The group structure of dynamical transformations between quantum reference frames

TL;DR

The paper establishes a group-theoretic foundation for quantum reference frame transformations by constructing a relational (4D) and a dynamical (7D) Lie algebra of canonical transformations acting on the joint phase space of QRFs. It shows that QRF transformations close under a group law, with the relational algebra $\mathcal{R}(4)$ as a building block and a dynamical extension $\mathcal{D}(7)$ governing time-dependent transformations, including a BCH-type composition rule. The κ and ħ parameters encode the noncommutative structure of QRFs, and the κ→0 limit recovers the classical centrally extended Galilei algebra, while a t=0 sector reveals an accidental (2+1) Poincaré symmetry. These results unify extended symmetry transformations with time evolution and relational observables, clarifying how quantum and dynamical features of reference frames modify standard Galilean invariance. The work opens avenues for higher-dimensional QRFs and rotations, enabling a comprehensive group-theoretical treatment of frame changes in quantum mechanics.

Abstract

Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part of. While such transformations were shown to be symmetries of the system's Hamiltonian, the question remained unanswered as to whether they enjoy a group structure, similar to that of the Galilei group relating classical reference frames in quantum mechanics. In this work, we identify the canonical transformations on the phase space of the quantum systems comprising the quantum reference frames, and show that these transformations close a group structure defined by a Lie algebra, which is different from the usual Galilei algebra of quantum mechanics. We further find that the elements of this new algebra are in fact the building blocks of the quantum reference frames transformations previously identified, which we recover. Finally, we show how the transformations between classical reference frames described by the standard Galilei group symmetries can be obtained from the group of transformations between quantum reference frames by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations.