# The group structure of dynamical transformations between quantum reference frames

Angel Ballesteros1, Flaminia Giacomini2, and Giulia Gubitosi3

1Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain
2Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada
3Dipartimento di Fisica Ettore Pancini, Università di Napoli Federico II, and INFN, Sezione di Napoli, Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy

### 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.

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### Cited by

[1] Flaminia Giacomini, "Spacetime Quantum Reference Frames and superpositions of proper times", Quantum 5, 508 (2021).

[2] Ismael L. Paiva, Augusto C. Lobo, and Eliahu Cohen, "Flow of time during energy measurements and the resulting time-energy uncertainty relations", Quantum 6, 683 (2022).

[3] Flaminia Giacomini and Achim Kempf, "Second-quantized Unruh-DeWitt detectors and their quantum reference frame transformations", Physical Review D 105 12, 125001 (2022).

[4] Shadi Ali Ahmad, Thomas D. Galley, Philipp A. Höhn, Maximilian P. E. Lock, and Alexander R. H. Smith, "Quantum Relativity of Subsystems", arXiv:2103.01232, Physical Review Letters 128 17, 170401 (2022).

[5] Giulia Gubitosi, Fedele Lizzi, José Javier Relancio, and Patrizia Vitale, "Double quantization", Physical Review D 105 12, 126013 (2022).

[6] Otto C.W. Kong, "Quantum frames of reference and the noncommutative values of observables", Results in Physics 31, 105033 (2021).

[7] Marion Mikusch, Luis C. Barbado, and Časlav Brukner, "Transformation of spin in quantum reference frames", Physical Review Research 3 4, 043138 (2021).

[8] Flaminia Giacomini and Časlav Brukner, "Quantum superposition of spacetimes obeys Einstein's equivalence principle", AVS Quantum Science 4 1, 015601 (2022).

[9] Sylvain Carrozza and Philipp A. Höhn, "Edge modes as reference frames and boundary actions from post-selection", Journal of High Energy Physics 2022 2, 172 (2022).

[10] Christophe Goeller, Philipp A. Hoehn, and Josh Kirklin, "Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance", arXiv:2206.01193.

[11] Sylvain Carrozza, Stefan Eccles, and Philipp A. Hoehn, "Edge modes as dynamical frames: charges from post-selection in generally covariant theories", arXiv:2205.00913.

[12] Anne-Catherine de la Hamette, Thomas D. Galley, Philipp A. Hoehn, Leon Loveridge, and Markus P. Mueller, "Perspective-neutral approach to quantum frame covariance for general symmetry groups", arXiv:2110.13824.

[13] Jianhao M. Yang, "Quantum Mechanics from Relational Properties, Part III: Path Integral Implementation", arXiv:1807.01583.

The above citations are from Crossref's cited-by service (last updated successfully 2022-07-05 19:23:20) and SAO/NASA ADS (last updated successfully 2022-07-05 19:23:21). The list may be incomplete as not all publishers provide suitable and complete citation data.