The group structure of dynamical transformations between quantum reference frames

Angel Ballesteros; Flaminia Giacomini; Giulia Gubitosi

doi:10.22331/q-2021-06-08-470

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

Published:	2021-06-08, volume 5, page 470
Eprint:	arXiv:2012.15769v2
Doi:	https://doi.org/10.22331/q-2021-06-08-470
Citation:	Quantum 5, 470 (2021).

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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.

► BibTeX data

@article{Ballesteros2021groupstructureof,
  doi = {10.22331/q-2021-06-08-470},
  url = {https://doi.org/10.22331/q-2021-06-08-470},
  title = {The group structure of dynamical transformations between quantum reference frames},
  author = {Ballesteros, Angel 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 = {5},
  pages = {470},
  month = jun,
  year = {2021}
}

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