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On the relation between perspective-neutral, algebraic, and effective quantum reference frames

Julian De Vuyst, Philipp A. Hoehn, and Artur Tsobanjan

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We show that these three approaches are, in fact, equivalent for ideal QRFs, distinguished by sharp orientations, which is the previous setting of the first two approaches. AbstractThe framework of internal quantum reference frames (QRFs) constitutes a universal toolset for dealing with symmetries in quantum theory and has led to new revelations in quantum gravity, gauge theories and foundational physics. We show that these three are equivalent in the case that the reference frame is defined using sharp orientation states. ``Relativity of Quantum Correlations: Invariant Quantities and Frame-Dependent Measures'' (2025).

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AbstractThe framework of internal quantum reference frames (QRFs) constitutes a universal toolset for dealing with symmetries in quantum theory and has led to new revelations in quantum gravity, gauge theories and foundational physics. Multiple approaches have emerged, sometimes differing in scope and the way symmetries are implemented, raising the question as to their relation. Here, we investigate the relation between three approaches to QRFs for gauge symmetries, namely the $effective$ semiclassical, $algebraic$, and $perspective-neutral$ (PN) approaches. Rather than constructing Hilbert spaces, as the PN approach, the effective approach is based on a quantum phase space parametrized by expectation values and fluctuations, while the emphasis of the algebraic approach is on the state space of complex linear functionals on a kinematical algebra. Nevertheless, external frame information is treated as gauge in all three formalisms, manifested in constraints on states and algebra. We show that these three approaches are, in fact, equivalent for ideal QRFs, distinguished by sharp orientations, which is the previous setting of the first two approaches. Our demonstration pertains to single constraints, including relativistic ones, and encompasses QRF changes. In particular, the QRF transformations of the PN framework agree semiclassically with those of the older effective approach, by which it was inspired. As a physical application, we explore the QRF covariance of uncertainties and fluctuations, which turn out to be frame dependent. This is particularly well-suited for the effective and algebraic approaches, for which these quantities form a natural basis. Finally, we pave the way towards extending these two approaches to non-ideal QRFs by studying the projection and gauge-fixing operations of the Page-Wootters formalism, built into the PN framework, on algebraic states.Featured image: Triangle showcasing the equivalence between the three approaches for ideal quantum reference frames. All three of them deal with systems characterised by the center in different ways.Popular summaryGeneral relativity has taught us that any coordinate system is fictitious. Open any atlas of the world and note down the coordinate of your home country, it will likely look something like (J,32). Obviously, such a label cannot have any meaning on itself. We might as well reshuffle the axes, giving a new coordinate label like (5,B) for the same points. Obviously, what does not change are the relations between them: the distance between your home country and Okinawa is indifferent to the labels attached to it. We thus obtain a redundancy. These redundancies are typically the consequence of gauge symmetries, manifesting themselves as one or more constraints $C_i = 0$. How to deal which such constraints in the quantum theory is essentially what quantum reference frames are about. Just as the atlas example, we define a reference frame and describe the remainder in terms of the relations between the two constituents. There are many different approaches to this idea, and all differ somewhat in the techniques involved and their range of applicability. In this paper, we look at three of them: -One which is based on both a Hilbert space and algebra of operators; -One which is purely algebraic; -One which describes quantum physics as an infinite-dimensional phase space. We show that these three are equivalent in the case that the reference frame is defined using sharp orientation states. We also look at possible generalisations for when they are unsharp. Furthermore, we study how transformations between chosen reference frames can affect the variances of position variables — they become frame-dependent. A particle which looks localised in one frame might not look localised from another frame.► BibTeX data@article{Vuyst2026relationbetween, doi = {10.22331/q-2026-08-20-2196}, url = {https://doi.org/10.22331/q-2026-08-20-2196}, title = {On the relation between perspective-neutral, algebraic, and effective quantum reference frames}, author = {Vuyst, Julian De and Hoehn, Philipp A. and Tsobanjan, Artur}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2196}, month = aug, year = {2026} }► References [1] Yakir Aharonov and Leonard Susskind. ``Charge Superselection Rule''. Phys. 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D 113, 124060 (2026). arXiv:2503.06667. https://doi.org/10.1103/2pn6-h5j4 arXiv:2503.06667 [130] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum Time''. Phys. Rev. D 92, 045033 (2015). arXiv:1504.04215. https://doi.org/10.1103/PhysRevD.92.045033 arXiv:1504.04215Cited byCould not fetch Crossref cited-by data during last attempt 2026-08-20 11:11:25: Could not fetch cited-by data for 10.22331/q-2026-08-20-2196 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-08-20 11:11:26: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractThe framework of internal quantum reference frames (QRFs) constitutes a universal toolset for dealing with symmetries in quantum theory and has led to new revelations in quantum gravity, gauge theories and foundational physics. Multiple approaches have emerged, sometimes differing in scope and the way symmetries are implemented, raising the question as to their relation. Here, we investigate the relation between three approaches to QRFs for gauge symmetries, namely the $effective$ semiclassical, $algebraic$, and $perspective-neutral$ (PN) approaches. Rather than constructing Hilbert spaces, as the PN approach, the effective approach is based on a quantum phase space parametrized by expectation values and fluctuations, while the emphasis of the algebraic approach is on the state space of complex linear functionals on a kinematical algebra. Nevertheless, external frame information is treated as gauge in all three formalisms, manifested in constraints on states and algebra. We show that these three approaches are, in fact, equivalent for ideal QRFs, distinguished by sharp orientations, which is the previous setting of the first two approaches. Our demonstration pertains to single constraints, including relativistic ones, and encompasses QRF changes. In particular, the QRF transformations of the PN framework agree semiclassically with those of the older effective approach, by which it was inspired. As a physical application, we explore the QRF covariance of uncertainties and fluctuations, which turn out to be frame dependent. This is particularly well-suited for the effective and algebraic approaches, for which these quantities form a natural basis. Finally, we pave the way towards extending these two approaches to non-ideal QRFs by studying the projection and gauge-fixing operations of the Page-Wootters formalism, built into the PN framework, on algebraic states.Featured image: Triangle showcasing the equivalence between the three approaches for ideal quantum reference frames. All three of them deal with systems characterised by the center in different ways.Popular summaryGeneral relativity has taught us that any coordinate system is fictitious. Open any atlas of the world and note down the coordinate of your home country, it will likely look something like (J,32). Obviously, such a label cannot have any meaning on itself. We might as well reshuffle the axes, giving a new coordinate label like (5,B) for the same points. Obviously, what does not change are the relations between them: the distance between your home country and Okinawa is indifferent to the labels attached to it. We thus obtain a redundancy. These redundancies are typically the consequence of gauge symmetries, manifesting themselves as one or more constraints $C_i = 0$. How to deal which such constraints in the quantum theory is essentially what quantum reference frames are about. Just as the atlas example, we define a reference frame and describe the remainder in terms of the relations between the two constituents. There are many different approaches to this idea, and all differ somewhat in the techniques involved and their range of applicability. In this paper, we look at three of them: -One which is based on both a Hilbert space and algebra of operators; -One which is purely algebraic; -One which describes quantum physics as an infinite-dimensional phase space. We show that these three are equivalent in the case that the reference frame is defined using sharp orientation states. We also look at possible generalisations for when they are unsharp. Furthermore, we study how transformations between chosen reference frames can affect the variances of position variables — they become frame-dependent. A particle which looks localised in one frame might not look localised from another frame.► BibTeX data@article{Vuyst2026relationbetween, doi = {10.22331/q-2026-08-20-2196}, url = {https://doi.org/10.22331/q-2026-08-20-2196}, title = {On the relation between perspective-neutral, algebraic, and effective quantum reference frames}, author = {Vuyst, Julian De and Hoehn, Philipp A. and Tsobanjan, Artur}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2196}, month = aug, year = {2026} }► References [1] Yakir Aharonov and Leonard Susskind. ``Charge Superselection Rule''. Phys. Rev. 155, 1428–1431 (1967). https://doi.org/10.1103/PhysRev.155.1428 [2] Yakir Aharonov and Leonard Susskind. ``Observability of the Sign Change of Spinors under 2pi Rotations''. Phys. Rev. 158, 1237–1238 (1967). https://doi.org/10.1103/PhysRev.158.1237 [3] Y. Aharonov and T. Kaufherr. ``Quantum frames of reference''. Phys. Rev. D 30, 368–385 (1984). https://doi.org/10.1103/PhysRevD.30.368 [4] Renato M. Angelo, Nicolas Brunner, Sandu Popescu, Anthony J. Short, and Paul Skrzypczyk. ``Physics within a quantum reference frame''. J. Phys. A: Math. Theor. 44, 145304 (2011). https://doi.org/10.1088/1751-8113/44/14/145304 [5] Carlo Rovelli. ``Quantum gravity''. Cambridge Monographs on Mathematical Physics.

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