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.
Highlights
In the standard description of quantum mechanics, the quantum states arising as solutions of the Schrodinger equation of a free particle are invariant under transformations between reference frames linked by Galilei transformations [1, 2]
In this work we have contributed to the understanding of the role of quantum reference frame (QRF) transformations as symmetries of quantum mechanical systems
It had been previously demonstrated that QRF transformations are extended symmetries of the free particle Hamiltonian, in the sense explained around Eq (2.10)
Summary
In the standard description of quantum mechanics, the quantum states arising as solutions of the Schrodinger equation of a free particle are invariant under transformations between reference frames linked by (centrally extended) Galilei transformations [1, 2]. The laws of transformation between different QRFs must somewhat generalize the usual Galilei transformations, in order to accommodate the quantum properties of the transformation parameters and the relational description of the phase space coordinates Thanks to these developments, the reference frames become less abstract entities, and can be identified with specific elements of the system. In this case we find that the Lie algebra of which the extended translation and boost generators are part is even larger, namely a 7-dimensional one, that will be called the dynamical Lie algebra for QRFs. Once more it only contains generators that had already been included in the QRF transformations of Sec. 2, in order to ensure that the latter define extended symmetry transformations of the Hamiltonian and in order to preserve the relational nature of the phase space coordinates. These works deal with finite-dimensional systems, and focus on a different question —understanding a specific QRF transformation as a symmetry of a given quantum system— to the one we answer here, i.e., finding that there exists a Lie group of inertial transformations for QRFs that generalizes the Galilean inertial transformations for the corresponding classical reference frames
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