but only as approximations. Under the traditional perspective of relativity symmetry being about spacetime reference frames, our task is then completed. We further look at the quantum version of Heisenberg–Weyl symmetry and perspectives on its relation with the subject matter and argue for the necessity of our background noncommutative geometric perspective for a conceptual consistent story.
Recent developments in quantum theory have focused attention on fundamental questions, in particular on whether it might be necessary to modify quantum mechanics to reconcile quantum gravity and general relativity. This book is based on a conference held in Oxford in the spring of 1984 to discuss quantum gravity. It brings together contributors who examine different aspects of the problem, including the experimental support for quantum mechanics, its strange and apparently paradoxical features, its underlying philosophy, and possible modifications to the theory.
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 [Quantum 4, 225 (2020)], 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.
Abstract In quantum mechanics courses, the concept of a reference frame is frequently hidden. In this article, we present a basic perspective to introduce the transformations of a reference frame based on unitary transformations. These unitary transformations are inspired in canonical transformation for reference frame from the perspective of classical mechanics. The manner we deal with this change in reference frame allows us to discuss some fundamental concepts in Schrödinger and Heisenberg pictures, as well as the ideas of passive and active description in quantum mechanics. These unitary transformations are inspired in a reference-frame canonical transformation from the perspective of classical mechanics. The approach we take for this reference-frame change allows for a discussion on fundamental concepts in Schrödinger and Heisenberg pictures, as well as the ideas of passive and active description in quantum mechanics. We apply this unitary transformation in five examples.