Quantum Galilei group as quantum reference frame transformations

Angel Ballesteros; Diego Fernandez-Silvestre; Flaminia Giacomini; Giulia Gubitosi

doi:10.22331/q-2025-12-10-1935

Quantum Galilei group as quantum reference frame transformations

Angel Ballesteros¹, Diego Fernandez-Silvestre², Flaminia Giacomini^3,4, and Giulia Gubitosi⁵

¹Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain.
²Departamento de Matemáticas y Computación, Universidad de Burgos, 09001 Burgos, Spain.
³Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland.
⁴Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133, Roma, Italy.
⁵Dipartimento di Fisica Ettore Pancini, Università di Napoli Federico II, and INFN, Sezione di Napoli, Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy.

Published:	2025-12-10, volume 9, page 1935
Editor:	Leon Loveridge
Eprint:	arXiv:2504.00569v3
Doi:	https://doi.org/10.22331/q-2025-12-10-1935
Citation:	Quantum 9, 1935 (2025).

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Abstract

Quantum groups have been widely explored as a tool to encode possible nontrivial generalisations of reference frame transformations, relevant in quantum gravity. In quantum information, it was found that the reference frames can be associated to quantum particles, leading to quantum reference frames transformations. The connection between these two frameworks is still unexplored, but if clarified it will lead to a more profound understanding of symmetries in quantum mechanics and quantum gravity.
Here, we establish a correspondence between quantum reference frame transformations and transformations generated by a quantum deformation of the Galilei group with commutative time, taken at first order in the quantum deformation parameter. This is found once the quantum group noncommutative transformation parameters are represented on the phase space of a quantum particle, and upon setting the quantum deformation parameter to be proportional to the inverse of the mass of the particle serving as the quantum reference frame. These results allow us to show that quantum reference frame transformations are physically relevant when the state of the quantum reference frame is in a quantum superposition of semiclassical states. We conjecture that the all-order quantum Galilei group describes quantum reference frame transformations between more general quantum states of the quantum reference frame.

► BibTeX data

@article{Ballesteros2025quantumgalileigroup,
  doi = {10.22331/q-2025-12-10-1935},
  url = {https://doi.org/10.22331/q-2025-12-10-1935},
  title = {Quantum {G}alilei group as quantum reference frame transformations},
  author = {Ballesteros, Angel and Fernandez-Silvestre, Diego and Giacomini, Flaminia and Gubitosi, Giulia},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1935},
  month = dec,
  year = {2025}
}

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[2] Vittorio D'Esposito, Giuseppe Fabiano, Domenico Frattulillo, and Flavio Mercati, "Doubly Quantum Mechanics", Quantum 9, 1721 (2025).

[3] Samuel Fedida and Jan Głowacki, "Foundations of Relational Quantum Field Theory I: Scalars", arXiv:2507.21601, (2025).

[4] Gaetano Fiore and Fedele Lizzi, "Mixed states for reference frames transformations", arXiv:2507.05758, (2025).

[5] Sébastien Christophe Garmier, Ladina Hausmann, and Esteban Castro-Ruiz, "The Perspectives of Non-Ideal Quantum Reference Frames", arXiv:2512.19343, (2025).

[6] Angel Ballesteros, Diego Fernandez-Silvestre, and Ivan Gutierrez-Sagredo, "Universal $T$-matrices for quantum Poincaré groups: contractions and quantum reference frames", arXiv:2604.01058, (2026).

[7] Ladina Hausmann and Renato Renner, "The Paradox of the Third Particle is classical", arXiv:2607.15351, (2026).

[8] Matthew J. Lake and Marek L. Miller, "How many degrees of freedom describe a quantum N-particle state?", arXiv:2607.16988, (2026).

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