{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T16:03:34Z","timestamp":1765382614546,"version":"3.40.4"},"reference-count":104,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T00:00:00Z","timestamp":1745452800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Agencia Estatal de Investigacion","award":["CNS2023-143760"],"award-info":[{"award-number":["CNS2023-143760"]}]},{"name":"Agencia Estatal de Investigacion","award":["PID2023-148373NB-I00"],"award-info":[{"award-number":["PID2023-148373NB-I00"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, we take the first steps towards a theory of ''Doubly'' Quantum Mechanics, a modification of Quantum Mechanics in which the geometrical configurations of physical systems, measurement apparata, and reference frame transformations are themselves quantized and described by ''geometry'' states in a Hilbert space. We develop the formalism for spin-1<\/mml:mn>2<\/mml:mn><\/mml:mfrac><\/mml:math> measurements by promoting the group of spatial rotations S<\/mml:mi>U<\/mml:mi>(<\/mml:mo>2<\/mml:mn>)<\/mml:mo><\/mml:math> to the quantum group S<\/mml:mi>U<\/mml:mi>q<\/mml:mi><\/mml:msub>(<\/mml:mo>2<\/mml:mn>)<\/mml:mo><\/mml:math> and generalizing the axioms of Quantum Theory in a covariant way. As a consequence of our axioms, the notion of probability becomes a self-adjoint operator acting on the Hilbert space of geometry states, hence acquiring novel non-classical features. After introducing a suitable class of semi-classical geometry states, which describe near-to-classical geometrical configurations of physical systems, we find that probability measurements are affected, in these configurations, by intrinsic uncertainties stemming from the quantum properties of S<\/mml:mi>U<\/mml:mi>q<\/mml:mi><\/mml:msub>(<\/mml:mo>2<\/mml:mn>)<\/mml:mo><\/mml:math>. This feature translates into an unavoidable fuzziness for observers attempting to align their reference frames by exchanging qubits, even when the number of exchanged qubits approaches infinity, contrary to the standard S<\/mml:mi>U<\/mml:mi>(<\/mml:mo>2<\/mml:mn>)<\/mml:mo><\/mml:math> case.<\/jats:p>","DOI":"10.22331\/q-2025-04-24-1721","type":"journal-article","created":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T16:06:56Z","timestamp":1745510816000},"page":"1721","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Doubly Quantum Mechanics"],"prefix":"10.22331","volume":"9","author":[{"given":"Vittorio","family":"D'Esposito","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica Ettore Pancini, Universit\u00e0 di Napoli ``Federico II'', Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy"},{"name":"INFN, Sezione di Napoli, I-80126 Napoli, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giuseppe","family":"Fabiano","sequence":"additional","affiliation":[{"name":"Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA"},{"name":"Department of Physics, University of California, Berkeley, CA 94720, USA"},{"name":"Centro Ricerche Enrico Fermi\u2014Museo Storico della Fisica e Centro Studi e Ricerche \u201cEnrico Fermi\u201d, Roma"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Domenico","family":"Frattulillo","sequence":"additional","affiliation":[{"name":"INFN, Sezione di Napoli, I-80126 Napoli, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Flavio","family":"Mercati","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidad de Burgos, 09001 Burgos, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,4,24]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Daniele Oriti, editor. ``Approaches to quantum gravity: Toward a new understanding of space, time and matter''. Cambridge University Press. (2009).","DOI":"10.1017\/CBO9780511575549"},{"key":"1","doi-asserted-by":"publisher","unstructured":"David J. Gross and Paul F. Mende. ``String Theory Beyond the Planck Scale''. Nucl. Phys. B 303, 407\u2013454 (1988).","DOI":"10.1016\/0550-3213(88)90390-2"},{"key":"2","doi-asserted-by":"publisher","unstructured":"D. Amati, M. Ciafaloni, and G. Veneziano. ``Classical and Quantum Gravity Effects from Planckian Energy Superstring Collisions''. Int. J. Mod. Phys. A 3, 1615\u20131661 (1988).","DOI":"10.1142\/S0217751X88000710"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Ronald J. Adler and David I. Santiago. ``On gravity and the uncertainty principle''. Mod. Phys. Lett. A 14, 1371 (1999). arXiv:gr-qc\/9904026.","DOI":"10.1142\/S0217732399001462"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Carlo Rovelli and Lee Smolin. ``Discreteness of area and volume in quantum gravity''. Nucl. Phys. B 442, 593\u2013622 (1995). arXiv:gr-qc\/9411005.","DOI":"10.1016\/0550-3213(95)00150-Q"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Abhay Ashtekar and Jerzy Lewandowski. ``Quantum theory of geometry. 1: Area operators''. Class. Quant. Grav. 14, A55\u2013A82 (1997). arXiv:gr-qc\/9602046.","DOI":"10.1088\/0264-9381\/14\/1A\/006"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Abhay Ashtekar and Jerzy Lewandowski. ``Quantum theory of geometry. 2. Volume operators''. Adv. Theor. Math. Phys. 1, 388\u2013429 (1998). arXiv:gr-qc\/9711031.","DOI":"10.4310\/ATMP.1997.v1.n2.a8"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Laurent Freidel, Etera R. Livine, and Carlo Rovelli. ``Spectra of length and area in (2+1) Lorentzian loop quantum gravity''. Class. Quant. Grav. 20, 1463\u20131478 (2003). arXiv:gr-qc\/0212077.","DOI":"10.1088\/0264-9381\/20\/8\/304"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Giovanni Amelino-Camelia, Giulia Gubitosi, and Flavio Mercati. ``Discreteness of area in noncommutative space''. Phys. Lett. B 676, 180\u2013183 (2009). arXiv:0812.3663.","DOI":"10.1016\/j.physletb.2009.04.045"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Roberto Percacci and Gian Paolo Vacca. ``Asymptotic Safety, Emergence and Minimal Length''. Class. Quant. Grav. 27, 245026 (2010). arXiv:1008.3621.","DOI":"10.1088\/0264-9381\/27\/24\/245026"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Hartland S. Snyder. ``Quantized space-time''. Phys. Rev. 71, 38\u201341 (1947).","DOI":"10.1103\/PhysRev.71.38"},{"key":"11","unstructured":"Shahn Majid. ``Algebraic approach to quantum gravity. II: Noncommutative spacetime'' (2006). arXiv:hep-th\/0604130."},{"key":"12","doi-asserted-by":"publisher","unstructured":"Richard J. Szabo. ``Quantum Gravity, Field Theory and Signatures of Noncommutative Spacetime''. Gen. Rel. Grav. 42, 1\u201329 (2010). arXiv:0906.2913.","DOI":"10.1007\/s10714-009-0897-4"},{"key":"13","doi-asserted-by":"publisher","unstructured":"A Pacho\u0142. ``Short review on noncommutative spacetimes''. Journal of Physics: Conference Series 442, 012039 (2013).","DOI":"10.1088\/1742-6596\/442\/1\/012039"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Shahn Majid. ``Foundations of quantum group theory''. Cambridge University Press. (1995).","DOI":"10.1017\/CBO9780511613104"},{"key":"15","doi-asserted-by":"publisher","unstructured":"G. Amelino-Camelia, John R. Ellis, N. E. Mavromatos, Dimitri V. Nanopoulos, and Subir Sarkar. ``Tests of quantum gravity from observations of gamma-ray bursts''. 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D 107, 045014 (2023). arXiv:2208.12083.","DOI":"10.1103\/PhysRevD.107.045014"},{"key":"36","doi-asserted-by":"publisher","unstructured":"Thomas D. Galley, Flaminia Giacomini, and John H. Selby. ``Any consistent coupling between classical gravity and quantum matter is fundamentally irreversible''. Quantum 7, 1142 (2023). arXiv:2301.10261.","DOI":"10.22331\/q-2023-10-16-1142"},{"key":"37","unstructured":"Lin-Qing Chen and Flaminia Giacomini. ``Quantum effects in gravity beyond the Newton potential from a delocalised quantum source'' (2024). arXiv:2402.10288."},{"key":"38","unstructured":"Anne-Catherine de la Hamette, Viktoria Kabel, and \u010caslav Brukner. ``What an event is not: unravelling the identity of events in quantum theory and gravity'' (2024). arXiv:2404.00159."},{"key":"39","doi-asserted-by":"publisher","unstructured":"Lakshay Goel, Everett A. Patterson, Mar\u00eda Rosa Preciado-Rivas, Mahdi Torabian, Robert B. Mann, and Niayesh Afshordi. ``Accelerated detector in a superposed spacetime''. Phys. Rev. D 111, 025015 (2025). arXiv:2409.06818.","DOI":"10.1103\/PhysRevD.111.025015"},{"key":"40","doi-asserted-by":"publisher","unstructured":"T. Kovachy, P. Asenbaum, C. Overstreet, C. A. Donnelly, S. M. Dickerson, A. Sugarbaker, J. M. Hogan, and M. A. Kasevich. ``Quantum superposition at the half-metre scale''. Nature 528, 530\u2013533 (2015).","DOI":"10.1038\/nature16155"},{"key":"41","doi-asserted-by":"publisher","unstructured":"G. Rosi, G. D'Amico, L. Cacciapuoti, F. Sorrentino, M. Prevedelli, M. Zych, C. Brukner, and G. M. Tino. ``Quantum test of the equivalence principle for atoms in superpositions of internal energy eigenstates''. Nature Commun. 8, 5529 (2017). arXiv:1704.02296.","DOI":"10.1038\/ncomms15529"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Chiara Marletto and Vlatko Vedral. ``Gravitationally-induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity''. Phys. Rev. Lett. 119, 240402 (2017). arXiv:1707.06036.","DOI":"10.1103\/PhysRevLett.119.240402"},{"key":"43","doi-asserted-by":"publisher","unstructured":"Sougato Bose, Anupam Mazumdar, Gavin W. Morley, Hendrik Ulbricht, Marko Toro\u0161, Mauro Paternostro, Andrew Geraci, Peter Barker, M. S. Kim, and Gerard Milburn. ``Spin Entanglement Witness for Quantum Gravity''. Phys. Rev. Lett. 119, 240401 (2017). arXiv:1707.06050.","DOI":"10.1103\/PhysRevLett.119.240401"},{"key":"44","doi-asserted-by":"publisher","unstructured":"Magdalena Zych, Fabio Costa, Igor Pikovski, and \u010caslav Brukner. ``Bell\u2019s theorem for temporal order''. 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Nature 591, 225\u2013228 (2021). arXiv:2009.09546.","DOI":"10.1038\/s41586-021-03250-7"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Carlo Cepollaro and Flaminia Giacomini. ``Quantum generalisation of Einstein\u2019s equivalence principle can be verified with entangled clocks as quantum reference frames''. Class. Quant. Grav. 41, 185009 (2024). arXiv:2112.03303.","DOI":"10.1088\/1361-6382\/ad6d26"},{"key":"49","doi-asserted-by":"publisher","unstructured":"Marios Christodoulou, Andrea Di Biagio, Markus Aspelmeyer, \u010caslav Brukner, Carlo Rovelli, and Richard Howl. ``Locally Mediated Entanglement in Linearized Quantum Gravity''. Phys. Rev. Lett. 130, 100202 (2023). arXiv:2202.03368.","DOI":"10.1103\/PhysRevLett.130.100202"},{"key":"50","doi-asserted-by":"publisher","unstructured":"Chris Overstreet, Joseph Curti, Minjeong Kim, Peter Asenbaum, Mark A. Kasevich, and Flaminia Giacomini. ``Inference of gravitational field superposition from quantum measurements''. Phys. Rev. D 108, 084038 (2023). arXiv:2209.02214.","DOI":"10.1103\/PhysRevD.108.084038"},{"key":"51","doi-asserted-by":"publisher","unstructured":"Michele Arzano, Vittorio D'Esposito, and Giulia Gubitosi. ``Fundamental decoherence from quantum spacetime''. Commun. Phys. 6, 242 (2023). arXiv:2208.14119.","DOI":"10.1038\/s42005-023-01159-3"},{"key":"52","doi-asserted-by":"publisher","unstructured":"Giovanni Amelino-Camelia, Vittorio D'Esposito, Giuseppe Fabiano, Domenico Frattulillo, Philipp A. H\u00f6hn, and Flavio Mercati. ``Quantum Euler angles and agency-dependent space-time''. PTEP 2024, 033A01 (2024). arXiv:2211.11347.","DOI":"10.1093\/ptep\/ptae015"},{"key":"53","doi-asserted-by":"publisher","unstructured":"Giovanni Amelino-Camelia, Giuseppe Fabiano, and Domenico Frattulillo. ``Total momentum and other Noether charges for particles interacting in a quantum spacetime''. Symmetry 17, 227 (2025). arXiv:2302.08569.","DOI":"10.3390\/sym17020227"},{"key":"54","doi-asserted-by":"publisher","unstructured":"Giovanni Amelino-Camelia. ``Relativity in space-times with short distance structure governed by an observer independent (Planckian) length scale''. Int. J. Mod. Phys. D 11, 35\u201360 (2002). arXiv:gr-qc\/0012051.","DOI":"10.1142\/S0218271802001330"},{"key":"55","doi-asserted-by":"publisher","unstructured":"N. R. Bruno, G. Amelino-Camelia, and J. Kowalski-Glikman. ``Deformed boost transformations that saturate at the Planck scale''. Phys. Lett. B 522, 133\u2013138 (2001). arXiv:hep-th\/0107039.","DOI":"10.1016\/S0370-2693(01)01264-3"},{"key":"56","doi-asserted-by":"publisher","unstructured":"J. Kowalski-Glikman and S. Nowak. ``Doubly special relativity theories as different bases of $\\kappa$-Poincar\u00e9 algebra''. Phys. Lett. B 539, 126\u2013132 (2002). arXiv:hep-th\/0203040.","DOI":"10.1016\/S0370-2693(02)02063-4"},{"key":"57","doi-asserted-by":"publisher","unstructured":"Giovanni Amelino-Camelia. ``Doubly special relativity''. Nature 418, 34\u201335 (2002). arXiv:gr-qc\/0207049.","DOI":"10.1038\/418034a"},{"key":"58","doi-asserted-by":"publisher","unstructured":"Jerzy Kowalski-Glikman. ``De sitter space as an arena for doubly special relativity''. Phys. Lett. B 547, 291\u2013296 (2002). arXiv:hep-th\/0207279.","DOI":"10.1016\/S0370-2693(02)02762-4"},{"key":"59","doi-asserted-by":"publisher","unstructured":"Jerzy Kowalski-Glikman. ``Introduction to doubly special relativity''. Lect. Notes Phys. 669, 131\u2013159 (2005). arXiv:hep-th\/0405273.","DOI":"10.1007\/11377306_5"},{"key":"60","doi-asserted-by":"publisher","unstructured":"Giovanni Amelino-Camelia, Giulia Gubitosi, Antonino Marciano, Pierre Martinetti, and Flavio Mercati. ``A No-pure-boost uncertainty principle from spacetime noncommutativity''. Phys. Lett. B 671, 298\u2013302 (2009). arXiv:0707.1863.","DOI":"10.1016\/j.physletb.2008.12.032"},{"key":"61","doi-asserted-by":"publisher","unstructured":"Giovanni Amelino-Camelia, Laurent Freidel, Jerzy Kowalski-Glikman, and Lee Smolin. ``The principle of relative locality''. Phys. Rev. D 84, 084010 (2011). arXiv:1101.0931.","DOI":"10.1103\/PhysRevD.84.084010"},{"key":"62","doi-asserted-by":"publisher","unstructured":"Giovanni Amelino-Camelia. ``On the fate of Lorentz symmetry in relative-locality momentum spaces''. Phys. Rev. D 85, 084034 (2012). arXiv:1110.5081.","DOI":"10.1103\/PhysRevD.85.084034"},{"key":"63","doi-asserted-by":"publisher","unstructured":"Giovanni Amelino-Camelia, Domenico Frattulillo, Giulia Gubitosi, Giacomo Rosati, and Suzana Bedi\u0107. ``Phenomenology of DSR-relativistic in-vacuo dispersion in FLRW spacetime''. JCAP 01, 070 (2024). arXiv:2307.05428.","DOI":"10.1088\/1475-7516\/2024\/01\/070"},{"key":"64","doi-asserted-by":"publisher","unstructured":"Gaetano Fiore and Julius Wess. ``Full twisted Poincar\u00e9 symmetry and QFT on Moyal-Weyl spaces''. Phys. Rev. D 75, 105022 (2007). arXiv:hep-th\/0701078.","DOI":"10.1103\/PhysRevD.75.105022"},{"key":"65","doi-asserted-by":"publisher","unstructured":"Richard J. Szabo. ``Quantum field theory on noncommutative spaces''. Phys. Rept. 378, 207\u2013299 (2003). arXiv:hep-th\/0109162.","DOI":"10.1016\/S0370-1573(03)00059-0"},{"key":"66","doi-asserted-by":"publisher","unstructured":"Fedele Lizzi and Flavio Mercati. ``$\\kappa$-Poincar\u00e9-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory''. Phys. Rev. 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Res. 3, 043138 (2021). arXiv:2103.05022.","DOI":"10.1103\/PhysRevResearch.3.043138"},{"key":"71","doi-asserted-by":"publisher","unstructured":"Seth Major and Lee Smolin. ``Quantum deformation of quantum gravity''. Nucl. Phys. B 473, 267\u2013290 (1996). arXiv:gr-qc\/9512020.","DOI":"10.1016\/0550-3213(96)00259-3"},{"key":"72","doi-asserted-by":"publisher","unstructured":"Laurent Freidel and Kirill Krasnov. ``Spin foam models and the classical action principle''. Adv. Theor. Math. 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